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Finding the Critical Damping Coefficient cr

The critical damping coefficient is a special case where the damping coefficient returns the suspension system back to its original position with no oscillations. To find the Critical Damping Coefficient we need to evaluate the equations that define damped harmonic motion.

Initial Conditions
The SKY's suspension consists of the four short and long double A-arm (SLA) suspended corners. One for each wheel, and each corner is an independent system consisting of a spring, a damper (shock) and the suspended mass. The initial conditions are with the chassis at equilibrium and the corner mass resting on the spring. An input to the suspension such as a bump in the road will cause a the spring to compression to absorb the energy of the bump and the damper by its design will resist any change in motion. These forces are negative values.

Using the Newton's first law of motion:

Force = m a
m = mass
a = acceleration

and defining the forces about one corner of the SKY's suspension we find: Ft = m a
Ft = total forces

The spring force is defined as:
Fs = -K x
k = spring constant (Hooke's Law)
x = distance traveled

The damper force is defined as:
Fd = -c v
c = damper coefficient
v = damper shaft velocity

Summing the forces:
Ft = Fd + Fs
Fd = -c v
Fs = -K x
m a = (-c v) + (-K x)
m a - (c v) - (K x) = 0

Rewriting the above expression as a differential equation and replacing c with cr for the critical damping coefficient we have:

m d2x/dt - cr dx/dt - k x = 0

For more information about the solutions for above equation and linear homogeneous second order differential equations click

linear homogeneous second order differential equations

The above equation can be solved using the characteristic equation

a r2 + b r + c = 0

which we all recognize as a second order polynomial equation whose solution can be found using the quadratic formula.

r = [(-b) +/- (b2 - 4ac)1/2] / 4ac

For the case of critical damping cr the discriminant is equal to zero.

cr2 - 4 m k = 0
cr2 = 4 m k
cr = 2 (m k)1/2

Rewriting the above and factoring out the angular frequency term ω

where: ω = (k / m)1/2

cr/m = [2 (m k)1/2]/m
cr/m = 2 (m)1/2 (k)1/2 (m)-1
cr/m = 2 (m)-1/2 (k)1/2
cr/m = 2 (k / m)1/2
cr = 2 m (k / m)1/2
cr = 2 m ω

And again rewriting the above in terms of the more common frequency of cycles per second (Hz) f

Where: f = ω/(2π)

cr/(2π) = 2 m [ω/(2π)]
cr = 4π m f

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