By denoting the amplitudes of velocity and acceleration of the mass of a viscously damped system subjected to a harmonic base motion, \(y(t)\), as shown in Fig. 3.14, as \(\dot{X}\) and \(\ddot{X}\), respectively, find expressions for the ratios \(\frac{\dot{X}}{\omega_{n} Y}\) and \(\frac{\ddot{X}}{\omega_{n}^{2} Y}\) in terms of \(r\) and \(\zeta\). The nondimensional ratios \(\frac{\dot{X}}{\omega_{n} Y}\) and \(\frac{\ddot{X}}{\omega_{n}^{2} Y}\) are called the velocity and acceleration frequency responses of the mass, respectively.

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00000 m +x +X +y m y(t) = Y sin wt k(xy) c(xy) Base (a) K FIGURE 3.14 Base excitation. (b) +x