Fig. 6. Numerical Coulombic free vibration decay of a flexible body Figure 6 is the numerical nonlinear transient response waveform for two beams that are damped by friction. The decay is linear as would be expected. However this waveform consists of two separated harmonics. The first harmonic is that of the beams during sliding, acting near frictionless, and with the friction component acting as predominantly damping and not as a stiffness shift. The second harmonic is that of when the layers are in stiction from when the system has damped out and now the layers act as a single beam. For the first harmonic the natural frequency is around 135.69Hz while the second harmonic is approximately 256.40Hz. This is expected and predicted as there were only two beams which results in the stiffness between stiction and frictionless is to be nearly 4X (2n) and since ωn = (k/m) 0.5. It should also be noted that there is little decay in the second harmonic as this is purely hysteretic and the defined material damping was extremely low. It has been shown that the bilinear nature has little effect on the shape of the overall waveform. Figure 7 is a comparison between an ESL representation of three layers with a bilinear material property derived from the analytical stiffness model and a steady-state sine wave in the form of x(t)=Xsinωt. In the zoomed view (on the right-hand side), there are some noticeable nonlinearities seen but do not have much of an effect on the actual waveform profile. It should be noted that from π/2 to π, of the first cycle, the waveform corrects itself and is not a permanent alteration that is cumulative. Attention should also be directed at the waveform comparison when t=0.03s. The waveforms begin to diverge in their profiles. This is a result of the second harmonic, similar to that in Figure 6. Fig. 7. Waveform comparison for ANSYS bilinear Coulombic free vibration decay and steady-state sinusoidal excitation first harmonic second harmonic 297
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