chaotic vibrations of a nonlinear viscoelastic beam

Chaos. Solrrons & Fmctols Vol. 7, No. 2, pp. 151-163, 1996

Pergamon Copyright 0 19% Ekvier Science Lfd

Printed in Great Britain. All rights reserved 0960-0779/96 $15.0(1 + 0.00

0960-0779(95)00097-6

Chaotic Vibrations of a Nonlinear Viscoelastic Beam

J. ARGYRIS

Institute for Computer Applications, University of Stuttgart, D-70569 Stuttgart, Germany

V. BELUBEKIAN and N. OVAKIMYAN

DAAD Scholars at the Institute for Computer Applications, University of Stuttgart, D-70569 Stuttgart, Germany

and

M. MINASYAN

Faculty of Mechanics, Yerevan State University, Yerevan 375049, Armenia

(Accepted 6 October 1995)

Abstract-The differential equation of motion of a nonlinear viscoelastic beam is established and is based on a novel and sophisticated stress-strain law for polymers. Applying this equation we examine a periodically forced oscillation of such a simply supported beam and search for possible chaotic responses. To this purpose we establish the Holmes-Melnikov boundary for the system. All further investigations are developed by means of a computer simulation. In this connection the authors examine critically the PoincarB mapping and the Lyapunov exponent techniques and distinguish in this way between chaotic and regular motion. A set of control parameters of the equation is found, for which either a chaotic or a regular motion can be generated, depending on the initial conditions and the corresponding basins of attraction. Thus, in this particular case two attractors of completely different nature-regular and chaotic, respectively-coexist in the phase space. The basins of attraction of the two attractors for a fixed instant of time are plotted, and appear to possess a very complex fractal geometry.

1. OSCILLATIONS OF A BEAM AND THE DUFFING EQUATION

Oscillations of an axially compressed nonlinear viscoelastic beam subject to a periodic transverse force are analysed critically in the range of small strains. Such oscillations of a fundamentally simple system demonstrate impressively complex responses of great variety, including chaotic ones [l-3]. Such phenomena were also observed experimentally. To analyse this problem we introduce a novel constituent model of a viscoelastic beam, which in comparison with the previous ones, reproduces more adequately the effects of viscosity. Previous investigations of such phenomena were based on the equation of motion of an axially compressed elastic beam. Ignoring for a moment the influence of an external excitation, and the effect of damping, the motion of an elastic beam, obeying the Euler-Bernoulli assumption (i.e., the neglect of the shear strain effects) is represented by the following equation:

&4w + p a*w a*w o ax4

-++A-= eff ax2 at2

151

152 J. ARGYRIS et al.

Here w is the deflection, E the Young’s modulus, p the density, A the area and I the moment of inertia of the cross-section of the beam. Peti represents the effective axial force in the beam. A harmonic transverse excitation with frequency w, and a viscous damping force ci+ are now assumed to act on the system. In extension of (1) the equation of motion is then usually written in the form:

Eza4w + p ax4

eff a*w + ca2w - - ax2 at + PA $ = fcos(wet). (2)

For a simply supported beam the boundary conditions are

w(0, t) = w(l, t) = 0 $ a2w 1 I

zz- =(). (3) x=0 ax2 x=i

Applying next the Galerkin procedure we obtain a modification of the classical Duffing equation in the form:

L’ - uy + by3 + cj = dcos(w,t), (4)

where only a single mode is retained. Here the linear stiffness coefficient is negative, i.e., a > 0, if the compressive axial force exceeds the first Eulerian buckling load. A great number of other physical processes can be described by (4). Numerous studies of the Duffing equation are quoted in the literature (see, for example, [l-5]). However, we notice, that the viscous damping in equation (2) is introduced somewhat artificially. Our aim is hence to establish a more precise equation of motion for a nonlinear viscoelastic beam and to reduce it subsequently by the Galerkin method to an ordinary differential equation. Such a system will then be investigated numerically for possible chaotic responses.

2. A MODEL OF A VISCOELASTIC BODY ACCORDING TO AMBARTSUMYAN AND MINASYAN

Careful interpretation of the experimental data reduced from axial testing of polymer materials [6-91 as also of bones [lo] reveals the following characteristic features of the loading response of polymer materials.

(i) Applying a sequence of different constant velocities of deformation, we observe that the appertaining stress-deformation diagrams represent a family of curves, which can be generated from the diagrams of quasi-static tests by means of linear transforma- tions.

(ii) The initial tangent modulus of the diagrams depends on the velocity of deformation, see, for example, Fig. 1.

