Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Forced Oscillations: Derivation

Description

This lecture covers the derivation of forced oscillations in the context of an harmonic oscillator, focusing on the solution of a second-order linear differential equation with a time-dependent function. The lecture explores the general solution, the method to find the particular solution, and the concept of phase difference between the excitation and the response.

Official source

Related concepts (26)

Oscillation

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.

Quasiperiodicity

Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpredictability that does not lend itself to precise measurement. It is different from the mathematical concept of an almost periodic function, which has increasing regularity over multiple periods. Climate oscillations that appear to follow a regular pattern but which do not have a fixed period are called quasiperiodic.

Vibration

Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The oscillations may be periodic, such as the motion of a pendulum, or random, such as the movement of a tire on a gravel road. Vibration can be desirable: for example, the motion of a tuning fork, the reed in a woodwind instrument or harmonica, a mobile phone, or the cone of a loudspeaker. In many cases, however, vibration is undesirable, wasting energy and creating unwanted sound.

Natural frequency

Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all parts of the system move sinusoidally with that same frequency). If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency.

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .

Related lectures (49)

Oscillations libres, ressort

Explores free undamped oscillations in a harmonic oscillator and their graphical representation over time.

Harmonic Oscillator II: Damped, Forced, Beats, Resonance

Explores stability, resonance, damping effects, and forced oscillations in harmonic systems.

Path Integral Representation of Propagators

Covers the path integral representation of propagators for free particles and harmonic oscillators.

Forced Oscillator: Transient Regime Analysis

Explores forced oscillator analysis, emphasizing resonance and the transition from transient to permanent regimes.

Forced Harmonic Oscillator: Resonance