By Dieter Schramm, Manfred Hiller, Roberto Bardini
The authors learn intimately the basics and mathematical descriptions of the dynamics of cars. during this context assorted degrees of complexity could be offered, beginning with simple single-track types as much as advanced three-d multi-body versions. a specific concentration is at the strategy of developing mathematical versions at the foundation of actual automobiles and the validation of simulation effects. The equipment provided are defined intimately by way of chosen program scenarios.
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Extra info for Vehicle Dynamics: Modeling and Simulation
Example text
16 Definition of the second CARDAN rotation Fig. 17 Definition of the third CARDAN rotation Kinematic CARDAN equations The kinematic CARDAN equations provide the relationship between the angular velocity x of the body-fixed system ðx3 ; y3 ; z3 Þ relative to the initial system ðx0 ; y0 ; z0 Þ and the time derivatives of the CARDAN _ h; _ u. _ From Fig. 5 Rotational Motion 39 Fig. 18 Kinematic CARDAN equations _ x ¼ w_ þ h_ þ u: ð2:84Þ This vector equation can be interpreted in the initial system ðx0 ; y0 ; z0 Þ as 0 x ¼ w_ 0 ez0 þ h_ 0 ey1 þ u_ 0 ex2 : ð2:85Þ The components of x in the spatially-fixed system ðx0 ; y0 ; z0 Þ are extracted from this in the following: 3 2 0 xx 4 0 xy 5 ¼ 4 0 0 1 xz 20 À sin w cos w 0 32 3 cos w cos h w_ sin w cos h 54 h_ 5: À sin h u_ ð2:86Þ The components of x in body-fixed systems ðx3 ; y3 ; z3 Þ, commonly referred to as the kinematic CARDAN equations, read as follows: 23 3 2 À sin h xx 4 3 xy 5 ¼ 4 cos h sin u 3 cos h cos u xz 0 cos u À sin u 32 3 1 w_ 0 54 h_ 5: 0 u_ ð2:87Þ Singular Position: h ¼ p2 ; sin h ¼ 1; cos h ¼ 0.
This pair ~ b ¼ ðuðtÞ; bðtÞÞ ð2:60Þ is called the ‘‘rotational displacement pair’’ ~ b, and is comprised of the axis of rotation uðtÞ and a corresponding angle of rotation bðtÞ, (Hiller 1981). g. h; w and u, just as it is with EULER or CARDAN angles (Hiller 1983). In the following sections, the method of the rotational displacement pair ~ b ¼ ðuðtÞ; bðtÞÞ will be discussed as well as the method of the CARDAN angles, an especially popular approach in automotive engineering. 1 gives a summary of possible parameterizations, each having either three or four parameters.
E. fGi ¼ 6 (Fig. 1). 1 Topological Modelling Two topological principles can be distinguished in kinematic chains: open and closed. Open kinematic chains—tree structure The path from one body to each other body in an open kinematic chain with tree structure is unique. Within this tree, a preceding body together with its preceding joint, respectively, is assigned to each body. Choosing one body in the tree as a reference body, one gets nG ¼ nB ð3:1Þ with nG number of joints nB number of bodies (not including the reference body) D.
