12 Simulating Nonlinear Beating Phenomena Induced by Dry Friction in Dynamic Systems 95 12.3 Analysis In the case where the system is being driven with an excitation force, Fex, one technique to obtain the displacement and velocities is to use a numerical approach using implicit time-integration schemes such as Newark, Wilson, and RungeKutta (RK4) methods to get the displacements and velocities of a nonlinear system. A time-integration scheme based on Runge-Kutta is developed in MATLAB to determine the displacement and velocities in both systems. The simulation was completed over a 10-second interval and with a 0.01 time discretization for 10 seconds. The velocities of the masses are shown in Fig. 12.2 where the “beating” phenomena is observed in the viscous damped, but not the friction-damped system. For both models, the response of the displacement has a similar pattern to the velocity, with differences due to magnitudes. A frequency domain analysis of the displacement in both systems is shown in Fig. 12.3. Figure 12.3a shows sub-harmonic interactions in the spectra. Analysis was done with a short-time Fourier transform (STFT) with six Gaussian windowing functions. 12.3.1 Numerical Integration Scheme Beating phenomena was observed in the viscous-damped case. There also appears to be some energy transfer between the harmonics in the case of viscous damping. This can be seen clearly in Fig. 12.3a where the peak frequency at 0.03 Hz is transferring its energy into the other modes of the system. Beat phenomena occurred here when the difference between the damped frequencies, ω = ω 1 − ω2 , is quite small. ω 1 represents the frequency with the highest amplitude for m 1 an d ω 2 is the frequency with the highest amplitude for m2. ωvisc wa s 0.001. This is different from the friction dissipated model, which at the given parameters did not demonstrate “beating” phenomena. ωBW wa s0.03. The other observation made from Fig. 12.3a is the presence of closely spaced frequencies near the peak frequency, ω1. Fig. 12.2 Input excitation force shown in (a ) with amplitude of 75 and T = 0.1. The viscous-damped system velocities (˙x) are shown in (b) and the friction-damped velocities (˙x) are shown in (c ) Fig. 12.3 STFT of the displacement, x2, for (a ) viscous-damped system and (b ) friction-damped system with Bouc-Wen friction element with maximum frequency of 0.2 Hz
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