Page 189 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Properties of Vibrating Systems Chap. 6
176

We now reverse the order of loading, in which case the total work done is

Because the work done in the two cases must be equal, we find that

Example 6.2-1
Figure 6.2-1 shows an overhanging beam with P first applied at 1 and then at 2. In
Fig. 6.2-l(a), the deflection at 2 is
3^2 ^ ^21^
In Fig. 6.2-l(b), the deflection at 1 is
yi = «12^

Because a ,2 = «21^ >’1 equal y2, i.e., for a linear system, the deflection at 2, due
to a load at 1, is equal to the deflection at 1 when the same load is applied at 2.

(a)

(b)

Figure 6.2-1.

6.3 STIFFNESS INFLUENCE COEFFICIENTS
The stiffness matrix written in terms of the influence coefficients k^j is
ki, k,2
^22 ^23 (6.3-1)
"32
The elements of the stiffness matrix have the following interpretation. If
jCj = 1.0 and X2 = X3 = 0, the forces at 1, 2, and 3 that are required to maintain
this displacement according to Eq. (6.3-1) are /C21, and k^i in the first column.
Similarly, the forces /j, / 2, and required to maintain the displacement configu­