PHYS2250 Chap.11 Oscillations, Damping and Resonance
Oscillations, Damping and Resonance
Read the physics from each differential-equation term
Simple harmonic motion follows from a linear restoring acceleration and exchanges kinetic with potential energy. Adding viscous damping changes the characteristic roots and separates underdamped, critically damped and overdamped returns.
Periodic driving produces a steady response at the drive frequency, with damping limiting and broadening the resonance peak.
Reasoning route and exam method
Simple harmonic motion occurs when acceleration is proportional to displacement and directed toward equilibrium: x¨ = −ω₀²x.
Its solutions are sinusoidal, with displacement and velocity one quarter-cycle out of phase and acceleration opposite displacement; this relationship belongs in the diagnostic checklist for the chapter. In a spring-mass system ω₀ = √(k/m). Energy alternates between kinetic and spring potential while their sum remains constant in the ideal model; this is a useful checkpoint before any numerical substitution.
The phase diagram is an ellipse in displacement-velocity space, making the constant-energy relation visible; the same result should survive the relevant limiting-case test.
Adding viscous damping produces mx¨ + bx˙ + kx = 0; that distinction determines which quantities may be combined.
The discriminant separates underdamped oscillation, critical damping and overdamped return; this observation fixes the physical interpretation of the algebra. Critical damping is the fastest return without oscillation; the statement supplies a direct check on the final result.
Adding a periodic drive produces a steady response at the driving frequency after transients decay; this condition should remain visible beside the governing equation. Resonance is not infinite in a real damped system; damping limits the amplitude and shifts the peak.
A 0.50 kg mass on a spring of constant 72 N m⁻¹ oscillates with amplitude 0.080 m. The angular frequency is ω₀ = √(72/0.50) = 12.0 rad s⁻¹.
The period is 2π/ω₀ = 0.524 s. Maximum speed occurs at equilibrium and equals ω₀A = 0.960 m s⁻¹. Total energy is ½kA² = ½(72)(0.080²) = 0.230 J. At an endpoint speed is zero, but acceleration magnitude is greatest.
ONE CYCLE Phase cues. At equilibrium: speed maximum, acceleration zero. At an endpoint: speed zero, restoring acceleration maximum.
Over a cycle, average kinetic and potential energies are equal.
The response amplitude peaks near the natural frequency. Lighter damping produces a taller, narrower resonance, while heavier damping lowers and broadens the peak.
Natural frequency belongs to the system; driving frequency belongs to the applied force. Resonance compares them.
What this chapter covers
- 01
simple harmonic motion
- 02
phase and energy
- 03
damped oscillator roots
- 04
forced response
- 05
resonance curves
Worked example · free
Characterise a spring oscillator
- 3Calculate natural angular frequency and period.
- 2Find maximum speed.
- 3Calculate total energy.
Key terms
- Critical Damping
- The fastest non-oscillatory return to equilibrium.
- Resonance
- Enhanced steady response when drive frequency lies near a natural frequency.
- Natural Frequency
- The characteristic frequency of free oscillation set by system inertia and restoring properties.
Oscillations, Damping and Resonance FAQ
What distinguishes natural and driving frequency?
Natural frequency belongs to the unforced system parameters, while driving frequency is imposed by the external periodic force and controls the steady-state response after transients decay.
How does damping alter resonance?
Greater damping lowers and broadens the response peak and shortens the transient, preventing the ideal divergence while changing the phase relation between drive and response.
Where are kinetic and potential energy greatest?
Kinetic energy is greatest at equilibrium; spring potential energy is greatest at the displacement endpoints, and their sum remains constant only in the undamped ideal system.
Exam move
Connect the equation coefficients to restoring, damping and driving effects. Sketch displacement, velocity, acceleration and the phase portrait for one complete cycle. Mark equilibrium crossings and endpoints, then state which of speed, acceleration, kinetic energy and potential energy is extreme at each.
Solve the characteristic equation for examples on both sides of critical damping and connect the roots to the visible time response. Compare resonance curves while changing only the damping coefficient, recording peak height, width and phase behaviour. For a driven case, separate the transient natural response from the steady response at the driving frequency.
Finish by checking the undamped limit and the zero-drive limit, because both should recover earlier models without changing the meaning of the initial conditions. Use the same initial displacement across comparisons so differences in the curves come from damping and driving rather than an unnoticed change in starting state.
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