Half Car Simulink Model

Half-Car Model John O’Donnell, Barry O’Donnell, David Rogers

Co ntents Introduction ........................................................................................... 3 Development ........................................................................................... 4 Assumptions ........................................................................................ 4 Development Process ............................................................................. 5 Final Model ......................................................................................... 9 Verification ....................................................................................... 11 Simulation and Testing of Model .............................................................. 12 Effect of Natural Frequency on Performance/Comfort ................................ 12 Testing ............................................................................................. 13 1.

Speed Bump: ........................................................................... 13

2.

Pothole: ................................................................................. 14

3.

Repeating Speed Bumps:............................................................ 14

Results ............................................................................................. 15 1.

Speed Bump: ........................................................................... 15

2.

Pothole: ................................................................................. 15

3.

Repeating Speed Bumps:............................................................ 15

Problems with Creating Speed Bumps ..................................................... 15 Verification ....................................................................................... 16 Conclusion ........................................................................................... 17 Bibliography......................................................................................... 17

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Appendix ............................................................................................. 18 [1] Derivations:............................................................................... 18 [2]Calculations of stiffness and damping coefficients for suspension: ....... 20 [4] Inputs used to create speed bump. .................................................. 21 [5] Resulting speed bump. ................................................................. 21 [6] Values for the two input steps. ...................................................... 21 [7] Displacement at 2.5km/hr ............................................................. 22 [8] Angle at 2.5km/hr ....................................................................... 22 [9]Displacement at front and rear at 2.5km/hr ....................................... 22 [10] Displacement at 5km/hr. ............................................................. 23 [11] Angle at 5km/hr. ....................................................................... 23 [12] Displacement at 10km/hr. ........................................................... 24 [13] Angle at 10km/hr. ..................................................................... 24 [14] Displacement at 20km/hr. ........................................................... 25 [15] Angle at 20km/hr. ..................................................................... 25 [16] Displacement at 30km/hr. ........................................................... 26 [17] Angle at 30km/hr. ..................................................................... 26 [18] Displacement at 40km/hr. ........................................................... 27 [19] Angle at 40km/hr. ..................................................................... 27 [20] Displacement for a pothole at 5km/hr. ........................................... 28 [21] Angle for a pothole at 5km/hr. ..................................................... 28 [22] Calculated values to be inputted into the pulse generators. ............... 29 [23] Inputs used to create series of speed bumps. .................................. 29 [24] Resulting speed bumps. .............................................................. 29 [25] Displacement for repeating speed bumps at 2.5km/hr. ...................... 30 [26] Angle for repeating speed bumps at 2.5km/hr. ................................ 30 [27] Input signals for a sample time of 10 seconds at 10km/hr. ................ 31 [28] Resulting speed bump. ............................................................... 31 [29]Displacement with insufficient sampling. ....................................... 32 [30] Angle with insufficient sampling. ................................................ 32 [31]Signal Builder. .......................................................................... 33 [32] Input signals and resulting bump for the table top bump creation. ...... 33 [33] Responses found by A. Kader et al. .............................................. 34

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Introductio n T h e a i m o f t h i s p r o j e c t w a s t o m o d e l a „ h a l f c a r ‟ s ys t e m . T h i s s ys t e m , w h i c h i s a l s o k n o w n a s a b i c yc l e m o d e l , c a n b e m o d e l l e d i n a n u m b e r o f w a ys u s i n g v a r i o u s p r o g r a m s o r m a t h e m a t i c a l m e t h o d s . The two main methods that were most suited to t he project were Simulink and SimMechanics. Both of these methods were available on the MATLAB program. SimMechanics is a more graphical approach that allows the program to create the relevant formulae, derived from the inputs selected by the operator, wher eas Simulink provides a more mathematical approach than practical. The Simulink application in the MATLAB program was used for this project. This approach was decided upon because it can be verified using mathematic formulae based on first principles of me chanics of machines, which is unavailable for other methods, like the SimMechanics method, for example. Providing these formulae were sound and that none of the major forces were neglected, the model produced would be accurate. The validity of the model wo uld be difficult to prove for the SimMechanics method. W i t h t h e m e t h o d s e l e c t e d , t h e s ys t e m w a s f i r s t a n a l ys e d t o d e t e r m i n e the main forces affecting the response of the model. Equations were then derived to simulate these forces and a mathematical model w as constructed. This model was then developed for the Simulink program, w h e r e v a r i o u s i n p u t s w e r e a p p l i e d t o a n a l ys e t h e r e s p o n s e o f t h e s ys t e m .

