Day 25 - Resonance#

Announcements#

• Midterm 1 is still being graded

• Homework 6 is due Friday

• Homework 7 is posted, due next Friday

• No office hours today

Seminars this week#

Most of MSU folks are at APS Global Physics Summit

Reminders#

We started to solve the forced harmonic oscillator equation:

We examined the case of a sinusoidal driving force:

There’s a complimentary case where the driving force is a sine wave:

Reminders#

We combined the two equations into a complex equation using these identities:

The resulting equation is:

Notice that there’s a homogeneous part ($z_h$) and a particular part ($z_p$).

Reminders#

The homogeneous part is the solution we’ve found before with the general solution:

where $r=-\beta \pm i\sqrt{{\omega }_0^2-{\beta }^2}$. In the case of a weakly damped oscillator (${\beta }^2<{\omega }_0^2$), we have:

These solutions die out as $t\to \mathrm{\infty }$. They are called transient solutions.

Solving the particular part#

The particular part is the solution to the driven harmonic oscillator equation:

Assume a sinusoidal solution (frequency, $\omega$) of the form: $$z_p(t)=C{e}^{i\omega t}$$

where $C$ is a complex number. Then, we have:

Amplitude of the particular solution#

We want to convert this to polar form:

where $A$ and $\delta$ are real numbers. We use the complex form to compute the magnitude of the amplitude:

Clicker Question 24-5#

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

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

Finding the phase#

With,

then we can compare the complex forms:

Both $f_0$ and $A$ are real numbers, so the phase $\delta$ is the same phase as the complex number:

The Particular Solution#

Let’s return to the particular solution:

So we get solutions to both driven oscillators:

These are the steady-state solutions.

They persist as $t\to \mathrm{\infty }$ and oscillate at the driving frequency $\omega$.

The Full Solution#

Here, $x_h(t)$ is the transient solution and $x_p(t)$ is the steady-state solution.

For weakly damped oscillators, the transient solution can be written in the form:

where $A_{tr}$ and ${\delta }_{tr}$ are real numbers and are the amplitude and phase of the transient solution. Both are determined by the initial conditions.

where $A$ and $\delta$ are real numbers and are the amplitude and phase of the steady-state solution.

The Full Solution#

The transient plus the steady-state solution is the full solution:

As $t\to \mathrm{\infty }$, the transient solution dies out and the steady-state solution persists.

where

Resonance#

The amplitude of the steady-state solution is:

We change ${\omega }_0$ and observe how the amplitude changes.

Achieving resonance#

The denominator of the equation controls the amplitude:

Case 1: Tune ${\omega }_0$ to be close to $\omega$. Car Radio tuning

With ${\omega }_0=\omega$, the amplitude is:

Achieving resonance#

The denominator of the equation controls the amplitude:

Case 2: Tune $\omega$ to be close to ${\omega }_0$. Pushing a swing

Find the $\omega$ that maximizes the amplitude by taking the derivative with respect to $\omega$: