Question 2.3.1: Use the frequency response approach to compute the amplitude......

First, write equation (2.2) with the forcing function modeled as a complex exponential:

m \ddot{x}(t)+k x(t)=F_0 e^{j \omega t}

Dividing by the mass, m, yields the monic form

\ddot{x}(t)+\omega_n^2 x(t)=f_0 e^{j \omega t}

Assume a particular solution of the exponential form given in equation (2.49) and substitute into the last expression to get

x_p(t)=X e^{j \omega t}     (2.49)

\left(-\omega^2+\omega_n^2\right) X e^{j \omega t}=f_0 X e^{j \omega t}

Solving for X yields

X=\frac{f_0}{\omega_n^2-\omega^2}

This is in perfect agreement with equation (2.7) derived using the cosine representation of the forcing function and (2.21) derived using the sine representation. This also agrees with solution given in equation (2.36) for the damped case, by setting \zeta=0 in that expression.

x_p(t)=\frac{f_0}{\omega_n^2-\omega^2} \cos \omega t     (2.7)

X=\frac{f_0}{\omega_n^2-\omega^2}      (2.21)

x_p(t)=\overbrace{\frac{f_0}{\sqrt{\left(\omega_n^2-\omega^2\right)^2+\left(2 \zeta \omega_n \omega\right)^2}}}^X \cos (\omega t-\overbrace{\tan ^{-1} \frac{2 \zeta \omega_n \omega}{\omega_n^2-\omega^2}}^\theta)       (2.36)