research article on the flexural-torsional vibration...

Research ArticleOn the Flexural-Torsional Vibration and Stability of BeamsSubjected to Axial Load and End Moment

M Tahmaseb Towliat Kashani Supun Jayasinghe and Seyed M Hashemi

Department of Aerospace Engineering Ryerson University Toronto ON Canada M5B 2K3

Correspondence should be addressed to Seyed M Hashemi smhashemryersonca

Received 21 January 2014 Revised 19 June 2014 Accepted 19 June 2014 Published 13 October 2014

Academic Editor Reza Jazar

Copyright copy 2014 M Tahmaseb Towliat Kashani et al 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

The free vibration of beams subjected to a constant axial load and end moment and various boundary conditions is examinedBased on the Euler-Bernoulli bending and St Venant torsion beam theories the differential equations governing coupledflexural-torsional vibrations and stability of a uniform slender isotropic homogeneous and linearly elastic beam undergoinglinear harmonic vibration are first reviewedThe existing formulations are then briefly discussed and a conventional finite elementmethod (FEM) is developed Exploiting the MATLAB-based code the resulting linear Eigenvalue problem is then solved todetermine the Eigensolutions (ie natural frequencies and modes) of illustrative examples exhibiting geometric bending-torsioncoupling Various classical boundary conditions are considered and the FEM frequency results are validated against those obtainedfrom a commercial software (ANSYS) and the data available in the literature Tensile axial force is found to increase naturalfrequencies indicating beam stiffening However when a force and an endmoment are acting in combination themoment reducesthe stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axialforce compared to the moment A buckling analysis of the beam is also carried out to determine the critical buckling end momentand axial compressive force

1 Introduction

Beams are important and versatile structural elements asmany civil mechanical and aerospace structures are com-monly modeled as preloaded beams or beam assemblies[1] In the past many researchers have investigated thevibration and stability of various beam structures subjectedto diverse loading and boundary conditions As a result thegoverning differential equations of motion for various beamconfigurations have been already developed Therefore frompast literatures it is known that the tensile and compressiveaxial loads affect the flexural stiffness of beams and increaseor decrease the uncoupled bending frequencies and criticalbuckling loads of such elements Apart from the above vari-ous coupled vibrational behaviors of beam structures causedby different geometric [2ndash10] and material couplings [11ndash20] have also been thoroughly studied and their associatedfindings have been well established The coupled vibrationsand stability of axially loaded or centrifugally stiffened beam

elements have also been investigated [21ndash31] However theliterature on the effects of combined axial load and endmoment and the couplings resulting from the endmoment onthe stability and vibration characteristics of beams are scarce[32ndash38]

Hashemi and Richard [2] used the frequency dependentdynamic finite element (DFE) method a hybrid methoddeveloped in [3] that combines the accuracy of analyticalmethods to the versatility of numerical methods to conducta vibration analysis of beams that are geometrically coupledin bending and torsion An exact method has been usedto determine the flexural-torsional vibration characteristicsof a uniform beam with single cross-sectional symmetry byDokumaci [4] The classical finite element method (FEM)was used by Mei [5] to study the coupled vibration of thinwalled beams with an open section This study included theeffects of warping stiffness The flexural-torsional vibrationof a uniform beam was studied by Tanaka and Bercin [6] bydetermining the exact solution of the governing differential

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 153532 11 pageshttpdxdoiorg1011552014153532

2 Shock and Vibration

equations An analytical dynamic stiffness matrix (DSM)solutionwas formulated by Banerjee [7] for coupled bending-torsion beam elements Banerjee et al [8] also developed anexact DSM to investigate the vibration of an Euler-Bernoullithin walled beam and exploited the Wittrick and Williams[9] root finding algorithm to extract the Eigensolutions Onceagain Banerjee and Su [10] used the DSM to conduct a freetransverse and lateral vibration analysis of a beam coupledwith torsion

Hashemi and Roach [11] also formulated a DFE solu-tion for the free vibration of an extension-torsion cou-pled composite beam A quasi-exact DFE formulation forthe free vibration analysis of a three layered sandwichbeam consisting of a thick soft low strength and densitycore and two-face layers made of high strength materialwas developed by Hashemi and Adique [12] Bornemanand Hashemi [13] developed a DFE for the free vibrationanalysis of bending-torsion coupled laminated compositewing beams Bannerjee and his coworkers have used thefrequency-dependent DSMmethod for the vibration analysisof isotropic [14] sandwich [15ndash17] and composite [18] beamsBorneman et al [19] also used theDSMmethod to investigatethe vibrational characteristics of a doubly coupled (materialand geometric) defective composite beam Furthermore aDSM formulation was presented by Hallauer and Liu [20]to determine the vibrational characteristics and generalizedmasses of an aircraft wingmodelled as a series of three simplebeams

Hashemi and his coauthors [21] developed a DFE formu-lation to analyse the free vibration of centrifugally stiffened(rotating) beams Furthermore Hashemi and Richard [22]formed aDFE solution for the free vibration analysis of axiallyloaded bending-torsion coupled beams An axially loadedisotropic Timoshenko beam coupled in bending and torsionwas studied by Banerjee and Williams [23] Leung [24]developed an exact DSM of a thin walled beam Once againan analytical solution was formulated by Banerjee and Fisher[25] to model a uniform axially loaded cantilevered beamwith flexural-torsional coupling as a result of noncoincidentshear and mass centers The effects of warping have beenneglected in this study Jun et al [26] examined the coupledflexural-torsional vibration of an axially loaded thin walledbeam with monosymmetrical cross sections by includingeffects of warping The effect of axial load has also beenpreviously studied by Murthy and Neogy [27] for clampedand pinned boundary conditions as well as by Gellert andGluck [28] for cantilevered boundary conditions Bokaian[29] determined the natural frequencies of a uniform singlespan beam subjected to a constant tensile axial load for vari-ous boundary conditions The same author also investigatedthe vibrational characteristics of a uniform single span beamfor ten different end conditions when a constant compressiveaxial load is applied [30] Shaker [31] conducted a modalanalysis to determine the effect of axial load on the modeshapes and natural frequencies of beams

It has been established byChen andAtsuta [39] that trans-verse bending and torsion is coupled by static end momentsand that flexural-torsional buckling is comprised of thistransverse flexure and axial torsion Analytical investigations

on the influence of axial loads and end moments on thevibration of beams have been previously reported by Joshiand Suryanarayan for a simply supported case [32] and otherboundary conditions [33] Joshi and Suryanarayan [34] alsostudied analytically the flexural-torsional instability of thinwalled beams subjected to axial loads and end momentsThe same authors examined the coupled bending-torsionvibration of a deep rectangular beam that is initially stressedas a result of the application of moments that vary along thespan [35] Pavlovic and Kozic [36] developed a closed formanalytical solution to investigate the effects of end momentson a simply supported thin walled beam FurthermorePavlovic et al [37] also formulated the analytical solutions tostudy a simply supported thin walled beam subjected to thecombined action of an axial force and end moment

The reliability and accuracy of such modal analysisresults depend on the method implemented A variety ofapproaches for the analysis of these structures have beenproposed and implemented in the past including analyticalsemianalytical and numerical methods Amongst these theclassical FEM method where beam element matrices areevaluated from assumed fixed (namely polynomial) shapefunctions has beenwidely used by investigatorsThis practiceresults in approximate equations in the form of mass andstatic stiffness matrices A deviation from the conventionalFEM formulation would pay dividends if improved accuracyof results can be obtained by using shape functions otherthan polynomials This is the case when the homogeneoussolution of the pertinent differential equation is availablefor the development of each of the element matrices Forstatic analysis the use of the homogeneous solution of thedifferential equation yields the exact stiffness matrix and loadvector for a beam element [40] The use of other alternativeshape functions in dynamic FEM formulations has also beenexplored

The dynamic stiffness matrix (DSM) method offers abetter alternative particularly when higher frequencies andbetter accuracies of results are required It relies on asingle frequency-dependent matrix which has both massand stiffness properties The use of a DSM in vibrationanalysis is well established [14ndash18] Obviously the methodgives more accurate results because it exploits the exactmember theory The matrix is obtained by directly solvingthe governing differential equation Other methods have alsobeen explored and reported which are more or less similarto those explained previously Some of them were developedto treat a particular configuration and group of mechanicalsystems

A thorough investigation of the existing conventionalFEM and alternative DSM methods in beam vibrations hasled to the development of the DFE approach [3] DFE bridgesthe gap between the standard FEM and the exact DSMmethods by advantageously exploiting the generality of FEMand the very precise frequency calculations provided byDSMapproach From this point of view the method retains thephysical aspect of analytical or semianalytical approaches andthe power of a numerical method It has been shown thatDFE is an efficient tool for handling periodical structures orsystems composed of several identical substructures [3]

Shock and Vibration 3

All methods stated in the above-mentioned referenceshave their inherent advantages and shortcomings Thereexists a class of problems for which an exact solution canbe obtained Nevertheless in most cases finding a generalexact solution for the normal modes and frequencies ofthe system would be cumbersome if not impossible andwould involve complicated mathematical procedures Thusrecourse would bemade to one of approximate solutions suchas the Rayleigh-Ritz method [41 42] or Galerkinrsquos method[40]

