COUPLED FLEXURAL TORSIONAL VIBRATION AND STABILITY ANALYSIS OF PRE-LOADED BEAMS USING CONVENTIONAL AND DYNAMIC FINITE ELEMENT METHODS

Transcription

1 COUPLED FLEXURAL TORSIONAL VIBRATION AND STABILITY ANALYSIS OF PRE-LOADED BEAMS USING CONVENTIONAL AND DYNAMIC FINITE ELEMENT METHODS by Heenkenda Jayasinghe, B. Eng Aeronautica Engineering City University London, United Kingdom, A thesis presented to Ryerson University in partia fufiment of the requirements for the degree of Master of Appied Science in the Program of Aerospace Engineering Toronto, Ontario, Canada, 3 Heenkenda Jayasinghe 3

2 Author s Decaration AUTHOR S DECLARATION FOR ELECTRONIC SUBMISSION OF A THESIS I hereby decare that I am the soe author of this thesis or dissertation. I authorize Ryerson University to end this thesis or dissertation to other institutions or individuas for the purpose of schoary research. I further authorize Ryerson University to reproduce this thesis or dissertation by photocopying or by other means, in tota or in part, at the request of other institutions or individuas for the purpose of schoary research. I understand that my thesis may be made eectronicay avaiabe to the pubic. ii

3 Abstract Dynamic Finite Eement (DFE) and conventiona finite eement formuations are deveoped to study the fexura torsiona vibration and stabiity of an isotropic, homogeneous and ineary eastic pre-oaded beam subjected to an axia oad and end-moment. Various cassica boundary conditions are considered. Eementary Euer Bernoui bending and St. Venant torsion beam theories were used as a starting point to deveop the governing equations and the finite eement soutions. The noninear Eigenvaue probem resuted from the DFE method was soved using a program code written in MATLAB and the natura frequencies and mode shapes of the system were determined from the Eigenvaues and Eigenvectors, respectivey. Simiary, a inear Eigenvaue probem was formuated and soved using a MATLAB code for the conventiona FEM method. The conventiona FEM resuts were vaidated against those avaiabe in the iterature and ANSYS simuations and the DFE resuts were compared with the FEM resuts. The resuts confirmed that tensie forces increased the natura frequencies, which indicates beam stiffening. On the contrary, compressive forces reduced the natura frequencies, suggesting a reduction in beam stiffness. Simiary, when an end-moment was appied the stiffness of the beam and the natura frequencies diminished. More importanty, when a force and an end-moment were acting in combination, the resuts depended on the direction and magnitude of the axia force. Nevertheess, the stiffness of the beam is more sensitive to the changes in the magnitude and direction of the axia force compared to the moment. A bucking anaysis of the beam was aso carried out to determine the critica bucking end-moment and axia compressive force. iii

4 Acknowedgement I woud ike express my sincere gratitude to Prof. Seyed M. Hashemi for his continuous patience, wisdom, and guidance throughout my research, as we as for his motivation and enthusiasm when I needed them. I woud aso ike to thank my famiy and friends for a of their unwavering support aong this journey. iv

5 Contents List of Tabes... vi List of Figures... vii List of Appendices... ix. Introduction.... Conventiona Finite Eement Anaysis FEM Numerica Tests The Dynamic Finite Eement (DFE) Method The Dynamic Stiffness Matrix method The conventiona FEM method based on poynomia interpoation functions The frequency- dependent Dynamic Finite Eement method based on trigonometric shape functions DFE Numerica Tests Concuding Remarks and Recommendations for Future Works Appendix A: Shape Functions Appendix B: ANSYS Mode v

6 List of Tabes Tabe : Comparison between the exact and FEM resuts for the first three natura frequencies at P = and M =... 8 Tabe : Convergence of the fifth natura frequency... 9 Tabe 3: First natura frequencies for cantiever boundary condition (C F) when force and moment are appied; a 4-eement FEM mode is used... Tabe 4: Fundamenta frequencies for camped camped boundary condition (C C), when force and end-moment are appied; a 4-eement FEM mode is used... Tabe 5: First natura frequencies for pinned pinned boundary condition (P P) when force and moment are appied; a 4-eement FEM mode is used... Tabe 6: First natura frequencies for pinned camped boundary condition (P C) when force and moment are appied; a 4-eement FEM mode is used... 3 Tabe 7: Critica bucking moment for cantievered boundary condition (C - F) with varying compressive force... 3 Tabe 8: Critica bucking compressive force for cantievered boundary condition (C - F) with varying end-moment... 4 Tabe 9: First natura frequencies for cantiever beam (C F) when force and moment are appied... 4 Tabe : First natura frequencies for camped camped boundary condition (C C) when force and moment are appied... 4 Tabe : First natura frequencies for pinned pinned boundary condition (P P) when force and moment are appied... 4 vi

7 Tabe : First natura frequencies for pinned camped boundary condition (P C) when force and moment are appied Tabe 3: Critica bucking moment for cantievered boundary condition (C - F) with varying compressive force Tabe 4: Critica bucking compressive force for cantievered boundary condition (C - F) with varying end-moment List of Figures Figure : Beam with axia oad and end-moment appied at x= and x=l... 9 Figure : System discretized using eements with 3 degrees of freedom per node... Figure 3: Convergence anaysis for conventiona FEM for cantievered beam... Figure 4:Variation of natura frequencies when tensie force and end-moment is appied for cantievered (C-F) boundary condition... 4 Figure 5: Variation of natura frequencies when tensie force and end-moment is appied for camped camped (C-C) boundary condition... 5 Figure 6: Variation of natura frequencies when tensie force and end-moment is appied for pinned pinned (P-P) boundary condition... 5 Figure 7: Variation of natura frequencies when tensie force and end-moment is appied for camped pinned (C-P) boundary condition... 6 Figure 8: Variation of critica bucking compressive force with end-moment... 6 Figure 9: Variation of critica bucking end-moment with tensie and compressive force... 7 Figure : Bending components of mode shapes... 7 Figure : Torsiona components of mode shapes... 8 Figure : Convergence anaysis for DFE method for cantievered beam... 4 vii

8 Figure 3: Comparison of convergence efficiency between DFE method and conventiona FEM for cantievered beam... 4 Figure 4: Variation of natura frequencies when tensie force and end-moment is appied for cantievered (C-F) boundary condition Figure 5: Variation of natura frequencies when tensie force and end-moment is appied for camped camped (C-C) boundary condition Figure 6: Variation of natura frequencies when tensie force and end-moment is appied for pinned pinned (P-P) boundary condition Figure 7: Variation of natura frequencies when tensie force and end-moment is appied for pinned camped (P-C) boundary condition Figure 8: Variation of critica bucking end-moment with axia force for cantievered (C-F) boundary condition Figure 9: Variation of critica bucking compressive force with end-moment for cantievered (C- F) boundary condition Figure : Bending component of mode shapes of the system Figure : Torsiona component of mode shapes of the system Figure : Cubic Hermite shape functions for bending and inear shape functions for torsion used in conventiona FEM Figure 3: Frequency dependent trigonometric bending and torsion shape functions presented in [] and [3] used in the DFE method Figure 4: Dependency on the frequency of the first bending shape function [, 3] used in the DFE method viii

9 Figure 5: Dependency on the frequency of the second bending shape function [, 3] used in the DFE method Figure 6: Dependency on the frequency of the third bending shape function [, 3] used in the DFE method Figure 7: Dependency on the frequency of the fourth bending shape function [, 3] used in the DFE method Figure 8: Dependency on the frequency of the first torsiona shape function [, 3] used in the DFE method Figure 9: Dependency on the frequency of the second torsiona shape function [, 3] used in the DFE method Figure 3: Beam meshed in ANSYS using SOLID 87 eements Figure 3: First couped bending torsion mode shape from ANSYS Figure 3: Second couped bending torsion mode shape from ANSYS Figure 33: Third couped bending torsion mode shape from ANSYS Figure 34: Fourth couped bending torsion mode shape from ANSYS Figure 35: Fifth couped bending torsion mode shape from ANSYS List of Appendices Appendix A: Shape Functions...53 Appendix B: ANSYS Mode.57 ix

