Modal analysis

The formula

Every linear structure obeys one matrix equation of motion:

M·ẍ + C·ẋ + K·x = f(t)

Reading: mass times acceleration, plus damping times velocity, plus stiffness times displacement, equals the applied force. M, C, and K are square matrices — one row and column per degree of freedom. For a finite element model that's thousands to millions of DOF.

Drop the damping and the forcing to find the free-vibration behavior, assume a sinusoidal motion x = φ·sin(ωt), and the equation collapses to a generalized eigenvalue problem:

( K − ω²·M )·φ = 0

Reading: a nonzero solution φ exists only when the bracket is singular. The values of ω² that make it singular are the eigenvalues; the φ that go with them are the eigenvectors.

ω² = eigenvalue   →   fₙ = ω / (2π)   (natural frequency, Hz)
φ  = eigenvector  →   mode shape (the deflected pattern)

Reading: each eigenvalue is a squared natural frequency; each eigenvector is the shape the structure takes when ringing at that frequency.

The eigenvectors are orthogonal through the mass and stiffness matrices:

φᵢᵀ·M·φⱼ = 0   and   φᵢᵀ·K·φⱼ = 0   for i ≠ j

Reading: distinct modes don't share energy — that's what lets you treat each one as an independent single-DOF oscillator and add them back up (modal superposition).

For a single-DOF system the whole thing reduces to what everyone memorizes:

fₙ = (1 / 2π)·√(k / m)

Where you meet it

FEA pre-test on the bench. Before a bracket, a payload adapter, or an avionics tray goes on a shaker, someone runs an eigenvalue solve in Nastran, Ansys, or Abaqus and hands you a mode table: mode 1 at 87 Hz, first bending; mode 2 at 210 Hz, first torsion. That table is what you check your random-vibration input spectrum against.
Tap test at the test stand. Instrumented hammer, one accelerometer, a two-channel analyzer. You whack the part, the analyzer computes the frequency response function (FRF) from force to acceleration, and the peaks are your measured natural frequencies ^[5]. This is experimental modal analysis (EMA), and it's how you find out whether the FEA model was telling the truth.
Test-analysis correlation on a review board. The whole reason you did both is this meeting. Analysis said 87 Hz; the tap test said 79 Hz. Now you argue about whether the boundary condition in the model was too stiff, whether a bolted joint is softer than modeled, or whether you fat-fingered a material modulus. Programs live and die on getting the modes to line up.
Resonance avoidance in rotating machinery. A turbopump, a fan, a gearbox running at 6,000 rpm puts a forcing tooth-pass line right where a blade mode might sit. Modal analysis is how you keep the operating line out of a Campbell-diagram crossing before the hardware self-destructs.

How it works

The eigenvalue solve gives you frequencies and shapes but not amplitudes. A mode shape is a relative pattern — the ratios between points are fixed, the overall scale is arbitrary. Solvers pick a scale by convention, almost always mass-normalization: scale each φ so that φᵀ·M·φ = 1. Do that and the modal stiffness comes out as ω², and modal superposition math gets clean. Do not read absolute displacement off a raw mode shape; it means nothing until you drive it with a real force.

One fine-print clause on the superposition itself: the mode-by-mode decoupling strictly requires the damping to be proportional — classical Rayleigh damping, a C that the same undamped modes diagonalize — or absent entirely. Light, roughly uniform damping in bolted metal structures is close enough that the assumption holds in practice; but concentrated dampers or heavily damped joints couple the modal equations, produce complex modes, and break the clean one-SDOF-oscillator-per-mode picture.

Damping is the honest weak point. The eigenvalue problem above is undamped — C is nowhere in it. That's fine because for lightly damped metal structures the natural frequencies barely move. But damping is exactly what controls how big the response gets at resonance, and it does not fall out of an FEA eigen-solve. You measure it. The standard bench method is half-power bandwidth ^[6]: find a resonant peak in the FRF, drop 3 dB (a factor of 1/√2 in amplitude), read the two frequencies where the curve crosses that level, and

ζ = (f₂ − f₁) / (2·fₙ)

A mode at 100 Hz with 2% damping shows a half-power band 4 Hz wide. If your peaks are closer together than their bandwidths, the half-power method smears them and lies to you — that's its limit of validity ^[6].

The mistake people make is trusting a frequency count without checking modal mass, or effective mass. A model spits out fifty modes below 500 Hz, but if you sum the effective modal mass, maybe six of them actually participate in the direction you care about. The rest are local — a lug flapping, a wire bracket buzzing — and carry almost no mass. Chasing a 140 Hz mode that has 0.1% of the total mass is wasted effort; it will never show up in the base-driven test. Always ask "how much mass is in this mode," not just "what frequency."

Two more traps. First, boundary conditions dominate everything. A cantilever clamped in the model but bolted in reality will always test lower than it analyzed, because a bolted joint is a spring, not a wall — that's most of the 87-vs-79 Hz argument on the review board. Second, modal superposition assumes linearity. Gaps, friction, rattling joints, and materials that soften with amplitude break it, and a structure that sounds one frequency at a light tap can drop to another under a hard hit. When the tap test won't repeat, suspect a nonlinearity before you suspect the analyzer.

The validity envelope, stated plainly: modal analysis is exact for a linear, time-invariant structure. Real hardware is close enough to linear at small amplitude that the modes are useful, and useless enough at large amplitude that you verify on a shaker before you fly anything.

History

The spine of this is an eigenvalue problem, and eigenvalue problems in vibration go back to Lord Rayleigh — John William Strutt — who laid out the energy method for finding a structure's lowest natural frequency in The Theory of Sound, volume I in 1877 and volume II in 1878 ^[1]^[2]. Rayleigh's trick was to guess the mode shape, set the peak kinetic and potential energy in a cycle equal, and solve for the frequency; the ratio that falls out is still called the Rayleigh quotient, and it's the ancestor of every modern eigen-solver ^[1]. He was working strings, bars, beams, membranes, and plates by hand, decades before anyone could form a stiffness matrix numerically.

The matrix machinery had to wait for computers, and the experimental side had to wait for one specific algorithm. In April 1965 James Cooley of IBM's Watson Research Center and John Tukey of Princeton published "An Algorithm for the Machine Calculation of Complex Fourier Series" in Mathematics of Computation — the Fast Fourier Transform ^[3]^[4]. The identity underneath it was older; Gauss had it around 1805 for interpolating asteroid orbits, and Danielson and Lanczos rediscovered a piece of it in 1942 ^[3]. But Cooley and Tukey made it practical on a digital machine, and that's what turned a tap test into an instrument. Before the FFT you could not compute an FRF fast enough to be useful; after it, you could hammer a part, watch the spectrum, and read the modes off the screen. The late-1960s and 1970s measurement effort built the whole method of extracting a modal model from measured FRFs on top of that one algorithm ^[5].

Related tools

/tools/vibration-natural-freq
/tools/spring-rate
/tools/beam-deflection
/tools/moment-of-inertia-shapes
/tools/q-factor-bandwidth
/tools/frequency-period

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →