nonlinear vibrations of dynamical systems with a general form of

7/30/2019 Nonlinear Vibrations of Dynamical Systems With a General Form Of

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19 August 2002

Physics Letters A 301 (2002) 6573

www.elsevier.com/locate/pla

Nonlinear vibrations of dynamical systems with a general form ofpiecewise-linear viscous damping by incremental harmonic

balance method

L. Xu a,, M.W. Lu a, Q. Cao b

a Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR Chinab Department of Mathematics and Physics, Shandong University of Technology, Jinan 250061, PR China

Received 6 December 2001; received in revised form 25 June 2002; accepted 2 July 2002

Communicated by C.R. Doering

Abstract

Incremental harmonic balance (IHB) method for computation of periodic solutions of nonlinear dynamical systems is

extended here for analysis of a class of periodically excited systems with a general form of piecewise-linear viscous damping

characteristics, with an explicit formulation being derived , which is in many respects distinctively advantageous over classical

approaches, and especially excels in performing parametric studies as frequency response property. Numerical simulation of

a specific periodically excited oscillator of the considered type is effectively carried out by the IHB scheme and the results

compare very well with direct numerical integration. The formulation derived here can readily be combined with the existing

IHB scheme designed for treating systems with only piecewise-linear stiffness in analyzing complex dynamical behavior as

bifurcation and chaos of more general piecewise-linear systems. 2002 Elsevier Science B.V. All rights reserved.

Keywords: Incremental harmonic balance; Piecewise linear; Frequency response; Phase plane

1. Introduction

Nonlinear problems occur in both the subject of natural science and of engineering technology [1]. Nowadays,

elements with nonlinear characteristics have been widely used to obtain better performance in engineering practice,and many of these elements may be modeled as possessing piecewise-linear (PWL) or piecewise-nonlinear

characteristics. For instance, a wide range of structural systems of practical interest possess piecewise-linear

stiffness or damping, such as structures with different tension and compression module, mechanical systems having

clearances, and compliant offshore structures and articulated mooring towers under certain constrained conditions,

etc. In other fields of electronics, biology, economy, etc., the theoretical models of many nonlinear dynamical

problems are also found to be systems with piecewise linearity [2,3].

* Corresponding author.

E-mail address: [email protected] (L. Xu).

0375-9601/02/$ see front matter

2002 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 9 6 0 - X
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66 L. Xu et al. / Physics Letters A 301 (2002) 6573

As the piecewise-linear systems widely exist in engineering practice, with the investigation of their dynamical

properties being conducive to engineering design and application, increasing attention is given to such systems

in the research of nonlinear dynamical systems and much has been studied fruitfully in this regard yet. It has

already been revealed that systems with a piecewise linearity may exhibit very complex dynamical behavior [46].For example, the famous Chuas circuit, which is a very simple autonomous electrical system with a 3-segment

piecewise-linear resistor as the only nonlinear element, has been shown to exhibit chaos with a macroscopic

double-scroll structure [7]. Some higher-dimensional piecewise-linear circuits have been found to display

hyperchaos [8,9].

On the other hand, though it is well-known that the free vibrations of piecewise-linear dynamical systems can

be solved exactly by the so-called seam-sewing method, it is yet a great challenge to obtain a closed-form solution

for an excited steady-state vibration even for such simple nonlinear systems. The disadvantages of analyzing

piecewise-linear systems using various classical perturbation methods such as PoincarLindstedt, multiple scale,

or KrylovBogoliubovMitropolski method are obvious [10,11], for one thing, they are regarded to be valid only

when the nonlinearity of the considered dynamical system is weak. The harmonic balance method (Galerkins

procedure) can deal with systems with strong nonlinearity, whereas the set of nonlinear algebraic equations in terms

of Fourier coefficients should be reformulated when more harmonic terms are taken in order to get more accurateperiodic solutions, which inevitably hinders the adaptability or efficiency of computer numerical simulation. Other

numerical approaches, as the fourth order RungeKutta integration method which can give precise transient and

steady state response for given initial conditions, is usually time-consuming for performing parametric studies such

as obtaining a wide range frequency response, especially when the rate of convergence is low.

