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Transverse vibrations of beams with variable moment of inertia and an intermediate support
Applied Acoustics 17 (1984) 255-260
Transverse Vibrations of Beams with Variable Moment of Inertia and an Intermediate Support
Silvia Alvarez and Patricio A. A. Laura Institute of Applied Mechanics, Puerto Belgrano Naval Base (Argentina) (Received: 4 May, 1983)
SUMMARY
The above problem is tackled using the Ritz method and approximating the deHection.['unction by means of polynomial co-ordinate functions which satisfy the essential boundary conditions. The presence of an axial force and concentrated masses is considered. Fundamental.frequeno' toe]: .ficients are determined in the case of a rather complex structure taking into account d(fferent types of boundary conditions.
INTRODUCTI ON Thorough studies are available in the literature on vibrating beams with intermediate supports.l-4 The case of a beam with a discontinuous moment of inertia has been treated recently.S The main purpose of the present study is to demonstrate the generality o f the approach presented in reference 1 in the case of a rather complex structural element (Fig. 1) and to study experimentally the convergence of the eigenvalues as the number of terms in the approximating function is increased. 255 Applied Acoustics 0003-682X/84/$03.00 © Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
256
Silvia Alvarez, Patricio A, A. Laura
M1
M2 $
S
P
X ~s2
R1 £1
II-
R2 E
-IP
Fig. I. Transversely vibrating system under investigation. AN APPROXIMATE SOLUTION The maximum strain and kinetic energies of the system are given by the expressions: Ureax = ½El(x)
fOL2(d2WX~2
\ - d ~ - ] d£ +
fL" (d2W~ 2 +½El(x)., \ ~ ] d£ + 2 ~22 \ d.f j ]~=L+ ½S
Tin.~= ½0A(x)e) 2
;
W 2 d£ +
+½0A(x)efl
W2d£
½EI(x)
;£3 (d2W~ 2
2 \d~-,] d£
1 l (dW~ 2 + ~ - ~ - \ ~ - ] [~=o
\~-j
d.f
f/
W 2 d2
½pA(x)e) 2
(l(a))
+ ½M~02 W2 .e= ~, + ½M2¢02W 21.~:.~2
3
(l(b)) where Umax is the maximum strain energy and Tmax is the maximum kinetic energy. The displacement amplitude function, W, is approximated
Transverse vibrations of beams with variable moment of inertia
257
in terms of a summation of co-ordinate functions: I
W(x) ~ Wo(x) = ) ' A#i(x)
(2)
i=1
where each qJi(x) satisfies, at least, the essential boundary conditions. Substituting eqn. (2) into eqns(l) and requiring the minimization condition: ? ¢3Ai [Urea x - Tm.x] = 0 (3) one obtains a linear system of homogeneous equations in the Ai's. From the non-triviality condition a frequency determinant is generated. It is convenient to use the dimensionless variable x = Y/E, resulting in the co-ordinate functions, making I = 2 in eqn. (2): W ( x ) = W a ( x ) = ml(O~15 x5 -{- 0~14x 4 + 0~13x3 -{- 0~12 x2 "+" 0~1 i X )
(4)
+ A z ( ~ z 6 X 6 --I-0~25x5 + 0~24x 4 -Jr-x 3)
The %'s are determined (see Appendix) by requiring: A--First Co-ordinate Function
dW., W.,(0)=0 W.,(L,) = 0 W.,(1)=0
EI 2 d2W.,
d x x=o= ~bl L
dx2 x=o (5)
dW., dx
El 2 d2W.,
= t=-¢2
/~
dx2 x=l
B--Second Co-ordinate Function W._.(L,) W.,(0) = 00 Wo:(l) = 0
dW. dx -" ~=
=-~b2 1
El z dzW.. /~
dx2 x=
} (6) i
The beam height is defined, in the case of the structural system shown in Fig. 1. by the following equations:
