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This study provides a quasi-three-dimensional (quasi-3D) dynamic model for rotating pre-twisted functionally graded (FG) blades based on the three-dimensional elasticity shell theory and Carrera unified formulation. The material properties of the pre-twisted FG blades alter continuously in the direction of the thickness. The boundary conditions (clamped-end and free-edge) are realized by utilizing the well-known penalty method. Initial centrifugal stress field due to the high rotating speed is determined by two centrifugal stress solving methods, direct integration of centrifugal forces (DICF) and static analysis under centrifugal forces (SACF). The modified Fourier spectral approach is utilized to construct the displacement variables of the higher-order configurations. Eigenvalue problems are obtained by using the Hamilton's principle on the base of the kinetic, strain, centrifugal potential and boundary potential energy. The results of the theoretical model display a good agreement with the reference data and FEM results. Parametric studies are then performed to assess the applicability of the centrifugal stress solving methods and investigate the effects of the centrifugal shear stress, rotating speed, pre-twisted angle, penalty factors as well as the volume fraction index on the vibration characteristics of the pre-twisted FG blades. It is expected that the quasi-3D solutions for the rotating pre-twisted FG blades will serve as benchmarks in future researches.

The nonlinear vibrations of a water-filled circular cylindrical shell subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated experimentally and numerically by using a seamless aluminium sample. The experimental boundary conditions are close to simply supported edges. The presence of exact one-to-one internal resonance, giving rise to a travelling wave response around the shell circumference and non-stationary vibrations, is experimentally observed and the nonlinear response is numerically reproduced. The travelling wave is measured by means of state-of-the-art laser Doppler vibrometers applied to multiple points on the structure simultaneously. Chaos is detected in the frequency region where the travelling wave response is present. The reduced-order model is based on the Novozhilov nonlinear shell theory retaining in-plane inertia and the nonlinear equations of motion are numerically studied (i) by using a code based on arclength continuation method that allows bifurcation analysis in case of stationary vibrations, (ii) by a continuation code based on direct integration and Poincare maps that evaluates also the maximum Lyapunov exponent in case of non-stationary vibrations. The comparison of experimental and numerical results is particularly satisfactory.

An efficient method based on quadratic performance index and Lyapunov's second method is developed for the vibration reduction optimization of frame structure rotating at a constant angular velocity. Four optimization formulations and algorithms are studied for both low and high angular velocity. These formulations consider both linear vs nonlinear dynamic models as well as direct integration and Lypunov method to evaluate performance index. Three numerical examples are presented to compare the effectiveness of the four algorithms. Numerical results show that all the four optimization methods can reduce vibration at low rotating speed and find very similar optimal designs. However, at high rotating speed, dynamic model of rotating structure must consider nonlinear axial strain in order to obtain correct dynamic response. Since the Lyapunov's second method is not applicable for the nonlinear model and the NFTI (nonlinear dynamic model with nonlinear axial strain and time integration) method is very inefficient, the present study develops a perturbation method, linearizes the nonlinear dynamic model and establishes the LNFLA(linearized dynamic model with nonlinear axial strain, the Lyapunov's second method). Numerical examples show the optimization method LNFLA provides an efficient numerical method for vibration reduction of rotating frame structure with both low and high rotating speed.

A response sensitivity-based approach is presented for identifying the local damages in isotropic plate structures from the measured structural dynamic responses. The local damage is simulated by a reduction in the elemental Young's modulus of the plate. In the forward analysis, the forced vibration responses of the plate under external force are obtained from Newmark direct integration. In the inverse analysis, a response sensitivity-based finite element model updating approach is used to identify local damages of the plate in time domain. The damage identification results are obtained iteratively with the penalty function method with Tikhonov regularization using the measured structural dynamic responses. Two numerical examples are investigated to illustrate the correctness and efficiency of the proposed method. Both single damage and multiple damages cases are studied. The effects of measurement noise and measurement point on the identification results are investigated. Studies in this paper indicate that the proposed method is efficient and robust for both single and multiple damages for plate structures. Good identified results can be obtained from the short time histories of a few number of measurement points.

