Conclusions: Numerical Simulations and Experimental Data

Figures 42 and 43 present the calculated and desired acceleration power spectral density of the platform. These figures indicate that the overall technique, going from a desired acceleration power spectral density profile to a controller position profile, does provide the desired effect – providing an experimental acceleration power spectral density that estimates the desired profile. However, note that the overall acceleration response between 4 Hz to 5 Hz is lower than at the lower frequencies. This is due to the controller limitation of only being able to move the table to a position every 30 milliseconds. A faster position table controller would allow more energy at the higher frequencies and enable a better fit to the ideal acceleration power spectral density profile. Also, note the variability of the experimental acceleration response. Longer sampling times (e.g., much greater than 5 minutes) and additional input signal variation (e.g., position signals longer than 130 seconds) would be needed to obtain a smoother acceleration power spectral density.

Tables 4, 5, and 6 and Figures 44 and 45 present data on the number of times the steel ball (i.e., the impactor) made contact, or the number of hits, with either end block A or B (see Figure 38). From these tables and figures, it is clear that the numerical model of the system does provide the same type of behavior of the real system, especially when the distance between end blocks is such that many hits occur per minute; however, a few differences can be noted. First, from the data tables, it is clear that for the numerical simulations the number of hits on the end blocks are nearly the same between A and B for each time period of evaluation; however, from the experimental data, the number of hits on the end blocks tend to be different. This is most likely attributed to the

construction of the experimental test assembly. Although care was given to ensure the steel rails were straight and level, in actuality, the rails were slightly bent (in both longitudinal and transverse directions); therefore, preventing a truly level set of rails with a constant separation distance on which the steel ball could travel.

Also, note that the numerical simulations actually predicted a lower hit count than was measured. This is most likely attributed to how the impactor contact with the end blocks was modeled in the simulations. First, it was assumed that the impact between the steel ball (impactor) and the end blocks was perfectly elastic (i.e., no kinetic energy was lost during the impact – a coefficient of restitution of 1.0). Clearly, in actuality this type of impact is not physically possible and that some energy during the impact must be lost to the system. Secondly, to help with numerical stability, it

was assumed that the steel ball would “stick” to the end blocks (i.e., have the same position and speed as the impacting end block) if the steel ball’s speed after an impact with an end block was nearly zero and the end block was accelerating in a direction towards the opposite end block. The steel ball would then be “released” from the end block once the end block started to accelerate in a direction away from the opposite end block, either transitioning from a positive to a negative acceleration, or transitioning from a negative to a positive acceleration. Without the “stick” to the end block assumption, the tolerances of the ordinary differential equation solvers would, in some impacts, allow the steel ball to move through the end blocks, ending up on the opposite side of the end block, which is clearly not physical. However, the tolerances chosen to implement the “stick” assumption may have been set too large (i.e., the speed of the steel ball after impact was too high to “stick”); therefore, causing a lower simulated number of hits with the end blocks.

Finally, the largest factor in the discrepancy between numerical simulation and experimental results was most likely how the dynamics of the end blocks contributed to the overall response of the steel ball. It was assumed that the duration of the impact was of such short duration that the dynamics of the end blocks were negligible (i.e., the end blocks act as rigid walls during the impact). It was also assumed that the dynamic response of the end blocks after each impact would dissipate (become negligible) before another impact could occur, and it was assumed that the ground motion did not excite the dynamics of the end blocks. Through observation of the test assembly, it was easy to verify that the ground motion (platform motion) did excite the end blocks; however, the overall effect of the dynamics of the end blocks could not be easily verified by direct observation. So, the modeling assumptions of the impact dynamics between the steel ball and the end blocks most likely had a large effect on the number of simulated hits being less than actually measured.

Another possible contributor to the differences in the numerical and experimental results may be due to the dynamics of the position table foundation and of the raised platform. It was assumed in the numerical simulations that the dynamics of both the foundation and the raised platform could be neglected; however, in the experimental acceleration results, high frequency dynamics from the platform and foundations could actually be observed.