The chassis and the car body subsequently will be treated as one single rigid body.
Effects like torsion or other deformations of the chassis will not be taken into account. The chassis can freely move in space. To describe its position and ori- entation in space, a vehicle-fixed reference frameKV ẳfOV;xV;yV;zVg will be introduced. The chassis-fixed reference point OV is located in the vehicle center plane, between the front wheels, on the level of the wheel center (DIN1994). The x-axis points into the direction of the longitudinal axis of the vehicle, they-axis into the lateral direction of the vehicle to the left, and thez-axis upwards in the vertical direction.
For the complete spatial description of the chassis, it is suitable to use for the translational part the three components ExV;EyV;EzV of the position vector rV expressed in coordinates of the inertial system, as well as the three CARDAN angles wV (yaw angle), hV (pitch angle), and uV (roll angle). The introduced reference frames are shown in Fig.5.1. The orientation of the vehicle-fixed system with respect to the inertial system is uniquely determined by the CARDAN angles.
The spatial motion can be illustrated, by starting with an initial position, where the orientation of the inertial system and the vehicle system coincide. Then the vehicle successively executes the three rotations of the CARDAN angles around the well- defined axes, as follows (Fig.5.2):
D. Schramm et al.,Vehicle Dynamics, DOI: 10.1007/978-3-540-36045-2_5, Springer-Verlag Berlin Heidelberg 2014
93
OV
Fig. 5.1 Reference frames for the description of the vehicle body
3.
2= 3=
1.
driving direction
1
1.
2.
3 = 3.
2.
3 =
1.
2.
3. 1= y2 2
= 1
Fig. 5.2 Definition of the CARDAN angles
• The system xE;yE;zE turns into system x1;y1;z1 with z1ẳzE, by rotating around thezE-axis with the yaw anglewV:
• The system x1;y1;z1 turns into system x2;y2;z2 with y2=y1, by rotating around they1-axis with the pitch anglehV:
• The system x2;y2;z2 turns into system xV;yV;zV with xV ẳx2, by rotating around thex2-axis with the roll angleuV.
For the transformation of the coordinates of an arbitrary vectorVri, given in components of the vehicle system (Fig.5.1), into the inertial system, the following relation holds:
E
VriẳETVV
Vri; ð5:1ị
with
E
Vri vector in coordinates of the inertial system,
V
Vri vector in coordinates of the vehicle system,
ETV transformation matrix vehicle!inertial system
The calculation of the transformation matrix in Eq. (5.1), based on CARDAN angles as rotational parameters, can be found in (Hiller1983). Omitting the index
‘‘V’’at the CARDAN angles one gets:
ETV ẳ
chcw sushcwcusw cushcwþsusw chsw sushswþcucw cushswsucw
sh such cuch
2 4
3
5: ð5:2ị
For the inverse transformation from the inertial to the vehicle system this follows, using the transposed transformation matrixVTE (see also Chap.2). The inverse (transposed) transformation is of great importance here, since all vectors and tensors, used for the calculation of the equations of motion are given in components of the vehicle system.
Based on CARDAN angles as kinematic parameters for the description of spatial rotations, also the corresponding angular velocities and angular accelera- tions, respectively, can be expressed with the help of the time derivatives of the CARDAN angles, and one gets the well-known kinematic CARDAN equations (Hiller1983). In the context of this book, the required kinematic equations for the angular velocity and angular acceleration, respectively, of the vehicle, will be expressed in components of the vehicle system (compare Sect.2.55):
VxV ẳ
sinh 0 1 coshsinu cosu 0 coshcosu sinu 0 2
4
3 5
w_ h_ u_ 2 4
3
5; ð5:3ị
Vx_V ẳ
sinh 0 1 coshsinu cosu 0 coshcosu sinu 0 2
64
3 75
w€
€h u€ 2 64
3 75
þ
chh_ 0 0 shsuh_þchcuu_ suu_ 0 shcuh_chsuu_ cuu_ 0 2
64
3 75
w_ h_ u_ 2 64
3 75:
ð5:4ị
The upper left index in Eqs. (5.3) and (5.4) indicates the coordinate system in which the components are given, the lower right index ‘‘V’’ states that one is actually regarding the motion of the vehicle system. An index on the lower left side indicates in case of a relative motion the reference system; in the case that the reference system is the inertial system, the index is omitted. Using the angular velocity and the angular acceleration of the vehicle, one can now calculate the absolute velocity and the absolute acceleration, respectively, of arbitrary vehicle fixed points:
_
riẳr_VỵxVVri; ð5:5ị
€riẳ€rVỵx_VVriỵxVðxVVriị: ð5:6ị For the statement of the equations of motion of the vehicle model, one has to calculate mainly kinematic expressions, like velocities or accelerations, which are
‘‘physical’’ vectors, to be represented in components in arbitrarily chosen coor- dinate systems. In case of the vehicle model under consideration, it is suitable to represent all vectors in the vehicle-fixed system. In this system, a large number of position vectors are constant, while others (e.g. in the area of wheel suspensions) can be transformed by simple transformations of the vehicle system.