Day 23 - Driven Oscillations

Day 23 - Driven Oscillations#

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Resonance in a driven pendulum system. \(\longrightarrow\)

Source: Wikipedia

Welcome Richard Hallstein!#

Announcements#

  • HW 5 given blanket extension to this Friday.

  • HW 6 still due this Friday, but will be extended if needed.

  • DC will return Wednesday

  • Mihir will lead workshop session for HW 6 on Friday

Reminders#

We solved the damped harmonic oscillator equation:

\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = 0\]

We chose a solution (ansatz) of the form

\[x(t) = C_1 e^{r t} + C_2 e^{r t}\]

and computed the roots of the characteristic equation:

\[r^2 + 2 \beta r + \omega_0^2 = 0\]

We found the roots to be:

\[r = -\beta \pm \sqrt{\beta^2 - \omega_0^2}\]

Weak Damping#

We found that when \(\beta^2 < \omega_0^2\), the roots are complex:

\[r = -\beta \pm i \sqrt{\omega_0^2 - \beta^2}\]

This means that the solution is oscillatory:

\[x(t) = e^{-\beta t} \left( C_1 \cos(\sqrt{\omega_0^2 - \beta^2} t) + C_2 \sin(\sqrt{\omega_0^2 - \beta^2} t) \right)\]

The solution is a damped oscillation with frequency \(\omega_1 = \sqrt{\omega_0^2 - \beta^2}\).

Strong Damping#

When \(\beta^2 > \omega_0^2\), the roots are real:

\[r = -\beta \pm \sqrt{\beta^2 - \omega_0^2}\]

This means that the solution is not oscillatory: $\(x(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}\)\( where \)r_1 = -\beta + \sqrt{\beta^2 - \omega_0^2} < 0\( and \)r_2 = -\beta - \sqrt{\beta^2 - \omega_0^2} < 0$.

The solution is the sum of two exponentials with different decay rates.

Critical Damping#

When \(\beta^2 = \omega_0^2\), the roots are real and equal (repeated roots): $\(r = -\beta\)$

This means that the solution is not oscillatory, but also that our ansatz is not sufficient. The correct form of the solution is:

\[x(t) = (C_1 + C_2 t) e^{-\beta t}\]

In most cases, we will work with weak damping.

Clicker Question 23-1#

What do we expect the phase space diagram (\(x\) vs \(\dot{x}\)) to look like for a weakly damped harmonic oscillator?

  1. A set of ellipses

  2. A set of spirals

  3. Depends on how weak the damping is

  4. Depends on the total energy

  5. More than one of the above

Clicker Question 23-2#

The driven harmonic oscillator equation is:

\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = f(t)\]

with \(w_0^2 = k/m\) and \(2\beta = b/m\). What is the dimension of the driving force \(f(t)\)?

  1. Force (Newtons, N)

  2. Force per unit second (N/s)

  3. Force per unit length (N/m)

  4. Force per unit mass (N/kg)

Clicker Question 24-3#

The driven harmonic oscillator equation is:

\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = f(t).\]

This ODE is a ________ differential equation.

  1. linear

  2. nonlinear

  3. first-order

  4. second-order

  5. more than one of the above

Example: Sinusoidal Driving Force#

Let \(f(t) = f_0 \cos(\omega t)\), so that the driven harmonic oscillator equation is:

\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = f_0 \cos(\omega t)\]

Note: \(\omega \neq \omega_0\)

Note that if the driving follows a sine wave, then we have:

\[\ddot{y} + 2 \beta \dot{y} + \omega_0^2 y = f_0 \sin(\omega t)\]

Interesting, \(e^{i \omega t} = \cos(\omega t) + i \sin(\omega t)\), let try to work with \(z(t) = x(t) + i y(t).\)

Clicker Question 24-4#

We found that the square amplitude of the driven harmonic oscillator is:

\[A^2 = \dfrac{f_0^2}{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2}\]

When is the amplitude of the driven oscillator maximized?

  1. When the driving frequency (\(\omega\)) is far from the natural frequency (\(\omega_0\))

  2. When the driving frequency (\(\omega\)) is close to the natural frequency (\(\omega_0\))

  3. When the damping (\(2\beta\)) is weak

  4. When the damping (\(2\beta\)) is strong

  5. Some combination of the above

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