10 . Introduction Beams are important and commony used structures, since many components of airborne vehices, such as wings and heicopter bades coud be modeed as a simpe beam or as a series of beams during the preiminary design stages. Additionay, mutipe ayers of beams coud be used to mode sandwich/composite materias and deamination probems. Such an aircraft structura component is exposed to a arge range of vibrationa frequencies during its operationa ifetime. Therefore, it is of utmost importance to study the vibrationa characteristics, such as the fundamenta frequencies and mode shapes of a system in order to avoid resonance. Thus, prior to manufacturing, most components undergo moda anayses which woud aow the designers to investigate the effects of various oading and boundary conditions the part woud be subjected to during its operation, and understand the effect of those on the vibrationa characteristics of the component. Using these resuts, the engineers and designers coud then ater and optimize the geometry of the system and the materias used, to gain a favorabe outcome. They woud aso be abe to determine the most suitabe ocations to add supports and areas that require additiona reinforcements. Thus, moda anaysis woud ensure that the natura frequencies of the component is maintained within an accepted range. In many aerospace appications, the structura beam eements are in a state of preoad or pre stress due to the appication of axia oads and end-moments (e.g. heicopter rotor bades and structura eements attached through semi rigid connections). The centrifuga force acting on the bades coud be modeed as an axia force. The presence of such axia oads and end-moments can affect the vibrationa and stabiity characteristics of the beam and as such it is worthwhie to investigate. Therefore, in this study a moda anaysis wi be carried out to investigate the stabiity

11 and vibration of a simpe Euer Bernoui beam under various boundary conditions subjected to axia force and end- moments. Many researchers have studied the free vibration of isotropic, sandwiched, composite and thinwaed beams subjected to various oading configurations, using numerous techniques. Hashemi and Richard [] deveoped a new Dynamic Finite Eement (DFE) formuation to anayse the free vibration of bending torsion couped beams. The fexura axis of the beam studied by Hashemi et a. is not coincident with the inertia axis. Hashemi and Roach [] aso formuated a DFE for the free vibration of an extension torsion couped composite beam. A quasi exact DFE formuation, for the free vibration anaysis of a three ayered sandwich beam consisting of a thick, soft, ow strength and density core as we as two face ayers made of high strength materia, was deveoped by Hashemi and Adique [3]. Borneman and Hashemi [4] deveoped a DFE for the free vibration anaysis of a bending torsion couped aminated composite wing beam. The effects of shear deformation and rotary inertia were negected in this study, but were accounted for in another study by Hashemi and Pereira [4]. The exact method has been used to determine the fexura torsiona vibrationa characteristics of a uniform beam with singe cross sectiona symmetry by Dokumaci et a [5]. The cassica finite eement method was used by Mei [6] to study the couped vibration of thin waed beams with open section. This study incuded the effects of warping stiffness. The free vibration of an open, variabe cross section I-beam was investigated by Wekezer [7]. The fexura torsiona vibration of a uniform beam was studied by Tanaka et a, [8], by determining the exact soution of the governing differentia equations. Bannerjee et a. have used the Dynamic Stiffness Matrix Method (DSM), which utiizes the genera cosed form soution of the governing differentia equations of motion to generate the frequency

12 dependent stiffness matrix, for vibrationa anaysis of isotropic [9,], sandwich [,,3] and composite [4] beams in the past. Banerjee et a. [5] aso deveoped an exact DSM to investigate the vibration of an Euer Bernoui thin waed beam and expoited the Wittrick Wiiams [6] root finding agorithm to arrive at the Eigensoutions. Bannerjee and Su [7] ater used the DSM to conduct a free, transverse and atera vibration anaysis of a beam couped with torsion. Borneman et a. [8] aso used the DSM method to investigate the vibrationa characteristics of a douby couped (materia and geometric) composite beam. Friberg [9] and Leung [] deveoped an exact DSM of a thin waed beam. The Dynamic Stiffness method was used by Haauer et a. [] to determine the vibrationa characteristics and generaized masses of an aircraft wing modeed as a series of three simpe beams. The axia oad and end-moment affect the stabiity and vibrationa characteristics of the beam. The effect of the axia oad on the transverse vibrationa characteristics of beams has been we estabished []. Hashemi and Richard [3] conducted a vibrationa anaysis on an axiay oaded bending torsion geometricay couped beam using the DFE method. Banerjee et a. [4] studied the couped fexura torsiona vibration of an axiay oaded Timoshenko beam. Anaytica soutions were deveoped by Bannerjee et a. [5, 6] to mode a uniform, axiay oaded, cantievered beam with fexura torsiona geometric couping as a resut of non-coincident shear and mass centers. The effects of warping has been negected in these studies. Jun et a. [7] studied the couped fexura torsiona vibration of an axiay oaded, thin waed beam with monosymmetrica cross sections and accounted for the effects of warping. The effect of axia oad has aso been previousy studied by Murthy and Neogy [8] for camped and pinned boundary conditions as we as by Gaert and Guck [9] for the cantievered boundary condition. Bokaian [3] studied the natura frequencies of a uniform singe span beam subjected to a constant tensie 3

13 axia oad for various boundary conditions. The same author aso investigated the vibrationa characteristics of a uniform singe span beam for ten different end conditions when a constant compressive axia oad is appied [3]. Shaker et a. [3] conducted a moda anaysis to determine the effect of axia oad on the mode shapes and natura frequencies of beams. It has aso been found by Chen and Astuta [33] that transverse bending and torsion is couped by static end-moments and that fexura torsiona bucking is comprised of this transverse fexure and torsion. Anaytica investigations on the infuence of axia oads and end-moments on the vibration of beams have been previousy reported by Joshi and Suryanarayan for a simpy supported case [34] as we as for various boundary conditions [35]. Joshi and Suryanarayan [36] aso studied anayticay, the fexura torsiona instabiity of thin waed beams subjected to axia oads and end-moments. The same authors studied the couped bending torsion vibration of a deep rectanguar beam that was initiay stressed due to the appication of moments varying aong the span of the beam [37]. Pavovic and Kosic [38] deveoped a cosed form anaytica soution to investigate the effects of end-moments on a simpy supported thin waed beam. Pavovic et a. [39] aso formuated the anaytica soutions to study a simpy supported thin waed beam subjected to the combined action of an axia force and end-moment. The reiabiity and accuracy of such moda anaysis resuts depends on the method impemented. There are severa anaytica, semi-anaytica and numerica methods that coud be used to carry out the moda anaysis. A methods mentioned in the above references have their inherent advantages and disadvantages. Athough there exists a cass of probems for which an exact soution can be easiy obtained, in most cases an exact soution for the norma modes and frequencies of the system woud be intractabe. Exact methods such as DSM are capabe of using just one eement matrix to produce the exact resuts for the vibrationa characteristics of a beam. 4

14 Nevertheess, the appicabiity of such anaytica methods is imited to simpe and specia cases. With every change made to the system configuration, the equations shoud be reformuated and it is difficut to use anaytica methods to mode probems with variations in geometry and materia properties. Thus, recourse woud be made to one of the many approximate soutions such as the Rayeigh-Ritz method and Gaerkin's method [4]. The conventiona Finite Eement Method (FEM), which uses the Gaerkin method of weighted residuas, is widey used for moda anaysis and is very popuar among researchers since it is convenient and adaptabe to many compex systems incuding systems consisting of materia and geometric variations. Geometric variations are modeed as stepped, piecewise uniform configurations. In this method, the system is discretized once the weak form of the governing differentia equations is obtained by appying the Gaerkin method of weighted residuas. Cubic Hermite approximations are frequenty used for transverse fexura dispacements and inear approximations for torsion. This resuts in the eement mass and stiffness matrices, which are independent of the natura frequency. Assembing the eement mass and stiffness matrices creates a inear Eigenvaue probem of which, the Eigenvaues and Eigenvectors give the natura frequencies and mode shapes of the system, respectivey. The DFE method [, 3] is a hybrid frequency-dependent approximate soution method which is more accurate than the conventiona FEM and, unike the DSM, is adaptabe to many compex configurations. It aows for a reduced mesh size and its formuation is quite simiar to that of the conventiona FEM. The Gaerkin method of weighted residuas is aso used in the DFE method to arrive at the weak integra form of the governing differentia equations, after which the DFE formuation process deviates from the cassica FEM. Instead of using poynomia and inear shape functions to approximate the fexura and torsiona dispacements, respectivey, the DFE method 5

