Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 493905, 7 pages http://dx.doi.org/10.1155/2013/493905 Research Article Free Vibration Analysis of a Thin-Walled Beam with Shear Sensitive Material K. B. Bozdogan 1 and D. Ozturk 2 1 Kirklareli University, 39000 Kirklareli, Turkey 2 Ege University, 35000 ˙ Izmir, Turkey Correspondence should be addressed to K. B. Bozdogan; [email protected] Received 25 June 2013; Accepted 27 August 2013 Academic Editor: Usik Lee Copyright © 2013 K. B. Bozdogan and D. Ozturk. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper presents a method for a free vibration analysis of a thin-walled beam of doubly asymmetric cross section filled with shear sensitive material. In the study, first of all, a dynamic transfer matrix method was obtained for planar shear flexure and torsional motion. en, uncoupled angular frequencies were obtained by using dynamic element transfer matrices and boundary conditions. Coupled frequencies were obtained by the well-known two-dimensional approaches. At the end of the study, a sample taken from the literature was solved, and the results were evaluated in order to test the convenience of the method. 1. Introduction In the last two decades research on the dynamics of beams has grown enormously. ere are numerous studies [1–29] on the bending-torsion coupled beam. In the beams, the elastic center and the center of mass are not coincident, so the trans- lational and torsional modes are inherently coupled as a result of this offset. Rafezy and Howson [24] proposed an exact dynamic stiffness matrix approach for the three-dimensional, bimaterial beam of doubly asymmetric cross-section. e beam comprises a thin-walled outer layer that encloses and works compositely with its shear sensitive core material. A dynamic transfer matrix method for the free vibration analysis of a thin-walled beam of doubly asymmetric cross- section filled with shear sensitive materials is suggested in this study. e following assumptions are made in this study: the behaviour of the material is linear elastic, small displacement theory is valid, and the dynamic coupling effect of structure caused by the eccentricity between the center of shear rigidity and the flexural rigidity center is ignored in analysis. 2. Analysis 2.1. Physical Model. Figure 1 shows a uniform, three dimen- sional beam of length . It has a doubly asymmetric cross-section comprising a thin-walled outer layer that encloses shear sensitive material [24]. e outer layer may have the form of an open or closed section that is assumed to provide warping and Saint-Venant rigidity, while the core materials provide Saint-Venant and shear rigidity. ese assumptions lead to a model in which a typical cross-section has independent centers of flexure, shear, and mass denoted by , , and , respectively [24]. For convenience, the origin of the coordinate system is located at the centre of flexure. gives the result that the axis of elastic flexure coincides with the -axis of the member. e - and -axes are subsequently aligned with the principle axes of the cross-section. e locations of the points and in the coordinate system Oxy are given by ( , ) and ( , ), respectively. e resulting elastic shear and mass axes then run parallel to the -axis through ( , ) and ( , ) respectively. When the elastic axis of the beam does not coincide, the lateral and torsional motion of the beam will always be coupled in one or more planes [24]. 2.2. Element Transfer Matrices for Planar Motion. e gov- erning equations for th element of uncoupled thin-walled shear sensitive beam can be written as 2 ( , ) 2 + 4 ( , ) 4 − () 2 ( , ) 2 = 0,
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 493905 7 pageshttpdxdoiorg1011552013493905
Research ArticleFree Vibration Analysis of a Thin-Walled Beam with ShearSensitive Material
K B Bozdogan1 and D Ozturk2
1 Kirklareli University 39000 Kirklareli Turkey2 Ege University 35000 Izmir Turkey
Correspondence should be addressed to K B Bozdogan kbbozdoganyahoocomtr
Received 25 June 2013 Accepted 27 August 2013
Academic Editor Usik Lee
Copyright copy 2013 K B Bozdogan and D Ozturk This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
This paper presents amethod for a free vibration analysis of a thin-walled beam of doubly asymmetric cross section filled with shearsensitive material In the study first of all a dynamic transfer matrix method was obtained for planar shear flexure and torsionalmotionThen uncoupled angular frequencies were obtained by using dynamic element transfer matrices and boundary conditionsCoupled frequencies were obtained by the well-known two-dimensional approaches At the end of the study a sample taken fromthe literature was solved and the results were evaluated in order to test the convenience of the method
1 Introduction
In the last two decades research on the dynamics of beamshas grown enormouslyThere are numerous studies [1ndash29] onthe bending-torsion coupled beam In the beams the elasticcenter and the center of mass are not coincident so the trans-lational and torsionalmodes are inherently coupled as a resultof this offset Rafezy and Howson [24] proposed an exactdynamic stiffnessmatrix approach for the three-dimensionalbimaterial beam of doubly asymmetric cross-section Thebeam comprises a thin-walled outer layer that encloses andworks compositely with its shear sensitive core material
A dynamic transfer matrix method for the free vibrationanalysis of a thin-walled beam of doubly asymmetric cross-section filledwith shear sensitivematerials is suggested in thisstudy The following assumptions are made in this study thebehaviour of the material is linear elastic small displacementtheory is valid and the dynamic coupling effect of structurecaused by the eccentricity between the center of shear rigidityand the flexural rigidity center is ignored in analysis
2 Analysis
21 Physical Model Figure 1 shows a uniform three dimen-sional beam of length 119871 It has a doubly asymmetriccross-section comprising a thin-walled outer layer that
encloses shear sensitive material [24] The outer layer mayhave the form of an open or closed section that is assumedto provide warping and Saint-Venant rigidity while the corematerials provide Saint-Venant and shear rigidity Theseassumptions lead to a model in which a typical cross-sectionhas independent centers of flexure shear and mass denotedby 119874 119878 and 119862 respectively [24] For convenience the originof the coordinate system is located at the centre of flexure 119874gives the result that the axis of elastic flexure coincides withthe 119911-axis of the memberThe 119909- and 119910-axes are subsequentlyaligned with the principle axes of the cross-section Thelocations of the points 119878 and 119862 in the coordinate system Oxyare given by 119878(119909
119904 119910119904) and119862(119909
119888 119910119888) respectivelyThe resulting
elastic shear and mass axes then run parallel to the 119911-axisthrough (119909
119904 119910119904) and (119909
119888 119910119888) respectively When the elastic
axis of the beam does not coincide the lateral and torsionalmotion of the beam will always be coupled in one or moreplanes [24]
22 Element Transfer Matrices for Planar Motion The gov-erning equations for 119894th element of uncoupled thin-walledshear sensitive beam can be written as
120588119894
1205972
119880119894(119911119894 119905)
1205971199052+ 119864119868119909119894
1205974
119880119894(119911119894 119905)
1205971199114
119894
minus (119866119860)119909119894
1205972
119880119894(119911119894 119905)
1205971199112
119894
= 0
2 Mathematical Problems in Engineering
120593
O
y
y
Cxc
x
xs
yz
S
Mass axis
Elastic flexure axis
z
(a)
O
y
yCxc
xs S
u(z t)
(zt)
S998400998400
C998400998400
120593(z t)
yc
ys
x
xO998400
S998400
C998400
(b)
Figure 1 Typical thin-walled beam [24]
120588119894
1205972
119881119894(119911119894 119905)
1205971199052+ 119864119868119910119894
1205974
119881119894(119911119894 119905)
1205971199114
119894
minus (119866119860)119910119894
1205972
119881119894(119911119894 119905)
1205971199112
119894
= 0
1205881198941199032
119898
1205972
Ψ119894(119911119894 119905)
1205971199052+ 119864119868119908119894
1205974
Ψ119894(119911119894 119905)
1205971199114
119894
minus (119866119869)119900119894
1205972
Ψ119894(119911119894 119905)
1205971199112
119894
= 0
(1)
where 119866119869119900119894
= 119866119905119894119869119905119894+ 119866119869119888119894
119864119868119909119894and 119864119868
119910119894are the flexural rigidity of the 119894th segment
in the 119909-119911 and 119910-119911 planes respectively and 119866119905119894119869119905119894and 119864119868
119908119894
are the Saint-Venant and warping torsion rigidity of the 119894thsegment about 119874 where 119868
119908is the warping moment of inertia
or warping constant 119866119860119909119894and 119866119860
119910119894are the effective shear
rigidities of the core material of the 119894th segment in 119909 and 119910
directions respectively and119866119869119900119894is the Saint-Venant torsional
rigidity of the core material about 119874 120588119894are the mass per unit
length of the 119894th segment and 119903119898is the polar mass radius of
gyration of cross section [24]If a sinusoidal variation of 119880 119881 and 120595 with circular
frequency 120596 is assumed then
119880119894(119911119894 119905) = 119906
119894sin (120596
119909119905)
119881119894(119911119894 119905) = V
119894sin (120596
119910119905)
120595119894(119911119894 119905) = 120579
119894sin (120596
120579119905)
(2)
where 119906119894 V119894 and 120579
119894are the amplitudes of the sinusoidally
varying displacementSubstituting (2) in (1) results are
1198894
119906119894
1198891199114
119894
minus(119866119860)119909119894
(119864119868)119909119894
1198892
119906119894
1198891199112
119894
minus119901119894
(119864119868)119909119894
1205962
119909119906119894= 0
1198894V119894
1198891199114
119894
minus
(119866119860)119910119894
(119864119868)119910119894
1198892V119894
1198891199112
119894
minus119901119894
(119864119868)119910119894
1205962
119910V119894= 0
1198894
120579119894
1198891199114
119894
minus(119866119869)119900119894
(119864119868)119908119894
1198892
120579119894
1198891205792
119894
minus1199011198941199032
119898
(119864119868)119908119894
1205962
120579120579119894= 0
(3)
When (3) is solved with respect to 119911119894 119906119894(119911119894) V119894(119911119894) and 120579
119894(119911119894)
can be obtained as follows
119906119894(119911119894) = 1198881cosh (119886
119909119894119911119894) + 1198882sinh (119886
119909119894119911119894)
+ 1198883cos (119887119909119894119911119894) + 1198884sin (119887119909119894119911119894)
(4)
V119894(119911119894) = 1198885cosh (119886
119910119894119911119894) + 1198886sinh (119886
119910119894119911119894)
+ 1198887cos (119887
119910119894119911119894) + 1198888sin (119887119910119894119911119894)
(5)
120579119894(119911119894) = 1198889cosh (119886
120579119894119911119894) + 11988810sinh (119886
120579119894119911119894)
+ 11988811cos (119887120579119894119911119894) + 11988812sin (119887120579119894119911119894)
(6)
where 120572119909119894 120572119910119894 120572120579119894 119887119909119894 119887119910119894 and 119887
120579119894can be calculated as follows
119886119909119894
= radic119904119909119894
+ 119901119909119894
2 119886
119910119894= radic
119904119910119894
+ 119901119910119894
2
119886120579119894
= radic119904120579119894
+ 119901120579119894
2 119887
119909119894= radic
minus119904119909119894
+ 119901119909119894
2
119887119910119894
= radicminus119904119910119894
+ 119901119910119894
2 119887
120579119894= radic
minus119904120579119894
+ 119901120579119894
2
119901119909119894
= radic(119866119860119909119894
119864119868119909119894
)
2
+ 4 lowast1199011198941205962
119909
(119864119868)119909119894
119901119910119894
= radic(
119866119860119910119894
119864119868119910119894
)
2
+ 4 lowast
1199011198941205962
119910
(119864119868)119910119894
Mathematical Problems in Engineering 3
119901120579119894
= radic(119866119869119900119894
119864119868119908119894
)
2
+ 4 lowast1199011198941205962
120579
(119864119868)119908119894
119904119909119894
=(119866119860)119909119894
(119864119868)119909119894
119904119910119894
=
(119866119860)119910119894
(119864119868)119910119894
119904120579119894
=(119866119869)119900119894
(119864119868)119908119894
(7)
By using (4) (5) and (6) the rotation angles in 119909 and 119910
directions (1199061015840
119894 V1015840119894) rate of twist (120579
1015840
119894) bending moments in
119909 and 119910 directions (119872119909119894119872119910119894) and bimoment (119872
119908119894) shear
forces in 119909 and 119910 directions (119881119909119894 119881119910119894) and torque (119872
119905119894) for
119894th element can be obtained as follows
119889119906119894(119911119894)
119889119911119894
= 1198881119886119909119894sinh (119886
119909119894119911119894) + 1198882119886119909119894cosh (119886
119909119894119911119894)
minus 1198883119887119909119894sin (119887119909119894119911119894) + 1198884119887119909119894cos (119887119909119894119911119894)
(8)
119889V119894(119911119894)
119889119911119894
= 1198885119886119910119894sinh (119886
119910119894119911119894) + 1198886119886119910119894cosh (119886
119910119894119911119894)
minus 1198887119887119910119894sin (119887119910119894119911119894) + 1198888119887119910119894cos (119887
119910119894119911119894)
(9)
119889120579119894(119911119894)
119889119911119894
= 1198889119886120579119894sinh (119886
120579119894119911119894) + 11988810119886120579119894cosh (119886
120579119894119911119894)
minus 11988811119887120579119894sin (119887120579119894119911119894) + 11988812119887120579119894cos (119887120579119894119911119894)
(10)
119872119909119894
(119911119894) = 119864119868
119909119894
1198892
119906119894(119911119894)
1198891199112
119894
= 119864119868119909119894
[11988811198862
119909119894cosh (119886
119909119894119911119894) + 11988821198862
119909119894sinh (119886
119909119894119911119894)
minus11988831198872
119909119894cos (119887119909119894119911119894) minus 11988841198872
119909119894sin (119887119909119894119911119894)]
(11)
119872119910119894
(119911119894) = 119864119868
119910119894
1198892V119894(119911119894)
1198891199112
119894
= 119864119868119910119894
[11988851198862
119910119894cosh (119886
119910119894119911119894) + 11988861198862
119910119894sinh (119886
119910119894119911119894)
minus11988871198872
119910119894cos (119887
119910119894119911119894) minus 11988881198872
119910119894sin (119887119910119894119911119894)]
(12)
119872119908119894
(119911119894) = 119864119868
119908119894
1198892
120579119894(119911119894)
1198891199112
119894
= 119864119868119908119894
[11988891198862
120579119894cosh (119886
120579119894119911119894) + 119888101198862
120579119894sinh (119886
120579119894119911119894)
minus119888111198872
120579119894cos (119887120579119894119911119894) minus 119888121198872
120579119894sin (119887120579119894119911119894)]
(13)
119881119909119894
(119911119894) = 119864119868
119909119894
1198893
119906119894(119911119894)
1198891199113
119894
minus (119866119860)119909119894
119889119906119894(119911119894)
119889119911119894
= [1198641198681199091198941198863
119909119894sinh (119886
119909119894119911119894) minus (119866119860)
119909119894119886119909119894sinh (119886
119909119894119911119894)] 1198881
+ [1198641198681199091198941198863
119909119894cosh (119886
