physics//mechanics//vibration//modal analysis

Modal analysis is the computation or measurement of a structure's natural frequencies and mode shapes, and it exists so that a designer learns before welding which speeds of a motor, a fan or a propeller would make the structure resonate. It is routine in any mechanical design that carries rotating machinery: pump frames, turbine foundations, vehicle bodies, drone arms. A structure of masses and stiffnesses obeys \(M\ddot x+Kx=0\); looking for motions in which every point oscillates at the same frequency gives

Modal analysis is the computation or measurement of a structure's natural frequencies and mode shapes, and it exists so that a designer learns before welding which speeds of a motor, a fan or a propeller would make the structure resonate. It is routine in any mechanical design that carries rotating machinery: pump frames, turbine foundations, vehicle bodies, drone arms. A structure of masses and stiffnesses obeys Mx¨+Kx=0M\ddot x+Kx=0Mx¨+Kx=0; looking for motions in which every point oscillates at the same frequency gives

Kv=ω2Mv,Kv=\omega^2 Mv,Kv=ω2Mv,

a generalized eigenvalue problem. Each solution is a mode: ω\omegaω is its natural frequency and vvv its mode shape, how much and in which direction each point of the structure moves when that mode alone is excited (the first bending of a beam, the twisting of a frame, the breathing of a casing).

It is done two ways. On a model, a finite-element mesh of the part supplies MMM and KKK with thousands of degrees of freedom and a solver returns the lowest few dozen modes, which are the ones that matter. On the real object, an instrumented hammer taps the structure while accelerometers record its answer, or a shaker drives it through a sweep; the measured impulse responses give the frequencies, the shapes and the damping, which models predict worst. Engineers compare the two to correct the model.

A mode is a property of the structure, and it only causes trouble when something drives it.

The result of a modal analysis is a list of frequencies to keep away from every excitation in the machine, with a margin, and a list of shapes that says where to stiffen, where to add mass and where to mount a sensor.

The shape decides where to look. An accelerometer placed at a node of a mode, a point that does not move in that mode, is blind to it; a stiffener placed where the mode bends most raises its frequency for the least added metal.

The analysis is linear. It describes small vibrations about equilibrium with damping added as a few percent per mode; bolted joints that slip, cracks that open and close or rubbing parts make the real structure behave differently at large amplitude.

The mathematics is the same decomposition as the modes of any linear system, applied to a mechanical one; what to do when an excitation lands near a mode anyway is resonance, and reading the vibration of a machine in service is vibration analysis.