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Similarly, µ 2 is found<br />
⎧ ⎫T ⎡ ⎤ ⎧ ⎫<br />
⎨ϕ 12 ⎬<br />
µ 2 = ⎣ m 11 m 12<br />
⎨ϕ ⎦ 12 ⎬<br />
⎩<br />
ϕ<br />
⎭<br />
22 m 21 m<br />
⎩<br />
22 ϕ<br />
⎭<br />
22<br />
{ } ⎡ ⎤ ⎧ ⎫<br />
= ϕ 12 ϕ ⎣ m 11 m 12<br />
⎨ϕ ⎦ 12 ⎬<br />
22<br />
m<br />
⎩<br />
22 ϕ<br />
⎭<br />
22<br />
m 21<br />
}<br />
=<br />
{ϕ ⎧ ⎫<br />
⎨ϕ 12 ⎬<br />
12 m 11 + ϕ 22 m 21 ϕ 12 m 12 + ϕ 22 m 22<br />
⎩<br />
ϕ<br />
⎭<br />
22<br />
= ϕ 12 (ϕ 12 m 11 + ϕ 22 m 21 ) + ϕ 22 (ϕ 12 m 12 + ϕ 22 m 22 )<br />
Now that µ 1 , µ 2 are obtained, the mass normalized shape vectors are found. They are called Φ 1 , Φ 2<br />
⎧ ⎫<br />
⎨ϕ 11 ⎬<br />
⎧ ⎫<br />
Φ 1 = ϕ ⎩<br />
ϕ<br />
⎭<br />
1 21<br />
⎨ √µ1<br />
ϕ 11 ⎬<br />
√ = √ =<br />
µ1 µ1 ⎩ √µ1<br />
ϕ 21 ⎭<br />
Similarly<br />
Φ 2 = ϕ 2<br />
√<br />
µ2<br />
=<br />
⎧<br />
⎨<br />
⎩<br />
ϕ 12<br />
ϕ 22<br />
⎫<br />
⎬<br />
⎭<br />
√<br />
µ2<br />
=<br />
⎧<br />
⎨<br />
õ2<br />
ϕ 12<br />
⎫<br />
⎬<br />
⎩ √µ2<br />
ϕ 22 ⎭<br />
1.5 Step 5, obtain the modal transformation matrix Φ<br />
The modal transformation matrix is the 2 × 2 matrix made of of Φ 1 , Φ 2 in each of its columns<br />
[Φ] = [Φ 1 Φ 2 ]<br />
⎡<br />
= ⎣<br />
⎤<br />
õ1<br />
ϕ 11 √µ2<br />
ϕ 12<br />
⎦<br />
õ1<br />
ϕ 21 √µ2<br />
ϕ 22<br />
Now the [Φ] is found, the transformation from the normal coordinates {x} to modal coordinates, which is<br />
called {η} is found<br />
{x} = [Φ] {η}<br />
⎧ ⎫ ⎡<br />
⎨x 1 (t) ⎬ ϕ 11<br />
⎩<br />
x 2 (t)<br />
⎭ = ⎣<br />
⎤<br />
õ1 õ2<br />
ϕ 12<br />
⎦<br />
õ1<br />
ϕ 21 √µ2<br />
ϕ 22<br />
⎧ ⎫<br />
⎨η 1 (t) ⎬<br />
⎩<br />
η 2 (t)<br />
⎭<br />
The transformation from modal coordinates back to normal coordinates is<br />
{η} = [Φ] −1 {x}<br />
⎧ ⎫ ⎡<br />
⎨η 1 (t) ⎬ ϕ 11<br />
⎩<br />
η 2 (t)<br />
⎭ = ⎣<br />
⎤<br />
õ1 õ2<br />
ϕ 12<br />
⎦<br />
õ1<br />
ϕ 21 √µ2<br />
ϕ 22<br />
−1 ⎧ ⎫<br />
⎨ x 1 (t) ⎬<br />
⎩<br />
x 2 (t)<br />
⎭<br />
7