Natural frequencies and mode shapes of non-homogeneous (deterministic and stochastic) rods and beams are studied. The solution is based on the functional perturbation method (FPM). The frequencies and mode shapes are considered as functionals of the non-homogeneous properties. The natural frequency and mode shape of the k th order is obtained analytically to any desired degree of accuracy. Once the functional derivatives (with respect to the non-uniform property) have been found, the solution for any morphology is obtained by direct integration without resolving the differential equation. Several examples with different non-homogeneous properties are solved and compared with exact solutions as an accuracy check. The FPM accuracy range for the frequency ω and the mode shape is less than 1% even for high heterogeneities. In the stochastic case the accuracy of the natural frequencies depends on the stochastic information used/given, on the correlation distance (roughly the “grain size”), on the function around which the perturbation is executed, and on whether we are interested in the properties of ω or of ω 2 . Moreover, all frequency modes have the same response to heterogeneity as long as their wave length is of the order of the heterogeneity's characteristic distance. In addition, the heterogeneity effect on the average natural frequencies is minimal for the fundamental mode, and may serve as a design tool.

Three numerical solution techniques for nonlinear eardrum-type oscillations with an even function in the restoring force are studied. The first suggested technique is named the target function technique. In the technique, the second zeros of the target function is the circular frequency of motion. The second suggested technique is named the multiple-parameter technique, and the involved parameters are evaluated from the governing equation, the initial conditions, and properties of motion. The third suggested technique is named the direct integration technique. All techniques depend on the computer computation significantly and provide very accurate numerical results. Finally, numerical examples are given.

The seal characteristic is an important factor affecting the performance of the rotor system. The nonlinear model of the rotor–seal system is established using Muszynska's nonlinear seal forces. An efficient and high-precision direct integration scheme is presented based on the 2 N type algorithm for the computation of exponential matrices. The nonlinear phenomena in the unbalanced rotor–seal system are investigated using the adopted model and numerical integration method. The influence of the seal on the nonlinear characteristics of the rotor system is analyzed by the bifurcation diagrams and Poincare maps. Various nonlinear phenomena in the rotor–seal system, such as periodic motion, double-periodic motion, quasi-periodic motion and Hopf bifurcation are investigated and the stability is judged by Floquet theory and bifurcation theorem. The influence of parameters on the critical instability speed of the balanced rotor–seal system is also included. The high-precision direct integration method is effectively applied to the nonlinear numerical analysis of the rotor–seal system. The scheme has high precision and a large time step may be adopted to save computing resources.

The non-linear dynamic behavior of a rotor–bearing system is analyzed based on a continuum model. The finite element method is adopted in the analysis. Emphasis is given on the so-called “oil whip phenomena” which might lead to the failure of the rotor system. The dynamic response of the system in an unbalanced condition is approached by the direct integration method and mode superposition method. It is found that a typical “oil whip phenomenon” is successfully produced. Furthermore, the bifurcation behavior of the oil whip phenomenon that is much concerned by present non-linear dynamics is analyzed. The rotor–bearing system is also examined by the simple discrete model. Significant differences are found between these two models. It is suggested that a careful examination should be made in modelling such non-linear dynamic behavior of the rotor system.

This paper examines local parametric vibrations in the stay cables of a cable-stayed bridge. The natural frequencies of the global modes are obtained by using a three-dimensional FE model. The global motions generated by (1) sinusoidal excitations using exciter, (2) a traffic loading, and (3) an earthquake are analyzed by using the modal analysis method or the direct integration method. The local vibration of stay cable is calculated by using a model in which inclined cable is subjected to time-varying displacement at one support during global motions. This paper describes the properties of the local vibrations in stay cables under these dynamic loadings by using an existing cable-stayed bridge.

The dynamic behavior of multi-span non-uniform beams transversed by a moving load at a constant and variable velocity is investigated. The continuous beam is modelled using Bernoulli–Euler beam theory. The solution is obtained by using both the modal analysis method and the direct integration method. The natural frequencies and mode shapes used in the solution of this problem are obtained exactly by deriving the exact dynamic stiffness matrices for any polynomial variation of the cross-section along the beam using the exact element method. The mode shapes are expressed as infinite polynomial series. Using the exact mode shapes yields the exact solution for general variation of the beam section in case of constant and variable velocity. Numerical examples are presented in order to demonstrate the accuracy and the effectiveness of the present study, and the results are compared to previously published results.

