Journal of Vibration Testing and System Dynamics

Volume 2 Issue 3 September 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 3, September 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(3) (2018) 187-207

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

A Modified Newmark Scheme for Simulating Dynamical Behavior of MDOF Nonlinear Systems S. D. Yu†, M. Fadaee Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3 Submission Info Communicated by S.C. Suh Received 21 December 2017 Accepted 2 April 2018 Available online 1 October 2018 Keywords Super-harmonic resonance Nonlinear oscillators Coupled nonlinear dynamical system Chaos numerical integration methods

Abstract The Newmark integration scheme, originally developed for simulating responses of linear dynamical systems, is modified in this paper to effectively model nonlinearities through introduction of the incremental displacements and the effective mass, damping and stiffness matrices. Based on the results of comparisons with the analytical solutions and the Runge-Kutta (RK) method for well-known nonlinear oscillators, the proposed scheme is found to capture accurately dynamical behaviors of nonlinear systems such as amplitude-dependent stiffening or softening natural frequencies, jump phenomena, superharmonic resonances, sub-harmonic resonances, and even chaos and bifurcations. The proposed scheme is more efficient than the RK method for large scale nonlinear systems with some type of sparsity for which a targeted algebraic equations solver can be employed to speed up the solution for each time step. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Nonlinear dynamical systems having multiple degrees of freedom (MDOF) are encountered in many fields of engineering, e.g., in vibration utilization [1]. Understanding the dynamic behaviour of a large DOF nonlinear dynamical system is important in dynamic design and smooth operations of many mechanical, electrical and electro-mechanical systems. Approximate analytical methods such as the well-known harmonic balance method [2], various perturbation methods [3, 4] and the multiple scales method [5, 6] have been employed to determine steady state responses of simple nonlinear dynamical systems, obtain the critical conditions for dynamic instability, and reveal typical nonlinear phenomena such as super-harmonic resonances, sub-harmonic resonances, combinational resonances, jump phenomena, etc. For general and large scale nonlinear systems, computational methods are widely used to determine their dynamical behaviour in the time domain. In computational dynamics, the entire time interval of interest is subdivided into a number of small and often uniform time steps. Within each time step, † 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.2018.09.001

188

S. D. Yu, M. Fadaee / Journal of Vibration Testing and System Dynamics 2(3) (2018) 187–207

various approximate integration schemes, explicit or implicit, used to relate the kinematical quantities at the end to those at the beginning of the time step, can be adopted to obtain a set of recursive equations for determining the state variables at the end of the step from the state variables at the beginning of the step with the help of some kinematical assumption within the time step. For example, the well-known implicit Newmark scheme assumes a linear acceleration in each time step. The explicit Runge-Kutta method of various orders of accuracy has been used widely in the literature to obtain numerical solutions for small scale dynamical systems [7]. The RK method possesses high degree of accuracy and high robustness through the use of four-point, five-point or six-point Simpson’s rule within a time step. Unfortunately, for dynamical systems with a large number of DOFs, the RK methods are computationally inefficient. Erlicher et al. [8] presented a comprehensive review of various numerical integration schemes for linear and nonlinear equations and proposed a generalized-α method, based on the Newmark integration scheme and a relaxation factor, for use in obtaining a numerical solution to second order system with positive definite constant mass matrix and positive semi-definite constant damping matrix, and with displacement-dependent nonlinear restoration forces. To handle the nonlinear terms, Erlicher et al. introduced a three-stage iteration scheme to approximate the nonlinear functions within a time step. Because of the iterations, the efficiency of the generalized-α method is similar to the RK methods in computational efficiency. The Newmark integration scheme has been successfully used to efficiently handle various nonsmooth nonlinearities in MDOF dynamical systems including dry fiction [9] and gap activated piecewise linear nonlinear springs [10]. In this paper, an implicit computational method is proposed to simulate behaviour of MDOF dynamical systems with smooth nonlinearities of varying degrees - weak, moderate or strong. The method is developed from the combination of the implicit Newmark integration scheme, the adoption of incremental displacements as the independent state variables, and the introduction of the effective mass/damping/stiffness matrices, which account for all types of nonlinearities in an MDOF system. Compared to the computational schemes available in the literature, the proposed method appears to meet all conceivable requirements in simulating dynamic behaviour of nonlinear systems with any number of DOFs. Numerical results obtained for various nonlinear problems indicate the proposed scheme is accurate and efficient.

2 Description of the proposed method The equations of motion of an n-DOF dynamical system may be written in a general form as ¨ q, ˙ q;t) = Q(t) ∀t ∈ [t0 ,t f ], F (q,

(1)

¨ q˙ and q are the vectors of the generalized accelerations, velocities and displacements, where t is time; q, respectively; Q(t) is the generalized force vector; F defines explicitly a set of continuous and continuously differentiable functions of the three state vectors. ¨ q˙ and q, or F = M(t)q¨ + For a linear dynamical system, each element in F is linearly dependent on q, C(t)q˙ + K(t)q, where M, C and K are the well-known mass, damping and stiffness matrices, respectively. For a chain-like nonlinear dynamical system of n masses, shown in Fig. 1, connected together by n + 1 ˙ + Fk (q), here nonlinear dampers and nonlinear springs , vector F may be written as F = Mq¨ + Fc (q) ˙ is the nonlinear damping force vector and Fk (q) is the nonlinear spring force vector. Fc (q) To solve the nonlinear problem, the time interval of interest, t ∈ [t0 ,t f ], is discretized into N time steps: [t0 ,t1 ] , [t1 ,t2 ], ..., and [tN−1 ,t f ], here, h is the time step, and t j = t0 + jh, j = 1, 2, . . ., N. If the state of the dynamical system at t = t j is determined, the state of the system at t = t j+1 may be determined by solving the following algebraic equations


x1

x2

Q1

Q2

m1

m2

xi

....

mi

xn

xn-1

Qi ....

189

Qn-1

Qn

mn-1

mn

Fig. 1 An n-DOF discrete system subjected to nonlinear springs and dampers.

F (q¨ j+1 , q˙ j+1 , q j+1 ;t j+1 ) = Q j+1 ,

(2)

where subscript j + 1 indicates that the quantities are evaluated at time t j+1 . From the Newmark integration scheme, the displacement, velocity an acceleration vectors at the beginning and end of a time step [t j ,t j+1 ] are related by the following two vector equations q j+1 = q j + hq˙ j + (0.5 − β ) h2 q¨ j + β h2 q¨ j+1 ,

(3)

q˙ j+1 = q˙ j + (1 − γ )hq¨ j + γ hq¨ j+1 ,

(4)

where β and γ are the Newmark coefficients. The following ranges of parameters should be observed for numerical stability β ≥ γ /2 ≥ 0.25. To extend the Newmark scheme for nonlinear dynamical problems, each of the following three vectors q j+1 , q˙ j+1 and q¨ j+1 , is split into two vectors as follows q˙ j+1 = q˙ j + Δq˙ j+1 ,

q j+1 = q j + Δq j+1,

q¨ j+1 = q¨ j + Δq¨ j+1 .

(5)

From Eqs. (3), (4) and (5), the incremental velocity and acceleration vectors, Δq˙ j+1 and Δq¨ j+1 , may be determined from the incremental displacement vector Δq j+1 as follows

γ γ γ Δq j+1 − q˙ j + (1 − )hq¨ j , βh β 2β 1 1 1 q˙ j − q¨ j . Δq¨ j+1 = 2 Δq j+1 − βh βh 2β Δq˙ j+1 =

(6) (7)

During a typical time interval [t j ,t j+1 ], F, as a continuous and continuously differentiable vector function of displacements, velocities and accelerations, can be expanded into the following Taylor series in the vicinity of (q j , q˙ j , q¨ j ) F j+1 = F j + M j Δq¨ j+1 + C j Δq˙ j+1 + K j Δq j+1 + O(Δq¨ j+12 , Δq˙ j+1 2 , Δq j+1 2 ).

(8)

According to Chiba and Kako [11], the error of the computed displacements by means of the Newmark integration scheme is in the order of O(h2 ) for γ = 0.5 and O(h) for γ > 0.5. The truncation error associated with the proposed treatment of nonlinear terms is compatible with the accuracy of the Newmark integration scheme. If a sufficiently small time step is used, the higher-order quantity term O(·) is small and negligible. As a result, one obtains the following simplified relationship F j+1 = F j + M j Δq¨ j+1 + C j Δq˙ j+1 + K j Δq j+1 , where F j = F| t j+1

q j , q˙ j, q¨ j

,

Kj =

∂F | ∂ qT tqj+1, q˙ j

, ¨j j, q

Cj =

∂F | ∂ q˙ T tqj+1, q˙ j

, ¨j j, q

Mj =

(9)

∂F | ∂ q¨ T tqj+1, q˙ j

. ¨j j, q


190

With the above simplification, Eq. (2) is rewritten as F j + M j Δq¨ j+1 + C j Δq˙ j+1 + K j Δq j+1 = Q j+1 .

(10)

Substituting Eqs. (6) and (8) into Eq. (10), one obtains K∗j+1 Δq j+1 = Q∗j+1 ,

(11)

where 1 γ C j + K j, Mj + 2 βh βh 1 1 γ γ )hq¨ j } q¨ j } − C j {− q˙ j + (1 − Q∗j+1 = Q j+1 − F j − M j {− q˙ j − βh 2β β 2β 1 1 γ γ − 1)hq¨ j }. q¨ j } + C j { q˙ j + ( = Q j+1 − F j + M j { q˙ j + βh 2β β 2β

K∗j+1 =

To start the recursive loop, the initial state of the dynamical system must be specified. It is a common practice to assume the initial displacements and velocities, and determine the initial accelerations by solving the following algebraic equations F (q¨ 0 , q˙ 0 , q0 ; 0) = Q0 .

(12)

With the initial state of the dynamical system determined, Eq. 3 can be solved progressively in any duration of interest using a proper time step and an appropriate linear algebraic equation solver. If K∗j+1 possesses some type of sparsity (e.g., banded or skyline), a targeted algebraic solver can be employed to solve Eq. 3 efficiently for Δq j+1 . Once the incremental displacements are found, the total displacement, velocity and acceleration vectors, q¨ j+1 , q˙ j+1 , and q j+1 , may be obtained using Eqs. (5), (6) and (7).

3 Case studies Dynamical behavior of some well-known nonlinear problems is investigated in this section using the proposed scheme with the two Newmark coefficients β = 0.5 and γ = 0.5 for the benefit of enhanced accuracy. Numerical results are compared with the solutions available in the literature and those obtained in this paper using the RK4 method to validate the proposed scheme for accuracy, robustness and appropriateness. 3.1

An SDOF Duffing oscillator

The equation of motion of an SDOF damped Duffing oscillator, as shown in Fig. 2, subjected to a harmonic excitation may be written as mx¨ + cx˙ + kx + k x3 = F0 sin ω t.

(13)

˜ here ωn = k/m an Are f = g/ωn2 g = 9.81 m/s2 , Introducing the transformations τ = ωn t and x = Are f x, the above equation may be written in the non-dimensional form as follows x˜ + 2ζ x˜ + x˜ + δ x˜3 = f0 sin λ τ ,

(14) √ where ζ = c/2 km; δ = k A2re f /k; f0 = F0 /kAre f ; λ = ω /ωn . Since the nonlinear term is a function of displacement only, the apparent force and the apparent stiffness, damping and mass associated with the cubic displacement nonlinearity may be obtained easily. They are provided below for reference Fj = x˜ j + δ x˜3j , M j = 1,C j = 2ζ , K j = 1 + 3δ x˜2j .

(15)


191

Nonlinear Duffing Spring

k,k’ F0 sinωt

m c Fig. 2 A damped Duffing oscillator under a harmonic excitation.

It can be seen that the apparent stiffness matrix is always positive for positive δ . The nonlinear term in the Duffing oscillator is known to have hardening effects. For the SDOF oscillator, Eq. 3 is reduced to a simple scalar equation containing a single unknown Δx˜ j+1 . For given initial displacement and velocity, and zero initial excitation, one can solve the original differential equation at time τ = 0 for the initial acceleration. As a test for convergence and accuracy, the undamped free vibration of the Duffing oscillator by setting ζ = 0 and f0 = 0 in Eq. (14) was computed using the proposed method, and compared with the following exact analytical solution: x˜ = ± x˜20 − x˜2 − x˜20 − 0.5δ x˜4 − x˜40 .

(16)

In this undamped free vibration test, the following nonlinear strength parameter and the initial conditions are used: δ = 0.1, x˜0 = 1, x˜0 = 5. Because of the absence of damping, a periodic response is established. In the phase plane, the motion is characterized by a closed curve. The numerical phase diagrams were obtained using the proposed scheme for four different time steps h = 2π /32, 2π /64, 2π /128, and 2π /256. Because of the small hardening spring effects, the period of motion is slightly smaller than 2π . Direct comparisons in Fig. 3 indicate that small but visible errors exist for h = 2π /32. However, as the time step is reduced, the numerical solutions and the analytical solution become indistinguishable. When damping is present, Johannessen [13] derived an exact analytical solution for damped free vibration of a Duffing oscillator and presented results for ζ = 0.04, 0.4, δ = 1, x˜0 = 0.5 and x˜0 = 0. Numerical results, obtained using the proposed method with a time step h = 2π /512 are compared with the analytical solution in Fig. 4. Again excellent agreement was achieved. When both damping and harmonic excitations are present in the Duffing oscillator, the RK4 method is used to provide an independent benchmark solution. To validate the proposed scheme for dynamic behavior of Duffing oscillators under harmonic excitations, numerical solutions are obtained for ζ = 0.05, f0 = 1.0, δ = 0.1(weak nonlinearity) and δ = 1(strong nonlinearity), with λ varying in [0, 4] at a fine increment of 0.002. For each set of numerical simulation, the response of the Duffing oscillator was computed from zero initial displacement and zero initial velocity using the RK4 method and the proposed method for a dimensionless duration of 80π with the same time step of h = 2π /512. A steady state is reached when the time elapsed from the beginning exceeds 72π . In each simulation, the maximum and minimum displacements of the steady state motion were recorded, and used to determine the mean amplitude using A˜ = (x˜max − x˜min )/2. The frequency response curves, obtained using the RK4 method and the proposed method, are compared Fig. 5. It can be seen that the two sets of results are in excellent agreement. For a Duffing oscillator of hardening stiffness (δ > 0), there is a jump phenomenon, widely known and well explained in the literature (e.g., Kalmar-Nagy and Balachandran [14]). In the principal


192

8

8

Exact Analytical Proposed Method (h=2π/32)

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-4

-2

0

2


6

Velocity

Velocity

6

-8

4

-4

-2

(a) Time step = 2π /32 8

2

2 Velocity

Velocity

4

0

0

-2

-2

-4

-4

-6

-6

-4

-2

0 Displacement

(c) Time step = 2π /128

4

2

Exact Analytical Proposed Method (h=2π/256

6

4

-8

2

(b) Time step = 2π /64 8


6

0 Displacement

Displacement

4

-8

-4

-2

0 Displacement

2

4

(d) Time step = 2π /256

Fig. 3 Comparison of free vibration of an undamped Duffing oscillator computed using the proposed method and various time steps with the exact analytical solution in the phase plane.

resonance zone of the frequency response curves, there exist two stable period-1 responses at the excitation frequency but with significantly different amplitudes – one on the “high” branch, and the other on the “low” branch. What lies between the two branches is an unstable motion, which cannot be numerically realized. It is not possible to know a priori which amplitude will prevail for a specified excitation frequency in the resonant zone. However, experimentally, the high branch of amplitudes can be obtained by up-tuning the excitation frequencies from well below the principal resonant frequency. The low branch of amplitudes can be realized by down-tuning the excitation frequencies from well above the principal resonant frequency. Because of the stiffening effect, the lower and higher jump frequencies are above the linear natural frequency (δ = 0). For a Duffing oscillator of softening stiffness (δ < 0), the process of finding the high and low branches of amplitude is reversed. Both jump frequencies are lower than the linear natural frequency (δ = 0). Since unstable motion of a dynamical system cannot be obtained from numerical simulations, steady state responses of a Duffing oscillator to harmonic excitations with frequencies varying between the two jump frequencies are unpredictable. To capture the well-known jump phenomenon, the amplitudes of steady state responses for a given excitation frequency, arrived from a set of 121 different initial conditions arbitrarily chosen from the domain x˜0 ∈ [−1, 1] and x˜0 ∈ [−1, 1], are found and recorded. In this

S. D. Yu, M. Fadaee / Journal of Vibration Testing and System Dynamics 2(3) (2018) 187–207 0.5

0.5

ζ=0.4

ζ=0.04

0.4

0.4

193

0.3 0.2 Displacement

Displacement

0.3 0.2 0.1

0.1 0 -0.1 -0.2

0

-0.3 -0.1 -0.2

-0.4 0

5

10 Time

15

-0.5

20

0

20

40

60

80

100

Time

(a) Damping ratio ζ = 0.4

(b) Damping ratio ζ = 0.04

Fig. 4 Free vibration of a damped Duffing oscillator with different damping ratios. 4

2

Proposed Method (δ=0.1) RK4 Method: (δ=0.1)

3.5

Proposed Method (δ=1.0) RK4 Method (δ=1.0)

1.8 1.6

3 Mean Amplitude

Mean Amplitude

1.4 2.5 2 1.5

1.2 1 0.8 0.6

1

0.4 0.5 0

0.2 0

0.5

1

1.5 2 2.5 3 Frequency Parameter λ

3.5

4

(a) FRC of a Duffing oscillator (delta = 0.1)

4.5

0

0

0.5

1

1.5 2 2.5 3 Frequency Parameter λ

3.5

4

4.5

(b) FRC of a Duffing oscillator (delta = 1.0)

Fig. 5 Comparison of frequency response curves of a damped Duffing oscillator (ζ = 0.05, f0 = 1.0, x˜0 = 0, x˜0 = 0).

verification test, for each excitation frequency in the range [0.5, 2.5] with fixed parameters ζ = 0.02, δ = 1, and f0 = 0.06, a total of 121 amplitudes of steady state vibration were obtained numerically using. The so-computed amplitude are compared with those first-order approximate analytical solutions given by Kalmar-Nagy and Balachandran [14]. In the approximate solution, Kalmar-Nagy and Balachandran introduced a small parameter ε to characterize the weak damping, weak nonlinearity and weak excitation, and solved the following Duffing equation x˜ + 2εζ x˜ + x˜ + εγ x˜3 = ε F cos Ωτ .

(17)

With the small parameter is set to be ε = 0.2, the three other parameters in the above equation are ζ = 0.1, γ = 5, and F = 0.3. The amplitudes of the steady state responses, computed using the proposed scheme and normalized to F as was done in Ref. [14], are shown in Fig. 6. It can be seen clearly that the jump phenomenon in connection with the hardening Duffing oscillator is captured precisely. The entire jump loop is defined by the high branch of amplitudes from S2 to S3 , downward jump from S3 down to S4 , the low branch of

194


Normalized Mean Displacement Amplitude

4 S3

3.5

Proposed Method

3 2.5

S2

2 1.5 1

S1

0.5 0

S4 0.5

1

1.5 Frequency Parameter λ

2

2.5

Fig. 6 Frequency response curves and jump phenomenon for a damped Duffing oscillator (ζ = 0.02, δ = 1.0, f0 = 0.06).

amplitudes from S4 to S1 , and the upward jump from S1 to S2 . The positions of points of interest are S1 (1.13, 0.89), S2 (1.13, 2.32), S3 (1.38, 3.65) and S4 (1.38, 0.27), obtained from the proposed method, are in good agreement with those reported in Ref. [14]: S1 (1.12, 1.23), S2 (1.12, 2.23), S3 (1.85, 5) and S4 (1.85, 0.3). The upward jump frequencies match quite well. However, there is some discrepancy in the downward jump. The reason for this discrepancy has to do with the fact that the approximate solution was derived under the assumption that the nonlinearity is weak. In the example plot of reference [14], the net nonlinear coefficient δ is 1.0 (ε = 0.2 and γ = 5), which is moderately large. Introduction of a small parameter into the equation of motion makes it easy to derive the hierarchical equations and obtain a corresponding analytical solution at each order. However, findings cannot be generalized or extended to beyond the valid range of weak nonlinearity. If δ A3 / A + δ A3 is used to evaluate the contribution of the cubic nonlinear term to the overall spring force, the nonlinear term at S3 is actually 93% of the total spring force for δ = 1. Therefore, the nonlinearity for δ = 1 cannot be considered to be weak. On the other hand, the proposed numerical scheme is valid for both weak and strong nonlinearities. To understand the effects of the nonlinearity, the frequencies response curves were plotted in Fig. 7 from the results obtained using the proposed scheme for four nonlinearity ratios: δ = 0.1, 0.5, 1, and 5 (ζ = 0.05 and f0 = 1.0). For each excitation frequency, 121 sets of responses corresponding to the 121 sets of initial conditions randomly picked from the region x˜0 ∈ [−1, 1] and x˜0 ∈ [−1, 1] were computed for a non-dimensional time duration of 80π . Compared to the linear scenario δ = 0 (resonance at λ ≈ 1 with a peak displacement amplitude of 1/2ζ or 10 for ζ = 0.05), presence of nonlinearity with hardening spring reduces the resonant amplitudes and shifts the occurrence of principal resonant frequency upward, and induces super-harmonic resonances of several orders. The downward jump frequencies are shifted from 1.38 for weak nonlinearity (δ = 0.1) to 2.95 for very strong nonlinearity (δ = 5). As the nonlinearity become stronger, the super-harmonic responses become more populated and more prominent. 3.2

A simple pendulum

The equation of motion of an undamped pendulum shown in Fig. 8a may be written as Io

d2θ + mgLG sin θ = 0, dt 2

(18)

where θ is the angular position of the pendulum measured counter clockwise with reference to the stable equilibrium position; m is the mass of the pendulum; Io is the mass moment of inertia of the pendulum


3

δ=0.1

δ=0.5

2.5

3.5

Mean Displacement Amplitude


4

3 2.5 2 1.5 1

2 1.5 1 0.5

0.5 0

195

0

0.5

1

1.5

2

2.5

3

3.5

0

4

0

0.5

1

Frequency Parameter λ

1.5

2

2.5

3

3.5

4


(a) δ = 0.1

(b) δ = 0.5

2.5

1.6

δ=1.0

δ=5.0

2



1.4

1.5

1

0.5

1.2 1 0.8 0.6 0.4 0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0


0.5

1

1.5

2

2.5

3

3.5

4


(c) δ = 1

(d) δ = 5

Fig. 7 Frequency response curves for a damped Duffing oscillator with weak and strong nonlinearities (ζ = 0.05, f0 = 1.0).

about the axis of rotation; LG is the distance between the mass centre and the axis of rotation; g is the gravitational acceleration. Introducing τ = ωn t, ωn = mgLG /I0 , Eq. (18) may be rewritten as d2θ + sin θ = 0. dτ 2

(19)

For free vibration, the pendulum motion is dependent only on the initial conditions (θ0 , θ0 ). Integrating the above equation once, one obtains 2 θ = ± θ0 + 2 (cos θ − cos θ0 ), (20) where prime indicates the derivatives with respect to τ . = Setting θ = π and θ = 0 in Eq. (20), the critical initial angular velocity is found to θ0,cr ± 2 (1 + cos θ0 ). For an initial angle −π < θ0 < π , the pendulum motion is oscillatory about the stable equilibrium position θ = 0, bounded by −π ≤ θ ≤ π if the initial angular velocity is less than the critical angular velocity. The motion becomes unbounded if the initial angular velocity is greater than the critical velocity.

196


Proposed Method (nfft = 64) Exact Analytical

1

O LG

Velocity

0.5

g

0 -0.5 -1

Ѳ

-1.5 -1.5

-1

-0.5 0 0.5 Angular Displacement

(a)

1

1.5

(b)

Fig. 8 (a) A simple pendulum in the gravitational field, (b) free vibration in the phase plane. 2.5 2

Angular Velocity (rad/s)

1.5 1 0.5

σ=0.1, nfft=512 σ=0.5, nfft=512 σ=0.95, nfft=512 σ=1.00, nfft=512 σ=1.05, nfft=512 σ=-1.05, nfft=512

0 -0.5 -1 -1.5 -2 -2.5 -20

-15

-10

-5

0

5

10

15

20

Angular Displacement (rad)

Fig. 9 Responses of a pendulum to various initial angular velocities in the phase plane.

The apparent force, the apparent mass, damping and stiffness are Fj = sin θ j , M j = 1, C j = 0, K j = cos θ j .

(21)

Incorporating the above terms into Eq. 3, one can solve the highly nonlinear vibration problem = ±2. For the initial conditions θ0 = 0 and easily. For θ0 = 0, the critical angular velocity is θ0,cr θ0 = 0.5θ0,cr , the resulting motion has an amplitude of 60◦ . Numerical results were obtained using the proposed method with a moderate time step h = 2π /64, and are compared in Fig. 8b with the analytical solution. It can be seen that the two solutions match perfectly. To check the validity of the proposed numerical scheme in simulating the global behaviour of the pendulum with bounded and unbounded motions, a fine time step h = 2π /512 was used to compute


Linear Spring

x1 Nonlinear

Duffing Spring

k m1

x2

197

Linear Spring

k

k,k’ m2

Fig. 10 An undamped 2-DOF nonlinear oscillator. the responses of the pendulum to a range of initial conditions defined by θ0 = 0 and θ0 = σ θ0,cr with σ = 0.1, 0.5, 0.95, 1.0, 1.05 and -1.05. The computed motions in the phase plane are shown in Fig. 9 for small amplitude oscillations (σ = 0.1), strong nonlinear oscillations (σ = 0.5, 0.95), and unbounded rotations (σ = ±1.05).

3.3

Free vibration of a 2-DOF nonlinear system

Free vibration of a 2-DOF nonlinear system shown in Fig. 10 is studied. The system consists of two masses, two linear springs and one nonlinear spring. The equations of motion of the 2-DOF system are 2k −k x1 k (x1 − x2 )3 m0 x¨1 + =− , (22) x¨2 x2 −k 2k 0m k (x2 − x1 )3 where k is the linear stiffness coefficient of the three active springs; k is the nonlinear stiffness coefficient of the spring connecting the two masses. Cveticanin [15] presented an exact analytical solution for free vibration of the coupled nonlinear system, which will be used as a benchmark solution for the verification purpose. For this 2-DOF system, the apparent force vector, the apparent mass/damping/stiffness matrices are kˆ (x1, j − x2, j )3 m0 3kˆ (x1, j − x2, j )2 −3kˆ (x2, j − x1, j )2 , C j = 0, K j = , Mj = . (23) Fj = − ˆ 0m −3kˆ (x1, j − x2, j )2 3kˆ (x2, j − x1, j )2 k (x2, j − x1, j )3 1 √ 2π The responses of the 2-DOF nonlinear system were computed using the proposed scheme with h = 512 3 2π for m = 1, k = kˆ = 1, x1,0 = 1.0, x2,0 = 0.5, and x˙1,0 = x˙2,0 = 0. Here √ , the natural period of the second 3

mode of vibration of the linearized system, is used as a time scale. The time domain responses of the two masses are plotted in Fig. 11. Careful examinations of the results in the two figures indicate that they are identical to those in Ref. [15]. 3.4

Forced vibration of a damped 2-DOF nonlinear system

In this section, the dynamical behaviour of a 2-DOF system, shown in Fig. 12, is investigated. The system consists of two masses whose motions are translational in the horizontal direction and subjected to three linear dampers of identical coefficient c and three nonlinear springs of identical linear stiffness coefficient k and identical cubic nonlinear stiffness with coefficient k . Under the two harmonic excitations of identical frequency and phase, the equations of motion of the system may be written as

(24) m [m] ˜ {x} ¨ + c [c] ˜ {x} ˙ + k k˜ {x} + k {F} = f˜ F0 sin ω t, where 3

−0.5 x1 + (x1 − x2 )3 10 2 −1 x1 ˜ ˜ , f = , {F} = . [m] ˜ = , [c] ˜ = k = , {x} = x2 1.0 01 −1 2 (x2 − x1 )3 + x32

S. D. Yu, M. Fadaee / Journal of Vibration Testing and System Dynamics 2(3) (2018) 187–207 1

1

0.8

0.8

0.6

0.6

0.4

0.4 Displacement x2

Displacement x1

198

0.2 0 -0.2

0.2 0 -0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

0

20

40

60

80

100

-1

0

20

40

60

80

100

Time (s)

Time (s)

(a)

(b)

Fig. 11 Motion of an undamped 2-DOF nonlinear oscillator. x2

x1 k,k’

F01 sinωt

k,k’

m1 c

F02 sinωt

k,k’

m2 c

c

Fig. 12 A damped 2-DOF nonlinear oscillator under harmonic excitations.

