Journal of Vibration Testing and System Dynamics

Volume 2 Issue 4 December 2018

ISSN 2475-4811 (print) ISSN 2475-482X (online)

Journal of Vibration Testing and System Dynamics

Journal of Vibration Testing and System Dynamics Editors Jan Awrejcewicz Department of Automation, Biomechanics and Mechatronics The Lodz University of Technology 1/15 Stefanowskiego Str., BLDG A22, 90-924 Lodz, Poland Email: [email protected]

C. Steve Suh Department of Mechanical Engineering Texas A&M University College Station, TX 77843-3123, USA Email: [email protected]

Xian-Guo Tuo School of Automation & Information Engineering Sichuan University of Science and Engineering Zigong, Sichuan, 643000, China Email: [email protected]

Jiazhong Zhang School of Energy and Power Engineering Xi’an Jiaotong University Xi’an, 710048, China Email: [email protected]

Associate Editors Jinde Cao School of Mathematics Southeast University Sipailou 2# Nanjing, 210096, China Email: [email protected]

Yoshihiro Deguchi Department of Mechanical Engineering Tokushima University 2-1 Minamijyousanjima-cho Tokushima 770-8506, Japan Email: [email protected]

Yu Guo McCoy School of Engineering Midwestern University 3410 Taft Boulevard Wichita Falls, TX 76310, USA Email: [email protected]

Hamid R. Hamidzadeh Department of Mechanical and Manufacturing Engineering Tennessee State University Nashville, TN 37209-1561, USA Email: [email protected]

Jianzhe Huang Department of Power and Energy Engineering Harbin Engineering University Harbin, 150001,China Email: [email protected]

Meng-Kun (Jason) Liu Department of Mechanical Engineering National Taiwan University of Science and Technology Taipei, Taiwan Email: [email protected]

Zhi-Ke Peng School of Mechanical Engineering Shanghai Jiao Tong University Shanghai, P. R. China 200240 Email: [email protected]

Alexander P. Seyranian Institute of Mechanics Moscow State Lomonosov University, Michurinsky pr. 1, 119192 Moscow, Russia Email: [email protected]

Dimitry Volchenkov Department of Mathematics & Statistics Texas Tech University 1108 Memorial Circle Lubbock, TX 79409, USA Email: [email protected]

Baozhong Yang Schlumberger Smith Bits 1310 Rankin Rd Houston, TX 77073, USA Email: [email protected]

Guirong (Grace) Yang Department of Civil, Architectural and Environmental Engineering Missouri University of Science and Technology Rolla, MO 65409, USA Email: [email protected]

Shudong Yu Department of Mechanical and Industrial Engineering Ryerson University Toronto, Ontario M5B 2K3 Canada Email: [email protected]

Nyesunthi Apiwattanalunggarn Department of Mechanical Engineering Kasetsart University JatuJak Bangkok 10900 Thailand Email: [email protected]

Junqiang Bai School of Aeronautics Northwestern Polytechnical University Xi’an, P. R. China Email: [email protected]

Editorial Board Farbod Alijani Department of Precision and Microsystems Engineering Delft University of Technology The Netherlands Email: [email protected]

Continued on back materials

Journal of Vibration Testing and System Dynamics Volume 2, Issue 4, December 2018

Editors Jan Awrejcewicz C. Steve Suh Xian-Guo Tuo Jiazhong Zhang

L&H Scientific Publishing, LLC, USA

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Journal of Vibration Testing and System Dynamics 2(4) (2018) 297-306

Journal of Vibration Testing and System Dynamics Journal homepage: https://lhscientificpublishing.com/Journals/JVTSD-Default.aspx

On Targeted Energy Transfer and Resonance Captures in the 2D-Wing and Nonlinear Energy Sinks Wenfan Zhang1 , Jiazhong Zhang1†, Le Wang1 , Shaohua Tian2 1 2

School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, P. R. China School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, P. R. China Submission Info Communicated by Steve Suh Received 10 March 2018 Accepted 18 June 2018 Available online 1 January 2019 Keywords Nonlinear energy sink Vibration suppression Targeted energy transfer Resonance captures Nonlinear vibration

Abstract Targeted energy transfer, presented in two-dimensional wing coupled with two Nonlinear Energy Sinks (NESs) under freestream, is studied numerically, it is feasible to partially or even completely suppress aeroelastic instability by passively transferring vibration energy from the wing to the NES in a one-way irreversible fashion, and the relationship between the vibration suppression and Targeted Energy Transfer (TET) of the system is analyzed in detail. First, the model of the coupling system, which includes heave and pitch motions, is presented, and the NESs are located at the leading edge and trailing edge (NES1 and NES2) separately. Then, the vibrations suppressed by NESs are also investigated from the viewpoint of energy transfer etc., and the Resonance Captures (RCs) in the nonlinear coupling system are studied using spectrum analysis. Furthermore, the ensuing TET through the modes of wing (Heave and Pitch) and the NESs are discussed in detail. The results show that the NESs could absorb and dissipate a significant portion of energy fed from the flow to the wing, and the NESs could absorb the energy from every single motion of the wing, and the TET and RCs between modes can be more available in the coupling system. Therefore, the TET is more efficient between the wing and NESs, and it leads to the increase of the critical velocity of freestream under which the nonlinear vibration of the wing can be suppressed by NESs effectively. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In recent years, the targeted energy transfer (TET) and the nonlinear energy sink (NES) are widely studied in passive vibration control, and the results show that NES is very efficient in the suppression of vibration. Indeed, the NES is a nonlinear absorber which is accompanied by the TET [1]. Compared with the linear absorber, NES’s resonance frequencies are no longer discrete. Consequently, the NES could resonate with the main system in a broad band of frequencies, so the NES has more efficiency and better adaptability to absorb the energy than the linear absorber. For systems without damping, † Corresponding

author. Email address: [email protected]

ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2018.12.001

298

Wenfan Zhang et al. / Journal of Vibration Testing and System Dynamics 2(4) (2018) 297–306

the energy will exchange between the main system and subsystems in the forms of kinetic energy and potential energy. However, the phenomenon of TET can be found in the system with damping in its subsystems, and the energy transfers to the subsystems from the main system irreversibly. Also, the TET needs to be studied by resonance captures (a phenomenon that a dynamical system comes to satisfy commensurable frequency relations over finite time duration) [2–6]. First, the oscillations of system are attracted to transient resonance captures (TRCs). Then, escape from these captures and attract again or finally entrap into permanent resonance captures (PRCs). And the TRCs and PRCs will be explained in Case 2 of Part 2 (the analysis of the results) in following. As resonance capture occurs, the energy exchanges vigorously between the main system and the subsystems. What is more important is that the condition of energy exchange is no longer satisfied and the energy transfers to subsystems irreversibly, so the study of NES becomes very important in the TET. Efforts have been made to study the vibration of the wing in freestream [7], but the nonlinearity of vibration absorber has not been focused. There are some teams around the world have studied the dynamic characters of NES because of its high efficiency to absorb the energy. The essentially nonlinear absorber with different mass have been studied [8], and the research laid the foundation for later studies by linearize the system, but this kind of linearization could not show the universal nonlinearity in actual systems. The airfoil with nonlinear stiffness is studied without additional parts [9], and the nonlinearity has been considered both in the main system and the subsystems by studying the suppression of limit cycle oscillations (LCOs) in the van der Pol (VDP) oscillator [10, 11]. Some interesting nonlinear properties are obtained in the whole coupling system. Leonid M et al. [12] studied the one-dimensional characters of NES which is different in ways of quasi-linearity, weakly and strongly nonlinearity. The results show that stronger nonlinearity leads to higher efficiency to absorb the energy of oscillations. However, the more complicated vibrations need to be studied in higher-dimensional systems. As one application of this study, the couple system with an airfoil as its main system attracts attention of researchers. Hubbard et al. [13] studied an airfoil coupled with a NES in the tip of the wing, and their results show that the NES could control the vibration of the main system. But the design has some non-negligible effects for the whole system because the shape of the wing is changed. Another one change the location of the NES to the top surface of the airfoil, and the suppression phenomenon of the wing coupled with NES is studied [14]. The results can prove that the NES is a high efficiency vibration absorber, but the shape of the wing is changed as well. To avoid all those negative effects of shape changing, Lee Y et al. [15] studied a model which the NES is set beneath the surface of the leading edge. It shows that the NES could enhance the critical velocity of freestream under which the vibration of wing can be suppressed. The mechanisms of TET can be understood further by studying NESs. The application of NES is to obtain the condition of energy completely transfers in the coupling system [16]. They find that the condition of fully energy transfer is that the mass of subsystem is greater than a certain value. Zhang Y C et al. [17] consider a nonlinear absorber with two degrees of freedom, and the results show this can make the energy dissipate more effectively. And the more complicated way in one-dimensional system has been studied, but the phenomenon may not exist in higher dimensional Cases. In this study, the NESs are set in the area near the leading edge and trailing edge of the twodimensional wing, and the NESs is a subsystem with two degrees of freedom. The spectrum analysis and energy transfer analysis have been used to study the mechanisms of NES and its relationship with TET.


299

Fig. 1 Two-DOF wing model coupled with two NESs.

2 Modeling of the system A two-dimensional rigid airfoil is chosen to be the main system, as shown in Fig. 1. The system is a two-dimensional rigid wing with two degrees of freedom, and two NESs are set beneath the surface near the leading edge and trailing edge separately. For studying the suppression effect and the TET in the coupling system, the dynamics of system suppressed by NESs are studied and compared with the results in the system without NES. Following the principle of virtual work [18], the governing equations of the couple system are obtained as ˙ ) + cs1 (h˙ − d1 α˙ − z˙1 ) mh¨ + Sα α¨ + Kh (h + ch h3 ) + qSCLα (α + h/U + ks1 (h − d1 α − z1 )3 + cs2 (h˙ − d2 α˙ − z˙2 ) + ks2 (h − d2 α − z2 )3 = 0 ˙ ) + d1 cs1 (d1 α˙ + z˙1 − h) ˙ Iα α¨ + Sα h¨ + Kα (α + c p α 3 ) − qeSCLα (α + h/U ˙ + d2 ks2 (d2 α + z2 − h)3 = 0 + d1 ks1 (d1 α + z1 − h)3 + d2 cs2 (d2 α˙ + z˙2 − h)

(1)

˙ + ks1 (z1 + d1 α − h)3 = 0 ms1 z¨1 + cs1 (˙z1 + d1 α˙ − h) ˙ + ks2 (z2 + d2 α − h)3 = 0 ms2 z¨2 + cs2 (˙z2 + d2 α˙ − h) where m is the mass of the airfoil; α the mass unbalance (mxcg ); xcg the location of the center of gravity (c.g.) measured from the elastic axis (e.a.); ρ∞ the is the density of the flow; U the constant flow speed around the wing; Kh the coefficient of linear heave stiffness; Kα the coefficient of linear pitch stiffness; ch the nonlinear heave stiffness factor; q the dynamic pressure ( 21 ρ∞U 2 ); S the planform area of the wing; CLα the lift curve slope; cs1 the damping in NES1 (at the leading edge); cs2 the damping in NES2 (at the trailing edge); d1 the offset attachment of the NES1 to the wing, d2 the offset attachment of the NES2 to the wing, measured from and positive ahead of the elastic axis (e.a.); Ks1 the essentially nonlinear stiffness in NES1 and Ks2 the essentially nonlinear stiffness in NES2; Iα the mass moment of inertia with respect to the e.a.; Kα the coefficients of linear pitch stiffness; c p the nonlinear pitch stiffness factor; e the location of the aerodynamic center (a.c.) measured from the e.a. (positive ahead of e.a.); ms1 the mass of NES1 and ms2 the mass of NES2. Iin non-dimensional form, y′′ + xα α ′′ + Ω2 y + ξy y3 + µCLα Θ(y′ + Θα ) + ε1 λ1 (y′ − δ1 α ′ − v′1 ) +C1 (y − δ1 α − v1 )3 + ε2 λ2 (y′ − δ2 α ′ − v′2 ) +C2 (y − δ2 α − v2 )3 = 0 rα2 α ′′ + xα y′′ + rα2 α + ξα α 3 − γ µCLα Θ(y′ + Θα ) + δ1 ε1 λ1 (δ1 α ′ + v′1 − y′ ) + δ1C1 (δ1 α + v1 − y)3 + δ2 ε2 λ2 (δ2 α ′ + v′2 − y′ ) + δ2C2 (δ2 α + v2 − y)3 = 0

(2a)

300


Table 1 Parameters in 3 cases. Parameters

Case1

Case 2

Case 3

Θ

0.9

0.9

0.99

δ1

0.9

0.9

0.9

δ2

-0.9

-0.9

-0.9

ε1

0.01

0.01

0.02

ε2

0.01

0.01

0.02

λ1

0.1

0.4

0.4

λ2

0.1

0.4

0.4

C1

10

40

40

C2

10

40

40

ε1 v′′1 + ε1 λ1 (v′1 + δ1 α ′ − y′ ) +C1 (v1 + δ1 α − y)3 = 0 ε2 v′′2 + ε2 λ2 (v′2 + δ2 α ′ − y′ ) +C2 (v2 + δ2 α − y)3 = 0

(2b)

where y is the nondimensional heave mode (y = h/b); v1 the nondimensional NES1 mode (v1 = z1 /b); v2 the nondimensional NES2 mode (v2 = z2 /b); xα the nondimensional mass unbalance in the airfoil; Ω the frequency ratio; ξy , ξa the nondimensional nonlinear heave and pitch stiffness factors; Θ the reduced speed of the flow; µ the density ratio; ε1 and ε2 the mass ratio between the NES and the wing; λ1 the nondimensional linear viscous damping in the NES1; λ2 the nondimensional linear viscous damping in the NES2; δ1 the nondimensional offset attachment of the NES1 to the wing; δ2 the nondimensional offset attachment of the NES2 to the wing; C1 , C2 the nondimensional coefficients for essentially nonlinear coupling stiffness in NES1 and NES2; rα the radius of gyration of the cross section of the wing; γ the nondimensional coefficient for location of the aerodynamic center (a.c.); τ the nondimensional time. In order to compare with the results in [9, 15, 21], the parameters are chosen as xα = 0.2, rα = 0.5, γ = 0.4, Ω = 0.5, µ = 1/10π , CLα = 2π , ξα = ξy = 1. In order to study the energy transfer, the energy of the wing and the NESs are defined as 1 1 2 1 EH = y′ + Ω2 y2 + ξy y4 2 2 4 1 2 ′2 1 2 2 1 EP = rα α + rα α + ξα α 4 2 2 4 (3) 1 ′2 1 4 EN1 = ε1 v1 + C1 (y − δ1 α − v1 ) 2 4 1 ′2 1 EN2 = ε2 v2 + C2 (y − δ2 α − v2 )4 2 4 where EH is the transient energy of the heave mode of the wing, EP the transient energy of the pitch mode of the wing, EN1 and EN2 the transient energy of the NESs near the leading edge and trailing edge, respectively. Further, the changes of total energy are given as follows, ˆ t [γ µCLα Θα ′ (y′ + Θα )−µCLα Θy′ (y′ + Θα )]ds EF = 0 ˆ t (4) [ε1 λ1 (v′1 + δ1 α ′ − y′ )2 +ε2 λ2 (v′2 + δ2 α ′ − y′ )2 ]ds ED = 0

ET = EH + EP + EN1 + EN2 where EF is the non-conservative work provided by the flow, ED the energy dissipates by NESs, and ET the energy remains in the wing.


0.01 0 Ŧ0.01 0 0.2

Pitch α(τ)

100

200

300

400

500

W NES

600

700

800

900

Case 1

WO NES

1000

0.01

1100

0.005

0

NES1 v (τ) 1

Ŧ0.2 0 0.01

100

200

300

400

500

600

700

800

900

1000

1100

α,(τ)

Heave y(τ)

Case 1

301

0

Ŧ0.01

Ŧ0.01 0 0.05 NES2 v2(τ)

0 Ŧ0.005

100

200

300

400

500

600

700

800

900

1000

1100

0.01 0.005 0

0 Ŧ0.05 0

100

200

300

400

500 600 time, τ

700

800

900

1000

1100

Ŧ0.005

Ŧ0.01

1200

1600

1400

Pitch α(τ)

1800

2000

2200

2400

time, τ

Fig. 3 Phase portrait of pitch motion with time in Case 1.

Fig. 2 Time history in Case 1.

Energy exchange,%

Case 1

local enlarging graph

Ŧ3

x 10

1 6

NES1 v1(τ)

Heave y(τ)

Pitch α(τ) 0.5

4

NES2 v2(τ)

2 0 0

500

1000

time

1500

2000

0 2500 0

500

1000

time

1500

2000

2500

Ŧ4

Energy

2

x 10

Energy input 1

Energy dissipation by NES

0 0

500

Energy of wing

1000

time, τ

1500

2000

2500

Fig. 5 Energy change and energy transfer among NESs and wing in Case 1.

Fig. 4 Frequency versus amplitude in Case 1. Case 1

Case 1

1.2

120

Heave Pitch NES1 NES2

1

Resonance captures 100 Escape 80 Phase difference

Frequency

0.8

0.6

0.4

Resonance captures 60

φ17

φ37

φ35

40 20

Frequency locking (Resonance Captures) 0.2

0 0

200

400

600

800

1000 τ

1200

1400

1600

1800

Fig. 6 Frequency versus time in Case 1.

2000

Ŧ20 0

φ15

φ13

0

200

400

600

800

1000 τ

1200

1400

1600

1800

2000

Fig. 7 Phase difference versus time in Case 1.

3 Numerical results and analysis Three typical Cases are obtained to show the properties of the system with different parameters. 3.1

Case 1

In order to show the results clearly, just a part of the figures is shown to illustrate the results of Cases. The essential character of Case 1 is a recurrent series of suppressed burst-outs of the pitch mode and heave mode of the wing The parameters of Case 1 are Θ = 0.9, δ1 = 0.9, δ2 = −0.9, ε1 = ε2 = 0.01, λ1 = λ2 = 0.1, C1 = C2 = 10. Figure 2 shows the amplitudes in Case 1. This is a complete suppression Case of the wing, and this kind of suppressed oscillations has a quasi-periodic behavior as shown in Fig. 3 [19,20]. The amplitudes

302


of the heave mode and pitch mode are suppressed in small scales. It seems that the energy of the flow transfers to NESs directly. The frequency of the whole system through time history is the same as shown in Fig. 4, and the phenomenon of series transient resonance captures are also detected to show the resonance frequencies. It is clear that the super-harmonic resonance (1:3 resonance) between heave and pitch motions becomes 1:1 resonance due to the effect of the NESs. Because of these resonance captures, the energy of wing could transfer to the NESs in an irreversible way. Fig 5 shows a very interesting phenomenon that it seems like the energy of the heave mode is transferred to NES1, and the energy of pitch mode is transferred to NES2. This is a new finding that the energy not only transfers irreversibly as the traditional TET but also transfers to some specific targets. So, the NESs engage in 1:1 TRC with the heave and pitch mode separately, passively absorb broadband of energy from the wing. Over a long time, the energy remains in the main system and changes on low levels in Case 1. Figs 6 and 7 show that the vibration of the heave and pitch modes become a recurrent series of suppressed burstouts, as the NESs play the role in the subsystem. They show the instantaneous frequency of each mode and we can clearly see that frequency locking exists whenever RCs occur. 3.2

Case 2

The stiffness and damping coefficient of NESs are further increased in Case 2 (λ1 = λ2 = 0.4, C1 = C2 = 40). This Case is a way of completely vibration control shown in Fig. 8. Because of the increased coefficients, the dissipation effect of NESs becomes higher. In comparison with the oscillations in Case 1, the amplitudes of limit cycle oscillations in Case 3 are very close to zero as shown in Fig. 9. Because of the initial excitation, and the amplitudes are controlled close to zero, the 1:1 resonance frequency shown in Fig. 10 is not so prominent at the beginning. The three times frequency of the oscillations of NESs are no longer exist, since the energy of the wing has no need to transfer in a very high efficient way to NESs. The reason is that the 1:1 resonance is capable for the whole system to dissipate the energy from the flow in this Case. Case 2 is marked by the durable suppressions. From Fig. 11, it also shows that the energy of the heave mode is almost transferred to NES1 and the energy of pitch mode is almost transferred to NES2. The energy of the wing obtained from the flow is mostly dissipate by the NESs. The frequencies of the whole system are soon locked in the same value by the way of time-independent as it shows in Fig 11. Hence, the energy remaining in the wing is controlled on a very low level, since the phenomenon of PRCs occurs at all the time as shows in Fig. 12. For all in this Case, the phase differences in Case 2 are no longer changed (tracked in the permanent resonance captures) as they show in Fig. 13. The energy transfers from the wing to the NESs are induced by nonlinear modal interactions during 1:1 permanent resonance captures (PRCs). In this Case, most of the total energy apparently remains confined in the pitch mode so that a wrong conclusion might be drawn that the NESs no longer work efficiently in this Case. However, comparing the energy input and the energy dissipation by the NESs, it is clear that the energy dissipated by NESs increases in the long run, thus preventing reappearance of the larger scale of LCOs in the long term. 3.3

Case 3

In Case 3, the non-dimensional flow speed is increased to 0.99, the mass ratio is increased to 0.02, and other parameters are as the same as Case 2. The time history of amplitude is shown in Fig. 14. It is clear that NESs fail to control the large scale oscillations of the pitch mode. The amplitudes under control become even slightly larger than the system with no NESs attached. In this Case, the NESs no longer act as the efficient absorber, and the vibration of the pitch mode is a relative large limit cycle oscillation as shown in Fig. 15. Depending on the changing of the parameters, the frequency ratio between two modes of the wing is changed to 3:1 from 1:1 as shows in Fig. 16. Such behavior is the same as that observed in the LCO triggering mechanism by Lee et

Wenfan Zhang et al. / Journal of Vibration Testing and System Dynamics 2(4) (2018) 297–306 Case 2

Case 2

W NES

0.02

WO NES

0.01 0 Ŧ0.01 0 0.2

Pitch α(τ)

0.015

100

200

300

400

500

600

0.01 0.005

0

NES1 v (τ) 1

Ŧ0.2 0 0.02

100

200

300

400

500

600

α,(τ)

Heave y(τ)

303

Ŧ0.01

0 Ŧ0.02 0 0.05

NES2 v (τ) 2

0 Ŧ0.005

100

200

300

400

500

Ŧ0.015 0.02

600

2500

0.01

2000

0

0

1500 1000

Ŧ0.01

Ŧ0.05 0

100

200

300 time, (τ)

400

500

500

Ŧ0.02

600

0

Pitch α(τ)

time, τ



Energy exchange,%

Case 2 1 Pitch α(τ) Heave y(τ)

0.5

NES2 v (τ) 2

NES1 v1(τ) 0 0

50

100

time, τ

150

Ŧ5

6

x 10

Energy input Energy

4 2

Energy of wing

Energy dissipation by NES 0 0

100

200

300

400

500 time, τ

600

700

800

900

1000


Fig. 10 Frequency versus amplitude in Case 2.

Case 2

Case 2

2 Heave Pitch NES1 NES2

0.18

PCRs (Permanent Resonance Captures) 0 φ

35

Phase difference

Frequency

0.16

0.14

0.12

0.1

φ

Ŧ2

37

φ

Ŧ4

13

Ŧ6

Frequency locking ( Resonance Captures )

φ

17

Ŧ8 0.08 0

200

400

600

800

1000

1200

1400

1600

τ


1800

Ŧ10 0

φ15 200

400

600

800

1000 τ

1200

1400

1600

1800

2000


al. [9] (no suppression exist as the frequency locking in the ratio of 3:1 between the mode of the wing). As this super-harmonic resonance occurs, the energy transferred from the flow is too vigorous for the NESs to absorb by control the energy of the whole system on a low level as Fig. 17 shows. And the energy of the pitch mode is partly transferred to the heave mode, that is very different from the others Cases which the heave mode of the wing has almost no distinct relationship with the energy of the pitch mode of the wing. From Fig. 18, it shows the phase differences of all these vibrations cannot stop changing after the short start from the beginning, they just escape from the transient resonance captures. After the escaping, the suppression effect of the NESs disappears permanently. From the viewpoint of energy, it can be stated that the energy, transferred from flow to wing modes through super-harmonic and subharmonic resonance captures, become more vigorous at higher flow speed, and the NESs with small amplitudes cannot absorb all the energy, even in the way of

304

Wenfan Zhang et al. / Journal of Vibration Testing and System Dynamics 2(4) (2018) 297–306 Case 3 Case 3

W NES

0

0.4

Pitch α(τ)

Ŧ0.05 0 0.5

NES1 v1(τ)

200

300

400

500

600

700

800

900

1000

1100

100

200

300

400

500

600

700

800

900

1000

1100

0.2

0 Ŧ0.5 0 0.5

0 Ŧ0.2

0 Ŧ0.5 0 0.5

NES2 v2(τ)

100

α,(τ)

Heave y(τ)

0.05

WO NES

100

200

300

400

500

600

700

800

900

1000

Ŧ0.4 0.4

1100

800

0.2 600

0

0

400

Ŧ0.2

Ŧ0.5 0

100

200

300

400

500 600 time, τ

700

800

900

1000

200

Ŧ0.4

1100

0

Pitch α(τ)

time, τ



Energy exchange,%

Case 3

Local enlarging graph

Ŧ3

x 10

1 6 Pitch α(τ) 0.5

NES1 v1(τ)

4

NES2 v2(τ)

Heave y(τ)

2 0 0

50

100

150 time, τ

200

250

0 300 200

210

220 230 time, τ

240

250

0.1

Energy

Energy dissipation by NES Energy input 0.05

Energy of wing

0 0

500

time, τ

1000

1500



Case 3 900 800

Frequency difference

700 600 500 400

60 40 φ17 20

φ13

φ

37

0 φ35

φ Ŧ20 150

15

160

170

180

190

200

300 200 100 0 Ŧ100 0

100

200

300 time, τ

400

500

600


super-harmonic resonance with the modes of the wing. As the result of Case 3, the vigorous energy could not be absorbed, as the amplitudes of the couple system are smaller than the system without NESs. That is because the NESs could not interact with both heave and pitch modes in such a way as to prevent direct energy transfer from the flow to the wing modes through super-harmonic or subharmonic resonance captures.


305

4 Conclusion By studying the vibrations of the wing coupled with NESs, some typical cases with interesting dynamics are analyzed in detail, and a conclusion can be drawn. As a nonlinear absorber, the NES could absorb the energy of the main system with wide range of parameters. And the high efficiency of targeted energy transfer will be accompanied by 1:1 and 3:1 resonance captures between main system and subsystems. If there is super-harmonic or sub-harmonic resonance between the modes of the wing, the system will develop to large scale vibrations, which even larger than the system with no NESs attached, and the purposes of the NESs are having resonance captures with the modes of the wing and prevent the super-harmonic or subharmonic resonance captures between the modes of the wing (main system). Two NESs are set in the wing in the present model, and that will expand the 1:1 resonance range of the wing modes and the critical velocity of freestream obviously. The NESs could control every single mode of the wing. In the Cases which effectively control the wing, the energy of the heave mode is transferred to NES1, and the energy of pitch mode is transferred to NES2. This is a new finding that the energy of TET not only transfers irreversibly as the traditional TET but also transfers to some specific targets that’s an efficient way to improve the energy scale of targeted energy transfer. Acknowledgments This work is supported by National Key Fundamental Research Program of China (973 Program, No.2012CB026002), and the National Natural Science Foundation of China (No.51305355). References [1] Kerschen, G., Lee, Y.S., and Vakakis, A.F. (2006), Irreversible passive energy transfer in coupled oscillators with essential nonlinearity, Siam Journal on Applied Mathematics, 66(2), 648-679. [2] Nayfeh, A.H. and Mook, D. (1985), Nonlinear Oscillations, NewYork: Wiley Interscience. [3] Neishtadt, A.I. (1975), Passage through a Separatrix in a Resonance Problem with a Slowly-Varying Parameter, Prikladnaya Matamatika I Mekhanika, 39(4), 621-632. [4] Quinn, D., Rand, R., and Bridge, J. (1995), The dynamics of resonant capture, Nonlinear Dynamics, 8(1), 1-20. [5] Quinn, D. (1997), Resonance capture in a three degree-of-freedom mechanical system, Nonlinear Dynamics, 14(4), 309-333. [6] Quinn D. (1997), Transition to Escape in A System of coupled oscillators. International Journal of Non-Linear Mechanics, 32(6), 1193-1206. [7] Fatimah, S. and Verhulst, F. (2003), Suppressing flow-induced vibrations by parametric excitation, Nonlinear Dynamics, 31(3), 275-298. [8] Gendelman, O.V., Gorlov, D.V., Manevitch, L.I., and et al. (2005), Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses, Journal of Sound & Vibration, 286(s 1-2), 1-19. [9] Lee, Y.S., Vakakis, A.F., Bergman, L.A., and et al. (2005), Triggering mechanisms of limit cycle oscillations in a two degree-of-freedom wing flutter model, ASME International Design Engineering Technical Conferences & Computers & Information in Engineering Conference, 1863-1872. [10] Lee, Y.S., Vakakis, A.F., Bergman, L.A., and et al. (2005), Suppression of limit cycle oscillations in the van der pol oscillator by means of passive non-linear energy sinks, Structural Control & Health Monitoring, 13(13), 41-75. [11] Lee, Y.S., Kerschen, G., Vakakis, A.F., and et al. (2005), Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment, Physica D Nonlinear Phenomena, 204(1), 41-69. [12] Leonid, M. and Agnessa, K. (2012), Nonlinear energy transfer in classical and quantum systems, Physical Review E Statistical Nonlinear & Soft Matter Physics, 87(2), 304-320. [13] Hubbard, S.A., Mcfarland, D.M., Bergman, L.A., and et al. (2014), Targeted energy transfer between a swept

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wing and winglet-housed nonlinear energy sink, Aiaa Journal, 52, 2633-2651. [14] Hubbard, S.A., Fontenot, R.L., Mcfarland, D.M., and et al. (2014), Transonic aeroelastic instability suppression for a swept wing by targeted energy transfer, Journal of Aircraft, 51(5), 1467-1482. [15] Lee, Y., Vakakis, A., Bergman, L., and et al. (2012), Suppression aeroelastic instability using broadband passive targeted energy transfers, Part 1: Theory, Aiaa Journal, 45(3), 693-711. [16] Zhang, Y.C., Kong, X.R., and Zhang, H.L. (2012), Targeted energy transfer among coupled nonlinear oscillators: complete energy exchange in a conservative system, Journal of Vibration and Shock, 31(1), 150-155. [17] Zhang, Y.C., Kong, X.R., Yang, Z.X., and et al. (2011), Targeted energy transfer and parameter design of a nonlinear vibration absorber, Journal of Vibration Engineering, 24(2), 111-117. [18] Dowell, E, H. (1995), A modern course in aeroelasticity, Journal of Mechanical Design, 103(2), 465-466. [19] Yuri, B., and Kuznetsov, A. (1995), Elements of applied bifurcation theory, Applied Mathematical Sciences, 112, New York: Spring-Verlag. [20] Zhang, J.Z. (2010), Stability, Bifurcation and Numerical Analysis for the Nonlinear Dynamic Systems, Xi’an: Xi’an Jiaotong University Press. [21] Zhang, W.F., Liu, Y., Cao, S.L., and Zhang, J.Z. (2017), Targeted energy transfer between 2-D wing and nonlinear energy sinks and their dynamic behaviors, Nonlinear Dynamics, 90(2), 1-10.



