Half Car Suspension Model Computer Modelling A computer model of a half car suspension model using Simulink in Matlab
Robert Tallent Thomas Byrne Fintan Monahan 12/1/2009
Table of Contents TABLE OF CONTENTS........................................................................................................................................ 2 1.
2.
3.
4.
INTRODUCTION ...................................................................................................................................... 3 1.1.
HISTORY ................................................................................................................................................. 3
1.2.
PROPERTIES ............................................................................................................................................. 3
1.3.
REQUIREMENTS ........................................................................................................................................ 4
1.4.
INDUSTRY................................................................................................................................................ 4
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
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
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 2
nd
Law of Motion
∑ Two forces acting: Damping Force and Spring Force (Hooke’s Law)
̇ ̇ Where C is the damping coefficient and ̇ ̇ is the rate
Figure 2.1: Spring Mass Damper System
of change of distance
Where K is the spring constant and is the distance For the purposes of simplicity let
̇ ̇ ̇ ̈
Subbing in Newton’s second Law of Motion (where acceleration is )
̈ ̇ Manipulating to get
̈ ̇
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2.2. ½ Car Model Figure 2.2: Graphical Image of Car to be modelled
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 Angle
& Rate of Change of Pitch Angle ̇)
, Pitch
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 .
( ̇ ̇ ) Inversely for the rear suspension the as and ̇ increase and ̇ decrease ( ̇ ̇ ) Page 6 of 15
To calculate the height of the beam from the road and the rate of change of height From Newton’s 2
nd
Law of Motion
Therefore:
∑ Subbing in Equations for the forces acting upward on both the front and rear suspension and taking into account the weight of the car
̈ Where
̈ is the acceleration
Manipulating to get
̈ An Integrator is then applied to get the rate of change of height
∫̈ ̇ A second Integrator is applied to get the height of the beam from the road
∫̇
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2.3. Angle and rate of change of pitch From Newton’s 2
nd
Law of Motion for rotation
̈ Where
is Moment of Inertia and ̈ is the angular acceleration ∑
Moment due to front suspension
Moment due to rear suspension
Subbing into Newton’s 2
nd
Law of Motion for rotation
̈ Manipulating to get
̈ An Integrator is then applied to get the rate of change of pitch angle
∫ ̈ ̇ A second Integrator is applied to get the pitch angle
∫ ̇
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3. Main System and Subsystems In 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:
Figure 3.1
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3.2. Rear Spring Mass Damper System
Figure 3.2
3.3. Pitch Angle Outputs
Figure 3.3
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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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
Figure 3.4
3.5. Road Height Input
Figure 3.5
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3.6. Complete System
Figure 3.6
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4. Outputs 4.1. Road Height
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
4.6. Force on Front Suspension
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