Modal Analysis
Mẍ(t) + Kx(t) = 0
1
Calculate M −1/2
2
Calculate the mass normalized stiffness matrix: K̃ = M −1/2 KM −1/2
3
Solve eigenvalue problem for K̃ to determine ωi2 and vi
4
Normalize vi and form matrix P = [v1 v2 ]
(P := matrix of eigenvectors)
5
Determine S = M −1/2 P and S −1 = P T M 1/2
(S := matrix of mode shapes)
6
Calculate the modal ICs: r(0) = S −1 x0 ; ṙ(0) = S −1 ẋ0
7
Substitute the modal ICs into eq’ns to get sol’n in modal coordinate r(t)
8
Finally, get the sol’n x(t) = Sr(t)
11/26
OKTAV
Mechanical Vibrations
Modal Analysis
MDOF
12/26
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Mechanical Vibrations
Modal Analysis
MDOF
12/26
OKTAV
Mechanical Vibrations
Modal Analysis
MDOF
13/26
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13/26
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Modal Analysis
MDOF
13/26
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Mechanical Vibrations
Modal Analysis
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14/26
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14/26
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15/26
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15/26
OKTAV
Mechanical Vibrations
Modal Analysis
MDOF
15/26
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Mechanical Vibrations
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16/26
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Ex:
16/26
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Mechanical Vibrations
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16/26
OKTAV
Mechanical Vibrations
Modal Analysis
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Solve the n-DOF system for n = 3 using modal analysis
16/26
OKTAV
Mechanical Vibrations
Modal Analysis
MDOF
Ex:
Solve the n-DOF system for n = 3 using modal analysis
Use the values m1 = m2 = m3 = 4 kg and k1 = k2 = k3 = 4 N/m
16/26
OKTAV
Mechanical Vibrations
Modal Analysis
MDOF
Ex:
Solve the n-DOF system for n = 3 using modal analysis
Use the values m1 = m2 = m3 = 4 kg and k1 = k2 = k3 = 4 N/m
ICs x1 (0) = 1 m and all other initial displacements and velocities zero
16/26
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Mechanical Vibrations
Modal Analysis
MDOF
17/26
OKTAV
Mechanical Vibrations
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MDOF
17/26
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18/26
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18/26
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19/26
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19/26
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20/26
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20/26
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21/26
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21/26
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22/26
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22/26
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23/26
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23/26
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24/26
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24/26
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25/26
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25/26
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26/26
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MDOF
26/26
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26/26
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