The title should be concise but informative

Manuscript submitted to Theoretical & Applied Mechanics

Dynamic analysis of special vehicles S. Tengler, 1)

1, a)

A. Harlecki,

1, b)

University of Bielsko-Biala, Department of Mechanics,

Willowa 2, 43-309 Bielsko-Biała, Poland a) E-mail: [email protected] b) E-mail: [email protected]

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 are an example of these types of vehicles. Algorithms for generating the equations of motion have been formulated by homogenous transformations and Lagrange’s equations. 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. Keywords dynamics, special vehicles, high gravity center, Lagrange’s equations

Determining position and orientation of system bodies Figure 1.

Defining joints between the bodies

Generating and solving equations of motion

The main stages of the analysis of dynamics of multibody systems

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 were used. Authors of the proposed method use joint coordinates basing on the approach applied in robotics[3]. 2. Mathematical formalism 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 figure 2) is determined by the transformation matrix: R ( , ,  ) d 3 x1  , T   3 x3 01x 3 1  

(1)

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

Xˆ i 1

{i}

d Yˆi 1

Zˆ i







Yˆi

Xˆ i

1. Introduction

Figure 2. Location of the body {i} in relation to the body {i-1}

In the case of designing special vehicles with high gravity center, an important issue is a phase of preparation and examination of a virtual vehicle prototype in scope of the analysis of its dynamics. For needs of the dynamics 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 figure 1).

Using Lagrange’s equation formalism equations of motion are derived[1,2,3]. Constraint equations are usually complementary to the equations of motions. The system of differential equations, which can be presented in a matrix form, is obtained as the result:

  Dr  f t , q, q  At , q q ,  T   w D q

(2)

1

2

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

cabin (1dof) car body (0 dof) frame (6 dof)

rear axle (2 dof)

left wheel (2 dof)

right wheel (2 dof) left wheel (1 dof) right wheel (1 dof)

Figure 3. Tree structure of the vehicle model

3. Vehicle model 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 were distinguished: frame, cabin, car body, front and rear axle, wheels connected by 6 spring-damper elements, respectively (see figure 4). 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 figure 3). The vehicle model in question has 17 degrees of freedom.

For each body of the vehicle model the vector of generalized coordinates was determined:

q f  [ x f , y f , z f , f , f , f ]T

-

frame; q c  [ c ]T - cabin; q b  {Ø} - car T body fixed to frame; q a ,i  [ z a ,i , a,i ] - axles f f f T ( i  1 front, i  2 rear); q w,k  [ w,k , w,k ] ,

q rw,k  [ wr ,k ]T - wheels ( f - front, r - rear; k  1 right, k  2 left).

Figure 4. The vehicle model

2

front axle (2 dof)

Manuscript submitted to Theoretical & Applied Mechanics

As the result, the generalized coordinates for the vehicle model can be presented in the form of the vector: T T T T q  q f T , q cT , qbT , q a,1T , q a,2T , q wf ,1 , q wf ,2 , q rw,1 , q rw,2   

the road tests on the professional track. To evaluate an error of calculations the following criterion was taken into account [3]:

T

(3) 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 equations (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 𝑓

courses of steer angles 𝜓𝑤,𝑘 of the front wheels in the case of different maneuvers. To determine reaction forces the Fiala model of tires[4] was used.

E

X O 100% , X

where:

(4)

tk

tk

0

0

X   x(t )dt , O   o(t )dt

- values

obtained from experimental measurements and calculations, respectively; 𝑡𝑘 - simulation period. 4.1. The lane change The lane change maneuver was made at the constant vehicle velocity of 50 km/h. Preservation of this value was assured by suitable course of drive torques applied to the rear wheels determined by using the PID controller.

4. Computer simulations In this paper, some results of two types of computer simulations: „a lane change” and „negotiating a circular track” were presented. To perform computer simulations, the authors’ computation package, called MBSolver, was used. This constantly updated package, in comparison to its previous version described in the paper[3], offers new possibilities both in scope of calculations (Fiala and Pacejka tire models[4], PID controller steering vehicle trajectory and velocity) and visualization (see figure 5).

a)

2 1

3

4 Figure 5. Examples of screenshots in the case of visualization

b)

of the vehicle model motion while simulating a lane change and negotiating a circular track

Figure 6. a) Course of steer angles of the front wheels,

b) Courses of lateral displacement of the origin

The computer simulation results were compared with the experimental results obtained from measurements performed during

of the frame coordinate system (1-measurements, 2-calculations) and frame yaw angle (3-measurements, 4-calculations) 3

4

The course of steer angles of the front wheels was assumed as an input function (see figure 6a). The results concerning courses of two exemplary parameters of the vehicle motion were presented in figure 6b. The analysis confirmed good compliance of the results of the computer simulations and the measurements (the relative error E in the both cases was about 3%).

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 was presented in figure 7b. Good compliance of the both kinds of results can be observed also in this case (the relative error E was about 5%).

4.2. Negotiating the circular track

The performed simulations showed that the relative errors, defined according to the assumed criterion, were relatively small. They confirmed correctness of the proposed method of dynamics 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 basic constructional parameters of an analyzed vehicle at which it loses its stability.

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.

References

a)

1

2

b) Figure 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)

Course of lateral velocity 𝑦𝑓̇ of this point was obtained from measurements (see figure 7a). In the calculations, preservation of this course was assured by suitable course of the steer angles of the front wheels determined by using PID controller. The PID controller was 4

5. Summary

1. Szczotka M., Tengler S.: 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. 2. Szczotka M., Tengler S., Wojciech S.: Application of joint co-ordinates and homogenous transformations to modelling of car dynamics, Proc. of 8th Conference on Dynamical Systems Theory and Applications, Lodz, 2005. 3. Warwas K., Tengler S.: Computer program for the simulation of physical structures based on the joint co-ordinates and homogenous transformations, ISBN 83-916420-5-4, Cracow, 2009. 4. MSC.ADAMS, Using Handling Force Models, Documentation. Acknowledgements This paper was presented at 11th Conference on “Dynamical Systems – Theory and Applications”, Lodz, Poland, 2011. Investigation was supported by National Science Centre in Cracow under doctoral research grant 0630/B/T02/2011/40.