Introducing the following variable transformations: τ = ω0t and {x} = A {x}, ˜ here ω0 = A = F0 /k, the equations of motion are non-dimensionalized and are rewritten as

˜ {x} ˜ + k˜ {x} ˜ + δ { f }s = f˜ sin λ τ , [m] ˜ {x} ˜ + 2ζ [c]

k/m and

(25)

√ where the damping ratio ζ = c/2 km, the nonlinear strength parameter δ = k A2 k. Using the proposed scheme, the apparent force vector and the apparent mass/damping/stiffness matrices are 3 x˜1 + (x˜1 − x˜2 )3 10 2 −1 , Cj = , , Mj = Fj = 01 −1 2 (x˜2 − x˜1 )3 + x˜32 (26) 2 2 −1 x˜1 + (x˜1, j − x˜2, j )2 − (x˜2, j − x˜1, j )2 +3 . Kj = −1 2 − (x˜2, j − x˜1, j )2 (x˜2, j − x˜1, j )2 + x˜22 Numerical results were obtained for the responses of the two masses under the harmonic excitations with frequency parameters varying from 0 to 3.5 at an increment 0.01. The damping ratio was taken to be 0.05. For specified values of λ and δ , a time step of h = 2π /512 was used to determine 121 sets of the numerical solutions corresponding to initial displacements and initial velocities randomly drawn from a rectangular region bounded by [−1, 1] for x˜1 , x˜2 and x˜1 , x˜2 . Each set of solution is 112π in duration. Without consideration of the spring nonlinearity, the system has two natural frequencies: 1.0 and 1.732. The steady state response is considered established after 50 cycles or 100π from the initial moment. Four different values of nonlinear strength parameter were used to cover linear δ = 0, weakly nonlinear δ = 0.1, moderately nonlinear δ = 0.5 and strongly nonlinear δ = 1.0 scenarios. Numerical results are plotted in Fig. 13 for mass 1, and in Fig. 14 for mass 2. It can be seen clearly that the amplitudes of the steady state responses of both masses jump between two branches during the two primary resonant


2.5

Mass 1 (ζ=0.05, δ=0.0)

1.5

1

0.5

0

Mass 1 (ζ=0.05, δ=0.1)

2

Mean Amplitude

Mean Amplitude

2

1.5

1

0.5

0

0.5

1

1.5 2 Frequency Parameter λ

2.5

3

0

3.5

0

0.5

1

1.6

2

2.5

3

3.5

3

3.5

(b) 1.4

Mass 1 (ζ=0.05, δ=0.5)

1.4

Mass 1 (ζ=0.05, δ=1.0)

1.2

1.2

1 Mean Amplitude

Mean Amplitude

1.5


(a)

1 0.8 0.6

0.8 0.6 0.4

0.4

0.2

0.2 0

199

0

0.5

1


(c)

2.5

3

3.5

0

0

0.5

1


2.5

(d)

Fig. 13 Frequency response curves of a damped 2-DOF oscillator for various degrees of nonlinearities: Mass 1.

regions. The width of each primary resonant zone increases with the nonlinearity strength. Although the steady state motion is stable in the primary resonant zone, it is not possible to predict a priori which of the two amplitudes will be realized. However, examinations of the 121 sets of amplitude data in each primary resonant zone indicate that motions at the lower amplitudes occur more frequently. In a way similar to an SDOF Duffing oscillator, noticeable super-harmonic resonances, for 3ω ≈ ωn,1 or λ ≈ 1/3 = 0.333 and 3ω ≈ ωn,2 or λ ≈ 1.732/3 = 0.577, also occur for the coupled systems when moderate and strong nonlinearities are present in the system. To verify that the two super-harmonic spikes present in in Fig. 13d and Fig. 14d indeed correspond to the two different vibration modes, the time domain responses of the two masses are obtained with zero initial conditions for δ = 1.0 and λ = 0.36 and 0.66. The excitations acting on the two masses and the steady state motions of the two masses and plotted together in Fig. 15 for the first super-harmonic resonance at λ = 0.36 and in Fig. 16 for the second super-harmonic resonance at λ = 0.66. It can be seen that the 3ω component dominates the response of the first mass and is significant in the response of the second mass for both super-harmonic resonances.


200 3

2.5

Mass 2 (ζ=0.05, δ=0.0)

2

2

Mean Amplitude

Mean Amplitude

2.5

1.5 1

1.5

1

0.5

0.5 0

Mass 2 (ζ=0.05, δ=0.1)

0

0.5

1


2.5

3

0

3.5

0

0.5

1

(a) 1.8

1.6

Mass 2 (ζ=0.05, δ=0.5)

3

3.5

3

3.5

Mass 2 (ζ=0.05, δ=1.0)

1.4

1.4

1.2

1.2

Mean Amplitude

Mean Amplitude

2.5

(b)

1.6

1 0.8 0.6

1 0.8 0.6 0.4

0.4

0.2

0.2 0


0

0.5

1

1.5

2

2.5

3

3.5


(c)

0

0

0.5

1


2.5

(d)

Fig. 14 Frequency response curves of a damped 2-DOF Duffing oscillator for various degrees of nonlinearities: Mass 2.

3.5

Chaotic behavior of a Duffing oscillator

In this section, the following equation of motion of a SDOF Duffing oscillator with cubic and quintic nonlinear restoration forces is solved numerically using the Runge-Kutta method and the proposed scheme (27) x¨ + δ x˙ + ω 2o x + β x3 + α x5 = f cos(ω t + θ ). The above type oscillator has been studied extensively in the literature (e.g., Cai and Yang [16]). To verify the proposed scheme for bifurcations and chaos, the following parameters, identical to those of Cai and Yang, are used: α = −0.1, β = 1.12, ω 2o = −1, δ = 0.5, θ = 1, ω = 1. The excitation strength parameter f is allowed to vary from 0.3 to 0.5 with an increment of 0.0005. For each excitation strength f , the response of the oscillator to the harmonic excitation was simulated using the RK4 and the modified Newmark scheme with a fine time step of h = T /512, here T = 2π /ω , for a total length of 600T . The steady state response is considered to be reached after 300 excitation cycles or an elapsed time of 300T from the initial time. From t = 300T , the displacements and velocities are sampled and recorded after every excitation period. The corresponding bifurcation diagrams are shown in Fig. First of all, the proposed scheme and the RK4 method yields the identical bifurcation plots. These two plots are in good agreement with those of Cai and Yang [16]. It can be seen that the Duffing oscillator


Force F1(t) Mass 1 (ζ=0.05, δ=1.0, δ=0.36)

1

1

0.5

0.5 Displacement

Displacement

1.5

0

-0.5

-1

-1

315

320

325 330 335 340 Non-dimensional Time τ

345

350

-1.5 310

355

Force F2(t) Mass 2 (ζ=0.05, δ=1.0, δ=0.36)

0

-0.5

-1.5 310

201

315

320

325

330

335

340

345

350

355

Non-dimensional Time τ

(a)

(b)

Fig. 15 Time responses of two masses in the damped 2-DOF Duffing oscillator at the first super-harmonic resonant frequency. 1.5

Force F1(t) Mass 1 (ζ=0.05, δ=1.0, δ=0.66)

1

1

0.5

0.5 Displacement

Displacement

1.5

0

0

-0.5

-0.5

-1

-1

-1.5 310

315

320

325 330 335 340 Non-dimensional Time τ

345

350

355

Force F2(t) Mass 2 (ζ=0.05, δ=1.0, δ=0.66)

-1.5 310

315

320

(a)

325

330

335

340

345

350

355

Non-dimensional Time τ

(b)

Fig. 16 Time responses of two masses in the damped 2-DOF Duffing oscillator at the second super-harmonic resonant frequency.

undergoes period-1 motion for f < 0.337, and a period-doubling bifurcation at f = 0.337, and a process from period-doubling bifurcation to chaos beyond f = 0.377. 3.6

Chaotic behavior of a parametrically excited pendulum

In this section, a parametrically excited pendulum whose motion is governed by the following differential equation (28) x¨ + δ x˙ + (α + Q0 cos Ωt) sin x = 0 is studied using the proposed scheme. Here δ is the damping coefficient, α is the stiffness; Q0 is the amplitude of the parametric excitation; and Ω is the frequency of the parametric excitation. This problem was thoroughly investigated by Guo and Luo [17] by means of a mapping scheme. Unfortunately, their results on structures of periodic solutions (stability and periodicity) cannot be directly as an independent benchmark to validate the proposed numerical scheme because (i) only stable motion from a prescribed initial state can be obtained from the proposed numerical scheme, and (ii) the long


202

2

RK4 Method

1.5

1.5

1

1 Response x

Response x

2

0.5

0.5

0

0

-0.5

-0.5

-1

0.3

0.35

0.4 Excitation Strength f

0.45

(a)

0.5

Proposed Method

-1

0.3

0.35

0.4 Excitation Strength f

0.45

0.5

(b)

Fig. 17 Bifurcation diagrams in the ( f , x) plane: (a) RK4 method and (b) the proposed scheme.

(a) Node velocities for initial state (0, 0.05)

(b) Node velocities for initial state (0, −0.05)

Fig. 18 Bifurcation diagrams of pendulum motion sampled at excitation frequency (δ = 0.1, α = 4, Ω = 2).

term behavior of the parametrically excited pendulum are sensitive or very sensitive to the initial conditions, depending on the system parameters. For this reason, the 4th order Runge-Kutta method was employed to provide independent solutions for the following fixed system parameters: δ = 0.1, α = 4, Ω = 2 and a varying excitation amplitude Q0 ∈ [0, 40] as a bifurcation parameter. Using the proposed numerical scheme, the effective inertial, damping and stiffness matrices along with the effective load for the parametrically excited pendulum are Fj = x¨ j + δ x˙ j + (α + Q0 cos Ωt j+1 ) sin x j , M j = 1, C j = δ , K j = (α + Q0 cos Ωt j+1 ) cos x j .

(29)

To investigate the long term behavior of the pendulum, excited vertically by a sinusoidal parametric excitation with varying amplitudes varying in the interval Q0 ⊂ [0, 40] with a small increment of 0.005, simulations were performed using the proposed numerical scheme and a time step of h = T /512, here T = 2π /Ω = π for (Q0 ) j = 0.005 j, j = 1, 2, ..., 8000. Among the 16, 000 total simulations, 8000 of them start from the initial state x0 = 0 and x˙0 = 0.05, and the other 8000 start from the initial state x0 = 0 and x˙0 = −0.05. The pendulum is initially placed at its stable equilibrium position at x0 = 0. A small non-zero initial velocity of 0.05 in the counterclockwise direction and -0.05 in the clockwise


203

Q0=4.5, ω=2

4

Excitation

2 0 -2 -4 -6

0

10

20

30 Time (s)

40

50

60

(a) Parametric excitation of the first cycles (Q0 = 4.5, ω = 2) Proposed Method for Initial Conditions (0,0.05) RK4 Method for Initial Conditions (0,0.05)

Proposed Method for Initial Condition (0,-0.05) RK4 Method for Initial Condition (0,-0.05)

0.4 Angular Displacement Normalized to π

Angular Displacement Normalized to π

0.4

0.2

0

-0.2

0.2

0

-0.2

-0.4

-0.4 0

10

20

30

40

50

60

Time (s)

(b) Responses of the first 20 cycles in time domain with initial condition (0, 0.05)

0

10

20

30 Time (s)

40

50

60

(c) Responses of the first 20 cycles in time domain with initial condition (0, −0.05)

Fig. 19 Motion of pendulum in the first 20 cycle (δ = 0.1, α = 4, Ω = 2, Q0 = 4.5): (a) excitation, (b) pendulum at initial state (0, 0.05), (c) pendulum at initial state (0, −0.05).

direction is used a seed value to start the motion of the pendulum for the parametric excitation. For each of the 16, 000 sets of responses, the simulation length is 600T consisting of 307, 200 discrete pendulum states. For the given damping, the steady state motion is considered to be reached at tss = 300T from the initial moment. The node velocities x˙k at the following discrete times tk = tss + kT , k = 0, 1, 2, ..., 300 were recorded and shown in Fig. 18a for the responses initiated from (0, 0.05), and Fig. 18b for the responses initiated from (0, −0.05). In each of the two sub-plots of Fig. 18, there are 301 node velocities corresponding to each Q0 . An examination of the bifurcation chart indicates that pendulum node velocities are completely symmetric with respect to x = 0, which is in agreement with the conclusion of Guo and Luo [17]. However, it should be pointed out here that the lower negative branch of period-1 motion for Q0 ∈ (2, 5.6) can only be materialized through a negative initial seed velocity. For small excitation amplitudes Q0 ∈ (0, 2), the vertically applied parametric excitations do not induce long-term pendulum motion. A static motion of x = 0 is the result. As the excitation

204


Proposed Method RK4 Method

3


1

2 0

1 -1

0 -2

-1 -3

-2 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

(a) Responses of the first 600 cycles in phase domain with initial condition (0, 0.05)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

(b) Responses of the first 600 cycles in phase domain with initial condition (0, −0.05)

Fig. 20 Motion of pendulum in the phase plane for the first 600 cycles(δ = 0.1, α = 4, Ω = 2, Q0 = 4.5): (a) pendulum at initial state (0, 0.05), (c) pendulum at initial state (0, −0.05).

amplitudes reach 2.0 and beyond, the long term response is a stable period-1 motion. As it can be seen from Fig. 18, parametrically excited pendulum expressed by Eq. (2), shows very complex and dynamically rich behavior. The system experiences both periodic and chaotic motion for different ranges of excitation amplitudes. To validate stable period-1 motions for Q0 ∈ (2, 5.6), the responses of the pendulum for the initial conditions (0, 0.05) and (0, −0.05), and the following parametric excitation Q(t) = 4.5 cos 2t, were computed using the 4th order Runge-Kutta method and the proposed numerical scheme with the same time step h = π /512. The angular displacements of the pendulum starting from two different states are compared in Fig. 19 for the first 20 excitation cycles. The eventual long-term periodic motions of the pendulum or the limiting circles, started from (0, 0.05) and (0, -0.05), respectively, are shown in the phase plane in Fig. 20 for a duration of 600 excitation cycles. It can be seen that one periodic motion is simply the mirror image of the other with respect to x = 0 in the phase plane. Good agreement between the RK-4 method and the proposed method is observed. According to the bifurcation chart in Fig. 18, for Q0 = 20, the node velocities sampled at the excitation period vary considerably in a wide range; the pendulum motion is chaotic. The responses of the pendulum, computed using the proposed method and the RK4 method, are compared in Fig. 21a for the response starting from (0, 0.05) and Fig. 21b for the response starting from (0, −0.05). The responses, obtained using the two different approaches, deviate from each other soon after the application of the excitation. The responses of the pendulum, obtained the proposed approach, are also shown in the phase plane in Fig. 22 for the entire motion and in Fig. 23 with the normalized node displacements limited to [−6, 6] for a clearer view of the phase trajectories of the chaotic motion of the pendulum. From the traditional definition of chaotic motion, the responses of the pendulum starting from the two almost identical initial states, were obtained using the proposed approach. Results shown in Fig. 24a and Fig. 24b clearly indicate that the responses of the pendulum become chaotic soon after the first time that the pendulum passes through the unstable equilibrium position x = π .


205

Q0=20, ω=2

20

Excitation

10

0

-10

-20 0

10

20

30 Time (s)

40

50

60

(a) Parametric excitation of the first 20 cycles (Q0 = 20, ω = 2) 40

Proposed Method for Initial Condition (0,0.05) RK4 Method for Initial Condition (0,0.05) Angular Displacement Normalized to π


60

40

20

0

-20

-40

0

500

1000 Time (s)

1500

2000

(b) Responses of the first 600 cycles in time domain with initial condition (0, 0.05)


20

0

-20

-40

-60

0

500

1000

1500

2000

Time (s)

(c) Responses of the first 600 cycles in time domain with initial condition (0, −0.05)

Fig. 21 Chaotic motion of pendulum in the time domain for the first 600 cycles(δ = 0.1, α = 4, Ω = 2, Q0 = 20): (a) pendulum at initial state (0, 0.05), (c) pendulum at initial state (0, −0.05).

4 Conclusions Ordinary and chaotic behaviours of nonlinear dynamical systems are investigated using the modified Newmark scheme. The proposed scheme captures accurately and efficiently all nonlinear phenomena including stiffness hardening, amplitude jump, and super-harmonic resonances. For large scale nonlinear dynamical systems with hundreds or thousands of degrees of freedom, the proposed scheme is expected to outperform the RK in computational efficiency because the effective inertia/damping/stiffness matrices accounting all types of nonlinearities encountered in engineering systems possess some type of sparsity (banded, skyline). The proposed scheme takes full advantage of the sparsity and yields a speedier solution than the RK method. In addition, the proposed method can be easily extended to deal with non-smooth nonlinearities in connection with the linear complementarity problem algorithm. It is anticipated that the proposed approach will gain popularity in the engineering nonlinear dynamics community because of these advantages.


206

Proposed Method for Initial Condition (0,0.05)

Proposed Method for Initial Condition (0,-0.05)

10

5

Angular Velocity (r/s)

Angular Velocity (r/s)

10

0

-5

5

0

-5

-10

-10 -60

-40

-20

0

20

40

-60

60

-40

-20

0

20

40

60



(a) Responses of the first 600 cycles in phase domain initial condition (0, 0.05)

(b) Responses of the first 600 cycles in phase domain initial condition (0, −0.05)

Fig. 22 Chaotic motion of pendulum in the phase plane for the first 600 cycles, (δ = 0.1, α = 4, Ω = 2, Q0 = 20,): (a) pendulum at initial state (0, 0.05), (b) pendulum at initial state (0, −0.05). Proposed Method for Initial Condition (0,0.05)

Proposed Method for Initial Condition (0,-0.05)

10 Angular Displacement Normalized to π


10

5

0

-5

5

0

-5

-10

-10 -6

-4

-2

0

2

4

6

(a) Responses of the first 600 cycles in phase domain initial condition (0, 0.05)

-6

-4

-2

0

2

4

6

(b) Responses of the first 600 cycles in phase domain initial condition (0, −0.05)

Fig. 23 Detailed view of chaotic motion of pendulum in the phase plane for the first 600 cycles (δ = 0.1, α = 4, Ω = 2, Q0 = 20): (a) pendulum at initial state (0, 0.05), (b) pendulum at initial state (0, −0.05).

References [1] Wen, B.C., Li, Y.N., Zhang, Y.M., and Song, Z.W. (2005), Vibration Utilization Engineering Science Publisher, Beijing. [2] Bolotin, V.V. (1964), The dynamic stability of elastic systems, Golden San Francisco, Holden-Day. [3] Minorsky, N. (1962), Nonlinear Oscillations, D. Van Nostrand Co. Inc., Princeton, New Jersey. [4] Nayfeh, A.H. (1985), Problems in Perturbation, John Wiley & Sons. [5] Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillations, John Wiley & Sons. [6] Nayfeh, A.H. (1981), Introduction to Perturbation Techniques, John Wiley &Sons. [7] Yu, S.D., Warwick, S.A., and Zhang, X. (2009), Nonlinear dynamics of a simplified engine-propeller system, Communications in Nonlinear Science and Numerical Simulation, 14(7), 3149-3169. [8] Erlicher, S., Bonaventura, L., and Bursi, O.S. (2002), The analysis of the generalized-α method for non-linear dynamic problems, Computational Mechanics, 28, 83-104. [9] Yu, S.D. and Wen, B.C. (2013), Vibration analysis of multiple degrees of freedom mechanical systems with

S. D. Yu, M. Fadaee / Journal of Vibration Testing and System Dynamics 2(3) (2018) 187–207 Initial State (0,0.05) Initial State (0,0.051)

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Initial State (0,-0.05) Initial State (0,-0.051)

4 Angular Displacement Normalized to π


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2 0 -2 -4 -6 -8

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(a) Responses from (0, 0.05) and (0, 0.051)

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50

60

(b) Responses from (0, −0.05) and (0, −0.051)

Fig. 24 Occurrence of rapid deviation of pendulum motion for the first 20 cycles (δ = 0.1, α = 4, Ω = 2, Q0 = 20): (a) initial states (0, 0.05) and (0, 0.051), (b) initial states (0, −0.05) and (0, −0.051). dry friction, Journal of Mechanical Engineering Science, 227(7), 1505-1514. [10] Yu, S.D. (2013), An efficient computational method for vibration of unsymmetric piecewise-linear dynamical systems with multiple degrees of freedom, Nonlinear Dynamics, 71(3), 493-504. [11] Chiba, F. and Kaka, T. (1999), Error analysis of Newmark’s method for the second order equation with inhomogeneous term, Workshop on MHD Computations: Study on Numerical Methods Related to Plasma Confinement, T. Kako and T. Watanabe, T. (eds.), National Inst. for Fusion Science, Nagoya (Japan), 120129. [12] Fadaee, M. and Yu, S.D. (2016), A numerical method for determining stick-slip motion of two-dimensional coulomb friction oscillators, Journal of Mechanical Engineering Science, 230(14), 2438-2448. [13] Johannessen, K. (2015), The Duffing oscillator with damping, Eur. J. Phys., 36, 065020. [14] Kalmar-Nagy, T. and Balachandran, B. (2011), The Duffing Equation: Nonlinear Oscillators and their Behaviour : Chapter 5: forced harmonic vibration of a Duffing oscillator with linear viscous damping, 1st ed., Edited by I. Kovacic and M. J. Brennan. John Wiley & Sons, Ltd. ISBN: 978-0-470-71549-9, 139-174. [15] Cveticanin, L. (2001), Vibrations of a coupled two-degree-of-freedom system, Journal of Sound and Vibration, 247(2), 279-292. [16] Cai, M.X. and Yang, J.P. (2006), Bifurcation of Periodic Orbits and Chaos in Duffing Equation, Acta Mathematicae Applicatae Sinica, English Series, 22(3), 495-508. [17] Guo, Y. and Luo, A.C.J. (2017), Complete bifurcation trees of a parametrically driven pendulum, Journal of Vibration Testing and System Dynamics, 1(2), 93-134.



Experimental Investigations of Energy Recovery from an Electromagnetic Pendulum Vibration Absorber Krzysztof Kecik†, Angelika Zaszczynska, Andrzej Mitura Department of Applied Mechanics, Mechanical Faculty Engineering, Lublin University of Technology, Nadbystrzycka 36 St., Lublin, Poland Submission Info Communicated by J.Z. Zhang Received 19 February 2018 Accepted 6 April 2018 Available online 1 October 2018 Keywords Experiment Energy recovery Pendulum Vibration mitigation

Abstract The paper presents an experimental study of a special non−linear low frequency system dedicated to vibration mitigation and energy recovery. The dual−function design was based on an autoparametric vibration system, which consists of an oscillator with an added pendulum vibration absorber. Its structure includes an energy harvesting device: a levitating magnet in a coil. The pendulum motion shows simultaneously the effects of vibration reduction and energy recovery. The influences of the magnet−coil configurations, and load resistances on vibration reduction and energy harvesting were studied in detail. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction At the beginning of the 21st century, humanity faces many problems [1]. The environmentalists pay attention to threats resulting from the climate change and limited fossil fuel resources [2, 3]. Scientists and engineers strive to create constructions and mechanisms with low energy consumptions. The energy demand can is supported by an increasing energy harvesting (EH). Nowadays, the recovered energy replace batteries in sensors and low power electronic systems in order to realize autonomous electronic applications [4]. Historically, the EH has been practiced in form of windmills, sailing ships, and waterwheels. The modern definition of EH is the process of extracting small amounts of energy (power) from the environment [5]. This technology has progressed rapidly, and is possible to use an energy harvester to power some MEMS devices using only ambient energy [6]. The term “environment” energy can be a vibrating structure, the radiant energy of the sunlight, even the thermal energy of a warm object. One of the well-known forms of energy harvesting in engineering applications is the solar technology, which uses the photo-voltaic conversion to generate electrical energy from the sunlight. Another popular form of EH is the vibration harvesting which has found use in many engineering fields. † 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.2018.09.002

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The electrical energy is converted by mainly three different methods that are most popular and extensively studied in the literature, based on piezoelectric, electromagnetic and electrostatic conversion mechanism. The electromagnetic energy harvesters use Faraday’s law on voltage induction. The effect causes change in the magnetic flux in a circuit and generates an electromotive force (EMF). Typically, this is achieved as a result of the relative motion between the magnet and a coil of wire. Generally, the magnitude of energy harvested from magnetic induction harvesters can range up to kilowatts [7]. A piezoelectric harvester based on mechanical strain, then piezoelectric material became electrically polarised. The recovered energy from piezoelectric systems can vary from microwatts to even watts [8, 9]. The electrostatic transduction mechanism uses capacitors to transfer mechanical energy into the electrical domain. Typically, these devices rely on an initial voltage source to create equal but opposite amount of charge on the plates of a capacitor. The magnitude of energy from such systems is usually on the order of microwatts. The vibration level, the harvester design and the coupling between the mechanical and electrical parts determine the energy recovering effectiveness. In the literature, different types of electromechanical coupling description are considered. In reference [10, 11], the simple mass - spring - damper model energy harvesting were used. The authors assumed that the coupling coefficient is approximately linearly related to the internal resistance of the coil. Based on the assumption the inductance was not included in electromagnetic description. The influence of the electrical circuit on the magnet motion as additional damping has been analysed (called electrical damping coefficient [11]). A promising means for increasing the effective operating bandwidth of EH is to utilise non−linearity [12]. The non−linarity can come from the construction and geometry of the EH devices (e.g. magnets). This problem is presented in reference [13], in which the investigations unveiled regions in the parameter space where non−linear coupling model is better than linear and regions in which it is the opposite. The non−linear model of a magnetic levitation harvester could be well represented by the Duffing’s equation. In references [14–16], systems with the levitation magnet are studied. The linear harvester models have been modified, instead of suspension with the linear spring, the authors proposed a magnetic system. The proper orientation of the magnet poles allows for levitation of the movable magnet. The obtained magnetic suspension leads to the non−linear Duffing’s model. In all three articles the numerical and experimental characteristics with the hardening effect (jump amplitude) were demonstrated. Reference [17] presented a harvester consisting of the magnets and spacers. The configuration of the magnets-spacers stack can change shape of a flux curves. Results showed that fixed magnets can influence on magnetic field distribution of movable stack. The authors suggested that to description non−linear electromechanical coupling coefficient the magnetic flux derivative should be taken. Whereas, in the paper [18] influence of initial distance between the coil and the movable magnet on recovered energy is studied experimentally. However, the system with the levitating magnet has only the one fixed magnet (lower). Four initial configuration of the coil - movable magnet were analyzed. The maximum electrical power was obtained, when the middle magnet and the upper coil’s end were overlapped. Interestingly, the optimal power values for the different magnet-coil configurations were observed at various excitation frequencies Another challenge in engineering is control and reduction of vibrations. One of the most popular method is application of the dynamical vibration absorber (DVA). The classical example of DVA is the pendulum vibration absorber (PVA), [19]. The PVA with the harvester device is a very interesting dynamical system. It connects two effects: vibration mitigation and energy recovery. This paper focuses on the possibility of energy recovering from a PVA. Inspired by the idea of EH and vibration mitigation, a new type of DVA called harvester−absorber system (HAS) is proposed. The motivating hypothesis for this paper was that electromagnetic non−linear coupling could be used to improve the performance of an energy harvester by broadening its frequency response. The electromagnetic coupling was changed by the different configuration of the magnet - coil system.


(a)

211

(b)

Fig. 1 (a) Photo and (b) scheme of the laboratory HAS system.

2 HAS system 2.1

Pendulum vibration absorber system

The HAS system consists of two crucial parts. The first is the main system or oscillator (I) in which vibration should be reduced. The second system is the special maglev pendulum (II). The laboratory rig and its scheme are presented in Fig. 1a and b, respectively. The pendulum is suspended on the main system (such a system is called an autoparametric vibration system in the literature [20]). The system works properly for low frequencies. The mitigation effect occurs if the pendulum swings. The oscillator’s suspension is made up of a linear spring (spring 1 in Fig. 1b) and a linear (or non−linear) damper. The oscillator moves on the sliding guides (slide bearing). The excitation of the HAS system is realized by a three-phase 1.5kW induction motor Sh 80X-4D with its own fan mounted on the shaft, the Hitachi L200 inverter to the motor control, and a special system which changes the rotational motion into the translation motion of the slider. The slider is connected to the oscillator by the linear spring (spring 2 in Fig. 1b). The change of the eccentric radius causes modification in the amplitude excitation level. Measurements of the pendulum swinging angle is realized by a rotary optoelectronic transducer MHK40 with a resolution 2π /1000, directly mounted on the pendulum axis of rotation. The pendulum is made of a colorless non-magnetic material in a tubular shape. A magnet moves inside a pendulum that is surrounded by a coil. The motion of the pendulum causes vibration mitigation of the main system and energy induction in the harvester. The system parameters are presented in Table 1. In all experiments, amplitude of the excitation was fixed to 0.03m. 2.2

Pseudo-maglev harvester

The Earnshaw theorem [21] states that use of only static ferromagnetism is impossible to stably levitate against the gravity. The magnetic levitation with a mechanical contact that provides stability is called pseudo-levitation. However, in the literature, the pseudo-levitation is often called levitation [22]. The detailed construction and the pseudo-maglev system are shown in Fig. 2. The harvester consists of two magnetic rings of 20mm in diameter and of 5mm in height. The pseudo-levitating cylindrical neodymium magnet is mounted in the pendulum tube. The motion of the magnet is limited by a repulsive force exerted by the opposing magnetic field of magnets placed

212


Table 1 Main parameters of the HAS system. Parameter

Value

Parameter

Value

oscillator (mass)

0.4kg

tube length

0.24m

pendulum (mass)

0.54kg

pendulum (damping)

0.0125Nms/rad

spring(1) (stiffness)

800N/m

load resistance

0-10kΩ

spring(2) (stiffness)

1000N/m

frequency excitation

0-25Hz

damper (damping)

10.5Ns/m

excitation amplitude

0-0.05m

(a)

(b)

Fig. 2 (a) Photo and (b) scheme of the pendulum, which plays role of a dynamic vibration absorber and electromagnetic harvester. Table 2 Parameters of the main harvester components. Coil parameter

Value

Magnet parameter

Value

coil resistance induction

1148Ω

type

NdFeB(N38)

1.46H

shape

cylinder

wire coils

12740

diameter

20mm

wire diameter

0.14mm

height

30mm

coil hight

50mm

mass

0.098kg

in the pendulum ends. The coil is wrapped on the outer pendulum surface with adjustable positions. The distance between the bottom and top fixed magnets can be set by the screw system. This allows modification of the magnetic suspension characteristics and the resonance shifting. The pseudo-maglev characteristic suspension can be found in the reference [23]. In order to reduce friction, the magnet and tube surfaces were sprayed by Teflon. In addition, the special air holes and gaps to air compression reduction are made on the pendulum surface. The coil and levitating magnet parameters are listened in Table 2. The harvester is connected to a data acquisition system consisting of the MicroDAQ module with a multi-core OMAP L137 DSP processor, a conditioning module, a power supply and a PC with C+ (Fig. 3c).


(a)

(b)

213

(c)

Fig. 3 The quasi-static tests of the harvester: (a) scheme of the test, (b) the Shimadzu machine with mounted the pendulum harvester and (c) a data acquisition system.

