Characterizing the dynamic stiffness of rubber bushings is traditionally achieved based on a hydro pulse machine. This method is relatively expensive and only capable of measuring the dynamic stiffness in translational directions. Alternatively, two Dynamic Substructuring based methods can be used to identify individual substructures (in this case, the rubber bushings). These two approaches additionally identify the dynamic stiffness in rotational DoFs while requiring only standard vibration testing equipment.

### Inverse Substructuring

The Inverse Substructuring method can be applied to an assembly of the rubber bushing connected to two substructures. This approach can be divided into three steps:

- FRF measurement on an assembly with Virtual Points before and after the bushing.
- Virtual Point Transformation (6 DoFs on each coupling point).
- Determination of the bushing stiffness from the off-diagonal terms.

The dynamic stiffness can be obtained by inverting the FRF matrix of the assembled system. The dynamic stiffness matrix has a block diagonal form, and the diagonal elements of the rubber insulator (I) overlap with the connecting substructures (A, B):

Let us have a close look at the dynamic stiffness matrix at the two interfaces (blue square) in this assembly:

$$\mathbf{Z}^{\mathrm{AIB}} = \begin{bmatrix}

\mathbf{Z}_{11}^\mathrm{A} + \mathbf{Z}_{11}^\mathrm{I} & \mathbf{Z}_{12}^\mathrm{I}\\

\mathbf{Z}_{21}^\mathrm{I} & \mathbf{Z}_{22}^\mathrm{B} + \mathbf{Z}_{22}^\mathrm{I}

\end{bmatrix}$$

Inverse Substructuring uses the fact that the off-diagonal terms $$\mathbf{Z}_{12}^\mathrm{I}$$, $$\mathbf{Z}_{21}^\mathrm{I}$$ are only a property of the rubber bushing. To come up with a model for the full isolator, a simple topology and a negligible mass of the rubber isolator is assumed:

$$\mathbf{Z}^\mathrm{I}_{11}\approx -\mathbf{Z}^\mathrm{I}_{12}\approx -\mathbf{Z}^\mathrm{I}_{21}\approx \mathbf{Z}^\mathrm{I}_{22}$$

Based on this assumption, evaluating one off-diagonal term determines the 6-DoF dynamic stiffness of the entire rubber bushing.

$$\mathbf{Z}^{\mathrm{I}} = \begin{bmatrix}

-\mathbf{Z}_{12}^\mathrm{I} & \mathbf{Z}_{12}^\mathrm{I}\\

\mathbf{Z}_{21}^\mathrm{I} & -\mathbf{Z}_{21}^\mathrm{I}

\end{bmatrix}$$

### Decoupling

On the other hand, the attached Transmission Simulators’ full decoupling is based on no assumptions other than linearity. This approach requires determining the FRFs of the attached Transmission Simulators. It can be split into five steps:

- Design, construction, and assembly of two Transmission Simulators and the rubber bushing.
- FRF measurement of the full assembly with Virtual Points before and after the bushing.
- FRF measurement – or numerical determination – of the Transmission Simulators
- Virtual Point Transformation (6 DoFs at each coupling point).
- Determination of the bushing stiffness using decoupling

A typical experimental test setup for determining the FRFs of a Transmission Simulator (in free-free condition) is given in the following picture:

After experimentally/numerically determining the dynamic stiffness of the Transmission Simulators $$\mathbf{Z}_{11}^\mathrm{A}$$, $$\mathbf{Z}_{22}^\mathrm{B}$$ one can decouple them from the measurement of the assembly.

$$\begin{bmatrix}

\mathbf{Z}_{11}^\mathrm{A}+\mathbf{Z}_{11}^\mathrm{I} & \mathbf{Z}_{12}^\mathrm{I}\\

\mathbf{Z}_{21}^\mathrm{I} & \mathbf{Z}_{22}^\mathrm{B}+\mathbf{Z}_{22}^\mathrm{I}

\end{bmatrix} – \begin{bmatrix}

\mathbf{Z}_{11}^\mathrm{A} & \mathbf{0}\\

\mathbf{0} & \mathbf{Z}_{22}^\mathrm{B}

\end{bmatrix} = \begin{bmatrix}

\mathbf{Z}_{11}^\mathrm{I} & \mathbf{Z}_{12}^\mathrm{I}\\

\mathbf{Z}_{21}^\mathrm{I} & \mathbf{Z}_{22}^\mathrm{I}

\end{bmatrix}$$

Based on decoupling, it is possible to evaluate the diagonal terms and compare them to the off-diagonal terms, which should be equal under the assumptions of Inverse Substructuring. Decoupling additionally considers mass effects which become more relevant with increasing frequencies.

### Summary

Both substructuring methods (Inverse Substructuring & decoupling) were validated and successfully applied in commercial projects. By determining the 6-DoF frequency-dependent stiffness, one can avoid the necessity for a material model or a hydro pulse measurement. The different approaches for rubber bushing identification are summarized in the following overview:

**Practical benefits:**

- Both substructuring methods can identify dynamic stiffness values of rubber bushings in more than 3 DoFs and for higher frequencies than usually possible with an expensive hydro pulse testing machine.
- Based on Dynamic Substructuring, the identified bushing stiffnesses can be iteratively modified such that the response at an attached receiver is reduced.