Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 144 (2016) 406 – 413
12th International Conference on Vibration Problems, ICOVP 2015
Nonlinear Dynamics of Traveling Continua with low flexural stiffness under Parametric and Internal Resonances Bamadev Sahooa*, L.N. Pandab, Goutam Pohitc a
Mechanical Engineering Department, IIIT, Bhubaneswar, 751029, India Mechanical Engineering Department, CET, Bhubaneswar, 751003, India c Mechanical Engineering Department, Jadavpur University, Kolkata 700032, India b
Abstract This study focuses on the trivial state stability boundary, steady state periodic response and chaotic behavior in the transverse motion of an axially moving viscoelastic continuum subject to parametric excitation and internal resonance. This parametric excitation comes from harmonic fluctuations of the traveling speed. The motion is restricted by viscous damping as well as material damping. The derived nonlinear integro-partial-differential equation of motion is solved through analytic-numerical approach. Direct method of multiple scales is used to solve the nonlinear equation of motion. A continuation algorithm is used to find the steady state response. Furthermore, Runge-Kutta method is applied to find the dynamic behavior of the system. The stable periodic response and the unstable chaotic motions are identified using different tools including the time traces, Phase portraits, Poincare map, fast Fourier transforms. Evolution of maximum Lyapunov exponent is used to locate the range of system parameters for which chaotic behavior exists. © Published by Elsevier Ltd. This ©2016 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICOVP 2015. Peer-review under responsibility of the organizing committee of ICOVP 2015
Keywords: Parametric excitation; Internal resonance; Stability; Bifurcation; Chaos
1. Introduction The vast literature on axially moving continua vibration has been reviewed by Wickert and Mote [1] up to 1988. Oz
*
Corresponding author. Tel.: 0674-6636625. E-mail address:
[email protected],
[email protected]
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICOVP 2015
doi:10.1016/j.proeng.2016.05.150
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et.al [2, 3], Chin et.al [4] and Panda et.al [5, 6] investigated the principal parametric and combination parametric resonances of traveling beams and pipes conveying pulsating fluid respectively. Pakdemirli et. al [7] studied the transverse vibration of simply supported axially moving Euler-Bernoulli beam for infinite mode analysis and truncation to resonant modes. Recently a systematic research on travelling beam was pursued by Ponomareva et. al [8] and Ghayesh et.al [9] . The authors of the present paper have studied the stability, bifurcation of a traveling beam under single frequency parametric [10-13] and two frequency parametric [14-16] excitations in conjunction with internal resonance.
Fig. 1 Schematic diagram of an axially travelling beam with variable velocity.
2. Formulation of the problem In the present work, an axially travelling simply supported Euler-Bernoulli beam (Fig.1) is considered. The dimensionless integro-differential equation governing the transverse motion of the system including the external and internal damping [9] and geometric cubic nonlinearity [3] is given by w 2vwc vwc (v 2 1))wcc v 2f wcccc 2HD wcccc 2HP w
1
1 2 vl wcc³ wc2dx 2 0
(1)