In order to overcome the deficiency mentioned above, the incremental harmonic balance (IHB) method

has been brought up, originally for treating periodic structural vibrations of elastic systems [12]. As noted

in [13], the IHB method is particularly convenient for computer implementation. In this method, the periodic

solution of the nonlinear dynamical system is represented by a limited expansion of Fourier series by which

the nonlinear differential control equations of the considered system are transformed into a set of linearized

incremental algebraic equations in terms of the Fourier coefficients, and so only linear equations have to

be formed and solved iteratively in each incremental step, with the formulation being maintained whenchanging the number of harmonic terms in the Fourier series of the solutions. Compared with classical

approaches, the IHB method is remarkably effective in computer implementation for obtaining response with

a desired accuracy over a wide range of varying parameter, with both stable and unstable solutions being

traced directly. The IHB method has already been successfully applied to or adapted for a wide range

of dynamical systems ever since its proposition. In [13], solution diagrams of three types of van der Pol

oscillator are plotted using this method. In [14] and [15], the method is applied, respectively, to the analysis

of bifurcation and chaos of an escape equation model and an articulated loading platform with piecewise-

nonlinear stiffness. The IHB method also finds its way in some class of piecewise-linear dynamical systems. In

[16], by expansion of the sign nonlinearity for small increments, IHB method is modified to perform a multi-

harmonic frequency domain analysis of dry friction damped systems. In [17], periodic steady state solution

of a nonlinear oscillator subjected to periodic excitation is analyzed where the stiffness of the system is of

the nature of unsymmetrical piecewise linearity. In [18], the IHB method is further extended to the periodic

vibrations of nonlinear systems with a general form of piecewise-linear restoring characteristics, which is of

great significance as many structural and mechanical systems of practical interest possess a piecewise-linear

stiffness.

In this Letter, the IHB computation scheme for a class of single degree-of-freedom systems with a general

form of piecewise-linear viscous damping force is derived, and numeric simulations of an oscillator of the

considered type are carried out in the meanwhile, showing great accuracy compared with the result of direct

numeric integration method. As many systems possess both stiffness and damping nonlinearity, the formulation

in the Letter can readily be combined with that of [18] in analyzing complex dynamical behavior as bifurcation

and chaos of such systems.


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L. Xu et al. / Physics Letters A 301 (2002) 6573 67

2. Dynamical system with piecewise-linear viscous damping of a general form

The dynamic model considered here is a single DOF vibration system with the equation of motion written as

(1)md2x

dt2+ h

dx

dt

+ k0x = q(t),

where m, k0, t and x refer to the mass, linear spring stiffness, time, and displacement, respectively. Assume that

the dynamical system is continuous only in defined subspaces Xi (i = 1, 2, . . . , n ) of the global state space X, thepiecewise-linear viscous damping force h(dx/dt) can then be denoted generally as the following form so as to be

qualified for a wide range of damping cases:

(2)h

dx

dt

= c0

dx

dt+ H

dx

dt

,

where H ( d xd t

)

=cic0

d xd t

, x

Xi , c0

=0 and ci

+c0 denotes the overall damping coefficient with respect to the

dampers which function in each subspace Xi (i = 1, 2, . . . , n ).Take the piecewise-linear oscillator in Fig. 1 as an example, the control equation of harmonically excited motion

is

(3)mx + hx+ k0x = Fcos t ,where the piecewise-linear damping force may be written as

(4)h

x=

(c0 + c1)x, x < d1,c0x, d1 x d2,(c0 + c1)x, x > d 2,

H

x=

c1c0

x, x < d1,0, d1 x d2,c1c0

x, x > d 2.