(iii) The difference between initial responses is larger at small deformation velocities (about lop3 s-l) and smaller at large deformation velocities (about lo3 s-l).

(iv) In the course of experiments with varying deformation velocities the stress decreases, but the deformations continue to grow.

When considering deformation processes and specific stress and strain states, which are lower than the permissible elastic states of a corresponding quasi-static test, the residual plastic deformations are assumed by us to be either non-existent or negligible. In the light of our current knowledge the features (i to iv) describe adequately the response of viscoelastic materials. Ambartsumyan and Minasyan [ll-131 proposed a constitutive model of a viscoelastic body, which in contradiction to the previously adopted models of Foicht,

Motion of nonlinear viscoelastic beam

Fig. 1. Initial tangent modulus.

Maxwell, Boltzmann et al., incorporates ideally the aforementioned characteristic features (i)-(iv). This proposed stress-strain model is given by

0 = Eoe[l + v(l+l”)l, (5)

where E. is the Young’s modulus as recorded in a static test, l/a is a characteristic retardation time, m is a constant and q(e) is a monotonically increasing function, for which ~(0) = 0. We usually apply for polymers a value of m in the range of 0 < m s 1, but for metals we may also observe m > 1. Note that polymers are more sensitive to small deformation velocities of about 10-l s-l. We confine ourselves here to materials for which, following removal of stresses, no residual plastic deformations are observed. This allows us to adopt the Hookean relation for unloading processes:

u = E+ (6)

Combining the two formulae (5) and (6) into a single expression we write

u = E,s[l + v(I+lm)O(c, e)]

where 8 is the Heaviside function:

(7)

8 = 1

1 when rb > 0 0 when &SC =s 0. (8)

Furthermore, we set approximately V(X) = x, which proves sufficiently accurate for the specification of many polymers even at relatively large deformation velocities.

3. EQUATION OF MOTION FOR A NONLINEAR VISCOELASTIC BEAM

We establish the equation of motion for the beam by applying the procedure of Argyris [2, 31 with the only difference being that instead of Hooke’s law we introduce the stress-strain relationship (7).

In the following we assume a linear small strain theory. If M is the bending moment, P

1.54 J. ARGYRIS et al.

the axial compressive and Q the shear force in the beam, the equation for the force and moment equilibrium can be written immediately as

aQ - = pA 9~ ap o

ax ----) -= at* ax

(9)

and

-aM - ax Q - P,,$.

Assuming for simplicity that the elementary theory of bending (the Bernoulli hypothesis) holds, i.e., that the effect of the shear strains can be neglected, the following relation applies for the bending strain E and the deflection w

E = -zd’“. ax*

(11)

For the bending moment we have

M = 0-z dz I A

(12)

where a, must be calculated from (7). In this way we obtain

(13)

where y depends on the shape of the cross-section, e.g., for a rectangular cross-section we obtain: y = 3b/(m + 3)(/z/2&)” (h and b are the height and width of the beam). Thus, we deduce from equations (9), (10) and (13):

El-$-{$-[I + yl~l”B($, s)]} + Pee$ + pAf$- = 0. (14)

The Pen can be calculated approximately from:

Assuming this force to be constant over the length of the beam, the modified equation (14) becomes:

+[P-$-$$-)2dx]s+pA$=O. (16)

If we also impose a transversal harmonic excitation on the system, we are led to a dynamic model of an externally excited beam subject to a compressive force P. The pertinent relation is a partial integro-differential equation in terms of the deflection w(x, f).

+ [p - +(($)2dx]$ + PAZ = fcos(w,t). (17)