15 utiizes frequency-dependent basis/shape functions presented in [] and [3] to approximate the dispacements. The trigonometric shape functions have been obtained in [] and [3] by taking the exact soutions for the differentia equations of motion governing the uncouped vibrations of a uniform beam. The noda approximations of dispacement is found using the frequency-dependent shape functions obtained when the soutions to the above governing differentia equations are empoyed as basis functions. Subsequenty, discretization of the beam is carried out simiar to the conventiona FEM, which eads to the frequency-dependent dynamic eement stiffness matrices. When these eement matrices are assembed, the frequency-dependent goba dynamic stiffness matrix is found. In order to find the natura frequencies of the system, the frequency, ω, is swept to search for particuar frequencies that woud make the determinant of the goba stiffness matrix go to zero. The corresponding eigenvector provides the mode shapes of the system. To the best of the author s knowedge, a conventiona or dynamic finite eement formuation has not yet been deveoped to mode the geometricay couped fexura torsiona free vibration of an Euer Bernoui beam subjected to an axia force and an end-moment simutaneousy. Therefore, in what foows, a cassica finite eement soution and a DFE formuation are presented to investigate the stabiity and fexura torsiona vibration of a simpe Euer Bernoui beam subjected to an axia oad and an end-moment. The effects of the axia oad and end-moment on the stiffness and natura frequencies of the beam for various cassica boundary conditions are examined. Program codes were written for the FEM and DFE methods using MATLAB and the resuts for the FEM code were vaidated using ANSYS commercia software. Subsequenty, the DFE code resuts were compared for accuracy using the resuts produced for the same mathematica mode, by the FEM code. 6

16 The purpose of deveoping a DFE soution for a beam mode that is axiay oaded and geometricay couped due to an end-moment thus inducing fexura torsiona vibration, is that it coud be then used as a powerfu too to quicky investigate the vibrationa and stabiity characteristics of numerous aerospace appications that are modeed as beams or assembages of severa identica beams (panar or space frames) to a very high degree of accuracy at the preiminary design stage. It is very important to take in to account the couping effects in vibration and response cacuations of these types of structures. The DFE beam mode coud be convenienty used to study periodica structures or more compex assembies made of severa identica substructures (beams) that has the same dynamic stiffness components and frequency characteristics. Given the magnitude of aerospace components that are axiay oaded and bending-torsion couped that coud be represented as beams to an acceptabe degree of accuracy during the preiminary design stages, such as heicopter, propeer, compressor and turbine bades, the fact that engineers and designers coud arrive at an acceptabe bapark for the vibrationa characteristics within a fraction of the time, especiay for higher modes, using an extremey coarser mesh in comparison to conventiona FEM is a massive advantage as it avoids the difficuty of having to sove a very arge Eigenvaue probem. For such aerospace components the couped bending-torsiona frequencies and mode shapes are crucia for aeroeastic cacuations. The fact that any mode number coud be investigated using dynamic eements regardess of the tota number of degrees of freedom of the goba system, is aso an advantage. Therefore, in the second chapter the probem that is studied wi be expicity defined and the theoretica aspects invoved in deveoping the conventiona FEM method wi be expained in detai commencing from the governing differentia equations of motion and extending to the point 7

17 where the inear Eigenvaue probem is deveoped. The eement mass and stiffness matrices resuting due to the discretization process wi aso be presented here. Next, the numerica test resuts obtained from the conventiona FEM anaysis and a verification of the FEM resuts using ANSYS simuations woud be presented. Comments wi be made on the accuracy and efficiency of the soution. In the third and penutimate chapter, the Dynamic Finite Eement formuation process woud be presented and its distinctions from the conventiona FEM method woud be eaborated. The process eading to the deveopment of the couped and uncouped eement stiffness matrices and the frequency dependent noninear Eigenvaue probem wi be discussed in detai. This wi be foowed by the resuts of the numerica tests performed using the DFE method which wi be compared for accuracy and efficiency of convergence with the conventiona FEM resuts. Finay, concusions wi be drawn upon the resuts presented previousy and the benefits and appications of the deveoped DFE beam mode woud be discussed. Additionay, this chapter wi aso be comprised of comments on the extendibiity of the current work to incorporate other effects that were not considered in this study such as warping, geometric noninearity and variation of materia properties that coud be usefu for future works. 8

18 . Conventiona Finite Eement Anaysis Consider a ineary eastic, homogeneous, isotropic beam subjected to an end-moment, M, and an axia oad, P, undergoing inear vibrations. Euer Bernoui bending and St. Venant torsion beam theories are used to derive the governing differentia equations of motion and a cassica finite eement soution is deveoped. As can be observed from equations () and () beow, the system is couped by the end-moments, M. The end-moments act about the z-axis (agwise), however, bending in the x-y pane (agwise) is not considered and bending occurs in the x-z pane (fapwise). Thus, the end-moments acting in the agwise direction introduces torsion to the system and creates fexura-torsiona couping. Figure (beow) iustrates the geometry of the studied system. Figure : Beam with axia oad and end-moment appied at x= and x=l The two governing differentia equations of the beam are as foows. EIw '''' Pw '' M Aw '' () '' PI P '' '' GJ Mw I P () A where w stands for the transverse fexura dispacement and θ represents the torsiona dispacement. The derivatives with respect to the ength of the beam and time are denoted with a 9

19 prime ( ) and a dot (.), respectivey. In equations () and () the appied moment and force are shown as M and P, respectivey. EI and GJ in the above equations are the Euer bending and St. Venant torsion stiffness terms, respectivey. The cross-sectiona area of the beam is denoted by A. The mass density is represented by ρ and Ip stands for the poar moment of inertia of the beam. In order to eiminate the time dependency in equations () and (), simpe harmonic vibration is considered and the foowing transformations are used to describe the transverse and torsiona dispacements. w( x, t) Wˆ sin( t) (3) ( x, t) ˆsin( t) (4) where ω is the circuar frequency and t, is the time. W and θ are the transverse and torsiona dispacement ampitudes, respectivey. Upon substituting equations (3) and (4), equations () and () becomes, ˆ '''' ˆ '' ˆ'' EIW PW M A Wˆ (5) ˆ'' ˆ'' ˆ '' GJ PI ˆ P MW IPA (6) The Gaerkin method of weighted residuas [4] is empoyed to deveop the integra form of the above equations. f L '''' '' '' ( ) (7) W W EIW PW M A W dx t L '' PIP '' '' ( P ) (8) A W GJ MW I dx

20 where δw and δθ (i.e. weighting functions) represent the transverse and torsiona virtua dispacements, respectivey. Performing integration by parts on equations (7) and (8) eads to the weak integra form of the governing equations, written as: f L ( '' '' ' ' ' ' ''' ' ' L '' ' L ) [( ) ] [( ) ] (9) W EIW W PW W M W A W W dx EIW PW M W EIW W PI PI Wt GJ MW I dx GJ MW A A L ( ' ' P ' ' ' ' ) [( ' P ' ' ) ] L P () Expressions (9) and () aso satisfy the principe of virtua work. W W W () INT EXT where, W () EXT and thus, WINT Wf Wt (3) The tota virtua work, interna virtua work and externa virtua work are denoted by W, W INT and W EXT, respectivey. The resuting shear force S(x), bending moment M(x), and torsiona torque T(x), defined as: M ( x) '' EIW (4)

