Modal analysis

Equation of motion is of the form:

m x+c x+kx=0

Neglecting ‘c’ in order to find the Eigenvalues and Eigenvectors.m x+kx=0

x=Xsinωt

x=ωXcosωt

x=−ω2 Xsinωt

[M ] (−ω2 X )+ [K ] ( X )=0

( [M ] (ω2 X )−[K ] (X )=0)÷ [K ]

[K ]−1 [M ]ω2 (X )− (X )=0

1

ω2 λ=1

ω2

[K ]−1 [M ] (X )= λ (X )

[K ]−1 [M ] – Dynamic matrix

[K ]−1 [M ]=P=[1 −0.0004 0 0 00 0 0 0 1

0.0001 1 −0.0631 0 00 −0.0055 −0.7016 0.9999 00 −0.0055 −0.7098 −0.0117 0

]λ1=0.6482×10−4

λ2=0.0003×10−4

λ3=0

λ4=0

λ5=0.0083×10−4

[P ]T [M ] [P ] Y +[P]T [K ] [P ]Y +[P ]T [C ] [P ] Y=0

C=[α [M ]+β [K ]]

Α=0.025 β=0.023

Generalized mass matrix

[P ]T [M ] [P ]

[P ]T [M ] [P ]=[24 0 0 0 00 68.0287 0 0 00 0 557.8206 0 00 0 0 13.1491 00 0 0 0 0.1250

]Generalized stiffness matrix

[P ]T [K ] [P ]Y

[P ]T [K ] [P ]Y=[0 0 0 0 00 0.0263×1011 0 0 00 0 −2.1954×1011 0 00 0 0 4.4823×1011 00 0 0 0 0

]Generalized damping matrix

[P ]T [C ] [P ]

[P ]T [C ] [P ]=[0 0 0 0 00 0.0060×1010 0 0 00 0 −0.5049×1010 0 00 0 0 1.0309×1010 00 0 0 0 0

]Uncoupled Equations

24 y1+0 y1+0 y1=0

68.0287 y2+0.0263×1011 y2+0.0060×1010 y2=0

557.8206 y3−2.1954×1011 y3−0.5049×1010 y3=0

13.1491 y4+4.4823×1011 y4+1.0309×1010 y4=0

0.1250 y5+0 y5+0 y5=0

Equation 2

ωn=√ k effmeff =√ 0.0263×1011

68.0287

ωn=6218.126 rad /s

cc=2mωn=2×68.0287×6218.126

cc=846013.35Ns /rad

Equation 3

ωn=√ k effmeff =√−2.1954×1011

557.8206

ωn=19838.53 irad /s

cc=2mωn=2×557.8206×19838.53 i

cc=2.21×107Ns/m

Equation 4

ωn=√ k effmeff =√ 4.4823×1011

13.1491

ωn=184630.70 rad / s

cc=2mωn=2×13.1491×184630.70

cc=4855455.07Ns/m