Spring Rate Theory
Motion Ratio and Lever Arms
The springs do not act directly at the wheel, they act through the lower control arm. Since the wheel is essentially pressing on a lever tied to the spring, the wheel gets an advantage. The spring is also not generally mounted exactly at the vertical. So we have to correct for that.
The force on the spring is the spring rate multiplied by the amount the lower spring perch rises vertically multiplied by the cosine of the spring angle from vertical.
The wheel force is the effective spring rate at the wheel multiplied by the vertical movement of the wheel.
Assuming small vertical motions, geometric similarity gives us:
Balancing the moments on the control arm:
Solving for and substituting into the first equation:
Using equation 3
Then equation 2
gives us the wheel rate in terms of the suspension geometry and the spring rate.
is called the “motion ratio”.
But we really can’t easily measure wheel rate directly. But we can measure the sprung weight and the amount of droop we have before the spring takes its share of the sprung weight. So,
where is the Sprung Weight and is the droop. Note that this assumes that at full droop the spring is fully extended, and has no preload.
Suspension Frequency Calculation
Another common measure of a suspension is its natural frequency. If a suspensions natural frequency is too high, the ride is perceived as being harsh, if is too low motion sickness can ensue. Typical road cars aim for suspension frequencies between 1 and 2 Hz. These frequencies are difficult to achieve with cars with limited suspension travel.
The suspension is similar to a damped harmonic oscillator. With no damping the natural frequency is
where is in Newtons per meter and is in kg. is in radians per second. This equation shows an interesting effect. Remember the the wheel rate, , is a function of the weight (mass) of the car and the suspension travel. The mass of the car in the denominator cancels out the mass in the numerator showing that suspension frequency is solely a function of suspension travel. This is the same as saying that a pendulum’s natural frequency is only dependent on its length, and not on its mass.
For further details see:
Competition Car Suspension Design Construction and Tuning, Allan Staniforth, 1999
Good coverage of suspension tuning, not as much design information as I would like, but this deserves a place on the bookshelf of anyone who wants to build their own car.