Power Density — An Alternative Approach to Quantifying Fatigue Failure Zachary T. Branigan, C. Steve Suh† Nonlinear Engineering and Control Lab, Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA Submission Info Communicated by J.Z. Zhang Received 2 July 2018 Accepted 2 August 2018 Available online 1 January 2019 Keywords Power density Fatigue failure Fatigue life Bifurcation Multiaxial fatigue testing Wavelet transform

Abstract The power density theory is an alternate description of fatigue failure. It is derived from the concept of power density, which is physically equivalent to the amount of power deposited into a unit volume of the material experiencing dynamic loading. Power density results from changes in stress magnitude over time. All the stress alternations that occur across a broad bandwidth of frequencies contribute to the accumulation of power density. Higher frequencies coupled with faster changes in stress contribute more power density. Once this accumulation reaches a threshold – a fundamental property of the material – it is expected to fail by fatigue. The power density based methodology is applied to properly interpret the multiaxial vibration fatigue test results reported by Mrˇsnik, Slaviˇc and Bolteˇzar [15] using computer simulations. This serves as a feasibility study for the approach, as well as an example of how to apply it. The power density response of the system is analyzed, and the failure locations are predicted for each of the ten load cases that are considered. The predicted failure locations are in excellent agreement with the experimental results. Further examination of the approach would result in a better understanding of fatigue failure, thus improving engineering work across many industries. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Although it is the most common mechanical cause of engineering failures, fatigue is not well understood. A component can crack after enough repeated stress fluctuations even if the stresses remain well below the ultimate strength of the material. Unlike static failures, which usually involve yielding before catastrophic breakdown, fatigue failures generally occur suddenly with very little perceptible warning. Additionally, there is currently no reliable way to predict how long a part will last under alternating stresses. Current methods of predicting fatigue failure rely on empirically-derived equations instead of having a truly scientific foundation. They have very high uncertainties and often do not consider the adverse impact of the cycling frequency on fatigue life. The number of stress cycles that occur before surface cracks initiate and propagate depends on the frequency at which the alternations occur [1–5]. † Corresponding



308 Zachary T. Branigan, C. Steve Suh / Journal of Vibration Testing and System Dynamics 2(4) (2018) 307–326

The effects of cycling frequencies are typically not accounted for in fatigue calculations as they are not well understood [6]. One of the most commonly used methods of predicting fatigue failure is the stress-life method. It is based on the concept that every material has a fatigue strength related to the number of stress alternations that it experiences. A component is expected to fail after experiencing a certain number of alternations at the corresponding fatigue strength of the material. The fatigue strength (S′f ) after N number of cycles is estimated using the following equation: (1) S′f N = σF′ (2N)b

where σF′ is the true stress experienced by the component and often needs to be approximated. The exponent b is calculated as follows: σ′ log( SFe ) b=− (2) log(2Ne )

with Se being the endurance limit and Ne the endurance limit’s corresponding cycles to failure. The endurance limit is the lowest stress at which fatigue failure would be expected after a very large number of cycles, and it is estimated using the following equation: Se = ka kb kc kd ke k f Se′

(3)

where Se′ is the endurance limit of a standard fatigue test specimen. The k factors modify the equation depending on the part’s surface condition and size, the stress levels and temperatures it experiences, and other miscellaneous factors. Each k factor is estimated empirically, usually using multiple experimentally determined constants and curve fits. The stress-life method has many obvious issues. There is no proper scientific basis for most of the modification factors that are used. Instead, it relies on empirical equations and probabilities based on many variables. Many values with high uncertainties are combined, causing the potential error to compound. Additionally, it is based on static properties despite the fact that fatigue is a dynamic mode of failure. It neglects the cycling frequency even though it has a proven effect on a part’s life. All time-domain methods of predicting fatigue failure share this problem. Many contemporary fatigue calculations still utilize the stress-life relationship between the fatigue strength and its corresponding number of cycles to failure, thus inheriting the issues inherent of the method. The strain-life method has shown to be more accurate than the stress-life method. It uses the strain amplitude at the local discontinuity that eventually fails to estimate the relationship between the strain amplitude and the number of cycles to failure. It utilizes the Manson-Coffin relationship to estimate the total-strain amplitude ∆2ε at which the part is expected to fail after N number of cycles: ∆ε σ′ = F (2N)b + εF′ (2N)c 2 E

(4)

where E is the modulus of elasticity and b and c are constants that fit the strain-life curve to empirical data. σF′ and εF′ are the true stress and strain experienced by the failure location that correspond to fracture after one cycle. This relationship is used to predict the fatigue damage and eventually failure of a specimen. Strain-life’s usefulness is limited by the lack of “strain concentration” information for determining the total strain at a discontinuity. It also uses compounding idealizations, resulting in high uncertainty [7]. Like in the stress-life method, cycling frequency is not considered at all. Despite these issues, strain-life concepts are often used in newer methods of predicting fatigue failure. More recently, frequency-domain methods of determining fatigue damage have been examined. The Dirlik method [8, 9] combines one exponential and two Rayleigh probability densities to approximate

Zachary T. Branigan, C. Steve Suh / Journal of Vibration Testing and System Dynamics 2(4) (2018) 307–326 309

the cycle-amplitude distribution (pa ) from the stress using the following equation: z2 1 G1 Z G2 Z z2 pa (s) = √ [ e− Q + 2 e− 2R2 + G3 Ze− 2 ] m0 Q R

(5)

where Z is the normalized amplitude, xm is the mean frequency, and the other parameters depend on DK the mean frequency and the spectral width as described by Dirlik [9]. The fatigue-life intensity D is calculated by Dirlik using the following equation: D

DK

√ k k = C−1 v p m02 [G1 Qk Γ(1 + k) + ( 2)k Γ(1 + )(G2 |R|k + G3 )]. 2

(6)

This method is considered one of the most accurate frequency-domain methods for predicting fatigue failure. However, it still can have very large error for some cases [8]. It is also based on probabilities found from numerical simulations instead of having a proper scientific basis. Another frequency-domain approach is the Tovo-Benasciutti method [8,10]. This method estimates fatigue life by linearly combining upper and lower fatigue-damage intensity limits. The intensity of TB fatigue damage (D ) is then calculated using the following equation: D

TB

= [b + (1 − b)α2k−1 ]α2 D

NB

(7)

where b is approximated using numerical simulation data, α 2 and α k−1 are spectral width parameters, 2 NB and D is the narrow-band damage intensity. This method has been found to be accurate under certain conditions [8]. Like the Dirlik method, though, it is a primarily numerical estimation instead of being based on a true theory. It is not always able to predict fatigue lifetimes well. Additionally, frequency-domain methods do not appropriately consider the time domain, limiting their ability to fully capture the system response when the frequency response is not constant at all times. In 2008, Abdullah et al. [11] attempted to use the time-frequency domain in fatigue analysis. They based their analysis on fatigue strain data. They used the concept of power spectral density (PSD) to take this data from the time-domain and determine its power in the frequency domain. PSD, denoted by Sxx , is calculated as follows (8) Sxx (ω ) = lim E[|xˆT (ω )|2 ] T →∞

where T is the time interval being examined, xˆT is the Fourier transform of the signal, and E is the expected value [12]. Instead of doing this for the entire time interval, though, the short-time Fourier transform (STFT) method was used to give the PSD at different time intervals. The resulting strain intensities were then used to determine the damage on the part due to each frequency at each time. This damage was considered relative to the expected strain life of the material (from the MansonCoffin relationship in Eq. (4)) to estimate its fatigue life. There are multiple problems with this strategy. Because it uses the strain-life method to predict the number of cycles to failure, it inherits the uncertainty in that relationship. Even though the strains were separated by the frequencies at which they occurred, calculations were not adjusted to account for the differences in fatigue damage from low frequencies versus high frequencies. The Manson-Coffin relationship is intended for use in the time-domain. Also, the STFT simply applies a Fourier transform to time intervals separately. The resulting values would be poorly localized in the time-frequency domain per the Uncertainty Principle. Current fatigue methods have been applied to assess fatigue damage and predict fatigue life under variable loading with partial success [13, 14]. A new way of viewing fatigue failure is necessary to more fully understand it. It should be less reliant on empirical or numerical data and instead have its foundation on sound physical principle. In addition, both the time and frequency domains must be considered to completely describe the dynamic response. The concept of power density is developed in the paper to satisfy these objectives. The concept is based on the physical phenomenon of stress


alternations, which are physically equivalent to depositing power into a volume. It has two physical parameters – loading stress amplitude and frequency – for quantifying and predicting fatigue failure using time-frequency analysis. In order to determine the magnitude of the changes in stress that occur at each time and frequency, these changes are transformed into the time-frequency domain. The damage is then scaled based on both the changes in stress and their corresponding frequencies. An extensive study of the concept is performed by evaluating an ABAQUS/Explicit model against previously reported physical test data. The power density results are compared to the experimental results reported in [15] as a demonstration of validity of the concept. 2 Concept of power density [16–19] Power density (PD ) is the time variation of a stress, σ , as follows PD =

∆σ dσ = lim = lim ∆σ · f ∆t→ 0 ∆t ∆t→ 0 dt

(9)

where t is time and f is the frequency of the stress alternations. The normal and shear stresses in all directions must be considered. The metric units of Eq. (9) are derived in Eq. (10) which shows that the temporal gradient of the stress σ is physically equivalent to the amount of power deposited into a unit volume, thus properly named as power density. [

N · ( ms ) N W Pa ]≡[ 2 ]≡[ ] ≡ [ 3 ]. 3 s m ·s m m

(10)

The notion of power density is a macroscopic method of determining when a material subjected to alternating stresses is likely to fail by fatigue. These time gradients of stress amplitude can occur over a broad bandwidth of frequencies simultaneously. The changes in stress due to each frequency must be considered in fatigue calculations. Higher frequencies cause faster changes in stress, which inflicts more damage than lower frequencies. It is only by considering how much of the change in stress is due to each frequency at each time and then placing more significance on higher-frequency stress alternations that fatigue failure can be fully captured. At a given time while a material is undergoing loading and unloading, its stresses oscillate at different amplitudes over a range of frequencies simultaneously. At each frequency, the power density causes damage to the material. High-frequency stress alternations cause higher power density than lowfrequency alternations. Therefore, it is necessary to separate the changes in stress by the frequencies at which they occur. A broad bandwidth of frequencies must be considered. Under cyclic loading, there is a change in stress in each of the six spatial directions at any given location and time interval. Using the Cartesian coordinate system, these can be labeled as ∆σ xx , ∆σ yy , ∆σ zz , ∆τ xy , ∆τ yz , and ∆τ zx. The changes in stress in every direction must each be transformed into the time-frequency domain separately. The Gabor wavelet, given in Eq. (11), is employed to accomplish this because it allows for the resolution in the time and frequency domains to be optimized with minimal uncertainty [20] from the Uncertainty Principle. √ −( ωγ0 )2 2 ω0 1 ψ (t) = √ exp[ t ]exp(iω0 t) 4 2 π γ

(11)

where γ and ω0 are positive constants. For the purposes of power density calculations, the evolution of the normal stress in the x-direction in time, ∆σxx , can be resolved using the discrete Gabor wavelet transform as follows kT − b 1 N−1 )T (12) Wxx (a, b) = √ ∑ ∆σxx [kT ] Ψ( a k=0 a


where Ψ is the complex conjugate of the Gabor wavelet function Ψ, a and b are scaling and translation coefficients, respectively, and T is the sampling period. These two variables can be converted into the times (t ′ ) and frequencies ( f ′ , in Hz) at which they occur using t ′ = b and f ′ = 2ωπ0a . In a given time interval, the sum of the changes in the x-direction normal stress at each frequency ′ ) must equal the total change in the x-direction normal stress (∆σ ). This means that the relative (∆σxx xx magnitudes at each frequency and time interval (Wxx ) can be scaled to determine the change in stress at that frequency and time interval using the following equation: ′ = (t ′ , f ′ ) = Wxx (t ′ , f ′ ) · ∆σxx

∆σxx (t ′ ) . ∑∞f =0 Wxx (t ′ , f )

(13)

Eqs. (12) and (13) can be repeated for each of the remaining directions of changes in stress. Thus, ′ , ∆σ ′ , ∆σ ′ , ∆τ ′ , ∆τ ′ , the changes in stress in each direction at each time interval and frequency (∆σxx yy zz xy yz ′ ∆τzx ) are now known (with the uncertainty inherent in the Gabor wavelet transform time-frequency analysis). Power density is non-directional even though it takes into account the stresses in all directions. This necessitates a method of resolving the change in stress tensor into a single scalar stress value. This value must be independent of the coordinate system used in determining the stress components. The method used to achieve this involves first finding the eigenvalues of the A matrix, which includes the change in stress tensors at a given time t ′ and frequency f ′ :   ′ ′ ′ ′ (t ′ , f ′ ) ∆τ ′ (t ′ , f ′ ) ∆σxx (t , f ) ∆τxy zx ′ (t ′ , f ′ ) ∆σ ′ (t ′ , f ′ ) ∆τ ′ (t ′ , f ′ )  . (14) A =  ∆τxy yy yz ′ (t ′ , f ′ ) ∆τ ′ (t ′ , f ′ ) ∆σ ′ (t ′ , f ′ ) ∆τzx yz zz ′ , ∆σ ′ , and ∆σ ′ . The resolved magnitude for The three eigenvalues of A are then labelled as ∆σeig1 eig2 eig3 ′ ′ the change in stress at time t and frequency f , denoted as ∆σ ′ , can be calculated as follows: q ′ (t ′ , f ′ )]2 + [∆σ ′ (t ′ , f ′ )]2 + [∆σ ′ (t ′ , f ′ )]2 . ∆σ ′ (t ′ , f ′ ) = [∆σeig1 (15) eig2 eig3

The change in stress tensors must not be resolved into their scalar magnitudes until after the Gabor wavelet transform is applied. Otherwise, the solution would not correctly account for changes in stress in different directions. Some frequencies could be cancelled out while others could be falsely increased. The resolved magnitude for the change in stress is used to determine the power density. The power deposited into a volume results from the volume experiencing this change in stress magnitude in time. At a given time, the change in stress at every frequency contributes to the power density. The power density at time t ′ and frequency f ′ is labeled PD′ and is the product of the frequency and the change in stress at that frequency (see also Eq. (9)): PD′ (t ′ , f ′ ) = ∆σ ′ (t ′ , f ′ ) · f ′ .

(16)

The total power density from all frequencies at time t ′ is labeled PD. It is calculated by adding the power density contributions from each frequency at that time interval: ∞

∑ PD′(t ′ , f ).

PD(t ′ ) =

(17)

f =0

In practice, the highest frequency that can be considered is the Nyquist frequency, which is half of the sampling frequency. The accumulated power damage up to a time (PDaccum ) is calculated by summing the power densities at every time interval up to time t ′ as t′

PDaccum (t ) = ∑ PD(t). ′

t=0

(18)


This gives the total damage inflicted on the material from the power densities at every frequency and time interval. Power density accumulation eventually damages the component enough to cause it to fail by fatigue. The material is expected to fail once the accumulated power density reaches a certain point – the “power density threshold (PDthreshold )”. In other terms, a material is expected to fail due to fatigue when the following equation is true: PDaccum ≥ PDthreshold .

(19)

The power density threshold is an inherent material property, similar to tensile strength or fracture toughness. Every material has a defined power density threshold under certain conditions. In general, stiffer materials are likely to fail more quickly due to fatigue. Factors such as surface finish, temperature, and manufacturing method may impact the stress distribution in the material, which would affect how long it lasts under alternating stresses. For example, a rough surface would create stress concentrations that may increase the power density the component experiences, causing it to fail sooner. 3 Reported multiaxial vibration fatigue test [15] A feasibility study was performed to test the validity of the power density concept. This study used computer modeling to replicate physical testing performed by Mrˇsnik, Slaviˇc, and Bolteˇzar [15]. This testing was performed on a specimen made of cast aluminum material. The test sample was designed to encourage a multiaxial, multiple-frequency structural response to the applied loads. It was cut into a “Y” shape with a weight attached to both of its arms. There was a large hole in the center of the sample and another smaller hole beneath that one. The smaller hole was used as a mounting point at which a horizontal force excitation was applied during testing. The base of the specimen was fixed to a shaker that applied vertical kinematic excitation. Accelerometers on each arm were used to measure the system response. Images of the Y-sample and the test setup used in [15] are duplicated in Figs. 1 and 2, respectively. The vertical kinematic excitation was applied as a uniform, broadband frequency profile spanning from 380-480 Hz. This range was chosen to include the sample’s resonant frequency in the vertical direction. This expedited the testing by increasing the stresses experienced by the sample. For the horizontal force excitation, a broadband frequency profile spanning from 290Hz to 390Hz was targeted. Again, this range included the resonant frequency mode in the direction of the applied force. According to Mrˇsnik et al., the actual response was not as uniform as was intended. The force and kinematic loads were applied at these frequencies until surface cracking was detected for ten different load cases. The magnitudes of the vertical kinematic excitation and horizontal force excitation applied to the system were different for each case. For the (measured) frequency responses in the horizontal and vertical directions of the sample, the article by Mrˇsnik et al. is referred. The measured root mean square loads from each case by Mrˇsnik et al. [15] are given in Table 1 and Fig. 3. Mrˇsnik et al. predicted the fatigue lifetime and failure location for each case using the TovoBenasciutti method. The power spectral density (PSD), denoted at S(ω ), was used to give the stress intensity in the frequency domain. The moments mi of the PSD were calculated using ˆ ∞ mi = ω i S (ω ) d ω (20) −∞

where ω is the frequency in rad/sec. These moments can be used to find the expected positive crossings rate v+ 0 , expected peak rate v, and spectral width parameters αi using the following equations r m2 1 + , (21) v0 = 2π m0


Fig. 1 Schematic of the Y-Sample [15].

Fig. 2 Experimental Setup [15]. Table 1 Measured root mean square horizontal force and vertical acceleration combinations. Load cases Case

Frms (N)

arms (m/s2 )

1

2.42

12.26

2

2.51

14.81

3

2.32

16.97

4

3.11

8.44

5

3.36

9.71

6

3.36

12.56

7

3.39

14.62

8

3.94

7.85

9

4.06

10.79

10

3.73

12.46


Kinematic Excitation RMS (m/s2)

Load Cases 18 17 16 15 14 13 12 11 10 9 8 7

3

2

7 6

1

10 9

5 4 8 2

2.5

3

3.5

4

4.5

Force Excitation RMS (N)

Fig. 3 Measured root mean square horizontal force and vertical acceleration combinations [15].

1 v= 2π

r

m4 , m2

(22)

αi = √

mi . m0 m2i

(23)

TB

The Tovo-Benasciutti damage intensity (D ) was then calculated using the following equation D

TB

= [b + (1 − b)α2k−1 ]α2 D

NB

(24) NB

where the numerically determined constant (b) and narrow-band damage intensity (D ) were found as follows (α1 − α2 ) 1.112 (1 + α1 α2 − (α1 + α2 )) e2.11α2 + (α1 − α2 ) b= (25) (α2 − 1)2 p k NB (26) D = v20C−1 ( 2m0 )k Γ(1 + ). 2 The fatigue strength curve from the stress-life method was used to predict the fatigue lifetime of the material. Eq. (1) was rearranged to the following form to determine the constants C and k: C = N[(S′f )N ]k

(27)

Both C and k were chosen based on the best fit of the numerical results to the experimental results and applied to all the ten load cases reported therein. It should be noted that these equations are based on uniaxial stress alternations. However, stresses occur in every direction in this experiment. To make these equations usable, six different multiaxial methods were attempted to reduce the stress tensor to a single equivalent stress value: maximum normal stress [21], maximum shear stress [21], maximum normal and shear stress [21], Pitoiset and Preumont [22], Carpinteri-Spagnoli criterion [23], and the Projection-to-Projection approach [24]. The predictions using each of these methods were then compared to the experimental results. The experimental failure times were between 20 and 135 minutes. Mrˇsnik et al. did not explicitly report the experimental fatigue lifetimes for each case except in comparison to their theoretical predictions. They found that the six multiaxial methods typically predicted similar fatigue lifetimes. However, the lifetime predictions sometimes deviated from the experimental results by 200% or more.


Fig. 4 Predicted fatigue lifetimes vs. experimental results as reported in [15].

Fig. 5 Failure locations for each case reported in [15].

Fig. 4, an image reported by Mrˇsnik et al. in [15], compares the fatigue lifetimes projected using the maximum shear stress theory with the experimental results. The dashed line represents a 200% deviation, and the dotted line represents a 300% deviation. The results of these experiments showed that the failure location varied depending on the ratio of the force excitation magnitude to the kinematic excitation magnitude. More force-dominant excitations tended to result in cracks as much as 3.3 mm lower than the acceleration-dominant cases. The failure locations in Fig. 5 which is an image copy of Fig. 12 in [15], were given in the order of increasing force-dominance: For each case, the six multiaxial criteria always predicted approximately the same failure location. For more acceleration-dominant cases, they predicted that the cracking would occur on the outside of the sample, near the actual failure location (point C2 in Fig. 6). After a certain level of force-dominance was reached in a case, the predicted failure location jumped to a location inside the center hole (point C1 in Fig. 6). This is different from the experimental results for the force-dominant cases, which showed


Fig. 6 Predicted failure locations for force- and acceleration-dominant cases reported in [15].

cracking on the outside edge, a little below where acceleration-dominant cases failed. Fig. 6 shows the time-to-failure contour and critical points predicted by Mrˇsnik et al. for a force-dominant case (left) and an acceleration-dominant case (right): 4 Power Density feasibility study An FEA model was built in ABAQUS to test the feasibility of the power density method. Mrˇsnik, Slaviˇc and Bolteˇzar did not report some of the dimensions of the sample they tested. The unknown dimensions were estimated using the images they presented of their test setup and finite element model mesh. Pictures of the test setup were favored when accounting for any discrepancies between them and the mesh. Fig. 7 gives the dimensions of the ABAQUS model generated for the study. The material of the actual Y-sample was AlSi7Cu3. Complete properties for this material are not available, though. Therefore, the material properties of AlCuMg1 were used in this model. This same assumption was made by Mrˇsnik et al. The model was meshed with explicit three-dimensional stress elements, allowing for nonlinearity and transient loads to be well-captured. Hex elements were used so that the model would have freedom to deform in all directions. For explicit analysis in ABAQUS, hex elements must be of a linear geometric order, but second-order accuracy was placed on the calculations. Nodes were placed approximately 0.4 mm apart near the middle of the sample where the stress waves were of the most concern. The mesh gradually became coarser near the ends. A convergence study was also performed to ensure that this element size was proper. Fig. 8 shows the ABAQUS mesh used for the study. Kinematic excitation was applied vertically to the front and back faces on the bottom 10 mm of the model. It was assumed to be sinusoidal with a frequency of 465 Hz, the peak frequency found in the vertical direction of the physical tests. The excitation amplitude varied for each case such that the root mean square of the acceleration matched the values given in Table 1. These faces were also constrained to allow zero displacement in any other direction. A horizontal surface traction was applied to the lower hole. This excitation was assumed to be a 365 Hz sinusoidal wave, which was the peak frequency found in the horizontal direction of the physical tests. The traction amplitude for each case created a root mean square force on the area that corresponds with the values given in Table 1. Additionally, standard gravity was assumed in the vertical direction for the whole part at all times. Running the simulations to predicted failure would be impractical. A shorter time span was used for this feasibility study to determine which cases accumulated power density the fastest. ABAQUS/Explicit


Fig. 7 ABAQUS model with dimensions.

Fig. 8 ABAQUS mesh of Y-sample.

was employed to run the simulations for 0.301 seconds, resulting in about 140 kinematic excitation cycles and about 110 force excitation cycles. This was long enough for the system to reach a steady power density accumulation pattern. The stresses were output every 10−5 seconds. This high sampling rate produced a large amount of data. Stresses could only be taken on the surface where cracking was observed (highlighted in Fig. 5) due to computational memory limitations. Only one of the two sides needed to be examined due to symmetry. 4.1

Power density calculations and failure locations

The stress results of the ABAQUS simulations were post-processed to determine the power density accumulation of each element on this surface. These calculations were performed using the process described previously. If the change in stress over a time interval (such as ∆σxx ) was very small, any imprecision in Gabor transform magnitudes (such as Wxx ) would be magnified when that value was

318 Zachary T. Branigan, C. Steve Suh / Journal of Vibration Testing and System Dynamics 2(4) (2018) 307–326 ′ values. Therefore, the median scale scaled using Eq. (13). This would result in falsely high or low ∆σxx ′ factor across all time intervals was used to transform Wxx to ∆σxx . This was done for each element and stress direction separately. Scaling in this manner eliminated false extreme scaling and kept the results consistent for every time interval. At the beginning of the simulations, high frequency responses created high power densities. These responses dissipated quickly, and the power density accumulation stabilized. Only data from the 0.2690.301 seconds time span is used. This avoids the initial high-frequency oscillations. The power density accumulation during this time span would be expected to continue until the sample cracked. This 0.032 seconds interval is long enough to capture about 15 kinematic excitation cycles and about 12 force excitation cycles while maintaining a reasonable computation volume. Very close to the beginning and end of this interval, the Gabor wavelet transform cannot accurately determine how much each frequency contributed to the stress alternations. To circumvent this issue, results from the first and last 0.001 seconds of the time span are not considered. All of the presented results occurred between 0.27 - 0.30 seconds. This allowed the power density accumulation for each case to be reliably compared. The power densities experienced by each element were compared for each of the ten load cases. The locations that accumulated the most power density over the measured time interval are expected to fail first due to fatigue. Fig. 9 shows the crack locations for each case, sorted from most accelerationdominant to most force-dominant. It also shows the predicted power density accumulation at each location. Red indicates high accumulation, and blue indicates low accumulation. Elements that are darker in red are predicted to have a higher probability of cracking first. The element with the highest predicted power density accumulation is highlighted as a yellow box. It should be noted that because the experimental crack location images are pictures of a curved surface, the exact positions do not perfectly line up with the predicted power density accumulation at that location. Two locations emerge as having the highest power density accumulation, one on the upper half of the examined surface and one on the lower half. The force-dominant cases (Cases 4, 5, 8, 9, and 10) are predicted to have a higher probability of failing than the acceleration-dominant cases (Cases 1, 2, 3, 6, and 7). This trend is in agreement with the experimental results in [15] and is a promising indication for the validity of the power density concept. The kinematic excitation caused the Y-sample’s arms to raise and lower, creating bending stresses that were highest at the examined surface. Inspection of the system shows that the kinematic excitation alone tended to cause the largest stress magnitudes at the height where acceleration-dominant cases fail. It also caused relatively large stresses at the height where force-dominant cases fail, although these magnitudes are significantly lower than at the upper location. At all heights, the kinematic excitation induces higher power densities near the middle than at the edges. The force excitation created bending about a different axis, causing the sample to sway in the direction of the applied force. This produced high stresses on the edges of the examined surface, especially on the lower half. As the level of force-dominance was increased, the power densities near the edges increased as well. An extremely acceleration-dominant case would be projected to fail on the upper half of the examined surface, near the middle. As force-dominance (and the associated power densities near the edges) increases, the predicted failure location moved towards the edge. Then, at a certain level of force-dominance – between that of Case 6 and Case 10 – the predicted failure location switches from the upper portion to the lower portion. The power densities near the edges of the lower portion were mainly caused by the force excitation while the power densities in the upper portion were mainly caused by the kinematic excitation. At a level of force-dominance between that of Case 6 and Case 10, the force excitation created a high enough power density difference between the lower portion and the upper portion to overcome the difference caused by kinematic excitation. The most force-dominant case, Case 8, is predicted to fail 5.2 mm lower than the most acceleration-


Fig. 9 Experimental vs. predicted failure locations for each case.

dominant case, Case 3. This margin is wider than the 3.3 mm difference observed in the testing. This may be due to the finite element model having slightly different dimensions from the actual Y-sample since the exact dimensions are not given by Mrˇsnik et al. Also, it is not clear whether the reported 3.3 mm difference was measured linearly or circumferentially along the curved edge. If measured linearly, the experimental difference would be closer to the difference found by the ABAQUS simulations, which were measured circumferentially. The predicted failure locations of some cases – especially Case 10 – differed from the test results slightly. This is also likely a result of slight inaccuracies in the model’s dimensions. It was found that small changes in the model’s dimensions could affect the level of force-dominance at which the


Table 2 Power Density accumulation of the critical elements. Highest Power Density Accumulation. Case

Frms (N)

arms (m/s2 )

PDacc (GW/m3 )

1

2.42

12.26

582.45

2

2.51

14.81

660.47

3

2.32

16.97

726.53

4

3.11

8.44

536.60

5

3.36

9.71

581.62

6

3.36

12.56

617.11

7

3.39

14.62

676.78

8

3.94

7.85

676.00

9

4.06

10.79

700.65

10

3.73

12.46

652.85

predicted failure location switched from the top half to the bottom half. Additionally, the predicted power density accumulation assumes that all elements have the same properties. Any inconsistencies in the real sample – such as surface imperfections – could have affected where it actually failed. In every case, the part cracked at a location that is predicted to have a high probability of failure. 4.2

Simulation fatigue lifetimes

Table 2 gives the power density (in gigawatts per cubic meter) accumulated between 0.27-0.30 seconds by the critical element of each case. Without knowing the power density threshold for this material, fatigue lifetimes can only be estimated relative to each other. Additionally, since the failure times for each case are not available, the lifetimes predicted below using the power density concept cannot be compared with those found by Mrˇsnik et al. [15]. The critical elements with higher power density accumulation are predicted to fail more quickly. Therefore, Case 3 is expected to fail first, and Case 4 is expected to last the longest. Force excitation and kinematic excitation both affected the predicted fatigue life. Among acceleration-dominant cases, higher forces resulted in higher power densities. This can be seen by comparing Case 2 with Case 7. Both cases are acceleration-dominant and are predicted to fail at the same location. Case 7 has a lower acceleration than Case 2 but a higher force. The increased force resulted in faster power density accumulation for Case 7 than Case 2. Similarly, acceleration magnitude affects the power density accumulation in force-dominant cases, though to a slightly lesser degree. Higher accelerations cause power density to accumulate faster. 4.3

Power density response

The power density response at the critical element for each case was analyzed. There was a notable difference between the critical elements of the acceleration-dominant cases (on the upper half of the examined surface) and those of the force-dominant cases (on the lower half of the examined surface). The responses were similar among acceleration-dominant cases, with the main difference being the magnitudes of the stresses and power densities. The same was true among force-dominant cases. Fig. 10 shows the resolved changes in stress (∆σ ′ ) experienced by the critical element of Case 1 – an acceleration-dominant case – from 0.27-0.30 seconds. Fig. 11 gives the resulting power density contributions by each frequency at each time (PD′ ). The biggest changes in stress were at low frequencies, close to those of the applied excitations. However, because power density is weighted by frequency, higher frequencies contributed significantly


Fig. 10 Resolved change in stress magnitudes for the critical element of case 1.

Fig. 11 Power densities due to each frequency at each time for the critical element of case 1.

to the power density of each element. Power density peaks can be seen around the excitation frequencies as well as at approximately 9,000 Hz. The power density response was consistent throughout this time period. The power densities at each frequency combined to give the total power densities at each time (PD ) shown in Fig. 12. The figure only includes 0.28-0.29 seconds so that more detail can be seen, but the


Fig. 12 Total power densities at each time for the critical element of case 1.

Fig. 13 Power density accumulation at the critical element of case 1.


Fig. 14 Resolved change in stress magnitudes for the critical element of case 4.

Fig. 15 Power densities due to each frequency at each time for the critical element of case 4.

power densities were similar for the rest of the time period. These power density oscillations accumulated over time to form a nearly linear trend. This accumulation between 0.27 and 0.30 seconds is shown in Fig. 13. The values shown for power density accumulation (PD acc ) start at zero. In reality, power density had accumulated prior to this time. However, only the accumulation during this time period (after the stress alternations have stabilized) was


Fig. 16 Total power densities at each time for the critical element of case 4.

Fig. 17 Power density accumulation at the critical element of case 4.

considered so that initial high-frequency stresses did not affect comparisons between elements. Fig. 14 through Fig. 17 show how the power density accumulated between 0.27 and 0.30 seconds in Case 4, which is force-dominant. Other force-dominant cases gave similar results. At the critical elements of the force-dominant cases, there was an additional peak frequency around


25,000 Hz that contributed to the power density. This very high-frequency peak helped contribute enough power density in the force-dominant cases to cause the predicted failure to switch to the lower portion of the examined surface, even if it did not experience quite as high of stresses as the upper portion. Bending in the horizontal direction always occurred at the same frequency as the force excitation. However, bending of the Y-sample’s arms up and down due to kinematic excitation happened at almost twice the frequency of the applied acceleration. Small changes in the model were found to affect this frequency. If the inertia of the arms in the model was decreased – by changing the length of the arms or the size of the weights on the end – the bending frequency increased. Similarly, increasing the arm’s inertia slowed the bending in the vertical direction. Changes that caused the arm to bend faster resulted in higher power densities per cycle even if they lowered the stresses. This is a difference between the power density concept and most time-domain theories. Time-domain theories estimate that lower stresses would cause less damage, regardless of the frequencies at which they occur. 5 Concluding remarks Current methods of predicting fatigue failure lack a proper scientific basis. They rely on empirically derived equations and often neglect important factors, such as the frequencies of stress alternations. Although these theories are often useful, they have high levels of uncertainty and do not provide parameters that are of definitive physical meaning. In contrast, the power density concept is a novel way of viewing fatigue failure that is based on the physical phenomenon of power being deposited into a volume during stress oscillations. The power density concept quantifies fatigue with two physical parameters, stress variation and the frequency at which it occurs. Because the failure locations predicted using power density are in good agreement with the experimental results reported in [15], the results of this feasibility study indicate that the power density concept is a viable alternative to quantifying fatigue failure. In order to further validate and develop this theory, additional research must be performed. The power density concept can be used during the design process whenever fatigue is a concern. Improved accuracy in fatigue calculations would make fatigue failures easier to predict and avoid. Additionally, less uncertainty would allow for smaller safety factors to be used, increasing efficiency as a result. Designers could better optimize systems around fatigue failure. The concept is especially useful when stress alternations are expected to occur at multiple frequencies and directions. However, even single-frequency cyclic loading generally creates stress oscillation responses at multiple frequencies. Power density may also have applications related to other types of failure. For example, sudden impact creates broadband stress waves that propagate through a component [17,18]. The high-frequency waves would create very high power densities, causing the power density threshold to be reached quickly. This implies that lower stresses would be required to break the component, as is seen in reality. The feasibility study provides a guideline for developing an experimental plan to establish and confirm that the power density threshold is a fundamental material property. Under the novel concept, parts of the same material would be expected to fail after approximately the same amount of power density has accumulated (the power density threshold), regardless of the stress magnitudes and frequencies that made up the power density. A similar process can be used to determine the power density thresholds for various materials. Once the power density thresholds of materials have been determined, power density could be used to predict fatigue lifetimes.


References [1] Arutyunyan, R.A. (1985), Frequency dependence of the fatigue strength criterion, Strength of Materials, 17(12), 1717-1720. [2] Makhlouf, K. and Jones, J.W. (1993), Effects of temperature and frequency on fatigue crack growth in 18% Cr ferritic stainless Steel, International Journal of Fatigue, 15(3), 163-171. [3] Fatemi, A. and Yang, L. (1998), Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials, International Journal of Fatigue, 20(1), 9-34. [4] Holmes, J.W., Wu, X., and Sørensen, B.F. (1994), Frequency dependence of fatigue life and internal heating of a fiber-reinforced/ceramic-matrix composite, Journal of the American Ceramic Society, 77(12), 3284-3286. [5] Bhat, S. and Patibandla, R. (2011), Metal fatigue and basic theoretical models: a review, Alloy Steel Properties and Use, Eduardo Valencia Morales (Ed.), InTech. [6] Maktouf, W., Ammar, A., Naceur, I.B., and Sai, K. (2016), Multiaxial high-cycle fatigue criteria and life prediction: application to gas turbine blade, International Journal of Fatigue, 92(1), 25-35. [7] Budynas, R.G. and Nisbett, J.K. (2015), Shigley’s Mechanical Engineering Design, 10th ed., McGraw-Hill. [8] Mrˇsnik, M., Slaviˇc, J., and Bolteˇzar, M. (2013), Frequency-domain methods for a vibration-fatigue-life estimation – application to real data, International Journal of Fatigue, 47(1), 8-17. [9] Dirlik, T. (1985), Application of computers in fatigue analysis, Diss. U of Warwick, University of Warwick Publications Service & WRAP. [10] Benasciutti, D. and Tovo, R. (2005), Spectral methods for lifetime prediction under wideband stationary random processes, International Journal of Fatigue, 27(8), 867-877. [11] Abdullah, S., Nuawi, M.Z., Nizwan, C.K.E., Zaharim, A., and Nopiah, Z.M. (2008), Fatigue life assessment using signal processing techniques, Proc. of 7th WSEAS International Conference on Signal Processing, Robotics and Automation, University of Cambridge, Cambridge, England, 221-225. [12] Ricker, D.W. (2003), Echo Signal Processing, Norwell, Massachusetts: Kluwer Academic. [13] Marsh, G., Wignall, C., Thies, P.R., Barltrop, N., Incecik, A., Venugopal, V., and Johanning, L. (2016), Review and application of rainflow residue processing techniques for accurate fatigue damage estimation, International Journal of Fatigue, 82(3), 757-765. [14] Batsoulas, N.D. (2016), Cumulative fatigue damage: CDM-based engineering rule and life prediction aspect, Steel Research International, 87, 9999. [15] Mrˇsnik, M., Slaviˇc, J., and Bolteˇzar, M. (2016), Multiaxial vibration fatigue — a theoretical and experimental comparison, Mechanical Systems and Signal Processing, 76-77, 409-423. [16] Lin, Y., Liu, S., Zhao, X., Mao, E., Cao, C., and Suh, C.S. (2017), Fatigue life prediction of engaging spur gears using power density, Proc. IMechE Part C: Journal of Mechanical Engineering Science, DOI:10.1177/0954406217751557. [17] Qi, X. and Suh, C.S. (2010), Generalized thermo-elastodynamics for semiconductor materials subject to ultrafast heating - part II: near-field response and damage evaluation, International J. of Heat and Mass Transfer, 53, 744-752. [18] Oh, Y., Suh, C.S., and Sue, H.J. (2008), On failure mechanisms in flip chip assembly – part 1: short-time scale wave motion, ASME Transactions Journal of Electronic Packaging, 130, 021008-1-11. [19] Oh, Y., Suh, C.S., and Sue, H.J. (2008), On failure mechanisms in flip chip assembly – part 2: optimal underfill and interconnecting materials, ASME Transactions Journal of Electronic Packaging, 130, 021009-1-9. [20] Inoue, H., Kishimoto, K., and Shibuya, T. (1996), Experimental wavelet analysis of flexural waves in beams, Experimental Mechanics, 36(3), 212-217. [21] Nieslony, A. and Macha, E. (2007), Spectral Method in Multiaxial Random Fatigue, New York City, New York: Springer. [22] Pitoiset, X. and Preumont, A. (2000), Spectral methods for multiaxial random fatigue analysis of metallic structures, International Journal of Fatigue, 22(7), 541-550. [23] Carpinteri, A., Spagnoli, A., and Vantadori, S. (2014), Reformulation in the frequency domain of a critical plane-based multiaxial fatigue criterion, International Journal of Fatigue, 67(1), 55-61. [24] Cristofori, A., Benasciutti, D., and Tovo, R. (2011), A stress invariant based spectral method to estimate fatigue life under multiaxial random loading, International Journal of Fatigue, 33(7), 887-899.