A quadratic isoparametric finite element formulation based on the first order shear deformation theory is presented for the free vibration and transient response analysis of multiple delaminated doubly curved composite shells subjected to a hygrothermal environment. A simple multiple delamination model developed by the authors earlier is employed to take care of any number/size of delamination located anywhere in the laminate. The analysis takes into account the lamina material properties at elevated moisture concentration and temperature. Newmark's direct integration scheme is used to solve the dynamic equation of equilibrium at every timestep during the transient analysis. Several numerical examples are considered and the results are compared with those available in the literature. The results show a reduction in the fundamental frequency with an increase in the percentage of uniform moisture content as well as temperature for simply supported antisymmetric crossply and angleply laminates for any size of delamination considered. The central dynamic displacements and the stresses at the center of laminates are observed to increase due to the effect of moisture/temperature subjected to the suddenly applied uniform pulse loading

In this paper, a method is proposed for modelling large deflection beam response involving multiple vibration modes. Significant savings in computational time can be obtained compared with the direct integration non-linear finite element method. The deflections from a number of static non-linear finite element test cases are transformed into modal co-ordinates using the modes of the underlying linear system. Regression analysis is then used to find the unknown coupled non-linear modal stiffness coefficients. The inclusion of finite element derived modal masses, and an arbitrary damping model completes the governing non-linear equations of motion. The response of the beam to excitation of an arbitrary nature may then be found using time domain numerical integration of the reduced set of equations. The work presented here extends upon the work of previous researchers to include non-linearly coupled multi-modal response. The particular benefits of this approach are that no linearization is imposed, and that almost any commercial finite element package may be employed without modification. The proposed method is applied to the case of a homogeneous isotropic beam. Fully simply supported and fully clamped boundary conditions are considered. For the free vibration case, results are compared to those of previous researchers. For the case of steady-state harmonic excitation, results are compared with the direct integration non-linear finite element method using ABAQUS. In all cases, excellent agreement is obtained.

The present work investigates the dynamics of a class of two-degree-of-freedom oscillators with cubic non-linearity in the restoring forces. These oscillators are under the action of an external load including constant and harmonic components. Initially, a perturbation analysis is applied to the equations of motion, demonstrating the effect of the asymmetry induced by the constant loading component on the classical 1:1 and 1:3 internal resonances, as well as on the possibility of the appearance of a first order 1:2 internal resonance. Next, sets of slow-flow equations governing the amplitudes and phases of vibration are derived for the special case of no internal resonance and for the most complicated case corresponding to 1:1 internal resonance. The analytical findings are then complemented by numerical results, obtained by examining the dynamics of a two-degree-of-freedom mechanical system. First, the effect of certain system parameters on the existence and stability of constant and periodic solutions of the slow-flow equations is illustrated by presenting a sequence of response diagrams. Finally, the dynamics of the system used as an example is investigated further by direct integration of the slow-flow equations. This shows the existence of a period-doubling sequence culminating into a continual interchange between quasiperiodic and chaotic response. It also demonstrates a new transition scenario from phase-locked to phase-entrained and drift response.

In this paper, vibration control of a rotating Euler–Bernoulli beam is considered. It is assumed that the fixed–free elastic beam is attached to the servomotor which uses PD control to achieve the desired angular rotation, at the same time, the shear force measured at the root of the beam is used as a feedback to control the beam tip vibration. Mode summation techniques and Laplace domain synthesis techniques are used to analyze the system. Parametric transfer functions relating beam tip motion to the desired rotation and beam rotation to the desired input rotation are obtained. One parameter is the frequency ratio between the natural frequency of the beam and the frequency of the control system, second parameter is the ratio between the shear force feedback gain and the control damping. Stability conditions with respect to these parameters are given. The effect of these parameters on the rotational motion and the beam tip vibration is discussed. Some values of these parameters that make possible desired rotational motion with suppressed tip vibration are suggested. Analysis and results of this work can also be applied to the bending strain feedback and tip velocity feedback control of a rotating Euler–Bernoulli beam.

The dynamic behavior of the multi-span continuous beam traversed by a moving mass at a constant velocity is investigated, in which it is assumed that each span of the continuous beam obeys uniform Euler–Bernoulli beam theory. The solution to this system is simply obtained by using both eigenfunction expansion or the modal analysis method and the direct integration method in combination. The effects of the inertia and the moving velocity of the load on the dynamic response of the continuous beam are evaluated for three kinds of continuous beams having uniform span length.

