Schematic and Equations

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



Figure 1.
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