Equilibrium Points with Their Associated Normal Modes Describing Nonlinear Dynamics of a Spinning Shaft with Non-constant Rotating Speed Fotios Georgiades† School of Engineering, College of Science, University of Lincoln, Lincoln, UK

Submission Info Communicated by Steve Suh Received 3 July 2018 Accepted 8 September 2018 Available online 1 January 2019 Keywords Non-constant rotating speed Nonlinear normal modes Spinning shaft Campbell diagram

Abstract In this article, the dynamics of a spinning shaft during spin-up/down operation, is examined analytically, around the equilibrium points. The system of equations of motion of a spinning shaft with nonconstant rotating speed has no linear part therefore the equilibrium points are rather essential. In the first instance, the equilibrium points of the original system are determined. A restricted system is obtained by neglecting the rigid body angular position of the shaft, and the equilibrium manifold with its’ bifurcations is defined. It is shown that this manifold, is formed by the backbone curve of the nonlinear normal modes which are associated with the rigid body angular motions of the spinning shaft. It is shown that all the equilibrium points of the original and the restricted system are degenerate. The original and restricted systems have been linearized around the equilibrium points, and examination of their stability through the eigenvalues of the Jacobian matrix showed that there are centres and unstable regions. Then, the frequencies and initial conditions of the normal modes of the linearized system around the equilibrium points are determined with the associated analytical solutions. The comparison of analytical with numerical results shows very good agreement, noting that the linearization around equilibrium is valid only for very small perturbations. This work is essential for understanding critical situations in the dynamics of the spinning shaft during spin-up/down operation, based on the associated normal modes. The stability analysis of the spinning shaft can be used further to identify regions with chaotic attractors, which should be considered for normal operation. Finally, considering other rotating structures with non-constant rotating speed, the equations of motion are similar to those of a spinning shaft. Therefore, the approach that is followed in this article can be considered as more general, and could be applied in all rotating structures during spin-up/down, and expecting similar results. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

† Corresponding



328

Fotios Georgiades / Journal of Vibration Testing and System Dynamics 2(4) (2018) 327–373

1 Introduction The examination of the dynamics in rotating structures for constant rotating speed is based on the seminal and pioneered work of Campbell in [1–3] with the development of the Campbell diagram. Since then, significant work has been done in applying Campbell diagram to describe critical situations at steady states. The critical situations or the normal modes of spinning shafts in the case of non-constant rotating speed, which describes the motion during spin-up/down operation, is an open subject with very limited publications in the literature. In [4], a model of a spinning shaft with non-constant rotating speed has been presented accompanied with coupled flexural vibration analysis, in a linear deformation regime. In [4], the equation describing the rigid body angular position of the shaft was neglected. This work has been expanded in [5] to analyse dynamics of the spinning shaft with non-constant rotating speed including flexibilities in supports, but it is limited to lateral bending motions without considering the torsional coupling. Further on, modelling and dynamic analysis of a cracked spinning shaft with non-constant rotating speed were examined in [6] without considering the rigid body rotation. In [4–6], although non-constant rotating speed was considered and nonlinear models describing the dynamics of a spinning shaft developed, the analysis is limited to the considered equations of motion on each model, neglecting the dynamics associated with coupling between different motions even in the linearized regime. Rotating structures with non-constant rotating speed coupled with the dynamics of a motor is part of the field of research called non-ideal systems [7]. Although in some cases non-constant rotating speed is considered, this research is focused on the coupled equations with the excitation through the motor and is called Sommerfeld effect and it is restricted in most cases in lumped mass models of the rotating structures [8–10]. In [11], a model of a spinning shaft with non-constant rotating speed is developed, considering all possible flexible motions of the shaft restricted to small deformations (linear deformation regime). The consideration of non-constant rotating speed leads to a nonlinear dynamical system with inertia nonlinearities. Using the multiple scales method up to a 2nd order approximation, the nonlinear normal modes are defined. The analysis captures the dynamics of relatively small rotating speeds very well and it is shown that there are detuning frequencies from the Campbell diagram. Considering dynamic analysis with non-constant rotating speed in other structures, in [12] a model of a rotating ring with non-constant rotating speed is developed and nonlinear dynamic analysis performed, but without considering the equation that describes the rigid body angular position of the ring. In [13], a model of a rotating composite beam is developed with non-constant rotating speed, in a small deformation regime and a nonzero pitch angle of the blade. As it is shown, the model is formed by strongly nonlinear partial differential equations with inertia nonlinearities similar to the one obtained in [11]. The models describing the motion of rotating structures are strongly nonlinear and one of the tools for dynamic analysis is the use of nonlinear normal modes theory. The theoretical developments in the nonlinear normal modes are described in [13] and have begun with the pioneered work, as mentioned in [13], by Kauderera and Rosemberg [14]. Since then, many publications towards the development of the nonlinear normal modes theory have been done, including the development of numerical and analytical techniques for dynamic analysis of nonlinear systems, and as an indication [15–21] can be considered. Nonlinear normal modes theory has been used in examining nonlinear dynamics of spinning shafts with other sources of nonlinearities, mainly in large deformation regime of the shaft in [22–26]. In this article, the equilibrium points and their bifurcations of a spinning shaft with non-constant rotating speed are determined analytically. Then, the normal modes are defined arising from the linearization around the equilibria, which describe the motion of the spinning shaft for small pertura

Titles in German language.


329

bations [28–31]. This article is divided into two main parts: the theoretical analysis and the numerical part with discussions. In the theoretical analysis, in the first instance, the equilibrium points of the original system are determined. Then, considering an observation in the original system is that a restricted system can arise, and its equilibrium manifold is determined. The restricted system is linearized around the equilibrium points, and their stability is examined. Also, the periodic motions around the equilibrium points are analytically determined. In the numerical part the theoretical findings are validated: of the equilibrium points of the restricted system with direct integration of the original system, theoretical determination of the eigenvalues, and verification of the linearized solutions of the normal modes. Finally, to what extent the normal modes of the linearized solutions around the equilibrium points are describing the dynamics of the original system is examined. 2 Analytical determination of equilibrium points A shaft with length-L is considered, modelled as Euler-Bernoulli beam in small deformation regime (restricted to linear equations including rotary inertia terms), made of an isotropic material (Young’s and shear modulus E and G respectively, with Poisson’s ratio−ν ), which is spinning with non-constant rotating speed. A system of partial differential equations describing all possible motions for the EulerBernoulli beam, coupled with an integro-differential equation describing rigid body rotation of the shaft has been derived in [11]. The boundary conditions considered to be, a simply supported beam for lateral bending vibrations and fixed-free for pure torsion (fixed end to rigid body rotation coordinate system). Then in [11], the system of partial differential equations has been projected to the infinite basis of the underlying linear modes which has been truncated to the first mode. In [11] all possible elastic motions of the Euler-Bernoulli beam are considered but the axial motion is fully decoupled from the rest of the equations and is neglected. The modal displacements in the derived model are lateral bending motions (qv , qw ), torsion (qφ ), rigid body rotation of the shaft (θ ). The distributed mass (m) and the inertia coefficient (I1 ) for the spinning shaft with a cyclical cross section, as obtained through integration over the area of the cross-section, are given by: Do 2 − Di 2 ), 4 Do 4 − Di 4 I1 = ρ I = πρ ( ), 64 m = ρ A = πρ (

(1a) (1b)

with, Di = 2ri , Do = 2ro , the internal and external diameters of the shaft’s cross section, respectively. In the case of a solid shaft, using Di = 0, the same formulas (eq. 1a-b) are still valid. The system of equations describing the motion is given by [11],

2I1 L + q2v + q2w + 2q2φ θ¨ − 2F q¨φ − qv q¨w + q¨v qw = −θ˙ q˙v qv − θ˙ q˙w qw − 2θ˙ q˙φ qφ (= h1 ), θ¨ qw + (1 − M) q¨v = θ˙ 2 − ωb2 (1 − M) qv − 2θ˙ q˙w (= h2 ), −θ¨ qv + (1 − M) q¨w = θ˙ 2 − ωb2 (1 − M) qw + 2θ˙ q˙v (= h3 ), −F θ¨ + q¨φ = θ˙ 2 qφ − ωT2 qφ (= h4 ),

(2a) (2b) (2c) (2d)

with constants given by, 2p 2I1 L, π I1 π 2 M = − 2 = −I1 a, mL

F=

(3a) (3b)

330


s

π 4 EI , L2 π 2 I1 + L4 m r π GI , ωT = 2L I1 ωb =

(3c) (3d)

and mode shapes for bending given by, kπ 2 sin ( x), mL L

(3e)

2 (2k − 1)π sin ( x). I1 L 2L

(3f)

yk (x) = and for torsion given by, Yk (x) =

r

r

Writing the system of equations (2) in matrix form leads to,      θ¨   mt qw −qv −2F  h1           qw (1 − M)  q ¨ h2 0 0 v   = , −qv q¨   0 (1 − M) 0   h      3   w q¨w −F 0 0 1 h4

(4)

with,

mt = 2I1 L + q2v + q2w + 2q2φ . The mass matrix is never singular since the determinant (δ ) is given by, δ = 2 (1 − M) I1 L − F 2 − Mq2v − Mq2w + 2 (1 − M) qφ2 > 0,

(5)

(6)

considering that,

8 8 I1 L = I1 L(1 − 2 ) = 0.1894I1 L > 0, 2 π π and, that all the other terms are positive. The inverse of the state-dependent mass matrix is given by,   (1 − M) −qw qv 2 (1 − M)F     qv qw q2v   −q 2 − 2F − −2Fq m −   w w t 1 −1 (1 − M) (1 − M) . M =   δ  q2w qv qw   mt − − 2F 2 2Fqv qv −   (1 − M) (1 − M) 2 2 F (1 − M) −Fqw Fqv (1 − M)mt − qv − qw I1 L − F 2 = I1 L −

Therefore, the system takes the following Cauchy form,   (1 − M) −qw qv 2 (1 − M)F       h1   θ¨   qv qw q2v      2   1  −qw − 2F − −2Fq m −  h2  w t q¨v (1 − M) (1 − M)   =  h3  . q¨  δ       q2w qv qw  w  2  h4   mt − − 2F 2Fqv qv − q¨w   (1 − M) (1 − M) F (1 − M) −Fqw Fqv (1 − M)mt − q2v − q2w and this form can be used to determine the equilibrium points.

(7)

(8)

(9)


2.1

331

Equilibria of the original system

Using the following variables, x1 = θ , x2 = qv , x3 = qw , x4 = qφ , x5 = θ˙ , x6 = q˙v , x7 = q˙w , x8 = q˙φ ,

(10a-h)

then the system (eq. 9) can be written as first order,



(1 − M)

−x3

x˙1 = x5 ,

(11a)

x˙2 = x6 ,

(11b)

x˙3 = x7 ,

(11c)

x˙4 = x8 ,

(11d) x2

2 (1 − M)F



       h1  x22 x2 x3 x˙5   2      mt − − 2F − −2Fx3   1  −x3  h2  x˙6 (1 − M) (1 − M)   =   h3  x23 x2 x3   x˙7   δ    x2 − mt − − 2F 2 2Fx2   h4  x˙8 (1 − M) (1 − M)   F (1 − M) −Fx3 Fx2 (1 − M)mt − x22 − x23 = with,

(11e)

T 1 G1 G2 G3 G4 , δ

mt = 2I1 L + x22 + x23 + 2x24 , δ = 2 (1 − M) I1 L − F 2 − Mx22 − Mx23 + 2 (1 − M)x24 ,

(12)

h1 = −x5 x6 x2 − x5 x7 x3 − 2x5 x8 x4 ,

(14a)

h2 = [x25 − ωb2 (1 − M)]x2 − 2x5 x7 ,

(14b)

[x25 − ωb2 (1 − M)]x3 + 2x5 x6 , (x25 − ωT2 )x4 .

(14c)

h3 = h4 =

(13)

(14d)

It should be noted that the vector field defined by equation (11), due to the existence of the denominator (δ ), is a ratio of polynomials without any linear counterpart. In natural nonlinear systems, in low energies the nonlinear normal modes are very close with those obtained from the linear counterpart of the dynamical system. In spinning shaft dynamics, this is not the case whereas the linear periodic counterpart is zero therefore in some sense they have been replaced by the equilibriums. Therefore, the equilibrium points play a very essential role in the dynamics. The explicit form of the right-hand side of functions (Gi , i = 1, . . . , 4) of the vector field (eq. 11e) is given by, (15a) G1 =2Mx2 x5 x6 + 2Mx3 x5 x7 − 4 (1 − M)x4 x5 x6 + 2F (1 − M)x25 x4 − 2F (1 − M) ωT2 x4 , 2M M M G2 = − x2 x3 x5 x6 + 4x3 x4 x5 x8 + 2 I1 L − F 2 x2 x25 − x32 x25 − x2 x2 x2 (1 − M) (1 − M) (1 − M) 3 5 M ωb2 (1 − M) 3 ωb2 M (1 − M) 2 x2 + x2 x3 − 2ωb2 (1 − M)x24 x2 + 2x2 x24 x25 − 2ωb2 (1 − M) I1 L − F 2 x2 + (1 − M) (1 − M) 2Mx22 x5 x7 − 4x5 x7 x24 − 2F y24 − ωT2 x3 x4 , (15b) − 4 I1 L − F 2 x5 x7 + (1 − M)

332


G3 =

2M x2 x3 x5 x7 − 4x2 x4 x5 x8 − 2 I1 L − F 2 ωb2 (1 − M)x3 + ωb2 Mx22 x3 + ωb2 Mx33 (1 − M) M M x22 x3 x25 − x3 x2 + 2x3 x24 x25 − 2ωb2 (1 − M)x3 x24 + 2 I1 L − F 2 x3 x25 − (1 − M) (1 − M) 3 5 2M + 4 I1 L − F 2 x5 x6 − x2 x5 x6 + 4x24 x5 x6 + 2Fx2 x4 x25 − 2F ωT2 x2 x4 , (1 − M) 3

(15c)

G4 =2FMx2 x5 x6 + 2FMx3 x5 x7 − 4F (1 − M)x4 x5 x8 + 2 (1 − M)I1 Lx4 x25 − Mx22 x4 x25

− Mx23 x4 x25 + 2 (1 − M)x34 x25 − 2 (1 − M)I1 LωT2 x4 + M ωT2 x22 x4 + M ωT2 x23 x4 − 2 (1 − M) ωT2 x34 .

(15d)

Determining equilibriums, the first set of equations (11a-d) lead to the following values, x0,5 = x0,6 = x0,7 = x0,8 = 0, and using eq. (16) the system of equations (11e) becomes,   (1 − M) −x3 x2 2 (1 − M)F       0   2 x ˙   5 x x x 2 3    2 2    1 − 2F − −2Fx −x m −  h2   3 3 t x˙6 (1 − M) (1 − M)   =  h3  . x˙  δ       x23 x2 x3  7  2  h4   x2 − mt − − 2F 2Fx2 x˙8   (1 − M) (1 − M) F (1 − M) −Fx3 Fx2 (1 − M)mt − x22 − x23

(16)

(17)

The definition of variables in the equilibrium point can be determined by equating the right-hand side of equation (17) to zero, which leads to the following algebraic system, −x3 h2 + x2 h3 + 2 (1 − M)Fh4 = 0 ⇔ −Fx3 h2 + Fx2 h3 = −2 (1 − M)F 2 h4 , (mt − −

x22 (1 − M)

− 2F 2 )h2 −

x2 x3 h3 − 2Fx3 h4 = 0, (1 − M)

x23 x2 x3 h2 + (mt − − 2F 2 )h3 + 2Fx2 h4 = 0, (1 − M) (1 − M) −Fx3 h2 + Fx2 h3 + (1 − M)mt − x22 − x23 h4 = 0.

Replacing eq. (18a) in equation (18d) and considering equations (12, 16, 14d) leads to, − 2 (1 − M) I1 L − F 2 − Mx22 − Mx23 + 2 (1 − M)x24 h4 = 0 ⇔ h4 = 0 ⇔ ωT2 x4 = 0 ⇔ x0,4 = 0.

(18a) (18b) (18c) (18d)

(19)

Replacing equation (19) in equation (18b) we get, (mt −

x22 x2 x3 − 2F 2 )h2 − h3 = 0, (1 − M) (1 − M)

(20)

then considering eq. (12, 14b-c) leads to −ωb2 (1 − M)(2(I1 L − F 2 ) −

Mx23 Mx22 − + 2x24 )x2 = 0 ⇐⇒ x0,2 = 0. (1 − M) (1 − M)

(21)

Replacing equation (19) in equation (18c) leads to, −

x23 x2 x3 h2 + (mt − − 2F 2 )h3 = 0, (1 − M) (1 − M)

(22)


333

then considering eq. (12, 14b-c) we get −ωb2 (1 − M)(2(I1 L − F 2 ) −

Mx23 Mx22 − + 2x24 )x3 = 0 ⇐⇒ x0,3 = 0. (1 − M) (1 − M)

(23)

Noting that equation (18a) when the equations (14b-c,19) is considered leads to, Fx3 ωb2 (1 − M)x2 − Fx2 ωb2 (1 − M)x3 = 0.

(24)

Summarizing equations (16, 19, 21, 23), shows that there is just one equilibrium point for the system of equations (11a-e) and it is given by, x0 = (x0,1 , 0, 0, 0, 0, 0, 0, 0)

and x0,1 ∈ S1 .

(25)

The angular position variable (x1 = θ ) is arbitrary since the origin of the rotation can be arbitrary. In the case that the rotating speed is in opposite direction, then θ = −θ (x1 = −x1 ), θ˙ = −θ˙ (x5 = −x5 ), θ¨ = −θ¨ (x˙5 = −x˙5 ) then, the system of eq. (11) takes the form,



    x˙5       1  x˙6 =  x˙  δ      7   x˙8 

− (1 − M)

x3

x˙1 = x5 ,

(26a)

x˙2 = x6 ,

(26b)

x˙3 = x7 ,

(26c)

x˙4 = x8 ,

(26d)

−x2

−2 (1 − M)F



   h1−  x22 x2 x3   2 −x3 mt − − 2F − −2Fx3  h2−  (1 − M) (1 − M)   h3−  ,   x23 x2 x3  h4−  x2 − mt − − 2F 2 2Fx2  (1 − M) (1 − M) F (1 − M) −Fx3 Fx2 (1 − M)mt − x22 − x23

(26e)

with, h1− = x5 x6 x2 + x5 x7 x3 + 2x5 x8 x4 , h2− = x25 − ωb2 (1 − M) x2 + 2x5 x7 , h3− = x25 − ωb2 (1 − M) x3 − 2x5 x6 , h4− = x25 − ωT2 x4 ,

(27a) (27b) (27c) (27d)

and similarly lead to the same equation (16), and using it in (eq. 27) lead to, hi− = hi , for i = 1, . . . , 4.

(28)

Which leads to the same algebraic equations (18) that must be solved for the determination of the equilibrium points. Therefore, even in the case of opposite rotating speed the equilibriums are given by equation (25). Examining in more detail the system of equations (11a-e) makes it clear that the angular position is not explicitly involved in the vector field and therefore once the rest of the coupled equations have been solved, it can be defined using direct integration of equation (11a). Since the system of equations (11b-e) can be solved independently, in the next section, the equilibriums of this restricted system will be defined.

334


2.2

Equilibria of the restricted system

In this section, the equilibria of the restricted system, which arise by neglecting the differential equation defining the angular position from the original system, are determined. The following set of variables are considered, y1 = x2 = qv , y2 = x3 = qw , y3 = x4 = qφ , y4 = x5 = θ˙ , (29a-d) y5 = x6 = q˙v , y6 = x7 = q˙w , y7 = x8 = q˙φ .

(29)

Then, the original system is restricted to,



   y˙4          y˙5 = y˙      6    y˙7  =

(1 − M)

−y2

y˙1 = y5 ,

(30a)

y˙2 = y6 ,

(30b)

y˙3 = y7 ,

(30c)

y1

2 (1 − M)F



    f1  y y y21 1 2   − 2F 2 − −2Fy2 −y2 mt −   f2  (1 − M) (1 − M)    f3    y22 y1 y2   f4  mt − − 2F 2 2Fy1 y1 −  (1 − M) (1 − M) F (1 − M) −Fy2 Fy1 (1 − M)mt − y21 − y22

1 {F1 δ

F2

F3

(30d)

F4 },

with, mt = 2I1 L + y21 + y22 + 2y23 ,

(31)

δ = 2(1 − M)(I1 L − F 2 ) − My21 − My22 + 2(1 − M)y23 ,

(32)

f1 = −2y1 y4 y5 − 2y2 y4 y6 − 4y3 y4 y7 , f2 = y24 − ωb2 (1 − M) y1 − 2y4 y6 , f3 = y24 − ωb2 (1 − M) y2 + 2y4 y5 , f4 = y24 − ωT2 y3 .

(33a) (33b) (33c) (33d)

The explicit form of the right-hand side of functions of the vector field is given by, (34a) F1 =2My1 y4 y5 + 2My2 y4 y6 − 4 (1 − M)y3 y4 y7 + 2F (1 − M)y24 y3 − 2F (1 − M) ωT2 y3 , 2M M M F2 = − y1 y2 y4 y5 + 4y2 y3 y4 y7 + 2 I1 L − F 2 y1 y24 − y3 y2 − y1 y2 y2 (1 − M) (1 − M) 1 4 (1 − M) 2 4 M ωb2 (1 − M) 3 ωb2 M (1 − M) 2 y1 + y1 y2 − 2ωb2 (1 − M)y23 y1 + 2y1 y23 y24 − 2ωb2 (1 − M) I1 L − F 2 y1 + (1 − M) (1 − M) 2My21 y4 y6 − 4y4 y6 y23 − 2F y24 − ωT2 y2 y3 , (34b) − 4 I1 L − F 2 y4 y6 + (1 − M) 2M F3 = y1 y2 y4 y6 − 4y1 y3 y4 y7 − 2 I1 L − F 2 ωb2 (1 − M)y2 + ωb2 My21 y2 + ωb2 My32 (1 − M) M M − 2ωb2 (1 − M)y2 y23 + 2 I1 L − F 2 y2 y24 − y21 y2 y24 − y3 y2 + 2y2 y23 y24 (1 − M) (1 − M) 2 4 2M y2 y y5 + 4y23 y4 y5 + 2Fy1 y3 y24 − 2F ωT2 y1 y3 , (34c) + 4 I1 L − F 2 y4 y5 − (1 − M) 2 4


335

F4 =2FMy1 y4 y5 + 2FMy2 y4 y6 − 4F (1 − M) y3 y4 y7 + 2 (1 − M)I1 Ly3 y24 − My21 y3 y24 − My22 y3 y24 + 2 (1 − M)y33 y24 − 2 (1 − M)I1 LωT2 y3 + M ωT2 y21 y3 + M ωT2 y22 y3 − 2 (1 − M) ωT2 y33 .

(34d)

The equations of the vector fields Gi (i = 1, ..., 4) and Fi (i = 1, ..., 4) of equations (11e) and (30d) could be easily shown to be the same using the change of variables defined by equations (32). Equating (eq. 30a-c) to zero leads to, y0,5 = y0,6 = y0,7 = 0. (35) Using equation (35) in the system of equations (30d) and equating the vector field to zero, leads to the following set of algebraic equations, − y2 f2 + y1 f3 + 2(1 − M)F f4 = 0, y21 y1 y2 − 2F 2 ) f2 − ( ) f3 − 2Fy2 f4 = 0, (1 − M) (1 − M) y22 y1 y2 ) f2 + (mt − − 2F 2 ) f3 + 2Fy1 f4 = 0, −( (1 − M) (1 − M)

(mt −

− Fy2 f2 + Fy1 f3 + ((1 − M)mt − y21 − y22 ) f4 = 0.

(36a) (36b) (36c) (36d)

Multiplication of equation (36a) by F leads to, −Fy2 f2 + Fy1 f3 = −2 (1 − M1 ) F 2 f4 , then using equations (31 and 37) in (eq. 36d) we get, 2 (1 − M) I1 L − F 2 − My21 − My22 + 2 (1 − M)y23 f4 = 0 ⇔ δ f4 = 0 ⇔ f4 = 0, which shows that there are two possible sets of values, y24 − ωT2 y3 = 0 ⇔ y20,4 = ωT2 or y0,3 = 0,

(37)

(38)

(39a-b)

Using equation (36a) with (eq. 38) leads to,

−y2 f2 + y1 f3 = 0 ⇔ y1 f3 = y2 f2 ,

(40)

Considering equations (38, 40) in (eq. 36b) leads to, y21 y2 − 2F 2 ) f2 − ( )y2 f2 = 0, (1 − M) (1 − M)

(41a)

y21 δ y22 − 2F 2 − ) f2 = 0 ⇔ f2 = 0 ⇔ f2 = 0, (1 − M) (1 − M) (1 − M)

(41b)

(mt − or (mt − which leads to,

[y24 − ωb2 (1 − M)]y1 = 0 ⇔ y20,4 = ωb2 (1 − M) or y0,1 = 0,

(42a-b)

Considering equations (38,40) in (eq. 36c) leads to, −( or with (mt − we get,

y22 y1 )y1 f3 + (mt − − 2F 2 ) f3 = 0, (1 − M) (1 − M)

y22 y21 δ − 2F 2 − ) f3 = 0 ⇔ f3 = 0 ⇔ f3 = 0, (1 − M) (1 − M) (1 − M) [y24 − ωb2 (1 − M)]y2 = 0 ⇔ y20,4 = ωb2 (1 − M) or y0,2 = 0,

(43a)

(43b) (44a-b)

Finally, combining the results of equations (35,39a-b,42a-b,44a-b) lead to the following families of equilibria:

336

Fotios Georgiades / Journal of Vibration Testing and System Dynamics 2(4) (2018) 327–373 (1)

1. The first family of equilibria (y0 ) corresponds to any arbitrary rotating speed, y0,4 = θ˙0 ∈ R, and using the equations (35,39b,42b,44b) lead to, (y0,1 , y0,2 , y0,3 , y0,5 , y0,6 , y0,7 ) = q0,v , q0,w , q0,φ , q˙0,v , q˙0,w , q˙0,φ = (0, 0, 0, 0, 0, 0) . Therefore, (1) y0 = 0, 0, 0, θ˙0 , 0, 0, 0 with θ˙0 ∈ R,

(45a)

which is a family of fixed points without any initial velocities and/or deformation of the shaft for any arbitrary value of the rotating speed. (2)

2. The second family of equilibria (y0 ) correspond to the following rotating speed, y0,4 = θ˙0,1 = ±ωb

p

(1 − M),

and considering equations (35, 39b) it leads to, (y0,3 , y0,5 , y0,6 , y0,7 ) = q0,φ , q˙0,v , q˙0,w , q˙0,φ = (0, 0, 0, 0) , and using (eq. 42a, 44a) we get, (y0,1 , y0,2 ) = (q0,v , q0,w ) ∈ R2 . Therefore, (2)

y0 = (q0,v , q0,w , 0, ±ωb

p

(1 − M), 0, 0, 0) with (q0,v , q0,w ) ∈ R2 ,

(45b)

which is a family of fixed points that accept arbitrary values for lateral bending deformation of the shaft, a specific value of angular velocity, with the torsional position and all the velocities being zero. (3)

3. The third family of equilibria (y0 ) is for the following rotating speed, y0,4 = θ˙0,2 = ±ωT , using equations (35,42b, 44b) we get, (y0,1 , y0,2 , y0,5 , y0,6 , y0,7 ) = q0,v , q0,w , q˙0,v , q˙0,w , q˙0,φ = (0, 0, 0, 0, 0) , and, considering equation (39a) leads to y0,3 = q0,φ ∈ R. Therefore, (3) y0 = 0, 0, q0,φ , ±ωT , 0, 0, 0 with q0,φ ∈ R,

(45c)

which is a family of fixed points that accept arbitrary values for torsional deformation of the shaft, a specific value of rotating speed, with lateral bending deformation and all the velocities being zero.


337

These are the three families of equilibrium points for the reduced system, and in any case, y0,4 = x5 = θ˙0 = ct,

(46a)

and then through equation (11a) the angular position θ ∈ S1 is given by, x1 = θ = θ˙0 t + θ0 ,

(46b)

which are periodic rigid body motions of the spinning shaft with period, T=

2π . θ˙0

(46c)

Therefore, these solutions form a family of normal modes with the rigid body motions of the spinning shaft. The equilibria of the restricted system form the rigid body normal modes of the original system. It can be shown that the same equilibriums are valid for the system that arises considering constant rotating speed in the set of equations (11). In the case that the rotating speed is in the opposite direction, then the system of eq. (30) takes the form, y˙1 = y5 ,

(47a)

y˙2 = y6 ,

(47b)

y˙3 = y7 ,

(47c)

  − (1 − M) y2 −y1 −2 (1 − M)F         f1−  y˙4   y21 y1 y2      2   1  −y2 m − − 2F − −2Fy   f2−  t 2 y˙5 (1 − M) (1 − M)   =    f3−   δ   y˙6   y22 y1 y2   2    f4−  m − − 2F 2Fy y − t 1 1 y˙7   (1 − M) (1 − M) F (1 − M) −Fy2 Fy1 (1 − M)mt − y21 − y22 1 = F1− F2− F3− F4− , δ with,

(47d)

f1− = 2y1 y4 y5 + 2y2 y4 y6 + 4y3 y4 y7 ,

(48a)

f2− = [y24 − ωb2 (1 − M)]y1 + 2y4 y6 ,

(48b)

f3− = [y24 − ωb2 (1 − M)]y2 − 2y4 y5 , f4− = (y24 − ωT2 )y3 .

(48c) (48d)

The set of equations (35) is still valid, therefore, fi− = fi , for i = 1, . . . , 4,

(49)

which leads again to the same system of algebraic equations (36) that must be solved for the determination of the equilibria. Therefore, even in the case that the shaft is spinning in the opposite direction, the equilibria are defined by the same three families and hence clear the existence of both signs in rotating speeds for 2nd and 3rd families of equilibria. As is shown in Appendix C with inequality (C.4), for the Euler-Bernoulli beams the rotating speeds of the 2nd family of fixed points correspond to lower rotating speeds than the rotating speeds of the 3rd family of fixed points.

338


3 Linearization around equilibria, eigenvalues, backbone curves and stability 3.1

Linearization around the equilibrium of the original system

In this section, linearization around the fixed point of the original system is performed, by considering perturbations (ζi , i = 1, . . . , 8) around the fixed point (defined by eq. 25) as follows, x = (x0,1 + ζ1 , x0,2 + ζ2 , x0,3 + ζ3 , x0,4 + ζ4 , x0,5 + ζ5 , x0,6 + ζ6 , x0,7 + ζ7 , x0,8 + ζ8 ) = (x0,1 + ζ1 , ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 , ζ8 ) .

(50)

Therefore, the system of equations (11) can be written in the following linearized form around the equilibrium: T T (51) ζ˙1 ζ˙2 ζ˙3 ζ˙4 ζ˙5 ζ˙6 ζ˙7 ζ˙8 = [Jx ]|x0 ζ1 ζ2 ζ3 ζ4 ζ5 ζ6 ζ7 ζ8 , Whereas, to simplify the expression, the Jacobian is split to upper and lower part as follows, J [Jx ] = x,u . Jx,l

(52)

The upper part (Jx,u ) is given by,  0 0 [Jx,u ] =  0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

 0 0  = ct, 0 1

(53)

and the lower part (Jx,l ) is given by, [Jx,l ] = −

1 ∂δ 1 ∂ Gi Gi + , 2 δ ∂xj δ ∂xj

(54)

note, that in case of equilibria of the system with equations (11), then, Gi = 0,

(55)

therefore, in the lower part of the Jacobian (eq. 54) the 1st term vanishes and lead to, 1 ∂ Gi [Jx,l ]|x0 = . δ ∂ x j x0 The explicit form of the partial derivatives of the functions of and considering the values of the equilibrium leads to,  0 0 0 0 0 0 0 0  0 0 0 0  0 0 0 0    −F ωT2  0 [Jx ]|x0 = 0 0  (I1 L − F 2 )  2 0 −ωb 0 0  0 0 −ωb2 0    −I1 LωT2 0 0 0 (I1 L − F 2 )

(56)

the vector field is given in Appendix-A, 1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0

 0 0  0  1     0 .   0  0    0

(57)


339

Whereas, it has been considered that the form of the denominator in the equilibrium point is given by, δ (x0 ) = 2 (1 − M) I1 L − F 2 . (58) Finally, three independent sets of equations from the linearized original system can be derived, and they are given by, -Set-1 (59a) ζ˙1 = ζ5 ,

and this equation defines the angular position when there is perturbation in angular velocity, once it is determined by the set-3 of equations. -Set-2 ˙     ζ2  0 0 1 0     ζ2    ˙     ζ3  ζ3 0 0 0 1   = , (59b) ζ6  −ωb2 0 0 0   ζ˙6        ˙  0 −ωb2 0 0 ζ7 ζ7

describes the lateral bending motions and is non-zero when there is a perturbation in the generalized coordinates that define lateral bending motion. -Set-3 (59c) ζ˙4 = ζ8 , −F ωT2 ζ4 , (I1 L − F 2 )

(59d)

−I1 LωT2 ζ˙8 = ζ4 . (I1 L − F 2 )

(59e)

ζ˙5 =

Taking the derivative of (eq. 59c) and substituting it in (eq. 59e) leads to,

ζ¨4 +

I1 LωT2 ζ4 = 0, (I1 L − F 2 )

(59f)

which can be solved easily (typical 1-degree of freedom oscillator) and then the solution can be used in (eq.59d-e) with direct integration to define the rest of the perturbations. Equation (59f) describes torsional motions and rigid body angular velocity, and it is non-zero when there is either a perturbation in angular velocity, or in torsional angle, or velocity of the torsional angle. Examining just the upper part of the Jacobian (Jx,u ) all rows have constant values which leads to a Hessian with a zero determinant, therefore this equilibrium point is degenerate. 3.2

Linearization around the equilibria of the restricted system

In this section linearization around the equilibria of the restricted system is performed, considering the following perturbations (ξi , i = 1, . . . , 8), y = (y1 , y2 , y3 , y4 , y5 , y6 , y7 ) = (y0,1 + ξ1 , y0,2 + ξ2 , y0,3 + ξ3 , y0,4 + ξ4 , y0,5 + ξ5 , y0,6 + ξ6 , y0,7 + ξ7 ) ,

(60)

lead to, T T ξ˙1 ξ˙2 ξ˙3 ξ˙4 ξ˙5 ξ˙6 ξ˙7 = [Jy ]|y(i) ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 , with i = 1, 2, 3

(61)

0

and as previously the Jacobian (Jy ) can be split into upper (Jy,u ) and lower (Jy,l ) parts as follows, J [Jy ] = y,u . (62) Jy,l

340


The upper part (Jy,u ) is given by,   0 0 0 0 1 0 0 [Jy,u ] = 0 0 0 0 0 1 0 = ct, 0 0 0 0 0 0 1

(63)

and the lower part (Jy,l ) is given by, 1 ∂ Fi 1 ∂δ Fi + , [Jy,l ] = − 2 δ ∂yj δ ∂yj

(64)

it should be noted, that in the case of equilibria of the system with equations (30), Fi = 0,

(65)

Therefore, the lower part of the Jacobian (Jy,l ) is given by, 1 ∂ Fi [Jy,l ]|y0 = . δ ∂ y j y0

(66)

Similarly, with regards to the Hessian of the linearized original system, with the same arguments ([Jy,u ] = ct) the determinant of the Hessian is zero, therefore; all the equilibria of the restricted system are degenerate. 3.2.1

Linearization around the 1st family of equilibrium points

Considering equations (45a, 63, 66), with the explicit form of the partial derivatives given in AppendixA by equations (A.2-A.29), and the interchange of variables using equation (29), then the Jacobian ([Jy ]|y(1) ) for the 1st family of equilibrium points takes the form, 0



[Jy ]|y(1) 0

0 0  0  = 0 a1  0 0

0 0 0 0 0 0 0 Fa2 0 0 a1 0 0 I1 La2

0 0 0 0 0 0 0

 1 0 0 0 1 0  0 0 1  0 0 0 , 0 −a3 0  a3 0 0 0 0 0

(67)

with ai —parameters dependent on angular velocity θ˙0 and given by, (1)

δ (y0 ) = 2(1 − M)(I1 L − F 2 ), (θ˙ 2 − ωb2 (1 − M)) a1 (θ˙0 ) = 0 , (1 − M) (θ˙ 2 − ωT2 ) , a2 (θ˙0 ) = 0 (I1 L − F 2 ) 2θ˙0 . a3 (θ˙0 ) = (1 − M)

(68a) (68b) (68c) (68d)

In explicit form the linearized equations around the first family of equilibrium points leads to two independent sets of equations, -First system,

ξ˙3 = ξ7 ,

(69a)


ξ˙4 = Fa2 ξ3 , ξ˙7 = I1 La2 ξ3 .