119909119894119911119894)
minus(119866119860)119909119894119886119909119894cosh (119886
119909119894119911119894)] 1198882
+ [(119864119868)1199091198941198873
119909119894sin (119887119909119894119911119894) + (119866119860)
119909119894119887119909119894sin (119887119909119894119911119894)] 1198883
+ [minus(119864119868)1199091198941198873
119909119894cos (119887119909119894119911119894)
minus(119866119860)119909119894119887119909119894cos (119887119909119894119911119894)] 1198884
(14)
119881119910119894
(119911119894) = 119864119868
119910119894
1198893V119894(119911119894)
1198891199113
119894
minus (119866119860)119910119894
119889V119894(119911119894)
119889119911119894
= [1198641198681199101198941198863
119910119894sinh (119886
119910119894119911119894) minus (119866119860)
119910119894119886119910119894sinh (119886
119910119894119911119894)] 1198885
+ [1198641198681199101198941198863
119910119894cosh (119886
119910119894119911119894)
minus(119866119860)119910119894119886119910119894cosh (119886
119910119894119911119894)] 1198886
+ [(119864119868)1199101198941198873
119910119894sin (119887119910119894119911119894) + (119866119860)
119910119894119887119910119894sin (119887119910119894119911119894)] 1198887
+ [minus(119864119868)1199101198941198873
119910119894cos (119887
119910119894119911119894)
minus(119866119860)119910119894119887119910119894cos (119887
119910119894119911119894)] 1198888
(15)
119872119905119894(119911119894) = 119864119868
119908119894
1198893
120579119894(119911119894)
1198891199113
119894
minus (119866119869)119900119894
119889120579119894(119911119894)
119889119911119894
= [1198641198681205791198941198863
120579119894sinh (119886
120579119894119911119894) minus (119866119869)
119900119894119886120579119894sinh (119886
120579119894119911119894)] 1198889
+ [1198641198681199081198941198863
120579119894cosh (119886
120579119894119911119894)
minus(119866119869)119894119900119886120579119894cosh (119886
120579119894119911119894)] 11988810
+ [(119864119868)1199081198941198873
120579119894sin (119887120579119894119911119894) + (119866119869)
119900119894119887120579119894sin (119887120579119894119911119894)] 11988811
+ [minus(119864119868)1199081198941198873
120579119894cos (119887120579119894119911119894)
minus(119866119869)119900119894119887119910119894cos (119887120579119894119911119894)] 11988812
(16)
The following equation shows thematrix form of (4) (8) (11)and (14)
[[[
[
119906119894(119911119894)
1199061015840
119894(119911119894)
119872119909119894
(119911119894)
119881119909119894
(119911119894)
]]]
]
= 119860119909119894
(119911119894)
[[[
[
1198881
1198882
1198883
1198884
]]]
]
(17)
4 Mathematical Problems in Engineering
For the 119910 direction the following shows the matrix form of(5) (9) (12) and (15)
[[[
[
V119894(119911119894)
V1015840119894(119911119894)
119872119910119894
(119911119894)
119881119910119894
(119911119894)
]]]
]
= 119860119910119894
(119911119894)
[[[
[
1198885
1198886
1198887
1198888
]]]
]
(18)
Similarly torsional motion can be written
[[[
[
120579119894(119911119894)
1205791015840
119894(119911119894)
119872119908119894
(119911119894)
119872119905119894(119911119894)
]]]
]
= 119860120579119894(119911119894)
[[[
[
1198889
11988810
11988811
11988812
]]]
]
(19)
At the initial point of the 119894th element (17) (18) and (19) canbe written as follows
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
= 119860119909119894
(0)
[[[
[
1198881
1198882
1198883
1198884
]]]
]
(20)
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
= 119860119910119894
(0)
[[[
[
1198885
1198886
1198887
1198888
]]]
]
(21)
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
= 119860120579119894(0)
[[[
[
1198889
11988810
11988811
11988812
]]]
]
(22)
When vector 119888 is solved from (20) and is substituted in (17)the following is obtained
[[[
[
119906119894(119911119894)
1199061015840
119894(119911119894)
119872119909119894
(119911119894)
119881119909119894
(119911119894)
]]]
]
= 119860119909119894
(119911119894) 119860119909119894(0)minus1
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
(23)
For 119911119894= 119897119894 (23) can be written as
[[[
[
119906119894(119897119894)
1199061015840
119894(119897119894)
119872119909119894
(119897119894)
119881119909119894
(119897119894)
]]]
]
= 119879119909119894
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
(24)
where 119879119909119894is the element dynamic transfer matrix of the 119894th
elementFor the 119910 direction (23) and (24) can be written as
follows
[[[
[
V119894(119911119894)
V1015840119894(119911119894)
119872119910119894
(119911119894)
119881119910119894
(119911119894)
]]]
]
= 119860119910119894
(119911119894) 119860119910119894(0)minus1
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
(25)
[[[
[
V119894(119897119894)
V1015840119894(119897119894)
119872119910119894
(119897119894)
119881119910119894
(119897119894)
]]]
]
= 119879119910119894
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
(26)
Similarly rotation motion can be written in equations asfollows
[[[
[
120579119894(119911119894)
1205791015840
119894(119911119894)
119872119908119894
(119911119894)
119872119905119894(119911119894)
]]]
]
= 119860120579119894(119911119894) 119860120579119894(0)minus1
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
(27)
[[[
[
120579119894(119897119894)
1205791015840
119894(119897119894)
119872119908119894
(119897119894)
119872119905119894(119897119894)
]]]
]
= 119879120579119894
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
(28)
If (24) is written successively the displacementsmdashinternalforces relationship between the initial part and end of thebeammdashcan be found as follows
[[[
[
119906end1199061015840
end119872119909end
119881119909end
]]]
]
= 119879119909119899
119879119909(119899minus1)
sdot sdot sdot 11987911990921198791199091
[[[
[
119906initial1199061015840
initial119872119909initial
119881119909initial
]]]
]
= 119905119909
[[[
[
119906initial1199061015840
initial119872119909initial
119881119909initial
]]]
]
(29)
For 119910 and rotation motion (29) can be written as follows
[[[
[
VendV1015840end
119872119910end
119881119910end
]]]
]
= 119879119910119899
119879119910(119899minus1)
sdot sdot sdot 11987911991021198791199101
[[[
[
VinitialV1015840initial
119872119910initial
119881119910initial
]]]
]
= 119905119910
[[[
[
VinitialV1015840initial
119872119910initial
119881119910initial
]]]
]
[[[
[
120579end1205791015840
end119872119908end
119872119905end
]]]
]
= 119879120579119899119879120579(119899minus1)
sdot sdot sdot 11987912057921198791205791
[[[
[
120579initial1205791015840
initial119872119908initial
119872119905initial
]]]
]
= 119905120579
[[[
[
120579initial1205791015840
initial119872119908initial
119872119905initial
]]]
]
(30)
The eigenvalue equation for a thin-walled beam filled withshear sensitive material can be established using (29) (30)and the specific boundary conditions are as follows
(1) Clamped-Free 119891119909
= 119905119909(3 3)lowast
119905119909(4 4) minus 119905
119909(3 4)lowast
119905119909(4
3) = 0 119891119910
= 119905119910(3 3)lowast
119905119910(4 4) minus 119905
119910(3 4)lowast
119905119910(4 3) =
0 119891120579= 119905120579(3 3)lowast
119905120579(4 4) minus 119905
120579(3 4)lowast
119905120579(4 3) = 0
(2) Clamped-Clamped 119891119909
= 119905119909(1 3)lowast
119905119909(2 4) minus 119905
119909(1
4)lowast
119905119909(2 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(2 4) minus 119905
119910(1 4)lowast
119905119910(2
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(2 4) minus 119905
120579(1 4)lowast
119905120579(2 3) = 0
Mathematical Problems in Engineering 5
A B C DC S
xx
c
xs
O
y
1m3m 3m
Figure 2 The doubly asymmetric continuous channel section and the cross section of beam of example 2 with warping allowed at B C andD but fully constrained at A
(3) Simply-Simply 119891119909
= 119905119909(1 2)lowast
119905119909(3 4) minus 119905
119909(3 2)lowast
119905119909(1
4) = 0 119891119910
= 119905119910(1 2)lowast
119905119910(3 4) minus 119905
119910(3 2)lowast
119905119910(1 4) =
0 119891120579= 119905120579(1 2)lowast
119905120579(3 4) minus 119905
120579(3 2)lowast
119905120579(1 4) = 0
(4) Free-Free 119891119909
= 119905119909(3 1)lowast
119905119909(4 2) minus 119905
119909(3 2)lowast
119905119909(4 1) =
0 119891119910= 119905119910(3 1)lowast
119905119910(4 2) minus 119905
119910(3 2)lowast
119905119910(4 1) = 0 119891
120579=
119905120579(3 1)lowast
119905120579(4 2) minus 119905
120579(3 2)lowast
119905120579(4 1) = 0
(5) Clamped-Simply 119891119909
= 119905119909(1 3)lowast
119905119909(3 4) minus 119905
119909(1
4)lowast
119905119909(3 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(3 4) minus 119905
119910(1 4)lowast
119905119910(3
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(3 4) minus 119905
120579(1 4)lowast
119905120579(3 3) = 0
In frequency equations the values of 120596 which set thedeterminant to zero are the uncoupled angular frequencies
23 Coupled Frequencies Ignoring the dynamic couplingeffect of structure caused by the eccentricity between thecenter of shear rigidity and the geometric center the coupledfrequencies of the shear torsional beam can be obtained byusing uncoupled frequencies and the well-known equation asfollows [28]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120596(119894)2
119895minus 120596(119894)2
1199090 minus119910
119888120596(119894)2
119895
0 120596(119894)2
119895minus 120596(119894)2
119910119909119888120596(119894)2
119895
minus119910119888120596(119894)2
119895119909119888120596(119894)2
1198951199032
119898(120596(119894)2
119895minus 120596(119894)2
120579)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(119895 = 1 2 3) (119894 = 1 2 3 )
(31)
3 Procedure of Computation
A program that considers the method presented in this studyas a basis has been prepared in MATLAB and the operationstages are presented below
(1) element dynamic Transfer matrices are calculated foreach element by using (24) (26) and (28)
(2) System dynamic transfer matrices (see (29)ndash(30)) areobtained with the help of element transfer matrices
(3) The angular frequencies of uncoupled vibrations areobtained by using the boundary conditions
(4) The coupled angular frequencies are found by using(31)
4 A Numerical Example
In this part of the study two numerical examples were solvedby a program written in MATLAB to validate the presentedmethod The results are compared with those given in theliterature
41 Numerical Example 1 The first example considers thebeam studied by Tanaka and Bercin [11] A typical uniformthin-walled beam has a length of 15m with a doublyasymmetric cross section The properties of the cross sectionare as follows
119909119888= 002316 119910
119888= 002625 120588 = 1947 kgm 119903
2
119898=
30303lowast
10minus3m2
119864119868119909
= 73480Nm2 119864119868119910
= 16680Nm2 119864119868119908
=
2364Nm4 and1198661198690= 1081Nm2
The first three coupled natural frequencies of the beam arecalculated by the presented method and compared with theresults by Tanaka and Bercin [11] and Rafezy and Howson[24] in Table 1 for clamped-free (C-F) and simply-simply (S-S) boundary conditions
42 Numerical Example 2 A typical continuous beam witha doubly asymmetric cross section is considered in thisexample (Figure 2)
The beam comprises a thin-walled outer layer and a shearcore with the following properties between support points Aand B The typical uniform thin-walled beam has a length of15m with a doubly asymmetric cross section The propertiesof the cross section are as follows
119909119904
= 008 119910119904
= 003 119909119888
= 005 119910119888
= 002 120588 =
20 kgm 1199032
119898= 0008m2
119864119868119909
= 216lowast
106Nm2 119864119868
119910= 173
lowast
106Nm2 119866
119905119869119905=
3200Nm2119864119868119908
= 14lowast
103Nm4 119866119860
119909= 600000N 119866119860
119910=
600000N and 119866119869119888= 3800Nm2
The shear core is omitted between points B and D wherethe cross-sectional properties remain unchanged except that119866119860119909
= 119866119860119910
= 119866119869119888= 0 and the small change in 120588 has been
ignored
6 Mathematical Problems in Engineering
Table 1 Coupled natural frequencies for the beam of example 1
Natural frequencies (Hz)
BC Proposed method Tanaka and Bercin [11] Rafezy and Howson [24]1198911
1198912
1198913
1198911
1198912
1198913
1198911
1198912
1198913
C-F 1717 2731 5910 1703 2758 5925 1717 2731 5910S-S 4471 7514 16487 4148 7412 16411 4471 7514 16487
Table 2 Coupled natural frequencies of the continuous beam ofexample 2
Frequencynumber This study Rafezy and Howson[24] Difference ()
1 6906 6940 minus0492 19763 19796 minus0173 35461 33836 480
The first three coupled natural frequencies of the beamare calculated by the presented method and compared withthe results of Rafezy and Howson [24] in Table 2
The main source of error between the proposed methodand Rafezy and Howson methods is the eccentricity betweenthe center of shear stiffness and flexural stiffness which wasnot taken into account in the proposed method
5 Conclusions
This paper presents a method for a free vibration analysisof a thin-walled beam of doubly asymmetric cross sectionfilled with shear sensitive material In the study first of alla dynamic transfer matrix method was obtained for planarshear flexure and torsional motionThen uncoupled angularfrequencies were obtained by using dynamic element transfermatrices and boundary conditions Coupled frequencieswereobtained by the well-known two-dimensional approachesIt was observed from the sample taken from the literaturethat the presented method gave sufficient results The errormargin of the proposed method is shown to be less than 5Themain source of error is the eccentricity between the centerof shear stiffness and flexural stiffness which was not takeninto account in the proposed method
The transfer matrix method is an efficient and computer-ized method which also provides a fast and practical solutionsince the dimension of thematrix for the elements and systemnever changes Because of this the proposedmethod is simpleand accurate enough to be used both at the concept designstage and for final analyses
References
[1] F Y Cheng ldquoVibrations of timoshenko beams and frameworksrdquoJournal of Structural Engineering vol 96 no 3 pp 551ndash571 1970
[2] C Mei ldquoCoupled vibrations of thin-walled beams of opensection using the finite element methodrdquo International Journalof Mechanical Sciences vol 12 no 10 pp 883ndash891 1970
[3] W LHallauer andR Y L Liu ldquoBeambending-torsion dynamicstiffness method for calculation of exact vibration modesrdquo
Journal of Sound and Vibration vol 85 no 1 pp 105ndash113 1982[4] E Dokumaci ldquoAn exact solution for coupled bending and tor-
sion vibrations of uniform beams having single cross-sectionalsymmetryrdquo Journal of Sound and Vibration vol 119 no 3 pp443ndash449 1987
[5] R H Gutierrez and P A A Laura ldquoApproximate analysis ofcoupled flexural-torsional vibrations of a beam of non-uniformcross-section using the optimized rayleigh methodrdquo Journal ofSound and Vibration vol 114 no 2 pp 393ndash397 1987
[6] J R Banerjee ldquoCoupled bending-torsional dynamic stiffnessmatrix for beam elementsrdquo International Journal for NumericalMethods in Engineering vol 28 no 6 pp 1283ndash1298 1989
[7] J R Banerjee and F W Williams ldquoCoupled bending-torsionaldynamic stiffness matrix for timoshenko beam elementsrdquoCom-puters and Structures vol 42 no 3 pp 301ndash310 1992
