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The transformation from modal coordinates back to normal coordinates is<br />
However, [Φ] −1 = [Φ] T [M] therefore<br />
{η} = [Φ] −1 {x}<br />
⎧ ⎫ ⎡<br />
⎤−1 ⎧ ⎫<br />
⎨η 1 (t) ⎬<br />
⎩<br />
η 2 (t)<br />
⎭ = 0.925 0.380 ⎨ x<br />
⎣ ⎦ 1 (t) ⎬<br />
−0.219 0.534<br />
⎩<br />
x 2 (t)<br />
⎭<br />
{η} = [Φ] T [M] {x}<br />
⎧ ⎫ ⎡<br />
⎤T ⎡ ⎤ ⎧ ⎫<br />
⎨η 1 (t) ⎬<br />
⎩<br />
η 2 (t)<br />
⎭ = 0.925 0.380<br />
⎣ ⎦ ⎣ 1 0 ⎨x ⎦ 1 (t) ⎬<br />
−0.219 0.534 0 3<br />
⎩<br />
x 2 (t)<br />
⎭<br />
⎡<br />
⎤ ⎧ ⎫<br />
0.925 −0.657 ⎨x = ⎣ ⎦ 1 (t) ⎬<br />
0.38 1.6<br />
⎩<br />
x 2 (t)<br />
⎭<br />
The next step is to apply this transformation to the original equations of motion in order to decouple them.<br />
Applying step 6 results in<br />
⎡ ⎤ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫<br />
⎣ 1 0 ⎨η 1<br />
⎦<br />
′′ ⎬<br />
0 1<br />
⎩<br />
η 2<br />
′′ ⎭ + ⎣ ω2 1 0 ⎨η ⎦ 1 ⎬ ⎨<br />
0 ω2<br />
2 ⎩<br />
η<br />
⎭ = 0 ⎬<br />
[Φ]T ⎩<br />
2 sin (5t)<br />
⎭<br />
⎡ ⎤ ⎧ ⎫ ⎡<br />
⎤ ⎧ ⎫ ⎡<br />
⎤T ⎧ ⎫<br />
⎣ 1 0 ⎨η 1<br />
⎦<br />
′′ ⎬<br />
0 1<br />
⎩<br />
η 2<br />
′′ ⎭ + ⎣ 1.8642 0 ⎨η ⎦ 1 ⎬<br />
0 0.438 2 ⎩<br />
η<br />
⎭ = 0.925 0.380 ⎨ 0 ⎬<br />
⎣ ⎦<br />
2 −0.219 0.534<br />
⎩<br />
sin (5t)<br />
⎭<br />
⎡ ⎤ ⎧ ⎫ ⎡<br />
⎤ ⎧ ⎫ ⎧ ⎫<br />
⎣ 1 0 ⎨η 1<br />
⎦<br />
′′ ⎬<br />
0 1<br />
⎩<br />
η ′′ ⎭ + 3. 47 0 ⎨η ⎣ ⎦ 1 ⎬ ⎨<br />
0 0.192<br />
⎩<br />
η<br />
⎭ = −0.219 sin (5t) ⎬<br />
⎩<br />
2 0.534 sin (5t)<br />
⎭<br />
2<br />
The EOM are now decoupled and each EOM can be solved easily as follows<br />
η ′′<br />
1 (t) + 3.47η 1 (t) = −0.219 sin (5t)<br />
η ′′<br />
2 (t) + 0.192η 2 (t) = 0.534 sin (5t)<br />
To solve these EOM’s, the initial conditions in normal coordinates must be transformed to modal coordinates<br />
using the above transformation rules<br />
⎧ ⎫ ⎡<br />
⎤ ⎧ ⎫<br />
⎨η 1 (0) ⎬<br />
⎩<br />
η 2 (0)<br />
⎭ = 0.925 −0.657 ⎨x ⎣ ⎦ 1 (0) ⎬<br />
0.38 1.6<br />
⎩<br />
x 2 (0)<br />
⎭<br />
⎧ ⎫ ⎡<br />
⎤ ⎧ ⎫<br />
⎨η 1 (0) ⎬<br />
⎩<br />
η 2 (0)<br />
⎭ = 0.925 −0.657 ⎨0⎬<br />
⎣ ⎦<br />
0.38 1.6<br />
⎩<br />
1<br />
⎭<br />
⎧ ⎫<br />
⎨−0.657⎬<br />
=<br />
⎩<br />
1.6<br />
⎭<br />
13