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IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999

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PAPER

Special Section on Nonlinear Theory and Its Applications

Stress Wave Propagation in One-Dimensionally Coupled Stick-Slip Pendulums∗ Takashi HIKIHARA† , Member, Yohsuke KONDO†† , Nonmember, and Yoshisuke UEDA† , Member

SUMMARY In this paper, the stress wave propagation in a coupled pendulum system with friction force is discussed experimentally and numerically. The coupled system is analogous to the one dimensional fault dynamics model in seismicity. However, we will not intend to discuss about the geophysical feature of the system. The system has rich characteristics of the spatiotemporal stress wave propagation effected by nonlinear friction force. The relation between the wave propagation and the vibration of the pendulum is mainly discussed on the standpoint of nonlinear coupled system. key words: nonlinear wave, friction, stick-slip, chaos

1.

Fig. 1 One-dimensionally coupled stick-slip pendulums (analogous to Burridge-Knopoff model).

Introduction

The stick-slip instabilities in coupled pendulums with friction force in laboratory experiments might be analogous to seismicity [1], [2]. Recently the seismological spatio-temporal complexity has been widely interested based on the studies in coupled nonlinear structures [3], [4]. There also exist these types of structures in mechanical apparatus. As for the system, even in a single oscillator with friction force, the dynamics which includes the stick-slip motion, might be chaotic [5], [6]. Once the oscillators are coupled spatially, the vibration begins to propagate through the distributed structure. That is, the wave arises. As shown in [7], [8], the nonlinear coupled system has a possibility to appear a solitary wave. It is well known that the solitary wave propagation induces the various vibration in the each oscillator. Burridge and Knopoff proposed 1-D coupled oscillator model with spatially homogeneous fracture strength and periodic boundary condition for the seismicity [9] as shown in Fig. 1. The similar experimental system was setup in our laboratory to find the feature of the system behavior and the onset of the wave propagation. We are going to discuss the dynamics of the system based on the experimental results. Especially, the system under the sinusoidal plate motion is significant as the globally excited coupled nonlinear system. The situation does not coincide with the seismological Manuscript received November 30, 1998. Manuscript revised March 16, 1999. † The authors are with the School of Electrical Engineering, Kyoto University, Sakyo-ku, Kyoto-shi, 606–8501 Japan. †† The author is with the Department of Electrical Engineering, Kansai University, Suita-shi, 564–8680 Japan. ∗ This paper was presented at NOLTA98.

problem. However, we can point out that the mechanical elements have the similar structure. Therefore, it is not lack of reality. Moreover, the mathematical model of the system is proposed to describe the experimental results. Based on the experimental and numerical results, the instability of standing wave depending on the amplitude of the excitation is suggested. These results will be a clue to grasp the complex dynamics which arises in the coupled oscillators with friction force. 2.

Experimental System

The experimental setup of one-dimensionally coupled stick-slip pendulum is shown in Fig. 2. The system consists of eight blocks with mass m (= 41.0g), interconnected by spring with the neighboring blocks. The spring constants kc are shown in Table 1. The blocks are forced sinusoidally through leaf springs attached to a rigid upper plate which is connected to electromagnetic shaker. The spring constant, the natural frequency and the damping coefficient of the leaf springs are shown in Table 2. They are used in their elastic region. Therefore, the strain is in proportion to the force which is loaded between the plate and block. The surface of the plate on which the blocks are supported is relatively rough because it is covered by sand paper. The static friction coefficient between the blocks and the plate is given in Table 3. In the experiment, the number of blocks are eight and they are numbered from left to right as shown in Fig. 2. Once the force worked on a block exceeds the maximum static friction force, the block begins to slip. The static friction force on blocks drops instantly to dynamic friction force. Under the slow periodical motion of upper plate,

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Spring constant kc of coupling spring.

Spring Number 1 2 3 4 5 6 7 Ave.

Table 2 Beam Number 1 2 3 4 5 6 7 8 Ave.

Spring Constant [N/m] 26.2 26.4 24.3 25.4 27.0 25.4 27.3 26.0

Fig. 2

Experimental system of coupled stick-slip pendulums.

Spring constant of leaf springs.

