Mechanical and Electrical Vibrations

The constant-coefficient second-order equation models oscillation. A mass on a spring, a pendulum at small amplitude, the current in a radio tuning circuit, and the sway of a bridge all obey the same equation. Newton's law for a spring-mass-damper produces it; the discriminant of its characteristic equation splits free motion into the undamped, underdamped, critically damped, and overdamped regimes; a periodic driving force exposes resonance; and a series RLC circuit satisfies an identical equation, so every mechanical result transfers to circuits by relabeling the constants.

The spring-mass-damper equation

Consider a mass $m$ on a spring, displaced $u(t)$ from equilibrium (positive downward). Four forces act on it.

• Gravity $w=mg$, downward and constant.
• Spring force $F_s=-k(L+u)$ by Hooke's law, where $L$ is the static stretch and $k>0$ the spring constant; it always restores toward the natural length.
• Damping force $F_d=-\gamma u^{\prime }$, opposing the velocity, with damping constant $\gamma >0$ (viscous damping from a dashpot, air resistance, or internal friction).
• External force $F(t)$, an applied or driving force.

At equilibrium the mass hangs at rest, so $mg-kL=0$. Newton's law $m{u}^{\prime \prime }=w+F_s+F_d+F$ then simplifies, since the constant gravity and static-stretch terms cancel, to the equation of motion

a second-order linear constant-coefficient equation, with initial conditions $u(0)=u_0$ (initial position) and $u^{\prime }(0)=v_0$ (initial velocity).

Free undamped vibration

With no drive and no damping ($F=0$, $\gamma =0$), equation (M) is $m{u}^{\prime \prime }+ku=0$. The characteristic roots are purely imaginary $r=\pm i{\omega }_0$, where

is the natural frequency. The general solution is $u=A\cos ({\omega }_{0}t)+B\sin ({\omega }_{0}t)$, which can be written as a single shifted cosine

Only the amplitude and phase depend on initial conditions; the frequency ${\omega }_0$ is set by the physical constants $k$ and $m$ alone. A stiffer spring or a lighter mass oscillates faster.

Free damped vibration

Restore damping ($\gamma >0$, still $F=0$): $m{u}^{\prime \prime }+\gamma u^{\prime }+ku=0$, with characteristic roots

Because $\gamma ,k,m>0$, both roots have negative real part, so every solution decays to zero: damping dissipates the energy and the motion dies out. The discriminant ${\gamma }^2-4km$ splits the behavior into three regimes.

• Underdamped (${\gamma }^2<4km$): complex roots, so $u=R{e}^{-\gamma t/(2m)}\cos (\mu t-\delta )$ with $\mu =\sqrt{4km-{\gamma }^2}/(2m)$. The motion oscillates inside a decaying envelope $\pm R{e}^{-\gamma t/(2m)}$.
• Critically damped (${\gamma }^2=4km$): a repeated real root, so $u=(A+Bt)e^{-\gamma t/(2m)}$. The fastest return to equilibrium without oscillating.
• Overdamped (${\gamma }^2>4km$): two distinct negative real roots, so $u=A{e}^{r_{1}t}+B{e}^{r_{2}t}$. The mass creeps back to equilibrium without oscillating, more slowly than critical.

In the underdamped case the motion is not periodic (the envelope shrinks), but $\mu$ still sets the spacing of the peaks. It is called the quasi-frequency and $T_d=2\pi /\mu$ the quasi-period. Comparing to the undamped frequency,

for small damping, so light damping lowers the frequency only slightly while still forcing the amplitude to decay. The controlling quantity is the dimensionless ratio ${\gamma }^2/(4km)$, not $\gamma$ alone; whether damping is large or small is judged against $4km$.

Forced vibration and resonance

Now drive the system with a periodic force $F(t)=F_0\cos (\omega t)$:

By the nonhomogeneous structure, the solution is $u=u_c(t)+U(t)$. Since the roots of the damped homogeneous equation have negative real part, $u_c\to 0$: it is the transient, present only long enough to satisfy initial conditions. The particular solution $U(t)=R\cos (\omega t-\delta )$ oscillates at the driving frequency $\omega$ and persists; it is the steady-state or forced response.

Substituting $U=R\cos (\omega t-\delta )$ into (F) and solving for the amplitude gives

The amplitude depends strongly on how close the drive frequency $\omega$ is to the natural frequency ${\omega }_0$. At low frequency ($\omega \to 0$) it approaches the static deflection $F_0/k$; at high frequency ($\omega \to \infty$) it falls to zero; in between it peaks. Setting $\mathbf{d}R/\mathbf{d}\omega =0$ locates the peak at

slightly below ${\omega }_0$, with maximum amplitude

for small $\gamma$. As the damping shrinks, $R_{\max }$ grows without bound: a lightly damped system driven near its natural frequency responds enormously to a small force. This is resonance.

Resonance is a design concern in both directions. In structures (bridges, buildings, machinery) it must be avoided or damped, because a modest periodic input at the wrong frequency can drive destructive amplitudes. In instruments (radio tuners, seismographs) it is exploited, using a sharp resonant peak to select or amplify a weak signal at a chosen frequency.

The transient-plus-steady-state split is visible in any driven solution: an initial stretch where the decaying $u_c$ distorts the motion, settling into a clean oscillation at the driving frequency.

Beats

With no damping, a drive at a frequency $\omega$ near but not equal to ${\omega }_0$ produces neither a decaying transient nor unbounded resonance, but beats: a fast oscillation whose amplitude is itself slowly modulated.

The electrical analogy

A series circuit with inductance $L$, resistance $R$, capacitance $C$, and impressed voltage $E(t)$ obeys Kirchhoff's voltage law: the applied voltage equals the sum of the drops across the three elements. With charge $Q$ on the capacitor and current $I=Q^{\prime }$, the drops are $RI$ (resistor), $Q/C$ (capacitor), and $L{I}^{\prime }$ (inductor), giving

Equation (E) has exactly the form of the mechanical equation (M). Every result above transfers by matching the roles term by term.

The correspondence is exact: the two are the same initial value problem with relabeled constants. Solving the second-order constant-coefficient equation once yields the behavior of every system it models: mechanical oscillators, electric circuits, and, with the same three damping regimes and the same resonance peak, many others.¹²³