three dimensional free vibration of pretwisted beams a method for analyzing the free vibration of pretwisted structural members of arbitrary but uniform cross section is presented this method is based on three dimensional elasticity theory and three dimensional finite element formulation the equations of elasticity and the governing equations of motion are obtained using a rotating coordinate system that rotates along with the cross section the relationship between the fixed cartesian coordinate system and the rotating coordinate system is based on the pretwist angle rate this method can capture warping poisson effects and shear effects hamilton s principle is used to derive the equations of motion that in turn are used to obtain the natural frequencies and mode shapes results showing the relationship between the rate of pretwist and the natural frequencies of prismatic members with rectangular cross sections and different lengths are presented the results also show coupling between the two bending modes and coupling between the axial and torsional modes if the axis of rotation and the centroid of the cross section are not coincident coupling among all four basic modes and higher modes may occur this behavior is difficult or sometimes impossible to capture with simpler technical theories because the method is based on three dimensional elasticity theory it can accurately capture the behavior of long or short beams thin or thick sections arbitrary geometry and inhomogeneous laminated anisotropic materials

rigid body concept for geometric nonlinear analysis of 3d frames plates and shells based on the updated lagrangian formulation abstract in the nonlinear analysis of elastic structures the displacement increments generated at each incremental step can be decomposed into two components as the rigid displacements and natural deformations based on the updated lagrangian ul formulation the geometric stiffness matrix kg is derived for a 3d rigid beam element from the virtual work equation using a rigid displacement field further by treating the three node triangular plate element tpe as the composition of three rigid beams lying along the three sides the kg matrix for the tpe can be assembled from those of the rigid beams the idea for the ul type incremental iterative nonlinear analysis is that if the rigid rotation effects are fully taken into account at each stage of analysis then the remaining effects of natural deformations can be treated using the small deformation linearized theory the present approach is featured by the fact that the formulation is simple the expressions are explicit and all kinds of actions are considered in the stiffness matrices the robustness of the procedure is demonstrated in the solution of several benchmark problems involving the postbuckling response

semilinear differential inclusions via weak topologies abstract the paper deals with the multivalued initial value problem x a t x x f t x for a a t a b x a x 0 in a separable reflexive banach space e the nonlinearity f is weakly upper semicontinuous in x and the investigation includes the case when both a and f have a superlinear growth in x we prove the existence of local and global classical solutions in the sobolev space w 1 p a b e with 1 p introducing a suitably defined lyapunov like function we are able to investigate the topological structure of the solution set our main tool is a continuation principle in frechet spaces and we prove the required pushing condition in two different ways some examples complete the discussion

two point b v p for multivalued equations with weakly regular r h s abstract a two point boundary value problem associated to a semilinear multivalued evolution equation is investigated in reflexive and separable banach spaces to this aim an original method is proposed based on the use of weak topologies and on a suitable continuation principle in frechet spaces lyapunov like functions are introduced for proving the required transversality condition the linear part can also depend on the state variable x and the discussion comprises the cases of a nonlinearity with sublinear growth in x or of a noncompact valued one some applications are given to the study of periodic and floquet boundary value problems of partial integro differential equations and inclusions appearing in dispersal population models comparisons are included with recent related achievements

asymptotic boundary value problems in banach spaces abstract a continuation principle is given for solving boundary value problems on arbitrary possibly infinite intervals to caratheodory differential inclusions in banach spaces for this aim the appropriate fixed point index is defined to condensing decomposable multivalued operators in frechet spaces this index extends and unifies the one for compact maps in andres et al trans amer math soc 351 1999 4861 4903 as well as the one for operators in banach spaces in bader ph d thesis university of munich 1995 as an application we prove the existence of an entirely bounded solution of a semilinear evolution inclusion

