VIBRATION ANALYSIS OF PRE-TWISTED ROTATING BEAMS. Tolga YILDIRIM

Transcription

1 VIBRATION ANALYSIS OF PRE-TWISTED ROTATING BEAMS B Tolga YILDIRIM A Dissertation Submitted to the Graduate School in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Department: Mechanical Engineering Major: Mechanical Engineering Izmir Institute of Technolog Izmir, Turke Ma, 3

2 We approve the thesis of Tolga YILDIRIM Date of Signature Bülent Yardımoğlu Asst. Professor of Mechanical Engineering Thesis Advisor H. Seçil Altundağ Artem Asst. Professor of Mechanical Engineering Engin Aktaş Asst. Professor of Civil Engineering M. Barış Özerdem Assoc. Professor and Chairman of the Department of Mechanical Engineering ii

3 ACKNOWLEDGEMENTS The present work developed during m activit as a research assistant in the Department of Mechanical Engineering at Izmir Institute of Technolog, Izmir. First and foremost, I wish to thank m advisor Asst. Prof. Dr. Bülent Yardimoglu, who made it possible for me to write this thesis. I consider mself luck to have two valuable ears working with him. For creating a pleasant working atmosphere I would like to thank the facult members and research assistants at the Department of Mechanical Engineering. For their patience and friendships special thanks to the research assistants Ahmet Kaan Tokso, Nurdan Yildirim, Pinar Ilker Alkan, Farah Tatar, Adil Caner Sener, Abdullah Berkan Erdogmus and Orhan Ekren, with whom I shared the same office at different times. Last but not least, I would like to thank the administrative personnel Deniz Demirli and Acin Bermant Ercan, from whom in one wa or another I received help during m activit as a research assistant in the Department of Mechanical Engineering at Izmir Institute of Technolog. Izmir, Ma 3 Tolga Yıldırım iii

4 ABSTRACT A new linearl pretwisted rotating Timoshenko beam element, which has two nodes and four degrees of freedom per node, is developed and subsequentl used for vibration analsis of pretwisted beams with uniform rectangular crosssection. First, displacement functions based on two coupled displacement fields (the polnomial coefficients are coupled through consideration of the differential equations of equilibrium) are derived for pretwisted beams. Net, the stiffness and mass matrices of the finite element model are obtained b using the energ epressions. Finall, the natural frequencies of pretwisted rotating Timoshenko beams are obtained and compared with previousl published both theoretical and eperimental results to confirm the accurac and efficienc of the present model. The new pretwisted Timoshenko beam element has good convergence characteristics and ecellent agreement is found with the previous studies. iv

5 ÖZ İki düğümlü ve sekiz serbestlik dereceli eni bir doğrusal burulmuş dönen Timoshenko çubuğu sonlu elemanı geliştirilmiş ve düzgün dikdörtgen kesitli önburulmalı çubukların titreşim analizinde kullanılmıştır. İlk olarak, anal erdeğiştirmeleri iki düzlemde bağlaşık olan önburulmalı çubuklar için erdeğiştirme fonksionları (polinom sabitleri, sözü geçen diferansiel denge denklemlerinde bağlaşık olan) türetilmiştir. Sonra, enerji ifadeleri kullanılarak sonlu eleman modelinin kütle ve direngenlik matrisleri elde edilmiştir. Son olarak da oluşturulan modelin doğruluğunu ve eterliğini kanıtlamak için önburulmalı dönen Timoshenko çubukların doğal frekansları elde edilmiş ve daha önce aınlanmış teorik ve denesel sonuçlarla karşılaştırılmıştır. Oluşturulan eni önburulmalı Timoshenko çubuk elemanı ii akınsama karakteristiği ve önceki çalışmalarla mükemmel bir uuşma göstermiştir. v

6 TABLE OF CONTENTS LIST OF FIGURES vii LIST OF TABLES viii NOMENCLATURE i Chapter. INTRODUCTION Chapter. THEORY Euler-Bernoulli beam theor Timoshenko beam theor Kinematics Strain displacement relations Equilibrium equations Constitutive equations Equations for pretwisted Timoshenko beam Chapter 3. FINITE ELEMENT VIBRATION ANALYSIS Introduction Finite element vibration analsis Mass and stiffness matrices of the finite element Numerical integration Assembling of the element matrices Determination of the natural frequencies Chapter 4. RESULTS AND DISCUSSION Simpl supported untwisted beam Cantilever pretwisted beam Cantilever pretwisted beam (various twist angle, length and breadth to depth ratio) Untwisted rotating cantilever beam Twisted rotating cantilever beam Chapter 5. CONCLUSION REFERENCES vi

7 LIST OF FIGURES Figure. Steam turbine Figure. Tpical blade cracks Figure.3 Schematic view of a part of a steam turbine Figure.4 Pretwisted beam model Figure. Kinematics of an Euler-Bernoulli beam Figure. Direct stress distribution acting on the beam face Figure.3 Direct and shear stress distributions on the beam cross-section... 8 Figure.4 Force and moment equilibrium of the beam Figure.5 Kinematics of the Timoshenko beam theor Figure.6 Application of uniform traction along -direction Figure 3. Description of the finite element Figure 3. Uniforml pretwisted constant cross-sectional beam Figure 3.3 Finite element model Figure 3.4 Model of rotation effect Figure 4. Frequenc ratio vs twist angle. Length 3.48 cm, breadth.54 cm, b/h = 8/ Figure 4. Frequenc ratio vs twist angle. Length 3.48 cm, breadth.54 cm, b/h = 4/ Figure 4.3 Frequenc ratio vs twist angle. Length 3.48 cm, breadth.54 cm, b/h = / Figure 4.4 Frequenc ratio vs twist angle. Length 7.6 cm, breadth.54 cm, b/h = 8/ Figure 4.5 Frequenc ratio vs twist angle. Length 5.4 cm, breadth.54 cm, b/h = 8/ Figure 4.6 Frequenc ratio vs twist angle. Length 5.8 cm, breadth.54 cm, b/h = 8/ vii

8 LIST OF TABLES Table 3. Gauss points and weights for 4-point Gaussian quadrature... 3 Table 3. Connectivit table Table 4. Comparison of coupled bending-bending frequencies of an untwisted, simple supported rectangular cross-section beam. 33 Table 4. Convergence pattern and comparison of the frequencies of a cantilever pretwisted uniform Timoshenko beam (Hz) Table 4.3 Two groups of cantilever beams for the analsis of the effect of various parameters on the natural frequencies Table 4.4 Result I for the first group of beams Table 4.5 Result II for the first group of beams Table 4.6 Result III for the first group of beams Table 4.7 Result I for the second group of beams Table 4.8 Result II for the second group of beams Table 4.9 Result III for the second group of beams Table 4. Comparison of bending frequencies of an untwisted rotating cantilever beam Table 4. Comparison of natural frequencies of a twisted rotating cantilever beam viii

