Question 3.2.2: Another common excitation in vibration is a constant force t......
Another common excitation in vibration is a constant force that is applied for a short period of time and then removed. A rough model of such a force is given in Figure 3.8. Calculate the response of an underdamped system to this excitation.
This pulse-like loading can be written as a combination of step functions calculated in Example 3.2.1, as illustrated in Figure 3.8. The response of a single-degree-of-freedom system to F(t)=F_1(t)+F_2(t) is just the sum of the response to F_1(t) and the response of F_2(t), because the system is linear. First, consider the response of an underdamped system to F_1(t). This response is just that calculated in Example 3.2.1 for t_0=0 and given by equation (3.17). Next, consider the response of the system to F_2(t). This is just the response given by equation (3.15) with F_0 replaced by -F_0 and t_0 replaced by t_1. Hence, subtracting equation (3.15) from equation (3.17) yields the result that the response to the pulse of Figure 3.8 is
x(t)=\frac{F_0}{k}-\frac{F_0}{k \sqrt{1-\zeta^2}} e^{-\zeta \omega_n\left(t-t_0\right)} \cos \left[\omega_d\left(t-t_0\right)-\theta\right] \quad t \geq t_0 (3.15)
x(t)=\frac{F_0}{k}-\frac{F_0}{k \sqrt{1-\zeta^2}} e^{-\zeta \omega_n t} \cos \left(\omega_d t-\theta\right) (3.17)
x(t)=\frac{F_0 e^{-\zeta \omega_n t}}{k \sqrt{1-\zeta^2}}\left\{e^{\zeta \omega_n t_1} \cos \left[\omega_d\left(t-t_1\right)-\theta\right]-\cos \left(\omega_d t-\theta\right)\right\}, \quad t>t_1
where \theta is as defined in equation (3.16). A plot of this response is given in Figure 3.9 for different pulse widths t_1. Note that the response is much different for t_1>\pi / \omega_n and has a maximum magnitude of about five times the maximum magnitude of the time response for t_1<\pi / \omega_n. Also, note that the steady-state response (i.e., the response for large time) is zero in this case.
\theta=\tan ^{-1} \frac{\zeta}{\sqrt{1-\zeta^2}} (3.16)