[8] J R Banerjee and F W Williams ldquoAn exact dynamic stiffnessmatrix for coupled extensional-torsional vibration of structuralmembersrdquo Computers and Structures vol 50 no 2 pp 161ndash1661994
[9] X Chen and K K Tamma ldquoDynamic response of elastic thin-walled structures influenced by coupling effectsrdquo Computersand Structures vol 51 no 1 pp 91ndash105 1994
[10] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996
[11] M Tanaka and A N Bercin ldquoFree vibration solution foruniform beams of nonsymmetrical cross section using Math-ematicardquo Computers and Structures vol 71 no 1 pp 1ndash8 1999
[12] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000
[13] R D Ambrosini J D Riera and R F Danesi ldquoA modifiedVlasov theory for dynamic analysis of thin-walled and variableopen section beamsrdquo Engineering Structures vol 22 no 8 pp890ndash900 2000
[14] L P Kollar ldquoFlexural-torsional vibration of open section com-posite beams with shear deformationrdquo International Journal ofSolids and Structures vol 38 no 42-43 pp 7543ndash7558 2001
[15] Y Matsui and T Hayashikawa ldquoDynamic stiffness analysis fortorsional vibration of continuous beamswith thin-walled cross-sectionrdquo Journal of Sound and Vibration vol 243 no 2 pp 301ndash316 2001
[16] V H Cortınez and M T Piovan ldquoVibration and buckling ofcomposite thin-walled beams with shear deformabilityrdquo Journalof Sound and Vibration vol 258 no 4 pp 701ndash723 2002
[17] A Arpaci S E Bozdag and E Sunbuloglu ldquoTriply coupledvibrations of thin-walled open cross-section beams includingrotary inertia effectsrdquo Journal of Sound and Vibration vol 260no 5 pp 889ndash900 2003
Mathematical Problems in Engineering 7
[18] M Y Kim H T Yun and N I Kim ldquoExact dynamic and staticelement stiffness matrices of nonsymmetric thin-walled beam-columnsrdquo Computers and Structures vol 81 no 14 pp 1425ndash1448 2003
[19] L Jun L Wanyou S Rongying and H Hongxing ldquoCoupledbending and torsional vibration of nonsymmetrical axiallyloaded thin-walled Bernoulli-Euler beamsrdquoMechanics ResearchCommunications vol 31 no 6 pp 697ndash711 2004
[20] L Jun H Hongxing S Rongying and J Xianding ldquoDynamicresponse of axially loaded monosymmetrical thin-walledBernoulli-Euler beamsrdquo Thin-Walled Structures vol 42 no 12pp 1689ndash1707 2004
[21] F Mohri L Azrar and M Potier-Ferry ldquoVibration analysisof buckled thin-walled beams with open sectionsrdquo Journal ofSound and Vibration vol 275 no 1-2 pp 434ndash446 2004
[22] A Prokic ldquoOn triply coupled vibrations of thin-walled beamswith arbitrary cross-sectionrdquo Journal of Sound and Vibrationvol 279 no 3ndash5 pp 723ndash737 2005
[23] M O Kaya and O O Ozgumus ldquoFlexural-torsional-coupledvibration analysis of axially loaded closed-section compositeTimoshenko beam by using DTMrdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 495ndash506 2007
[24] B Rafezy and W P Howson ldquoExact dynamic stiffness matrixfor a thin-walled beam of doubly asymmetric cross-sectionfilled with shear sensitive materialrdquo International Journal forNumerical Methods in Engineering vol 69 no 13 pp 2758ndash2779 2007
[25] H H Chen andKMHsiao ldquoCoupled axial-torsional vibrationof thin-walled Z-section beam induced by boundary condi-tionsrdquoThin-Walled Structures vol 45 no 6 pp 573ndash583 2007
[26] D Ambrosini ldquoOn free vibration of nonsymmetrical thin-walled beamsrdquoThin-Walled Structures vol 47 no 6-7 pp 629ndash636 2009
[27] G M Voros ldquoOn coupled bending-torsional vibrations ofbeams with initial loadsrdquoMechanics Research Communicationsvol 36 no 5 pp 603ndash611 2009
[28] B Rafezy and W P Howson ldquoExact natural frequencies of athree-dimensional shear-torsion beamwith doubly asymmetriccross-section using a two-dimensional approachrdquo Journal ofSound and Vibration vol 295 no 3ndash5 pp 1044ndash1059 2006
[29] F de Borbon and D Ambrosini ldquoOn free vibration analysis ofthin-walled beams axially loadedrdquo Thin-Walled Structures vol48 no 12 pp 915ndash920 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
120593
O
y
y
Cxc
x
xs
yz
S
Mass axis
Elastic flexure axis
z
(a)
O
y
yCxc
xs S
u(z t)
(zt)
S998400998400
C998400998400
120593(z t)
yc
ys
x
xO998400
S998400
C998400
(b)
Figure 1 Typical thin-walled beam [24]
120588119894
1205972
119881119894(119911119894 119905)
1205971199052+ 119864119868119910119894
1205974
119881119894(119911119894 119905)
1205971199114
119894
minus (119866119860)119910119894
1205972
119881119894(119911119894 119905)
1205971199112
119894
= 0
1205881198941199032
119898
1205972
Ψ119894(119911119894 119905)
1205971199052+ 119864119868119908119894
1205974
Ψ119894(119911119894 119905)
1205971199114
119894
minus (119866119869)119900119894
1205972
Ψ119894(119911119894 119905)
1205971199112
119894
= 0
(1)
where 119866119869119900119894
= 119866119905119894119869119905119894+ 119866119869119888119894
119864119868119909119894and 119864119868
119910119894are the flexural rigidity of the 119894th segment
in the 119909-119911 and 119910-119911 planes respectively and 119866119905119894119869119905119894and 119864119868
119908119894
are the Saint-Venant and warping torsion rigidity of the 119894thsegment about 119874 where 119868
119908is the warping moment of inertia
or warping constant 119866119860119909119894and 119866119860
119910119894are the effective shear
rigidities of the core material of the 119894th segment in 119909 and 119910
directions respectively and119866119869119900119894is the Saint-Venant torsional
rigidity of the core material about 119874 120588119894are the mass per unit
length of the 119894th segment and 119903119898is the polar mass radius of
gyration of cross section [24]If a sinusoidal variation of 119880 119881 and 120595 with circular
frequency 120596 is assumed then
119880119894(119911119894 119905) = 119906
119894sin (120596
119909119905)
119881119894(119911119894 119905) = V
119894sin (120596
119910119905)
120595119894(119911119894 119905) = 120579
119894sin (120596
120579119905)
(2)
where 119906119894 V119894 and 120579
119894are the amplitudes of the sinusoidally
varying displacementSubstituting (2) in (1) results are
1198894
119906119894
1198891199114
119894
minus(119866119860)119909119894
(119864119868)119909119894
1198892
119906119894
1198891199112
119894
minus119901119894
(119864119868)119909119894
1205962
119909119906119894= 0
1198894V119894
1198891199114
119894
minus
(119866119860)119910119894
(119864119868)119910119894
1198892V119894
1198891199112
119894
minus119901119894
(119864119868)119910119894
1205962
119910V119894= 0
1198894
120579119894
1198891199114
119894
minus(119866119869)119900119894
(119864119868)119908119894
1198892
120579119894
1198891205792
119894
minus1199011198941199032
119898
(119864119868)119908119894
1205962
120579120579119894= 0
(3)
When (3) is solved with respect to 119911119894 119906119894(119911119894) V119894(119911119894) and 120579
119894(119911119894)
can be obtained as follows
119906119894(119911119894) = 1198881cosh (119886
119909119894119911119894) + 1198882sinh (119886
119909119894119911119894)
+ 1198883cos (119887119909119894119911119894) + 1198884sin (119887119909119894119911119894)
(4)
V119894(119911119894) = 1198885cosh (119886
119910119894119911119894) + 1198886sinh (119886
119910119894119911119894)
+ 1198887cos (119887
119910119894119911119894) + 1198888sin (119887119910119894119911119894)
(5)
120579119894(119911119894) = 1198889cosh (119886
120579119894119911119894) + 11988810sinh (119886
120579119894119911119894)
+ 11988811cos (119887120579119894119911119894) + 11988812sin (119887120579119894119911119894)
(6)
where 120572119909119894 120572119910119894 120572120579119894 119887119909119894 119887119910119894 and 119887
120579119894can be calculated as follows
119886119909119894
= radic119904119909119894
+ 119901119909119894
2 119886
119910119894= radic
119904119910119894
+ 119901119910119894
2
119886120579119894
= radic119904120579119894
+ 119901120579119894
2 119887
119909119894= radic
minus119904119909119894
+ 119901119909119894
2
119887119910119894
= radicminus119904119910119894
+ 119901119910119894
2 119887
120579119894= radic
minus119904120579119894
+ 119901120579119894
2
119901119909119894
= radic(119866119860119909119894
119864119868119909119894
)
2
+ 4 lowast1199011198941205962
119909
(119864119868)119909119894
119901119910119894
= radic(
119866119860119910119894
119864119868119910119894
)
2
+ 4 lowast
1199011198941205962
119910
(119864119868)119910119894
Mathematical Problems in Engineering 3
119901120579119894
= radic(119866119869119900119894
119864119868119908119894
)
2
+ 4 lowast1199011198941205962
120579
(119864119868)119908119894
119904119909119894
=(119866119860)119909119894
(119864119868)119909119894
119904119910119894
=
(119866119860)119910119894
(119864119868)119910119894
119904120579119894
=(119866119869)119900119894
(119864119868)119908119894
(7)
By using (4) (5) and (6) the rotation angles in 119909 and 119910
directions (1199061015840
119894 V1015840119894) rate of twist (120579
1015840
119894) bending moments in
119909 and 119910 directions (119872119909119894119872119910119894) and bimoment (119872
119908119894) shear
forces in 119909 and 119910 directions (119881119909119894 119881119910119894) and torque (119872
119905119894) for
119894th element can be obtained as follows
119889119906119894(119911119894)
119889119911119894
= 1198881119886119909119894sinh (119886
119909119894119911119894) + 1198882119886119909119894cosh (119886
119909119894119911119894)
minus 1198883119887119909119894sin (119887119909119894119911119894) + 1198884119887119909119894cos (119887119909119894119911119894)
(8)
119889V119894(119911119894)
119889119911119894
= 1198885119886119910119894sinh (119886
119910119894119911119894) + 1198886119886119910119894cosh (119886
119910119894119911119894)
minus 1198887119887119910119894sin (119887119910119894119911119894) + 1198888119887119910119894cos (119887
119910119894119911119894)
(9)
119889120579119894(119911119894)
119889119911119894
= 1198889119886120579119894sinh (119886
120579119894119911119894) + 11988810119886120579119894cosh (119886
120579119894119911119894)
minus 11988811119887120579119894sin (119887120579119894119911119894) + 11988812119887120579119894cos (119887120579119894119911119894)
(10)
119872119909119894
(119911119894) = 119864119868
119909119894
1198892
119906119894(119911119894)
1198891199112
119894
= 119864119868119909119894
[11988811198862
119909119894cosh (119886
119909119894119911119894) + 11988821198862
119909119894sinh (119886
119909119894119911119894)
minus11988831198872
119909119894cos (119887119909119894119911119894) minus 11988841198872
119909119894sin (119887119909119894119911119894)]
(11)
119872119910119894
(119911119894) = 119864119868
119910119894
1198892V119894(119911119894)
1198891199112
119894
= 119864119868119910119894
[11988851198862
119910119894cosh (119886
119910119894119911119894) + 11988861198862
119910119894sinh (119886
119910119894119911119894)
minus11988871198872
119910119894cos (119887
119910119894119911119894) minus 11988881198872
119910119894sin (119887119910119894119911119894)]
(12)
119872119908119894
(119911119894) = 119864119868
119908119894
1198892
120579119894(119911119894)
1198891199112
119894
= 119864119868119908119894
[11988891198862
120579119894cosh (119886
120579119894119911119894) + 119888101198862
120579119894sinh (119886
120579119894119911119894)
minus119888111198872
120579119894cos (119887120579119894119911119894) minus 119888121198872
120579119894sin (119887120579119894119911119894)]
(13)
119881119909119894
(119911119894) = 119864119868
119909119894
1198893
119906119894(119911119894)
1198891199113
119894
minus (119866119860)119909119894
119889119906119894(119911119894)
119889119911119894
= [1198641198681199091198941198863
119909119894sinh (119886
119909119894119911119894) minus (119866119860)
119909119894119886119909119894sinh (119886
119909119894119911119894)] 1198881
+ [1198641198681199091198941198863
119909119894cosh (119886
119909119894119911119894)
minus(119866119860)119909119894119886119909119894cosh (119886
119909119894119911119894)] 1198882
+ [(119864119868)1199091198941198873
119909119894sin (119887119909119894119911119894) + (119866119860)
119909119894119887119909119894sin (119887119909119894119911119894)] 1198883
+ [minus(119864119868)1199091198941198873
119909119894cos (119887119909119894119911119894)
minus(119866119860)119909119894119887119909119894cos (119887119909119894119911119894)] 1198884
(14)
119881119910119894
(119911119894) = 119864119868
119910119894
1198893V119894(119911119894)
1198891199113
119894
minus (119866119860)119910119894
119889V119894(119911119894)
119889119911119894
= [1198641198681199101198941198863
119910119894sinh (119886
119910119894119911119894) minus (119866119860)
119910119894119886119910119894sinh (119886
119910119894119911119894)] 1198885
+ [1198641198681199101198941198863
119910119894cosh (119886
119910119894119911119894)
minus(119866119860)119910119894119886119910119894cosh (119886
119910119894119911119894)] 1198886
+ [(119864119868)1199101198941198873
119910119894sin (119887119910119894119911119894) + (119866119860)
119910119894119887119910119894sin (119887119910119894119911119894)] 1198887
+ [minus(119864119868)1199101198941198873
119910119894cos (119887
119910119894119911119894)
minus(119866119860)119910119894119887119910119894cos (119887
119910119894119911119894)] 1198888
(15)
119872119905119894(119911119894) = 119864119868
119908119894
1198893
120579119894(119911119894)
1198891199113
119894
minus (119866119869)119900119894
119889120579119894(119911119894)
119889119911119894
= [1198641198681205791198941198863
120579119894sinh (119886
120579119894119911119894) minus (119866119869)
119900119894119886120579119894sinh (119886
120579119894119911119894)] 1198889
+ [1198641198681199081198941198863
120579119894cosh (119886
120579119894119911119894)
minus(119866119869)119894119900119886120579119894cosh (119886
120579119894119911119894)] 11988810
+ [(119864119868)1199081198941198873
120579119894sin (119887120579119894119911119894) + (119866119869)
119900119894119887120579119894sin (119887120579119894119911119894)] 11988811
+ [minus(119864119868)1199081198941198873
120579119894cos (119887120579119894119911119894)
minus(119866119869)119900119894119887119910119894cos (119887120579119894119911119894)] 11988812
(16)
The following equation shows thematrix form of (4) (8) (11)and (14)
[[[
[
119906119894(119911119894)
1199061015840
119894(119911119894)
119872119909119894
(119911119894)
119881119909119894
(119911119894)
]]]
]
= 119860119909119894
(119911119894)
[[[
[
1198881
1198882
1198883
1198884
]]]
]
(17)
4 Mathematical Problems in Engineering
For the 119910 direction the following shows the matrix form of(5) (9) (12) and (15)
[[[
[
V119894(119911119894)
V1015840119894(119911119894)
119872119910119894
(119911119894)
119881119910119894
(119911119894)
]]]
]
= 119860119910119894
(119911119894)
[[[
[
1198885
1198886
1198887
1198888
]]]
]
(18)
Similarly torsional motion can be written
[[[
[
120579119894(119911119894)
1205791015840
119894(119911119894)
119872119908119894
(119911119894)
119872119905119894(119911119894)
]]]
]
= 119860120579119894(119911119894)
[[[
[
1198889
11988810
11988811
11988812
]]]
]
(19)
At the initial point of the 119894th element (17) (18) and (19) canbe written as follows
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
= 119860119909119894
(0)
[[[
[
1198881
1198882
1198883
1198884
]]]
]
(20)
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
= 119860119910119894
(0)
[[[
[
1198885
1198886
1198887
1198888
]]]