The conventional finite element method (FEM) whichuses the Galerkin method of weighted residuals [40] iswidely accepted and used for structural analysis The FEMis adaptable to many complex systems including those withmaterial and geometric variations for example nonuniformgeometry However it seems that a FEM-based thoroughinvestigation of the geometrically coupled flexural-torsionalvibration of beams simultaneously subjected to both axialforce and end moment and the coupling effects caused byend moments have not been reported in the open litera-ture Therefore in what follows a classical FEM solutionof the above problem is presented and the static bucklingstability and the coupled free vibrations of preloaded thin(Euler-Bernoulli) beams subjected to various classical endconditions (with ends free to warp) are investigated Theaxial load and end moment are varied and their effects onthe beam stiffness and natural frequencies are examinedIt is worth noting that in addition to the conventionalbeam stiffness and mass matrices [38 40] the presentedformulation is characterized by a geometric stiffness matrixwhich includes uncoupled and coupled components Theuncoupled component in turn is formed by two parts oneassociatedwith the flexure of axially loaded beams [38 43 44]and another relative to the torsion However to the best ofauthorsrsquo knowledge the coupling geometric stiffness matrixresulting from the end moment has not been reported in theliterature The presented FEM formulation is applicable tothe members composed of closed sections (eg rectangularor square hollow sections) where the torsional rigidity (119866119869)

is very large compared with the warping rigidity (119864Γ) withends free to warp that is state of uniform torsion wherethe twist rate is constant along the span The presented FEMformulation however can also be extended to thin walledbeams with open cross sections where torsion-related warp-ing effects cannot be neglected

2 Theory

Consider a linearly elastic homogeneous isotropic beamsubjected to an end moment 119872 and an axial load 119875undergoing linear vibrations Euler-Bernoulli bending andSt Venant torsion beam theories are used to derive thegoverning differential equations of motion and a classicalfinite element solution is developed The end moments actabout the 119911-axis (lagwise) however bending in the 119909-119910plane (lagwise) is not considered and bending occurs in the119909-119911 plane (flapwise) Thus the end moments acting in thelagwise direction introduce torsion to the system and create

P P

M M

xx = Lx = 0

z

t

h

Figure 1 Beam with axial load and end moment applied at 119909 = 0

and 119909 = 119871

flexural-torsional coupling Figure 1 illustrates the geometryof the studied system where 119871 ℎ and 119905 stand for the beamrsquoslength width and height respectively

The two governing differential equations of the beam areas follows

1198641198681199081015840101584010158401015840

+ 11987511990810158401015840+ 119872120579

10158401015840minus 120588119860 = 0 (1)

11986611986912057910158401015840+

119875119868119875

11986012057910158401015840+ 119872119908

10158401015840minus 120588119868119875

120579 = 0 (2)

where 119908 stands for the transverse flexural displacement and120579 represents the torsional displacement The derivatives withrespect to the length of the beam and time are denoted by aprime (1015840) and a dot (sdot) respectively In (1) and (2) the appliedmoment and force are shown as 119872 and 119875 respectivelyThe cross-sectional area of the beam is denoted by 119860 Themass density is represented by 120588 and 119868

119875stands for the polar

moment of inertia of the beam The beamrsquos torsional rigidity(119866119869) is assumed to be very large compared with its warpingrigidity (119864Γ) and ends are free to warp that is state ofuniform torsion As can be observed from (1) and (2) thesystem is coupled by the end moments119872

In order to eliminate the time dependency in (1) and (2)simple harmonic vibration is considered and the followingtransformations are used to describe the transverse andtorsional displacements

119908 (119909 119905) = sin (120596119905) (3)

120579 (119909 119905) = 120579 sin (120596119905) (4)

where 120596 is the circular frequency and 119905 is the time and120579 are the transverse and torsional displacement amplitudesrespectivelyUpon substituting (3) and (4) (1) and (2) become

1198641198681015840101584010158401015840

+ 11987510158401015840+ 11987211991111991112057910158401015840+ 120588119860120596

2 = 0 (5)

11986611986912057910158401015840+ 11987511986811987512057910158401015840+ 11987211991111991110158401015840+ 1205881198681198751198601205962120579 = 0 (6)


The Galerkin method of weighted residuals [40] is employedto develop the integral form of the above equations where theldquohatrdquo signs have been dropped for the sake of simplicity

119882119891= int119871

0

120575119882(1198641198681198821015840101584010158401015840

+ 11987511988210158401015840+ 11987211991111991112057910158401015840+ 120588119860120596

2119882)119889119909 = 0

(7)

119882119905= int119871

0

120575120579 (11986611986912057910158401015840+

119875119868119875

11986012057910158401015840+ 11987211991111991111988210158401015840+ 1205881198681198751205962120579) 119889119909 = 0

(8)

where 120575119882 and 120575120579 (ie weighting functions) represent thetransverse and torsional virtual displacements respectivelyPerforming integration by parts on (7) and (8) leads to theweak integral form of the governing equations written as

119882119891= int119871

0

(1198641198681198821015840101584012057511988210158401015840minus 119875119882

10158401205751198821015840+ 119872120579

10158401205751198821015840

+1205881198601205962119882120575119882)119889119909 +

[(119864119868119882

101584010158401015840+ 119875119882

1015840+ 119872120579

1015840) 120575119882]

119871

0

minus[(119864119868119882

10158401015840)1205751198821015840]119871

0= 0

(9)

119882119905= int119871

0

(11986611986912057910158401205751205791015840+

119875119868119875

11986012057910158401205751205791015840+ 119872119882

10158401205751205791015840minus 1205881198681198751205962120579120575120579) 119889119909

minus[(119866119869120579

1015840+

119875119868119875

1198601205791015840+ 119872119882

1015840)120575120579]119871

0

= 0

(10)

Expressions (9) and (10) also satisfy the principle of virtualwork

119882 = 119882INT minus 119882EXT = 0 (11)

where

119882EXT = 0 (12)

and thus

119882INT = 119882119891+ 119882119905 (13)

The total virtual work internal virtual work and externalvirtual work are denoted by119882119882INT and119882EXT respectivelyThe resulting shear force 119878(119909) bending moment 119872(119909) andtorsional torque 119879(119909) defined as

119872(119909) = minus11986411986811988210158401015840 (14)

119878 (119909) = 119864119868119882101584010158401015840

+ 1198721205791015840+ 119875119882

1015840 (15)

119879 (119909) = 1198661198691205791015840+

119875119868119875

1198601205791015840+ 119872119882

1015840 (16)

are zero at the free end and the displacements are set to zeroat the fixed boundaries As a result the bracketed boundaryterms in expressions (9) and (10) vanish for all boundaryconditions The system is then discretized using elements

1 2

120579j+1

120579j

WjW998400j

NE + 1NEWj+1W998400j+1

Figure 2 System discretized using elements with 3 degrees offreedom per node

with 2 nodes and three DOF per node as shown in Figure 2such that

119882 = 119882INT =

No of Elementssum119896=1

119882119896

=


119882119896

119891+ 119882119896

119905

(17)

Nodal DOFs are lateral displacement 119908 rotation (ie slope)1199081015840 and torsional displacement 120579 The classical finite ele-ment formulation is developed using cubic Hermite typepolynomial approximations for bending displacement (18)and linear approximations for torsional displacements (19)introduced in the weak integral form of the governingequations such that for a two-node three-degree-of-freedomper node element

119908 (119909) = ⟨1 119909 1199092 1199093⟩119862

1 (18)

120579 (119909) = ⟨1 119909⟩1198622 (19)

In (18) and (19) above 1198621and 119862

2are columns vectors of

unknown constant coefficientsThe vectors of nodal displace-ment for bending and torsion are shown below

119882119899 =

1198821

11988210158401

1198822

11988210158402

=[[[

[

1 0 0 0

0 1 0 0

1 119897 1198972 1198973

0 1 2119897 31198972

]]]

]

1198621 = [119875

119899119908]1198621 (20)

120579119899 =

1205791

1205792

= [1 0

1 119897] 1198622 = [119875

119899119905] 1198622 (21)

Thus

119882(119909) = ⟨1 119909 1199092 1199093⟩ [119875119899119908

]minus1

119882119899 = ⟨119873 (119909)⟩ 119882

119899

(22)

120579 (119909) = ⟨1 119909⟩ [119875119899119905]minus1

120579119899 = ⟨119871 (119909)⟩120579

119899 (23)

where ⟨119873(119909)⟩ and ⟨119871(119909)⟩ are both row vectors comprisingcubic and linear shape functions for bending and torsionrespectively The cubic shape functions 119873

1 1198732 1198733 and 119873

4

are

1198731(119909) =

21199093

1198973minus

31199092

1198972+ 1 119873

2(119909) =

1199093

1198972minus

21199092

119897+ 119909

1198733(119909) =

minus21199093

1198973+

31199092

1198972 119873

4(119909) =

1199093

1198972minus

1199092

119897

(24)

The linear shape functions 1198711and 119871

2are defined as

1198711(119909) = 1 minus

119909

119897 119871

2(119909) =

119909

119897 (25)