3 Results and discussions 3.1

Harvester quasi-static analysis

The electromagnetic coupling influences significantly the induction effect. Generally, it means a fraction of magnetic flux produced by the current in the coil. The value depends on many factors: the separation distance between the magnet and coil, the length of the coil, the electromagnetic drag force, etc. To show the electromagnetic coupling influence, one can start by examining how the induced current in the coil is related to the magnet position in the coil. The quasi-static tests have been performed on the universal tensile testing Shimadzu machine in the positive and then negative directions (it means that the magnet was moved by a triangular signal), without the fixed magnets. The magnet has been moving in the coil with a low constant speed. During tests, five cycles of the excitation were applied. The scheme and the experiment are presented in Fig. 3. The selected results of the induced current versus the magnet position from a quasi-static tests for different magnet speeds and load resistances are presented in Fig. 4a and b, respectively. The quasi-static tests have been performed with the selected velocities 0.0083m/s (black line), 0.0117m/s (blue line) and 0.0167m/s (red line, see Fig. 4a). Similar tests have been done for different resistance loads 1.15kΩ (black line), 4kΩ (blue line) and 6kΩ (red line, see Fig. 4b). The vertical dashed line means the coil position (beginning and end). The maximal value of induced current of about 0.44mA is reached when the magnet is close to the coil end, while the best load resistance is close to the coil resistance (≈ 1.15kΩ). It can be seen that the induced current is a strongly non−linear function of the magnet position. Doubling the magnet velocity causes a twofold increase in the current. This means that the induced current linearly proportional to the magnet speed. The maximal recovered power from the quasi-static tests is about 0.45mW. 3.2

Coil position influence

Based on the results presented in Fig. 4, three configurations of the coil position in the HAS system were studied. The levitating magnet was in the static equilibrium (the gravitational and magnetic forces are equal). In the first configuration no. 1, the centre of the movable magnet and the coil center were in the same position - overlapping (Fig. 5, configuration no. 1). In second configuration no. 2 the magnet centre is set close to the coil end (Fig. 5, configuration no. 2). In the last configuration, the coil was set to be 50mm below the magnet (configuration no. 3). The distance between the fixed magnets was constant and equals 100mm. The height of the coil including housing was 50mm.


214

0.5

0.5

0.25

Current (mA)

Current (mA)

0.25

0.0083(m/s) 0.0117(m/s) 0.0167(m/s)

0

1.15(k ) 4(k ) 6(k )

0

-0.25

-0.25

-0.5 -75 -60 -45 -30 -15

0

15

30

Displacement (mm)

(a)

45

60

75

-0.5 -75 -60 -45 -30 -15

0

15

30

45

60

75

Displacement (mm)

(b)

Fig. 4 The relation between induced current and magnet position: (a) for the different magnet speeds and fixed load resistance 1.15kΩ and (b) different load resistances and fixed magnet speed 0.0117m/s.

Fig. 5 Configurations of the harvester: the coil centre coincides with the magnet centre (configuration no. 1), the magnet centre coincides withe the coil end (configuration no. 2) and the coil is below the magnet (configuration no. 3).

The coil position changes can be interpreted as the shifting of the vertical dashed line in Fig. 4. The three configurations of the pendulum - harvester device were installed on the laboratory rig. These configurations were tested during the pendulum oscillations (active PVA) and in a rest (inactive PVA). 3.3

Frequency response

Generally, resonance is a phenomenon in which a vibrating system or external force drives another system to oscillate with a great amplitude at a specific frequency. Usually, at resonant frequencies the energy harvesting is most effective (higher velocity). On another word, the high vibration level in DVA is unwanted. Therefore, to find a compromise between the two effects is very difficult [24]. The pendulum with harvester device shows the most effective energy harvesting when the pendulum is rest. Activation of the pendulum (swinging) causes reduction in vibration of the main system, but the energy recovery is reduced. The key question that arises is whether it is for all magnet-coil configurations?


(a)

215

(b)

(c)

Fig. 6 The resonance curves: (a) the oscillator, (b) the pendulum and (c) recovered energy.

Firstly, the parametric resonance region has been found. The experimental resonance curves for the oscillator, the pendulum and the recovered power versus excitation frequencies are shown in Fig. 6a-c, respectively. These curves show the maximal displacements of the oscillator and the pendulum at a load resistance of 1.15kΩ. The black line corresponds to configuration no. 1, the red line to configuration no. 2 and the blue line to configuration no. 3. The solid lines denote the solution in which the pendulum oscillates (active), while the dashed-dotted lines mean the pendulum is in rest (inactive). Coil positions influence the inertial mass momentum and the pendulum gravity centre. Therefore, the frequency shifts in the resonance curves are observed. We can see that the pendulum oscillation causes the reduction in vibration of the main system. The frequency range where the pendulum vibrates is narrow and depends on the coil positions, generally lying between 1.875Hz and 2.750Hz. A higher amplitude of the pendulum vibration is observed for configuration no. 3. This is caused by the higher mass moment of inertia. The vibration mitigation is very similar in all configurations. The RMS recovered power has been calculated for a 10 second time period and is shown in Fig. 6c. The recovery energy is the most effective in configuration no. 2. Compared to configuration no. 1 we observed a twofold increase in the recovered energy. As expected, configuration no. 3 is the least effective. The minimal EH for the active pendulum equals 2.5mW for the configurations no. 2, 1.1mW for the configuration no. 1 and 0.05mW for configuration no. 3.

216


(a)

(b)

Fig. 7 Influence (a) of the load resistances on RMS current and (b) RMS power at configuration no. 1.

(a)

(b)

Fig. 8 Influence (a) of the load resistances on RMS current and (b) RMS power at configuration no. 2.

(a)

(b)

Fig. 9 Influence (a) of lthe oad resistances on RMS current and (b) RMS power at configuration no. 3.


(a)

217

(b)

Fig. 10 Influence of the load resistance: (a) on pendulum angle and (b) oscillator displacement.

3.4

Load resistance influence

This section presents the load resistance influence on the induced current and the recovered energy. The load resistance means equivalent resistance of the device attached to the output of a given system. The influences of the load resistance on the RMS current and recovered power for all configurations are shown in Figs. 7-10. All results obtained for the fixed frequency 2.25Hz and the amplitude 0.03m. The red lines indicate the responses of the system with an active PVA, while the black line with inactive PVA. Both resonance curves, show decreasing tendency of the recovered current with increasing load resistance. However, the recovered energy shows that the RMS power reaches a maximum value around the load resistance 1.5kΩ. This means that the load resistances should be set close to the coil resistance. Analysing the second configuration, some interesting results can be observed for low load resistances 0 − 0.5kΩ. In this region, the induced current as well as the recovered power are higher for the active pendulum (Fig. 8). This means that the activation of the pendulum can improve EH. The maximum recovered power is 2.7mW for active PVA, and about 5mW for inactive PVA. Configuration no. 2 is the best from the EH point of view. Note that the PVA usually works in the low frequency region. Therefore, this result is very important. For higher load resistances, the PVA activation reduces the EH as much 50%. The third configuration is the least promising (Fig. 9). The maximal recovered energy is 0.36mW, or 0.06mW for the activated PVA. In comparison with the most effective coil setting (configuration no. 2), the recovered energy is even fifteen times smaller. As it can be seen, the load resistances strongly influence on the recovered energy level, especially if the PVA works. However, the load resistance practically not influences on the vibration mitigation (the main system and pendulum oscillations). The relationship is clearly shown in Figs. 10a and b, obtained for parameters: frequency 2.25Hz, amplitude 0.03m of the excitations, and the load resistances 1.15kΩ. The exemplary time series for the three configurations of the magnet - coil configuration are shown in Fig. 11. The results for the inactive PVA are shown in Fig. 11a, while the results for the active PVA are presented in Fig. 11b. The main difference between both figures is that activation of the PVA increases the period of magnet vibration.

218


(a)

(b)

Fig. 11 Time series for the three magnet-coil configurations: (a) the inactive PVA, (b) the PVA active. The parameters are: frequency 2.25Hz and load resistances 1.15kΩ.

4 Conclusions and final remarks This paper presents experimental investigations of the pendulum vibration absorber applied for energy harvesting. Originally, the system is dedicated to the vibration mitigation by the energy transfer from the main system to the pendulum. However, the special construction of the pendulum by adding the electromagnetic harvester device allows for energy recovery. The harvester system does not influence the effectiveness of the vibration mitigation. The proper tuning of the electromagnetic system can change the effective energy harvesting level. The quasi-static analysis of the harvester device shows a strongly non-linear electromechanical coupling between the coil and magnet. The maximal value of the current is reached close to the ends of the coil. The experimental work in this paper investigates the harvester design parameters: the magnet position and the load resistance. Measurements of different configurations have been performed in order to investigate the induced power response and to optimize the harvester design. The most promising configurations is when the magnet’s centre is near the coil’s end. The maximal recovered energy is about 5mW for an inactive PVA and about 3mW for an active PVA. Additionally, in low frequency range, the PVA at configuration no. 2 can increase in energy recovery. Therefore, the position of the coil plays important role in design optimization. The load resistances influence on the recovered energy level, especially if the PVA is active and not influence on the vibration mitigation. The next step will be to analyze different configuration of the stack set of the magnets, the stack set of the magnets with stoppers, and a set of a coils.

References [1] International Energy Agency (IEA) (2012), Energy Policies of IEA Countries: The United Kingdom, 2012 Review, Paris: IEA Publications. [2] Judkins, R.R., Fulkerson, W., and Sanghvi, M.K. (1993), The dilema of fossil fuel use and global climate change, Energy & Fuels, 7, 12-22. [3] Sorrell, S. (2015), Reducing energy demand: A review of issues, challenges and approaches, Renewable and Sustainable Energy Reviews, 47, 74-82. [4] Saha, C.R., O’Donnell, T., Wang, N., and McCloskey, P. (2008), Electromagnetic generator for harvesting energy from human motion, Sensors and Actuators A, 147, 248-253.


219

[5] Chen, Z., Guo, B., Yang, Y., and Cheng, C. (2014), Metamaterials-based enhanced energy harvesting: A review, Physica B Condensed Matter, 438, 1-8. [6] Klub, H. (2011), High efficient micro electromechanical capacitive transducer for kinetic energy harvesting, Ph.D. Dissertation, Albert Ludwigs University Freiburg. [7] Joyce, S. (2011), Development of an electromagnetic energy harvester for monitoring wind turbine blades, MSc Thesis, Virginia Tech, Blacksburg. [8] Bebby, S., Tudor, M., and White, N. (2006), Energy harvesting vibration sources for microsystems applications, Measurement Science and Technology, 17, R175-R195. [9] Jonnalagadda, A. (2007), Magnetic induction systems to harvest energy from mechanical vibrations, Massachusetts Institute Engineering, Massachusetts. [10] Williams, C.B. and Yates, R.B. (1996), Analysis of a micro-electric generator for microsystems, Sensors and Actuators A, 52(1-3), 8-11. [11] Stephen, N.G. (2006), On energy harvesting from ambient vibration, Journal of Sound and Vibration, 293(12), 409-425. [12] Yildirim, T., Ghayesh, M.H., Li, W., and Alici, G. (2017), A review on performance enhancement techniques for ambient vibration energy harvesters, Renewable and Sustainable Energy Reviews, 71, 435-449. [13] Owens, B.A.M. and Man, B.P. (2012), Linear and nonlinear electromagnetic coupling models in vibrationbased energy harvesting, Journal of Sound and Vibration, 331, 922-937. [14] Olaru, R., Gherca, R., and Petrescu, C. (2014), Analysis of a vibration energy harvester using permanent magnets, Revue Roumaine Des Sciences Techniques, 52(2), 131-140. [15] Mann, B.P. and Sims, N.D. (2009), Energy harvesting from the nonlinear oscillations of magnetic levitation, Journal of Sound and Vibration, 319(1-2), 515-530. [16] Kecik, K., Mitura, A., Lenci, S., and Warminski, J. (2017), Energy harvesting from a magnetic levitation system, International Journal of Structural Stability and Dynamics, 94, 200-206. [17] Saravia, C.M., Ramirez, J.M., and Gatti, C.D. (2017), A hybrid numerical-analytical approach for modeling levitation based vibration energy harvesters, Sensors and Actuators A, 257, 20-29. [18] Al-Halhouli, A., Kloub, H.A., Wegnerc, M., and Buttgenbach, A. (2015), Design and experimental investigations of magentic energy harvester at low resonance frequency, International Journal of Energy and Environmental Engineering, 10(1), 69-78. [19] Kecik, K., Mitura, A., Sado, D., and Warminski, J. (2015), Magnetorheological damping and semi-active control of an autoparametric vibration absorber, Meccanica, 49(8), 1887-1900. [20] Kecik, K. and Warminski, J. (2011), Dynamics of an autoparametric pendulum-like system with a nonlinear semiactive suspension, Mathematical Problems in Engineering, Article ID 451047, 1-15. [21] Earnshaw, S. (1842), On the nature of the molecular forces which regulate the constitution of the luminiferous ether, Transactions of the Cambridge Philosophical Society, 7, 97-112. [22] Soares, S.M.P., Ferreira, J.A.F., Simoes, J.A.O., Pascoal, R., Torrao, J., Xue, X., and Furlani, E.P. (2016), Magnetic levitation-based electromagnetic energy harvesting: a semi-analytical non-linear model for energy transduction, Scientific Reports, 6, Article ID 18579, 1-9. [23] Kecik, K., Mitura, A., Warminski, J., and Lenci, S. (2018), Foldover effect and energy output from a nonlinear pseudo-maglev harvester, AIP Conference Proceedings, 1922(1), 1-7. [24] Kecik, K., Brzeski, P., and Perlikowski, P. (2017), Non-Linear dynamics and optimization of a harvesterabsorber system, International Journal of Structural Stability and Dynamics, 17(5), 1-15.



Huge Size Structure Damage Localization and Severity Prediction: Numerical Modeling, Simulation and SVM Regression Method Gang Jiang1 , Yiming Deng2†, Lili Liu3 , Canghai Liu1 , Zihong Liu1 , Yong Jiang1 1

2

3

Manufacturing Process Testing Key Lab of the Ministry of Education, Southwest University of Science and Technology, Mianyang, Sichuan, 621010, China Nondestructive Evaluation Laboratory, Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA School of Material Science and Engineering, Southwest University of Science and Technology, Mianyang, Sichuan, 621010, China Submission Info Communicated by S.C. Suh Received 28 June 2017 Accepted 17 August 2017 Available online 1 October 2018 Keywords Structure nondestructive testing (SNNT) Structural health monitoring (SHM) PROE modeling and ADAMS simulation Support vector machine (SVM) Pattern recognition

Abstract In systematic identification for real bridge engineering structural damage evaluation, lack of “negative samples” is the main reason for misjudgment and false classification. Aimed at this problem, the paper proposed a new kinetic-parameters-analysis-based method of damage identification for bridge structures, using both numerical simulation and real experiments under controlled lab conditions. PROE and ADAMS were adopted to build structural models and integrate with simulation experiments under virtual force, frequency response data were gathered and used as “simulation datasets”. In real structural experiments, accelerators and advanced signal acquisition equipments were used to collect signals from real structures hit by real force. Signals gathered by equipments were used as “real datasets”, corresponded with “simulation datasets”. After that, features were extracted from these two kinds of datasets. Authors found that numerical simulation models were not always accurate, while real model had its own advantages and disadvantages. To fill this gap, relationships between simulation and actual measurements were investigated in this paper. Finally, Support Vector Machine (SVM) method was used to perform pattern recognition experiments and showed its good performance on structure damage identification. The proposed method is scalable and can be extended to a bigger structure, such as the entire bridge faults diagnosis. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Joints and beams are main parts of a suspension bridge. They will bear complex vibration and heavy loads. It is necessary to acquire accurate and real-time health status of them so as to prevent failure † 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.2018.09.003

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Gang Jiang et al. / Journal of Vibration Testing and System Dynamics 2(3) (2018) 221–237

Fig. 1 Bridge disasters: I-35 Bridge in USA, JIUJIANG Bride in China, SANGSU Bridge in Korea, Quebec Bridge in Canada.

and prolong their service lives. During long-term service period, joints loose and beams damage will lead to bridge broken disasters. In 2011, the I-35W Bridge in Mississippi, USA, collapsed [1]. It’s not the only disaster in the world, there are many other famous bridge accidents, such as Quebec Bridge in Canada, collapsed in 1916, Florida-Ohio Kanauga Bridge, in USA broken in 1967, Seoul SANGSU Bridge in Korea, fallen in 1994, JIUJIANG Bridge in Guang Dong province, China, collapsed in 2007. Structural Health Monitoring (SHM) for bridges has been widely researched. Many scholars focused on evaluation practices and procedures utilized in bridge health status inspection [2]. Generally speaking, there are two mainly kinds of techniques used to evaluate the condition of bridge structures: (1) Micro computational mechanics technique, which was called flexibility-based approach. (2) Macro dynamics technique, which was called vibration-based approach. (1) Flexibility-based approach. In earlier research, Zhao and Dewolf [3] found modal flexibility was very sensitive to structure damage and used it as damage descriptor. Pandey and Biswas [4] researched a damage prediction method based on modal flexibility changes. Earlier study of Toksoy and Aktan [5] showed a promising technique and do some experiments on a highway bridge. Zhang [6] and Parketal [7] tried to do some works on the uniform load surface (ULS). Catbas [8] suggested that curvature has some advantages after compare analysis. Bernal [9] found Damage locating vector (DLV) approach can get zero magnitudes. Gao [10] used DLV to do some pattern recognition experiments on truss structure. Bernal [11] proposed a Stochastic-Dynamic-DLV(SDDLV) technique after compare analysis. Koo [12] finished some development works based on modal flexibility. For thin plate structures, Kazemi [13] proposed a two-phase method. Sung [14] developed a damage-induced inter-storey deflection (DIID) method. Flexibility-based approach is useful but several shortcomings also. It has no obvious relationship between damage and damage features, has noise vulnerable characteristic, and need an intact finite element model. We have to solve these problems first, which are very difficult and complex. (2) Vibration-based approach. The vibration-based approach was called Vibration-Based Damage Detection (VBDD) methods, which has been widely applied by using various vibration features, such as natural frequencies, modal damping ratio, and modal flexibility. Farrar, Doebling and Nix [15] explained that changes in dynamic responses can be expected to occur in the time domain as well as frequency domain. W. Fan and P. Qiao [16] compared VBDD with Non-Destructive Testing (NDT) methods and found its advantages in localization of unknown damage. Rucka [17] studied a kind of cantilever damage detection method based on wavelet technique. In order to identify multiple cracks, Moradi [18] developed an evolutionary technique. Nassar [19] investigated natural frequency effects on structure features. Chen [20] studied crack characters position and their effects. In earlier research work of Lee [21], analyzing the effects of some Eigen-characters of cantilever crack were included. Seyedpoor [22] adpoted a two-stage technique in structure multiple damages identification experiment. Naseralavi [23] calculated the damage position and extents of a cantilever beam.


223

However, we found that most existing research works were mainly studied by numerical simulation and lab experiments. A few reports describe their practical validity for actual bridges, which are likely to be subject to budget limitations and service conditions which prevent the relevant authorities from granting permission to apply damage to the bridge. The reason is that actual bridge building costs were too expensive for common scholar researchers to do destructive tests. Without special permission, nobody could change any bridge’s screws tightness degrees, cut a wound on a beam, or take off a cantilever, so as to make the bridge falling down and obtain “negative” samples of an actual collapse or broken bridge. Without “negative” samples, any research efforts on dataset completely consisted of “positive” samples, would be meaningless. As a result, theory and practice are out of touch, research work of pure numerical simulation or under simple lab conditions has few real effects in actual bridge structure health monitoring. The objective of this paper was to find a way to connect virtual simulation model with an actual bridge, so as to get “positive” samples as well as “negative” samples more similar with actual data of bridges under numerical simulation and lab conditions. Main contents of this paper were organized as: (1) provided a 3-dimension model of an actual bridge by PROE software. (2) built a experiment objective by reference of a part of bridge structures and did actual vibration tests. (3) created a numerical model by ADAMS software and did simulation vibration tests. (4) found the relationship between actual model and simulation model, so as that we could get “negative” samples similar with an actual bridge by building virtual model with PROE-ADAMS method. (5) finally, in order to prove the effectiveness of our method, we used Support Vector Machine (SVM) to do pattern recognition tests at the end of our experiments.

2 Pattern recognition: support vector machine (SVM) Support Vector Machine was proposed to solve 2-class classification originally. The most important concept is the Hyper Plane [24, 25]. SVM is a hyper plane learning algorithm. In order to apply SVM learning algorithms, we need to find a kind of function whose capacity can be computed firstly. Give a hyper planes defined as w · x ) + b = 0, (w

w ∈ R N , b ∈ R.

(1)

We can design a decision functions as follows: w · x ) + b]. f (xx) = sgn[(w

(2)

The optimal hyper plane in 2-dimension space can be shown as figure 2. To create optimal hyper plane, maximization of distance between two decision planes is needed. 1 w||2 ), w)] = min( ||w min[T (w 2 w · x ) + b] ≥ 1, i = 1, 2, ..., n. s.t. yi · [(w

(3)

The optimization problem with inequation constraints can be solved by Lagrange multipliers method. Give Lagrange formulation as 1 w||2 − ∑ αi {yi · [(xxi · w ) + b] − 1}. w, b, α ) = ||w L(w 2 i=1

(4)

Based on saddle point theory, the derivatives of formula L with respect to the primal variables must

224


Fig. 2 Optimal Hyper plane (2-dimension) of SVM.

be vanished, which can be expressed as follow: ⎧ n ∂ L(α , b, w ) ⎪ ⎪ = w − yi αi x i = 0, ⎪ ∑ ⎨ ∂w i=1 n ⎪ ∂ L(α , b, w ) ⎪ ⎪ = ∑ yi αi = 0. ⎩ ∂b i=1

(5)

⎧ n ⎪ ⎪ w = ⎪ ∑ yi αi x i , ⎨

Leads to:

i=1

(6)

n ⎪ ⎪ ⎪ ⎩ ∑ yi αi = 0. i=1

The solution vector thus has an expansion in terms of a subset of the training patterns. Those patterns whose Lagrange multipliers are non-zero were called Support Vectors (SVs). By using Karush-Kuhn-Tucker complementarily conditions (KKT conditions, which can be found in Vapnik [24], Cristianini [25]

αi · {yi · [(xxi · w ) + b] − 1} = 0,

i = 1, 2, ..., n.

(7)

Those Support Vectors lie on the margin. All remaining examples of the training sets are irrelevant: they do not play any role during optimization process. By substituting formula (6) into formula (4), we can eliminate the primal variables and get the Wolfe dual optimization problem. We aim at finding multipliers which can satisfy conditions as follows: n

max[W (α )] = max[ ∑ αi − i=1

1 n αi α j yi y j (xi · x j )], 2 i,∑ j=1

n

s.t.

(8)

∑ αi yi = 0,

i=1

αi ≥ 0, i = 1, 2, ..., n. Using formula (7), parameter b can be calculated. After that, the hyper plane decision function can be written as follows: n

f (xx) = sgn[ ∑ αi yi · (xx · x i ) + b]. i=1

(9)


225

Fig. 3 Map input space to a feature space and convert nonlinear model to a linear model.

To solve nonlinear problem, we need a transform function Φ, to map input space into a feature space, whose dimension maybe higher than original space, as figure 3 shows. If Φ satisfies Mercer condition [24], it can be used as a kernel function K (xx · x i ), which can substitute (xx · x i ). Kernel functions often used are linear kernel, polynomial kernel, radial basis function kernel and sigmoid kernel, as follows: K(xx, x i ) =< x · x i >, K(xx, x i ) = (< x · x i > +1)d , K(xx, x i ) = exp −((xx − x i )/σ )2 , K(xx, x i ) = tanh(v < x · x i > +c).

(10) (11) (12) (13)

Salvador [26] developed an online for SHM method based on SVM technique. Xiaoma [27] used Least Square Support Vector Machine (LS-SVM) to do damage localization experiments. Rivera-Castillo [28] proposed a SHM approach based on optical scanning data and SVM method. After building a timevarying Auto-Regressive Moving Average (ARMA) model, ZHANG [29] developed a structural damage identification approach, and used SVM to find structure damage positions.

3 Experiments and discussion FOUR-RIVER Bridge in China is now the highest bridge in the world (The distance from deck to river is 560 meter), as well as the longest mountain suspension bridge of the world (1365 meter). We tried to do some research work on FOUR-RIVER Bridge’s structure by using actual lab experiments structure and virtual simulation model designed by PROE and ADAMS software. 3.1

PROE modeling

Based on engineering drawings (total number is 119) and parameters calculation forms, we designed a 3D model of engineering parts and whole bridge structure by PROE software (wildfire 5.0 edition). The bridge model was too big for us to do simulation directly, as well as that we could not do real destructive test on the actual bridge, such as cutting a wound on a beam, loosen or taking off some screws to make the bridge fall down, and so on, so as to collect “negative” samples data. So, it seemed that there’s no way for us to build the connection between actual bridge and 3D model. How to build the connection and to find the mapping relationship between “virtual” and “actual”, so that we could get more similar results and deviation as little as possible by putting theory research method into actual bridge testing practice?

226


Fig. 4 FOUR-RIVER Bridge in China.

Fig. 5 Construction process of FOUR-RIVER Bridge.

Fig. 6 3D model designed by PROE. Table 1 Parameters of SD14N14 accelerometer.

3.2

Parameter No.

Parameter Name

Parameter Value

1

Sensitivity (mV/g)

100

2

Dynamic Scope (g)

50

3

Low frequency response (Hz)

0.3

4

High frequency response (Hz)

10000

5

Resonance frequency (Hz)

30000

6

Accuracy (g)

0.0002

Actual and virtual model of a structure part

We selected a typical structure part of the whole bridge to do contrast experiment between virtual simulation model and actual structure. Once the relationship between them could be found, regularities and rules of their features could be analysed out also. We can always find a way to generalize our method to a bigger structure part and even to the whole actual bridge. We manufactured an actual experiment structure using stainless steel pipes, based on precise size parameters of 3D model in PROE software. After that, we installed three sensors on JOINT1 position so as to collect x, y, z-direction vibration signals of that point. The sensors were SD14N14. It’s a high sensitive vibration accelerometer which could be used to gather signals in frequency band from 0.3 to 10000 Hz. Features of SD14N14 were listed in Table 1.


227

Joint2 Pole2 Beam1 Joint1 Hit point

Fig. 7 Truss structure of suspension bridge(Real size model).

3.3

Beam2

Pole1 Wound

Fig. 8 A typical structure 3D model of truss beam.

Actual structure vibration test

We built up an experiment system, used a steel ball as the hit force by rolling down from a slope designed by a smooth pipe. The pipe can be used to restrict the ball’s rolling path and direction, so as to control the hit force magnitude and direction as well as the hit point accurately. We used an ECON PREMAX instrument as the vibration signal collector, and used a computer to save original data so that we can use these data to do simulation analysis by MATLAB software in the next experiment stage. Sampling frequency of ECON PREMAX 1000 collector is 12000Hz. Based on Nyquist-Shannon sampling theory, it can be used for mechanical high frequency vibration signal acquisition whose frequency is lower than 6000Hz. A steel ball was utilized in our actual hit experiment as the hit force source by rolling down from our pipe-slope. Based on classical physical knowledge: 1 2 mv = mgh. 2 Where:

(14)

m — steel ball’s mass, m = 3kg; v — steel ball’s velocity; g — acceleration of gravity; h — height of steel ball rolling down; F · dt = mv,

(15)

Where:

F — hit force; dt — time of steel ball hit the objective; We found that dt = 0.25 seconds in our experiment system by using high-speed camera. This parameter would be applied in numerical simulation in ADAMS in the next experiment stage. It’s easy to calculate the relationship between F and height and control the hit force by adjusting the height of steel ball rolling down by the following formula, h=

1 Fdt 2 ( ) . 2g m

(16)

In our experiment, we found that when F > 0 Newton force, the objective was easily to be damaged by the steel ball. So, hit force F was fixed at 20N in actual experiment as well as in ADAMS numerical simulation. We did four kinds of tests as follows, each kind with 10 times. A dataset with “4–by-10” size was built successfully.

228


Sensor X

Joint2

Sensor Y

Pole2

Sensor Z Joint1 Hit point Pole1

Beam1

Wound

Beam2

Fig. 9 Actual experiment objective and steel ball used as hit force.

ECON Premax Signal collector Computer (save data)

Experiment objective

Pipe slope (ball rolling down)

Fig. 10 Experiment system.

Fig. 11 ECON PREMAX 1000 signal collector.

(1) Health status (No. 1-10 samples of dataset, classification label was “1”): the objective structure was health, without any wounds or bolts loose. (2) Bolts loose (No. 11-20 samples of dataset, classification label was “2”): bolts and screws were loose at Joint1 and Joint2 position. (3) Wound status (No. 21-30 samples of dataset, classification label was “3”): there was a wound at the position, as Figure 9 shows. (4) Compound damage (No. 31-40 samples of dataset, classification label was “4”): there was a wound at that position as well as bolts loose at two Joints. For the four kinds status listed above, time domain signal of x, y, z-direction collected by ECON PREMAX could be plotted by MATLAB software. We also used MATLAB to distil vibration frequency of these signals by FFT algorithm. Time domain signals and frequency domain features of some samples were shown as follows:


229

Fig. 12 Actual structure’s signal feature in time domain and frequency domain (No.3(health), No.15(loose), No.22(wound), No.37(loose+ wound) samples in dataset).


230

Fig. 12 Continued.

Beam1

Joint2

Pole2 Joint1 Hit point Pole1 Wound Beam2

Ground

Fig. 13 ADAMS model with markers and joints.

3.4

Fig. 14 Rigid body transformed into flexible body.

ADAMS simulation

After actual structure vibration test experiment, 3D model built in PROE software was imported into ADAMS software for dynamic analysis. We added markers (ADAMS special term, which generally means connect points or force act points), and established constraint relations among structure parts by joints. Each joint’s lock status could be adjusted separately so as to simulate various structure loose situations. As figure 13 shows. For pole1, pole2, beam1 and beam2, we set materials as stainless steel, which was the same as actual structure experiment objective. For each part, set parameters as table 2 and transformed these rigid body into flexible body. It’s very important to set parameters carefully, because incorrect parameter settings would result in incorrect results which could not match the actual structure well, even failed to make the objective flexible. Figure 14 showed the results of completely transform of all parts.


231

Table 2 Transform parameters of flexible body.

Material

Pole1

Pole1(wound)

Pole2

Beam1

Beam2

Stainless

Stainless

Stainless

Stainless

Stainless

Mass(kg)

0.01374

0.01367

0.01374

0.3931

1.1078

Node count

370

370

370

622

3453

Mode count

24

30

24

24

48

Elem. Type

Solid

Solid

Solid

Solid

Solid

Elem. Shape

Tetrahe-dral

Tetrahe-dral

Tetrahe-dral

Tetrahe-dral

Tetrahe-dral

Elem. Order

Linear

Linear

Linear

Linear

Quadra-tic

Edge Shape

Straight

Straight

Straight

Straight

Mixed

Elem. Size

20mm

20mm

20mm

20mm

10mm

Min Size

5mm

5mm

5mm

5mm

0.2mm

Growth Rate

1.5

1.5

1.5

1.5

1.5

Ang. / Elem.

