Single Degree of Freedom Vibration Calculator

The single degree of freedom model

A mass, a spring and a damper make up the simplest picture of machine vibration. An engine on its mounts, an instrument on springs or one wheel of a vehicle is studied with this model as a first approximation. Three numbers define the system: mass m, stiffness k and damping coefficient c.

Damping ratio

Whether the system oscillates is decided by the damping ratio: ζ = c / (2 × √(k × m)). Below 1 the system oscillates as it decays, at exactly 1 it returns as fast as possible without oscillating, and above 1 it returns slowly. The quantity 2 × √(k × m) in the denominator is the critical damping.

Forced vibration

Under a sinusoidal force the amplitude is the static deflection times the magnification factor: X = M × F_0 / k. The magnification factor depends on the frequency ratio (r = f / f_n) and on the damping ratio. At resonance, where r = 1, M = 1 / (2 × ζ). A system with a damping ratio of 0.1 vibrates at five times its static deflection at resonance.

Vibration isolation

Transmissibility is the share of the force that reaches the base. Once the frequency ratio exceeds √2 it falls below 1 and isolation begins. That is why isolators are chosen soft enough to put the natural frequency well below the running frequency.

Example

A 10 kg mass on a 4 N/mm spring with a 40 N·s/m damper has a critical damping of 400 N·s/m, a damping ratio of 0.1 and a natural frequency of 3.18 Hz. Forced at twice the natural frequency (6.37 Hz) with an amplitude of 100 N, it moves 8.3 mm and has a transmissibility of 0.36, so 36 percent of the force reaches the base.

Limits

The model is linear and covers motion in one direction. Real machines have six degrees of freedom, several natural frequencies and mounts whose properties change with frequency. Treat the results as a first approximation.