Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 619

Since the generalized masses M i of Eq. (10.72) are defined in terms of the mode shapes
which contain an arbitrary constant, proper selection of the constant will permit M i to be de-
fined (scaled) such that M i = 1. Then Eq. (10.79), appears as

2

q i + 2 ζ i ω i ˙ q i + ω

i q i =

P i (10.80)

This equation is readily solved.

The homogeneous form of Eq. (10.80) has the solution

A 1 e α 1 t A 2 e α 2 t

q i = +

with

ζ i ω i ± 4

i = ω i − ζ i ± ζ

2 −

1

1

i i i

α 1 , 2 = 2 ζ ω − 4 ω −

when

ζ i < 1 ,q i is underdamped and oscillates

ζ i > 1 ,q i is overdamped and exponentially decays

ζ i = 1 ,q i is critically damped, C i = C cr

10.4.2 Summary of the Modal Superposition Method

The dynamic response of any linear vibration system modeled with n d DOF can be com-
puted using the modal superposition procedure just described. The procedure can be sum-
marized in the following steps.

STEP 1. Formulation of Equations of Motion
For the class of problems considered here, the equations of motion can be expressed in
the matrix form

M V + KV = P (10.81)

) = V 0

V

with initial conditions V ( 0 ) = V 0 and ( 0 .

STEP 2. Free Vibration Analysis
The mode shape vectors φ i and corresponding natural frequencies ω i are computed by

solving the generalized eigenvalue problem

2

K φ i = ω

i M φ i

(10.82)

STEP 3. Compute Generalized Mass and Loading
The generalized mass M i and generalized force P i for the i th mode are computed
using

φ i Φ T MΦ M i = M φ i or M n d =

(10.83)

P i = φ i P = Φ T P

or P

for all modes of interest. If the scaled mode shapes are used, M n d =

.

I