Page 207 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

194 Properties of Vibrating Systems Chap. 6
geneous equation and are as follows:

Floor 0) j = 0.1495vT 7^ CO2 = 0A45l^k/m i03 = 0.7307V^

</>i(x) 4>^ix)
10 1.0000 1.0000 1.0000
9 0.9777 0.8019 0.4662
8 0.9336 0.4451 -0.3165
7 0.8686 0.0000 -0.9303
6 0.7840 -0.4451 -1.0473
5 0.6822 -0.8019 -0.6052
4 0.5650 - 1.0000 1.6010
3 0.4352 - 1.0000 0.8398
2 0.2954 -0.8019 1.0711
1 0.1495 -0.4451 0.7307
0 0.0000 0.0000 0.0000
The equation of motion of the building due to ground motion u^it) is
MX T CX + KX =

where 1 is a unit vector and A' is a 10 X 1 vector. Using the three given modes, we

make the transformation
X = Pq

where P is a 10 X 3 matrix and ^ is a 3 X 1 vector, i.e.,
^3(-^i)
^3(-^2)
P = Q

Prcmultiplying by P \ we obtain
P^MPq + P'CPq + P^KPq = -P^M\U(^{t)
and by assuming C to be a proportional damping matrix, the foregoing equation
results in three uncoupled equations:
10
+ C||(/| + k^^q^ = L I^,4>i{x,)
i= 1
10
'”22<?2 + i'22<i'2 + ^22<?2 = - “o(0 L ">>2(-^,)

+ Cjjcr, + = -«„(0 E m,03(jr,)
/-I
where m,-, c„, and k¿¿ are generalized mass, generalized damping, and generalized
stiffness. The qj(t) are then independently solved from each of the foregoing equa­
tions. The displacement of any floor must be found from the equation X = Pq