Mechanical Vibrations

A rigorous model of free and forced mass–spring motion, damping regimes, resonance, and physical interpretation.

Amplitude, period, and phase shift describe a repeating graph. A wave repeats across the axis. A vertical measure marks amplitude, a horizontal bracket marks one period, and a second faint wave shows how a phase shift moves the cycle.

Precise definition

A linear mass–spring–damper model is $mx''+cx'+kx=F(t)$, with mass $m>0$, damping coefficient $c\ge0$, spring constant $k>0$, displacement $x(t)$, and applied force $F$. The unforced characteristic equation is $mr^2+cr+k=0$.

Notation and mathematical language

The natural frequency without damping is $\omega_0=\sqrt{k/m}$. The damping ratio is $\zeta=c/(2\sqrt{mk})$. Underdamping has $\zeta<1$, critical damping $\zeta=1$, and overdamping $\zeta>1$. Initial displacement and velocity specify the motion.

Conceptual picture

The spring stores potential energy $kx^2/2$, the mass stores kinetic energy $m(x')^2/2$, and damping removes energy at rate $c(x')^2$. Complex roots produce decaying oscillations; real roots produce non-oscillatory return under the linear model.

Fully worked example

Interpretation and application

Suspensions, buildings, instruments, and sensors use vibration models. A computed resonance frequency is conditional on linearity and parameter estimates; structural safety requires uncertainty, multiple modes, nonlinearities, and validated engineering standards beyond this single-degree model.