Driven damped harmonic oscillators: resonance, phase lag, and real‑world demonstrations (MIT OCW)

MIT OpenCourseWare presents a lecture on driven damped harmonic oscillators. The instructor derives the steady‑state response to a harmonic driving torque, explains how the amplitude and phase lag depend on the drive frequency, and highlights transient decay through the homogeneous solution. Real‑world demonstrations including a pendulum, a mass on a cart, and resonance in a glass are discussed to connect the math with physical intuition. This post summarizes the key ideas and results from the lecture, which also introduces the important distinction between natural and driven frequencies and sets the stage for building toward many‑body oscillatory systems.

Overview of Driven Damped Oscillators

The lecture analyzes a single harmonic oscillator with damping and an external harmonic drive. The driving torque is tau_drive = D0 cos(omega_D t), and the equation of motion is theta'' + gamma theta' + omega_0^2 theta = F0 cos(omega_D t), with gamma encoding drag and omega_0 the natural angular frequency without drag. By moving to a complex representation Z(t) and proposing a steady state Z(t) = A e^{i(omega_D t - delta)}, the instructor derives the steady‑state amplitude A and phase lag delta as functions of drive frequency omega_D. The key results are A = F0 / sqrt[(omega_0^2 - omega_D^2)^2 + (gamma^2 omega_D^2)] and tan(delta) = (gamma omega_D)/(omega_0^2 - omega_D^2). The approach also emphasizes the role of the homogeneous solution, which decays in time and represents transient behavior before the system settles into the steady state.

Beyond the mathematics, the talk discusses physical interpretations, limiting cases, and several demonstrations. The slow drive limit (omega_D -> 0) yields a steady displacement proportional to the drive amplitude, while the fast drive limit (omega_D -> infinity) decouples and the response becomes small and out of phase with the drive. At drive frequencies near the natural frequency, resonance occurs, producing large amplitudes when damping is small. The lecture then connects these ideas to real experiments such as a pendulum driven by a moving support, a mass on a cart with a driving mechanism, and a glass resonance demonstration at the end, illustrating how resonance manifests in everyday life and in engineered systems.

Key Takeaways

• Steady‑state response dominates after transients die away.
• Amplitude and phase lag depend on drive frequency, damping, and natural frequency.
• Resonance occurs near the natural frequency and is strongly affected by the damping parameter Q = omega_0/gamma.

These concepts pave the way toward multiple coupled oscillators and wave phenomena, which the course will explore in subsequent sessions.