# SVT Decoupling#

pyFBS has implemented the novel SVD-based approach for interface reduction in LM-FBS. The SVT is the first engineering tool to tackle the issue of flexible interfaces in lightly damped system for experimental frequency-based substructuring. The methodology can be applied without any knowledge of system geometry and treat efficiently measurement error by combining reduction, filtering and regularization in a single transformation step.

Note

Download example showing a substructure decoupling application with SVT: `19_FBS_decoupling_SVT.ipynb`

## Example Datasets and 3D view#

Load the required predefined datasets and open the 3D viewer in the background as already shown in 3D Display. Also for decoupling, a subplot representation, as already presented in Coupling, can be used.

## Experimental model#

Load the experimental measurements file from the example datasets.

```
exp_A = r"./lab_testbench/Measurements/Y_A.p"
exp_B = r"./lab_testbench/Measurements/Y_B.p"
exp_AB = r"./lab_testbench/Measurements/Y_AB.p"
freq, _Y_A_exp = np.load(exp_A, allow_pickle = True)
_, _Y_B_exp = np.load(exp_B, allow_pickle = True)
_, _Y_AB_exp = np.load(exp_AB, allow_pickle = True)
Y_A_exp = np.transpose(_Y_A_exp, (2, 0, 1))
Y_B_exp = np.transpose(_Y_B_exp, (2, 0, 1))
Y_AB_exp = np.transpose(_Y_AB_exp, (2, 0, 1))
```

## Singular vector transformation#

The SVT can be performed directly on the measured data. Make sure that input and output DoFs involved in the transformation are the same in the subsystem to be decoupled (B) and the assembled system (AB). Furthermore, the same reduction spaces must be used for both the systems (B and AB) in order to guarantee a proper compatibility and equilibrium. The reduced singular subspaces are extracted from the subsystem B and a number of 6 DoFs is used.

```
k = 6
svt = pyFBS.SVT(df_chn_B,df_imp_B,freq,Y_B_exp,[1,10],k)
```

Apply the defined SVT to systems B and AB:

```
_,_,FRF_B_sv= svt.apply_SVT(df_chn_B,df_imp_B,freq,Y_B_exp)
_,_,FRF_AB_sv= svt.apply_SVT(df_chn_AB,df_imp_AB,freq,Y_AB_exp)
```

## LM-FBS Decoupling#

The uncoupled global admittance is constructed using the transformed FRF datasets and imposing the decoupling operation through the minus sign.

```
Y_AB_un = np.zeros((len(freq),2*k+6,2*k+6),dtype = complex)
Y_AB_un[:,0:2*k,0:2*k] = FRF_AB_sv
Y_AB_un[:,2*k:,2*k:] = -1*FRF_B_sv
plt.spy(np.abs(Y_AB_un[100])) # display at arbitrary frequency to check for shape
```

The compatibility and the equilibrium conditions are defined via the signed Boolean matrices.

```
Bu = np.zeros((k,2*k+6))
Bu[:k,0:k] = 1*np.eye(k)
Bu[:k,2*k:2*k+6] = -1*np.eye(k)
Bf = Bu
```

Apply the LM-FBS based on the defined compatibility and equilibrium conditions.

```
Y_int = Bu @ Y_AB_un @ Bf.T
Y_A_dec = Y_AB_un - Y_AB_un @ Bf.T @ np.linalg.pinv(Y_int) @ Bu @ Y_AB_un
```

## Results#

First extract the FRFs at the reference DoFs:

```
arr_ = [6,7,8,9,10,11]
Y_A_LMFBS = Y_A_dec[:,arr_,:][:,:,arr_]
Y_A_ref = Y_A_exp
```

The decoupled and the reference results for A can be compared:

**Thatâ€™s a wrap!**

Want to know more, see a potential application? Contact us at info.pyfbs@gmail.com!