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242 Multibody Systems Approach to Vehicle Dynamics
demonstrated with Body 2 the following equations would be generated for
all the bodies:
∑ { F} m { A G }
∑{ M } [ I ] { 3} [ 3] [ I ] { 3}
∑
∑{ M } [ I ] { 4} [ 4] [ I ] { 4}
∑
2 1 2 2 1
∑ { M } [ I ] { 2 [ 2 [ I {
/ / } / ] / ] / 2}
/
G2 12 2 2 2 1 2 1 2 2 2 2 1 2
∑ { F} m { A G }
3 1 3 3 1
G3 13 / 3 3/ 3 1/ 3 1/ 3 3 3/ 3 1/
3
{ F} m { A G }
4 1 4 4 1
G4 14 / 4 4/ 4 1/ 4 1/ 4 4 4/ 4 1/
4
{ F} m { A G }
5 1 5 5 1
(4.340)
(4.341)
(4.342)
(4.343)
(4.344)
(4.345)
(4.346)
∑{ M } [ I ] { 5} [ 5] [ I ] { 5}
G5 15 / 5 5/ 5 1/ 5 1/ 5 5 5/ 5 1/
5
(4.347)
The equations of motion above yield 24 equations leaving one further
equation to be derived to solve the 25 unknowns. The final equation allows
us to formulate the scalar F s with the appropriate magnitude and sign to
represent the force acting along the strut. Referring back to the discussion
of spring and damper forces in Chapter 3 we are reminded that for a linear
formulation based on the spring stiffness k, free length L, and the damping
coefficient c, of the damper we can formulate the force using
F s k(L |R CI |) c VR CI (4.348)
The term (L |R CI |) represents the deflection of the spring relative to the
free length. The term VR CI represents the radial line of sight velocity. This
is effectively the magnitude of the velocity vector {V CI } 1 given a sign so
that VR CI is negative when points C and I are approaching each other in
bump and is positive when separating in rebound. The result of this is that
the component of spring force is positive when the spring is compressed
and the damper force component is positive during bump movement.
4.10.6 Geometry analysis
The preceding use of vectors to carry out three-dimensional velocity, acceleration,
static force and dynamic force analyses of the double wishbone
suspension system should have provided the reader with an insight into the
computational work performed by an MBS program during the solution
phase. An important aspect of this is that in all the preceding analyses the
geometry has been assumed fixed throughout the solution. This is in fact
not fully representative of the problem. For example, for the static analysis
the damper acting between C and I is assumed to be locked so that although