Chaotic vibration of beams

7/27/2019 Chaotic vibration of beams

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European Journal of Mechanics A/Solids 24 (2005) 944956

Chaotic response of a large deflection beamand effect of the second order mode

Qiang Han , Xiangfeng Zheng

Department of Mechanics, College of Traffic and Communications, South China University of Technology, Guangzhou 510640, China

Received 19 October 2004; accepted 12 April 2005

Available online 13 June 2005

Abstract

This study intends to investigate the dynamic behavior of a nonlinear elastic beam of large deflection. Using the Galerkin

principle, the dynamic nonlinear governing equations are derived based on the single and double mode methods. Two different

kinds of nonlinear dynamic equations are obtained with the variation of the dimension and loading parameters. The chaotic

critical conditions are given by Melnikov function method for the single mode model. The chaotic motion is investigated and

the comparison between single and double mode models is carried out. The results show that the single mode method usually

used may lead to incorrect conclusions in some conditions, and instead the double mode or higher order mode method should

be used. Finally, the applicable condition of the single mode method is analyzed.

2005 Elsevier SAS. All rights reserved.

Keywords: Galerkin principle; Melnikov function; Double mode method; Poincare map

1. Introduction

Comprehensive understanding of the vibratory characteristics of the beams is extremely important to the researchers and

engineers in the field of aeronautics and astronautics, nuclear power and mechanical engineering. In theoretically anglicizing

the dynamics of the beams, the effects of systematic nonlinearities must be taken into consideration especially for the cases

of large deflections. The sources of nonlinearity of vibration systems are generally considered as due to the following aspects:

(1) the physical nonlinearity, (2) the geometric nonlinearity and, (3) the nonlinearity of boundary conditions. In solid mechanics,

the chaotic response of buckled beams is studied by many researchers, and this phenomenon has been more fully understood

(Holms and Marsden, 1981; Moon and Shaw, 1983; Paniela, 1994; Keragiozov and Keoagiozova, 1995). The forced response

of a nearly square plate, the nonlinear dynamics of a shallow arch, and the chaotic motion of a circular plate and the cylindricalshell are examples of some of the recent studies in mechanical and structural systems (Yang and Sethna, 1992; Winmin et al.,

1994; Han et al., 1998, 1999). Yet, few archival publications dealt with the chaotic motion based on the multi-mode method.

Especially, the single mode method is usually used and the difference between the single and double mode methods has not

been attended to. Recently, chaos of systems with finite degrees of freedom has attracted attention and stimulated studies.The

BubnovGalerkin method is applied to reduce partial differential equations governing the dynamics of flexible plates and shells

to a discrete system with finite degrees of freedom (Awrejcewicz and Krysko, 2003). Chaotic dynamics in four dimensional

* Corresponding author.

E-mail address: [email protected] (Q. Han).

0997-7538/$ see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.euromechsol.2005.04.003


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Q. Han, X. Zheng / European Journal of Mechanics A/Solids 24 (2005) 944956 945

self-excited systems with Coulomb-like friction is also studied, The obtained Melnikovs function yields thresholds for two

types stickslip onsets of chaos (Awrejcewicz and Sendkowski, 2004).

The goal of the present research focuses on the nonlinear dynamics of an elastic beam and the difference between the single

and double mode methods. Large deflection of the beam is taken into consideration. Equations of motion are derived with

both single and double modes. Nonlinear behavior of the beam, such as chaos, is studied. The criteria for chaos of the beam

is developed and the chaotic behavior of the transverse vibration of the beam is investigated through a numerical analysis.Results generated by single and double mode models are compared and the differences of the two models is identified and

analyzed.

2. Governing equations

As shown in Fig. 1, a pinned beam of length l is subjected to a lateral load p(x,t), and a longitudinal load P. The forces on

the unit element of the beam are shown in Fig. 2.

The lateral and the longitudinal dynamic equations of the beam can be given in the following form.

A ds2w

t2 =Q cos

Nsin

Q +Q

sds cos+

sds+N+

N

sds sin+

sds

0w

tds + p(x,t) ds, (1)

A2u

t2ds = Ncos Q sin +

N+ N

sds

cos

+

sds

+

Q + Qs

ds

sin

+

sds

, (2)

where, ds = dx/ cos , w denotes the lateral displacement, u the axial displacement, the material density, 0 the dampingcoefficient, A the cross section area, Q the shear force in the beam, N the axial force in the beam, the rotational angle of

(a) (b)

Fig. 1. A pinned beam.

Fig. 2. Forces on the unit element of the beam.