h(x)=h2(l- 7x) h(x) = h I h(x)=h2[l+y(x- 1)]
O<_x
(7)
Silvia Alvarez, Patricio A. A. Laura
258
One then obtains the following expressions for the cross-sectional areas and moments of inertia. For: O<_x <_L 2 A ( x ) = A2(I - 7x)
where A 2 = b2h 2 wherel2 -
l(x)=12(1-Tx)3
b2h~
12
For: L 2 _< x_< L~ A ( x ) : A 1 = bh 1
bh¢ 12
l ( x ) : 11 -
For:Ls<_x<_l
A(x)=A2[I + 7 ( x - 1)l l(x)
:
1211 +
7(x
-
1 )]3
NUMERICAL RESULTS In
the
present
study
fundamental
frequency
coefficients
f~--=
( x / ~ l / ' E I ~ ) o ~ £ 2 have been obtained for 7--0.65. Accordingly:
A2 - 1.14 Al
12 - 1.5 I1
Table 1 shows frequency values for two cases. In the first situation the intermediate support is placed at the center (L~ =0"5) whilst, in the second situation, L~ = L 2 =0.2. The effect of an axial force and the presence of concentrated masses have also been considered. A very slight change in the determined eigenvalue is observed when the intermediate support is placed at the center, as the number of co-ordinate functions is increased. On the other hand, the variation is of considerable magnitude when the support is placed at x = L 2. The present approach is quite simple and straightforward. It may be extended to the case of forced vibrations. On the other hand, it possesses the advantage that the entire algorithm can be easily programmed and implemented on a microcomputer.
Transverse vibrations of beams with variable moment of inertia
259
TABLE ! Fundamental Frequency Coefficients for Different Combinations of Parameters, see Fig. I (7 =0.65). Note: M* = M,/M,.; M~ = M2/M ,, where M,. = p(Al£ 3 + A2£2)
S[~2EI1
Values o[f~ = N/ Ell ¢°£z 0
M* = M~ = 0 Lj =0"5 L 2 =0"2 L 3 =0.8 x I =0"25 x 2 = 0"75
73.66 73.63
10 (one term) (two terms)
76.57 76.54
M* = M * =0.10
61.49 61.47
63.93 63 "90
M* = M* = 0
40.45 40.44
44.98 44.97
Ml* = M* = 0.10
34.24 34-23
38"07 38.07
M* = M* = 0
37.98 34.29
40.33 36.90
34.46 31.98
36.59 34.38
22-90 21.93
26.52 25.45
20.42 19"79
23.64 22.99
L 1 = L 2 = 0-2 M* = M * = O . I O L 3 =0"8 x I =0.25 M* = MJ' = 0 x_, = 0.75 M* = M~' =0.10
(°i
4,'~ = , ~ = o
(Rigidly clamped)
,~', = 4'~ =
(Simply supported)
4,', = q~'., = o
4", = 4,~ =
REFERENCES
I. R. D. Blevins, Formulas.Jbr naturalfrequen~ T andmode shape, Van N o s t r a n d Reinhold Co., New York, 1979. 2. D . J . G o r m a n , Free lateral vibration analysis of d o u b l e - s p a n u n i f o r m beams, Int. Journal o[" Mechanical Engineering Sciences, 16 (I 974), pp. 345 5 I. 3. D . J . G o r m a n a n d R. K. Sharma, Vibration frequencies a n d m o d a l shapes for multi-span beams with u n i f o r m l y spaced supports, Ottawa University, O n t a r i o , C a n a d a , Report a n d Conference No. 740330-1, 1974. 4. P . A . A . Laura, P. Verniere de lrassar a n d G. Ficcadenti, A note on transverse vibrations of c o n t i n u o u s beams subject to an axial force and carrying concentrated masses, Journal of Sound and Vibration, 86(2) (1983), pp. 279-84. 5. S. Alvarez a n d P. A. A. Laura, F u n d a m e n t a l frequency of an elastically restrained beam with d i s c o n t i n u o u s m o m e n t o f inertia a n d an intermediate support, Journal o/Sound and Vibration, 85(3)(1982), pp. 148-50.