Natural Frequency [Hz] 33.78 33.98 34.76 32.22 34.57 34.76 31.44 35.54 33.9

Spring Constant [N/m] 24.6 24.4 19.3 21.4 19.0 24.4 21.3 23.6 22.3

Damping Coefficent 0.940 0.720 0.780 0.840 0.940 0.900 0.800 0.802 0.84 Fig. 3

Table 3 Block Number 1 2 3 4 5 6 7 8

Measurement of the force between blocks.

Friction coefficient. Static Friction Coefficient 0.416 0.423 0.417 0.405 0.428 0.414 0.407 0.418

the motion of block is initiated. When the blocks are in the synchronous motion, the strain of spring between blocks are constant. On the other hand, if two neighboring blocks move out of phase, the strain of the coupling spring appears. In the experimental system the strain between the blocks can be measured by strain gages attached on the leaf springs as shown in Fig. 3. The displacement of a block can also be measured by a transparent type laser displacement sensor, Keyence VG-300, as shown in Fig. 4. In our experiment only a displacement of block # 3 is under watching during the experiment. The analog data are converted into digital data by 12 bit A/D converter. Both ends of the one-dimensionally coupled system are set as free ends. The upper plate is sinusoidally moved by an electromagnetic shaker EMIC 512-A, which is a voice coil type actuator. The shaker signal is driven by a power amplifier. Then the amplitude of the shaker is in proportion to the voltage. It was confirmed experimentally. The relation is given by (Amplitude) [mm]
47 × (Input Voltage) [V] around 3 to 4 Hz. In the experimental system, the upper plate is excited sinusoidally. It causes the force worked on the block through the leaf springs. The leaf springs have the role to excite the blocks simultaneously and to mea-

Fig. 4

Measurement of the block displacement.

sure the imbalance of the force between the blocks. 3.

Experimental Result

The relation between the stick-slip vibration of block and the stress wave propagation is discussed in this section based on the experimental results. 3.1 Vibration of Block and Force Relation At first, the experiment was performed with removing the springs between blocks to measure the relation between the force and the displacement. Figure 5 shows a vibration of the block # 3 without any connection to the neighboring blocks. The upper plate is sinusoidally driven at the frequency f = 3.2 Hz and the amplitude of excitation voltage V = 6.6 V. Then the blocks are un-

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Fig. 5

Displacement vibration of block #3. Fig. 7 Slip force of leaf spring depending on displacement for the block # 3.

Fig. 6

Force vibration of leaf spring #3.

der a global excitation through the leaf springs. Then the blocks will move under the force depending on the external force and the friction force between the block and the lower plate. However, the friction gives it the sticking motion occasionally during the long term experiment. Once the motion becomes sticking state, the state keeps for a while. On the other hand the leaf spring between the block # 3 and the upper plate shows the elastic displacement as shown in Fig. 6. The displacement is transformed easily to the force acting on the beam when the leaf springs are in the elastic region. The figures show that the block # 3 is driven periodically and moves periodically when the block is slipping. However, when the block shows the stick motion, it causes the change of the force wave form. It implies the leaf beam is driven by upper plate with another end clipped. Figure 7 is the force-displacement relation of the block # 3 at f = 3.2 Hz and V = 6.6 V. It was obtained based on Figs. 5 and 6. The loci is simple under the periodical movement. However, when the block show the sticking motion, the loci is biased to forward direction and have the different shape. 3.2 Spatio-Temporal Stress Wave Propagation Under the periodical excitation, the stress wave appears in the coupled system. In the experimental system, the stress is caused by the relative displacement between the blocks. It can be measured by the subtraction between the output of strain gages attached on the neighboring leaf springs. The system is discrete spatially. However, we will show the relation of the force to spatial axis and time continuously by polynomial approximation by Mathematica or Matlab for easy to grasp the wave propagation. At first, Fig. 8 shows