45

45

45

45

45

Shell Thick

1.0mm

1.0mm

1.0mm

1.0mm

1.0mm

Gave a pulse force F at the “Hit Position” shown in Figure13. Force F could be defined as follow formula based on actual structure parameters. F = step(time, 0, 20, 0.25, 0)

(17)

In formula (17), “20” meant that pulse force peak value was 20 Newton, “0.25” meant that total time of force acting on objective position was 0.25 seconds. Three acceleration components of joint1 could be calculated and saved to data file, which could be used to do numerical analysis in MATLAB software. By adjusting status of joint1 and joint2 and selecting pole1 from health to wound, we finished simulation under similar situation of actual structure experiment: health, loose, wound, compound (loose + wound) , as figure 15 shows. Contrast experiments were applied between actual structure experiment and ADAMS virtual simulation. Defined relative error as: (18) E = (P − P )/P ∗ 100%, Where:

P—Actual FFT value; P —ADAMS model’s FFT value; E—Relative error value of each FFT dot. We found a few rules in experiments: (1) Health and loose status: ADAMS simulation results was more similar with actual structure experiments than wound and compound damage status. Dot number (error> 5% or >10%) of the former was about 10% of the latter. (2) Wound and compound status: the difference between ADAMS simulation results and actual structure experiments mainly existed on the lower frequency band. (3) All of four statuses: relative errors between ADAMS simulation results and actual structure experiments were very small, about 99.85% FFT value errors were lower than 5%. There were several reasons for rules listed above: (1) Material difference: Actual structure was not produced by pure stainless steel, maybe it contained some impurities. However, simulation model was set as stainless steel in ADAMS. (2) Machining errors: As a kind of mechanical parts, it was inevitable that some errors existed in actual structures, such as the hole drilling, wall thickness of steel tube, welding procedures, screws and blots tightness, and so on.

232


Fig. 15 ADAMS simulation model’s signal feature in time domain and frequency domain(health, loose, wound, loose+ wound).


233

Fig. 15 Continued.

(3) Connecting fixed-parts vibration: We must fixed our experiment object on an immovable part, such as a table, a heavy board, a cantilever, and so on. Actually, all these “immovable” parts were not really “immovable”. They vibrated also. As a result, their vibration frequency were collected by our ECON PREMAX signal collector. However, in ADAMS software, it’s well known that the GROUND part was really “immovable” in simulation tests. (4) Parameter settings: Maybe our real structure geometric parameters were not total equal with virtual simulation model in ADAMS. 3.5

Pattern recognition based on SVM

Our pattern recognition experiments were applied on a computer, with MATLAB R2014a edition software, 32-bits Microsoft Windows 7 version system, 4GB DDR memory and Intel i7-4702MQ 2.20 GHz CPU. The SVM classification algorithm introduced in section 2 was programmed in MATLAB language. Original data were three components (Fx, Fy, Fz) of FFT results of actual structure experiment signals collected by ECON PREMAX. It was a 60000 –by-3 matrix. Due to the symmetry of FFT algorithm, we only need the front half of the matrix, which dimension was 30000-by-3. So we had 40 matrices of 30000-by-3 size. In order to use these 30000-by-3 matrices as input data, we must transform each of them into a vector. Each sample, which was also a input vector in SVM algorithm, was organized as follows: x(i) = [F F x(1 : j), F y(1 : j), F z(1 : j), Label]

(19)

Where: i = 1, 2, . . . , 40, j = 30000. It’s a row vector with 90001 elements. Dataset was organized as: X = [xx(1); x (2); . . . ; x (40)]

(20)

It’s a matrix, row = 40, column = 90001. Normalize SVM input data is essential. Gave a normalized equation defined as follows: X NE =

vmax − vmin ) × (x − xmin + vmin . xmax − xmin

(21)

234


Fig. 16 Relative error distribution of actual structure and ADAMS simulation model (health, loose, wound, loose+ wound).

Generally speaking, normalize all input data in [−1, 1] was a good choice. So parameters in formula (37) should be set as vmax = 1, vmin = 1. Selected radial basis function as kernel function: (22) K(xx, x i ) = exp −((xx − x i )/σ )2 Selected C-SVC as classification model, parameter C = 1. Set kernel parameter σ = 1, ε = 0.028, tolerance of termination criterion δ = 0.001. We selected 30 samples from dataset randomly, and used them as training samples. Another 10 samples were testing data. They were No. 1, 3, 7, 12, 16, 25, 27, 32, 36, 39 samples in original dataset. Training output results shown that there were 20 Support Vectors (SVs), contained 10 border SVs. Based on training model composed of these parameters, we did prediction experiments, tried to forecast testing samples’ class labels, i.e. structure health status. Predicted labels of testing samples were [1, 2,1, 2, 2, 3, 3, 4, 4, 4], with 90% accuracy. Pattern recognition experiments shown good performance of SVM in structure health status diagnosis application. How to improve classification accuracy is our next step task.


235

Table 3 Error distribution analysis (40 samples of dataset, contain health, loose, wound, loose+ wound). Sample No.

Class Label

Damage Status

Total Dot

Error >5% Dot

Error >10% Dot

Number

Percent (%)

Number

Percent (%)

1

1

Health

90000

15

0.0167

3

0.0033

2

1

Health

90000

14

0.0156

3

0.0033

3

1

Health

90000

18

0.0200

5

0.0056

4

1

Health

90000

14

0.0156

0

0

5

1

Health

90000

14

0.0156

2

0.0022

6

1

Health

90000

16

0.0178

5

0.0056

7

1

Health

90000

19

0.0211

6

0.0067

8

1

Health

90000

21

0.0233

4

0.0044

9

1

Health

90000

15

0.0167

1

0.0011

10

1

Health

90000

14

0.0156

2

0.0022

11

2

Loose

90000

27

0.0300

1

0.0011

12

2

Loose

90000

23

0.0256

4

0.0044

13

2

Loose

90000

27

0.0300

4

0.0044

14

2

Loose

90000

28

0.0311

4

0.0044

15

2

Loose

90000

30

0.0333

10

0.0111

16

2

Loose

90000

24

0.0267

3

0.0033

17

2

Loose

90000

30

0.0333

4

0.0044

18

2

Loose

90000

29

0.0322

5

0.0056

19

2

Loose

90000

27

0.0300

3

0.0033

20

2

Loose

90000

22

0.0244

4

0.0044

21

3

Wound

90000

230

0.2556

61

0.0678

22

3

Wound

90000

245

0.2722

71

0.0789

23

3

Wound

90000

245

0.2722

69

0.0767

24

3

Wound

90000

220

0.2444

66

0.0733

25

3

Wound

90000

259

0.2878

70

0.0778

26

3

Wound

90000

259

0.2878

75

0.0833

27

3

Wound

90000

212

0.2356

67

0.0744

28

3

Wound

90000

245

0.2722

79

0.0878

29

3

Wound

90000

254

0.2822

70

0.0778

30

3

Wound

90000

251

0.2789

73

0.0811

31

4

Compound

90000

143

0.1589

49

0.0544

32

4

Compound

90000

136

0.1511

50

0.0556

33

4

Compound

90000

137

0.1522

49

0.0544

34

4

Compound

90000

151

0.1678

44

0.0489

35

4

Compound

90000

138

0.1533

50

0.0556

36

4

Compound

90000

140

0.1556

53

0.0589

37

4

Compound

90000

139

0.1544

42

0.0467

38

4

Compound

90000

137

0.1522

46

0.0511

39

4

Compound

90000

137

0.1522

46

0.0511

40

4

Compound

90000

138

0.1533

52

0.0578

236


4 Conclusion This paper proposed a kind of technique named as PROE-ADAMS-SVM for bridge structure joints loose and poles damage identification. It was outlined firstly that theory studies would normally be inaccessibly because of lacking of “Negative samples”. Authors decided to use a part of an actual bridge to do contrast experiment between actual structure and virtual simulation model. Simulation model was built by PROE and ADAMS software. Vibration signals of actual structure were collected by SD14N14 accelerators and ECON PREMAX signal collector. Virtual vibration was applied in ADAMS and vibration signals were calculated and saved by ADAMS. Authors distilled frequency of these two kinds of vibration signals, analyzed their difference, and found some error distribution rules and analyzed reasons of these errors. Finally, a 40-by-90000 size sample datasets was built up, and Support Vector Machine was adopted to do pattern recognition experiments on actual structure’s health status: Health, Loose, Wound, Compound damage (Loose + Wound). Experiment results showed that PROE-ADAMS-SVM method was useful and applicable for structure damage diagnosis, and could be extent to bigger parts even a whole huge suspension bridge’s structure damage analysis.

Acknowledgments Authors would like to thank the supports from Innovation Team Key Fund of Sichuan Provincial Department of Education (16TD0016) and Key Project of Science and Technology Plan of Mianyang City (KJ20170507).

References [1] Alampalli, S. and Rehm, K.C. (2011), Impact of I-35W bridge failure on state transportation agency bridge inspection and evaluation programs, Structures Congress, 1019-1026. [2] Wang, T., Celik, O., and Catbas, F.N. (2016), Damage detection of a bridge model based on operational dynamic strain measurements, Advances in Structural Engineering, 19(9), 1379-1389. [3] Zhao, J. and Dewolf, J.T. (1999), Sensitivity study for vibration parameters used in damage detection, Journal of Structural Engineering, 125, 410-416. [4] Pandey, A.K. and Biswas, M. (1994), Damage detection in structures using changes inflexibility, Journal of Sound and Vibration, 169, 3-17. [5] Toksoy, T. and Aktan, A.E. (1994), Bridge condition assessment by modal flexibility, Experimental Mechanics, 34, 271-278. [6] Zhang, Z. and Aktan, A.E.(1998), Application of modal flexibility and its derivatives in structural identification, Research in Nondestructive Evaluation, 10, 43-61. [7] Park, H., Koo, K., and Yun, C. (2007), Modal flexibility-based damage detection technique of steel beam by dynamic strain measurements using FBG sensors, Steel Structures, 7, 11-18. [8] Catbas, F.N., Gul, M., and Burkett, J.L. (2008), Damage assessment using flexibility and flexibility-based curvature for structural health monitoring, Smart Materials and Structures, 17, 1-12. [9] Bernal, D. (2002), Load vectors for damage localization, Journal of Engineering Mechanics, 128, 7-14. [10] Gao, Y., Spencer, B.F., and Bernal, D. (2007), Experimental verification of the flexibility-based damage locating vector method, Journal of Engineering Mechanics, 133, 1043-1049. [11] Bernal, D. (2004), Load vectors for damage location in systems identified from operational loads, Journal of Engineering Mechanics, 136, 31-39. [12] Koo, K., Lee, J., Yun, C., and Kim, J. (2008), Damage detection in beam-like structures using deflections obtained by modal flexibility matrices, Smart Structures and Systems, 4, 605-628. [13] Kazemi, S., Rahai, A.R., Daneshmand, F., and Fooladi, A. (2011), Implementation of modal flexibility variation method and genetically trained ANNs in fault identification, Ocean Engineering, 38, 774-781. [14] Sung, S.H., Koo, K.Y., and Jung, H.J. (2014), Modal flexibility-based damage detection of cantilever beamtype structures using base line modification, Journal of Sound and Vibration, 333, 4123-4138. [15] Farrar, C.R., Doebling, S.W., and Nix, D.A. (2001), Vibration-based structural damage identification, Philo-


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sophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences, 359, 131-149. [16] Fan, W. and Qiao, P. (2011), Vibration-based damage identification methods: a review and comparative study, Structural Health Monitoring, 10(1), 83-111. [17] Rucka, M. (2011), Damage detection in beams using wavelet transform on higher vibration modes, Journal of Theoretical and Applied Mechanics, 49(2), 399-417. [18] Moradi, S.H. and Kargozarfard, M. (2013), On multiple crack detection in beam structures, Journal of Mechanical Science and Technology, 27, 47-55. [19] Nassar, M., Matbuly, M.S., and Ragb, O. (2013), Vibration analysis of structural elements using differential quadrature method, Journal of Advance Research, 4, 93-102. [20] Chen, L.H., Sun, Y., and Zhang, W. (2012), Study of vibration characteristics of cantilever rectangular plate with side crack, Applied Mechanics and Materials, 226-228(5), 113-118. [21] Lee, H.L. and Chang, W.J. (2012), Dynamic response of a cracked atomic force microscope cantilever used for nanomachining, Nanoscale research letters, 7, 131-137. [22] Seyedpoor, S.M. (2012), Two stage method for structural damage detection using a modal strain energy based index and particle swarm optimization, International Journal of Non-Linear Mechanics, 47(1), 1-8. [23] Naseralavi, S.S., Fadaee, M.J., and Salajegheh, J. (2011), Subset solving algorithm: a novel sensitivity-based method for damage detection of structures, Applied Mathematical Modeling, 35(5), 2232-2252. [24] Vapnik, V.N. (1998), Statistical Learning Theory, New York: Wiley Publishing House. [25] Cristianini, N. and Taylor, J.S. (2000), An Introduction to Support Vector Machines and Other Kernel-based Learning Methods, England: Cambridge University Press. [26] Villegas, S., Li, X.O., and Yu, W. (2015), Detection of building structure damage with support vector machine, 12th International Conference on Networking, Sensing and Control, 619-624. [27] Dong, X., Wei, B., Sun, Q., and Hou, X. (2010), Investigation on damage self-diagnosis of fiber smart structures based on LS-SVM, 2nd IEEE International Conference on Information Management and Engineering, 626-638. [28] Rivera-Castillo, J., Rivas-López, M., Nieto-Hipolito, J.I., and et al. (2014), Structural Health Monitoring based on Optical Scanning Systems and SVM. 23rd International Symposium on Industrial Electronics (ISIE), 1961-1966. [29] Zhang, X.Z., Yao, W.J., and Tian, F. (2012), Structural damage identification based on time-varying arma model and support vector machine, Journal of Basic Science and Engineering, 21(6), 1094-1102.



A Low Cost Device for Excessive Vibration Detection in Electric Motors M. Sundin1 , A. Babaei2 , S. Paudyal2 , C. Yang2†, N. Kaabouch1 1 2

Department of Electrical Engineering, University of North Dakota, Grand Forks, ND 58202-7165, USA Department of Mechanical Engineering, University of North Dakota, Grand Forks, ND 58202-8359, USA Submission Info Communicated by S.C. Suh Received 1 January 2018 Accepted 25 February 2018 Available online 1 October 2018 Keywords Fault diagnosis Smart systems Embedded system Signal processing Vibration measurement Condition monitoring

Abstract Health monitoring and fault diagnosis are essential to ensuring reliable operation of machinery in industry. In this paper, we present the design of a low cost device for sensing mechanical vibrations and detecting excessive vibration. Advantages of the device include cost effectiveness and simplicity of the design. Piezoelectric-based sensor, light emitting diodes, resistors, and liquid crystal display, and an Arduino board are the main components of the device. The effectiveness of designed device was tested on simple electric motors, such as fan and drill motors. This vibration measurement device can be used to monitor machinery health by detecting unwanted oscillations and subsequent potential hazards.

©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Electric motors are important machines that have a wide range of applications from household appliances to industrial machineries. In industry, they are critical components as any kind of failure can impact the operation and productivity of the industry. Thus, the condition of motors needs be monitored carefully to reduce the downtime and maintenance cost. A reliable real-time fault identification device would increase the life of these systems and effectively mitigate potential failure. Various types of condition monitoring techniques based on acoustic emission, temperature, and vibration have been proposed. Selecting the technique based on the application, economy, simplicity is key to ensuring the best result while monitoring the system. The repetitive motion described by vibrations can be measured and analyzed using various systems. Some machines can tolerate extremely high-level of vibrations (like rock crushers), while others can be damaged even by low-level vibrations. An acceptable level of vibration does not cause any permanent † Corresponding


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damage or significant reduction in the life of the machine; however, repetitive vibrations can be damaging and can lead to a complete failure. There are several methods for determining acceptable levels of vibrations, including establishing a baseline by keeping recorded vibration data over time, monitoring vibrations on identical machines and observing any outlier data being shown by one of the machines, or contacting the manufacturer of the machine when vibration data are collected for determining proper function [1]. Every component of a rotary machine generates a vibration signal which can be used for monitoring the health of the machine and identifying the fault by analyzing the amplitude and frequency of the vibration signal. In order to analyze signals, one has to first sense the signal which can be done by installing a sensor in the proximity of the device to be monitored. For larger industrial machines accelerometers are often used to pick up vibrations. An accelerometer produces an electrical signal proportional to the vibration of the component attached to this sensor. The signal can then be processed into either an acceleration waveform or spectrum by applying Fourier Transform or other techniques. For rotating machines, vibration readings are generally taken through each bearing housing in the horizontal, axial, and vertical directions. They are measured with the accelerometer perpendicular to the surface. Because using hand to hold the accelerometer, the contact is not tight enough to get accurate measurements, each accelerometer should be mounted securely either magnetically, glued, or screwed onto the surface. Predictive maintenance is required to eliminate unnecessary and unscheduled downtime and in situations when it is anticipated that a fault in one machine would affect many other sections of the operation or bring production to a complete halt while repairs are being made. The monitoring of a machine can be continuous or periodic depending on factors such as its importance to the operation of the business, how much it would cost to repair, and how likely a failure of the machine is to injure workers [2–5]. Extraction of features, health assessment, and diagnostic approaches implemented for conditionbased maintenance have been the subject of interest of several decades [6, 7]. Vibration based fault identification technique is one of the popular tools used for fault identification. A number of techniques based on signal processing have been proposed. These techniques include Fast Fourier Transform [5–9], Wavelet Analysis [10], Discrete Wavelet Transform [11], and fuzzy logic inference systems to automate and diagnose the fault [12, 13]. Principal component analysis [14, 15] and K nearest neighbor based rotating machine fault detection using vibration features have also been proposed to predict the fault faster and earlier [16–20]. Conventional systems of vibration measurement are costly and require an expert to analyze and interpret the acquired vibration signals. In this paper, we propose a simple and low cost device that monitors the vibration of electric motors using a sensor, a microcontroller, and a display system. These components are simple and readily available in the market, making the implementation cheaper, faster, and easier. The remainder of the paper is organized as follows. The detail of the device design along with the hardware and the flowchart of the algorithm is described in Section 2. Experimental results and discussion are presented in Section 3, followed by a conclusion in Section 4.

2 Methodology The proposed device includes a piezoelectric-based sensor, a LED light, and an evaluation board (Arduino Uno board). Vibrations are first measured by the piezoelectric-based sensor which converts them to voltage values. The signal is then fed to the microcontroller for processing and determining whether the vibration exceeds the acceptable level. This is done by converting the analog signal, voltage, to a digital value using an analog to digital converter. When the amplitude of vibration acceleration exceeds a specific threshold set by the user depending on the specific electric motor, the LED light turns on,


241

Fig. 1 A view of the used sensor, LDT0-028K.

Fig. 2 Arduino Uno Board used to build the device.

Fig. 3 Flowchart of the basic circuit.

and the health condition of the electric motor is displayed as “extensive vibration” on the LCD screen. The sensor, LDT0, is a flexible component comprising a 28 μ m thick piezoelectric PVDF polymer film with screen-printed Ag-ink electrodes, laminated to a 0.125 mm polyester substrate, and fitted with two crimped contacts. As the piezo film is displaced from the mechanical neutral axis, bending creates very high strain within the piezo-polymer and therefore high voltages are generated [24]. The frequency response of the sensor is [1Hz, 10 kHz] which is more than what is needed for this type of application. Figure 1 shows the piezoelectric sensor used [24]. An Arduino Uno evaluation board is chosen because it is more user friendly and less expensive than other boards. The model selected is shown in Fig. 2. This Arduino board is a printed circuit board containing a microcontroller and other circuits that enable a fast design and testing. As shown in Fig. 2, the board contains several analog and digital inputs, a microcontroller, and a USB connector to connect the board to a computer for programming

242


Fig. 4 SainSmart LCD.

Fig. 5 LCD is wired to Arduino board.

the microcontroller. Other components used include an LED and an LCD. The circuit is wired by placing a 1MΩ resistor in parallel with the vibration sensor, then connecting two wires from each leg of the resistor, one going to ground GND and the other connected to analog input A1. The resistor is used in the circuit to prevent large voltage spikes which can potentially damage the analog inputs of the Arduino board. The LED is inserted directly into the Arduino board with the positive end being placed at digital pin 13 and the negative end run to ground. The algorithm is first written for a simple device without the LCD. Fig. 3 illustrates this algorithm for the basic program. To display the health condition of the electric motor based on the amplitude of vibration acceleration, a SainSmart LCD is added to the circuit as shown in Fig. 4 and Fig. 5. The pins of the SainSmart LCD board can either be soldered to wires then connected to the Arduino, or as an easier method, the 4 pins labeled as GND, Vcc, SDA and SLC on the back of the board can be wired respectively to the GDN, +5v, and Analog A4 and A5 ports on the Arduino Uno board. The flowchart for this advanced circuit with the LCD is shown in Fig. 6.

3 Results and discussion To test the developed device, an electric drill and a fan are used to generate vibrations. However, in normal conditions the amplitude of vibration acceleration of these two systems is below the threshold


Fig. 6 Flowchart of the advanced circuit.

Fig. 7 Serial monitor without any vibrations.

Fig. 8 Serial plotter without any vibrations.

243

244


Fig. 9 Serial monitor with vibration.

Fig. 10 Serial plotter with vibration

Fig. 11 LCD displays that vibration is within the normal level.

that triggers the LED or generates the “Excessive Vibration” message on the LCD screen. Therefore, the speeds of the motor and the fan are increased to high values in order to get high vibrations and test the developed device. Display screen is set to show excessive vibration message when the sensor senses a value that exceeds the threshold and switches to normal message when the vibration is under the threshold. Examples of results are shown in Figs. 7 through 14. When no vibration is detected, the value of the serial monitor is zero, and the serial plotter shows a flat continuous waveform at 0 volts. Figs. 7 and 8 show the serial monitor and serial plotter, respectively. The program is written to continuously monitor the analog input connected to the sensor, and compare the digital value corresponding to the voltage being sensed with a threshold value. For testing purposes, we set the digital threshold to 1000. When the digital value resulting from the sensed voltage is lower than 1000, the LED remains in the LOW/OFF state. However, when the digital value exceeds the threshold, the LED lights up for 20 seconds and then turn off, and the program runs again. Once the device begins to sense vibration, the serial plotter and monitor begin to show values and waveforms


245

Fig. 12 Serial plotter when LED triggered.

Fig. 13 Arduino with LED Illuminated.

Fig. 14 LCD displays that vibration level is exceeded the threshold value.

as shown in Figs. 9 and 10. The serial plotter shows the values of the voltage changing as the vibration is implied. Fig. 9 shows that there is no vibration above the threshold for the electric drill and fan in normal conditions. Fig. 10 shows very low spikes corresponding to normal but irregular vibrations below the threshold level of the fan and electric drill. In this case of low amplitude of vibration under the threshold value, the LED is not trigged, the condition of the electric motor is healthy and the message “Normal Vibration” is displayed on the LCD screen as shown in Fig.11. These results also demonstrate that the device is functioning properly and

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the algorithm is performing as expected. Fig.12 shows an example of results in the case of excessive motor vibrations. As one can see, the amplitudes of the spikes are irregular and variable ranging from 0 to 1100. The spikes that exceed the threshold correspond to excessive, vibrations of the motor. In this case of Fig. 13, the LED illuminates as shown in Fig. 12. When the speed of the motor exceeds a certain value, the motor experiences excessive vibrations, which means that the condition of electric motor is unhealthy. In this case, a message of “Excessive Vibration” is displayed on the LCD screen as shown in Fig. 14.

4 Conclusions and future work Monitoring machines on a consistent basis to detect vibration changes is important because it is almost impossible to tell from the first measurement whether the motor is failing or operating normally. Once an abnormal variation in the vibration signal is detected, preventative maintenance can be performed. Conventional systems to measure motor vibrations and identify faults are costly and require an expert to analyze the acquired vibration signals. In this paper, we propose a simple and low cost device that monitors the vibrations in real-time of electric motors. The device uses a piezoelectric sensor, a microcontroller, and a display system. These components are simple and readily available in the market, making the implementation cheaper, faster, and easier. Future research will take into consideration other characteristics of the vibration signals such as the distribution of frequencies and develop algorithms including fuzzy logic and deep learning to enhance the accuracy of detection, classification and prediction, and efficiency for different faults.

References [1] Niu, G., Yang, B.S., and Pecht, M. (2010), Development of an optimized condition-based maintenance system by data fusion and reliability-centered maintenance, Reliability Engineering & System Safety, 95(7), 786-796. [2] TungTran, V. and Yang, B. (2012), An intelligent condition-based maintenance platform for rotating machinery, Journal Expert Systems with Applications, 39(3), 2977–298. [3] Attoui, I., Boutasseta, N., Fergani, N., Oudjani, B., and Deliou, A. (2015), Vibration-based bearing fault diagnosis by an integrated DWT-FFT approach and an adaptive neuro-fuzzy inference system, 3rd International Conference on Control, Engineering & Information Technology (CEIT). [4] Seryasat, O.R., Honarvar, F., and Rahmani, A. (2010), Multi-fault diagnosis of ball bearing using FFT, wavelet energy entropy mean and root mean square (RMS), In Systems Man and Cybernetics (SMC), 2010 IEEE International Conference on, 4295-4299. [5] Nejadpak, A. and Yang, C. (2016), A vibration-based diagnostic tool for analysis of superimposed failures in electric machines, Proceedings of IEEE International Conference on Electro Information Technology (EIT), Grand Forks, ND, 2016, 0324-0329, DOI: 10.1109/EIT.2016.7535260. [6] Plante, T., Nejadpak, A., and Yang, C. (2015), Faults detection and failures prediction using vibration analysis, Proceedings of IEEE Autotestcon, 2015, National Harbor, MD, 227-231, DOI: 10.1109/AUTEST.2015.7356493. [7] Broch, J.T. (1980), Mechanical vibration and shock measurements, Br¨ uel & Kjær. [8] Tsypkin, M. (2013), Induction motor condition monitoring: Vibration analysis technique-a twice line frequency component as a diagnostic tool, In Electric Machines & Drives Conference (IEMDC), IEEE International 2013 May 12, 117-124. [9] Patil, S. and Gaikwad, J. (2013), Vibration analysis of electrical rotating machines using FFT: a method of predictive maintenance, In: 2013 fourth international conference on computing, communications and networking technologies (ICCCNT), Tiruchengode, India, 4–6 July 2013, pp.1–6, New York: IEEE. [10] Atoui, I., Meradi, H., and Boulkroune, R. (2013), Fault detection and diagnosis in rotating machinery by vibration monitoring using FFT and Wavelet techniques, In: 2013 8th international workshop on systems, signal processing and their applications (WoSSPA), Algiers, Algeria, 12–15 May 2013, 401-406, New York: IEEE.


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[11] Bajric, R., Zuber, N., Skrimpas, G., and Mijatovic, N. (2016), Feature extraction using discrete wavelet transform for gear fault diagnosis of wind turbine gearbox, Shock and Vibration, 2016, Article ID 6748469, 10 pages, 2016, DOI:10.1155/2016/6748469. [12] Adhikari, S., Sinha, N., and Dorendrajit, T. (2016), Fuzzy logic based on-line fault detection and classification in transmission line, SpringerPlus, 5(1), 1002, DOI:10.1186/s40064-016-2669-4. [13] Klomjit, J. and Ngaopitakkul, A. (2016), Selection of proper input pattern in fuzzy logic algorithm for classifying the fault type in underground distribution system, Region 10 Conference (TENCON) 2016 IEEE, 2650-2655, ISSN: 2159-3450. [14] Plante, T., Stanley, L., Nejadpak, A., and Yang, C. (2016), Vibration Based Fault Detection using Principal Component Analysis, Proceedings of IEEE Autotestcon 2016, Anaheim, CA, USA, Sept. 12-15, 2016. [15] Sinha, J. and Elbhbah, K. (2013), A future possibility of vibration based condition monitoring of rotating machines, Mechanical Systems and Signal Processing, 34(1), 231-40. [16] Nejadpak, A. and Yang, C. (2017), Misalignment and unbalance faults detection and identification using KNN analysis, Proceedings of CANCAM 2017, Victoria, BC, May 28-June 1, 2017, pp. 0324-0329, DOI: 10.1109/EIT.2016.7535260. [17] He, Q. and Wang, J. (2007), Fault detection using the k-nearest neighbor rule for semiconductor manufacturing processes, IEEE Transactions on Semiconductor Manufacturing, 20(4), 345-354, Nov. 2007, DOI: 10.1109/TSM.2007.907607. [18] Johnson, J. and Yadav, A. (2016), Fault detection and classification technique for HVDC transmission lines using KNN, International Conference on ICT for Sustainable Development ICT4SD. [19] Pandya, D., Upadhyay, S., and Harsha, S. (2013), Fault diagnosis of rolling element bearing with intrinsic mode function of acoustic emission data using APF-KNN, Expert Systems with Applications, 40(10), 41374145. [20] Li, C., René-Vinicio, S., Grover, Z., Mariela, C., and Diego, C. (2016), Fault diagnosis for rotating machinery using vibration measurement deep statistical feature learning, Sensors, 16(6), 895. [21] Elbhbah, K. and Sinha, J. (2013), Vibration-based condition monitoring of rotating machines using a machine composite spectrum, Journal of Sound and Vibration, 332(11), 2831-2845, [22] Fundamentals-of-piezoelectric-shock-and-vibration-sensors, (2011). [23] Nejadpak, A. and Yang, C. (2017), Fast Unbalancing of Rotating Machines by Combination of Computer Vision and Vibration Data Analysis, Journal of Vibration Testing and System Dynamics, 1(4), 343-352, DOI: 10.5890/JVTSD.2017.12.005. [24] https://www.parallax.com/sites/default/files/downloads/605-00004-Piezo-Film-Datasheet.pdf.



A New Application of the Normal Form Description to a N-Dimensional Dynamical Systems Attending the Conditions of a Hopf Bifurcation Vin´ıcius B. Silva1†, João P. Vieira2 , Edson D. Leonel1,3 1

2

3

Departamento de F´ısica, UNESP - Universidade Estadual Paulista, Av. 24A, 1515 - Bela Vista, 13506-900, Rio Claro, SP, Brazil Departamento de Matem´ atica, UNESP - Universidade Estadual Paulista, Av. 24A, 1515 - Bela Vista, 13506-900, Rio Claro, SP, Brazil Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy Submission Info Communicated by S. Lenci Received 5 January 2018 Accepted 26 May 2018 Available online 1 October 2018 Keywords Normal form Center manifold Bifurcation theory Nonlinear dynamics

Abstract In this paper we prove the following theorem: consider a N-dimensional dynamical system that is reduced to its center manifold. If it is proved the system satisfies the conditions of a Hopf bifurcation theorem then the original system of differential equations describing the dynamics can be rewritten in a simpler analytical expression that preserves the phase space topology. The theorem proposed and proven effectively reduces the work done to obtain the normal form for the class of dynamical systems with the occurrence of Hopf bifurcation.