The dimensional scheme [13] is adopted here. Dot denotes derivatives with respect to time and prime denotes derivatives with respect to spatial derivative x. νf, νl, D , P and : are dimensionless flexural stiffness, longitudinal stiffness, material damping, viscous damping and frequency respectively. Reordering the transverse displacement with the relation, w H w# , where H 1 and putting it in the equation of motion Eq.(1), the system is converted into a weakly nonlinear [3]. For convenience, the superscript “#” is removed, and the weakly nonlinear equation of motion becomes w 2vwc vwc (v 2 1))wcc v 2f wcccc 2HD wcccc 2HP w
1 2 1 2 H v wcc ³ wc dx 2 l 0
With boundary conditions w 0, t w 1, t wcc 0, t wcc 1, t 0
(2)
(3)
The variable velocity of the beam is assumed to be v v0 H v sin :t
(4)
where v0 is mean velocity, H v is the amplitude and : is the frequency of the harmonically varying components. 3. Method of analysis The method of multiple scales is directly applied to (2-4) to obtain two complex variable modulation equations for amplitudes and phase angles of the first two interacting modes. The frequency detuning parameters V 1 and V 2 are introduced to define internal resonance and principal parametric resonance of first mode respectively. Z2
3Z1 HV1, and : 2Z1 HV 2
(5)
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Adopting similar steps and using the equations in [10, 14] and applying the solvability conditions, one gets the reduced equations for the modulation of amplitudes and phases as p1c
-1q1 S1R p13 p1q12 S1I p12q1 q13 S2 R p1 p22 p1q22 S 2 I q1 p22 q1q22
g1R p12 p2 p2 q12 2 p1q1q2 g1I 2 p1q1 p2 p12q2 q12q2 P C1R p1
(6)
P C1I q1 D e1R p1 D e1I q1 K1R p1 K1I q1 K 2 R p2 K 2 I q2
q1c -1 p1 S1I p13 p1q12 S1R p12q1 q13 S2 R q1 p22 q1q22 S 2 I p1 p22 p1q22 g1R 2 p1q1 p2 p12 q2 q12 q2 g1I 2 p1q1q2 p12 p2 p2q12 K1R q1 K1I p1
(7)
K 2 R q2 K 2 I p2 P C1R q1 P C1I p1 D e1R q1 D e1I p1 p2c
-2 q2 S4 R p23 p2 q22 S 4 I q23 p22q2 S3 R p12 p2 p2q12
S3 I p12 q2 q12 q2 g 2 R p13 3 p1q12 g 2 I q13 3 p12q1
(8)
K3 R p1 K3 I q1 P C2 R p2 P C2 I q2 D e2 R p2 D e2 I q2 q2c
-2 p2 S4 R q23 p22q2 S4 I p23 p2q22 S3 R p12q2 q12q2 S3 I p12 p2 p2 q12 g 2 R q13 3 p12q1 g 2 I p13 3 p1q12
(9)
K3 R q1 K 3 I p1 P C2 R q2 P C2 I p2 D e2 R q2 D e2 I p2
Where -1
0.5V 2 and -2
1.5V 2 V1 , and Si , gi , Ki , Ci and ei are nonlinear interaction co-efficient [10].
For evaluating stability, the above equations (6-9) are perturbed yielding [ J c ] , the Jacobian matrix, whose eigenvalues determine the stability and bifurcation of the system. This perturbed equations may be represented as
^'p1c 'q1c 'p2c 'q2c `
T
> J c @^'p1 'q1 'p2 'q2`
T
(10)
The normalized reduced equations (6)-(9) are solved numerically by either using a continuation algorithm to determine the steady state curves, their stability and bifurcation or direct time integration to plot various dynamic characteristic motions with variation of control parameters 4. Results and discussions For the present case of principal parametric resonance of the first mode in presence of 3:1 internal resonance is considered. The natural frequencies of the traveling system are numerically evaluated at different mean velocities ( v0 ) with flexural stiffness v f 0.07 by simultaneous solution of dispersive relation and support condition [3, 1016]. From this numerical calculation it is found that 3:1 internal resonance occurs between the first two modes corresponding to the non dimensional mean velocity is v0 0.8220 . For further analysis, a specific value of mean traveling velocity is considered i.e. v0 0.9436 for which Z1 0.6785 , Z2 2.9593 and the corresponding internal detuning parameter σ1=92.39. The book keeping parameter is taken as ε =0.01.
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Fig. 2. Trivial state stability boundary for principal parametric resonance of first mode for a traveling continuum with (a, b) v f 0.07 . Values of the nondimensional damping parameters (P , D ) are indicated on the curves.