Motion of nonlinear viscoelastic beam 155

This equation differs from (2) only through the term

~~~/m~(-$~ $)>

which expresses the nonlinear dissipative effect of our material model.

4. REDUCTION OF EQUATION (17) TO AN ORDINARY DIFFERENTIAL EQUATION

So far our problem is that of equation (17) with the boundary conditions (3). In order to reduce the problem to an ODE we apply the Bubnov-Galerkin procedure. To establish this transformation we adopt in the usual way the following single mode expression for w:

w(x, t) = F(t) sin (TX//). (18)

Applying the standard procedure we obtain:

P + (w; - wf,,)F + Alkj”W(f, k) + k3F3 = Q,COSWJ, (19)

where

w; = D?r4/(ph14), 0; = PT-?/(phP), k2 = 3h202 0

A = 30&n + 3) + Sr2((m + 3)/2)/S((m + 3)/2). PO)

Here l? is the standard I function. We consider here only the case when the axial force is large enough to generate buckling, i.e.,

w’, > 0;. (21)

To obtain a more compact expression we introduce the new variables

z= tn, y2k Q = Qoknm3, p=hl ) b-2k-,?I $= 2 2 WN - w,,. n

With this new notation (19) simplifies to (22)

j; - Y + Pl,‘l”f3 y, j) + y3 = Qcosvt (23)

where dots define differentiation with respect to t. The equation (23) differs from that of Duffing only through the expression of the dissipative term /3ijl”‘6(y, 3). The reduced equation for free oscillations becomes

L’ - Y + PIPI”~(Y, jl) + Y3 = 0

and differs also slightly from the corresponding Duffing ODE

(24)

x - x + cl + x3 = 0. (25)

Nevertheless, the solutions of the two systems are fundamentally different. The solution of the Duffing expression (25) possesses in specific cases one of the following regular characteristics.

1. The disturbances attenuate without oscillations, Fig. 2(a). 2. The disturbances attenuate after a single zero crossing, Fig. 2(b). 3. The disturbances attenuate after an infinite number of zero crossings, Fig. 2(c).

The numerical investigation of equation (24) demonstrates in contrast, that Case 3 is not possible for the refined viscoelastic system. Instead, attenuation of disturbances is possible


(4 (b) (4 Fig. 2.(a)-(d). Attenuative modes of oscillation for the Duffing equation.

after a limited number of oscillations, greater than 1, Fig. 2(d). Indeed, such behaviour may be expected, since with increasing 1~1, ]jl the dissipative term of equation (23) grows at a lower rate than the other terms. This indicates, that initial conditions are possible, for which a motion is generated, in the initial phase similar to that of Fig. 2(c), i.e., the motion corresponds to one of small linear damping. On the other hand, when ly 1, /j/ decrease, the damping decreases at a lower rate than the other terms. As a result, after some initial (or transient phase) the motion corresponds to that with a large linear damping, Fig. 2(a) and (b).

5. THE HOLMES-MELNIKOV BOUNDARY

Next we propose to establish the Holmes-Melnikov boundary [l-3, 141 for the viscoelastic system of equation (24). First we have to set up the equation of a homoclinic orbit of the associated Hamiltonian system, i.e., the undamped unforced oscillator. Thus, for /3 = 0, Q = 0 equation (24) is reduced to

j-y+y3=o. (26)

To construct the homoclinic orbit of equation (26) we apply the method of Holmes [l-3, 141. In this way we find

y = 1/2/cash r, p = -(j/2/cash r) tanh t. (27)

Here p denotes the j coordinate in the phase plane. The Melnikov function derives from P-31

MC&O) = -B/O pm+l ydt -t Q I

m pcosY(t + Eg)dr. (28) -cc -a.

The second integral in (28) is listed in [15]. The first can easily be established for an arbitrary m. Thus, the Melnikov function becomes ultimately

M(q) = -/32"i2 r2(m/2 + 1) IY(m + 2)

- (Qav/cosh (av/2)) sin vro.

In this way the Holmes-Melnikov boundary is given by

Q> 2"'2/?I'2(m/2 + l)cosh(zrv/2)

' wT(m + 2)

6. COMPUTER EXPERIMENTS AND POINCARI? MAPPING

(29)

(30)

We return now to the main purpose of this paper and proceed to find possible chaotic solutions of (23). To distinguish a chaotic response from a regular one, the Poincare

Motion of nonlinear viscoelastic beam 1.57

mapping technique proves, as is well known, very informative. To eliminate the undesir- able information of the transient phase we start recording after a sufficiently long initial interval of, say, 5000 oscillations. Following each set of 27r/v units of time the state of the system is examined in the phase plane y , j. According to classical theory we must then observe in the phase plane for a regular regime either a closed curve or a limited set of points. On the other hand, in a chaotic motion a great number of points are generated, forming a kind of cloud. In the course of motion these points, reproducing the evolution of the state of the system, will be seen to concentrate on a restricted domain of the phase plane y, j,. Our computer experiments were recorded following an initial interval or transient phase extending over 5000 periods of the applied external force. The figures (see, e.g., Fig. 3(a)-(d)) reproduce 5000 points in each chaotic PoincarC map and appertain to values of the parameters /3 = 0.5, Q = 35; 50; 85, 90 in (23). The capacity dimensions D,