the case of standing stress wave in the experimental system. The experimental condition is at f = 3.45 Hz and V = 6.2 V. The light color denotes the expanding state (sparse) and the dark the compressing state (dense). There appears 3 nodes. The space axis implies the position between the blocks. They are always moves according to the motion of the system. However, we are going to keeps it at the constant coordinate position. It makes easy to grasp the stress wave in the system. At the condition, every block moves periodically in anti-direction between the neighboring blocks. When the voltage is swept at the same frequency, the slightly disordered standing wave appears as shown in Fig. 9. It was obtained at V = 6.8 V. It seems there are 2 nodes. Both sides are stuck and the blocks # 3 to 5 gather. Sometimes the block # 5 and 6 vibrates in anti-direction. It seems that the stressed state locally propagates through some blocks. When we increase the voltage of vibration to V = 7.0 V, the temporal recursive feature disappears and the spatial creep motion can be seen in Fig. 10. At the condition f = 3.45 Hz and V = 7.2 V, the sparse state and dense state interconnect through the blocks as shown in Fig. 11. It implies the appearance of the stress wave in the coupled system. Figure 12 shows the reflecting wave at the frequency f = 3.4 Hz. In this case, the stress wave propagates clearly from the space # 7 to 1. The direction of wave is against to the case in Fig. 11. In these spatio-temporal experimental data, we can see the bifurcation of the wave pattern. For example, the change of nodes in standing waves and the appearance of the traveling waves. In the nonlinearly coupled system, many waves coexist at the same setting condition. Some are stable and others unstable [10]. In the experiments, the state appears depending on the setting condition and the initial condition. Then it is difficult to reproduce the same state exactly. The most important point is that the we can see one of the stable states coexisting in the system by the similar bifurcation. The velocity of the stress wave propagation can be measured by the former results. It depends on the

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(a)

(a)

(b)

(b)

Fig. 8 Standing wave at f = 3.45 Hz and V = 6.2 V; (a) contour and (b) 3D plot.

Fig. 10 Bifurcated standing wave at f = 3.45 Hz and V = 7.0 V; (a) contour and (b) 3D plot.

(a) (a)

(b) Fig. 9 Temporally slightly disordered standing wave at f = 3.45 Hz and V = 6.8 V; (a) contour and (b) 3D plot.

amplitude of excitation as shown in Fig. 13. It is the typical feature of the nonlinear waves. The velocity dependence implies the waves appeared in the experimental system has a feature of solitary wave. However, there cannot appear the soliton in such a dissipative system. In the coupled nonlinear oscillators, for example the array of Josephson Junction, it has already

(b) Fig. 11 Spatio-temporal chaotic wave propagation at f = 3.45 Hz and V = 7.2 V; (a) contour and (b) 3D plot.

known that the localized oscillation, the propagation and the chaotic propagation appear under the external force and dissipation [11]–[14]. The results obtained in this paper semm to be a new example of the dynamics in the non-integrable dissipative nonlinear distributed system.

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Parameter settings.

Parameter α β µ µ0 m kc kp

Value 0.5 100 0.02 1.0 0.5 40 30

pendulums is applied; d2 x1 =kc (x2 − x1 ) dt2 −kp (x1 − B sin ωt) − F (x˙ 1 ) 2 d xi m 2 =kc (xi+1 − 2xi + xi−1 ) dt −kp (xi − B sin ωt) − F (x˙ i ) 2 d xn m 2 =kc (xn−1 − xn ) dt −kp (xn − B sin ωt) − F (x˙ n ) m

(a)

(b) Fig. 12 Reflecting wave at f = 3.4 Hz and V = 7.0 V; (a) contour and (b) 3D plot.

4.

Wave velocity depending on amplitude of external

Numerical Model and Simulated Results

There are many types of model for the friction characteristics. Based on them, the wave propagation in the coupled system with friction has been discussed [4], [5]. They give us the detail information about bifurcation, population dynamics and wave propagation. However, most of them is not clearly confirmed in experiments. In this section we are try to use a simple model for the globally excited coupled pendulums. The following mathematical model for the coupled

(2)

(3)

where xi denotes the displacement of #i pendulum, t the time variable, B and ω the amplitude and the frequency of the external periodical forcing, respectively. The forcing implies the periodical motion of the upper plate in the experimental system. kc and kp denote the spring constant of the coupling spring and the leaf spring. The friction between the blocks and the basement is given by a function F (x). ˙ The identification of the friction function has been one of the difficult research topics in the mechanical field. Here we introduce the following function; ˙ ){µx˙ + µ0 sign(x)} ˙ F (x) ˙ = (1 + αe−β|x|

Fig. 13 force.

