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Transverse vibrations of a tapered beam embedded in a non-homogeneous winkler foundation
Applied Acoustics 26 (1989) 67-72
Technical Note Transverse Vibrations of a Tapered Beam Embedded in a Non-homogeneous Winkler Foundation
A BS TRA C T The present study deals with the determination of the fundamental frequency of transverse vibration of the structural elements described in the title. The fundamental mode shape is approximated by means of a polynomial coordinate function which contains an exponential optimization parameter 'n' which allows for minimization of the fundamental frequency parameter.
INTRODUCTION The present study deals with the determination of the frequency of vibration of the structural system shown in Fig. 1 using Rayleigh's optimization procedure. 1 Apparently no studies are available on the title problem with the exception of the investigation reported in Ref. 2. The approximate approach presented in this Note makes it possible to obtain natural frequencies of embedded systems of more complex characteristics than the one considered in Ref. 2, for instance the case of linearly varying thickness.
A P P R O X I M A T E S O L U T I O N OF T H E P R O B L E M U N D E R INVESTIGATION In the case of normal modes of vibration one makes: v(x, t) = V(x). e iw'
(1)
67 Applied Acoustics 0003-682X/89/$03.50 O 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
68
C. P. Filipich, M. J. Sonenblum, E. A. Gil
P<"
>P
~,
T
~.L
L Fig. I. Vibrating structural system studied in the present investigation.
and following Ref. 3 it is convenient to make:
(2)
V(x) ~_ Va(x) = x" + A x 3 + B x 2 + C x + D
where n is Rayleigh's optimization parameter and x = X / L . The constants A, B, C and D are determined substituting eqn (2) in the governing b o u n d a r y conditions (see Fig. 1): [V" - VoV']x_-o = 0
Ev]x_-o=O [ B V " + v 1V']x= 1 = 0
(3)
[v]~=~ = 0
k.
1//\\.,///\~ t/2
~
L/2
,~
Fig. 2. Vibrating beam of discontinuously varying cross section.
69
Transverse vibrations of a tapered beam
Proceeding now with the determination of Rayleigh's quotient given by: Us + p 2 0 p + Uw = [2x2(n)
2 , PoAo
n
=w,L e- 0 =
(4)
rk
where: 20s = Jo ~p(x). (V") 2 dx + v o g'~(O) + v~ V'2(1) 2 Up = .f~ ( V')2 dx 2Uw = k
V2 dx + ~
V2 dx
Tk = f l g(x) V2 dx
3o
B = E111
k,o2 = ko L4
Edo
Edo
golo
P1A1
kaL 4
k'l2
Y = PoAo
v ,IL Vo,l = Eolo
k'12 = El11
p2= pL2
(>o)
~ = k--~o
cp(x) = (1 + ( h l / h o - 1)x)3 g(x) = 1 + '(ht/h o - 1). x
one determines numerically the value of the parameter 'n' which minimizes eqn (4). The eigenvalue can be improved by constructing a summation of polynomial coordinate functions and the higher optimized eigenvalues may also be determined by minimization with respect to the parameter 'n' as shown in Refs 4 and 5. N U M E R I C A L RESULTS Tables 1 and 2 contain frequency coefficients for several tapered beams elements as a function of the governing parameters. The following combinations of boundary conditions have been considered: ---clamped-clamped ---clamped-simply supported --simply supported-clamped --simply supported-simply supported
C. P. Filipich, M. J. Sonenblum, E. A. Gil
70
TABLE 1 Fundamental Frequency Coefficients and Values of the Optimization Parameter for Several Combinations of the Boundary Conditions and of the Governing Geometric and Mechanical Parameters
Structural ~vstem
Tensile force
Compressive force
0
3
5
2
5
C l a m p e d - - s i m p l y supported 1"0 k~2 = 25 vo ~ oo k'~ = 0 v 1 = 0 0"5 ct = 0.5
15'71 n =4.33 12.72 n = 10.96
16-77 n = 4.21 14.74 n=8"84
17.44 n=4"13 15.92 n=7"84
14.95 n = 4"41 11.14 n=12"77
13.74 n = 4-56 8.16 n=15-51
1"0
15"96 n=4.16 13'40 n = 9"56
17"00 n=4"06 15'29 n=7-77
17-65 n=3'99 16.41 n=6"96
15-22 n=4"24 11"93 n=11.24
14"05 n=4"37 9"23 n=14"12
22.72 n=4"10 16.76 n=8-31
23"50 n = 4"09 18-24 n=8"00
24.00 n = 4"09 19-16 n=7"82
22.18 n = 4" 11 15"69 n=8"54
21"36 n = 4.12 13'92 n=8"94
22.72 n = 4"03 17"04 n = 7-84
23-50 n = 4"02 18"49 n = 7-57
24-01 n = 4"02 19'39 n = 7'41
22"19 n = 4.04 16-00 n = 8'05
21"36 n = 4.04 14'28 n = 8-41
C l a m p e d - - s i m p l y supported k ~~ = 0
Vo--" oo
k~ = 25 = 0"5
v1 - 0
0-5 1.0
Clamped-clamped k~~ = 25 k'~ = 0 ct = 0.5
vo ~ vI ~ o v
0'5 1"0
Clamped-clamped ~ = 0
Vo ~
k'~ = 25 = 0"5
v I --*
As an addition corresponding presented
0"5
to the results presented
to a beam
in Table
in Ref. 2 the frequency
of discontinuously
3. E n d s
varying
elastically
restrained
accuracy
achieved
cross
against
coefficients section rotation
are are
considered. Judgifig reasonably
from expect
the that
excellent
the results obtained
possess, at least, acceptable
engineering
in
Ref.
