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
(FEA) on other hand can be used for solving such problems by using available computer programs such as
described in [15-16].
2. Methodology
The Finite Element method is used to solve physical problems in engineering analysis and design. The
idealization of physical problem to mathematical model requires certain assumptions that together lead to
differential equation governing the mathematical model. The FEA solves this mathematical model. Here the
mass matrix formulation is carried out. A two-element cantilever beam is used in order to develop the consistent
mass matrix. The six degrees of freedom lumped mass is used for constructing the lumped mass matrix The
global mass matrix is built up as an assemblage of element mass matrices. A method analogous to static
condensation, Guyan reduction, is used to reduce the size of the two-element cantilever problem. The model is
then solved for its eigenvalues using Guyan reduction.
Element Stiffness Matrix
The element stiffness matrix is developed by using basic strength of materials techniques to analyse the forces
required to displace each degree of freedom a unit value in the positive direction:
Beam Node Definitions
The two-beam elements are made identical, with the same E, I and length; the global stiffness matrix can then be
rewritten as:
⎡ 24 –12 6 ⎤
0
k
g
⎢ 3
1
⎢
⎢ 0
= ⎢
⎢ –12
⎢ 3
1
⎢ 6
⎢ 2
⎣ 1
8
1
– 6
2
1
2
1
3
1
– 6
2
1
12
3
1
– 6
2
1
2
1 ⎥
2 ⎥
⎥
1 ⎥
– 6⎥
2
1 ⎥
4 ⎥
⎥
1 ⎦
For solving the dynamics of the cantilever beam, a mass matrix is developed to complete the equations of
motion. For a beam finite element, there are a number of different mass matrix formulations as discussed in [13-
15] out of which Consistent mass – distributed mass is used.
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