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Develo pment A s s u mp t i o n s T h e m o d e l i s b a s e d o n t w o l i n e a r s ys t e m s f o r f r o n t a n d r e a r s u s p e n s i o n . T h e s e s ys t e m s a r e w e l l e s t a b l i s h e d a n d r e a d i l y p r o v e n . T h e b o d y o f t h e car is represented by a mass connected to the front and rear suspension s ys t e m s . T h e d a m p i n g c o - e f f i c i e n t a n d s p r i n g s t i f f n e s s o f b o t h t h e t yr e and the main suspension element are taken into account in the model. T h e s e p a r a m e t e r s , t o g e t h e r w i t h t h e m a s s e s o f t h e w h e e l s a n d t h e b o d y, the inertia of the mass are included in the project as the major forces involved with the model. Some minor forces have been neglected from the model to reduce t he complexity of the mathematical model. These minor forces include a s p e c t s l i k e f l e x i n t h e b o d y, m o v e m e n t i n t h e b u s h e s o r b a c k l a s h f r o m g e a r s ys t e m s , e t c . T h e s e f o r c e s a r e s m a l l a n d t h e i r e f f e c t o n t h e v e h i c l e is minimal and hence they are left out. The mounting method for the suspension is also neglected which is one of the more important minor force elements. The model produced from the project assumes the suspension is linear when in a real vehicle the suspension would be fixed to a component connecting the wheel to the body of the vehicle, a wishbone is an example of this component. This was neglected so the parameters of the wishbone could be neglected. The mass, dimensions and inertia of the wishbone would need to be sought. While the mass and dimen sions can be simply obtained, the inertia would be a very complex mathematical operation. Neglecting this aspect reduced the complexity of the model w i t h o u t l o s i n g m u c h r e a l i s m . B y l e a v i n g t h i s e l e m e n t o u t o f t h e s ys t e m t h e v e h i c l e c a n j u s t b e a s s u m e d t o h a v e a s i m p l e r s u s p e n s i o n l a yo u t . The bump stops have also been left out of the model because including them would have involved a complex process that could not easily be proven. When the suspension travel runs out and the bump stop on the s u s p e n s i o n h i t s t h e b u m p s t o p o f t h e c a r b o d y t h e s u s p e n s i o n s ys t e m has no effect whatsoever on the response of the vehicle, effectively making the vehicle rigid, except for the relatively minor behaviour of t h e t yr e s . T h i s n o n - l i n e a r p a r a m e t e r w o u l d b e v e r y d i f f i c u l t t o q u a n t i f y and prove. The problem becomes more complex when both suspension elements hit the bump stops and the non -linear parameter changes from the condition where just one hits the bump stop. For the reasons outlined above, the minor forces and the excee dingly complex forces have been neglected. This means the model produced is valid within certain conditions: 4

1. 2. 3. 4. 5. 6.

Suspension does not run out of travel Suspension connected in a linear manner All bearings and fixings are fully rigid All bearings and fixings are frictionless All masses have no flex (i.e. no spring stiffness) The parameters do not change during operation of the vehicle

D e v e l o p me n t P r o c e s s

Figure 1: Half-car model.

U s i n g f i r s t p r i n c i p l e s a n d t h e s ys t e m s h o w n i n F i g u r e 1 a b o v e 2 equations were derived by equating the upward forces with the downward forces. Equation 1 relates the mass of the body of the car (ms) and the upwards acceleration of the car at the centre of gravity ( ) with the coefficients of the spri ngs (ks1), (ks2) and dampers (cs1), (cs2) in the suspension and their respective displacements and velocities. Equation 2 relates the moment of inertia of the car (I s) and the angular acceleration ( ) with the difference in the moments at the f r o n t o f t h e c a r a n d t h e r e a r o f t h e c a r . [ 1] Equation 1: Equation 2: Rearranging the above equations in order to find

and

we get:

Equation 3: 5

Equation 4: In the development of the model, common blocks of terms were noticed in equations 3 and 4. In order to reduce work load and maintain s i m p l i c i t y w i t h i n t h e m o d e l , s e v e r a l s u b s ys t e m s w e r e c r e a t e d w i t h i n the Simulink package. The common blocks and their corresponding s u b s ys t e m a r e s h o w n b e l o w i n F i g u r e 2 .