natural vibration of pre twisted shear deformable beam systems subject to multiple kinds of initial stresses free vibration and buckling of pre twisted beams exhibit interesting coupling phenomena between compression moments and torque and have been the subject of extensive research due to their importance as models of wind turbines and helicopter rotor blades the paper investigates the influence of multiple kinds of initial stresses due to compression shears moments and torque on the natural vibration of pre twisted straight beam based on the timoshenko theory the derivation begins with the three dimensional green strain tensor the nonlinear part of the strain tensor is expressed as a product of displacement gradient to derive the strain energy due to initial stresses the frenet formulae in differential geometry are employed to treat the pre twist the strain energy due to elasticity and the linear kinetic energy are obtained in classical sense from the variational principle the governing equations and the associated natural boundary conditions are derived it is noted that the first mode increases together with the pre twisted angle but the second decreases seeming to close the first two modes together for natural frequencies and compressions the gaps close monotonically as the angle of twist increases for natural frequencies and buckling compressions however unlike natural frequencies and compressions the closeness is not monotonic for buckling shears moments and torques

free vibration analysis of a flat plate using the hierarchical finite element method abstract the hierarchical finite element method is used to determined the natural frequencies and modes of a flat rectangular plate ten different boundary conditions including free edges and point supports are considered in this paper extensive results are presented for each case including the variation of frequency with the aspect ratio and the poisson ratio and these are shown to be in very good agreement with the work of other investigators this confirms both the applicability and accuracy of solution of the hfem to problem of this type

a finite element for vibration analysis of twisted blades based on beam theory on presente une methode d elements finis de discretisation en segments de poutre d aubes tournantes pretordues en employant la methode de deplacement de matrice on developpe les proprietes de rigidite et de masse on obtient de maniere systematique la matrice de rigidite structurale et l effet du mouvement de rotation du rotor sur la matrice de rigidite a partir des fonctions potentielles et d energie cinetique

dynamic analysis of thin walled closed section beams the present paper considers the problem of dynamic behavior of thin walled beams of arbitrary closed cross section by means of an exact solution starting from benscoter s theory the differential equations of motion are derived by postulating the principle of the virtual work due to a variation of displacements in the case of simply supported thin walled beam a closed form solution for the coupled natural frequencies of free harmonic vibrations was derived the method is illustrated by examples and results are compared with analytical results analyzed by vlasov s theory as well as with fem results

development of an exact dynamic stiffness matrix for free vibration analysis of a twisted timoshenko beam abstract an exact dynamic stiffness matrix for a twisted timoshenko beam is developed in this paper in order to investigate its free vibration characteristics first the governing differential equations of motion and the associated natural boundary conditions of a twisted timoshenko beam undergoing free natural vibration are derived using hamilton s principle the inclusion of a given pretwist together with the effects of shear deformation and rotatory inertia gives rise in free vibration to four coupled second order partial differential equations of motion involving bending displacements and bending rotations in two planes for harmonic oscillation these four partial differential equations are combined into an eighth order ordinary differential equation which is identically satisfied by all components of bending displacements and bending rotations this difficult task has become possible only with the help of symbolic computation next the exact solution of the differential equation is obtained in completely general form in terms of eight arbitrary constants this is followed by application of boundary conditions for displacements and forces the procedure leads to the formation of the dynamics stiffness matrix of the twisted timoshenko beam relating harmonically varying forces with harmonically varying displacements at its ends the resulting dynamic stiffness matrix is used with particular reference to the wittrick williams algorithm to compute the natural frequencies and mode shapes of a twisted timoshenko beam with cantilever end condition the exact results from the present theory are compared with numerically simulated results using simpler theories and some conclusions are drawn

on continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals soit x t f t x t t j x s ou jcr est un intervalle donne f j r n r n est une fonction continue et s un sous ensemble de c j r n l espace des fonctions continues de j dans r n on considere l equation operateur associee x t x ou t est un operateur non lineaire discontinu dans son domaine

vibrations of elastically restrained nonuniform beams with arbitrary pretwist the coupled governing differential equations with general elastic boundary conditions for coupled bending bending vibration of a pretwisted nonuniform beam are derived by using hamilton s principle the e eld transfer matrix for the general system is expressed in integration form as the elements of the matrix can be integrated analytically the exact e eld matrix is therefore obtained the transfer matrix method is presented to study the vibrations of any nonuniform cross sectional beam with arbitrary variation of pretwist angle the frequency equation is derived and expressed in terms of the transfer matrix several limiting cases are examined the corresponding results are compared with previous work in the literature finally the ine uence of elastically restrained boundary conditions and uniform and nonuniform pretwists on the natural frequencies is investigated

