Mechanical Vibrations

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.

Conditions and key results

The model assumes a linear spring, viscous damping, one-dimensional motion, and parameters constant over the displacement range. Resonance conclusions depend on damping and the measured output: the undamped displacement amplitude becomes unbounded at exact forcing frequency, whereas damping produces a finite peak.

A reliable strategy

1. Define positive displacement, equilibrium, units, and the parameters $m,c,k,F$.
2. Solve the homogeneous characteristic equation and classify the damping regime.
3. Find a particular solution for the forcing, avoiding resonance conflicts, then add the homogeneous response.
4. Apply initial data and check units, limiting behaviour, and the original force balance.

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.

Common mistakes

Verification and reasonableness

• Substitute $x,x',x''$ into the force equation.
• Check $x(0)$ and $x'(0)$ independently and attach units.
• For free damped motion, verify the energy derivative is $-c(x')^2\le0$.

Practice

1. For $m=1,k=9,c=0$, find $\omega_0$.
2. Classify $m=1,c=4,k=4$.
3. What happens to free motion when $c>0$ and roots have negative real parts?

Further deduction

In undamped forced motion $x''+\omega_0^2x=F_0\cos\omega t$, a nonresonant particular solution has amplitude $F_0/|\omega_0^2-\omega^2|$. At $\omega=\omega_0$, the usual cosine trial duplicates a homogeneous solution and the response contains $t\sin\omega_0t$, whose envelope grows linearly. Damping changes both the peak frequency and amplitude, so the undamped formula must not be carried over unchanged.

For free motion, multiply $mx''+cx'+kx=0$ by $x'$. Then $d[m(x')^2/2+kx^2/2]/dt=-c(x')^2$. With $c>0$, mechanical energy never increases. This identity independently confirms decay but does not specify whether the return oscillates; the characteristic roots supply that distinction between underdamped and non-oscillatory regimes.

Related topics

• Second-Order Linear Differential Equations
• Characteristic Equation
• Undetermined Coefficients