Figure 2: Subsystem blocks.

E q u a t i n g t h e s e s u b s ys t e m s a s a b o v e i t i s p o s s i b l e t o r e d u c e e q u a t i o n s 3 and 4. Equation 5:

Equation 6: When modelling in Simulink began we first asked ourselves what is it w e a r e t r yi n g t o f i n d ? I t w a s d e c i d e d t h a t t h e f i n a l g o a l s w e r e t o b e t h e displacement at the centre of gravity of the car and the angle of tilt of t h e c a r . T w o m a i n a p p r o a c h e s w e r e t a k e n w h e n t r yi n g t o d e v e l o p t h e mathematical model and are discussed below.

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The first involved modelling out equations 5 and 6 directly however this resulted in a huge error due to a n u m b e r o f d i f f e r e n t i a t i o n s . T h e m o d e l w a s a l s o i n c r e d i b l y d i f f i c u l t t o f o l l o w a n d m o d i f i c a t i o n o f t h e s ys t e m t o r e d u c e t h e e r r o r p r o v e d n e a r i m p o s s i b l e d u e t o i t s c o m p l e x i t y. T h i s m o d e l i s s h o w n i n F i g u r e 3 b e l o w .

Figure 3: First attempt at half-car model.

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The second approach involved taking the alr eady established quarter car model, which was derived in class, and expanding it in order to find the displacement and angle of tilt at the centre of gravity of the car. The quarter car model is shown below in Figure 4 and is derived from the spring-mass-damper equation. Equation 5:

where yr is the displacement of the road.

The full quarter car model consists of two of these in series where the output is the displacement of the car body at that point.

Figure 4: Quarter-car model.

Using equations 1 and 2, the half -car model was developed. As discussed earlier, Equation 1 relates the displacement outputs of the f r o n t s ys t e m a n d r e a r s ys t e m w i t h t h e o v e r a l l d i s p l a c e m e n t o f t h e c a r a t the centre of gravity, while Equation 2 relates the resulting moments at the front and rear of the car in order to establish the pitch. When fully developed (the full half -car model is shown later in Figure 6 ) t h e r e w a s s t i l l a s m a l l e r r o r w i t h i n t h e s ys t e m w h i c h c a u s e d a ramping effect in the outputs. The source of this error was found to be the differentiation in the suspension section of the quarter car model which caused a small error which was then added by a number of i n t e g r a t i o n s l a t e r o n i n t h e s ys t e m . I n o r d e r t o r e c t i f y t h i s , i n s t e a d o f differentiating the incoming displacement due to the unsprung mass ( yu ), t he vel ocit y ( ) was taken as a second out put from the wheel and t yr e s ys t e m a n d i n s e r t e d i n t o t h e s u s p e n s i o n s ys t e m a s a s e c o n d i n p u t . This is shown below in Figure 5.

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Figure 5: Removal of error.

Final Model The final half-car model is shown below in Figure 6. As discussed e a r l i e r , i t c o n s i s t s o f t w o q u a r t e r - c a r m o d e l s , t h e c o m m o n s u b s ys t e m blocks as shown in Figure 2, and the formation of equations 5 and 6. The input shown below is a sim ple step however more complicated inputs were used to obtain the results. The transport delay is used to vary the velocity of the car, with the input to the rear of the car being d e l a ye d b e h i n d t h a t o f t h e f r o n t o f t h e c a r . T h e t w o q u a r t e r - c a r s ys t e m s a r e u s e d t o e v a l u a t e t h e d i s p l a c e m e n t s ( and velocities ) of the sprung and unsprung masses at the front and the rear of the car. These values are then fed i n t o t h e s u b s ys t e m b l o c k s a s d i s c u s s e d e a r l i e r w h i c h r e s u l t i n t h e equations A, B, C and D from earlier. Equation 5:

