Forced Oscillations and Resonance

The Intuitive Picture

Imagine pushing a child on a swing. If you push at random times, the swing will not go very high. However, if you push exactly when the swing is moving away from you, your small force adds energy efficiently to the system. This is resonance. In a forced oscillator, the external force does work on the mass. When the driving frequency matches the system's natural tendency to oscillate, the force and the velocity of the mass remain in phase for much of the cycle, leading to maximum energy transfer. Damping acts as a brake, limiting the amplitude. Without damping, the amplitude would grow indefinitely at resonance. With damping, the amplitude reaches a finite maximum value determined by the strength of the damping. The phase of the motion relative to the drive also changes: at low frequencies, the mass moves in step with the force; at high frequencies, it moves in opposition; and at resonance, it lags by exactly \(90^\circ\).

The Steady-State Solution

Consider a mass \(m\) attached to a spring of stiffness \(k\) and subject to viscous damping coefficient \(c\). The system is driven by a force \(F(t) = F_0 \cos(\omega t)\). The equation of motion is \(m\ddot{x} + c\dot{x} + kx = F_0 \cos(\omega t)\). We seek a solution of the form \(x(t) = A \cos(\omega t - \phi)\), where \(A\) is the amplitude and \(\phi\) is the phase lag. Substituting this trial solution into the differential equation and matching coefficients of \(\cos(\omega t)\) and \(\sin(\omega t)\) yields the amplitude and phase given in the cited derivation D-061.

Here, \(\omega_0 = \sqrt{k/m}\) is the natural frequency and \(\zeta = \frac{c}{2\sqrt{mk}}\) is the damping ratio. The denominator represents the "impedance" of the system. The term \((\omega_0^2-\omega^2)\) reflects the restoring force and inertia mismatch, while \(2\zeta\omega_0\omega\) reflects the dissipative effect. The phase lag is given by:

Note that the phase is defined such that \(x(t)\) lags the driving force. At \(\omega = 0\), \(\phi = 0\). As \(\omega \to \infty\), \(\phi \to \pi\). The transition occurs smoothly through \(\pi/2\) at \(\omega = \omega_0\).

Resonance Frequency and Amplitude

Resonance is defined as the frequency at which the amplitude \(A(\omega)\) is maximized. To find this, we minimize the denominator of \(A(\omega)\) with respect to \(\omega\). Let \(D(\omega) = (\omega_0^2-\omega^2)^2+(2\zeta\omega_0\omega)^2\). Setting \(\frac{dD}{d\omega} = 0\) and solving for \(\omega\) yields the resonant frequency:

This result is valid only for \(\zeta < 1/\sqrt{2}\). If the damping is too high (\(\zeta \geq 1/\sqrt{2}\)), the amplitude is monotonically decreasing with frequency, and no peak exists. At the resonant frequency, the amplitude reaches its maximum value:

For light damping, the resonant frequency is very close to the natural frequency \(\omega_0\). The peak amplitude is inversely proportional to the damping coefficient \(c\). This explains why lightly damped systems exhibit sharp, high-amplitude resonance peaks, while heavily damped systems show broad, low-amplitude responses.

Phase Behavior and Energy Transfer

The phase lag \(\phi(\omega)\) provides a sensitive indicator of the system's state. At \(\omega = \omega_0\), the phase lag is exactly \(\pi/2\), regardless of the damping ratio. This is a robust feature of linear systems. Physically, this means the velocity \(\dot{x}(t)\) is in phase with the driving force \(F(t)\). Since power delivered is \(P = F \dot{x}\), the average power transfer is maximized when the force and velocity are in phase. At frequencies far from resonance, the phase is either near \(0\) or near \(\pi\), meaning the force and velocity are nearly out of phase, resulting in low average power transfer. The phase curve is a smooth, monotonic function that transitions from \(0\) to \(\pi\) as \(\omega\) increases from \(0\) to \(\infty\). This behavior is universal for all linear, time-invariant second-order systems.