Spring brake

Understanding damping ratios and the notion of critical damping

- CLAUDE ROUELLE

‘The damper is the brake of the spring.’ In this article we will review that theory, as well as the pros and cons of some oversimplified quarter-car simulation.

If you would put a mass on a spring with no damper and push the mass down by 60mm (as seen on the green trace in Figure 1), then release your force, the mass will oscillate indefinitely. We should say nearly indefinitely because ultimately the movement will stop, mainly due to the spring material internal resistance, but also because of air friction. Note that in this example you allow the spring to work in tension and move not only down but also up by 60mm. As we will see later that is not the case in most car suspension.

The mass position, as a function of time, can be described by a sinusoid called harmonic motion. The equations are as follow: z=Z_0 e^(-ζω_n t) sin (ω_D t+φ)

With the undamped period being ω_N=√(K/m) in rad/sec and the undamped frequency f =1/2π √(K/m) in Hertz (Hz)

The critical damping coefficient is C_crit=2√Km in N/(m/sec)

The damping ratio is ζ=C/C_crit =C/ (2√Km), which is dimensionless.

The damped natural period is ω_D=ω_N √(1-ζ^2 ) in rad/sec, the damped frequency is f =1/2π √(K/m) √(1-ζ^2 )in Hertz (Hz) with Z0 being the initial amplitude (in metres), t the time in seconds, φ the phase difference (in radian), K the spring rate (in N/m), m the mass (in kg), and C the damping in N/(m/sec).