9 NOMENCLATURE { a } a,a,a, a3 A b b c,b,b, b3,c, c { d } d E G h,d, d I, I I independent coefficient vector polnomial coefficients of the linear displacement in z plane cross-sectional area of the beam polnomial coefficients of the linear displacement in z plane breadth of the beam polnomial coefficients of the angular displacement about the ais dependent coefficient vector polnomial coefficients of the angular displacement about the ais modulus of elasticit modulus of rigidit depth of the beam area moments of inertia of the cross-section about and aes product moment of inertia of the cross-section about - I, I aes area moments of inertia of the cross-section about '' and k [ Ke] L m() z mo M, [ M e] P() z { q e } M '' aes shear coefficient element stiffness matri length of the beam mass of the beam according to the analsed nodal coordinate total mass of the beam bending moments about and aes mass matri Aial force element displacement vector i

10 [ Se ] T u U v V V, V w,, z zel θ θ θ ρ φ φ ψ, ψ geometric stiffness matri kinetic energ linear displacement in z plane strain energ linear displacement in z plane strain energ due to aial force shear forces in and direction rotational speed principal aes through the centroid at root section principal aes through the centroid at an section co-ordinate distance measured along beam co-ordinate of the element from the hub twist angle per unit length angular displacement about the ais angular displacement about the ais densit initial pretwist angle of the finite element pretwist angle of the finite element shear angles about and aes Ω fundamental natural circular frequenc of a untwisted beam Ω natural circular frequenc of a pretwisted beam (. ) differentiation with respect to time ( ) differentiation with respect to z

11 Chapter INTRODUCTION The da that Dr. Gustaf Patrik de Laval, a Swedish engineer presented his marine steam turbine to the World Columbian Eposition in 893, marks the beginning of the era of high speeds in rotating machiner. In the 9 s the turbine industr designed machines to operate at substantiall higher loads and at speeds above the lowest critical speed, and this introduced the modern-da rotor dnamics problems. Figure. Steam turbine Since the advent of steam turbines and their application in various sectors of industr, it is a common eperience that blade failures are a major cause of breakdown in these machines. Blade failures due to fatigue are predominantl vibration related. When a rotor blade passes across the nozzles of the stator, it eperiences fluctuating lift and moment forces repeatedl at a frequenc given b the number of nozzles multiplied b the speed of the machine. The blades are ver fleible structural members, in the sense that a significant number of their natural frequencies can be in the region of possible nozzle ecitation frequencies.

12 Figure. Tpical blade cracks It is ver important for manufacturers of turbo machiner components to know the natural frequencies of the rotor blades, because the have to make sure that the turbine on which the blade is to be mounted does not have some of the same natural frequencies as the rotor blade. Otherwise, a resonance ma occur in the whole structure of the turbine, leading to undamped vibrations, which ma eventuall wreck the whole turbine. Figure.3 Schematic view of a part of a steam turbine

13 A single free standing blade can be considered as a pretwisted cantilever beam with a rectangular cross-section. Vibration characteristics of such a blade are alwas coupled between the two bending modes in the flapwise and chordwise directions and the torsion mode. The problem is also complicated b several second order effects such as shear deformations, rotar inertia, fiber bending in torsion, warping of the cross-section, root fiing and Coriolis accelerations. Figure.4 Pretwisted beam model Man researchers analzed uniform and twisted Timoshenko beams using different techniques: Eact solutions of Timoshenko s equation for simple supported uniform beams were given b Anderson []. The general equations of motion of a pretwisted cantilever blade were derived b Carnegie []. Then Carnegie [3] etended his stud for the general equations of motion of a pretwisted cantilever blade allowing for torsion bending, rotar inertia and deflections due to shear. Dawson et al. [4] found the natural frequencies of pretwisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotar inertia b the numerical integration of a set of first order simultaneous differential equations. The also made some eperiments in order to obtain the natural frequencies for beams of various breadth to depth ratios and lengths ranging from 3 to in and pretwist angle in the range -9. Gupta and Rao [5] used the finite element method to determine the natural frequencies of uniforml pretwisted tapered cantilever beams. Subrahmanam et al. [6] applied the Reissner method and the total potential energ approach to calculate the natural frequencies and mode shapes of pretwisted cantilever blading including shear deformation and rotar inertia. Rosen [7] presented a surve paper as an etensive bibliograph on the structural and dnamic aspects of pretwisted beams. Chen and Keer [8] studied the transverse vibration problems of a rotating twisted Timoshenko beam under aial loading and spinning about aial ais, and 3

14 investigated the effects of the twist angle, rotational speed, and aial force on natural frequencies b finite element method. Chen and Ho [9] introduced the differential transform to solve the free vibration problems of a rotating twisted Timoshenko beam under aial loading. Lin et al. [] derived the coupled governing differential equations and the general elastic boundar conditions for the coupled bending-bending forced vibration of a nonuniform pretwisted Timoshenko beam b Hamilton s principle. The used a modified transfer matri method to stud the dnamic behavior of a Timoshenko beam with arbitrar pretwist. Banerjee [] developed a dnamic stiffness matri and used for free vibration analsis of a twisted beam. Rao and Gupta [] derived the stiffness and mass matrices of a rotating twisted and tapered Timoshenko beam element, and calculated the first four natural frequencies and mode shapes in bending-bending mode for cantilever beams. Naraanaswami and Adelman [3] showed that a straightforward energ minimization ields the correct stiffness matri in displacement formulations when transverse shear effects are included. The also stated that in an finite element displacement formulation where transverse shear deformations are to be included, it is essential that the rotation of the normal (and not the derivative of transverse displacement) be retained as a nodal degree of freedom. Dawe [4] presented a Timoshenko beam finite element that has three nodes and two degrees of freedom per node, which are the lateral deflection and the cross-sectional rotation. The element properties were based on a coupled displacement field; the lateral deflection was interpolated as a quintic polnomial function and the cross-sectional rotation was linked to the deflection b specifing satisfaction of the moment equilibrium equation within the element. The effect of rotar inertia was included in lumped form at the nodes. Subrahmanam et al. [5] analsed the lateral vibrations of a uniform rotating blade using Reissner and the total potential energ methods. Another vibration analsis of rotating pretwisted blades have been done b Yoo et al. [6] The main purpose of this stud is to create a new finite element model that shows a better convergence character and more accurate results with respect to the other finite element formulations in the literature to determine the natural frequencies of the blade structure. In order to reach this purpose, a new finite element model as an etension of Dawe s stud to pretwisted Timoshenko beam 4