Silvia Alvarez, Patricio A. A. Laura
260
APPENDIX--EXPRESSIONS
DEFINING
T H E ~ifs
~11 = 2qf, ~12 = 1
~t3 = - 1 -2q5' 1 - ~xl,,- ~15
g~(-44,; -124,'14,~-44q- l) + L3(8~'1 + 404)'~b~ + 18q5'2 + 3) + L 2 ( - 14~b~ - 2)
L~(-6q52-
1) + L~(14O~ + 2) + L~(-8(;b 2 - 1) t
t
t
L1 ( - 4q51 -- 28~b2(D 1 )
L ~ ( - 6 q 5 2 - 1) + L~(14~b~ + 2 ) + L 3 ( - 8 q ~ - 1)
L~(4~'~ + 12q~'1~. + 4 ~ + 1) + L ~ ( - 6~b'~ - 244)'~q~. - lOq~2 - 2) + L12(64)~, + 1) L ~ ( - 6 q ) ~ - 1) + L4(14~b~ + 2) + L 3 ( - 8 q ) ~ - 1) +
L,(2~b'1 + 12q~'~q5~) L ~ ( - 6 ~ b ~ - 1) + L4(14~b~ + 2) + L 3 ( - 8 0 ~ - 1)
where: t
~1 =
~1E12 E
f
~2 =
~2E12 E
5 L6( - 1 4 ~ - 2) + L,(24q~ 2¢ + 3) + L3( - lOq~ - 1)
L~(6~b~ + 1) + L~(-24q6~ - 3) + L3(18q)~ + 2 ) :~2~ = L6(8~b~ + 1) + L ~ ( - 18q~2 - 2) + L4(lO~b~ + 1) L ~ ( - 6 q ~ - 1) + L~(14q~'2 + 2) + L ~ ( - 8 q ~ - 1) ~:~ = L ~ ( 8 G + l) + L ~ ( - 1 8 G - 2) + L?(10,G + l)
Transverse Vibrations of Beams with Variable Moment of Inertia and an Intermediate Support
Silvia Alvarez and Patricio A. A. Laura Institute of Applied Mechanics, Puerto Belgrano Naval Base (Argentina) (Received: 4 May, 1983)
SUMMARY
The above problem is tackled using the Ritz method and approximating the deHection.['unction by means of polynomial co-ordinate functions which satisfy the essential boundary conditions. The presence of an axial force and concentrated masses is considered. Fundamental.frequeno' toe]: .ficients are determined in the case of a rather complex structure taking into account d(fferent types of boundary conditions.
INTRODUCTI ON Thorough studies are available in the literature on vibrating beams with intermediate supports.l-4 The case of a beam with a discontinuous moment of inertia has been treated recently.S The main purpose of the present study is to demonstrate the generality o f the approach presented in reference 1 in the case of a rather complex structural element (Fig. 1) and to study experimentally the convergence of the eigenvalues as the number of terms in the approximating function is increased. 255 Applied Acoustics 0003-682X/84/$03.00 © Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
256
Silvia Alvarez, Patricio A, A. Laura
M1
M2 $
S
P
X ~s2
R1 £1
II-
R2 E
-IP
Fig. I. Transversely vibrating system under investigation. AN APPROXIMATE SOLUTION The maximum strain and kinetic energies of the system are given by the expressions: Ureax = ½El(x)
fOL2(d2WX~2
\ - d ~ - ] d£ +
fL" (d2W~ 2 +½El(x)., \ ~ ] d£ + 2 ~22 \ d.f j ]~=L+ ½S
Tin.~= ½0A(x)e) 2
;
W 2 d£ +
+½0A(x)efl
W2d£
½EI(x)
;£3 (d2W~ 2
2 \d~-,] d£
1 l (dW~ 2 + ~ - ~ - \ ~ - ] [~=o
\~-j
d.f
f/
W 2 d2
½pA(x)e) 2
(l(a))
+ ½M~02 W2 .e= ~, + ½M2¢02W 21.~:.~2