in the present
2, o n e
may
investigation
precision.
REFERENCES 1. Bert, C. W., U s e o f s y m m e t r y in a p p l y i n g the R a y l e i g h - S c h m i d t m e t h o d to static a n d f r e e - v i b r a t i o n p r o b l e m s . Industrial Mathematics, J. Industrial Math. Soc., 34 (1984) 65-7. 2. F i l i p i c h , C. P., L a u r a , P. A. A., S o n e n b l u m , M. & Gil, E., T r a n s v e r s e v i b r a t i o n s o f a n o n - u n i f o r m b e a m s u b j e c t to an axial f o r c e e m b e d d e d in a n o n - h o m o g e n e o u s W i n k l e r f o u n d a t i o n . Institute of Applied Mechanics, Puerto Belgrano Naval Base, 8111 A r g e n t i n a , P u b l i c a t i o n N o . 87-29, 1987.
TABLE 2 F u n d a m e n t a l Frequency Coefficients and Values o f the O p t i m i z a t i o n P a r a m e t e r for Several C o m b i n a t i o n s o f the B o u n d a r y C o n d i t i o n s a n d o f the G o v e r n i n g G e o m e t r i c a n d M e c h a n i c a l Parameters
Structural system
Simply s u p p o r t e d - c l a m p e d k~~ = 25 vo = 0 k'~=O vl-,~ 0c =~0'5 Simply s u p p o r t e d - c l a m p e d 12 ko = 0 vo = 0 k'~ = 25 vl~oo ~t= 0'5 Simply s u p p o r t e d - s i m p l y supported k~2 = 25 vo = 0 k'l~ = 0 vl=0 ct = 0-5 Simply s u p p o r t e d - s i m p l y supported r2 ko = 0 Vo=0 k I = 25 vI = 0 ct = 0'5
?•p2 1.0 0.5
1.0 0"5
1.0 0"5
1"0 0'5
Compressive force
Tensile force 0
3
5
2
5
15"96 n=3"95 11"03 n=7"55
17.00 n=4"03 12"79 n=7"57
17"65 n=4"08 13"84 n=7"59
15"23 n=3"90 9"68 n=7'52
14"06 n=3'83 7"20 n=7'49
15"72 n=3-84 10-87 n=6"71
16"77 n=3-92 12.66 n=6"80
17.44 n=3"96 13.72 n=6"85
14.97 n=3"79 9"50 n=6"65
13.77 n=3"72 6"95 n=6"55
10"48 n=4.21 8"08 n=10"32
I 1"81 n=4-20 10-32 n=8'27
12"62 n=4'19 11"57 n=7'41
9.49 n=4"22 6-11 n=12'31
7"78 n=4"23 <0
10.48 n=3.88 8"36 n=7'08
11.81 n=3"88 10-51 n=6"10
12.62 n=3'89 11.72 n=5-70
9.50 n=3.87 6-53 n=8"24
7.78 n=3.86 1-11 n=11-41
TABLE 3 V i b r a t i n g Beam o f D i s c o n t i n u o u s l y Varying Cross Section with E n d s Elastically R e s t r a i n e d Against R o t a t i o n
Structural system
kb2 = 25 k'; = 25 ~t = 0"5
Vo= 1 vl=l
~ 1'0
0-5
k '=25 k'? = 25
vo = 10 v I = 10
ct = 0"5 k~2 = 25
k'] = 25 ct = 0"5
1"0 0"5
Vo=100 vl=l~
1"0 0-5
2
Tensile force
Compressive force
0
3
5
2
5
12"59 n = 4"04 11.32 n=10"33
13'72 n = 4"03 13.07 n=9-61
14"42 n = 4"03 14"11 n=9.19
11"78 n = 4-04 9.98 n=10"88
10"45 n = 4-04 7"51 n=11"79
18"01 n = 4"05 15-10 n=9"32
18"85 n = 4"05 16"62 n=8"91
19'39 n = 4"04 17'55 n=8"66
17"43 n = 4'06 13"99 n=9"62
16'52 n --- 4"07 12"13 n=10"ll
22" 18 n = 4"06 17-04 n=9"35
22"95 n = 4"06 18-59 n=8'94
23.45 n = 4"05 19'54 n=8-68
21.65 n = 4-07 15"92 n=9'66
20"83 n = 4"08 14"05 n=10'17
72
C. P. Filipich, M. J. Sonenblum, E. A. Gil
