Question

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 \(10 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).

Short answer

Answer: The response function of a single degree of freedom system depends on the damping ratio (ζ). For different values of ζ, the response function can be divided into three scenarios: 1. Underdamped (\(\zeta < 1\)): \(x(t)=e^{-\zeta\omega_nt}\left[x_0\cos(\omega_d t)+\left(\frac{v_0+\zeta\omega_nx_0}{\omega_d}\right)\sin(\omega_d t)\right]\), where \(\omega_d=\omega_n\sqrt{1-\zeta^2}\). 2. Critically damped (\(\zeta = 1\)): \(x(t)=e^{-\omega_nt}\left[x_0+(v_0+\omega_n x_0)t\right]\) 3. Overdamped (\(\zeta > 1\)): \(x(t)=e^{-\zeta\omega_nt}\left[C_1e^{\omega_n\sqrt{\zeta^2-1}t}+C_2 e^{-\omega_n \sqrt{\zeta^2-1}t}\right]\), where \(C_1\) and \(C_2\) are coefficients determined from the initial conditions.

Step-by-step solution

Setup the equation of motion

For a single degree of freedom system with mass (\(m\)), spring stiffness (\(k\)) and damping coefficient (\(c\)), the equation of motion can be written as: \(m \ddot{x} + c \dot{x} + k x = 0\) We are given that \(m = 4\text{ kg}\), \(k = 6\text{ N/m}\), and the initial displacement \(x_0 = 2\text{ m}\) and velocity \(v_0 = 4\text{ m/s}\).

Determine the natural frequency and damping ratio

Compute the natural frequency (\(\omega_n\)) and damping ratio (\(\zeta\)) for the given system. They are calculated as follows: \(\omega_n = \sqrt{\frac{k}{m}}\) \(\zeta = \frac{c}{2 m \omega_n}\)

Solve the equation of motion for different damping coefficients

For each damping coefficient, we can now solve the equation of motion to compute the response of the system. Since we have 3 different damping coefficients: \(1\text{ Ns/m}\), \(5\text{ Ns/m}\), \(10\text{ Ns/m}\), we need to compute the response for each case.

Determine the response function

Depending on the value of the damping ratio \(\zeta\), the response function \(x(t)\) will have different forms: 1. For \(\zeta < 1\), the system is underdamped, and the response function is: \(x(t)=e^{-\zeta\omega_nt}\left[x_0\cos(\omega_d t)+\left(\frac{v_0+\zeta\omega_nx_0}{\omega_d}\right)\sin(\omega_d t)\right]\) where \(\omega_d=\omega_n\sqrt{1-\zeta^2}\) is the damped natural frequency. 2. For \(\zeta = 1\), the system is critically damped, and the response function is: \(x(t)=e^{-\omega_nt}\left[x_0+(v_0+\omega_n x_0)t\right]\) 3. For \(\zeta > 1\), the system is overdamped, and the response function is: \(x(t)=e^{-\zeta\omega_nt}\left[C_1e^{\omega_n\sqrt{\zeta^2-1}t}+C_2 e^{-\omega_n \sqrt{\zeta^2-1}t}\right]\) where \(C_1\) and \(C_2\) are coefficients determined from the initial conditions.

Calculate the response and plot the history

For each damping coefficient, compute the response function using the appropriate formula from Step 4. Then, plot the response history for each case (a), (b), and (c). For (a) \(c=1\text{ Ns/m}\), compute the response and plot the history using the appropriate formula. For (b) \(c=5\text{ Ns/m}\), compute the response and plot the history using the appropriate formula. For (c) \(c=10\text{ Ns/m}\), compute the response and plot the history using the appropriate formula.

Key concepts

Equation of Motion

In a single degree of freedom system, the equation of motion describes how the system responds to external forces or displacements over time. It effectively acts as a mathematical model of the system's physical behavior. The basic form of the equation of motion for such a system is:\[m \ddot{x} + c \dot{x} + k x = 0\]Where:

• \(m\) is the mass of the system.
• \(c\) is the damping coefficient, representing the resistance to motion.
• \(k\) is the stiffness of the spring, which dictates how much the system can stretch or compress.
• \(x\), \(\dot{x}\), and \(\ddot{x}\) are the displacement, velocity, and acceleration, respectively.

This equation shows us that the motion is determined by the balance between inertial forces caused by mass, resistive forces due to damping, and restorative forces from the spring. This classic form of the equation of motion assumes that there are no external forces acting on the system, making it a homogeneous equation.

Natural Frequency

The natural frequency is a critical property of any oscillatory system, indicating how it tends to vibrate when not externally forced. It is the frequency at which the system naturally oscillates if disturbed. For our single degree of freedom system, the natural frequency \(\omega_n\) is computed using the formula:\[\omega_n = \sqrt{\frac{k}{m}}\]Here:

• \(k\) is the spring stiffness, affecting how responsive the system is to displacements.
• \(m\) is the mass, influencing the inertia of the system.

The natural frequency, \(\omega_n\), is crucial because it helps in determining the system's response characteristics. If the frequency of any external force matches this natural frequency, the system can experience resonance, resulting in large amplitude oscillations which can potentially lead to system failure.

Damping Ratio

Damping ratio \(\zeta\) is a dimensionless measure of how oscillations in a system decay after a disturbance. It gives us a sense of how the damping affects the vibrating system:\[\zeta = \frac{c}{2m\omega_n}\]Where:

• \(c\) is the damping coefficient.
• \(m\) is the mass.
• \(\omega_n\) is the natural frequency.

The damping ratio plays a pivotal role in classifying the system's response. Essentially, it indicates whether the system will vibrate freely and then gradually stop over time (underdamped), reach its rest position without oscillating (critically damped), or return to rest slowly without oscillations (overdamped). Understanding this ratio helps us in predicting and controlling the behavior of the system after a disturbance.

Underdamped, Critically Damped, Overdamped Systems

A system's damping ratio defines its response type, which can be underdamped, critically damped, or overdamped. These terms relate to the nature and speed of the system's return to equilibrium after a disturbance.**Underdamped Systems**When the damping ratio \(\zeta < 1\), the system is underdamped. In this case, the system oscillates before eventually coming to rest. This is typical for systems designed for maximum responsiveness, like some suspension systems in vehicles, where we also value a quick stop.**Critically Damped Systems**For \(\zeta = 1\), the system is critically damped. It returns to rest quickly without overshooting the equilibrium position. This type is favorable in precision systems, such as some doors or instruments, where we want the fastest response without oscillation.**Overdamped Systems**When \(\zeta > 1\), the system is considered overdamped. The return to the equilibrium comes without oscillation but more slowly than the critically damped system. Overdamped systems are used where slowing down motion is more important than speed, as seen in heavy doors or certain automatic control systems.Understanding these damping modes allows engineers to design the appropriate type of damping for the desired application, balancing responsiveness and stability.