free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method abstract starting from the governing differential equations of motion in free vibration the dynamic stiffness matrix of a uniform rotating bernoulli euler beam is derived using the frobenius method of solution in power series the derivation includes the presence of an axial force at the outboard end of the beam in addition to the existence of the usual centrifugal force arising from the rotational motion this makes the general assembly of dynamic stiffness matrices of several elements possible so that a non uniform or tapered rotating beam can be analyzed for its free vibration characteristics by idealizing it as an assemblage of many uniform rotating beams the application of the derived dynamic stiffness matrix is demonstrated by investigating the free vibration characteristics of uniform and non uniform tapered rotating beams with particular reference to the wittrick williams algorithm the results from the present theory are compared with published results it is shown that the proposed dynamic stiffness method offers an accurate and effective method of free vibration analysis of rotating beams

hierarchical finite element analysis of the vibration of membranes abstract a hierachical finite element for the vibration of membranes is presented the element has the ability to be joined to neighbouring elements and the numbers of hierarchical terms are allowed to vary in both directions of the element co ordinate axes the element transverse displacement is described by four linear shape functions plus a variable number of hierarchical functions which are forms of legendre orthogonal polynomials the four nodal displacements and the amplitudes of the hierarchical functions on the edges and in the interior of the element are used as generalized co ordinates inter element compatibility is achieved by matching the generalized co ordinates at the nodes and edges shared by elements results are obtained for simply supported square and l shaped membranes comparisons are made with exact solutions for the square membrane and with highly accurate approximate and linear finite element solutions for the l shaped membrane the results for the square membrane confirm that the solutions always converge from above to the exact values as the numbers of hierarchical terms are increased and highly accurate answers are obtained depite the use of a very few hierarchical terms the results of the l shaped membrane show that the hierarchical finite element solutions are largely more accurate than the linear finite element solutions despite the use of fewer system degrees of freedom

vibration of thin pre twisted helical beams abstract a helical beam has non zero curvature and tortuosity when there is a pre twist the frenet triad is not necessary coincident with the principal triad when an initially straight rod undergoing a large deformation it will be curved and twisted during each deformation sequence the purpose of the paper is to establish the governing equations and the associated natural boundary conditions for a pre twisted helical beam by variational principles and differential geometry although circular helix is taken as example the formulation is valid for inhomogeneous and non uniform parameters along the centerline that variable curvature and tortuosity can be treated without difficulties the method can be used to treat helical anisotropy if the pre twist rate of a ring is 0 5 we show a thick mobius ring for the first time the solution of the natural vibration problem is through chebyshev discretization and clenshaw curtis integration by means of the galerkin formulation

on the use of consistent approximations in the solution of semi infinite optimization and optimal control problems we consider a pair consisting of an optimization problem and its optimality function p and define consistency of approximating problem optimality function pairs p n n to p in terms of the epigraphical convergence of the p n to p and the hypographical convergence of the optimality functions n to we then show that standard discretization techniques decompose semi infinite optimization and optimal control problems into families of finite dimensional problems which together with associated optimality functions are consistent discretizations to the original problems we then present two types of techniques for using consistent approximations in obtaining an approximate solution of the original problems the first is a filter type technique similar to that used in conjunction with penalty functions the second one is an adaptive discretization technique that can be viewed as an implementation of a conceptual algorithm for solving the original problems

asymptotic boundary value problems for second order differential systems abstract topological methods are developed for the solvability of vector second order boundary value problems on noncompact intervals the solutions are located in given sets and enjoy prescribed properties the main theorem is supplied by two illustrating examples

diagonally modified conditional gradient methods for input constrained optimal control problems many iterative methods for constrained minimization problems conform to the general scheme u i 1 in a i omega f u i u 1 in omega where omega is a set of feasible vectors in a banach space x f is a payoff function whose minimum is sought in omega and a i is a map with range in the set of subsets of omega if omega i f i is a sequence of related problems that approximate omega f in some sense with omega i subset omega i 1 subset omega then the corresponding diagonal modification of the original algorithm generates iterates via the recursion u i 1 in a i omega i f i u i u 1 in omega 1 if omega i f i is properly selected the diagonal modification can compute approximate solutions for omega f efficiently in circumstances where the original algorithm is difficult or impossible to implement for omega f in particular this happens for certain gradient related descent methods and increasingly r

