0 2 4 6 8 10 -20 0 20 Time, sec w(t), lb Figures 3a. Realization of a band-limited white noise excitation. 0 2 4 6 8 10 -10 0 10 Time, sec x(t), in Figure 3b. Realization of displacement response of the system shown in Figure 1 to the excitation shown above. 0 2 4 6 8 10 -2 0 2 Time, sec v(t), in/sec Figure 3c. Realization of velocity response of the system shown in Figure 1 to the excitation shown above. Smoluchowski (1916) and Furth (1917) independently generalized Einstein’s analysis and performed Brownian motion experiments to verify the predictions of the theory. Smoluchowski was first to write a form of the diffusion equation (later known as Fokker-Planck equation) for systems in which a displacement-related force restrains the mass. He wrote the diffusion equation governing the PDF of the response of a single-degree-of-freedom (SDF) system connected to a fixed boundary via a linear spring and viscous damper. A schematic of such a system is shown in Figure 4. But the equation that he developed, and its solution, govern only the response of an over-damped SDF system, i.e., a system with relatively high damping – one that executes a non-oscillatory response in free vibrations caused by non-zero initial conditions. The overdamped solution is less interesting to structural dynamicists than the lightly damped one because most practical structural dynamic systems are lightly damped. Figure 4. Schematic of the system considered by Smoluchowski in his random vibration analysis. m c x W(t) k 109
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