Forced Oscillation

Steady response of an oscillator driven by an external periodic force.

Conceptual context

A forced oscillation occurs when an oscillator is not left to evolve freely but receives an external periodic excitation. The system no longer chooses the final rhythm by itself: in steady state it responds at the frequency imposed by the source. The relevant question is not only whether it vibrates, but how much it vibrates and how late it responds relative to the applied force.

This leaf studies the permanent response of a linear oscillator with mass $m$, stiffness $k$, and damping $b$. The source provides an amplitude $F_0$ and an angular frequency $\omega_f$. The steady output is summarized by an amplitude $X$, a phase lag $\delta$, and a time position $x$. The derivatives $\dot{x}$ and $\ddot{x}$ appear because the full equation includes velocity and acceleration.

🟢 Essential level

The essential idea is that an external force can impose a rhythm on the system. If an object attached to a spring is pushed periodically, its final response does not necessarily keep its free natural frequency; it synchronizes with the driving force. What changes as the driving rhythm changes is the amplitude obtained and the lag of the response.

At low frequency, the system often behaves almost as if the force were applied slowly: stiffness dominates. At high frequency, inertia dominates and the system cannot follow easily. Between those behaviors, a large-response region may appear, which is studied more specifically in resonance. Here the goal is to construct the general steady response.

Phase lag is as important as amplitude. Two systems may vibrate with similar amplitudes and still respond with different delays. That delay indicates how stiffness, inertia, and dissipation share the response during a cycle.

A quick self-check before calculating: predict whether stiffness or inertia should dominate the response at the given driving frequency, then verify that the computed amplitude and phase lag agree with that expectation.

🔵 Formal level

Basic formulas

The dynamic model is written as a balance among inertia, dissipation, elasticity, and external force:

Forced oscillator equation

Once the free transient has decayed, the steady-state amplitude is computed with:

Steady-state amplitude

This relation does not merely say that more force produces more motion. The denominator expresses the system dynamic opposition. The elastic term $k$ competes with the inertial term involving $m$ and the imposed frequency, while $b$ introduces velocity-proportional dissipation.

Real student questions

A common question is whether the response uses the natural frequency. In forced steady state, the response uses the external force frequency. Another question is why phase lag appears. It appears because inertia, dissipation, and stiffness do not respond at the same time during the cycle.

Students also ask whether the amplitude formula includes the transient. It does not: it describes the permanent part after free motion has faded.

Final synthesis

Forced oscillation teaches how to move from an external periodic source to a steady response with amplitude, phase, and time signal. The final competence is deciding how imposed frequency modifies response without confusing this general analysis with the special case of resonance.