]
(21)
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
= 119860120579119894(0)
[[[
[
1198889
11988810
11988811
11988812
]]]
]
(22)
When vector 119888 is solved from (20) and is substituted in (17)the following is obtained
[[[
[
119906119894(119911119894)
1199061015840
119894(119911119894)
119872119909119894
(119911119894)
119881119909119894
(119911119894)
]]]
]
= 119860119909119894
(119911119894) 119860119909119894(0)minus1
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
(23)
For 119911119894= 119897119894 (23) can be written as
[[[
[
119906119894(119897119894)
1199061015840
119894(119897119894)
119872119909119894
(119897119894)
119881119909119894
(119897119894)
]]]
]
= 119879119909119894
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
(24)
where 119879119909119894is the element dynamic transfer matrix of the 119894th
elementFor the 119910 direction (23) and (24) can be written as
follows
[[[
[
V119894(119911119894)
V1015840119894(119911119894)
119872119910119894
(119911119894)
119881119910119894
(119911119894)
]]]
]
= 119860119910119894
(119911119894) 119860119910119894(0)minus1
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
(25)
[[[
[
V119894(119897119894)
V1015840119894(119897119894)
119872119910119894
(119897119894)
119881119910119894
(119897119894)
]]]
]
= 119879119910119894
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
(26)
Similarly rotation motion can be written in equations asfollows
[[[
[
120579119894(119911119894)
1205791015840
119894(119911119894)
119872119908119894
(119911119894)
119872119905119894(119911119894)
]]]
]
= 119860120579119894(119911119894) 119860120579119894(0)minus1
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
(27)
[[[
[
120579119894(119897119894)
1205791015840
119894(119897119894)
119872119908119894
(119897119894)
119872119905119894(119897119894)
]]]
]
= 119879120579119894
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
(28)
If (24) is written successively the displacementsmdashinternalforces relationship between the initial part and end of thebeammdashcan be found as follows
[[[
[
119906end1199061015840
end119872119909end
119881119909end
]]]
]
= 119879119909119899
119879119909(119899minus1)
sdot sdot sdot 11987911990921198791199091
[[[
[
119906initial1199061015840
initial119872119909initial
119881119909initial
]]]
]
= 119905119909
[[[
[
119906initial1199061015840
initial119872119909initial
119881119909initial
]]]
]
(29)
For 119910 and rotation motion (29) can be written as follows
[[[
[
VendV1015840end
119872119910end
119881119910end
]]]
]
= 119879119910119899
119879119910(119899minus1)
sdot sdot sdot 11987911991021198791199101
[[[
[
VinitialV1015840initial
119872119910initial
119881119910initial
]]]
]
= 119905119910
[[[
[
VinitialV1015840initial
119872119910initial
119881119910initial
]]]
]
[[[
[
120579end1205791015840
end119872119908end
119872119905end
]]]
]
= 119879120579119899119879120579(119899minus1)
sdot sdot sdot 11987912057921198791205791
[[[
[
120579initial1205791015840
initial119872119908initial
119872119905initial
]]]
]
= 119905120579
[[[
[
120579initial1205791015840
initial119872119908initial
119872119905initial
]]]
]
(30)
The eigenvalue equation for a thin-walled beam filled withshear sensitive material can be established using (29) (30)and the specific boundary conditions are as follows
(1) Clamped-Free 119891119909
= 119905119909(3 3)lowast
119905119909(4 4) minus 119905
119909(3 4)lowast
119905119909(4
3) = 0 119891119910
= 119905119910(3 3)lowast
119905119910(4 4) minus 119905
119910(3 4)lowast
119905119910(4 3) =
0 119891120579= 119905120579(3 3)lowast
119905120579(4 4) minus 119905
120579(3 4)lowast
119905120579(4 3) = 0
(2) Clamped-Clamped 119891119909
= 119905119909(1 3)lowast
119905119909(2 4) minus 119905
119909(1
4)lowast
119905119909(2 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(2 4) minus 119905
119910(1 4)lowast
119905119910(2
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(2 4) minus 119905
120579(1 4)lowast
119905120579(2 3) = 0
Mathematical Problems in Engineering 5
A B C DC S
xx
c
xs
O
y
1m3m 3m
Figure 2 The doubly asymmetric continuous channel section and the cross section of beam of example 2 with warping allowed at B C andD but fully constrained at A
(3) Simply-Simply 119891119909
= 119905119909(1 2)lowast
119905119909(3 4) minus 119905
119909(3 2)lowast
119905119909(1
4) = 0 119891119910
= 119905119910(1 2)lowast
119905119910(3 4) minus 119905
119910(3 2)lowast
119905119910(1 4) =
0 119891120579= 119905120579(1 2)lowast
119905120579(3 4) minus 119905
120579(3 2)lowast
119905120579(1 4) = 0
(4) Free-Free 119891119909
= 119905119909(3 1)lowast
119905119909(4 2) minus 119905
119909(3 2)lowast
119905119909(4 1) =
0 119891119910= 119905119910(3 1)lowast
119905119910(4 2) minus 119905
119910(3 2)lowast
119905119910(4 1) = 0 119891
120579=
119905120579(3 1)lowast
119905120579(4 2) minus 119905
120579(3 2)lowast
119905120579(4 1) = 0
(5) Clamped-Simply 119891119909
= 119905119909(1 3)lowast
119905119909(3 4) minus 119905
119909(1
4)lowast
119905119909(3 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(3 4) minus 119905
119910(1 4)lowast
119905119910(3
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(3 4) minus 119905
120579(1 4)lowast
119905120579(3 3) = 0
In frequency equations the values of 120596 which set thedeterminant to zero are the uncoupled angular frequencies
23 Coupled Frequencies Ignoring the dynamic couplingeffect of structure caused by the eccentricity between thecenter of shear rigidity and the geometric center the coupledfrequencies of the shear torsional beam can be obtained byusing uncoupled frequencies and the well-known equation asfollows [28]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120596(119894)2
119895minus 120596(119894)2
1199090 minus119910
119888120596(119894)2
119895
0 120596(119894)2
119895minus 120596(119894)2
119910119909119888120596(119894)2
119895
minus119910119888120596(119894)2
119895119909119888120596(119894)2
1198951199032
119898(120596(119894)2
119895minus 120596(119894)2
120579)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(119895 = 1 2 3) (119894 = 1 2 3 )
(31)
3 Procedure of Computation
A program that considers the method presented in this studyas a basis has been prepared in MATLAB and the operationstages are presented below
(1) element dynamic Transfer matrices are calculated foreach element by using (24) (26) and (28)
(2) System dynamic transfer matrices (see (29)ndash(30)) areobtained with the help of element transfer matrices
(3) The angular frequencies of uncoupled vibrations areobtained by using the boundary conditions
(4) The coupled angular frequencies are found by using(31)
4 A Numerical Example
In this part of the study two numerical examples were solvedby a program written in MATLAB to validate the presentedmethod The results are compared with those given in theliterature
41 Numerical Example 1 The first example considers thebeam studied by Tanaka and Bercin [11] A typical uniformthin-walled beam has a length of 15m with a doublyasymmetric cross section The properties of the cross sectionare as follows
119909119888= 002316 119910
119888= 002625 120588 = 1947 kgm 119903
2
119898=
30303lowast
10minus3m2
119864119868119909
= 73480Nm2 119864119868119910
= 16680Nm2 119864119868119908
=
2364Nm4 and1198661198690= 1081Nm2
The first three coupled natural frequencies of the beam arecalculated by the presented method and compared with theresults by Tanaka and Bercin [11] and Rafezy and Howson[24] in Table 1 for clamped-free (C-F) and simply-simply (S-S) boundary conditions
42 Numerical Example 2 A typical continuous beam witha doubly asymmetric cross section is considered in thisexample (Figure 2)
The beam comprises a thin-walled outer layer and a shearcore with the following properties between support points Aand B The typical uniform thin-walled beam has a length of15m with a doubly asymmetric cross section The propertiesof the cross section are as follows
119909119904
= 008 119910119904
= 003 119909119888
= 005 119910119888
= 002 120588 =
20 kgm 1199032
119898= 0008m2
119864119868119909
= 216lowast
106Nm2 119864119868
119910= 173
lowast
106Nm2 119866
119905119869119905=
3200Nm2119864119868119908
= 14lowast
103Nm4 119866119860
119909= 600000N 119866119860
119910=
600000N and 119866119869119888= 3800Nm2
The shear core is omitted between points B and D wherethe cross-sectional properties remain unchanged except that119866119860119909
= 119866119860119910
= 119866119869119888= 0 and the small change in 120588 has been
ignored
6 Mathematical Problems in Engineering
Table 1 Coupled natural frequencies for the beam of example 1
Natural frequencies (Hz)
BC Proposed method Tanaka and Bercin [11] Rafezy and Howson [24]1198911
1198912
1198913
1198911
1198912
1198913
1198911
1198912
1198913
C-F 1717 2731 5910 1703 2758 5925 1717 2731 5910S-S 4471 7514 16487 4148 7412 16411 4471 7514 16487
Table 2 Coupled natural frequencies of the continuous beam ofexample 2
Frequencynumber This study Rafezy and Howson[24] Difference ()
1 6906 6940 minus0492 19763 19796 minus0173 35461 33836 480
The first three coupled natural frequencies of the beamare calculated by the presented method and compared withthe results of Rafezy and Howson [24] in Table 2
The main source of error between the proposed methodand Rafezy and Howson methods is the eccentricity betweenthe center of shear stiffness and flexural stiffness which wasnot taken into account in the proposed method
5 Conclusions
This paper presents a method for a free vibration analysisof a thin-walled beam of doubly asymmetric cross sectionfilled with shear sensitive material In the study first of alla dynamic transfer matrix method was obtained for planarshear flexure and torsional motionThen uncoupled angularfrequencies were obtained by using dynamic element transfermatrices and boundary conditions Coupled frequencieswereobtained by the well-known two-dimensional approachesIt was observed from the sample taken from the literaturethat the presented method gave sufficient results The errormargin of the proposed method is shown to be less than 5Themain source of error is the eccentricity between the centerof shear stiffness and flexural stiffness which was not takeninto account in the proposed method
The transfer matrix method is an efficient and computer-ized method which also provides a fast and practical solutionsince the dimension of thematrix for the elements and systemnever changes Because of this the proposedmethod is simpleand accurate enough to be used both at the concept designstage and for final analyses
References
[1] F Y Cheng ldquoVibrations of timoshenko beams and frameworksrdquoJournal of Structural Engineering vol 96 no 3 pp 551ndash571 1970
[2] C Mei ldquoCoupled vibrations of thin-walled beams of opensection using the finite element methodrdquo International Journalof Mechanical Sciences vol 12 no 10 pp 883ndash891 1970
[3] W LHallauer andR Y L Liu ldquoBeambending-torsion dynamicstiffness method for calculation of exact vibration modesrdquo
Journal of Sound and Vibration vol 85 no 1 pp 105ndash113 1982[4] E Dokumaci ldquoAn exact solution for coupled bending and tor-
sion vibrations of uniform beams having single cross-sectionalsymmetryrdquo Journal of Sound and Vibration vol 119 no 3 pp443ndash449 1987
[5] R H Gutierrez and P A A Laura ldquoApproximate analysis ofcoupled flexural-torsional vibrations of a beam of non-uniformcross-section using the optimized rayleigh methodrdquo Journal ofSound and Vibration vol 114 no 2 pp 393ndash397 1987
[6] J R Banerjee ldquoCoupled bending-torsional dynamic stiffnessmatrix for beam elementsrdquo International Journal for NumericalMethods in Engineering vol 28 no 6 pp 1283ndash1298 1989
[7] J R Banerjee and F W Williams ldquoCoupled bending-torsionaldynamic stiffness matrix for timoshenko beam elementsrdquoCom-puters and Structures vol 42 no 3 pp 301ndash310 1992
[8] J R Banerjee and F W Williams ldquoAn exact dynamic stiffnessmatrix for coupled extensional-torsional vibration of structuralmembersrdquo Computers and Structures vol 50 no 2 pp 161ndash1661994
[9] X Chen and K K Tamma ldquoDynamic response of elastic thin-walled structures influenced by coupling effectsrdquo Computersand Structures vol 51 no 1 pp 91ndash105 1994
[10] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996
[11] M Tanaka and A N Bercin ldquoFree vibration solution foruniform beams of nonsymmetrical cross section using Math-ematicardquo Computers and Structures vol 71 no 1 pp 1ndash8 1999
[12] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000
[13] R D Ambrosini J D Riera and R F Danesi ldquoA modifiedVlasov theory for dynamic analysis of thin-walled and variableopen section beamsrdquo Engineering Structures vol 22 no 8 pp890ndash900 2000
[14] L P Kollar ldquoFlexural-torsional vibration of open section com-posite beams with shear deformationrdquo International Journal ofSolids and Structures vol 38 no 42-43 pp 7543ndash7558 2001
[15] Y Matsui and T Hayashikawa ldquoDynamic stiffness analysis fortorsional vibration of continuous beamswith thin-walled cross-sectionrdquo Journal of Sound and Vibration vol 243 no 2 pp 301ndash316 2001
[16] V H Cortınez and M T Piovan ldquoVibration and buckling ofcomposite thin-walled beams with shear deformabilityrdquo Journalof Sound and Vibration vol 258 no 4 pp 701ndash723 2002
[17] A Arpaci S E Bozdag and E Sunbuloglu ldquoTriply coupledvibrations of thin-walled open cross-section beams includingrotary inertia effectsrdquo Journal of Sound and Vibration vol 260no 5 pp 889ndash900 2003
Mathematical Problems in Engineering 7
[18] M Y Kim H T Yun and N I Kim ldquoExact dynamic and staticelement stiffness matrices of nonsymmetric thin-walled beam-columnsrdquo Computers and Structures vol 81 no 14 pp 1425ndash1448 2003
[19] L Jun L Wanyou S Rongying and H Hongxing ldquoCoupledbending and torsional vibration of nonsymmetrical axiallyloaded thin-walled Bernoulli-Euler beamsrdquoMechanics ResearchCommunications vol 31 no 6 pp 697ndash711 2004
[20] L Jun H Hongxing S Rongying and J Xianding ldquoDynamicresponse of axially loaded monosymmetrical thin-walledBernoulli-Euler beamsrdquo Thin-Walled Structures vol 42 no 12pp 1689ndash1707 2004
[21] F Mohri L Azrar and M Potier-Ferry ldquoVibration analysisof buckled thin-walled beams with open sectionsrdquo Journal ofSound and Vibration vol 275 no 1-2 pp 434ndash446 2004
[22] A Prokic ldquoOn triply coupled vibrations of thin-walled beamswith arbitrary cross-sectionrdquo Journal of Sound and Vibrationvol 279 no 3ndash5 pp 723ndash737 2005
[23] M O Kaya and O O Ozgumus ldquoFlexural-torsional-coupledvibration analysis of axially loaded closed-section compositeTimoshenko beam by using DTMrdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 495ndash506 2007
[24] B Rafezy and W P Howson ldquoExact dynamic stiffness matrixfor a thin-walled beam of doubly asymmetric cross-sectionfilled with shear sensitive materialrdquo International Journal forNumerical Methods in Engineering vol 69 no 13 pp 2758ndash2779 2007
[25] H H Chen andKMHsiao ldquoCoupled axial-torsional vibrationof thin-walled Z-section beam induced by boundary condi-tionsrdquoThin-Walled Structures vol 45 no 6 pp 573ndash583 2007
[26] D Ambrosini ldquoOn free vibration of nonsymmetrical thin-walled beamsrdquoThin-Walled Structures vol 47 no 6-7 pp 629ndash636 2009
[27] G M Voros ldquoOn coupled bending-torsional vibrations ofbeams with initial loadsrdquoMechanics Research Communicationsvol 36 no 5 pp 603ndash611 2009
