Vehicle Dynamics Assignment: Half-Car Model Analysis

This document presents a comprehensive solution to a vehicle dynamics assignment focusing on a half-car model. The assignment involves modeling a vehicle's response to varying road surfaces using a 2D model comprising the vehicle body and unsprung masses at the front and rear. The solution employs spring and damper elements to represent tire stiffness and suspension, simulating the vehicle's behavior on a test rig. The methodology includes applying Newton's second law to formulate linear differential equations, representing the system in matrix form, and solving these equations using MATLAB's Simscape model. The results section presents the reaction forces and compression of the front and rear wheels when encountering road bumps. The discussion highlights the model's sensitivity, the application of Hooke's law, and the impact of initial values on the simulation's accuracy. References to relevant literature are also included. The assignment aims to analyze the effects of speed bumps with varying profiles on different vehicles.

INSTITUTIONAL AFFILIATION
ADVANCED DYNAMICS WITH APPLIED COMPUTER
MODELLING
2018/2019
ASSIGNMENT 2
VEHICLE DYNAMICS
STUDENT NAME
STUDENT REGISTRATION NUMBER
DATE OF SUBMISSION
2019

Vehicle Dynamics
INTRODUCTION
A half-car model is a 2D vehicle model used to model the vehicle’s response when driving over
various road surfaces. The model consists of the body of the vehicle, the unsprung mass at the
front, and the unsprung mass at the rear [1]. The tire stiffness in this design is modelled using the
spring and damper while the body accounts for the mass.
ASSUMPTIONS
(i) Car is tested on a test-rig and not on an actual road.
(ii) Road displacement are enforced when the vertical displacements are applied on the
lower end of the spring representing the tire.
(iii) The angular displacement of the car body may cause a change in the horizontal
distance between the attachment points.
The vehicle parameters for a medium-sized car are based on a 50/50 front-rear weight
distribution based on the use of similar spring and damper values at the front and rear are as
shown in the table below [2],
These parameters are measured for the entire car body. The suspension values and the unsprung
mass values are measured at the corner of the vehicle and hence are not an aggregate value of the
entire vehicle system.

Vehicle Dynamics
SYSTEM MODELLING
The linear suspension for the half car suspension model is obtained by applying Newton’s
second law of motion formulating linear differential equations to describe the dynamics of the
passive half car suspension [3],
Using the free body diagram for the half car model,
The rear tire and the front tire are considered rather stiff and when the tire hits a different road
profile, an upward force is transmitted to the wheels in a vertical acceleration of the wheels [4].
The suspension elements tend to dampen the vertical force for the transmission of the car body.
Representing the equations in matrix form, we obtain,

Vehicle Dynamics
Solving the matrix equations using the default parameters gives [5],
METHODOLOGY
Assuming 4 degrees of freedom on the half car vehicle dynamic model, the random road profile
is implemented over the given sample time and the length of the road pitch is certain.
Considering the road profile and modelling it with the simscape model [6],

Vehicle Dynamics
RESULTS
Running the simulation with the Simscape MATLAB models, the variables were uploaded on the
workspace as defined in the system parameter payload section. The default initial conditions are
specified in the introduction section [7].
For the front wheel, the reaction forces on hitting the road bumps were demonstrated as,
For the rear wheel, once the simulation hits the road bumps, the reaction forces are estimated as,

Vehicle Dynamics
When the vertical dynamics are evaluated in the simulation at different road bump heights, the
compression of the front and rear wheels are estimated as,
DISCUSSION
The simscape model was chosen because it represented the half car model using the mass spring
damper models for each wheel. The sensitivity of the model is measured using the forces
estimated in the design for the system dynamics. According to Hooke’s law, it can be stated that
force exerted on the spring is proportional to its change in length. Compression of the suspension
in his simulation is proportional to the corresponding forces shown in the figures in the results
section. The possible reason of this change in pitch and bounce is the exchange of force and
momentum between the front and rear suspensions [2]-[8].
The system is able to focus on the effect of the road profile in terms of the damping effect on the
spring and damper circuit at the tire. The error computed from the compression of the Mass
spring damper system below the chassis mass determines how sensitive the system is to the

Vehicle Dynamics
model design. Another plausible reason of this behavior is the inaccurate implementation of
initial values and car specifications. Solving this problem requires cumbersome trial and error
method.
REFERENCES
[1]. AE (Ames Engineering) (2009) http://www.amesengineering.com/amestxr.htm, accessed
on 22-01-2009.
[2]. ASTM (1996) Standard Test Method for Measuring Longitudinal Profile of Traveled
Surfaces with an Accelerometer Established Inertial Profiling Reference. Annual Book of
ASTM standards Vol. 04.03. E950-96.
[3]. Connon, W.H. (2000) Determining vehicle sensitivity to changes in test – course
roughness. IEST 46th Annual Technical Meeting and Exposition. pp 30–37.
[4]. Dodds, C.J. and Robson, J.D. (1973) The description of the road profile roughness.
Journal of Sound and Vibration, 31, 175–183. FCT (Face Construction Technologies,
Inc.) 2009. http://www.prestostore.com/cgibin/pro16.pl?ref=dipstick&pg=11398,
accessed on 22-01-2009
Gawade, T.R., Mukherjee, S. and Mohan, D. (2004) Wheel lift-off and ride comfort of
three-wheeled vehicle over bump. IE (I) Journal-MC, 85, 79–83.
[5]. Gonzalez, A., O’Brien, E.J., Li, Y.Y. and Cashell, K. (2007). The use of acceleration
measurements to estimate road roughness. Vehicle System Dynamics, 46 (6),
483–499.
Michael, W.S. and Steven, M.K. (1996) The little book of profiling. University of
Michigan, 2–5, 40, 41.
[6]. Min, B.H. and Jeong, W.U. (1994) Design method of test road profile for vehicle
accelerated durability test. Journal of KSAE, 2 (1), 128–141.
[7]. Ngwangwa, H.M., Heyns, P.S., Labuschagne, F.J.J., Kululanga, G.K. (2009)
Reconstruction of road defects and road roughness classification using vehicle
responses with Artificial Neural Networks simulation. Journal of
Terramechanics, 47 (2), 97–111.
[8]. Ngwangwa, H.M., Heyns, P.S., Labuschagne, K.F.J.J. and Kululanga, G.K. 2008. 27th
Southern African Transport Conference (SATC), pp. 312–319.