15 is derived. Elastic and geometric stiffness and mass matrices of the element are obtained and used to reach the natural frequencies of the structure. The results of our stud show us that an ecellent agreement with the previous studies has obtained. 5

16 Chapter THEORY There are two beam theories when dealing with transverse vibrations of prismatic beams:. Euler-Bernoulli beam theor (Classical beam theor). Timoshenko beam theor... Euler-Bernoulli beam theor The Euler-Bernoulli beam equation arises from a combination of 4 distinct subsets of beam theor [3]: Kinematic Constitutive 3. Force resultant 4. Equilibrium Kinematics describes how the beam s deflections are tracked. Out-ofplane displacement w, the distance the beam s neutral plane moves from its resting position, is usuall accompanied b a rotation of the beam s neutral plane, defined as θ, and b a rotation of the beam s cross-section, χ. Figure. Kinematics of an Euler-Bernoulli beam 6

17 What we reall need to know is the displacement in the -direction across a beam cross-section, u (, ), from which we can find the direct strain ε (, ) b the equation, du ε = d (.) To do so requires that we make a few assumptions on just how a beam cross-section rotates. For the Euler-Bernoulli beam, the assumptions were given b Kirchoff and dictate how the normals behave (normals are lines perpendicular to the beam s neutral plane and are thus embedded in the beam s cross-sections). Kirchoff Assumptions. Normals remain straight (the do not bend). Normals remain unstrecthed (the keep the same length) 3. Normals remain normal (the alwas make a right angle to the neutral plane) With the normals straight and unstretched, we can safel assume that there is negligible strain in the direction. Along with normals remaining normal to the neutral plane, we can make the and dependance in ( ) simple geometric epression, u, eplicit via a u ( ) = χ( ), (.) With eplicit dependance in u, we can find the direct strain throughout the beam, dχ ε (, ) = (.3) d 7

18 Finall, again with normals alwas normal, we can tie the cross-section rotation χ to the neutral plane rotation θ, and eventuall to the beam s displacement w, χ = θ = dw d (.4) The Constitutive equation describes how the direct stress σ and direct strain ε within the beam are related. Direct means perpendicular to a beam crosssection; if we were to cut the beam at a given location, we would find a distribution of direct stress acting on the beam face. Figure. Direct stress distribution acting on the beam face Beam theor tpicall uses the simple one-dimensional Hooke s equation, (, ) Eε(, ) σ = (.5) It can be noted that the stress and strain are functions of the entire beam cross-section (i.e. the can var with ). Force resultants are a convenient means for tracking the important stresses in a beam. The are analogous to the moments and forces of statics theor, in that their influence is felt throughout the beam (as opposed to just a local effect). Their convenience lies in them being onl functions of, whereas stresses in the beam are functions of and. If we were to cut a beam at a point, we would find a distribution of direct stresses σ ( ) and shear stresses ( ) σ, Figure.3 Direct and shear stress distributions on the beam cross-section 8

19 Each little portion of direct stress acting on the cross-section creates a moment about the neutral plane (=). Summing these individual moments over the area of the cross-section is the definition of the moment resultant M, () ( ) M = σ, ddz (.6) where z is the coordinate pointing in the direction of the beam width (out of page). Summing the shear stresses on the cross-section is the definition of the shear resultant V, () σ ( ) V =, ddz (.7) There is one more force resultant that we can define for completeness. The sum of all direct stresses acting on the cross-section is known as N, () ( ) N = σ, ddz (.8) N () is the total direct force within the beam at some point, et it does not pla a role in (linear) beam theor since it does not cause a displacement w. Instead, it plas a role in the aial displacement of rods and bars. B inverting the definitions of the force resultants, we can find the direct stress distribution in the beam due to bending, (, ) σ = M I (.9) It is obvious that the bending stress in beam theor is linear through the beam thickness. The maimum bending stress occurs at the point furthest awa from the neutral ais, =c, σ ma Mc = I (.) 9

20 What about the other non-linear direct stresses shown acting on the beam cross-section? The average value of the direct stress is contained in N and does not contribute to beam theor. The remaining stresses (after the average and linear parts are subtracted awa) are self-equilibrating stresses. B a somewhat circular argument, the are self-equilibrating precisel because the do not contribute to M or N, and therefore the do not pla a global role. On the contrar, selfequilibrating loads are confined to have onl a localized effect as mandated b Saint Venant s Principle. [Saint-Venant's Principle can be stated as follows: If a set of selfequilibrating loads are applied on a bod over an area of characteristic dimension d, the internal stresses resulting from these loads are onl significant over a portion of the bod of approimate characteristic dimension d. Note that this principle is rather vague, as it deals with ``approimate'' characteristic dimensions. It allows qualitative rather than quantitative conclusions to be drawn.] The Equilibrium equations describe how the beam carries eternal pressure loads with its internal stresses. Rather than deal with these stresses themselves, it is chosen to work with the resultants since the are functions of onl (and not of ). To enforce equilibrium, consider the balance of forces and moments acting on a small section of beam, Figure.4 Force and moment equilibrium of the beam Equilibrium in the direction gives the equation for the shear resultant V, dv d = p (.)

21 Moment equilibrium about a point on the right side of the beam gives the equation for the moment resultant M, dm = V d (.) It can be noted that the pressure load p does not contribute to the moment equilibrium equation. The outcome of each these segments is summarized here: dw Kinematics: χ = θ = d Constitutive: σ (, ) = Eε(, ) Resultants: M () σ (, ) Equilibrium: = ddz () σ ( ) V =, ddz dm = V d dv d = p To relate the beam s out-of-plane displacement w to its pressure loading p, the results of the 4 beam sub-categories are combined in the order shown, Kinematics -> Constitutive -> Resultants -> Equilibrium = Beam Equation This hierarch will be demonstrated b working backwards. First, the two equilibrium equations are combined to eliminate V, d d M = p (.3) Net the moment resultant M is replaced with its definition in terms of the direct stress σ, d d [ ddz] = p σ (.4)