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Let us neglect the effect of the lateral motion on the longitudinal motion, the longitudinal dynamic equation can be simplified

as follows

2u

t2 E

2u

x2= 0. (13)

The corresponding boundary conditions are

u(0, t) = 0, EA u(l,t)x

= P(t). (14)

Let us neglect the longitudinal inertia effect in Eq. (13), which means N(x,t) EAu/x = P(t) = P0, Eq. (12b) canbe rewritten as follows.

A2w

t2+ D(w) + 0

w

t p(x,t) = 0, (15)

where

D(w) = P02w

x2+ EJ

4w

x4 3EJ

2w

x2

3 11EJ

2w

x2

3w

x3

w

x

12

(P0 + 3EA) 2w

x2+ 6EJ 4w

x4 21EJ

2wx2

3wx

2. (16)

Using the Galerkin principle and the multi- mode method w(x,t) = nr=1 r (x)qr (t), where functions r (x)(r = 1, 2, . . . , n ) must satisfy the boundary conditions of the beam, we can obtain the equations in terms of the time-dependentvariables.

l0

A

nr=1

r (x)qr (t) + D

nr=1

r (x)qr (t)

s (x) dx +

l0

n

r=1r (x)qr (t)

p(x,t)

s (x) dx = 0,

s = 1, 2, . . . , n . (17)

3. The single and double mode methods

3.1. The single mode method

Let us consider the beam subjected to simple harmonic excitation, p(x,t) = f sin (x/l) cos t. Substituting 1(x) =sin (x/l) into Eq. (17), we can obtain the following equation for the single mode model.

q1(t ) +

Aq1(t)

21

16

EJ 8

Al8q51 (t) +

5

4

EJ 6

Al 6+ 1

2

P04

Al4+ 3

8

E 4

l 4

q31 (t)

+

EJ 4

Al 4 P0

2

Al2

q1(t)

f cos t

A= 0, (18)

where 0 =

, is a small parameter. It is convenient to introduce the following non-dimensional quantities.

= 2

EJ

Al 4t, 1 = q1

A

J, = l

2

2

A

EJ,

= l2

2

AEJ, f = f l

4

4EJ

A

J, P0 =

P0

EA.

(19)

One obtains the non-dimensional equation for the single mode model.

1 + 1 + 31 + 51 (f cos 1) = 0, (20)where

=1

l2A

2J P0,

=1

2 P0

5

4

2J

l2A +3

8

,

= 21

16

4J2

l4A2< 0. (21)


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948 Q. Han, X. Zheng / European Journal of Mechanics A/Solids 24 (2005) 944956

3.2. The double mode method

If substituting 1(x) = sin(x/l) and 2(x) = sin(2x/l) into Eq. (17), we can obtain the following equations for thedouble mode model.

q1(t) +

A q1(t) 21

16

EJ 8

Al8 q

5

1 (t) + 651

8

EJ 8

Al 8 q

2

2 (t) 5

4

EJ 6

Al 6 +1

2

P04

Al4 +3

8

E 4

l 4

q

3

1 (t)

+

22 EJ 6

Al 6q22 (t) + 3

E 4

l 4q22 (t) 252

EJ 8

Al 8q42 (t) + 4

P04

Al4q22 (t) +

EJ 4

Al 4 P0

2

Al2

q1(t)

f cos tA

= 0, (22a)

q2(t) +

Aq2(t) 336

EJ 8

Al 8q52 (t) +

315 EJ

8

Al8q21 (t) 80

EJ 6

Al 6+ 8 P0

4

Al4+ 6 E

4

l 4

q32 (t)

+

28 EJ 6

Al 6q21 (t) + 3

E 4

l 4q21 (t) 21

EJ 8

Al 8q41 (t) + 4

P04

Al 4q21 (t) + 16

EJ 4

Al 4 4 P0

2

Al2

q2(t) = 0. (22b)

Introducing the following non-dimensional quantities.

= 2

EJ

Al 4t, 1 = q1

A

J, 2 = q2

A

J, = l

2

2

A

EJ,

= l2

2

AEJ, f = f l

4

4EJ

A

J, P0 =

P0

EA.

(23)

One obtains the non-dimensional equations for the double mode model.

1 + m11 + m21 + m3122 + m4142 + m531 + m631 22 + m751 + m8 = 0, (24a)2 + n12 + n32 + n3221 + n4241 + n532 + n632 21 + n752 = 0, (24b)

where

m1 = , m2 = 1 l2

A 2J

P0, m3 = 4 P0 22 2

Jl2A

+ 3,

m4 = 252 4J2

l4A2, m5 =

1

2P0

5

4

2J

l2A+ 3

8, m6 =

651

8

4J2

l4A2,

m7 = 21

16

4J2

l4A2, m8 = f cos , n1 = 2, n2 = 4

l2A

JP0 + 16 2, (25)

n3 = 4 2 P0 28 4J

l2A+ 3 2, n4 = 21

6J2

l4A2, n5 = 8 2 P0 80

4J

l2A+ 6 2,

n6 = 315 6J2

l4A2, n7 = 336

6J2

l4A2.