3. Laura, P. A. A. & Cortinez, V. H., Transverse vibrations of a cantilever beam carrying a concentrated mass at its free end and subjected to a variable axial force. J. Sound and Vibration, 103 (1985) 596-9. 4. Laura, P. A. A. & Cortinez, V. H., Optimization of eigenvalues when using the Galerkin method. American Inst. of Chem. Engineers J., 32 (1986) 1025-6. 5. Filipich, C. P. & Rosales, M. B., A variant of Rayleigh's method applied to Timoshenko beams embedded in a Winkler-Pasternak medium. Department of Engineering U.N.S., Bahia Blanca 8000 Argentina, Publication 87-2, 1987.
C. P. Filipich Mechanical Systems Analysis Group, Facultad Regional Bahia Blanca and Universidad Nacional del Sur, 8000 Bahia Blanca, Argentina M. J. Sonenblum & E. A. Gil Mechanical Systems Analysis Group, Facultad Regional Bahia Blanca, Argentina
(Received 17 August 1987; accepted 19 April 1988)
Technical Note Transverse Vibrations of a Tapered Beam Embedded in a Non-homogeneous Winkler Foundation
A BS TRA C T The present study deals with the determination of the fundamental frequency of transverse vibration of the structural elements described in the title. The fundamental mode shape is approximated by means of a polynomial coordinate function which contains an exponential optimization parameter 'n' which allows for minimization of the fundamental frequency parameter.
INTRODUCTION The present study deals with the determination of the frequency of vibration of the structural system shown in Fig. 1 using Rayleigh's optimization procedure. 1 Apparently no studies are available on the title problem with the exception of the investigation reported in Ref. 2. The approximate approach presented in this Note makes it possible to obtain natural frequencies of embedded systems of more complex characteristics than the one considered in Ref. 2, for instance the case of linearly varying thickness.
A P P R O X I M A T E S O L U T I O N OF T H E P R O B L E M U N D E R INVESTIGATION In the case of normal modes of vibration one makes: v(x, t) = V(x). e iw'
(1)
67 Applied Acoustics 0003-682X/89/$03.50 O 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
68
C. P. Filipich, M. J. Sonenblum, E. A. Gil
P<"
>P
~,
T
~.L
L Fig. I. Vibrating structural system studied in the present investigation.
and following Ref. 3 it is convenient to make:
(2)
V(x) ~_ Va(x) = x" + A x 3 + B x 2 + C x + D
where n is Rayleigh's optimization parameter and x = X / L . The constants A, B, C and D are determined substituting eqn (2) in the governing b o u n d a r y conditions (see Fig. 1): [V" - VoV']x_-o = 0
Ev]x_-o=O [ B V " + v 1V']x= 1 = 0
(3)
[v]~=~ = 0
k.
1//\\.,///\~ t/2
~
L/2
,~
Fig. 2. Vibrating beam of discontinuously varying cross section.
