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While we have concentrated on finding the response of the output variable (v0 in our example), we can of course find the response curves of any other system variables that might be of interest (see Fig. 7-7). The displacement x 0 of the mass is obtained from

X 0 = X 0 (0) + t V 0 dt = Kfist + Kfisr(e-tfr-1) (7-23) where we have taken the initial displacement X 0 (0) to be zero. Note that after a transient period during which e-t/r is dying out, the displacement becomes asymptotic to the straight line Kfis(t-r). (If the force lis is left on, a translational damper must sooner or later encounter mechanical stops and cause the mass to stall. However, the rotational version allows continuous unimpeded motion such that the output angular displacement can actually "approach infinity" as indicated by the equations.)

A mathematical description of system behavior as given by the graphs of Fig. 7-7 should generally be interpreted in physical terms as a means of checking the plausibility of the results and reinforcing our intuitive feelings about the system. We might put it this way: The suddenly applied force fis causes a sudden acceleration of the mass M; however, as the acceleration acts over time and produces some velocity, the dashpot develops an opposing force which reduces the net accelerating force available. As the velocity builds up, the dashpot force approaches the external driving force more closely, and the acceleration approaches zero. The system thus approaches asymptotically a terminal velocity given by the ratio of applied force