OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
Consistent mass
The consistent mass matrix for a beam element is a filled matrix. The filled matrix can be combined with other
consistent mass matrices of other elements of the structure, in the same manner as the element stiffness matrices
are combined, to yield the final global mass matrix. The element consistent mass matrix for a prismatic beam is,
with mass per unit length m and length l.
m
e
=
ml
420
⎡ 156
⎢
221
⎢
⎢ 54
⎢
⎣−131
221
41
2
131
2
− 31
54
131
156
− 221
−131⎤
2
− 31
⎥
⎥
− 221⎥
2 ⎥
41 ⎦
Assuming the two elements have the same properties and lengths, the global mass matrix becomes:
2
2
⎡ 156m1 22m1 54m1 −13m1
0 0 ⎤
⎢ 2
3
2
3
⎥
⎢ 22m1 4m1 13m1 − 3m1 0 0 ⎥
⎢
2
2
1 54m1 13m1 312m1 0 54m1 −13m1
⎥
mg
= ⎢
2
3
3
2
3
⎥
420 ⎢−13m1
− 3m1 0 8m1 13m1 − 3m1 ⎥
⎢
2
2
0 0 54m1 13m1 156m1 − 22m1 ⎥
⎢
⎥
2
3
2
3
⎢⎣
0 0 −13m1
− 3m1 − 22m1 4m1 ⎥⎦
Taking into account the two constrained degrees of freedom at the built in end, we can eliminate the first two
rows and columns:
m
g
=
1
420
⎡ 312m1
⎢
⎢
⎢
⎢
⎣
0
54m1
2
−13m1
0
8m1
3
13m1
2
3
− 3m1
54m1
13m1
2
156m1
2
− 22m1
−13m1
− 3m1
− 22m1
4m1
3
3
2
2
⎤
⎥
⎥
⎥
⎥
⎦
Having the mass and stiffness matrices allows us to solve the eigenvalue problem for the homogeneous
equations of motion:
mg ž + kgz
= [ 0]
It is better to reduce the 4x4 problem down to 2x2 size. Therefore, the Guyan reduction [15] will be used to
reduce the size of the problem.
Two element cantilever eigen value closed form solution using guyan reduction
Repeating the rearranged global stiffness matrix from the static run,
⎡ 24
– 12
⎢
0
3
3
1
1
⎢
8 – 6
⎢ 0
2
k = ⎢
1 1
g
EI
⎢ – 12 – 6 12
⎢ 3
2
3
1 1 1
⎢ 6 2 – 6
⎢
2
2
⎣ 1 1 1
6
2
1
2
1
– 6
2
1
4
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Equation of motion is: mss
x + kssx = [0]
392