10.5 Dynamic Stiffness Matrix for Flexure and …
411
= −
m
m
(λL 4 )
4 E I
L
3
4
(δ 3 ) 4
= −
958
95.8
(1.94)
4 E I
12.2 3 (δ 3 ) 4
= −0.078E I (δ 3 ) 4
b 4 = λL 4 = 0.97 ×
12.2
6.1
= 1.94
Once the dynamic stiffness matrix for the individual element has been obtained,
they can now be assembled to form the dynamic overall stiffness matrix for the frame
[K ] = E I
⎡
⎣
0.544 0
0.113
0 0.819
0
0.113 0 0.05295
⎤
⎦
Therefore, the determinant of [K] is
|K | = E I × 0.01313 = 8.28 × 10
7
× 0.01313 = 1.087 × 10
6
Calculations may be started with a fresh value of b 2 , and the above steps are to
be repeated. Natural frequency lies in between two values, where the determinant
changes from positive to negative.
10.6 Dynamic Stiffness Matrix of a Bar Undergoing Axial
Deformation
We approach the problem in the same manner, as we have done in the case of bending
member in Article 10.4.
For a bar subjected to axial deformation due to its distributed mass, we can write
the following equation based on Eq. (8.14) as
U = A 1 sin λ a x + A 2 cos λ a x
(10.52)
where
λ a =
p
2
ρ
E
(10.53)