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8 Free Vibration Analysis of Continuous Systems
Problem 8.2 Determine the natural frequencies and mode shapes for a uniform beam
having both ends elastically restrained in rotation, the coefficient of elastic restraint
being R.
The end conditions for this problem are
At
x = 0, Y = 0 and R
dY
dx
= E I
d
2 Y
dx 2
(a)
and at
x = L , Y = 0 and R
dY
dx
= −E I
d
2 Y
dx 2
(b)
Substituting the conditions given by Eq. (a) into Eq. (8.85), we get
C 2 + C 4 = 0
( c )
C 1 + C 3 =
E I λ
R
[−C 2 + C 4 ]
(d)
C 1 sin μ + C 2 cos μ + C 3 sinh μ + C 4 cosh μ = 0
( e )
where
μ = λL
(f)
and
R[C 1 cos μ − C 2 sin μ + C 3 cosh μ + C 4 sinh μ]
= −
E I μ
L
[−C 1 sin μ − C 2 cos μ + C 3 sinh μ + C 4 cosh μ]
(g)
Combining Eqs. (c), (d) and (e), we get
C 1
C 2
=
cosh μ − cos μ + 2ϕμ sinh μ
sin μ − sinh μ
(h)
where
φ =
E I
RL
(i)
8 Free Vibration Analysis of Continuous Systems
Problem 8.2 Determine the natural frequencies and mode shapes for a uniform beam
having both ends elastically restrained in rotation, the coefficient of elastic restraint
being R.
The end conditions for this problem are
At
x = 0, Y = 0 and R
dY
dx
= E I
d
2 Y
dx 2
(a)
and at
x = L , Y = 0 and R
dY
dx
= −E I
d
2 Y
dx 2
(b)
Substituting the conditions given by Eq. (a) into Eq. (8.85), we get
C 2 + C 4 = 0
( c )
C 1 + C 3 =
E I λ
R
[−C 2 + C 4 ]
(d)
C 1 sin μ + C 2 cos μ + C 3 sinh μ + C 4 cosh μ = 0
( e )
where
μ = λL
(f)
and
R[C 1 cos μ − C 2 sin μ + C 3 cosh μ + C 4 sinh μ]
= −
E I μ
L
[−C 1 sin μ − C 2 cos μ + C 3 sinh μ + C 4 cosh μ]
(g)
Combining Eqs. (c), (d) and (e), we get
C 1
C 2
=
cosh μ − cos μ + 2ϕμ sinh μ
sin μ − sinh μ
(h)
where
φ =
E I
RL
(i)