Meng31301_b_1(1).pdf

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Orthogonality and Modes

Regardless of the type of vibration being considered, the matrix form the equation of motion may be expressed in the form: M¨ y + Ky = 0 Recapping, assuming simple harmonic motion, y = Yejωt , this may be written as KY = ω 2 MY which is the eigenvalue/vector problem defining the natural frequencies and mode shapes. For the lth natural frequency/mode shapes pair we can write: KY l = ωl2 MY l Pre-multiplying by the transpose of the mth eigenvector solution gives: YTm KY l = ωl2 YTm MY l

(1)

Alternatively we write the equation for the mth solution and pre-multiplying by the transpose of the lth eigenvector solution to give: 2 YTl MY m YTl KY m = ωm

Using AB = (BT AT )T and noting that K and M are symmetric and that CT = DT is the same as C = D, this may be rewritten as 2 YTm MYl YTm KYl = ωm

(2)

Assuming that the natural frequencies are distinct, subtracting (1) from (2) gives Y Tm MYl = 0 where l != m Mode shapes are orthogonal with respect to the mass matrix. Substituting back into (2) gives YTm KY l = 0 where l != m Mode shapes are also orthogonal with respect to the stiffness matrix. For the case where l = m, equation 2 may be rewritten as YTl KYl = ωl2 Y Tl MYl

!

ωl2 =

kl ml

where the scalars kl and ml are given by kl = Y Tl KY l and ml = YTl MY l . Defining the modal matrix, Φ, as a collection of all the mode shape column vectors Yi:  

Φ=  Y1



... ...  . . . Yi . . .   ... ... 20