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

- Similarly, the effective damping
`C`is ${c}_{0}$ multiplied by a factor:${S}_{w}=\left({q}_{0}+{q}_{1}{\left|\dot{w}\right|}^{{}^{{q}_{2}}}\right)$

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: $Ky+C\dot{w}$