Half Car Suspension Model

Half Car Suspension ModelComputer Modelling

A computer model of a half car suspension model using Simulink in Matlab

Robert TallentThomas ByrneFintan Monahan12/1/2009

Table of ContentsTABLE OF CONTENTS..................................................................................................................................... 2

1. INTRODUCTION................................................................................................................................... 3

1.1. HISTORY................................................................................................................................................3

1.2. PROPERTIES............................................................................................................................................3

1.3. REQUIREMENTS......................................................................................................................................4

1.4. INDUSTRY..............................................................................................................................................4

2. EQUATIONS AND MODEL..................................................................................................................... 5

2.1. ¼ CAR MODEL.......................................................................................................................................5

2.2. ½ CAR MODEL.......................................................................................................................................6

2.3. ANGLE AND RATE OF CHANGE OF PITCH.......................................................................................................8

3. MAIN SYSTEM AND SUBSYSTEMS......................................................................................................... 9

3.1. FRONT SPRING MASS DAMPER SYSTEM:.....................................................................................................9

3.2. REAR SPRING MASS DAMPER SYSTEM......................................................................................................10

3.3. PITCH ANGLE OUTPUTS..........................................................................................................................10

3.4. RELATIVE POSITION OUTPUT...................................................................................................................11

3.5. ROAD HEIGHT INPUT.............................................................................................................................11

3.6. COMPLETE SYSTEM................................................................................................................................12

4. OUTPUTS........................................................................................................................................... 13

4.1. ROAD HEIGHT......................................................................................................................................13

4.2. PITCH ANGLE........................................................................................................................................13

4.3. HEIGHT AT FRONT AXLE.........................................................................................................................14

4.4. HEIGHT AT REAR AXLE...........................................................................................................................14

4.5. FORCE ON REAR SUSPENSION..................................................................................................................15

4.6. FORCE ON FRONT SUSPENSION................................................................................................................15

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1. Introduction

1.1. History

Computer modelling was originally used during World War II to simulate various battle

scenarios and since then its growth and development has evolved hand in hand with the

computer. Two particular spurts in its development could be noted during the 1980’s with

the introduction of the personal computer, where people who were up till then without

access to a computer, could own one and write code. Secondly, the development of

computer generated images for the film and game industries has contributed massively to

the ability of a computer model to be represented graphically, making it much easier for the

user to input data and interpret results.

1.2. Properties

The system we are observing is that of a half car suspension model including independent

front and rear vertical suspension. This is achieved by modelling each suspension

component as a spring mass damper system.

The system in question can be said to display the following properties:

Deterministic: This means the behaviour of the system is predictable and not

random, meaning that given a particular set of inputs, the output will always be the

same.

Continuous: This describes the fact that a change in the input results in a change in

the output and the proportionality of the input and output is quantifiable by some

mathematical function.

Dynamic: This describes the time dependence of a point in the model’s position in its

ambient space formalised mathematically

Local: Given the specifications of a modern computer, it would be more than capable

of performing this calculation in an acceptable time locally, without the requirement

of a networked system of computers

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1.3. Requirements

A good car suspension system should have satisfactory road holding ability, while still

providing comfort when riding over bumps and holes in the road. When the car is

experiencing any road disturbance (i.e. pot holes, cracks, and uneven pavement), the car

body should not have large oscillations, and the oscillations should dissipate quickly. This

can be achieved by investigating comfort vs. performance by changing the damping values

for a given step input (for example a car coming out of a pothole). In general, sports and

racing cars would adopt a stiffer suspension to increase overall car performance, whereas

commercial vehicles would have a softer more oscillatory response to provide comfort for

the driver.

1.4. Industry

Performance of an accurate model of this system allows us to test various parameters of the

suspension and alter them as necessary in order achieve the above requirements. Building a

computer model of this system to obtain required results would take a fraction of the time

necessary to carry out the mathematical calculations by hand, which outlines its vital

importance in industry, and in particular the automotive industry, due to the large amount

of control systems integrated in a modern vehicle. Preparing computer models of each of

these control systems allows engineers to make sure all components of a given system will

work in synergy before the physical building of the vehicle takes place.

This leads to a large improvement in the performance of a vehicle, without a large increase

in its cost.

