# 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.
• $\stackrel{˙}{y}$ and $\stackrel{˙}{w}$ 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:

${S}_{y}=\left({p}_{0}+{p}_{1}{|y|}^{{}^{{p}_{2}}}\right)$

• Similarly, the effective damping C is ${c}_{0}$ multiplied by a factor:

${S}_{w}=\left({q}_{0}+{q}_{1}{|\stackrel{˙}{w}|}^{{}^{{q}_{2}}}\right)$

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: $Ky+C\stackrel{˙}{w}$