Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 618

in which φ ij is the j th element in φ i and m jk is the element in the j th row and the k th column
of M . Substitution of Eq. (10.73) or (10.74) into Eq. (10.69) provides the complete physical
response V of an arbitrarily loaded n d DOF system.
If the scaled mode shapes are used to form R , then the coordinate transformation of Eq.
(10.69) results in

2
i q i

q + Λq = P or q i + ω = P i (10.75)

where Λ is the diagonal matrix of Eq. ( 10.66b). That i s, the diagonal elements are the
eigenvalues of the system. Also, P

R T P and P i = φ i

= P . The solution for each uncoupled

equation is

t

sin ω i t

ω i +

1
ω i

q i = q i ( 0 ) cos ω i t +˙ q i ( 0 ) P i (τ ) sin ω i ( t − τ) d τ (10.76)

0

where the initial conditions are obtained in the same way as in Eq. (10.73) or (10.74), with
Φ − 1 as the inverse of the matrix containing the scaled mode shapes.
Recall that the modal superposition solution is formed from mode shapes that have
unknown amplitudes. However, in spite of this indeterminate characteristic of the mode
shapes, the general forced response for a dynamic system is fully determined. This can be
shown by multiplying the mode shapes φ i by a constant and then noting from the formula
for q i of Eq. (10.73) that this constant cancels out in Eq. (10.69).
It has been demonstrated here that the modal superposition method can be used to
compute the vibration responses of systems subjected to arbitrary loading. It can also be
used to find the steady-state sinusoidal response, in which case the effect of damping is
very important, especially if the exciting frequency is one of the natural frequencies of the
system. In general, the principal coordinates for M and K are not able to decouple the
damping matrix C in a viscous damping model of the form

M V C V + + KV = P (10.77)

because Φ T CΦ is not a diagonal matrix unless C is in the form of proportional damping ,
i.e., C are constants. Hence, the equations of motion for a

damped system usually cannot be decoupled using the normal modes of the correspond-
ing undamped system. One way of incorporating damping into the modal superposition
method is the use of modal damping . Since the governing equation of motion in principal
coordinates

= α M + β K , where α and β

is similar to the equations of motion for a single-DOF spring
mass system, it would seem reasonable that damping can be assigned to each mode leading
to equations of the form of a single DOF system with a dashpot. That is, assign damping
C i to the i th mode to form the equations

( M i q i + K i q i = P i )

M i q i + C i q i ˙ + K i q i = P i (10.78)

ζ

which replaces Eq. (10.72). It is more common to assign a “damping ratio”

i to each mode

with the resulting modal equations appearing as

2

q i + 2 ζ i ω i ˙ q i + ω

i q i =

P i / M i (10.79)

2 √ M i K i .
Modal damping is one form of proportional damping. The assignment of damping should
be based on test data if available or else on engineering judgment and previous experience
with similar systems.

where ζ = C i /( 2 M i ω ) = C i / C cr with C cr , the critical damping ,defined as C cr =

i i