
Now that we can determine the natural frequencies and mode shapes for our structures, we can build the modal matrix $[\Phi]$. As we'll see in the next lecture, this matrix is the key to uncoupling the equations of motion.

We define the modal matrix as a matrix whose columns are the vectors of mode shapes. Building on the modal analysis we completed in the previous lecture; we can construct the modal matrix as follows,

$$
\begin{bmatrix}
\hat{\Phi}
\end{bmatrix} =
\begin{bmatrix}
\hat{\phi}_{1,1} & \hat{\phi}_{1,2} & \hat{\phi}_{1,3}\\
\hat{\phi}_{2,1} & \hat{\phi}_{2,2} & \hat{\phi}_{2,3}\\
\hat{\phi}_{3,1} & \hat{\phi}_{3,2} & \hat{\phi}_{3,3}\\
\end{bmatrix} =
\begin{bmatrix}
1 & 1 & 1\\
1.59426742 & 0.09297207 & -2.07875654\\
1.81496643 & -0.65183625 & 1.28348058\\
\end{bmatrix}
\tag{1}
$$

Note that the modal matrix above is designated as $[\hat{\Phi}]$ rather than $[\Phi]$; the modal matrix constructed from the raw eigenvectors is denoted $[\hat{\Phi}]$. This is because the eigenvectors originally obtained are of arbitrary length. **The important information they represent is the ratio of displacements**, not the actual displacement values themselves. As such they may be scaled or normalised to be to a consistent scale. The symbol $[\Phi]$ is used to represent the normalised modal matrix.

There are a number of simple approaches that may be used, however a common and convenient approach is to normalise them such that,

$$
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
M
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix} =
\begin{bmatrix}
I
\end{bmatrix}
\tag{2}
$$

where $[\Phi]$ is a matrix constructed from the normalised eigenvectors and $[I]$ is the identity matrix. This is known as **mass normalisation** and it will become apparent later why we refer to it as convenient. Mass normalisation is achieved by scaling each column $\{\hat{\phi}\}_i$ by a scale factor as follows,

$$
\{\phi\}_i = \frac{\{\hat{\phi}\}_i}{\sqrt{\{\hat{\phi}\}_i^T[M]\{\hat{\phi}\}_i}}
\tag{3}
$$

where $\{\hat{\phi}\}_i$ is the $i^{th}$ column of the pre-normalised modal matrix and $\{\phi\}_i$ is the $i^{th}$ column after normalisation. If the mass matrix is diagonal (as is the case here), the scale factor can be reduced to,

$$
\phi_{i,j} = \frac{\hat{\phi}_{i,j}}{\sqrt{\sum_{k=1}^n m_k\:\hat{\phi}_{kj}^2}}
\tag{4}
$$

where $\phi_{i,j}$ is the $i^{th}$ component of modal vector (column) `j` in matrix $[\Phi]$. So for our 3 story structure we have for example,

$$
\phi_{1,1} = \frac{\hat{\phi}_{1,1}}{\sqrt{m_1\:\hat{\phi}_{1,1}^2 + m_2\:\hat{\phi}_{2,1}^2 + m_3\:\hat{\phi}_{3,1}^2}}
\tag{5}
$$

Now, let's calculate the mass normalised modal matrix for our three story structure.

### Calculating the mass normalised modal matrix

First we'll define the masses and mode shape vectors calculated previously.

```python
#Define masses and the mass matrix
m1 = 21145.85
m2 = 20773.5
m3 = 20476.5
M = np.matrix([[m1,0,0],[0, m2,0],[0,0,m3]])

#Define the previously calculated mode shape vectors
eVecA1 = np.array([[1],[1.59426742],[1.81496643]])
eVecB1 = np.array([[1],[0.09297207],[-0.65183625]])
eVecC1 = np.array([[1],[-2.07875654],[1.28348058]])
```

Next we can unpack the individual values from the mode shape vectors.

```python
#Pre-normalised modal matrix elements
pHat11 = eVecA1[0]
pHat21 = eVecA1[1]
pHat31 = eVecA1[2]
pHat12 = eVecB1[0]
pHat22 = eVecB1[1]
pHat32 = eVecB1[2]
pHat13 = eVecC1[0]
pHat23 = eVecC1[1]
pHat33 = eVecC1[2]
```

Our pre-normalised modal matrix is therefore,

```python
M_preNorm = np.matrix([[pHat11[0],pHat21[0],pHat31[0]],
  [pHat12[0],pHat22[0],pHat32[0]],
  [pHat13[0],pHat23[0],pHat33[0] ]]).T
print('The pre-normalised modal matrix is:')
print(M_preNorm)
```

$$
\begin{bmatrix}
1 & 1 & 1\\
1.59426742 & 0.09297207 & -2.07875654\\
1.81496643 & -0.65183625 & 1.28348058\\
\end{bmatrix}
\tag{6}
$$

Next we can scale each element of the modal matrix to produce the mass normalised modal matrix.

```python
#Mass normalised modal matrix elements
phi11 = pHat11/math.sqrt(m1*pHat11**2 + m2*pHat21**2 + m3*pHat31**2)
phi21 = pHat21/math.sqrt(m1*pHat11**2 + m2*pHat21**2 + m3*pHat31**2)
phi31 = pHat31/math.sqrt(m1*pHat11**2 + m2*pHat21**2 + m3*pHat31**2)

phi12 = pHat12/math.sqrt(m1*pHat12**2 + m2*pHat22**2 + m3*pHat32**2)
phi22 = pHat22/math.sqrt(m1*pHat12**2 + m2*pHat22**2 + m3*pHat32**2)
phi32 = pHat32/math.sqrt(m1*pHat12**2 + m2*pHat22**2 + m3*pHat32**2)

phi13 = pHat13/math.sqrt(m1*pHat13**2 + m2*pHat23**2 + m3*pHat33**2)
phi23 = pHat23/math.sqrt(m1*pHat13**2 + m2*pHat23**2 + m3*pHat33**2)
phi33 = pHat33/math.sqrt(m1*pHat13**2 + m2*pHat23**2 + m3*pHat33**2)

Phi = np.matrix([[phi11[0], phi12[0], phi13[0]],
                 [phi21[0], phi22[0], phi23[0]],
                 [phi31[0], phi32[0], phi33[0]]]) #Mass normalised modal matrix
print('Mass normalised modal matrix is:')
print(Phi)
```

$$
\begin{bmatrix}
0.00265937 & 0.00577103 & 0.00262936\\
0.00423975 & 0.00053654 & -0.00546579\\
0.00482667 & -0.00376177 & 0.00337473\\
\end{bmatrix}
\tag{7}
$$

Finally we can confirm that our normalisation was successful by evaluating,

$$
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
M
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix} =
\begin{bmatrix}
I
\end{bmatrix}
\tag{8}
$$

```python
#Evaluating
print(np.matrix.round(Phi.transpose()*M*Phi,3))
```

$$
\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{bmatrix}
\tag{9}
$$

With the exception of minor rounding errors, we can see that our normalisation was successful. The fact that we have shown $[\Phi]^T[M][\Phi] = [I]$ is confirmation of the orthogonality properties of the mode shape vectors. We will explore orthogonality and how we make use of it in the next lecture.


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