This discretizing process leads to the element stiffness andmass and coupling matrices which when assembled together


within the FEM code written inMATLABwould result in thelinear Eigenvalue problem shown in

⟨120575119882119899⟩ (119870 minus 120596

2119872) 119882

119899 = 0

det (119870 minus 1205962119872) = 0

(26)

where 119870 stands for the global stiffness matrix which is acollection of all the element stiffness matrices The globalmass matrix is symbolized by 119872

Matrix (27a) shown below is the element mass matrix[119898]119896 and matrices (27b) through (27f) are the uncoupled

coupled and geometric element stiffness matrices Whenmatrices (27b) through (27f) are assembled together the finalelement stiffness matrix would resultThis is shown asmatrix(27g) Consider

[119898]119896=

[[[[[[[[[[[[[[[[[[

[

156119898119897

420

221198981198972

4200

54119898119897

420

minus131198981198972

4200

41198981198973

4200

131198981198972

420

minus31198981198973

4200

120588119868119875119897

30 0

120588119868119875119897

6156119898119897

420

minus221198981198972

4200

Sym41198981198973

4200

120588119868119875119897

3

]]]]]]]]]]]]]]]]]]

]

(27a)

where 119897 stands for the element length and 119898 representsthe element mass per unit length The element uncoupledbending stiffness matrix [119896

119861] is shown below

[119896119861] =

[[[[[[[[[[

[

12119864119868

11989736119864119868

1198972minus12119864119868

11989736119864119868

1198972

4119864119868

119897

minus6119864119868

11989722119864119868

119897

Sym12119864119868

1198973minus6119864119868

1198972

4119864119868

119897

]]]]]]]]]]

]

(27b)

The final element stiffness matrix is modified due to thepresence of the end moment and axial load which con-tributes the [119896]geometric matrix [119896]torsion matrix bending-torsion coupling stiffness matrix [119896

119861119879]119888 and the torsion-

bending coupling stiffness matrix [119896119879119861

]119888 These are added to

the bending stiffness matrix [119896119861] above to form the final

element stiffness matrix [119896]119896 The geometric and torsion

stiffness matrices contributed by the axial load 119875 are shownbelow

[119896]geometric = 119875

[[[[[[[[[[[[

[

6

5119897

1

10

minus6

5119897

1

10

2119897

15

minus1

10

minus119897

30

Sym6

5119897

minus1

10

2119897

15

]]]]]]]]]]]]

]

(27c)

[119896]torsion = (119866119869

119897+

119875119868119875

119860119897) [

1 minus1

minus1 1] (27d)

The bending-torsion and torsion-bending coupling stiffnessmatrices introduced by the end moment119872 are as follows

[119896119861119879

]119888=

119872

119897[

1 0 minus1 0

minus1 0 1 0] (27e)

[119896119879119861

]119888=

119872

119897

[[[

[

1 minus1

0 0

minus1 1

0 0

]]]

]

(27f)

Therefore the final element stiffness matrix which is acollection of the five submatrices takes the following form

[119896]119896=

[[[[[[[[[[[[[[[[[[

[

12119864119868

1198973minus

6119875

5119897

6119864119868

1198972minus

119875

10

119872

119897

minus12119864119868

1198973+

6119875

5119897

6119864119868

1198972minus

119875

10

minus119872

1198974119864119868

119897minus

2119875119897

150

minus6119864119868

1198972+

119875

10

2119864119868

119897+

119875119897

300

119866119869

119897+

119875119868119875

119860119897

minus119872

1198970

minus119866119869

119897minus

119875119868119875

119860119897

Sym12119864119868

1198973minus

6119875

5119897

minus6119864119868

1198972+

119875

10

119872

119897

4119864119868

119897minus

2119875119897

150

119866119869

119897+

119875119868119875

119860119897

]]]]]]]]]]]]]]]]]]

]

(27g)


Table 1 FEM results versus exact data for the first three natural frequencies at 119875 = 0 and119872 = 0

Boundary conditionNatural frequencies (Hz) at P = 0 andM = 0

Mode 1 Mode 2 Mode 3Exact [38] FEM (40 elements) Exact [38] FEM (40 elements) Exact [38] FEM (40 elements)

C-F 2556 2556 15995 15995 44858 44858C-C 16266 16266 44858 44858 87970 87970P-P 7175 7175 28718 28718 64633 64633P-C 11209 11209 36351 36351 75611 75611

The solution to the linear Eigenvalue problem in (26) isachieved by determining the Eigenvalues and Eigenvectorsusing a FEM code developed in MATLAB Various classicalboundary conditions are also applied within the MATLABcode Thus the natural frequencies and mode shapes of thebeam are evaluated

It is worth noting that if the beamrsquos warping rigidity (119864Γ)

is large compared with its torsional rigidity (119866119869) then torsionequation (2) changes to a 4th-order differential equation sim-ilar in form to the bending equation (1)While the above FEMformulation is developed for the state of uniform torsion inthat case following the above-presented procedure and usingcubic interpolation functions similar to (24) instead of linearones (25) to express torsional displacements the presentedformulation can be extended to also include the warpingeffects The development of such formulation however isbeyond the scope of this paper

3 Numerical Results

In this section the validity and practical applicability ofthe presented FEM procedure are demonstrated throughconsideration of various examples A generic beam made ofstructural steel subjected to different combinations of axialload end moment and end conditions is first investigatedIn a second case study the comparison is made betweenthe FEM frequency results and limited experimental dataavailable in the open literature

At first let us consider a beam made of structural steel(119864 = 200GPa and 120588 = 7800 kgm3) with a length of 8mwidth of 04m and depth of 02m The first stage of thenumerical tests was to validate the developed FEM codeDue to the lack of analytical results for the preloaded casesthe accuracy of the natural frequency values from the codewas established by comparing with the analytical data foran unloaded beam Table 1 includes the results for the firstthree natural frequencies for various boundary conditions(with ends free to warp) using the exact method [38] anda 40-element FEM mesh As it can be observed the resultsproduced by the exact and the FEMmethods are identical andas such the FEM code generates accurate results

The FEM mesh size was chosen based on a convergencetest as is presented in Figure 3 For this a cantilever beamsubjected to 185MN of tensile force and 921MNsdotm of endmoment was used With a focus on higher frequenciesthe fifth mode in this case the FEM convergence was

0002004006008

01012014016018

0 50 100 150 200 250

Erro

r (

)

Number of elements

Figure 3 FEM convergence analysis the fifth natural frequency ofpreloaded cantilevered beam

investigated by increasing the numbers of elements from10 to 1000 as there were no analytical results available forsimilar cases The fifth natural frequency was observed toremain at 88243Hz when 200 or more elements were usedThus for the given configuration the 88243Hz value is takenas the exact reference result for the 5th natural frequencyHowever since it is apparent that a 40-element mesh resultsin an error less than 001 percent such a mesh size wasconsidered as reasonable for further studies Furthermore itis important to note that convergence of the results for thefirst four fundamental frequencies was also checked and itwas observed that these results converged to the analyticalresult with an even smaller number of elements

The accuracy of the results produced by the presentedFEM method was also verified using a prestressed modalanalysis conducted using ANSYS-14 commercial softwarewhere SOLID-187 elements were used The SOLID-187 ele-ment is a higher order 3D 10-node element capable of 6degrees of freedom (3 translations and 3 rotations) per node

Tables 2 3 4 and 5 present the FEM fundamentalfrequencies for the beam subjected to different combina-tions of tensile force and end moment and various bound-ary conditions cantilevered (C-F) clamped-clamped (C-C)pinned-pinned (P-P) and pinned-clamped (P-C) respec-tively Table 2 also includes a comparison of the results fromthe developed FEM solution with those from the ANSYSsoftware which shows a maximum error of 13 percent at119875 = 0 and 119872 = 921MN sdot mThese errors may be attributedto the fact that the ANSYS software incorporates the effects


Table 2 Fundamental frequencies for cantilever beam subjected to force and end moment

C-FEnd moment

0 (MNsdotm) 614 (MNsdotm) 921 (MNsdotm)Fundamental frequency (Hz)

Force (MN) ANSYS FEM code 40 elements ANSYS FEM code 40 elements ANSYS FEM code 40 elements0 2555 2556 2241 2234 1749 1727062 2883 2884 2620 2614 2233 2216123 3168 3169 2939 2934 2614 2600185 3421 3422 3217 3213 2934 2922

Table 3 Fundamental frequencies for clamped-clamped boundarycondition (C-C) when force and end moment are applied a 40-element FEMmodel is used

C-C End moment (MNsdotm)0 614 921

Force (MN) FEM fundamental frequency (Hz)0 16266 16141 15984062 16413 16290 16134123 16559 16437 16283185 16703 16582 16430

Table 4 Fundamental frequencies for pinned-pinned boundarycondition (P-P) when force and end moment are applied a 40-element FEMmodel is used

P-P End moment (MNsdotm)0 614 921


Table 5 Fundamental frequencies for pinned-clamped boundarycondition (P-C) when force and end moment are applied a 40-element FEMmodel is used

