To explain why an amplification effect is seen for the higher preload levels, the system was approximated by the simple lumped parameter two degree of freedom system shown in Fig. 10. The larger mass, M1, represents the circular plate while the smaller mass, M2, represents the square plate. The bolts are represented by a piecewise linear spring and damper. The impact force is applied to the center of M1, as in the real system. The position of the large mass is denoted as x, while the position of the smaller mass is denoted as y. For this simple model, the values for M1 and M2 were chosen to be 4kg and 1kg, respectively. The value of C was chosen based on a proportional damping estimate and is always one thousandth of the value of K. The values of K were varied in order to simulate stiffening in the bolts. The system has a free-free boundary condition. The frequency response functions between the input force and the two plates’ responses were calculated and plotted. The frequency response of the system for K=0.5N/m is shown in Fig. 11. The result when the value of K is raised to 10N/m is shown in Fig. 12. The result for K=30N/m is shown in Fig. 13. For all three responses, two modes of vibration can be observed. At DC, the rigid body mode occurs. The response of both bodies overlaps for a region, and then as the larger body M1 begins to enter an anti-resonance, the motion is larger on the smaller body M2. In the anti-resonance region, the motion of the larger body goes to nearly zero and only the smaller body responds. In this vibration phenomenon, one body is excited, yet the largest motion occurs in the other body. After this region, the second mode of vibration, which consists of asynchronous motion between the bodies, can be observed. Shortly after this peak, the motion of the larger body begins to dominate. In the region around the anti-resonance, the smaller body acts as a tuned mass vibration absorber. The three plots show that the frequency bounds for each of these regions are highly dependent on the value of K, and, subsequently, the preload in the bolts. For the lowest stiffness, which simulates very low preload, the region in which motion is amplified on the smaller plate is almost completely below 1 rad/s. When the stiffness is raised to 10N/m, the region stretches from less than 1 rad/s to nearly 4.5 rad/s. For the highest preload value of 30N/m, the region lasts from less than 1 rad/s to nearly 8 rad/s. It should also be noted that the width of the anti-resonance in the response of the large plate, where the amplification of motion is the greatest, grows with the stiffness value. In general, as preload increases, the range where the motion of the smaller plate is greater than that of the larger plate increases as well. This result qualitatively agrees with the amplification of motion across the interface for higher preload values that was observed in the experimental data. Fig. 10 Lumped Parameter Two Degree of Freedom System Model 575
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