Bifurcation & Chaos in Structural Dynamics: A Comparison ...bifurcation and chaos in structural...

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1 Bifurcation & Chaos in Structural Dynamics: A Comparison of the Hilber-Hughes-Taylor- α and the Wang-Atluri (WA) Algorithms Xuechuan Wang*1, Weicheng Pei **, and Satya N. Atluri *2 Texas Tech University, Lubbock, TX, 79415 Center for Advanced Research in the Engineering Sciences Abstract The WA algorithms (Computational Mechanics, Vol. 59, No. 5, pp. 861-876, 2017) are rederived as being based on optimal-feedback-accelerated Picard iteration, wherein the solution vectors at any time t in a finitely large time interval 1 i i t t t + are corrected by a weighted (with a matrix λ ) integral of the error from i t to t . The 3 WA algorithms are rederived in detail, based on 3 different approximate solutions to the Euler-Lagrange equations for λ . The interval 1 ( ) i i t t + in the 3 WA algorithms can be several hundred times larger than the increment ( t ) required in the HHT- α , for the same stability and accuracy. Moreover, the WA algorithms-2, 3 do not require the inversion of the tangent stiffness matrix, as is required in HHT- α . It is found that WA algorithms-1, 2, 3 (especially WA algorithm-2) are far more superior to HHT- α and ode-45 in terms of computational speed, accuracy, and convergence. Keywords: Nonlinear structural dynamics; bifurcation and chaos; HHT- α algorithm; Wang-Atluri algorithms *1 Visiting Scholar from Northwestern Polytechnical University, and Graduate Student in Texas Tech University, Corresponding Author, Email: [email protected] ** Visiting Scholar, from Beihang University *2 Director, Center for Advanced Research in the Engineering Sciences

Transcript of Bifurcation & Chaos in Structural Dynamics: A Comparison ...bifurcation and chaos in structural...

Page 1: Bifurcation & Chaos in Structural Dynamics: A Comparison ...bifurcation and chaos in structural dynamics. The WA algorithms are also superior to the optimized ODE45 algorithm in MATLAB,

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Bifurcation & Chaos in Structural Dynamics: A Comparison of the Hilber-Hughes-Taylor-α and the Wang-Atluri (WA) Algorithms

Xuechuan Wang*1, Weicheng Pei**, and Satya N. Atluri*2

Texas Tech University, Lubbock, TX, 79415 Center for Advanced Research in the Engineering Sciences

Abstract

The WA algorithms (Computational Mechanics, Vol. 59, No. 5, pp. 861-876, 2017) are rederived as being based on optimal-feedback-accelerated Picard iteration, wherein the solution vectors at any time t in a finitely large time interval 1i it t t +≤ ≤ are corrected by a weighted (with a matrix λ ) integral of the error from it to t . The 3 WA algorithms are rederived in detail, based on 3 different approximate solutions to the Euler-Lagrange equations for λ . The interval 1( )i it t+ − in the 3 WA algorithms can be several hundred times larger than the increment ( t∆ ) required in the HHT-α , for the same stability and accuracy. Moreover, the WA algorithms-2, 3 do not require the inversion of the tangent stiffness matrix, as is required in HHT-α . It is found that WA algorithms-1, 2, 3 (especially WA algorithm-2) are far more superior to HHT-α and ode-45 in terms of computational speed, accuracy, and convergence.

Keywords: Nonlinear structural dynamics; bifurcation and chaos; HHT-α algorithm; Wang-Atluri algorithms

*1 Visiting Scholar from Northwestern Polytechnical University, and Graduate Student in Texas Tech University, Corresponding Author, Email: [email protected]

** Visiting Scholar, from Beihang University

*2 Director, Center for Advanced Research in the Engineering Sciences

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1. Introduction

Nonlinearities in the mechanics of structures could arise from various sources such as large deformations, large rotations, nonlinearities of the material, damping, and boundary conditions, etc. [1] They are so common in real life that the nonlinear behaviors can be found everywhere ranging from civil engineering, automobile manufactory, aircraft design, to robotic dynamics, and on-orbit maneuvers of spacecraft. With the development of lighter, more flexible and multifunctional materials and structures, the effect of nonlinearity will become more significant in these areas. Linearization was once widely used to simplify nonlinear problems, but in the meantime, it was also realized that the basic principles of linear analysis are not valid even in some cases of weak nonlinearity. The nonlinear dynamical system is characterized by some unique and complex phenomena, including bifurcation, jump phenomena, snap-through, and chaos, that the linear system cannot exhibit [2]. For weakly nonlinear problems, the perturbation method has been successfully used to obtain asymptotically analytical approximations of the solutions. However, for strongly nonlinear dynamical systems with complex forms, the solution can only be obtained using numerical integration methods, in the current state of mathematics.

The investigation of structural vibration helps to enhance the reliability and performance of structures, as well as reveal potential failures. With proper structural models, one can use numerical simulation as an alternative to experiment, and to save a lot of time and expense. In literature, the methods for numerically solving a system of nonlinear ordinary differential equations in structural dynamics are roughly divided into explicit and implicit methods. The finite difference methods are a broad class of explicit methods involving central difference method, Runge-Kutta methods and Taylor series expansion schemes [3]. These methods are very easy to implement, but often lack stability in computations using large size steps. The implementation of an implicit method results in nonlinear algebraic equations of unknown states, which are usually solved with Newton-Raphson methods. For some typical implicit methods, such as the Newmark-β methods [4] and the Hilber-Hughes-Taylor-α method [5], the stability is guaranteed only for linear problems. However, for nonlinear problems, the stability is doubtful and often depends on the specific problem and the step size adopted in the method. With relatively large size steps, some implicit methods could become so unstable and inaccurate that they are obviously unsuited for numerical integration to find the long-term nonlinear dynamical behaviors [6]. Even though, in the analysis of vibrating structures by a finite element spatial discretization method, for the integration of the semi-discrete nonlinear time ODEs, schemes such as the central difference, the Newmark and the Hilber-Hughes-Taylor-α (HHT-α ) methods are still widely used in industrial software without reliable proofs showing that the nonlinear behaviors including bifurcation and chaos can be accurately predicted by these methods.

This paper concerns the phenomena of bifurcation and chaos in structural dynamics, governed by a semi-discrete system of equations in time, after a spatial discretization has been carried out. Analyses of structural dynamics problems have been carried out extensively in the past, by using the Newmark- β and its slight modification with added damping, the Hilber-Hughes-Taylor- α methods, wherein Taylor series expansions of the displacement and velocity vectors are used over a very small time step t∆ , around the generic time it . On the other hand, the Wang-Atluri algorithms, which involve finitely large time intervals

( 1i it t+ − ) were first introduced in [7], and applied to orbital mechanics problems in [8]. In orbital mechanics problems, the WA algorithms were proved to be the best in literature so far, in terms of computational time, accuracy, and speed of convergence. These algorithms are reinterpreted in the present paper as being derived from highly optimal feedback accelerated Picard iteration methods, wherein the solution vector consisting of displacements ( x ) and velocities ( ′x ) at any time t , 1i it t t +≤ ≤ , is corrected by an optimally

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weighted (with the weighting matrix λ ) integral of the error in the solution vector from it to t . By

assuming orthogonal polynomials for the solution vector over ( 1i it t+ − ), and using collocation points mit

( 1, 2,...,m M= ) within ( 1i it t+ − ), we use the optimally accelerated Picard-iteration from it to mit , to

reduce the semi-discrete nonlinear ODEs to algebraic equations. We present 3 approximations to solve the Euler-Lagrange equations for the optimal weight functions λ . Thus we present 3 algorithms denoted as Wang-Atluri algorithms-1, 2, 3. In these algorithms, the time interval ( 1i it t+ − ) can be several hundred times larger than the t∆ used in the HHT-α method, for the same stable and accurate performance. Even the distance between the collocation points 1m m

i it t+ − within the interval 1i it t+ − can be several times larger than the t∆ used in the HHT-α method. In addition, the WA algorithms-2 and 3 do not involve the inversion of a Jacobian (tangent stiffness) matrix, as is the case in the HHT-α method. It is shown that all the WA algorithms-1, 2, 3 (especially the WA algorithm-2) are far more superior to the HHT-α algorithm in computational speed, accuracy, and speed of convergence, in the presently considered problems of bifurcation and chaos in structural dynamics. The WA algorithms are also superior to the optimized ODE45 algorithm in MATLAB, in terms of speed and accuracy. The WA algorithms-2 may be considered as a significant improvement to the state of science in nonlinear structural dynamics.

