THEORETICAL & APPLIED MECHANICS LETTERS 2, 043014 (2012)

Dynamics of vehicles with high gravity centre

Szymon Tengler,a) and AndrzeiHarleckib)

University of Bielsko-Biala, Department of Mechanics, Willowa 2, 43-309 Bielsko-Biala, Poland (Received 5 June 2012; accepted 24 June 2012; published online 10 July 2012)

Abstract The vehicles with high gravity centre are more prone to roll over. The paper deals with a method of dynamics analysis of fire engines which is an example of these types of vehicle. Algorithms for generating the equations of motion have been formulated by homogenous transformations and Lagrange's equation. The model presented in this article consists of a system of rigid bodies connected one with another forming an open kinematic chain. Road maneuvers such as a lane change and negotiating a circular track have been presented as the main simulations when a car loses its stability. The method has been verified by comparing numerical results with results obtained by experimental measurements performed during road tests. © 2012 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1204314]

Keywords dynamics, vehicles, high gravity center, Lagrange's equation

In the case of designing vehicles with high gravity center, an important issue is a phase of preparation and examination of a virtual vehicle prototype in view of the analysis of its dynamics.1'2 For the needs of the dynamic analysis, vehicles with high gravity centre can be modeled as multibody systems in a form of open kinematic chains. Generally, the analysis of dynamics of such systems can be divided into three basic stages (see Fig. 1).

In order to determine a location, thus a position and an orientation, of particular bodies constituting the multibody system, generalized coordinates which can be divided into two types: absolute and joint coordinates based. Authors of the proposed method use joint coordinates based on the approach applied in robotics.4

Determining position and orientation of system bodies

Defining joints between the bodies

Generating and solving equations of motion

Fig. 1. The main stages of the analysis of dynamics of multibody systems.

{i - 1} A Zi-1

Fig. 2. Location of the body {¿} in relation to the body {¿-1}.

a) Corresponding author. Email: stengler@ath.bielsko.pl.

b) Email: aharlecki@ath.bielsko.pl.

In mathematical formalism based on homogenous transformations, the position and the orientation of a body in relation to the preceding body in the structure of an open kinematic chain (see Fig. 2) are determined by the transformation matrix

#3x3 d3xl

0ix3 1

where R and d are rotation matrix and position vector of the body {i} in relation to the preceding body {i-1}, respectively.

Using Lagrange's equation formalism equations of motion are derived.3'4 Constraint equations are usually complementary to the equations of motion. The system of differential equations, which can be presented in a matrix form, is obtained as the result

A (t, q) q - Dr = f (t, q, q), d T q = w,

where t is the time; q, q and q are vectors of generalized coordinates, velocities and accelerations, respectively; A is the mass matrix; f is the vector of generalized forces and derivatives of kinetic energy, potential energy, Rayleigh's dissipation function; r is the vector of unknown reaction forces; D is the matrix of coefficients corresponding to particular reaction forces; w is the vector of right sides of constraint equations.

The subject of this paper is a fire engine of typical construction, which is an example of a vehicle with high gravity centre. In the vehicle model, assumed in the form of open kinematic chain, 9 subsystems as rigid bodies are distinguished: frame, cabin, car body, front and rear axle, wheels connected by 6 spring-damper elements, respectively (see Fig. 3).

The structure of the discussed chain can be considered as a tree structure, in which each body has a determined number of degrees of freedom (dof) in relation to the preceding body (see Fig. 4). The vehicle model in question has 17 degrees of freedom.

Fig. 3. The vehicle model.

Fig. 4. Tree structure of the vehicle model.

For each body of the vehicle model the vector of generalized coordinates is determined: qf = [xf, yf ,Zf ,df, ^f]T — frame; qc = [0C]T — cabin; qh = {0} — car body fixed to frame; qM = [zM]T —

axles (i = 1 front, i = 2 rear); qWk = kk]T,

qW k = [^W k]T — wheels (f — front, r — rear; k = 1 right, k = 2 left).

As the result, the generalized coordinates for the vehicle model can be presented in the form

r f f r r ]

q = qf, qc, qb, qa ,i, qa , 2, qw , 1, qw , 2 , qw ,1 , qw , 2 J

In the analyzed model a rear drive system was taken into account. Its operation was modeled by drive torques applied to the rear wheels of the vehicle, according to the rule of the open symmetric differential gear. The generalized forces included in the Eqs. (2)

result from driving torques and reaction forces of the road surface which acted on the vehicle wheels (taking a selected model of tires into account). The constraint equations result from courses of steer angles k of the front wheels in the case of different maneuvers. To determine reaction forces the Fiala model of tires5 is used.

