# 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:

where

* stiffness* or

*spring rate*in

* contact depth* or

*compresion distance*in

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* *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

*supported by the wheel:*

**weight**- 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
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.**center of mass** - 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:

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:

where

### 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:

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:

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:

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:

In under-damped suspensions (*natural frequency*:

### 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:

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:

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

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:

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*.