The nonlinear vibration of a buckled beam is very representative in nonlinear structural dynamics and has attracted a number of researchers [11, 12]. Commonly the dynamics of a buckled beam can be well modelled by finite element, boundary element, or meshless methods. However, it is shown in [13] that the clamped-clamped buckled beam model can also be well discretized into a four-mode approximation using the spatial Galerkin method. In this paper, both the Hilber-Hughes-Taylor-α and Wang-Atluri algorithms are used to investigate the nonlinear vibrations of a buckled beam which may exhibit bifurcation, jump phenomena, and chaos. The validity of the four-mode buckled beam model is first examined, and the nonlinear dynamical behaviors such as period doubling process and chaotic motion are then investigated. The numerical results obtained by Wang-Atluri method are compared with those by Hilber-Hughes-Taylor-α method to evaluate the performance of these two methods. It is shown that the WA algorithms are far more superior to the HHT-α algorithms in terms of accuracy, computational time, and convergence speed, in problems involving bifurcation and chaos in nonlinear structural dynamics.

2. An Elucidation of HHT-α and WA Algorithms

In structural dynamics modeling, the vibration of a structure is often described by a system of second-order nonlinear ordinary differential equations after spatial discretization has already been carried out, where x is the vector of generalized displacements, and ′′x are the accelerations, M is the mass matrix, C is the damping matrix, N is the vector of nonlinear restoring forces which may depend on x and the velocity vector ′x , and F is the external force.

( , ) ( )t′′ ′ ′+ + =Mx Cx N x x F . (1)

Eq. (1) can be further rewritten as a system of nonlinear first order ODEs:

211 1 1

2 2 1 2( , ) ( )t− − −

′ = ′ − − +

xxx M Cx M N x x M F

, (2)

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where N denotes the nonlinear terms, 1x and 2x represent displacement and velocity vector of the

motion. Eq. (2) may be considered as the “mixed-form” equivalent of Eq. (1), wherein 1x is the

displacement vector and 2x is the velocity vector.

2.1 HHT-α Algorithm

Based on the theory of Taylor series, the discretized displacement 1( )it t+ ∆x (where t∆ is infinitesimally small) can be approximated [as done by Newmark (4)] by

2 31 1 1 1 1

2 31 1 1 1

1 1( ) ( ) ( ) ( ) ( )2 3!1( ) ( ) ( ) ( )2

i i i i i

i i i i

t t t t t t t t t

t t t t t t tβ

′ ′′ ′′′+ ∆ = + ∆ + ∆ + ∆ +

′ ′′ ′′′= + ∆ + ∆ + ∆

x x x x x

x x x x

, (3)

where 16

β ≥ . Similarly, using Taylor series expansion, 2 ( )it t+ ∆x can be approximated [as done by

Newmark (4)] by

2

2 2 2 2

22 2 2

1( ) ( ) ( ) ( )2

( ) ( ) ( )

i i i i

i i i

t t t t t t t

t t t t tγ

′ ′′+ ∆ = + ∆ + ∆ +

′ ′′= + ∆ + ∆

x x x x

x x x

, (5)

where 12

γ ≥ .

Since t∆ is very small, 1 ( )it′′′x in the last term of Eq. (3) can be approximated as

( ) [ ( ) ( )]i i it t t t t′′′ ′′ ′′= + ∆ − ∆x x x . (6)

Substituting Eq. (6) into Eq. (3) leads to

2 21 1 1 1 1 1

2 21 1 1 1

1( ) ( ) ( ) ( ) [ ( ) ( )]21( ) ( ) ( ) ( ) ( )2

i i i i i i

i i i i

t t t t t t t t t t t

t t t t t t t t

β

β β

′ ′′ ′′ ′′+ ∆ = + ∆ + ∆ + + ∆ − ∆

′ ′′ ′′= + ∆ + − ∆ + + ∆ ∆

x x x x x x

x x x x. (7)

Further, 2 ( )it′′x in the last term of Eq. (5) is approximated by

2 2 2( ) [ ( ) ( )]i i it t t t t′′ ′ ′= + ∆ − ∆x x x . (8)

Substituting it into Eq. (5) leads to

2 2 2 2 2

2 2 2

( ) ( ) ( ) [ ( ) ( )]

( ) (1 ) ( ) ( )i i i i i

i i i

t t t t t t t t t

t t t t t t

γ

γ γ

′ ′ ′+ ∆ = + ∆ + + ∆ − ∆

′ ′= + − ∆ + + ∆ ∆

x x x x x

x x x. (9)

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From Eq. (2), we have

1 2

1 1 12 1 2 1 2

( ) ( )

( ) ( ) ( ) [ ( ), ( )] ( )i i

i i i i i i

t t t t

t t t t t t t t t t t t− − −

′ + ∆ = + ∆ ′′′ + ∆ = + ∆ = − + ∆ − + ∆ + ∆ + + ∆

x x

x x M Cx M N x x M F. (10)

By substituting Eq. (10) into Eqs. (7, 9), a system of nonlinear algebraic equations about 1( )it t+ ∆x and

2 ( )it t+ ∆x are obtained as

[ ]

1 122 2

1 1 1 1 11 2

1 12

2 2 2 11 2

( ) ( )1( ) ( ) ( ) ( ) ( ) 02 ( ), ( )

( ) ( )( ) ( ) (1 ) ( )

[ ( ), ( )]

i ii i i i

i i

i ii i i

i i

t t t tt t t t t t t t

t t t t

t t t tt t t t t

t t t t

β β

γ γ

− −

− −

+ ∆ − + ∆ ′ ′′+ ∆ − − ∆ − − ∆ + ∆ = + + ∆ + ∆

+ ∆ − + ∆′+ ∆ − − − ∆ + + + ∆ + ∆

M Cx M Fx x x x

M N x x

M Cx M Fx x x

M N x x0t

∆ =

(11)

In HHT- α method, the proceeding algorithms of Newmark- β method are slightly modified by replacing

1( )it t+ ∆x and 2 ( )it t+ ∆x in Eq. (10) with 1( )it tα+ ∆x and 2 ( )it tα+ ∆x respectively, where

1 1 1 1

2 2 2 2

( ) ( ) [ ( ) ( )]( ) ( ) [ ( ) ( )]

i i i i

i i i i

t t t t t t tt t t t t t t

α αα α

+ ∆ = + ∆ + + ∆ − + ∆ = + ∆ + + ∆ −

x x x xx x x x

. (12)

Therefore, Eq. (10) is modified as

1 2

11 1 12 1 2

2

( ) ( )( )

( ) ( ) ( ) ( )( )

i i

ii i i i

i

t t t tt t

t t t t t t t tt t

αα

α αα

− − −

′ + ∆ = + ∆ + ∆ ′′′ + ∆ = + ∆ = − + ∆ − + + ∆ + ∆

x xx

x x M Cx M N M Fx

. (13)

Substituting Eq. (13) into Eqs. (7, 9) leads to

[ ]

1 122 2

1 1 1 1 11 2

1 12

2 2 2 11 2

( ) ( )1( ) ( ) ( ) ( ) ( ) 02 ( ), ( )

( ) ( )( ) ( ) (1 ) ( )

[ ( ), (

i ii i i i

i i

i ii i i

i i

t t t tt t t t t t t t

t t t t

t t t tt t t t t

t t t t

α αβ β

α α

α αγ γ

α α

− −

− −

+ ∆ − + ∆ ′ ′′+ ∆ − − ∆ − − ∆ + ∆ = + + ∆ + ∆

+ ∆ − + ∆′+ ∆ − − − ∆ ++ + ∆ + ∆

M Cx M Fx x x x

M N x x

M Cx M Fx x x

M N x x0

)]t

∆ =

(14)

The parameters in HHT-α method are optional. Normally they are selected as 1 22αγ −

= , 21

2αβ − =

and 1 ,03

α ∈ − .