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2. Equations and Model

2.1. ¼ Car Model

Newton’s 2nd Law of Motion

∑ Force=m.a

Two forces acting: Damping Force and Spring Force (Hooke’s

Law)

ForceDamping=−C .( v−u)

Where C is the damping coefficient and ( v−u) is the rate of

change of distance

ForceSpring=−K .(v−u)

Where K is the spring constant and (v−u) is the distance

For the purposes of simplicity let

˙z=(v−u)

z=(v−u)

Subbing in Newton’s second Law of Motion (where acceleration is z)

m . z=−K . z−C . z

Manipulating to get

z+Cm. z+ K

m. z=0

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Figure 2.1: Spring Mass Damper System

2.2. ½ Car Model

For a half car model is more complex as the model now has two spring mass damper

systems and are connected by a beam which adds elements to the model (Inertia I yy, Pitch

Angle θ & Rate of Change of Pitch Angle θ)

In order to calculate the Pitch Angle and Rate of Change of Pitch Angle the mass of the

system is not required rather the moment of inertia is used; therefore the equation above

to model a ¼ car suspension system cannot be used.

For a ½ car suspension model the centre of gravity of the beam (of length L) is located at a

point x from the front suspension. In order to calculate the upward force acting on the front

suspension the pitch angle and rate of change of pitch angle must be taken into account. As

the pitch angle increases and rate of change of pitch angle increases so too does the height

z and the rate of change of height z.

FFront=KFront ( x . z−θ )+CFront ( (L−x ) . z−θ )

Inversely for the rear suspension the as θ and θ increase z and z decrease

FRear=−KRear ( x . z+θ )−CRear ( (L−x ) . z+θ )

To calculate the height of the beam from the road and the rate of change of height

From Newton’s 2nd Law of Motion

Force=m .a

Therefore:

m .a=∑ Forces

Subbing in Equations for the forces acting upward on both the front and rear suspension

and taking into account the weight of the car

MassBody× z=FFront+FRear−(Mass¿¿ Body×Gravity)¿

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Figure 2.2: Graphical Image of Car to be modelled

Where z is the acceleration

Manipulating to get

z=FFront+FRearMassBody

−Gravity

An Integrator is then applied to get the rate of change of height

∫ z= z

A second Integrator is applied to get the height of the beam from the road

∫ z=z

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2.3. Angle and rate of change of pitch

From Newton’s 2nd Law of Motion for rotation

I yy θ=Torque

Where I yy is Moment of Inertia and θ is the angular acceleration

Torque=∑Moments

Moment due to front suspension

M Front=−x . FFront

Moment due to rear suspension

M Rear=(L−x ) .FRear

Subbing into Newton’s 2nd Law of Motion for rotation

I yy θ=M Front+MRear

Manipulating to get

θ=M Front+M Rear

I yy

An Integrator is then applied to get the rate of change of pitch angle

∫ θ=θ

A second Integrator is applied to get the pitch angle

∫ θ=θ

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3. Main System and SubsystemsIn deciding our final design for the model we chose to use a series of subsystems to

represent our different equations. With the proper labelling and layout this method allows

for an easy to follow and changeable overall system. We began with the Front and Rear

suspension systems which can be seen below in Figure 3.1 and Figure 3.2. It is these two

subsystems that provide the varying forces applied to the front and rear of our half-car. This

is the point at which we introduce spring stiffness, the damping factors and also the lengths

from both the front and rear of the wheelbase to the centre of gravity. As can be seen there

are two outputs from each of these subsystems, the first leads to the vital subsystem which

gives us our pitch angle and change in pitch angle. This system is shown in Figure 3.3 on the

next page.

3.1. Front Spring Mass Damper System:

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Figure 3.1

3.2. Rear Spring Mass Damper System

3.3. Pitch Angle Outputs

This is quite a simple system, the only variable being the Body Inertia. From this we can then

use the integrator functions Theta and Delta Theta to turn these values into the outputs we

require. These outputs are then looped back and used as one of the inputs for our front and

rear suspension systems. Our other system is the bounce and change in bounce values. The

output here will once again be looped back to be used as our input for our initial suspension

systems. This subsystem is illustrated below in Figure 3.4, it can be seen here that this

subsystem is dependant on body mass, acceleration due to gravity and the road height.

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Figure 3.2

Figure 3.3

This variable road height makes up our final subsystem. As shown in Figure 3.5, we have

combined two step inputs to simulate the bump in the road. A sine wave is then fed into the

resulting input to give the signal. Finally a transport delay is introduced, brought in to give

the delay between the front and rear. As well as determining the road height, it is here that

we can vary the speed of the car by adjusting the time delay. When all our subsystems are

combined with the appropriate addition and subtraction links we find our final overall

system, Figure 3.6. Through scopes we can then assess the desired outputs as discussed

earlier and in our Results and Conclusions below.

3.4. Relative Position Output

3.5. Road Height Input

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Figure 3.4

Figure 3.5

3.6. Complete System

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4. Outputs

4.1. Road Height

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Figure 3.6

4.2. Pitch Angle

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4.3. Height at Front Axle

4.4. Height at Rear Axle

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4.5. Force on Rear Suspension

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4.6. Force on Front Suspension

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