P-C End moment (MNsdotm)0 614 921


of shear deformation and rotary inertia into the calculationsThe data tabulated in Tables 2 through 5 are also graphicallypresented in Figures 4 5 6 and 7 The critical bucklingend moments and compressive forces were also determinedfor the cantilevered boundary condition and the results areshown in Tables 6 and 7 respectively These tabulated dataare also graphically presented in Figure 8 Finally Figures 9and 10 depict the bending and torsional components of the

Table 6 Critical buckling end moment versus compressive forcecantilevered boundary condition

Force (MN) Buckling moment (MNsdotm)minus185 3900minus123 7750minus062 10600 1228062 1376123 1557185 1695

0

05

1

15

2

25

3

35

4

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)

M = 0M = 614MNmiddotmM = 921MNmiddotm

Figure 4 Variation of the fundamental frequency with tensile forceand end moment cantilevered (C-F) boundary condition

first five natural modes for the preloaded (119875 = 185MN and119872 = 921MN sdotm) cantilevered beam

Consider now the axially loaded industrial aluminumbeam (119864 = 70GPa and 120588 = 2700 kgm3) with a length of1495mm and rectangular cross section of 50mm in widthand 10mmin depth as reported by Laux (2012) [45] Thefirst three natural frequencies of the beam subjected todifferent axial loads are calculated using the presented FEMformulation and are presented in Table 8 along with ANSYSresults and experimental data reported in [45] Different


159

16

161

162

163

164

165

166

167

168

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)


Figure 5 Variation of the fundamental frequency with tensile forceand end moment clamped-clamped (C-C) boundary condition

64

66

68

7

72

74

76

78

8

82

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)


Figure 6 Variation of the fundamental frequency with tensile forceand end moment pinned-pinned (P-P) boundary condition

mesh sizes are used and as can be seen from the table for thefirst and second natural frequencies a 5-element coursemeshand a 40-element fine mesh both lead to almost identicalresults For the 3rd natural frequency however there is aslight difference of 05 between the results obtained fromthe two meshes The FEM and ANSYS results are found tobe in excellent agreement with the experimental data witha maximum difference of less than 1 For the case in handhowever there is no experimental buckling data available forcomparison (refer to Table 9)

106

108

11

112

114

116

118

12

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)


Figure 7 Variation of the fundamental frequency with tensile forceand end moment clamped-pinned (C-P) boundary condition

Table 7 Critical buckling compressive force versus end momentcantilevered boundary condition

Moment (MNsdotm) Buckling force (MN)0 minus2057307 minus1900614 minus1750921 minus0900

4 Discussion and Conclusion

The bending-torsion vibration and buckling of beams sub-jected to axial load and endmoment were revisited Neglect-ing the shear deformation rotary inertia andwarping effectsthe differential equations of motion coupled by the endmoment were discussed Exploiting the cubic and linearinterpolation functions for bending and torsional displace-ments respectively together with theGalerkin-typeweightedresidual method a finite element is developed Frequencyand stability analyses of two illustrative examples are carriedout and the FEM results are compared with those obtainedfrom ANSYS software and the data available in the literatureThe presented FEM results showed excellent agreement withthose obtained from ANSYS and experimental data Asexpected tensile axial load increases the natural frequenciesof the beam indicating an increase in the stiffness of thebeam for all classical boundary conditions When only theend moment is applied the natural frequencies reduce forall boundary conditions indicating a reduction in stiffnessof the beam If the end moment is held constant and thetensile load is increased the natural frequencies increaseindicating an increase in the beam stiffness Conversely if thetensile load is held constant and the endmoment is increasedthe beam stiffness reduces A compressive axial load has


0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1

End moment (MNmiddotm)

02468

1012141618

0 1 2 3Axial force (MN)

minus3 minus2 minus1

Buck

ling

mom

ent (

MNmiddotm

)

Figure 8 Variation of critical buckling compressive force with end moment

Table 8 Comparison between the experimental and FEM natural frequencies versus axial load for an aluminum fixed-fixed beam

Axial load (N) 1st nat freq (Hz) 2nd nat freq (Hz) 3rd nat freq (Hz)Exp [45] FEM (540) ANSYS (540) Exp [45] FEM (540) ANSYS (540) Exp [45] FEM (540) ANSYS (540)

1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t

Distance from base of the beam

1st2nd3rd

4th5th

minus15

minus1

minus05

minus2

Figure 9 Bending component of the first five natural modespreloaded cantilevered beam

the opposite effect and the critical buckling moment reduceswith a progressive increase in the compressive load appliedThe coupled vibration of the beam however is found to bepredominantly flexural in the first few natural frequencies(the first three for the case studied here) and torsion becomespredominant in a higher natural frequency

Finally it is worth noting that the presented FEM isdesigned to account for the axial load end moment orcombined effects automatically In contrast carrying outa similar analysis using the FEM-based software available

0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2

Figure 10 Torsional components of the first five natural modespreloaded cantilevered beam

(ANSYS eg) needs special attention as the preloads shouldbe introduced in the model through a prestressed analysisFurthermore contrary to the analytical formulations thepresented FEM formulation can also be readily extendedto the dynamic and stability analyses of (non-)uniformpreloaded layered beams and beam structures to includethe warping effects neglected in the present study and toanalyze beam-like structural components exhibiting othertypes of coupled behaviour for example geometric coupling


Table 9 Critical buckling force versus end moment fixed-fixedaluminum beam

Moment (Nsdotm) Buckling Force (N)FEM (540) ANSYS (40)

0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35

noncoincident elastic and inertial axes or material couplingsin composite beams

Disclosure

This paper presents the results of a recent research conductedby the graduate students (M Tahmaseb Towliat Kashani andSupun Jayasinghe) under the supervision of the third author(Seyed M Hashemi) M Tahmaseb Towliat Kashani andSupun Jayasinghe are coauthors The authors also declarethat the present paper in its present form has not beenpublished nor has it been submitted for publication in anyother journals

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was supported in part by NSERCDiscovery GrantPartial support was also provided by Ryerson University

References

[1] S Timoshenko Vibration Problems in Engineering Van Nos-trand Reinhold New York NY USA 1964

[2] 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

[3] S M Hashemi Free vibrational analysis of rotating beam-likestructures a dynamic finite element approach [PhD thesis]Department ofMechanical Engineering Laval University Que-bec Canada 1998

[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] 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

[6] M Tanaka and A N Bercin ldquoFree vibration solution for uni-form beams of nonsymmetrical cross section using Mathemat-icardquo Computers and Structures vol 71 no 1 pp 1ndash8 1999

[7] J R Banerjee ldquoCoupled bending-torsional dynamic stiffnessmatrix for beam elementsrdquo International Journal for NumericalMethods in Engineering vol 28 no 6 pp 1283ndash1298 1989

[8] 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

[9] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[10] J R Banerjee andH Su ldquoFree transverse and lateral vibration ofbeams with torsional couplingrdquo Journal of Aerospace Engineer-ing vol 19 no 1 pp 13ndash20 2006

[11] S M Hashemi and A Roach ldquoA dynamic finite element for thefree vibration analysis of extension-torsion coupled compositebeamsrdquoMathematics in Engineering Science and Aerospace vol1 no 3 pp 221ndash239 2010

[12] S M Hashemi and E J Adique ldquoA quasi-exact dynamic finiteelement for free vibration analysis of sandwich beamsrdquo AppliedComposite Materials vol 17 no 2 pp 259ndash269 2010

[13] S Borneman and S M Hashemi ldquoA dynamic finite element forthe vibration analysis of tapered composite beamsrdquo in Proceed-ing of the 5th Canadian-International Composite ConferenceVancouver Canada 2005

[14] J R Banerjee and F W Williams ldquoFree vibration of compositebeamsmdashan exact method using symbolic computationrdquo Journalof Aircraft vol 32 no 3 pp 636ndash642 1995

[15] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers amp Structures vol 81 no18-19 pp 1915ndash1922 2003

[16] J R Banerjee and A J Sobey ldquoDynamic stiffness formulationand free vibration analysis of a three-layered sandwich beamrdquoInternational Journal of Solids and Structures vol 42 no 8 pp2181ndash2197 2005

[17] J R Banerjee C W Cheung R Morishima M Perera andJ Njuguna ldquoFree vibration of a three-layered sandwich beamusing the dynamic stiffness method and experimentrdquo Interna-tional Journal of Solids and Structures vol 44 no 22-23 pp7543ndash7563 2007

[18] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[19] S R Borneman SMHashemiand andHAlighanbari ldquoVibra-tion analysis of doubly coupled cracked composite beamsan exact dynamic stiffness matrixrdquo International Review ofAerospace Engineering vol 1 no 3 pp 298ndash309 2008

[20] W LHallauer andR Y L Liu ldquoBeambending-torsion dynamicstiffness method for calculation of exact vibration modesrdquoJournal of Sound and Vibration vol 85 no 1 pp 105ndash113 1982

[21] S M Hashemi M J Richard and G Dhatt ldquoA new DynamicFinite Element (DFE) formulation for lateral free vibrationsof Euler-Bernoulli spinning beams using trigonometric shapefunctionsrdquo Journal of Sound and Vibration vol 220 no 4 pp601ndash624 1999

[22] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers and Structures vol 77 no 6 pp 711ndash7242000