1 Introduction In the investigation of the qualitative behavior of the flow of solutions near equilibrium, one of the most important tools is the normal form theory. It corresponds to the simplest analytical expression that a dynamical system can be written without changing the phase space topology of the original given system [1, 2]. Although, as already discussed in [3, 4], the computation of normal forms for dynamical systems is not a simple task once, the methods applied to obtain them usually involve matrix operations that are tough to be taken without the possibility of committing mistakes [4, 5]. Our attempt in this paper is to fill up this gap. As the main result of this article we present a theorem that effectively reduces the work done to obtain the normal form for the class of dynamical systems attending the conditions of Hopf bifurcation. The theorem says that for a given N-dimensional dynamical system reduced to its two-dimensional center manifold, if it is proved the system satisfies the non-degeneracy, transversality and, non-hyperbol† 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.2018.09.005

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icity conditions of Hopf bifurcation theorem, then the original system of differential equations describing the dynamics can be rewritten in the simple analytical expression given by { y˙1 = α (µ )y1 − β (µ )y2 + (ay1 − by2 )((y1 )2 + (y2 )2 ), (1) y˙2 = α (µ )y2 + β (µ )y1 + (ay2 + by1 )((y1 )2 + (y2 )2 ), where y1 and y2 are dynamical variables, α is a control parameter, a, b and β are constants. Our main tools in this paper are the center manifold theory [6] that allows us to reduce the dimension of the dynamical system and the normal form theory for the vector fields described by differential equations [7] to compute the analytical expression above. This paper is organized as follows. The proof of the main theorem is given in Section 2 while conclusions are made in Section 3.

2 The main result The following theorem is the main result of this article: Theorem 1. Given any N-dimensional dynamical system x˙ = f (x, µ ), x ∈ ℜn , µ ∈ ℜ,

(2)

with smooth f , having for all sufficiently small µ the fixed point x0 = 0 with eigenvalues λ1,2 = α (µ ) ± iβ (µ ), λ3 , λ4 , . . . , λn . If for µ = 0 the following conditions are satisfied: 1.

α (0) = 0, β (0) ̸= 0 (non − hyperbolicity). 2.

d α (µ ) µ =0 ̸= 0 (transversality). dµ

3. l1 (0)a ̸= 0

(non-degeneracy),

(3)

then, the dynamical system (2) reduced to its two-dimensional center manifold is rewritten in the normal form { y˙1 = α (µ )y1 − β (µ )y2 + (ay1 − by2 )((y1 )2 + (y2 )2 ), (4) y˙2 = α (µ )y2 + β (µ )y1 + (ay2 + by1 )((y1 )2 + (y2 )2 ). The theorem tells that for a generic N-dimensional dynamical system, if it is proved the system satisfies the conditions of the Hopf bifurcation theorem, then its normal form is known to be described by (4). The proof of this theorem is arranged as follows. First, the center manifold theory is applied to reduce the dimension of the given dynamical system. From the knowledge of this, the normal form a

l1 is the first Lyapunov coefficient whose expression is given by l1 (0) =

with fxy =

∂ 2 fµ ∂x ∂y

1 1 ( fxxx + fxyy + gxxy + gyyy ) + ( fxy ( fxx + fyy ) − gxy (gxx + gyy ) − fxx gxx + fyy gyy ) 16 16β (0)

µ =0 (0, 0) and so on.


251

theory is then applied to “simplify” t he nonlinear system by removing as much nonlinearity as possible. Proof. Consider a generic dynamical system x˙ = f (x, µ ), x ∈ ℜn , µ ∈ ℜ,

(5)

with a smooth function f . By the Implicit Function Theorem, the dynamical system posses an unique fixed point x0 in some neighborhood of the origin for all sufficiently small |µ | at which it is possible to perform a coordinate shift, placing the fixed point at the origin. Thus, the smooth function f can be expressed in terms of a Taylor series expansion given by f = f (x, µ ) = f (x0 , µ ) +

df 1 d2 f (x0 , µ )x + (x0 , µ )x2 + . . . dx 2! dx2

Although as long as f (x0 , µ ) = 0, this implies the system can be written as f (x, µ ) = A(µ )x + F(x, µ ),

(6)

where A(µ ) is the Jacobian matrix and F(x, µ ) is a smooth function whose terms have Taylor expansions in x starting with at least quadratic terms. Let λ1 , λ2 , λ3 , . . ., λn be the eigenvalues of the characteristic equation for the system (6) and λ1 and λ2 be the complex conjugate eigenvalues such that

λ1 = α (µ ) + iβ (µ ), λ2 = α (µ ) − iβ (µ ),

(7)

where for µ = 0 the following conditions are satisfied: 1.

α (0) = 0, β (0) ̸= 0 (non − hyperbolicity). 2.

d α (µ ) µ =0 ̸= 0 (transversality). dµ

3. l1 (0) ̸= 0

(non-degeneracy).

(8)

Then, the Jacobian matrix of (6) written in the canonical real (Jordan) form, J, is given by      α (µ ) −β (µ ) 0 ... 0 w1 g˜1 (w1 , w2 , . . . , wn )  β (µ ) α (µ ) 0 ... 0   w2   g˜2 (w1 , w2 , . . . , wn )        0   w3   g˜3 (w1 , w2 , . . . , wn )  0 λ 0 0 3 Jw + g(w) ˜ =  + .  ..  ..    .. .. . . ..   .      . . . . 0 . 0

0

. . . . . . λn

wn

gñ (w1 , w2 , . . . , wn )

From the knowledge of J, it is applied the center manifold theorem to reduce dimension of the dynamical system. This theorem is established as follows. Theorem 2. There is a locally defined smooth N-dimensional invariant manifold W c of (6) that is tangent to the linear eigenspace of the Jacobian matrix A(µ ) at x = x0 . The manifold W c is called the center manifold. Let ϕ t denote the flow associated with (6). Then, there is a neighborhood U of x0 , such that if ϕ t x ∈ U for all t ≥ 0 (t ≤ 0), then ϕ t x → W c for t → +∞ (t → −∞).

252


As already discussed and proved in details in Refs. [6,8], a derivation of the dynamics on the center manifold W c for (6) leads us to [ ] [ ][ ] [ ] u˙1 α (µ ) −β (µ ) u1 g˜u1 (u1 , u2 ) + , (9) = β (µ ) α (µ ) u2 u˙2 g˜u2 (u1 , u2 ) where g˜u1 (u1 , u2 ) and g˜u2 (u1 , u2 ) are the nonlinear terms with order ≥ 2. Since the canonical real (Jordan) form of the Jacobian matrix of Eq.(6) is obtained and the dynamics on the center manifold is known (10), we have now dedicated our effort to determine the normal form of the dynamical system that exhibits the conditions to observe a Hopf bifurcation. Here the process to obtain the normal form consists in: (i) write a simplified expression for the Jordan matrix on the center manifold and; (ii) apply the normal form theory to obtain the simplified expression searched. 2.1

A simplification for the Jordan matrix J

Here, we propose an alternative expression for the Jordan matrix J that consists in decomposing it ˆ into a sum of a symmetric and skew-symmetric matrix through the action of the operator residue, V, on J. This decomposition will simplify the computation of the normal form searched. Lemma 3. Let J be the canonical real (Jordan) matrix of a dynamical system that exhibits a Hopf ˆ on J leads to bifurcation. The action of the operator Residue, V, ( ) ( ) α (µ ) 0 0 −β (µ ) ˆ VJ = + , (10) 0 α (µ ) β (µ ) 0 (

where

α˜ = is the α˜ -reduced matrix and

( J˜ =

α (µ ) 0 0 α (µ )

)

) 0 −β (µ ) , β (µ ) 0

(11)

(12)

˜ is J-reduced matrix. Proof. The action of operator residue on J is expressed by ˆ = 1 (J + JT ) + 1 (J − JT ). VJ 2 2 This lead to, ˆ =1 VJ 2

(

Therefore,

( ˆ = VJ

2.2

2α ( µ ) 0 0 2α ( µ )

)

1 + 2

(

) 0 −2β (µ ) . 2β ( µ ) 0

) ( ) α (µ ) 0 0 −β (µ ) + . β (µ ) 0 0 α (µ )

(13)

Computation of the normal form

From the knowledge of α˜ and J,˜ follows the normal form theorem. Theorem 4. Let u˙ = f(u) be Cr system of differential equations with f (u0 ) = 0 and D f (u0 )u = J. Choose a complement Gk for L (H (k) ) in (H k ), so that L (H (k) ) ⊕ G(k) = H (k) .

(14)

Then, there is an analytic change of coordinates in the neighborhood of the origin which transforms the system u˙ = f (u) into y˙ = g(y) = g(1) (y) + g(2) (y)+ . . . + g(r) (y) + Rr with J(y) = g(1) (y) and Rr = O(|y|r ).


253

The detailed proof of this theorem 3 is found in [6]. Remark 1 (i). The normal form theory is a powerful approach to construct a simplified analytical expression for a nonlinear dynamical system. It is based upon the computation of nonlinear transformations which reduce the coupling among the terms in the original dynamical system. Although, here the computation is done by applying the approach on α˜ and J˜ matrices. Remark 2 (ii). In the computation of the normal form, the first-order normal form problem consists of looking at g(1) (y). However, according to the theorem 3 g(1) (y) = J(y). Based on this, our work will be only limited to determine g(2) (y) and, g(3) (y). Lemma 5. The α˜ -reduced matrix does not bring contributions to the normal form of dynamical systems which exhibit conditions to observe a Hopf bifurcation. Proof. For the second-order normal form problem, the basis for H (2) consists of 6 monomials ] [ ] [ ] [ ] [ ] [ ] [ y1 y2 (y2 )2 0 0 0 (y1 )2 , , , , , . 0 0 0 (y1 )2 y1 y2 (y2 )2 These monomials are represented respectively by ⃗m1 , ⃗m2 , . . . ,⃗m6 . In this case, the functions ⃗g(2) and ⃗h(2) are elements from the space of H (2) . To find the contribution of the α˜ -reduced matrix to the normal form, it is necessary to obtain firstly the complementary sub-space G(2) . At the space of H (2) , whose basis is composed by the 6 monomials ⃗m j , Lα˜ (H (2) ) can be represented by   α (µ ) 0 0 0 0 0  0 α (µ ) 0 0 0 0     0 0 α (µ ) 0 0 0  (2)  . Lα˜ (H ) =  0 0 α (µ ) 0 0   0   0 0 0 0 α (µ ) 0  0 0 0 0 0 α (µ ) Note however the determinant of L (H (2) ) is non-zero. Therefore when the determinant of L (H (k) ) admits non-zero values, this implies that the k-order terms can be eliminated from the normal form. Since the normal form does not have second-order terms, we repeat the process for the third-order. In this new case, the basis of H (3) is now composed by 8 monomials [ ] [ ][ ] [ ] (y1 )3 (y1 )2 y2 y1 (y2 )2 (y2 )3 , , , , 0 0 0 0 [

] [ ] [ ] [ ] 0 0 0 0 , , , . (y1 )3 (y1 )2 y2 y1 (y2 )2 (y2 )3

Therefore, the Lα˜ (H (3) ) is given by   0 0 0 0 0 0 2α (µ ) 0  0 2α ( µ ) 0 0 0 0 0 0     0 0 2α ( µ ) 0 0 0 0 0     0 0 0 2α ( µ ) 0 0 0 0  (3) .  Lα˜ (H ) =   0 0 0 0 2 α ( µ ) 0 0 0    0 0 0 0 0 2α ( µ ) 0 0     0 0 0 0 0 0 2α ( µ ) 0  0 0 0 0 0 0 0 2α ( µ )

254


Notice again that the determinant of L (H (3) ) is also non-zero. Therefore the third-order terms can be eliminated from the normal form also. Hence, The α˜ -reduced matrix does not bring contributions to the normal form of dynamical systems that exhibit the Hopf bifurcation. ˜ Lemma 6. Only the J-reduced matrix brings contributions to the normal form of dynamical systems that exhibit a Hopf bifurcation. ˜ Proof. Let us now consider the J-reduced matrix. For the second-order normal form problem, the (2) basis for H consists of 6 monomials [ ] [ ] [ ] [ ] [ ] ] [ 0 (y1 )2 y1 y2 (y2 )2 0 0 , , , , . , 0 0 (y1 )2 y1 y2 (y2 )2 0 ˜ To determine the contributions of J-reduced matrix for normal form the complementary sub-space (2) must be obtained. At the H space, whose basis is composed by the 6 monomials ⃗m j , LJ˜(H (2) ) can be represented by the following matrix   0 β 0 β 0 0  −2β 0 2β 0 β 0     0 −β 0 0 0 β  (2)  . LJ˜(H ) =   − β 0 0 0 β 0    0 −β 0 −2β 0 2β  0 0 −β 0 −β 0

G(2)

The determinant of L (H (2) ) is non-zero. The determinant of L (H (2) ) admits non-zero values, this implies the second-order terms can be eliminated from the normal form. Since the normal form does not have second-order terms, we repeat the process for the third-order. In this new case, the basis of H (3) is now composed by 8 monomials [ ] [ ][ ] [ ] (y1 )3 (y1 )2 y2 y1 (y2 )2 (y2 )3 , , , , 0 0 0 0 [ ] [ ] [ ] [ ] 0 0 0 0 , , , . (y1 )3 (y1 )2 y2 y1 (y2 )2 (y2 )3 The LJ˜(H (3) ) is given by



−β  0   0   0 (3) LJ˜(H ) =   0   0   0 0

0 −β 0 0 0 0 0 0

0 0 −β 0 0 0 0 0

 0 0 β 0 0 0 −3β 0 2β 0   0 0 −2β 0 3β   −β 0 0 −β 0  . 0 6β 0 −6β 0   0 0 2β 0 −2β   0 0 0 0 0  0 0 0 0 0

The third order terms cannot be eliminated since det(LJ˜(H (3) )) is zero. Hence, the only contribu˜ tions for the normal form expression come from the J-reduced matrix and are of the third-order. (3) Although, as long as det(LJ˜(H )) is zero, then there is a complementary sub-space G(3) to be determined. Once (LJ˜(H (3) ) is rank 6 and the rank of (H (3) ) is 8, the null-space G(3) must be given by a linear combination of two vectors that can be represented as ( 3 ) ( 2 ) y1 + y1 y22 −y1 y2 − y32 (3) g =a 2 +b , (15) y1 y2 + y32 y31 + y1 y22


255

where a and b are constants. Therefore, according to the theorem 3, for the N-dimensional system (6) reduced to its two-dimensional center manifold W c and, satisfying the conditions from the theorem of Hopf bifurcation, its normal form is given by { y˙1 = α (µ )y1 − β (µ )y2 + (ay1 − by2 )((y1 )2 + (y2 )2 ), (16) y˙2 = α (µ )y2 + β (µ )y1 + (ay2 + by1 )((y1 )2 + (y2 )2 ). This result ends the proof of the theorem 1. As a consequence, follows its corollary. Corollary 7. Suppose the N-dimensional system x˙ = f (x, µ ), x ∈ ℜn , µ ∈ ℜ,

(17)

and, x0 ∈ ℜn its fixed point. At µ = 0, the dynamical system (17) is said to exhibit a Hopf bifurcation. Suppose further that for µ < µ0 (µ > µ0 ), (17) has a pair of complex-conjugate eigenvalues with positive real part and, for µ > µ0 (µ < µ0 ), (17) has a pair of complex-conjugate eigenvalues with negative real part. Then, 1. For l1 < 0, which happens when a < 0, the fixed point x0 is said to be asymptotically stable at µ = µ0 . Although, at µ > µ0 (µ < µ0 ) an unique stable (unstable) curve of periodic solutions bifurcates from the unstable (stable) fixed point. In this case, the dynamical system exhibits the so called supercritical Hopf bifurcation. 2. For l1 > 0, which happens when a > 0, the fixed point x0 is said to be unstable at µ = µ0 . However, at µ < µ0 (µ > µ0 ) an unique unstable (stable) curve of periodic solutions bifurcates from the stable (unstable) fixed point. In this case, the dynamical system exhibits the so called subcritical Hopf bifurcation. 3. For l1 = 0, nothing can be said about the dynamics of (17). The corollary above establishes whether the dynamical system in study exhibits the Supercritical or the Subcritical cases of the Hopf bifurcation.

3 Conclusion The development of this work allowed us to prove for a N-dimensional dynamical system described by differential equations that its normal form can be easily obtained, once the dynamical system in study is reduced to its two-dimensional center manifold W c and it is verified the conditions from the theorem of Hopf bifurcation. In general, the normal form theory is one of the most important tools in the study of the qualitative behavior of orbit structures of vector fields near equilibrium. The reason is that it can be applied to investigate families of different equations dependent on parameters whose variation leads to changes in the topological structure of the spatial space. Based in this scenario, the present work could make a certain effort, once it was proposed and proved a theorem that guarantees the normal form for a given dynamical system reduced to its center manifold, if it is proved that it satisfies the conditions of the Hopf bifurcation theorem. In this way, the main result of this article which it is described by theorem 1, provides an interesting and efficient way to determining the normal form for a class of dynamical systems that exhibits a Hopf bifurcation.

Acknowledgements V.B.S. acknowledges FAPESP (2015/23142-0) for financial support. EDL acknowledges support from CNPq (303707/2015-1) and FAPESP (2012/23688-5). The authors acknowledge fruitful discussions with Professor Jo˜ ao Paulo Cerri.

256


References [1] Betancur, H.A. (2010), Bifurcaciones B´ asicas y Formas Normales, Universidad Nacional de Colombia: Colombia. [2] Verri, A.J. (2013), Estabilidade Global e Bifurca¸c˜ ao de Hopf em um modelo de HIV baseado em Sistemas do Tipo Lotka-Volterra, Universidade Estadual Paulista: Rio Claro. [3] Chow, S.N., Drachman, B., and Wang, D. (1990), Computation of normal forms, Journal of Computation and Applied Mathematics, 29, 129-143. [4] Altman, E.J. (1993), Normal form analysis of Chua’s circuit with applications for trajectory recognition, IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 40, 675-682. [5] Cushman, R. and Sanders, J. (1987), Splitting Algorithm for Nilpotent Normal form, MSI Cornell Univ.: New York. [6] Marsden, J.E. and McCracken, M. (1976), The Hopf bifurcation and its applications, Springer-Verlag: New York. [7] Monteiro, L.H. (2002), Sistemas dinˆ amicos, Livraria da F´ısica: Rio de Janeiro. [8] Kuznetsov, Y.A. (1998), Elements of applied bifurcation theory, Springer: New York.



Nonlinear Dynamics of a Reduced Cracked Rotor K. Lu1,3,4 , Y. Lu1 , B. C. Zhou1 , W. Jian1 , Y. F. Yang1†, Y. L. Jin2 , Y. S. Chen3 1 2 3 4

Institute of Vibration Engineering, Northwestern Polytechnical University, Xi’an, 710072, P. R. China School of Aeronautics and Astronautics, Sichuan University, 610065, P. R. China School of Astronautics, Harbin Institute of Technology, Harbin 150001, P. R. China College of Engineering, The University of Iowa, Iowa City, IA 52242, USA Submission Info Communicated by S.C. Suh Received 20 May 2018 Accepted 20 June 2018 Available online 1 October 2018 Keywords POD method Rotor system POM energy Crack

Abstract The transient proper orthogonal decomposition (TPOD) method is applied for order reduction in the rotor system in this paper. A 26DOFs rotor model with crack is established by the Newton’s second law, and the dynamical behaviors (bifurcation diagram, amplitude frequency curve, etc.) of the crack fault are discussed. The optimal reduced model can be provided by the proper orthogonal mode (POM) energy method, the TPOD method is applied to reduce the original system to a three-DOFs one at a certain speed corresponds to the maximum energy. The efficiency of the TPOD method is verified via comparing with the bifurcation diagram of the original and reduced rotor system. The order reduction method provides qualitative analysis to study the reduced model of the high-dimensional rotor system. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Most actual engineering systems are complex, high-dimensional and nonlinear. They are very expensive to carry out the numerical calculations and difficult to study the qualitative analysis. So the study on order reduction methods has become one central issue of concerns in research area of nonlinear dynamics. The order reduction methods include the center manifold method [1,2], the inertial manifold method [3,4], the Lyapunov-Schmidt (L-S) method [5], the Galerkin method [6], the POD method [7,8] and other order reduction methods [9], were summarized by Rega and Steindl in their applied studies of nonlinear dynamics [10, 11]. The POD method is an effective and powerful method for analysis of data aimed at gaining low dimensional approximate descriptions of a high dimensional system. The POD method has been widely used in the high-dimensional systems. On the basis of the order reduction accuracy, the POD method can not only reduce the DOFs of a system greatly but also improve the computational efficiency [12]. It can be applied in a variety of areas, for example, fluid mechanics [13], signal processing [14], image † Corresponding


258

K. Lu et al. / Journal of Vibration Testing and System Dynamics 2(3) (2018) 257–269

processing [15], optimal design [16], structure dynamics [17,18], chemical reaction control [19] and ocean engineering [20], the POD method can also be used to solve optimal control problems [21]. However, the reduced order modes (ROMs) obtained by the POD method usually lack robustness when the system parameters change. Some modified POD methods were proposed to resolve the parametric robustness: the global POD method [22], the local POD method [23], the adaptive POD method, etc. [24]. Many researchers propose the modified POD for order reduction in the high-dimensional and nonlinear rotor system. The POD method was applied in the rotor-bearing system and the singularity and primary vibration analysis of the reduced model were discussed [25]. Lu and Yu [26] proposed the modified nonlinear POD method, which was applied in the 23-DOFs rotor system with looseness at one end, the efficiency and accuracy of the order reduction method was verified via the comparison of the bifurcation diagram and the relative error analysis. Lu and Jin [27] applied the TPOD method to reduce a 7-DOFs rotor system supported by a pair of ball bearings with pedestal looseness to a 2-DOFs one. The topological structure comparison of bifurcation verified the efficiency of the order reduction method, and the stability analysis of the reduced model was studied. The POM energy method was used to confirm the DOF number of the reduced rotor system obtained by the TPOD method, and the structure order reduction method was also used to compare with the TPOD method so that to verify the efficiency of the TPOD method [28, 29]. The aim of this paper is to apply the TPOD method in the rotor system from the qualitative aspect. A 26-DOFs rotor model with crack fault is established by the Newton’s second law. The bifurcation and frequency behaviors of the crack of the rotor system are discussed. The POM energy is applied to confirm the dimension of the reduced model. The POD and POM energy methods are applied in the rotor-bearing system to obtain the optimal reduced model, the efficiency of the order reduction method is verified based on the qualitative analysis (bifurcation behavior). Finally, the conclusions outlooks are drawn.

2 The rotor-bearing model with crack fault In this section, the crack fault will be introduced briefly first, and then a 26-DOFs rotor model supported by a pair of liquid-film bearings with crack fault is established by the Newton’s second law. 2.1

The crack fault model

The schematic diagram of the cracked section is shown in Fig.1. α is the depth of crack, β is the intersection angle between the unbalance and the crack normal vector. Law of open and close describes the effects of crack intersection angle to the shaft stiffness. As usual, the crack model contains three kinds of models: the first is the cosine model; the second is the square wave model and the third is the model integrates cosine and square wave. In Fig.1, the initial intersection angle of the cracked normal vector and x axis is 0. We consider the case of gravity dominance, the open and close function of crack f (ϕ ) can be expressed as the function of rotating angle ω t. k is the cracked stiffness of rotating shaft, Δkξ and Δkη are the stiffness variations of cracked normal and tangential vectors. The stiffness of rotating shaft with crack is expressed as: Δkξ cos2 ω t + Δkη sin2 ω t (Δkξ − Δkη ) sin ω t cos ω t k 0 kx kxy (1) = − f (ϕ ) kyx ky 0 k (Δkξ − Δkη ) sin ω t cos ω t Δkξ sin2 ω t + Δkη cos2 ω t The expression of f (ϕ ) is shown in formula (2), this model can select the square wave and cosine model based on the depth of the crack respectively.


259

Fig. 1 Schematic diagram of the cracked section. m m m m

Fy

k

Fx

k

k

k

m m k k

m

k

m

m

m m m

k

k

k

k

o o o o o o o o o o

o

m

k

o o

Crack

Fig. 2 Rotor model with crack fault.

⎧ ⎨ 1 + 2 cos (ϕ ) − 2 cos (3ϕ ) + 2 (5ϕ ) − . . . 3π 5π f (ϕ ) = 2 π ⎩ (1 + cos ϕ ) /2 2.2

a/R ≤ 0.5,

(2)

a/R > 0.5.

The establishment of the rotor model

As is shown in Fig.2, it is a 26-DOFs rotor model with crack. Here we assume that the axial and torsional vibrations of the system and the gyroscopic moment are neglectful. oi (i = 1, ..., 13) are the geometric centers of the discs. mi (i = 1, ..., 13) are the equivalent lumped masses. ci (i = 1, ..., 13) are the equivalent damping coefficients at the position of the lumped masses. ki (i = 1, 2, . . . , 12) are the equivalent stiffness of the corresponding discs. Assume that the crack occurs on the shaft between the first and second disc. The rotor model is supported by a pair of liquid-film bearings on both ends. The parameters in details are expressed as follows: m1 = m13 = 7kg, m2 = m12 = 6kg, m3 = m11 = 5kg, m4 = m10 = 4kg, m5 = m9 = 3kg, m6 = m8 = 2kg, m7 = 1kg, R = 30mm, c1 = c13 = 350Ns/m, c2 = c12 = 320Ns/m, c3 = c11 = 290Ns/m, c = 0.11mm, c4 = c10 = 260Ns/m, c5 = c9 = 230Ns/m, c6 = c8 = 200Ns/m, c7 = 170Ns/m, π ki = 2.3 × 107 N/m (i = 1, 2, . . . , 12) , L = 25mm, β3 = β11 = , E3 = E11 = 0.08mm, μ = 0.018Pa · s 4 Ei = 0 and βi = 0, i = 1, 2, . . . , 13, i = 3, 11. 2.3

The dynamical equation of the rotor system

The dynamical equation is established by the Newton’s second law, and it is expressed as formula (3). For the convenience of the calculation, we will use the dimensionless form of the equation (3), and the

260


dimensionless process and the equation will be shown in details. m1 X¨1 + c1 X˙1 + k1 (X1 − X2 ) − f (ϕ )[(Δkξ cos2 ω t + Δkη sin2 ω t)(X1 − X2 ) +(Δkξ − Δkη ) sin ω t cos ω t(Y1 −Y2 )] = Fx (X1 ,Y1 , X˙1 , Y˙1 ) + m1 E1 ω 2 cos(ω t + β1 ),

m1Y¨1 + c1Y˙1 + k1 (Y1 −Y2 ) − f (ϕ )[(Δkξ − Δkη ) sin ω t cos ω t(X1 − X2 ) +(Δkξ sin2 ω t + Δkη cos2 ω t)(Y1 −Y2 )] = Fy (X1 ,Y1 , X˙1 , Y˙1 ) + m1 E1 ω 2 sin(ω t + β1 ) − m1 g, m2 X¨2 + c2 x˙2 + k1 (X2 − X1) + k2 (X2 − X3 ) − f (ϕ )[(Δkξ cos2 ω t + Δkη sin2 ω t)(X2 − X1 ) +(Δkξ − Δkη ) sin ω t cos ω t(Y2 −Y1 )] = m2 E2 ω 2 cos(ω t + β2 ), m2Y¨2 + c2Y˙2 + k1 (Y2 −Y1 ) + k2 (Y2 −Y3 ) − f (ϕ )[(Δkξ − Δkη ) sin ω t cos ω t(X2 − X1 ) +(Δkξ sin2 ω t + Δkη cos2 ω t)(Y2 −Y1 )] = m2 E2 ω 2 sin(ω t + β2 ) − m2 g, m j X¨ j + c j X˙ j + k j−1 (X j − X j−1) + k j (X j − X j+1 ) = m j E j ω 2 cos(ω t + β j ),

(3)

( j = 3, 4, . . . , 12) m jY¨ j + c jY˙ j + k j−1 (Y j −Y j−1 ) + k j (Y j −Y j+1 ) = m j E j ω 2 sin(ω t + β j ) − m j g, m13 X¨13 + c13 X˙13 + k12 (X13 − X12 ) = Fx (X13 ,Y13 , X˙13 , Y˙13 ) + m13 E13 ω 2 cos(ω t + β13 ), m13Y¨13 + c13Y˙13 + k12 (Y13 −Y12 ) = Fy (X13 ,Y13 , X˙13 , Y˙13 ) + m13 E13 ω 2 sin(ω t + β13 ) − m13 g. The dimensionless process is as follows: Xi Yi dxi dyi d x˙i d y˙i , yi = , x˙i = , y˙i = , xï = , yï = c c dτ dτ dτ dτ 2 2 Fy m1 cω m13 cω Fx μω RL R 2 L 2 , M13 = , fx = , fy = , s = ( ) ( ) . M1 = sP sP sP sP P c 2R where fx and fy are dimensionless nonlinear oil-film force, α is directional components of bearing nonlinear oil-film force. c is bearing clearance, s is the Sommerfeld number. μ is lubricating oil viscosity, L is bearing length, R is bearing radius, ω is external excitation, P is the loading, and τ is the dimensionless time. 1 1 1 c1 x˙1 + k1 (x1 − x2 ) − f (ϕ )[(Δkξ cos2 τ + Δkη sin2 τ )(x1 − x2 ) x¨1 + m1 ω m1 ω 2 m1 ω 2 1 1 fx (x1 , y1 , x˙1 , y˙1 ) + E1 cos(τ + β1 ), +(Δkξ − Δkη ) sin τ cos τ (y1 − y2 )] = M1 c 1 1 1 c1 y˙1 + k1 (y1 − y2 ) − f (ϕ )[(Δkξ − Δkη ) sin τ cos τ (x1 − x2 ) y¨1 + m1 ω m1 ω 2 m1 ω 2 1 1 1 fy (x1 , y1 , x˙1 , y˙1 ) + E1 sin(τ + β1 ) − 2 g, +(Δkξ sin2 τ + Δkη cos2 τ )(y1 − y2 )] = M1 c ω c 1 1 1 1 c2 x˙2 + k1 (x2 − x1 ) + k2 (x2 − x3 ) − f (ϕ )[(Δkξ cos2 τ + Δkη sin2 τ )(x2 − x1 ) x¨2 + m2 ω m2 ω 2 m2 ω 2 m2 ω 2 1 +(Δkξ − Δkη ) sin τ cos τ (y2 − y1 )] = E2 cos(τ + β2 ), c 1 1 1 1 c2 y˙2 + k1 (y2 − y1 ) + k2 (y2 − y3 ) − f (ϕ )[(Δkξ − Δkη ) sin τ cos τ (x2 − x1 ) y¨2 + 2 2 m2 ω m2 ω m2 ω m2 ω 2 1 1 +(Δkξ sin2 τ + Δkη cos2 τ )(y2 − y1 )] = E2 sin(τ + β2 ) − 2 g, c ω c ⎫ 1 1 1 1 ⎪ ⎪ E c j x˙ j + k (x − x ) + k (x − x ) = cos( τ + β ), x¨ j + j−1 j j−1 j j j+1 j j ⎬ m jω m jω 2 m jω 2 c ( j = 3, 4, . . . , 12) 1 1 1 1 1 ⎪ ⎪ c j y˙ j + k j−1 (y j − y j−1 ) + k j (y j − y j+1 ) = E j sin(τ + β j ) − 2 g;⎭ y¨ j + m jω m jω 2 m jω 2 c ω c (4a)

τ = ω t, xi =


1 1 1 k12 (x13 − x12 ) = fx (x13 , y13 , x˙13 , y˙13 ) + E13 cos(τ + β13 ), 2 m13 ω m13 ω M13 c (4b) 1 1 1 1 1 y¨13 + E g. c13 y˙13 + k (y − y ) = f (x , y , x ˙ , y ˙ ) + sin( τ + β ) − 12 13 12 y 13 13 13 13 13 13 m13 ω m13 ω 2 M13 c ω 2c The nonlinear oil-film force [30] of x and y directions can be found as follows:

˙ 2 ]1/2 3xV (x, y, α ) − G(x, y, α ) sin α − 2S(x, y, α ) cos α [(x − 2y) ˙ 2 + (y + 2x) fx =− . (5) fy 3yV (x, y, α ) + G(x, y, α ) cos α − 2S(x, y, α ) sin α 1 − x2 − y2 x¨13 +

1

261

c13 x˙13 +

The corresponding parameters are expressed as follows: y + 2x˙ π π y + 2x˙ ) − sgn( ) − sgn(y + 2x), ˙ α = arctan( x − 2y˙ 2 x − 2y˙ 2 y cos α − x sin α 2 π [ + arctan ], G(x, y, α ) = 2 2 1/2 2 (1 − x − y ) (1 − x2 − y2 )1/2 2 + (y cos α − x sin α )G(x, y, α ) V (x, y, α ) = , 1 − x2 − y2 x cos α + y sin α . S(x, y, α ) = 1 − (x cos α + y sin α )2 The dimensionless equation of (4) can be written as equation (6), in which z = (x1 , y1 , · · · , x13 , y13 )T , ¯ ¯f are shown as above: the expression of c¯ , k, ¯ + ¯f. z¨ = −¯cz˙ − kz

(6)

To facilitate the theory analysis, we implement the Taylor series expansion of the oil-film force, thus α can be rewritten as: ˙ − 2y) ˙ y + 2x˙ π (y + 2x)(x π y + 2x˙ − ( )− ( ). (7) α = arctan x − 2y˙ 2 |y + 2x| ˙ · |x − 2y| ˙ 2 |y + 2x| ˙ For the convenience of calculation, formula (6) is written as (8) briefly: ¨ = −CZ ˙ − KZ + F. Z

(8)

In formula (8), C is the damping matrix, K is the stiffness matrix, Fis the force vector, which includes oil-film and external excitation etc. Z = [z1 z2 , ..., z26 ]T corresponds to [x1 y1 , ..., x13 y13 ]T in formula (6).