The present work investigates dynamics of a gear-pair system involving backlash and time-dependent mesh stiffness. In addition, the system is under the action of external excitation, caused by torsional moments and gear geometry errors. First, the equation of motion is established in a strongly non-linear form. Then, the emphasis is laid on a specific forcing frequency range, corresponding to conditions of simultaneous fundamental parametric resonance and principal external resonance. For these conditions, several types of periodic steady state response are identified and determined by employing suitable methodologies, including techniques applicable to piecewise linear systems and to oscillators with time-periodic coefficients. Moreover, these methodologies are complemented by appropriate procedures revealing the stability properties of the located periodic solutions. In the second part of the work, numerical results are presented. These results verify the validity and effectiveness of the new analytical methodology and provide information on the gear-pair dynamics. First, series of typical response diagrams are obtained, illustrating the effect of the mesh stiffness variation, the damping and the forcing parameters on the gear-pair periodic response. These response diagrams are accompanied by results obtained with direct integration of the equation of motion. In this way, it is demonstrated that for some parameter combinations, the dynamical system examined can exhibit more complicated and irregular response, including crises and intermittent chaos.

This paper is concerned with laminated plates mounted on a rotating disk under accelerating conditions. Interfacial friction is included at specified locations on the plate. The effects of in-plane loads and non-linear Coriolis forces are included. An eight noded isoparametric element is derived for the analysis. Under the action of combined accelerating conditions and Coriolis forces, it is shown that the blade is subjected to shock forces. Employing the Newmark direct integration method, the results obtained are presented.

This paper presents a method of vibration control for a beam carrying a mass at its tip subjected to earthquakes. A vibration isolation mechanism consisting of a gear train for the beam is presented. Theoretical analysis for the beam is developed, and to validate the method and analysis, experimental tests are carried out for a model of the present mechanism. It is clarified that the vibration displacements and the moments in the beam are suppressed significantly in comparison with a general beam without the mechanism. A method of optimal design has also been presented, and numerical calculations have been carried out for the beam with actual sizes. In the present beam, energies for controlling vibration are not required, because the earthquake force is utilized as a control force. Therefore, the structure using the beam has advantages as compared with the structure having an active vibration control system.

A method of modal analysis is proposed in this paper to investigate the forced vibration of multi-span Timoshenko beams. The ratio of the radius of gyration of the cross-section to one span length is defined as a parameterr. The effect ofron the first modal frequency of a beam is studied. A concentrated force traversing on the beam is used as an example. The effects of span number, rotary inertia and shear deformation on the maximum moment, the maximum deflection and the critical velocity of a beam are examined. The results are compared with those of a multi-span Bernoulli–Euler beam.

The analysis of the Timoshenko beam traversed by uniform partially distributed moving masses is carried out. Equations of motion are solved by using a finite difference based algorithm. The beam response, and the distribution of the shear force and bending moment along the beam have been computed. The effects of shear deformation, rotary inertia and the length of load distribution on the vibration of the beam have been analyzed. The results compare well with those reported in the literature for the limiting case in which the length of the load distribution reduces to zero and also the effects of shear deformation and rotary inertia are neglected.

Even though chaotic vibrations have been observed in many structural mechanics systems, their analysis has almost always been limited to single-degree-of-freedom (SDOF) approximations A typical example is the magnetoelastic beam studied by Moon and Holmes [1], which is reported to be the first experimental evidence of chaotic vibrations in structural mechanics. However, the authors have not come across any detailed structural analysis of the system. The present paper reports a structural dynamic analysis of the problem through a finite element formulation and the integration of the resulting equations of motion by a variable time stepping Newmark method (trapezoidal rule). The solution scheme has built-in algorithms for equilibrium iteration of the non-linear forces and check of the temporal solution trajectory. It is shown that the direct integration and mode superposition schemes are equally applicable for problems with chaotic response. The authors have the following conclusions: (1) the SDOF approximation with a high accuracy integration scheme may not reveal the regime of chaos even coarsely; (2) the manifestation of chaos is significantly influenced by the higher modes; (3) a spatially discrete model which represents the beam accurately could reveal regimes of chaos reasonably well even with second order schemes such as the trapezoidal rule, but it is essential for the model to be fine enough to represent the motion in higher modes accurately; (4) computationally efficient methods such as the mode superposition method, with an adequate number of modes included, could give accurate solutions to vibration problems involving chaos. (C) 1996 Academic Press Limited

The dynamic response of imperfect (initially deformed) rectangular plates has been previously investigated both experimentally and theoretically. The theoretical approach used was an asymptotic method that required simplifying assumptions. To eliminate the need for these assumptions, a direct integration approach has been used. The results confirm the asymptotic method results on the modal interaction between forced and parametric resonances, explain experimental results on the modal interaction between resonances of the same nature (forced–forced and parametric-parametric) and show the existence of coalescence phenomena (two different modes vibrating at the same frequency), with the particular characteristic that the dominating mode transfers energy to the one outside its instability zone.