[28] B Rafezy and W P Howson ldquoExact natural frequencies of athree-dimensional shear-torsion beamwith doubly asymmetriccross-section using a two-dimensional approachrdquo Journal ofSound and Vibration vol 295 no 3ndash5 pp 1044ndash1059 2006
[29] F de Borbon and D Ambrosini ldquoOn free vibration analysis ofthin-walled beams axially loadedrdquo Thin-Walled Structures vol48 no 12 pp 915ndash920 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
119901120579119894
= radic(119866119869119900119894
119864119868119908119894
)
2
+ 4 lowast1199011198941205962
120579
(119864119868)119908119894
119904119909119894
=(119866119860)119909119894
(119864119868)119909119894
119904119910119894
=
(119866119860)119910119894
(119864119868)119910119894
119904120579119894
=(119866119869)119900119894
(119864119868)119908119894
(7)
By using (4) (5) and (6) the rotation angles in 119909 and 119910
directions (1199061015840
119894 V1015840119894) rate of twist (120579
1015840
119894) bending moments in
119909 and 119910 directions (119872119909119894119872119910119894) and bimoment (119872
119908119894) shear
forces in 119909 and 119910 directions (119881119909119894 119881119910119894) and torque (119872
119905119894) for
119894th element can be obtained as follows
119889119906119894(119911119894)
119889119911119894
= 1198881119886119909119894sinh (119886
119909119894119911119894) + 1198882119886119909119894cosh (119886
119909119894119911119894)
minus 1198883119887119909119894sin (119887119909119894119911119894) + 1198884119887119909119894cos (119887119909119894119911119894)
(8)
119889V119894(119911119894)
119889119911119894
= 1198885119886119910119894sinh (119886
119910119894119911119894) + 1198886119886119910119894cosh (119886
119910119894119911119894)
minus 1198887119887119910119894sin (119887119910119894119911119894) + 1198888119887119910119894cos (119887
119910119894119911119894)
(9)
119889120579119894(119911119894)
119889119911119894
= 1198889119886120579119894sinh (119886
120579119894119911119894) + 11988810119886120579119894cosh (119886
120579119894119911119894)
minus 11988811119887120579119894sin (119887120579119894119911119894) + 11988812119887120579119894cos (119887120579119894119911119894)
(10)
119872119909119894
(119911119894) = 119864119868
119909119894
1198892
119906119894(119911119894)
1198891199112
119894
= 119864119868119909119894
[11988811198862
119909119894cosh (119886
119909119894119911119894) + 11988821198862
119909119894sinh (119886
119909119894119911119894)
minus11988831198872
119909119894cos (119887119909119894119911119894) minus 11988841198872
119909119894sin (119887119909119894119911119894)]
(11)
119872119910119894
(119911119894) = 119864119868
119910119894
1198892V119894(119911119894)
1198891199112
119894
= 119864119868119910119894
[11988851198862
119910119894cosh (119886
119910119894119911119894) + 11988861198862
119910119894sinh (119886
119910119894119911119894)
minus11988871198872
119910119894cos (119887
119910119894119911119894) minus 11988881198872
119910119894sin (119887119910119894119911119894)]
(12)
119872119908119894
(119911119894) = 119864119868
119908119894
1198892
120579119894(119911119894)
1198891199112
119894
= 119864119868119908119894
[11988891198862
120579119894cosh (119886
120579119894119911119894) + 119888101198862
120579119894sinh (119886
120579119894119911119894)
minus119888111198872
120579119894cos (119887120579119894119911119894) minus 119888121198872
120579119894sin (119887120579119894119911119894)]
(13)
119881119909119894
(119911119894) = 119864119868
119909119894
1198893
119906119894(119911119894)
1198891199113
119894
minus (119866119860)119909119894
119889119906119894(119911119894)
119889119911119894
= [1198641198681199091198941198863
119909119894sinh (119886
119909119894119911119894) minus (119866119860)
119909119894119886119909119894sinh (119886
119909119894119911119894)] 1198881
+ [1198641198681199091198941198863
119909119894cosh (119886
119909119894119911119894)
minus(119866119860)119909119894119886119909119894cosh (119886
119909119894119911119894)] 1198882
+ [(119864119868)1199091198941198873
119909119894sin (119887119909119894119911119894) + (119866119860)
119909119894119887119909119894sin (119887119909119894119911119894)] 1198883
+ [minus(119864119868)1199091198941198873
119909119894cos (119887119909119894119911119894)
minus(119866119860)119909119894119887119909119894cos (119887119909119894119911119894)] 1198884
(14)
119881119910119894
(119911119894) = 119864119868
119910119894
1198893V119894(119911119894)
1198891199113
119894
minus (119866119860)119910119894
119889V119894(119911119894)
119889119911119894
= [1198641198681199101198941198863
119910119894sinh (119886
119910119894119911119894) minus (119866119860)
119910119894119886119910119894sinh (119886
119910119894119911119894)] 1198885
+ [1198641198681199101198941198863
119910119894cosh (119886
119910119894119911119894)
minus(119866119860)119910119894119886119910119894cosh (119886
119910119894119911119894)] 1198886
+ [(119864119868)1199101198941198873
119910119894sin (119887119910119894119911119894) + (119866119860)
119910119894119887119910119894sin (119887119910119894119911119894)] 1198887
+ [minus(119864119868)1199101198941198873
119910119894cos (119887
119910119894119911119894)
minus(119866119860)119910119894119887119910119894cos (119887
119910119894119911119894)] 1198888
(15)
119872119905119894(119911119894) = 119864119868
119908119894
1198893
120579119894(119911119894)
1198891199113
119894
minus (119866119869)119900119894
119889120579119894(119911119894)
119889119911119894
= [1198641198681205791198941198863
120579119894sinh (119886
120579119894119911119894) minus (119866119869)
119900119894119886120579119894sinh (119886
120579119894119911119894)] 1198889
+ [1198641198681199081198941198863
120579119894cosh (119886
120579119894119911119894)
minus(119866119869)119894119900119886120579119894cosh (119886
120579119894119911119894)] 11988810
+ [(119864119868)1199081198941198873
120579119894sin (119887120579119894119911119894) + (119866119869)
119900119894119887120579119894sin (119887120579119894119911119894)] 11988811
+ [minus(119864119868)1199081198941198873
120579119894cos (119887120579119894119911119894)
minus(119866119869)119900119894119887119910119894cos (119887120579119894119911119894)] 11988812
(16)
The following equation shows thematrix form of (4) (8) (11)and (14)
[[[
[
119906119894(119911119894)
1199061015840
119894(119911119894)
119872119909119894
(119911119894)
119881119909119894
(119911119894)
]]]
]
= 119860119909119894
(119911119894)
[[[
[
1198881
1198882
1198883
1198884
]]]
]
(17)
4 Mathematical Problems in Engineering
For the 119910 direction the following shows the matrix form of(5) (9) (12) and (15)
[[[
[
V119894(119911119894)
V1015840119894(119911119894)
119872119910119894
(119911119894)
119881119910119894
(119911119894)
]]]
]
= 119860119910119894
(119911119894)
[[[
[
1198885
1198886
1198887
1198888
]]]
]
(18)
Similarly torsional motion can be written
[[[
[
120579119894(119911119894)
1205791015840
119894(119911119894)
119872119908119894
(119911119894)
119872119905119894(119911119894)
]]]
]
= 119860120579119894(119911119894)
[[[
[
1198889
11988810
11988811
11988812
]]]
]
(19)
At the initial point of the 119894th element (17) (18) and (19) canbe written as follows
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
= 119860119909119894
(0)
[[[
[
1198881
1198882
1198883
1198884
]]]
]
(20)
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
= 119860119910119894
(0)
[[[
[
1198885
1198886
1198887
1198888
]]]
]
(21)
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
= 119860120579119894(0)
[[[
[
1198889
11988810
11988811
11988812
]]]
]
(22)
When vector 119888 is solved from (20) and is substituted in (17)the following is obtained
[[[
[
119906119894(119911119894)
1199061015840
119894(119911119894)
119872119909119894
(119911119894)
119881119909119894
(119911119894)
]]]
]
= 119860119909119894
(119911119894) 119860119909119894(0)minus1
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
(23)
For 119911119894= 119897119894 (23) can be written as
[[[
[
119906119894(119897119894)
1199061015840
119894(119897119894)
119872119909119894
(119897119894)
119881119909119894
(119897119894)
]]]
]
= 119879119909119894
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
(24)
where 119879119909119894is the element dynamic transfer matrix of the 119894th
elementFor the 119910 direction (23) and (24) can be written as
follows
[[[
[
V119894(119911119894)
V1015840119894(119911119894)
119872119910119894
(119911119894)
119881119910119894
(119911119894)
]]]
]
= 119860119910119894
(119911119894) 119860119910119894(0)minus1
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
(25)
[[[
[
V119894(119897119894)
V1015840119894(119897119894)
119872119910119894
(119897119894)
119881119910119894
(119897119894)
]]]
]
= 119879119910119894
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
(26)
Similarly rotation motion can be written in equations asfollows
[[[
[
120579119894(119911119894)
1205791015840
119894(119911119894)
119872119908119894
(119911119894)
119872119905119894(119911119894)
]]]
]
= 119860120579119894(119911119894) 119860120579119894(0)minus1
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
(27)
[[[
[
120579119894(119897119894)
1205791015840
119894(119897119894)
119872119908119894
(119897119894)
119872119905119894(119897119894)
]]]
]
= 119879120579119894
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
(28)
If (24) is written successively the displacementsmdashinternalforces relationship between the initial part and end of thebeammdashcan be found as follows
[[[
[
119906end1199061015840
end119872119909end
119881119909end
]]]
]
= 119879119909119899
119879119909(119899minus1)
sdot sdot sdot 11987911990921198791199091
[[[
[
119906initial1199061015840
initial119872119909initial
119881119909initial
]]]
]
= 119905119909
[[[
[
119906initial1199061015840
initial119872119909initial
119881119909initial
]]]
]
(29)
For 119910 and rotation motion (29) can be written as follows
[[[
[
VendV1015840end
119872119910end
119881119910end
]]]
]
= 119879119910119899
119879119910(119899minus1)
sdot sdot sdot 11987911991021198791199101
[[[
[
VinitialV1015840initial
119872119910initial
119881119910initial
]]]
]
= 119905119910
[[[
[
VinitialV1015840initial
119872119910initial
119881119910initial
]]]
]
[[[
[
120579end1205791015840
end119872119908end
119872119905end
]]]
]
= 119879120579119899119879120579(119899minus1)
sdot sdot sdot 11987912057921198791205791
[[[
[
120579initial1205791015840
initial119872119908initial
119872119905initial
]]]
]
= 119905120579
[[[
[
120579initial1205791015840
initial119872119908initial
119872119905initial
]]]
]
(30)
The eigenvalue equation for a thin-walled beam filled withshear sensitive material can be established using (29) (30)and the specific boundary conditions are as follows
(1) Clamped-Free 119891119909
= 119905119909(3 3)lowast
119905119909(4 4) minus 119905
119909(3 4)lowast
119905119909(4
3) = 0 119891119910
= 119905119910(3 3)lowast
119905119910(4 4) minus 119905
119910(3 4)lowast
119905119910(4 3) =
0 119891120579= 119905120579(3 3)lowast
119905120579(4 4) minus 119905
120579(3 4)lowast
119905120579(4 3) = 0
(2) Clamped-Clamped 119891119909
= 119905119909(1 3)lowast
119905119909(2 4) minus 119905
119909(1
4)lowast
119905119909(2 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(2 4) minus 119905
119910(1 4)lowast
119905119910(2
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(2 4) minus 119905
120579(1 4)lowast
119905120579(2 3) = 0
Mathematical Problems in Engineering 5
A B C DC S
xx
c
xs
O
y
1m3m 3m
Figure 2 The doubly asymmetric continuous channel section and the cross section of beam of example 2 with warping allowed at B C andD but fully constrained at A
(3) Simply-Simply 119891119909
= 119905119909(1 2)lowast
119905119909(3 4) minus 119905
119909(3 2)lowast
119905119909(1
4) = 0 119891119910
= 119905119910(1 2)lowast
119905119910(3 4) minus 119905
119910(3 2)lowast
119905119910(1 4) =
0 119891120579= 119905120579(1 2)lowast
119905120579(3 4) minus 119905
120579(3 2)lowast
119905120579(1 4) = 0
(4) Free-Free 119891119909
= 119905119909(3 1)lowast
119905119909(4 2) minus 119905
119909(3 2)lowast
119905119909(4 1) =
0 119891119910= 119905119910(3 1)lowast
119905119910(4 2) minus 119905
119910(3 2)lowast
119905119910(4 1) = 0 119891
120579=
119905120579(3 1)lowast
119905120579(4 2) minus 119905
120579(3 2)lowast
119905120579(4 1) = 0
(5) Clamped-Simply 119891119909
= 119905119909(1 3)lowast
119905119909(3 4) minus 119905
119909(1
4)lowast
119905119909(3 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(3 4) minus 119905
119910(1 4)lowast
119905119910(3
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(3 4) minus 119905
120579(1 4)lowast
119905120579(3 3) = 0
In frequency equations the values of 120596 which set thedeterminant to zero are the uncoupled angular frequencies
23 Coupled Frequencies Ignoring the dynamic couplingeffect of structure caused by the eccentricity between thecenter of shear rigidity and the geometric center the coupledfrequencies of the shear torsional beam can be obtained byusing uncoupled frequencies and the well-known equation asfollows [28]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120596(119894)2
119895minus 120596(119894)2
1199090 minus119910
119888120596(119894)2
119895
0 120596(119894)2
119895minus 120596(119894)2
119910119909119888120596(119894)2
119895
minus119910119888120596(119894)2
119895119909119888120596(119894)2
1198951199032
119898(120596(119894)2
119895minus 120596(119894)2
120579)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(119895 = 1 2 3) (119894 = 1 2 3 )
(31)
3 Procedure of Computation
A program that considers the method presented in this studyas a basis has been prepared in MATLAB and the operationstages are presented below
(1) element dynamic Transfer matrices are calculated foreach element by using (24) (26) and (28)
(2) System dynamic transfer matrices (see (29)ndash(30)) areobtained with the help of element transfer matrices
(3) The angular frequencies of uncoupled vibrations areobtained by using the boundary conditions
(4) The coupled angular frequencies are found by using(31)
4 A Numerical Example
In this part of the study two numerical examples were solvedby a program written in MATLAB to validate the presentedmethod The results are compared with those given in theliterature
41 Numerical Example 1 The first example considers thebeam studied by Tanaka and Bercin [11] A typical uniformthin-walled beam has a length of 15m with a doublyasymmetric cross section The properties of the cross sectionare as follows
119909119888= 002316 119910
119888= 002625 120588 = 1947 kgm 119903
2
119898=
30303lowast
10minus3m2
119864119868119909
= 73480Nm2 119864119868119910
= 16680Nm2 119864119868119908
=
2364Nm4 and1198661198690= 1081Nm2
The first three coupled natural frequencies of the beam arecalculated by the presented method and compared with theresults by Tanaka and Bercin [11] and Rafezy and Howson[24] in Table 1 for clamped-free (C-F) and simply-simply (S-S) boundary conditions
42 Numerical Example 2 A typical continuous beam witha doubly asymmetric cross section is considered in thisexample (Figure 2)
The beam comprises a thin-walled outer layer and a shearcore with the following properties between support points Aand B The typical uniform thin-walled beam has a length of15m with a doubly asymmetric cross section The propertiesof the cross section are as follows
119909119904
= 008 119910119904
= 003 119909119888
= 005 119910119888
= 002 120588 =
20 kgm 1199032