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Eqs. (12, 14) provide a system of nonlinear algebraic equations about 1( )it t+ ∆x and 2 ( )it t+ ∆x that can be solved for using Newton-Raphson method, where the Jacobian matrix and its inversion has to be calculated in each iteration step. The Newton-Raphson iteration formula for solving Eq. (14) is written as

1

11 11

2 2

( ) ( )( )

( ) ( )

j jj ji iHHT HHTj j

i i

t t t tt t t t

+−

+

+ ∆ + ∆= − + ∆ + ∆

x xJ R

x x, (15)

where 1 1

22 21 1 1 1 1

1 2

1 12

2 2 2 11

( ) ( )1( ) ( ) ( ) ( ) ( )2 ( ), ( )

( ) ( )( ) ( ) (1 ) ( )

[ (

ji ij

i i i i j ji ij

HHTj

i iji i i j

i

t t t tt t t t t t t t

t t t t

t t t tt t t t t

t

α αβ β

α α

α αγ γ

− −

− −

+ ∆ − + ∆ ′ ′′+ ∆ − − ∆ − − ∆ + ∆ + + ∆ + ∆ =+ ∆ − + ∆′+ ∆ − − − ∆ +

+

M Cx M Fx x x x

M N x xR

M Cx M Fx x x

M N x 2), ( )]ji

tt t tα α

∆ + ∆ + ∆ x

(16)

The Jacobian matrix of jHHTR with respect to 1 ( )j

it t+ ∆x and 2 ( )jit t+ ∆x is derived as

1 1 11 2

1 1 11 2

1 (1 )[ ( )] (1 ) [ ( )](1 )[ ( )] 1 (1 ) [ ( )]

j j j jj i iHHT j j j j

i i

t t t tt t t t

β α α β α αγ α α γ α α

− − −

− − −

+ + ∂ ∂ + ∆ + + ∂ ∂ + ∆= + ∂ ∂ + ∆ + + + ∂ ∂ + ∆

M N x M C M N xJ

M N x M C M N x. (17)

2.2 Wang-Atluri Algorithms Based on Optimal Error-Feedback Accelerated Picard Iteration Concepts

Let 1u and 2u be the trial functions for 1x and 2x in a finitely large local time interval 1[ , ]i it t t +∈ . By substituting them into Eq. (2), the error residual function is obtained as

211 1 1

2 2 1 2

( )( , ) ( )

tt− − −

−′ = + ′ + −

uuR

u M Cu M N u u M F, 1[ , ]i it t t +∈ . (18)

To optimally correct the approximate solution at t , a simple mechanism of an optimally weighted feedback of the error is adopted herein, which has the expression:

21 1 11 1 1

2 2 2 2 1 2

( ) ( )( )

( ) ( ) ( , ) ( )i

t

tc

t td

t t tτ τ

− − −

−′ = + + ′ + − ∫

uu u uλ

u u u M Cu M N u u M F, 1[ , ]i it t t +∈ ,

(19)

where the subscript c on the left-hand side indicates the “correction”. In Eq. (19), ( )τλ , a matrix, is the set of optima weighting functions for the feedback of the error residual ( )tR . Eq. (19) indicates that the solution vector 1 2[ ( ); ( )]t tu u at any time t in the interval 1i it t t +≤ ≤ is corrected by an optimally

weighted error residual from time it to t .

The iteration formula of the original Picard iteration method [14, 15] can be regarded as a special case of Eq. (19), where ( )τλ is simply selected as the negative unit matrix ( )τ = −λ I . Although the original

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Picard iteration method may converge, it is not the most efficient approach, since ( )τ = −λ I is selected too roughly. The derivation of the optimal ( )τλ is, however, as follows.

For 1u and 2u in the neighborhood of the true solutions, i.e. 1 1 1true δ= +u u u , 2 2 2

true δ= +u u u , λ is expected to be optimally determined so that

1 1

2 2

( ) ( )( ) ( )

true

truec

t tt t

=

u uu u

. (20)

It means that the variation of Eq. (19) should equals to be zero if 1 1true=u u , 2 2

true=u u , which leads to:

2 21 1 1 11 1 1 1 1 1

2 2 2 22 2( ) ( )i i

t t

t tc

dt t

δ δ δ τ δ− − − − − −

− −′ ′ = + + + + = ′ ′+ − + − ∫ ∫

u uu u u uλ λ 0

u u u uM Cu M N M F M Cu M N M F, (21)

Since 1u and 2u in Eq. (21) are supposed to be the true solutions, we have

211 1 1

2 2 ( )t− − −

−′ + = ′ + −

uu0

u M Cu M N M F. (22)

Therefore, Eq. (21) leads to:

21 1 11 1 1

2 2 2 2

1 1

2 2

( )

( )

i

i

t

tc

t

t

dt

d

δ δ δ τ

δ δ τ

− − −

−′ = + + ′ + −

′= + + − + =

uu u uλ

u u u M Cu M N M F

u uI λ λ λJ 0

u u

, (23)

where

1 1 11 2( ) ( )− − −

− = ∂ ∂ + ∂ ∂

0 IJ

M N u M C M N u. (24)

In Eq. (23), there are variations both inside and outside the integral over time, thus the variations are made to be zero separately by enforcing the weighting function matrix λ to satisfy the following conditions:

( )t + =λ I 0 , and ( , ) ( , ) ( )t tτ τ τ′− + =λ λ J 0 , [ , ]it tτ ∈ (25)

Obviously, λ is related to 1u , 2u , which are supposed to be 1trueu and 2

trueu . However, the true solutions

are not known in advance. Considering that, we calculated λ in the present paper using approximated solutions instead of the true solutions in the implementation of the algorithms, which are labeled as WA algorithms for convenience.

2.3 Large Time Interval 1[ , ]i it t + Orthogonal Polynomial Collocation

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Further, the following weak formulation of Eq. (19) can be established, using a matrix of test functions ( )tv .

1 1 21 1 11 1 1

2 2 2 2 1 2

( ) ( )( ) ( )

( ) ( ) ( , ) ( )i i

i i i

t t t

t t tc

t tt dt t d dt

t t tτ+ +

− − −

−′ = + + ′ + − ∫ ∫ ∫

uu u uv v λ

u u u M Cu M N u u M F.

(26)

In this formula, ( )tv are test functions, the same as those in the classical weighted residual methods. Let ( )tv be a diagonal matrix [ , ,...]diag v v=v , and v be Dirac Delta function for a group of collocation

points mit , in the finite large time interval it to 1it + . For simplicity, the collocation points m

it are simply

denoted as mt in the following parts.

( )mv t tδ= − , 1[ , ]m i it t t +∈ , 1, 2,...,m M= , (27)

then Eq. (26) becomes

21 1 11 1 1

2 2 2 2

( ) ( )( ) ( )

m

i

tm m

tm mc

t td

t tτ

− − −

−′ = + + ′ + − ∫

uu u uλ

u u u M Cu M N M F, 1[ , ]m i it t t +∈ , 1,...,m M= . (28)

Eq. (20) may be interpreted as the correction of the error at each collocation point mt , with the error residual

being optimally weighted by λ .

By using a set of orthogonal polynomials 0 1 2 , , ,...Tφ φ φ=Φ as basis functions, the trial functions 1u and

2u can be constructed as

1, 1, ,0

N

e e n nn

u a φ=

=∑ , and 2, 2, ,0

N

e e n nn

u a φ=

=∑ , (29)

Where 1,eu and 2,eu are elements of 1u and 2u respectively. 1, ,e na and 2, ,e na are coefficients to be determined. From Eq. (29), we have

, 1 , 2 , , ,0 , ,1 , ,[ ( ), ( ),..., ( )] [ , ,..., ]T Tp e p e p e M p e p e p e Nu t u t u t a a a= B , 1, 2p = , (30)

and

, 1 , 2 , , ,0 , ,1 , ,[ ( ), ( ),..., ( )] [ , ,..., ]T Tp e p e p e M p e p e p e Nu t u t u t a a a′ ′ ′ = LB , 1, 2p = , (31)

where

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0 1 1 1 1

0 2 1 2 2

0 1 ( 1)

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

N

N

M M N M N M

t t tt t t

t t t

φ φ φφ φ φ

φ φ φ+ ×

=

B

,

0 1 1 1 1

0 2 1 2 2

0 1 ( 1)

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

N

N

M M N M N M

t t t

t t t

t t t

φ φ φ

φ φ φ

φ φ φ+ ×

′ ′ ′ ′ ′ ′

= ′ ′ ′

LB

. (32)

Normally, the number of colocation points M is selected as the same as the number of basis functions

1N + . It can be found that the values of ,p eu ′ at collocation points are related with those of ,p eu by

1, 1 , 2 , , 1 , 2 ,[ ( ), ( ),..., ( )] ( ) [ ( ), ( ),..., ( )]T T

p e p e p e M p e p e p e Mu t u t u t u t u t u t−′ ′ ′ = LB B . (33)

In an analogous way, the values of ,i

t

p etu dτ∫ at collocation points mt , 1, 2,...,m M= can also be obtained

from those of ,p eu through the following transformation.

