Overview

Topic 1 - Establishing our independent parameters

We will examine the problem of improving the ride and handling characteristics of a particular car.  We might have freedom to change the spring rates in the suspension but for the particular car that we have been assigned to, we might not be free to change the weight of its chassis and body, although in a wider project of designing a completely new car, these might, in fact, be ‘free’ independent parameters. In our situation, since we are not allowed to modify them, we would consider them to be ‘fixed’ independent parameters. 

Next we will examine the dependent parameters which can be used to evaluate our solution.  In the example just given, regarding the ride quality of the car, these might be things like smoothness of ride, minimal oscillations, and low interior noise.

Then we will examine the ‘free’ independent parameters, which are the ones that we have the ability to change within the scope of our project.  In our example, these might be the rates of the springs, the damping provided by the shock absorber, and the unsprung weight of the suspension.

The next step is for us to analyse the relationships between the independent parameters and the resulting dependent parameters.  This can be accomplished by working with the parametric relationships from fundamental principles.  When we understand the relationships well enough, sometimes we can develop computer routines which can be used to iteratively analyse the effect of all of the independent parameters on the dependent parameters, and thus develop our way towards a solution.

However, to understand how such an analysis would be developed, it is best that we work through a parametric analysis just using fundamentals.  To examine how a parametric study can be used to guide us, we will use a real-world problem of changing the stiffness of a vehicle’s springs to improve the cornering performance of a small vehicle intended for automotive speed competitions, such as time trials or races.

© P Hylton

Topic 1 - Application

The following table gives us the “fixed” independent parameters.  That is, those that we could change if we were designing a completely new car.  But in our case, these are fixed, so the only things we are free to change are the spring rates, so these parameters in the table are all fixed for our example.

Basic data table for the vehicle

Wheelbase (distance between the front and rear wheel centres): 261.6 cm, vehicle centre of mass is 38.1 cm above the ground.

Topic 2 - Establishing our dependent parameters

Let us consider a dependent parameter which might be used to quantify what is ‘good’ or ‘bad’ in terms of the vehicle’s cornering performance. Ride Frequencies are the natural frequencies related to the mass and stiffness of the suspension of a vehicle and are used as a measure of suspension effectiveness in race cars.  Thus, this is a viable dependent parameter to use for our example.

We need a starting point for the desired front and rear ride frequencies of the car. Optimally, we might determine our target based on experience gained during the development of similar cars we have worked with.  Failing that, we can refer to the work of experts in the field, such as Milliken and Milliken (1995) whose text on race car vehicle dynamics, although dated, is still an industry standard.  Millikens’ rule of guidance for ride frequency is 1.6 - 2.0 hz (remember the unit hertz, hz = cycles/second) for non-aerodynamic cars (i.e. cars without wings and ground effects, like our example) or 3.0 - 5.0 hz for aerodynamic cars (i.e. a pure race car using wings and/or ground effects).

As an aside, why would we want the frequency to be higher for pure race cars? Recall from vibration theory that the natural frequency can be calculated from the square root of stiffness over mass (ω_n = [k/m]^1/2).  Pure race cars tend to be lighter (lower m) and stiffer (higher k) than non-race cars.  Reducing m and increasing k results in a higher natural frequency.

The Millikens also indicate that the front frequency should be slightly higher than the rear.  So for our problem, let us choose a starting point of 1.9 hz at the front and 1.7 hz at the rear, based upon their recommendations.  Therefore, we will use the ride frequency as our dependent variable and work on developing a relationship between it and our free independent parameters (spring rates for the springs in our suspension), working around the fixed independent parameters from the car’s data table.

Topic 2 - Application

Let us now calculate the total spring rate that we want each end of the car to have from ω = [k/m]^1/2.    Rearranging, we get:  k = ω²m.

We would like to choose k to be the desired spring rate for the front of the car, where m would be the mass carried at the front, and ω would be the target frequency.  Remember that to change the weight/force (Newtons) to mass (kg) we divide the force by the acceleration of gravity (9.8 metres/second²).

Thus we could determine the target spring rate at each corner of the car, k (i.e. the spring rate we would like to try to achieve) for the front axle and the rear axle.  Note that we would use one half of the weight on each wheel.

${\text{k}}_{\text{FrontTarget}}={\text{ω}}^2\text{m}$ 

$={\left[\right(\text{1.9 cycles/sec}\left)\right(\text{2π radian/cycle}\left)\right]}^2$      $\frac{\text{4360 Newtons (1/2)}}{\text{(}{\text{9.8 metre/sec}}^2\text{)}}$  

$=31700\text{ Newton/metre}$  

${\text{k}}_{\text{RearTarget}}={\text{ω}}^2\text{m}$ 

$={\left[\right(\text{1.7 cycles/sec}\left)\right(\text{2π radian/cycle}\left)\right]}^2$      $\frac{\text{5173 Newtons (1/2)}}{\text{(}{\text{9.8 metre/sec}}^2\text{)}}$  

$=30100\text{ Newton/metre}$  

If the values we just calculated are the total spring rate that we want at a given corner of the car, then it is a combination of the suspension spring rate and the spring rate of the tyres.  We could get stiffness data for our particular tyre from the literature.  A reasonable value that we could use is 700000 Newton/metre.