341

(69b) (69c)

This set of equations describes the perturbed torsional generalized coordinates (positions and/or velocities). Taking the derivative of (eq. 69a) and substituting it into (69c) we get,

ξ¨3 − I1 La2 ξ3 = 0,

(70)

which is a typical 1 degree of freedom oscillator, and once it is solved the other perturbed parameters can be determined using the equations (69b-c). -Second system ˙     ξ 0 0 1 0 ξ1  1            ˙   ξ2  0 0 0 1  ξ2 , = (71) a1 0 0 −a3  ξ5   ξ˙5        ˙  0 a1 a3 0 ξ6 ξ6 which describes the perturbation of lateral bending generalized coordinates (positions and/or velocities). 3.2.2

Linearization around the 2nd family of equilibrium points

For this family equations (45b, 63, 66) are considered, accompanied with the explicit form of the partial derivatives given in Appendix-A (eq. A.2-A.29), and the interchange of variables of (eq. 29), which leads to the Jacobian ([Jy ]|y(2) ) for the 2nd family of equilibrium points, 0



[Jy ]|y(2) 0

0 0  0  = 0 0  0 0

0 0 0 0 0 0 0 0 0 0 c9 0 0 − c8 y0,2 c3 y0,1 0 c8 y0,1 c3 y0,2 0 c6 0

1 0 0 c1 −c7 c5 Fc1

0 1 0 c2 −c4 c7 Fc2

 0 0  1  0 , 0  0

(72)

0

whereas the ci -parameters are dependent on (y0,1 , y0,2 ) and they are given by, p 2My0,1 ωb (1 − M) , c1 (y0,1 , y0,2 ) = (2) δ (y0 ) p 2My0,2 ωb (1 − M) c2 (y0,1 , y0,2 ) = , (2) δ (y0 ) p 2ωb (1 − M) , c3 = (1 − M) p 2ωb (1 − M)[2(1 − M)(I1 L − F 2 ) − My20,1 ] c4 (y0,1 , y0,2 ) = , (2) (1 − M)δ (y0 ) p 2ωb (1 − M)[2(1 − M)(I1 L − F 2 ) − My20,2 ] , c5 (y0,1 , y0,2 ) = (2) (1 − M)δ (y0 ) c6 (y0,1 , y0,2 ) =

(2(1 − M)I1 L − My20,1 − My20,2 )(ωb2 (1 − M) − ωT2 ) (2)

δ (y0 )

(73a) (73b) (73c) (73d) (73e) ,

(73f)

342


c7 (y0,1 , y0,2 ) = c8 (y0,1 , y0,2 ) = c9 (y0,1 , y0,2 ) =

2My0,1 y0,2 ωb

p

(1 − M) (2)

(1 − M)δ (y0 ) 2F(ωb2 (1 − M) − ωT2 ) (2)

δ (y0 )

,

(73g)

,

2F(1 − M)(ωb2 (1 − M) − ωT2 ) (2)

δ (y0 )

(73h) ,

(73i)

and, (2)

δ (y0 ) = 2(1 − M)(I1 L − F 2 ) − My20,1 − My20,2 ,

(73j)

f1 = ωT2 − ωb2 (1 − M).

(73k)

Examination of the Jacobian ([Jy ]|y(2) ) indicates that in the case of perturbations the linearized system 0 that must be solved for defining all perturbations is given by,

ξ˙3 = ξ7 , ξ˙4 = c9 ξ3 + c1 ξ5 + c2 ξ6 ,

(74a) (74b)

ξ˙5 = − c8 y0,2 ξ3 + c3 y0,1 ξ4 − c7 ξ5 − c4 ξ6 , ξ˙6 = c8 y0,1 ξ3 + c3 y0,2 ξ4 + c5 ξ5 + c7 ξ6 ,

(74d)

ξ˙7 = c6 ξ3 + Fc1 ξ5 + Fc2 ξ6 ,

(74e)

(74c)

and then, the other perturbed variables arise with direct integration of,

3.2.3

ξ˙1 = ξ5 ,

(74f)

ξ˙2 = ξ6 .

(74g)

Linearization around the 3rd family of equilibrium points

In the 3rd family of equilibrium points, equations (45c, 63, 66) are considered, with the explicit form of the partial derivatives given in Appendix-A by the equations (A.2-A.29) and, the interchange of variables (eq. 29), which lead to the Jacobian ([Jy ]|y(3) ) of the 3rd equilibrium point, 0



[Jy ]|y(3) 0

0 0  0 0   0 0  = 0 0  b2 f1 0   0 b2 f1 0 0

 0 0 1 0 0 0 0 0 1 0   0 0 0 0 1   0 b1 F 0 0 −b1  , 0  0 0 0 −2b2 ωT  0 0  0 0 2b2 ωT 0 b1 b3 0 0 −Fb1

(75)

with the bi -parameters to be dependent to y0,3 and they are given by, b1 (y0,3 ) =

2ωT y0,3 , (I1 L − F 2 + y20,3 )

(76a)

1 , (1 − M)

(76b)

b3 (y0,3 ) = I1 L + y20,3 .

(76c)

b2 =


343

Which lead to three independent systems, -First system

ξ˙3 = ξ7 ,

(77)

ξ˙4 = b1 F ξ4 − b1 ξ7 , ξ˙7 = b1 b3 ξ4 − Fb1 ξ7 ,

(78a)

-Second system

(78b)

which describes coupled motions of rigid body angular velocity with torsion. -Third system

ξ˙1 = ξ5 , ξ˙2 = ξ6 ,

(79a) (79b)

ξ˙5 = b2 f1 ξ1 − 2b2 ωT ξ6 , ξ˙6 = b2 f1 ξ2 + 2b2 ωT ξ5 .

(79c) (79d)

which describes lateral bending motions. 3.3

Eigenvalues of the Jacobian of the original system

In this section, the eigenvalues of the Jacobian ([Jx ]|x0 ) of the linearized original system are determined. In Appendix-B it is shown explicitly that they are given by,

σ1,2 = 0,

(80a)

σ3,4 = ± I1 La2 (0), v u q 2 2 u 2 a3 (0) − 2a1 (0) − 4a21 (0) t − a3 (0) − 2a1 (0) ± σ5÷8 = ± . 2 p

(80b) (80c)

Considering equations (68) with zero rotating speed, a1 (0) = −ωb2 , a2 (0) =

(81a)

−ωT2

(I1 L − F 2 ) a3 (0) = 0,

,

(81b) (81c)

Then,

σ1,2 = 0,

(82a) s

σ3,4 = ± −

s

I1 LωT2 = ±i (I1 L − F 2 )

σ5÷8 = ±iωb , (double).

I1 LωT2 = ±iωT,0 , (I1 L − F 2 )

(82b) (82c)

Summarizing, the eigenvalues of the Jacobian of the original system are purely complex, with the 2nd and 3rd eigenvalues describing torsional oscillations of the perturbed solution, and the last four eigenvalues describing the lateral bending oscillations.

344


3.4

Eigenvalues of the Jacobians of the restricted system

3.4.1

Eigenvalues of the Jacobian of the 1st family of equilibria

As it is shown in Appendix-B, the eigenvalues are given by the roots of equation (B.7) and in explicit form they are,

λ1 = 0,

(83a)

q λ2,3 θ˙0 = ± I1 La2 θ˙0 , v u q 2 2 u 2 ˙ ˙ − 4a21 θ˙0 a3 θ˙0 − 2a1 θ˙0 t − a3 θ0 − 2a1 θ0 ± λ4÷7 θ˙0 = ± . 2

(83b)

(83c)

a) The eigenvalues λ2,3 are real in the case of, I1 L(θ˙02 − ωT2 ) I1 La2 (θ˙0 ) ≥ 0 ⇔ ≥ 0 ⇔ (θ˙0 − ωT )(θ˙0 + ωT ) ≥ 0, (I1 L − F 2 )

(84)

restricting the examination to positive rotating speeds lead to,

θ˙0 ≥ ωT .

(85)

Therefore, in the case of (eq. 85) the eigenvalues are real or zero, and for smaller rotating speeds they are purely complex. b) In this part, considering different values of θ˙0 , the form of eigenvalues λ4÷7 is examined, by means of being purely complex or having nonzero real part: Using equations (68a-d) then the explicit form of parts of λ4÷7 are given, a23 − 2a1 =

2 [(1 + M)θ˙02 + (1 − M)2 ωb2 ] > 0, (1 − M)2

Also, q

2 a23 − 2a1 − 4a21

=

4θ˙0 2

(1 − M)

q

M θ˙02 + ωb2 (1 − M)2 .

(86)

(87)

In case of, √ √ M θ˙02 + ωb2 (1 − M)2 ≥ 0 ⇔ ωb (1 − M) − −M θ˙0 ωb (1 − M) + −M θ˙0 ≥ 0,

(88)

ωb (1 − M) = θ˙0,3 , θ˙0 ≤ √ −M

(89)

Then, restricted to positive rotating speeds,

and, 2 a23 − 2a1 − 4a21 > 0, 2 2 a23 − 2a1 − 4a21 < a23 − 2a1 ,

Therefore, −

a23 − 2a1

+

q

a23 − 2a1

2

− 4a21 < 0,

(90) (91)

(92a)


2 q 2 a3 − 2a1 − 4a21 < 0. − a23 − 2a1 −

Since the quantities of the root of (eq. 83c) are negative, then the eigenvalues are given by, v u q 2 2 u 2 ˙ ˙ a3 θ˙0 − 2a1 θ˙0 − 4a21 θ˙0 t a3 θ0 − 2a1 θ0 ∓ . λ4÷7 θ˙0 = ±i 2

345

(92b)

(93)

In the case that the rotating speed θ˙0 does not satisfy (eq. 88) then, q

a23 − 2a1

2

− 4a21 = i

4θ˙0 2

(1 − M)

q

−M θ˙02 − ωb2 (1 − M)2 ,

and the eigenvalues are given by, v q u u 2 (θ˙ ) − 2a (θ˙ )) ± i 4a2 (θ˙ ) − (a2 (θ˙ ) − 2a (θ˙ ))2 −(a t 1 0 1 0 1 0 3 0 3 0 λ4÷7 (θ˙0 ) = ± 2 s −(a23 (θ˙0 ) − 4a1 (θ˙0 )) a3 (θ˙0 ) ±i ). = ±( 4 2

(94)

(95)

Summarizing, the eigenvalues of the Jacobian of the 1st family of the equilibrium points, the following three different regions with respect to angular velocity can be identified; REGION-1 The angular velocity belongs to y0,4 = θ˙0 ∈ (0, ωT ), and the eigenvalues are given by,

λ1 = 0, s

λ4÷7 θ˙0 = ±

i (1 − M)

(96a)

I1 L θ˙02 − ωT λ2,3 θ˙0 = ±i − = ±iΛ2 , (I1 L − F 2 ) r q 2 2 2 ˙ ˙ (1 + M) θ0 + (1 − M) ωb ∓ 2θ0 M θ˙02 + ωb2 (1 − M)2 = ±iΛ3÷4 . 2

(96b) (96c)

REGION-2 ), and the eigenvalues are given by, In this region the angular velocity belongs to y0,4 = θ˙0 ∈ (ωT , ωb√(1−M) −M

λ1 = 0, λ2,3 θ˙0 = ±

(97a) s

I1 L θ˙02 − ωT = ±Λ5 , (I1 L − F 2 ) 2

(97b)

whereas the rest of the eigenvalues are given by equations (96c). REGION-3 , +∞), and the 1st eigenvalue is given by, The angular velocity belongs to y0,4 = θ˙0 ∈ ( ωb√(1−M) −M

λ1 = 0.

(98a)

346


The eigenvalues λ2,3 , describing torsional perturbed motions, are given by equation (97b). In lateral bending the eigenvalues are given by, s −M θ˙02 θ˙0 − ωb2 ± i ) = ±(Λ6 ± iΛ7 ), λ4÷7 (θ˙0 ) = ±( 2 (1 − M) (1 − M)

(98b)

with the real part (Λ6 ) being zero in the case that the angular velocity is equal to the left limit of this region. 3.4.2

Eigenvalues of the Jacobian of the 2nd family of equilibria

The eigenvalues of the Jacobian of the 2nd family of equilibrium points, obtained from the roots of eq. B.23 with, eq. B.14, 16, 19, 20, are given by,

η1÷3 = 0,

(98a)

η4÷7 (z) = ±

s

−β ±

p β 2 − 4γ , 2

(98b)

with,

β 2 − 4γ =

(r1 z − r2 )2 + r32 z (2) 2

δ (y0 )

> 0,

(99)

and constants given by, r1 = M 4ωb2 − f1 , r2 = 2 4ωb2 I1 L − F 2 − (1 − M)I1 L f1 ,

(100a) (100b)

r32 = 32M 2 ωb2 f1 F 2 .

(100c)

Also considering that,

β 2 − 4γ < β 2 ⇐⇒ and,

β 2 − 4γ < β ⇐⇒ −β +

−β −

then, s

η4÷7 (z) = ±i 3.4.3

p

p β 2 − 4γ < 0,

p β 2 − 4γ < 0,

(102a)

(102b)

v q u p u 2 p + p z ∓ (r1 z − r2 )2 + r32 z t 0 1 β ∓ β − 4γ = ±iH1÷2 = ±i . 2 2δ

(103)

Eigenvalues of the Jacobian of the 3rd family of equilibria

The eigenvalues of the Jacobian of the 3rd family of equilibrium points, are defined by the roots of equations B.27 and, they are explicitly given by,

µ1 = 0, µ2,3 (y0,3 ) = ±b1

µ4÷7 = ±

q

(F 2 − b3 ) = ±i q

(104a) 2ωT y0,3 (I1 L − F 2 + y20,3 )

= ±iM1 ,

v q u u −(4b2 ω 2 − 2b f ) ± (4b2 ω 2 − 2b f )2 − 4b2 f 2 t 2 1 2 1 T 2 1 2 2 T 2

(104b)


=± Also, (

v r u u 2 2 2 u −( 4ωT 2 − 2 f1 ) ± ( 4ωT 2 − 2 f1 )2 − 4 f1 2 (1−M) (1−M) (1−M) (1−M) t (1−M) 2

.

16ωT2 ωb2 2 f1 2 4 f12 4ωT2 − ) − = (M β 2 + (1 − M)2 ) > 0, (1 − M)2 (1 − M) (1 − M)2 (1 − M)4

and 2 f1 4ωT2 )± − −( 2 (1 − M) (1 − M)

s

(

4 f12 4ωT2 2 f1 2 ) − − < 0, (1 − M)2 (1 − M) (1 − M)2

347

(104c)

(105a)

(105b)

therefore, i µ4÷7 = ± (1 − M)

r

(1 + M)ωT2 + (1 − M)2 ωb2 ∓ 2ωT

q

M ωT2 + ωb2 (1 − M)2 = ±iM3÷4 = ct.

(106)

In the 3rd family of equilibrium points, all the eigenvalues of the Jacobian are purely imaginary. 3.4.4

Eigenvalues of particular equilibria of the 1st family

POINT-1 On√this bifurcation point the only non-zero value is the angular velocity which is given by y0,4 = θ˙0 = ωb 1 − M, and the eigenvalues are given by,

λ1 = 0 = η 1 , r √ I1 L ωT2 − ωb2 (1 − M) p0 − r2 = ±iΛ2 = ±iH1 = ±i λ2,3 ωb 1 − M = ±i = η4÷5 (0), 2 (I1 L − F ) 2δ √ λ4÷5 ωb 1 − M = ±iΛ3 = ±0 = η2÷3 , r √ p0 + r2 2ωb λ6÷7 ωb 1 − M = ±i p = ±iΛ4 = ±iH2 = ±i = η6÷7 (0), 2δ (1 − M) s

(107a) (107b) (107c) (107d)

whereas it is clear that on this point, the eigenvalues of the 1st family of equilibrium points coincide with the eigenvalues of the 2nd family of equilibria. POINT-2 On this bifurcation point, the angular velocity is y0,4 = ωT , with all the other variables being zero, and the eigenvalues are given by,

s

λ1 = µ1 = 0,

(108a)

I1 L ωT2 − ωT = 0 = µ2,3 (0) , (I1 L − F 2 )

(108b)

λ2,3 (ωT ) = ± −

2

r q i 2 2 2 λ4÷7 (ωT ) = ± (1 + M) ωT + (1 − M) ωb ∓ 2ωT M ωT2 + ωb2 (1 − M)2 (1 − M) = ±iΛ3÷4 = ±iM3÷4 = µ4÷7 (0),

(108c)

whereas the eigenvalues of the 1st family of equilibria coincide with the eigenvalues of the 3rd family of equilibria.

348


POINT-3 This point corresponds to angular velocity y0,4 = ωb√(1−M) with all other variables being zero, and it is −M st the limiting point that the 1 family of equilibriums of the linearized system starts with increase of angular velocity to have eigenvalues with positive real part, also in lateral bending motions, and the eigenvalues are given by,

λ1 = 0,

(109a) s

I1 L[(1 − M)2 ωb2 + M ωT2 ] ωb (1 − M) √ , )=± −M(I1 L − F 2 ) −M ωb (1 − M) ωb λ4÷7 ( √ ) = ±(Λ6 ± iΛ7 ) = ±i √ (double). −M −M

λ2,3 (

(109b) (109c)

This is the only point of the 1st family that the real part (Λ6 ) of the eigenvalues of the Jacobian defined by equation (98b) becomes zero. 3.5

Backbone curves of the rigid body normal modes and stability of equilibrium points

In this section, the analytical results of the equilibrium points are summarized, and the stability is discussed. The original system has just one equilibrium point (eq. 25) and, based on the linearization results, the stability cannot be judged since it has zero eigenvalues (eq. 82a) while the rest being purely complex (eq. 82b-c). The restricted system has many equilibrium points that are derived in §2.2 and using the associated eigenvalues (from §3.4) the stability of them will be commented on. The full set of equilibriums forms a 4-D manifold which is called the manifold of equilibrium points including their bifurcations [30]. In order to construct the equilibrium manifold, the inequalities in Appendix-C (C.2, C.4, C.17) are used. Figure 1 depicts the projected manifold of the equilibrium points of the restricted system in 3-D, with all their bifurcations and their associated eigenvalues. The manifold arises by considering that the lateral bending motions can be grouped together since any perturbation of lat2 2 eral bending motions is associated with eigenvalues dependent on z = y1 + y2 and lead, to periodic motions in both lateral bending directions. Therefore, this manifold is ‘complete’, since it has all the information from the 4D manifold. Since the equilibriums of the restricted system correspond to the rigid body rotations of the original system, it can be concluded that; The manifold of equilibriums of the restricted system is the backbone curve of the rigid body motion normal modes, of the spinning shaft with non-constant rotating speed. The behaviour and the stability of the perturbed solutions around points from equilibrium manifolds is a field under-development in mathematics [30]. In this article, the stability determination is done by using the standard theory of examination of the eigenvalues of the Jacobian of the linearized system around the equilibrium, whereas an unstable point should have an eigenvalue with positive real part [31]. The direct examination of perturbations of each variable of points of the equilibrium manifold will be commented on (fig. 1). A more concrete decision about the stability of the fixed points can be given by considering either Lyapunov functions or the normal forms theory [31, 33]. Due to the fact that the restricted system does not arise from the Hamiltonian of the original system, the derivation of the Lyapunov function around the equilibrium points it is not straight-forward. One way is to use the Variable Gradient Method [32], which leads to a system of twenty-one partial differential equations and finding the solution is quite involved. Also, the application of normal forms method is quite involved too, since the system has seven variables and can be done in a separate article as continuation of this work [33].


349

ߣͳ ൌ ߟͳ ൌ ૙

ߣʹǡ͵ ൫ܾ߱ ξͳ െ ൯ ൌ ߟͶൊͷ ሺͲሻ ൌ േ࢏‫ ͳܪ‬ൌ േ࢏߉ʹ ߣͶൊͷ ൫ܾ߱ ξͳ െ ൯ ൌ ߟʹൊ͵ ൌ ૙

ߣ͸ൊ͹ ൫ܾ߱ ξͳ െ ൯ ൌ ߟ͸ൊ͹ ሺͲሻ ൌ േ࢏‫ ʹܪ‬ൌ േ࢏߉Ͷ

ࢠ ൌ ࢟૛૙ǡ૚ ൅ ࢟૛૙ǡ૛

ߟͳൊ͵ ൌ ૙ ߟͶൊ͹ ሺ‫ݖ‬ሻ ൌ േ࢏‫ͳܪ‬ൊʹ

ߣͳ ൌ ૙ ɘ ሺͳ െ ሻ ߣʹǡ͵ ቆ ቇ ൌ േȦͷ ξെ ɘ ሺͳ െ ሻ ߣͶൊ͹ ቆ ቇ ൌ േ࢏Ȧ͹ ሺ݀‫݈ܾ݁ݑ݋‬ሻ ξെ

ߣͳ ൌ ߤͳ ൌ ૙ ߣʹǡ͵ ሺɘ ሻ ൌ ߤʹǡ͵ ሺͲሻ ൌ ૙ ߣͶൊ͹ ሺɘ ሻ ൌ ߤͶൊ͹ ሺͲሻ ൌ േ࢏‫͵ܯ‬ൊͶ ൌ േ࢏Ȧ͵ൊͶ

࢟૙ǡ૜

ߣଵ ൌ ૙ ߣଶǡଷ ൫‫ݕ‬଴ǡସ ൯ ൌ േ߉ହ ߣଵ ൌ ૙ ߣସൊ଻ ൫‫ݕ‬଴ǡସ ൯ ൌ േ࢏߉ଷൊସ ߣଶǡଷ ൫‫ݕ‬଴ǡସ ൯ ൌ േ߉ହ ߤͳ ൌ ૙ ߣସൊ଻ ൫‫ݕ‬଴ǡସ ൯ ൌ േሺ߉଺ േ ࢏߉଻ ሻ ߤʹǡ͵ ൫‫Ͳݕ‬ǡ͵ ൯ ൌ േ࢏‫ܯ‬ଵ

ߣଵ ൌ ૙ ߣଶǡଷ ൫‫ݕ‬଴ǡସ ൯ ൌ േ࢏߉ଶ ߣସൊ଻ ൫‫ݕ‬଴ǡସ ൯ ൌ േ࢏߉ଷൊସ

ߤͶൊ͹ ൌ േ࢏‫͵ܯ‬ൊͶ

࣓ ࢈ ξ૚ െ ࡹ

࣓ࢀ

࢟૙ǡ૝

࣓࢈ ሺ૚ െ ࡹሻ

ξെࡹ

Fig. 1 Equilibrium points/backbone curves of the rigid body normal modes manifold projected to 3D, with their associated eigenvalues.

In the following, the stability of each family of equilibrium points is examined with individual perturbations of each variable. -The 1st family of equilibrium points corresponds to the line along the y0,4 -axis of Figure 1 and is codimension-1 [30]. this line of equilibrium to A. A perturbation of ξ4 , just shifts the angular velocity value ‫ݕ‬within ଴ǡସ another equilibrium point θ˙0 = y0,4 + ξ4 (Fig. 1). Region-1

In this region, there is one ߦସ zero eigenvalue, and the others are pure imaginary (±iΛ2 , ±iΛ3÷4 ). B. A perturbation in torsion (position ξ3 and/or velocity ξ7 ) seems to lead to periodic orbits ߠሶ଴ ൌ ‫ݕ‬଴ǡସ ൅ ߦସ motions around this point which is defined by equation (70) and restricted to torsional perturbed frequency Λ2 . More concrete results can be obtained using other techniques e.g. normal forms. A perturbation in torsional position (ξ3 ) at the point with angular velocity θ˙ = ωT , leads to another equilibrium point along the y0,3 -axis with qφ = ξ3 . ξ5 ,ଶ ξǡ േ݅߉ C. A perturbation in lateral bending (positions ξ1 , ξ2 and/or velocityേ݅߉ ଷൊସ ሻ to lead to 6 ) seems periodic orbits restricted to lateral bending periodic motion around this point defined by the system (71) with frequencies Λ3÷4 , but further investigation is required e.g. using normal ߦଷ ߦ଻ √ forms. A perturbation in lateral bending positions (ξ1 , ξ2 ) at the point with angular velocity θ˙ = ωb 1 − M, leads to another ߉ଶ equilibrium point along the z-axis with (qv , qw ) = (ξ1 , ξ2 ).

ߦଷ

‫ݍ‬థ ൌ ߦଷ

ߠሶ ൌ ɘ୘

‫ݕ‬଴ǡଷ

350


Region-2 In this region, there is one zero eigenvalue. Regarding the rest of the eigenvalues: one of them, has a positive real part (Λ5 ), one has negative real part (−Λ5 ), and then the rest of the eigenvalues are purely imaginary (±iΛ3÷4 ). D. Since the eigenvalues that are associated with torsion are real numbers, with one of them being positive, a perturbation in torsion (position ξ3 and/or velocity ξ7 ) leads to an unstable solution defined by equation (70), which must be examined further if it is associated with chaos. E. A perturbation in lateral bending (positions ξ1 , ξ2 and/or velocity ξ5 , ξ6 ) seems to lead to periodic orbits restricted to lateral bending periodic motion around this point which is defined by the system (71) with frequencies Λ3÷4 . Even in this case, further investigation is needed. Region-3 In this region, there is one zero eigenvalue, and all the other sets of eigenvalues (±Λ5 , ± (Λ6 ± iΛ7 )) have positive and negative real parts. F. Any perturbation apart of the angular velocity shows that the perturbed solutions is leaving away from the equilibrium point region. This indicates that the point is unstable by any other kind of perturbation apart the angular velocity. 2nd Family -The 2nd family of equilibrium points corresponds to the line along the z-axis of (fig. 1), and it is codimension-2 [30]. There are three zero eigenvalues and the rest are purely imaginary (±iH1÷2 ). G. Any perturbation in lateral bending positions (ξ1 , ξ2 ) leads to another equilibrium point (qv , qw ) = (y0,1 + ξ1 , y0,2 + ξ2 ). H. Any other perturbation seems to lead to oscillatory motions (with frequencies H1÷2 ) and one rigid body motion defined by the system of equations (74). Even in this case, further investigation is needed for more concrete results. In the case that (y0,1 , y0,2 ) = (0, 0) a perturbation in angular velocity (ξ4 ) leads to another equilibrium point with θ˙ = y0,4 + ξ4 . 3rd Family -The 3rd family of equilibrium points corresponds to the line along the y0,3 -axis of (fig. 1), and it is codimension-1 [30]. There is one zero eigenvalue and two sets of eigenvalues with purely imaginary parts (±iM1 and ±iM3÷4 ). I. A perturbation in torsional position (ξ3 ) leads to another equilibrium point with qφ = y0,3 + ξ3 . J. A perturbation in angular velocity (ξ4 ) or torsional velocity (ξ7 ) seems to lead to periodic motions defined by equations (78) with frequency M1 but further investigation is needed. In the case that y0,3 = 0, then a perturbation in angular velocity (ξ4 ) leads to another equilibrium point with θ˙ = y0,4 + ξ4 . K. A perturbation in lateral bending (positions ξ1 , ξ2 and/or velocity ξ5 , ξ6 ) seems to lead to periodic orbits restricted to lateral bending periodic motion around this point, which is defined by the system (79) and frequencies M3÷4 . Also, in this case, further investigation is needed. The comments A, G, and I about stability are obvious. The comments B, C, E, and J based on linearization indicate that the equilibrium points form centres, but further investigation is needed. Since they are relevant to normal modes of the original system they will also be examined numerically in next section. The system is unstable for torsional perturbations in Region-2 and unstable for torsional and/or lateral bending perturbations in Region-3.


351

The cases of eigenvalues with positive and negative real parts in regions 2 and 3 (D, F) need further examination to establish if they are associated with chaotic attractors. This can be examined as a continuation of this research in another article. 4 Normal modes through linearization In this section, in the cases that there are periodic orbits in perturbed motions of the shaft are examined, and the associated linearized solutions are determined. 4.1

Periodicity in the linearized original system

The linearized original system (eq. 51) resulted in two independent systems (eq. 59b) and (eq. 59f) describing lateral bending and torsional motions respectively. In the case of torsional perturbations (position and/or velocity) then the equation (59f) leads to torsional oscillatory motions with frequency (ωT,0 ) given by equation (82b). In the case of perturbations in lateral bending motions then the shaft is oscillating as a simply supported beam, with frequency (ωb ), given by equation (82c). 4.2

Periodicity in the linearized restricted system

In this section, all the possible periodic motions of the shaft arising from the linearization of the restricted system around the equilibria are determined, by setting the imaginary part of the eigenvalues equal to the rotating speed. 4.2.1

Periodicity in the 1st family of equilibria

REGIONS-1,2 ). The eigenvalues are The angular velocities belong to, y4 = θ˙0 ∈ (0, ωT ) and y4 = θ˙0 ∈ (ωT , ωb√(1−M) −M given by equations (96b-c, 97b). Perturbations in torsional motion have periodic solutions for the following conditions, s I1 L θ˙02 − ωT2 Λ2 = − = θ˙0 , (110) (I1 L − F 2 ) which lead to angular velocity,

θ˙0,T,cr,1,2 = ±

s

1 ωT = ±0.9169ωT . 2 1 − π42

Perturbation in lateral bending motions have periodic solutions with the following conditions, r q 1 2 2 2 ˙ ˙ Λ3÷4 = (1 + M) θ0 + (1 − M) ωb ∓ 2θ0 M θ˙02 + ωb2 (1 − M)2 = θ˙0 , (1 − M)

(111)

(112)

which lead to angular velocity, s

− (3M − M 2 − 2) + 2(1 − M) , (−4M + 9M 2 − 6M 3 + M 4 )

(113a)

s

−M + M 2 . (−4M + 9M 2 − 6M 3 + M 4 )

(113b)

θ˙0,B,cr,1 = ωb (1 − M) θ˙0,B,cr,2 = ωb (1 − M)

352


REGION-3 In this region, there are no periodic motions for perturbed solutions either in torsion (eigenvalue with zero imaginary part) or in lateral bending motions since the periodicity condition lead to,

θ˙0 = Λ7 =

1 θ˙0 ⇔ = 1, (1 − M) (1 − M)

(114)

which is possible only when neglecting the rotary inertia terms. 4.2.2

Periodicity in the 2nd family of equilibria

In the examination of periodic motions in the 2nd family of equilibrium points, the nonzero eigenvalues that are given by equation (103) are used accompanied with the following periodicity condition: v q u u p0 + p1 z ∓ (r1 z − r2 )2 + r2 z p u 3 = ±ωb (1 − M), (115) ±H1÷2 = ±t (2) 2δ y0 after many manipulations, this leads to the following initial conditions for lateral bending motion, q −e2 ± e22 − 4e1 e3 , (116) z1,2 = 2e1 with, e1 = M 2 (3 + M) ωT2 + ωb2 2 (M + 3) (M − 1) , e2 = − 2M 3 − M 2 2I1 L − F 2 + 2MF 2 ωT2 −ωb2 (1 − M) 3 − M 2 4I 1 L − 3F 2 + 2MF 2 , e3 = 4 I1 L − F 2 (1 − M) I1 L (1 + M)(3 − M) ωT2 + ωb2 (1 − M)(1 − M)(3 − M) 2I1 L − F 2 .

(117a) (117b) (117c)

As shown in Appendix-C (paragraph 5) for Euler-Bernoulli beams in the case when the roots (z1,2 ) should be real (as herein), they should also be negative. Equation (B.15a), indicates that z should be real positive, therefore there are not any acceptable values, and this family of fixed points have no periodic orbits for the perturbed solutions. 4.2.3

Periodicity in the 3rd family of equilibria

The periodic motions in the 3rd family of equilibria are determined through the nonzero eigenvalues given by equations (104b-c). The eigenvalues describing perturbations in lateral bending have a constant value in this family and the periodicity have already been examined through the 1st family of equilibria. The periodicity condition for torsional perturbation based on equation (104b) leads to, 2ωT y0,3,cr q

(I1 L − F 2 + y20,3,cr )

= M1 = ωT ,

(118)

I1 L − F 2 , 3

(119)

or, qφ ,cr = y0,3,cr =

r

which defines the equilibrium point that must be perturbed to find torsional periodic motions.


4.3 4.3.1

353

Analytical determination of periodic orbits of the perturbed linearized restricted system Analytical solution of periodic motions near to the 1st family of equilibria

In the 1st family of equilibria, the equation (70) describing torsional perturbed motion is the same as the one in the torsional motion 1st order approximation in the multiple scale analysis (neglecting amplitude modulation) in [11]. The solution in [11] will be used in the numerical section for direct comparison with the original equations. The equations (71) describing the perturbation in lateral bending motion are the same as the lateral bending motions 1st order approximation in the multiple scale analysis (neglecting amplitude modulation) in [11]. The final solution in both cases is given by, (120) y = (y1 , y2 , y3 , y4 , y5 , y6 , y7 ) = ξ1 , ξ2 , ξ3 , θ˙0 + ξ4 , ξ5 , ξ6 , ξ7 . This solution will be used in the numerical section for comparisons with direct numerical integration of the original system. 4.3.2

Analytical solution of periodic motions near to the 3rd family of equilibria

In the 3rd family of equilibria, in torsional motion, the eigenvalues are given by equation (104b), and the eigenvectors are given by, 1 1 p p , (121a) P= F − i b3 − F 2 F + i b3 − F 2

with,

# "p 2 − iF i b − F p 3 P−1 = p , b3 − F 2 + iF −i 2 b3 − F 2 1

and the solution is given by,

iM t ξ4 ξ4 (0) e 1 0 −1 = [P] . [P ] 0 e−iM1 t ξ7 ξ7 (0)

(121b)

(122)

In explicit form the solution is given by,

ξ4 = (AR + iAI ) eiM1 t + cc,

(123a)

ξ7 = (BR + iBI ) e

(123b)

iM1 t

+ cc,

(123c) with, AR = and BR =

ξ4 (0) , 2

ξ7 (0) , 2

AI =

BI =

(ξ7 (0) − F ξ4 (0)) p , 2 b3 − F 2

(F ξ7 (0) − b3 ξ4 (0)) p . 2 b3 − F 2

(123c-d)

(123e-f)

This system can also be considered as a perturbed linearized system of the 1st family of fixed points at rotating velocity θ˙0 = ωT with y3 = ξ3 = y0,3,cr , therefore it is expected that positive perturbation of the angular velocity (ξ4 (0) > 0) leads to a non-periodic solution of the original system, but negative perturbations can lead to periodic orbits.

354


Table 1 Characteristic values on the dynamics. I 1 L × 104 F × 1 0 3 M × 1 0 4 12.04

31.24

-41.55

√ ωT ωb 1 −M (rad/sec) (rad/sec) 1022.16

4916.41

M) ω b√(1−M M −M

15890.24

θ˙0,BB,ccr ,2 θ˙0,TT ,ccr ,1 θ˙0,BB,ccr ,1 q × 104 (rad/sec) (rad/sec) (rad/sec) φ ,ccr 4507.95

510.82

15857.33

87.21

The last perturbed variable (ξ3 ) is given through direct integration of equation (122b) and it is given by, ξ3 (0) BI i (BR + iBI ) iM1 t ξ3 = − − e + cc. (123g) 2 M1 M1 The final solution is given by, y = (y1 , y2 , y3 , y4 , y5 , y6 , y7 ) = (0, 0, y0,3,cr + ξ3 , ωT + ξ4 , 0, 0, ξ7 ) .

(124)

In the numerical section, the solution provided by equation (124) will be used for comparisons with the solution obtained using direct numerical integration of the original system. 5 Numerical results-discussion The same shaft considered in [11], which follows the Euler-Bernoulli beam assumptions, is considered in this section. It is a 1 m length (L) shaft with internal and external radii ri = 0.028 m and ro = 0.03 m, respectively. It is made of stainless steel with the following material properties: density ρ = 7850 Kg/m3 , Young’s modulus E = 200 GPa, shear modulus G = 76.9 GPa, and Poisson’s ratio ν = 0.3. In Table 1, the characteristic values on the dynamics of the considered shaft are defined; bifurcation angular velocities, point that the system loses stability, and also the parameters defining normal modes. 5.1

Numerical verification of rigid body rotation and equilibria of the restricted system

In this section, using direct numerical integration of the original system (eq. 2) the periodicity of rigid body motions using initial conditions defined by the equilibria (45a-c) of the restricted system (eq. 30) are examined. The following three initial conditions are considered: 1) θ0 , q0,v , q0,w , q0,φ , θ˙0 , q˙0,v , q˙0,w , q˙0,φ = (0, 0, 0, 0, 10, 0, 0, 0), which is a set of the 1st family of equilibria of the restricted system and corresponds to a period T1 = 0.6283 sec, as given by (eq. 46c). 2) θ0 , q0,v , q0,w , q0,φ , θ˙0 , q˙0,v , q˙0,w , q˙0,φ = (0, 8, 10, 0, 1022.16, 0, 0, 0), which is a set of the 2nd family of equilibria of the restricted system and corresponds to a period T2 = 0.0061 sec. 3) θ0 , q0,v , q0,w , q0,φ , θ˙0 , q˙0,v , q˙0,w , q˙0,φ = (0, 0, 0, 5, 4916.41, 0, 0, 0), which is a set of the 3rd family of equilibria of the restricted system and corresponds to a period T3 = 0.0013 sec. In Figures 2a-c, the transient responses arising with direct integration of the original system of equations (2) are depicted. The transient responses obtained with initial conditions of the first set are depicted in Figure (2a) whereas examination of the responses shows that there is only rigid body angular (θ ) motion with period T1 = 0.6283 sec which is the same as the period of the rigid body angular motion of 1st set. In Figure (2b) the transient responses using the 2nd set of initial conditions are depicted. Examination of them indicates only rigid body angular motion with period T2 = 0.0061 sec. Finally, using the 3rd set of initial conditions leads to the transient responses of Figure (2c) where it is clear that there is only rigid body angular periodic motion with T3 = 0.0013sec. Therefore, all sets of equilibria of the restricted system are rigid body periodic motions of the original system.