1 2 1 1, , , , 1 , 2 ,[ , ,..., ] ( ) [ ( ), ( ),..., ( )]M

i i i

t t t T Tp e p e p e p e p e p e Mt t t

u d u d u d u t u t u tτ τ τ − −=∫ ∫ ∫ L B B , (34)

where

1 1 1

2 2 2

0 1

0 11

0 2( 1)

i i i

i i i

M M M

i i i

t t t

Nt t t

t t t

Nt t t

t t t

Nt t t N M

d d d

d d d

d d d

φ τ φ τ φ τ

φ τ φ τ φ τ

φ τ φ τ φ τ

+ ×

=

∫ ∫ ∫

∫ ∫ ∫

∫ ∫ ∫

L B

. (35)

A schematic presentation of the large time collocation is given in Fig. 1, wherein the time interval 1( )i it t+ −

can be several hundreds of times larger than the step size t∆ of HHT-α algorithm. In each time interval

it to 1it + , M collocation points are selected to interpolate the approximated solution. In this paper, the two

ends of the time interval are selected as the first and the last collocation points, i.e. 1i it t= , 1

Mi it t += . It

should be noted that even the time step between two neighboring collocation points can be larger than the time step t∆ used in the HHT algorithm.

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10

it1it − 1it + 2it +

it 1it +1it

2it 1M

it− M

it

Knowns Unknowns Unknowns

1

2

( )( )

i

i

tt

xx

1

2

( )( )

xx

mimi

tt

1 1

2 1

( )( )

xx

i

i

tt+

+

May be the used in the HHT- Algorithmt∆ α

Fig. 1 Schematic presentation of large time collocation in the interval 1i it t t +≤ ≤

2.4 Wang-Atluri (WA) Algorithms

2.4.1. WA Algorithm-1

Although we have already obtained the iterative formula (28) through collocation, the matrix of generalized Lagrange multipliers λ in it still remain a puzzle. The possibility of directly solving the constraint equations (25) for λ is unlikely for most nonlinear cases. Fortunately, there is an approach to bypass this dilemma.

Differentiating Eq. (19) leads to

1 1

2 2

( ) ( )( ) ( ) ( )

( ) ( ) i

t

tc

t td d t t dt tdt dt t

τ τ ∂

= + + ∂ ∫

u u λλ R Ru u

, (36)

where ( )τR is the residual function:

211 1 1

2 2

( )( )( )

( ) ( ) ( ) ( )ττ

ττ τ τ τ− − −

−′ = + ′ + −

uuR

u M Cu M N M F. (37)

According to Eq. (25) and using the theory of Magnus series, it can be proved [17] that ( )t = −λ I , and

( )tt

∂= −

∂λ J λ . Substituting them into Eq. (36), we have

2 21 11 1 1 1 1 1

2 22 2

( )( )

( ) i

t

tc

td t dtdt

τ− − − − − −

− −′ = − − + ′+ − + − ∫

u uu uJ λ

u uM Cu M N M F M Cu M N M F. (38)

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Noting that 21 1 11 1 1

2 2 22 ( )i

t

tc

dt

τ− − −

−′ + = − ′ + − ∫

uu u uλ

u u uM Cu M N M F according to the optimal

error-feedback iteration formula (19), Eq. (38) is rewritten as

21 1 11 1 1

2 2 22

( )( )

( ) c c

td ttdt − − −

− = − − − + −

uu u uJ

u u uM Cu M N M F. (39)

After rearrangement, it is rewritten as

21 1 11 1 1

2 2 22

( ) ( ) ( )( ) ( )

( ) ( ) ( )c c

t t td t tt t tdt − − −

− + = − + + −

uu u uJ J

u u uM Cu M N M F. (40)

Eq. (40) can be regarded as equivalent to Eq. (19). By collocating in the local time interval [ ]1,i it t + , an

algebraic iterative formula is obtained as.

21 1 11 1 1

2 222

( )( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( )( )

mm m mm m

m mm m mcm c

tt t tt t

t tt t tt− − −

′ − + = − + + − ′

uu u uJ J

u uM Cu M N M Fu, (41)

where 1, 2,...m M= , [ ]1,m i it t t +∈ , and 1i it t+ − is a finite large time interval.

After rearranging the sequential of the collocation equations and using the relationship in Eq. (33), Eq. (41) is rewritten as

21 11 1 1

2 22

( )c

− − −

− + = − + + −

UU UE J J

U UM CU M N M F

, (42)

where

,1 1 ,1 2 ,1 ,2 1 ,2 2 ,2( ) ( ) ... ( ) ( ) ( ) ... ( ) ...T

p p p p M p p p Mu t u t u t u t u t u t = U , 1, 2p = . (43)

The configuration of the matrices E , J , M , C , N , F are provided in Appendix C.

It should be noted that the initial conditions are not incorporated in Eq. (42). For that, without loss of generality, we usually select the first collocation point at the initial boundary, of which the values , 1( )p eu t

are given. By doing that, Eq. (42) becomes overdetermined, thus it is necessary to drop excess collocation equations at time 1t . Finally, Eq. (42) is modified as

21 1 111 1 1

2 2 2 2

( )rr r

r r

c

−− − −

− = − + + + −

UU U UE J E

U U U M CU M N M F

, (44)

The symbol [ ]r• denotes the remained matrix after the ( 1)lM th+ rows and columns in [ ]• are dropped,

0,1, 2,...l = . If [ ]r• is a vector, we just need to remove the ( 1)lM th+ rows in [ ]• .

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A flow diagram is presented in Fig. 2 to illustrate the WA algorithm-1.

Constant Matrices Construct , , ,

Initialization Let be the starting guesses: 0

pUpU

Correction Get the corrected solution using Eq. (44)( )p cU

Correction Norm

( )p c p pε = −U U U

Stopping Criterion 1 , where is the toleranceε δ≤ δ

Yes

Finished

No Stopping Criterion 2

maxi i<

No

Stop

Yes

Increment1i i= +

State Update( )p p c=U U

E

Incorporating Initial Condition Drop excess collocation equations at in Eq. (44) 1t

Evaluation of Varying Matrices The restoring force vector: in Eq. (42) the Jacobian matrix: in Eq. (42)J

N

C M F

Fig. 2 Flow chart overview of the WA algorithm-1

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13

2.4.2. WA Algorithm-2

According to Eq. (25), the Lagrange multipliers can be approximated by Taylor series as

20 1 2( ) ( ) ( ) ...t tτ τ τ= + − + − +λ T T T , (45)

where 0 = −T I , 1 ( )t= −T J , 22 ( ) 2t= −T J , and so on. Substituting it into Eq. (28), the correctional

formula is obtained as

221 1 121 1 1

2 2 2 2

( ) ( ) ( )[ ( )( ) ( ) ]( ) ( ) 2

m

i

tm m

tm mc

t t tt t t dt t

τ τ τ− − −

−′ = + − − − − − + + ′ + − ∫

uu u uJI Ju u u M Cu M N M F

(46)

In implementations, ( )τλ is commonly approximated by truncated Taylor series. The simplest could be the zeroth order approximation

( )τ = −λ I , (47)

or the first order approximation

( ) ( )( )t tτ τ= − − −λ I J . (48)

Higher order approximations are possible, but they are rarely used in practice considering the computational complexity.