3
(l(b)) where Umax is the maximum strain energy and Tmax is the maximum kinetic energy. The displacement amplitude function, W, is approximated
Transverse vibrations of beams with variable moment of inertia
257
in terms of a summation of co-ordinate functions: I
W(x) ~ Wo(x) = ) ' A#i(x)
(2)
i=1
where each qJi(x) satisfies, at least, the essential boundary conditions. Substituting eqn. (2) into eqns(l) and requiring the minimization condition: ? ¢3Ai [Urea x - Tm.x] = 0 (3) one obtains a linear system of homogeneous equations in the Ai's. From the non-triviality condition a frequency determinant is generated. It is convenient to use the dimensionless variable x = Y/E, resulting in the co-ordinate functions, making I = 2 in eqn. (2): W ( x ) = W a ( x ) = ml(O~15 x5 -{- 0~14x 4 + 0~13x3 -{- 0~12 x2 "+" 0~1 i X )
(4)
+ A z ( ~ z 6 X 6 --I-0~25x5 + 0~24x 4 -Jr-x 3)
The %'s are determined (see Appendix) by requiring: A--First Co-ordinate Function
dW., W.,(0)=0 W.,(L,) = 0 W.,(1)=0
EI 2 d2W.,
d x x=o= ~bl L
dx2 x=o (5)
dW., dx
El 2 d2W.,
= t=-¢2
/~
dx2 x=l
B--Second Co-ordinate Function W._.(L,) W.,(0) = 00 Wo:(l) = 0
dW. dx -" ~=
=-~b2 1
El z dzW.. /~
dx2 x=
} (6) i
The beam height is defined, in the case of the structural system shown in Fig. 1. by the following equations:
h(x)=h2(l- 7x) h(x) = h I h(x)=h2[l+y(x- 1)]
O<_x
(7)
Silvia Alvarez, Patricio A. A. Laura
258
One then obtains the following expressions for the cross-sectional areas and moments of inertia. For: O<_x <_L 2 A ( x ) = A2(I - 7x)
where A 2 = b2h 2 wherel2 -
l(x)=12(1-Tx)3
b2h~
12
For: L 2 _< x_< L~ A ( x ) : A 1 = bh 1
bh¢ 12
l ( x ) : 11 -
For:Ls<_x<_l
A(x)=A2[I + 7 ( x - 1)l l(x)
:
1211 +
7(x
-
1 )]3
NUMERICAL RESULTS In
the
present
study
fundamental
frequency
coefficients
f~--=
( x / ~ l / ' E I ~ ) o ~ £ 2 have been obtained for 7--0.65. Accordingly:
A2 - 1.14 Al
12 - 1.5 I1
Table 1 shows frequency values for two cases. In the first situation the intermediate support is placed at the center (L~ =0"5) whilst, in the second situation, L~ = L 2 =0.2. The effect of an axial force and the presence of concentrated masses have also been considered. A very slight change in the determined eigenvalue is observed when the intermediate support is placed at the center, as the number of co-ordinate functions is increased. On the other hand, the variation is of considerable magnitude when the support is placed at x = L 2. The present approach is quite simple and straightforward. It may be extended to the case of forced vibrations. On the other hand, it possesses the advantage that the entire algorithm can be easily programmed and implemented on a microcomputer.