It should be noted here that the piecewise linear function H(dx/dt) is homogeneous with respect to dx/dt, that

is, for any constant C, it holds that H (C d xd t ) = CH ( d xd t ).The periodic exciting force q(t) can be expanded as a limited Fourier series up to N harmonic terms

(5)q(t) = f02

+N

n=1(fn cos nt+ gn sin nt),

where f0, fn and gn (n = 1, 2, . . . , N ) are force component amplitudes of the corresponding harmonic terms.By letting a new time scale = t, frequency ratio =

k0/m, damping ratio = c0

2

mk0, and noting that

H ( dxd

) = H ( d xd

), Eq. (1) is transformed into

(6)

2 d2x

d2 + 2 dx

d + x + 2 Hdx

d=

1

k0 q().

Fig. 1. Oscillator with piecewise-linear damping.


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3. IHB scheme

With regard to the piecewise-linear differential system (6), by a NewtonRaphson procedure, assuming that

x0( ) stands for an initially approximated vibrating state corresponding to the excitation parameters 0 and q0,a neighboring state may be denoted by

(7)x() = x0( ) + x( ), q ( ) = q0( ) + q( ), = 0 + ,where x(), q() and are small increments.

Correspondingly, the piecewise-linear function H ( dxd

) may be expressed by a first order Taylor expansion as

(8)H

dx

d

= H

dx0

d

+ H

dx0

d

dx

d.

By substituting expressions (7), (8) into Eq. (6) and neglecting the nonlinear terms of the small increments, (6)

becomes linearized as

(9)20d2x

d2+ 2 0

dx

d+ x + 2 0H

dx0

d

dx

d= R + S+ Q,

where

R =

20d2x0

d2+ 2 0

dx0

d+ x0 + 2 0H

dx0

d

1

k0q0

,

(10)S= 20d2x0

d2 2 dx0

d 2 H

dx0

d

, Q = q/k0.

R is the corrective term which goes to zero when the solution is reached.

Though Eq. (9) is linear, there are variable coefficients due to piecewise linearity of the damping force and

thus does not seem feasible to be solved directly, hence a Galerkin procedure is carried out as follows. Both the

approximate initial periodic solution and its small increment may be expressed as

(11)x0 =a0

2+

Nn=1

(an cos n+ bn sin n ), x =a0

2+

Nn=1

(an cos n+ bn sin n),

where N is the number of harmonic terms taken in the limited Fourier series, as in (5). By taking ans, bns as

the generalized coordinates, it is derived from Eq. (9) that

(12)

2

0

20

d2x

d2+ 2 0

dx

d+ x + 2 0H

dx0

d dx

d (x)d=2

0

(R + S + Q)(x)d,

which is equivalent to a system of 2N+ 1 linearized equations with the ans and bns being variables

(13)Ca = R + S + Qwhere

a = [a0, a1, . . . , aN, b1, b2, . . . , bN]T, a = [a0, a1, . . . , aN, b1, b2, . . . , bN]T,

(14)C =

C11 C12C21 C22

, R =

R1R2

, S =

S1S2

, Q =

Q1Q2

.


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The explicit expressions for elements of the above matrices are worked out as follows:

C11ij = jij

1 j220 + CNL11ij (i = 0, 1, . . . , N , j = 0, 1, . . . , N ) ,

C12ij = 2j ij 0 + CNL

12ij (i = 0, 1, . . . , N , j = 1, . . . , N ) ,C21ij = 2j ij 0 + CNL21ij (i = 1, 2, . . . , N , j = 0, 1, . . . , N ) ,C22ij = ij

1 j220

+ CNL22ij (i = 1, 2, . . . , N , j = 1, . . . , N ) ,R1i = i

1 i220

ai + 2i 0bi

fi

k0

+ RNL1i (i = 0, 1, . . . , N ) ,

R2i =

1 i220

bi 2i 0ai gi

k0

+ RNL2i (i = 1, 2, . . . , N ) ,

S1i = 2i(i0ai bi ) + SNL1i (i = 0, 1, . . . , N ) ,S2i = 2i(i0bi + ai ) + SNL2i (i = 1, 2, . . . , N ) ,

Q1i = 1k0

i fi (i = 0, 1, . . ., N), Q2i = 1k0

gi (i = 1, 2, . . . , N ) ,

where

n =

1, n = 0,1/2, n = 0, ij =

1, i = j,0, i = j.