An analysis is presented on the dynamics of a two-degree-of-freedom non-linear mechanical oscillator. The model consists of a rigid body which is supported by non-linear springs and can exhibit both vertical translation and rocking motions. Attention is focused on the response of the system under external excitation of the translation only, close to conditions of subharmonic resonance of order three. For the cases where the two linear natural frequencies of the system are almost identical, approximate periodic motions are determined by applying the method of multiple time scales. It is shown that besides linear and non-linear single-mode response, two-mode response is also possible, due to the 1:1 internal resonance. Some important aspects of the dynamics when no internal resonance is activated are also investigated. Next, the analytical findings the effect of the system parameters on the existence and stability of the predicted motions. The results reveal patterns of appearance of the periodic motions, providing valuable help in the efforts to eliminate them and achieve proper vibration isolation. Finally, direct integration of the original equations of motion shows that complex responses of the system can coexist with the analytically predicted motions, within the frequency ranges of interest.

A co-rotational finite element formulation of slender curved beam element is presented to investigate the effect of member initial curvature on the dynamic behaviour of planar flexible mechanisms. The Euler-Bernoulli hypothesis and the initial curvature are properly considered for the kinematics of curved beam. The nodal co-ordinates, incremental displacements and rotations, velocities, accelerations and the equations of motion of the system are defined in terms of a fixed global co-ordinate system, while the total strains in the beam element are measured in element co-ordinates which are constructed at the current configuration of the beam element. The element equations are constructed first in the element co-ordinate system and then transformed to the global co-ordinate system by using standard procedures. Both the deformation nodal forces and the inertia nodal forces of the beam element are systematically derived by consistent linearization of the non-linear beam theory in the element co-ordinates. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the non-linear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed element and to investigate the effect of the initial curvature on the dynamic response of the flexible mechanisms.

An approach is presented deriving analytical stability and bifurcation conditions for systems with periodically varying coefficients. The method is based on apoint mapping(period to period mapping) representation of the system's dynamics. An algorithm is employed to obtain an analytical expression for the point mapping and its dependence on the system's parameters. The algorithm is devised to derive the coefficients of a multinominal expansion of the point mapping up to an arbitrary order in terms of the state variables and of the parameters. Analytical stability and bifurcation condition are then formulated and expressed as functional relations between the parameters. To demonstrate the application of the method, the parametric stability of Mathieu's equation and of a two-degree of freedom system are investigated. The results obtained by the proposed approach are compared to those obtained by perturbation analysis and by direct integration which we considered to the “exact solution”. It is shown that, unlike perturbation analysis, the proposed method provides very accurate solution even forlarge valuesof the parameters. If an expansion of the point mapping in terms of a small parameter is performed the method is equivalent to perturbation analysis. Moreover, it is demonstrated that the method can be easily applied to multiple-degree-of-freedom systems using the same framework. This feature is an important advantage since most of the existing analysis methods apply mainly to single-degree-of-freedom systems and their extension to higher dimensions is difficult and computationally cumbersome.

Based on the physical properties and sectional dimensions of a conventional channel beam, a new channel beam (called the “iso-moment-of-inertia” beam), with area moment of inertia about the unsymmetrical principal axis of the cross-section equal to that about the symmetrical one, is obtained. The dynamic responses of the new channel beam subjected to a moving load are then investigated. It is found that the dynamic behaviors of coupled vibration and uncoupled vibration of the iso-moment-of-inertia beam are very different. The degree of divergence has something to do with the moving load speed and the effects of shear deformation, rotatory inertia, and warping torsion.

An integral formulation approach for the characterization of a displacement sensitive non-linear damping force is proposed. This method is based on the transformation of the displacement sensitivity characteristic of the damper into a velocity sensitivity characteristic, and then using an integral formulation to obtain the damping force over the velocity range. The response characteristics of a single-degree-of-freedom (DOF) isolator with displacement sensitive non-linear damper is obtained using the proposed formulation. A local equivalent linearization technique based on energy similarity, as well as direct integration is used to obtain simulation results. The transmitted force-time history as well as system transmissibility are compared with previous studies employing the traditional non-integral approach for damping force formulation. Simulation results are also compared with available experimental results to demonstrate the effectiveness of the proposed method. It is shown that for displacement dependent dampers, the damper force estimated using an integral formulation is the correct and accurate approach for simulation purposes.

