A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\) and a viscous damper whose coefficient is \(2 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). (a) Determine the response of the horizontally configured system if the mass is displaced 1 meter to the right and released from rest. Plot and label the response history of the system. (b) Determine the response and plot its history if the damping coefficient is \(8 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).

Question: Calculate and compare the response of a single degree of freedom system with an initial displacement of 1 m, mass of 2 kg, and spring stiffness of 4 N/m when subjected to (a) a viscous damper coefficient of 2 N·s/m and (b) a viscous damper coefficient of 8 N·s/m. Briefly describe the differences in the response histories. Answer: For case (a) with a viscous damper coefficient of 2 N·s/m, the system is underdamped, and its response is given by the equation: $$x(t) = e^{-\frac{\sqrt{2}}{4}t}( \cos\left(\frac{\sqrt{15}}{2}t\right) + \frac{\sqrt{2}}{\sqrt{15}}\sin\left(\frac{\sqrt{15}}{2} t\right))$$ For case (b) with a viscous damper coefficient of 8 N·s/m, the system is critically damped, and its response is given by the equation: $$x(t) = (1 + \sqrt{2}t)e^{-\sqrt{2}t}$$ The underdamped system (case a) exhibits oscillatory behavior, with the amplitude of oscillations decaying over time due to damping, whereas the critically damped system (case b) returns to the equilibrium position as quickly as possible without any oscillations.