Small Oscillations of a One-Degree-of-Freedom System

Small oscillations occur near a stable equilibrium when deviations are small enough for the equations of motion to be linearized. A one-degree-of-freedom system is described by one generalized coordinate.

Undamped free oscillations

The standard linear equation is:

$$m\ddot x+kx=0.$$

The natural circular frequency is:

$$\omega_n=\sqrt{\frac{k}{m}},$$

and the period is $T=2\pi/\omega_n$.

Free-oscillation response

The solution may be written $x=C_1\cos\omega_nt+C_2\sin\omega_nt$ or $x=A\cos(\omega_nt+\varphi)$. Amplitude and initial phase are determined from the initial conditions.

Viscous damping

With linear resistance $c\dot x$:

$$m\ddot x+c\dot x+kx=0.$$

The type of motion depends on damping relative to its critical value. With light damping, the system undergoes decaying oscillations.

Forced oscillations

For harmonic excitation $F_0\cos\Omega t$:

$$m\ddot x+c\dot x+kx=F_0\cos\Omega t.$$

The steady-state response has excitation frequency $\Omega$, while its amplitude depends on frequency ratio and damping.

Resonance

In an ideal undamped system, harmonic excitation at $\Omega=\omega_n$ produces resonant growth of amplitude. With damping, the amplitude remains finite and the frequency-response maximum shifts depending on damping.

Linearization near equilibrium

For a general system, the coordinate is measured from stable equilibrium and only first-order terms in the small deviation are retained. The result has the form $m_{eq}\ddot q+c_{eq}\dot q+k_{eq}q=Q(t)$.

Example

For $m=2$ kg and $k=50$ N/m without damping, $\omega_n=\sqrt{50/2}=5$ rad/s and $T=2\pi/5\approx1.26$ s.

Common mistakes

• confusing circular frequency in rad/s with ordinary frequency $f=\omega/(2\pi)$ in hertz;
• using a linear small-oscillation model for large deviations without checking validity;
• ignoring damping when estimating resonant amplitude of a real system;
• using a physical mass directly instead of equivalent inertia for a compound mechanism without derivation.