effective diagonalization strategies for the solution of a class of optimal design problems it is shown that it is possible to obtain optimal diagonalization strategies for the discretization of semiinfinite minimax optimal design problems both exact and approximate methods for the computation of these optimal diagonalization strategies are proposed the algorithms for computing approximate diagonalization strategies yield very good approximations in much less computing time than needed to compute an optimal diagonalization strategy exactly the proposed diagonalization strategies can be implemented by using estimation schemes to obtain approximations to the various quantities which determine an optimal strategy experimental results involving the solution of optimal loop shaping problems for multivariable linear feedback systems show that the use of these implementable strategies leads to considerable savings in computer time over alternative approaches

vibration of skew plates resting on point supports abstract this paper deals with the vibrations of skew plates resting on point supports the spline element method is used to demonstrate the accuracy of the method several examples are solved and results are compared with those obtained by other approximate methods the effects of skew angles and locations of point supports on natural frequencies of isotropic skew plates are also investigated

natural frequencies of a pre twisted blade in a centrifugal force field abstract in this paper starting with the thin shell theory the governing partial differential equation of motion for the transverse deflection of a rotating pre twisted plate is derived strain displacement relationships include the effect of warping of the cross section due to twist bend coupling effect introduced as a result of pre twist in the plate of non circular rectangular cross section then the equation of motion thus derived is used to formulate the free vibration of a typical turbo machinery cantilevered airfoil blade by considering it as a plate of an equivalent rectangular cross section subjected to a quasi static load due to centrifugal force field the analytical derivation considers both the stress stiffening as well as stress softening effects of the centrifugal forces on the spinning airfoil the partial differential equation governing the flexural motion of the plate is transformed into a matrix eigenvalue form using a rayleigh ritz technique the plate deformations are represented by a set of admissible sinusoidal trial functions which fully satisfy all the clamped end constrains as well as the free edge boundary conditions the results of the analytical model exhibit an excellent agreement with the previously published test data both for thin and thick plate geometries and even in highly twisted configurations the results of the eigenvalue solution are presented in a non dimensional form for plates of varying aspect ratios and different amounts of pre twist in the plate the numerical results are directly applicable in determining the static and running frequencies of typical blades used in turbo machinery

a spectral tchebychev solution for three dimensional dynamics of curved beams under mixed boundary conditions abstract this paper presents the application of the spectral tchebychev st technique for solution of three dimensional dynamics of curved beams structures having variable and arbitrary cross section under mixed boundary conditions to accurately capture the vibrational behavior of curved structures a three dimensional 3d solution approach is required since these structures generally exhibit coupled motions in this study the integral boundary value problem ibvp governing the dynamics of the curved structures is found using extended hamilton s principle where the strain energy is expressed using 3d linear elasticity equation to solve the ibvp numerically the 3d spectral tchebychev 3d st approach is used to evaluate the integral and derivative operations defined by the ibvp and to render the complex geometry into an equivalent straight beam with rectangular cross section a series of coordinate transformations are applied to validate and assess the performance of the presented solution approach two case studies are performed i curved beam with rectangular cross section ii curved and pretwisted beam with airfoil cross section in both cases the results natural frequencies and mode shapes are also found using a finite element fe solution approach it is shown that the difference in predicted natural frequencies are less than 1 and the mode shapes are in excellent agreement based on the modal assurance criteria mac analyses however the presented spectral tchebychev solution approach significantly reduces the computational burden therefore it can be concluded that the presented solution approach can capture the 3d vibrational behavior of curved beams as accurately as an fe solution but for a fraction of the computational cost

stochastic vibration of axially loaded monosymmetric timoshenko thin walled beam abstract an analytical method is presented to perform the flexure torsion coupled stochastic response analysis of monosymmetric axially loaded timoshenko thin walled beam subjected to various kinds of concentrated and distributed stochastic excitations with stationary and ergodic properties the effects of warping stiffness axial force shear deformation and rotary inertia are included in the present formulations first the damped general governing differential equations of motion of axially loaded timoshenko thin walled beam are developed and the free vibration analysis is performed once the natural frequencies and mode shapes are obtained mode superposition method in conjunction with receptance method is used to compute the mean square displacement response of the axially loaded thin walled beam finally the method is illustrated by its application to two test examples to investigate the effects of warping stiffness axial force shear deformation and rotary inertia on the stochastic response of the thin walled beams