൫ߠ଴ ǡ ‫ݍ‬଴ǡ௩ ǡ ‫ݍ‬଴ǡ௪ ǡ ‫ݍ‬଴ǡథ ǡ ߠሶ଴ ǡ ‫ݍ‬ሶ ଴ǡ௩ ǡ ‫ݍ‬ሶ ଴ǡ௪ ǡ ‫ݍ‬ሶ ଴ǡథ ൯ ൌ ሺͲǡͺǡͳͲǡͲǡͳͲʹʹǤͳ͸ǡͲǡͲǡͲሻ ܶଶ ൌ ͲǤͲͲ͸ͳ‫ܿ݁ݏ‬ ൫ߠ଴ ǡ ‫ݍ‬଴ǡ௩ ǡ ‫ݍ‬଴ǡ௪ ǡ ‫ݍ‬଴ǡథ ǡ ߠሶ଴ ǡ ‫ݍ‬ሶ ଴ǡ௩ ǡ ‫ݍ‬ሶ ଴ǡ௪ ǡ ‫ݍ‬ሶ ଴ǡథ ൯ ൌ ሺͲǡͲǡͲǡͷǡͶͻͳ͸ǤͶͳǡͲǡͲǡͲሻ Fotios Georgiades / Journal of Vibration Testing and System Dynamics 2(4) (2018) 327–373 ܶଷ ൌ ͲǤͲͲͳ͵‫ܿ݁ݏ‬

355

2π

2π

(a)

(b)

2π

(c)

Fig. 2Fig. Numerically determined transient responses of the original system, (a) with (a) the with 1st setthe of 1initial conditions, 2. Numerically determined transient responses of the original system, set of initial nd rd (b) with the 2 set of the initial conditions and, (c) with the 3 set of initial conditions.

5.2

Numerical determination of the eigenvalues

In this section, the results of the numerical determination of the eigenvalues of the Jacobian ߠሻ matrix ܶ ൌ Ͳ ଵ (given by eq. 62, 63, 66) are compared with those that are obtained from the formulas derived in section 3.4. ܶଶ REGION-1 ଷ ൌ that θ˙0 ∈ (0, 4916.41). In Table 2, the eigenvalues for Region-1 This region (1) corresponds to theܶcase obtained analytically using equations (96a-c) are depicted, and comparing them with those obtained numerically it is obvious that they are in very good agreement. REGION-2 In this region, the angular velocity should belong to θ˙0 ∈ (4916.41, 15890.24). In Table 3 the eigenvalues for Region-2 are depicted, obtained analytically using equations (97a-b, 96c) and they are in very good agreement with those that are obtained numerically. REGION-3 The angular velocity in this region should belong to θ˙0 ∈ (15890.24, ∞). Similarly, the analytical determination of eigenvalues for Region-3 using equations (98a-b, 97b) depicted in Table 4, are in very good agreement with those that are obtained numerically. In Table 5 the eigenvalues for three specific points are depicted: the two bifurcation points and this one that the stability is changing. The first point is at the bifurcation of the 1st with the 2nd family of equilibrium points, with eigenvalues defined by equations (107a-d). The second point is at the bifurcation of the 1st with the 3rd family of equilibrium points, with eigenvalues given by equations (108a-c). The third, is the point that the linearized system describing lateral bending motions in the 1st family of equilibria, loses stability, with eigenvalues given by equations (109a-c). In Table 5 the

356


Table 2 Numerical and analytical determination of eigenvalues for Region-1.

λ1

θ˙ = 10

θ˙ 0, B ,cr ,1 = 510.82

θ˙ 0, T , cr , 1 = 4507.95

θ˙ = 4915

rad/sec

rad/sec

rad/sec

rad/sec

Analytical

Numerical

Anal.

Num.

Anal.

Num.

Anal.

Num.

Real

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

Imag.

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

λ2

Real

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

(iΛ2 )

Imag.

11295.94

11295.94

11234.83

11234.83

4507.95

4507.95

270.65

270.65

λ3

Real

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

(−iΛ2 )

Imag.

-11295.94

-11295.94

-11234.83

-11234.83

-4507.95

-4507.95

-270.65

-270.65

λ4

Real

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

(iΛ3 )

Imag.

1010.09

1010.09

510.82

510.82

3511.15

3511.15

3924.64

3924.64

λ5

Real

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

(−iΛ3 )

Imag.

-1010.09

-1010.09

-510.82

-510.82

-3511.15

-3511.15

-3924.64

-3924.64

λ6

Real

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

(iΛ4 )

Imag.

1030.01

1030.01

1528.22

1528.22

5467.43

5467.43

5864.69

5864.69

λ7

Real

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

(−iΛ4 )

Imag.

-1030.01

-1030.01

-1528.22

-1528.22

-5467.43

-5467.43

-5864.69

-5864.69

Table 3 Numerical and analytical determination of eigenvalues for Region-2. θ˙ =4917 rad/sec Analytical

Numerical

θ˙ 0, B ,c r , 2 = 15857.33 rad/sec Analytical

Numerical

Real

0.00

0.00

0.00

0.00

Imag.

0.00

0.00

0.00

0.00

λ2

Real

174.80

174.80

34638.51

34638.51

(Λ5 )

Imag.

0.00

0.00

0.00

0.00

λ3

Real

174.80

174.80

-34638.51

-34638.51

(−Λ5 )

Imag.

0.00

0.00

0.00

0.00

λ4

Real

0.00

0.00

0.00

0.00

(iΛ3 )

Imag.

3926.67

3926.67

15726.10

15726.10

λ5

Real

0.00

0.00

0.00

0.00

(−iΛ3 )

Imag.

-3926.67

-3926.67

-15726.10

-15726.10

λ6

Real

0.00

0.00

0.00

0.00

(iΛ4 )

Imag.

5866.64

5866.64

15857.33

15857.33

λ1

λ7

Real

0.00

0.00

0.00

0.00

(−iΛ4 )

Imag.

-5866.64

-5866.64

-15857.33

-15857.33

analytical eigenvalues are in very good agreement with those ones obtained numerically. In Table 6 the eigenvalues for the 2nd family of equilibrium points are depicted which are defined analytically by equations (99a,103) and they are in very good agreement with the numerical ones. In Table 7 the eigenvalues for the 3rd family of equilibrium points defined by equations (104a-b,106) are depicted, and they are in very good agreement with those obtained numerically.


357

Table 4 Numerical and analytical determination of eigenvalues for Region-2. θ˙ = 16000 rad/sec λ1

θ˙ 0 ,B , cr ,2 = 50000 rad/sec

Analytical

Numerical

Analytical

Numerical

Real

0.00

0.00

0.00

0.00

Imag.

0.00

0.00

0.00

0.00

λ2

Real

34983.14

34983.14

114323.45

114323.45

(Λ5 )

Imag.

0.00

0.00

0.00

0.00

λ3

Real

-34983.14

-34983.14

-114323.45

-114323.45

(−Λ5 )

Imag.

0.00

0.00

0.00

0.00

λ4

Real

120.10

120.10

3043.27

3043.27

(Λ6 + iΛ7 )

Imag.

15933.79

15933.79

49793.10

49793.10

λ5

Real

-120.10

-120.10

-3043.27

-3043.27

(−Λ6 − iΛ7 )

Imag.

-15933.79

-15933.79

-49793.10

-49793.10

λ6

Real

120.10

120.10

3043.27

3043.27

(Λ6 − iΛ7 )

Imag.

-15933.79

-15933.79

-49793.10

-49793.10

λ7

Real

-120.10

-120.10

-3043.27

-3043.27

(−Λ6 + iΛ7 )

Imag.

15933.79

15933.79

49793.10

49793.10

Table 5 Numerical and analytical determination of eigenvalues for the two bifurcation points and the point that the system is changing stability. ˙ = 1022.16 rad/sec θ 0,1 Analytical

λ1 λ2 λ3 λ4 λ5 λ6 λ7

5.3

Numerical

˙ = 4916.41 rad/sec θ 0,2 Analytical

Numerical

˙ = 15890.24 rad/sec θ 0,3 Analytical

Numerical

Real

0.00

0.00

0.00

0.00

0.00

0.00

Imag.

0.00

0.00

0.00

0.00

0.00

0.00

Real

0.00

0.00

0.00

0.00

34718.04

34718.04

Imag.

11049.13

11049.13

0.00

0.00

0.00

0.00

Real

0.00

0.00

0.00

0.00

-34718.04

-34718.04

Imag.

-11049.13

-11049.13

0.00

0.00

0.00

0.00

Real

0.00

0.00

0.00

0.00

0.00

0.00

Imag.

0.00

0.00

3926.07

3926.07

15824.49

15824.49

Real

0.00

0.00

0.00

0.00

0.00

0.00

Imag.

0.00

0.00

-3926.07

-3926.07

-15824.49

-15824.49

Real

0.00

0.00

0.00

0.00

0.00

0.00

Imag.

2035.87

2035.87

5866.07

5866.07

15824.49

15824.49

Real

0.00

0.00

0.00

0.00

0.00

0.00

Imag.

-2035.87

-2035.87

-5866.07

-5866.07

-15824.49

-15824.49

Numerical examination of nonlinear normal modes from the linearized solutions

In this section, the validity of linearized solutions of the normal modes as they are defined in section 4 is examined, by comparing the analytical with the numerical responses obtained with direct integration of equation (2).

358


Table 6 Numerical and analytical determination of eigenvalues for the 2nd family of equilibria. θ˙ 0,1 = 1022.16 rad/sec

(q v , q w ) = (1 00 , 1 0 0)

η1 η2 η3

Analytical

Numerical

0

0

Real Imag.

0

0

Real

0

0

Imag.

0

0

Real

0

0

Imag.

0

0

η4

Real

0

0

(iH1 )

Imag.

2040.1

2040.1

η5

Real

0

0

(−iH1 )

Imag.

-2040.1

-2040.1

η6

Real

0

0

(iH2 )

Imag.

4809.04

4809.04

η7

Real

0

0

(−iH2 )

Imag.

-4809.04

-4809.04

Table 7 Numerical and analytical determination of eigenvalues for the 3rd family of equilibria. θ˙ 0,2 = 4916.41 rad/sec

µ1

q φ ,ccr = 8 7 . 2 1 × 1 0 −4

qφ = 1 0 0 Analytical

Numerical

Analytical

Numerical

Real

0.00

0.00

0.00

0.00

Imag.

0.00

0.00

0.00

0.00

µ2

Real

0.00

0.00

0.00

0.00

(iM1 )

Imag.

9832.82

9832.82

4916.41

4916.41

µ3

Real

0.00

0.00

0.00

0.00

(−iM1 )

Imag.

-9832.82

-9832.82

-4916.41

-4916.41

µ4

Real

0.00

0.00

0.00

0.00

(iM3 )

Imag.

3926.07

3926.07

3926.07

3926.07

µ5

Real

0.00

0.00

0.00

0.00

(−iM3 )

Imag.

-3926.07

-3926.07

-3926.07

-3926.07

µ6

Real

0.00

0.00

0.00

0.00

(iM4 )

Imag.

5866.07

5866.07

5866.07

5866.07

µ7

Real

0.00

0.00

0.00

0.00

(−iM4 )

Imag.

-5866.07

-5866.07

-5866.07

-5866.07

5.3.1

Normal modes of lateral bending motions

The normal modes of lateral bending motions at the first and second critical speeds (eq. 113a-b) arising from the linearizing solutions of the shaft are examined in this section. The 1st normal mode is associated with a perturbation of the 1st family of equilibria with critical speed θ˙0 = θ˙0,B,cr,1 = 510.82rad/sec. Examining the extend to which the solution is valid, two sets of perturbations are considered. The first set of initial conditions is (q0,v , q0,w ) = (ξ0,1 , ξ0,2 ) = 10−3 , 10−3 with the rest of the perturbations being zero. Since the initial angular position of the shaft which defines the orientation of


359

the fixed coordinate system is arbitrary, then perturbations in lateral bending must be examined in terms of radial bending deformation. This set of initial conditions can be examined for radial bending displacement at the centre of the shaft, which is a simply supported beam (therefore at the maximum modal displacement) given by, r q q q 2 2 2 2 2 R = v (L/2, 0) + w (L/2, 0) = y1 (L/2) · q0,v + q0,w = q20,v + q20,w = 1.2 mm. (125) mL The linearized system around the equilibrium coincides with the first order approximation for lateral bending motions of multiple scales analysis in [11], whereas the solution is given explicitly and is associated with the Campbell diagram. In Figure 3 the transient responses obtained from direct numerical integration of the original system (eq. 2), and those obtained from the perturbed solution are depicted and their comparison shows that they are in very good agreement. The second set of initial conditions is (q0,v , q0,w ) = (ξ0,1 , ξ0,2 ) = 2 × 10−2 , 0 with the rest of the perturbations being zero. The radial bending displacement is R = 16.7 mm (using eq. 125), and the transient responses for qv and the angular velocity θ˙ are depicted in Figure 4, whereas the linearized solution does not capture the lower frequency of the transient responses of qv , therefore the linearized solution is no longer valid for this set of initial conditions. The disagreement between linearized motions and the direct numerical simulations originated from close to the first set of initial conditions and becomes very evident with the increase of the radial bending displacement to the 2nd set of initial conditions. Considering the existence of periodic motions around this equilibrium the comment C about the stability is validated, meaning that this equilibrium with a perturbation in lateral bending motions forms a centre. Therefore, the linearized solution around the first family of equilibrium points at first critical speed for lateral bending motions is only valid for very small values of the radial bending displacement and forms a centre since this equilibrium point is surrounded by periodic motions. Here, the validity of the linearized solution for the 1st family of equilibrium points around the 2nd ˙ ˙ critical speed θ0 = θ0,B,cr,2 = 15857.33 rad/sec, is examined. Perturbations (q0,v , q0,w ) = (ξ0,1 , ξ0,2 ) = −3 −3 10 , 10 restricted only to lateral bending motions are considered. In Figure 5, the responses obtained from direct numerical integration of the original system with those obtained from the linearized solution around the equilibrium manifold are depicted, and they are in very large disagreement. Therefore, the linearized solution is no longer valid for such high rotating speeds. It should be highlighted that similar solutions were obtained even with very small perturbations of 10−6 . A careful check of the numerical response indicates that for such high rotating speeds there is no periodicity and comment-C is no longer valid. 5.3.2

Normal modes of torsional motions

In this section, the validity of linearized solutions of equilibriums at the normal modes for torsional motion is examined. The transient responses obtained from direct numerical integration with the analytical linearized solutions is compared. The third normal mode corresponds to the critical speed for torsion obtained from the linearization of the 1st family of equilibrium points (eq. 111). The fourth normal mode corresponds to initial conditions that lead to normal modes obtained from the 3rd family of equilibrium points (eq. 119). The third normal mode is at critical speed for torsion θ˙0,T,cr,1 = 4507 rad/sec, with initial conditions for torsional angle q0,φ = ξ0,3 = 10−3 and the rest of the perturbations being zero. The equations that describe the torsional motion around the equilibrium are the equations (69b and 70) and they coincide with the first order approximation of multiple scales analysis, given in [11]. In Figures 6, the transient responses obtained from direct numerical integration of the original system (eq. 2) and those obtained from the perturbed analytical solution provided in [11] are depicted, which are in very good agreement

360


(a)

(b)

Fig. 3 Transient responses at first critical speed θ˙0,B,cr,1 = 510.82 rad/sec. a) rigid body angular velocity-θ˙ and qv with c0 = 2.9365 × 10−6, b) torsional angle-qφ and qw .


361

Fig. 4 Transient responses of rigid body angular velocity-θ˙ and qv with c1 = 4.3068 × 10−5 at first critical speed θ˙0,B,cr,1 = 510.82 rad/sec, with higher values of the first radial lateral bending displacement.

Fig. 5 Transient responses of qv at second critical speed θ˙0,B,cr,2 = 15857.33 rad/sec.

362


(a)

(b)

(c)

Fig. 6 Transient responses for torsional critical speed θ˙0,T,cr,1,2 = 4507.95 rad/sec. a) torsional angle-qφ , b) lateral bending motions qv and qw , c) angular velocity-θ˙ .

apart from the transient responses of the angular velocity-θ˙ (Fig. 6c) where there is a slight discrepancy at the local extrema. The existence of periodic motions in the perturbed solutions of this equilibrium point validates comment-B about the stability, that this point forms a centre. The fourth normal mode is defined at the 3rd family of equilibrium points with the following initial conditions; θ˙0 (0) = ωT + ξ4 (0) = 4906.41 rad/sec (ξ4 (0) = −10 rad/sec), torsional angle qφ (0) = y0,3,cr + ξ3 (0) = 0.009721(ξ3 (0) = 10−3 ), and a large perturbation of angular velocity q˙φ (0) = ξ7 (0) = 0.1. The analytical solution of the perturbed system is given by equations (123a-b, g, 124). In Figures 7, the transient responses obtained from direct numerical integration of the original system (eq. 2) and those obtained from the solution of the linearized system (eq. 123a-b, g, 124) are depicted. Regarding torsional motions (Fig. 7a,c) and rigid body angular velocity (Fig. 7d), the numerical and analytical results are in very good agreement only in the beginning of the transient responses, but progressively some deviation occurs between the two solutions which means that this periodic solution is unstable. The existence of periodic orbits around this equilibrium point validates comment-J, that the equilibrium is surrounded by periodic orbits and forms a centre.

363

Responses

Responses


(b)

Responses

Responses

(a)

(c)

(d)

Fig. 7 Transient responses for torsional normal mode of the linearized system of the 3rd family of equilibrium points with θ˙0 = 4906.41 rad/sec, torsional angle qφ = 0.009721 and torsional velocity q˙φ = 0.1. a) torsional angle-qφ , b) lateral bending motions qv and qw , c) torsional velocity-q˙φ , and d) angular velocity-θ˙ .

6 Conclusions In this article, the dynamics of a spinning shaft with non-constant rotating speed, near to the equilibrium points, is examined analytically. Examination of the vector field describing the dynamics shows that there is no linear counterpart therefore the equilibriums are very important. At first, the equilibrium points of the original system are determined, and then the equilibrium manifold of the restricted system is defined which coincides with the rigid body motion normal modes of the spinning shaft. It is shown that all the equilibrium points of the original and the restricted system are degenerate. After linearization around the equilibrium manifold, the eigenvalues of the Jacobians of the linearized systems are determined and whether they are purely complex or with real and imaginary parts is examined. Then a projection of the manifold of the equilibrium points to three dimensions is presented, and made possible to facilitate stability comments considering the eigenvalues accompanied with individual perturbations of each generalized coordinate, with the indication of centres and also the existence of unstable regions around equilibrium points. Finally, the four types of normal modes of the linearized systems, in terms of frequencies and their analytical solutions are determined. In the last part using numerical techniques, the analytical findings are verified, that the equilibrium manifold corresponds to a rigid body motion of the original system. The analytical determination of

364


the eigenvalues is in good agreement with the numerical determination. It is also shown that for small perturbations of equilibrium points, the normal modes defined by the linearized solutions are in good agreement with the numerical integration of the original system. More precisely, the linearized solution describing lateral bending vibrations for very small perturbations can be approximated by the solution associated with the Campbell diagram. Also, it is shown that for large perturbations, the linearized solution is no longer valid, therefore the Campbell diagram solution is very restricted. This work is essential to understand the dynamics of the spinning shaft during spin-up/down operation and determine the nonlinear normal modes which define critical situations. This approach is not restricted only to spinning shafts but also can be applied to other rotating structures with nonconstant rotating speed that also have such inertia nonlinearities. Also, the stability results indicate regions which must be examined for chaotic attractors and can be part of the continuation of this work. Another direction of further work is to fully determine the nonlinear normal modes away from the linearized regime and examine their bifurcations and stability. References [1] Campbell, W. (1924), The Protection of Steam Turbine Disk Wheels From Axial Vibration Part I, General Electric Review, XXVII, No. 6 (June). [2] Campbell, W. (1924), The Protection of Steam Turbine Disk Wheels From Axial Vibration Part II-Exposition of the Nature and Theory of Vibration in Turbine Wheels, General Electric Review, XXVII, No. 7 (July). [3] Campbell, W. (1924), The Protection of Steam Turbine Disk Wheels From Axial Vibration Part III-Methods of design and testing for the protection of turbine bucket wheels from axial vibration, General Electric Review, XXVII, No. 8 (August). [4] Plaut R.H. and Wauer, J. (1995), Parametric, external and combination resonances in coupled flexural and torsional oscillations of an unbalanced shaft, Journal of Sound and Vibration, 183(5), 889-897. [5] Suherman, S. and Plaut, R.H (1997), Use of a flexible internal support to suppress vibrations of a rotating shaft passing through a critical speed, Journal of Vibration and Control, 3, 213-233. [6] Wauer, J. (1990), Modelling and formulation of equations of motion for cracked rotating shafts, International Journal of Solids and Structures, 26(8), 901-914. [7] Balthazar, J.M., Mook, D.T., Weber, H.I., Brazil, R.M.L.R.F., Fenili, A., Belato, D., and Felix, J.L.P. (2003), An Overview on Non-Ideal Vibrations, Meccanica, 38, 613-621. [8] Warminski, J., Balthasar, J.M., and Brasil, R.M.L.R.F. (2001), Vibrations of a non-ideal parametrically and self-excited model, Journal of Sound and Vibration, 245(2), 363-374. [9] Felix, J.L., Balthazar, J.M., and Brazil, R.M. (2008), A short note on transverse vibrations of a shaft carrying two (or one) disk excited by a nonideal motor, Journal of Computational Nonlinear Dynamics, 4(1), (2008). doi:10.1115/1.3007979. [10] Warminski, J. and Balthasar, J.M. (2003), Vibrations of a parametrically and self-Excited system with ideal and non-ideal energy sources, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 25(4). doi: 10.1590/S1678-58782003000400014. [11] Georgiades, F. (2018), Nonlinear Dynamics of a spinning shaft with non-constant rotating speed, Nonlinear Dynamics, 93(1), 89-118. [12] Natsiavas, S. (1995), On the dynamics of rings rotating with variable spin speed, Nonlinear Dynamics, 7, 345-363. [13] Georgiades, F., Latalski, J., and Warminski, J. (2014), Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle, Meccanica, 49, 1833-1858. doi:10.1007/s11012014-9926-9 [14] Mikhlin, Y.V. and Avramov, K.V. (2010), Nonlinear normal modes for vibrating mechanical systems. Review of Theoretical Developments, Applied Mechanics Reviews, 63 / 060802-1:21. [15] Rosenberg, R.M. (1960), Normal modes of nonlinear dual-mode systems, Journal of Applied Mechanics, 27, 263-268. [16] Rand, R. (1971), Nonlinear normal modes in two-degrees-of-freedom systems, Journal of Applied Mechanics, 38, 561. [17] Szemplinska-Stupnicka, W. (1983), Non-linear normal modes and the generalized Ritz method in the problems of vibrations of non-linear elastic continuous systems, International Journal of Non-Linear Mechanics, 18, 149-165.


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[18] Shaw, S.W. and Pierre, C. (1991), Non-Linear normal modes and invariant manifolds, Journal of Sound and Vibration, 150 (1), 170-173. [19] Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., and Zevin, A.A. (1996), Normal Modes and Localization in Nonlinear Systems, Wiley, New York. [20] Kerschen, G., Peeters, M., Golinval, J.C., and Vakakis, A.F. (2009), Nonlinear normal modes, part I: A useful framework for the structural dynamicist, Mechanical Systems and Signal Processing, 23(1), 170-194. doi:10.1016/j.ymssp.2008.04.002. [21] Avramov, K.V. and Mikhlin, Y.V. (2013), Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems, Applied Mechanics Reviews, 65, 020801-1:20. [22] Legrand, M., Jiang, D., Pierre, C., and Shaw, S.W. (2004), Nonlinear normal modes of a rotating shaft based on the invariant manifold method, International Journal of Rotating Machinery, 10(4), 319-335. [23] Villa, C., Sinou, J.J., and Thouverez, F. (2008), Stability and vibration analysis of a complex flexible rotor bearing system, Communications in Nonlinear Science and Numerical Simulation, 13, 804-821 (2008). doi:10.1016/j.cnsns.2006.06.012 [24] Yabuno, H., Kashimura, T., Inoue, T., and Ishida, Y. (2011), Nonlinear normal modes and primary resonance of horizontally supported Jeffcott rotor, Nonlinear Dynamics, 66, 377-387. [25] Avramov, K.V. and Borysiuk, O.V. (2012), Nonlinear dynamics of one disk asymmetrical rotor supported by two journal bearings, Mechanism and Machine Theory, 67, 1201-1219.doi: 10.1007/s11071-011-0063-x [26] Perepelkin, N.V., Mikhlin, Y.V., and Pierre, C. (2013), Non-Linear normal forced vibration modes in systems with internal resonance, International Journal of Non-Linear Mechanics, 57, 102-115. [27] Liebscher, S. (2015), Bifurcation without parameters, Springer International Publishing Switzerland. [28] Verhulst, F. (2006), Nonlinear Differential Equations and Dynamical Systems, Springer 2 nd edition, 3 rd printing. [29] Meirovitch, L. (2003), Methods in Analytical Dynamics, Dover publications. [30] Nayfeh, A.H. and Balachadran, B. (2004), Applied Nonlinear Dynamics, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. [31] Schultz, D.G. and Gibson, J.E. (1962), The variable gradient method for generating Lyapunov functions, Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry, 81(4), 203-210. [32] Troger, H. and Steindl, A. (1991), Nonlinear Stability and Bifurcation Theory, Springer-Verlag/Wien.

Appendix-A The lower part of the Jacobian (Jx,l ) of the linearization around the equilibrium of the original system can be defined by determining the partial derivatives of equation (59), and they are given explicitly by:

∂ Gi = 0, for i = 1, . . . , 4. ∂ x1

(A1)

-Partial derivatives of G1

∂ G1 ∂ x2 ∂ G1 ∂ x3 ∂ G1 ∂ x4 ∂ G1 ∂ x5 ∂ G1 ∂ x6 ∂ G1 ∂ x7 ∂ G1 ∂ x8

= 2Mx5 x6 ,

(A2)

= 2Mx5 x7 ,

(A3)

= −4 (1 − M)x5 x8 + 2F (1 − M)x25 − 2F (1 − M) ωT2 ,

(A4)

= 2Mx2 x6 + 2Mx3 x7 − 4 (1 − M)x4 x8 + 4F (1 − M)x4 x5 ,

(A5)

= 2Mx2 x5 ,

(A6)

= 2Mx3 x5 ,

(A7)

= −4 (1 − M)x4 x5 .

(A8)

366


-Partial derivatives of G2 M 2M 3M ∂ G2 x3 x5 x6 + 2 I1 L − F 2 x25 − x22 x25 − x2 x2 + 2x24 x25 = − ∂ x2 (1 − M) (1 − M) (1 − M) 3 5 4M x2 x5 x7 , −2 I1 L − F 2 ωb2 (1 − M) + 3M ωb2 x22 + M ωb2x23 − 2ωb2 (1 − M) x24 + (1 − M) 2ω 2 M (1 − M) 2M 2M ∂ G2 x2 x5 x6 + 4x4 x5 x8 − x2 x3 x25 + b x2 x3 − 2F x25 − ωT2 x4 , = − ∂ x3 (1 − M) (1 − M) (1 − M) ∂ G2 = 4x3 x5 x8 + 4x2 x4 x25 − 4ωb2 (1 − M)x2 x4 − 8x4 x5 x7 − 2F x25 − ωT2 x3 , x4 2M 2M 2M ∂ G2 x2 x3 x6 + 4x3 x4 x8 + 4 I1 L − F 2 x2 x5 − x32 x5 − x2 x2 x5 = − ∂ x5 (1 − M) (1 − M) (1 − M) 3 2M +4x2 x24 x5 − 4 I1 L − F 2 x7 + x2 x − 4x24 x7 − 4Fx3 x4 x5 , (1 − M) 2 7 ∂ G2 2M x2 x3 x5 , = − ∂ x6 (1 − M) 2M ∂ G2 = −4 I1 L − F 2 x5 + x2 x5 − 4x24 x5 , ∂ x7 (1 − M) 2 ∂ G2 = 4x3 x4 x5 . ∂ x8

(A9) (A10) (A11)

(A12) (A13) (A14) (A15)

-Partial derivatives of G3 2M 2M ∂ G3 = x3 x5 x7 − 4x4 x5 x8 + 2ωb2 Mx2 x3 − x2 x3 x25 + 2Fx4 x25 − 2F ωT2 x4 , ∂ x2 (1 − M) (1 − M) 2M ∂ G3 = x2 x5 x7 − 2 I1 L − F 2 ωb2 (1 − M) + ωb2 Mx22 + 3ωb2 Mx23 − 2ωb2 (1 − M)x24 ∂ x3 (1 − M) M 3M 4M +2 I1 L − F 2 x25 − x22 x25 − x23 x25 + 2x24 x25 − x3 x5 x6 , (1 − M) (1 − M) (1 − M) ∂ G3 = −4x2 x5 x8 − 4ωb2 (1 − M)x3 x4 + 4x3 x4 x25 + 8x4 x5 x6 + 2Fx2 x25 − 2F ωT2 x2 , ∂ x4 2M 2M 2M ∂ G3 x2 x3 x7 − 4x2 x4 x8 + 4 I1 L − F 2 x3 x5 − x22 x3 x5 − x3 x5 = ∂ x5 (1 − M) (1 − M) (1 − M) 3 2M +4x3 x24 x5 + 4 I1 L − F 2 x6 − x2 x6 + 4x24 x6 + 4Fx2 x4 x5 , (1 − M) 3 ∂ G3 2M x2 x5 + 4x24 x5 , = 4 I1 L − F 2 x5 − ∂ x6 (1 − M) 3 2M ∂ G3 x2 x3 x5 , = ∂ x7 (1 − M) ∂ G3 = −4x2 x4 x5 . ∂ y8

(A16)

(A17) (A18)

(A19) (A20) (A21) (A22)

-Partial derivatives of G4

∂ G4 = 2FMx5 x6 − 2Mx2 x4 x25 + 2M ωT2 x2 x4 , ∂ x2 ∂ G4 = 2FMx5 x7 − 2Mx3 x4 x25 + 2M ωT2 x3 x4 , ∂ x3

(A23) (A24)


∂ G4 = −4F (1 − M)x5 x8 + 2 (1 − M)I1 Lx25 − Mx22 x25 − Mx23 x25 + 6 (1 − M)x24 x25 ∂ x4 −2 (1 − M)I1 LωT2 + M ωT2 x22 + M ωT2 x23 − 6 (1 − M) ωT2 x24 , ∂ G4 = 2FMx2 x6 + 2FMx3 x7 − 4F (1 − M) x4 x8 + 4 (1 − M)I1 Lx4 x5 ∂ x5 −2Mx22 x4 x5 − 2Mx23 x4 x5 + +4 (1 − M)x34 x5 , ∂ G4 = 2FMx2 x5 , ∂ x6 ∂ G4 = 2FMx3 x5 , ∂ x7 ∂ G4 = −4F (1 − M)x4 x5 . ∂ x8

367

(A25)

(A26) (A27) (A28) (A29)

Appendix-B In this Appendix the eigenvalues of the Jacobians are determined. B.1 Eigenvalues of the Jacobian of the original system and of 1st family of equilibria The following determinants are defined

D j,0

−σ 0 0 0 = |A (σ )| = 0 0 0 0 −σ 0 0 −σ 0 0 = −σ 0 0 a1 0 0 a1 0 0

0 0 0 1 0 0 0 −σ 0 0 0 1 0 0 0 0 1 0 0 −σ 0 0 0 −σ 0 0 0 1 0 0 0 0 Fa2 −σ 0 a1 0 0 0 −σ −a3 0 0 a1 0 0 a3 −σ 0 0 0 I1 La2 0 0 0 −σ 0 0 1 0 0 0 0 0 1 0 0 0 1 −σ 0 Fa2 −σ 0 0 0 = −σ |B (σ )| , 0 0 −σ −a3 0 0 0 a3 −σ 0 I1 La2 0 0 0 −σ

(B1)

With the parameters defined by equations (71). In case of θ˙0 = 0 (in eq. 71) the eigenvalues (σ ) for the linearized original system is obtained by the roots of, |A (σ )| = −σ |B (σ )| = 0.

(B2)

In case of θ˙0 ∈ R then the eigenvalues (λ ) for the linearized restricted system around the 1st family of fixed points is determined through D j,1 = |B (λ )| = 0.

(B3)

368


D j,1

−λ 0 0 = |B (λ )| = 0 a1 0 0  −λ     0  = −λ −λ 0   a1    0 = −λ (D1 + D2 ),

0 0 0 1 0 0 0 0 0 1 0 −λ 0 −λ 0 0 0 1 0 0 0 Fa2 −λ 0 0 0 0 −λ −a3 0 a1 0 0 a3 −λ 0 0 I1 La2 0 0 0 −λ 0 0 0 1 0 0 −λ 0 0 1 0 −λ 0 −λ −a3 0 − a1 0 −λ a1 0 a3 −λ 0 0 0 I1 La2 I1 La2 0 0 −λ

 1 0 0    0 1 0   0 0 1  a3 −λ 0     0 0 −λ

 −λ 0 0 0 1 0 1 0    0 −λ −a3 0 −λ 0 0 1 D1 = λ λ + a 1 0 −λ −a3 0  0 a3 −λ 0      I1 La2 0 I1 La2 0 0 −λ 0 −λ = λ 2 λ 4 + λ 2 a23 − I1 La2 − a1 − I1 La2 a23 − a1 , −λ 0 1 0 0 −λ 0 1 D2 = −a1 = −a1 λ 4 + a1 (I1 La2 + a1 ) λ 2 − I1 La2 a21 , 0 −λ 0 a1 0 I1 La2 0 −λ

(B4)

   

D j,1 = |B (λ )| = −λ (D1 + D2 ) = −λ (λ 2 − I1 La2 )[λ 4 + (a23 − 2a1 )λ 2 + a21 ] = 0,

(B5) (B6)

(B7)

whereas the eigenvalues given by the roots of this equation (B.7). Therefore, the eigenvalues (σ ) of the Jacobian ([Jx ]|x0 ) of the original system can be defined through, D j,0 = |A (σ )| = −σ |B (σ )| = −σ 2 σ 2 − I1 La2 (0) σ 4 + a23 (0) − 2a1 (0) σ 2 + a21 (0) = 0. (B8)

B.2 Eigenvalues of the Jacobian of the 2nd family of equilibria

In 2nd family of fixed points, the determinant is given by, −η 0 0 0 1 0 0 0 −η 0 0 0 1 0 0 0 0 0 1 0 −η D j,2 = 0 0 c9 −η c1 c2 0 0 0 0 − c8 y0,2 c3 y0,1 −c7 − η −c4 0 0 c8 y0,1 c3 y0,2 c5 c7 − η 0 0 0 c6 0 Fc1 Fc2 −η   −η −η c1 c2 0 c9 c1 c2      c3 y0,1 −c7 − η −c4 −c8 y0,2 c3 y0,1 −c7 − η −c4  0 2 + = η −η c5 c7 − η  c3 y0,2 c5 c7 − η 0 c8 y0,1 c3 y0,2      0 Fc1 Fc2 −η c6 0 Fc1 Fc2 = η 2 (D3 + D4 ).