With the zeroth order approximation of ( )τλ , Eq. (45) becomes

21 1 11 1 1

2 2 2 2

( ) ( )( ) ( )

m

i

tm m

tm mc

t td

t tτ

− − −

−′ = − + ′ + − ∫

uu u uu u u M Cu M N M F

. (49)

With the first order approximation of ( )τλ , Eq. (45) becomes

21 1 11 1 1

2 2 2 2

( ) ( )[ ( )( )]

( ) ( )m

i

tm mm mt

m mc

t tt t d

t tτ τ

− − −

−′ = + − − − + ′ + − ∫

uu u uI J

u u u M Cu M N M F. (50)

where 1, 2,...m M= , [ ]1,m i it t t +∈ , and 1i it t+ − is a finite large time interval. By separating the terms

involving mt and τ , Eq. (50) leads to

1 1

2 2

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )m m m

i i i

t t tm mm m mt t t

m mc

t td t t d t d

t tτ τ τ τ τ τ τ

= − + −

∫ ∫ ∫

u uR J R J R

u u, (51)

After some rearrangements, Eq. (51) can be rewritten as

1 1

2 2c

= − + −

U UHR JTHR JHTR

U U , (52)

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14

where H is the transformation matrix corresponding to integral, and T is the matrix related with time t . The configurations of matrices H , T , J and R are provided in Appendix C. Fig. 3 shows the flow chart of WA algorithm-2

Constant Matrices Construct , , , , ,

Evaluation of Varying Matrices The Jacobian matrix: in Eq. (52) and the residual matrix: in Eq. (52)

J

Initialization Let be the starting guesses: 0

pUpU

Correction Get the corrected solution using Eq. (52)( )p cU

Correction Norm

( )p c p pε = −U U U

Stopping Criterion 1 , where is the toleranceε δ≤ δ

Yes

Finished

No Stopping Criterion 2

maxi i<

No

Stop

Yes

Increment1i i= +

State Update( )p p c=U U

H

R

E TM C F

Fig. 3 Flow chart overview of the WA algorithm-2

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15

2.4.3 WA Algorithm-3

Considering Eq. (38), if the Lagrange multiplier λ is approximated by Taylor series, we have

2 21 10 11 1 1 1 1

2 22 2

( )( ) [ ( ) ...]

( ) i

t

tc

td t t dtdt

τ τ− − − − −

− −′ = − − + − + + ′+ − + − ∫

u uu uJ T T

u uM Cu M N F M Cu M N M F(53)

If λ is simply approximated by 0T , Eq. (53) becomes

2 21 101 1 1 1 1 1

2 22 2

( )( )

( ) i

t

tc

td t dtdt

τ− − − − − −

− −′ = − − + ′+ − + − ∫

u uu uJ T

u uM Cu M N M F M Cu M N M F.

(54)

Since 0 = −T I , Eq. (54) is further rewritten as

2 21 11 1 1 1 1 1

2 22 2

( )( )

( ) i

t

tc

td t dtdt

τ− − − − − −

− −′ = − + + ′+ − + − ∫

u uu uJ

u uM Cu M N M F M Cu M N M F (55)

By using pu ( 1, 2p = ) as trial functions and making collocations, the WA algorithm-3 is obtained from Eq. (55) as

2 21 11 1 1 1 1 1

22 22

( )( )( )

( ) ( ) ( )( )

m

i

tmmm t

m m mm c

ttt d

t t ttτ

− − − − − −

′ − −′ = − + + ′+ − + − ′ ∫

u uu uJ

uM Cu M N M F M Cu M N M Fu

(56)

where ( )p mt′u and the integrals can be obtained as is stated in Eqs. (35, 34). After rearrangements, Eq. (56) can be written as

211 1 1

2 2c− − −

− = − + + −

UUE JHR

U M CU M N M F

. (57)

The configurations of matrices E , M , C , N , J , H , and R are provided in Appendix C.

Note that the initial conditions are not guaranteed by Eq. (57), just like Eq. (42) in WA algorithm-1, thus it should be further modified as

21 11 1 1

2 22

rr dr d

c c− − −

− = − + − + −

UU UE JHR E

U UM CU M N M F

, (58)

The symbol [ ]r• denotes the remained matrix after the ( 1)lM th+ rows and columns in [ ]• are dropped,

0,1, 2,...l = . If [ ]r• is a vector, we just need to remove the ( 1)lM th+ rows in [ ]• . The symbol [ ]d•

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16

denotes the part of [ ]• that were dropped to obtain [ ]r• . The flow chart of this algorithm is provided in Fig. 4

Constant Matrices Construct , , , , ,

Evaluation of Varying Matrices The restoring force vector: in Eq. (57) the Jacobian matrix: in Eq. (57) and the residual matrix: in Eq. (57)

J

Initialization Let be the starting guesses: 0

pUpU

Correction Get the corrected solution using Eq. (58)( )p cU

Correction Norm

( )p c p pε = −U U U

Stopping Criterion 1 , where is the toleranceε δ≤ δ

Yes

Finished

No Stopping Criterion 2

maxi i<

No

Stop

Yes

Increment1i i= +

State Update( )p p c=U U

N

Incorporating Initial Condition Drop excess collocation equations at in Eq. (58) 1t

R

H E TM C F

Fig. 4 Flow chart overview of the WA algorithm-3

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3. Nonlinear Vibrations of a Buckled Beam

The governing equation of a buckled one-dimensional Bernoulli beam is written in non-dimensional form as

1 2

0

1 ( ) cos2

ivw w Pw cw w w dx F x t′′ ′′ ′+ + + − = Ω∫ , (59)

BC’s: 0w w′= = at 0x = and 1x = ,

where w is the transverse displacement of the beam, and P is the axial load on the beam. ( )F x is the transverse distributed load on the beam and Ω is the frequency of the applied load ( )F x . The overdot denotes the derivative with respect to time t , while the prime denotes the derivative with respect to the spatial coordinate x .

To solve the preceding partial differential equation, we first assume the spatial modes:

1

( , ) ( ) ( , ) ( ) ( ) ( )N

s s n nn

w x t w x v x t w x x q tφ=

= + = +∑ , (60)

where sw is the static postbuckling displacement, ( , )v x t is the superposed dynamic response around the

buckled configuration. ( )n xφ are the mode shapes of vibration around the buckled configuration, and

( )nq t are the amplitudes of ( )n xφ .

The buckled configuration ( )sw x can be obtained by first solving the static buckling problem where the time derivatives and the dynamic load are removed in Eq. (59).

1 2

0

1 02

ivw Pw w w dx′′ ′′ ′+ − =∫ . (61)

Mathematically, there could be various buckled mode shapes, depending on the corresponding load. However, in structural mechanics, the first buckled mode shape is mostly of importance, from which ( )sw x is obtained as

. (62)

The nondimensional transverse displacement b at the midspan of the beam is related to the load P via

2 24( )cb P P π= − , (63)

where cP is the critical load corresponding to the first Euler buckled mode, namely 24cP π= .

Substituting the assumed solution of ( , )w x t into the governing equation and dropping all the nonlinear, damping, and forcing terms, we have the following linear partial differential equation that can be tackled using the linear vibration mode theory.

1( ) (1 cos 2 )2sw x b xπ= −

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12 2 3

04 2 cos 2 sin 2 0ivv v v b x v xdxπ π π π′′ ′+ + − =∫ . (64)

By assuming ( , ) ( ) i tv x t x e ωφ= and substituting it into Eq. (64), the mode shape is obtained as

1 1 2 1 3 2 4 2 5( ) ( sin cos sinh cosh ) cos 2h px c s x c s x c s x c s x c xφ φ φ π= + = + + + + , (65)

where 2 2 2 1/21,2 ( 2 4 )s π π ω= ± + + , and 5c should satisfy the following equation:

12 4 2 2 3

5 0(2 ) 2 sin 2hb c b xdxπ ω π φ π′− = ∫ . (66)

Using the boundary conditions and Eq. (66), the mode shapes ( )n xφ can be obtained.

The resulting linear vibration mode shapes ( )n xφ are used to construct the solution ( , )v x t . Using the

multi-mode Galerkin discretization, where the weighting functions are the same as the trial functions ( )n xφ , the partial differential equation (59) is then reduced to a system of coupled Duffing equations.

2

, , ,cos

N N

n n n n nij i j nijk i j k ni j i j k

q q cq b A q q B q q q f tω+ = − + + + Ω∑ ∑ , 1, 2,...,n N= . (67)

In this paper, four modes are retained in the reduced model. The buckling level is selected as 4b = . Correspondingly the natural frequencies of the four linear vibration modes are obtained as 1 30.7067ω = ,

2 44.3627ω = , 3 108.3322ω = , and 4 182.1178ω = . The parameters nijA , and nijkB are provided in Appendix A and B.

