Page 303 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

290 Vibration of Continuous Systems Chap. 9

Figure 9.7-1. Repeated structure
for difference equation analysis.

lateral shear stiffness of k Ib/in., as modeled in Fig. 9.7-1. By applying the method
of difference equations to such structures, simple analytical equations for the
natural frequencies and mode shapes can be found.
By referring to Fig. 9.7-1, the equation of motion for the nih mass is
rnXn = + 1 - ^ n ) - ~ ^ (9-7-1)

which for harmonic motion can be represented in terms of the amplitudes as

2 1 X„_, = Q (9.7-2)
2k
The solution to this equation is found by substituting

(9.7-3)
which leads to the relationship
-h e -ifi

1 - = cos P
2k
^ 2 ( 1 - cos/3) = 4 s i n 2 | (9.7-4)
The general solution for is
= A cos pn B sin /3n (9.7-5)

where A and B are evalauted from the boundary conditions.

Boundary conditions. The difference equation (9.7-2) is restricted to
1 < n < (N - 1) and must be extended to n = 0 and n = N by the boundary
conditions.