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228 Multibody Systems Approach to Vehicle Dynamics
This results in the following set of 20 unknowns that must be found to solve
for static equilibrium:
F A31x , F A31y , F A31z
F B31x , F B31y , F B31z
F D43x , F D43y , F D43z
F E21x , F E21y , F E21z
F F21x , F F21y , F F21z
F G24x , F G24y , F G24z
f S1 , f S2
The problem can be solved by setting up the equations of equilibrium for
Bodies 2, 3 and 4. The use of scale factors to model the forces acting along
Body 5 and the strut, Bodies 6 and 7, means that these bodies cannot be
used to generate any useful equations to solve the problem. Thus we could
generate 18 equations as follows:
For Body 2 summing forces and taking moments about point G gives
∑{F 2 } 1 {0} 1 (4.261)
∑{M G2 } 1 {0} 1 (4.262)
For Body 3 summing forces and taking moments about point D gives
∑{F 3 } 1 {0} 1 (4.263)
∑{M D3 } 1 {0} 1 (4.264)
For Body 4 summing forces and taking moments about point G gives
∑{F 4 } 1 {0} 1 (4.265)
∑{M G4 } 1 {0} 1 (4.266)
This leaves us with the requirement to generate another two equations for
solution. The answer comes from a more considered study of the connections
or mounts between the upper and lower wishbones and the ground
part. Four possible MBS modelling solutions are shown in Figure 4.74.
In Figure 4.74(a) the wishbone is mounted using two bush force elements.
Using this configuration the wishbone is mounted on an elastic foundation
and the body has 6 rigid body degrees of freedom relative to the part on
which it is mounted, which for this example is a non-moving ground part.
If the actual wishbone is mounted on the vehicle in this way this would be
the MBS modelling solution of choice if as discussed earlier the simulation
aimed to produce accurate predictions of the mount reaction forces. The
movement of the wishbone relative to the part on which it is mounted is
controlled by the compliance in the bushes. This typically would allow relatively
little resistance to rotation about an axis through the bushes, while
strongly resisting motion in the other 5 degrees of freedom.
In Figure 4.74(b) the wishbone is constrained by a spherical joint at each
bush location. Each spherical joint constrains 3 degrees of freedom. This is
in fact equivalent to our vector-based model shown as a free-body diagram
in Figure 4.73 where we currently have three constraint reaction forces at
each of our mount locations A, B, E and F. The problem with this approach