[23] J R Banerjee and F W Williams ldquoCoupled bending-torsionaldynamic stiffness matrix of an axially loaded timoshenko beamelementrdquo International Journal of Solids and Structures vol 31no 6 pp 749ndash762 1994

[24] A Y T Leung ldquoNatural shape functions of a compressedVlasovelementrdquoThin-Walled Structures vol 11 no 5 pp 431ndash438 1991

[25] J R Banerjee and S A Fisher ldquoCoupled bending-torsionaldynamic stiffness matrix for axially loaded beam elementsrdquoInternational Journal for Numerical Methods in Engineering vol33 no 4 pp 739ndash751 1992

[26] L Jun S Rongying H Hongxing and J Xianding ldquoCoupledbending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effectsrdquo Applied Acoustics vol65 no 2 pp 153ndash170 2004

[27] M Murthy and J Neogy ldquoDetermination of fundamentalnatural frequencies of axially loaded columns and framesrdquoJournal of the Institution of Engineers (India) Civil EngineeringDivision vol 49 pp 203ndash212 1969

[28] M Gellert and J Gluck ldquoThe influence of axial load oneigen-frequencies of a vibrating lateral restraint cantileverrdquoInternational Journal of Mechanical Sciences vol 14 no 11 pp723ndash728 1972

[29] A Bokaian ldquoNatural frequencies of beams under compressiveaxial loadsrdquo Journal of Sound and Vibration vol 126 no 1 pp49ndash65 1988

[30] A Bokaian ldquoNatural frequencies of beams under tensile axialloadsrdquo Journal of Sound and Vibration vol 142 no 3 pp 481ndash498 1990

[31] F J Shaker ldquoEffect of axial load onmode shapes and frequenciesof beamsrdquo NASA Technical Note TN D-8109 1975

[32] A Joshi and S Suryanarayan ldquoCoupled flexural-torsionalvibration of beams in the presence of static axial loads and endmomentsrdquo Journal of Sound and Vibration vol 92 no 4 pp583ndash589 1984

[33] A Joshi and S Suryanarayan ldquoUnified analytical solution forvarious boundary conditions for the coupled flexural-torsionalvibration of beams subjected to axial loads and end momentsrdquoJournal of Sound and Vibration vol 129 no 2 pp 313ndash326 1989

[34] A Joshi and S Suryanarayan ldquoA unified solution for variousboundary conditions for the coupled flexural-torsional instabil-ity of closed thin-walled beam-columnsrdquo International Journalof Solids and Structures vol 20 no 2 pp 167ndash178 1984

[35] A Joshi and S Suryanarayan ldquoIterative method for coupledflexural-torsional vibration of initially stressed beamsrdquo Journalof Sound and Vibration vol 146 no 1 pp 81ndash92 1991

[36] R Pavlovic and P Kozic ldquoAlmost sure stability of the thin-walled beam subjected to end momentsrdquo Theoretical andApplied Mechanics vol 30 no 3 pp 193ndash207 2003

[37] R Pavlovic P Kozic P Rajkovic and I Pavlovic ldquoDynamicstability of a thin-walled beam subjected to axial loads and endmomentsrdquo Journal of Sound and Vibration vol 301 no 3ndash5 pp690ndash700 2007

[38] J S Przemieniecki Theory of Matrix Structural AnalysisMcGraw-Hill New York NY USA 1968

[39] W F Chen and T Atsuta Theory of Beam Columns SpaceBehaviour and Design Volume McGraw-Hill New York NYUSA 1977

[40] K J Bathe Finite Element Procedures Prentice Hall New YorkNY USA 1996

[41] LMeirovitchAnalyticalMethods inVibrationsTheMacMillan1967

[42] L Meirovitch Computational Methods in Structural DynamicsSijthoff amp Noordhoff 1980

[43] F Y ChenMatrix Analysis of Structural Dynamics Applicationsand Earthquake Engineering Marcel Decker New York NYUSA 2001

[44] J S Wu Analytical and Numerical Methods for VibrationAnalysis John Wiley amp Sons Singapore 2013

[45] S Laux Estimation of axial load in timber beams using resonancefrequency analysis [MS thesis] Chalmers University of Tech-nology Gothenburg Sweden 2012

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation



DistributedSensor Networks



equations An analytical dynamic stiffness matrix (DSM)solutionwas formulated by Banerjee [7] for coupled bending-torsion beam elements Banerjee et al [8] also developed anexact DSM to investigate the vibration of an Euler-Bernoullithin walled beam and exploited the Wittrick and Williams[9] root finding algorithm to extract the Eigensolutions Onceagain Banerjee and Su [10] used the DSM to conduct a freetransverse and lateral vibration analysis of a beam coupledwith torsion

Hashemi and Roach [11] also formulated a DFE solu-tion for the free vibration of an extension-torsion cou-pled composite beam A quasi-exact DFE formulation forthe free vibration analysis of a three layered sandwichbeam consisting of a thick soft low strength and densitycore and two-face layers made of high strength materialwas developed by Hashemi and Adique [12] Bornemanand Hashemi [13] developed a DFE for the free vibrationanalysis of bending-torsion coupled laminated compositewing beams Bannerjee and his coworkers have used thefrequency-dependent DSMmethod for the vibration analysisof isotropic [14] sandwich [15ndash17] and composite [18] beamsBorneman et al [19] also used theDSMmethod to investigatethe vibrational characteristics of a doubly coupled (materialand geometric) defective composite beam Furthermore aDSM formulation was presented by Hallauer and Liu [20]to determine the vibrational characteristics and generalizedmasses of an aircraft wingmodelled as a series of three simplebeams

Hashemi and his coauthors [21] developed a DFE formu-lation to analyse the free vibration of centrifugally stiffened(rotating) beams Furthermore Hashemi and Richard [22]formed aDFE solution for the free vibration analysis of axiallyloaded bending-torsion coupled beams An axially loadedisotropic Timoshenko beam coupled in bending and torsionwas studied by Banerjee and Williams [23] Leung [24]developed an exact DSM of a thin walled beam Once againan analytical solution was formulated by Banerjee and Fisher[25] to model a uniform axially loaded cantilevered beamwith flexural-torsional coupling as a result of noncoincidentshear and mass centers The effects of warping have beenneglected in this study Jun et al [26] examined the coupledflexural-torsional vibration of an axially loaded thin walledbeam with monosymmetrical cross sections by includingeffects of warping The effect of axial load has also beenpreviously studied by Murthy and Neogy [27] for clampedand pinned boundary conditions as well as by Gellert andGluck [28] for cantilevered boundary conditions Bokaian[29] determined the natural frequencies of a uniform singlespan beam subjected to a constant tensile axial load for vari-ous boundary conditions The same author also investigatedthe vibrational characteristics of a uniform single span beamfor ten different end conditions when a constant compressiveaxial load is applied [30] Shaker [31] conducted a modalanalysis to determine the effect of axial load on the modeshapes and natural frequencies of beams

It has been established byChen andAtsuta [39] that trans-verse bending and torsion is coupled by static end momentsand that flexural-torsional buckling is comprised of thistransverse flexure and axial torsion Analytical investigations

on the influence of axial loads and end moments on thevibration of beams have been previously reported by Joshiand Suryanarayan for a simply supported case [32] and otherboundary conditions [33] Joshi and Suryanarayan [34] alsostudied analytically the flexural-torsional instability of thinwalled beams subjected to axial loads and end momentsThe same authors examined the coupled bending-torsionvibration of a deep rectangular beam that is initially stressedas a result of the application of moments that vary along thespan [35] Pavlovic and Kozic [36] developed a closed formanalytical solution to investigate the effects of end momentson a simply supported thin walled beam FurthermorePavlovic et al [37] also formulated the analytical solutions tostudy a simply supported thin walled beam subjected to thecombined action of an axial force and end moment

The reliability and accuracy of such modal analysisresults depend on the method implemented A variety ofapproaches for the analysis of these structures have beenproposed and implemented in the past including analyticalsemianalytical and numerical methods Amongst these theclassical FEM method where beam element matrices areevaluated from assumed fixed (namely polynomial) shapefunctions has beenwidely used by investigatorsThis practiceresults in approximate equations in the form of mass andstatic stiffness matrices A deviation from the conventionalFEM formulation would pay dividends if improved accuracyof results can be obtained by using shape functions otherthan polynomials This is the case when the homogeneoussolution of the pertinent differential equation is availablefor the development of each of the element matrices Forstatic analysis the use of the homogeneous solution of thedifferential equation yields the exact stiffness matrix and loadvector for a beam element [40] The use of other alternativeshape functions in dynamic FEM formulations has also beenexplored

The dynamic stiffness matrix (DSM) method offers abetter alternative particularly when higher frequencies andbetter accuracies of results are required It relies on asingle frequency-dependent matrix which has both massand stiffness properties The use of a DSM in vibrationanalysis is well established [14ndash18] Obviously the methodgives more accurate results because it exploits the exactmember theory The matrix is obtained by directly solvingthe governing differential equation Other methods have alsobeen explored and reported which are more or less similarto those explained previously Some of them were developedto treat a particular configuration and group of mechanicalsystems

