Dissertation JC

8/3/2019 Dissertation JC

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School of Engineering, Technology and

Maritime Operations

Development of a dynamic vehicle model for the

prediction of subsystem and component loading

Draft Report

Name: Jack Clisby

Supervisor: Dr Christian Matthews

Date: 03 December 2010



Development of a dynamic vehicle model for the prediction of subsystem and component loading

Jack Clisby 382862 2

Contents

1 Introduction................................................................................................................................ 3

2 Aim ............................................................................................................................................ 4

3 Objectives .................................................................................................................................. 4

4 Literature review ........................................................................................................................ 5

4.1 SAE vehicle axis system ..................................................................................................... 5

4.2 Quarter car modelling ......................................................................................................... 5

4.2.1 Damper ....................................................................................................................... 7

4.2.2 Spring ......................................................................................................................... 9

4.2.3 Newton’s Equation of Motion ..................................................................................... 9

5 Dymola Modelling ................................................................................................................... 10

6 Conclusion ............................................................................................................................... 11

7 Further work ............................................................................................................................ 11

References ....................................................................................................................................... 12





1 Introduction

The Formula Student series is aimed at undergraduate students who wish to improve their knowledge

of Engineering applications. The students are required to design, build and race a single seat racing

car. The car must adhere to strict rules as published by the Society of Automotive Engineers (SAE),the rules are not to restrict design but to ensure safety and encourage solution Engineering. The nature

of the rules denotes that FS cars generally weigh 150 – 300 Kg with an average power output of

70kW. The competition has 3 entry levels. Class 1 requires a fully functional racing car, Class 2

requires a conceptually designed car and Class 1A requires a fully functional hybrid or alternative

energy car.

Liverpool John Moores University (LJMU) has been competing in the FS competition since 2008.

LJMU competed in class 1 for the first time at FS 2010; the car weighed 281 Kg and had a predicted

power output of 65kW. The team completed all events and finished 21 st Over All (OA).

To produce a competitive car for 2011 a strict weight budget has been defined by the team. Reducing

weight improves the handling characteristics and power/weight ratio. Main areas of weight reduction

are aimed at chassis, suspension and drivetrain assembly. These areas have been targeted, as they have

received little to no analysis in previous designs. Accurate analytical calculations and modelling

techniques have not been utilised and materials not critically refined in the target areas.





2 Aim

The aim of this report is to establish a vehicle model that can accurately predict subsystem and

component loading of a Formula Student (FS) racing car suspension system. The accurate prediction

of loading conditions enables components to be critically designed thus improving the performance of the vehicle.

3 Objectives

To build a functional dynamic model using Dymola© software. Dymola© is a comprehensive solution

for physical modelling of multi-domain systems to support hardware and control system design and

optimisation (1). The construction of this model will enable the prediction of forces and torques that

are transmitted through the suspension components. The focus will be on the A-arm structure to ensurethat maximum refinement and weight reduction is achieved. Cornering and weight transfer loads are

permanently transferred through the A-arm. Failure of the A-arm causes a situation where the vehicle

is not moveable under its own power and therefore would result in a Did Not Finish (DNF) at

competition.

Validation of the model is achieved with the use of CarSim® a vehicle simulation package. The

CarSim® package will validate the cornering and weight transfer loads that the vehicle is expected to

produce. Further validation will be achieved with dynamic testing on the vehicle. A physical quarter

car model utilising strain gauges will provide physical evidence of the calculated forces and torques

experienced at the A-arm. The validation is crucial in the development of computer based simulation

to ensure that the input parameters are behaving as expected and the results achieved quantifiable.

During the construction of the simulation model linear components will be utilised for simplicity.

Through development non-linear characteristics will be built into the model to improve accuracy

between the simulated and physical model. Specifically a non-linear damping model will replace the

linear model as this has been identified as the largest simplification during quarter car modelling (2).

Kinematics and compliance of the A-arm system will also be considered during the comparison

process.





4 Literature review

4.1 SAE vehicle axis system

Figure 4-1, SAE vehicle axis system (9) Figure 4-2, SAE tyre axis system (9)

The use of the SAE vehicle axis system enables a sign convention to be applied to the simulation.

Utilising a standard enables the transfer of data between different simulation packages to be consistent

and to allow for comparison to other authors work that have utilised the SAE axis system.

