Oscillations

Simple harmonic motion, damped oscillations, forced vibration, and mechanical resonance.

General idea of the topic

An oscillation is a motion that repeats about an equilibrium position. It appears whenever there is a restoring force that tends to return the system to equilibrium and an inertia that makes it overshoot. The basic model is simple harmonic motion, with a force proportional to the displacement.

The block starts from simple harmonic motion and applies it to the mass-spring system and the simple pendulum, adds the damping that dissipates energy, studies the physical pendulum and its advanced version, and ends with forced oscillations and resonance.

What you will truly understand

By the end you will be able to compute period, frequency and energy of simple oscillators, describe how damping changes the motion and explain resonance when an external force acts at the natural frequency.

You will also know when the small-oscillation approximation is valid.

Recommended learning route

  1. Simple Harmonic Motion: equation of motion, energy and phase of simple harmonic motion.
  2. Mass-Spring Oscillator: spring with a mass, period T = 2π√(m/k) and stored energy.
  3. Simple Pendulum: pendulum with small angle, period T = 2π√(l/g) and limits of the approximation.
  4. Damping: viscous friction forces and underdamped, critical and overdamped regimes.
  5. Physical Pendulum: extended body oscillating about an axis, with its moment of inertia.
  6. Advanced Physical Pendulum: physical pendulum with corrections and cases departing from the ideal model.
  7. Forced Oscillations and Resonance: response to an external periodic force, amplitude and resonance.

Physical criterion you must acquire

The essential criterion is to recognize the equation x'' + ω²x = 0 in each system: identifying ω² as the ratio of effective stiffness to effective inertia gives the period directly. With damping, the friction must be compared with the critical value; with an external force, with the natural frequency.

Connection with real problems

  • clocks, pendulums and tuning forks.
  • vehicle suspensions and shock absorbers.
  • buildings and bridges under vibration.
  • circuits and waves.

Final goal of the topic

The final goal is to model any oscillator, compute period, amplitude and energy, and predict its response to damping and external forces.

Topics in this section

  • Simple Harmonic Motion

    Differential equation and analytical solutions of simple harmonic motion in springs and pendulums.

  • Mass-Spring Oscillator

    Linear mass-spring oscillator for modeling period, phase, and energy transfer in classical harmonic motion.

  • Simple Pendulum

    Small-angle oscillation of a point mass. Theory, formulas, worked examples and interactive graphs.

  • Damping

    Underdamped, critically damped, and overdamped regimes in real dissipative oscillators.

  • Physical Pendulum

    Oscillations of a rigid body about a fixed axis using the small-angle approximation.

  • Advanced Physical Pendulum

    Advanced physical pendulum model with inertia, mass, and gravity; includes Kater's reversible pendulum.

  • Forced Oscillations and Resonance

    Periodic steady-state response to harmonic external drives, resonance curves, and Q factor.