Dynamics cheat sheet

However, [Φ] −1 = [Φ] T [M] therefore
{η} = [Φ] T [M] {x}
⎧ ⎫ ⎡ ⎤
⎨η 1 (t) ⎬ √µ1
ϕ 11 √µ2
ϕ 12
⎩
η 2 (t)
⎭ = ⎣ ⎦
õ1
ϕ 21 √µ2
ϕ 22
⎤ ⎧ ⎫
⎣ m 11 m 12
⎨x ⎦ 1 (t) ⎬
m 21 m
⎩
22 x 2 (t)
⎭
T ⎡
The next step is to apply this transformation to the original equations of motion in order to decouple them
1.6 Step 6, applying modal transformation to decouple the original equations of motion
The EOM in normal coordinates is
⎡
⎣ m 11 m 12
m 21
⎤ ⎧
⎨
⎦
m
⎩
22
x ′′
1
x ′′
2
⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫
⎬
⎭ + ⎣ k 11 k 12
⎨x ⎦ 1 ⎬ ⎨
k
⎩
22 x
⎭ = f 1 (t) ⎬
⎩
2 f 2 (t)
⎭
Applying the above modal transformation {x} = [Φ] {η} on the above results in
⎡ ⎤ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫
⎣ m 11 m 12
⎨η 1
⎦ ′′ ⎬
[Φ]
m 21 m
⎩
22 η ′′ ⎭ + ⎣ k 11 k 12
⎨η ⎦ 1 ⎬ ⎨
[Φ]
k 21 k
⎩
22 η
⎭ = f 1 (t) ⎬
⎩
2 f 2 (t)
⎭
pre-multiplying by [Φ] T results in
⎡ ⎤ ⎧
[Φ] T ⎣ m 11 m 12
⎨
⎦ [Φ]
m
⎩
22
m 21
η 1
′′
η 2
′′
2
k 21
⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫
⎬
⎭ + [Φ]T ⎣ k 11 k 12
⎨η ⎦ 1 ⎬ ⎨
[Φ]
k
⎩
22 η
⎭ = f 1 (t) ⎬
[Φ]T ⎩
2 f 2 (t)
⎭
⎡ ⎤
⎡ ⎤
The result of [Φ] T ⎣ m 11 m 12
⎦ [Φ] will always be ⎣ 1 0 ⎦. This is because mass normalized shape vectors
m 21 m 22 0 1
are used. If the shape functions were not mass normalized, then the diagonal values will not be 1 as shown.
⎡ ⎤
⎡ ⎤
The result of [Φ] T ⎣ k 11 k 12
⎦ [Φ] will be ⎣ ω2 1 0
⎦.
k 21 k 22 0 ω2
2
⎧ ⎫ ⎧ ⎫
⎨
Let the result of [Φ] T f 1 (t) ⎬ ⎨ ˜f
⎩
f 2 (t)
⎭ be 1 (t) ⎬
,Therefore, in modal coordinates the original EOM becomes
⎩ ˜f 2 (t)
⎭
⎡ ⎤ ⎧
⎣ 1 0 ⎨
⎦
0 1
⎩
η 1
′′
η 2
′′
k 21
⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫
⎬
⎭ + ⎣ ω2 1 0 ⎨η ⎦ 1 ⎬ ⎨ ˜f
0 ω2
2 ⎩
η
⎭ = 1 (t) ⎬
⎩
2
˜f 2 (t)
⎭
The EOM are now decouples and each can be solved as follows
η ′′
1 (t) + ω 2 1η 1 (t) = ˜f 1 (t)
η ′′
2 (t) + ω 2 2η 2 (t) = ˜f 2 (t)
To solve these EOM’s, the initial conditions in normal coordinates must be transformed to modal coordinates
using the above transformation rules
{η (0)} = [Φ] T [M] {x (0)}
{
η ′ (0) } = [Φ] T [M] { x ′ (0) }
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