(4 10 r

01 -1s -13 -10 4 -5 -3 0 3 5 B ID

si

6

5

3

1

-1 . .

Fig. 3.(a) and (b). Caption on p. 158


ol -20 -15 -10 -5 0 5 10 15 20 25 30

G

0 -20 -15 -10 -5 cl 5 10 IS 20 25 30

Y

Fig. 3. (a) PoincarC map for Q = 35; (b) Poincark map for Q = 50; (c) PoincarC map for Q = 8.5; (d) PoincarC map for Q = 90.

of the attractors are calculated approximately according to Argyris [3] and are found to be in the range of CO.701, 0.9021. Assuming for p = 0.5 another set of values of the parameter Q, extending over the values 1.5; 1.5; 25; 30; 89; 91; 95; 110, we observe regular domains, represented by sets of a limited number of points, say, 2, 4, or 6 or closed curves. The reader will note that on the axis of amplitude or frequency the chaotic domains are interlaced with regular ones. For the construction of the Poincare maps we applied the Runge-Kutta procedure with an automatically adjustable step. In all our numerical experiments we assumed m = l/4, which is representative of many viscoelastic materials [N-13].

Motion of nonlinear viscoelastic beam 154,

7. ON THE LYAPUNOV EXPONENTS

Another criterion, which we applied to locate domains of chaotic motion is that of the Lyapunov’s exponents. In the present case there are two exponents, but of interest is the largest one. The sign of the exponent demonstrates at a glance, that for a positive sign initially neighbouring trajectories diverge exponentially, while for a negative sign the trajectories converge. Following the exposition in Argyris [3] we note, that for a negative exponent, the motion converges after a transient phase towards a regular periodic or quasiperiodic pattern. If the Lyapunov exponent is positive and the motion takes place in a limited domain of the phase plane, then the response can never evolve as a regular one; thus chaotic behaviour then takes place. In Fig. 4 we have plotted for Y = 1, the dependence of the Lyapunov exponents on the amplitude of the external axial force. Similarly, Fig. 5 reproduces the dependence on the frequency for Q = 90. As before, our computer experiments reproduce the response, following an initial interval of time. In this way we exclude the influence of the transient phase. The figures demonstrate that chaotic responses are interlaced with responses of regular motion. Note that the results obtained by Poincare mapping are confirmed by the Lyapunov exponents. Also, the parameters corresponding to chaotic motion satisfy the Holmes-Melnikov criterion (30). The calcula- tion of the Lyapunov exponents is based on the Runge-Kutta technique with a constant step.

8. DEPENDENCE OF THE RESPONSE OF THE BEAM ON INITIAL CONDITIONS

Consider (23) and the special case, when Q = 3, Y = 1. We observe that the criterion of Holmes-Melnikov is satisfied in this case. However, the behaviour of the beam proves strongly dependent on initial conditions. Figure 6 demonstrates a Poincare map for the initial conditions y. = 1.1, PO = 0.22. The map shows in this case, that a chaotic motion takes place, which is also confirmed by the positive sign of the Lyapunov exponent o = 0.9. On the other hand, for the initial conditions y, = 1.0, 3 = 1.1 the Poincare map is reduced

Fig. 4. Lyapunov exponent versus amplitude of exciting force.


-25

-50 0 2 4 6 8 10

I/

Fig. 5. Lyapunov exponent versus frequency of exciting force.

4 -3 -a -I 0 1 2 3 u

j,

Fig. 6. Poincart map for Q = 3.

to 7 points. This indicates, that in this case a limit cycle exists with period 14n. The negative sign of the largest Lyapunov exponent o = -0.22 also confirms the regularity of the motion. In Fig. 7, the relative topological position of chaotic and regular attractors is shown; the regular one is denoted by boxes. In the case of chaotic motion the system resides on the Poincare map in a limited domain of the phase plane, which is usually denoted as the ‘strange attractor’. In our special case the strange attractor coexists in the phase plane with a regular motion, namely with an orbit of period 7. The coexistence of different attractors in the phase plane is well-known. However, the coexistence of two attractors of completely different characteristics-regular and chaotic-to the best of our


3L -u -3 -2 -1 0 1 2 3 9

jl

Fig. 7. Relative topological position of the two attractors

