FINITE ELEMENT METHOD
Handout 14  MME 615: Advanced Vibration
Kumar V. Singh
 1 
Introduction to Finite Element (Computational Framework: Energy Method)
Powerful numerical technique that uses energy methods, calculus of variation and
interpolation schemes in solving large boundary value problems in which a continuous
structure is divided into well defined substructures (finite elements) having elemental
material and geometrical parameters
These elements are defined by nodes which connect the neighboring elements, possesses
degree of freedom (translation and/or rotation) and at which the boundary conditions and
loads are applied.
Extremely useful for approximating the governing equations of complicated, large and
complex geometry structures
Provides a computational framework which can address wide range of static and
dynamics problems
Energy associated with the continuous and discrete structures
Kinetic energy of the continuous bar or beam:
2
0
,
1
2
L
bar
beam
u x t
T
T
A x
x
dx
t
Potential energy of the continuum bar:
2
0
,
1
2
L
bar
u x t
U
E x A x
dx
x
Potential energy of the continuum beam:
2
0
,
1
2
L
beam
w x t
U
E x I x
dx
x
x
Kinetic energy of the discrete mass:
2
1
2
mass
i
i
T
m u
Potential energy of the discrete stiffness:
2
1
2
springs
i
i
U
k u
Nonconservative work done due to external force
,
x t
f
:
0
,
,
L
W
x t
u x t dx
f
Find the Total Kinetic energy of the system, potential energy of the system and nonconservative
forces and apply Lagrange method:
i
i
i
i
d
T
T
U
F
dt
u
u
u
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FINITE ELEMENT METHOD
Handout 14  MME 615: Advanced Vibration
Kumar V. Singh
 2 
Main idea:
Figure 8.1 [1]
A structure is divided into small pieces (finite elements) with nodes which connects with
neighboring elements and represents the motion of element.
Describe the physical parameters of each element
Assemble the elements at the nodes in forming a approximated discrete system
representing whole structure
Solve the system of equations involving unknown quantities at the nodes (e.g.,
displacements, slope, strains, stresses, natural frequencies, mode shapes etc.)
FEM FORMULATION FOR AXIALLY VIBRATING BAR
Consider an axially loaded bar as shown in the figure here. The
Potential energy
of the
axially loaded bar corresponding to the
exact solution
u(x)
can be obtained as,
2
0
1
2
L
du
U
EA
dx
dx
,
And the work done due to distributed forces can be
obtained as,
0
( )
L
W
F x u(x)dx
These quantities associated with the axially loaded bar can also be computed by choosing a
suitable
approximated solution
w(x)
as follows,
2
0
0
1
2
( )
L
L
dw
U w
EA
dx
dx
W
w
F x w(x)dx
x
L
F
)
(
,
x
A
E
y
)
(
x
u
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 Fall '11
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 Finite Element Method, Kumar V. Singh

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