4.2 Quarter car modelling

Vehicle dynamic modelling is based on the principles of system dynamics. Simple spring mass (SM)

and spring mass damper (SMD) systems are found in many applications with numerous requirements

requiring motion and/or vibration control. A vehicle is a good example of dynamic systems as the

vehicle ride characteristics are directly related to the motions of systems due to inputs over a time

response.

Modern suspension systems are constructed using a complex mechanical linkage. The frequencies of

the systems are governed by a spring and/or viscous damper. The complexity due to non linearity and

compliance of the 3 dimensional (3D) system often results in simple lumped mass models being

created, these can be used to calculate the dynamic functions quickly and easily. The 2 degrees of

freedom (2DOF) quarter car model is the most commonly used lumped mass system. The quarter car

system has been criticised by some authors for the level of accuracy that can be achieved from the

model (2). However it has been widely proven that the quarter car model is accurate for basic

evaluation of body bounce and wheel hop using linear components (3) - (6).





The models accuracy can be improved through introducing non-linear components into the model for

more ‘realistic’ data (2) - (6). The quarter car model is constructed of four components, the un-sprung

mass, sprung mass, spring damper and Tyre. The un-sprung mass is the mass of the components

not supported by the spring damper see Figure 4-4, it is shown as a lumped mass. The sprung mass

is the mass of the vehicle supported by the spring damper see Figure 4-4 and is shown as a

lumped mass. The Spring damper is shown as a spring and damper in parallel. The tyre

is shown as a simple spring to simulate the tyre wall stiffness. This simplified model is used in the

linear quarter car model as often the effects of damping in the tyre can be neglected due to its visco-

elastic nature (7). The road excitations are determined by . There are various methods of

interpreting road data as an input. For the purpose of this paper, a sinusoidal input is generated at

varying frequencies and amplitudes. The sinusoidal input is often used as a method for testing car

models as inputs. A frequency sweep between 1.6 – 24.0 Hz at 0.001m of amplitude (2) will provide

sufficient data to visualise and characterise the quarter car model. The displacement of the un-sprung

and sprung mass is denoted by and respectivly.

Figure 4-3, spring mass damper model (7) Figure 4-4, Sprung and Un-sprung mass (8)





4.2.1 Damper

A damper is commonly used in the automotive suspension system due to its tuneable characteristics

which enable the control of motion. Critical damping is defined as the minimum amount of

viscous damping required in a linear system to bring the displaced mass back to equilibrium without

overshoot (6). The inherent nature of dampers creates a non-linear curve when oscillated at a high

frequency and amplitude. The non-linear damper is more complex to model than a linear

representation and research has shown that for quarter car models the linear damping characteristics

provide sufficient data for initial design stages, which outweighs the inaccuracies (2).

The LJMU10 dampers are FOX 5.0 DHX Figure 4-5, specifically designed for mountain bikes. They

are a gas damper that utilise the pressure of the chamber to control the bump and rebound. The damper

has two main adjusters that alter the resistive force acting on the bump and rebound. The rebound

adjuster opens an orifice that allows the gas to return at a higher rate. The Bump simply increases the

gas pressure in the cylinder and therefore increases resistance to bump forces.

Figure 4-5, Internals of FOX DHX 5.0 oil and air damper (8)

To implement the damper into the quarter car model the damping constant is required. This is

achieved through the use of Roehig 3VS damper dyno. Aurok Ltd who completed testing on behalf of

LJMU ran the damper through 10 tests. Each test constituted different settings to ensure a full map of

the damper characteristics. Aurok define damper settings as fully clockwise = 0 clicks (maximum

stiffness). Bump clicks are always listed first then rebound and then bottom out. Bottom out is stated

as complete revolutions. LJMU10 damper settings for Silverstone 2010 were 04/14-3. These settings

will be the focus of analysis and used to calculate the damping constant. The damping constant iscalculated below in [Equation 1.

[Equation 1] damping constant





Figure 4-6, Linear damping constant for FOX DHX 5.0

Figure 4-6 shows data from testing which clearly shows the non-linear characteristics of the damper.

Milliken (8) suggests that research on vehicles has proven that the wheel velocity in bump is higher

than the velocities in rebound by about a factor of two, the damper is designed with corresponding

asymmetry so the rebound coefficient is about twice that of bump. This and other factors such as

frictional stiction or hysteresis could be a cause of non-linearity. Newton’s second law of motion

governs the quarter car model. The damping component of the quarter car model is shown by

[Equation 2.