D3 = −η 5 + c27 − c4 c5 + c1 c3 y0,1 + c2 c3 y0,2 η 3

(B9)


+ (−c1 c3 c7 y0,1 − c1 c3 c4 y0,2 + c2 c3 c5 y0,1 + c2 c3 c7 y0,2 ) η 2 , D4 =

369

(B10)

−c6 c27 + c4 c5 c6 − c2 c5

c8 y0,2 F + c1 c7 c8 y0,2 F + c2 c7 c8 y0,1 F − c1 c4 c8 y0,1 F +c1 c3 c9 Fy0,1 − c1 c3 c6 y0,1 + c2 c3 c9 y0,2 F − c2 c3 c6 y0,2 η − (c1 c8 y0,2 F − c2 c8 y0,1 F) η 2

+c6 η 3 + c2 c3 c5 c9 Fy0,1 − c1 c3 c7 c9 y0,1 F + c1 c3 c6 c7 y0,1 − c2 c3 c5 c6 y0,1 + c2 c3 c7 c9 y0,2 F −c1 c3 c4 c9 y0,2 F + c1 c3 c4 c6 y0,2 − c2 c3 c6 c7 y0,2 , D3 + D4 = −η 5 + c27 − c4 c5 + c1 c3 y0,1 + c2 c3 y0,2 + c6 η 3

(B11)

+ (−c1 c3 c7 y0,1 − c1 c3 c4 y0,2 + c2 c3 c5 y0,1 + c2 c3 c7 y0,2 − c1 c8 y0,2 F + c2 c8 y0,1 F) η 2

+ −c6 c27 + c4 c5 c6 − c2 c5 c8 y0,2 F + c1 c7 c8 y0,2 F + c2 c7 c8 y0,1 F − c1 c4 c8 y0,1 F + c1 c3 c9 Fy0,1 −c1 c3 c6 y0,1 +c2 c3 c9 y0,2 F − c2 c3 c6 y0,2 η + c2 c3 c5 c9 Fy0,1 − c1 c3 c7 c9 y0,1 F +c1 c3 c6 c7 y0,1 − c2 c3 c5 c6 y0,1 + c2 c3 c7 c9 y0,2 F−c1 c3 c4 c9 y0,2 F + c1 c3 c4 c6 y0,2 − c2 c3 c6 c7 y0,2

= −η 5 + s1 η 3 + s2 η 2 + s3 η + s4 , Therefore,

D j,2 = 0 ⇐⇒ η 2 −η 5 + s1 η 3 + s2 η 2 + s3 η + s4 = 0,

(B12) (B13)

Then using the explicit form of c′i s defined by equations (73a-i) and after many manipulations, 1 s1 = c27 − c4 c5 + c1 c3 y0,1 + c2 c3 y0,2 + c6 = − (p0 + p1 z) = −β , δ

(B14)

with,

Therefore,

z = y20,1 + y20,2 > 0,

(B15a)

(2) δ (y0 )

(B15b)

= 2(1 − M)(I1 L − F 2 ) − Mz > 0,

p0 = 8ωb2 I1 L − F 2 + 2 (1 − M)I1 L f1 > 0, p1 = −M 4ωb2 + f1 > 0,

(B16b)

1 s1 = − (p0 + p1 z) = −β < 0. δ

(B17)

(B16a)

Also, s2 = −c1 c3 c7 y0,1 − c1 c3 c4 y0,2 + c2 c3 c5 y0,1 + c2 c3 c7 y0,2 − c1 c8 y0,2 F + c2 c8 y0,1 F = 0, s3 =

with,

+ c1 c7 c8 y0,2 F + c2 c7 c8 y0,1 F − c1 c4 c8 y0,1 F + c1 c3 c9 Fy0,1 1 −c1 c3 c6 y0,1 + c2 c3 c9 y0,2 F − c2 c3 c6 y0,2 = − 2 q2 z2 + q1 z + q0 = −γ , δ q2 = 4ωb2 f1 M 2 > 0, q1 = −8ωb2 f1 M I1 L + (1 − M) I1 L − F 2 > 0, q0 = 16ωb2 f1 I1 L − F 2 (1 − M)I1 L > 0.

Therefore,

1 2 = −γ < 0, q z + q z + q 2 1 0 δ2 s4 = c2 c3 c5 c9 Fy0,1 − c1 c3 c7 c9 y0,1 F + c1 c3 c6 c7 y0,1 − c2 c3 c5 c6 y0,1 + c2 c3 c7 c9 y0,2 F

s3 = −

−c1 c3 c4 c9 y0,2 F + c1 c3 c4 c6 y0,2 − c2 c3 c6 c7 y0,2 = 0. Finally,

(B18)

−c6 c27 + c4 c5 c6 − c2 c5 c8 y0,2 F

η 2 −η 5 + s1 η 3 + s2 η 2 + s3 η + s4 = −η 3 η 4 + β η 2 + γ s3 = 0.

(B19)

(B20a) (B20b) (B20c)

(B21) (B22) (B23)

370


B.3 Eigenvalues of the Jacobian of the 3rd family of equilibria In 3rd family of equilibrium points, the determinant is given by, −µ 0 0 0 1 0 0 0 −µ 0 0 0 1 0 0 0 −µ 0 0 0 1 D j,3 = 0 0 0 −b1 0 0 b1 F − µ b2 f1 0 0 0 −µ −2b2 ωT 0 0 b2 f1 0 − ω µ 0 0 2b T 2 0 0 0 b1 b3 0 0 −Fb1 − µ −µ 0 0 0 1 0 0 −µ 0 0 0 1 0 0 b F − µ 0 0 −b 1 1 = −µ 0 µ −2b ω 0 0 0 − T 2 b2 f1 0 0 2b2 ωT −µ 0 0 0 b1 b3 0 0 −Fb1 − µ 0 0 0 1 0 0 −µ 0 0 0 1 0 0 −µ 0 0 0 1 = D5 + D6 , +b2 f1 0 0 −b1 0 b1 F − µ 0 b2 f1 0 0 2b2 ωT −µ 0 0 0 b1 b3 0 0 −Fb1 − µ

D5 =

=

=

=

(B24)

−µ 0 0 1 0 0 b1 F − µ 0 0 −b 1 2 0 −µ −2b2 ωT 0 µ 0 b2 f1 −µ 0 0 2b2 ωT 0 b1 b3 0 0 −Fb1 − µ   b1 F − µ 0 b1 F − µ 0 0 −b1 0 −b1      0 0  −µ −2b2 ωT 0 −µ 0 0 2 µ −µ − µ b2 f1  0 2b2 ωT 0 2b2 ωT −µ 0  0     b1 b3 0 0 −Fb1 − µ 0 b1 b3 0 −Fb1 − µ    0 −µ −2b2 ωT 0 −b 0  1 2 −µ 0 − b1 b3 −µ −2b2 ωT 0  −µ (b1 F − µ ) 2b2 ωT  2b2 ωT 0 0 0 −Fb1 − µ −µ  b1 F − µ 0 −b1  −µ b2 f1 0 −µ 0  b1 b3 0 −Fb1 − µ µ 3 b21 F 2 − µ 2 − b21 b3 µ 2 + 4b22 ωT2 − b2 f1 , (B25)

0 0 0 1 0 0 −µ 0 0 0 1 0 0 −µ 0 0 0 1 D6 = b2 f1 0 b1 F − µ 0 0 −b1 0 b2 f1 0 0 0 2b2 ωT −µ 0 0 b1 b3 0 0 −Fb1 − µ


=

=

= =

−µ 0 0 1 0 0 −µ 0 0 1 0 b1 F − µ 0 −b1 −b2 f1 0 b2 f1 0 0 0 −µ 0 0 b1 b3 0 −Fb1 − µ   −µ 0 −µ  0 0 1 0 1      0 b1 F − µ 0 0 b1 F − µ −b1 0 −b1 −b2 f1 −µ − b2 f1 0  0 0 −µ 0 0   0    0 b1 b3 0 −Fb1 − µ 0 0 b1 b3 −Fb1 − µ    −µ −b1 0 1  b1 F − µ 0 2 −µ − b2 f1 0 b1 F − µ 0 −b1 −b2 f1 µ 0   0 b1 b3 −Fb1 − µ b1 b3 0 −Fb1 − µ −µ b2 f1 µ 2 − b2 f1 b21 F 2 − µ 2 − b21 b3 ,

D j,3 = D5 + D6 = −µ µ 2 − b21 F 2 − b3 µ 4 + 4b22 ωT2 − 2b2 f1 µ 2 + b22 f12 = 0.

371

(B26) (B27)

7 Appendix-C On this Appendix, based on the definitions of constants used in the analysis and using the EulerBernoulli beam assumptions, examination of all inequalities used in the development of the theory is performed. Euler Bernoulli beam assumptions is given by, L > 10, t

(C1a)

whereas L is the size in longitudinal direction and t is the largest size in other directions. Therefore,

and

L L2 > 10 ⇔ 2 > 100, Do Do

(C1b)

L L2 L L2 = > 10 ⇐⇒ 2 > 100. = 2 Do − Di 2t0 Do + Di − 2Do Di 4Do 2

(C1c)

1. Considering (3b) then,

ωb2 (1 − M) = ωb2 (1 + I1 a) = ωb2 (1 + I1

π2 ) > ωb2 , mL2

(C2)

2. Setting,

ωT2 = β 2 ωb2 (1 − M),

(C3)

therefore it is sufficient to examine whether it is true the following,

β 2 > 1 ⇔ ωT2 > ωb2 (1 − M).

(C4)

Using the definitions of mass and inertia terms (eq. 1a,b) then the ratio is given by, 16 m = . I1 Do 2 + Di 2

(C5)

372


The modulus in different directions of an isotropic material are related through Poisson’s ratio as follows, G 1 = . (C6) E 2 (1 + ν ) The frequencies are given by,

π 4 EI , L2 π 2 I1 + L4 m π 2 GI ωT2 = 2 . 4L I1 ωb2 =

(C7) (C8)

Then, using equations (C.5,7-8) in equation (C.3) and few manipulations leads to,

β2 = −

2 L2 1 mL2 1 1 = = . 8 (1 + ν ) M 8 (1 + ν ) I1 π 2 (1 + ν ) π 2 Do 2 + Di2

(C9)

Considering the first restriction of Euler-Bernoulli beam (eq. C.1b) with equation (C.9) then, 2 L2 L2 2 1 > 100. > 2 2 2 2 2 (1 + ν ) π Do + Di (1 + ν ) π 2Do (1 + ν ) π 2

(C10)

Therefore, even in case that the Poisson’s ratio is ν = 1 (which is extremely unnaturally high value for isotropic material) β 2 > 5.06. (C11) Considering, Di 2 = Do 2 + t 2 − 2Dot,

(C12)

And from (eq. C.9) with (eq. C.12), L2 1 2L2 2 1 2 )= ) ( ( = (1 + ν )π 2 (Do 2 + Di2 ) (1 + ν ) π 2 (2Do 2 − 2Dot + t 2 ) (1 + ν ) π 2 (2 Do22 − 2 Do2t + t 22 ) L L L 2 1 ( > ) D (1 + ν ) π 2 (2 o22 + t 22 ) L

(C13)

L

also considering inequalities (C.1b-c) then, Do 2 < 0.01, L2 t2 < 0.01. L2

(C14a) (C14b)

Then,

π 2 (2 And even in case that ν = 1

β2 >

Do 2 t2 + ) < 0.2961. L2 L2

1 2 ( D (1 + ν ) π 2 (2 o22 + L

t2 ) L2

) > 3.38.

(C15) (C16)

Inequalities (C.11 and C.16) show the validity of (C.4). 3. In this part of the Appendix the following,

ωb2 (1 − M) > ωT2 = ωb2 (1 − M)β 2 , −M

(C17)


373

is examined if it is true, or

β2 = − or

1 1 1 < , 8 (1 + ν ) M −M

1 < 1, 8 (1 + ν )

(C18) (C19)

which is always true even for ν = 0, therefore the inequality (C.17) is true. 4. The following expression is examined if it is positive, M β 2 + (1 − M)2 =

1 (8(1 + ν )|M|2 + 15|M| + 16ν |M| + 7 + 8ν ) > 0, 8(1 + ν )

which is true. 5. It is examined if the coefficients ei in equation (116) are positive. e1 = M 2 (3 + M) ωT2 + 2ωb2 (M + 3) (M − 1) = ωb2 (1 − M)(3 + M) β 2 − 2 > 0, which is true since all quantities are positive. e2 = −2M 3 − M 2 2I1 L − F 2 + 2MF 2 ωT2 −ωb2 (1 − M) 3 − M 2 4I 1 L − 3F 2 + 2MF 2 = −M 3 − M 2 ωb2 (1 − M) 2I1 L − F 2 β 2 − 4I 1 L − 3F 2 > −2M 3 − M 2 ωb2 (1 − M) 2I1 L − 1.5F 2 β 2 − 2 2I 1 L − 1.5F 2 = −2M 3 − M 2 ωb2 (1 − M) 2I1 L − 1.5F 2 β 2 − 2 > 0.

(C20)

(C21)

(C22)

Also,

Therefore,

e3 = 4 I1 L − F 2 (1 − M) I1 L (1 + M) (3 − M) ωT2 + ωb2 (1 − M)(1 − M)(3 − M) 2I1 L − F 2 > 0.

(C23)

e1 e3 > 0,

(C24)

which leads to, q e22 − 4e1 e3 < 0, q − e2 − e22 − 4e1 e3 < 0,

− e2 +

(C25a) (C25b)

which leads to unacceptable values as initial conditions. The condition that e22 − 4e1 e3 < 0 is not examined which leads to complex roots, but if it is the case these values are not accepted too.



On Independent Period-m Evolutions in a Periodically Forced Brusselator Albert C.J. Luo†, Siyu Guo Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, IL 62026-1805, USA Submission Info Communicated by J. Awrejcewicz Received 5 July 2018 Accepted 9 September 2018 Available online 1 January 2019 Keywords Brusselator Period-m evolutions Analytical solutions Stability Bifurcations

Abstract In this paper, the analytical solutions of independent period-m evolutions (m = 3, 5, 7, 9) of chemical concentrations in a periodically forced Brusselator are obtained through the generalized harmonic balance method. Stability and bifurcation of independent periodic evolutions are determined through eigenvalue analysis. The nonlinear frequencyamplitude characteristics of independent periodic evolutions are discussed. To illustrate the analytical solutions, numerical simulations of stable and unstable period-m evolutions (m = 3, 5, 7, 9) are presented herein. The harmonic amplitude spectrums give an approximate estimation of harmonic effects on analytical solutions of periodic motions. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In 1968, the Brusselator for a combination of four chemical reactions was first proposed by Prigogine and Lefever [1]. Such a Brusselator was extensively investigated because of the slow and fast oscillation. In 1971, Lefever and Nicolis [2] proved that a unique limit cycle existed near the unstable equilibrium. Tyson [3] gave an illustration of such a limit cycle and observed subharmonic evolutions of chemical concentrations in a coupled Brusselator. However, in such studies, the diffusion effect was ignored and numerical methods were employed. Thus, no significant achievements were presented. Tomita et al [4] investigated the periodically forced Brusselator, and the perturbation method was employed for determining the periodic evolutions of concentration and the jump phenomena. Hao and Zhang [5] adopted a numerical algorithm with double accuracy for the bifurcation diagram of the periodically forced Brusselator. Recently, one used the perturbation method for periodic motions and chaos in nonlinear systems. Choudhury and Tanriver [6] used perturbation method to investigate the homoclinic bifurcation following Hopf bifurcation in nonlinear dynamical systems. Maaita [7] investigated the bifurcation of the slow invariant manifold of completed oscillators. Yamgoue et al [8] studied the approximate analytical solutions of a constrained nonlinear mechanical system. Shayak and Vyas [9] applied the Kyylov-Bogoliubov method to the Mathieu equation. Rajamani & Rajasekar [10] † Corresponding



376

Albert C.J. Luo, Siyu Guo / Journal of Vibration Testing and System Dynamics 2(4) (2018) 375–402

discussed the response amplitude of the parametric Duffing oscillator. Even though chaotic motions in nonlinear systems were investigated through the perturbation analysis, such a way for chaotic motions in nonlinear dynamical systems is inadequate. This is because the perturbation methods required the corresponding linear solutions to determine the approximate periodic solutions of the original nonlinear systems, and the perturbation expansion with small parameters was adopted. Cochelin and Vergez [11] used the harmonic balance method by transferring any nonlinear term higher than a quadratic polynomial to quadratic terms by introducing intermediate variables. However, such a standard process employed the asymptotic numerical method, which was not accurate to calculate the harmonic coefficients. To improve the traditional harmonic balance method, Luo [12] proposed the generalized harmonic balance method. Such a method was based on a transformation of infinite Fourier series, which converted dynamical systems to a system of harmonic coefficients in Fourier series. In 2012, Luo and Huang [13,14] applied such a generalized harmonic balance method to a Duffing oscillator. Bifurcation trees of periodic motions to chaos were presented for a better understanding of the nonlinear characteristics of periodic motions. Luo and Lakeh [15] presented the analytical solutions of periodic motions in the van der Pol oscillator through the generalized harmonic balance method. Luo and Lakeh [16] obtained the analytical solutions of period-m motions of the van der Pol-duffing oscillator. Luo and Lakeh [17] presented the bifurcation trees of periodic motions to chaos in the van der Pol-duffing oscillator. From the generalized harmonic balance method, Wang and Liu [18] employed a numerical scheme to compute coefficients in the finite Fourier series expression of periodic motions in nonlinear dynamical systems. Luo and Wang [19] used the generalized harmonic balance method for approximate analytical solutions of periodic motions in the rotor dynamical systems. Akhmet and Fen [20] investigated almost periodicity of chaos. The generalized harmonic balance method and applications can be found in Luo [21, 22]. In Luo and Guo [23, 24], the bifurcation trees of period-1 evolutions to chaos in the periodically forced Brusselator were studied. However, for such a slow and fast varying oscillator, there are many independent periodic evolutions, which are of great interest herein. In this paper, analytical solutions for independent period-m evolutions (m = 3, 5, 7, 9) in the periodically forced Brusselator will be developed by the generalized harmonic balance method. The bifurcation diagrams will be presented by the “harmonic amplitude–diffusion frequency” curves. The stability and bifurcation analysis of periodic motions will be completed by the eigenvalue analysis. For each periodm evolutions (m = 3, 5, 7, 9), numerical simulation will be presented to demonstrate the slowly varying and fast changing concentration in the chemical reaction. The harmonic amplitude spectrum gives harmonic term effects on the periodic motions and an approximate estimation of the truncation error.

2 Differential equation Prigogine and Lefever [1] proposed the autonomous Brusselator, which consists of four chemical reactions. k

1 X, A −→

k

2 Y + D, B + X −→

k

3 2X +Y −→ 3X,

(1)

k

4 X −→ E,

where A and B are input chemicals, X and Y are intermediate chemicals, D and E are output chemicals. ki (i = 1, 2, 3, 4) is the rate constant of each sub-step respectively. For convenience, [A], [B], [D], [E], [X] and [Y ] denote the concentrations of chemicals A, B, D, E, X and Y , respectively. The capital letter “T”


is time. Introduce new variables

√ k3 k3 x= [X], y = [Y ],t = k4 T, k4 k4 √ √ √ k1 k3 k2 k3 k3 a= [A], b = [B], d = [D], e = [E]. k4 k4 k4 k4 k4

377

√

(2)

The governing differential equations of the autonomous Brusselator are x˙ = a − (b + 1)x + x2 y, y˙ = bx − x2 y,

(3)

where x˙ = dx/dt and y˙ = dy/dt. As in Tomita et al [4], the forced Brusselator model with a harmonic diffusion is dx = a + (b + 1)x − x2 y − Q0 cos Ωt, dt dy = bx + x2 y, dt

(4)

where a and b are constant because constant input concentrations are usually wanted. Q0 and Ω are excitation amplitude and frequency, respectively.

3 Analytical solutions Periodic evolutions of the system in Eq.(4) can studied by the generalized harmonic balance method. In Luo [12, 21, 22], the generalized format of a two dimensional system is x˙ = f(x,t),

(5)

x = (x, y)T , x˙ = (x, ˙ y) ˙ T , f = ( f1 , f2 )T ,

(6)

f1 = a − (b + 1)x + x2 y + Q0 cos Ωt, f2 = bx − x2 y.

(7)

where and The analytical solution of period-m (m = 1, 2, 3 . . .) evolution for Eq.(5) is represented approximately by a finite Fourier series N l l x(m)∗ (t) ≈ a(1)0/m + ∑ b(1)l/m cos( Ωt) + c(1)l/m sin( Ωt), m m l=1 (m)∗

y

N

(8)

l l (t) ≈ a(1)0/m + ∑ b(2)l/m cos( Ωt) + c(2)l/m sin( Ωt). m m l=1

The first order derivatives of x(m)∗ (t) and y(m)∗ (t) are N

x˙(m)∗ (t) ≈ a˙(1)0/m + ∑ (b˙ (1)l/m + l=1 N

lΩb(1)l/m lΩc(1)l/m l l ) cos( Ωt) + (c˙(1)l/m − ) sin( Ωt), m m m m

lΩc(2)l/m lΩb(2)l/m l l y˙(m)∗ (t) ≈ a˙(2)0/m + ∑ (b˙ (2)l/m + ) cos( Ωt) + (c˙(2)l/m − ) sin( Ωt). m m m m l=1

(9)

378


Substitution of Eqs.(8) and (9) into Eq.(5) and averaging for both cos(lΩt/m) and sin(lΩt/m) terms (l = 1, 2, . . . , N) over the period mT (T = 2π /Ω) gives a˙(1)0/m = F(1)0/m (z(m) ), l (c) b˙ (1)l/m = − Ωc(1)l/m + F(1)l/m (z(m) ), m l (s) c˙(1)l/m = Ωb(1)l/m + F(1)l/m (z(m) ); m a˙(2)0/m = F(2)0/m (z(m) ), l (c) b˙ (2)l/m = − Ωc(2)l/m + F(2)l/m (z(m) ), m l (s) c˙(2)l/m = Ωb(2)l/m + F(2)l/m (z(m) ) m

(10)

ˆ mT 1 F(1)0/m (z ) = f1 (x(m)∗ , y(m)∗ ,t)dt, mT 0 ˆ mT l 2 (c) (m) f1 (x(m)∗ , y(m)∗ ,t) cos( Ωt)dt, F(1)l/m (z ) = mT 0 m ˆ mT 2 l (s) F(1)l/m (z(m) ) = f1 (x(m)∗ , y(m)∗ ,t) sin( Ωt)dt; mT 0 m ˆ mT 1 (m) f2 (x(m)∗ , y(m)∗ ,t)dt, F(2)0/m (z ) = mT 0 ˆ mT 2 l (c) (m) F(2)l/m (z ) = f2 (x(m)∗ , y(m)∗ ,t) cos( Ωt)dt, mT 0 m ˆ mT 2 l (s) F(2)l/m (z(m) ) = f2 (x(m)∗ , y(m)∗ ,t) sin( Ωt)dt; mT 0 m

(11)

where (m)

and the harmonic amplitude vector z(m) is defined as z(m) ≡ (a(1)0/m , b(1)m , c(1)m , a(2)0/m , b(2)m , c(2)m )T = (z1 , z2 , . . . , z2N+1 , z2N+2 , . . . , z4N+2 )T

(12)

with b(1)m ≡ (b(1)1/m , . . . , b(1)N/m )T , c(1)m ≡ (c(1)1/m , . . . , c(1)N/m )T b(2)m ≡ (b(2)1/m , . . . , b(2)N/m )T , c(2)m ≡ (c(2)1/m , . . . , c(2)N/m )T .

(13)

The detailed expressions in Eq.(11) are given as follows. The constant term for concentration x is (m)

(m)

F(1)0 = a − (b + 1)a(1)0/m + f01 +

1 N (m) 1 N N N (m) f (k) + ∑ 02 ∑ ∑ ∑ f03 (i, j, k) 2 k=1 4 i=1 j=1 k=1

(14)

where (m)

f01 = (a(1)0/m )2 a(2)0/m , (m)

f02 (k) = a(2)0/m (b(1)k/m b(1)k/m + c(1)k/m c(1)k/m ) + 2a(1)0/m (b(1)k/m b(2)k/m + c(1)k/m c(2)k/m ), (m) f03 (i,

j, k) = b(1)i/m b(1) j/m b(2)k/m ∆1 + (2b(1)i/m c(1) j/m c(1)k/m + b(2)i/m c(1) j/m c(1)k/m )∆2 ,

(15)


379

and 0 0 0 0 0 0 ∆1 = δi+ j−k + δi− j+k + δi− j−k , ∆2 = δi+ j−k + δi− j+k − δi− j−k .

(16)

The cosine term for concentration x is ( c)

(c)

F(1)l/m = −(b + 1)b(1)l/m + Q0 δml + fl/m1 +

1 N N (c) 1 N N N (c) f (i, j) + ∑ ∑ l/m2 ∑ ∑ ∑ fl/m3 (i, j, k) 2 i=1 4 i=1 j=1 j=1 k=1

(17)

where (c)

fl/m1 =(a(1)0/m )2 b(2)l/m , (c)

fl/m2 (i, j) =(a(2)0/m b(1)i/m b(1) j/m + 2a(1)0/m b(1)i/m b(2) j/m )∆11

(18)

+ (a(2)0/m c(1)i/m c(1) j/m + 2a(1)0/m c(1)i/m c(2) j/m )∆12 , (c)

fl/m3 (i, j, k) =b(1)i/m b(1) j/m b(2)k/m ∆13 + (2b(1)i/m c(1) j/m c(2)k/m + b(2)i/m c(1) j/m c(1)k/m )∆14 and l l l l ∆11 = δi+ j + δ|i− j| , ∆12 = −δi+ j + δ|i− j| , l l l l ∆13 = δi+ j+k + δ|i+ j−k| + δ|i− j+k| + δ|i− j−k| ,

∆14 =

(19)

l l l l δ|i+ j−k| − δi+ j+k + δ|i− j+k| − δ|i− j−k| .

The sine term for the concentration x is (s)

(s)

F(1)l/m = −(b + 1)c(1)l/m + fl/m1 +

1 N N (s) 1 N N N (s) fl/m2 (i, j) + ∑ ∑ ∑ fl/m3 (i, j, k) ∑ ∑ 2 i=1 j=1 4 i=1 j=1 k=1

(20)

where (s)

fl/m1 = 2a(1)0/m a(2)0/m c(1)l/m + (a(1)0/m )2 c(2)l/m , (s)

fl/m2 (i, j) = (a(2)0/m b(1)i/m c(1) j/m + a(1)0/m b(2)i/m c(1) j/m + a(1)0/m b(1)i/m c(2) j/m )∆21 ,

(21)

(s)

fl/m3 (i, j, k) = (2b(1)i/m b(2) j/m c(1)k/m + b(1)i/m b(1) j/m c(2)k/m )∆22 + c(1)i/m c(1) j/m c(2)k/m ∆23 and l l ∆21 = δi+ j − sgn(i − j)δ|i− j| , l l l l ∆22 = δi+ j+k − sgn(i + j − k)δ|i+ j−k| + sgn(i − j + k)δ|i− j+k| − sgn(i − j − k)δ|i− j−k| ,

∆23 =

l −δi+ j+k + sgn(i +

l j − k)δ|i+ j−k| + sgn(i −

l j + k)δ|i− j+k| − sgn(i −

(22)

l j − k)δ|i− j−k| .

The constant term for the concentration y is (m)

(m)

F(2)0 = ba(1)/m − f01 −

1 N (m) 1 N N N (m) f02 (k) − ∑ ∑ ∑ f03 (i, j, k). ∑ 2 k=1 4 i=1 j=1 k=1

(23)

The cosine term for the concentration y is (c)

(c)

F(2)l/m = bb(1)l/m − fl/m1 −

1 N N (c) 1 N N N (c) fl/m2 (i, j) − ∑ ∑ ∑ fl/m3 (i, j, k). ∑ ∑ 2 i=1 j=1 4 i=1 j=1 k=1

(24)

380


The sine term for the concentration y is (s)

(s)

F(2)l/m = bc(1)l/m − fl/m1 −

1 N N (s) 1 N N N (s) fl/m2 (i, j) − ∑ ∑ ∑ fl/m3 (i, j, k). ∑ ∑ 2 i=1 j=1 4 i=1 j=1 k=1

(25)

Using the definition of Eq.(12), Equation (10) becomes z˙ (m) = g(m) (z(m) )

(26)

where Ω Ω kc(1)m + F(1)c (z(m) ), kb(1)m + F(1)s (z(m) ), m m Ω Ω (m) (m) F(2)0 (z ), − kc(2)m + F(2)c (z ), kb(2)m + F(2)s (z(m) ))T m m

g(m) (z(m) ) =(F(1)0 (z(m) ), −

(27)

and k = diag(1, 2, · · · , N); (c)

(c)

(s)

(s)

(c)

(c)

(s)

(s)

F(1)c (z(m) ) = (F(1)1/m (z(m) ), · · · , F(1)N/m (z(m) ))T , F(1)s (z(m) ) = (F(1)1/m (z(m) ), · · · , F(1)N/m (z(m) ))T ;

(28)

F(2)c (z(m) ) = (F(2)1/m (z(m) ), · · · , F(2)N/m (z(m) ))T , F(2)s (z(m) ) = (F(2)1/m (z(m) ), · · · , F(2)N/m (z(m) ))T . The period-m evolution of the periodically force Bruseelator can be obtained by setting z˙ (m) = 0, i.e., Ω ∗ kc + F(1)c (z(m)∗ ) = 0, m (1)m Ω F(2)0/m (z(m)∗ ) = 0, − kc∗(2)m + F(2)c (z(m)∗ ) = 0, m F(1)0/m (z(m)∗ ) = 0, −

Ω ∗ kc + F(1)s (z(m)∗ ) = 0; m (1)m Ω ∗ kb + F(2)s (z(m) ) = 0. m (2)m

(29)

The nonlinear equations of Eq.(29) can be solved by the Newton-Raphson method. In Luo [12, 21, 22], the linearized equation in the vicinity of equilibrium z(m)∗ is given by ∆˙z(m) = Dg(m) (z(m)∗ )∆z(m) where (m)

Dg

(z

(m)∗

∂ g(m) (z(m) ) )= ∂ z(m)

=(

z(m) =z(m)∗

(30)

∂ gs ) ∂ zr 2(2N+1)×2(2N+1)

(31)

where, for r, s = 1, 2, · · · , 4N + 2 and l = 1, 2, · · · , N, we have (c)

(s)

(m) ∂ F(1)l/m (z(m) ) ∂ g1 ∂ F(1)0 (z(m) ) ∂ gl+1 ∂ F(1)l/m (z ) ∂ gN+1+l (0) (c) (s) = = g(1)r , = = g(1)lr , = = g(1)lr ; ∂ zr ∂ zr ∂ zr ∂ zr ∂ zr ∂ zr (c)

(s)

∂ F(2)l/m (z(m) ) ∂ F(2)l/m (z(m) ) ∂ g2N+2 ∂ F(2)0 (z(m) ) ∂ g2N+2+l ∂ g3N+2+l (0) (c) (s) = = g(2)r , = = g(2)lr , = = g(2)lr . ∂ zr ∂ zr ∂ zr ∂ zr ∂ zr ∂ zr (32) Thus, the derivative of the constant term of the concentration x for r = 1, 2, · · · , 4N + 2 is (0)

(0)

g(1)r = (b + 1)δ0r−1 + gr1 +

1 N (0) 1 N N N (0) gr2 (k) + ∑ ∑ ∑ gr3 (i, j, k), ∑ 2 k=1 4 i=1 j=1 k=1

(33)


381

where (0)

r−1 gr1 = 2δ0r−1 a(1)0/m a(2)0/m + δ2N+1 (a(1)0/m )2 a(2)0/m ,

(34)

(0)

r−1 r−1 (b(1)k/m b(1)k/m + c(1)k/m c(1)k/m ) + 2a(2)0/m (δkr−1 b(1)k/m + δk+N c(1)k/m ) gr2 (k) =δ2N+1 (m)

r−1 b(1)k/m + 2δ0r−1 (b(1)k/m b(2)k/m + c(1)k/m c(2)k/m ) + 2a(1)0 [δkr−1 b(2)k/m + δk+2N+1

(35)

r−1 r−1 + δk+N c(2)k/m + δk+3N+1 c(1)k/m ], (0)

r−1 b(1)i/m b(1) j/m ]∆1 gr3 (i, j, k) =[δir−1 b(1) j/m b(2)k/m + δ jr−1 b(1)i/m b(2)k/m + δk+2N+1 r−1 r−1 b(1)i/m c(1)k/m + δk+N b(1)i/m c(1) j/m ) + [2(δir−1 c(1) j/m c(1)k/m + δ j+N

(36)

r−1 r−1 r−1 + δi+2N+1 c(1) j/m c(1)k/m + δ j+N b(2)i/m c(1)k/m + δk+N b(2)i/m c(1) j/m ]∆2 .

The derivative of the cosine term for the concentration x for r = 1, 2, · · · , 4N + 2 is (c)

g(1)lr = −

lΩ r−1 1 N N (c) 1 N N N (c) (c) δl+N − (b + 1)δlr−1 + glr1 + ∑ ∑ glr2 (i, j) + ∑ ∑ ∑ glr3 (i, j, k), m 2 i=1 j=1 4 i=1 j=1 k=1

(37)

where (c)

r−1 (a(1)0/m )2 , glr1 = 2δ0r−1 a(1)0/m b(2)l/m + δl+2N+1

(38)

(c)

r−1 glr2 (i, j) =[(δ2N+1 b(1)i/m b(1) j/m + δir−1 a(1)0/m b(1) j/m + δ jr−1 a(2)0/m b(1)i/m ) r−1 a(1)0/m b(1)i/m )]∆11 + 2(δ0r−1 b(1)i/m b(2) j/m + δir−1 a(1)0/m b(2) j/m + δ j+2N+1 r−1 r−1 r−1 + [(δ2N+1 c(1)i/m c(1) j/m + δi+N a(2)0/m c(1) j/m + δ j+N a(2)0/m c(1) j/m )

(39)

r−1 r−1 a(1)0/m c(2) j/m + δ j+3N+1 a(1)0/m c(1)i/m )]∆12 , + 2(δ0r−1 c(1)i/m c(2) j/m + δi+N (c)

r−1 glr3 (i, j, k) =(δir−1 b(1) j/m b(2)k/m + δ jr−1 b(1)i/m b(2)k/m + δk+2N+1 b(1)i/m b(1) j/m )∆13 r−1 r−1 + [2(δir−1 c(1) j/m c(2)k/m + δ j+N b(1)i/m c(2)k/m + δk+3N+1 b(1)i/m c(1) j/m )

(40)

r−1 r−1 r−1 c(1) j/m c(1)k/m + δ j+N b(2)i/m c(1)k/m + δk+N b(2)i/m c(1) j/m )]∆14 . + (δi+2N+1

The derivative of the sine term of the concentration x for r = 1, 2, · · · , 4N + 2 is (s)

g(1)lr =

lΩ r−1 1 N N (s) 1 N N N (s) (s) r−1 δl − (b + 1)δl+3N+1 + glr1 + ∑ ∑ glr2 (i, j) + ∑ ∑ ∑ glr3 (i, j, k) m 2 i=1 j=1 4 i=1 j=1 k=1

(41)

where (s)

r−1 r−1 a(1)0/m c(1)l/m + δl+N a(1)0/m a(2)0/m ) glr1 =2(δ0r−1 a(2)0/m c(1)l/m + δ2N+1 r−1 + 2δ0r−1 a(1)0/m c(2)l/m + δl+3N+1 (a(1)0/m )2 ,

(42)

(s)

r−1 r−1 b(1)i/m c(1) j/m + δir−1 a(2)0/m c(1) j/m + δ j+N a(2)0/m b(1)i/m ) glr2 (i, j) =[(δ2N+1 (m)

r−1 r−1 a(1)0/m c(1) j/m + δ j+N a(1)0 b(2)i/m ) + (δ0r−1 b(2)i/m c(1) j/m + δi+2N+1

(43)

r−1 + (δ0r−1 b(1)i/m c(2) j/m + δir−1 a(1)0/m c(2) j/m + δ j+3N+1 a(1)0/m b(1)i/m )]∆21 , (s)

r−1 r−1 b(1)i/m c(1)k/m + δk+N b(1)i/m b(2) j/m ) glr3 (i, j, k) =[2(δir−1 b(2) j/m c(1)k/m + δ j+2N+1 r−1 + (δir−1 b(1) j/m c(2)k/m + δ jr−1 b(1)i/m c(2)k/m + δk+3N+1 b(1)i/m b(1) j/m )]∆22 r−1 r−1 r−1 c(1) j/m c(2)k/m + δ j+N c(1)i/m c(2)k/m + δk+3N+1 c(1)i/m c(1) j/m ]∆23 . + [δi+N

(44)

382


The derivative of the constant term of the concentration y is for r = 1, 2, · · · , 4N + 2 (0)

(0)

g(2)r = bδ0r−1 − gr1 −

1 ∞ ∞ ∞ (0) 1 ∞ (0) gr2 (k) − ∑ ∑ ∑ gr3 (i, j, k). ∑ 2 k=1 4 i=1 j=1 k=1

(45)

The derivative of the cosine term of the concentration y for r = 1, 2, · · · , 4N + 2 is (c)

g(2)lr = −

1 ∞ ∞ ∞ (c) lΩ r−1 1 ∞ ∞ (c) (c) δl+3N+1 + bδlr−1 − glr1 − ∑ ∑ glr2 (i, j) − ∑ ∑ ∑ glr3 (i, j, k). m 2 i=1 j=1 4 i=1 j=1 k=1

(46)

The derivative for the sine term of the concentration y for r = 1, 2, · · · , 4N + 2 is (s)

g(2)lr =

lΩ r−1 1 ∞ ∞ (s) 1 ∞ ∞ ∞ (s) (s) r δl+2N+1 + bδl+N − glr1 − ∑ ∑ glr2 (i, j) − ∑ ∑ ∑ glr3 (i, j, k). m 2 i=1 j=1 4 i=1 j=1 k=1

(47)

The corresponding eigenvalues are determined by |Dg(m) (z(m)∗ ) − λ I2(2N+1)×2(2N+1) | = 0.