Equation 6: By comparing equations 5 and 6 above to Figure 6 below, it is clear what is happening in the green blocks to the right of the diagram. E q u a t i o n 5 s u m s t h e f o u r s u b s ys t e m s A , B , C a n d D , m u l t i p l i e s t h e m b y -1 and then divides by m s to get the upwards acceleration of the car. By i n t e g r a t i n g t h i s t w i c e w e c a n g e t t h e u p w a r d s d i s p l a c e m e n t . S i m i l a r l y, Equation 6 sums blocks B with D and A with C and multiplies them by l 2 a n d l 1 r e s p e c t i v e l y. T h e r e s u l t i s d i v i d e d b y t h e m o m e n t o f i n e r t i a I s t o g i v e t h e a n g u l a r a c c e l e r a t i o n a t t h e c e n t r e o f g r a v i t y. B y i n t e g r a t i n g this twice the pitch of the car is found. 9

Figure 6: Final half-car model.

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Verification A case study was used to confirm the equations and methodology necessary for the project. The case study was „A half car model for d yn a m i c a n a l ys i s o f v e h i c l e s w i t h r a n d o m p a r a m e t e r s ‟ b y W . G o a e t a l . [reference] The authors also used a half car model but the damping e f f e c t o f t h e t yr e s w e r e n e g l e c t e d . T h i s f o r c e w a s i n c l u d e d i n t h e current project. The aim of the above study was to take into account the uncertainty of t h e p a r a m e t e r s u s e d i n t h e d yn a m i c a n a l ys i s o f v e h i c l e s . T h e parameters referred to are mass, spring stiffness, damping v alues, etc. Their concept is that although these values are accurate and reliable when the vehicle is first manufactured they can become unreliable after a period of time of operation of the vehicle. For example, the spring s t i f f n e s s a n d d a m p i n g c o - e f f i c i e n t o f t h e t yr e a r e d e p e n d e n t o n t h e p r e s s u r e i n s i d e t h e t yr e w h i c h c h a n g e s o v e r a r e l a t i v e l y s h o r t a m o u n t of time. T h e d a m p e r i n s i d e t h e m a i n s u s p e n s i o n s ys t e m i s a n e x a m p l e o f t h e parameter changing over a longer period of time. The damper uses an orifice with a small diameter to restrict the fluid flow converting k i n e t i c e n e r g y t o t h e r m a l e n e r g y. A s t h e f l u i d p a s s e s t h r o u g h t h e orifice continuously when the vehicle moves it gradually wears down the orifice, changing the damper co -efficient of the suspension gradually over a large period of time. The paper concludes from the results that the uncertainty of these parameters has an effect on the response of the vehicle, i.e. the natural frequency of the system changes with the changing parameters. This means that as the natural frequency shifts higher or lower than the designed value the vehicle will become uncomfortable for the passengers.

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Simula tion a nd Testing o f Mo del E f f e c t o f N a t u r a l F r e q u e n c y o n P e r f o r ma n c e / C o mf o r t The set value for the natural frequency for passenger vehicles is in the range between 1.5 to 2.1Hz. 1.4Hz is generally the natural frequency of the human body and therefore is the most uncomfortable value to choose because the oscillations of the vehicle will be „in tune‟ with the passenger, i.e. under repeated oscillations at 1.4Hz the displacement felt at the passenger‟s seat will be amplified and gradually increase, throwing the person from the seat. This could only be described as „uncomfortable‟ from the passenger‟s point of vi ew. Therefore a value h i g h e r t h a n t h i s i s n e e d e d f o r t h e m o d e l . ( 1 . 5 - 1 . 6 H z a r e v a l u e s t yp i c a l of buses and other low performance vehicles where passenger comfort i s a n e c e s s i t y) . A value of 2.2Hz is common among Formula One cars where the passenger‟s comfort is not an issue and the performance of the suspension is vital (i.e. maximum grip levels, etc.), hence a value less than this is required. 1.8Hz was chosen to provide a response that is a compromise between comfort and performance. Values for the spring stiffness constant and damping coefficient in the suspension were f o u n d u s i n g t h e v a l u e o f 1 . 8 H z a s t h e n a t u r a l f r e q u e n c y. [ 3] This gave the values as shown in Figure 7 below: ku1

200,000

ks1

3240

ku2

200,000

ks2

3240

cu1

500

cs1

2520

cu2

500

cs2

2520

Figure 4: Values for spring stiffness and damping coefficients in the tyre and suspension.

