Simple lateral vehicle dynamics model - arranging equations for solving
I have run into a problem when trying to model a simplified vehicle that has three degrees of freedom – roll, vertical and lateral motions. It is represented by a single mass with two links that model the suspension behaviour. The links connect the tyre-to-ground contact points (tc) to the body at the ‘roll centres’ (rc). The motion of the points tc and rc are described by known polynomial curves that are a function of the body position relative to ground, so the link lengths and angles to ground vary according to the motion of the body. However, rc is always directly below cg.
The suspension between the body and the tc points consists of three elements that are described by the body position and velocity relative to ground. Forces are also transmitted to the body through the two links and being massless these are calculated using static force and moment balance equations.
The input to the model is a lateral force (FY) but this force is shared between the two tc points in proportion to the vertical forces at these points. 544
I have set up the differential equations for the body in the usual way and have written the equations for the motion of the links. The problem I have is how to formulate the link force and moment equations so that they are solved using a Matlab ODE at the same time as the body motion.
The free body diagram for the left link is below, as an example. The suspension force FZSL is described in terms of the body motion, so I think this should be solvable. The points rc and tc are also described by the body motion and, consequently, FZRCL and FYRCL are scaled in proportion to the angle between the link and the ground (that is to say, their resultant is parallel to the link but their magnitudes are unknown). So, the remaining four forces need to be solved, but these are directly a function of FYL, which is a function of FZL, through the statics equations.
The problem I seem to have is finding a way of arranging the static equations for each link so that they are written in terms of the body motion. However, each attempt I make to rearrange them fails to do this. I perhaps have too many unknowns and not enough solvable equations.
The body equations are:
$$ma_Y=FYRC_L+FYRC_R$$
$$ma_Z=FZS_L+FZS_R+FZRC_L+FZRC_R–FZW$$
$$I_X\ddotθ_X=FZS_L \cdot tcy_L-FZS_R \cdot tcy_R+FYRC_L \cdot (cgz-rcz_L)+FYRC_R \cdot (cgz-rcz_R)$$
The equations for the left link are (taking moments about rc):
$$FY_L+FYRC_L=0$$
$$FZ_L+FZS_L+FZRC_L=0$$
$$FY_L \cdot rcz_L+FZS_L \cdot tcy_L+FZ_L \cdot tcy_L=0$$
The equations for the right link are similar.
The input to the system is FY, and:
$$FY_L=FY \cdot (FZ_L/(FZ_L+FZ_R ))$$
$$FY_R=FY-FZ_L$$
Sorry for the longwinded post but I’d be very grateful for any direction you could give.
Thanks. 101