Problem \#2 (20 pts): Complex frequency response of a SDOF system
A single degree of freedom spring-mass-damper system shown in Figure 1 is subject to sinusoidal loading \( p_{0} \sin (\Omega t) \). If \( k_{1}=1880 \mathrm{~N} / \mathrm{m}, k_{2}=1000 \mathrm{~N} / \mathrm{m}, m=20 \mathrm{~kg} \), and \( c= \) \( 24 \mathrm{Ns} / \mathrm{m} \) :
a) Calculate the natural frequency \( \omega_{n} \) in \( \mathrm{rad} / \mathrm{s} \) and the damping ratio \( \zeta \). Start from a free body diagram and write the equation of motion before you calculate \( \omega_{n} \) and \( \zeta \).
b) Calculate the complex frequency response \( H \) using the nondimensional form found in Problem 1 part d for a forcing frequency of \( \Omega=3 \mathrm{rad} / \mathrm{s}, 5 \mathrm{rad} / \mathrm{s} \), and \( 8 \mathrm{rad} / \mathrm{s} \).
c) Figure 2 shows the Bode plot, CO-QUAD plot, and Nyquist plot of the system in Figure 1. On the graph, mark the points \( \mathrm{A}, \mathrm{B} \), and C that correspond to the forcing frequencies \( \Omega=3 \mathrm{rad} / \mathrm{s}, 5 \mathrm{rad} / \mathrm{s} \), and \( 8 \mathrm{rad} / \mathrm{s} \), respectively.
Figure 1 SDOF system with sinusoidal loading.
Figure 2 Bode, CO-QUAD, and Nyquist Plots for a SDOF system
