R. Yamapi Laboratoire de Mécanique, Faculté des Sciences, Université de Yaoundé I, B.P. 812, Yaoundé, Cameroun e-mail: [email protected]

J. B. Chabi Orou Institut de Mathématiques et de Sciences Physiques, B.P. 613 Porto-Novo, Bénin

P. Woafo1 Laboratoire de Mécanique, Faculté des Sciences, Université de Yaoundé I, B.P. 812, Yaoundé, Cameroun e-mail: [email protected]

1

Effects of Discontinuity of Elasticity and Damping on the Dynamics of an Electromechanical Transducer In this paper, we study numerically the effects of discontinuity in elasticity and damping on the dynamics of an electromechanical transducer. Frequency-response curves of oscillatory states are obtained. Bifurcation structures and transitions to nonperiodic or chaotic motion are found. 关DOI: 10.1115/1.1888589兴

Introduction

Models with discontinuous parameters are frequently used in many engineering systems. Recent studies have shown that such systems may exhibit various types of behaviors 关1–8兴, resulting in abrupt changes of the damping and stiffness coefficients, characterized by many unusual and complicated features. For example, in Refs. 关1,2兴 the long time response of a harmonically excited mechanical system involving piecewise linear stiffness was not necessarily periodic. Some interesting results have been obtained recently for oscillators with bi-linear stiffness. Maeczawa et al. 关3兴 have used a Fourier series expansion approach for determining periodic solutions under general periodic excitation. Another approximate approach, combining the harmonic balance method with the fast Fourier transform technique, was presented by Choi et al. 关4兴. Experimental data, together with an analytical procedure for determining periodic steady state solutions of the system under harmonic excitation, were presented by Masri et al. 关5,6兴. For oscillators with bi-linear damping and stiffness, Natsiavas considered the problem in Ref. 关7兴. He analyzed the existence and the stability for single-crossing, periodic solutions, using a formulation which is appropriate for general piecewise linear systems. It was found that there are sets of parameters for which the system undergoes a set of bifurcations which many lead to the loss of stability of periodic solutions and the appearance of chaotic responses. Our aim in this paper is to analyze numerically the effects of discontinuity in elasticity and damping on the behavior of an electromechanical transducer 共see Fig. 1兲, which consists of a forced electrical oscillator coupled to a mechanical oscillator. The paper is organized as follows. In the next section, we describe the physical model and give the resulting equations of motion. We consider in Sec. 3 the regular behaviors of the model using a direct integration of the equations of motion. We find and analyze the response curves of the electromechanical model when some particular parameters are varied. In Sec. 4, some bifurcation structures and transitions to chaotic motion are analyzed. We conclude in the last section.

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coupling between both parts is realized through the electromagnetic force due to a permanent magnet. It creates a Laplace force in the mechanical part and the Lenz electromotive voltage in the electrical part. The electrical part of the system consists of a resistor R, an inductor L, a condenser C, and a sinusoidal voltage source e共␶⬘兲 = v0 cos ⍀␶⬘ 共v0 and ⍀ being respectively the amplitude and frequency, and ␶⬘ the time兲, all connected in series. The mechanical part is composed of a mobile beam which can move along the zជ axis on both sides. The rod T which has the similar motion is bound to a mobile beam with a spring. During the oscillations, the mechanical part passes through the two regions delimited by z ⬍ zc and z ⬎ zc, with different damping and elasticity coefficients. The model presented in Fig. 1 can be seen as a representative of macro- or micro-electromechanical devices used for cutting, drilling, and other machine processes such as loudspeaker 关9兴. It is also equivalent to an electromechanical vibration absorber 关10兴. Using the electrical and mechanical laws, and taking into account the contributions of the Laplace force and the Lenz electromotive voltage, the system is described by the following set of differential equations: Lq¨ + Rq˙ +

1 q + Blz˙ = v0 cos ⍀␶⬘ , C

mz¨ + g共z,z˙兲 − Blq˙ = 0,

共1兲

with

Description of the Physical Model

The device shown in Fig. 1 is an electromechanical transducer with discontinuous parameters. It is composed of an electrical part coupled to a mechanical one, all governed by two oscillators. The 1 Corresponding author. Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 8, 2003; final revision, September 8, 2004. Associate Editor: R. Ohayon.