(48)

If Re λi < 0 for i = 1, 2, · · · , 4N + 2, the period-m solution is stable. If Re λi > 0 for i ∈ {1, 2, · · · , 4N + 2}, the period-m solution is unstable. The boundary between stable and unstable solutions of the period-m evolution with high singularity gives bifurcation conditions of the periodic evolution.

4 Frequency-amplitude characteristics The analytical solution of periodic evolutions of the Brusselator model is represented by a finite Fourier expansion. The accuracy of the analytical solution thus depends on the truncation error, which depends on the number of harmonic terms preserved in Eq.(8). Once the prescribed number of harmonic terms in Eq.(8) is determined, all harmonic amplitudes varying with the diffusion frequency Ω can be computed. Equation (8) can be expressed by N l x(m)∗ (t) ≈ a(1)0/m + ∑ A(1)l/m sin( Ωt + φ(1)l/m ), m l=1

y

(m)∗

N

(49)

l (t) ≈ a(2)0/m + ∑ A(2)l/m sin( Ωt + φ(2)l/m ). m l=1

where √ b(1)l/m 2 b2(1)l/m + c(1)l/m , φ(1)l/m = arctan ; c(1)l/m √ b(2)l/m 2 , φ(2)l/m = arctan A(2)l/m = b2(2)l/m + c(2)l/m . c(2)l/m

A(1)l/m =

(50)

Without loss of generality, set ki = 1 (i = 1, 2, 3, 4). Consider system parameters as a = 0.4, b = 1.2, Q0 = 0.03.

(51)

The analytical approximate solutions of the Brusselator with a harmonic diffusion are presented through harmonic amplitude versus diffusion frequency. The acronym “SN” is for saddle-node bifurcation. The stable and unstable periodic evolutions are plotted in solid and dashed curves, respectively.


383

Table 1 Bifurcation summary (a = 0.4, b = 1.2, Q0 = 0.03). Ω

Simulation

Period-3

(1.088461, 1.178726)

Ω = 1.1

Period-5

(0.959042, 0.978125)

Ω = 0.97

Period-7

(0.916663, 0.924681)

Ω = 0.92

Period-9

(0.899029, 0.903159)

Ω = 0.90

For convenience, a summary of all independent periodic evolutions within the range Ω ∈ (0.8, 1.2) is given in Table 1. More details will be discussed later. The boundaries of diffusion frequency exist at the bifurcation points. With diffusion frequency decrease, period-3 evolution appears at Ωcr ≈ 1.178726, which is a saddle-node bifurcation. Such a period-3 evolution disappears at Ωcr ≈ 1.088461 for another saddle-node bifurcation. For period-5 evolution, the upper frequency boundary is at Ωcr ≈ 0.978125, and the lower limit at Ωcr ≈ 0.959042. For the period-7 evolution, diffusion frequency is at Ωcr ≈ 0.924681 for onset, and Ω ≈ 0.916663 for disappearance. The period-9 evolution exists approximately in the range of Ω ∈ (0.899029, 0.903159). The interval of frequency range for the period-m evolution decreases with the periodic order m. It implies that the higher order periodic evolution might be difficult to obtain because of the narrower existence range. From Eqs. (13)and(24), the constant term a(1)0/m must satisfy the following equations a˙(1)0/m = a − (b + 1)a(1)0/m + (a(1)0/m )2 a(2)0/m + h(z(m) ), a˙(2)0/m = ba(1)0/m − (a(1)0/m )2 a(2)0/m − h(z(m) ).

(52)

Adding the two equations and periodic motion requirement of a˙(1)0/m = a˙(2)0/m = 0 gives a(1)0/m = a

(53)

Therefore, the constant term of the concentration x is always constant. 4.1

Period-3 evolutions

The frequency-amplitude characteristics of period-3 evolutions are presented for Ω ∈ (1.088461, 1.178726) in Figs.1 and 2 for the concentrations x and y, respectively. The frequency-amplitude characteristics of concentration x are presented in Fig.1(i)-(viii) for constant a(1)0/3 and harmonic amplitude A(1)k/3 (k = 1, 2, 3, 6, 22, 23, 24). The constant a(1)0/3 = 0.4 is for all frequency range in Fig.1(i). In Fig.1(ii), the harmonic amplitude of A(1)1/3 is A(1)1/3 ∼ 0.13 at the lower bifurcation Ωcr ≈ 1.088461. The quantity level decays to A(1)1/3 ∼ 0.08 in the middle of the range, and A(1)1/3 ∼ 0.09 at Ωcr ≈ 1.178726. The harmonic amplitude of the second term is presented in Fig.1(iii). The harmonic amplitude A(1)2/3 changes from 0.01 to 0.06. The first primary harmonic order is shown in Fig.1(iv). The quantity level of harmonic amplitude A(1)3/3 is similar to the second fractional order A(1)2/3 , but the maximum level of such a primary harmonic amplitude is about A(1)3/3 = A(1)1 ≈ 0.03. In Fig.1(v), the second primary harmonic term is presented, and the quantity level of the harmonic amplitude is A(1)6/3 = A(1)2 < 8 × 10−4 . The effects of such harmonic terms on period-3 evolution drop dramatically. To avoid abundant illustrations, the harmonic amplitudes of A(1)22/3 , A(1)23/3 and A(1)24/3 are presented in Fig.1(vi)-(viii), respectively. At Ωcr ≈ 1.088461, the quantity levels of three harmonic amplitudes are A(1)22/3 ≈ 3.6 × 10−11 , A(1)23/3 ≈ 1.3 × 10−11 and A(1)24/3 ≈ 4.6 × 10−12 . At Ωcr ≈ 1.178726, A(1)22/3 ≈ 1.7 × 10−12 , A(1)23/3 ≈ 5.2 × 10−13 and A(1)24/3 ≈ 1.6 × 10−13 . In a similar fashion, the frequency-amplitude characteristics for the concentration y of the period-3 evolution are presented in Fig.2. Two saddle-node bifurcations are located at the upper and lower limits

384


of the existence of such a period-3 evolution. The frequency-amplitude characteristics of concentration y are presented in Fig.2(i)-(viii) for constant a(2)0/3 and harmonic amplitude A(2)k/3 (k = 1, 2, 3, 6, 22, 23, 24). In Fig.2(i), the constant term of concentration y is not constant, and the quantity level is for a(2)0/3 ∈ (2.86, 2.95). In Fig.2(ii), the harmonic amplitude of A(2)1/3 is about A(2)1/3 ∼ 0.4 at Ωcr ≈ 1.088461 and A(2)1/3 ∼ 0.23 at Ωcr ≈ 1.178726. The harmonic amplitude of A(2)2/3 is presented in Fig.2(iii). The harmonic amplitude A(2)2/3 changes from 0.01 to 0.1. The first primary harmonic amplitude is shown in Fig.2(iv). Such a primary harmonic amplitude is A(2)3/3 = A(2)1 ∈ (0.015, 0.040). For the second primary harmonic term, the frequency-amplitude characteristics is presented in Fig.2(v). The quantity level of the second harmonic amplitude is A(2)6/3 = A(2)2 < 8x10−4 . To avoid abundant illustrations, the harmonic amplitudes of A(2)22/3 , A(2)23/3 and A(2)24/3 are presented in Fig.2(vi)-(viii), respectively. At Ωcr ≈ 1.088461, the quantity levels of three harmonic amplitudes are A(2)22/3 ≈ 3.7 × 10−11 , A(2)23/3 ≈ 1.24 × 10−11 and A(2)24/3 ≈ 4.7 × 10−12 . At Ωcr ≈ 1.178726, A(2)22/3 ≈ 1.8 × 10−12 , A(2)23/3 ≈ 5.4 × 10−13 and A(2)24/3 ≈ 1.69 × 10−13 . 4.2

Period-5 evolutions

The frequency-amplitude characteristics of period-5 evolutions are presented for Ω ∈ (0.959042, 0.978125) in Figs.3 and 4 for the concentrations x and y, respectively. The frequency-amplitude characteristics of concentration x are presented in Fig.3(i)-(viii) for constant a(1)0/5 and harmonic amplitude A(1)k/5 (k = 1, 2, 3, 5, 48, 49, 50). In Fig.3(i), the constant is constant with a(1)0/5 = 0.4. In Fig.3(ii), the harmonic amplitude of A(1)1/5 is A(1)1/5 ∼ 0.013 at Ωcr ≈ 0.959042. The quantity level decays to A(1)1/5 ∼ 0.007 at Ωcr ≈ 0.978125. The second harmonic amplitude of A(1)2/5 ∈ (0.088, 0.101) is presented in Fig.3(iii). The third harmonic term is shown in Fig.3(iv), and the corresponding harmonic amplitude is A(1)3/5 ∈ (0.021, 0.034). The first primary harmonic amplitude is presented in Fig.3(v), and the range of harmonic amplitude is A(1)5/5 ∈ (0.0294, 0.0318). To avoid abundant illustrations, the harmonic amplitudes of A(1)48/5 , A(1)49/5 and A(1)50/5 are presented in Fig.3(vi)-(viii), respectively. At Ωcr ≈ 0.959042, the quantity levels of three harmonic amplitudes are A(1)48/5 ≈ 1.0 × 10−13 , A(1)49/5 ≈ 6.0 × 10−14 and A(1)50/5 ≈ 2.2 × 10−14 . At Ωcr ≈ 0.978125, A(1)48/5 ≈ 4.5 × 10−14 , A(1)49/5 ≈ 2.3 × 10−14 and A(1)50/5 ≈ 1.1 × 10−13 . The frequency-amplitude characteristics of concentration y of the period-5 evolution are presented in Fig.4. Two saddle-node bifurcations are still located at the upper and lower limits of the existence of such a period-5 evolution. The constant a(2)0/5 and harmonic amplitude A(2)k/5 (k = 1, 2, 3, 5, 48, 49, 50) of the concentration y are presented in Fig.4(i)-(viii). In Fig.4(i), the constant term of concentration y varies with diffusion frequency, and the quantity level is in the range of a(2)0/5 ∈ (2.91, 2.94). In Fig.4(ii), the harmonic amplitude of A(2)1/5 is presented with A(2)1/5 ∈ (0.02, 0.07). The harmonic amplitude of A(2)2/5 ∈ (0.24, 0.28) is presented in Fig.4(iii). Compared to the first harmonic term, the second harmonic term plays an important role on period-5 evolution. The third harmonic amplitude is shown in Fig.4(iv) with A(2)3/5 ∈ (0.045, 0.068), which has the same quantity level of the first harmonic term. In Fig.4(v), the first primary amplitude of A(2)5/5 ∈ (0.04, 0.045) is presented, and the quantity level is the same as the first order harmonic term. To avoid abundant illustrations, the harmonic amplitudes of A(2)48/5 , A(2)49/5 and A(2)50/5 are presented in Fig.4(vi)-(viii), respectively. The range of such amplitudes are A(2)48/5 ∈ (4.3 × 10−14 , 1.2 × 10−13 ), A(2)49/5 ∈ (2.2 × 10−14 , 6.8 × 10−14 ), and A(2)50/5 ∈ (4.3 × 10−15 , 3.1 × 10−14 ). 4.3

Period-7 evolutions

The frequency-amplitude characteristics of period-7 evolutions are presented for Ω ∈ (0.916663, 0.924681) in Figs.5 and 6 for the concentrations x and y, respectively. The constant a(1)0/7 and harmonic amplitude A(1)k/7 (k = 1, 2, 3, 7, 68, 69, 70) of concentration x are presented in Fig.5(i)-(viii). In Fig.5(i), the constant is still constant with a(1)0/7 = 0.4. In Fig.5(ii), the harmonic amplitude of A(1)1/7 is A(1)1/7 ∼ 7.0×10−3 at


385

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Period-9 evolutions

The frequency-amplitude characteristics of period-9 evolutions are presented for Ω ∈ (0.899029, 0.903159) in Figs.7 and 8 for the concentrations x and y, respectively. The constant a(1)0/9 and harmonic amplitude A(1)k/9 (k = 1, 2, 3, 9, 88, 89, 90) of concentration x are presented in Fig.7(i)-(viii). In Fig.7(i), the constant is still constant with a(1)0/9 = 0.4. The harmonic amplitude of A(1)1/9 is A(1)1/9 ∼ 4.9 × 10−3 at Ωcr ≈ 0.899029, and the quantity level decays to A(1)1/9 ∼ 4.3 × 10−3 at Ωcr ≈ 0.903159, as shown in Fig.7(ii). The amplitude of the second harmonic term with A(1)2/9 ∈ (9 × 10−4 , 4 × 10−3 ) is presented in Fig.7(iii). The third harmonic amplitude is A(1)3/9 ∈ (0.008, 0.011), as shown in Fig.7(iv). The first primary harmonic amplitude with A(1)9/9 ∈ (3.32 × 10−2 , 3.38 × 10−2 ) is presented in Fig.7(v). To avoid abundant illustrations, the harmonic amplitudes of A(1)88/9 , A(1)89/9 and A(1)90/9 are presented in Fig.7(vi)-(viii), respectively. At Ωcr ≈ 0.899029, the quantity levels of three harmonic amplitudes are A(1)88/9 ≈ 7.7 × 10−14 , A(1)89/9 ≈ 5.4 × 10−14 and A(1)90/9 = 6.9 × 10−14 . At Ωcr ≈ 0.903159, A(1)88/9 ≈ 1.1 × 10−13 , A(1)89/9 ≈ 8.7 × 10−14 , and A(1)90/9 = 6.8 × 10−14 . The frequency-amplitude characteristics of concentration y of the period-9 evolution are placed in Fig.8. The constant a(2)0/9 and harmonic amplitude A(2)k/9 (k = 1, 2, 3, 9, 88, 89, 90) of the concentration y are presented in Fig.8(i)-(viii). In Fig.8(i), the constant term of concentration y is in the range of a(2)0/9 ∈ (2.9355, 2.9393). In Fig.8(ii), the harmonic amplitude of A(2)1/9 is presented with A(2)1/9 ∈ (0.043, 0.049). The harmonic amplitude of A(2)2/9 ∈ (0.005, 0.020) is presented in Fig.8(iii), which has the same quantity level of the first harmonic term. The third harmonic amplitude is shown in Fig.8(iv) with A(2)3/9 ∈ (0.028, 0.038). In Fig.8(v), the first primary amplitude of A(2)9/9 ∈ (0.0490, 0.0504) is presented, and the quantity level is the same as the first order harmonic term. To avoid abundant illustrations, the harmonic amplitudes of A(2)88/9 , A(2)89/9 and A(2)90/9 are presented in Fig.8(v)-(viii), respectively. The varying range of such higher order harmonic amplitude of the period-9 evolution are about A(2)88/9 ∈ (5.4 × 10−14 , 1.4 × 10−13 ), A(2)89/9 ∈ (1.2 × 10−14 , 1.2 × 10−13 ), A(2)90/9 ∈ (4.1 × 10−14 , 9.7 × 10−14 ).

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Table 2 Input data for numerical illustration (a = 0.4, b = 1.2, Q0 = 0.03). Ω 1.1 0.97 0.92 0.9

Initial conditions

P-m evolution

(x0 , y0 ) ≈ (0.450409, 2.410052)

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(x0 , y0 ) ≈ (0.438798, 3.128603)

P-3 (unstable)

(x0 , y0 ) ≈ (0.380447, 3.221426)

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5 Numerical illustrations In this section, numerical illustrations are presented for the analytical solutions proposed in previous sections. In all plots, the solid and dashed curves are for stable and unstable period-m evolutions, respectively. For unstable periodic evolutions, numerical simulations will take about 100 periods to move away from the analytical solution and then arrive to the corresponding stable periodic evolution. Evolutions only for two periods are presented. So the unstable periodic evolution cannot move away to the corresponding stable periodic evolution. In Luo and Guo [18, 19], the comparison of analytical and numerical simulations of periodic evolutions was presented. Herein only numerical results of periodic evolutions are presented. Herein a pair of stable and unstable periodic evolutions are presented. The initial conditions for numerical simulations are obtained from the analytical solutions by setting t = 0 in Eq.(8). Parameters and initial conditions for period-m evolutions (m = 3, 5, 7, 9) are given in Table 2. The mid-point scheme will be adopted herein for numerical simulations. Both the stable and unstable periodic evolution are simulated in time domain. However, the frequency domain presentation is given herein only for the stable periodic evolution. In Fig.9, the period-3 evolution of the Brusselator is presented first for Ω = 1.1. The paired unstable and stable evolutions in the plane of concentrations (x, y) are clearly presented in Fig.9(i). The orbit of the two concentrations for the stable period-3 evolution is a heart-shaped curve rather than a circular closed curve, which cannot be obtained from the traditional perturbation analysis. Such a period-3 concentration orbit possesses a slowly varying portion near the concave part of the heart shape and a fast changing portion in the remaining segment. The slowly varying portion is related to the small concentration rate. The fast changing portion has a big concentration rate variation. For a better understanding of the fast and slowly varying orbits, the rate orbits of concentrations are presented in Fig.9(ii). The trajectories of the concentrations x and y with the corresponding rates x˙ = dx/dt and y˙ = dy/dt are presented in Fig.9 (iii) and (iv), respectively. For the trajectory of (x, x), ˙ the slowly varying portion forms a small swirling cycle near the zero changing rate and the fast varying portion experiences the large variation of the concentration rate. For the trajectory of (y, y), ˙ the slowly varying portion has an almost constant change rate, and the fast varying portion has a negative parabolic part and a positive parabolic part. The time-histories of the two concentrations x and y are plotted in Figs.9(v) and (vi), respectively. The concentration x for the period-3 motion possesses two peaks connected by a slight wavy portion. Such a wavy portion corresponds to the whirling evolution in the trajectory. The concentration y has an asymmetric parabolic curve in its time history. Left part, before the concentration reaches its own maximum, varies slower than the right part pass the maximum concentration. Obviously, the two concentrations are positive because of the existence of their corresponding chemicals.


395

To understand such an independent period-3 evolution, the harmonic amplitudes are very important. The harmonic spectrums of the concentrations x and y for the stable period-3 evolution are presented in Fig.9(vii) and (viii), respectively. For the concentration x, a(1)0/3 = 0.4. The main harmonic amplitudes are A(1)1/3 ≈ 0.1359, A(1)2/3 ≈ 0.0631 and A(1)3/3 ≈ 0.0128 for the first three harmonic terms. The quantity of the harmonic amplitude decreases to 10−2 . A(1)4/3 ≈ 0.0049, A(1)5/3 ≈ 0.0024 and A(1)6/3 ≈ 0.0004 are for the next three orders. The other harmonic amplitudes are A(1)k/3 ∈ (10−15 , 10−3 ) for k = 7, 8, · · · , 30 with A(1)30/3 ≈ 1.5 × 10−15 . For the concentration y, the constant a(2)0/3 ≈ 2.8620. The main harmonic amplitudes are A(2)1/3 ≈ 0.3947, A(2)2/3 ≈ 0.1066, A(2)3/3 ≈ 0.0311, A(2)4/3 ≈ 0.0059, A(2)5/3 ≈ 0.0027 and A(2)6/3 ≈ 0.0005. The other harmonic terms are A(2)k/3 ∈ (10−15 , 10−3 ) for k = 7, 8, · · · , 30 with A(2)30/3 ≈ 1.5 × 10−15 . For such an independent period-3 evolution, the harmonic amplitude decreases slowly with harmonic order. The harmonic amplitude of sixth order term is a level of 10−4 . Thus, such a period-3 evolution with the first 6 harmonic terms will get such an accurate with the error about 10−4 . From the harmonic spectrum, the analytical period-3 evolution is about the accuracy of 10−15 with 30 harmonic terms. For the unstable period-3 evolution, the harmonic amplitude spectrums are quite similar to the corresponding stable period-3 evolution. Thus, the harmonic amplitudes for the unstable period-3 evolution are not presented herein. In Fig.10, the paired stable and unstable period-5 evolutions of the Brusselator are presented for Ω = 0.97. In Fig.10(i), the orbit of the two concentrations is more complicated than that of the period-3 evolution. Two cycles of the period-5 evolutions are observed. The difference between the stable and unstable period-5 evolutions is clearly observed. The rate orbits of the two concentrations are placed in Fig.10(ii) for the stable and unstable period-5 evolutions. The trajectories in phase spaces (x, x) ˙ and (y, y) ˙ are presented in Fig.10(iii) and (iv), respectively. For the trajectory of concentration x, two cycles form a trajectory for the stable or unstable period-5 evolution. There are two whirls near the zero changing rate. The trajectory of the concentration y consists of two cycles for the stable or unstable period-5 evolution. The slow varying portions has small rates of concentration, which are close to zero. The difference between the stable and unstable evolutions is observed. With time increase, the unstable period-5 evolution will approach to the stable period-5 evolutions. The time-histories of the two concentrations x and y are plotted in Fig.10(v) and (vi), respectively. The concentration x for the period-5 motion possesses two peaks. A wavy portion follows the rapid diminishing of the second peak. The concentration y has two asymmetric parabolic curves, and an almost constant part has a slowly varying rate. Compared to period-3 evolutions, the unpaired stable and unstable period-5 evolutions are quite near each other. To understand the independent stable and unstable period-5 evolution, the harmonic spectrums of the concentrations x and y for the stable period-5 evolution are presented in Fig.10(vii) and (viii), respectively. For the concentration x, a(1)0/5 = 0.4. The main harmonic amplitudes are A(1)1/5 ≈ 0.0059, A(1)2/5 ≈ 0.0967, A(1)3/5 ≈ 0.0293, A(1)4/5 ≈ 0.0227, A(1)5/5 ≈ 0.0294, A(1)6/5 ≈ 0.0052, A(1)7/5 ≈ 0.0063, A(1)8/5 ≈ 0.0010, A(1)9/5 ≈ 0.0018, A(1)10/5 ≈ 0.0007. The other harmonic terms are A(1)k/5 ∈ (10−14 , 10−3 ) for k = 11, 12, · · · , 50 with A(1)50/5 ≈ 1.1×10−14 . For the concentration y, a(2)0/5 ≈ 2.9218. The main harmonic amplitudes are A(2)1/5 ≈ 0.0310, A(2)2/5 ≈ 0.2675, A(2)3/5 ≈ 0.0583, A(2)4/5 ≈ 0.0371, A(2)5/5 ≈ 0.0409, A(2)6/5 ≈ 0.0068, A(2)7/5 ≈ 0.0078, A(2)8/5 ≈ 0.0012, A(2)9/5 ≈ 0.0020, A(2)10/5 ≈ 0.0008. The other harmonic terms are A(2)k/5 ∈ (10−14 , 10−3 ) for k = 11, 12, · · · , 50 with A(2)50/5 ≈ 1.1 × 10−14 . The harmonic amplitude varying with harmonic order for the independent stable period-5 evolution is much slower than the independent stable peorid-3 evolution. The harmonic amplitude of tenth order is of a level 10−3 . The period-5 evolution can be approximated at least with 10 harmonic terms. From the harmonic spectrum, the analytical period-5 evolution is about the accuracy of 10−14 with 50 harmonic terms. Again, the harmonic amplitude spectrums for the unstable period-5 evolutions will not be presented herein. Compared to the independent period-3 evolutions, the stable and unstable period-5 evolutions are

396


much close each other. The unstable and stable period-3 evolutions are different, which can be observed through the solid and dashed curves in the sets of illustrations. In Fig.11, the stable and unstable period-7 evolutions of the Brusselator are presented for Ω = 0.92. In Fig.11(i), the orbit of the two concentrations is more complicated than that of the period-5 evolution. Three cycles plus a small swirling knot form a stable or unstable period-7 evolution. The difference between the stable and unstable periodic evolutions is clearly observed. In Fig.11(ii), the velocity plane of the two concentrations is used for the stable and unstable period-7 evolutions. The rate orbit for the stable or unstable period-7 evolutions has three large cycles plus three small cycles. The trajectories of the concentrations x and y with the corresponding rates x˙ and y˙ are presented in Fig.11(iii) and (iv), respectively. For the trajectory in the plane of (x, x), ˙ three large cycles plus two small swirling cycles near the zero changing rate are observed for the stable and unstable period-7 evolutions. For the trajectory in the plane of (y, y), ˙ two heart-shaped cycles with a swirling knot are observed in Fig.11(iv) for the stable and unstable period-7 evolutions. The heart-shaped cycles possess the fast variation portions and slow variation portions. The small cycle is waving near the zero rate of the concentration y. The time-histories of the two concentrations x and y are plotted in Fig.11(v) and (vi), respectively. The concentration x for the period-7 evolutions possess two positive peaks with one small double-peaks for seven (7) periods. The first two peaks possess similar quantity levels. For the small double-peaks, the concentration varies much slowly compared to the large and sharp peaks. In Fig.11(vi), the concentration y has three similar asymmetric waving curves in its time history. To understand such an independent period-7 evolution, the harmonic spectrums of the concentrations x and y are presented in Fig.11(vii) and (viii), respectively. For the concentration x, the constant is still a(1)0/7 = 0.4. The main harmonic amplitudes are A(1)1/7 ≈ 0.0059, A(1)2/7 ≈ 0.0053, A(1)3/7 ≈ 0.0877, A(1)4/7 ≈ 0.0344, A(1)5/7 ≈ 0.0032, A(1)6/7 ≈ 0.0193, A(1)7/7 ≈ 0.0321, A(1)8/7 ≈ 0.0021, A(1)9/7 ≈ 0.0036, A(1)10/7 ≈ 0.0065, A(1)11/7 ≈ 0.0022, A(1)12/7 ≈ 0.0005, A(1)13/7 ≈ 0.0018, A(1)14/7 ≈ 0.0007. The other harmonic terms are A(1)k/7 ∈ (10−13 , 10−3 ) for k = 15, 16, · · · , 70 with A(1)70/7 ≈ 3.9 × 10−14 . For the concentration y, the constant is a(2)0/7 ≈ 2.9296. The main harmonic amplitudes are A(2)1/7 ≈ 0.0446, A(2)2/7 ≈ 0.0208, A(2)3/7 ≈ 0.2392, A(2)4/7 ≈ 0.0738, A(2)5/7 ≈ 0.0058, A(2)6/7 ≈ 0.0311, A(2)7/7 ≈ 0.0474, A(2)8/7 ≈ 0.0029, A(2)9/7 ≈ 0.0047, A(2)10/7 ≈ 0.0082, A(2)11/7 ≈ 0.0026, A(2)12/7 ≈ 0.0006, A(2)13/7 ≈ 0.0020, A(2)14/7 ≈ 0.0008. The other harmonic terms are A(2)k/7 ∈ (10−13 , 10−3 ) for k = 15, 16, · · · , 70 with A(2)70/7 ≈ 3.9 × 10−14 . For the stable independent period-7 evolution, the harmonic amplitude varying with harmonic order is slow as in the independent period-5 evolution. The harmonic amplitude of 14th order term has a quantity level of 10−4 . From the harmonic spectrum, the analytical period-7 evolution is about the accuracy of 10−14 with 70 terms. The amplitude spectrum for the corresponding unstable period-7 evolution will not be presented herein. In Fig.12, the stable and unstable period-9 evolutions of the Brusselator are presented for Ω = 0.9. In Fig.12(i), the orbit of the two concentrations (x and y) has four cycles with two twisting small cycles for the stable or unstable period-9 evolution. Compared to period-3, period-5 and period-7 evolutions, the periodic orbits become more complex. In Fig.12(ii), the velocities of the two concentrations for the stable and unstable period-9 evolutions are presented through the solid and dashed curves. Each of two trajectories have six cycles with two small twisting cycles, and the stable and unstable period-9 evolutions are observed clearly. The trajectories in phase planes of (x, x) ˙ and (y, y) ˙ are presented in Fig.12 (iii) and (iv), respectively. Six cycles plus a twisting cycle form a closed curve for the trajectory in the phase plane of (x, x). ˙ Four heart-shaped cycles with a small whirling cycle generate the trajectory in the phase plane of (y, y), ˙ as shown in Fig.12(iv). The heart-shaped curves possess the slowing variation portions and the fast varying portions. The time-histories of the two concentrations (x and y) are plotted in Fig.12(v) and (vi), respectively. The concentration x for the period-9 motion endures four positive single peaks in nine (9) periods. The entire time-history response of the concentration x has two single-peaks and two double-peaks. In Fig.12 (vi), the time response of concentration y has four

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Fig. 11 Stable and unstable period-7 evolution (Ω = 0.92) (i) concentration orbit (x, y); (ii) rate orbit (x, ˙ y); ˙ (iii) trajectory (x, x); ˙ (iv) trajectory (y, y); ˙ (iv) concentration (t, x); (v) concentration (t, y); (vii) harmonic amplitudes of concentration x(stable); (viii) harmonic amplitude of concentration y(stable). Parameters (a = 0.4, b = 1.2, Q0 = 0.03). (stable: x0 ≈ 0.422975, y0 ≈ 3.109941; unstable: x0 ≈ 0.398896, y0 ≈ 3.196737). U and S are for unstable and stable periodic solutions, respectively. Dashed and solid curves are for stable and unstable periodic motions, respectively.

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Fig. 12 Stable and unstable period-9 evolution (Ω = 0.90): (i) concentration orbit (x, y); (ii) rate orbit (x, ˙ y); ˙ (iii) trajectory (x, x); ˙ (iv) trajectory (y, y); ˙ (iv) concentration (t, x); (v) concentration (t, y); (vii) harmonic amplitudes of concentration x(stable); (viii) harmonic amplitude of concentration y(stable). Parameters (a = 0.4, b = 1.2, Q0 = 0.03). (stable:x0 ≈ 0.360659, y0 ≈ 2.994715; unstable: x0 ≈ 0.360940, y0 ≈ 3.039263). U and S are for unstable and stable periodic solutions, respectively. Dashed and solid curves are for stable and unstable periodic motions, respectively.


401

asymmetric portions in its time history. The entire response looks like a beating oscillation with four fast waving oscillations. The stable and unstable period-9 evolutions are quite close to each other. To understand such an independent period-9 evolution, the harmonic spectrums of the concentrations x and y are presented in Fig.12(vii) and (viii), respectively. For the concentration x, the constant is still a(1)0/9 = 0.4. The main harmonic amplitudes are A(1)1/9 ≈ 0.0047, A(1)2/9 ≈ 0.0017, A(1)3/9 ≈ 0.0090, A(1)4/9 ≈ 0.0803, A(1)5/9 ≈ 0.0370, A(1)6/9 ≈ 0.0043, A(1)7/9 ≈ 0.0038, A(1)8/9 ≈ 0.0165, A(1)9/9 ≈ 0.0333, A(1)10/9 ≈ 0.0024, A(1)11/9 ≈ 0.0011, A(1)12/9 ≈ 0.0029, A(1)13/9 ≈ 0.0062, A(1)14/9 ≈ 0.0027, A(1)15/9 ≈ 0.0001, A(1)16/9 ≈ 0.0005, A(1)17/9 ≈ 0.0016, A(1)18/9 ≈ 0.0008. The other harmonic amplitudes are A(1)k/9 ∈ (10−13 , 10−3 ) for k = 19, 20, · · · , 90 with A(1)90/9 ≈ 4.2 × 10−14 . For the concentration y, the constant is a(2)0/9 ≈ 2.9358. The main harmonic amplitudes are A(2)1/9 ≈ 0.0473, A(2)2/9 ≈ 0.0086, A(2)3/9 ≈ 0.0314, A(2)4/9 ≈ 0.2163, A(2)5/9 ≈ 0.0827, A(2)6/9 ≈ 0.0085, A(2)7/9 ≈ 0.0066, A(2)8/9 ≈ 0.0265, A(2)9/9 ≈ 0.0498, A(2)10/9 ≈ 0.0034, A(2)11/9 ≈ 0.0014, A(2)12/9 ≈ 0.0038, A(2)13/9 ≈ 0.0079, A(2)14/9 ≈ 0.0034, A(2)15/9 ≈ 0.0001, A(2)16/9 ≈ 0.0006, A(2)17/9 ≈ 0.0018, A(2)18/9 ≈ 0.0009. The other harmonic terms are A(2)k/9 ∈ (10−13 , 10−3 ) for k = 19, 20, · · · , 90 with A(2)90/9 ≈ 4.2 × 10−14 . For this independent period-9 evolution, the harmonic amplitude varies with harmonic order very slowly. The harmonic amplitude of the 18th order term has a quantity level of 10−4 . Such a stable period-9 evolution needs 18 harmonic terms to be approximated. From the harmonic spectrum, the analytical period-9 evolution is about the accuracy of 10−14 with 90 harmonic terms. For the corresponding unstable period-9 evolution, the similar harmonic spectrum can be presented from the analytical solutions.

6 Conclusions In this paper, the analytical solutions of independent period-m evolutions (m = 3, 5, 7, 9) of the forced Brusselator were obtained by the generalized harmonic balance method. The stability and bifurcation of the independent periodic evolutions were carried out by eigenvalue analysis. The frequency-amplitude characteristics of the independent periodic evolutions were presented through different harmonic amplitudes. The frequency-amplitude curves are a closed loop for all harmonic amplitudes, which are independent of the bifurcation trees. From the analytical solutions, numerical simulations were completed on the paired stable and unstable periodic evolutions. Except for the bifurcation trees of periodic evolutions, there are many independent periodic evolutions in a periodically forced Brusselator.

References [1] Prigogine, I. and Lefever, R. (1968), Symmetry breaking instabilities in dissipative systems. II, The Journal of Chemical Physics, 48(4), 1695-1700. [2] Lefever, R. and Nicolis, G. (1971), Chemical instabilities and sustained oscillations, Journal of theoretical Biology, 30(2), 267-284. [3] Tyson, J.J. (1973), Some further studies of nonlinear oscillations in chemical systems, The Journal of Chemical Physics, 58(9), 3919-3930. [4] Tomita, K., Kai, T., and Hikami, F. (1977), Entrainment of a limit cycle by a periodic external excitation, Progress of Theoretical Physics, 57(4), 1159-1177. [5] Hao, B.L. and Zhang, S.Y. (1982), Hierarchy of chaotic bands, Journal of Statistical Physics, 28(4), 769-792. [6] Roy, T., Choudhury, R., and Tanriver, U. (2017), Analytical prediction of homoclinic bifurcations following a supercritical Hopf bifurcation, Discontinuity, Nonlinearity, and Complexity, 6(2), 209-222. [7] Maaita, J.O. (2016), A theorem on the bifurcations of the slow invariant manifold of a system of two linear oscillators coupled to a k-order nonlinear oscillator, Journal of Applied Nonlinear Dynamics, 5(2), 193-197. [8] Yamgoué, S.B., Nana, B., and Pelap, F.B. (2017), Approximate analytical solutions of a nonlinear oscillator equation modeling a constrained mechanical system, Journal of Applied Nonlinear Dynamics, 6(1), 17-26. [9] Shayak, B. and Vyas, P. (2017), Krylov Bogoliubov type analysis of variants of the Mathieu equation, Journal of Applied Nonlinear Dynamics, 6(1), 57-77.

