How suspensions work

A suspension is essentially a damped spring producing opposing force when being compressed. Springs sustain the weight of the vehicle. Dampers oppose the spring movement, dissipating their energy and preventing them to bounce without control.

The force produced by the springs depends on the distance they are compressed and it's given by Hooke's Law:

$$F=-Kx$$

where

$K$ is the spring's stiffness or spring rate in $N/m$, and

$x$ is the contact depth or compresion distance in $m$.

The force produced by the dampers depend on how fast the suspension is being compressed or elongated (contact speed), opposing the movement.

When a wheel is lifted from the ground the suspension produces no force. At the slightest contact possible it also won't produce any force. As the spring gets more compressed, more force is produced proportionally to the contact depth:

The compression limit is the suspension distance. Beyond this point the spring has reached its maximum force and cannot compress further. A hard contact with the rigid body is produced.

The slope of the force line is given by the stiffness $K$. The more stiffness, more steep slope.

The suspension position is the contact depth where the spring force matches exactly the force applied on the spring. In vehicles this force is typically caused by the weight supported by the wheel:

The suspension force is calculated as:

$$suspensionForce = {stiffness}\cdot{contactDepth} + {damper}\cdot{contactSpeed}$$

When the suspension is not moving the contact speed is 0. This happens when the vehicle is either resting, cruising at constant speed or under constant non-vertical acceleration. The suspension position for a specific wheel can then be calculated as:

$$suspensionPosition = \frac{weight \cdot{gravity}}{stiffness}$$

where $weight$ is the actual weight being supported by that wheel.

Studying the oscillating behavior

The suspension properties can be studied from the point of view of the oscillating behavior (Harmonic oscillator). The associated concepts are used to study the reactions of the suspension in different situations.

Given the force produced by the suspension at a specific steady state (contact speed = 0) the equivalent sprung mass value for studying that situation can be calculated as:

$$sprungMass = \frac{suspensionForce}{gravity}$$

When the vehicle is at rest, cruising at constant speed or under constant non-vertical acceleration the sum of the sprung masses of all the wheels matches the mass of the vehicle exactly.

Using the sprung mass you can calculate the natural frequency for the spring under that load. The natural frequency is the speed at which the spring can respond to changes in load:

$$naturalFrequency = \sqrt{\frac{stiffness}{sprungMass}}$$

The natural frequency defines the oscillating behavior of the suspension. A typical family car is set up to exhibit a natural frequency somewhere between 5 and 10.

The sprung mass is also used for studying the damping behavior, that is, the rate at which the suspension dissipates the energy stored at the spring. We can calculate the damping ratio for learning whether the suspension will be under-damped, over-damped or critically-damped:

$$dampingRatio = \frac{damper}{2 \sqrt{stiffness \cdot{sprungMass}}}$$

A damping ratio greater than 1.0 means over-damping (sluggish suspension), a value of exactly 1.0 is critically-damped, and a value less than 1.0 is under-damped (bouncy suspension). Values for realistic vehicles are in the range of 0.2 and 0.6. The damper rate that targets a specific damping ratio can be calculated by rearranging the equation above:

$$damper = dampingRatio \cdot{2} \sqrt{stiffness \cdot{sprungMass}}$$

In under-damped suspensions ($dampingRatio < 1$) the frequency at which the system oscillates is different than the natural frequency:

$$underdampedFrequency = naturalFrequency \cdot{\sqrt{1-dampingRatio^2}}$$

Suspension and simulation steps

Another interesting concept for the simulation is the number of simulating updates that will occur during each spring oscillation. This number is given by the alpha ratio:

$$alpha = \frac{1}{timestep} \sqrt{\frac{sprungMass}{stiffness}}$$

Applying the Nyquist theorem we can deduct that a physically correct simulation should have alpha >= 2. Smaller values won't likely cause noticeable artifacts, only the produced forces won't be physically precise.

Application to real vehicles

Suspensions in real vehicles don't have constant frequency and damping ratio at all times. You may calculate and study the oscillating behavior in specific situations separately: at rest, accelerating, braking... Weight transfer on some of these situations actually affects the behavior of the suspension. That's the challenge of configuring suspensions in real vehicles: you have to find a good balance for most situations.

If the vehicle is under constant non-vertical acceleration (accelerating / braking / cornering) the weight is redistributed among the wheels. Wheels will be supporting more or less load than in rest position. This effectively modifies the oscillating properties of the suspensions at those specific situations, thus having different reactions. For instance, imagine a racing car heavily braking at the end of a long straight before entering a slow curve. If that part of the track is a bumpy surface then the suspension must be set up properly for ensuring correct handling while braking over the bumps. Another example is the downforce caused by aerodynamic surfaces. Suspension will have different behavior on high speeds due to the extra sustained load. Studying the oscillating behavior of the suspension in this detail is critical for setting up racing cars that react properly on every situation.

The most important facts in a vehicle suspension are:

When the suspension is not moving the dampers have no effect. This happens when the vehicle is at rest, cruising at constant speed, or under constant non-vertical acceleration. Otherwise, weight transfers occur among the suspensions. The springs should be strong enough for sustaining the weight of the vehicle preventing the suspension to reach its limits on all situations, including weight transfers.

When the formulas for the Harmonic Oscillator (above) are applied to vehicles in those situations they yield a surprising result:

$${\begin{cases} naturalFrequency = \sqrt{\frac{stiffness}{sprungMass}}\\ sprungMass = \frac{suspensionForce}{gravity}\\ suspensionForce = stiffness \cdot{contactDepth} \end{cases}} $$

$$\Longrightarrow naturalFrequency = \sqrt{\frac{gravity}{contactDepth}}$$

So the most important factor for the frequency of the suspension is the contact depth! Not the spring rate, not even the sprung mass! The frequency of the suspension will vary on the different situations (accelerating, braking, cornering...) according to the contacth depth. Note that this contact depth includes any pre-compression of the spring inside the suspension strut.

This is where dynamic suspension systems kick in. Some high-end modern cars have electronically controlled suspensions which dynamically modify the suspension properties in order to... (guess what?) keeping the contact depth (also ride height) constant at all situations. This preserves the original properties of the suspension when accelerating, braking, cornering, or carrying variable cargo or passengers.

Vehicle Physics Pro includes the component VPDynamicSuspension that modifies the suspension properties in order to preserve a given contact depth.