Free vibration occurs when a system oscillates due to an initial disturbance without external forces. Single degree of freedom systems are fundamental models used to analyse vibration behaviour in engineering.
Free vibration occurs when a system oscillates after an initial disturbance without external forces. In SDOF systems, motion is governed by mass, stiffness and damping characteristics.
This page explains simple harmonic motion, natural frequency, damping, and the behaviour of underdamped, critically damped, and overdamped systems.
Symbols.....
Simple Harmonic Motion.....
Natural Frequency.....
Damped Vibrations.....
Critical Damping.....
Underdamped oscillations.....
Overdamped oscillations.....
Logarithmic Decrement.....
Angular Vibrations
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x = displacement (m) dx/dt = velocity (m/s) d2x/dt2 = acceleration (m/s2) θ = angular displacement (rad) dθ/dt = angular velocity (rad/s) d2θ/dt2 = angular acceleration (rad/s²) m = mass (kg) |
I = moment of inertia (kg·m²) k = spring stiffness (N/m) c = damping coefficient (Ns/m) cc = critical damping coefficient ζ = damping ratio = c / (2mωn) ω = angular velocity (rad/s) ωn = natural frequency (rad/s) |
Simple harmonic motion occurs when the restoring force is proportional to displacement and directed towards equilibrium.
x(t) = A cos(ωt + φ)
The acceleration is always directed towards the equilibrium position and is proportional to displacement.
t = 2π / ω
n = ω / (2π)
The natural frequency is the frequency at which a system oscillates when no damping is present.
ωn = √(k / m)
n = (1 / 2π) √(k / m)
The equation of motion for an undamped system is:
m (d2x / dt2) + kx = 0
Damping introduces a force proportional to velocity, reducing oscillation amplitude over time.
m (d2x / dt2) + c (dx / dt) + kx = 0
Standard form:
(d2x / dt2) + 2ζωn(dx/dt) + ωn2x = 0
Critical damping is the minimum damping required to prevent oscillation and return the system to equilibrium quickly.
cc = 2√(km)
When ζ < 1, the system oscillates with gradually decreasing amplitude.
ωd = ωn √(1 − ζ2)
When ζ > 1, the system returns to equilibrium without oscillating, but more slowly than critical damping.
The logarithmic decrement measures how quickly vibration amplitude decreases.
δ = ln(x1 / x2)
ζ = δ / √(4π2 + δ2)
Angular vibration systems involve rotational motion and behave similarly to linear systems.
I (d2θ / dt2) + c (dθ/dt) + kθ = 0
Free vibration occurs when a system oscillates after an initial disturbance without any external force acting on it.
An SDOF system has one independent motion coordinate, typically represented by a mass-spring-damper model.
Natural frequency is the frequency at which a system vibrates when no damping or external forces are present.
The damping ratio measures how quickly oscillations decay and determines whether a system is underdamped, critically damped or overdamped.
Links on Free Vibrations
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