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Fig. 1 Electromechanical transducer with discontinuity

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Fig. 2 Effects of elasticity on the frequency-response curves yi max of the mechanical oscillator versus w with the parameters ␥1 = 0.1; ␥2 = 1.5; ␭1 = 0.2; ␭2 = 0.6; w2 = 1.2; ␥0 = 1.2; E0 = 5.0; w = 1.3; and w1 = 1.0

g共z,z˙兲 =

再

for z 艋 zc ,

c1z˙ + k1z

c2z˙ + k2z + k2⬘zc for z ⬎ zc ,

冎

E0 =

␥1 =

q = Q0x,

t = w e␶ ,

␭2 =

Bl , mwe

w2e =

R , 2Lwe

␥2 =

c1 , 2mwe

␭1 =

Bl2 , LweQ0

k2⬘ , m

␥0 =

c2⬘ , 2mwe

w2 =

wm , we

⬘= wm

Journal of Vibration and Acoustics

1 , LC

wm =

w0 =

v0 , LQ0

w=

⍀ , we

the equations of motion 共1兲 are reduced to the following set of nondimensional differential equations:

where l is the length of the electrical wire inside the magnetic field B and the two mobile rods supporting the beam, ci and ki 共i = 1 , 2兲, are the damping and elasticity coefficients, respectively. The dot over a quantity denotes the time derivative. c2 = c1 + c⬘2 and k2 = k1 + k⬘2, where the quantities c⬘2 and k⬘2 measure the differences between the damping and elasticity coefficients in the region z ⬎ zc and z ⬍ zc. q denotes the instantaneous electrical charge of the condenser and z denotes the displacement of the mobile beam. Introducing the following normalizations and changes of variables,

z = ly,

Fig. 3 Effects of damping on the frequency-response curves yi max of the mechanical oscillator versus w with the parameters of Fig. 2 and w0 = 3.0

k1 , m

⬘ wm , we

x¨ + 2␥1x˙ + x + ␭1y˙ = E0 cos wt, y¨ + f共y,y˙ 兲 − ␭2x˙ = 0, with f共y,y˙ 兲 =

再

2␥2y˙ + w22y

共2兲 for y 艋 y c ,

2共␥2 + ␥0兲y˙ +

共w22

+

w20兲y

−

w20y c ,

for y ⬎ y c ,

冎

where we fix y c = 1. Now, ␥0 and w0 are the measures of the discontinuities of the dimensionless damping and elasticity coefficients, respectively.

3

Behavior of the Electromechanical Model

With the discontinuous parameters, new interesting phenomena can appear. Due to the complexity of the resulting equations of motion, analytical investigations are very difficult and we only concentrate on the numerical analysis of the behavior of the electromechanical model when some particular parameters evolve, in particular when the frequency w of the external excitation, the perturbation coefficients ␥0 and w0 vary. The results are obtained by solving numerically, with the Runge-Kutta algorithm, the equations of motion 共2兲. We present in Fig. 2 the numerical frequencyresponse curves for y ⬎ y c 关y 2 max in Fig. 2共a兲 being the highest DECEMBER 2005, Vol. 127 / 589

Fig. 4 Amplitude-response curves xi max and yi max vs. w0 with the parameters of Fig. 2

Fig. 5 Amplitude-response curves xi max and yi max vs. w0 with the parameters of Fig. 2

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Fig. 6 Effects of elasticity on the phase lane of the mechanical oscillator with the parameters of Fig. 2 and ␥0 = 0.1. „a… w0 = 0.2, „b… w0 = 2.2, „c… w0 = 4.2; „d… w0 = 15.2

Fig. 7 Effects of damping on the phase plane of the mechanical oscillator with the parameters of Fig. 2 and w0 = 0.2. „a… ␥0 = 0.1, „b… ␥0 = 2.1, „c… ␥0 = 4.2; „d… ␥0 = 15.2

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Fig. 8 Bifurcation diagram showing the coordinate y vs. ␥0 with the parameters of Fig. 6 and ␥0 = 0.1 and E0 = 3.0

values of y above y c兴 and for y 艋 y c 关y 1 max in Fig. 2共b兲 being the highest values of y at the lower side of y c兴 when the frequency w varies, for different values of the coefficients w0. The increase of w0 reinforces the asymmetry of the oscillations since y 1 max becomes larger and larger than y 2 max. The same structure appears when the discontinuity in the damping coefficient is considered 共see Fig. 3兲. We also plot after numerical simulations of the equations of motion the amplitudes xi max and y i max when the discontinuity parameters ␥0 and w0 evolve. Figure 4 presents the amplitude-response curves xi max and y i max vs. w0 for several values of ␥0, while for Fig. 5 we present the amplitude-response curves xi max and y i max vs. ␥0 for three values of w0. It is found that as w0 increases, y 2 max decrease; on the contrary, when ␥0 increases, both y i max decrease while both xi max increase. The evolution of the phase portraits of the mechanical oscillator is presented in Figs. 6 and 7 as ␥0 and w0 vary. Our investigation shows that the electromechanical transducer exhibits a regular behavior only for small values of the discontinuity coefficients ␥0 and w0.