22 The constitutive relation is used to eliminate σ in favour of the strain ε, and then kinematics is used to replace ε in favour of the normal displacement w, d d [ E ddz] = p ε d d dχ E d ddz = p d d d w E d ddz = p (.5) As a final step, recognizing that the integral over is the definition of the beam s area moment of inertia I, I = ddz (.6) allows us to arrive at the Euler-Bernoulli beam equation, d d EI d w = d p (.7).. Timoshenko beam theor Fleural wave speeds are much lower than the speed of either longitudinal or torsional waves. Therefore fleural wavelengths which are less than ten times the cross-sectional dimensions of the beam will occur at much lower frequencies. This situation occurs when analsing deep beams at low frequencies and slender beams at higher frequencies. In these cases, deformation due to transverse shear and kinetic energ due to rotation of the cross-section become important. In developing energ epressions which include both shear deformation and rotar inertia, the assumption that plane sections which are normal to the undeformed centroidal ais remain plane after bending, will be retained. However, it will no longer be assumed that these sections remain normal to the deformed ais []. The classical Euler-Bernoulli theor predicts the frequencies of fleural vibration of the lower modes of slender beams with adequate precision. However, because in this theor the effects of transverse shear deformation and rotar

23 inertia are neglected the errors associated with it become increasingl large as the beam depth increases and as the wavelength of vibration decreases. Timoshenko, a highl qualified engineer from Russia but had worked for an US turbine compan Westinghouse, had made the corrections to the classical beam theor and developed the energ epressions which include both shear deformation and rotar inertia effects... Kinematics We consider a prismatic beam, smmetric cross-section with respect to (w.r.t.) z-ais (Figure.5). Figure.5 Kinematics of the Timoshenko Beam Theor Appl (, z h / ) T z = traction (Figure.6), uniform along -direction, so that the applied transverse load will be, g () = bt (,h / ) z (.8) 3

24 z b g() Figure.6 Application of uniform traction along -direction Assumptions () Plane sections such as ab, originall normal to the centerline of the beam in the undeformed geometr, remain plane but not necessaril normal to the centerline in the deformed state. () The cross-sections do not stretch or shorten, i.e., the are assumed to act like rigid surfaces. (3) All displacements and strains are small, i.e., h w <<, ε = / ( + u ) ij u i, j j, i Assumption () implies that; () u z = w or ε (.9) zz = u z, z = Assumption () implies that there eist constant (through the thickness) shear strains, i.e., γ z = γ z () (.) Now, writing down the shear strain in the -z plane 4

25 γ z = u, z + u z, = u, z + w, (.) Now, solving Equation (.) for u, z u, z = γ z w, (.) and integrating w.r.t. z ( γ w ) f () u = z z, + (.3) Evaluating u at the centerline z = we have; ( z = ) = f ( ) u( ) u, = (.4) where u () denotes displacement in the -direction of an point on the centerline. Replacing () u() f = into Equation (.3) we have; (, z) u( ) + z( w ) u z, = γ (.5) where we note that γ z = γ z () and w = w(). Now introducing a variable called the bending rotation, θ (), we can write θ () = γ z w, (.6) from which γ z = w, + θ (.7) and Equation (.5) becomes ( z) = u zθ u, + (.8) 5

26 and u z ( z) w( ), (.9) Equations (.8) and (.9) represent the components of the displacement vector {} { } u u z u = of the Timoshenko beam. (It is noted as before that u =, i.e., all deformations along -ais are neglected)... Strain Displacement Relations The onl nonzero strains are; ε = u = u + zθ = ε + zk,,, (.3) where ε = u, (centerline aial strain) and k θ, = (bending curvature) (.3) and the transverse shear strain γ z = w, + θ (.3) Hooke s Law (Stress-Strain Relations) ( σ ) σ + = Eε + ν σ zz τ z = Gγ z (.33) (.34) The underlined term is generall neglected since it is much smaller than the first term. We then have: 6

27 ( ε + zk ) = E( u z ) σ = + E ε = E, θ, ( w θ ) (.35) τ G γ = G, + (.36) z = z..3 Equilibrium Equations The following integrals are defined: N V M = A = A = τ A σ z σ da da zda (.37) (.38) (.39) where N is the aial force, V is the shear force and M is the bending moment. The application of Principles of Virtual Work results the following equilibrium equations in the range < < L for the Timoshenko beam: N, = V, + g = M, V = (.4) (.4) (.4) These three equations can be simplified further, If we differentiate Equation (.4) w.r.t. and substitute Equation (.4) into Equation (.4): M, + g = (bending) (.43) N, = (aial) (.44) also, Equation (.4) gives: V M, = (bending) (.45) 7

28 ..4 Constitutive Equations With reference to Equation (.37), σ da = E( u, + zθ ) N =, N A A, da = EAu = EAε (.46) σ zda = Ez( u, + zθ ) M =, A M = EIk = EIθ, I = A z da A Shear force: Substituting Equation (.36) into Equation (.38) ields da (.47) V = A Gγ z da k GAγ z (.48) where GA = Shear rigidit k = nondimensional coefficient, referred to as a shear correction factor. Bending Equilibrium Equations in terms of the Kinematic Variables of Timoshenko Beam Theor Substituting the constitutive Equations (.47) and (.48) into Equations (.4) and (.4) gives: d d d d [ k GA( w + )] + g() =, θ (.49) ( EIθ ) k GA( w + θ ),, = (.5) 8

29 Assuming GA and EI are constant, the above two equations can be readil reduced to a single equation in terms of w onl, i.e., ( 4) EI EIw + g g k GA =, (.5) The strain energ stored in the element is the sum of the energies due to bending and shear deformation; which is given b U =.5 σ ε dv +. 5 τ γ dv (.5) V V The kinetic energ of the straight beam consists of kinetic energ of translation and kinetic energ of rotation which is epressed as T L =.5 ρ Aw& d +.5 ρiθ& d (.53) L.3 Equations for pretwisted Timoshenko beam as [3]; The elastic potential energ of the pretwisted Timoshenko beam is given U L { E( I ' + I θ ' θ ' + I θ ' ) + kag( u' θ ) + ( v' ) )} =.5 θ θ dz (.54) where, the smbol represents differentiation with respect to z which is the longitudinal ais of the beam. The kinetic energ of the pretwisted thick beam is given as follows [3]; T L { A( u + v& ) + ( I θ& + I θ& θ& I & )} =.5 ρ & + θ dz (.55) 9

30 The differential equations of motion of the pretwisted beam with uniform rectangular cross-section are given as follows [3, 4]; d dz d dz ( M ) V = I ρ & θ d dz, ( ) = ( V ) ρau& d dz =, ( V ) ρav M V I ρ & θ (.56,.57) = & (.58,.59) where M θ = EI + EI θ, M EI + EI θ = θ (.6,.6) V = kagψ, V kagψ = (.6,.63) in which ψ = v θ, ψ = u θ (.64,.65) In the above equations; M and M represents bending moments about and aes, V and V represents shear forces in and directions, represents shear angles about and aes. ψ and ψ