A thorough investigation of the existing conventionalFEM and alternative DSM methods in beam vibrations hasled to the development of the DFE approach [3] DFE bridgesthe gap between the standard FEM and the exact DSMmethods by advantageously exploiting the generality of FEMand the very precise frequency calculations provided byDSMapproach From this point of view the method retains thephysical aspect of analytical or semianalytical approaches andthe power of a numerical method It has been shown thatDFE is an efficient tool for handling periodical structures orsystems composed of several identical substructures [3]





2 Theory


P P

M M

xx = Lx = 0

z

t

h


and 119909 = 119871



1198641198681199081015840101584010158401015840

+ 11987511990810158401015840+ 119872120579

10158401015840minus 120588119860 = 0 (1)

11986611986912057910158401015840+

119875119868119875

11986012057910158401015840+ 119872119908

10158401015840minus 120588119868119875

120579 = 0 (2)





119908 (119909 119905) = sin (120596119905) (3)

120579 (119909 119905) = 120579 sin (120596119905) (4)


1198641198681015840101584010158401015840

+ 11987510158401015840+ 11987211991111991112057910158401015840+ 120588119860120596

2 = 0 (5)

11986611986912057910158401015840+ 11987511986811987512057910158401015840+ 11987211991111991110158401015840+ 1205881198681198751198601205962120579 = 0 (6)



119882119891= int119871

0

120575119882(1198641198681198821015840101584010158401015840

+ 11987511988210158401015840+ 11987211991111991112057910158401015840+ 120588119860120596

2119882)119889119909 = 0

(7)

119882119905= int119871

0

120575120579 (11986611986912057910158401015840+

119875119868119875

11986012057910158401015840+ 11987211991111991111988210158401015840+ 1205881198681198751205962120579) 119889119909 = 0

(8)


119882119891= int119871

0

(1198641198681198821015840101584012057511988210158401015840minus 119875119882

10158401205751198821015840+ 119872120579

10158401205751198821015840

+1205881198601205962119882120575119882)119889119909 +

[(119864119868119882

101584010158401015840+ 119875119882

1015840+ 119872120579

1015840) 120575119882]

119871

0

minus[(119864119868119882

10158401015840)1205751198821015840]119871

0= 0

(9)

119882119905= int119871

0

(11986611986912057910158401205751205791015840+

119875119868119875

11986012057910158401205751205791015840+ 119872119882

10158401205751205791015840minus 1205881198681198751205962120579120575120579) 119889119909

minus[(119866119869120579

1015840+

119875119868119875

1198601205791015840+ 119872119882

1015840)120575120579]119871

0

= 0

(10)


119882 = 119882INT minus 119882EXT = 0 (11)

where

119882EXT = 0 (12)

and thus

119882INT = 119882119891+ 119882119905 (13)


119872(119909) = minus11986411986811988210158401015840 (14)

119878 (119909) = 119864119868119882101584010158401015840

+ 1198721205791015840+ 119875119882

1015840 (15)

119879 (119909) = 1198661198691205791015840+

119875119868119875

1198601205791015840+ 119872119882

1015840 (16)


1 2

120579j+1

120579j

WjW998400j

NE + 1NEWj+1W998400j+1



119882 = 119882INT =


119882119896

=


119882119896

119891+ 119882119896

119905

(17)


119908 (119909) = ⟨1 119909 1199092 1199093⟩119862

1 (18)

120579 (119909) = ⟨1 119909⟩1198622 (19)

In (18) and (19) above 1198621and 119862



119882119899 =

1198821

11988210158401

1198822

11988210158402

=[[[

[

1 0 0 0

0 1 0 0

1 119897 1198972 1198973

0 1 2119897 31198972

]]]

]

1198621 = [119875

119899119908]1198621 (20)

120579119899 =

1205791

1205792

= [1 0

1 119897] 1198622 = [119875

119899119905] 1198622 (21)

Thus

119882(119909) = ⟨1 119909 1199092 1199093⟩ [119875119899119908

]minus1

119882119899 = ⟨119873 (119909)⟩ 119882

119899

(22)

120579 (119909) = ⟨1 119909⟩ [119875119899119905]minus1

120579119899 = ⟨119871 (119909)⟩120579

119899 (23)


1 1198732 1198733 and 119873

4

are

1198731(119909) =

21199093

1198973minus

31199092

1198972+ 1 119873

2(119909) =

1199093

1198972minus

21199092

119897+ 119909

1198733(119909) =

minus21199093

1198973+

31199092

1198972 119873

4(119909) =

1199093

1198972minus

1199092

119897

(24)


2are defined as

1198711(119909) = 1 minus

119909

119897 119871

2(119909) =

119909

119897 (25)




⟨120575119882119899⟩ (119870 minus 120596

2119872) 119882

119899 = 0

det (119870 minus 1205962119872) = 0

(26)




[119898]119896=

[[[[[[[[[[[[[[[[[[

[

156119898119897

420

221198981198972

4200

54119898119897

420

minus131198981198972

4200

41198981198973

4200

131198981198972

420

minus31198981198973

4200

120588119868119875119897

30 0

120588119868119875119897

6156119898119897

420

minus221198981198972

4200

Sym41198981198973

4200

120588119868119875119897

3

]]]]]]]]]]]]]]]]]]

]

(27a)



[119896119861] =

[[[[[[[[[[

[

12119864119868

11989736119864119868

1198972minus12119864119868

11989736119864119868

1198972

4119864119868

119897

minus6119864119868

11989722119864119868

119897

Sym12119864119868

1198973minus6119864119868

1198972

4119864119868

119897

]]]]]]]]]]

]

(27b)


119861119879]119888 and the torsion-






[119896]geometric = 119875

[[[[[[[[[[[[

[

6

5119897

1

10

minus6

5119897

1

10

2119897

15

minus1

10

minus119897

30

Sym6

5119897

minus1

10

2119897

15

]]]]]]]]]]]]

]

(27c)

[119896]torsion = (119866119869

119897+

119875119868119875

119860119897) [

1 minus1

minus1 1] (27d)


[119896119861119879

]119888=

119872

119897[

1 0 minus1 0

minus1 0 1 0] (27e)

[119896119879119861

]119888=

119872

119897

[[[

[

1 minus1

0 0

minus1 1

0 0

]]]

]

(27f)


[119896]119896=

[[[[[[[[[[[[[[[[[[

[

12119864119868

1198973minus

6119875

5119897

6119864119868

1198972minus

119875

10

119872

119897

minus12119864119868

1198973+

6119875

5119897

6119864119868

1198972minus

119875

10

minus119872

1198974119864119868

119897minus

2119875119897

150

minus6119864119868

1198972+

119875

10

2119864119868

119897+

119875119897

300

119866119869

119897+

119875119868119875

119860119897

minus119872

1198970

minus119866119869

119897minus

119875119868119875

119860119897

Sym12119864119868

1198973minus

6119875

5119897

minus6119864119868

1198972+

119875

10

119872

119897

4119864119868

119897minus

2119875119897

150

119866119869

119897+

119875119868119875

119860119897

]]]]]]]]]]]]]]]]]]

]

(27g)





C-F 2556 2556 15995 15995 44858 44858C-C 16266 16266 44858 44858 87970 87970P-P 7175 7175 28718 28718 64633 64633P-C 11209 11209 36351 36351 75611 75611




3 Numerical Results




0002004006008

01012014016018

0 50 100 150 200 250

Erro

r (

)

Number of elements







C-FEnd moment















0

05

1

15

2

25

3

35

4

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)






159

16

161

162

163

164

165

166

167

168

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)



64

66

68

7

72

74

76

78

8

82

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)




106

108

11

112

114

116

118

12

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)








0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1


02468

1012141618



Buck

ling

mom

ent (

MNmiddotm

)




1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t


1st2nd3rd

4th5th

minus15

minus1

minus05

minus2




0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2






0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













2 Theory


P P

M M

xx = Lx = 0

z

t

h


and 119909 = 119871



1198641198681199081015840101584010158401015840

+ 11987511990810158401015840+ 119872120579

10158401015840minus 120588119860 = 0 (1)

11986611986912057910158401015840+

119875119868119875

11986012057910158401015840+ 119872119908

10158401015840minus 120588119868119875

120579 = 0 (2)





119908 (119909 119905) = sin (120596119905) (3)

120579 (119909 119905) = 120579 sin (120596119905) (4)


1198641198681015840101584010158401015840

+ 11987510158401015840+ 11987211991111991112057910158401015840+ 120588119860120596

2 = 0 (5)

11986611986912057910158401015840+ 11987511986811987512057910158401015840+ 11987211991111991110158401015840+ 1205881198681198751198601205962120579 = 0 (6)



119882119891= int119871

0

120575119882(1198641198681198821015840101584010158401015840

+ 11987511988210158401015840+ 11987211991111991112057910158401015840+ 120588119860120596

2119882)119889119909 = 0

(7)

119882119905= int119871

0

120575120579 (11986611986912057910158401015840+

119875119868119875

11986012057910158401015840+ 11987211991111991111988210158401015840+ 1205881198681198751205962120579) 119889119909 = 0

(8)


