RES.8-005 | Fall 2012 | Undergraduate

Vibrations and Waves Problem Solving

Problem Solving Videos

Driven Harmonic Oscillators

About this Video

 

In this session we first give advice on how, in general, one approaches the solving of “physics problems.” We then consider three very different oscillating systems, show how in each the equation of motion can be derived and then solve these equations to obtain the motion of the oscillator.

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Driven Harmonic Oscillators

Problem 1

A torsional oscillator comprises a cylinder with moment of inertia, \(I\), hanging from a light rod with torsional spring constant, \(\kappa\). The cylinder also experiences a drag torque equal to \(-\mu \dot \theta\), when moving with angular velocity \(\dot \theta\). The top of the rod is driven with angular displacement \(\phi(t) = \phi_0 \cos{\omega t}\).

  1. Find the steady-state solution for \(\theta(t)\).
  2. Plot the amplitude \(A(\omega) \) and phase \(\delta(\omega)\) of your solution for \(\theta(t)\) in (1) as a function of \(\omega\). For your plot, assume that the natural frequency of oscillation of the system \(\omega_0 = 1\), and plot three curves on the same plot with \(\frac{\mu}{I} = 0.25\), 1 and 2. Label your curves to distinguish the three cases.

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\begin{align*} \theta(t) &= \frac{\omega_0^2\,\phi_0}{\sqrt{\left(\omega_0^2 - \omega^2\right)^2 + \gamma^2\,\omega^2}}\,\cos\left({\omega\,t - \arctan{\left(\frac{\gamma\,\omega}{\left(\omega_0^2 -\omega^2\right)}\right)}}\right) \end{align*} \begin{align*} \text{where} \hspace{5mm} \gamma =\mu/I \hspace{5mm} \text{and} \hspace{5mm} \omega_0^2 = \kappa/I \end{align*}

\begin{align*} & A(\omega) = \frac{\omega_0^2\,\phi_0}{\sqrt{\left(\omega_0^2 - \omega^2\right)^2 + \gamma^2\,\omega^2}}\hspace{1mm}, \hspace{4mm} & \tan{\delta(\omega)} = \frac{\gamma\,\omega}{\left(\omega_0^2 -\omega^2\right)} \end{align*}

Figure3_1

Problem 2

A capacitor (of capacitance C), a resistor (of resistance R) and an inductor (of inductance L) are connected to an AC voltage source \(V = V_0 \sin(\omega t)\) starting at \(t=0\) as shown in the diagram below.

Figure 3_2

Assuming that both the current and the charge of the capacitor are initially zero, determine the expression for \(V_C(t\ge0)\) with \(\omega=\omega_0=\dfrac{1}{\sqrt{LC}}\) and \(L < 4R^2C\).

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\begin{align*} V_C(t\ge0) = V_{0}[-\dfrac{\omega_{0}}{\omega’}e^{-\frac{\gamma t}{2}} \sin(\omega’t) + \sin(\omega_{0}t)] \end{align*} \begin{align*} \text{where} \hspace{5mm} \omega’ = \sqrt{\frac{1}{LC} - \frac{1}{4R^2C^2}} \hspace{5mm} \text{and} \hspace{5mm} \gamma = \frac{1}{RC} \end{align*}

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