Issue 62
J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25
'
f
u kG kGA
where
. In this way, the equation of motion can be expressed as follows
2
EI
I
EI
I
f
IV
''
I u Au
u f
EIu
f
(14)
kG
kG kGA kGA
d m at d L , we have
Considering this is a problem of free vibration with mass addition as discontinuity
2
EI
I
IV
I u Au ''
u m u x L
(15)
EIu
0
d
d
kG
kG
x u u , x Lx and n T t t t f
in which u , x , and t are dimensionless parameters, Eqn. (15) is given by
Assuming
2
2
x
EI
I
x
2
2
4
x IV
I
f u A f u '' x n n
f u m f u d x n
EI u
(16)
1 0
x n
4
2
kG
kG
L
L
L
d
Dividing everything by 2
x n A f , Eqn. (16) is described as
4
2
f
m
EI
E I kG A
I
x
u
u u
u
IV
n
d
u
(17)
1 1 ''
1 0
4 2
2
2
kG
A L
AL f
L
Af
d
n
n
Furthermore, according to Blevins [59]:
2 4 L
1 4
(18)
2
4
Af
EI
n
Thus, the equation of motion can be given by
2 f I n
2
m
L
E I kG A
4
u
u u
u x
IV
d
d
u
1 1 ''
0
(19)
4
2
kGA AL
L
L
and 2 / I A R (radius of rotation of the cross section), we have the dimensionless motion
/ E Gk
It is known that
equation of the Timoshenko beam:
f R m 2 2
2
L
4
u
2 1 1
'' R u u
u x
IV
n
d
d
(20)
u
0
4
2
kG AL
L
L
1/ L .
where
Neglecting the terms underlined in Eqn. (20), the dimensionless beam motion equation is obtained on the Euler-Bernoulli theory that considers a kinematic hypothesis formulated without considering the shear and the rotational beam effects. In Eqn. (21), the motion of the Euler-Bernoulli beam is described.
2
m
L
4
u x
IV u u
d
d
0
(21)
4
AL
L
353
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