In this paper, some results of two types of computer simulations: "a lane change" and "negotiating a circular track" are presented. To perform computer simulations, the computation package of the authors, called MBSolver, is used. This constantly updated package, in comparison to its previous version described in the doctoral dissertation,4 offers new possibilities both in scope of calculations (Fiala and Pacejka tire models,5 PID controller steering vehicle trajectory and velocity) and visualization (see Fig. 5).

The computer simulation results are compared with the experimental results obtained from measurements performed during the road tests on the professional track. To evaluate an error of calculations the following criterion is taken into account4

• 100%,

where X = f^ x(t) d t and O = f^ o(t) d t are values obtained from experimental measurements and calculations, respectively.

The lane change maneuver is made at the constant vehicle velocity of 50 km/h. Preservation of this value is assured by suitable course of drive torques applied to the rear wheels determined by using the PID controller.

The course of steer angles of the front wheels is assumed as an input function (see Fig. 6(a)). The results concerning courses of two exemplary parameters of the vehicle motion are presented in Fig. 6(b). The analysis

043014-3 Dynamics of vehicles with high gravity centre

Theor. Appl. Mech. Lett. 2, 043014 (2012)

Fig. 5. Examples of screenshots in the case of visualization of the vehicle model motion while simulating a lane change and negotiating a circular track.

2 3 4 5

0.25 0.20 0.15 0.10 0.05 0

0 1 2 3 4 5

Fig. 6. (a) Course of steer angles of the front wheels; (b) Courses of lateral displacement of the origin of the frame coordinate system (1—measurements, 2—calculations) and frame yaw angle (3—measurements, 4—calculations).

15 t/s (a)

)r\ Û

- ! ; J s-j j 2

\.............................j...........................-

0 5 10 15 20 25 30

t/s (b)

Fig. 7. (a) Course of lateral velocity of the origin of frame coordinate system; (b) Courses of steer angles of the front wheels (1—measurements, 2—PID controller).

confirm good compliance of the results of the computer simulations and the measurements (the relative error E in both cases is about 3%).

In the case of this simulation, the origin of the frame coordinate system moved along a circular trajectory with a radius of 22.5 m.

Course of lateral velocity of this point is obtained from measurements (see Fig. 7(a)). In the calculations, preservation of this course is assured by suitable course of the steer angles of the front wheels determined by using PID controller. The PID controller is also used to steer course of angles of the front wheels simulating, in such a way, the vehicle driver trying to keep the predetermined trajectory. The calculation result compared with the measurement result is presented in Fig. 7(b). Good compliance of both kinds of results can be observed also in this case (the relative error E is about 5%).

The performed simulations show that the relative errors, defined according to the assumed criterion, are relatively small. They confirm correctness of the proposed method of dynamic analysis. In the authors' opinion, conclusions resulting from the computer simulations of vehicle motion can provide essential guidelines for designers. It should be emphasized that these simulations will enable to determine limit values of ba-

sic constructional parameters of an analyzed vehicle at which it loses its stability.

This paper was presented at the 11th Conference on Dynamical Systems—Theory and Applications, December 58, 2011, Lodz, Poland. This work was supported by National Science Centre in Cracow under doctoral research grant 0630/B/T02/2011/40.

1. R. Andrzejewski, and J. Awrejcewicz, Nonlinear Dynamics of Wheeled Vehicle (Springer Verlag, Berlin, 2005).

2. J. Awrejcewicz, Classical Mechanics, Dynamics (Springer Verlag, New York, 2012).

3. M. Szczotka, and S. Tengler, Numerical Effectiveness of Models and Methods of Integration of the Equations of Motion of a Car, Differential Equations and Nonlinear Mechanics, Article ID 49157, (2007).

4. S. Tengler, Analysis of Dynamics of Special Vehicles with High Gravity Centre, [Ph.D. thesis], Faculty of Mechanical Engineering and Computer Science, University of Bielsko-Biala, (2012).

5. MSC. ADAMS Documentation, Using Handling Force Models.