31 Chapter 3 FINITE ELEMENT VIBRATION ANALYSIS 3. Introduction The Finite Element Method (FEM) is a numerical procedure that can be used to obtain solutions to a large class of engineering problems involving stress analsis, heat transfer, electromagnetism, fluid flow and vibration and acoustics. In FEM, a comple region defining a continuum is discretized into simple geometric shapes called finite elements (see Figure 3.). The material properties and the governing relationships are considered over these elements and epressed in terms of unknown values at element corners, called nodes. An assembl process, dul considering the loading and constraints, results in a set of equations. Solution of these equations gives us the approimate behaviour of the continuum. Figure 3. Description of the finite element

32 Basic ideas of the FEM originated from advances in aircraft structural analsis. The origin of the modern FEM ma be traced back to the earl th centur, when some investigators approimated and modelled elastic continua using discrete equivalent elastic bars. However, Courant has been credited with being the first person to develop the FEM. He used piecewise polnomial interpolation over triangular subregions to investigate torsion problems in a paper published in 943. The net significant step in the utilisation of Finite Element Method was taken b Boeing. In the 95 s Boeing, followed b others, used triangular stress elements to model airplane wings. But the term finite element was first coined and used b Clough in 96. And since its inception, the literature on finite element applications has grown eponentiall, and toda there are numerous journals that are primaril devoted to the theor and application of the method. 3. Finite element vibration analsis Here are the steps in finite element vibration analsis:. Discrete and select element tpe. Select a displacement function 3. Derive element stiffness and mass matrices 4. Assemble the element matrices and introduce BC s 5. Solve the eigenvalue problem and obtain the natural frequencies A uniforml pretwisted constant cross-sectional beam is shown in Figure 3.. Differential equations of the motion of pretwisted Timoshenko beam with uniform rectangular cross-section are given in the preceding chapter. The finite element model derived here is based on eplicit satisfaction of the homogeneous form of Equations ( ). In Equations ( ), eliminating three parameters from the set {u, v, θ, θ } in turn gives, 4 d u 4 dz d v d θ d θ =, =, =, = dz dz dz ( )

33 The Equations ( ) result in an element with constant shear forces along its length, linear variation of moments, quadratic variation of cross-sectional rotations and cubic variation of transverse displacements. Therefore, the general solutions of these four equations are chosen as polnomials in z as follows: 3 () z = a + a z + a z a u o + 3z 3 () z = b + b z + b z b v o + θ θ 3z () z = c + c z c o + z () z = d + d z d o + z (3.5) (3.6) (3.7) (3.8) z L Figure 3. Uniforml pretwisted constant cross-sectional beam Using homogeneous form of Equations ( ), the relationships are obtained linking u, v, θ and θ in the form; d dz d dz ( EI θ + EI θ ) + kag ( v θ ) = ( EI θ + EI θ ) + kag( u θ ) = (3.9) (3.) The area moments of inertia of the cross-section should be noted as follows: I () z = I φ () z I sin φ () z cos + 3

34 I I () z = I φ () z I sin φ () z cos + () z =.5 ( I I ) sin φ () z (3.) where φ () z φ + θ z = B using the Equations (3.5-3.) the coefficients c, c, c, d, d and d can be epressed in terms of the coefficients a, a, a, a 3, b, b, b and b 3 b equating coefficients of the powers of z. This procedure ields: c o = +, a + a3 + b + 3b 4b3 c 5a3 + b + 6b3 =, c = 3b 3 (3.) d d a + 7a + 8a3 + b + b3 =, a + 9a3 + 5b3 =, d = 3a 3 where E = kag I = 6 E kag I 6E ( I + I ) + I kag = E I ' 3 kag 4 = 6 E kag 6E ( I + I ) + I kag = 6E I ' 5 kag (3.3) = 6E I ' 6 kag = E I ' 7 kag 4

35 5 ( ) I kag E I I kag E + + = I kag = 6E 9 in which ( ) () z I I I φ θ sin = ( ) () z I I I φ θ sin = (3.4) ( ) () z I I I φ θ cos = It is convenient to epress the Equation (3.) in the matri form: {} [ ]{} a B d = or in open form = b b b b a a a a d d d c c c (3.5) where {a} and {d} are named as independent and dependent coefficient vectors, respectivel. Similar procedure was applied to untwisted Timoshenko beam b Naraanaswami and Adelman [3] and Dawe [4]. 3.3 Mass and stiffness matrices of the finite element The new Timoshenko beam finite element has two nodes and four degrees of freedom per node, namel, two transverse deflections and two rotations (Figure 3.3). The element displacement vector can be written as:

36 6 { } { } T e v u v u q θ θ θ θ = (3.6) Figure 3.3 Finite element model Then, b using Equations ( ) and (3.), {q e } can be epressed in terms of the independent coefficient vector as follows: { } [ ]{} a C q e = (3.7) ( ) ( ) ( ) ( ) ( ) ( ) = b b b b a a a a L L L L L L L L L L L L L L v u v u θ θ θ θ The linear and angular displacement functions can be written b using the independent and dependent coefficient vectors, respectivel, as follows: () [ ]{} [ ]{ } a z z z a P z u u 3 = = (3.8) () [ ]{} [ ]{ } a z z z a P z v v 3 = = (3.9) () [ ]{ } [ ]{ } d z z d P z = = θ θ (3.) θ u v θ z

37 θ () z [ P ]{ d} = [ z z ]{ d} = θ (3.) Now, Equations (3.) and (3.) can be epressed b using Equation (3.5) as follows: () z [ P ] [ B]{} a θ = θ () z [ P ] [ B]{ a} θ = θ (3.) (3.3) The elastic potential energ of the finite element in Figure 3.3 is written as [3]: U L { E( I ' + I θ ' θ ' + I θ ' ) + kag( u' θ ) + ( v' ) )} =.5 θ θ dz (3.4) where, the smbol represents differentiation with respect to z. Substituting Equations (3.), (3.8), (3.9), (3.) and (3.3) into Equation (3.4) gives U =. 5 T { q } [ K ] q } e e { e (3.5) where [K e ] is the element stiffness matri given b L { } T [ K e ] = [ C] [ k][ C] dz (3.6) in which T T T T T [] k = [ B] E I ()[ z P ] [ P θ ] I ()[ z P ] [ P ] I ()[ z ( P ] [ P ] [ P ] [ P ]) θ + θ θ + θ θ + θ θ T T T T T + kag{ ([ P ] [ P u u ] + [ P ] [ P v v ]) + ([ B] ([ Pθ ] [ P ] [ P ] [ P ])[ B] )} θ + θ θ T T T T T kag{ ([ B] ([ P ] [ P ] + [ P ] [ P ] ) + ( [ P ] [ P ] + [ P ] [ P ])[ B] )} (3.7) { }[ B] θ u θ v u θ v θ 7

