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.
- k0 and k1 represent the bushing rubber stiffness.
- k2 is used to control the stiffness at large velocities.
- c0 produces the roll-off observed in the experimental data at low velocities.
- c1 accounts for the relaxation of the bushing impact force.
- c2 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.
- ˙y and ˙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
k0
multiplied by a factor:
Sy=(p0+p1|y|p2)
- Similarly, the effective damping C is
c0
multiplied by a factor:
Sw=(q0+q1|˙w|q2)
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: Ky+C˙w