Damped Oscillations

Definition

If the amplitude of oscillations of an oscillator is reduced by the application of an external force, the oscillator and its motion are said to be damped. Periodic oscillations of gradually decreasing amplitude are called damped harmonic oscillations, and the oscillator is called a damped harmonic oscillator.

For example, the motion of a simple pendulum dies out eventually, as air exerts a viscous force on the pendulum and there may be some friction at the support.

The standard arrangement

Diagram (Fig. 5.13): a block of mass $m$ hangs from a spring and can oscillate vertically. From the block a rod extends downwards to a vane that is submerged in a liquid in a vessel. As the vane moves up and down, the liquid exerts a drag force on it and hence on the whole oscillating system. The mechanical energy of the block–spring system decreases with time, as energy is transferred to thermal energy of the liquid and the vane.

Setting up the equation

The damping force $F_d$ depends on the nature of the surrounding medium and is directly proportional to the speed $v$ of the vane and the block: $$F_d=-bv$$ where $b$ is the damping constant and the negative sign indicates that $F_d$ opposes the velocity.

For spring constant $k$, the force on the block from the spring is $$F_s=-kx$$

Assuming that the gravitational force on the block is negligible compared with $F_d$ and $F_s$, the total force acting on the mass at any time $t$ is $$F=F_d+F_s$$ $$\therefore ma=-bv-kx$$ $$\therefore ma+bv+kx=0$$ $$\therefore m\dfrac{d^2x}{dt^2}+b\dfrac{dx}{dt}+kx=0$$

The solution

The solution of this equation describes the motion of the block under a damping force proportional to the speed. It is found to be of the form $$x=A\,e^{-bt/2m}\cos(\omega't+\phi)$$

• $A\,e^{-bt/2m}$ is the amplitude of the damped harmonic oscillations; it decreases exponentially with time.
• The factor $\cos(\omega't+\phi)$ shows that the motion is still an S.H.M.

The angular frequency of the damped oscillator is $$\omega'=\sqrt{\dfrac{k}{m}-\left(\dfrac{b}{2m}\right)^2}$$ and the period of oscillation is $$T=\dfrac{2\pi}{\omega'}=\dfrac{2\pi}{\sqrt{\dfrac{k}{m}-\left(\dfrac{b}{2m}\right)^2}}$$

Effect of damping

Damping increases the period (it slows down the motion) and decreases the amplitude.

Graph

Graph (Fig. 5.14): displacement plotted against time. The curve is a cosine curve whose successive peaks get smaller and smaller. The peaks on either side lie on the two exponential envelopes $x=+A\,e^{-bt/2m}$ and $x=-A\,e^{-bt/2m}$, which close in on the time axis as $t$ increases, so the oscillation gradually dies away.

Standard results

• Damped harmonic oscillations: periodic oscillations of gradually decreasing amplitude
• $F_d=-bv$; $b$ is the damping constant
• $m\dfrac{d^2x}{dt^2}+b\dfrac{dx}{dt}+kx=0$
• $x=A\,e^{-bt/2m}\cos(\omega't+\phi)$, amplitude $A\,e^{-bt/2m}$
• $\omega'=\sqrt{\dfrac{k}{m}-\left(\dfrac{b}{2m}\right)^2}$, $T=\dfrac{2\pi}{\omega'}$
• Damping increases the period and decreases the amplitude