38 In order to eamine the effect of rotational speed on the natural frequencies, the sstem shown in Figure 3.4 is considered. The strain energ due to aial force can be written as follows: V = l ()( z u + v ) P dz (3.8) where () z = m() z w ( z z) P el + (3.9) and m () z m ( z z) = µ (3.3) o el + in which m o is the total mass of the beam and µ is the mass/unit length of the beam. Substituting the derivations of Equations (3.8) and (3.9) with the Equations (3.9) and (3.3) into (3.8) gives: V =. 5 T { q } [ S ]{ q } e e e (3.3) where [ S e ] is the element geometric stiffness matri given b ( ) dz [ ] l = u u v v C T T T [ S ] [ C] P()[ z P ] [ P ] + [ P ] [ P ] e (3.3) The kinetic energ of the pretwisted thick beam is given as follows [3]: L { A ( u + v& ) + ( I θ& + I θ& θ& I & )} T=.5 ρ & + θ dz (3.33) 8

39 where the use of the overdot is a compact notation for differentiation with respect to time. Substituting Equations (3.), (3.8), (3.9), (3.) and (3.3) into Equation (3.33) gives T =.5 T { q& } [ M ]{ q& } e e e (3.34) w Hub Finite element under consideration z z e l r L Figure 3.4 Model for rotation effect where [M e ] is the element mass matri given b L { } T [ M ] = [ C] [ m][ C] e dz (3.35) in which T T [ m ] = ρa( [ P& ] [ P& ] + [ P& ] [ P& ] ) + ρ u u v v T {[ B] ( I ()[ z P & T ] [ P& T T T θ θ ] + I ()[ z ( P& ] [ P& θ θ ] + [ P& ] [ P& ]) + θ θ I ()[ z P& θ ] [ P& θ ])[ B] } (3.36) 9

40 3.4 Numerical integration In order to compute [ K e ], [ S ] and [ e ] e M in the Equations (3.6), (3.3) and (3.35), Gauss-Legendre 4-point numerical integration is used. The n-point approimation is given b the following formula [9]; I = f () ξ dξ w f ( ξ ) + w f ( ξ ) w f n ( ξ ) n (3.37) where w, w, w 3 and w 4 are the weights and ξ, ξ, ξ 3 and ξ 4 are the sampling points or Gauss points. The idea behind Gaussian quadrature is to select the n Gauss points and n weights such that Equation (3.37) provides an eact answer for polnomials f () ξ of as large a degree as possible. The Gauss points and weights for 4-point Gauss-Legendre numerical integration is given in Table 3.. In our analsis the integration starts from to the length of the finite element, so a modification should be needed for the Gauss points and weights [8]. I = w i b f a i= = n () ξ dξ w f ( ξ ) ( b a) i w i ξ = ( a + b) ( b a) + i ξ i i (3.38) (3.39) (3.4) Point number Gauss points Weights Table 3. Gauss points and weights for 4-point Gaussian quadrature 3

41 3.5. Assembling of the element matrices The global mass and stiffness matrices are obtained b assembling the element matrices given in Equations (3.6), (3.3) and (3.35). The assembling process is carried out b the computer program developed in MatLAB. The connectivit table for the element solution is given in Table 3. to give an idea about how the computer connects the element matrices. Ever node in an element has both a local coordinate and a global coordinate. Element Local Coordinates Number Global Coordinates Table 3. Connectivit Table 3.6. Determination of the natural frequencies B using the well-known procedures of vibration analsis, the eigenvalue problem can be given as [7]; ([ K] Ω [ M ]){} q = (3.4) 3

42 where [ K ] and [ M ] are global stiffness (geometric stiffness matri included) and mass matrices, respectivel, and { q } is global displacement vector, and Ω is the natural circular frequenc. The eigenvalue problem given in Equation (3.4) is solved b using computer programs developed in MatLAB. 3

43 Chapter 4 RESULTS AND DISCUSSION In order to validate the proposed finite element model for the vibration analsis of pretwisted Timoshenko beam, various numerical results are obtained and compared with available solutions in the published literature. 4.. Simpl supported untwisted beam The first eample to be considered is the case of lateral vibrations of a non-rotating untwisted rectangular cross-section beam with both ends simpl supported. In Table 4., comparison of the analtical results obtained from closedform solution derived b Anderson [], finite element solution with and 4 elements given b Chen and Keer [8] and the present model with elements is made. Ecellent agreement is observed. The phsical properties of the beam are given in Table 4.. Chen, Keer FEM [8] Anderson Mode Analtical [] elements 4 elements Present FEM elements Difference between Analtical and Present % 4,78 5,4 4,87 3,99,69 333,46 334,44 333,7 33,, ,49 459, 454,86 45,55,65 4,38 7,4 7,3 995,8,46 5 6,7 9,63 9,9,76,4 Data: length of beam =.6 cm, width = 5.8 cm, thickness = 5.4 cm, shear coefficient = 5/6, E = 6.8 Gpa, G = 79.3 Gpa, mass densit = 786 kg/m 3. Table 4. Comparison of coupled bending-bending frequencies of an untwisted, simple supported rectangular cross-section beam 33

44 4.. Cantilever pretwisted beam (twist angle = 45 ) The second eample is concerned with a cantilever pretwisted beam treated eperimentall b Carnegie [] and b theoretical means b Lin et al. [] and Subrahmanam et al. [6]. The properties of the beam are given in Table 4.. To show efficienc and convergence of the proposed model, the first four frequencies of the second eample are calculated. For comparison, the present results as well as those given b other investigators are tabulated in Table 4.. It is observed that the agreement between the present results and results of the other investigators is ver good. The natural frequencies calculated b the proposed model converge ver rapidl. Even when the number of the element is onl, the present fundamental frequenc is converged. Number of element Mode number ,3 465,5 87,7 9,6 4 6,3 37,3 4,7 6, 6 6, 33,6 986,3 79, 8 6,9 39,3 965,4 78,3 6,8 37,3 956, 8,9 6,8 36,3 95, 85,4 4 6,8 35,6 948,3 88, 6 6,8 35,3 946,6 9, 8 6,8 35, 945,4 9,7 6,8 34,8 944,5 93, Lin et al. [] 6,7 3,9 97, 75, Subrahmanam et al. [6] 6, 35, 955, 4,7 Subrahmanam et al. [6] 6,9 34,7 937, 5, Carnegie [] 59, 9, 9,, Data: length of beam = 5.4 cm, breadth =.54 cm, depth =.77 cm, shear coefficient = , E = 6.85 Gpa, G = 8.74 Gpa, mass densit = kg/m 3, twist angle = 45. Table 4.. Convergence pattern and comparison of the frequencies of a cantilever pretwisted uniform Timoshenko beam (Hz). 34

