# Motion ratio, wheel rate and ride frequency

> Motion ratio is spring travel per unit of wheel travel. How it sets wheel rate (spring rate times motion ratio squared), ride rate with the tire, and ride frequency, worked through for the opensus Formula SAE example.

Motion ratio is how far the spring (or damper) moves for each millimetre the wheel moves. Because the ratio applies once to the spring's travel and once to its force, the stiffness the wheel feels, the **wheel rate**, is the spring rate times the motion ratio squared.

## Formulas

$$\text{MR} = \frac{dL_{\text{spring}}}{ds_{\text{wheel}}} \qquad k_{\text{wheel}} = k_{\text{spring}}\,\text{MR}^2$$

The tire is a spring in series, so the **ride rate** at the contact patch is

$$k_{\text{ride}} = \frac{k_{\text{wheel}}\,k_{\text{tire}}}{k_{\text{wheel}} + k_{\text{tire}}}$$

and the corner's **ride frequency**, with \(m\) the sprung mass on that corner, is

$$f = \frac{1}{2\pi}\sqrt{\frac{k_{\text{ride}}}{m}}$$

Use N/m and kg in the last formula (multiply N/mm by 1000).

## Worked example: the Formula SAE front

The [Formula SAE example](/docs/examples/fsae/) drives a 60 N/mm coilover through a pushrod and rocker. opensus measures the motion ratio at ride height as **0.674**:

$$k_{\text{wheel}} = 60 \times 0.674^2 = 27.2\ \text{N/mm}$$

With a 120 N/mm tire:

$$k_{\text{ride}} = \frac{27.2 \times 120}{27.2 + 120} = 22.2\ \text{N/mm}$$

Each front corner carries 58 kg of sprung mass:

$$f = \frac{1}{2\pi}\sqrt{\frac{22\,190}{58}} = 3.11\ \text{Hz}$$

All three match the **Ride, at design** section of the readout. A 2.6 to 3.5 Hz ride frequency is typical of a winged FSAE car; road cars are usually between 1 and 1.5 Hz.

## Rising and falling rate

A rocker's geometry makes the motion ratio change with travel. If it rises in bump, the wheel rate rises faster than the motion ratio itself (it goes with the square). **Kinematics → Heave → Motion ratio** plots it across travel. Pushrod and rocker angles are the usual way to tune it.

## Damping

The same squared ratio applies to the damper: the damping the wheel feels is the damper's coefficient times its motion ratio squared. The readout shows the resulting **damping ratio** in bump and rebound, and **Open 3D detail and damper curves** on any spring or damper lets you draw its force-velocity curve directly.

[Open in opensus](https://opensus.app/app/?example=fsae)

Source: https://opensus.app/docs/motion-ratio/ (last updated 2026-11-07)