119898= 0008m2
119864119868119909
= 216lowast
106Nm2 119864119868
119910= 173
lowast
106Nm2 119866
119905119869119905=
3200Nm2119864119868119908
= 14lowast
103Nm4 119866119860
119909= 600000N 119866119860
119910=
600000N and 119866119869119888= 3800Nm2
The shear core is omitted between points B and D wherethe cross-sectional properties remain unchanged except that119866119860119909
= 119866119860119910
= 119866119869119888= 0 and the small change in 120588 has been
ignored
6 Mathematical Problems in Engineering
Table 1 Coupled natural frequencies for the beam of example 1
Natural frequencies (Hz)
BC Proposed method Tanaka and Bercin [11] Rafezy and Howson [24]1198911
1198912
1198913
1198911
1198912
1198913
1198911
1198912
1198913
C-F 1717 2731 5910 1703 2758 5925 1717 2731 5910S-S 4471 7514 16487 4148 7412 16411 4471 7514 16487
Table 2 Coupled natural frequencies of the continuous beam ofexample 2
Frequencynumber This study Rafezy and Howson[24] Difference ()
1 6906 6940 minus0492 19763 19796 minus0173 35461 33836 480
The first three coupled natural frequencies of the beamare calculated by the presented method and compared withthe results of Rafezy and Howson [24] in Table 2
The main source of error between the proposed methodand Rafezy and Howson methods is the eccentricity betweenthe center of shear stiffness and flexural stiffness which wasnot taken into account in the proposed method
5 Conclusions
This paper presents a method for a free vibration analysisof a thin-walled beam of doubly asymmetric cross sectionfilled with shear sensitive material In the study first of alla dynamic transfer matrix method was obtained for planarshear flexure and torsional motionThen uncoupled angularfrequencies were obtained by using dynamic element transfermatrices and boundary conditions Coupled frequencieswereobtained by the well-known two-dimensional approachesIt was observed from the sample taken from the literaturethat the presented method gave sufficient results The errormargin of the proposed method is shown to be less than 5Themain source of error is the eccentricity between the centerof shear stiffness and flexural stiffness which was not takeninto account in the proposed method
The transfer matrix method is an efficient and computer-ized method which also provides a fast and practical solutionsince the dimension of thematrix for the elements and systemnever changes Because of this the proposedmethod is simpleand accurate enough to be used both at the concept designstage and for final analyses
References
[1] F Y Cheng ldquoVibrations of timoshenko beams and frameworksrdquoJournal of Structural Engineering vol 96 no 3 pp 551ndash571 1970
[2] C Mei ldquoCoupled vibrations of thin-walled beams of opensection using the finite element methodrdquo International Journalof Mechanical Sciences vol 12 no 10 pp 883ndash891 1970
[3] W LHallauer andR Y L Liu ldquoBeambending-torsion dynamicstiffness method for calculation of exact vibration modesrdquo
Journal of Sound and Vibration vol 85 no 1 pp 105ndash113 1982[4] E Dokumaci ldquoAn exact solution for coupled bending and tor-
sion vibrations of uniform beams having single cross-sectionalsymmetryrdquo Journal of Sound and Vibration vol 119 no 3 pp443ndash449 1987
[5] R H Gutierrez and P A A Laura ldquoApproximate analysis ofcoupled flexural-torsional vibrations of a beam of non-uniformcross-section using the optimized rayleigh methodrdquo Journal ofSound and Vibration vol 114 no 2 pp 393ndash397 1987
[6] J R Banerjee ldquoCoupled bending-torsional dynamic stiffnessmatrix for beam elementsrdquo International Journal for NumericalMethods in Engineering vol 28 no 6 pp 1283ndash1298 1989
[7] J R Banerjee and F W Williams ldquoCoupled bending-torsionaldynamic stiffness matrix for timoshenko beam elementsrdquoCom-puters and Structures vol 42 no 3 pp 301ndash310 1992
[8] J R Banerjee and F W Williams ldquoAn exact dynamic stiffnessmatrix for coupled extensional-torsional vibration of structuralmembersrdquo Computers and Structures vol 50 no 2 pp 161ndash1661994
[9] X Chen and K K Tamma ldquoDynamic response of elastic thin-walled structures influenced by coupling effectsrdquo Computersand Structures vol 51 no 1 pp 91ndash105 1994
[10] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996
[11] M Tanaka and A N Bercin ldquoFree vibration solution foruniform beams of nonsymmetrical cross section using Math-ematicardquo Computers and Structures vol 71 no 1 pp 1ndash8 1999
[12] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000
[13] R D Ambrosini J D Riera and R F Danesi ldquoA modifiedVlasov theory for dynamic analysis of thin-walled and variableopen section beamsrdquo Engineering Structures vol 22 no 8 pp890ndash900 2000
[14] L P Kollar ldquoFlexural-torsional vibration of open section com-posite beams with shear deformationrdquo International Journal ofSolids and Structures vol 38 no 42-43 pp 7543ndash7558 2001
[15] Y Matsui and T Hayashikawa ldquoDynamic stiffness analysis fortorsional vibration of continuous beamswith thin-walled cross-sectionrdquo Journal of Sound and Vibration vol 243 no 2 pp 301ndash316 2001
[16] V H Cortınez and M T Piovan ldquoVibration and buckling ofcomposite thin-walled beams with shear deformabilityrdquo Journalof Sound and Vibration vol 258 no 4 pp 701ndash723 2002
[17] A Arpaci S E Bozdag and E Sunbuloglu ldquoTriply coupledvibrations of thin-walled open cross-section beams includingrotary inertia effectsrdquo Journal of Sound and Vibration vol 260no 5 pp 889ndash900 2003
Mathematical Problems in Engineering 7
[18] M Y Kim H T Yun and N I Kim ldquoExact dynamic and staticelement stiffness matrices of nonsymmetric thin-walled beam-columnsrdquo Computers and Structures vol 81 no 14 pp 1425ndash1448 2003
[19] L Jun L Wanyou S Rongying and H Hongxing ldquoCoupledbending and torsional vibration of nonsymmetrical axiallyloaded thin-walled Bernoulli-Euler beamsrdquoMechanics ResearchCommunications vol 31 no 6 pp 697ndash711 2004
[20] L Jun H Hongxing S Rongying and J Xianding ldquoDynamicresponse of axially loaded monosymmetrical thin-walledBernoulli-Euler beamsrdquo Thin-Walled Structures vol 42 no 12pp 1689ndash1707 2004
[21] F Mohri L Azrar and M Potier-Ferry ldquoVibration analysisof buckled thin-walled beams with open sectionsrdquo Journal ofSound and Vibration vol 275 no 1-2 pp 434ndash446 2004
[22] A Prokic ldquoOn triply coupled vibrations of thin-walled beamswith arbitrary cross-sectionrdquo Journal of Sound and Vibrationvol 279 no 3ndash5 pp 723ndash737 2005
[23] M O Kaya and O O Ozgumus ldquoFlexural-torsional-coupledvibration analysis of axially loaded closed-section compositeTimoshenko beam by using DTMrdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 495ndash506 2007
[24] B Rafezy and W P Howson ldquoExact dynamic stiffness matrixfor a thin-walled beam of doubly asymmetric cross-sectionfilled with shear sensitive materialrdquo International Journal forNumerical Methods in Engineering vol 69 no 13 pp 2758ndash2779 2007
[25] H H Chen andKMHsiao ldquoCoupled axial-torsional vibrationof thin-walled Z-section beam induced by boundary condi-tionsrdquoThin-Walled Structures vol 45 no 6 pp 573ndash583 2007
[26] D Ambrosini ldquoOn free vibration of nonsymmetrical thin-walled beamsrdquoThin-Walled Structures vol 47 no 6-7 pp 629ndash636 2009
[27] G M Voros ldquoOn coupled bending-torsional vibrations ofbeams with initial loadsrdquoMechanics Research Communicationsvol 36 no 5 pp 603ndash611 2009
[28] B Rafezy and W P Howson ldquoExact natural frequencies of athree-dimensional shear-torsion beamwith doubly asymmetriccross-section using a two-dimensional approachrdquo Journal ofSound and Vibration vol 295 no 3ndash5 pp 1044ndash1059 2006
[29] F de Borbon and D Ambrosini ldquoOn free vibration analysis ofthin-walled beams axially loadedrdquo Thin-Walled Structures vol48 no 12 pp 915ndash920 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
For the 119910 direction the following shows the matrix form of(5) (9) (12) and (15)
[[[
[
V119894(119911119894)
V1015840119894(119911119894)
119872119910119894
(119911119894)
119881119910119894
(119911119894)
]]]
]
= 119860119910119894
(119911119894)
[[[
[
1198885
1198886
1198887
1198888
]]]
]
(18)
Similarly torsional motion can be written
[[[
[
120579119894(119911119894)
1205791015840
119894(119911119894)
119872119908119894
(119911119894)
119872119905119894(119911119894)
]]]
]
= 119860120579119894(119911119894)
[[[
[
1198889
11988810
11988811
11988812
]]]
]
(19)
At the initial point of the 119894th element (17) (18) and (19) canbe written as follows
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
= 119860119909119894
(0)
[[[
[
1198881
1198882
1198883
1198884
]]]
]
(20)
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
= 119860119910119894
(0)
[[[
[
1198885
1198886
1198887
1198888
]]]
]
(21)
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
= 119860120579119894(0)
[[[
[
1198889
11988810
11988811
11988812
]]]
]
(22)
When vector 119888 is solved from (20) and is substituted in (17)the following is obtained
[[[
[
119906119894(119911119894)
1199061015840
119894(119911119894)
119872119909119894
(119911119894)
119881119909119894
(119911119894)
]]]
]
= 119860119909119894
(119911119894) 119860119909119894(0)minus1
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
(23)
For 119911119894= 119897119894 (23) can be written as
[[[
[
119906119894(119897119894)
1199061015840
119894(119897119894)
119872119909119894
(119897119894)
119881119909119894
(119897119894)
]]]
]
= 119879119909119894
[[[
[
119906119894(0)
1199061015840
119894(0)
119872119909119894
(0)
119881119909119894
(0)
]]]
]
(24)
where 119879119909119894is the element dynamic transfer matrix of the 119894th
elementFor the 119910 direction (23) and (24) can be written as
follows
[[[
[
V119894(119911119894)
V1015840119894(119911119894)
119872119910119894
(119911119894)
119881119910119894
(119911119894)
]]]
]
= 119860119910119894
(119911119894) 119860119910119894(0)minus1
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
(25)
[[[
[
V119894(119897119894)
V1015840119894(119897119894)
119872119910119894
(119897119894)
119881119910119894
(119897119894)
]]]
]
= 119879119910119894
[[[
[
V119894(0)
V1015840119894(0)
119872119910119894
(0)
119881119910119894
(0)
]]]
]
(26)
Similarly rotation motion can be written in equations asfollows
[[[
[
120579119894(119911119894)
1205791015840
119894(119911119894)
119872119908119894
(119911119894)
119872119905119894(119911119894)
]]]
]
= 119860120579119894(119911119894) 119860120579119894(0)minus1
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
(27)
[[[
[
120579119894(119897119894)
1205791015840
119894(119897119894)
119872119908119894
(119897119894)
119872119905119894(119897119894)
]]]
]
= 119879120579119894
[[[
[
120579119894(0)
1205791015840
119894(0)
119872119908119894
(0)
119872119905119894(0)
]]]
]
(28)
If (24) is written successively the displacementsmdashinternalforces relationship between the initial part and end of thebeammdashcan be found as follows
[[[
[
119906end1199061015840
end119872119909end
119881119909end
]]]
]
= 119879119909119899
119879119909(119899minus1)
sdot sdot sdot 11987911990921198791199091
[[[
[
119906initial1199061015840
initial119872119909initial
119881119909initial
]]]
]
= 119905119909
[[[
[
119906initial1199061015840
initial119872119909initial
119881119909initial
]]]
]
(29)
For 119910 and rotation motion (29) can be written as follows
[[[
[
VendV1015840end
119872119910end
119881119910end
]]]
]
= 119879119910119899
119879119910(119899minus1)
sdot sdot sdot 11987911991021198791199101
[[[
[
VinitialV1015840initial
119872119910initial
119881119910initial
]]]
]
= 119905119910
[[[
[
VinitialV1015840initial
119872119910initial
119881119910initial
]]]
]
[[[
[
120579end1205791015840
end119872119908end
119872119905end
]]]
]
= 119879120579119899119879120579(119899minus1)
sdot sdot sdot 11987912057921198791205791
[[[
[
120579initial1205791015840
initial119872119908initial
119872119905initial
]]]
]
= 119905120579
[[[
[
120579initial1205791015840
initial119872119908initial
119872119905initial
]]]
]
(30)
The eigenvalue equation for a thin-walled beam filled withshear sensitive material can be established using (29) (30)and the specific boundary conditions are as follows
(1) Clamped-Free 119891119909
= 119905119909(3 3)lowast
119905119909(4 4) minus 119905
119909(3 4)lowast
119905119909(4
3) = 0 119891119910
= 119905119910(3 3)lowast
119905119910(4 4) minus 119905
119910(3 4)lowast
119905119910(4 3) =
0 119891120579= 119905120579(3 3)lowast
119905120579(4 4) minus 119905
120579(3 4)lowast
119905120579(4 3) = 0
(2) Clamped-Clamped 119891119909
= 119905119909(1 3)lowast
119905119909(2 4) minus 119905
119909(1
4)lowast
119905119909(2 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(2 4) minus 119905
119910(1 4)lowast
119905119910(2
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(2 4) minus 119905
120579(1 4)lowast
119905120579(2 3) = 0
Mathematical Problems in Engineering 5
A B C DC S
xx
c
xs
O
y
1m3m 3m
Figure 2 The doubly asymmetric continuous channel section and the cross section of beam of example 2 with warping allowed at B C andD but fully constrained at A
(3) Simply-Simply 119891119909
= 119905119909(1 2)lowast
119905119909(3 4) minus 119905
119909(3 2)lowast
119905119909(1
4) = 0 119891119910
= 119905119910(1 2)lowast
119905119910(3 4) minus 119905
119910(3 2)lowast
119905119910(1 4) =
0 119891120579= 119905120579(1 2)lowast
119905120579(3 4) minus 119905
120579(3 2)lowast
119905120579(1 4) = 0
(4) Free-Free 119891119909
= 119905119909(3 1)lowast
119905119909(4 2) minus 119905
119909(3 2)lowast
119905119909(4 1) =
0 119891119910= 119905119910(3 1)lowast
119905119910(4 2) minus 119905
119910(3 2)lowast
119905119910(4 1) = 0 119891
120579=
119905120579(3 1)lowast
119905120579(4 2) minus 119905
120579(3 2)lowast
119905120579(4 1) = 0
(5) Clamped-Simply 119891119909
= 119905119909(1 3)lowast
119905119909(3 4) minus 119905
119909(1
4)lowast
119905119909(3 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(3 4) minus 119905
119910(1 4)lowast
119905119910(3
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(3 4) minus 119905
120579(1 4)lowast
119905120579(3 3) = 0
In frequency equations the values of 120596 which set thedeterminant to zero are the uncoupled angular frequencies
23 Coupled Frequencies Ignoring the dynamic couplingeffect of structure caused by the eccentricity between thecenter of shear rigidity and the geometric center the coupledfrequencies of the shear torsional beam can be obtained byusing uncoupled frequencies and the well-known equation asfollows [28]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120596(119894)2
119895minus 120596(119894)2
1199090 minus119910
119888120596(119894)2
119895
0 120596(119894)2
119895minus 120596(119894)2
119910119909119888120596(119894)2
119895
minus119910119888120596(119894)2
119895119909119888120596(119894)2
1198951199032
119898(120596(119894)2
119895minus 120596(119894)2
120579)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(119895 = 1 2 3) (119894 = 1 2 3 )
(31)
3 Procedure of Computation
A program that considers the method presented in this studyas a basis has been prepared in MATLAB and the operationstages are presented below
(1) element dynamic Transfer matrices are calculated foreach element by using (24) (26) and (28)
(2) System dynamic transfer matrices (see (29)ndash(30)) areobtained with the help of element transfer matrices
(3) The angular frequencies of uncoupled vibrations areobtained by using the boundary conditions