Topic 3 - Finding our desired independent parameter value

The springs representing the suspension and the tyre would be in series, and series springs can be combined using the relationship shown here:

Combining springs

Each axle of the vehicle will have a linkage ratio based on the geometry of the suspension.  This ratio is based on the fact that relative to a fixed point on the chassis, the effective spring rate at the wheel is not the spring rate of the mechanical spring due to the geometry, as described in the following:

K_T is the spring rate of the tyre.  K_S would be the spring rate of the actual physical spring in the suspension.

The angle of rotation of the linkage is either     θ = Dx/D   or    θ = Dy/d

Let f_S be the force in the spring, K_S and let F_Z be the force in spring K_T representing the entire suspension to ground.

The rotational stiffness created by the springs is

${\text{K}}_{\text{Rotational}}=\text{M/θ}={\text{F}}_{\text{Z}}\text{D/θ}={\text{f}}_{\text{S}}\text{d/θ}$  

${\text{F}}_{\text{Z}}\text{D/(Δx/D)}={\text{f}}_{\text{S}}\text{d}/\text{(Δy/d)}$ 

$({\text{F}}_{\text{Z}}/\text{Δx}){\text{D}}^2=({\text{f}}_{\text{S}}/\text{Δy}){\text{d}}^2$ 

$\left({\text{K}}_{\text{Suspension}}\right){\text{D}}^2=\left({\text{K}}_{\text{Spring}}\right){\text{d}}^2$ 

${\text{K}}_{\text{Suspension}}=\left({\text{K}}_{\text{Spring}}\right)[{\text{d}}^2/{\text{D}}^2]\text{ where d/D is called the linkage ratio (LR)}$ 

${\text{K}}_{\text{Spring}}={\text{K}}_{\text{Suspension}}/{\text{(LR)}}^2$ 

Recall the linkage ratios were fixed independent variables, given in our car’s data table.

Topic 3 - Application

We can determine what the spring rate of the suspension needs to be, knowing the overall desired rate and the spring rate of the tyre. 

For the front suspension:

$1/31700=1/700000+1/{\text{K}}_{\text{suspension}}$ 

${\text{K}}_{\text{suspension}}=33204\text{ Newton/metre}$  

And for the rear suspension:

$1/30100=1/700000+1/{\text{K}}_{\text{suspension}}$ 

${\text{K}}_{\text{suspension}}=31450\text{ Newton/metre}$  

We can now calculate spring rates for the springs that we actually want to install in our car (remember, those were our free independent parameters), in order to provide the ride frequencies (remember, those were our dependent parameter, and we had selected target values that we wanted to achieve).

For the front suspension:

${\text{K}}_{\text{FrontSpring}}=\frac{33204\text{ Newton/metre}}{{\text{(.726)}}^2}=63000\text{ Newton/metre}$ 

And for the rear suspension:

${\text{K}}_{\text{RearSpring}}=\frac{31450\text{ Newton/metre}}{{\text{(.962)}}^2}=34000\text{ Newton/metre}$ 

Topic 4 - Expanding our use of parametric relationships

We have now fully established the relationship between our free independent parameters (the spring rate of the suspension springs), the fixed independent parameters (the attributes of the vehicle which we do not have freedom to change at this time, and the dependent (or target) parameters (the ride frequencies that we picked to be our measure of the vehicle’s cornering capability).

Having established this relationship, if we wanted, we could now develop a computer routine that would calculate the ride frequencies for any spring rate we might choose (or for that matter, if we varied any of the other independent parameters, such as suspension linkage ratio or vehicle weight.  That might make our new tool even more valuable in the future for preliminary design of a new car.

Although this example was most likely not in your area of expertise, hopefully, this will aid you in seeing how a parametric relationship can be developed for systems which are more familiar to you.

Additional insight into the vehicle dynamics of our example car and the next logical step in the analysis of its suspension is given in the next topic of this module.

Summary

• The concept of dependent and independent (both fixed and free) parameters were examined.
• The vehicle dynamic analysis of a performance car suspension was studied as an example for developing parametric relationships between the three types of parameters.
• Once relationships of this type are developed, they can be used to build data spreadsheets or computer programmes which can be used in future analyses.
• The method for analysing the spring rates of vehicle suspension as free independent parameters was examined, for a pre-determined car which with fixed parameters involving its overall characteristics.

Reference and bibliography

Gillespie, T. (2000) Fundamentals of Vehicle Dynamics.  Detroit USA: SAE International.

Milliken, W. & Milliken, D. (1995) Race Car Vehicle Dynamics. Detroit, USA: SAE International.

Milliken, W. (2006) Equations of Motion.  Cambridge, USA: Bentley.

Seward, D. (2014) Race Car Design.  London: MacMillan.