[Equation 2] damping force

y = 18.925x - 4.2401

-250

-200

-150

-100

-50

0

50

100

150

200

250

300

-15 -10 -5 0 5 10 15

F o r c e ( l b s )

Velocity (in/sec)

04_14_3 FOX DHX 5.0

04_14_3 FOX DHX 5.0

Linear (04_14_3 FOX DHX5.0)

Rebound

Bump





4.2.2 Spring

The spring in the system is to create a stiffness and reaction to the oscillating forces. It is a linear

system. The spring component of the quarter car model is shown in [Equation 3.

[Equation 3] Spring force

4.2.3 Newton’s Equation of Motion

By combing all of the elements of the quarter car model the equation of motion can be generated.

The governing differential equations of motion of the linear quarter car model in Figure 4-3 are:

[Equation 4] equation of motion for the Un-sprung mass

[Equation 5] equation of motion for the sprung mass

Where is the sprung mass and the un-sprung mass. is the damping constant , and are

the spring stiffness and tyre stiffness respectively. , and and are the displacements of the

Road input, un-sprung mass and sprung mass respectively.





5 Dymola Modelling

Figure 5-1, Dymola generated quarter car model

The model generated in Dymola Figure 5-1 is generated using the initial condition of the vehicle. The

simulation of the model creates the output seen in Figure 5-2.

Figure 5-2, comparison of un-sprung and sprung acceleration

This initial data from Dymola is showing the frequency response of the vehicle. Further analysis will

generate the natural frequency of the system. This can be calculated and serve as a validation of the

system.





6 Conclusion

The linear quarter car model will act as the basis for the development of non-linear quarter car model.

The linear model is a good representation of basic characteristics of the vehicle. However many non-

linear components and compliances in the connections create very non-linear data. To accuratelypredict the forces and torques experienced in the suspension components a non-linear model is

required. Many authors have researched and written papers on the subject and agree with the statement

above. The use of Dymola® enables for easy and efficient quarter car modelling that can then be

extended into a full car model. However this data is only useful if validated. The validation can be

proved mathematically and with the use of pre validated models in CarSim®.

7 Further work

Further development of the quarter car model to gain a full understanding and effects on the dynamic

response of the parameters. The ultimate aim is to create a model that can be utilised in a full car

model and be an accurate representation of the dynamic response of the vehicle. The Dymola quarter

car model will be validated using the mathematical model. Knowing this validation will enable the

Dymola model to be refined with non-linear components.





References

1. Claytex services LTD. Dymola Multi-Engineering Modeling and Simulation. claytex.com. [Online]

Claytex, 2010. [Cited: 09 11 2010.] http://www.claytex.com/dymola/.

2. An insight into linear quarter car model accuracy. Maher, Damien and Young, Paul. Dublin :

iFirst, 2010, Vols. 1-18. 1744-5159.

3. SAE J670e. Vehicle dynamics terminology. Warrendale : Society of Automotive Engineers, 1976.

4. Reduced-order modelling and parameter esimation for a quarter-car suspension system. Kim, C

and Ro, P I. D, Raleigh : IMechE, 2000, Vol. 214.

5. Estimation of suspension parameters. Majjad, R. Hertzstr : IEEE, 1997.

6. Gillespie, Thomas D. Fundamentals of vehicle dynamics. Warrendale : SAE, 1992. 1-56091-199-9.

7. Root mean square optimization criterion for vibration behaviour of linear quarter car using

analytical methods. Nakhaie Jazar, G, Alkhatib, R and Golnaraghi, M F. 6, Waterloo : Vehicle

system dynamics, 2006, Vol. 44. 1744-5159.

8. Milliken, William F and Milliken, Douglas L. Race car vehicle dynamics. Warrendale : SAE,

1995. 1-56091-526-9.

9. Pink Bike. [Online] [Cited: 03 12 2010.] http://photo.pinkbike.com/photo/932/pbpic932317.jpg.

10. A stochastic quarter-car model for dynamic analysis of vehicle with uncertain parmaeters. Gao,

Wei, Zhang, Nong and Dai, Jun. 12, Xi'an and Sydney : Taylor and Francis Group, 2007, Vol. 46.

1744-5159.

11. Nonlinear adaptive control of active suspensions. Alleyne, Andrew and Hedrick, J Karl. 1,

Illinois : IEEE, 1994, Vol. 3.

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