# Sensitivity analysis with Stiffness Injection

## What is stiffness injection?

The Stiffness Injection (SI) method consists of changing the stiffness of one or more components of a physically existing assembly, introducing extra (virtual) spring and/or damper elements. This is possible using the Virtual Point Transformation and Dynamic Substructuring technologies. Stiffness injection is applied to a component in an assembly, where all transfer paths are already measured and defined. With SI, you add and/or remove stiffness between the DoFs on the active and passive sides of a component. As a result, you calculate new assembly FRFs using Dynamic Substructuring.

## Why use stiffness injection?

You usually perform Stiffness Injection at a later stage of a vehicle development process. This is because SI consists of modifying the full-vehicle FRFs by adding and/or removing stiffness to a single component. For this reason, you need the assembled system physically available, with all transfer paths defined and measured.

Theoretically, you can apply SI to any component, between its active and passive sides. Practically, SI is mainly applied to bushings, where you add or remove stiffness to make them stiffer or softer. You use the stiffness injection technology for two main reasons:

1. To optimize the stiffness of one element of the already existing assembly.
2. To study the effects of the durability of bushings (likely the final bushing stiffness will differ from the original one).

## Stiffness injection: the math behind it

Stiffness Injection consists of introducing spring or damper elements between the DoFs on the active and passive sides of a component. In particular, you add virtual springs and/or dampers acting in parallel to the existing element.

In the picture above, you can check the notation used. A is the active component, B is the passive component, and C is the component on which stiffness will be injected. We begin with a measured FRF matrix of the assembled system $$\mathbf{Y}^{\mathrm{ABC}}$$. The virtual spring will apply forces $$\lambda$$ to both sides of component C,

$$\mathbf{u}=\mathbf{Y}^{\mathrm{ABC}} (\mathbf{f-B^\mathrm{T}\boldsymbol{\lambda}})$$

whereis the boolean matrix corresponding to these interfaces. The forces $$\lambda$$ are given by the dynamic stiffness of the virtual spring

$$\boldsymbol{\boldsymbol{}\lambda}=\mathbf{Z}_{\mathrm{mod}}^{\mathrm{C}}\mathbf{Bu}$$

For a simple spring-damper element, this dynamic stiffness is given by

$$\mathbf{Z}_{\mathrm{mod}}^{\mathrm{C}}=\mathbf{K}_\mathrm{{mod}}+\mathrm{j\omega}\mathbf{D}_\mathrm{{mod}}$$

We can solve the equations to find the spring forces $$\lambda$$ for any given applied force f and apply them to the assembly.

$$\mathbf{\lambda}=((\mathbf{Z}_{\mathrm{mod}}^{\mathrm{C}})^{-1}+\mathbf{BY}^{\mathrm{ABC}}\mathbf{B}^\mathrm{T})^{-1}\mathbf{BY}^{\mathrm{ABC}}\mathbf{f}$$

This gives us the modified FRF matrix of the assembly, which we can use to make predictions of the target responses in the modified system.

$$\widehat{\mathbf{Y}}^{\mathrm{ABC}}=\mathbf{Y}^{\mathrm{ABC}}-\mathbf{B}^\mathrm{T}((\mathbf{Z}_{\mathrm{mod}}^{\mathrm{C}})^{-1}+\mathbf{BY}^{\mathrm{ABC}}\mathbf{B}^\mathrm{T})^{-1}\mathbf{BY}^{\mathrm{ABC}}$$

If you are in an early development stage, you can apply stiffness injection without too many limitations. At a later development stage, the geometries and components are mostly fixed, so changes are only limited to material properties. In this case, the stiffness modifications can be simplified as proportional stiffness changes on every DoF, expressed by the formula

$$\mathbf{Z}_{\mathrm{mod}}^{\mathrm{C}}=\eta \mathbf{Z}_{\mathrm{orig}}^{\mathrm{C}}$$

where $$\eta$$ is a scalar that expresses the relative change of the modification compared to the original stiffness.

