Damped and Forced Oscillations

This lesson covers forced oscillations and the phenomenon of resonance. When a periodic driving force acts on a damped oscillator, the steady-state response depends critically on the driving frequency.

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1 The Driven Oscillator Equation

Consider a damped oscillator subject to an external periodic driving force $F_0\cos(\Omega t)$F0​cos(Ωt):

$$m\ddot{x} + b\dot{x} + kx = F_0\cos(\Omega t)$$mx¨+bx˙+kx=F0​cos(Ωt)

or equivalently,

$$\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = \frac{F_0}{m}\cos(\Omega t)$$x¨+2γx˙+ω02​x=mF0​​cos(Ωt)

where $\Omega$Ω is the driving angular frequency and $\omega_0 = \sqrt{k/m}$ω0​=k/m​ is the natural frequency.

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2 Steady-State Solution

After initial transients die out (due to damping), the system settles into a steady-state oscillation at the driving frequency $\Omega$Ω:

$$x = A(\Omega)\cos(\Omega t - \delta)$$x=A(Ω)cos(Ωt−δ)

where:

Amplitude:

$$A(\Omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \Omega^2)^2 + (2\gamma\Omega)^2}}$$A(Ω)=(ω02​−Ω2)2+(2γΩ)2​F0​/m​

Phase lag:

$$\tan\delta = \frac{2\gamma\Omega}{\omega_0^2 - \Omega^2}$$tanδ=ω02​−Ω22γΩ​

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3 Resonance

Resonance occurs when the amplitude of the steady-state response is maximised. The resonance condition depends on the level of damping.

For the displacement amplitude:

The maximum of $A(\Omega)$A(Ω) occurs at:

$$\Omega_{\text{res}} = \sqrt{\omega_0^2 - 2\gamma^2}$$Ωres​=ω02​−2γ2​

(provided $\omega_0^2 > 2\gamma^2$ω02​>2γ2; otherwise the maximum is at $\Omega = 0$Ω=0).