Q: A simple pendulum of length $L$ is released from a small angle. What is the linearized equation of motion for its angular displacement $\theta(t)$?

$\ddot{\theta} + \frac{g}{L} \theta = 0$

Q: Given a damped oscillator with undamped natural frequency $\omega$ and damping ratio $\xi$, what is the formula for the damped natural frequency $\omega_d$?

$\omega_d = \omega \sqrt{1 - \xi^2}$

Q: Which of the following is NOT a condition for free linear vibration of a 1-DOF system?

The equation of motion must include a third-order derivative.

Q: Which of the following is a real-world example of energy dissipation due to vibration?

A car's shock absorber converting vibration energy into heat.

Q: In the context of vibration measurement using digital images, what does the pixel displacement between two frames represent?

The physical displacement of the object, which can be converted to real units using a reference scale.

Q: Which of the following best describes the least-square method in the context of vibration analysis?

A mathematical technique to fit a model to experimental data by minimizing the sum of squared errors.

Q: A mass-spring-damper system has $k = 10$ N/m, $T_d = 0.75$ s, and a logarithmic decrement $\delta = 0.38$. What is the damping ratio $\xi$?

$\xi = \frac{0.38}{\sqrt{0.38^2 + 4\pi^2}} \approx 0.06$

Question 1

Which of the following best describes an oscillation in the context of mechanics?

Accepted Answer

A repetitive deviation of a system from its reference state, generally the static equilibrium position.

Question 2

The Tacoma Narrows Bridge collapse is a classic example of which physical phenomenon?

Accepted Answer

Resonance due to oscillatory motion.

Question 3

A mass-spring system with mass $m$ and spring constant $k$ has an undamped natural frequency $\omega$. Which formula correctly gives $\omega$?

Accepted Answer

$\omega = \sqrt{\frac{k}{m}}$

Question 4

Which of the following is a correct expression for the total mechanical energy $E$ of an undamped mass-spring system at any time $t$?

Accepted Answer

$E = \frac{m}{2} \dot{x}^2(t) + \frac{k}{2} x^2(t)$

Question 5

A simple pendulum of length $L$ is released from a small angle. What is the linearized equation of motion for its angular displacement $	heta(t)$?

Accepted Answer

$\ddot{	heta} + \frac{g}{L} 	heta = 0$

Question 6

If two springs with stiffness $k_1$ and $k_2$ are connected in parallel to a mass $m$, what is the equivalent stiffness $k_{eq}$?

Accepted Answer

$k_{eq} = k_1 + k_2$

Question 7

A viscous damper exerts a force $\vec{F}_d$ on a mass moving with velocity $\vec{v}$. Which equation correctly describes this force?

Accepted Answer

$\vec{F}_d = -c \vec{v}$

Question 8

For a damped mass-spring system, the equation of motion is $m \ddot{x} + c \dot{x} + k x = 0$. What is the damping ratio $\xi$?

Accepted Answer

$\xi = \frac{c}{2m\omega}$

Question 9

Which regime is characterized by $\xi < 1$ in a damped oscillator?

Accepted Answer

Underdamped

Question 10

A car's suspension is designed to return to equilibrium as quickly as possible after hitting a bump, without oscillating. What type of damping is ideal?

Accepted Answer

Critical damping ($\xi = 1$)

Question 11

Given a damped oscillator with undamped natural frequency $\omega$ and damping ratio $\xi$, what is the formula for the damped natural frequency $\omega_d$?

Accepted Answer

$\omega_d = \omega \sqrt{1 - \xi^2}$

Question 12

Which of the following is NOT a condition for free linear vibration of a 1-DOF system?

Accepted Answer

The equation of motion must include a third-order derivative.

Question 13

The logarithmic decrement $\delta$ is used to determine which property of a damped oscillator?

Accepted Answer

Damping ratio $\xi$

Question 14

If a system's displacement is given by $x(t) = 2 \cos(3t) + 4 \cos(5t)$, what is the period of the resulting motion?

Accepted Answer

$T = 2\pi$

Question 15

Which of the following is a real-world example of energy dissipation due to vibration?

Accepted Answer

A car's shock absorber converting vibration energy into heat.

Question 16

A mattress is modeled as a set of $N$ parallel springs, each with stiffness $k_i$. What is the equivalent stiffness $k_{eq}$?

Accepted Answer

$k_{eq} = \sum_{i=1}^N k_i$

Question 17

In the context of vibration measurement using digital images, what does the pixel displacement between two frames represent?

Accepted Answer

The physical displacement of the object, which can be converted to real units using a reference scale.

Question 18

Which of the following best describes the least-square method in the context of vibration analysis?

Accepted Answer

A mathematical technique to fit a model to experimental data by minimizing the sum of squared errors.

Question 19

A mass-spring-damper system has $k = 10$ N/m, $T_d = 0.75$ s, and a logarithmic decrement $\delta = 0.38$. What is the damping ratio $\xi$?

Accepted Answer

$\xi = \frac{0.38}{\sqrt{0.38^2 + 4\pi^2}} \approx 0.06$

Oscillations and Vibrations: Physics of Periodic Motion and Damping

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