21 S( x) EIW M PW ''' ' ' (5) PI P T( x) GJ MW A ' ' ' (6) are zero at the free end and the dispacements are set to zero at the fixed boundaries. As a resut, the bracketed boundary terms in expressions (9) and () vanish for a boundary conditions. The system is then discretized using eements with nodes and three DOF per node as shown in Figure beow such that, (7) No. ofeements k No. ofeements k k INT k k f t W W W W W Figure : System discretized using eements with 3 degrees of freedom per node Noda DOF s are atera dispacement w, rotation (i.e. sope) w and torsiona dispacement θ. The cassica finite eement formuation is deveoped using cubic Hermite type poynomia approximations for bending dispacement (equation 8) and inear approximations for torsiona dispacements (equation 9) introduced in the weak integra form of the governing equations such that for a two node, three degree-of-freedom per node eement, 3 ( x) x x x C w (8) ( x) x C (9)

22 3 In equations (8) and (9) above C and C are coumns vectors of unknown constant coefficients. The vectors of noda dispacement for bending and torsion are shown beow., 3 ' ' 3 C P C W W W W W w n n (), C P C t n n () Thus, n n w n W x N W P x x x x W ) ( ) (, 3 () and n n t n x L P x x ) ( ] [ ) (, (3) where <N(x)> and <L(x)> are both row vectors consisting of cubic and inear shape functions for bending and torsion, respectivey. The cubic shape functions N, N, N3 and N4 are, 3 ) ( 3 3 x x x N (4) x x x x N 3 ) ( ) ( x x x N x x x N 3 4 ) (

23 The inear shape functions L and L are defined as, L ( x) L ( x) x x (5) A graphica representation of the shape functions described in equations (4) and (5) above coud be seen in Figure in Appendix A. This discretizing process eads to the eement stiffness, mass and couping matrices which when assembed together within the FEM code written in MATLAB woud resut in the inear Eigenvaue probem shown in equation (6) beow. W n K M W det( K M ) n (6) where K stands for the goba stiffness matrix, which is a coection of a the eement stiffness matrices. The goba mass matrix is symboized by M. Matrix (6-a) shown beow is the eement mass matrix, [m] k, and matrices (6-b) through (6-f) are the uncouped, couped and geometric eement stiffness matrices. When matrices (6-b) through (6-f) are assembed together, the fina eement stiffness matrix woud resut. This is shown as matrix (6-g). 4

24 [ m] k 56m 4 m 4 3 4m 4 Sym. I P 3 54m 4 3m 4 56m 4 3m 4 3 3m 4 m 4 3 4m 4 I P 6 I P 3 (6-a) where stands for the eement ength and m represents the eement mass per unit ength. The eement uncouped bending stiffness matrix, [kb], is shown beow. [ k B EI 3 ] 6EI 4EI Sym. EI 3 6EI EI 3 6EI EI 6EI 4EI (6-b) The fina eement stiffness matrix is modified due to the presence of the end-moment and axia oad which contributes the [k]geometric matrix, [k]torsion matrix, bending torsion couping stiffness matrix, [kbt]c, and the torsion bending couping stiffness matrix, [ktb]c. These are added to the bending stiffness matrix, [kb], above, to form the fina eement stiffness matrix, [k] k. The geometric and torsion stiffness matrices contributed by the axia oad P are shown beow. 5

25 ] [ Sym P k geometric (6-c) ] [ A PI GJ k P torsion (6-d) The bending torsion and torsion bending couping stiffness matrices introduced by the endmoment M are as foows. ] [ M k c BT (6-e) ] [ M k c TB (6-f) Therefore, the fina eement stiffness matrix, which is a coection of the five sub matrices, takes the foowing form.

26 [ k] k EI 3 6P 5 6EI P 4EI P 5 Sym. M GJ PI P A EI 6P 3 5 6EI P M EI 6P 3 5 6EI P EI P 3 6EI P 4EI P 5 M GJ PI A M GJ PI P A P (6-g) The soution to the inear Eigenvaue probem in equation (6) is achieved by determining the Eigenvaues and Eigenvectors using a FEM code deveoped in MATLAB. Various cassica boundary conditions are aso appied within the MATLAB code. Thus, the natura frequencies and mode shapes of the beam are evauated. 7

27 . FEM Numerica Tests A Stee beam (E= GPa and d=78 kg/m 3 ), having a ength of 8m, width of.4m and depth of.m was studied. The first stage of the numerica tests was to vaidate the resuts obtained using the deveoped FEM code with known exact resuts. Due to the ack of anaytica resuts for the probem containing an axia oad and end-moment, the accuracy of the natura frequency vaues from the code were compared with the anaytica resuts for a beam without any force or moment. Tabe beow incude the resuts for the first three natura frequencies for various boundary condition types using the exact [4] and FEM methods. Tabe : Comparison between the exact and FEM resuts for the first three natura frequencies at P = and M = Boundary Condition C - F C - C P - P P - C Natura Frequencies (Hz) at P = and M = Mode Mode Mode 3 Exact [4] FEM (4 eements) Exact [4] FEM (4eements) Exact [4] FEM (4eements) From Tabe above it can be observed that the resuts produced by the exact method and the FEM method are identica and as such the FEM code generates accurate resuts. In order to determine the rate of convergence of the FEM method, the fifth natura frequency of a cantievered beam was used. Since couping caused by end-moment introduces more error to the system, the highest possibe oading configuration with.85 MN of tensie force and 9. MN.m of end-moment was appied. The convergence was verified by using different numbers of eements 8

28 to generate the mesh and finding the east number of eements that woud yied acceptabe resuts. Once again, as there were no anaytica resuts avaiabe for simiar cases, the FEM code was used to determine the exact resuts by progressivey increasing the number of eements used from to. Tabe beow incudes these resuts and it can be observed that the vaue for the fifth natura frequency remains at Hz when or more eements are used. Thus, as the resut has converged and it does not change even when eements are utiized, Hz is taken to be the exact vaue for the 5 th natura frequency when the cantievered beam is subjected to a tensie force of.85 MN and end-moment of 9. MN.m. However, since 4 eements is sufficient to obtain a resut with an error ess than. percent, 4 eements were considered as a reasonabe number of eements for the FEM method. Convergence of the resuts for the first four fundamenta frequencies were aso checked and it was observed that these resuts converged to the anaytica resut with esser number of eements. Tabe : Convergence of the fifth natura frequency No. of Eements Mode 5 (Hz)

29 Error (%) Figure 3 beow is a graphica representation of the reduction in percentage error between the FEM and exact resuts as the number of eements is increased. It iustrates that the resut for the fifth natura frequency of the beam subjected to the above mentioned oading and boundary conditions woud converge to the exact resut when eements are used. It aso depicts that 4 eements are sufficient to attain a percentage error beow. percent No. of Eements Figure 3: Convergence anaysis for conventiona FEM for cantievered beam A pre-stressed moda anaysis was conducted using ANSYS 4 to simuate the probem and to further vaidate the FEM code resuts. For FEM meshing of the beam, SOLID 87 eements were used (see Figure 3 in Appendix B). The SOLID 87 eement is a higher order, 3D, node eement capabe of 6 degrees of freedom (3 transations and 3 rotations) per node. A tota of eements were used for the meshing process in ANSYS. The first natura frequency of the beam was determined for a cassica boundary conditions when subjected to an axia tensie oad and an end-moment. The accuracy of the resuts produced by the proposed FEM method was checked using ANSYS commercia software for the cantievered case and the resuts are incuded in Tabe 3 beow.

