Mechanical and electrical oscillators

A car rolling over a bump, a building swaying in wind, a machine shaking its foundation, a pedestrian bridge bouncing under footsteps, a radio tuning to a station: all are oscillators, and nearly all are described by the same second-order equation. This chapter applies Lesson 2 to the mass-spring-damper and its electrical twin, the RLC circuit. The goal is engineering fluency: given a system, find its natural frequency and damping ratio, predict how it responds to a disturbance, and predict how it responds to a steady vibration, including at resonance.

These are the fundamentals of vibration engineering, suspension design, earthquake engineering, vibration isolation and filter design, and the vocabulary here (natural frequency, damping ratio, resonance, amplification) is shared across mechanical, civil and electrical engineering.

The mass-spring-damper

A mass m on a spring of stiffness k, with a viscous damper of coefficient c, driven by a force F(t), obeys Newton's second law:

The spring pushes back in proportion to displacement; the damper resists in proportion to velocity. This "single degree of freedom" model is a surprisingly good description of many real systems: a car body on its suspension (the quarter-car model), a water tower swaying, a machine on rubber mounts, the first mode of a building.

Dividing by m puts the equation in the standard form used throughout vibration engineering:

• \(\omega_n\) is the undamped natural frequency, in rad/s. In hertz, \(f_n = \omega_n/2\pi\); the period is \(T_n = 2\pi/\omega_n\).
• \(\zeta\) (zeta) is the damping ratio, the actual damping divided by the critical damping \(c_c = 2\sqrt{km}\).

Two numbers characterize the whole system, which is what makes this form so useful.

Free vibration

With \(F = 0\), the characteristic equation is \(r^2 + 2\zeta\omega_nr + \omega_n^2 = 0\), with roots \(r = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2 - 1}\). The three cases of Lesson 2 become three kinds of physical behavior:

• Underdamped, \(\zeta < 1\): complex roots; the system oscillates at the damped natural frequency \(\omega_d = \omega_n\sqrt{1 - \zeta^2}\) with amplitude decaying as \(e^{-\zeta\omega_nt}\). Almost all structures and most machines are in this range, often with \(\zeta\) of only 0.01 to 0.05.
• Critically damped, \(\zeta = 1\): a repeated root; the fastest return to rest without overshoot.
• Overdamped, \(\zeta > 1\): two real roots; a slow, sluggish return without oscillation.

Measuring damping: the logarithmic decrement

Damping is hard to calculate but easy to measure. Give the system a tap and record the free vibration. Successive peaks decay by a constant ratio, and the natural log of that ratio is the logarithmic decrement:

Forced vibration

Now apply a steady harmonic force \(F(t) = F_0\cos\omega t\), from an unbalanced rotating machine, wind gusts, footsteps or road roughness. After the transient dies away, the steady-state response is a cosine at the forcing frequency, with amplitude X and a phase lag. Writing the frequency ratio \(r = \omega/\omega_n\) and the static deflection \(X_{st} = F_0/k\):

This dynamic amplification factor is the most important curve in vibration engineering.

Three regions matter:

• \(r \ll 1\) (slow forcing): \(X \approx X_{st}\). The system follows the force as if it were static.
• \(r \approx 1\) (resonance): \(X \approx X_{st}/(2\zeta)\). With \(\zeta = 0.02\), the response is 25 times the static deflection. This is why lightly damped structures must not be excited near their natural frequencies.
• \(r \gg 1\) (fast forcing): \(X\) becomes small; the mass cannot keep up. This is the basis of vibration isolation: mount a machine on springs soft enough that its running frequency is well above the mounting's natural frequency.

Resonance in the real world

Two well-known bridges illustrate how seriously engineers take dynamics. When London's Millennium Bridge opened in June 2000, it swayed sideways under crowds. Pedestrians unconsciously adjusted their steps to the sway, synchronizing with the bridge's lateral natural frequency and feeding energy into it. The bridge was closed two days after opening and reopened in 2002 after dampers were installed to raise its damping. The 1940 collapse of the Tacoma Narrows Bridge is often described as resonance, but it was a more complex aeroelastic instability (flutter), in which the wind and the bridge's twisting motion fed each other. Both cases led to permanent changes in how long, light structures are designed and checked for dynamic behavior.

The RLC circuit

A series circuit with an inductor L, a resistor R and a capacitor C, driven by a voltage E(t), obeys Kirchhoff's voltage law for the charge q on the capacitor:

This is the same equation as the mass-spring-damper, with \(L \leftrightarrow m\), \(R \leftrightarrow c\), \(1/C \leftrightarrow k\). So everything above carries over directly:

The analogy is more than a curiosity. Engineers once built electrical circuits to simulate mechanical systems (analog computers), and today the same transfer-function mathematics (Lesson 4) is used to analyze and control both.