knowledge, appears to be rarely noticed. Thus, the three-dimensional phase space y, j, t contains two distinct subdomains: the so-called basins of attraction. The trajectories, starting from the points of one of these domains generate quasi-periodic motions, while trajectories, emanating from the points of the other domain demonstrate chaotic behaviour. The relative disposition of the two domains is plotted in Fig. 8 for the cross-section r = 0 of the phase space. The two basins of attraction are associated with a regular or chaotic motion, respectively. Since a very large number of trajectories must be evaluated for possible chaotic responses, we require a criterion to distinguish rapidly between chaos and regular motions. Fortunately, the analysis of Poincare mappings demonstrates in the

-9 -a -7 -6 -5 -u -3 -2 -1 0 1 2 3 u 5 6 7

-;I”

Fig. 8. Basins of attraction

162 J. ARGYRIS et 01.

Fig. 9. Basins of attraction, extract.

present case, that for regular motion the ultimate pattern is reached within a very short interval of, say, 3 to 4 periods of the external force. Thus, we assumed in our research, that it is sufficient to record the response only for a small interval of, say, 40 periods of the external force. In order to specify the nature of the basins of attraction a rectangular domain in the plane of initial conditions bounded by -2.1 s y. s 4.1, -8.1 c j. 6 8.1 was chosen and divided into 114151 boxes. If the centre of the box was found to appertain to a regular basin, the whole box was assumed to belong to this category and was marked in black. In this way Fig. 8 provides a rough idea of the location of the respective basins of attraction. To ascertain the reliability of our methodology we reconsidered with finer precision a small area of Fig. 8. In fact, the area of each box is in the present case 14 times smaller than in Fig. 8 (see Fig. 9).

We observe that the more accurate calculations confirm, in principle, the results of the first rough computations. However, at the same time we discover a more intricate structure of the regular basin. Thus, the dark strips, which appear to be solid in Fig. 8 are seen to consist of several thinner strips, see Fig. 9. To verify the display of Fig. 8 a sample of 25 sets of initial conditions was randomly chosen and the corresponding responses recorded over a sufficiently long interval of time on a Poincare map. The results, emanating from chaotic and regular basins of attraction confirmed our preceeding computations.

Acknowledgements-The authors extend their gratitude to the DAAD for the scholarships granted to W. Belubekian and N. Ovakimyan, which enabled them to partake in the present research. Special thanks are also addressed to Gerhard Frik for computational support. We are aiso grateful to Mrs M. Parsons for her support during the whole process of the creation of this paper.

REFERENCES

1. F. C. Moon, Chaotic Vibrations-An Introduction for Applied Scientists and Engineers. John Wiley L Sons, New York (1987).

2. J. Argyris, G. Faust and M. Haase, An adventure in chaos, in Computer Methods in Applied Mechanics and Engineering. North Holland, Amsterdam (1991).

3. J. Argyris, G. Faust and M. Haase, An Exploration of Chaos. Vol. 91, Nos. 1-3, pp. 997-1091. North Holland, Amsterdam (1994).


4. Y. Ueda, Explosion of strange attractors exhibited by Duffing’s equation, in Nonlinear Dyanmics, edited by R. H. G. Helleman, pp. 422-434. New York Academy of Sciences, New York (1980).

5. Y. Ueda, Steady motions exhibited by Duffing’s equation-A picture book of regular and chaotic motions (1980).

6. H. Kolsky, Proc. Phys. Sot. B 62, 359 (1950). 7. V. Victorov, I. Dobrovolski and G. Shapiro, Mechanica Polimerov, Vol. 1. Riga (1967). 8. S. Kokoshvili, V. Tamuzh and G. Shapiro, Mechanica Polimerov, Vol. 2, Riga (1970). 9. P. M. Ogibalov, N. I. Malinin, Y. P. Netrebko and B. P. Kishkin, Construction of Polymers, Vols 1 and 2.

Moscow (1972). 10. G. Herrmann and H. Liebovitz, Mechanics of Bone Fracture, pp. 772-836. Academic Press, New York

(1970). 11. S. A. Ambartsumyan and M. M. Minasyan, On the model of bodies with their mechanical properties

depending on the strain rate. Int. J. Nonlinear Mech. 21(l), 27-36 (1986). 12. S. A. Ambartsumyan and M. M. Minasyan, Bending vibrations of beams with their mechanical properties

depending on strain rate. Mekhanika. Mezhvuzovisky Sbornik 5, 85-95 (1986). 13. S. A. Ambartsumyan and M. M. Minasyan, On a Nonlinear Model of Viscoelastic Body, No. 4, pp. 165-172.

MT-I, Moscow (1991). 14. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields.

Springer, New York (1983). 15. I. S. Gradstein and I. M. Ryzhik, Table of Integrals, Series and Products. Academic Press, New York (1965). 16. J. Argyris, G. Faust and M. Haase, Routes to chaos and turbulence-A computational introduction.

Philosophical Transactions A344, 207-234 (1993). 17. F. C. Moon, Chaotic and Fractal Dynamics-An Introduction for Applied Scientists and Engineers. John Wiley

& Sons, New York (1992).

chaotic vibrations of a nonlinear viscoelastic beam

Documents