4. Numerical Results and Discussion

Considering the discretized nonlinear system (67), several types of nonlinear resonances of the buckled beam may occur due to the external harmonic excitation. The primary resonance is mostly observed when the excitation frequency Ω is close to one of the mode frequencies nω , which often leads to a periodic motion of large amplitude in that mode. For the existence of quadratic nonlinearities and cubic nonlinearities, the subharmonic resonances and superharmonic resonances may also occur for Ω that has an integer relationship to nω .

In the following, the resonance of the buckled beam under harmonic excitations is investigated. The frequency Ω is selected close to the natural frequency of the first vibration mode 1ω . The external force is supposed to be uniform over the length of the beam, thus ( )F x in Eq. (59) is constant. Through Galerkin discretization, the forces imposed on the four linear vibration modes are obtained as 1 0.850654f F= − ,

2 0f = , 3 0.309884f F= , and 4 0f = .

In the numerical simulations, the force-sweep as well as the frequency-sweep processes are used to obtain an overview of the nonlinear dynamical behaviors of the buckled beam when subjected to a primary

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resonance excitation of its first vibration mode. Considering that the magnitudes of the coefficients 2nω ,

nijbA , and nijkB are roughly between 310 and 410 in the restoring forces, the amplitude of the external

force F is set to vary between 40F = and 600F = in the force-sweep process . In the frequency-sweep process, F is fixed at 400F = , while the frequency Ω is varied between 30Ω = and 28Ω = .

There are abundant types of basis functions for choice in literature. In this paper, the basis functions used in WA method is selected as Chebyshev polynomials of the first kind and the collocation points are selected as Chebyshev-Gauss-Lobatto nodes [15].

4.1 Nonlinear Dynamical Behavior including Bifurcation and Chaos

With the discretized equations derived above, the nonlinear vibrations of a buckled beam are first investigated under uniform harmonic excitations, in which the frequency of the external load ( ) cosF x tΩ is 30Ω = . A force sweeping approach is adopted herein to capture the bifurcation phenomenon. For simplicity, a periodic motion is referred to as period- n motion if its period is nT , where 2T π= Ω . It is shown in Fig. 5 that a period-one motion is obtained for 40F = . Fig. 5 (a) is the Poincare map that indicates the evolution of the displacement 1( )q t and the velocity 1( )q t from the initial values to the final state, which is referred to as the sink. It can be found that the transient process of period-one motion is rather simple. Since the sink is the only attractor in this area, all the points around it are attracted to it asymptotically.

As the excitation force F increases, the first period-doubling bifurcation occurs at about 420F = . A typical period-two motion is presented in Fig. 6 for 440F = . In Fig. 6 (a), the Poincare maps of the transient process are obtained using WA algorithms of various integrating accuracy. It is found that the two sinks standing for the period-two motion exist in a “strange attractor”. This means that the transient response of this motion is chaotic-like. For the existence of sinks, the motion will eventually settle down to a steady limit cycle oscillation. However, it could be difficult to determine the exact time when the motion settles down if the transient stage is prolonged. The accurate integration of the system will also be a challenge, because the trajectory in the chaotic regime can easily diverge even with a very small integration error.

By further increasing the force amplitude, the second period-doubling occurs at about 454F = . A period-four motion is presented in Fig. 7 for 460F = . It can be observed in Fig. 7 (a) that the four sinks also coexist with the chaotic attractor, thus the transient motion is also chaotic. By comparing the chaotic regimes in Fig. 7 (a) and Fig. 7 (a), there is not much difference before and after the bifurcation occurs. Therefore, it seems that the chaotic attractor is barely affected by the bifurcation of periodic motions. The next bifurcation occurs at 461F = , leading to the period-eight motion. The corresponding Poincare map, phase portrait and response curve are plotted in Fig. 8.

The results in Figs. 4-7 can be obtained by both the HHT-α and WA algorithms. However, the time step size has to be selected very small to accurately obtain the steady periodic motions by using the HHT-α algorithm, of which the computational time will be much more prolonged. On the contrary, the step size of WA algorithms can be selected relatively very large, but it still easily achieve high accuracy in predicting these limit cycle oscillations. It will be shown in the next subsection that the WA algorithm has a far better performance than the HHT-α method in terms of accuracy, computational time, and speed of convergence, in predicting the nonlinear dynamical responses of the buckled beam involving bifurcation and chaos.

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Fig. 5 (a)

Fig. 5 (b)

Fig. 5 (a) The Poincare map for 40F = , (b) The period-1 limit cycle oscillation. Same results for HHT-α and WA algorithms, but with different computational performances, as is discussed in subsection 4.2.

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Fig. 6 (a)

Fig. 6 (b)

Fig. 6 (a) The Poincare map for 440F = , (b) The period-2 limit cycle oscillation. Same results for HHT-α and WA algorithms, but with different computational performances, as is discussed in subsection 4.2.

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Fig. 7 (a)

Fig. 7 (b)

Fig. 7 (a) The Poincare map for 460F = , (b) The period-4 limit cycle oscillation. Same results for HHT-α and WA algorithms, but with different computational performances, as is discussed in subsection 4.2.

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Fig. 8 (a)

Fig. 8 (b)

Fig. 8 (a) The Poincare map for 461F = , (b) The period-8 limit cycle oscillation. Same results for HHT-α and WA algorithms, but with different computational performances, as is discussed in subsection 4.2.

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Fig. 9 (a)

Fig. 9 (b)

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Fig. 9 (c)

Fig. 9 (d)

Fig. 9 The same chaotic motion revealed by HHT-α (with very small step size) and WA algorithms. (a) Phase portrait, (b) time responses, (c) Poincare map, (d) FFT curves.

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Fig. 10 (a)

Fig. 10 (b)

Fig. 10 Bifurcation diagrams obtained by (a) force-sweeping, (b) frequency-sweeping process. Same results obtained for HHT-α (with very small step size) and WA algorithms.

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As the period-doubling bifurcation proceeds, more sinks appear in the chaotic regime, and eventually the motion becomes completely chaotic. In Fig. 9, the chaotic motion is presented through phase portrait, response curve, Poincare map, and FFT curve. Similar period-doubling route to chaos is also observed for varied excitation frequency with the amplitude fixed at 400F = . The bifurcation diagrams obtained through force-sweeping process and frequency-sweeping process are plotted in Fig. 10.

In predicting chaotic motion, the trajectories obtained using the considered two methods diverge after a short time. However, the Poincare maps and the FFT curves coincides very well, which shows that the same chaotic motion is revealed by these two methods.

4.2 Comparison Between the HHT-α and WA Algorithms

The discretized model is solved using both the HHT-α and WA methods. It is found that both methods can predict the limit cycle oscillations and chaos. However, the computational performances of these two methods are very different. In the analysis below, various step sizes are used to test the stability of the algorithms. It is found through simulations that the WA method converges for both periodic and chaotic motions with step sizes as large as 5T , where T is the period of the first vibration mode, 12T π ω= . The largest step size of HHT-α method depends on the nonlinear algebraic equations (NAEs) solver. By using Newton-Raphson method, the step size should be no greater than 20T , otherwise the solution of nonlinear equilibrium equations of HHT-α may not be found.

We tried to reveal the steady periodic motions of the buckled beam under different external excitations, using HHT-α and WA method separately. After the motion settles down, the extremes of the displacement

1( )q t obtained by these two methods are recorded in Table 1. The exact values are assumed to be provided by ode45, and they are fully consistent with the results obtained by WA method, with the time step size being 5T . The number of collocation points used in WA method is 7N = . Increasing the collocation points can further improve the accuracy of this method.

The HHT-α method fails to work for 5t T∆ = and 10T , because the Newton-Raphson iteration

scheme cannot converge for such large steps. With the step size being 50T , the HHT-α method provides steady periodic motions, although they are quite erroneous compared to the exact ones. Particularly, in the parameter region where the period-8 motion dominates, the HHT-α method only provides period-4 motion, which is due to the integrating inaccuracy. By shortening the step size, the accuracy is significantly improved. However, there are still some observable discrepancies between the results of HHT-α and the exact ones, even with the step size in HHT being as small as 1000t T∆ = . In the simulations, different values of α are used. In the case of 0.1α = − , a moderate numerical damping is included, which could filter the high-frequency component of motion and avoid divergence. For 0α = , no numerical damping is introduced and the HHT-α method becomes the average acceleration method. By comparing the values of extremes with the exact ones, table 1 indicates that the periodic motions are better predicted with 0α = than with 0.1α = − . This result is reasonable since the numerical damping introduced in HHT-α method makes the system behaves like that extra damping exists in the simulation.