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Testing T h r e e d i f f e r e n t t yp e s o f r o a d d i s p l a c e m e n t s w e r e t e s t e d , a s p e e d b u m p , a pothole and repeating speed bumps. 1. Speed Bump: A speed bump of a width of 1.5m was chosen. This was found from research and closely matches speed bumps located in our area. Width of speed bump = 1.5m Various components needed to be calculated for each speed. As the speed increases the time taken to trav el over the speed bump decreases so the time delay between the front of the car and the back of the car has to be calculated. Also in terms of the signal being applied to the circuit, the length of time that the speed bump affects the car decreases, so the period of the wave decreases and new frequencies need to be calculated that are to be inputted into the signal generator for each speed. Speeds of 2.5km per hour ranging up to 40km/hr were chosen because realistically cars would never travel over a speed bump travelling over 40km/hr for fear of causing damage to the car and the discomfort it w o u l d c a u s e . [ 3] The speed bump is formed from a signal generator and two step signals as shown below in Figure 7:

Figure 8: Formation of speed bump.

A signal generator emits a sine wave with an amplitude of 15cm. Two step signals are then introduced. The first step signal is sent to correspond to the start of the positive side of the sign wave. The next 13

step has an initial value equal to that the final value of the first step i n p u t . [ 4] I t i s d e l a ye d t o o n e h a l f o f t h e p e r i o d o f t h e o s c i l l a t i o n o f the wave to create the speed bump. [5] When isolating the positive half of a sine wave to create the speed bump, the different values to be applied to the step inputs neede d to be resolved. As the speed changes, the frequency changes so new positions a l o n g t h e w a v e n e e d t o b e c a l c u l a t e d . [ 6] 2. Pothole: A 20cm deep pothole with a width of 75cm was modelled with a car travelling at 5km/hr. 3. Repeating Speed Bumps: Inputs are creat ed to simulate a series of speed bumps close together. The circuit to create this road condition is shown below. It consists of a signal generator creating a sine wave with a 15cm amplitude. Two pulse generators are added with a phase difference between th em, as shown below in Figure 8, which results in the elimination of the bottom h a l v e s o f t h e s i n e w a v e . T h i s c r e a t e s a s e r i e s o f s p e e d b u m p s . [ 2 3] [ 2 4]

Figure 9: Formation of repeating speed bumps. [22]

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Results 1. Speed Bump: R e s u l t s f o r 2 . 5 k m / h r . [ 7] [ 8] [ 9] R e s u l t s f o r 5 k m / h r . [ 1 0] [ 1 1] R e s u l t s f o r 1 0 k m / h r . [ 1 2] [ 1 3] R e s u l t s f o r 2 0 k m / h r . [ 1 4] [ 1 5] R e s u l t s f o r 3 0 k m / h r . [ 1 6] [ 1 7] R e s u l t s f o r 4 0 k m / h r . [ 1 8] [ 1 9]

