Vibrations: Everything Is a Spring and a Mass

Pluck a guitar string and it rings at one pitch, no matter how hard you pluck it. Push a child on a swing in time with the swing and a gentle shove builds a big arc. Drive over a washboard road at the wrong speed and the whole car shudders. Different things, one idea: almost every object has a frequency it prefers to move at, and it responds wildly to being pushed at that rhythm.

A vibrations graduate sees this everywhere. A phone buzzing on a table, a washing machine walking across the floor, a skyscraper swaying in a typhoon, a tall bridge that wobbles under a crowd: each is a mass, a spring and some friction, and each has a number that says how badly it will misbehave.

Everything is a spring and a mass

Take anything that springs back when displaced: a diving board, a building, a guitar string, a car on its tyres. Close up, it behaves like a mass on a spring. Newton's law says that the spring pushes back in proportion to how far you stretch it, so the acceleration is proportional to the displacement and points the other way. That gives a motion that repeats, at a frequency that depends on only two things.

Stiffer means faster; heavier means slower. Divide by 2π to get hertz. Worked example: a 1 kg mass on a 100 N/m spring has ω = 10 rad/s, so f = 1.59 Hz, one swing every 0.63 s. A car body of about 400 kg per wheel on a 20,000 N/m spring comes to 1.1 Hz, roughly a walking pace, which is why a car feels like it is floating. Four times the stiffness, or a quarter of the mass, doubles the frequency, because of the square root.

The size of the pluck doesn't appear in the formula. The frequency is a property of the object itself, which is what makes it so useful.

Damping: where the energy goes

Real things stop ringing because friction, air and bending metal turn motion into heat. That is damping. Engineers measure it with a single dimensionless number, the damping ratio ζ (zeta): the actual damping divided by the critical damping, the exact amount that just prevents any overshoot.

• ζ below 1 (underdamped): it overshoots and rings, each swing smaller. A guitar string has tiny ζ and rings for seconds; a car's shock absorbers are set well below 1 so the wheels still follow the road.
• ζ = 1 (critical): the quickest return to rest with no overshoot. A good door closer aims here.
• ζ above 1 (overdamped): no overshoot, but a slow, syrupy creep. For large ζ it behaves like a first-order system, a single exponential with a time constant, which is exactly what the next subject uses to describe cruise control.

For ζ below 1 each swing is smaller than the last by a fixed factor. At ζ = 0.05 each swing is 73% of the one before, so it rings for dozens of cycles. At ζ = 0.2 it is 28%: three swings and it's gone.

Forced vibration and resonance

Now push the mass back and forth at a steady frequency. At first the mass just follows the push. If you push very fast, it barely moves, because there isn't time. But push at the natural frequency and each push arrives just as the mass is moving the right way, so the energy adds up cycle after cycle until damping drains energy as fast as you put it in. That is resonance.

The amplitude, as a multiple of what a steady push of the same size would do, is the famous curve:

At r = 1 the first bracket vanishes and the amplification is 1/(2ζ). A lightly damped structure with ζ = 0.02 moves 25 times further than a static push; with ζ = 0.5 barely more than a static push. So damping, not the size of the push, decides how bad resonance is. The motion also shifts from moving with the push (below resonance) to moving against it (above), with a 90° lag exactly at resonance.

Isolation: soft mounts and the √2 rule

Sometimes the push is something you want to keep away from a structure: an engine shaking a car body, a washing machine's drum shaking a floor, a seismometer ignoring the ground. The trick is to put a spring between them. The fraction of shaking that gets through is called transmissibility, and it has a surprising shape: below r = √2 (about 1.41) the mount makes things worse, and only above it does isolation begin. Above √2, more damping lets more through at high frequency.

So the rule: make the mount's natural frequency well below the frequency you want to block. To cut a 30 Hz engine's force to 10%, with light damping you need r ≈ 3.3, so a mount with a natural frequency near 9 Hz. In practice that means a soft spring that sags a few millimetres under the load. The price is that every time the machine starts or stops it sweeps through the mount's natural frequency, so you want some damping to survive the crossing.

Modes: a structure is a choir of springs

A single mass has one natural frequency. A string, beam, plate or aircraft wing is a continuous chain of masses and springs, so it has many: a mode is one pattern of motion in which every part moves in step at one shared frequency. The lowest mode is the whole string bowing out. The second has a still point in the middle (a node), the third has two, and so on. Pluck the string and you excite several at once; that blend is the timbre of the instrument.

For an ideal string the modes are whole-number multiples of the lowest, a harmonic series. A guitar's low E at 82.41 Hz also rings at 164.8, 247.2 and so on. A beam fixed at one end (a diving board, a wing, a skyscraper) is different: its second mode is 6.27 times the first and its third 17.5 times, so the overtones are not neat multiples. Engineers find these modes first, then keep forces away from them.