45 4.3. Cantilever pretwisted beam (various twist angle, length, breadth to depth ratio) This eample is considered to evaluate the present finite element formulation for the effects of related parameters (e.g. twist angle, length, breadth to depth ratio) on the natural frequencies of the pretwisted cantilever Timoshenko beams treated eperimentall b Dawson et al. [4]. The natural frequencies are prescribed in terms of the frequenc ratio Ω / Ω, where Ω is the natural frequenc of pretwisted beam and Ω is the fundamental natural frequenc of untwisted beam. The natural frequenc ratios for the first five modes of vibration are obtained for two groups of cantilever beams (Table 4.3) of uniform rectangular cross-section b using elements. Length 7,6 cm 5,4 cm 3,48 cm 5,8 cm 8/ b/d 4/ / Table 4.3. Two groups of cantilever beams for the analsis of the effect of various parameters on the natural frequencies. First group includes the sets of beams of breadth.54 m and length.348 m and various breadth to depth ratios and pretwist angle in the range - 9. The results for first group are shown in Tables 4.4, 4.5, 4.6 and in Figures 4., 4., and 4.3. Second group contains the sets of beams of breadth.54 m and breadth to depth ratio 8/ and length ranging from.76 m to.58 m and pretwist angle in the range -9. The results for second group are shown in Figures 4., 4.4, 4.5, and 4.6. It can easil be checked out from the Figures of the second group that the natural frequencies increase as the beam length decreases. 35

46 Mode Numbers Twist Frequenc Ratio (b/d=8/), Length= in Angle 3 4 5, 6,3 8, 7,5 34,3 3, 5,3 9,4 7, 34, 6, 4,,7 6, 33, 9, 3,4,8 6,3 3,7, 6, 8, 7,7 35, 3, 5, 9,4 7, 34,6 6, 4,,7 6,5 33,6 9, 3,,9 6,8 3,, 6,4 7,7 7, 34, 3, 5,3 9, 6,5 33,4 6, 4,,3 5,7 3,3 9, 3,4,4 6, 3,8, 6, 7,8 7,3 34, 3, 5, 9, 6,7 33,7 6, 4,3,6 6, 3,7 9, 3,3, 6,5 3, Analsis Tpes Present ( Elements) Chen, Keer [8] (5 Elements) Dawson et al. [4] Chen, Ho [9] Table 4.4 Result I for the first group of beams 36

47 Frequenc Ratio 5 5 Present Chen, Keer [8] Dawson et al. [4] Chen, Ho [9] Pretwist Angle Figure 4.. Frequenc ratio vs twist angle. Length 3.48 cm, breadth.54 cm, b/h = 8/. 37

48 Mode Numbers Twist Frequenc Ratio (b/d= 4/), Length= in Angle 3 4 5, 4, 6,3 7,5 4, 3, 3,7 6,7 6,6 5,4 6, 3,3 7,7 5, 7,9 9,,8 8,9 3,5 9,4, 4, 6, 7, 4,5 3,9 3,6 6,3 6,4 5, 6, 3, 7, 4,6 7,3 9,9,8 8,3 3, 8,9 Analsis Tpes Present Dawson et al. [4] Table 4.5 Result II for the first group of beams 35 3 Frequenc Ratio 5 5 Present Dawson et al. [4] Pretwist Angle Figure 4.. Frequenc ratio vs twist angle. Length 3.48 cm, breadth.54 cm, b/h = 4/. 38

49 Mode Numbers Twist Frequenc Ratio (b/d=/), Length= in Angle 3 4 5,, 6,, 7, 3,, 6,3,8 7,6 6,,9 6,5, 8,6 9,,8 6,9,4,,, 6,,6 6,3 3,,9 5,9,4 6,8 6,,7 6,,7 8, 9,,8 6,5 9,8 9,7 Analsis Tpes Present Dawson et al. [4] Table 4.6 Result III for the first group of beams 5 Frequenc Ratio 5 Present Dawson et al. [4] Pretwist Angle Figure 4.3. Frequenc ratio vs twist angle. Length 3.48 cm, breadth.54 cm, b/h = /. 39

50 Mode Numbers Twist Frequenc Ratio (b/d=8/), Length=3 in Angle 3 4 5, 6, 7,4 7, 33, 3, 5, 8,8 6,8 3,8 6, 4,,8 5,6 3,8 9, 3,3,8 5, 9,8, 6, 7, 6,7 3,7 3, 5, 8,6 6, 3,8 6, 4,,4 4,8 3,3 9, 3,,4 5,3 8,7 Analsis Results Present ( Elements) Dawson et al. [4] Table 4.7 Result I for the second group of beams Frequenc Ratio Present Dawson et al. [4] Pretwist Angle Figure 4.4. Frequenc ratio vs twist angle. Length 7.6 cm, breadth.54 cm, b/h = 8/. 4

51 Mode Numbers Twist Frequenc Ratio (b/d=8/), Length=6 in Angle 3 4 5, 6,3 7,8 7,5 34, 3, 5,3 9,3 7, 33,8 6, 4,,4 6, 3,9 9, 3,4,6 6, 3,4, 6, 7,4 7, 33,7 3, 5,3 8,7 6,7 33,3 6, 4,3,8 5,6 3,9 9, 3,3, 5,7 3,6 Analsis Results Present ( Elements) Dawson et al. [4] Table 4.8 Result II for the second group of beams Frequenc Ratio Pretwist Angle Present Dawson et al. [4] Figure 4.5. Frequenc ratio vs twist angle. Length 5.4 cm, breadth.54 cm, b/h = 8/. 4

52 Mode Numbers Twist Frequenc Ratio (b/d=8/), Length= in Angle 3 4 5, 6,3 8, 7,5 34,4 3, 5,3 9,5 7, 34, 6, 4,,8 6, 3,9 9, 3,4,8 6,5 3,6, 6, 8, 7,6 34,4 3, 5,4 9,7 7, 33,8 6, 4,3,9 6, 3,6 9, 3,4,8 6,9 3,3 Analsis Tpes Present ( Elements) Dawson et al. [4] Table 4.9 Result III for the second group of beams Frequenc Ratio Pretwist Angle Present Dawson et al. [4] Figure 4.6. Frequenc ratio vs twist angle. Length 5.8 cm, breadth.54 cm, b/h = 8/. 4