30 Tabe 3: First natura frequencies for cantiever boundary condition (C F) when force and moment are appied; a 4-eement FEM mode is used End-Moment C-F (MN.m) 6.4 (MN.m) 9. (MN.m) Fundamenta Frequency (Hz) Force FEM Code FEM Code FEM Code ANSYS ANSYS ANSYS (MN) (4 eements) (4 eements) (4 eements) As can be observed from Tabe 3, the deveoped FEM code yieds resuts that are coser to the exact resuts, compared to the ANSYS simuation. This coud be due to the shear and warping effects of the 3D eement (SOLID 87) used in ANSYS that were not accounted for in the code. Tabes 4 through 6 beow incude the resuts for the first natura frequency for different combinations of tensie force and end-moment for the camped camped, pinned pinned and pinned camped boundary conditions, respectivey.

31 Tabe 4: Fundamenta frequencies for camped camped boundary condition (C C), when force and end-moment are appied; a 4-eement FEM mode is used C-C (MN.m) End-Moment (MN.m) (MN.m) Force (MN) FEM Fundamenta Frequency (Hz) Tabe 5: First natura frequencies for pinned pinned boundary condition (P P) when force and moment are appied; a 4-eement FEM mode is used P-P (MN.m) End-Moment 6.4 (MN.m) 9. (MN.m) Force (MN) FEM Fundamenta Frequency (Hz)

32 Tabe 6: First natura frequencies for pinned camped boundary condition (P C) when force and moment are appied; a 4-eement FEM mode is used P-C (MN.m) End-Moment 6.4 (MN.m) 9. (MN.m) Force (MN) FEM Fundamenta Frequency (Hz) The critica bucking end-moments and compressive forces were aso determined for the cantievered boundary condition and the resuts are shown in Tabe 7 and Tabe 8 beow. Tabe 7: Critica bucking moment for cantievered boundary condition (C - F) with varying compressive force Force (MN) Bucking Moment (MN.m)

33 Tabe 8: Critica bucking compressive force for cantievered boundary condition (C - F) with varying end-moment Moment (MN.m) Bucking Force (MN) Figures 4 through 7 iustrate the variation of the first natura frequency when both tensie force and end-moment is acting on the beam, for various cassica boundary conditions. Figure 4:Variation of natura frequencies when tensie force and end-moment is appied for cantievered (C-F) boundary condition 4

34 Figure 5: Variation of natura frequencies when tensie force and end-moment is appied for camped camped (C-C) boundary condition Figure 6: Variation of natura frequencies when tensie force and end-moment is appied for pinned pinned (P-P) boundary condition 5

35 Figure 7: Variation of natura frequencies when tensie force and end-moment is appied for camped pinned (C-P) boundary condition Figure 8 depicts how the critica bucking compressive force changes when the end-moment is varied. Simiary, Figure 9 shows the fuctuation of the magnitude of the critica bucking moment with respect to the change in tensie and compressive axia force. Figure 8: Variation of critica bucking compressive force with end-moment 6

36 Bending Dispacement (m) Figure 9: Variation of critica bucking end-moment with tensie and compressive force In Figure and Figure, respectivey, the bending and torsiona mode shapes of the first five natura frequencies for the cantievered beam subjected to a tensie force of.85mn and endmoment of 9. MN.m are shown Mode Mode Mode 3 Mode 4 Mode Distance from Camped End (m) Figure : Bending components of mode shapes 7

37 Torsiona Dispacement (m) Mode Mode Mode 3 Mode 4 Mode Distance from Camped End (m) Figure : Torsiona components of mode shapes The frequency resuts of the deveoped FEM code showed good agreement with the ANSYS resuts and the error was ess than percent. As expected, tensie axia oad increased the natura frequencies of the beam thus, indicating and increase in the stiffness of the beam for a cassica boundary condition types. The increment in natura frequencies corresponding to the appication of higher vaues of tensie oad shows that the beam gets stiffer when the tensie oad is increased. When the end-moment is appied without a tensie force, the natura frequencies reduce again for a boundary conditions indicating that the moment aso causes a reduction in stiffness of the beam. Any increment in the magnitude of the appied end-moment essened the stiffness of the beam further. As can be seen from Tabes 3 through 6 and Figures 4 through 7, if the end-moment is hed constant and the tensie oad is increased, the natura frequencies increase indicating an increase in beam stiffness. However, on the contrary, if the tensie oad is hed constant and the end-moment is increased, the beam stiffness reduces. 8

38 A compressive axia oad has the opposite effect to that of a tensie oad, on the natura frequencies and stiffness of a beam. The resuts in Tabe 7 show that the critica bucking moment is.8 MN.m when no force is acting, for the cantievered boundary condition. However, when a compressive force is appied, the beam buckes at much ower magnitudes of the end-moment. Thus, the magnitude of the critica bucking moment reduces with a progressive increase in the compressive oad appied. In contrast, resuts in Tabe 7 show that when a simiar magnitude of tensie force is appied, the critica bucking moment increases. This further confirms the caim that tensie forces introduce additiona reinforcement to the structure, increasing its stiffness. More quantitativey, the critica bucking end-moment increases from 3.9 MN.m to 6.95MN.m simpy by reversing the direction of the appied force whie keeping the magnitude constant at.85mn. The above resuts are depicted graphicay in Figure 9. Simiary, Tabe 8 shows that the magnitude of the critica bucking compressive force reduces with increasing vaues of end-moment, further confirming that the end-moment has a detrimenta effect on the stiffness of the beam. This is shown in Figure 8. Figure and Figure, respectivey, depict the fexura and torsiona components of the mode shapes of the beam. As can be seen from these figures, the vibration of the beam is predominanty fexura in the first three natura frequencies. Torsion becomes predominant in the 4 th natura frequency. The 5 th natura frequency again becomes predominanty fexura. 9

39 3. The Dynamic Finite Eement (DFE) Method A ineary eastic, homogeneous and isotropic beam with the same dimensions, materia properties, oading configuration and boundary conditions as the beam used in the previous chapter is considered. 3. The Dynamic Stiffness Matrix method If the beam eement is considered to be uniform and homogeneous, thus, making materia and geometric properties such as EI, GJ and mass uniform throughout the beam, it is possibe to deveop a DSM formuation. This is not aways the case and materia non uniformity can make the DSM method inappicabe. However, the frequency dependent shape functions of the DFE method are found by using the exact soution to the uncouped governing equations as expansion terms. 3. The conventiona FEM method based on poynomia interpoation functions As seen in the previous chapter, the FEM method is very fexibe and convenient. Unike the DSM method, it coud be extended to incude systems consisting of materia and geometric non inearity making it a usefu method for compex probems. However, the FEM method uses a arge number of eements to achieve a reasonabe degree of accuracy especiay when the higher frequencies of vibration are of interest, thus, consuming more computationa overhead and increasing round off error. On the other hand the DFE method has been proven in the past [, 3] to produce exact resuts using just eement for uncouped systems. Highy accurate resuts for couped systems have been achieved using the DFE method with 5 or 6 eements [, 3]. The formuation of the equations for the FEM and DFE methods are the same up to the point of discretization (equation 7), using two-node, six degree-of-freedom eements (see Figure ). Beyond this point, the two 3

40 methods diverge in their formuation processes as the FEM method utiizes the cubic Hermite shape functions (equation 4) and inear shape functions (equation 5) for bending and torsion, respectivey, as opposed to the DFE method which uses trigonometric shape functions presented in [] and [3]. 3.3 The frequency- dependent Dynamic Finite Eement method based on trigonometric shape functions In this section, the fexura torsiona vibration and stabiity of a simpe Euer Bernoui beam wi be investigated using the DFE method. As mentioned previousy, the DFE method is a hybrid and intermediate method that combines the accuracy of the DSM method as we as the adaptabiity of the conventiona FEM method to obtain a better finite eement mode. Therefore, the starting point of the DFE formuation woud be the two discretized weak form equations from the conventiona FEM shown beow, after the bracketed boundary terms expained in the previous chapter vanished. x k j '' '' ' ' ' ' f ( ) ( ) x j W x EIW W PW W M W A W W dx (7) x k j ' ' PIP ' ' ' ' Wt ( x) ( GJ MW IP ) dx (8) x j A Two sets of integration by parts wi be carried out on the first two terms of the equation for fexure and one set of integration by parts wi be performed on the first two terms of the equation for torsion in order to obtain the foowing forms. xj x '''' ' j ' ' ' '' ''' ' (9) W k ( x f ) ( EIW W PW W A W W ) dx M W dx EIW W EI WW PWW xj xj 3