3 The discussion of the dynamical characteristics The dynamical characteristics (bifurcation and amplitude-frequency) of the rotor system with the cracked fault will be discussed in details. In Fig.3, it shows the bifurcation diagram of the rotor system without crack (Fig.3 (a)) and that with crack (Fig.3 (b)) when the speed varies from 200-900rad/s. It is clear that the period 2 bifurcation occurs in the rotor system with crack via comparing with the system without crack (Fig.3 (a)) when the speed is about 700rad/s. The crack fault causes the variation of the bifurcation behaviors. As is shown in Fig.4, the blue line represents the amplitude-frequency curve of the system without crack and the red line represents that of the system with crack. Obvious difference occurs between the blue and the red one, the natural frequency ωc changes when the crack occurs. When the rotating speed is about 1.8ωc (700rad/s), the amplitude of the vibration jumps to about twice relative to ωc rapidly, which indicates that the 1/2 sub-harmonic resonance occurs in the cracked system. The variation of the dynamical characteristic of the amplitude-frequency curve also verifies the period-2 bifurcation in the rotor system with crack.


262

1

Amplitude of z 1

0.5 0 -0.5 -1 200

300

400

500

600

700

800

900

Z(rad/s)

(a)

Amplitude of z1

1 0.5 0 -0.5 -1 200

400

600

900

Z (rad/s)

(b)

Fig. 3 The bifurcation diagrams of z1 DOF. (a) the rotor system without crack, (a) the rotor system with crack.

Fig. 4 The amplitude-frequency curves of z1 DOF of the rotor system with and without cracks.

Remark 1. The study on the dynamical characteristics of the bifurcation and amplitude-frequency indicates that the crack causes the 1/2 sub-harmonic resonance in the rotor system. The numerical results provide the theoretical guidance to avoid the resonance frequency in actual engineering system.


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4 The optimal reduced model based on the POM energy In this section, we will introduce the POM briefly, and the POM energy method based on the transient time series will be proposed. Then we will provide the optimal order reduction model of the original one at a certain speed of the rotor. 4.1

The POM energy based on the transient time series

POD is a procedure for simply extracting a basis for a modal decomposition from an ensemble of signals measured from the vibration structure [31]. The POMs have been used as empirical modes in the analysis of fluids and structures for modal projections of partial differential equations. In certain systems, the POMs have been seen to approximate linear modes [32]. The POMs have been shown to represent the optimal distributions of kinetic energy or power, and the proper orthogonal values (POVs) indicate the power associated with these principal distributions [33]. Application of POD to structures typically requires the displacements of a dynamical system at M DOFs. Denote these displacements as x1 (t) , x2 (t) , . . . , xM (t). When the displacements are sampled N times, we can get the matrix xi = (xi (t1 ) , xi (t2 ) , . . . , xi (tN ))T , i = 1, . . . , M. On the basis of the POD, the displacement histories are used to form an N × M order ensemble matrix X = [x1 , x2 , . . . , xM ]. Each row of X represents a point in the coordinate space at a particular instant in time. Then the correlation matrix R = N1 XT X can be formed, the order is M × M. The eigenvectors of R form an orthogonal basis, since R is real and symmetric. The POMs are the eigenvectors of R, and the eigenvalues are the POVs. The POMs can be applied in the actual engineering systems for order reduction to obtain the corresponding reduced model. The POMs have been treated as empirical modal bases for discretizing partial differential equations by the Galerkin projection in turbulence applications and more recently in structural dynamics [34]. Efforts to tie the POMs to the linear normal modes in vibrating systems include theoretical and numerical studies on discrete systems as well as experimental study using the measured strain gage signals [35]. The POMs can also be applied in the rotor system: Lu [27] used the first two order POMs to occupy 99.99% energy of the 7-DOFs rotor system and obtained the order reduction model via the TPOD method; the POM energy has also been applied to verify the physical significance of the TPOD method via comparing with the tradition POD method [28]. The energy of the POMs can be studied in Ref. [36] in details. 4.2

The optimal order reduction model

Although the POM energy has been applied in the rotor system to confirm the dimension of the reduced model, the method to obtain the optimal order reduction model is not clear. Since the case of the POM energy varies with the rotating speed of the rotor is not considered. The DOF selection method of the reduced model will be discussed in this section. The optimal order reduction model will be proposed based on the energy criterion of POM. The POM associated with the largest POV is the optimal vector to characterize the ensemble of snapshots. The POM with the second largest POV restricted to the space orthogonal to the first order POM is also the optimal vector to characterize the ensemble of snapshots, and so as follows. The energy contained in the data can be expressed by the POVs of the covariance matrix, i.e., the sum of the POVs, and the energy captured by the kth POM is expressed as λk / ∑i λi (λi ≥ 0). Choose an l-DOFs reduced model as an example in this section. We choose the first l order POMs relative to the first l largest POVs to observe the energy percentage captured by the POMs. The traditional POD method is compared with the TPOD method in Ref. [26] to verify the efficiency of the TPOD method, so we do not discuss the traditional POD method in details here. Through the numerical simulation, the transient time series displacement information of all DOFs was obtained in equal time interval, denoted by x1 (t) , x2 (t) , ..., xm (t), where m is the number of DOF. We

264


gain n equal time interval displacement series for each DOF, written as xi (t) = [xi (t1 ) , xi (t2 ) , · · · , xi (tn )]T , i = 1, 2, · · · , m, the time series form the matrix X = [x1 (t) , x2 (t) , · · · , xm (t)], and the order of X is n×m. So we get the correlation matrix T = 1n XT X with order m × m. The eigenvectors of the correlation matrix are ϕ1 , ϕ2 , ..., ϕm , which are called POMs, the corresponding eigenvalues are λ1 ≥ λ2 ≥ . . . ≥ λm ≥ 0, referred as POVs. We calculate the sum of eigenvalues λ1 , λ2 , · · · , λl to gain the energy of the first l order POMs, the energy is denoted by s in formula (9). l

m

i=1

j=1

s = ∑ λi / ∑ λ j × 100% (i = 1, 2, . . . , l; j = 1, 2, . . . , m; l ≤ m) .

(9)

The energy of the POMs varies with the rotating speed of the rotor. When the energy is approximate to 100%, the number of l is the optimal modal number. Then we will apply the order reduction method in the original rotor-bearing model and get the reduced one at the rotating speed relative to the largest energy. The numerical results will be discussed in section 5. Remark 2. The POMs are the eigenvectors of the correlation matrix. The POMs represent the energy that occupies the original high-dimensional system. The POM energy varies with the rotating speed of the rotor system. The highest energy (approximate to 100%) can provide the optimal order reduction model at the corresponding rotating speed of the rotor system. The order of the POM is the DOF number of the reduced model.

5 The efficiency of the order reduction methods The TPOD method will be applied in the rotor system with the crack fault. We will introduce the basic theory of the order reduction method first, and then the efficiency of the method is verified based on the qualitative aspect. 5.1

The TPOD method based on the qualitative aspect

Before we discuss the TPOD method, we should provide the basic theory formation of the traditional POD method. Then we will introduce the TPOD method in details. Finally the efficiency of the order reduction method will be studied. 5.1.1

The basic theory of the POD method

We present the mathematical formulation of the POD method here to follow the Ref. [12]. Let m (x,t) be a random field on the domain H. This field is resolved into the mean n (x) and f (x,t) varies with the time: m(x,t) = n(x) + f (x,t). (10) We give the definitions: ∗

∗

ˆ

( f ,g ) = H

f ∗ (x) g∗ (x) dH : The inner product in H 1

· : The average operation. · = (·, ·) 2 : The norm. |·| : The modulus The system lists a snapshot at time ti , f i (x) = f (x,ti ). The purpose of the POD is to obtain the most characteristic structure φ (x) of an ensemble of snapshots of the field f (x,t) [36]. This is equivalent to find the basis function φ (x) that maximizes the ensemble average of the inner products between f i (x) and φ (x), the expression is as follow: Maximize ( f i , φ )|2 , let ||φ ||2 = 1.

(11)


265

We define the constrain φ 2 = 1, which imposed to make the calculation consistent. The formula (11) means that φ is the best basic function if the field f is projected on it, then we make use of the Lagrange multiplier: (12) J[φ ] = |( f , φ )|2 − λ (||φ ||2 − 1). When the functional derivative is equal to 0, we can get the extremum. As Ref. [12], this condition can be changed into the following eigenvalue problem: ˆ i f (x) f i (x ) φ (x )dx = λ φ (x), (13) H

where f i (x) f i (x ) is the averaged auto-correlation function. As mentioned above, the orthogonal eigenfunctions φ p (x) of the equation (13) gives the solution of the optimization problem (11), φ p (x) are called the POMs and the corresponding eigenvalues λ p (λ p ≥ 0) are the POVs. The field f (x,t) can be used to decompose into the POMs:

∞

f (x,t) =

∑ a p (t)φ p (x).

(14)

p=1

The POD method is to find the POMs associated with the first largest proper orthogonal values (POV), i.e., to find the optimal vector to characterize the ensemble of snapshots. In formula (14), a p (t) is the uncorrelated function, i.e., a p (t)aq (t) = ω pq λ p , and are determined by a p (t) = ( f (x,t), φ p (x)). To summarize, if the displacements dk (t) of a discrete dynamical system with M DOFs are sampled N times and the M × N matrix is formed in formula (15): ⎤ ⎡ d1 (t1 ) . . . d1 (tN ) χ = ⎣ . . . . . . . . . ⎦ = [d (t1 ) . . . d (tN )] (15) dM (t1 ) . . . dM (tN ) then the POMs are the eigenvectors of V = N1 χ χ T and the corresponding eigenvalues are the POVs. A POV measures the relative energy of the system dynamics contained in the associated POM. 5.1.2

Introduction to the TPOD method

In general, through the equivalent transformation, the multiple-DOFs system can be written as: ¨ = −CZ ˙ − KZ + F(t, Z), Z

(16)

where Z is the vibration displacement vector. C is the equivalent damping matrix. K is the equivalent stiffness matrix. F(t, Z) is the nonlinear function about t and Z. F(t, Z) is denoted as F briefly in this paper. The construction processes are as follows: (1) Provide the initial conditions, proceed numerical simulation and obtain displacement information from transient process of various DOFs, which is denoted as z1 (t), z2 (t), ..., zm (t). m is the number of DOFs in the system, recording transient time interval displacement sequence of all DOFs zi (t) = (zi (t1 ), zi (t2 ), · · · , zi (tN ))T , i = 1, . . . , m. The number of time interval is N and the interval is equal to each other, the time interval is denoted as t1 ,t2 ...,tN . The time series form the matrix X = [z1 (t), z2 (t), ..., zm (t)], the order of X is N × m. The matrix X can be written in details as formula (17): ⎞ ⎛ z1 (t1 ) z2 (t1 ) · · · zm (t1 ) ⎜ z1 (t2 ) z2 (t2 ) · · · zm (t2 ) ⎟ ⎟ ⎜ (17) X=⎜ . .. .. ⎟ . .. ⎝ .. . . . ⎠ z1 (tN ) z2 (tN ) · · · zm (tN )


266

0.08

0.02

0.06

0.015

0.8 0.6 0.4 0.2 0

200

400

600

Z (rad/s)

(a)

800

1000

Third POM Energy

Second POM Energy

First POM energy

1

0.04 0.02 0

0

500

1000

0.01 0.005 0

0

Z (rad/s)

(b)

500

1000

Z (rad/s)

(c)

Fig. 5 The energy curve of the first three order POMs.

When calculating the correlation matrix T = XT X, the order of the matrix is m × m. The eigenvector of the matrix can therefore be denoted by ϕ1 , ϕ2 , ..., ϕm , and the corresponding eigenvalues are λ1 > λ2 > ... > λm . (2) Q is the matrix formed by the first n orders of T = XT X and it can also indicate that the matrix Q contains the first n largest eigenvalues of T. The order of Q is m × n, so we know that the order of QT Q is n × n. Taking the coordinates transformation on system coordinates Z, acquiring the new coordinates P, Z = QP, substituting Z into equation (16), then we can get the equation (18): QP¨ = −CQP˙ − KQP + F.

(18)

Due to the column vector of Q is orthogonal and nonzero, and the matrix QT Q is n order diagonal and full rank, hence there exits inverse matrix of QT Q. Taking (QT Q)−1 QT left multiplication on both sides of equation (18), we can obtain the formula (19): P¨ = −(QT Q)−1 QT CQP˙ − (QT Q)−1 QT KQP + (QT Q)−1 QT F.

(19)

Setting Cn = −(QT Q)−1 QT CQ, Kn = −(QT Q)−1 QT KQ, Fn = (QT Q)−1 QT F, then we get the equation (20): (20) P¨ = −Cn P˙ − Kn P + Fn . Similarly, the original system is transformed into the n-DOFs system by the TPOD method. 5.1.3

The efficiency of the order reduction method

Firstly, we will apply the POM energy method (it is discussed in section 4) to confirm the dimension of the reduced model. In Figs.5a-5c, the first three order POMs occupy principal energy of the original system. So the DOF number of the reduced model should be three. The energy is approximate to 100% when the speed is about 240rad/s, we can obtain the optimal order reduction model based on this speed. Secondly, we apply the TPOD method to reduce the original system to a 3-DOFs one. The initial conditions are zi = 0(i = 1, 2, . . . , 26), z˙i = 0.001(i = 1, 2, . . . , 26, i = 5, 6), z˙5 = 0.8, z˙6 = 0.5, the time step length is π /256. The time history of z1 under the initial conditions is expressed in Fig.6 when the speed is 240rad/s. It is clear that τ ∈ [0, 6π ] is the transient process, the time history is from 0 to 200π , the time history stops at 650 seconds, so the TPOD method in section 5.1.2 is applied to reduce the 26-DOFs system to a 3-DOFs one. As is shown in Fig.7, the bifurcation of the reduced system (Fig.7) reserves the dynamical characteristics of the original system (Fig.3 (b)). The bifurcation point and the topological structures of the reduced system agree well with the original one , so the efficiency of the TPOD method is verified via comparing the bifurcation diagrams based on the qualitative aspect.


267

Fig. 6 The time history of z1 under the initial conditions when the speed is 240rad/s. 1

Amplitude of z 1

0 -1 -2 -3 -4 200

300

400

500

600

700

800

900

Z(rad/s)

Fig. 7 The bifurcation diagram of the reduced system.

6 Conclusions The TPOD method has been applied in the rotor system supported by a pair of sliding bearings with the crack fault. A 26-DOFs rotor model with the crack fault has been established by the Newton’s second law. The bifurcation and amplitude-frequency characteristic of the crack has been studied, and we found that the crack could cause the 1/2 sub-harmonic resonance. The dimension of the reduced model has been confirmed by the POM energy method and the optimal order reduction model has been provided. We have used the TPOD method to reduce the original system to a three-DOFs one based on the qualitative aspect, and verified the efficiency of the order reduction method via the comparison between the original and the reduced system. The TPOD method can be used in the high-dimensional rotor system for order reduction to obtain the efficiently and accurately reduced models. Further studies on this subject will be carried out by the present authors in the two aspects as follows: the first is to analyze the bifurcation behaviors of 1/2 sub-harmonic resonance via the singularity theory; the second is to combine the POD and polynomial dimensional decomposition method [37] to study the dynamical characteristics of the rotor system with uncertainty.

References [1] Alho, A. and Uggla, C. (2015), Global dynamics and inflationary center manifold and slow-roll approximants, J. Math. Phys., 56, 012502. [2] Valls, C. (2015), Center problem in the center manifold for quadratic and cubic differential systems in R3 , Appl. Math. Comput., 251, 180-191. [3] Kazufumi, I. and Karl, K. (2008), Reduced-order optimal control based on approximate inertial manifolds

268


for nonlinear dynamical systems, SIAM J. Numer. Anal., 46, 2867-2891. [4] Marion, M. (1989), Approximate inertial manifolds for reaction–diffusion equations in high space dimension, Dyn. Differ. Equ., 1 245–267. [5] Guo, S.J. and Yan, S.L. (2016), Hopf bifurcation in a diffusive Lotka- Volterra type system with nonlocal delay effect, J. Differential Equations, 260, 781-817. [6] Marion, M. and Temam, R. (1989), Nonlinear Galerkin methods, SIAM J. Numer Anal., 5, 1139–1157. [7] Stefanescu, R., Sandu, A., and Navon, I.M. (2015), POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation, J. Comput. Physi., 295, 569-595. [8] Benaissa, B., Hocine, N.A., Belaidi, I., Hamrani, A., and Pettarin, V. (2016), Crack identification using model reduction based on proper orthogonal decomposition coupled with radial basis functions, Struc. Multidisc. Optim., DOI: 10.1007/s00158-016-1400-y. [9] Yang, H.L. and Radons, G. (2012), Geometry of inertial manifolds probed via a Lyapunov projection method, Phys. Rev. Lett., 108, 154101. [10] Rega, G. and Troger, H. (2005), Dimension reduction of dynamical systems: methods, models, applications, Nonlinear Dynamics, 41, 1–15. [11] Steindl, A. and Troger, H. (2001), Methods for dimension reduction and their application in nonlinear dynamics, Int. J. Solids Struct., 38, 2131–2147. [12] Holmes, P.J., Lumley, L., and Berkooz, G. (2012), Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press. [13] Lumley, J.L. (2001), Early work on fluid mechanics in the IC engine, Ann. Rev. Fluid Mech., 33, 319–38. [14] Vaccaro, R. (1991), SVD and Signal Processing II: Algorithms, analysis and applications, Elsevier Science Inc.. [15] Rosenfeld, A. and Kak, A.C. (1982), Digital Picture Processing, Academic Press, New York. [16] Oliveira, I. and Patera, A. (2007), Reduced-basis techniques for rapid reliable optimization of systems described by affnely parametrized coercive elliptic partial differential equations, Optimization and Engineering, 8, 43–65. [17] Georgiou, I.T. and Schwartz, I.B. (1999), Dynamics of large scale coupled structural mechanical systems: a singular perturbation proper orthogonal decomposition approach, SIAM J. Appl. Math., 59, 1178–1207. [18] Kerschen, G., Feeny, B.F., and Golinval, J.C. (2003), On the exploitation of chaos to build reduced-order models, Computer Methods in Applied Mechanics and Engineering, 192, 1785-1795. [19] Gay, D.H. and Ray, W.H. (1988), Application of singular value methods for identification and model based control of distributed parameter systems, In Proc. IFAC Workshop on Model Based Process Control, Atlanta, GA, June 1988, 95–102. [20] Preisendorfer, R.W. (1988), Principal component analysis in meteorology and oceanography, Elsevier, Amsterdam. [21] Hinze, M. and Volkwein, S. (2008), Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition, Computational Optimization and Applications, 39, 319-345. [22] Fossati, M. and Habashi, W.G. (2013), Multiparameter Analysis of aero-icing problems using proper orthogonal decomposition and multidimensional interpolation, AIAA Journal, 51, 946-960. [23] Terragni, F., Valero, E., and Vega, J.M. (2011), Local POD plus Galerkin projection in the unsteady lid-driven cavity problem, SIAM J. Sci. Comput., 33, 3538–3561. [24] Lieu, T. and Farhat, C. (2007), Adaptation of aeroelastic reduced-ordermodels and application to an F-16 configuration, AIAA Journal, 45, 1244–1257. [25] Lu, K., Chen, Y.S., Cao, Q.J., Hou, L., and Jin, Y.L. (2017), Bifurcation analysis of reduced rotor model based on nonlinear transient POD method, International Journal of Non-Linear Mechanics, 89, 83-92. [26] Lu, K., Yu, H., Chen, Y.S., Cao, Q.J., and Hou, L. (2015), A modified nonlinear POD method for order reduction based on transient time series, Nonlinear Dynamics, 79, 1195-1206. [27] Lu, K., Jin, Y.L., Chen, Y.S., Cao, Q.J., and Zhang, Z.Y. (2015), Stability analysis of reduced rotor pedestal looseness fault model, Nonlinear Dynamics, 82, 1611-1622. [28] Lu, K., Chen, Y.S., Jin, Y.L., and Hou, L. (2016), Application of the transient proper orthogonal decomposition method for order reduction of rotor systems with faults, Nonlinear Dynamics, 86, 1913-1926. [29] Lu, K., Lu, Z.Y., and Chen, Y.S. (2017), Comparative study of two order reduction methods for high-dimensional rotor systems, International Journal of Non-Linear Mechanics, DOI: 10.1016/j.ijnonlinmec.2017.09.006. [30] Adiletta, G., Guido, A.R., and Rossi, C. (1996), Chaotic motions of a rigid rotor in short journal bearings, Nonlinear Dynamics, 10, 251–269. [31] Riaz, M.S. and Feeny, B.F. (2003), Proper orthogonal modes of a beam sensed with strain gages, Journal of Vibration and Acoustics, 125, 129–131. [32] Han, S. and Feeny, B.F. (2003), Application of proper orthogonal decomposition to structural vibration


269

analysis, Mech. Syst. Signal Process, 17, 989-1001. [33] Feeny, B.F. and Kappagantu, R. (1998), On the physical interpretation of proper orthogonal modes in vibrations, Journal of Sound Vibration, 211, 607–616. [34] Sipcic, S.R. and Bengeudouar, A. (1996), Karhunen-Loeve decomposition in dynamical modeling, Sixth conference on Nonlinear Vibrations, Stability, and Dynamics of Structures, Blacksburg, VA. [35] Han, S. and Feeny, B.F. (2002), Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures, Journal of Sound and Vibration, 8, 19–40. [36] Kerschen, G., Golinval, J.C., Vakakis, A.F., and Bergman, L.A. (2005), The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview, Nonlinear Dynamics, 41, 147–169. [37] Lu, K. (2018), Statistical moment analysis of multi-degree of freedom dynamic system based on polynomial dimensional decomposition method, Nonlinear Dynamics, 10.1007/s11071-018-4303-1.



Dynamical Balance and Verification of a Rotor System Based on Sensitivity Analysis Zhong Luo1,2†, Yanhui Wei1,2 , Xiaojie Hou1,2 , Fei Wang1,2 1 2

School of Mechanical Engineering & Automation, Northeastern University, Shenyang, China Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of China, Northeastern University, Shenyang, Liaoning, China Submission Info Communicated by J.Z. Zhang Received 22 May 2018 Accepted 26 May 2018 Available online 1 October 2018 Keywords Rotor Three trial weight balancing method Modal balancing method Sensitivity

Abstract In allusion to the unbalanced vibration of rotor system, sensitivity analysis and balance method of rotor system were studied. Based on rotor dynamics, combined sensitivity analysis with the balancing methods of three trial weight and modal, a hybrid method was proposed which can be balanced without phase information. Firstly, the model of rotor was established by the finite element method and the correction plane is selected by the sensitivity analysis. Furthermore, the balance speed is determined according to the modal equilibrium theory and the weight calculation is conducted by the balancing method of three trial weight Finally, a dynamic balance experiment was carried out on the rotor model test station. The results of simulation and experiment show that the proposed balance method can not only effectively reduce the residual vibration of rotor, but avoid the blindness of selecting the correction plane and balance speed by the traditional balance method, which is expected to improve the balance efficiency. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The imbalance throughout the manufacturing process and installation of the rotor system is inevitable, furthermore, more than 75% of varieties of failures that occurred in aircraft engine, gas turbine and other large rotating machinery, are caused by imbalance. Today, the rotor system is developed towards the direction of high speed, large length-diameter ratio and high reliability, which require a simple and efficient dynamic balancing method, that has little damage on rotor system. According to the balancing principle, the method can be classified as influence coefficient method and modal balance method. During their process, the amplitude and phase angle of vibration should be measured respectively when speed balanced. However, the number of start is too much so that rotor passing by critical speed frequently effecting the life of rotor. Meanwhile, vibration phase is sensitive to vibration variations. † Corresponding


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Zhong Luo et al. / Journal of Vibration Testing and System Dynamics 2(3) (2018) 271–280

All of these give rise to it is not effective in the actually balancing process. Therefore, it is developing many methods of improving dynamic balancing based on the two balancing theory. Many researches have been conducted on the investigated investigation of to the rotor system’s dynamic balancing method. For example, Amit [1] provided a balancing method using amplitude subtraction, achieved to rotor system balancing according to the correlation of amplitude measurement at the site of support and correction plane. Preciado [2] provided a method of two plane flexible rotors balancing without phase measurement, but this method need more starting numbers. Gunter [3] used three trial weight balancing method to balance modal shape of rotor under critical speed. Nevertheless, rotor system stay in site of critical speed so long time during balancing process, it has influence reliability and service life of system. Huang [4] brought in modal ratio among measurement to balancing flexible rotor, using information of transient amplitude during rotor starting process, based on the three trial weight balancing method. But it needs so high experience of experimental operation that limited practical application and so on. Above methods have some limitations, to address these problems, Therefore, in order to improve the balance effect and confirm the influence of the imbalance on the rotor structure parameters, a variety of methods for calculating the sensitivity of the rotor system have also been developed. He [5] based on the finite element method, he derived the formula for calculating the sensitivity of the rotor’s unbalanced response to various parameters of the rotor -bearing system. Yang [6] proposed an improved transfer matrix method based on the fusion of transfer matrix and parameter matching. The rotor system was divided into multiple subsystems. The transfer matrix model was established for each subsystem, The system unbalance response equation was established with boundary conditions of each subsystem at the interface It can be easily used to analyze sensitivity of the rotor’s unbalanced response. In view of the vibration problem caused by the unbalanced rotor system, the rotor system model is established based on the finite element theory. According to the sensitivity analysis results of the rotor system unbalance quantity of the rotor system, combining the influence coefficient method and the modal balancing method, designing the reasonable balance method, determining the balance plane and balance speed., and the dynamic balancing of the rotor system is realized by the three trial weight balancing method, The validity of the method is verified by simulation and experiments.

2 Balancing techniques 2.1

Three trial weight balancing method

Take single plane balance as an example to illustrate the basic principle of the Three trial weight balancing method [7], Run the unbalanced rotor to the balanced speed and measure the original vibration then add the trial mass to the position 0◦ , 120◦ and 240◦ in succession, respectively to measure the vibration amplitude at the corresponding position. Finally calculate the trial weight and phase angle by the graphing method According to the three trial weight balancing method to balance the linear system, the following equation is established: Ak exp(iαk ) − A0 exp(iθk ) = CT exp[i(θk + β )],

k = 1, 2, 3,

(1)

√ where i = −1, A0 is original vibration amplitudes of the measuring point of the imbalance rotor system; α0 is original vibration phase angle of the measuring point of the imbalance rotor system; A1 , A2 and A3 are vibration amplitudes as adding the trial mass to the position 0◦ , 120◦ and 240◦ in succession; α1 , α2 and α3 are vibration phase angle of the measuring point as adding the trial mass to the position 0◦ , 120◦ , 240◦ in succession; C is influence coefficient of balancing plane and measuring plane; T is the trial weight; θ1 , θ2 and θ3 are the position of the trial weight add to the balancing plane; β is phase angle of the influence coefficient between the balancing plane and the measuring plane.