4. The Melnikov method for the single mode model

Using Eq. (20), one can easily obtain the corresponding unperturbed system for the single mode model.

= , = 3 5. (26)

Its Hamilton function is

H = 2

2+

22 +

44 +

66 = h. (27)

For the different values of H, they indicate the different curves in the phase portrait; its value is determined by the initial

conditions. If2

4 < 0 or 2

4 > 0 and < 0, the point (0, 0) is the only one fixed point of the system, because of

this we assume 2 4 > 0 and > 0 in the following analysis.


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4.1. < 0

If < 0, there are five fixed points, the points (0, 0) and (

( +

2 4)/(2 ), 0) are three hyperbolic-type saddlepoints, and the others, (

(

2 4 )/(2 ), 0), are center points.

Using Eq. (27) two homoclinic orbits can be given in the following form.

=

4ep[(ep + /2)2 (4/3)] ,

= 2

2

4 4

3 e2p

ep

[(ep + /2)2 (4/3)]3 ,

p = 2 + ln

2

4 4

3

.

(28)

The Melnikov function for the homoclinic orbits can be given as follows.

M(0) =

1 cos( + 0) + 2d = f 11 cos(0 + 0) + 12, (29)

where

11 =

211 + 212, 12 = 3

4

+

3

2

4 4

3

ln

/2 + 4 /3/2 4 /3

,

0 = arcsin12

11

, 11 = 2

ep

(ep + /2)2 43 sin d, (30)

12 =

2

ep(ep + /2)2 43

cos d.

If the Melnikov function has simple zero points, the stable and unstable manifolds intersect. The Poincare map has a horse-

shoes, so there exists the strange constant set, it is possible for the dissipative system to enter chaos. According to the Melnikov

method, one can get the chaotic critical condition

f

> 1cr =

12

11. (31)

Using Eq. (27) two heteroclinic orbits can also be given as follows.

=

+

2 4

2

ep + 2

2 4

2(e

p ( + 42 4 )/(2 ))2 3( +2 4 )2 4

,

= 3

+

2 4(2 4 )3

2

ep[ep 2 +

2 4 ]

[

2(ep ( + 4

2 4 )/(2 ) )2 3( +

2 4 )

2 4 ]3, (32)

p =

( +

2 4 )

2 4 .

The Melnikov function for the heteroclinic orbits can be given as follows.

M(0) = f 21 cos(0 + 0) + 22, (33)


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where

21 = 3

+

2 4

(

2 4 )32

221 + 222,

22 = 1

81

33

3

3

+2 4 2 4+ 3(3

2 16 )2

ln

3

2 4 +

+

2 43

2 4

+

2 4

,

12 =

ep[ ep /2 +

2 4 ][

2(ep ( + 4

2 4 )/(2 ))2 3( +

2 4 )

2 4 ]3cos d,

22 =

ep[ ep /2 +

2 4 ][

2(ep ( + 42 4)/(2 ))2 3( +

2 4 )2 4 ]3

sin d,

0 = arcsin

21221+222

.

(34)

According to the Melnikov method, one can get the chaotic critical condition

f

> 2cr =

22

21. (35)

4.2. > 0

If > 0, there are three fixed points, the point (0, 0) is a center point, and the others, (

(

2 4 )/(2 ), 0),are two hyperbolic-type saddle points. The heteroclinic orbits can be expressed by Eq. (32), and one can get the chaotic critical

condition that is identical with Eq. (35).

5. Numerical computations

In order to show the difference between the single and double mode methods, one can rewrite the nonlinear dynamic equa-

tions (20) and (24) in the following form.

+ + 3 + 5 (f cos 1) = 0, (36)

1 + m11 + m21 + m3122 + m4142 + m531 + m631 22 + m751 + m8 = 0, (37a)2 + n12 + n32 + n3221 + n4241 + n532 + n632 21 + n752 = 0. (37b)

The numerical computations are carried out using the following parameters.

A = 0.01 0.01 m2, l = 5 m, P0 = 1.0 103 N, = 1, E = 69.7 109 Pa, = 2.78 103 kg/m, = 3.966 102 kg (m s)1.

Fig. 3(a) is similar to Fig. 3(b), it corresponds to a chaotic motion. According to Figs. 3 and 4, |2(t)| |1(t)| and(t) 1(t), the results obtained from the single and double mode models are completely identical when f = 1500, = 0.2and = 1.8 if the other parameters are the same. That is to say, in these cases the single mode model analysis is sufficientand correct.

