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12 Finite Element Method in Vibration Analysis
Fig. 12.1 Torsional element
f = φ = α 1 + α 2 x
(12.1)
and
f = φ = [1 x]
α 1
α 2
(12.2)
Here, α 1 and α 2 are constants. Number of constants in a displacement function
should be equal to the nodal displacements. The nodal values are
φ 1
φ 2
=
1 0
1 L
α 1
α 2
or
α 1
α 2
=
1 0
1 L
− 1 φ 1
φ 2
(12.3)
Combining Eqs. (12.2) and (12.3), we get
f = φ = [1 x]
1 0
1 L
−1 φ 1
φ 2
or
f = φ = [N ] {X } e
(12.4)
where
[N ] =
1 −
x
L
x
L
(12.5)
and
{X } e =
φ 1
φ 2
(12.6)
[N] is called the interpolation matrix, or the shape function matrix. The equivalent
stress–strain relationship of the shaft is