Solved Examples of Underdamped Oscillations with Step-by-Step Solutions

📚 Quick Study Guide

• 🍎 An underdamped oscillation occurs when the damping force is small relative to the inertial force, allowing the system to oscillate with gradually decreasing amplitude.
• 📐 The general solution for an underdamped system is given by: $x(t) = Ae^{-\zeta \omega_n t} \cos(\omega_d t - \phi)$, where $A$ is the initial amplitude, $\zeta$ is the damping ratio, $\omega_n$ is the natural frequency, $\omega_d$ is the damped frequency, and $\phi$ is the phase angle.
• ⏱️ The damping ratio, $\zeta$, is defined as $\zeta = \frac{c}{2\sqrt{mk}}$, where $c$ is the damping coefficient, $m$ is the mass, and $k$ is the spring constant. For underdamped oscillations, $0 < \zeta < 1$.
• 📉 The damped frequency, $\omega_d$, is related to the natural frequency by $\omega_d = \omega_n \sqrt{1 - \zeta^2}$.
• 💡 The logarithmic decrement, $\delta$, quantifies the rate at which the amplitude decreases and is given by $\delta = \zeta \omega_n T = \frac{2\pi \zeta}{\sqrt{1-\zeta^2}}$, where $T$ is the period of oscillation.

🧪 Practice Quiz

1. Which of the following conditions defines an underdamped oscillation?

   1. $\zeta > 1$
   2. $\zeta = 1$
   3. $0 < \zeta < 1$
   4. $\zeta = 0$

2. The amplitude of an underdamped oscillation ____ with time.

   1. Increases linearly
   2. Remains constant
   3. Decreases exponentially
   4. Increases exponentially

3. What does $\omega_d$ represent in the equation $x(t) = Ae^{-\zeta \omega_n t} \cos(\omega_d t - \phi)$?

   1. Natural frequency
   2. Damping ratio
   3. Damped frequency
   4. Phase angle

4. Which parameter determines how quickly the oscillations decay in an underdamped system?

   1. Initial amplitude ($A$)
   2. Damping ratio ($\zeta$)
   3. Phase angle ($\phi$)
   4. Natural frequency ($\omega_n$)

5. If the damping ratio $\zeta$ is increased in an underdamped system, what happens to the damped frequency $\omega_d$?

   1. Increases
   2. Decreases
   3. Remains constant
   4. Becomes imaginary

6. In an underdamped system, the oscillations eventually stop because of:

   1. Increase in potential energy
   2. Conservation of energy
   3. Energy dissipation due to damping
   4. Increase in kinetic energy

7. What is the value of damping ratio for critically damped system?

   1. $\zeta < 1$
   2. $\zeta > 1$
   3. $\zeta = 1$
   4. $\zeta = 0$