Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 600

With substitution of Eq. (10.28) into Eq. (10.26), and premultiplication of the result by T T ,
the mass term in Eq. (10.26) appears as
T T M aa

T V a

M a 0

M 0 a M 00

= M aa − K 0 a K − 1

00 T M 0 a −

K 0 a K − 1

00 T M 00 K − 1 00 K 0 a V a

M a 0 K − 1

00 K 0 a +

M V a = (10.29a)

Similarly,

= K aa

00 K 0 a V a

T T KTV a K a 0 K − 1 − = KV a (10.29b)

Equation (10.26) becomes

M V a + KV a = 0 (10.30)

Equation (10.30) is the governing equation of m otio n with condensed dynamic DOF. In the
case of a lumped mass model, the mass matrix M may not be a diagonal matrix anymore
because of the matrix operations in Eq. (10.29a).

10.3 Free Vibration Analysis

The governing equations for the “free” motion of a structure are

M V + KV = 0 (10.31)

The motion is referred to as being free, since there are no applied loadings. Often in this
section, it will be assumed that M and K are the mass and stiffness matrices after the
boundary conditions are imposed and, if desired, Guyan reduction is performed.
By assuming harmonic motion ,

V = φ sin ω t (10.32)

and the corresponding mode shapes φ can be computed from the
generalized eigenvalue problem

ω

the natural frequencies

2 M φ

ω =

K φ

(10.33)

or

2 M

( K − ω ) φ = 0 (10.34)

Because φ is nontrivial,

2 M

| K − ω |= 0 (10.35)

2

or, with ω = λ ,

| K − λ M |= 0 (10.36)