(1)

(4)

This model is applied only for describing the shape of the friction-velocity relation. The more detail models are given by [2]. We modified the relation used in [3]. The friction function consists of the static and dynamic friction components. µ denotes the coefficient of the dynamic friction. α and µ0 implies the coefficient of the static one. The friction force has an discrete feature at x˙ = 0. The static friction disappears depending on β as the velocity increases. In the numerical simulation, the parameters are set as shown in Table 4. The parameters are adjusted to show the force-velocity relation obtained in the experiment. β gives the relaxation effect to the friction force. It has a similarity to the nonlinear friction force to the Rayleigh oscillator [15]. The some of the simulated results are shown in Figs. 14, 15, and 16. The typical wave propagation, standing wave, creep wave and reflecting wave, can also be seen in this numerical simulation as same as in the results of experiments. These numerical results at the amplitude 0.5, the standing wave bifurcates spatially and goes to complex states in time and space. At the frequency 4.0, the spatial static mode (standing wave) appears. Depending

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sibility to appear the typical waves in simulation. The nonlinearity of the friction characteristics in the model is essential to show the stress waves. As for the sticking and slipping motions, the model is not enough to examined. The parameter fitting is inevitable to grasp the bifurcation of waves in space and time. The details are our next subject. 5.

Fig. 14

Simulated stress standing wave (frequency : 4.0).

Concluding Remarks

In this paper we discussed the stress wave propagation in coupled stick-slip pendulums. This is the case of globally excited coupled system with nonlinear external force. Each excitation initiates the internal stress in the coupled system caused by the friction force. These types of experimental studies have never been seen. Therefore, it gives rich information about the stress wave in a coupled system. The model is analogous to the one dimensional fault dynamics model in seismicity. The obtained experimental results let us remember that the periodical forcing by tectonic plate initiates the slip motion of a block in the coupled system. The stress wave is a nonlinear wave which velocity is in proportion to the amplitude of the excitation. The friction characteristics was also discussed based on the experimental data. The system was modeled by a coupled differential equations with nonlinear friction force. The onset of the instability of standing wave was found both in experiments and numerical simulation. The bifurcation theory and the manifolds will explain these results hopefully. References

Fig. 15

Fig. 16

Spatial bifurcated standing wave (frequency : 3.95).

Spatio-temporal unstable wave (frequency : 3.5).

on the excitation frequency and the amplitude, the varieties of stress wave propagation are possible. Among them the chaotic wave propagation is liable to appear. The results implies the mathematical model has a pos-

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Tech. Rep., NLP98-75, 1998. [11] H. Konno, Chaos and soliton in engineering system, Applied Chaos, K.Aihara (in Japanese), ed., Science Co. Ltd., 1994. [12] K. Geist and W. Lauterborn, “Chaos upon soliton decay in a perturbed periodic Toda chain,” Physica D, vol.23, pp.374–380, 1986. [13] W.J. Yeh, O.G. Symko, and D.J. Zheng, “Chaos in long Josephson junction without external rf driving force,” Phys. Rev. B, vol.42, no.7, pp.4080–4087, 1990. [14] P.S. Lomdahl, O.H. Soerensen, and P.L. Christiansen, “Soliton excitations in Josephson tunnel junctions,” in Voupled Nonlinear Oscillations, eds. J. Chandra and A.C. Scott, North Holland, 1983, 43–67. [15] A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations, John Wiley and Sons, 1979.

Takashi Hikihara was born in Kyoto, Japan, in 1958. He received his B.E. degree from Kyoto Institute of Technology in 1982, M.E. and Dr. E. degree from Kyoto University in 1984, 1990, respectively. He is now an Associate Professor of School of Electrical Engineering, Kyoto University. Since 1998, he has been an Associate Editor of the Transactions of IEICE on Fundamentals of Electronics, Communications and Computer Science. He is a member of IEE Japan, IEEE, and APS. His research interests include nonlinear dynamics in electrical and mechanical field and magnetic levitation.

Yohsuke Kondo was born in Osaka, Japan, in 1973. He received his B.E. and M.E. degree from Kansai University in 1995, 1997 respectively. Since 1997, he has been with Hitach Kiden Co. Ltd. He had been engaged in the research project of nonlinear wave in stick-slip coupled system.

Yoshisuke Ueda was born in Kobe, Japan, in 1936. He received his B.E, M.E, and Dr.E. degrees in Electrical Engineering from Kyoto University in 1959, 1961, and 1964, respectively. He is now a Professor of Kyoto University. He was the Vise President of IEE, Japan from 1993 to 1994. He is a adviserly committee of many international journals related to nonlinear phenomenon. He is a member of IEE Japan and IEEE. His research interests are in nonlinear dynamics in electronic circuits and electric power system.