53 4.4 Untwisted rotating cantilever beam In this eample, the case of a rotating untwisted cantilever beam is considered. The first three natural frequencies has been determined and compared with the results of Subrahmanam and Kulkarni [5] and shown in Table 4.9. The properties of the beam is shown below: L = mm A = 8.58 mm ρ =.73 kg/cm 3 E = 6.85 Gpa I = mm 4 w = rad/sec r = mm G = 8.74 Gpa κ =.85 Mode Number I II III Present [5] Table 4. Comparison of bending frequencies of an untwisted rotating cantilever beam The average difference between the present results and the theoretical results [5] is onl.73 %, it is possible to sa that the created FE model shows accurate results even when the rotation effect is included. 4.5 Twisted rotating cantilever beam Lastl, the accurac of the present model needs to be confirmed for the twisted rotating cantilever case. The lowest two natural frequencies has been determined and compared with the results of Yoo et al [6] in Table

54 The properties of the beam is shown below: L = 5 mm a = mm (breadth) b = mm (thickness) ρ = 783 kg/m 3 E = 6.85 Gpa T = (ρal/(ei)) w = γ/t r = mm G = 8.74 Gpa κ =.85 θ = 45º γ First natural frequenc Second natural frequenc Present Reference[6] Present Reference[6].,763,763,9888,985.88,3,,9,3.763,963,357,5,53.645,3955,488,47,796 Table 4. Comparison of natural frequencies of a twisted rotating cantilever beam An average of 3% difference is observed, it can be also found that the natural frequencies obtained b the present modelling method are lower than those obtained in reference [6]. Thus, the present modeling method provides more accurate results. 44

55 Chapter 5 CONCLUSION A new linearl pretwisted rotating Timoshenko beam finite element, which has two nodes and four degrees of freedom per node, is developed and subsequentl used for vibration analsis of pretwisted beams with uniform rectangular cross-section. The finite element model developed is based on two displacement fields that couple the transverse and angular displacements in two planes b satisfing the coupled differential equations of static equilibrium. This procedure means that the rotar inertia term is ignored in the moment equilibrium equation within the element but the effect of rotar inertia will be included in lumped form at the nodes. The present model is verified for various parameters ( such as twist angle, length, breadth to depth ratio) in different range on the vibrations of the twisted beam treated eperimentall b Carnegie [] and Dawson et al. [4] and theoreticall b other investigators [, 6, 8, 9,, 5, 6] even with ten elements. The new pretwisted both non-rotating and rotating Timoshenko beam element has shown good convergence characteristics and ecellent agreement is found with the previous studies. The effects of pretwist angle, beam length and breadth to depth on the natural frequencies are also studied. 45

56 REFERENCES. R. A. ANDERSON 953 Journal of Applied Mechanics, Fleural vibrations in uniform beams according to the Timoshenko theor.. W. CARNEGIE 959 Proceeding of the Institution of Mechanical Engineers 73, Vibration of pre-twisted cantilever blading. 3. W. CARNEGIE 964 Journal of Mechanical Engineering Science 6, 5-9. Vibration of pre-twisted cantilever blading allowing for rotar inertia and shear deflection. 4. B. DAWSON, N. G. GHOSH and W. CARNEGIE 97 Journal of Mechanical Engineering Science 3, Effect of slenderness ratio on the natural frequencies of pre-twisted cantilever beams of uniform rectangular crosssection. 5. R. S. GUPTA and S. S. RAO 978 Journal of Sound and Vibration 56, 87-. Finite element eigenvalue analsis of tapered and twisted Timoshenko beams. doi:.6/jsvi K. B. SUBRAHMANYAM, S. V. KULKARNI and J. S. RAO 98 International Journal of Mechanical Sciences 3, Coupled bending-bending vibration of pre-twisted cantilever blading allowing for shear deflection and rotar inertia b the Reissner method. 7. A. ROSEN 99 Applied Mechanics Reviews 44, Structural and dnamic behavior of pretwisted rods and beams. 8. W. R. CHEN and L. M. KEER 993 Journal of Vibrations and Acoustics 5, Transverse vibrations of a rotating twisted Timoshenko beam under aial loading. 9. C. K. CHEN and S. H. HO 999, International Journal of Mechanical Science 4, Transverse vibration of a rotating twisted Timoshenko beams under aial loading using differential transform.. S. M. LIN, W. R. WANG and S. Y. LEE International Journal of Mechanical Science 43, The dnamic analsis of nonuniforml pretwisted Timoshenko beams with elastic boundar conditions. 46

57 . J. R. BANERJEE International Journal of Solids and Structures 38, Free vibration analsis of a twisted beam using the dnamic stiffness method.. S. S. RAO and R. S. GUPTA Journal of Sound and Vibration 4, 3-4. Finite element vibration analsis of rotating Timoshenko beams.. R. NARAYANASWAMI and H. M. ADELMAN 974 AIAA Journal, Inclusion of transverse shear deformation in finite element displacement formulations. 4. D. J. DAWE 978 Journal of Sound and Vibration 6, -. A finite element for the vibration analsis of Timoshenko beams. 5. K. B. SUBRAHMANYAM, S. V. KULKARNI and J.S. RAO 98 Mechanism and Machine Theor 7(4), Analsis of lateral vibrations of rotating cantilever blades allowing for shear deflection and rotar inertia b Reissner and potential energ methods. 6. H. H. YOO, J.Y. KWAK and J. CHUNG Journal of Sound and Vibration 4(5), Vibration Analsis of rotating pretwisted blades with a concentrated mass. 7. M. PETYT 99 Introduction to Finite Element Vibration Analsis. Cambridge: Cambridge Universit Press. 8. K. J. BATHE 996 Finite Element Procedures. New Jerse, Prentice Hall. 9. T. R. CHANDRUPATLA and A. D. BELEGUNDU 997 Introduction to Finite Elements in Engineering. New Jerse, Prentice-Hall, Inc.. W. T. THOMSON 988 Theor of Vibration and Applications. New Jerse, Prentice-Hall, Inc.. J. S. RAO 99 Turbomachine Blade Vibration. Wile Eastern Limited, New Delhi, India.. W. WEAVER, S. P. TIMOSHENKO, D. H. YOUNG 99 Vibration Problems in Engineering. John Wile & Sons, Ltd. 3. WWW Page of efunda, Inc. The ultimate online reference for engineers. ( 47