Oscillations

17.1 Simple harmonic oscillations

Terms

The defining equation

Simple harmonic motion (SHM) occurs when the acceleration is proportional to the displacement from a fixed point, and is always in the opposite direction, so directed towards that point:

$a = -\omega^2 x$

The minus sign is what makes it SHM: the acceleration always points back towards the equilibrium. A graph of $a$ against $x$ is a straight line through the origin with gradient $-\omega^2$.

Examples include a mass on a spring and a simple pendulum at small angles.

Solutions

If timing starts at the equilibrium position:

$x = x_0\sin\omega t \qquad v = v_0\cos\omega t$

where $v_0 = \omega x_0$ is the maximum speed. If timing starts at the maximum displacement, swap to $x = x_0\cos\omega t$.

The speed at any displacement is:

$v = \pm\omega\sqrt{x_0^2 - x^2}$

Graphs

• Velocity is the gradient of displacement against time, and it leads displacement by a quarter of a cycle ($\pi/2$).
• Acceleration is the gradient of velocity against time, and it is exactly opposite to the displacement ($\pi$ out of phase).

17.2 Energy in simple harmonic motion

During SHM, energy continually changes between kinetic and potential, while the total stays constant, if there is no damping.

• At equilibrium: all kinetic.
• At the extremes: all potential.

$E = \tfrac{1}{2}m\omega^2 x_0^2$

This comes from the maximum kinetic energy, $\tfrac{1}{2}mv_0^2$, with $v_0 = \omega x_0$. So the energy is proportional to the amplitude squared.

The kinetic energy at displacement $x$ is $\tfrac{1}{2}m\omega^2(x_0^2 - x^2)$, and the potential energy is $\tfrac{1}{2}m\omega^2x^2$.

17.3 Damped and forced oscillations, resonance

Damping

A resistive force acting on an oscillating system causes damping: it removes energy, so the amplitude falls over time.

Car suspensions are close to critically damped, so that the car settles quickly without bouncing.

Forced oscillations and resonance

A free oscillation happens at the system's natural frequency. In a forced oscillation, a periodic driving force makes the system oscillate at the driver's frequency.

Resonance is a maximum amplitude of oscillation, occurring when the driving frequency equals the system's natural frequency. Energy is then transferred from the driver most efficiently.

More damping makes the peak lower and broader, and it moves the peak slightly to a lower frequency.