The trivial state stability boundary shown in Figure 2 is plotted for the system parameters vl 40, v0 0.9436, Z1 0.6785, Z2 2.9593 , v f 0.0.07, and with different external and internal damping values ( P and D ).The region inside the boundary denotes instability. Higher values of damping have the effect of raising and narrowing the instability zones. For a particular value of external damping (say, D =0.4) the instability region spreads over a wider range of frequency detuning parameter V 2 and velocity fluctuation component v1 in the case of traveling beam with relatively lower flexural stiffness v f 0.07 ( represented in Figure 2(b)) compared with the case v f 0.2 ([10], Fig.2(c)]. 4.1. Stability and bifurcation of equilibrium solutions The normalized reduced equations (6-9) are used to obtain the nonlinear steady-state response of the system and also for the analysis of stability and bifurcations of the equilibrium solutions. Frequency response curves are obtained when the amplitudes of first and second modes are plotted against the variation in frequency detuning parameter V 2 for typical system parameters P 0.1, D 0, v1 10, vl 40 and V1 92.39 and are shown in Figure 3. The normal continuous lines in the figure represent stable equilibrium solution, the bold lines represent unstable foci, and the dotted lines denote saddles. If we compare the frequency response curves of the present case ( v f 0.07 ) with the similar curves of principal parametric resonance of first mode of a traveling beam with higher flexural stiffness ( v f 0.2 ) which was discussed in ([10], Fig.4]), we find many interesting qualitative and quantitative changes in system behavior. In the present case (Fig.3), the range for trivial state instability widens as compared to the case of v f 0.2 . The system experiences multiple number of jump phenomena at different saddle node bifurcation points like SN1 (V 2 144.198) , SN2 (V 2 132.002) and SN3 (V 2 153.028) as compared to single jump phenomenon at SN (V 2 301.873) ([10], Fig.4]). Besides this, the maximum amplitude decreases for both modes and saddle node bifurcation points occur at lower values of parametric frequency detuning parameter (V 2 ) .
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Fig. 3. Frequency response curves for the first and second modes when the first mode is parametrically excited for the system parameters P 0.1, D 0, v1 10, vl =40 , V1 92.39 and v f 0.07 .
Figure 4 illustrates the influence of internal damping on frequency response curves of traveling beams with different flexural stiffness values. Unlike the cases discussed in ([10], Fig.6]), in the present case, the number of equilibrium solution branches are limited to two only as shown in Figure 4. Here, the effect of nonlinearity due to internal resonance is suppressed when the material damping is introduced. Further, the amplitude of frequency response curves are significantly lower as depicted in Figure 4(a, b) compared to the amplitudes of frequency response curves [10].
Fig.4. Frequency response curves for the first and second modes when the first mode is parametrically excited for the system parameters P 0, D 0.001, v1 10, vl =40 , V1 92.39, (a, b) v f 0.07 (with introduction of material of damping).
4.2. Dynamic solutions Dynamic solutions of the system in the form of periodic, quasiperiodic and chaotic responses are evaluated in the vicinity of the bifurcation points and some typical results are presented here. The dynamic system responses are
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found to be initial point dependent which implies the influence of domain of attraction of the individual equilibrium solutions among the multi-branch equilibrium solutions caused due to nonlinearity, internal resonance and parametric resonances in the system. Near the Hopf bifurcation point H 3 of the upper non trivial stable branch (C-G) of the frequency response curve (Fig.3) corresponding to detuning parameter V 2 130.5201 and with P 0.1, D 0, v1 10 and V1 92.39 , the system exhibits initial chaotic response in both modes, then jump phenomena in first mode only and finally settles to periodic behavior in both modes as shown in two dimensional phase portraits (a, b) and time traces (c, d) of Figure 5.
Fig. 5. Phase portraits (a, b) and time histories (c, d) in the upper nontrivial stable branch of the frequency response curve Figure 3 for V 2 130.5201, P 0.1, v1 10, D 0, V1 92.39 .
Corresponding to another set of system parameters P 0.1, D 0, v1 15, V1 92.39 and at V 2 141.0201 , torus form of the phase portrait (a) , closed loop character of the Poincare map(d) depicts the quasiperiodic system response for first mode in Figure 6.
Fig. 6. Phase portraits (a, b) time history (c) and Poincare map (d) for V 2 141.0201, P
0.1, v1 15, D
0, V1
92.39.
Corresponds to a point at V 2 137.6799 where the system response changes to chaotic behavior in first as
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well as second mode as depicted by Poincare maps and FFT power spectra in Figure 7. This chaotic system behavior is illustrated (Fig.8) in terms of evolution of maximum Lyapunov exponent ( V ) with the variation in parametric frequency detuning parameter ( V 2 ). The maximum Lyapunov coefficient becomes positive for a range of V 2 (from 136.8799 to 146.8201) where multiple stable and unstable equilibrium solutions exist in close proximity and intersect each other. In such zones the dynamic behavior of the system is observed as chaotic.
Fig. 7: Poincare maps (a, b) and time history (c) and FFT power spectra (d) for V 2 137.6799, P
0.1, v1 15, D
0, V1
92.39.
Fig. 8: Evolution of maximum Lyapunov exponent ( V ) for principal parametric resonance of first mode for system parameters
P 0.1, D 0, v1 15 and V1 =92.39.
5. Conclusions
The effect of low bending stiffness of travelling continua is investigated in case of principal parametric resonances of the first two modes and internal resonance. For a specified value of damping parameter, the instability
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region grows with decrease in bending stiffness. The beam with relatively lower flexural stiffness experiences both qualitative and quantitative differences from the beam with relatively higher flexural stiffness when system parameters like velocity fluctuation component, internal and external damping values and internal frequency detuning parameter are varied. The qualitative differences include emergence of more number of Hopf and saddle node bifurcation points, and quantitative differences include appearance of additional zones of instability caused due to Hopf bifurcation points, variation in amplitudes of the system, occurrences of saddle node bifurcation points at lower and/or higher values of control parameter. The travelling system displays periodic, quasiperiodic and chaotic behavior with variation in system parameters. References [1] J.A. Wickert, C.D. Mote Jr., Current research on the vibration and stability of axially moving materials, Shock and Vib. Dig. 20(1988), 3-13. [2] H.R Oz, M. Pakdemirli, Vibrations of an axially moving beam with time-dependent velocity, J. Sound Vib. 227(2) , (1999), 239-257. [3] H.R Oz, M. Pakdemirli, H. Boyaci, Non-linear Vibrations of an axially moving beam with time-dependent velocity, Int. J. Non-Linear. Mech. 36, 227(2) (2001) 107-115. [4] C.M. Chin, A. H. Nayfeh, Three-to-one internal resonance in parametrically excited Hinged-clamped beams, Nonlinear Dyn. 20(1999)131-158. [5] L.N. Panda, R.C. Kar, Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances, Nonlinear Dyn. 49(2007) 9-30. [6] L. N. Panda, R.C. Kar, Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances, J. Sound Vib. 309(2008) 375-406. [7] M. Pakdemirli, H.R. Oz, Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations, J. Sound Vib. 311( 2008) 1052-1074. [8] S.V. Ponomareva, W.T. van Horssen, On the transversal vibration of an axially moving continuum with a time-varying velocity: Transient from string to beam behavior, J. Sound Vib. 325(2009) 959-973. [9] M.H. Ghayesh, M. Amabili, H. Farokhi, Coupled global dynamics of an axially moving viscoelastic beam, Int. J. Non-Linear. Mech. (2013)5154-74. [10] B. Sahoo, L. N. Panda, G. Pohit, Parametric and internal resonances of an axially moving beam with time dependent velocity. Model. Simul. Eng., volume 2013, Article ID 919517, 18 pp, (2013). [11] B. Sahoo, L. N. Panda, G. Pohit, Nonlinear Dynamics of an Euler-Bernoulli Beam with Parametric and Internal Resonances, Procedia Engineering 64(2013), 727-736. [12] B. Sahoo, L. N. Panda, G. Pohit, Stability and bifurcation analysis of an axially accelerating beam. In: Sinha, J.K. (ed.), Vibration Engineering and Technology of Machinery (Mechanisms and Machine Science 23), Springer International Publishing Switzerland, (2015) DOI 10.1007/978-3-319-09918-7_81 [13] B. Sahoo, L. N. Panda, G. Pohit, Combination parametric and internal resonances of an axially moving beam, Journal of Vibration Engineering and Technology, 3(2) (2015), 137-150. [14] B. Sahoo, L. N. Panda, G. Pohit, Two Frequency Parametric Excitation and Internal Resonance of a Moving Viscoelastic Beam, Nonlinear Dyn.82(4)(2015) 1721-1742. [15] B. Sahoo, L. N. Panda, G. Pohit, Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation, Int. J. Non-Linear. Mech. doi:10.1016/j.ijnonlinmec.2015.09.017. [16] B. Sahoo, L. N. Panda, G. Pohit, Nonlinear Dynamics of a Travelling Beam Subjected to Multi-frequency Parametric Excitation, Applied Mechanics and Materials 592(2014), 2076-2080.