finite element model for vibration analysis of pre twisted timoshenko beam abstract a new linearly pre twisted timoshenko beam finite element which has two nodes and four degrees of freedom per node is developed and subsequently used for coupled bending bending vibration analysis of pre twisted beams with uniform rectangular cross section first displacement functions based on two coupled displacement fields the polynomial coefficients are coupled through consideration of the differential equations of equilibrium are derived for pre twisted beams whose flexural displacements are coupled in two planes this approach helps to reduce the number of nodal variables next the stiffness and mass matrices of the finite element model are obtained by using the energy expressions finally the natural frequencies of pre twisted timoshenko beams are obtained and compared with previously published theoretical and experimental results to confirm the accuracy and efficiency of the present model excellent agreement is found with the previous studies also the new pre twisted timoshenko beam element has good convergence characteristics

the torsion extension coupling in pretwisted elastic beams abstract an explicit asymptotic formula is derived for the untwist of a pretwisted elastic beam subjected to homogeneous extension or equivalently for the longitudinal contraction produced by a torsional moment it is based on an asymptotic expansion of the three dimensional equations of linear elasticity the general result consists of two terms one due to lengthwise variation of the warping of the cross sections and another due to local rotations from bending when the elastic center is not on the axis of pretwist the result supports some recent approximate theories for pretwisted elastic beams

a spectral tchebychev technique for solving linear and nonlinear beam equations abstract this paper presents a spectral tchebychev technique for solving linear and nonlinear beam problems the technique uses tchebychev polynomials as spatial basis functions and applies galerkin s method to obtain the spatially discretized equations of motion unlike alternative techniques that require different admissible functions for each different set of boundary conditions the spectral tchebychev technique incorporates the boundary conditions into the derivation and thereby enables the utilization of the solution for any linear boundary conditions without re derivation furthermore the proposed technique produces symmetric system matrices for self adjoint problems in this work the spectral tchebychev solutions for euler bernoulli and timoshenko beams are derived the convergence and accuracy characteristics of the spectral tchebychev technique is studied by solving eigenvalue problems with different boundary conditions it is found that the convergence is exponential and a small number of polynomials is sufficient to obtain machine precision accuracy the application of the technique is demonstrated by solving 1 eigenvalue problems for tapered timoshenko beams with different boundary conditions taper ratios and beam lengths 2 an euler bernoulli beam problem with spatially and temporally varying forcing elastic boundary and damping 3 large deflection nonlinear euler bernoulli beam problems with different boundary conditions and 4 a micro beam problem with nonlinear electrostatic excitation the results obtained from the spectral tchebychev solutions are seen to be in excellent agreement with those presented in the literature

the saint venant problem and principle in elasticity abstract the traditional semi inverse solution method of the saint venant problem and the saint venant principle which were described in the euclidian space under the lagrange system formulation are updated to be solved in the symplectic space under the conservative hamiltonian system thus the saint venant problem and the saint venant principle have been unified by the direct method it is proved in the present paper that all the saint venant solutions can be obtained directly via the zero eigenvalue solutions and all their jordan normal forms of the corresponding hamiltonian operator matrix and the saint venant principle corresponds to neglect of all the non zero eigenvalue solutions in which the non zero eigenvalue gives the decay rate

finite element vibration analysis of rotating timoshenko beams abstract the stiffness and mass matrices of a rotating twisted and tapered beam element are derived the angle of twist breadth and depth are assumed to vary linearly along the length of beam the effects of shear deformation and rotary inertia are also considered in deriving the elemental matrices the first four natural frequencies and mode shapes in bending bending mode are calculated for cantilever beams the effects of twist offset speed of rotation and variation of depth and breadth taper ratios are studied

the axial torsional vibration of pretwisted beams we investigate the coupled axial torsional vibration of pretwisted beams the equations of motion governing the extension torsion and cross sectional warping of pretwisted beams are derived from hamiltons principle and the common assumptions used to simplify the equations are carefully examined through scaling analysis inconsistencies in previous works such as the neglect of spatial and time derivatives of torsion in the lagrangian functional are remedied giving rise to fourth order terms in the equations of motion that are significant for higher harmonics furthermore scaling analysis of the governing equations yields a set of objective criteria for checking the validity of the frequently misused assumption that the warping function under pretwist is locally identical to that of a prismatic beam we show that this simplification is responsible for the common poor prediction of axial resonance frequencies for pretwisted beams

