How simple 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 more weight is loaded on a wheel, the more compressed will be its spring (suspension position visibly lower).
• The less weight is loaded on a wheel, the less compressed will be its spring (suspension position visibly higher).
• If the center of mass of a vehicle is moved (load, passengers...) the weight will be redistributed along the wheels and their suspensions will be compressed / elongated as result of the new weight distribution.
• If the vehicle is accelerating, braking or cornering the weight will be temporarily shifted among the wheels, varying their suspension positions accordingly. For instance, accelerating makes a certain amount of weight (depending on the actual acceleration) to be transfered from front wheels to rear wheels. Similar effects happen when braking and cornering.
• Aerodynamic surfaces push the vehicle down with speed, increasing the load on the wheels and compressing their suspensions accordingly.

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 acceleration. The suspension position for a specific wheel may 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 may 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 effective sprung mass value for studying that situation may be calculated as:

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

When the vehicle is at rest, cruising at constant speed or under constant acceleration on a flat surface the sum of the sprung masses of all the wheels matches the mass of the vehicle exactly.

Using the sprung mass you may calculate the natural frequency for the spring under those conditions. The natural frequency is the rate 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. For example, a typical family car is set up to exhibit a natural frequency somewhere between 5 and 10.

The effective sprung mass may also be used for studying the damping behavior, that is, the rate at which the suspension dissipates the energy stored at the spring. We may 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 may 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 deduct that a physically correct simulation should have alpha >= 2. Smaller values means that the simulation sampling rate is not enough to simulate the given spring rate.

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: finding a good balance for most situations.

If the vehicle is under constant acceleration (accelerating / braking / cornering) the weight is redistributed among the wheels. Wheels will be supporting more or less load than in their positions at rest. This effectively modifies the oscillating properties of the suspensions at those specific situations, therefore producing 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 additional load. Studying the oscillating behavior of the suspension in this detail is critical for setting up racing cars that react properly on every situation.

We may summarize the role of the vehicle suspension as:

• Springs sustain the weight of the vehicle.
• Dampers dissipate the energy in the springs when the suspension moves (weight transfers, bumps).

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 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 single factor that defines the frequency of our suspension is the contact depth. The frequency of the suspension will vary on the different situations (accelerating, braking, cornering...) according to the contact depth. Note that this contact depth includes any pre-load of the spring inside the suspension strut.

As a result, configuring the vehicle suspension for a similar behavior in a broad range of conditions requires minimizing the changes in the contact depth in those conditions. There are several strategies for this:

• Harder springs and anti-roll bars, reducing weight transfers on accelerating, braking, cornering, etc.
• A lower center of mass reduces the weight transfers without modifying the suspension properties.
• Electronically controlled suspension which dynamically modify the suspension properties to preserve the contact depth (and the ride height) constant in all situations: accelerating, braking, cornering or carrying variable cargo or passengers.

Vehicle Physics Pro includes a variety of suspension components allowing different ways of configuring the suspension:

• VPWheelCollider component with standard spring and damper settings.
• VPAntiRollBar: links the suspension of the wheels in the same axle in order to control the lateral roll in curves.
• VPAdvancedDamper: configures dampers with bump / rebound parameters: slow bump, fast bump, slow rebound, fast rebound.
• VPDynamicSuspension: modifies the suspension spring in runtime order to preserve a given contact depth (or ride height).
• VPProgressiveSuspension: Modifies the suspension properties along the suspension travel. This allows simulating bump stops or leaf spring suspensions.