Figs. 57 show the difference between the single and double mode models. One or both of two equations, |2(t)| |1(t)|and (t) 1(t) are not true. Namely, in these cases the single mode method usually used may lead to incorrect conclusions.That is to say, in some cases we cannot use the single mode method to analyze the structures nonlinear response, and instead

the double mode or higher order mode method should be used.


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(a)

(b)

(c)

Fig. 3. Comparison between single and double mode models ( = 0.2 , f = 1500). (a) single mode model ; (b) double mode model 1;(c) double mode model 2.

The displacement modes used in this paper based on the single and double mode models can be expressed as follows.

w1(x,t) = (t) sin x

lfor the single mode model, (38)

w2(x,t) = 1(t) sin x

l + 2(t) sin2 x

l for the double mode model. (39)


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(a)

(b)

(c)

Fig. 4. Comparison between single and double mode models ( = 1.8 , f = 1500). (a) single mode model ; (b) double mode model 1;(c) double mode model 2 .

If the single mode model is correct, the following equations are true.

w1(x,t) w2(x , t ), (t ) 1(t),2(t) 1(t). (40)

When (t)

1(t) and

|2(t)

| |1(t)

|, w2(x,t) can be a better approximate solution than w1(x,t). If (t) is chaotic,

1(t) is also chaotic, and vice versa. In these cases the single mode method can be used to analyze the chaotic behavior.


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(a)

(b)

(c)


6. Conclusions

The single and double mode models of an elastic beam under simple harmonic excitation are presented. The dynamic

nonlinear governing equations are derived. Two different kinds of nonlinear dynamic equations are obtained with the variation

of the loading parameters. The chaotic critical conditions are given by Melnikov function method for the single mode model.

The chaotic motion is investigated and the comparison between single and double mode models is carried out using numerical

computations.


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(a)

(b)

(c)


Let us consider the effect of higher order modes on chaotic vibration. First, the partial differential equation governing the

transverse vibration of the beam is derived in this paper. Then, by means of the Galerkin approach, the partial differential

equation can be simplified into a single ordinary differential equation or a set of ordinary differential equations according to the

number of mode taken into account.

For the single mode model w1(x,t)

=(t)w1 (x) and for the double mode model w2(x,t)

=1(t)w

1 (x)

+2(t)w

2 (x),

when (t) 1(t) and |2(t)| |1(t)|, w2(x,t) may be a better approximate solution than w1(x,t). If (t) is chaotic,


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(a)

(b)

(c)


1(t) is also chaotic, and vice versa. This is the reason why the single mode method is usually used to analyze the non-

linear behavior. When one of the equations, (t) 1(t) and |2(t)| |1(t )|, is not true, the single mode methodusually used may lead to incorrect conclusions, and instead the double mode or higher order mode method should

be used. This shows that we should be careful to analyze nonlinear dynamic systems by means of the single mode

method.


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Acknowledgements

The authors wish to acknowledge, with thanks, the financial support from the National Natural Science Foundation of China

(10272046) and the National Natural Science Foundation of Guangdong Province (020858).

References

Awrejcewicz, J., Krysko, A.V., 2003. Analysis of complex parametric vibration of plates and shells using BubnovGalerkin approach. Arch.

Appl. Mech. 73, 495504.

Awrejcewicz, J., Sendkowski, D., 2004. How to predict stickslip chaos in R4. Phys. Lett. A 330, 371376.

Han, Q., Hu, H.Y., Yang, G.T., 1998. Chaotic motion of an elastic circular plate. Trans. Nanjing Univ. Aeronaut. Astronaut. 15, 206210.

Han, Q., Hu, H.Y., Yang, G.T., 1999. A study of chaotic motion in elastic cylindrical shells. Eur. J. Mech. A Solids 18, 351360.

Holms, P., Marsden, J., 1981. A partial differential equation with infinitely many periodic orbits: chaotic oscillation of a forced beam. Arch.

Rational Mech. Anal. 2, 135165.

Keragiozov, V., Keoagiozova, D., 1995. Chaotic phenomena in the dynamic buckling of an elastic-plastic column under an impact. Nonlinear

Dynamics 13, 116.

Moon, F.C., Shaw, S.W., 1983. Chaotic vibration of a beam with nonlinear boundary conditions. Non-Linear Mech. 18, 230240.

Paniela, D.B., 1994. Mathematical models used in studying the chaotic vibration of buckled beam. Mech. Res. Commun. 21, 189196.

Winmin, T., Namachchivaya, N.S., Malhotra, N., 1994. Non-linear dynamics of a shallow arch under periodic excitation II.1:1 internalresonance. Int. J. Non-Linear Mech. 29, 367385.

Yang, X.L., Sethna, P.R., 1992. Nonlinear forced vibrations of a nearly square plate-antisymmetric case. J. Sound Vib. 155, 413441.

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Chaotic vibration of beams