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230 Multibody Systems Approach to Vehicle Dynamics
mount with a spherical joint and modelling the other mount with an inline
joint primitive, as described earlier in Chapter 3. The inline primitive constrains
2 degrees of freedom to maintain the mount position on the axis
through the two mount locations. This constraint does not prevent translation
along the axis through the mounts, this ‘thrust’ being reacted by the
single spherical joint. Thus this selection of rigid constraints provides us
with a solution that is not overconstrained. Although the MBS approach
would best utilize the model with two bushes to predict the mount reaction
forces the model in Figure 4.74(d) provides us with an understanding of the
overconstraint problem and a methodology we can adapt to progress the
vector-based analytical solution.
If we return now to the analytical solution and consider the lower wishbone
Body 2, we can see in Figure 4.75 that a comparable approach to the use of
the MBS inline joint primitive constraint is to ensure that the line of action
of one of the mount reaction forces, say {F F21 } 1 , is perpendicular to the
axis E–F through the two wishbone mounts.
Thus we can derive the final two equations needed to progress the analytical
solution using the familiar approach with the vector dot product to constrain
the reaction force at the mount to be perpendicular to an axis through
the mounts at, say, point B for the upper wishbone and point F for the lower
wishbone:
{F F21 } 1 • {R EF } 1 0 (4.267)
{F B31 } 1 • {R AB } 1 0 (4.268)
Having established the 20 equations required for solution, it is possible to
set up the equations starting with the force equilibrium of Body 2:
∑{F 2 } 1 {0} 1 (4.269)
{F E21 } 1 {F F21 } 1 {F G24 } 1 {0} 1 (4.270)
⎡F
⎢
⎢
F
⎣⎢
F
E21x
E21y
E21z
⎤ ⎡FF21x⎤
⎡FG24x⎤
⎡0⎤
⎥ ⎢
F
⎥ ⎢
F21y
F
⎥
⎥
⎢ ⎥
⎢ G24y⎥
⎢
0
⎥
N
⎢ ⎥
⎦⎥
⎣⎢
FF21z⎦⎥
⎣⎢
FG24z⎦⎥
⎣⎢
0⎦⎥
(4.271)
The summation of forces in (4.271) leads to the first set of three equations:
Equation 1 F E21x F F21x F G24x 0 (4.272)
{F F21 } 1
SPHERICAL
INLINE
F
•
{F E21 } 1
•
E
Fig. 4.75
MBS approach
Analytical approach
Comparable MBS and analytical wishbone mounting models