69
Transverse vibrations of a tapered beam
Proceeding now with the determination of Rayleigh's quotient given by: Us + p 2 0 p + Uw = [2x2(n)
2 , PoAo
n
=w,L e- 0 =
(4)
rk
where: 20s = Jo ~p(x). (V") 2 dx + v o g'~(O) + v~ V'2(1) 2 Up = .f~ ( V')2 dx 2Uw = k
V2 dx + ~
V2 dx
Tk = f l g(x) V2 dx
3o
B = E111
k,o2 = ko L4
Edo
Edo
golo
P1A1
kaL 4
k'l2
Y = PoAo
v ,IL Vo,l = Eolo
k'12 = El11
p2= pL2
(>o)
~ = k--~o
cp(x) = (1 + ( h l / h o - 1)x)3 g(x) = 1 + '(ht/h o - 1). x
one determines numerically the value of the parameter 'n' which minimizes eqn (4). The eigenvalue can be improved by constructing a summation of polynomial coordinate functions and the higher optimized eigenvalues may also be determined by minimization with respect to the parameter 'n' as shown in Refs 4 and 5. N U M E R I C A L RESULTS Tables 1 and 2 contain frequency coefficients for several tapered beams elements as a function of the governing parameters. The following combinations of boundary conditions have been considered: ---clamped-clamped ---clamped-simply supported --simply supported-clamped --simply supported-simply supported
C. P. Filipich, M. J. Sonenblum, E. A. Gil
70
TABLE 1 Fundamental Frequency Coefficients and Values of the Optimization Parameter for Several Combinations of the Boundary Conditions and of the Governing Geometric and Mechanical Parameters
Structural ~vstem
Tensile force
Compressive force
0
3
5
2
5
C l a m p e d - - s i m p l y supported 1"0 k~2 = 25 vo ~ oo k'~ = 0 v 1 = 0 0"5 ct = 0.5
15'71 n =4.33 12.72 n = 10.96
16-77 n = 4.21 14.74 n=8"84
17.44 n=4"13 15.92 n=7"84
14.95 n = 4"41 11.14 n=12"77
13.74 n = 4-56 8.16 n=15-51
1"0
15"96 n=4.16 13'40 n = 9"56
17"00 n=4"06 15'29 n=7-77
17-65 n=3'99 16.41 n=6"96
15-22 n=4"24 11"93 n=11.24
14"05 n=4"37 9"23 n=14"12
22.72 n=4"10 16.76 n=8-31
23"50 n = 4"09 18-24 n=8"00
24.00 n = 4"09 19-16 n=7"82
22.18 n = 4" 11 15"69 n=8"54
21"36 n = 4.12 13'92 n=8"94
22.72 n = 4"03 17"04 n = 7-84
23-50 n = 4"02 18"49 n = 7-57
24-01 n = 4"02 19'39 n = 7'41
22"19 n = 4.04 16-00 n = 8'05
21"36 n = 4.04 14'28 n = 8-41
C l a m p e d - - s i m p l y supported k ~~ = 0
Vo--" oo
k~ = 25 = 0"5
v1 - 0
0-5 1.0
Clamped-clamped k~~ = 25 k'~ = 0 ct = 0.5
vo ~ vI ~ o v
0'5 1"0
Clamped-clamped ~ = 0
Vo ~
k'~ = 25 = 0"5
v I --*
As an addition corresponding presented
0"5
to the results presented
to a beam
in Table
in Ref. 2 the frequency
of discontinuously
3. E n d s
varying
elastically
restrained
accuracy
achieved
cross
against
coefficients section rotation
are are
considered. Judgifig reasonably
from expect
the that
excellent
the results obtained
possess, at least, acceptable
engineering
in
Ref.
in the present
2, o n e
may
investigation
precision.
REFERENCES 1. Bert, C. W., U s e o f s y m m e t r y in a p p l y i n g the R a y l e i g h - S c h m i d t m e t h o d to static a n d f r e e - v i b r a t i o n p r o b l e m s . Industrial Mathematics, J. Industrial Math. Soc., 34 (1984) 65-7. 2. F i l i p i c h , C. P., L a u r a , P. A. A., S o n e n b l u m , M. & Gil, E., T r a n s v e r s e v i b r a t i o n s o f a n o n - u n i f o r m b e a m s u b j e c t to an axial f o r c e e m b e d d e d in a n o n - h o m o g e n e o u s W i n k l e r f o u n d a t i o n . Institute of Applied Mechanics, Puerto Belgrano Naval Base, 8111 A r g e n t i n a , P u b l i c a t i o n N o . 87-29, 1987.