2. Pothole: R e s u l t s f o r 5 k m / h r . [ 2 0] [ 2 1]

3. Repeating Speed Bumps: R e s u l t s f o r 2 . 5 k m / h r . [ 2 5] [ 2 6]

P r o b l e ms w i t h C r e a t i n g S p e e d B u mp s For speeds of 10km/hr and over, the simulation time had to be reduced to 4 seconds to ensure the sine wave that creates the speed bump has enough sample points to create an accurate sine wave. If the simulation time was left at 10 seconds a jagged formation in the input waves f o r m e d i n s t e a d o f a s m o o t h s i n e w a v e . [ 2 7] T h i s i l l u s t r a t e s t h a t a n insufficient number of sample points can result in the wrong resultant s i g n a l o r i n t h i s c a s e n o s i g n a l a t a l l . [ 2 8] A s s u c h t h e r e i s n o t e n o u g h t i m e f o r t h e r e s p o n s e o f t h e c a r t o b e d i s p l a ye d . [ 2 9] [ 3 0 ] Another way to create more sample points on the sine wave was needed. Using the signal builder a sine wave with more sample points per period was made. However when the simulation was run, not enough sample points were stored in memory so only two seconds of t h e r e a c t i o n w a s d i s p l a ye d . [ 3 1] However, these results were still too inaccurate as more signals were being misinterpreted and a simpler way to input the speed bump was devised. A table top speed bump was created instead using four ramps but the frequencies needed to model the time taken to go over the speed bump were retained. The width of the speed bump was broken into one quarter of its length as the incline, a half length as the flat top and the 15

l a s t q u a r t e r a s t h e d e c l i n e . T h e s ys t e m u s e d i s s h o w n b e l o w i n F i g u r e 9.

Figure 10: Formation of table top speed bump.[32]

The table was applied to 10km/hr again to give a less erratic response. [ 1 2] [ 1 3] T h i s i n p u t w a s u s e d t o o b t a i n t h e 1 0 , 2 0 , 3 0 a n d 4 0 k m / h r results for the speed bump.

Verification F r o m m a n u f a c t u r e ‟ s d a t a a n d f r o m o t h e r a n a l ys i s p e r f o r m e d e l s e w h e r e , the simulations are accurate. It can be seen that the simulation responses in Simulink match closely with the responses found by Adam Kader et al. response closely matches the response in the simulations. The speed bumps modelled in the simulations were longer than the speed bump modelled by Kader which accounts for the two stage r e s p o n s e . [ 3 3]

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Co nclusion In conclusion, the half-car modelled is accurate within the parameters that were set out earlier in the paper. Drawbacks include the restrictions in modification of the model which other methods, such as the SimMechanics method, do not suffer from as well as a difficulty in immediate understanding of the model when examined. The SimMechanics method is based more in the visual element so it is easier to see the working parts of the model. However, the mathematical model does have the advantage in that it is backed up by already established formulae whereas the SimMechanics approach has a less stable mathematical base. It would be possible to expand the model to a full car, however with the added degrees of freedom, new formulae would need to be derived in order to establish the relationship between each moving part, as such, a SimMechanics approach would be more ideal if expansion were required.

Bibliog ra phy http://www.sccs.swarthmore.edu/users/06/adem/engin/e12/lab4/ http://espace.library.uq.edu.au/eserv/UQ:132217/B5.1.pdf

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Appendix [ 1] D e r i v a t i o n s : We know that: 1. 2.

3. where 4.

Figure 5: Spring-Mass-Damper System

A p p l yi n g t h i s t o t h e s ys t e m s h o w n i n F i g u r e 1 : 5. 6. M u l t i p l yi n g a c r o s s b y L w e g e t : 7.

for the unsprung mass.

8.

for the sprung mass.

The displacement of the front and rear suspension units can be found using these equations with the front having “1” in the subscript and the rear having “2” in the subscript. The two traces required are the position of the centre of gravity of the sprung mass and the angle of the sprung mass with the horizontal. We can find the position of the sprung mass at the centre of gravity by adding the total vertical forces at the rear with the total vertical forces at the front such that: 18

9.

The displacement of the unsprung masses (

are related to the

road displacement and are derived through equations 7 and 8 from above. The pitch angle of the sprung mass can be found similarly by find ing taking the moments created at the front and rear of the car about the c e n t r e o f g r a v i t y, s u c h t h a t : 10.

In the above equation when the sprung mass rotates about the centre of gravity the forces generated by the front and rear suspensions act in opposite directions. Figure 2 below shows the full half -car model.

Figure 6: Half-car model.