(4) The coupled angular frequencies are found by using(31)
4 A Numerical Example
In this part of the study two numerical examples were solvedby a program written in MATLAB to validate the presentedmethod The results are compared with those given in theliterature
41 Numerical Example 1 The first example considers thebeam studied by Tanaka and Bercin [11] A typical uniformthin-walled beam has a length of 15m with a doublyasymmetric cross section The properties of the cross sectionare as follows
119909119888= 002316 119910
119888= 002625 120588 = 1947 kgm 119903
2
119898=
30303lowast
10minus3m2
119864119868119909
= 73480Nm2 119864119868119910
= 16680Nm2 119864119868119908
=
2364Nm4 and1198661198690= 1081Nm2
The first three coupled natural frequencies of the beam arecalculated by the presented method and compared with theresults by Tanaka and Bercin [11] and Rafezy and Howson[24] in Table 1 for clamped-free (C-F) and simply-simply (S-S) boundary conditions
42 Numerical Example 2 A typical continuous beam witha doubly asymmetric cross section is considered in thisexample (Figure 2)
The beam comprises a thin-walled outer layer and a shearcore with the following properties between support points Aand B The typical uniform thin-walled beam has a length of15m with a doubly asymmetric cross section The propertiesof the cross section are as follows
119909119904
= 008 119910119904
= 003 119909119888
= 005 119910119888
= 002 120588 =
20 kgm 1199032
119898= 0008m2
119864119868119909
= 216lowast
106Nm2 119864119868
119910= 173
lowast
106Nm2 119866
119905119869119905=
3200Nm2119864119868119908
= 14lowast
103Nm4 119866119860
119909= 600000N 119866119860
119910=
600000N and 119866119869119888= 3800Nm2
The shear core is omitted between points B and D wherethe cross-sectional properties remain unchanged except that119866119860119909
= 119866119860119910
= 119866119869119888= 0 and the small change in 120588 has been
ignored
6 Mathematical Problems in Engineering
Table 1 Coupled natural frequencies for the beam of example 1
Natural frequencies (Hz)
BC Proposed method Tanaka and Bercin [11] Rafezy and Howson [24]1198911
1198912
1198913
1198911
1198912
1198913
1198911
1198912
1198913
C-F 1717 2731 5910 1703 2758 5925 1717 2731 5910S-S 4471 7514 16487 4148 7412 16411 4471 7514 16487
Table 2 Coupled natural frequencies of the continuous beam ofexample 2
Frequencynumber This study Rafezy and Howson[24] Difference ()
1 6906 6940 minus0492 19763 19796 minus0173 35461 33836 480
The first three coupled natural frequencies of the beamare calculated by the presented method and compared withthe results of Rafezy and Howson [24] in Table 2
The main source of error between the proposed methodand Rafezy and Howson methods is the eccentricity betweenthe center of shear stiffness and flexural stiffness which wasnot taken into account in the proposed method
5 Conclusions
This paper presents a method for a free vibration analysisof a thin-walled beam of doubly asymmetric cross sectionfilled with shear sensitive material In the study first of alla dynamic transfer matrix method was obtained for planarshear flexure and torsional motionThen uncoupled angularfrequencies were obtained by using dynamic element transfermatrices and boundary conditions Coupled frequencieswereobtained by the well-known two-dimensional approachesIt was observed from the sample taken from the literaturethat the presented method gave sufficient results The errormargin of the proposed method is shown to be less than 5Themain source of error is the eccentricity between the centerof shear stiffness and flexural stiffness which was not takeninto account in the proposed method
The transfer matrix method is an efficient and computer-ized method which also provides a fast and practical solutionsince the dimension of thematrix for the elements and systemnever changes Because of this the proposedmethod is simpleand accurate enough to be used both at the concept designstage and for final analyses
References
[1] F Y Cheng ldquoVibrations of timoshenko beams and frameworksrdquoJournal of Structural Engineering vol 96 no 3 pp 551ndash571 1970
[2] C Mei ldquoCoupled vibrations of thin-walled beams of opensection using the finite element methodrdquo International Journalof Mechanical Sciences vol 12 no 10 pp 883ndash891 1970
[3] W LHallauer andR Y L Liu ldquoBeambending-torsion dynamicstiffness method for calculation of exact vibration modesrdquo
Journal of Sound and Vibration vol 85 no 1 pp 105ndash113 1982[4] E Dokumaci ldquoAn exact solution for coupled bending and tor-
sion vibrations of uniform beams having single cross-sectionalsymmetryrdquo Journal of Sound and Vibration vol 119 no 3 pp443ndash449 1987
[5] R H Gutierrez and P A A Laura ldquoApproximate analysis ofcoupled flexural-torsional vibrations of a beam of non-uniformcross-section using the optimized rayleigh methodrdquo Journal ofSound and Vibration vol 114 no 2 pp 393ndash397 1987
[6] J R Banerjee ldquoCoupled bending-torsional dynamic stiffnessmatrix for beam elementsrdquo International Journal for NumericalMethods in Engineering vol 28 no 6 pp 1283ndash1298 1989
[7] J R Banerjee and F W Williams ldquoCoupled bending-torsionaldynamic stiffness matrix for timoshenko beam elementsrdquoCom-puters and Structures vol 42 no 3 pp 301ndash310 1992
[8] J R Banerjee and F W Williams ldquoAn exact dynamic stiffnessmatrix for coupled extensional-torsional vibration of structuralmembersrdquo Computers and Structures vol 50 no 2 pp 161ndash1661994
[9] X Chen and K K Tamma ldquoDynamic response of elastic thin-walled structures influenced by coupling effectsrdquo Computersand Structures vol 51 no 1 pp 91ndash105 1994
[10] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996
[11] M Tanaka and A N Bercin ldquoFree vibration solution foruniform beams of nonsymmetrical cross section using Math-ematicardquo Computers and Structures vol 71 no 1 pp 1ndash8 1999
[12] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000
[13] R D Ambrosini J D Riera and R F Danesi ldquoA modifiedVlasov theory for dynamic analysis of thin-walled and variableopen section beamsrdquo Engineering Structures vol 22 no 8 pp890ndash900 2000
[14] L P Kollar ldquoFlexural-torsional vibration of open section com-posite beams with shear deformationrdquo International Journal ofSolids and Structures vol 38 no 42-43 pp 7543ndash7558 2001
[15] Y Matsui and T Hayashikawa ldquoDynamic stiffness analysis fortorsional vibration of continuous beamswith thin-walled cross-sectionrdquo Journal of Sound and Vibration vol 243 no 2 pp 301ndash316 2001
[16] V H Cortınez and M T Piovan ldquoVibration and buckling ofcomposite thin-walled beams with shear deformabilityrdquo Journalof Sound and Vibration vol 258 no 4 pp 701ndash723 2002
[17] A Arpaci S E Bozdag and E Sunbuloglu ldquoTriply coupledvibrations of thin-walled open cross-section beams includingrotary inertia effectsrdquo Journal of Sound and Vibration vol 260no 5 pp 889ndash900 2003
Mathematical Problems in Engineering 7
[18] M Y Kim H T Yun and N I Kim ldquoExact dynamic and staticelement stiffness matrices of nonsymmetric thin-walled beam-columnsrdquo Computers and Structures vol 81 no 14 pp 1425ndash1448 2003
[19] L Jun L Wanyou S Rongying and H Hongxing ldquoCoupledbending and torsional vibration of nonsymmetrical axiallyloaded thin-walled Bernoulli-Euler beamsrdquoMechanics ResearchCommunications vol 31 no 6 pp 697ndash711 2004
[20] L Jun H Hongxing S Rongying and J Xianding ldquoDynamicresponse of axially loaded monosymmetrical thin-walledBernoulli-Euler beamsrdquo Thin-Walled Structures vol 42 no 12pp 1689ndash1707 2004
[21] F Mohri L Azrar and M Potier-Ferry ldquoVibration analysisof buckled thin-walled beams with open sectionsrdquo Journal ofSound and Vibration vol 275 no 1-2 pp 434ndash446 2004
[22] A Prokic ldquoOn triply coupled vibrations of thin-walled beamswith arbitrary cross-sectionrdquo Journal of Sound and Vibrationvol 279 no 3ndash5 pp 723ndash737 2005
[23] M O Kaya and O O Ozgumus ldquoFlexural-torsional-coupledvibration analysis of axially loaded closed-section compositeTimoshenko beam by using DTMrdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 495ndash506 2007
[24] B Rafezy and W P Howson ldquoExact dynamic stiffness matrixfor a thin-walled beam of doubly asymmetric cross-sectionfilled with shear sensitive materialrdquo International Journal forNumerical Methods in Engineering vol 69 no 13 pp 2758ndash2779 2007
[25] H H Chen andKMHsiao ldquoCoupled axial-torsional vibrationof thin-walled Z-section beam induced by boundary condi-tionsrdquoThin-Walled Structures vol 45 no 6 pp 573ndash583 2007
[26] D Ambrosini ldquoOn free vibration of nonsymmetrical thin-walled beamsrdquoThin-Walled Structures vol 47 no 6-7 pp 629ndash636 2009
[27] G M Voros ldquoOn coupled bending-torsional vibrations ofbeams with initial loadsrdquoMechanics Research Communicationsvol 36 no 5 pp 603ndash611 2009
[28] B Rafezy and W P Howson ldquoExact natural frequencies of athree-dimensional shear-torsion beamwith doubly asymmetriccross-section using a two-dimensional approachrdquo Journal ofSound and Vibration vol 295 no 3ndash5 pp 1044ndash1059 2006
[29] F de Borbon and D Ambrosini ldquoOn free vibration analysis ofthin-walled beams axially loadedrdquo Thin-Walled Structures vol48 no 12 pp 915ndash920 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
A B C DC S
xx
c
xs
O
y
1m3m 3m
Figure 2 The doubly asymmetric continuous channel section and the cross section of beam of example 2 with warping allowed at B C andD but fully constrained at A
(3) Simply-Simply 119891119909
= 119905119909(1 2)lowast
119905119909(3 4) minus 119905
119909(3 2)lowast
119905119909(1
4) = 0 119891119910
= 119905119910(1 2)lowast
119905119910(3 4) minus 119905
119910(3 2)lowast
119905119910(1 4) =
0 119891120579= 119905120579(1 2)lowast
119905120579(3 4) minus 119905
120579(3 2)lowast
119905120579(1 4) = 0
(4) Free-Free 119891119909
= 119905119909(3 1)lowast
119905119909(4 2) minus 119905
119909(3 2)lowast
119905119909(4 1) =
0 119891119910= 119905119910(3 1)lowast
119905119910(4 2) minus 119905
119910(3 2)lowast
119905119910(4 1) = 0 119891
120579=
119905120579(3 1)lowast
119905120579(4 2) minus 119905
120579(3 2)lowast
119905120579(4 1) = 0
(5) Clamped-Simply 119891119909
= 119905119909(1 3)lowast
119905119909(3 4) minus 119905
119909(1
4)lowast
119905119909(3 3) = 0 119891
119910= 119905119910(1 3)lowast
119905119910(3 4) minus 119905
119910(1 4)lowast
119905119910(3
3) = 0 119891120579= 119905120579(1 3)lowast
119905120579(3 4) minus 119905
120579(1 4)lowast
119905120579(3 3) = 0
In frequency equations the values of 120596 which set thedeterminant to zero are the uncoupled angular frequencies
23 Coupled Frequencies Ignoring the dynamic couplingeffect of structure caused by the eccentricity between thecenter of shear rigidity and the geometric center the coupledfrequencies of the shear torsional beam can be obtained byusing uncoupled frequencies and the well-known equation asfollows [28]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120596(119894)2
119895minus 120596(119894)2
1199090 minus119910
119888120596(119894)2
119895
0 120596(119894)2
119895minus 120596(119894)2
119910119909119888120596(119894)2
119895
minus119910119888120596(119894)2
119895119909119888120596(119894)2
1198951199032
119898(120596(119894)2
119895minus 120596(119894)2
120579)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(119895 = 1 2 3) (119894 = 1 2 3 )
(31)
3 Procedure of Computation
A program that considers the method presented in this studyas a basis has been prepared in MATLAB and the operationstages are presented below
(1) element dynamic Transfer matrices are calculated foreach element by using (24) (26) and (28)
(2) System dynamic transfer matrices (see (29)ndash(30)) areobtained with the help of element transfer matrices
(3) The angular frequencies of uncoupled vibrations areobtained by using the boundary conditions
(4) The coupled angular frequencies are found by using(31)
4 A Numerical Example
In this part of the study two numerical examples were solvedby a program written in MATLAB to validate the presentedmethod The results are compared with those given in theliterature
41 Numerical Example 1 The first example considers thebeam studied by Tanaka and Bercin [11] A typical uniformthin-walled beam has a length of 15m with a doublyasymmetric cross section The properties of the cross sectionare as follows
119909119888= 002316 119910
119888= 002625 120588 = 1947 kgm 119903
2
119898=
30303lowast
10minus3m2
119864119868119909
= 73480Nm2 119864119868119910
= 16680Nm2 119864119868119908
=
2364Nm4 and1198661198690= 1081Nm2
The first three coupled natural frequencies of the beam arecalculated by the presented method and compared with theresults by Tanaka and Bercin [11] and Rafezy and Howson[24] in Table 1 for clamped-free (C-F) and simply-simply (S-S) boundary conditions
42 Numerical Example 2 A typical continuous beam witha doubly asymmetric cross section is considered in thisexample (Figure 2)
The beam comprises a thin-walled outer layer and a shearcore with the following properties between support points Aand B The typical uniform thin-walled beam has a length of15m with a doubly asymmetric cross section The propertiesof the cross section are as follows
119909119904
= 008 119910119904
= 003 119909119888
= 005 119910119888
= 002 120588 =
20 kgm 1199032
119898= 0008m2
119864119868119909
= 216lowast
106Nm2 119864119868
119910= 173
lowast
106Nm2 119866
119905119869119905=
3200Nm2119864119868119908
= 14lowast
103Nm4 119866119860
119909= 600000N 119866119860
119910=
600000N and 119866119869119888= 3800Nm2
The shear core is omitted between points B and D wherethe cross-sectional properties remain unchanged except that119866119860119909
= 119866119860119910
= 119866119869119888= 0 and the small change in 120588 has been
ignored
6 Mathematical Problems in Engineering
Table 1 Coupled natural frequencies for the beam of example 1
Natural frequencies (Hz)
BC Proposed method Tanaka and Bercin [11] Rafezy and Howson [24]1198911
1198912
1198913
1198911
1198912
1198913
1198911
1198912
1198913
C-F 1717 2731 5910 1703 2758 5925 1717 2731 5910S-S 4471 7514 16487 4148 7412 16411 4471 7514 16487
Table 2 Coupled natural frequencies of the continuous beam ofexample 2
Frequencynumber This study Rafezy and Howson[24] Difference ()
1 6906 6940 minus0492 19763 19796 minus0173 35461 33836 480
The first three coupled natural frequencies of the beamare calculated by the presented method and compared withthe results of Rafezy and Howson [24] in Table 2
The main source of error between the proposed methodand Rafezy and Howson methods is the eccentricity betweenthe center of shear stiffness and flexural stiffness which wasnot taken into account in the proposed method
5 Conclusions
This paper presents a method for a free vibration analysisof a thin-walled beam of doubly asymmetric cross sectionfilled with shear sensitive material In the study first of alla dynamic transfer matrix method was obtained for planarshear flexure and torsional motionThen uncoupled angularfrequencies were obtained by using dynamic element transfermatrices and boundary conditions Coupled frequencieswereobtained by the well-known two-dimensional approachesIt was observed from the sample taken from the literaturethat the presented method gave sufficient results The errormargin of the proposed method is shown to be less than 5Themain source of error is the eccentricity between the centerof shear stiffness and flexural stiffness which was not takeninto account in the proposed method