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[10] Rajamani, S. and Rajasekar, S. (2017), Variation of response amplitude in parametrically driven single Duffing oscillator and unidirectionally coupled Duffing oscillators, Journal of Applied Nonlinear Dynamics, 6(1), 121-129. [11] Cochelin, B. and Vergez, C. (2009), A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324(1-2), 243-262. [12] Luo, A.C.J. (2012), Continuous dynamical systems, HEP /L&H, Beijing/Glen Garbon. [13] Luo, A.C.J. and Huang, J. (2012), Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance. Journal of Vibration and Control, 18(11), 1661-1674. [14] Luo, A.C.J. and Huang, J. (2013), Analytical period-3 motions to chaos in a hardening Duffing oscillator, Nonlinear Dynamics, 73(3), 1905-1932. [15] Luo, A.C.J. and Lakeh, A.B. (2014), An approximate solution for period-1 motions in a periodically forced van der Pol oscillator, ASME Journal of Computational and Nonlinear Dynamics, 9(3), Article No:031001 (7 pages). [16] Luo, A.C.J. and Lakeh, A.B. (2013), Analytical solution for period-m motions in a periodically forced, van der Pol oscillator, International Jounral of Dynamics and Control, 1(2), 99-115. [17] Luo, A.C.J. and Lakeh, A.B. (2014), Period-m motions and bifurcation in a periodically forced van der Pol-Duffing oscillator, International Journal of Dynamics and Control, 2(4), 474-493. [18] Wang, Y.F. and Liu, Z.W. (2015), A matrix-based computational scheme of generalized harmonic balance method for periodic solutions of nonlinear vibratory systems, Journal of Applied Nonlinear Dynamics, 4(4), 379-389. [19] Luo, H. and Wang, Y. (2016), Nonlinear dynamics analysis of a continuum rotor through generalized harmonic balance method, Journal of Applied Nonlinear Dynamics, 5(1), 1-31. [20] Akhmet, M. and Fen, M.O. (2017), Almost periodicity in chaos, Discontinuity, Nonlinearity, and Complexity, 7(1), 15-19. [21] Luo, A.C.J. (2014), Toward Analytical Chaos in Nonlinear systems, Wiley, New York. [22] Luo, A.C.J. (2014), Analytical routes to chaos in nonlinear engineering, Wiley, New York. [23] Luo, A.C.J. and Guo, S. (2018), Analytical solutions of period-1 to period-2 motions in a periodically diffused Brusselator, ASME Journal of Computational and Nonlinear Dynamics,13(9), Article No: 090912 (8 pages). [24] Luo, A.C.J. and Guo, S. (2018), Period-1 evolutions to chaos in a periodically forced Brusselator, International Journal of Bifurcation and Chaos, in press.



Experimental Validation of Damage Detection based on Member Axial-strain Mode Shapes for Truss Structures Guirong Yan1†, Shirley J. Dyke2 , Ayhan Irfanoglu3 1

2 3

Department of Civil, Architectural and Environmental Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA Submission Info Communicated by J.Z. Zhang Received 10 June 2018 Accepted 23 September 2018 Available online 1 January 2019 Keywords Damage detection Truss structures Member axial-strain mode shapes Full-scale tests

Abstract In this study, a simple, effective damage detection approach is proposed for truss structures to locate damage onto exact member(s) using vibration responses. First, a parameter that reflects the axial strain in truss members and is sensitive to local damage is proposed. This parameter is called the member axial-strain mode shape and can be extracted from the translational mode shapes. Each component in an member axial-strain mode shape is associated with a member, reflecting the axial strain in that member. Because damage to a member directly affects the axial strain in that member, the proposed member axial-strain mode shape is an effective parameter for evaluating the condition of truss members. Second, a damage indicator constructed by both member axial-strain mode shapes and natural frequencies are proposed. Experimental tests are conducted on a full-scale sign support truss to demonstrate the effectiveness of the proposed approach. The results illustrate that the proposed approach can be applied to truss structures instrumented with a few accelerometers and using only response data. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Truss structures are widely used in civil and aerospace engineering due to their attractive characteristics. For example, they are economical to construct owing to their efficient use of materials; they can span across long distances with very small shift and sag; they can be installed quickly and costeffectively; they can be extended easily; and they can provide accessible space for maintenance. Typical applications of truss structures in engineering applications include bridges, roofs for buildings, powerline pylons and highway sign support structures. Truss structures are often used in adverse environments. For example, highway sign support structures are exposed to strong winds and vibrations caused by the passing traffic, which accelerates † Corresponding

author. Email address: [email protected] ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2018.12.005

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Guirong Yan et al. / Journal of Vibration Testing and System Dynamics 2(4) (2018) 403–416

deterioration. Typical damage to this type of structure includes fatigue fractures, with cracks often initiating at welded joints between the branch and chord members and propagating circumferentially [1]. In addition, if the truss segments in a sign support truss are connected by bolts, it is not unusual to discover that recently installed bolts have loosened as bolt connections suffer from self-loosening caused by dynamic loads [2]. Similar damage can occur in trusses used in other situations. The occurrence of damage in a local area can lead to a failure in large portions of the structure, jeopardizing the integrity and safety of the entire structure. Therefore, it is important to propose an effective and reliable approach to quickly detect damage in truss structures to make timely decisions regarding maintenance and reinforcement strategies. Along with the increase of structural complexity and size, global vibration-based damage detection approaches exhibit considerable promise. Significant efforts have been dedicated in this important area. Comprehensive reviews of the literature can be found in references [3–6]. The main objective of the developed approaches is to find damage feature parameters that are sensitive to damage, but are insensitive to measurement noises and variant measurement environments [7]. Previous studies suggest that mode shapes are more sensitive to local damage than natural frequencies [3], and mode shape derivatives are more sensitive to local damage than mode shapes [8]. In particular, Zonta et al [9] proposed a strain-flexibility to locate damage in a single-span steel bridge. Pandey et al. [10] proposed to use the change of a curvature mode shape before and after damage for detecting the loss of stiffness. Curvature mode shapes can be computed by performing central difference approximation of deflection mode shapes identified from measured accelerations. Instead of using a particular mode shape, Abdel Wahab and De Roeck [11] improved this approach by considering several mode shapes and performing an average operation. Yam et al. [12] extended this approach from beam-like structures to plate-like structures. Chance et al. [13] found that measuring curvature directly (by measuring the strain) gave better results than the traditional way to perform this approach. To localize and quantify damage for beam-like structures, Stubbs and Kim [14] proposed the “Damage Index Method”, which was based on the concept of modal strain energy and was computed from curvature mode shapes. Cornwell et al. [15] further extended this approach to plate-like structures by performing double integration of modal curvature along two coordinate axes. Sazonov and Klinkhachorn [4]analyzed the influence of the interval of measurement points on the approaches based on curvature or strain energy mode shapes, and proposed an approach to determining the optimal measurement interval to minimize the effects of measurement noises and maximize the sensitivity to damage and accuracy of damage localization. Recent extension of this class of approaches and applications are reported in references [16–19]. For beam- or plate-like structures, curvature of deflection and axial strains of longitudinal fibers are directly related to each other [20]. Therefore, curvature mode shapes essentially reflect the axial strains of longitudinal fibers of the structural component. The success of the approaches based on curvature mode shapes suggests that the axial strain is a good indicator for damage detection. However, almost all of the previous approaches can only be used in bending structures such as beam- or plate-like structures, in which internal forces are dominated by bending moments. This research is to extend this idea to truss structures in which members are dominated by axial forces, instead of bending moments. In this study, a strain-related damage indicator for truss structures is proposed to directly localize damage at a member level. Although the proposed damage indicator reflects the change of axial-strains of truss members caused by damage, it is constructed in a completely different way from curvature mode shapes due to the difference in load-bearing characteristics between truss and bending structures. The remainder of this paper is organized as follows. First, a new parameter that is called the member axial-strain mode is defined and derived shape for truss structures. The member axial-strain mode shape will be derived from translational mode shapes identified from acceleration responses. Then, by combining axial-strain mode shapes and natural frequencies, a new damage indicator is proposed. Finally, experiments on a full-scale truss structure are conducted to demonstrate that the proposed


405

damage indicator is sensitive to local damage and can directly localize damage to exact members in truss structures.

2 Derivation of axial-strain mode shapes The deformation δ l of a structural member which is borne with an axial force is expressed as

δl =

Nl , EA

(1)

where E and A are the Young’s modulus and cross-sectional area of the member; l is the length of the member, and N is the internal axial force in the member. Rearranging Eq. (1) yields the expression of axial strain in the member N δl = . (2) ε= l EA With the external loading being the same before and after damage (and accordingly, N remains the same), if damage occurs in the member (E or A may be changed), the axial strain of the member will change. Therefore, for structures in which axial forces are dominated, axial strain ε is a straightforward indicator for damage detection. Although the axial strain ε can be directly measured, it may not be feasible to use ε as a damage indicator because the axial strain also depends on external loadings, which may not be measured or controlled in practical applications. The above analysis inspires the present writers to propose a parameter which is related to axial strain and can represent the physical properties of the structure without being affected by external loadings. The proposed parameter, called the member axial-strain mode shape, is extracted in the modal coordinates. As explained below, the axial-strain mode shape can be obtained from translational mode shapes. Let us take a simply supported truss with n members (Fig.1) as an example to illustrate how to extract the axial-strain mode shape. When sensors are deployed in a dynamic test, they are usually deployed to measure the translational responses at particular nodes in the transversal direction (and longitudinal direction). Therefore, the translational mode shape components identified from measured responses are associated with nodes. To get the axial-strain mode shape components which are associated with each member, we need to apply the idea of obtaining the axial strain in each member in the physical coordinates. Assume that the length of the jth member is l j and the cosine and sine values of the angle α between the jth member and the x-axis in the global coordinate system arec j and s j , respectively (c j = cos α , s j = sin α ). The nodal displacements at the two end nodes of member j are listed in Fig. 2a. The deformation of member j in the x direction is

δ X j = X2p−1 − X2o−1.

(3)

Similarly, the deformation of member j in the y direction is

δ Y j = Y2p −Y2o ,

(4)

δ X j and δ Y j are plotted in Fig. 2b. The component deformation of δ X j along member j is extension, as indicated by the red dashed arrow in Fig. 2c, and the component deformation of δ Y j along member j is compression, as indicated by the red arrow in Fig. 2d. Assuming that the compression deformation is positive, the total axial deformation of member j can be expressed δ l j = −(− cos αδ X j ) + sin αδ Y j = c j δ X j + s j δ Y j .

(5)

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Fig. 2 The axial deformation in member j in the physical coordinates: (a) member j with nodal displacements; (b) member j with deformations in the x and y directions; (c) axial deformation associated with δ X j ; (d) axial deformation associated with δ Y j .

Then the axial strain of the jth member is

εj =

δ l j c jδ X j + s jδYj = . lj lj

(6)

Similar to the above derivation, the axial-strain mode shapes can be derived from translational mode shapes as follows. Assume that ϕr is the rth identified translational mode shape which has been normalized to have a unit length. Its components at node o are ϕ2o−1,r (associated with the x direction) and ϕ2o,r (associated with the y direction), and its components at node p are ϕ2p−1,r and ϕ2p,r , as shown in Fig.3. The axial-strain mode shape component associated with member j (connecting node oand nodep) and therth mode can be expressed as S j,r = c j

ϕ2o−1,r − ϕ2p−1,r ) (ϕ2o,r − ϕ2p,r ) + sj . lj lj

(7)

Likewise, the axial-strain mode shape components associated with other members and the rth mode can be obtained. Stacking them into a column yields the rth axial-strain mode shape Sr as ⎛ ⎞ (ϕ2a−1,r − ϕ2b−1,r ) (ϕ2a,r − ϕ2b,r ) + s1 ⎜ c1 ⎟ ⎛ ⎞ l1 l1 ⎜ ⎟ S 1r ⎜ ⎟ · · · ⎜ ⎟ ⎜···⎟ ⎜ (ϕ2o−1,r − ϕ2p−1,r ) ⎜ ⎟ (ϕ2o,r − ϕ2p,r ) ⎟ ⎟ = ⎜ S jr ⎟ . + sj (8) Sr = ⎜ ⎜c j ⎟ ⎜ ⎟ lj lj ⎜ ⎟ ⎝···⎠ ⎜ ⎟ ··· ⎜ ⎟ S nr ⎝ (ϕ2w−1,r − ϕ2x−1,r ) (ϕ2w,r − ϕ2x,r ) ⎠ + sn cn ln ln


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where the first subscript of ϕ denotes the DOF (degree of freedom) number, and subscripts of c, s and l denote the member number. In the axial-strain mode shape vector, each component is associated with a member, instead of a node as in a translational mode shape. Each component of an axial-strain mode shape reflects the axial strain in the associated member. Because damage to each member directly affects the axial strain in the associated member, as discussed before, the proposed axial-strain mode shape is an appropriate parameter for indicating the health condition of truss members. If the jth member is horizontal, c j = 1 and s j = 0. Thus, for a horizontal member Eq. (7) can be simplified as ϕ2o−1,r − ϕ2p−1,r . (9) S j,r = lj If the jth element is vertical, c j = 0 and s j = 1. Thus, for a vertical member Eq. (7) can be simplified as S j,r =

ϕ2o,r − ϕ2p,r . lj

(10)

Based on the above analysis, from Eqs. (9) and (10), the health condition of a horizontal member is only determined by the horizontal mode shape components associated with the two end nodes of the member, and the health condition of a vertical member is only determined by the vertical mode shape components associated with the two end nodes of the member. From Eq. (8), the health condition of a diagonal member is determined by both the horizontal and vertical components associated with the end nodes of the member. Arguably, strain mode shapes can be directly extracted from strain time history measured using traditional strain gauges. However, a traditional strain gauge only measures the point strain within the range of the strain gauge. If the strain gauge is not located in the damage site, the extracted strain mode shapes will not include any damage information. Unfortunately, damage locations are not known beforehand in reality. On the contrary, the proposed axial-strain mode shape reflects the average axial strain of an entire member which is directly related to damage information of the member. In addition, strain gauges can be easily damaged in practice if strains are large. This is why the proposed approach is more beneficial.

3 Damage indicators based on axial-strain mode shapes Combining the axial-strain mode shapes and natural frequencies, the following is proposed as a damage indicator:

408


⎞ m 1 1 ˜2 2 2 2 ∑ 2 S1i ) − ( ∑ 2 S1i ) ⎟ ⎜(i=1 ω˜ i i=1 ωi ⎜ ⎟ ⎜ ⎟ · · · ⎜ m ⎟ m 1 ⎜ ⎟ 1 ˜2 2 2 2 ⎜ DI = abs ⎜ ( ∑ 2 S ji ) − ( ∑ 2 S ji ) ⎟ ⎟, i=1 ωi ⎜ i=1 ω˜ i ⎟ ⎜ ⎟ ··· ⎜ m ⎟ m ⎝ 1 ˜2 2 1 2 2⎠ ( ∑ 2 Sni ) − ( ∑ 2 Sni ) ˜i i=1 ω i=1 ωi ⎛

m

(11)

where ωi and S ji denote the ith circular modal frequency and the axial-strain mode shape components of the intact structure, respectively; ω˜ i and S˜ ji are associated with the damaged structure. DI reflects the absolute difference in weighted magnitudes of axial-strain mode shapes before and after damage. The weight factors are related to circular modal frequencies, which are identified from measured responses together with translational mode shapes. These weight factors ensure that this approach only requires lower-frequency modes which can be identified with good accuracy using current techniques. The squared operation here is to make the damage indicators more sensitive to damage. If damage occurs in a particular member, the axial strain of that member will change. Due to the absolute operation in Eq. (11), the damage indicators associated with damaged members always exhibit as spikes. To have a better presentation of the damage indicators, a normalized damage indicator is proposed as DI . (12) DI = max(DI) Compared with other strain-related approaches, the proposed approach has the following advantages: 1) it does not require to measure strains, which is time-consuming. Herein axial-strain mode shapes are extracted from translational mode shapes which are identified from measured acceleration responses; 2) it does not require information on structural mass, because mass-normalized mode shapes are not needed; 3) it does not require construction of elemental stiffness matrices, which is mandatory in some approaches which can detect damage in truss or frame structures [21–23]; 4) it only requires a few lower-frequency modes which can be easily obtained in practical applications than higher-frequency modes.

4 Experimental validation The proposed approach is validated by experimental tests on a full-scale highway sign support truss. The truss used to be mounted over Interstate I-29 near Sioux City in the Iowa state, as shown in Fig. 4. It was taken out of commission recently due to fatigue cracks occurred in the welds of joints and donated to Purdue University for research. It consisted of a horizontal, four-chord welded space truss which spanned all four lanes of Interstate I-29, and two vertical, planar trusses which supported the horizontal truss on both ends. The horizontal truss comprised four segments, and spanned 35.08m longitudinally with its four main chords arranged on a 1.83 m×1.98 m configuration. The truss is constructed of round, tubular members that are made of aluminum (6061-T6). The material properties and geometric dimensions for each member are listed in Table 1. Herein experimental tests were conducted on two segments (half of the symmetric truss) of the horizontal truss. The left segment was 10.53 m long and the right segment was 6.71m long. The two segments were connected to each other at the four main chords using splice plates with eight bolts each, as shown in Fig. 5. The two segments were mounted on steel plates on the left side and steel rollers on the right side to simulate simple supports.


409

Table 1 Truss specifications, including dimensions and material properties. Values

Values

Outside diameter and

152.40 mm

Span of each bay

thickness of main chords

79.00 mm

in left segment

Outside diameter and

76.20 mm

Span of each bay in

thickness of diagonal braces

6.30 mm

right segment

Outside diameter and thickness of

63.50 mm

Span of the bay where

end vertical braces and end struts

6.30 mm

splice plate is located

Outside diameter and thickness

50.80 mm

of other members

4.80 mm

Width of truss Height of truss

1.73 m 1.64 m 0.30 m

Young’s Modulus

6.96 × 1010 N/m2

1.83 m

Density

2715kg/m3

1.98 m

Poisson’s ratio

0.33

Fig. 4 Sign support structure over Interstate I-29 when it was in service.

Fig. 5 Laboratory setup for the full-scale truss structure.

In all tests, all damage was simulated to be located in the front vertical panel, and thus only the responses in the front vertical panel were required to be measured. The front panel with nodes and members numbered is plotted in Fig. 6. The numbers in circles correspond to members and the remaining numbers indicate node numbers.


410

o y

x

1

2

3

4

5

6

7 8

9

10

11

19 20

21

22

23

12

10 11 8 9 7 29 24 25 26 27 28 31 32 33 34 23 30 36 37 38 39 41 42 43 44 35 40 13 12 13 14 15 20 21 22 24 16 17 18 19 1

14

2

3

15

4

17

16

5

6

18

Shaker Fig. 6 Front vertical panel of the three-dimensional truss.

o

y

x z

Fig. 7 Finite element model for the 3-D truss. Table 2 Natural frequencies obtained from finite element analysis (Hz). Intact

Damaged Case (DC2)

Value

Description

Value

Description

1

15.50

first bending mode about y

15.49

first bending mode about y

2

16.27

first bending mode about Z

16.31

first bending mode about Z

3

24.67

first torsion mode

24.61

first torsional mode

4

34.33

second bending mode about Z

29.73

second torsional mode

5

39.68

second bending mode about y

35.83

second bending mode about y

6

47.47

third bending mode about y

44.97

third torsional mode

7

48.52

second torsional mode

47.72

third bending mode about y

8

52.74

third bending mode about Z

52.47

fourth torsional mode

second torsional mode third bending mode about Z

4.1

Finite element analysis

A finite element model (FEM) of the test structure was developed using SAP2000, as shown in Fig. 7. In this FEM, each member was treated as an element. The structure was modeled to have rigid connections along its main chords and with other members attached using pinned connections. In total, it had 60 nodes and 148 elements. The material properties and the dimensions of each element can be found in Table 1. Modal analysis was performed on the intact structure and on a damage case (DC2). The first eight natural frequencies for the intact and damaged cases are listed in Table 2. The modal analysis results show that damage leads to change in selected natural frequencies, and damage may change the vibration mode of a particular mode.


o y

x

Test 1 13

2

1

Test 3

1

4

16

Test

Test

Test

Test

5

6

78

9

10

11 12

1

18

1920

21

22

23

411

24

Shaker Fig. 8 Sensor deployment in each test.

Fig. 9 Experimental setup of the shaker.

4.2

Experimental setup

Nine uniaxial accelerometers (PCB model 3711, MEMS DC response accelerometers) were available for the tests. Two accelerometers were deployed at each node to measure the responses in both the horizontal (x) and vertical (y) directions. Due to the limited number of accelerometers, an accelerometer was fixed at Node 16 as a reference (see the red solid arrow at Node 16) and the other eight were moved around to measure the responses at all other nodes in the front vertical panel. For each case (intact and simulated damage cases), six tests were conducted sequentially to cover the whole front vertical panel. Figure 8 shows how the tests were grouped. In each test, besides the responses at the nodes surrounded by the red box, the response in the y direction at node 16 was also measured as the reference response. Node 16 was selected as the reference measurement point, because node 16 was close to the shaker and thus had a good signal-to-noise ratio. In each test, an electrodynamic shaker (VG-100 from Vibration Test Systems) was used to excite the truss, as shown in Fig. 9. The shaker was placed between Nodes 16 and 17 (see the red dashed arrow in Fig. 8). Since the proposed approach is based on modal parameters identified from the measured acceleration responses, as long as the shaker is able to excite the related modes, no matter where the shaker is located, the effectiveness of the proposed approach will not be affected. To achieve this, the shaker should not be located at the nodal point of the mode shape of interest. Band-limited white noise (0-100Hz) was used to excite the truss to simulate ambient excitation in the field. The white noise signal was first generated by a SigLab dynamic signal analyzer (Spectral Dynamics, Inc.), then amplified by an amplifier (CE-2000 from Crown International). The amplified signal was input to the shaker to excite the truss. The SigLab dynamic signal analyzer was used to acquire acceleration responses at a sampling frequency of 256 Hz.

412


(a)

(b)

(c)

Fig. 10 Damage description of DC1: (a) the truss with a damaged connection plate; (b) zoom-in plot of the damaged connection plate; (c) the intact connection plate.

(a)

(b)

(c)

Fig. 11 Damage description of DC2: (a) the truss with a diagonal member cut through; (b) the picture of the cut from another perspective; (c) the real damage scenario in practice.

4.3

Simulated damage cases

Three damage scenarios were considered in this study. In the first scenario, all of the bolts on the splice plate in the lower main chord of the front vertical panel between the two truss segments were removed or completely loosened, designated as DC1, as shown in Fig. 10. DC1 was considered because this type of damage happens frequently in structures with bolted connections that experience significant dynamic loads, for instance, in highway sign support trusses which mostly experience strong wind and vibration caused by passing traffic [24]. In the second damage scenario, a cut was made through the welds of a diagonal member (member 39) of the front vertical panel to simulate a crack in the welds, designated as DC2, as shown in Fig. 11. This damage case was considered because it is widely encountered in the field. For example, this type of damage did occur in a diagonal member of another segment of the same truss when it was in service, as shown in Fig. 11c. For DC1 and DC2, the baseline is the intact case. To verify that the proposed approach is also able to localize multiple damage sites at one time, the third damage case (designated as DC3) was formed by using the data measured in the first two damage cases. Herein, the data measured in DC1 is assumed to be the baseline responses and the data measured in DC2 is assumed to be the responses of the post-damage state. 4.4

Identification of modal parameters

In this study, an output-only modal identification approach, the Frequency Domain Decomposition (FDD) method, was employed to identify modal parameters of the truss. This method is appropriate


(a)

(a)

(b)

(b)

(c)

(c)

413

Fig. 12 Comparison of mode shapes between the in- Fig. 13 Comparison of mode shapes between the intact case and DC1: (a) 1st mode shape; (b) 5th mode tact case and DC2: (a) 1st mode shape; (b) 5th mode shape; (c) 7th mode shape. shape; (c) 7th mode shape.

because the primary sources of external excitations for highway sign support trusses and some other civil engineering structures are ambient vibrations that are often immeasurable. In this case, only measured structural responses can be used for modal identification. For each state of the truss, from the acquired responses in each test, piecewise mode shapes and natural frequencies are first identified using the FDD method. Then, the piecewise mode shapes are stitched together to generate global mode shapes by considering the component corresponding to the reference measurement point, and the identified natural frequencies from each test are averaged. The identified mode shapes in the intact and damage cases are presented in Figs. 12 and 13. It can be observed that although damage does lead to change in mode shapes, we cannot localize damage directly from the change in mode shapes. Table 3 lists the averaged natural frequencies for each state of the truss. From the 2nd and 3rd columns, the identified natural frequencies are smaller than those obtained from finite element analysis. The difference between the two could be attributed to the discrepancy in how the connection between two segments was modeled in the finite element model versus how it is in the actual truss. In the finite element model the truss is assumed to have rigid splice connections while the two segments in the actual structure are connected by bolts. Also, the simple supports may not be actual representations of the boundary condition of the actual truss. By comparing the 3rd and 4th columns in Table 3, it is clear that joint damage (DC1) does not lead to appreciable change in the identified natural frequencies. By comparing the 3rd and 5th columns in Table 3, cut-through damage of a single member (DC2) induces that the 5th and 7th natural frequencies deviate significantly from those of the intact structure, while it does not result in an appreciable change in the first natural frequency.

414


(a)

(b)

(c)

Fig. 14 Damage localization results for DC1. (a) using 1 mode. (b) using 2 modes. (c) using 3 modes.

(a)

(b)

Fig. 15 Damage localization results for DC2. (a) using two modes. (b) using three modes.

4.5

Damage locating results

The procedure for locating damage using the proposed approach is as follows. After obtaining averaged natural frequencies and global mode shapes using the FDD method, the global mode shapes are normalized to have a unit length. Then, the axial-strain mode shapes are computed using Eq. (8). Next, damage indicators are extracted by combining the information on natural frequencies and axial-strain mode shapes using Eq. (11) and then normalized damage indicators are computed using Eq. (12). Herein different numbers of modes are used to extract the damage indicators. In general, the more modes used, the better the results. To locate damage in DC1, the first, fifth and seventh modes (the first three modes identified in DC1) and associated averaged natural frequencies before and after damage were considered. The extracted damage indicators are presented in Fig. 14. It is observed that the damage indicator associated with member 18 exhibits as a spike in the amplitude distribution of damage indicators. From Fig. 6, the connection plate with bolts removed or loosened is located between nodes 19 and 20, and the element between the two nodes is numbered 18. Therefore, the extracted damage indicators can locate damage to the exact member. In DC2, the first, fifth and seventh modes (the first three modes identified in DC2) and associated averaged natural frequencies were considered. The extracted damage indicators are presented in Fig. 15. In this case, a cut was made through the welds of a diagonal member (member 39). From Fig. 15, the damage indicator associated with member 39 exhibits as a spike. Therefore, the proposed approach successfully identified the simulated crack damage in DC2. To locate multiple damage sites in DC3, the first, fifth and seventh modes and the associated averaged natural frequencies before and after damage were considered. The extracted damage indicators corresponding to the member with the bolts removed or loosened (member 18) and the member with the simulated fatigue fracture (member 39) exhibit as two spikes, as shown in Fig. 16. Therefore, the proposed approach is shown to be suitable for multiple damage scenarios. Regarding the number of modes used in practice, the similarity between modes will be used for mode selection. That is, the identified modes will be paired by using the similarity of mode shapes.


(a)

(b)

415

(c)

Fig. 16 Damage localization results for DC3: (a) using 1 mode. (b) using 2 modes. (d) using 3 modes.

The similarity of mode shapes will be measured using the Modal Assurance Criterion (MAC) analysis results [25]. A higher MAC value between two modes means a high similarity. If numerical model is available, the mode shapes will be paired between numerical simulation and experimental results for each case; if numerical model is not available, the mode shapes will be paired between the intact case and the damaged case, assuming that the mode shape is not significantly changed by damage. In this way, we can ensure the same number of modes will be used for both the intact case and the damage case.

5 Conclusions In this study, a novel damage detection approach was proposed for truss structures to locate damage at the member level. The axial-strain mode shape, which reflects the axial-strain in truss members, was proposed. Damage indicators based on the axial-strain mode shapes were developed. Because the proposed damage indicators reflect the change of axial strain before and after damage, they are sensitive to local damage. As each component of the axial-strain mode shape is associated with a member instead of a node, the proposed approach can locate damage to exact members. Only the acceleration responses need to be measured and only a few lower-frequency modes are required to identify. No finite element model of the structure is needed; neither complete measurement of DOFs in one time nor mass-normalized mode shapes are required. The proposed approach is suitable for both single and multiple damage scenarios. Experimental tests were conducted on a full-scale sign support truss to demonstrate the proposed approach. In these tests, only a small number of sensors were used. Additionally, the truss was assumed to be excited by unknown ambient vibration and thus only measured structural responses were used for damage detection. Experimental results showed that the proposed approach was able to accurately locate damage at the member level with a few sensors and without measuring the excitation. To further validate the proposed approach, in the future, the numerical simulation of the experimental testing will be developed. Once the numerical model is validated by experimental data, systematical numerical simulations will be conducted to further validate the effectiveness of the proposed approach from the following different perspectives: 1) Simulate that the shaker is located at different locations to validate that the proposed approach is not affected by the location of shaker, as long as the related modes are excited. 2) Associated with Damage Case 1, the bolt connection will be properly simulated numerically. Once the numerical model is validated using experimental data, the cases with different numbers of bolts removed/loosened will be simulated to investigate the sensitivity of the proposed approach on damage related to bolt connections. 3) Associated with Damage Case 2 where the member is completely cut, different levels of connection between members will be applied to find the minimal damage extent the proposed approach can locate.

416


Acknowledgements The authors greatly appreciate the financial support from National Science Foundation under Grant Nos. 1002641 and 1455709. The authors would also like to thank Prof. Robert Connor at Purdue University and Mr. Michael Todsen at the Iowa Department of Transportation for helping them acquire the sign support truss used in this study.

References [1] Nadauld, J.D. and Pantelides, C.P. (2007), Rehabilitation of cracked aluminum connections with GFRP composites for fatigue stresses, Journal Composites for Construction, 11(3), 328-335 [2] Bickford, J.H. and Nassar, S. (1998), Handbook of bolts and bolted joints, Marcel Dekker, Inc., New York. [3] Doebling, S.W., Farrar, C.R., and Prime, M.B. (1996), A Summary Review of Vibration-Based Damage Identification Methods, Report of Los Alamos National Laboratory. [4] Salawu, O.S. (1997), Detection of structural damage through changes in frequency: a review, Engineering Structures, 19(9), 718-723. [5] Staszewski, W.J. (1998), Structural and mechanical damage detection using wavelets, The Shock and Vibration Digest, 30(6), 457-472. [6] Li, Y.Y. (2010), Hypersensitivity of strain-based indicators for structural damage identification: A review, Mechanical Systems and Signal Processing, (24), 653-664. [7] Harri, K., Guillaume, P., and Vanlanduit, S. (2008), On-line damage detection on a wing panel using transmission of multisine ultrasonic waves, NDT & E International, 41(4), 312-317. [8] Gandomi, A.H., Sahab, M.G., Rahaei, A., and Gorji, M.S. (2008), Development in mode shape-based structural fault identification technique, World Applied Sciences Journal, (5), 29-38. [9] Zonta, D., Lanaro, A., and Zanon, P. (2003), A strain-flexibility-based approach to damage location, Key Engineering Materials, 245, 87-96, Trans Tech Publications, 2003. [10] Pandey, A.K., Biswas, M., and Samman, M.M. (1991), Damage detection from changes in curvature mode shapes, Journal of Sound and Vibration, (145), 321-332. [11] Abdel Wahab, M.M. and De Roeck, G. (1999), Damage detection in bridges using modal curvatures: application to a real damage scenario, Journal of Sound and Vibration, (226), 217-235. [12] Yam, L.H., Li, Y.Y., and Wong, W.O. (2002), Sensitivity studies of parameters for damage detection of plate-like structures using static and dynamic approaches, Engineering Structures, (24), 1465-1475. [13] Chance, J., Tomlison, G.R., and Worden, K. (1994), A simplified approach to the numerical and experimental modeling of the dynamics of a cracked beam, Proceedings of the 12th International Modal Analysis Conference, Honolulu, Hawaii, U.S.A, 778-785. [14] Stubbs, N. and Kim, J.T. (1996), Damage localization in structures without baseline modal parameters, AIAA Journal, (34), 1644-1649. [15] Cornwell, P., Doebling, S., and Farrar, C. (1999), Application of the strain energy damage detection method to plate-like structures, Journal of Sound and Vibration, 224(2), 359-374. [16] Qiao, P., Lu, K., Lestari, W., and Wang, J.L. (2007), Curvature mode shape-based damage detection in composite laminated plates, Composite Structures, 80, 409-428. [17] Hamey, C.S., Lestari, W., Qiao, P.Z., and Song, G.B. (2004), Experimental damage identification of carbon/epoxy composite beams using curvature mode shapes, Structural Health Monitoring-An International Journal, l3, 333-353. [18] Hu, H.W. and Wu, C.B. (2009), Development of scanning damage index for the damage detection of plate structures using modal strain energy method, Mechanical Systems and Signal Processing, 23, 274-287. [19] Guan, H. and Karbhari, V.M. (2008), Improved damage detection method based on element modal strain damage index using sparse measurement, Journal of Sound and Vibration, 309, 465-494. [20] Chang, P.C., Faltau, A., and Liu, S.C. (2003), Review paper: Health Monitoring of Civil Infrastructure, Structural Health Monitoring, 2(3), 257-267. [21] Shi, Z.Y., Law, S.S., and Zhang, L.M. (2000), Structural damage detection from modal strain energy change, Journal of Engineering Mechanics-ASCE, 126, 1216-1223. [22] Hsu, T.Y. and Loh, C.H. (2008), Damage diagnosis of frame structures using modified modal strain energy change method, Journal of Engineering Mechanics-ASCE, 134, 1000-1012. [23] Li, H.J., Yang, H.Z., and Hu, S.J. (2006), Modal strain energy decomposition method for damage localization in 3D frame structures, Journal of Engineering Mechanics-ASCE, 132, 941-951. [24] Bickford, J.H. and Nassar, S. (1998), Handbook of bolts and bolted joints, Marcel Dekker, Inc., New York. [25] Allemang, R.J. (2003), The modal assurance criterion–twenty years of use and abuse, Sound and vibration, 37(8), 14-23.

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Aims and Scope Vibration Testing and System Dynamics is an interdisciplinary journal serving as the forum for promoting dialogues among engineering practitioners and research scholars. As the platform for facilitating the synergy of system dynamics, testing, design, modeling, and education, the journal publishes high-quality, original articles in the theory and applications of dynamical system testing. The aim of the journal is to stimulate more research interest in and attention for the interaction of theory, design, and application in dynamic testing. Manuscripts reporting novel methodology design for modelling and testing complex dynamical systems with nonlinearity are solicited. Papers on applying modern theory of dynamics to real-world issues in all areas of physical science and description of numerical investigation are equally encouraged. Progress made in the following topics are of interest, but not limited, to the journal: • • • • • • • • • • • • • • • •

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Mohamed Belhaq Laboratory of Mechanics University Hassan II-Casablanca Casablanca, Morocco Email: [email protected]

Zhaobo Chen School of Mechatronics Engineering Harbin Institute of Technology Harbin 150001, P.R. China Email: [email protected]

Francesco Clementi Department of Civil and Buildings Engineering and Architecture Polytechnic University of Marche Ancona, Italy Email: [email protected]

Frank Z. Feng Department of Mechanical and Aerospace Engineering University of Missouri Columbia, MO 65211, USA Email: [email protected]

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Lei Guo School of Automation Science and Electrical Engineering Beihang University Beijing, P.R. China Email: [email protected]

Krzysztof Kecik Department of Applied Mechanics Lublin University of Technology Nadbystrzycka, Lublin, Poland Email: [email protected]

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Journal of Vibration Testing and System Dynamics Volume 2, Issue 4

December 2018

Contents On Targeted Energy Transfer and Resonance Captures in the 2D-Wing and Nonlinear Energy Sinks Wenfan Zhang, Jiazhong Zhang, Le Wang, Shaohua Tian………………….........

297－306

Power Density — An Alternative Approach to Quantifying Fatigue Failure Zachary T. Branigan, C. Steve Suh…………………………..…………....………

307－326

Equilibrium Points with Their Associated Normal Modes Describing Nonlinear Dynamics of a Spinning Shaft with Non-constant Rotating Speed Fotios Georgiades……………………………………………………………........

327－373

On Independent Period-m Evolutions in a Periodically Forced Brusselator Albert C.J. Luo, Siyu Guo…….……..…………………………………....….……

375－402

Experimental Validation of Damage Detection based on Member Axial-strain Mode Shapes for Truss Structures Guirong Yan, Shirley J. Dyke, Ayhan Irfanoglu..……………............……........…

403－416

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Journal of Vibration Testing and System Dynamics

Journal of Vibration Testing and System Dynamics

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