A single degree of freedom system is represented as a \(4 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(6 \mathrm{~N} / \mathrm{m}\) and a viscous damper whose coefficient is \(1 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). (a) Determine the response of the horizontally configured system if the mass is displaced 2 meters to the right and released with a velocity of \(4 \mathrm{~m} / \mathrm{sec}\). Plot and label the response history of the system. (b) Determine the response and plot its history if the damping coefficient is \(5 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). (c) Determine the response and plot its history if the damping coefficient is \(10 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).

To determine the response of the system, first write the equation of motion for the single degree of freedom system: \(m\ddot{x} + c\dot{x} + kx = 0\), where \(m\) is the mass, \(c\) is the damping coefficient, \(k\) is the spring stiffness, \(x\) is the displacement, \(\dot{x}\) is the velocity, and \(\ddot{x}\) is the acceleration. With the given values, the equation of motion is: \(4\ddot{x} + c\dot{x} + 6x = 0\)

Solve the equation of motion for three different damping coefficients: \(c_1 = 1\), \(c_2 = 5\), and \(c_3 = 10\). For each case, the equation of motion takes the form: - Case a: \(4\ddot{x} + 1\dot{x} + 6x = 0\) - Case b: \(4\ddot{x} + 5\dot{x} + 6x = 0\) - Case c: \(4\ddot{x} + 10\dot{x} + 6x = 0\) Solve the three equations for displacement \(x(t)\), considering the initial conditions: \(x(0) = 2 \mathrm{~m}\) and \(\dot{x}(0) = 4 \mathrm{~m}/\mathrm{sec}\).

Using the displacement expressions obtained in Step 2 for each damping coefficient, plot the response history of the system. Make sure to label the plots for each case (a), (b), and (c). To create these plots, you can use a graphing calculator, a spreadsheet program like Microsoft Excel or Google Sheets, or a programming language like Python or MATLAB. For each case, compute the displacement \(x(t)\) for a range of time values, typically from \(t = 0\) to a chosen time where the system has reached a steady state. Then, plot the displacement versus time for each case and compare the results to analyze the effect of the different damping coefficients on the system's response. It's important to remember that this is a qualitative exercise where the main goal is to understand the system's behavior under different damping conditions. The response will depend on the specific equations obtained in Step 2, but generally, the higher the damping coefficient, the faster the system stabilizes.