Transverse vibrations of beams with variable moment of inertia
259
TABLE ! Fundamental Frequency Coefficients for Different Combinations of Parameters, see Fig. I (7 =0.65). Note: M* = M,/M,.; M~ = M2/M ,, where M,. = p(Al£ 3 + A2£2)
S[~2EI1
Values o[f~ = N/ Ell ¢°£z 0
M* = M~ = 0 Lj =0"5 L 2 =0"2 L 3 =0.8 x I =0"25 x 2 = 0"75
73.66 73.63
10 (one term) (two terms)
76.57 76.54
M* = M * =0.10
61.49 61.47
63.93 63 "90
M* = M* = 0
40.45 40.44
44.98 44.97
Ml* = M* = 0.10
34.24 34-23
38"07 38.07
M* = M* = 0
37.98 34.29
40.33 36.90
34.46 31.98
36.59 34.38
22-90 21.93
26.52 25.45
20.42 19"79
23.64 22.99
L 1 = L 2 = 0-2 M* = M * = O . I O L 3 =0"8 x I =0.25 M* = MJ' = 0 x_, = 0.75 M* = M~' =0.10
(°i
4,'~ = , ~ = o
(Rigidly clamped)
,~', = 4'~ =
(Simply supported)
4,', = q~'., = o
4", = 4,~ =
REFERENCES
I. R. D. Blevins, Formulas.Jbr naturalfrequen~ T andmode shape, Van N o s t r a n d Reinhold Co., New York, 1979. 2. D . J . G o r m a n , Free lateral vibration analysis of d o u b l e - s p a n u n i f o r m beams, Int. Journal o[" Mechanical Engineering Sciences, 16 (I 974), pp. 345 5 I. 3. D . J . G o r m a n a n d R. K. Sharma, Vibration frequencies a n d m o d a l shapes for multi-span beams with u n i f o r m l y spaced supports, Ottawa University, O n t a r i o , C a n a d a , Report a n d Conference No. 740330-1, 1974. 4. P . A . A . Laura, P. Verniere de lrassar a n d G. Ficcadenti, A note on transverse vibrations of c o n t i n u o u s beams subject to an axial force and carrying concentrated masses, Journal of Sound and Vibration, 86(2) (1983), pp. 279-84. 5. S. Alvarez a n d P. A. A. Laura, F u n d a m e n t a l frequency of an elastically restrained beam with d i s c o n t i n u o u s m o m e n t o f inertia a n d an intermediate support, Journal o/Sound and Vibration, 85(3)(1982), pp. 148-50.
Silvia Alvarez, Patricio A. A. Laura
260
APPENDIX--EXPRESSIONS
DEFINING
T H E ~ifs
~11 = 2qf, ~12 = 1
~t3 = - 1 -2q5' 1 - ~xl,,- ~15
g~(-44,; -124,'14,~-44q- l) + L3(8~'1 + 404)'~b~ + 18q5'2 + 3) + L 2 ( - 14~b~ - 2)
L~(-6q52-
1) + L~(14O~ + 2) + L~(-8(;b 2 - 1) t
t
t
L1 ( - 4q51 -- 28~b2(D 1 )
L ~ ( - 6 q 5 2 - 1) + L~(14~b~ + 2 ) + L 3 ( - 8 q ~ - 1)
L~(4~'~ + 12q~'1~. + 4 ~ + 1) + L ~ ( - 6~b'~ - 244)'~q~. - lOq~2 - 2) + L12(64)~, + 1) L ~ ( - 6 q ) ~ - 1) + L4(14~b~ + 2) + L 3 ( - 8 q ) ~ - 1) +
L,(2~b'1 + 12q~'~q5~) L ~ ( - 6 ~ b ~ - 1) + L4(14~b~ + 2) + L 3 ( - 8 0 ~ - 1)
where: t
~1 =
~1E12 E
f
~2 =
~2E12 E
5 L6( - 1 4 ~ - 2) + L,(24q~ 2¢ + 3) + L3( - lOq~ - 1)
L~(6~b~ + 1) + L~(-24q6~ - 3) + L3(18q)~ + 2 ) :~2~ = L6(8~b~ + 1) + L ~ ( - 18q~2 - 2) + L4(lO~b~ + 1) L ~ ( - 6 q ~ - 1) + L~(14q~'2 + 2) + L ~ ( - 8 q ~ - 1) ~:~ = L ~ ( 8 G + l) + L ~ ( - 1 8 G - 2) + L?(10,G + l)