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Multibody systems simulation software 111
example, building a sensor into the model to stop the simulation should this
be about to occur allowing the analyst to investigate the problem further.
The various elastic bushes or mounts used throughout a suspension system to
isolate vibration may be represented initially by six linear uncoupled equations
based on stiffness and damping. As with a joint a bush connects two
parts using an I marker on one body and a J marker on another body. These
markers are normally taken to be coincident when setting up the model but
it will be seen from the formulation presented here that any initial offset,
either translational or rotational, would result in an initial preforce or torque
in the bush. This would be in addition to any initial value for these that the
user may care to define.
The general form of the equation for the forces and torques generated in the
bush is given in (3.47):
{F ij } j [k]{d ij } j [c]{v ij } j {f ij } j (3.47)
where
{F ij } j is a column matrix containing the components of the force and torque
acting on the I marker from the J marker
[k] is a square stiffness matrix where all off diagonal terms are zero
{d ij } j is a column matrix containing the components of the displacement
and rotation of the I marker relative to the J marker
[c] is a square damping matrix where all off diagonal terms are zero
{v ij } j is a column matrix of time derivatives of the terms in the {d ij } matrix
{f ij } j is a column matrix containing the components of the preforce and
pretorque applied to the I marker
Expanding equation (3.47) leads to the following set of uncoupled equations
presented in matrix form as
⎡Fx
⎤ ⎡kx
0 0 0 0 0 ⎤ ⎡dx
⎤ ⎡cx
0 0 0 0 0 ⎤ ⎡ vx
⎤
⎢
F
⎥ ⎢
y 0 ky
0 0 0 0
⎥ ⎢
d
⎥ ⎢
⎢ ⎥
y 0 c
⎥ ⎢ ⎥
⎢
⎥ ⎢ ⎥ ⎢
y 0 0 0 0 v
⎥ ⎢
y
⎥
⎢Fz
⎥ ⎢ 0 0 kz
0 0 0 ⎥ ⎢dz
⎥ ⎢ 0 0 cz
0 0 0 ⎥ ⎢ vz
⎥
⎢ ⎥⎢
⎥ ⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎢
Tx
⎥ ⎢
0 0 0 ktx
0 0
⎥ ⎢
rx
⎥ ⎢
0 0 0 ctx
0 0
⎥ ⎢
x ⎥
⎢T
⎥ ⎢
y 0 0 0 0 k ⎥
ty 0 ⎢ r ⎥ ⎢
⎥ ⎢
y 0 0 0 0 c ⎥
ty 0 y
⎣⎢
Tz
⎦⎥
⎣⎢
0 0 0 0 0 ktz
⎦⎥
⎢⎣
rz
⎦⎥
⎣⎢
0 0 0 0 0 ctz
⎦⎥
⎣⎢
z ⎦⎥
⎡ fx
⎤
⎢
f
⎥
⎢
y
⎥
⎢ fz
⎥
⎢ ⎥
⎢
tx
⎥
⎢t
⎥
y
⎣⎢
tz
⎦⎥
(3.48)
It should be noted that all the terms in (3.47) and (3.48) are referenced to
the J marker reference frame and that the equilibriating force and torque
acting on the J marker is determined from
{F ji } j {F ij } j (3.49)
{T ji } j {T ij } j {d ij } j {F ij } j (3.50)
The shape and construction of the bush will dictate the manner in which
the I marker and J marker are set up and orientated. This is illustrated in
Figure 3.34 where it can be seen that for the cylindrical bush where there are
no voids the radial stiffness is constant circumferentially. For this bush it is
only necessary to ensure the z-axes of the I and J markers are aligned with