It can be seen that the linear parts of the elements of C, R and S are roughly the same as in [18], while their

nonlinear parts are, respectively, expressed by:

CNL11ij = 2j i 02

0

H

dx0

d cos i sin j d , CNL12ij = 2j i 0

2

0

H

dx0

d cos i cos j d,

CNL21ij = 2j 02

0

H

dx0

d

sin i sin j d , CNL22ij = 2j 0

20

H

dx0

d

sin i cos j d,

RNL1i = 2i 02

0

H

dx0

d

cos i d , RNL2i = 2 0

20

H

dx0

d

sin i d,

SNL1i = 2i 2

0

H

dx0

d

cos i d , S NL2i = 2

20

H

dx0

d

sin i d.

The evaluation of these piecewise-linear integrals in programming can be achieved explicitly by a procedure usingbisection and interpolation method, which has been well expounded in [18]. Alternatively, powerful function of

modern mathematics software as MATLAB renders the direct integration of these integrals fast and precise enough

to serve the need.

Generally, in the study of forced vibration of dynamical systems, the excitation level is kept as constant, thus

Q = 0, and scheme (13) is reduced to(15)Ca = R + S.

As noted in [18], the -incrementation procedure for obtaining the frequency response curve of a dynamical

system may be carried out by incrementing from point to point, which implies that = 0 though the iteration


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process at every point, leading to the following equations

(16)C(i) a(i+1) = R(i) , a(i+1) = a(i) + a(i+1).

Being reevaluated in terms of the (i + 1)th amplitude vector a(i+1), the matrices C(i+1), R(i+1) are updated atevery increment.

From the derivation of the IHB computation scheme, it can be seen that N, that is, number of harmonics taken

in the limited Fouries series, is incorporated into the iteration process as an independent parameter, thus can be

conveniently varied according to the required precision, facilitating programming in computer simulation.

4. Numerical simulation

Taking the simple nonlinear oscillator shown in Fig. 1 as a numerical example, with stiffness k0 = 5.0 N/m, anddamping coefficients c0

=c1

=0.2 N s/m fixed, the IHB scheme as derived above for analyzing periodic motions

of a class of dynamical systems with a general form of piece-wise linear viscous damping is tested and evaluatedhere by some revealing numerical simulations.

One of the distinctive advantages of the IHB method over classical approaches is that it can be used with no

difficulty to obtain the response property of the considered dynamical system over a wide range of varying system

control parameter. The amplitudes of vibration may be expressed either as peak amplitudes per cycle or as the

norm of the harmonic components which is indicative of the total energy of the motion. The effect of various

system parameters, such as the influence of force amplitude, viscous damping and number of harmonics included

in the IHB scheme, on frequency response curves of the considered system with the clearance d1 = 0.8 m andd2 = 1.0 m can be systematically studied. By using peak amplitude and taking five harmonics, Fig. 2(a) shows thedependence of the frequency response at the force level F = 10 N on viscous damping = 0.05, 0.1 and 0.3, andthat at a viscous damping ratio of = 0.1 on various levels of force F = 2.0, 5.0 and 8.0 N is shown in Fig. 2(b).In the generation of the above curves, the IHB method uses an initial guess of the amplitude vector and iteratesuntil convergence is achieved, and the following rate of convergence is then remarkably improved by choosing the

initial guess of the amplitude vector to be the already converged vector from a neighboring frequency.