273

0q A0 A1 O

A3

120q

240q

A2 P

Fig. 1 Three-circle balance method for measuring rotor unbalance.

Since the vibration phase angle of the measuring point of the rotor system is not measured, the left side of Eq. (1) is the sum of the two vector and the direction is unknown, the right side of Eq. (1) is a vector with unknown magnitude and direction.In order to solving this equation requires some equivalent deformation of the equation Rotate each vector of Eq. (1) (θ0 +α0 ) clockwise: Ak exp[i(αk − α0 − θk )] − A0 exp(iθk ) = CT exp[i(β − α0 )],

k = 1, 2, 3.

(2)

Rotate each vector of Eq. (2) 180◦ clockwise and symmetry transforming with x axis −Ak exp[i(α0 + θk − αk )] + A0 exp(iθk ) = −CT exp[i(β − α0 )],

k = 1, 2, 3,

(3)

and Ak exp[i(π + α0 + θk − αk )] + A0 exp(iθk ) = CT exp[i(π + β − α0 )],

k = 1, 2, 3,

(4)

the left side of Eq. (4) is the sum of the two vector and the one size and direction is known the another direction is unknown. The solution of the equation can be used as a graph method. A point O is chosen as the center and A0 is a radius of the circle. At the positions of θ1 , θ2 , and θ3 in the circumference, draw three circles with the radius of A1 , A2 and A3 . P is three circle intersection OP right at the vector equation. The influence coefficient is C = |OP|/T . The balance vector of the rotor system is W = A0 /C, and the direction is in the same as OP. When balancing the specific rotor system, the specific balancing steps are: i Driving the rotor to the balance speed, measuring original vibration amplitudes of the measuring point of the imbalance rotor is A0 . ii Adding the trial mass T to the position 0◦ , 120◦ and 240◦ in succession and measuring vibration amplitudes of the measuring point of the rotor is A1 , A2 and A3 . iii Then take any point O on the plane as the center of the circle, and A0 as the radius to make a circle, for the convenience of description which is named as base circle as shown in Figure 1. At the position 0◦ , 120◦ and 240◦ in the circumference, draw three circles with the radius of A1 , A2 and A3 , and P is three circle intersection. iv Connecting OP, the size of the OP represents the amount of vibration caused by the imbalance, the direction of the OP means the direction that to reduce vibration amplitudes of the rotor should be added to the Correcting weight, and the size of the counterweight is W = A0 OP T .

274


s u s

K

[

o

s I1 s

K

[

D1

o I2 s

s

K

[

D2

o

s I3 s

K

[

D3

o Fig. 2 Unbalanced mode decomposition.

2.2

Modal balancing method

According to the vibration theory, the unbalance quantity of the rotor can be decomposed into many unbalanced components basis on the main vibration mode [8], which can be shown in Figure 2. Each component can only arouse one main vibration type, and the modes of each are orthogonal to each other. The balance of one of the modes will not affect the other modes. In theory, the rotor vibration can be completely eliminated by balancing each order of vibration mode one by one. As the rotor has infinite main vibration which can be shown in the following equation (5), it is impossible to achieve complete balance in fact, and the amplitude of the first several main modes is the largest, so only the vibration of the main modes before the balance is needed, and the vibration of the rotor will be very small. ˆ l 0, n = k, (5) m(s)φn (s)φk (s)ds = N n , n = k, 0 where Nn is the modulus or modal mass of mode n vibration. With the main modal orthogonality theory, each step modes of the imbalances u(s) can be expressed as ∞

u(s) =

∑ uneiα ϕn(s). n

(6)

n=1

Balancing the first three vibration mode of the rotor as an example, three correction planes are required, and the trial weights of T1, T2, and T3 are respectively added. After balancing, the vibration amplitude of the first three modes is zero, which can be shown in the following Eq. (7), if Eq. (7) is multiplied by ω 2 on both sides, then the left side of the equation is expressed as the work done by the three weights and the third-order vibration, respectively, on the unbalanced forces generated by the third-order primary vibration. That is, the work done by the unbalanced force of the correction weight is equal to the work done by the original unbalanced force of the rotor in the opposite direction, so


that the balance can be achieved. ⎧ ⎪ ⎨ϕ1 (s1 )T1 + ϕ1 (s2 )T2 + ϕ1 (s3 )T3 = −u1 N1 , ϕ2 (s1 )T1 + ϕ2 (s2 )T2 + ϕ2 (s3 )T3 = −u2 N2 , ⎪ ⎩ ϕ3 (s1 )T1 + ϕ3 (s2 )T2 + ϕ3 (s3 )T3 = −u3 N3 . 2.3

275

(7)

Hybrid balancing method

Combined the characteristics of the three trial weight balancing method with the mode balancing method developed a hybrid balancing method. Which only measuring the amplitude of the vibration of the rotor, the balance calculation is simple and balance different modes of vibration do not influence each other. It reduces the influence of the three trial weight balancing method on the vibration amplitude of the balance at multiple and multi plane balancing processes, and determines that the balance speed can only be selected near the critical speeds of each balance. If the working speed range of the rotor exceeds the critical speed of the multistage, the correction plane corresponding to the critical speed order should be selected correspondingly. When balancing the rotor with the hybrid balance method, if the correct surface can be selected based on the vibration mode of the rotor and the sensitivity of the disc pair imbalance, the balance result will be greatly improved.

3 Sensitivity analysis of rotor unbalance For the continuous differentiable multivariate function Y = Y (X1 , X2 , · · · , Xn ), when the independent variable produces increment ΔXi, the dependent variable will generate ΔY accordingly, and the degree of change caused by one variable to another can be expressed by its rate of change ΔY ΔXi. The above is the general concept of sensitivity. In order to facilitate computation, measurement and analysis, we can use the following equation to describe the effect of one parameter on another parameter. ∂Y Y × 100%. (8) S= ∂X X In which S is sensitivity, that is, sensitivity is defined as the ratio of the relative change of dependent variable to the relative change of independent variable. According to the definition of sensitivity, the vibration displacement response of rotor system along the axial position is regarded as dependent variable and the unbalance force of the rotor system is regarded as the independent variable [17,18]. If it is known that the unbalanced force F acts on the rotor system, through the equations of the rotor system, the vibration displacement response of the rotor system along each axial position YZ is solved. Therefore, the sensitivity SR of the unbalanced response at different positions on the rotor can be expressed as ΔYZ YZ . (9) SR = ΔF F In order to avoid the partial derivative matrix generated by each element, and to reduce the workload of computers and increase the computation speed, incremental “Δ” is instead of partial derivative “∂ ”. As long as ΔF is small enough, the sensitivity of each node’s imbalance response can be well solved. 4 Simulation verification For the purpose of verifying the validity of the balance method that we propose , a rotor system simulation model is established as shown in Figure 3. Assuming the same material for the shaft and


276

Plane 1

Plane 2

Plane 3

shaft

Plane 4

z o

y

x k1

c1

k2

c2

Support2

Support1

Fig. 3 Rotor system structure. Table 1 Structure parameters of rotor. Outside Inside Outside Inside Outside Axial Shaft Length Shaft Length Disk Width diameter diameter diameter diameter diameter position section l(mm) section l(mm) number B(mm) R(mm) r(mm) R(mm) r(mm) R(mm) L(mm) 1

71

35

0

6

134

70

56

1

300

11

214

2

16

35

0

7

68

70

56

2

300

11

315

3

86

38

0

8

11

70

0

3

300

11

449

4

11

70

0

9

55

40

0

4

300

11

583

5

30

70

56

10

61

40

0

Table 2 Sensitivity of plane and support to unbalanced response. Corresponding point

Disk 1 sensitivity (%)




Support 1

2.3

0.9

0.3

1.9

Support 2

1.4

0.7

1.5

5.2

Disk 1

1.7

0.9

0.9

0.5

Disk 2

1.3

0.9

1.7

2.0

Disk 3

0.9

1.0

2.7

3.8

Disk 4

4.4

2.8

3.4

8.8

plane, the modulus of elasticity E is 2.09 × 1011 Pa, and the density ρ is 7850kg/m3 , the poisson’s ratio is 0.3. The structural parameters of the rotor system as table 1 shows. Add an unbalanced weights 800g · mm∠70◦ to plane 1and 500g · mm∠130◦ to plane 2, respectively. Plane 1, plane 2, plane 3 and plane 4 can be used as correction plane, the two plane with the highest sensitivity are selected as the balance planes, according to the of sensitivity analysis results of the unbalance of the plane. Add 100g · mm∠60◦ unbalanced weights to each of the four balance planes, respectively, the corresponding increment of each node is collected, and then the sensitivity response of the balance plane to each node is derived as shown in Table 2. Acording to the calculation results, it’s known that the vibration effects of the rotor support at plane 1 and plane 4 is relatively large, being 2.3% and 5.2%, respectively.and the unbalanced quantities of the plane 4 is also sensitive to the self-vibration. In a comprehensive comparison , plane 1 and plane 4 are selected as correction surfaces in order to improve the efficiency . The modal analysis of the rotor system model can be used to obtain the first and second critical speeds of 2400r/min and 4600r/min, respectively. To verify the validity of the balance method, neglecting the influence of excessively large vibration


277

Table 3 Balancing record of simulation. Balance speed

Position

2400 r/min Correction plane 1 4800 r/min

Trial weight (g) — 30∠0◦ 30∠120◦ 30∠240◦ — 30∠0◦ 30∠120◦ 30∠240◦

Vibration of point 1 (mm) 0.1368 0.3078 0.4803 0.2916 0.2063 0.3642 0.5009 0.1776

Correction weight (g)

Correction plane 4

— 30∠0◦ 30∠120◦ 30∠240◦ — 30∠0◦ 30∠120◦ 30∠240◦

19.4∠272◦

0.14

%HIRUHEDODQFH $IWHUEDODQFH

0.15

Vibration Amplitude PP


Trial weight (g)

12.4∠296◦

0.2

0.1

0.05

0

Position

0

1000

2000

3000 4000 Rotor Speed (r/min)

Vibration of point 4 (mm) 0.1532 0.4541 0.3856 0.4796 0.1014 0.2924 0.4009 0.4677

Correction weight (g) 4.1∠93◦

7.8∠25◦


0.12 0.1 0.08 0.06 0.04 0.02 0

5000 6000

(a) Vibration response of support 1

0

1000

2000

3000

4000

Rotor Speed (r/min)

5000 6000

(b) Vibration response of support 2

0.3



0.2


0.25 0.2 0.15 0.1

0.05 0

0

1000

2000

3000

4000 Rotor Speed (r/min)

5000

(c) Vibration response of plane 1

6000


0.15

0.1

0.05

0

0

1000

2000

3000

4000

Rotor Speed (r/min)

5000 6000

(d) Vibration response of plane 4

Fig. 4 Vibration response before and after balancing at different axial positions.

of the model rotor on the stability of the rotor at the critical speed. Therefore, the first and second critical speeds is selected as the balance speed. The balancing steps are as described above, the balancing records are shown in Table 3. The vibration response of the rotor system at the support 1, the support 2, the correction plane1, and the correction plane4 , respectively, when the dynamic balance speed from 0 to 6000r/min is shown in Figure 4. Vibration response of each node at different speeds before and after balancing is shown in Figure 5 . According to the simulation balance, the vibration of the rotor system is effectively suppressed, and the vibration amplitudes reduction at the support and correction planes are all above 95%, which shows that the comprehensive balance method proposed in this paper is feasible.


278

1RGH

6SHHG˄UPLQ˅

(a) Before balancing

(b) After balancing

Fig. 5 Vibration response of each node at different speeds before and after balancing. Plane1 Plane2 Plane3 Plane4

Point1

Point 2

Fig. 6 Rotary test station and measuring point layout.

5 Experimental verification Experiments on a rotor system to further verify the effectiveness of the balance method. Rotary bench structure shown in Figure 6. The stiffness of the support is 2 × 106 N/m, Four disks on the shaft can be used as balance correction planes. Trial weights were added by connecting different quality screws and washers through threaded holes on the correction planes. A cDAQ9188 chassis, NI9229 acquisition card, and Lab VIEW acquisition software were used to establish a signal acquisition and analysis system to measure the unbalanced response of the rotor system. The eddy current displacement sensor selected for the test is CWY-DO-502 with a sensitivity of 4mV/um. Sensors are arranged horizontally and vertically at the measuring point, respectively. According to the principle of modal balancing, The dynamic balance at the critical speed of the rotor system has palpable vibration damping effect on the vibration response of the rotor system at any speed below the critical speed [3]. With the limitation of the rotation speed of the system, this experiment only balances at the first-order critical rotation speed. In order to accurately obtain the critical speed of the rotor system, Test the vibration response of the test bench at a speed of 1003000r/min, after sweep frequency processing, it can be concluded that the first critical speed of the rotary test station is 2400r/min. Because the vibration amplitude of the rotor system is small at the first critical speed, the short time stays will not cause great influence on the test station, So the balance speed is selected as 2400r/min. According to the sensitivity analysis of the turntables. Select disk 1 and disk 4 as the correction planes, and balance the rotor system according to the three trial weight balancing method. Balance data record is shown in Table 4. Due to the limitation of the balance plate of the rotor test station, the correction weight can’t be added according to the calculation results in the actual balancing process. Therefore, the correction weight added to the correction plane 1 and the correction plane 4 are 44.2g∠15◦ and 12.4g∠45◦


279

Table 4 Balancing record of experiment. Balance speed

Position

2400 r/min

Correction plane 1

Trial weight (g) — 68.3∠0◦ 68.3∠120◦ 68.3∠240◦

Vibration of point 1 (mm) 0.2075 0.1716 0.1691 0.2172

Correction weight (g)

Position

Trial weight (g)

44.2∠14◦

Correction plane 2

— 33.1∠0◦ 33.1∠120◦ 33.1∠240◦

Vibration of point 4 (mm) 0.1046 0.0772 0.1002 0.1291

Correction weight (g) 12.4∠40◦

Table 5 Vibration reduction of balancing. Position

Before/mm

After/mm

Vibration reduction rate

Point 1

0.2075

0.0256

87.64%

Point 2

0.1046

0.0262

75.48%

(a) Measuring point 1-Left support

(b) Measuring point 2-Right support

Fig. 7 Comparison of vibration amplitude before and after balancing of measurement points 1 and 2.

respectively. Before and after dynamic balancing, the vibration amplitude of the rotor system at 100-3000r/min speed is shown in Figure 7. The specific value is shown in Figure 5. It can be seen that after the dynamic balance of the rotor system, the vibration amplitude of each rotor speed decreases obviously, which proves that this method is effective. At the same time, the dual plane three trial weight balancing method requires the rotor system to start 7 times, thus limiting the application. Subsequent research can improve the balance method and make it widely used in balancing complex rotor systems.

6 Conclusions In this paper proposes a hybrid balance method combining the three trial weight balancing method and the modal balancing method, according to the results of the sensitivity analysis of the unbalance, selecting the plane which has the maximum sensitivity to the imbalance, detailed explanation of the balance process. The results of simulation and balance experiment show the effectiveness of the balancing method. The following conclusions can be drawn: i This balance method combined with the idea of modal balancing method needs to know the critical speed of rotor system and the amplitude of balancing speed without phase measurement. ii The balance method needs to calculate the sensitivity of each plane to the imbalance, and select the

280


most sensitive plane as the correction plane, which avoids the blindness of the previously selected correction plane and improves the balance rate and effect.

Acknowledgements This work was supported by the National Science Foundation of China under the grant number 11572082; the Fundamental Research Funds for the Central Universities of China under the grant numbers N170308028 and N160312001; and the Excellent Talents Support Program in Institutions of Higher Learning in Liaoning Province of China under the grant number LJQ2015038.

References [1] Preciado, D.E. and Bannister, R.H. (2002), Balancing of an experimental rotor without trial runs, International Journal of Rotating Machinery, 8(2), 99-108. [2] Gunter, E.J., Springer, H., and Humphris, R.R. (1982), Balancing of a multimass flexible rotor-bearing system without phase measurements, Energia Elettrica, 59(10), 383-389. [3] Huang, J.P., Ren, X.M., and Deng, W.Q. (2010), Novel modal balancing of flexible rotor by using the run-up amplitude, Journal of Mechanical Engineering, 46(5), 55-62. [4] He, Y.Z., Wang, Z., and Sun, J.W. (1997), Analysis of the unbalance response sensitivity of a rotor bearing system, Journal of Tsinghua University, 37(8), 9-11. [5] Yang, J.G. (2001), An improved transfer matrix method for calculating the unbalanced response and sensitivity of a rotor system, Journal of Mechanical Engineering, 37(6), 109-112. [6] Huang, J., Ren, X., Deng, W., and Liu, T. (2010), Two-plane balancing of flexible rotor based on accelerating unbalancing response data, Acta Aeronautica Et Astronautica Sinica, 31(2), 400-409. [7] Zhang, H.B. (2000), Aero Engine Design Manual Nineteenth, Aeronautical Industry Press: Beijing (in Chinese) [8] Yang, X.D., Zhang, W., Chen, L.Q., and et al. (2012), Dynamical analysis of axially moving plate by finite difference method, Nonlinear Dynamics, 67(2), 997-1006. [9] Chen, X., and Liao, M.F.(2017), Field balancing technology for low pressure rotors of high bypass ratio turbofan enginers, Journal of Aerospace Power, 32(4), 809-819. [10] Cao, H., Zhang, X., Chen, X. (2016), The concept and progress of intelligent spindles: A review, International Journal of Machine Tools & Manufacture, 112, 21-52. [11] Zhao, M., Lin, J., Xu, X., and et al. (2014), Multi-fault detection of rolling element bearings under harsh working condition using IMF-based adaptive envelope order analysis, Sensors, 14(11), 20320-20346. [12] Zhang, D., Liu, S., Liu, B., and et al. (2013), Investigation on bending fatigue failure of a micro-gear through finite element analysis, Engineering Failure Analysis, 31(6), 225-235. [13] Ma, H., Sun, W., Wang, X.J., and Wen, B.C. (2009), Characteristic analysis of looseness fault in rotor system, Journal of Northeastern University(Natural Science), 30(3), 400-404. [14] Bin, G.F., He, L.D., and Gao, J.J. (2013), High-speed dynamic balancing method for low pressure rotor of a large steam turbine based on modal shape analysis, Vibration and shock, 32(14), 88-92. [15] Han, Q. and Chu, F.(2015), Parametric instability of flexible rotor-bearing system under time-periodic base angular motions, Applied Mathematical Modelling, 39(15), 4511-4522. [16] Qin, Z., Yan, S., and Chu, F. (2014), Influence of clamp band joint on dynamic behavior of launching system in ascent flight, Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering, 228(1), 97-114. [17] Bhende, A.R. (2017), A new rotor balancing method using amplitude subtraction and its performance analysis with phase angle measurement-based rotor balancing method, Australian Journal of Mechanical Engineering, 2017(8), 1-7.



Multi-objective Optimization Design of Flap Sealing Valve Structure for Deep Sea Sediment Sampling Guangping Liu†, Yongping Jin, Youduo Peng, Buyan Wan National-Local Joint Engineering Laboratory of Marine Mineral Resources Exploration Equipment and Safety Technology, Hunan University of Science and Technology, Xiangtan, Hunan, P. R. China Submission Info Communicated by J.Z. Zhang Received 22 May 2018 Accepted 29 May 2018 Available online 1 October 2018 Keywords Flap seal valve Optimization design Finite element analysis Experimental research

Abstract Taking the deep sea sediments air-tight sampler flap seal valve as the research object, using the response surface method (RSM) and finite element analysis method to carry out the flap sealing valve geometry design parameter optimization design research. First of all, a set of eccentrically set flap seal valve is designed for the requirement of the pressurized seal of the marine deep sediment air-tight sampler, three-dimensional geometric model of the flap seal valve is built using Solidworks.Then, The flap sealing valve eccentricity angle θ , the valve cap upper end diameter D3 , and the valve cover length l as the design variables, the bottom end opening diameter D5 of the valve body and the material strength under a given safety factor are set as constraints, the maximum stress value and weight of the flap seal valve are set as optimization goals.The response surface model of the flap sealing valve eccentricity angle θ , the valve cap upper end diameter D3 , the valve cover length l and the flap sealing valve maximum stress and weight is constructed. Finally, The geometrical structure parameters of the flap sealing valve are optimized. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The ocean is a treasure trove of human common resources, rich in minerals, biology, oil and gas, and many other resources. However, deep sea sediments contain a large number of life communities such as microbes, which are important ways for humans to understand and study the evolution of deep-sea life and deep-sea environment changes [1–4]. In order to obtain the pressure-enhancing microorganisms in deep sea sediments, the development of an all-sea deep-seated airtight sampler must be carried out, and the flap sealing valve is an important part of the sea-deep sampler for deep sea sediments, Realization of hermetic sealing of deep-seated hermetic samplers is essential [5–10]. Since deep-sea sediment samplers are operated by submersible manipulators, in view of the constraints imposed by the manipulator load, flaps should be reduced as much as possible during the design process of the all-sea deep-deposited airtight sampler flap seal valves. the weight of the flap sealing valve should be reduced as much as † Corresponding


282

Guangping Liu et al. / Journal of Vibration Testing and System Dynamics 2(3) (2018) 281–290

possible. However, the flap sealing valve is in an ultra-high pressure environment. For safety needs, the maximum stress on the flap sealing valve should be reduced as much as possible. For this contradiction problem, in the design of the sealing performance of the flap sealing valve, the two different goals of the flap sealing valve weight and the maximum stress should be taken into consideration. Because the two targets are in contact with each other and conflict with each other, that is, excessive reduction of the weight of the flap sealing valve may cause the maximum stress value of the flap sealing valve to be increased. Therefore, we must conduct in-depth research on the organization and coordination of the two goals. In this paper, the all-sea deep-seated gas-tight sampler flap sealing valve is taken as the research object. Finite element analysis of the whole sea-deeper gas-tight sampler flap sealing valve is carried out using finite element software ANSYS [11–13]. An experimental design method was used to study the relationship between the eccentric angle θ of the flap sealing valve, the diameter of the upper end of the valve cap D3 , the length of the busbar of the valve cover l, and the weight and maximum stress of the flap sealing valve of the deep sea sediment sampling filter. The weight and maximum stress of the flap sealing valve are the optimization objectives. The geometric design parameters of the flap sealing valve are multi-objectively optimized.

2 Valve sealing valve structure parameters and numerical simulation method 2.1

Flap seal valve design

In this paper, the flap sealing valve required for the research project “Development of a deep-sea sediments airtight sampler” of the national key R&D program is studied. According to the technical specifications of the subject, the inner diameter of the sampling tube is D1 = 42mm, and the outer diameter is D2 = 45mm. In the preliminary design of this paper, the height of the flap sealing valve is h1 = 70mm, the bottom opening diameter of the valve body is D5 , the diameters of the upper and lower ends of the valve cover are D3 = 70mm, D4 = 46mm, and the width of the valve body is L = 194mm, taper θ of valve cover and horizontal plane. According to the design requirements, the material of the flap sealing valve is selected as stainless steel 316. The mechanical properties of stainless steel 316 are: tensile strength σb ≥ 520 MPa. To ensure the safety of the flap sealing valve, the safety factor is selected to be nb = 1.5. In this type of flap sealing valve device, the valve cover and the valve body are eccentrically arranged to ensure a reliable sealing of the valve cover. A large gap between the valve cover and the bonnet shaft is used to fit the two purposes, the bonnet automatically adjusts its position in the valve body’s conical surface and is not constrained by the shaft, which helps seal the bonnet [14,15]. The schematic diagram of the flap sealing valve is shown in Figure 1. The three-dimensional structure of the flap sealing valve is shown in Figure 2. For sampling tubes with an inner diameter D1 of the airtight sampler, if the bonnet is maintained in pressure from the outside to the inside, the bonnet sealing tension is F and the sampling water depth is h. 4F . (1) h≥ ρ gπ D21 When Eq.(1) is satisfied, the pressure difference between the inside and outside generated after the sampling is completed will open the valve cover and affect the seal. It is known that the inner diameter of the airtight sampler is D1 = 42mm. When sampling from 10000 to 11000m sea bottom, the sealing tension of the valve cover can be F 1.34 × 105 N. It is difficult to achieve the sealing force of the valve cover under without power. Therefore, the airtight sampler must use a bonnet seal from inside to outside. When the sampling of the airtight sampler is completed, the flap sealing valve is sealed from the


283

Fig. 1 Schematic diagram of flap seal valve.

Fig. 2 3D structure diagram of flap seal valve.

inside to the outside. At this time, the upper end of the flap sealing valve cap is subjected to a pressure of 110 MPa. Therefore, the stress on the valve cover of the flap valve is

σ=

F1 . A

(2)

The surface area of the flap sealing valve cover is. A = π (r32 + r42 + r3 l + r4 l). The length of the busbar for the valve cover l is l = h21 + (r32 − r42 ),

(3)

(4)

h1 is the height of the valve cover, since the stress on the valve cover of the flap valve should be less than the allowable stress of the material, so

σ ≤ [σ ].

(5)

The relationship between the diameter of the bottom opening of the valve body and the diameter of the bottom of the valve cover is (6) r4 cos θ = r5 .

284


Table 1 Flap sealing valve parameters. h1

D3

D4

θ

L

D5

l

70

70

46

6◦

194

45.7

50

Fig. 3 Body meshing.

Fig. 4 Bonnet meshing.

Fig. 5 Stress and deformation clouds of flap sealing valve.

It is known that the maximum coefficient of static friction between stainless steel and stainless steel under seawater lubrication conditions is 0.1, i.e., tan5.8◦ . It is known from Eq. (6) that increasing the eccentricity angle θ will make the opening diameter of the bottom end of the valve body smaller, and the sampling tube cannot enter the flap sealing valve. From (2) to (6), the range of the eccentricity angle θ can be changed from 5.8◦ to 10.2◦ . The eccentricity angle is set to θ = 6◦ . The preliminary design parameters of the flap sealing valve are shown in Table 1. 2.2

Finite element analysis of flap sealing valve

In order to facilitate the simulation analysis, the model structure is simplified during the construction of the finite element model of the flap sealing valve. Then, the valve body and valve cover of the flap sealing valve are meshed separately, as shown in Figs. 3 and 4. Before the finite element analysis, first of all, the material of the flap sealing valve needs to be defined. According to the requirements of the subject, the material of the flap sealing valve body and the flap sealing valve cap is defined as stainless steel 316, and the elastic modulus is 200 GPa., Poisson ratio is 0.3, the valve body and valve cover contact unit is set to Frictional unit, the valve body contact surface is set as the target surface, the valve cover contact surface is set as the contact surface, the friction coefficient between the two is set to 0.1; The lower end of the bonnet is constrained upwards, and the bonnet is in contact with the lower portion of the annular O-ring. The outer surface above the O-ring of the valve body and the upper surface of the bonnet are applied with a uniform pressure of 110 MPa. Figs. 5 and 6 show the stress and deformation cloud diagrams of the flap sealing valve and the


285

Fig. 6 Stress and deformation clouds of flap sealing valve cover. Table 2 Factor levels and coding. X1

X2

X3

+1

10◦

70

50

0

8◦

68

48

-1

6◦

66

46

Table 3 Test plan. RUN

X1

X2

X3

Weigh/kg

Maximum stress/MPa

1

1

-1

0

4.36

304.23

2

1

1

0

4.32

306.12

3

-1

1

1

4.74

290.54

4

1

1

1

4.38

309.95

5

0

0

0

4.50

296.17

6

-1

0

1

4.72

284.32

7

-1

1

0

4.73

285.03

8

1

0

1

4.34

302.45

9

1

0

0

4.28

301.26

10

-1

0

0

4.68

282.14

11

0

-1

-1

4.42

290.23

12

0

1

-1

4.46

292.18

13

-1

-1

0

4.64

280.03

14

0

-1

1

4.43

291.27

15

0

1

1

4.55

299.58

16

1

0

-1

4.30

304.73

17

-1

0

-1

4.59

281.98

valve cover in the deep sea and deep high pressure environment (110MPa). From Figs. 5 and 6, it can be seen that the maximum deformation of the flap sealing valve is 0.0179mm; the maximum stress is 284.16MPa, and the maximum stress occurs at the periphery surrounded by the sealing ring, the maximum deformation occurs at the top of the valve cover, and the overall stress distribution is relatively uniform, maximum stress value is less than the allowable stress of the material. It is safe, but it can be further optimized.

286


Table 4 Regression analysis results of flap sealing valve weight. Source

Sum of Squares

df

Mean square

F value

P-value Prob > F

Model

0.41

9

0.045

41.93

< 0.0001

X1

0.32

1

0.32

299.4

< 0.0001

X2

0.00056

1

0.00056

5.24

0.0560

X3

0.00094

1

0.00094

8.79

0.0210

X1 X2

0.00022

1

0.00022

2.10

0.1908

X1 X3

0.00061

1

0.00061

0.57

0.4745

X2 X3

0.00016

1

0.00016

1.50

0.2606

X12

0.00008

1

0.00008

0.80

0.4005

0.00001

1

0.00001

0.10

0.7555

0.00001

1

0.00001

1.19

0.3106

X22 X32 Residual

0.00075

7

0.00075

Lack of Fit

0.00007

3

0.00007 0

Pure Error

0

4

Cor Total

0.41

16

3 Response surface modeling and analysis 3.1

Experimental design

To study the influence of the design parameters of the flap sealing valve on the maximum stress and weight of the flap sealing valve, consider that the outer diameter D2 of the sampling tube and the opening diameter D5 of the bottom end of the valve body of the flap sealing valve match each other. Therefore, the design parameters of the flap sealing valve (eccentricity angle θ , diameter of the upper end of the valve cap D3 , length of the busbar of the valve cover l) are selected as design variables. The experimental design method was used to reveal the relationship between the eccentric angle θ of the flap sealing valve, the diameter of the upper end of the valve cap D3 , the length of the busbar l of the valve cover, and the maximum stress and weight of the flap sealing valve. So as to provide reference for multi-objective optimization design of flap sealing valve. Considering various methods of experimental design, this paper uses BOX-BEHNKEN experimental design to arrange the experimental scheme [16]. Prior to the experimental design, the range of variation of each design variable was determined by a single design variable factor experiment. The results of the single design variable factor experiment showed that: the eccentric angle θ of the flap sealing valve, the diameter of the upper end of the valve cap D3 , and the length of the busbar of the valve cover l. The maximum value and weight of the plate seal valve have an effect. The variation range of each design variable in the experiment was determined by a single design variable factor experiment (the θ , D3 and l are replaced by X1 , X2 , and X3 , respectively, for ease of analysis). The levels and codes of the factors are shown in Table 2, and the experimental scheme is shown in Table 3. 3.2

Analysis of experimental results

The design parameters of the sea-depth sediment flap sealing valve eccentric angle θ , the valve cap upper end diameter D3 , and the length of the valve cap’s busbar l were used to perform response surface regression analysis with the flap sealing valve weight and maximum stress as the response variables. The results are shown in Tables 4 and 5. From Tables 4 and 5, it can be seen that the regression analysis test is significant and can better reflect the relationship between the weight and maximum stress of the flap sealing valve and the


Table 5 Maximum stress regression analysis results for flap sealing valves. Source

Sum of Squares

df

Mean square

F value

P-value Prob > F

Model

1438.61

9

159.85

34.11

< 0.0001

X1

1242.91

1

1242.91

265.2

< 0.0001

X2

49.58

1

49.58

10.58

0.0140

X3

14.72

1

14.72

3.14

0.1196

X1 X2

1.43

1

1.43

0.31

0.5979

X1 X3

4.22

1

4.22

0.90

0.3742

X2 X3

20.54

1

20.54

4.38

0.0746

X12

0.34

1

0.34

0.072

0.7966

0.53

1

0.53

0.11

0.7469

X32

0.029

1

0.029

0.006

0.9391

X22 Residual

32.80

7

4.69

Lack of Fit

32.76

3

32.76 27.42

Pure Error

27.42

4

Cor Total

1471.41

16

Fig. 7 3-D response surface surface and contour map.

