project

Analysis of an Automotive suspension Brian Fett December 7, 1998 EGR 345 Dynamic Systems Modeling and Control Prof. Hugh Jack

Executive Summary

The following pages contain a detailed analysis of a automobile suspension. A mathematical model was made to help identify the effects of changing the struts on the automobile. To simplify things, the tires were given no mass, and the axle was also considered massless. With the help of Mathcad and Working Model 2D the values of the spring and dampening coefficients are easily modified and the outputs analyzed. Since this suspension is for a sports car, it should be very stiff and allow for very little oscillation to be felt by the driver. The value of the dampening coefficient was then modified to produce the desired output. This value was found to be 300 Nm/s. This combination of damper and spring will deliver the performance that our customers desire.

Introduction

The suspension of a modern automobile is a very complex mechanism. A bump in the road causes a displacement of the tire, thereby causing a force to be transmitted through the axle to the strut to the vehicle. Since the spring and dampening coefficients of the tire are already set, the strut is the only factor to modify. The system was analyzed using standard modeling techniques which include free body diagrams, and D’Alemberts theorem. From this point it is possible to substitute in varied values for the struts and determine the best possible configuration.

Problem review

System model Figure 1

The schematic diagram shown in Figure 1 represents the suspension system, where the mass is the weight of the car and the first spring damper pair is a shock, while the second pair is the tire itself. The force F, would represent the reaction force of the road. The suspension can be easily modified by letting a mass represent the vehicle and connecting it to the axle by a spring damper combination, representing the strut. A massless lever represents the axle, which links the strut to the road via the tire. A spring damper pair also represents the tire. The dampening coefficient is a fixed property of the rubber while the spring coefficient is a property of the tire pressure. Our goal is to determine the necessary spring and damper coefficients for an average sports car.

Design Objectives

Our design requires a full 12 cm of vertical travel in the suspension. Because the strut must support 340,240 kg of mass, the value of the spring constant is determined by Hookes’ law to be equal to 275 N/m. Our goal is to find a damper which will work with this chosen spring to absorb any bumps or road noise while minimizing any sort of oscillation. Such a combination will result in a comfortable ride.

Modeling and Analysis

As shown in Figure 1, a mass is attached to the lever through a spring-damper pair. As the lever deflects, forces are exerted through another spring-damper pair to produce the reaction force F. In order to make this model manageable, we will neglect the mass of the lever. We will also make the assumption that the lever moves through small angles. To begin the analysis we draw a free body diagram for the mass. (Fig.2)

Free body diagram of mass Figure 2

Using D’Alemberts law (Eq.1), we can sum the forces acting on the mass. Notice that the elongation of the spring and damper is written in terms of displacement (x₂). Also notice the basic ratio of the length to the point of interest divided by the total length of the lever, this eliminates the need for another displacement variable.

(1)

Now, with some rearranging we can put this equation into state variable form.

(2)

(3)

In order to put this equation (Eq.2) into state variable form, we must eliminate the derivatives from the right hand. This can be done by bringing the derivative over to the left-hand side and setting q₁` equal to it.

Therefore, we have our first state variable equation Eq.(4)

Let

Therefore

(4)

We can now solve q in terms of v₁ and substitute it into equation three to get a second state equation. (Eq.5)

(5)

Since we still have x₂ in Equations 4 and 5, we must develop an expression for x₂ in terms of x₃ and x₁ This can be achieved by writing equations for the forces of the spring-damper pairs. Since the lever is assumed to be massless we can use a simple ratio to determine the relationship between x₂, x₁ and x₃. The relationship is as follows.

Equations for the forces can now be written and put into the equality above. The compression between the lever and the mass and the lever and road are derived below (Equations 6 and 7 respectively).

(6)

(7)

Now these two equations will provide us with our third differential equation.

This equation can be further simplified, and all derivatives must be moved to the left of the equation.

Now we can define q₂ as the left hand side of this equation, and determine our second state equation. Eq.(8)

(8)

Now if we solve q₂ for x₂ we can substitute the equation into the previous state equations. And define our state variable definition in Mathcad.

We can now use these equations to approximate the behavior of the system using the Runga-Kutta method. In order to solve the system of equations the rkfixed function within Mathcad was used. The values used for this solution are found in Table 1 below.

Table 1

Parameter	Symbol	Value	Units
Spring stiffness	ks1	275	N/m
Dampening coefficient	kd1	300	Nm/s
Spring stiffness	ks2	100	N/m
Dampening coefficient	kd2	100	Nm/s
Distance from pivot	r1	0.2	m
Distance between spring-dampers	r2	0.3	m
Mass	M	10	kg
Input	x3	.2 sin (t)	m

The solution graph is shown in figure 5.

MathCAD solution Figure 5

Working model simulation

This same situation can be examined using a software package called Working Model 2D. Working Model 2D is a two-dimensional simulation program, which is capable of performing the complex integration needed to simulate this system. Figure 6 shows the model as drawn in working model.

Working Model System Figure 6

There are several assumptions that go into Working Model. The main assumption is the accuracy control, this can be set to almost any number of frames per second. The higher the value, the more accurate working model is. There is one drawback, however, and this is if the model is too accurate it displays all sorts of very small oscillations. This can be somewhat avoided by sacrificing the accuracy for results that can be interpreted. We settled for 500 frames per second. The second assumption is the mass of the lever. In the Mathcad calculations, the mass was neglected in order to make the equations manageable. Working Model 2D does not deal with massless members very well. Therefore I set the mass of the lever at .005 kg. The last assumption was that x₃is equal to zero. This was done so that there is an actual reaction in the spring damper pair. Without this assumption, Working Model 2D would allow gravity to pull the massless rectangle down, making the tension in the spring damper pair very close to nothing. The Working Model 2D solution is shown in Figure 7.

Working model solution Figure 7

The most accurate part of this data is in the first few seconds. Obviously, the system collapses under its own weight within a second. In reality, it wasn’t designed to do that, therefore the first few milliseconds are all we are interested in anyway. When the accuracy in Working Model 2D is increased you can better see the small oscillations that are within the large oscillation. These small oscillations should be able to be reduced if not eliminated by changing the damper and spring coefficients.

Summary

Overall, the new dampening and spring coefficients seem to function properly. We have a very stiff system with virtually no oscillation whatsoever. This will cause a slightly bumpy ride, but the vehicle should stick to the road pretty well. Other future modifications that could improve performance would be to select different tires for the car. This would change the tires dampening and spring coefficients. The other major possibility is to modify r1 & r2. This would requite a special suspention arm and could possibly result in another product for our company.