119882119891= int119871

0

(1198641198681198821015840101584012057511988210158401015840minus 119875119882

10158401205751198821015840+ 119872120579

10158401205751198821015840

+1205881198601205962119882120575119882)119889119909 +

[(119864119868119882

101584010158401015840+ 119875119882

1015840+ 119872120579

1015840) 120575119882]

119871

0

minus[(119864119868119882

10158401015840)1205751198821015840]119871

0= 0

(9)

119882119905= int119871

0

(11986611986912057910158401205751205791015840+

119875119868119875

11986012057910158401205751205791015840+ 119872119882

10158401205751205791015840minus 1205881198681198751205962120579120575120579) 119889119909

minus[(119866119869120579

1015840+

119875119868119875

1198601205791015840+ 119872119882

1015840)120575120579]119871

0

= 0

(10)


119882 = 119882INT minus 119882EXT = 0 (11)

where

119882EXT = 0 (12)

and thus

119882INT = 119882119891+ 119882119905 (13)


119872(119909) = minus11986411986811988210158401015840 (14)

119878 (119909) = 119864119868119882101584010158401015840

+ 1198721205791015840+ 119875119882

1015840 (15)

119879 (119909) = 1198661198691205791015840+

119875119868119875

1198601205791015840+ 119872119882

1015840 (16)


1 2

120579j+1

120579j

WjW998400j

NE + 1NEWj+1W998400j+1



119882 = 119882INT =


119882119896

=


119882119896

119891+ 119882119896

119905

(17)


119908 (119909) = ⟨1 119909 1199092 1199093⟩119862

1 (18)

120579 (119909) = ⟨1 119909⟩1198622 (19)

In (18) and (19) above 1198621and 119862



119882119899 =

1198821

11988210158401

1198822

11988210158402

=[[[

[

1 0 0 0

0 1 0 0

1 119897 1198972 1198973

0 1 2119897 31198972

]]]

]

1198621 = [119875

119899119908]1198621 (20)

120579119899 =

1205791

1205792

= [1 0

1 119897] 1198622 = [119875

119899119905] 1198622 (21)

Thus

119882(119909) = ⟨1 119909 1199092 1199093⟩ [119875119899119908

]minus1

119882119899 = ⟨119873 (119909)⟩ 119882

119899

(22)

120579 (119909) = ⟨1 119909⟩ [119875119899119905]minus1

120579119899 = ⟨119871 (119909)⟩120579

119899 (23)


1 1198732 1198733 and 119873

4

are

1198731(119909) =

21199093

1198973minus

31199092

1198972+ 1 119873

2(119909) =

1199093

1198972minus

21199092

119897+ 119909

1198733(119909) =

minus21199093

1198973+

31199092

1198972 119873

4(119909) =

1199093

1198972minus

1199092

119897

(24)


2are defined as

1198711(119909) = 1 minus

119909

119897 119871

2(119909) =

119909

119897 (25)




⟨120575119882119899⟩ (119870 minus 120596

2119872) 119882

119899 = 0

det (119870 minus 1205962119872) = 0

(26)




[119898]119896=

[[[[[[[[[[[[[[[[[[

[

156119898119897

420

221198981198972

4200

54119898119897

420

minus131198981198972

4200

41198981198973

4200

131198981198972

420

minus31198981198973

4200

120588119868119875119897

30 0

120588119868119875119897

6156119898119897

420

minus221198981198972

4200

Sym41198981198973

4200

120588119868119875119897

3

]]]]]]]]]]]]]]]]]]

]

(27a)



[119896119861] =

[[[[[[[[[[

[

12119864119868

11989736119864119868

1198972minus12119864119868

11989736119864119868

1198972

4119864119868

119897

minus6119864119868

11989722119864119868

119897

Sym12119864119868

1198973minus6119864119868

1198972

4119864119868

119897

]]]]]]]]]]

]

(27b)


119861119879]119888 and the torsion-






[119896]geometric = 119875

[[[[[[[[[[[[

[

6

5119897

1

10

minus6

5119897

1

10

2119897

15

minus1

10

minus119897

30

Sym6

5119897

minus1

10

2119897

15

]]]]]]]]]]]]

]

(27c)

[119896]torsion = (119866119869

119897+

119875119868119875

119860119897) [

1 minus1

minus1 1] (27d)


[119896119861119879

]119888=

119872

119897[

1 0 minus1 0

minus1 0 1 0] (27e)

[119896119879119861

]119888=

119872

119897

[[[

[

1 minus1

0 0

minus1 1

0 0

]]]

]

(27f)


[119896]119896=

[[[[[[[[[[[[[[[[[[

[

12119864119868

1198973minus

6119875

5119897

6119864119868

1198972minus

119875

10

119872

119897

minus12119864119868

1198973+

6119875

5119897

6119864119868

1198972minus

119875

10

minus119872

1198974119864119868

119897minus

2119875119897

150

minus6119864119868

1198972+

119875

10

2119864119868

119897+

119875119897

300

119866119869

119897+

119875119868119875

119860119897

minus119872

1198970

minus119866119869

119897minus

119875119868119875

119860119897

Sym12119864119868

1198973minus

6119875

5119897

minus6119864119868

1198972+

119875

10

119872

119897

4119864119868

119897minus

2119875119897

150

119866119869

119897+

119875119868119875

119860119897

]]]]]]]]]]]]]]]]]]

]

(27g)





C-F 2556 2556 15995 15995 44858 44858C-C 16266 16266 44858 44858 87970 87970P-P 7175 7175 28718 28718 64633 64633P-C 11209 11209 36351 36351 75611 75611




3 Numerical Results




0002004006008

01012014016018

0 50 100 150 200 250

Erro

r (

)

Number of elements







C-FEnd moment















0

05

1

15

2

25

3

35

4

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)






159

16

161

162

163

164

165

166

167

168

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)



64

66

68

7

72

74

76

78

8

82

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)




106

108

11

112

114

116

118

12

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)








0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1


02468

1012141618



Buck

ling

mom

ent (

MNmiddotm

)




1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t


1st2nd3rd

4th5th

minus15

minus1

minus05

minus2




0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2






0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119882119891= int119871

0

120575119882(1198641198681198821015840101584010158401015840

+ 11987511988210158401015840+ 11987211991111991112057910158401015840+ 120588119860120596

2119882)119889119909 = 0

(7)

119882119905= int119871

0

120575120579 (11986611986912057910158401015840+

119875119868119875

11986012057910158401015840+ 11987211991111991111988210158401015840+ 1205881198681198751205962120579) 119889119909 = 0

(8)


119882119891= int119871

0

(1198641198681198821015840101584012057511988210158401015840minus 119875119882

10158401205751198821015840+ 119872120579

10158401205751198821015840

+1205881198601205962119882120575119882)119889119909 +

[(119864119868119882

101584010158401015840+ 119875119882

1015840+ 119872120579

1015840) 120575119882]

119871

0

minus[(119864119868119882

10158401015840)1205751198821015840]119871

0= 0

(9)

119882119905= int119871

0

(11986611986912057910158401205751205791015840+

119875119868119875

11986012057910158401205751205791015840+ 119872119882

10158401205751205791015840minus 1205881198681198751205962120579120575120579) 119889119909

minus[(119866119869120579

1015840+

119875119868119875

1198601205791015840+ 119872119882

1015840)120575120579]119871

0

= 0

(10)


119882 = 119882INT minus 119882EXT = 0 (11)

where

119882EXT = 0 (12)

and thus

119882INT = 119882119891+ 119882119905 (13)


119872(119909) = minus11986411986811988210158401015840 (14)

119878 (119909) = 119864119868119882101584010158401015840

+ 1198721205791015840+ 119875119882

1015840 (15)

119879 (119909) = 1198661198691205791015840+

119875119868119875

1198601205791015840+ 119872119882

1015840 (16)


1 2

120579j+1

120579j

WjW998400j

NE + 1NEWj+1W998400j+1



119882 = 119882INT =


119882119896

=


119882119896

119891+ 119882119896

119905

(17)


119908 (119909) = ⟨1 119909 1199092 1199093⟩119862

1 (18)

120579 (119909) = ⟨1 119909⟩1198622 (19)

In (18) and (19) above 1198621and 119862



119882119899 =

1198821

11988210158401

1198822

11988210158402

=[[[

[

1 0 0 0

0 1 0 0

1 119897 1198972 1198973

0 1 2119897 31198972

]]]

]

1198621 = [119875

119899119908]1198621 (20)

120579119899 =

1205791

1205792

= [1 0

1 119897] 1198622 = [119875

119899119905] 1198622 (21)

Thus

119882(119909) = ⟨1 119909 1199092 1199093⟩ [119875119899119908

]minus1

119882119899 = ⟨119873 (119909)⟩ 119882

119899

(22)

120579 (119909) = ⟨1 119909⟩ [119875119899119905]minus1

120579119899 = ⟨119871 (119909)⟩120579

119899 (23)


1 1198732 1198733 and 119873

4

are

1198731(119909) =

21199093

1198973minus

31199092

1198972+ 1 119873

2(119909) =

1199093

1198972minus

21199092

119897+ 119909

1198733(119909) =

minus21199093

1198973+

31199092

1198972 119873

4(119909) =

1199093

1198972minus

1199092

119897

(24)


2are defined as

1198711(119909) = 1 minus

119909

119897 119871

2(119909) =

119909

119897 (25)