exact dynamic stiffness elements based on one dimensional higher order theories for free vibration analysis of solid and thin walled structures in this paper an exact dynamic stiffness formulation using one dimensional 1d higher order theories is presented and subsequently used to investigate the free vibration characteristics of solid and thin walled structures higher order kinematic fields are developed using the carrera unified formulation which allows for straightforward implementation of any order theory without the need for ad hoc formulations classical beam theories euler bernoulli and timoshenko are also captured from the formulation as degenerate cases the principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions an exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses the explicit terms of the dynamic stiffness matrices are also presented the resulting dynamic stiffness matrix is used with particular reference to the wittrick williams algorithm to carry out the free vibration analysis of solid and thin walled structures the accuracy of the theory is confirmed both by published literature and by extensive finite element solutions using the commercial code msc nastran

finite element theory for curved and twisted beams based on exact solutions for three dimensional solids part 1 beam concept and geometrically exact nonlinear formulation a geometrically exact and completely consistent finite element theory for curved and twisted beams is proposed it is based on the kinematical hypothesis generally formulated for large deformation and accounts for all kinds of deformation of a three dimensional solid translational and rotational displacements of the cross sections warping of their plane and distortion of their contours the principle of virtual work is applied in a straightforward manner to all non zero six components of the strain and stress tensors expressions are given for tangent matrices of elastic inertia and external forces and specific techniques for discretization and updating are developed for the analysis of beams in inertial and non inertial frames finally the numerical properties of the finite element models are demonstrated through examples c 1998 elsevier science s a all rights reserved

a contribution to the exact modal solution of in plane beam structures abstract the exact vibration modes and natural frequencies of planar structures and mechanisms comprised euler bernoulli beams are obtained by solving a transcendental non linear eigenvalue problem stated by the dynamic stiffness matrix dsm to solve this kind of problem the most employed technique is the wittrick williams algorithm developed in the early seventies by formulating a new type of eigenvalue problem which preserves the internal degrees of freedom for all members in the model the present study offers an alternative to the use of this algorithm the new proposed eigenvalue problem presents no poles so the roots of the problem can be found by any suitable iterative numerical method by avoiding a standard formulation for the dsm the local mode shapes are directly calculated and any extension to the beam theory can be easily incorporated it is shown that the method here adopted leads to exact solutions as confirmed by various examples extensions of the formulation are also given where rotary inertia end release skewed edges and rigid offsets are all included

dispersion relations of axisymmetric wave propagation in initially twisted bi material compounded cylinders abstract this paper investigates the dispersion relations of axisymmetric wave propagation in initially twisted bi material compound cylinders within the scope of the piecewise homogeneous body model utilizing 3d linearized wave propagation theory in an initially stressed body it is assumed that the constituents of the compound cylinder are isotropic and homogeneous the corresponding eigen value problem is formulated and the solution method for this problem as well as the algorithm for obtaining the numerical results is developed the numerical results illustrate the influence of the initial twisting of the compound cylinder on the dispersion of the axisymmetric wave propagation as presented and discussed in the present paper in particular it is established that as a result of the existence of the initial twisting at least in one constituent of the considered compound cylinder the axisymmetric longitudinal and torsional waves cannot propagate separately i e the new type axisymmetric waves which differ from the axisymmetric torsional and longitudinal waves must appear

free vibration analysis of a twisted beam using the dynamic stiffness method abstract an exact dynamics stiffness matrix is developed and subsequently used for free vibration analysis of a twisted beam whose flexural displacements are coupled in two planes first the governing differential equations of motion of the twisted beam undergoing free natural vibration are derived using hamilton s principle next the general solutions of these equations are obtained when the oscillatory motion of the beam is harmonic this is followed by application of boundary conditions for displacements and forces which essentially leads to the formation of the dynamics stiffness matrix of the twisted beam relating harmonically varying forces with harmonically varying displacements at its ends the resulting dynamic stiffness matrix is used in connection with the wittrick williams algorithm to compute natural frequencies and mode shapes of a twisted beam with cantilever end condition these are compared with previously published results to confirm the accuracy of the method and some conclusions are drawn