Three-Dimensional Analysis of Vehicle Stability Using Graph Theory

Abstract

Vehicle stability is a widely studied topic today. It is crucial that we develop a better understanding of one of the main problems of vehicular accident rate problems throughout the world: the rollover accident. The main goal for researchers is to determine a way to predict vehicle behaviour under a variety of circumstances. Davies method is a mathematical tool that allows the static and kinematic analysis of any kind of mechanisms, as well as we can find in vehicle suspensions. This method uses the Graph theory that enables kinematic chain representation by means of a graph for later analysis. In this paper we present the vehicle stability kinematic analysis using Graph theory, Screw theory and the Davies method.

Appendices

Appendix A. Wrench Parameters of the Mechanism

Constraints and forces	s i	$s_{0i}$
F x1	1 0 0	0	0	0
F y1	0 1 0	0	0	0
F z1	0 0 1	0	0	0
M y1	0 1 0	0	0	0
M z1	0 0 1	0	0	0
F z2	0 0 1	0	t cos θ 1	t sin θ 1
F z3	0 0 1	−L	t cos θ 4	t sin θ 4
F x4	1 0 0	−L	0	0
F y4	0 1 0	−L	0	0
F z4	0 0 1	−L	0	0
M y4	0 1 0	−L	0	0
M z4	0 0 1	−L	0	0
W	0 0 −1	−a 1	t/2	h
m a y	0 −1 0	−a 1	t/2	h

Appendix B. The Equations System from the Statics of the Mechanism

$$\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & t\cos \theta _{1} & t\cos \theta _{4} & 0 & 0 & 0 & 0 & 0 & | & { - {\raise0.7ex\hbox{$t$} \!\mathord{\left/ {\vphantom {t 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} & h \\ 0 & 0 & 0 & 1 & 0 & 0 & L & 0 & 0 & L & 1 & 0 & | & { - a_{1} } & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & - L & 0 & 0 & 1 & | & 0 & {a_{1} } \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & | & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & | & 0 & { - 1} \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & | & { - 1} & 0 \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {F_{{x1}} } \\ {F_{{y1}} } \\ {F_{{z1}} } \\ {M_{{y1}} } \\ {M_{{z1}} } \\ {F_{{z2}} } \\ {F_{{z3}} } \\ {F_{{x4}} } \\ {F_{{y4}} } \\ {F_{{z4}} } \\ {M_{{y4}} } \\ {M_{{z4}} } \\ {\text{--}} \\ {\text{W}} \\ {ma_{y} } \\ \end{array} } \right] = \left[ 0 \right]_{{6 \times 1}}$$