Oscillations in Mechanics: Energy, Damping, and Resonance

Simple harmonic motion (SHM)

Oscillatory motion about equilibrium caused by a restoring force proportional to displacement, producing sinusoidal position vs. time in the ideal case.

Kinetic energy (K) in SHM

Energy of motion of the oscillator: K = (1/2)mv^2; it is maximum at equilibrium and zero at turning points (ideal SHM).

Elastic potential energy (U_s)

Spring-stored potential energy relative to equilibrium: U_s = (1/2)kx^2; it is maximum at turning points and zero at equilibrium.

Total mechanical energy (E) in ideal SHM

Sum of kinetic and potential energies that remains constant without nonconservative forces: E = K + U_s.

Amplitude (A)

Maximum displacement from equilibrium in SHM; occurs at turning points x = ±A.

Undamped natural angular frequency (ω0) for mass–spring

The oscillator’s natural angular frequency without damping: ω0 = √(k/m).

Maximum speed in SHM (v_max)

Greatest speed occurs at equilibrium and satisfies v_max = ωA (with ω = √(k/m) for a mass–spring oscillator).

Speed–displacement relation in SHM

Speed at position x (without using time): v = ±ω√(A^2 − x^2), derived from energy conservation.

Turning point (in SHM)

Location where displacement is maximal (x = ±A) and the speed is zero; energy is entirely potential (ideal SHM).

Equilibrium position (in SHM)

Location x = 0 where the restoring force is zero; speed is maximum and spring potential energy is minimum (ideal SHM).

Energy exchange in SHM

In ideal SHM, kinetic energy and potential energy continuously trade off while the total mechanical energy stays constant.

Frequency doubling of energy graphs

Because K ∝ v^2 and U ∝ x^2, both K(t) and U(t) oscillate at twice the frequency of x(t) and never go negative.

Small-angle approximation (pendulum)

For small angular displacements (in radians), sinθ ≈ θ, making a simple pendulum approximately behave like SHM.

Pendulum angular frequency (small angles)

For a simple pendulum of length L (small angles), the SHM angular frequency is ω = √(g/L).

Small-angle pendulum potential energy (quadratic form)

Approximate gravitational potential energy relative to the bottom: U ≈ (1/2)mgLθ^2 (valid only for small angles).

Effective spring constant for a small-angle pendulum

Using x = Lθ, the potential energy becomes U ≈ (1/2)(mg/L)x^2, so k_eff = mg/L.

Damping

Energy loss mechanisms (e.g., friction/air resistance) that remove mechanical energy over time, typically reducing amplitude.

Viscous (linear) damping force

A common damping model where the force is proportional to velocity and opposite motion: F_d = −bv.

Damping rate (β)

Parameter setting exponential decay in a damped mass–spring oscillator: β = b/(2m).

Underdamped motion

Damped oscillation where the system still oscillates but with exponentially decreasing amplitude: x(t) = A0 e^(−βt) cos(ω_d t + φ).

Damped angular frequency (ω_d)

Oscillation frequency in the underdamped case: ω_d = √(ω0^2 − β^2), slightly less than ω0 for light damping.

Critical damping

Boundary case between oscillatory and non-oscillatory return, giving the fastest return to equilibrium without overshoot; occurs when β = ω0.

Critical damping coefficient (b_c)

Damping coefficient that produces critical damping for a mass–spring system: b_c = 2√(km).

Quality factor (Q)

Dimensionless measure of how lightly damped an oscillator is (light damping): Q = ω0/(2β); higher Q means slower energy loss and sharper resonance.

Driven oscillator amplitude response A(ω)

Steady-state amplitude for m x¨ + b x˙ + kx = F0 cos(ωt): A(ω) = F0 / √((k − mω^2)^2 + (bω)^2); peaks near resonance and stays finite when b > 0.