Damped harmonic motion

An oscillator undergoing damped harmonic motion is one, which, unlike a system undergoing simple harmonic motion, has external forces which slow the system down.

Damped harmonic motion

The damping force can come in many forms, although the most common is one which is proportional to the velocity of the oscillator. This creates a differential equation in the form

$ma+cv+kx=0$

$mx''+cx'+kx=0$

with a characteristic equation

$mr^{2}+cr+k=0$

Depending on the value of the discriminant (c² - 4mk), the characteristic equation can have two real, one real, or two complex solutions. These states are known as overdamping, critical damping, and underdamping respectively.

Underdamping

If the discriminant is negative, the solution will take the form

$x(t)=Ae^{-{\tfrac {c}{2m}}t}\cos({\tfrac {\sqrt {4mk-c^{2}}}{2m}}t-\varphi )=Ae^{-\gamma t}\cos(\omega _{1}t-\varphi ),\quad \omega _{1}={\sqrt {\omega _{0}^{\,2}-\gamma _{0}^{\,2}}}$

Underdamped systems will oscillate but the amplitude of the oscillations approaches zero with time.

Critical damping

Critically damped systems approach zero in the fastest possible time without oscillating. They are important in many engineering applications, as most shock absorbers are designed to be critically damped. They will follow the equation

$x(t)=(C_{1}+C_{2}t)e^{-{\frac {c}{2m}}t}=(C_{1}+C_{2}t)e^{-\omega _{0}t}$

C₁ and C₂ are determined by the initial conditions of the system.

$C_{1}=x(0),\quad C_{2}=v(0)+\omega _{0}x(0)$

Overdamping

Overdamped systems do not oscillate, but take more time to approach zero due to excessive damping. They follow the equation

$x(t)=C_{1}e^{{\tfrac {-c+{\sqrt {c^{2}-4mk}}}{2m}}t}+C_{2}e^{{\tfrac {-c-{\sqrt {c^{2}-4mk}}}{2m}}t}$

See also

• Simple harmonic motion
• Driven harmonic motion
• Ordinary differential equation on Math Wiki