41 xj x k PI '' p '' j PI ' ' ' p ' Wt x ( GJ I p ) dx MW dx GJ xj A xj A (3) Substituting, ξ = / in both equations above woud resuts in, k '''' '' ' '' ''' ' Wf W ( EIW PW A W ) d M W d EIW W EIWW PWW 3 3 (3) * '' PI p '' ' ' PI k ' p ' Wt ( GJ I ) p d MW d GJ A A ** The interpoation functions which respect the noda properties woud be the soutions to the (3) integra terms marked as (*) and (**). Thus, the non noda approximation of the soution functions, W and θ, and the test functions δw, and δθ, written in terms of generaised parameters are as foows. W P a W P a (33) f t t f P b P b (34) The basis functions of the approximation are shown beow. These basis function are the soutions to the characteristic equations (*) and (**). When the roots, α, β, and τ of the characteristic equations tend to zero, the resuting basis functions are simiar to that of a standard beam eement in the cassica FEM where fexure and torsion are approximated using cubic Hermite poynomias and inear functions, respectivey. P f sin cosh cos sinh sin cos ; ; ; 3 3 (35) The roots, α, β, and τ are defined as foows. P cos ;sin / (36) t 3

42 X B B 4AC B B 4AC X A A (37) where the constants, EI P ( A B C m ) 3 (38) Thus, the roots are, X X (39) and, I A p AGJ PI p (4) Repacing the generaized parameters, a, δa, b and δb in equations (33) and (34) with the noda variabes, W W W W, δw δw δw δw, θ θ and δθ δθ, and re-writing equations (33) and (34) wi resut in equations (4) and (4) beow. W P a W P a (4) n n f n n P b P b n n t n n t The matrices,[p n ] f and [P n ] t, are defined as, f (4) [ P ] n f cos( ) sin( ) sin( ) cos( ) [cosh( ) cos( )] ( ) [ sinh( ) sin( )] ( ) ( ) 3 3 ( ) [sinh( ) sin( )] 3 3 ( ) [ cosh( ) cos( )] 3 3 ( ) (4-a) 33

43 P sin( ) n ] cos( ) (4-b) [ t Thus, equations (33), (34) and the, [Pn]f, and [Pn]t, matrices above coud be combined in the foowing manner to construct noda approximations for fexura dispacement W(ξ) and torsion dispacement θ(ξ). W ( ) P ( ) P W N ( ) W (43) f n f n f n ( ) P ( ) P N ( ) (44) t n t n t n In equations (43) and (44), N(ξ) f, and N(ξ) t, are the frequency dependent trigonometric shape functions for fexure and torsion, respectivey. Equations (43) and (44) coud aso be re-written as, W ( ) ( ) Nw n (45) where, N f ( ) N f ( ) N3 f ( ) N4 f ( ) [ N] N t( ) Nt( ) (46) and ' ' T w WW W W (47) n The definitions of the frequency-dependent trigonometric shape functions for fexure obtained from [] and [3] are as foows. Athough, dependent on the frequency ω esewhere (see Figures 4 through 9 in Appendix A), the trigonometric shape functions for bending are independent of the frequency at the eement boundaries and as such, Nf =, Nf =, N3f =, and N4f =, at ξ = 34

44 , and Nf =, Nf =, N3f =, and N4f =, at ξ =. See Figure 3 in Appendix A for an iustration of fexura shape functions used for the DFE method. ( ) N f ( ) cos( ) cos( ( )) cosh( ) cos( ) cosh( ( )) Df (48) cosh( ) sin( ( )) sinh( ) *sin( )*sinh( ( )) N f ( ) cosh( ( )) sin( ) cosh( ) sin( ( )) sin( ) D f cos( ( )) sinh( ) cos( ) sinh( ( )) sinh( ) (49) ( ) N3 f ( ) cos( ( )) cos( ) cosh( ) cosh( ( )) cos( ) Df (5) cosh( ) sin( ) sinh( ) sin( ) sinh( ) N 4 f ( ) cosh( ) sin( ) sin( ( )) cosh( ) sin( ) D f cos( ) sinh( ) sinh( ( )) cos( ) sinh( ) (5) where, D f ( ) ( cos( ) cosh( )) sin( ) sinh( ) (5) The trigonometric shape functions for torsion presented in [] and [3] are shown beow. Simiary, these shape functions are aso independent of ω at the eement boundaries such that, Nt = and Nt =, at ξ = and Nt = and Nt =, at ξ =. See Figure 3 in Appendix A for a graphica representation of the torsiona shape functions utiized in the DFE method. 35

45 N t sin( ) ( ) cos( ) cos( ) (53) D N t t sin( ) ( ) (54) D t where, Dt sin( ) (55) Therefore, using equations (3), (3) and the shape functions (48) through (55), the eement stiffness matrix is obtained. The eement stiffness matrix, [K DS ] k, consists of two couped dynamic k stiffness matrices, [K DS ] BT,c uncouped dynamic stiffness matrices, [K DS ] k u as, [K DS ] u k. k, and [K DS ] TB,c, symboized coectivey as, [K DS ] k c, and four k k, [K DS ] u, [K DS ] u3, and [K DS ] u4, jointy denoted k The fina eement dynamic stiffness matrix,[k DS ] k, is determined by assembing these six couped and uncouped sub matrices as shown beow. k k n DS DS n c u W k w K K w (56) The goba dynamic stiffness matrix, [KDS], is then obtained by assembing a the eement stiffness matrices together. This process was performed using a DFE code written in MATLAB software which resuted in the non inear Eigenvaue probem shown in equation 57 beow. ( ) K (57) DS W n Matrices (57-a) through (57-d) beow are the four uncouped eement stiffness matrices mentioned above. 36

46 [ K ] DS k u EI 3 L N N N N N N N N N N N N N N N N N N N N N N N N N N N ' '' ' '' ' '' ' '' f f f f f 3 f f 4 f ' '' ' '' ' '' ' '' f f f f f 3 f f 4 f ' '' ' '' ' '' ' '' 3 f f 3 f f 3 f 3 f 3 f 4 f ' '' ' '' ' '' ' '' 4 f f 4 f N f N4 f N3 f N4 fn4 f (57-a) [ K ] ''' ''' ''' ''' N f N f N f N f N f N3 f N f N 4 f ''' ''' ''' ''' EI N f N f N f N f N f N3 f N f N4 f ''' ''' ''' ''' L N3 f N f N3 f N f N3 f N3 f N3 f N 4 f ''' ''' ''' ''' N4 f N f N4 f N f N4 f N3 f N4 f N 4 f k DS u 3 (57-b) [ K ] DS k u3 N N N N N N N N P N N N N N N N N L N N N N N N N N N N N N N N N N ' ' ' ' f f f f f 3 f f 4 f ' ' ' ' f f f f f 3 f f 4 f ' ' ' ' 3 f f 3 f f 3 f 3 f 3 f 4 f ' ' ' ' 4 f f 4 f f 4 f 3 f 4 f 4 f (57-c) ' ' k P t t t t DS u4 GJ ' ' L A NtN t NtNt [ K ] PI N N N N (57-d) Matrices (57-e) and (57-f) are the two couped stiffness matrices stated previousy. [ K ] k DS BT, c ' ' ' ' ' ' ' ' M N t N f N t N f N t N3 f N t N 4f ' ' ' ' ' ' ' ' d L Nt Nf Nt N f Nt N3 f Nt N4 f (57-e) [ K ] DS k TB, c ' ' ' ' N f N t N f N t ' ' ' ' M N f N t N f Nt ' ' ' ' L N3 fnt N3 f N d (57-f) t ' ' ' ' N4 f N t N4 f N t Various cassica boundary conditions were aso appied on the goba dynamic stiffness matrix within the MATLAB code. The natura frequencies of the system woud be the vaues of ω that woud yied a zero determinant for the goba dynamic stiffness matrix. This is obtained by sweeping the frequency domain using visua approximation to find particuar vaues of ω that 37

