Dynamics cheat sheet

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