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[ 2] C a l c u l a t i o n s o f s t i f f n e s s a n d d a m p i n g c o e f f i c i e n t s f o r s u s p e n s i o n : It is known from control theory that:

As such:

Using a damping ratio,

a n a t u r a l f r e q u e n c y,

and a mass

of 1000kg, the c and k values are found to be:

[ 3] T a b l e s h o w i n g t h e v a r i o u s s p e e d s a n d r e s u l t i n g f r e q u e n c i e s a n d time delays required. speed km/hr 2.5 5 10 15 20 25 30 35 40

speed m/hr 2500 5000 10000 15000 20000 25000 30000 35000 40000

speed m/s 0.694 1.389 2.778 4.167 5.556 6.944 8.333 9.722 11.111

period/2 1.08 0.54 0.27 0.18 0.135 0.108 0.09 0.08 0.0675

Frequency Hz 0.926 1.852 3.704 5.556 7.407 9.259 11.111 12.963 14.815

time delay sec 2.880 1.440 0.720 0.480 0.360 0.288 0.240 0.206 0.180

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[ 4] I n p u t s u s e d t o c r e a t e s p e e d b u m p .

[ 5] R e s u l t i n g s p e e d b u m p .

[ 6] V a l u e s f o r t h e t w o i n p u t s t e p s . sample time speed km/hr 5

step 1 1.08

step 2 1.35 21

10 15 20 25 30 35 40

0.54 0.36 0.27 0.216 0.18 0.15 0.135

0.68 0.45 0.34 0.27 0.23 0.19 0.17

[ 7] D i s p l a c e m e n t a t 2 . 5 k m / h r

[ 8] A n g l e a t 2 . 5 k m / h r

[ 9] D i s p l a c e m e n t a t f r o n t a n d r e a r a t 2 . 5 k m / h r

22

[ 1 0] D i s p l a c e m e n t a t 5 k m / h r .

[ 1 1] A n g l e a t 5 k m / h r .

23

[ 1 2] D i s p l a c e m e n t a t 1 0 k m / h r .

[ 1 3] A n g l e a t 1 0 k m / h r .

24

[ 1 4] D i s p l a c e m e n t a t 2 0 k m / h r .

[ 1 5] A n g l e a t 2 0 k m / h r .

25

[ 1 6] D i s p l a c e m e n t a t 3 0 k m / h r .

[ 1 7] A n g l e a t 3 0 k m / h r .

26

[ 1 8] D i s p l a c e m e n t a t 4 0 k m / h r .

[ 1 9] A n g l e a t 4 0 k m / h r .

27

[ 2 0] D i s p l a c e m e n t f o r a p o t h o l e a t 5 k m / h r .

[ 2 1] A n g l e f o r a p o t h o l e a t 5 k m / h r .

28

[ 2 2] C a l c u l a t e d v a l u e s t o b e i n p u t t e d i n t o t h e p u l s e g e n e r a t o r s . Pulse Pulse Generator Generator 1 2 speed Pulse phase Pulse km/hr Period width delay Period width 5 1.08 50 1.08 2.16 50 10 0.54 50 0.54 1.08 50 15 0.36 50 0.36 0.72 50 20 0.27 50 0.27 0.54 50 25 0.216 50 0.216 0.432 50 30 0.18 50 0.18 0.36 50 35 0.15 50 0.15 0.31 50 40 0.135 50 0.135 0.27 50

phase delay 0 0 0 0 0 0 0 0

[ 2 3] I n p u t s u s e d t o c r e a t e s e r i e s o f s p e e d b u m p s .

[ 2 4] R e s u l t i n g s p e e d b u m p s .

29

[ 2 5] D i s p l a c e m e n t f o r r e p e a t i n g s p e e d b u m p s a t 2 . 5 k m / h r .

[ 2 6] A n g l e f o r r e p e a t i n g s p e e d b u m p s a t 2 . 5 k m / h r .

30

[ 2 7] I n p u t s i g n a l s f o r a s a m p l e t i m e o f 1 0 s e c o n d s a t 1 0 k m / h r .

[ 2 8] R e s u l t i n g s p e e d b u m p .

31

[ 2 9] D i s p l a c e m e n t w i t h i n s u f f i c i e n t s a m p l i n g .

[ 3 0] A n g l e w i t h i n s u f f i c i e n t s a m p l i n g .

32

[ 3 1] S i g n a l B u i l d e r .

[ 3 2] I n p u t s i g n a l s a n d r e s u l t i n g b u m p f o r t h e t a b l e t o p b u m p c r e a t i o n .

33

[ 3 3] R e s p o n s e s f o u n d b y A . K a d e r e t a l .

34