Mode Acceleration Method
The mode acceleration method (MAM) will improve the accuracy of the responses; displacements and derivatives thereof such as element forces, stresses, etc., with respect to the mode displacement method (MDM) when a reduced set of mode shapes is used [Thom
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15.1 Introduction The mode acceleration method (MAM) will improve the accuracy of the responses ; displacements and derivatives thereof such as element forces, stresses, etc., with respect to the mode displacement method (MDM) when a reduced set of mode shapes is used [Thomson 98, McGowan 93, Craig 68,77]. The MDM is often called the mode superposition method. The MDM may only be used for linear dynamics systems. The MAM takes the truncated modes "statically" into account. Using the MAM, less modes may be taken into account compared to the MDM.
15.2 Decomposition of Flexibility and Mass Matrix 15.2.1 Decomposition of the Flexibility Matrix The undamped equations of motion of a multi-degrees of freedom linear dynamic system is written as [M]{x} + [K]{x}
= {F},
with •
• •
•
the mass matrix [K] the stiffness matrix {x} the physical degrees of freedom {F} the load vector of external forces [M]
J. Wijker, Mechanical Vibrations in Spacecraft Design © Springer-Verlag Berlin Heidelberg 2004
(15.1)
15 Mode Acceleration Method
314
The eigenvalue problem becomes (15.2)
the j-th eigenvalue of the eigenvalue problem the j-th eigenvector (mode shape) of the eigenvalue problem In general the eigenvectors are normalised with respect to the mass matrix [M] in such a way that (orthogonality relations)
(15.3)
with
0 ij
the Kronecker delta function.
With the general modal matrix [] = []T_[]T[MHcI>HMH{ = {O} .
(15.6)
After some manipulations we get ('Arl[{[K]-[MH]T
= {O} .
(15.7)
Premultiplying (15.7) by the modal matrix [cI>] we get [] ('A)-l [{[K]-[HM][]T = {O} .
This is equal to
(15.8)
15.2 Decomposition of Flexibility and Mass Matrix
315
(15.9)
The inverse of the nonsingular stiffness matrix is called the flexibility matrix [G] = [Kr' and therefore (15.10)
A linear undamped dynamic system is described by the following equation of motion
10OJ [020 003 The stiffness matrix [K] and the flexibility matrix [G] are [K]
=
15 -5 -lOJ -5 10 -5 and [Kr'= [G] [ -10 -5 25
=
[0.1800 0.1000 0.lOO0~ 0.1400 0.2200 0.1000 . 0.10000.10000.1000
The diagonal matrix of the system eigenvalues ('A) and the unit normalised associated eigenvectors or mode shapes [] , with [{[M][] = [I] , are calculated as ('A)
=
j
1.3397 0 0 0 8.3333 0 , [] [ o 0 18.6603
FinaIIy the flexibility matrix [G] [G] =
=
[0.4206 0.2402 0.8749J 0.5067 -0.4804 -0.1117 . 0.3212 004003 -0.2644
= [] (A)- , [] T
becomes :
0.1800 0.1000 0.1000~ 0.1400 0.2200 0.1000 . [ 0.1000 0.1000 0.1000
15.2.2 Decomposition of the Mass Matrix The modal matrix [] is normalised with respect to the mass matrix such that (15.4): [ {[M][ ]
= [I].
Premultiplying the previous equation with the modal matrix [] and postmultiplying the previous equation with [M] [] T the result will be:
15 Mode Acceleration Method
316
(15.11)
Using the orthogonality relations of the modal matrix with respect to the mass matrix we obtain:
= [I] .
[cI>][cI> {[M]
(15.12)
Thus finally the inverse of the mass matrix can be calculated as: [Mr
l
= [cI>][cI>{
(15.13)
.
The
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