Schematic and Equations

This model can be used for both dynamic and quasi-static tests.

The figure above-left shows a schematic of the bushing model where:

X is the input displacement provided to the bushing.
y and w are the internal states of the bushing.
$k_{0}$ and $k_{1}$ represent the bushing rubber stiffness.
$k_{2}$ is used to control the stiffness at large velocities.
$c_{0}$ produces the roll-off observed in the experimental data at low velocities.
$c_{1}$ accounts for the relaxation of the bushing impact force.
$c_{2}$ represents the viscous damping observed at large velocities.

The governing equations for this bushing are shown above-right where:

R is the cutoff frequency associated with a first order filter that acts on the input X.
x is the dynamic content of the bushing input X. This is the filter output.
$\dot{y}$ and $\dot{w}$ are the time derivatives of the internal states of the bushing y and w.
K is the effective stiffness of the bushing.
C is the effective damping of the bushing.
Spline (X) is the static force response of the bushing.

The effective stiffness K and effective damping C are dependent on nonlinear effects such as friction in the bushing material and other nonlinear behavior that cannot be easily represented physically.

The effective stiffness K is $k_{0}$ multiplied by a factor:
$S_{y} = (p_{0} + p_{1} {| y |}^{^{p_{2}}})$
Similarly, the effective damping C is $c_{0}$ multiplied by a factor:
$S_{w} = (q_{0} + q_{1} {| \dot{w} |}^{^{q_{2}}})$

The total force generated by the bushing is the sum of 2 forces:

Static force at the operating point: Spline (X)
Force due to the dynamic behavior of the bushing: $K y + C \dot{w}$