Page 202 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 6.8 Decoupling Forced Vibration Equations 189

6.8 DECOUPLING FORCED VIBRATION EQUATIONS
When the normal modes of the system are known, the modal matrix P or P can be
used to decouple the equations of motion. Consider the following general equation
of the forced undamped system:

MX KX = F (6.8-1)
By making the coordinate transformation X = PY, the foregoing equation be­
comes
MPY + KPY = F
Next, premultiply by the transpose P^ to obtain

(P^M F)y H- {P^KP)Y = P^F (6.8-2)

Because the products P^MP and P^KP are diagonal matrices due to orthogonal­
ity, the new equations in terms of Y are uncoupled and can be solved as a system
of 1 DOF. The original coordinates X can then be found from the transformation
equation
X=^PY (6.8-3)

Example 6.8-1
Consider the two-story building of Fig. 6.8-1 excited by a force F{t) at the top. Its
equation of motion is
2 0 -h k 3 -1
0 1 - 1 1
The normal modes of the homogeneous equation are
'0.5) . f-1
= (¡>2
1
from which the P matrix is assembled as
0.5 -1
P = 1 1

m
k
2m
2k
Figure 6.8-1.