(a) (b)

Fig. 2. Frequency response by the IHB method with different (a) damping ratio, (b) force level.


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Table 1

Amplitude components by the IHB method with different number of harmonics

NH a0 a1 a2 a3 a4 a5 b1 b2 b3 b4 b5

1 0.0000 1.7769 2.59122 0.0000 1.7729 0.0103 0.0332 2.5998 0.0042 0.05333 0.0000 1.7719 0.0105 0.0325 0.0030 0.0165 2.6017 0.0043 0.0538 0.0004 0.0008

(a) (b)

(c) (d)

Fig. 3. Numerical results by the IHB method with different number of harmonics. (a) Time history of displacement and velocity. (b) Phase plane

with NH = 1. (c) Phase plane with NH = 3. (d) Phase plane with NH = 5.

The accuracy of the IHB method is evaluated here by comparing the results against those of existing time domain

analysis. In this case, a fourth order RungeKutta numerical integration routine is used to provide an accurate

basis for comparison. By letting = 0.8, = 0.2 and F = 10.0 N, Table 1 shows the amplitude componentscomputed by the IHB scheme with different number of harmonics (NH = 1, 3, 5, respectively). The time historyof displacement and velocity generated by numerically integrating Eq. (3) until steady periodic state is achieved is

shown in Fig. 3(a), while Fig. 3(b)(d) compares the corresponding steady state solutions by the IHB scheme with

NH = 1, 3, 5 denoted by the solid line with the result of numeric integration, which is dotted at some discrete pointsin the same phase plane, showing the effect of varying the number of harmonics on the precision of the results.


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(a) (b)

Fig. 4. Frequency response and phase plane of the oscillator with multi-frequency external periodic excitation. (a) Frequency response. (b)Phase plane.

The IHB result by taking only one harmonic as shown in Fig. 3(b) is exactly the solution that would have been

obtained by the first order harmonic balance (HB) method or equivalent first order frequency domain techniques.

More harmonics are necessary to approximate more precisely the periodic motion. It can be seen from Fig. 3(d)

that the IHB result by taking five harmonics already provides a fairly accurate approximation in the case.

In the case of multi-frequency external periodic excitation, the IHB scheme also proves to be highly efficient.

With = 0.02, d1 = d2 = 1.0 m, Fig. 4(a) shows the frequency response of the considered system under excitation5cos t + 5cos2t + sin4t N. The steady state solution of the case at = 0.2 by the IHB scheme with fiveharmonics taken is plotted in Fig. 4(b), which is compared with the dotted points by the fourth order RungeKutta

numerical integration routine, showing great accuracy.

5. Conclusion

The incremental harmonic balance (IHB) method is successfully extended to a class of nonlinear dynamical

systems with a general form of piecewise-linear viscous damping characteristics for computing periodic solutions,

which is in many respects distinctively advantageous over classical approaches. Numerical simulation of a specific

periodically excited oscillator of the considered type is effectively carried out by the IHB scheme, with the effect

of various system parameters, such as the influence of force amplitude, viscous damping and number of harmonic

included in the IHB scheme, on frequency response curves of the considered system being systematically studied,

and the results compare very well with direct numerical integration. The formulation derived here can readily

be combined with the already published IHB scheme designed for treating systems with only piecewise-linearstiffness in analyzing complex dynamical behaviors as bifurcation and chaos [19,20] of more general piecewise-

linear systems.

References

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[3] K. Murali, M. Lakshmanan, L.O. Chua, IEEE Trans. Circuits Systems 41 (1994) 462.

[4] Y.S. Choi, S.T. Noah, J. Sound Vibration 121 (1988) 117.


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[5] S. Maezawa, H. Kumano, Y. Minakuchi, Bull. JSME 23 (1980) 68.

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[15] A. Raghothama, S. Narayanan, Ocean Engrg. 27 (2000) 1087.

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nonlinear vibrations of dynamical systems with a general form of

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