47 produce a zero determinant. The Eigenvectors corresponding to these natura frequencies provide the mode shapes of the system. 38

48 3.4 DFE Numerica Tests Figure beow depicts the convergence anaysis carried out for the 5 th natura frequency of a cantievered beam when a tensie force of.85 MN and end-moment of 9. MN.m was acting. As the DFE method yieds exact resuts with just one eement for a other oading configurations pertaining to this study, except for when an end-moment is appied, oading configurations with no moment were not considered for the convergence anaysis. Furthermore, the same oading conditions used for the convergence anaysis of the FEM method were utiized to ensure ease of comparison of the rates of convergence of the two methods. A comparison between the DFE method and conventiona FEM with regards to the efficiency in convergence is iustrated in Figure 3. For the 5 th natura frequency, the DFE method produces resuts with an error ess than. percent compared to the exact resut with just 5 eements or to an error ess than. percent using 8 eements. As can be seen from both Figure and Figure 3, if the number of eements in the DFE method is increased to, the percentage error woud be further reduced. In contrast, the FEM method requires at east eements to achieve an error ess than. percent for the 5 th natura frequency. Thus, even for higher natura frequencies the DFE method uses.5 times ess eements than the FEM method and as such the former is capabe of converging faster. It is important to note here that for the st, nd, 3 rd and 4 th natura frequencies, the DFE method produces much accurate resuts with even fewer eements. This can be observed from Tabe 9 beow which shows that the DFE method converges to the FEM resut for the st natura frequency with an error between.7 percent using just 5 eements, whereas the FEM method required 4 eements to achieve this. Thus, the FEM method used 8 times more eements than the DFE method. Tabe 9 aso shows that when no end-moment is acting, the DFE method is capabe of producing the exact resuts [4] for the uncouped case of P = and M = using just eement whereas the 39

49 Error (%) FEM requires 4 eements to achieve the same degree of accuracy. Since, the DFE method uses the exact soutions to the uncouped governing differentia equations as expansion terms (i.e. basis functions of approximation space) to obtain the frequency dependant shape functions that are ater utiized to approximate the bending and torsiona dispacements as we as rotation, the DFE method is capabe of producing the exact resuts with a singe eement if the system is uncouped. However, when the end-moment is appied, the DFE method requires up to 5 eements to generate resuts simiar to the FEM resuts, with an error ess than. percent. According to the third terms (couping terms) of both equations () and (), the end-moment, M, coupes the two governing differentia equations. When the dynamic shape functions that are based on the soutions to the uncouped systems are used to approximate the behaviour of a geometricay couped system when end-moment is appied, error is introduced in to the cacuations and thus, more than one eement is required to converge the resuts to a suitabe degree of accuracy No. of Eements Figure : Convergence anaysis for DFE method for cantievered beam 4

50 Error (%) DFE FEM No. of Eements Figure 3: Comparison of convergence efficiency between DFE method and conventiona FEM for cantievered beam Tabes 9 through show the resuts for the first natura frequency for cantievered, camped camped, pinned pinned and pinned camped boundary conditions, respectivey. Tabe 9: First natura frequencies for cantiever beam (C F) when force and moment are appied End-Moment C-F (MN.m) 6.4 (MN.m) 9. (MN.m) FEM Fundamenta Frequency (Hz) Force (MN) DFE ( eement) FEM (4 eements) DFE (5 eements) FEM (4 eements) Error (%) DFE (5 eements) FEM (4 eements) Error (%)

51 Tabe : First natura frequencies for camped camped boundary condition (C C) when force and moment are appied End-Moment C-C (MN.m) 6.4(MN.m) 9.(MN.m) FEM Fundamenta Frequency (Hz) Force (MN) DFE ( eement) FEM (4 eements) DFE (5 eements) FEM (4 eements) DFE (5 eements) FEM (4 eements) Tabe : First natura frequencies for pinned pinned boundary condition (P P) when force and moment are appied End-Moment P-P (MN.m) 6.4 (MN.m) 9. (MN.m) FEM Fundamenta Frequency (Hz) Force (MN) DFE ( eement) FEM (4 eements) DFE (5 eement) FEM (4 eements) DFE (5 eement) FEM (4 eements)

52 Tabe : First natura frequencies for pinned camped boundary condition (P C) when force and moment are appied End-Moment P-C (MN.m) 6.4 (MN.m) 9. (MN.m) FEM Fundamenta Frequency (Hz) Force (MN) DFE ( eement) FEM (4 eements) DFE (5 eements) FEM (4 eements) DFE (5 eements) FEM (4 eements) A bucking anaysis was aso carried out using the DFE method and the resuts for the critica bucking compressive force and end-moment are incuded in Tabe 3 and Tabe 4 beow. Tabe 3: Critica bucking moment for cantievered boundary condition (C - F) with varying compressive force Force (MN) Bucking Moment (MN.m) DFE (5 eement)

53 Fundamenta Natura Frequency (Hz) Tabe 4: Critica bucking compressive force for cantievered boundary condition (C - F) with varying end-moment Moment (MN.m) Bucking Force (MN) DFE (5 eements) Figures 4 through 7 beow are graphica representations of the resuts in Tabes 9 through. These figures iustrate the variation of the first fundamenta frequency with tensie axia force and end-moment M= M = 6.4 MN.m.5 M = 9. MN.m Axia Force (MN) Figure 4: Variation of natura frequencies when tensie force and end-moment is appied for cantievered (C-F) boundary condition 44

54 Fundamenta Natura Frequency (Hz) Fundamenta Natura Frequency (Hz) M= 6. M = 6.4 MN.m 5.8 M = 9. MN.m Axia Force (MN) Figure 5: Variation of natura frequencies when tensie force and end-moment is appied for camped camped (C-C) boundary condition M= M = 6.4 MN.m M = 9. MN.m Axia Force (MN) Figure 6: Variation of natura frequencies when tensie force and end-moment is appied for pinned pinned (P-P) boundary condition 45

55 Critica Bucking Moment (MN.m) Fundamenta Natura Frequency (Hz) M=. M = 6.4 MN.m.8 M = 9. MN.m Axia Force (MN) Figure 7: Variation of natura frequencies when tensie force and end-moment is appied for pinned camped (P-C) boundary condition Figure 8 beow iustrates how the critica bucking end-moment varies with axia force. Figure 9 depicts the fuctuation of the critica bucking compressive force with changing end-moment Axia Force (MN) Figure 8: Variation of critica bucking end-moment with axia force for cantievered (C-F) boundary condition 46

56 Bending Dispacement (m) Critica Bucking Force(MN) End Moment (MN.m) Figure 9: Variation of critica bucking compressive force with end-moment for cantievered (C-F) boundary condition Figure and Figure show the bending and torsiona components of mode shapes of a cantiever beam when a force of.85 MN and moment of 9. MN.m is appied, respectivey Mode Mode Mode 3 Mode 4 Mode Distance from Camped End (m) Figure : Bending component of mode shapes of the system 47