The limit cycles of period-4 and period-8 motions are obtained using WA method with 5t T∆ = . They

are compared with the results obtained from HHT-α method, where 0.1α = − and 50t T∆ = . It is shown in Fig. 11 that the HHT-α method cannot predict the true dynamical behavior with such a large

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step size. In Fig. 11 (a), an obvious discrepancy exists between the results of WA and HHT-α method, while in Fig. 11 (b), the HHT-α method gives a period-4 motion at where the period-8 motion should be revealed. In all, both the WA and the HHT-α method can stably integrate the nonlinear system of buckled beam. However, the step size of HHT-α method should be selected very small to obtain valid solutions.

Table 1 Extremes of the steady periodic motions obtained by HHT-α and WA methods

Methods Period-1 Period-2 Period-4 Period-8 WA ( 5T ) 0.6006 3.0983 2.9881 2.9752

HHT-α 0.1α = − 0α = 0.1α = − 0α = 0.1α = − 0α = 0.1α = − 0α = 5T \ \ \ \ \ \ \ \

10T \ \ \ \ \ \ \ \

50T 0.5626 0.5954 3.1282 3.1164 3.0015 2.9955 \ \

100T 0.5826 0.5995 3.1074 3.1025 2.9912 2.9898 \ 2.9828

500T 0.5972 0.6006 3.0994 3.0984 2.9884 2.9881 2.9770 2.9755

1000T 0.5989 0.6006 3.0988 3.0983 2.9882 2.9881 2.9760 2.9756

Fig. 11 (a)

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Fig. 11 (b)

Fig. 11 Comparisons between the limit cycle oscillations obtained by the WA ( 5t T∆ = ) and HHT-α

( 50t T∆ = , 0.1α = − ) algorithms.

Generally, the evaluation of computational accuracy is nontrivial for numerical methods, especially in nonlinear problems. Herein, the numerical solution of MATLAB built in ode45 function is used as the benchmark to estimate the computational error of HHT-α and WA methods. The absolute and the relative accuracies of ode45 are both set as 1 15E − . The transient responses of period-one, period-two, period-four and period-eight motions are obtained using HHT-α and WA methods respectively. Then by comparing with those obtained by ode45, the discrepancies of the numerical results are shown in Fig. 12. To evaluate the highest accuracy these two methods may achieve, the steps size of HHT method is selected as

1 4t e∆ = − , while that of WA method is 6 0.035t T∆ = = with 13 collocation points in each time step.

As is illustrated in subsection 4.1, the transient response of period-one motion is relatively simple and non-chaotic. It is reflected by the consistent numerical discrepancies over the simulation time in Fig. 12 (a). As is shown, the WA method achieves very high accuracy with respect to ode45. The discrepancies between them is five magnitudes lower than that between HHT-α and ode45. For the multiple-period motions, the numerical discrepancies accumulate exponentially for both HHT-α and WA algorithms before 10t = , as shown in Fig. 12 (b-d). This can be explained by the fact that chaotic regime exists around the periodic motion. It is shown that the accuracy of WA method is still much higher than the HHT-α method in the integration of transient chaotic motions. Although the results of WA method diverge from that of ode45 after 10t = , it does not mean that the WA method fails. Actually, the chaotic regime is so sensitive to the initial state that even an error occurred on machine precision could blow up and cause significant discrepancy in the final state.

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Fig. 12 (a)

Fig. 12 (b)

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Fig. 12 (c)

Fig. 12 (d)

Fig. 12 The computational error of the HHT-α and WA algorithms

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Table 2 lists the computational time, iteration steps and step sizes of HHT-α and WA algorithms. It is found that the WA method consumes only 20% of the computational time required by HHT-α method. This cost savings are expected to be much more dramatic for large order dynamical systems. The step size used in HHT-α is 0.0001, while that in WA is 0.035. According to the performances demonstrated in Fig. 12 and Table 2, the WA method is much more accurate and efficient than HHT-α method, in addition to requiring only much larger step sizes. For further comparison, the performance indexes of ode45 are also presented in Table 2. It is shown that the computational efficiency of the WA algorithm is far superior to that of ode45. Noting that the ode45 is a built-in function in MATLAB that has already been optimized, while the WA algorithms are implemented in MATLAB with a roughly designed program. Moreover, the WA algorithms can be easily realized using parallel programing, which will further improve the computational efficiency.

Table 2. Computational cost of HHT-α and WA method

Cases Computational Time (sec) Iteration Steps Step Size HHT-α WA ode45 HHT-α WA ode45 HHT-α WA ode45

Period-1 11.4 2.6 17.6 593468 5392 903797 0.0001 0.035 1.6e-05 Period-2 11.8 2.5 22.5 599664 5509 1022317 0.0001 0.035 1.5e-5 Period-4 11.6 2.6 21.2 599670 5608 1015945 0.0001 0.035 1.5e-5 Period-8 11.7 2.4 19.6 599629 5567 1015821 0.0001 0.035 1.5e-5

To investigate the energy conservation properties of the HHT-α and the WA method, a simple undamped duffing equation is used for demonstration.

3 0x x x′′ − + = .

The Hamiltonian energy of this system is

2 2 4

2 2 4x x xH′

= − + .

Starting from the initial state (0) 1.5x = , (0) 0x′ = , the system is integrated using the HHT-α , the WA and the ode45 methods. The step size of HHT-α method is 0.01t∆ = and the simulation is carried in the time interval [0,1000]t∈ . For the WA method, the step size is selected as 2t∆ = , with 32 collocation points in each step. The absolute and relative accuracies of ode45 are both set as 1 15E − . The computational errors of the Hamiltonian are recorded and plotted in Fig. 13. It can be seen that both the WA method and ode45 are much superior to the HHT-α method on energy conservation. Notably, the WA method behaves even better than ode45, with a negligible error of Hamiltonian of 1 13E − . Among all these methods, the HHT-α method with 0.1α = − is the worst on energy conservation, as can be seen in Fig. 13 (a). After dropping out the numerical damping by setting 0α = , the performance of HHT-α method is much improved, with the error of Hamiltonian being 1 5E − .

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Fig. 13 (a)

Fig. 13 (b)

Fig. 13 Computational error of Hamiltonian using the HHT and WA algorithms, and ode45

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4.3 Discussions of the relative performances of the 3 WA algorithms

Although the WA algorithms-1, 2, and 3 are derived from the same optimal error-feedback iteration concept, they have some differences in implementation and computational performances that the users of them should be aware of. In Figs. 1-3, the flow chart overview of these three algorithms are presented. In the implementations of WA algorithms-1 and 3, the initial conditions have to be incorporated in the iteration formula and the excess collocation equations need to be removed. In WA algorithm-2, the initial conditions are naturally satisfied, thus the implementation of it is more straightforward than WA algorithms-1 and 3.

The convergence speed of the WA algorithm-1 is supposed to be the fastest, since it is equivalent to Eq. (12), where the matrix of weighting functions λ is optimally derived. However, the WA algorithm-1 involves inversion of the Jacobian matrix ( )+E J , which is varying during the iteration process. For systems with high dimensions, computing the inversion of the Jacobian matrix could be very time consuming. In addition, if the Jacobian matrix become ill-conditioned during the iteration, the WA algorithm-1 may easily diverge. On the contrary, the WA algorithms-2 and 3 are free from inverting matrices during the iteration process.

Considering the buckled beam problem in this paper, the performances of the WA algorithms are compared with each other. The comparing results listed in table 3 are obtained during the simulation time of 200T , where T is the period of the first mode.

Table. 3 Comparison of WA algorithms

Method Largest Step Size Number of Iterative Steps Computational Time

WA algorithm-1 5T 18202 9s WA algorithm-2 5T 14765 6s WA algorithm-3 5T 16636 10s

As is shown in table 3, the largest step sizes of the WA algorithms are the same, while the number of iterative steps and computational time of WA algorithm-2 are the least among the three proposed algorithms. The computational errors are not provided herein because the WA algorithms behave similar in terms of accuracy, which is already presented in Fig. 12. Overall, although all the proposed WA algorithms can be efficiently applied to solving the nonlinear dynamical equations of the buckled beam, the WA algorithm-2 is the most recommended, not only because it incorporates the initial boundary conditions inherently and is free from inverting the Jacobian matrix, but also for that it has the best computational performance among the proposed method.