The transfer matrix method is an efficient and computer-ized method which also provides a fast and practical solutionsince the dimension of thematrix for the elements and systemnever changes Because of this the proposedmethod is simpleand accurate enough to be used both at the concept designstage and for final analyses
References
[1] F Y Cheng ldquoVibrations of timoshenko beams and frameworksrdquoJournal of Structural Engineering vol 96 no 3 pp 551ndash571 1970
[2] C Mei ldquoCoupled vibrations of thin-walled beams of opensection using the finite element methodrdquo International Journalof Mechanical Sciences vol 12 no 10 pp 883ndash891 1970
[3] W LHallauer andR Y L Liu ldquoBeambending-torsion dynamicstiffness method for calculation of exact vibration modesrdquo
Journal of Sound and Vibration vol 85 no 1 pp 105ndash113 1982[4] E Dokumaci ldquoAn exact solution for coupled bending and tor-
sion vibrations of uniform beams having single cross-sectionalsymmetryrdquo Journal of Sound and Vibration vol 119 no 3 pp443ndash449 1987
[5] R H Gutierrez and P A A Laura ldquoApproximate analysis ofcoupled flexural-torsional vibrations of a beam of non-uniformcross-section using the optimized rayleigh methodrdquo Journal ofSound and Vibration vol 114 no 2 pp 393ndash397 1987
[6] J R Banerjee ldquoCoupled bending-torsional dynamic stiffnessmatrix for beam elementsrdquo International Journal for NumericalMethods in Engineering vol 28 no 6 pp 1283ndash1298 1989
[7] J R Banerjee and F W Williams ldquoCoupled bending-torsionaldynamic stiffness matrix for timoshenko beam elementsrdquoCom-puters and Structures vol 42 no 3 pp 301ndash310 1992
[8] J R Banerjee and F W Williams ldquoAn exact dynamic stiffnessmatrix for coupled extensional-torsional vibration of structuralmembersrdquo Computers and Structures vol 50 no 2 pp 161ndash1661994
[9] X Chen and K K Tamma ldquoDynamic response of elastic thin-walled structures influenced by coupling effectsrdquo Computersand Structures vol 51 no 1 pp 91ndash105 1994
[10] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996
[11] M Tanaka and A N Bercin ldquoFree vibration solution foruniform beams of nonsymmetrical cross section using Math-ematicardquo Computers and Structures vol 71 no 1 pp 1ndash8 1999
[12] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000
[13] R D Ambrosini J D Riera and R F Danesi ldquoA modifiedVlasov theory for dynamic analysis of thin-walled and variableopen section beamsrdquo Engineering Structures vol 22 no 8 pp890ndash900 2000
[14] L P Kollar ldquoFlexural-torsional vibration of open section com-posite beams with shear deformationrdquo International Journal ofSolids and Structures vol 38 no 42-43 pp 7543ndash7558 2001
[15] Y Matsui and T Hayashikawa ldquoDynamic stiffness analysis fortorsional vibration of continuous beamswith thin-walled cross-sectionrdquo Journal of Sound and Vibration vol 243 no 2 pp 301ndash316 2001
[16] V H Cortınez and M T Piovan ldquoVibration and buckling ofcomposite thin-walled beams with shear deformabilityrdquo Journalof Sound and Vibration vol 258 no 4 pp 701ndash723 2002
[17] A Arpaci S E Bozdag and E Sunbuloglu ldquoTriply coupledvibrations of thin-walled open cross-section beams includingrotary inertia effectsrdquo Journal of Sound and Vibration vol 260no 5 pp 889ndash900 2003
Mathematical Problems in Engineering 7
[18] M Y Kim H T Yun and N I Kim ldquoExact dynamic and staticelement stiffness matrices of nonsymmetric thin-walled beam-columnsrdquo Computers and Structures vol 81 no 14 pp 1425ndash1448 2003
[19] L Jun L Wanyou S Rongying and H Hongxing ldquoCoupledbending and torsional vibration of nonsymmetrical axiallyloaded thin-walled Bernoulli-Euler beamsrdquoMechanics ResearchCommunications vol 31 no 6 pp 697ndash711 2004
[20] L Jun H Hongxing S Rongying and J Xianding ldquoDynamicresponse of axially loaded monosymmetrical thin-walledBernoulli-Euler beamsrdquo Thin-Walled Structures vol 42 no 12pp 1689ndash1707 2004
[21] F Mohri L Azrar and M Potier-Ferry ldquoVibration analysisof buckled thin-walled beams with open sectionsrdquo Journal ofSound and Vibration vol 275 no 1-2 pp 434ndash446 2004
[22] A Prokic ldquoOn triply coupled vibrations of thin-walled beamswith arbitrary cross-sectionrdquo Journal of Sound and Vibrationvol 279 no 3ndash5 pp 723ndash737 2005
[23] M O Kaya and O O Ozgumus ldquoFlexural-torsional-coupledvibration analysis of axially loaded closed-section compositeTimoshenko beam by using DTMrdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 495ndash506 2007
[24] B Rafezy and W P Howson ldquoExact dynamic stiffness matrixfor a thin-walled beam of doubly asymmetric cross-sectionfilled with shear sensitive materialrdquo International Journal forNumerical Methods in Engineering vol 69 no 13 pp 2758ndash2779 2007
[25] H H Chen andKMHsiao ldquoCoupled axial-torsional vibrationof thin-walled Z-section beam induced by boundary condi-tionsrdquoThin-Walled Structures vol 45 no 6 pp 573ndash583 2007
[26] D Ambrosini ldquoOn free vibration of nonsymmetrical thin-walled beamsrdquoThin-Walled Structures vol 47 no 6-7 pp 629ndash636 2009
[27] G M Voros ldquoOn coupled bending-torsional vibrations ofbeams with initial loadsrdquoMechanics Research Communicationsvol 36 no 5 pp 603ndash611 2009
[28] B Rafezy and W P Howson ldquoExact natural frequencies of athree-dimensional shear-torsion beamwith doubly asymmetriccross-section using a two-dimensional approachrdquo Journal ofSound and Vibration vol 295 no 3ndash5 pp 1044ndash1059 2006
[29] F de Borbon and D Ambrosini ldquoOn free vibration analysis ofthin-walled beams axially loadedrdquo Thin-Walled Structures vol48 no 12 pp 915ndash920 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Table 1 Coupled natural frequencies for the beam of example 1
Natural frequencies (Hz)
BC Proposed method Tanaka and Bercin [11] Rafezy and Howson [24]1198911
1198912
1198913
1198911
1198912
1198913
1198911
1198912
1198913
C-F 1717 2731 5910 1703 2758 5925 1717 2731 5910S-S 4471 7514 16487 4148 7412 16411 4471 7514 16487
Table 2 Coupled natural frequencies of the continuous beam ofexample 2
Frequencynumber This study Rafezy and Howson[24] Difference ()
1 6906 6940 minus0492 19763 19796 minus0173 35461 33836 480
The first three coupled natural frequencies of the beamare calculated by the presented method and compared withthe results of Rafezy and Howson [24] in Table 2
The main source of error between the proposed methodand Rafezy and Howson methods is the eccentricity betweenthe center of shear stiffness and flexural stiffness which wasnot taken into account in the proposed method
5 Conclusions
This paper presents a method for a free vibration analysisof a thin-walled beam of doubly asymmetric cross sectionfilled with shear sensitive material In the study first of alla dynamic transfer matrix method was obtained for planarshear flexure and torsional motionThen uncoupled angularfrequencies were obtained by using dynamic element transfermatrices and boundary conditions Coupled frequencieswereobtained by the well-known two-dimensional approachesIt was observed from the sample taken from the literaturethat the presented method gave sufficient results The errormargin of the proposed method is shown to be less than 5Themain source of error is the eccentricity between the centerof shear stiffness and flexural stiffness which was not takeninto account in the proposed method
The transfer matrix method is an efficient and computer-ized method which also provides a fast and practical solutionsince the dimension of thematrix for the elements and systemnever changes Because of this the proposedmethod is simpleand accurate enough to be used both at the concept designstage and for final analyses
References
[1] F Y Cheng ldquoVibrations of timoshenko beams and frameworksrdquoJournal of Structural Engineering vol 96 no 3 pp 551ndash571 1970
[2] C Mei ldquoCoupled vibrations of thin-walled beams of opensection using the finite element methodrdquo International Journalof Mechanical Sciences vol 12 no 10 pp 883ndash891 1970
[3] W LHallauer andR Y L Liu ldquoBeambending-torsion dynamicstiffness method for calculation of exact vibration modesrdquo
Journal of Sound and Vibration vol 85 no 1 pp 105ndash113 1982[4] E Dokumaci ldquoAn exact solution for coupled bending and tor-
sion vibrations of uniform beams having single cross-sectionalsymmetryrdquo Journal of Sound and Vibration vol 119 no 3 pp443ndash449 1987
[5] R H Gutierrez and P A A Laura ldquoApproximate analysis ofcoupled flexural-torsional vibrations of a beam of non-uniformcross-section using the optimized rayleigh methodrdquo Journal ofSound and Vibration vol 114 no 2 pp 393ndash397 1987
[6] J R Banerjee ldquoCoupled bending-torsional dynamic stiffnessmatrix for beam elementsrdquo International Journal for NumericalMethods in Engineering vol 28 no 6 pp 1283ndash1298 1989
[7] J R Banerjee and F W Williams ldquoCoupled bending-torsionaldynamic stiffness matrix for timoshenko beam elementsrdquoCom-puters and Structures vol 42 no 3 pp 301ndash310 1992
[8] J R Banerjee and F W Williams ldquoAn exact dynamic stiffnessmatrix for coupled extensional-torsional vibration of structuralmembersrdquo Computers and Structures vol 50 no 2 pp 161ndash1661994
[9] X Chen and K K Tamma ldquoDynamic response of elastic thin-walled structures influenced by coupling effectsrdquo Computersand Structures vol 51 no 1 pp 91ndash105 1994
[10] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996
[11] M Tanaka and A N Bercin ldquoFree vibration solution foruniform beams of nonsymmetrical cross section using Math-ematicardquo Computers and Structures vol 71 no 1 pp 1ndash8 1999
[12] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000
[13] R D Ambrosini J D Riera and R F Danesi ldquoA modifiedVlasov theory for dynamic analysis of thin-walled and variableopen section beamsrdquo Engineering Structures vol 22 no 8 pp890ndash900 2000
[14] L P Kollar ldquoFlexural-torsional vibration of open section com-posite beams with shear deformationrdquo International Journal ofSolids and Structures vol 38 no 42-43 pp 7543ndash7558 2001
[15] Y Matsui and T Hayashikawa ldquoDynamic stiffness analysis fortorsional vibration of continuous beamswith thin-walled cross-sectionrdquo Journal of Sound and Vibration vol 243 no 2 pp 301ndash316 2001
[16] V H Cortınez and M T Piovan ldquoVibration and buckling ofcomposite thin-walled beams with shear deformabilityrdquo Journalof Sound and Vibration vol 258 no 4 pp 701ndash723 2002
[17] A Arpaci S E Bozdag and E Sunbuloglu ldquoTriply coupledvibrations of thin-walled open cross-section beams includingrotary inertia effectsrdquo Journal of Sound and Vibration vol 260no 5 pp 889ndash900 2003
Mathematical Problems in Engineering 7
[18] M Y Kim H T Yun and N I Kim ldquoExact dynamic and staticelement stiffness matrices of nonsymmetric thin-walled beam-columnsrdquo Computers and Structures vol 81 no 14 pp 1425ndash1448 2003
[19] L Jun L Wanyou S Rongying and H Hongxing ldquoCoupledbending and torsional vibration of nonsymmetrical axiallyloaded thin-walled Bernoulli-Euler beamsrdquoMechanics ResearchCommunications vol 31 no 6 pp 697ndash711 2004
[20] L Jun H Hongxing S Rongying and J Xianding ldquoDynamicresponse of axially loaded monosymmetrical thin-walledBernoulli-Euler beamsrdquo Thin-Walled Structures vol 42 no 12pp 1689ndash1707 2004
[21] F Mohri L Azrar and M Potier-Ferry ldquoVibration analysisof buckled thin-walled beams with open sectionsrdquo Journal ofSound and Vibration vol 275 no 1-2 pp 434ndash446 2004
[22] A Prokic ldquoOn triply coupled vibrations of thin-walled beamswith arbitrary cross-sectionrdquo Journal of Sound and Vibrationvol 279 no 3ndash5 pp 723ndash737 2005
[23] M O Kaya and O O Ozgumus ldquoFlexural-torsional-coupledvibration analysis of axially loaded closed-section compositeTimoshenko beam by using DTMrdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 495ndash506 2007
[24] B Rafezy and W P Howson ldquoExact dynamic stiffness matrixfor a thin-walled beam of doubly asymmetric cross-sectionfilled with shear sensitive materialrdquo International Journal forNumerical Methods in Engineering vol 69 no 13 pp 2758ndash2779 2007
[25] H H Chen andKMHsiao ldquoCoupled axial-torsional vibrationof thin-walled Z-section beam induced by boundary condi-tionsrdquoThin-Walled Structures vol 45 no 6 pp 573ndash583 2007
[26] D Ambrosini ldquoOn free vibration of nonsymmetrical thin-walled beamsrdquoThin-Walled Structures vol 47 no 6-7 pp 629ndash636 2009
[27] G M Voros ldquoOn coupled bending-torsional vibrations ofbeams with initial loadsrdquoMechanics Research Communicationsvol 36 no 5 pp 603ndash611 2009
[28] B Rafezy and W P Howson ldquoExact natural frequencies of athree-dimensional shear-torsion beamwith doubly asymmetriccross-section using a two-dimensional approachrdquo Journal ofSound and Vibration vol 295 no 3ndash5 pp 1044ndash1059 2006
[29] F de Borbon and D Ambrosini ldquoOn free vibration analysis ofthin-walled beams axially loadedrdquo Thin-Walled Structures vol48 no 12 pp 915ndash920 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
[18] M Y Kim H T Yun and N I Kim ldquoExact dynamic and staticelement stiffness matrices of nonsymmetric thin-walled beam-columnsrdquo Computers and Structures vol 81 no 14 pp 1425ndash1448 2003
[19] L Jun L Wanyou S Rongying and H Hongxing ldquoCoupledbending and torsional vibration of nonsymmetrical axiallyloaded thin-walled Bernoulli-Euler beamsrdquoMechanics ResearchCommunications vol 31 no 6 pp 697ndash711 2004
[20] L Jun H Hongxing S Rongying and J Xianding ldquoDynamicresponse of axially loaded monosymmetrical thin-walledBernoulli-Euler beamsrdquo Thin-Walled Structures vol 42 no 12pp 1689ndash1707 2004
[21] F Mohri L Azrar and M Potier-Ferry ldquoVibration analysisof buckled thin-walled beams with open sectionsrdquo Journal ofSound and Vibration vol 275 no 1-2 pp 434ndash446 2004
[22] A Prokic ldquoOn triply coupled vibrations of thin-walled beamswith arbitrary cross-sectionrdquo Journal of Sound and Vibrationvol 279 no 3ndash5 pp 723ndash737 2005
[23] M O Kaya and O O Ozgumus ldquoFlexural-torsional-coupledvibration analysis of axially loaded closed-section compositeTimoshenko beam by using DTMrdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 495ndash506 2007
[24] B Rafezy and W P Howson ldquoExact dynamic stiffness matrixfor a thin-walled beam of doubly asymmetric cross-sectionfilled with shear sensitive materialrdquo International Journal forNumerical Methods in Engineering vol 69 no 13 pp 2758ndash2779 2007
[25] H H Chen andKMHsiao ldquoCoupled axial-torsional vibrationof thin-walled Z-section beam induced by boundary condi-tionsrdquoThin-Walled Structures vol 45 no 6 pp 573ndash583 2007
[26] D Ambrosini ldquoOn free vibration of nonsymmetrical thin-walled beamsrdquoThin-Walled Structures vol 47 no 6-7 pp 629ndash636 2009
[27] G M Voros ldquoOn coupled bending-torsional vibrations ofbeams with initial loadsrdquoMechanics Research Communicationsvol 36 no 5 pp 603ndash611 2009
[28] B Rafezy and W P Howson ldquoExact natural frequencies of athree-dimensional shear-torsion beamwith doubly asymmetriccross-section using a two-dimensional approachrdquo Journal ofSound and Vibration vol 295 no 3ndash5 pp 1044ndash1059 2006
[29] F de Borbon and D Ambrosini ldquoOn free vibration analysis ofthin-walled beams axially loadedrdquo Thin-Walled Structures vol48 no 12 pp 915ndash920 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Related Documents