⟨120575119882119899⟩ (119870 minus 120596

2119872) 119882

119899 = 0

det (119870 minus 1205962119872) = 0

(26)




[119898]119896=

[[[[[[[[[[[[[[[[[[

[

156119898119897

420

221198981198972

4200

54119898119897

420

minus131198981198972

4200

41198981198973

4200

131198981198972

420

minus31198981198973

4200

120588119868119875119897

30 0

120588119868119875119897

6156119898119897

420

minus221198981198972

4200

Sym41198981198973

4200

120588119868119875119897

3

]]]]]]]]]]]]]]]]]]

]

(27a)



[119896119861] =

[[[[[[[[[[

[

12119864119868

11989736119864119868

1198972minus12119864119868

11989736119864119868

1198972

4119864119868

119897

minus6119864119868

11989722119864119868

119897

Sym12119864119868

1198973minus6119864119868

1198972

4119864119868

119897

]]]]]]]]]]

]

(27b)


119861119879]119888 and the torsion-






[119896]geometric = 119875

[[[[[[[[[[[[

[

6

5119897

1

10

minus6

5119897

1

10

2119897

15

minus1

10

minus119897

30

Sym6

5119897

minus1

10

2119897

15

]]]]]]]]]]]]

]

(27c)

[119896]torsion = (119866119869

119897+

119875119868119875

119860119897) [

1 minus1

minus1 1] (27d)


[119896119861119879

]119888=

119872

119897[

1 0 minus1 0

minus1 0 1 0] (27e)

[119896119879119861

]119888=

119872

119897

[[[

[

1 minus1

0 0

minus1 1

0 0

]]]

]

(27f)


[119896]119896=

[[[[[[[[[[[[[[[[[[

[

12119864119868

1198973minus

6119875

5119897

6119864119868

1198972minus

119875

10

119872

119897

minus12119864119868

1198973+

6119875

5119897

6119864119868

1198972minus

119875

10

minus119872

1198974119864119868

119897minus

2119875119897

150

minus6119864119868

1198972+

119875

10

2119864119868

119897+

119875119897

300

119866119869

119897+

119875119868119875

119860119897

minus119872

1198970

minus119866119869

119897minus

119875119868119875

119860119897

Sym12119864119868

1198973minus

6119875

5119897

minus6119864119868

1198972+

119875

10

119872

119897

4119864119868

119897minus

2119875119897

150

119866119869

119897+

119875119868119875

119860119897

]]]]]]]]]]]]]]]]]]

]

(27g)





C-F 2556 2556 15995 15995 44858 44858C-C 16266 16266 44858 44858 87970 87970P-P 7175 7175 28718 28718 64633 64633P-C 11209 11209 36351 36351 75611 75611




3 Numerical Results




0002004006008

01012014016018

0 50 100 150 200 250

Erro

r (

)

Number of elements







C-FEnd moment















0

05

1

15

2

25

3

35

4

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)






159

16

161

162

163

164

165

166

167

168

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)



64

66

68

7

72

74

76

78

8

82

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)




106

108

11

112

114

116

118

12

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)








0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1


02468

1012141618



Buck

ling

mom

ent (

MNmiddotm

)




1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t


1st2nd3rd

4th5th

minus15

minus1

minus05

minus2




0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2






0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











⟨120575119882119899⟩ (119870 minus 120596

2119872) 119882

119899 = 0

det (119870 minus 1205962119872) = 0

(26)




[119898]119896=

[[[[[[[[[[[[[[[[[[

[

156119898119897

420

221198981198972

4200

54119898119897

420

minus131198981198972

4200

41198981198973

4200

131198981198972

420

minus31198981198973

4200

120588119868119875119897

30 0

120588119868119875119897

6156119898119897

420

minus221198981198972

4200

Sym41198981198973

4200

120588119868119875119897

3

]]]]]]]]]]]]]]]]]]

]

(27a)



[119896119861] =

[[[[[[[[[[

[

12119864119868

11989736119864119868

1198972minus12119864119868

11989736119864119868

1198972

4119864119868

119897

minus6119864119868

11989722119864119868

119897

Sym12119864119868

1198973minus6119864119868

1198972

4119864119868

119897

]]]]]]]]]]

]

(27b)


119861119879]119888 and the torsion-






[119896]geometric = 119875

[[[[[[[[[[[[

[

6

5119897

1

10

minus6

5119897

1

10

2119897

15

minus1

10

minus119897

30

Sym6

5119897

minus1

10

2119897

15

]]]]]]]]]]]]

]

(27c)

[119896]torsion = (119866119869

119897+

119875119868119875

119860119897) [

1 minus1

minus1 1] (27d)


[119896119861119879

]119888=

119872

119897[

1 0 minus1 0

minus1 0 1 0] (27e)

[119896119879119861

]119888=

119872

119897

[[[

[

1 minus1

0 0

minus1 1

0 0

]]]

]

(27f)


[119896]119896=

[[[[[[[[[[[[[[[[[[

[

12119864119868

1198973minus

6119875

5119897

6119864119868

1198972minus

119875

10

119872

119897

minus12119864119868

1198973+

6119875

5119897

6119864119868

1198972minus

119875

10

minus119872

1198974119864119868

119897minus

2119875119897

150

minus6119864119868

1198972+

119875

10

2119864119868

119897+

119875119897

300

119866119869

119897+

119875119868119875

119860119897

minus119872

1198970

minus119866119869

119897minus

119875119868119875

119860119897

Sym12119864119868

1198973minus

6119875

5119897

minus6119864119868

1198972+

119875

10

119872

119897

4119864119868

119897minus

2119875119897

150

119866119869

119897+

119875119868119875

119860119897

]]]]]]]]]]]]]]]]]]

]

(27g)





C-F 2556 2556 15995 15995 44858 44858C-C 16266 16266 44858 44858 87970 87970P-P 7175 7175 28718 28718 64633 64633P-C 11209 11209 36351 36351 75611 75611




3 Numerical Results




0002004006008

01012014016018

0 50 100 150 200 250

Erro

r (

)

Number of elements







C-FEnd moment















0

05

1

15

2

25

3

35

4

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)






159

16

161

162

163

164

165

166

167

168

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)



64

66

68

7

72

74

76

78

8

82

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)




106

108

11

112

114

116

118

12

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)








0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1


02468

1012141618



Buck

ling

mom

ent (

MNmiddotm

)




1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t


1st2nd3rd

4th5th

minus15

minus1

minus05

minus2




0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2






0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













C-F 2556 2556 15995 15995 44858 44858C-C 16266 16266 44858 44858 87970 87970P-P 7175 7175 28718 28718 64633 64633P-C 11209 11209 36351 36351 75611 75611




3 Numerical Results




0002004006008

01012014016018

0 50 100 150 200 250

Erro

r (

)

Number of elements







C-FEnd moment















0

05

1

15

2

25

3

35

4

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)






159

16

161

162

163

164

165

166

167

168

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)



64

66

68

7

72

74

76

78

8

82

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)




106

108

11

112

114

116

118

12

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)








0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1


02468

1012141618



Buck

ling

mom

ent (

MNmiddotm

)




1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t


1st2nd3rd

4th5th

minus15

minus1

minus05

minus2




0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2






0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











C-FEnd moment















0

05

1

15

2

25

3

35

4

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)






159

16

161

162

163

164

165

166

167

168

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)



64

66

68

7

72

74

76

78

8

82

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)




106

108

11

112

114

116

118

12

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)








0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1


02468

1012141618



Buck

ling

mom

ent (

MNmiddotm

)




1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t


1st2nd3rd

4th5th

minus15

minus1

minus05

minus2




0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2






0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










159

16

161

162

163

164

165

166

167

168

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)



64

66

68

7

72

74

76

78

8

82

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)




106

108

11

112

114

116

118

12

0 05 1 15 2

Fund

amen

tal f

requ

ency

(Hz)

Axial force (MN)








0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1


02468

1012141618



Buck

ling

mom

ent (

MNmiddotm

)




1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t


1st2nd3rd

4th5th

minus15

minus1

minus05

minus2




0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2






0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1 3 5 7 9 11

Buck

ling

forc

e (M

N)

minus25

minus2

minus15

minus05

minus1

minus1


02468

1012141618



Buck

ling

mom

ent (

MNmiddotm

)




1962 360 361 361 931 940940 941940 1770 18031791 180517954022 400 401 401 989 998999 998998 1840 18771858 187718666671 445 446 446 1066 10741074 10741074 1939 19651955 196719617750 464 465 465 1096 11051105 11061106 1976 20091997 201220009810 495 496 496 1149 11571156 11561155 2044 20742058 20742062

0

05

1

15

0 2 4 6 8 10

Nor

mal

ized

ben

ding

disp

lace

men

t


1st2nd3rd

4th5th

minus15

minus1

minus05

minus2




0

02

04

06

08

1

12

0 2 4 6 8 10

Nor

mal

ized

tors

iona

l disp

lace

men

t


1st2nd3rd

4th5th

minus04

minus02

minus2






0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0 708707 69850 669669 661100 551551 549150 354348 346205 4746 35


Disclosure




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article on the flexural-torsional vibration...

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