Page 206 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 6.10 Normal Mode Summation 193

22ZZZ2

!2y/ZZ2
Z///ZZ

ZZZZZZ
ZZZZZ2
ZZZZZZ
^ 2ZZZZ.
7ZZZZZ
ZZZZZZ.

Figure 6.10-1. Building displacement represented by normal modes,

where the term

_J_______________________ (6.10-6)
r, =

is called the mode participation factor.
In many cases, we are interested only in the maximum peak value of x^, in
which case, the following procedure has been found to give acceptable results. We
first find the maximum value of each qjit) and combine them in the form
U,lmax = + l <^3() <73, max (6.10-7)'^
Thus, the first mode response is supplemented by the square root of the sum of the
squares of the peaks for the higher modes. For the previous computation, a shock
spectrum for the particular excitation can be used to determine ^ax-
predominant excitation is about a higher frequency, the normal modes centering
about that frequency can be used.
Example 6.10-1
Consider the 10-story building of equal rigid floors and equal interstory stiffness. If
the foundation of the building undergoes horizontal translation determine the
response of the building.
Solution: We assume the normal modes of the building to be known. Given are the first
three normal modes, which have been computed from the undamped homo-

^The method is used by the shock and vibration groups in various industries and the military.