When it goes wrong: Tacoma Narrows and the wobbly bridge

Tacoma Narrows (1940). The usual textbook story says the wind shook the bridge at its natural frequency and resonance tore it down. That version is wrong. The bridge opened on 1 July 1940 and fell on 7 November, in a wind of roughly 65 km/h (reports range from 35 to 42 mph). The deck was slender and flexible in twisting, and on the day it went from bouncing to twisting about once every four seconds. In 1991 K. Y. Billah and R. H. Scanlan, in the American Journal of Physics, argued that the cause was aeroelastic flutter: the twisting itself changed the wind's forces on the deck so that they fed energy into the twist. This is a self-excited motion, with no steady external beat to resonate with. Once the wind passed a threshold speed, the aerodynamic effect acted like negative damping. Some details are still argued over, but "simple resonance" is not the accepted account.

London Millennium Bridge (2000). The footbridge opened on 10 June 2000 and was closed two days later after it swayed sideways under crowds. Walking pushes sideways on the deck about once per two steps. If the deck sways, people adjust their stride, and on this bridge their combined pushes ended up in phase with the sway, adding energy instead of removing it. The effect only began once the crowd passed a critical size. The Arup engineers' study showed that any bridge with a lateral frequency below about 1.3 Hz could suffer the same if enough people were on it. The fix was not to stiffen it but to add dampers (viscous and tuned mass dampers) and it reopened in early 2002. Engineers still debate how much of the effect is people falling into step with each other and how much is each person balancing against the sway.

Both stories share one mechanism: something that pushes in a way that reduces the effective damping. Once the damping reaches zero, any disturbance grows. Try it.

Tuned mass dampers: a second spring that eats the vibration

You can fight resonance by adding damping, or by adding a second mass-spring tuned to the first. A tuned mass damper is a small mass on a spring and damper, attached to the structure and tuned to its natural frequency. When the structure resonates, the small mass swings against it and absorbs energy. The best tuning, worked out by J. P. Den Hartog and still the starting point, is simple: for a mass ratio μ, tune the damper to 1/(1 + μ) of the building's frequency, with damping ratio √(3μ/(8(1+μ)³)).

Taipei 101 is the famous case: a 660-tonne steel sphere about 5.5 m across (728 short tons) hangs like a pendulum in the upper floors, and in Typhoon Soudelor in 2015 it swung about a metre. The makers claim it can reduce sway by up to 40% (a company figure). In the simulator below, a damper that weighs only 2% of the structure cuts the peak response from 50× to about 9×.

What a graduate just sees

• Everything with mass and springiness has a natural frequency, √(k/m), set by the object and not by how hard it is hit.
• Stiffer goes up, heavier goes down, and by the square root: four times the stiffness doubles the frequency.
• Damping ratio, not the force, decides how bad resonance is: the peak is about 1/(2ζ).
• Below √2 times its natural frequency a soft mount amplifies; above it, it protects.
• Any structure has a whole family of modes; engineers keep forces away from the low ones.
• Tacoma Narrows was flutter, a self-excited instability, not simple resonance.
• When something fails by growing vibration, look for what has made the damping effectively negative.
• If you can't remove the push, add damping (or a tuned damper) rather than making it stiffer.

Where it shows up next

• Control Systems (34/20): the equation of a mass with damping and a spring is the standard second-order system, with the same ζ and ω_n. Feedback is how you add damping by software: a flight controller does for a drone what a viscous damper does for a bridge.
• Machine Design (34/19): shafts and gearboxes have critical speeds at which they whip; turbines and rotors are balanced so their push stays small.
• Propulsion (34/21) and Fluid Mechanics (34/15): compressor and turbine blades and wings can flutter, the Tacoma mechanism at jet speeds.
• The real world: earthquake-resistant towers, car and aircraft ride, hard-drive heads, noise and comfort engineering, musical instrument design.
• Computer Science: analysing vibrations is Fourier analysis (see Fourier Transform, 12/12), and software simulating springs underlies physics engines, animation and robotics.

If you want to go deeper

• 📘 Textbook: Singiresu S. Rao, Mechanical Vibrations (Pearson): the most widely assigned undergraduate text, with many worked examples.
• 📘 Textbook: Daniel J. Inman, Engineering Vibration (Pearson): clearer on modelling and on MATLAB use. J. P. Den Hartog's Mechanical Vibrations (Dover) is the short classic that gave us the tuned damper.
• 📘 Gentler: The Feynman Lectures on Physics, Vol. I, ch. 21 "The Harmonic Oscillator" and ch. 23 "Resonance", free online.
• ▶️ Free course: MIT OpenCourseWare 2.003SC Engineering Dynamics (Fall 2011), an introduction to vibrations and dynamics of lumped mechanical models, with recitations on frequency response.
• 🌐 Easier first read in this library: Waves and Sound (13/13) and Fourier Transform (12/12).