## How to: Apply the stiffness injection technology

SI consists of introducing extra springs/dampers between the Virtual Points of the active and passive sides of a component. The superposition principle allows the expression of the new dynamical balance of the coupled system using Dynamic Substructuring. To apply SI, you need to perform the following steps:

1. Perform FRF testing (in DIRAC) of the entire assembly. Make sure to place sensors on the active and passive sides of the element you want to inject to perform the Virtual Point Transformation.
2. Update the vehicle model with stiffness injection (in COUPLE) and recompute the vehicle responses.

In addition, you can calculate blocked forces (on the active side, upstream to the SI location), perform SI and check the predictions (with component TPA) at the target locations.

## Example case

To better explain the topic, we will present an example. In this case, we take a full vehicle and apply stiffness injection to the bushings of the lower arms of the front suspension. In particular, we have selected two sets of bushings, located on the A-Arms and G-Arms, both on the left and right sides of the vehicle. In total, we apply stiffness injection to 4 bushings, all simultaneously. To do so, we need FRFs on the active and passive sides of each bushing, so 8 Virtual Points in total. A schematic representation is shown in the following picture.

After doing the FRF measurements (in DIRAC), we perform the Virtual Point Transformation to the 8 VPs. Before importing the model to COUPLE, it is convenient to create a custom present containing only the VP transformed channels and reference channels and the target sensors.

## How to: Perform stiffness injection in COUPLE

You can perform stiffness injection in COUPLE by following the next steps.

1. Import the full assembly model (VP transformed) and add it to the Design Area.
2. Link the VPs on the active side with the corresponding ones on the passive side.
3. Create synthetic stiffness models with the delta stiffness to inject.
4. Apply the injected stiffness to the links (using the variation studies feature).
5. Visualize the results.

### Step 1: Import and add the model

The first step consists of importing the VP-transformed FRF model into COUPLE, with all channels and interfaces correctly set up. After importing, you add the model to the design area.

To learn how to import models in COUPLE, check this link.

### Step 2: Link active and passive side VPs

The second step consists of linking each VP on the active side to the corresponding one on the passive side. By doing this, you create a connection in parallel to what is measured. This will become your virtual spring/damper after adding and removing stiffness.

### Step 3: Create the synthetic stiffness models

The third step consists of creating the synthetic stiffness models for all links and adding them to the library. To do so, you need to know the original component stiffness (in all the directions you want to apply SI) and calculate the delta stiffness to add/remove. In the example case, we apply SI to Tx, Ty and Tz of both the A-Arm and G-Arm bushings. We apply and remove stiffness in steps of 5%, from ±5% to ±40%.

The table below provides an example. For each bushing, we get the nominal stiffness in the three principal directions (in DIRAC) and calculate the delta stiffness to add/remove.

To learn how to create synthetic stiffness models, check this link.

 DoF Initial Stiffness [N/m] ±Δ5% [N/m] ±Δ10% [N/m] … ±Δ 40% [N/m] A-Arm Tx 3.76E7 ±1.88E6 ±3.76E6 … ±1.50E7 A-Arm Ty 3.76E7 ±1.88E6 ±3.76E6 … ±1.50E7 A-Arm Tz 1.40E6 ±7.00E4 ±1.40E5 … ±5.60E5 G-Arm Tx 8.01E5 ±4.01E4 ±8.01E4 … ±3.20E5 G-Arm Ty 1.37E6 ±6.85E4 ±1.37E5 … ±5.48E5 G-Arm Tz 1.36E5 ±6.80E3 ±1.36E4 … ±5.44E4

You then use the calculated delta stiffnesses to generate synthetic stiffness models in COUPLE, which you save in the library.

### Step 4: Apply the injected stiffness to the links

After creating all the synthetic stiffness models, you can add them to the corresponding links. To make the process efficient, you can use the variation study feature. You create a new variation study for the Stiffness Injection and add the four links as assembly parts to vary. You then add one variant for each synthetic stiffness model you have created and add the corresponding models.

To learn how to create variation studies, check this link.

### Step 5: Visualize the results

After creating all variants, you can go to the Analysis module and plot all the curves, to see how the nominal curve varies when adding or removing stiffness to the selected bushings.

To learn how to plot results, check this link.