57 Torsiona Dispacement (m) Mode Mode Mode 3 Mode 4 Mode Distance from Camped End (m) Figure : Torsiona component of mode shapes of the system The resuts obtained using the DFE method are in exceent agreement with the resuts of the conventiona FEM for a types of oading and cassica boundary conditions, as can be observed from Tabes 9 through. Once again it is evident that the natura frequency of the system increases when the tensie axia force is appied. Any increment in the magnitude of the tensie force further increases the natura frequency of the system suggesting of a simutaneous increment in the stiffness of the beam. This is further confirmed in Figure 8 and Tabe 3 as it coud be observed that a tensie force of.85 MN causes the critica bucking moment to increases to 7 MN.m. Requiring a arger moment to cause bucking indicates a stiffening of the beam. The DFE method further confirms that appication of an end-moment reduces the stiffness of the beam. If the magnitude of the end-moment is increased, the vaue of the fundamenta natura frequencies reduces proportionatey. This observation is supported by the resuts incuded in 48

58 Figure 9 and Tabe 4 which show that the higher the end-moment appied, the smaer the compressive force required to bucke the beam. If the beam buckes due to a sma compressive force, this means that it has ost stiffness due to the moment. Tabes 9 through aso incude the effect of the end-moment and tensie force when they are acting together. If the tensie force is kept constant and the end-moment is increased, the natura frequencies of the beam reduce. The opposite happens, if the end-moment is unchanged whie the tensie force is increased. A the above observations hod true for a cassica boundary conditions considered in this study. Graphica representations of the resuts in Tabes 9 through are shown in Figures 4 through 7, respectivey. It coud aso be observed that Figures 4 through 7 that were generated using the resuts of the DFE method are amost identica to Figures 4 through 7 produced using FEM resuts. Thus, the DFE resuts agree we with the findings from the FEM method. Resuts from the DFE method aso confirm that unike tensie force, appication of compressive force causes a reduction in the stiffness of the beam which is accompanied by a simutaneous reduction in the natura frequencies of the system. As seen from Figure 8 and Tabe 3, when no axia oad is appied, the critica bucking moment of the beam is.33 MN. However, when a compressive force of -.85 MN is acting, the critica bucking end-moment reduces to 3.9 MN.m, which is ess than the critica moment when no forces are acting whatsoever. The resuts in Figure 9 and Tabe 4 further confirm this. Figure 9 and Tabe 4 show that if the compressive force is reduced to -.9 MN a very arge bucking moment of 9. MN.m is required to cause eastic instabiity of the beam, however, if the compressive force is increased to -.6 MN the beam woud bucke even without the presence of an end-moment. These resuts are true for a boundary conditions and amost identica to the findings from the conventiona FE method. Thus, the 49

59 findings of the DFE method agree with the findings in the iterature that a compressive axia oad reduces the fundamenta frequencies as we as the stiffness of a beam. Figure iustrates that bending dispacement is predominant unti the fourth mode is reached. In Figure, the fourth mode shape shows very itte bending dispacement compared to the rest of the modes. However, in Figure the fourth mode shows significant magnitudes of torsiona dispacement. Thus, the fourth mode shape is the first major torsiona mode shape. The concusion that the st, nd 3 rd and 5 th modes are predominanty fexura is confirmed by the fact that these mode shapes show very itte torsiona dispacement in Figure. The mode shapes generated using the DFE method are very much identica to those produced by the FEM method seen in Figure and Figure. Figures 3 through 35 in the Appendix B depict the couped bending torsion mode shapes generated using ANSYS. Figure 34 from ANSYS, confirms that the fourth mode is the first predominanty torsiona mode. 5

60 4. Concuding Remarks and Recommendations for Future Works The study confirms the findings from past iterature that tensie forces increase the natura frequencies and stiffness of a beam whie compressive forces and end-moments reduces the natura frequencies and stiffness of a beam for a cassica boundary conditions. The resuts determined using the DFE method are in agreement with the resuts found from the FE method. Most importanty, this study demonstrates that the DFE method coud be extended to anayse systems exhibiting geometric coupings due to the presence of an end-moment. The Dynamic Finite Eement (DFE) method is a superconvergent method that requires significanty ess eements compared to the conventiona FEM method and as such is very efficient. If the system is uncouped, the DFE method is capabe of producing exact resuts with just a singe eement as it utiizes the exact soutions to the uncouped governing differentia equations as dynamic basis functions of approximation space to deveop the frequency dependent trigonometric shape functions. Even for a couped system, the DFE method requires much ess eements compared to the FE method. Another advantage of the DFE method is that, unike the conventiona FE method, it aows the user to determine if a natura frequency of the system exists within a given range of frequencies. This is very important for aerospace appications where the range of operating frequencies of a certain component is aready known, since the DFE method coud then be used to determine if a natura frequency of the component occurs within the range of its operationa frequencies. It woud aow the designers to eiminate or imit the risk of resonance. Apart from the inherent advantage of aowing for a much coarser mesh, thus resuting in a argey simpified Eigenvaue probem due to the esser number of eements compared to the conventiona FEM method, the DFE method is aso a powerfu too that coud be used during the preiminary design stages to arrive at natura 5

61 frequencies and mode shapes of a component with acceptabe precision within a very short time. Especiay during the eary design stages for aerospace components that coud be modeed as beams, such as heicopter, propeer and compressor bades or for structures such as panar or space frames that coud be represented by assembages of identica beams, the DFE beam mode woud be usefu as the couped natura frequencies of these components are necessary for aeroeastic cacuations. For future works, the effects of warping for torsion coud be incuded to take in to account thinwaed beams and beams with open cross sections. Furthermore, the effects of shear coud aso be investigated. The DFE beam mode presented here coud be used as a base on to which more compex features such as geometric noninearity and variation of materia properties are incorporated in the future. Athough, in this study the frequency domain was swept using visua inspection to determine the system natura frequencies, an improvement coud be made to the method by empoying the Wittrick Wiiams [6] root finding agorithm for determinant search, thus ensuring that no natura frequencies are missed. Furthermore, the DFE method coud be used to determine the vibration and stabiity of Timoshenko beams and ayered (hybrid) beams. Finay, in the future the DFE method coud aso be extended to carry out vibrationa anayses of two dimensiona eements such as pates. As the Dynamic Finite Eement method described in this study is not ony imited to beams, it coud be extended to pate structures which are more commony used to mode fuseage and wing skins. As the geometries and materia properties of these structures are very compex, future studies coud aso focus on deveoping robust DFE formuations of pate structures that coud be convenienty used to mode the skins of fuseages and wings during preiminary design stages. 5

62 N(ξ) N(ξ) 5. Appendix A: Shape Functions Nf Nf N3f N4f Nt Nt ξ Figure : Cubic Hermite shape functions for bending and inear shape functions for torsion used in conventiona FEM Nf Nf N3f N4f Nt Nt ξ Figure 3: Frequency dependent trigonometric bending and torsion shape functions presented in [] and [3] used in the DFE method 53

63 Nf (ξ) Nf (ξ) Nf (omega ) Nf (omega ) Nf (omega 3) ξ Figure 4: Dependency on the frequency of the first bending shape function [, 3] used in the DFE method Nf (omega ) Nf (omega ) Nf (omega 3) ξ Figure 5: Dependency on the frequency of the second bending shape function [, 3] used in the DFE method 54

64 N4f (ξ) N3f (ξ) N3f (omega ) N3f (omega ) N3f (omega 3) ξ Figure 6: Dependency on the frequency of the third bending shape function [, 3] used in the DFE method N4f (omega ) N4f (omega ) N4f (omega 3) ξ Figure 7: Dependency on the frequency of the fourth bending shape function [, 3] used in the DFE method 55

65 Nt (ξ) Nt (ξ) Nt (omega ) Nt (omega ) Nt (omega 3) ξ Figure 8: Dependency on the frequency of the first torsiona shape function [, 3] used in the DFE method Nt (omega ) Nt (omega ) Nt (omega 3) ξ Figure 9: Dependency on the frequency of the second torsiona shape function [, 3] used in the DFE method 56

66 6. Appendix B: ANSYS Mode Figure 3: Beam meshed in ANSYS using SOLID 87 eements Figure 3: First couped bending torsion mode shape from ANSYS 57

67 Figure 3: Second couped bending torsion mode shape from ANSYS Figure 33: Third couped bending torsion mode shape from ANSYS 58