Harmonic oscillator

Simple harmonic motion

• A displacement function $x(t)$ is said to describe simple harmonic motion iff it satisfies the differential equation $\frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=0$.
• $x(t)$ can be written as $x(t)=A\cos (\omega t+\varphi )$ (or $x(t)=A\sin (\omega t+\varphi )$, for a different choice of $\varphi$), i.e. the motion is sinusoidal in time.
  • $\varphi$ is the phase angle (in $\mathrm{rad}$)
  • $A$ is the amplitude (the maximum displacement from the equilibrium)
  • $\omega$ is the natural angular frequency (in $rad/s$)
  • $f=\frac{\omega }{2\pi }$ is the natural frequency (in $\mathrm{Hz}$)
  • $T=\frac{1}{f}$ is the period (in $\mathrm{s}$)
• $v(x)=\omega \sqrt{A^2-x^2}$ (velocity as a function of position)
• $v(t)=\frac{dx(t)}{dt}=-A\omega \sin (\omega t+\phi )$ (velocity as a function of time)
• $v_{\text{max}}=\pm A\omega$ is the maximum velocity

Mass-spring System

$m\ddot{x}+kx=0$

• $m$ is the mass of the oscillating body (in $\mathrm{kg}$)
• $k$ is the spring constant (related to the stiffness of the spring) (in $\mathrm{N}\cdot {\mathrm{m}}^{-1}$)
• $F=-kx$ is the restoring force (Hooke’s Law)
• $\omega =\sqrt{\frac{k}{m}}$
• $E=\frac{1}{2}k{x}^2+\frac{1}{2}m{v}^2$ is the total mechanical energy (in $\mathrm{J}$)
  • $\frac{1}{2}m{v}^2$ is the kinetic energy (in $\mathrm{J}$) (it’s total energy in the moment of equilibrium, $x=0$)
  • $\frac{1}{2}k{x}^2$ is the elastic potential energy (in $\mathrm{J}$) (it’s total energy in the monent of turning point, $v=0$)

Simple pendulum

• The oscillating body is the pendulum bob
• $\omega =\sqrt{\frac{g}{L}}$
• $L$ is the length of the pendulum
• $g$ is the acceleration due to gravity
• $\frac{mg}{L}$

Damped harmonic oscillator

$m\ddot{x}+b\dot{x}+kx=0$