TABLE 2 F u n d a m e n t a l Frequency Coefficients and Values o f the O p t i m i z a t i o n P a r a m e t e r for Several C o m b i n a t i o n s o f the B o u n d a r y C o n d i t i o n s a n d o f the G o v e r n i n g G e o m e t r i c a n d M e c h a n i c a l Parameters
Structural system
Simply s u p p o r t e d - c l a m p e d k~~ = 25 vo = 0 k'~=O vl-,~ 0c =~0'5 Simply s u p p o r t e d - c l a m p e d 12 ko = 0 vo = 0 k'~ = 25 vl~oo ~t= 0'5 Simply s u p p o r t e d - s i m p l y supported k~2 = 25 vo = 0 k'l~ = 0 vl=0 ct = 0-5 Simply s u p p o r t e d - s i m p l y supported r2 ko = 0 Vo=0 k I = 25 vI = 0 ct = 0'5
?•p2 1.0 0.5
1.0 0"5
1.0 0"5
1"0 0'5
Compressive force
Tensile force 0
3
5
2
5
15"96 n=3"95 11"03 n=7"55
17.00 n=4"03 12"79 n=7"57
17"65 n=4"08 13"84 n=7"59
15"23 n=3"90 9"68 n=7'52
14"06 n=3'83 7"20 n=7'49
15"72 n=3-84 10-87 n=6"71
16"77 n=3-92 12.66 n=6"80
17.44 n=3"96 13.72 n=6"85
14.97 n=3"79 9"50 n=6"65
13.77 n=3"72 6"95 n=6"55
10"48 n=4.21 8"08 n=10"32
I 1"81 n=4-20 10-32 n=8'27
12"62 n=4'19 11"57 n=7'41
9.49 n=4"22 6-11 n=12'31
7"78 n=4"23 <0
10.48 n=3.88 8"36 n=7'08
11.81 n=3"88 10-51 n=6"10
12.62 n=3'89 11.72 n=5-70
9.50 n=3.87 6-53 n=8"24
7.78 n=3.86 1-11 n=11-41
TABLE 3 V i b r a t i n g Beam o f D i s c o n t i n u o u s l y Varying Cross Section with E n d s Elastically R e s t r a i n e d Against R o t a t i o n
Structural system
kb2 = 25 k'; = 25 ~t = 0"5
Vo= 1 vl=l
~ 1'0
0-5
k '=25 k'? = 25
vo = 10 v I = 10
ct = 0"5 k~2 = 25
k'] = 25 ct = 0"5
1"0 0"5
Vo=100 vl=l~
1"0 0-5
2
Tensile force
Compressive force
0
3
5
2
5
12"59 n = 4"04 11.32 n=10"33
13'72 n = 4"03 13.07 n=9-61
14"42 n = 4"03 14"11 n=9.19
11"78 n = 4-04 9.98 n=10"88
10"45 n = 4-04 7"51 n=11"79
18"01 n = 4"05 15-10 n=9"32
18"85 n = 4"05 16"62 n=8"91
19'39 n = 4"04 17'55 n=8"66
17"43 n = 4'06 13"99 n=9"62
16'52 n --- 4"07 12"13 n=10"ll
22" 18 n = 4"06 17-04 n=9"35
22"95 n = 4"06 18-59 n=8'94
23.45 n = 4"05 19'54 n=8-68
21.65 n = 4-07 15"92 n=9'66
20"83 n = 4"08 14"05 n=10'17
72
C. P. Filipich, M. J. Sonenblum, E. A. Gil
3. Laura, P. A. A. & Cortinez, V. H., Transverse vibrations of a cantilever beam carrying a concentrated mass at its free end and subjected to a variable axial force. J. Sound and Vibration, 103 (1985) 596-9. 4. Laura, P. A. A. & Cortinez, V. H., Optimization of eigenvalues when using the Galerkin method. American Inst. of Chem. Engineers J., 32 (1986) 1025-6. 5. Filipich, C. P. & Rosales, M. B., A variant of Rayleigh's method applied to Timoshenko beams embedded in a Winkler-Pasternak medium. Department of Engineering U.N.S., Bahia Blanca 8000 Argentina, Publication 87-2, 1987.
C. P. Filipich Mechanical Systems Analysis Group, Facultad Regional Bahia Blanca and Universidad Nacional del Sur, 8000 Bahia Blanca, Argentina M. J. Sonenblum & E. A. Gil Mechanical Systems Analysis Group, Facultad Regional Bahia Blanca, Argentina
(Received 17 August 1987; accepted 19 April 1988)