Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 616

10.4 Forced Response

The response of a structural system to prescribed time-dependent loading P ( t ) involves the

solution of

M V + KV = P (10.68)

This is an ordinary differential equation in time that can be integrated directly. Textbooks
on vibrations or structural dynamics describe a variety of time integration techniques for
solving Eq. (10.68), some of which are presented in Section 10.5. For linear problems, how-
ever, the most frequently used technique in practice is the modal superposition method
that employs the free vibration responses. Natural frequencies and mode shapes are often
calculated at the start of a dynamic analysis of a linear structural system, so it is a rel-
atively simple procedure to compute the transient response using modal superposition.
This method will be described briefly here.

10.4.1 Modal Superposition Method
The equations of motion for the undamped systems studied thus far exhibit coupling for
various DOF. In some cases, the coupling is through the stiffness terms, e.g., Eq. (3) of
Example 10.6, while in other cases, the coupling is in the inertia terms, e.g., Eq. (5) of Example
10.5 (where neither the stiffness matrix K nor the mass matrix M is diagonal). The former is
usually called static or stiffness coupling and the latter is called dynamic coupling . By properly
choosing coordinates (DOF), a system can be rendered uncoupled. The process of expressing
the equations of motion in terms of different coordinates is called a coordinate transformation .
The set of coordinates for which the equations of motion are completely uncoupled is called
the principal, normal, modal ,or natural coordinates . The principal coordinates are useful in
simplifying response calculations.
It is possible to find principal coordinates for any linear system. Let V be the coordinates
in which the equations of motion are coupled, and q

q n d ] T

= [ q 1 q 2 ··· are the principal

coordinates, where n d is the number of DOF. Also, let

V = Rq (10.69)

Choose

R = [ φ 1 φ 2 ··· φ n d ] = Φ (10.70)

where φ i , the mode shapes, are solutions of

2

ω

i M φ i =

K φ i

Substitute Eq. (10.69) into Eq. (10.68), and premultiply by R T to obtain

R T MR q R T KRq R T P

+ =

(10.71)

From the orthogonality condition for mode shapes [Eq. (10.65a)],

Φ T MΦ Φ T KΦ R T MR R T KR = = M n d and = = K n d

in which M n d and K n d are the diagonal matrices of Eqs. (10.65b) and (10.65c). It is thus
concluded that use of Eq. (10.69) with R defined by Eq. (10.70), uncouples Eq. (10.68), and