287

288


Fig. 8 Relationship between the actual value of the weight and maximum stress of the flap sealing valve and the predicted value.

Fig. 9 Stress and deformation clouds of flap sealing valve.

eccentric angle θ of the flap sealing valve, the diameter of the upper end of the valve cap D3 , and the length of the busbar of the valve cover l. That is, by changing the eccentric angle θ of the flap sealing valve, the diameter of the upper end of the valve cap D3 , and the length l of the valve cover, the flap sealing valve weight and the maximum stress value can be effectively reduced. The eccentricity angle θ of the flap sealing valve, the diameter D3 of the upper end of the valve cap, the response length of the valve body length l to the weight and maximum stress of the flap sealing valve and their contours are shown in Fig. 7. Multivariate regression fitting was performed on each factor to obtain a regression model with the flap sealing valve W (X1 , X2 , X3 ) and the maximum stress F2 (X1 , X2 , X3 ) as the objective function: W = 4.48 − 0.17X1 + 0.025X2 + 0.033X3 − 0.021X1 X2 − 0.011X1 X3 + 0.018X2 X3 + 0.018X12 + 0.0005714X22 − 0.019X32 , F2 = 293.65 + 10.63X1 + 2.38X2 + 1.29X3 − 0.53X1 X2 − 0.9X1 X3 + 2.05X2 X3 − 0.35X12 + 0.39X22 − 0.092X32 .

4 Multi-objective optimization and analysis of structure parameters of flap sealing valve As shown in Fig.8, the actual value and predicted value of the weight and maximum stress of the flap sealing valve are distributed on a straight line. This model better reflects the weight and maximum stress of the flap sealing valve and the eccentric angle θ of the flap sealing valve. The model better reflects the relationship between the weight and maximum stress of the flap sealing valve and the eccentric angle θ of the flap sealing valve, the diameter of the upper end of the valve cap D3 , and the length of l the valve cover. The obtained regression model can better predict the change law of weight


289

Fig. 10 Stress and deformation clouds of flap sealing valve cover.

and maximum stress of the flap sealing valve with each parameter. Therefore, Therefore, the regression model can be used to determine the optimum eccentric angle θ of the flap sealing valve, the diameter of the upper end of the valve cap D3 , and the length of l the valve cover. According to the quadratic polynomial of the regression model, when the maximum stress of the flap sealing valve is less than the allowable stress of the material, the minimization of the weight of the flap sealing valve is the optimization goal, and the opening diameter D5 of the bottom end of the valve body are taken as the constraint condition, after optimization, The conditions for obtaining the minimum weight of the flap sealing valve are: X1 = 1, X2 = 1, and X3 = −1. At this time, the weight of the pressure-keeping container is 4.18kg, and the maximum stress is 292.34MPa, which translates into the actual parameter θ = 10◦ , D3 = 70, l = 46, and the weight of the original flap seal valve was reduced by 1.18%. The finite element analysis of the optimized flap sealing valve was performed using the Workbench module in the finite element software ANSYS. Its analysis structure agrees with the optimization results. Figures 9 and 10 show the stress and strain diagrams of the flap valve after optimization.

5 Conclusion In order to meet the needs of pressure-sealing and sealing of deep-seabed airtight samplers, a set of eccentrically-installed flap sealing valves was designed. The stress and strain of flap valve body and valve cover were analyzed using finite element analysis software ANSYS. Using the BOX-BEHNKEN test design and response surface methodology, a mathematical model was established for the relationship between the weight and maximum stress and eccentricity angle θ , the diameter of the upper end of the valve cap D3 , and the length of the valve cover l of the all-sea deep sediment flap sealing valve. The regression model can well reflect the relationship between the pressure holding container weight and the maximum stress and eccentricity angle θ , the diameter of the upper end of the valve cover D3 , and the length of the busbar of the valve cover l.

Acknowledgement This project was supported by the National Natural Science Foundation of China (Grant No.51705145, 51779092), and the National Key Research and Development Program of China (Grant No.2016YFC0300502, 2017YFC0307501).

290


References [1] Jiang, B.G. (2011), Study on Industrialization Development of Deep-Sea Strategic Resources Exploitation in China, Ocean University of China. [2] Joung, T.H., Lee, J.H., Nho, I.S., and et al. (2008), A study on the pressure vessel design, structural analysis and pressure test of a 6000 m depth-rated unmanned underwater vehicle, Ships & Offshore Structures, 3(3), 205-214. [3] Ng, R.K.H., Yousefpour, A., Uyema, M., and et al. (2002), Design, analysis, manufacture, and test of shallow water pressure vessels using E-Glass/Epoxy woven composite material for a semi-autonomous underwater vehicle, Journal of Solid State Electrochemistry, 16(2), 429-434. [4] Joung, T.H., Lee, J.H., Nho, I.S., and et al. (2009), A study on the design and manufacturing of a deepsea unmanned underwater vehicle based on structural reliability analysis, Ships & Offshore Structures, 4(1), 19-29. [5] Qin, H.W., Chen, Y., Gu, L.Y., and et al. (2009), The development of gas-tight sampling techniques, Journal of Tropical Oceanography, 28(4), 42-48. [6] Tan, F.L. and Zhang, S.Z. (2002), Introduction to coring technology by drilling of gas hydrates, Geological Science and Technology Information, 21(2), 97-99. [7] Li, X.G., Xu, J., and Xiao, X. (2013), High pressure adaptation of deep-sea microorganisms and biogeochemical cycles, Microbiology China, 40(1), 59-70. [8] Mian, H.H., Wang, G., Dar, U.A., and et al. (2013), Optimization of composite material system and lay-up to achieve minimum weight pressure vessel, Applied Composite Materials, 20(5), 873-889. [9] Paknahad, A. and Nourani, R. (2014), Mix model of FE method and IPSO algorithm for dome shape optimization of articulated pressure vessels considering the effect of non-geodesic trajectories, Journal of The Institution of Engineers (India): Series C, 95(2), 151-158. [10] Schultheiss, P., Holland, M., and Humphrey, G. (2009), Wireline coring and analysis under pressure: recent use and future developments of the hyacinth system, Scientific Drilling, 7(7), 2809-2818. [11] Ji, F.L., Wang, Y.J., Chen, W.S., and et al. (2009), The high-pressure vessel contour structure size optimization analysis based on SolidWorks simulation, Machinery, 36(8), 33-36. [12] Yun, T.S., Lee, C., Lee, J., and et al. (2011), A pressure core based characterization of hydrate-bearing sediments in the Ulleung Basin, Sea of Japan (East Sea), Journal of Geophysical Research Atmospheres, 116(B2), 1145-1160. [13] Liu, P., Song, W.J., You, Z.M., and et al. (2015), Optimization design and strength analysis of gas-tight container for deep-sea plankton based on ANSYS, Manufacturing Automation, 2015(20), 114-116. [14] Xia, F.S., Zhu, Z., and Dan, Y. (2010), Optimum design of the cylinder structure of high-pressure vessel, Journal of Xi’an Shiyou University (Natural Science Edition), 25(1), 81-83. [15] Wang, Y.W. (2005), Optimum design for the high pressure vessel, Machinery Design & Manufacture, 2005(12), 16-17. [16] Zhang, Z.Z., Han, C.L., and Li, C.W. (2011), Application of response surface method in experimental design and optimization, Journal of Henan Institute of Education (Natural Science Edition), 20(4), 34-37.



Inverse Problem for Degenerate Lotka-Volterra System of Three Equations Varadharaj Dinakar1†, Krishnan Balachandran2 1 2

Department of Mathematics, Central University of Tamilnadu, Thiruvarur, India - 610101 Department of Mathematics, Bharathiar University, Coimbatore, India - 641 046 Submission Info Communicated by J.Z. Zhang Received 21 April 2017 Accepted 12 June 2018 Available online 1 October 2018

Abstract We consider the degenerate Lotka-Volterra system with three equations in the linearized form. The internal observations with two measurements are allowed to obtain the stability result for the inverse problem consisting of simultaneously retrieving three coefficients in the given parabolic system with the help of Carleman estimates for the degenerate Lotka-Volterra system.

Keywords Stability Inverse problem Degenerate Lotka-Volterra model Carleman estimate


1 Introduction Certain population models in mathematical biology were first formulated in the field of autocatalytic chemical reactions by Lotka and later studied extensively by Volterra and henceforth they were commonly called as Lotka-Volterra systems. One among the notable works in this system was done by Grammaticos et al. [1] in which they studied the integrability of the system through the singularity analysis, the linear compatibility method and the Jacobi multiplier method. In the last few decades there have been great development in understanding the dynamics of the Lotka-Volterra system. A commendable outline of these results and applications to many biological fields were given by Hofbauer and Sigmund [2]. One can look into Pao [3, 4] for existence and other regularity conditions for the Lotka-Volterra system. In the context of inverse problems, eventhough few stability works were done for Lotka-Volterra system using various methods and numerical techniques, not much of the works are done by Carleman estimate technique. Sakthivel et al. [5] used the Carleman estimate to study the inverse problem of retrieving two smooth diffusion coefficients from the observation of two measurements. With this as a motivation, we study the inverse problem of determining three coefficients from only two observations where the linear Lotka-Volterra system possesses degenerate diffusivity. In this paper, we consider the inverse problems for the linearized Lotka-Volterra equation † Corresponding


292

V. Dinakar, K. Balachandran / Journal of Vibration Testing and System Dynamics 2(3) (2018) 291–296

⎫ ut = div(xα ∇u) − a1 (x)u − a2 (x)v − a3 (x)w + f1 (x), in QT := (0, T ) × Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ vt = div(xα ∇v) − a4 (x)v − a5 (x)w − a6 (x)u + f2 (x), in QT ⎪ ⎬ α wt = div(x ∇w) − a7 (x)w − a8 (x)u − a9 (x)v + f3 (x), in QT ⎪ ⎪ ⎪ ⎪ u(0, x) = v(0, x) = w(0, x) = 0, in Ω ⎪ ⎪ ⎪ ⎭ u(t, x) = v(t, x) = w(t, x) = 0, on Γ := (0, T ) × ∂ Ω

(1)

for Ω = (0, 1) and α ∈ [0, 2), with degenerate diffusion coefficients. The coefficients ai ∈ L∞ (Ω), i = 1, 2, . . . , 9 have respective lower bounds ri > 0 over the domain Ω, the source terms fi ∈ L2 (Ω), i = 1, 2, 3 and the solutions u, v, w ∈ C1 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H 2 (Ω)). Many population models describe spatial distribution of the species and these dispersions are sometimes modeled by degenerate parabolic equations where the diffusion coefficient vanishes at one extreme point of the domain. This produces significant technical challenges in the analysis of such equations. Such kind of differential equations with degenerate diffusion coefficients describe many phenomena in nature such as filtration, the Gauss curvature flow and the harmonic mean curvature flow in the porous medium equation, dynamics of several biological groups and so on. Carleman type inequalities are used to study several controllability and observability results of degenerate parabolic system in the last two decades along with existence and other regularity conditions in [6–9]. As far as inverse problem is concerned, a stationary potential for the Schrödinger equation is retrieved from a single boundary and an internal measurement was studied in [10]. Cannarsa et al. [11] studied the inverse problem of finding the source term in a single equation whereas Boutaayamou et al. [12] determined a single source term for the system of two equations with only one observation. In [13], interior degenerate coefficient is studied in an inverse source problem. Similar type of inverse source problem for a population model was discussed in [14]. The main result of our work can be given by the following theorem. Theorem 1. Let (u, v, w) and ( u, v, w) be the solutions of the system (1) associated with (a1 , a4 , a7 ) and (a1 , a4 , a7 ) respectively. Then there exists a positive constant C = C(Ω, ω , ri , β , θ , T ) such that, a1 − a1 L2 (Ω) + a4 − a4 L2 (Ω) + a7 − a7 L2 (Ω) ≤ C(vt − vt L2 (ω ) + wt − w t L2 (ω ) + u(θ , ·) − u(θ , ·)H 2 (Ω)

(2)

θ , ·)H 2 (Ω) ) +v(θ , ·) − v(θ , ·)H 2 (Ω) + w(θ , ·) − w( for ω Ω and some arbitrary θ ∈ (0, T ). The outline of this paper is as follows: In Section 2, we recall a Carleman estimate for the parabolic degenerate operator. This estimate is applied successfully in Section 3 to derive an estimate for the three coefficients with two observations for the Lotka-Volterra system.

2 Carleman Estimate In this section, we quote a Carleman type estimate for Lotka-Volterra degenerate system with two observations on a subdomain ω of Ω on the right-hand side of the estimate. The Carleman inequality for the case of degenerate diffusion operator is given in [6]. This result was achieved with the help of certain Hardy-type inequality. Before we proceed further, let us first define some regular functions.


For ω = (a, b) Ω, let us call ω0 := (κ1 , κ2 ) ω with κ1 = 2a+b 3 , κ2 = function ξ ∈ C2 (R) such that 0 ≤ ξ ≤ 1 and 1 if x ∈ (0, κ1 ), ξ (x) = 0 if x ∈ (κ2 , 1),

a+2b 3

293

and let us define a cut-off

so that ∇ξ = 0, x ∈ R \ ω0 . Let us introduce two functions 1 , ∀t ∈ (0, T ), (t(T − t))4 ψ (x) = x2−α − c1 , ∀x ∈ [0, 1], Θ(t) =

(3)

where c1 is chosen so that ψ < 0 in [0, 1]. It is easy to see that, ψx = (2 − α )x1−α , ψxx = (2 − α )(1 − α )x−α , |Θt | ≤ CΘ5/4 ≤ CΘ2 and that Θ → ∞ as t → 0+ or t → T − . Now set the weight function φ (t, x) = Θ(t)ψ (x) in QT . It is then easy to find that, for some s > 0, e2sφ → 0 as t → 0, T since ψ < 0. Consider the linear parabolic operator, for K ∈ L∞ (Ω), Ly := yt − div(xα ∇y) − Ky in Q.

(4)

Then the Carleman inequality for the above operator is given as follows Theorem 2. (Carleman estimate) For 0 ≤ α < 2 and T > 0, there exists two positive constants C and s0 so that for all s ≥ s0 , ˆ |yt |2 + |div(xα ∇y)|2 + sΘxα y2x + s3 Θ3 x2−α y2 ] dx dt I (y) := e2sφ [ sΘ Q ˆ ˆ 2sφ 2 e2sφ y2 dx dt) (5) ≤ C( e |Ly| dx dt + Q

Qω

for ω Ω and any y ∈ C1 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H 2 (Ω)). We would apply (5) to the modified system of (1) to attain our main result (2) in our next section.

3 Stability Result In this section, we establish a stability estimate using certain ideas from [15]. In proving these kinds of stability estimates, the global Carleman estimate obtained in Theorem 2.1 will play a crucial part along with certain energy estimates. Let (u, v, w) and ( u, v, w) be the solutions of the system (1) associated with (a1 , a4 , a7 ) and (a1 , a4 , a7 ) X = ∂t p1 , Y = ∂t p2 and Z = ∂t p3 respectively. Let us define p1 = u − u, p2 = v − v, p3 = w − w, ⎫ Xt = div(xα ∇X ) − a1 (x)X − a2 (x)Y − a3 (x)Z + f ut (x) in Q ⎪ ⎪ ⎪ ⎪ ⎪ Yt = div(xα ∇Y ) − a4 (x)Y − a5 (x)Z − a6 (x)X + g vt (x) in Q ⎪ ⎪ ⎬ α t (x) in Q (6) Zt = div(x ∇Z) − a7 (x)Z − a8 (x)X − a9 (x)Y + h w ⎪ ⎪ ⎪ ⎪ X (0, x) = Y (0, x) = Z(0, x) = 0 in Ω ⎪ ⎪ ⎪ ⎭ X (t, x) = Y (t, x) = Z(t, x) = 0 on Σ where f = a1 − a1 , g = a4 − a4 and h = a7 − a7 .

294


Now applying this Carleman estimate (5) to the equations in (6) we get ˆ t |2 + |X |2 + |Y |2 + |Z|2 dx dt I (X ) + I (Y ) + I (Z) ≤ C( e2sφ | f ut |2 + |g vt |2 + |h w ˆQ e2sφ |X |2 + |Y |2 + |Z|2 ) dx dt . + The term C

´

(7)

Qω 2sφ (|X |2 + |Y |2 + |Z|2 ) dx dt Qe

is absorbed by I (X ) + I (Y ) + I (Z). Thus, ˆ t |2 dx dt I (X ) + I (Y ) + I (Z) ≤ C( e2sφ | f ut |2 + |g vt |2 + |h w Q ˆ e2sφ (|X |2 + |Y |2 + |Z|2 ) dx dt). +

(8)

Qω

´ We try to estimate T := Qω e2sφ |X |2 dx dt by the localized observations of Y and Z. Further recalling ´ that a6 (x) ≥ r6 > 0, we can estimate the integral T = Qω e2sφ a6 ξ |X |2 dx dt. By the second equation of (6), we have ˆ e2sφ ξ (−Yt + div(xα ∇Y ) − a4Y − a5 Z + g vt )|X | dx dt T = Qω ˆ ˆ 2sφ e ξ (Yt )X dx dt + e2sφ ξ div(xα ∇Y )X dx dt = − Qω Qω ˆ ˆ 2sφ e ξ a4 XY dx dt − e2sφ ξ a5 X Z dx dt − Q Qω ˆ ω e2sφ ξ g vt X dx dt. + Qω

With the help of integration by parts and Young’s inequality, for sufficiently large s, we arrive at ˆ ˆ 2sφ α −2 2 sΘe x |Y | dx dt + e2sφ xα −2 |g vt |2 dx dt + I (X ) T ≤C Qω

which sums up to 1 |T | ≤ |T | ≤ C r6

Qω

ˆ

sΘe2sφ xα −2 (|Y |2 + |Z|2 ) dx dt + ‘Absorbed Terms’.

Qω

Now applying the global Carleman estimate (5) to the system (6), ˆ t |2 ) dx dt I (X ) + I (Y ) + I (Z) ≤ C( e2sφ (| f ut |2 + |g vt |2 + |h w ˆQ sΘe2sφ xα −2 (|Y |2 + |Z|2 ) dx dt). +

(9)

Qω

Recalling that φ → −∞ as t → 0 and using Young’s inequality, we get ˆ 1 ˆ θ ˆ 1 ∂ 2sφ (θ ,x) 2 ( e |X (θ , x)| dx = e2sφ (θ ,x) |X (θ , x)|2 dx)dt 0 0 0 ∂t ˆ (2sφt X 2 + 2X Xt )e2sφ dx dt = Q ˆ θ 1 (sΘ2 x2−α |X |2 + |Xt |2 )e2sφ dx dt ≤ sΘ Q ≤ I(X ),

(10)


295

where Qθ := (0, θ ) × (0, T ). Similarly we get ˆ

1 0

ˆ

1

0

which sums upto ˆ 1

2sφ (θ ,x)

e

0

e2sφ (θ ,x) |Y (θ , x)|2 dx ≤ I(Y ) e2sφ (θ ,x) |Z(θ , x)|2 dx ≤ I(Z)

ˆ

2

|X (θ , x)| dx +

1 0

2sφ (θ ,x)

e

2

|Y (θ , x)| dx +

≤ I (X ) + I (Y ) + I (Z) ˆ ˆ 2sφ 2 2 2 t | )dxdt + ≤ C( e (| f ut | + |g vt | + |h w Q

ˆ 0

(11)

1

e2sφ (θ ,x) |Z(θ , x)|2 dx

sΘe2sφ xα −2 (|Y |2 + |Z|2 )dxdt).

(12)

Qω

Since X (θ , ·) = div(xα ∇p1 (θ , ·)) − a1 p1 (θ , ·) − a2 p2 (θ , ·) − a3 p3 (θ , ·) − f u(θ , ·),

v(θ , ·), Y (θ , ·) = div(xα ∇p2 (θ , ·)) − a4 p2 (θ , ·) − a5 p3 (θ , ·) − a6 p1 (θ , ·) − g

θ , ·), Z(θ , ·) = div(xα ∇p3 (θ , ·)) − a7 p3 (θ , ·) − a8 p1 (θ , ·) − a9 p2 (θ , ·) − hw( we get that | f u(θ , ·)|2 ≤ C(|X (θ , ·)|2 + |div(xα ∇p1 (θ , ·)|2 + |p1 (θ , ·)|2 + |p2 (θ , ·)|2 + |p3 (θ , ·)|2 ) |g v(θ , ·)|2 ≤ C(|Y (θ , ·)|2 + |div(xα ∇p2 (θ , ·)|2 + |p1 (θ , ·)|2 + |p2 (θ , ·)|2 + |p3 (θ , ·)|2 )

|hw( θ , ·)|2 ≤ C(|Z(θ , ·)|2 + |div(xα ∇p3 (θ , ·)|2 + |p1 (θ , ·)|2 + |p2 (θ , ·)|2 + |p3 (θ , ·)|2 ) and so ˆ 1 e2sφ (θ ,x) |g v(θ , x)|2 dx + e2sφ (θ ,x) |hw( θ , x)|2 dx 0 0 0 ˆ ˆ 2sφ (θ ,x) 2 2 2 2sφ (θ ,x) α −2 (| f ut | + |g vt | + |h w t | ) dx dt + sΘe x (|Y |2 + |Z|2 ) dx dt ≤ C[ e ˆ

1

e2sφ (θ ,x) | f u(θ , x)|2 dx +

ˆ

ˆ

1

Q 1

Qω

e2sφ (θ ,x) (|div(xα ∇p1 (θ , ·))|2 + |div(xα ∇p2 (θ , ·))|2 + |div(xα ∇p3 (θ , ·))|2

+|p1 (θ , ·)|2 + |p2 (θ , ·)|2 + |p3 (θ , ·)|2 dx].

+

0

(13)

Since u ∈ H 1 (0, T ; H 2 (Ω)), ˆ

1

0

ˆ ≤ C[

e2sφ (θ ,x) | f |2 dx +

Qω ˆ 1

+

0

ˆ

2sφ (θ ,x) α −2

sΘe

x

1

0

e2sφ (θ ,x) |g|2 dx + 2

ˆ

1 0

e2sφ (θ ,x) |h|2 dx

2

(|Y | + |Z| ) dx dt

e2sφ (θ ,x) (|div(xα ∇p1 (θ , ·))|2 + |div(xα ∇p2 (θ , ·))|2 + |div(xα ∇p3 (θ , ·))|2

+|p1 (θ , ·)|2 + |p2 (θ , ·)|2 + |p3 (θ , ·)|2 ) dx]. Hence we arrive at our main result (2).

(14)

296


Corollary 3. Moreover, when θ ∈ (0, T ) is chosen so that (u, v, w)(θ , ·) = ( u, v, w)( θ , ·), the above result reduces to t L2 (ω ) ) a1 − a1 L2 (Ω) + a4 − a4 L2 (Ω) + a7 − a7 L2 (Ω) ≤ C(vt − vt L2 (ω ) + wt − w

(15)

which gives the bounds for the three coefficients in terms of two observations in the interior domain ω . Corollary 4. When the system possesses non-degenerate diffusion terms, the Carleman estimate is given by, for all s ≥ s0 ˆ ˆ ˆ |yt |2 + |div(k(x)∇y)|2 e2sφ [ + sη y2x + s3 φ 3 y2 ] dx dt ≤ C( e2sφ |Ly|2 dx dt + e2sφ s3 φ 3 y2 dx dt) (16) s η Q Q Qω for ω Ω and any y ∈ C1 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H 2 (Ω)). Note that η and φ are defined using Θ(t) along with some positive smooth function on Ω which vanishes in the boundary of Ω and has non-zero gradient over some interior subset ω0 ω . References [1] Grammaticos, B., Moulin-Ollagnier, J., Ramani, A., Strelcyn, J.M., and Wojciechowski, S. (1990), Integrals of quadratic ordinary differential equations in R3 : The Lotka-Volterra system, Physica A: Statistical Mechanics and its Applications, 163, 683-722. [2] Hofbauer, J. andSigmund, K. (1998), The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge. [3] Pao, C.V. (1992), Nonlinear Parabolic and Elliptic Equations, Plenum: New York. [4] Pao, C.V. (2004), Global Asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, Nonlinear Analysis: Real World Applications, 5, 91-104. [5] Sakthivel, K., Baranibalan, N., Kim, J.H., and Balachandran, K. (2010), Stability of diffusion coefficients in an inverse problem for the Lotka-Volterra competition system, Acta Applicandae Mathematicae, 111, 129-147. [6] Cannarsa, P. and Teresa, L.D. (2009), Controllability of 1-D coupled degenerate parabolic equations, Electronic Journal of Differential Equations, 2009(73), 1-21. [7] Cannarsa, P., Martinez, P., and Vancostenoble, J. (2005), Null controllability of degenerate heat equations, Advances in Differential Equations, 10, 153-190. [8] Cannarsa, P., Martinez, P., and Vancostenoble, J. (2008), Carleman estimate for a class of degenerate parabolic operators, SIAM Journal on Control and Optimization, 47, 1-19. [9] Du, R. and Xu, F. (2017), Null controllability of a coupled degenerate system with the first order terms, Journal of Dynamical and Control Systems, DOI:10.1007/s10883-016-9353-4. [10] Mercado, A., Osses, A., and Rosier, L., Inverse problems for the Schröinger equation via Carleman inequalities with degenerate weights, Inverse Problems, 015017. [11] Cannarsa, P., Tort, J., and Yamamoto, M. (2010), Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26, 105003 (20pp). [12] Boutaayamou, I., Hajjaj, A., and Maniar, L. (2014), Lipschitz stability for degenerate parabolic sysytems, Electronic Journal of Differential Equations, 2014(149), 1-15. [13] Boutaayamou, I., Fragnelli, G., and Maniar, L. (2016), Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions, Journal of Inverse and Ill-posed Problems, 24(3), 275-292. [14] Dinakar, V., Baranibalan, N., and Balachandran, K. (2017), Identification of source terms in a coupled age-structured population model with discontinuous diffusion coefficients, AIMS Mathematics, 2, 81-95. [15] Cristofol, M., Gaitan, P., Ramoul, H., and Yamamoto, M. (2011), Identification of two coefficients with data of one component for a nonlinear parabolic system, Applicable Analysis, 1-9. [16] Deng, Z.C. and Yang, L. (2011), An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation, Journal of Computational and Applied Mathematics, 235, 4404-4417. [17] Evans, L.C. (1998), Partial Differential Equations, AMS, Providence.

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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: • • • • • • • • • • • • • • • •

Vibration testing and design Dynamical systems and control Testing instrumentation and control Complex system dynamics in engineering Dynamic failure and fatigue theory Chemical dynamics and bio-systems Fluid dynamics and combustion Pattern dynamics Network dynamics Control signal synchronization and tracking Bio-mechanical systems and devices Structural and multi-body dynamics Flow or heat-induced vibration Mass and energy transfer dynamics Wave propagation and testing Acoustics

No length limitations for contributions are set, but only concisely written manuscripts are published. Brief papers are published on the basis of Technical Notes. Discussions of previous published papers are welcome.

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Journal of Vibration Testing and System Dynamics Volume 2, Issue 3

September 2018

Contents A Modified Newmark Scheme for Simulating Dynamical Behavior of MDOF Nonlinear Systems S. D. Yu, M. Fadaee………………………………………………………….........

187－207

Experimental Investigations of Energy Recovery from an Electromagnetic Pendulum Vibration Absorber Krzysztof Kecik, Angelika Zaszczynska, Andrzej Mitura…….…………....………

209－219

Huge Size Structure Damage Localization and Severity Prediction: Numerical Modeling, Simulation and SVM Regression Method Gang Jiang, Yiming Deng, Lili Liu, Canghai Liu, Zihong Liu, Yong Jiang…........

221－237

A Low Cost Device for Excessive Vibration Detection in Electric Motors M. Sundin, A. Babaei, S. Paudyal, C. Yang, N. Kaabouch…….……..…..….……

239－247

A New Application of the Normal Form Description to a N-Dimensional Dynamical Systems Attending the Conditions of a Hopf Bifurcation Vinícius B. Silva, João P. Vieira, Edson D. Leonel...……………......……........…

249－256

Nonlinear Dynamics of a Reduced Cracked Rotor K. Lu, Y. Lu, B.C. Zhou, W. Jian, Y.F. Yang, Y.L. Jin, Y.S. Chen........................…

257－269

Dynamical Balance and Verification of a Rotor System Based on Sensitivity Analysis Zhong Luo, YanhuiWei, Xiaojie Hou, Fei Wang…………………………………..

271－280

Multi-objective Optimization Design of Flap Sealing Valve Structure for Deep Sea Sediment Sampling Guangping Liu, Yongping Jin, Youduo Peng, Buyan Wan………………………..

281－290

Inverse Problem for Degenerate Lotka-Volterra System of Three Equations Varadharaj Dinakar, Krishnan Balachandran……………………………………

291－296

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Journal of Vibration Testing and System Dynamics

Journal of Vibration Testing and System Dynamics

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