5. Conclusion

The Wang-Atluri Algorithms are used to solve a nonlinear dynamical model of the clamped-clamped buckled beam. The numerical results show that the proposed method is very accurate and efficient in predicting nonlinear dynamical behaviors including bifurcation and chaos. Through force-sweep and frequency-sweep processes, the periodic motions and the period-doubling routes to chaos are successfully captured by the Wang-Atluri algorithms. It is also found in the simulations that the steady multiple-periodic

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motions are actually surrounded by chaotic motions in the neighborhood. It is the reason for the chaotic transient motions to be hardly predicted precisely.

The robustness of Wang-Atluri method is indicated by the highly accurate integration using a large step size 5t T∆ = . Compared with the Hilber-Hughes-Taylor-α method, the proposed method is superior on many aspects involving accuracy, efficiency, stability and energy conservation. It is shown in this paper that the Hilber-Hughes-Taylor-α method cannot accurately predict the periodic vibrations of the buckled beam unless extremely small step sizes are used. An obvious discrepancy can still be observed between the exact phase portrait and that provided by Hilber-Hughes-Taylor method-α even when a relatively small step size 100t T∆ = is used. The example of Duffing equation also indicates that the energy of the system is barely conserved using Hilber-Hughes-Taylor-α method after the simulation time 1000t = , while the error of energy in the results of Wang-Atluri algorithm is negligible.

Appendix A

1,1,1 124.905A = , 1,1,2 0A = , 1,1,3 162.554A = , 1,1,4 0A = , 1,2,2 171.098A = , 1,2,3 0A = ,

1,2,4 -141.515A = , 1,3,3 407.072A = , 1,3,4 0A = , 1,4,4 625.188A = ;

2,1,1 0A = , 2,1,2 342.195A = , 2,1,3 0A = , 2,1,4 -141.515A = , 2,2,2 0A = , 2,2,3 391.398A = , 2,2,4 0A = ,

2,3,3 0A = , 2,3,4 -161.863A = , 2,4,4 0A = ;

3,1,1 81.2768A = , 3,1,2 0A = , 3,1,3 814.145A = , 3,1,4 0A = , 3,2,2 195.699A = , 3,2,3 0A = ,

3,2,4 -161.863A = , 3,3,3 1264.72A = , 3,3,4 0A = , 3,4,4 715.082A = ;

4,1,1 0A = , 4,1,2 -141.515A = , 4,1,3 0A = , 4,1,4 1250.38A = , 4,2,2 0A = , 4,2,3 -161.863A = , 4,2,4 0A = ,

4,3,3 0A = , 4,3,4 1430.16A = , 4,4,4 0A = ;

Appendix B

1,1,1,1 -65.3795B = , 1,1,1,3 -79.2731B = , 1,1,2,2 -268.675B = , 1,1,2,4 222.22B = , 1,1,3,3 -600.138B = ,

1,1,4,4 -981.733B = , 1,2,2,3 -108.59B = , 1,2,3,4 89.8146B = , 1,3,3,3 -233.924B = , 1,3,4,4 -396.786B = ;

2,1,1,2 -268.675B = , 2,2,2,2 -1104.11B = , 2,1,2,3 -217.18B = , 2,2,3,3 -2378.47B = , 2,1,1,4 111.11B = ,

2,2,2,4 1369.81B = , 2,1,3,4 89.8146B = , 2,3,3,4 983.614B = , 2,2,4,4 -4412.05B = , 2,4,4,4 1668.42B = ;

3,1,1,1 -26.4244B = , 3,1,2,2 -108.59B = , 3,1,1,3 -600.138B = , 3,2,2,3 -2378.47B = , 3,1,3,3 -701.773B = ,

3,3,3,3 -5123.69B = , 3,1,2,4 89.8146B = , 3,2,3,4 1967.23B = , 3,1,4,4 -396.786B = , 3,3,4,4 -8690.89B = ;

4,1,1,2 111.11B = , 4,2,2,2 456.603B = , 4,1,2,3 89.8146B = , 4,2,3,3 983.614B = , 4,1,1,4 981.733B = − ,

4,2,2,4 -4412.05B = , 4,1,3,4 -793.572B = , 4,3,3,4 8690.89B = − , 4,2,4,4 5005.26B = , 4,4,4,4 -14741.6B = .

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Appendix C

Table 4 The Constant and Varying Matrices in the WA algorithms

Constant Matrices Varying Matrices 1

2 2 ( )L L−

×= ⊗E I LB B ,

M M×= ⊗M M I ,

M M×= ⊗C C I ,

1 2[ , ,..., ]TMt t t=t ,

ˆ ( )diag=t t ,

2 2L L×= ⊗t I t , 1 1

2 2 ( )L L− −

×= ⊗H I L B B .

ˆ( )=J J t , ( )=N N t ,

211 1

2 2− −

− = + + −

UUR E

U M CU M N F

( )=F F t

M is the number of collocation points in each time interval; L is the length of variable vector x in Eq. (1)

Acknowledgement

The authors thank Dr. Lawrence Schovanec, the President of Texas Tech University, for his support of this research, through the Presidential Chair and University Distinguished Professorship.

References

[1] Kerschen, G.; Worden, K.; Vakakis, A. F.; Golinval, J. C.: Past, present and future of nonlinear system identification in structural dynamics. Mechanical Systems and Signal Processing, vol. 2006, no. 20, pp. 505-592 (2006)

[2] Strogatz, S. H.: Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Addison-Wesley (1994)

[3] Fehlberg, E.: Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems. Technical report, NASA (1969)

[4] Newmark, N. M.: A method of computation for structural dynamics. Journal of the Engineering Mechanics Division, vol. 85, no. 3, pp.67-94 (1959)

[5] Hilber, H. M.; Hughes, T. J.; Taylor, R. L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering & Structural Dynamics, vol. 5, no. 3, pp. 283-292 (1977)

[6] Xie, Y. M.: An assessment of time integration schemes for nonlinear dynamic equations. Journal of Sound and Vibration, vol. 192, no. 1, pp. 321-331 (1996)

Page 37: Bifurcation & Chaos in Structural Dynamics: A Comparison ...bifurcation and chaos in structural dynamics. The WA algorithms are also superior to the optimized ODE45 algorithm in MATLAB,

37

[7] Wang, X.; Atluri, S. N.: A Novel Class of Highly Efficient & Accurate Time-Integrators in Nonlinear Computational Mechanics, Computational Mechanics, vol. 59, no. 5, pp. 861-876, (2017)

[8] Wang, X.; Atluri, S. N.: A New Feedback-Accelerated Picard Iteration Method for Orbital Propagation & Lambert’s Problem, Jounral of Guidance, Control, and Navigation, online.

[9] Dong, L.; Alotaibi, A.; Mohiuddine, S. A.; Atluri, S. N.: Computational Methods in Engineering: A Variety of Primal & Mixed Methods, with Global & Local Interpolations, for Well-Posed or Ill-Posed BCs. Computer Modeling in Engineering & Sciences, vol. 99, no. 1, pp. 1-85 (2014)

[10] Atluri, S. N.: Methods of Computer Modeling in Engineering & the Sciences, Volume I. Tech Science Press, Forsyth (2005)

[11] Kreider, W.; Nayfeh, A. H.: Experimental Investigation of Single-Mode Responses in a Fixed-Fixed Buckled Beam. Nonlinear Dynamics, vol. 15, no. 2, pp. 155-177 (1998)

[12] Tseng, W. Y.; Dugunji, J.: Nonlinear vibrations of a beam under harmonic excitation. Journal of Applied Mechanics, vol. 37, no. 2, pp. 292-297 (1970)

[13] Emam, S. A.; Nayfeh, A. H.: On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation. Nonlinear Dynamics, vol. 35, no. 1, pp. 1-17 (2004)

[14] Bai, X.; Junkins, J. L.: Modified Chebyshev-Picard Iteration Methods for Solution of Boundary Value Problems. The Journal of Astronautical Sciences, vol. 58, no. 4, pp. 615-642 (2011)

[15] Fukushima, T.: Picard iteration method, Chebyshev polynomial approximation, and global numerical integration of dynamical motions. The Astronomical Journal, vol. 113, no. 5, pp. 1909-1914 (1997)