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Structural Vibration and Dynamics 271

where ξ is the damping ratio at mode i, Equation 8.25 becomes:
i
z i +ξ ω2 z + ω 2 z i = p t (), i = 1 2, , …, n (8.26)
i ii i i
Equations in Equation 8.25 with modal damping, or in Equation 8.26, are called modal equa-
tions. These equations are uncoupled, second-order differential equations, that are much
easier to solve than the original dynamic equation which is a coupled system.
To recover u from z, apply the transformation in Equation 8.24 again, once z is obtained
from Equation 8.26.

Notes:
• Only the first few modes may be needed in constructing the modal matrix Φ (i.e.,
Φ could be an n × m rectangular matrix with m < n). Thus, significant reduction in
the size of the system can be achieved.
• Modal equations are best suited for structural vibration problems in which
higher modes are not important (i.e., for structural vibrations, but not for struc-
tures under impact or shock loadings).

8.4 Formulation for Frequency Response Analysis

For frequency response analysis (also referred to as harmonic response analysis), the
applied dynamic load is a sine or cosine function. In this case, the equation of motion is

Mu + Cu + Ku = Fsin (8.27)
ωt

Harmonic loading
8.4.1 Modal Method

In this approach, we apply the modal equations, that is

z i + 2ξ i ω z +ω 2 z i = p i sin ω ti = 1 2, , …, m (8.28)
ii i
These are uncoupled equations. The solutions for z are in the form:

p i ω 2
zt() = i sin(ω− θ i ) (8.29)
t
i
i ) + 2
1
( −η 22 ( ξ i )η i 2
where
 2 ξη i
i
 θ= arctan 1 2 , phase angle;
i
 −η i

 η= ωω i ;
i

 ξ= i c = i c , dammpingratio
i
  c c 2 m ω i