However, for the hard value of ␥0 and w0, the system undergoes bifurcations leading to the appearance of nonperiodic or chaotic responses as it appears in Figs. 6 and 7.

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Bifurcation Structures of the Model

The aim of this section is to find some bifurcation structures in the electromechanical model as the amplitude E0 and the perturbation coefficients ␥0 and w0 evolve. For this purpose, the periodic stroboscopic bifurcation diagram of the coordinate y is used to map the transitions 共the stroboscopic time period is T = 2␲ / w兲. Figure 8 shows a representative bifurcation diagram versus ␥0 with the appropriate set of physical parameters and small value of w0. This is to analyze the effects of the discontinuity in the damping coefficient on the electromechanical model. The following transitions are observed. As ␥0 increases from zero, the amplitude of the periodic oscillations exist until ␥0 = 6.8, when a tiny multiperiodic transition appears and the system passes into another periodic state. As ␥0 increases further, the periodic orbit bifurcates to

Fig. 9 Bifurcation diagram showing the coordinate y vs. w0 with the parameters of Fig. 8 and ␥0 = 10 and E0 = 5.0

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Fig. 10 Bifurcation diagram showing the coordinate y vs. E0 with the parameters of Fig. 8 and ␥0 = 10.0 and w0 = 0.3

a nonperiodic or chaotic orbit at ␥0 = 8.0, which remains with the increase of ␥0. In Fig. 9, the control parameter is the elasticity coefficient w0 but with high value of ␥0. We find that nonperiodic or chaotic motion appears for a small value of w0, since we consider the high value of ␥0. But when w0 becomes large, the transition from the nonperiodic or chaotic motion to the periodic oscillations occurs. Figure 10 presents other types of bifurcation sequences for E0 varying from 0 to 8.

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Conclusion

In this paper, we have considered the numerical study of an electromechanical transducer with discontinuity in parameters. Frequency-response curves are obtained and the effects of the discontinuity in damping and elasticity are discussed. Bifurcation diagrams showing transition from regular to nonperiodic or chaotic motion have been drawn. Because of the recent advances in the theory of nonlinear phenomena, it would be interesting to extend the study of the electromechanical transducers containing one or various nonlinear components. In particular, a self-sustained 共autonomous兲 electromechanical system such as the one analyzed in Ref. 关11兴, but with discontinuity in parameters, is under consideration.

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References 关1兴 Natsiavas, S., 1987, “Response and failure of fluide-filled tanks under base excitation,” Ph.D. thesis, GALCITSM, Report M6-26, California Institute of technology, Pasadena, CA. 关2兴 Natsiavas, S., and Babcock, C. D., 1988, “Behavior of unanchored fluid-filled tanks subjected to ground excitation,” ASME J. Appl. Mech., 55, pp. 654– 659. 关3兴 Maeczawa, S., Kumano, H., and Minakuchi, Y., 1980, “Forced vibrations in an unsymmetric piecewise-linear system excited by general periodic force functions,” Bull. JSME, 23, pp. 68–75. 关4兴 Choi, Y. S., and Noah, S. T., 1988, “Forced periodic vibration of unsymmetric piecewise-linear system,” J. Sound Vib., 121, pp. 117–126. 关5兴 Masci, S. F., 1978, “Analytical and experimental studies of a dynamic system with a gap,” ASME J. Mech. Des., 100, pp. 480–486. 关6兴 Masci, S. F., Mariamy, Y. A., and Anderson, J. C., 1981, “Dynamic response of a beam with a geometric non-linearity,” ASME J. Appl. Mech., 48, pp. 404–410. 关7兴 Natsiavas, S., 1990, “On the dynamics of oscillators with bilinear damping and stiffness,” Int. J. Non-Linear Mech., 25, pp. 535–554. 关8兴 Masri, S. F., and Caughey, T. K., 1966, “On the stability of the impact damper,” ASME J. Appl. Mech., 33, pp. 586–592. 关9兴 Olson, H. F., 1967, Acoustical Engineering, Van Nostrand, Princeton. 关10兴 Korenev, B. G., and Reznikov, L. M., 1997, Dynamics Vibration Absorbers, Wiley, New York. 关11兴 Chedjou, J. C., Woafo, P., and Domngang, S., 2001, “Shilnikov chaos and dynamics of a self-sustained electrome chanical transducer,” ASME J. Vibr. Acoust., 123, pp. 170–174.

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