Despite the rapid progress in numerical fluid dynamics, vortex shedding simulation is still a complicated process requiring its representation by empirical models. The applicability of these models is often restricted, by their inherent non-linearity, to simple beams. The complexities of most structural systems demand an improved generalization of the suggested models and a more representative method of analysis. In this paper, the finite element method in conjunction with Scanlan’s vortex shedding empirical model (used extensively for long-span bridges) is employed to formulate a plane frame element. Based on simple assumptions conforming with the attributes of flow-induced vibrations, the proposed approach is extended further to encompass a space frame element. The developed elements are then used to analyze elementary structures subjected to vortex shedding instability. The vortex shedding oscillatory response is determined by assembling and solving the constituent nonlinear equations of motion using a direct integration scheme and an iterative process. The results verify the validity of the proposed finite element procedure, which should prove to be a flexible means of vortex shedding analysis for more complex structure types.

The dynamics of two-degree-of-freedom oscillators including Rayleigh and Duffing type non-linearities is investigated. The method of multiple scales is first applied and a set of averaged equations is derived for cases of primary external resonance. These equations admit two types of constant solutions. The first type involves the directly excited mode only, while for some parameter combinations the non-linear damping terms excite the second mode also, even when no internal resonance is present. Stability and bifurcation analyses are then presented. Emphasis is placed on deriving explicit conditions on the system parameters that will lead to forms of the evolution equations which can be studied in detail, by utilizing results from the area of dynamical systems. To illustrate the effectiveness and accuracy of the analysis, numerical results are presented for a dynamic model of a specific practical system: namely, a three-parameter study is first carried over in order to reveal basic response features of a metal cutting system. Codimension one, two and three bifurcations are determined and their effect on the interaction and transition between the response modes is investigated. A representative sample of response diagrams and results from direct integration are also presented, providing an overall picture of the dynamics.

A method is presented to perform the deterministic and random vibration analysis of a Rayleigh-Timoshenko beam on an elastic foundation, which may simulate bridges, runways, rails, roadways, pipelines, etc. The Rayleigh-Timoshenko beam is assumed to be uniform, damped, with generalized boundary conditions and subjected to axial load; also the Rayleigh-Timoshenko beam includes the effect of shear deformation and rotatory inertia, which are usually important if the span-depth ratio of the beam is relatively small. Once the natural frequencies and mode shapes of the beam are obtained, modal analysis is used to compute the dynamic responses of the structure, such as the displacement and bending moment, and some statistical responses such as mean square values of the dynamic displacement and bending moment. These dynamic responses are quite useful for structural design, while the statistical dynamic responses play an important role in structural reliability analysis.

The influence of taper ratio, elastic root restraint, setting angle and rotational speed on the bending natural frequencies of a rotating non-uniform beam is investigated using a semi-exact numerical method. One observes that the influence of taper ratio on the second and third natural frequencies of a rotating beam with constant width and linearly varied depth and a double-tapered beam is greater than that of a beam with constant depth and linearly varied width. For a beam with rotational flexibility only, the first three natural frequencies of the rotating beam are greater than those of the non-rotating beam. The second and third natural frequencies of a rotating beam with translational flexibility are greater than those of the non-rotating beam. However, the fundamental natural frequency of the rotating beam can be less than that of the non-rotating beam. In particular, when the translational rigidity of the root is relatively low and the setting angle and rotational speed of the beam are relatively high, the value of the natural frequency becomes pure imaginary and the phenomenon of divergence instability occurs. The natural frequencies are decreased when the setting angle is increased, and the influence of the setting angle on the natural frequencies becomes very significant when the phenomenon of divergence instability is about to occur.

The dynamic finite element matrix equations are reduced to a scalar second order non linear differential equation, solved by using the direct integration method.

The frequency equation for a cantilever beam with a heavy mass at its free end is derived, taking into account shear and rotary inertia effects. Boundary conditions are altered to account for the discontinuities of the second and third derivatives of the displacement functions at the point of attachment of the extra mass, and the frequency equation is derived from the differential equation of uniform beam motion. Computer calculations for the first five modes for various mass and moment of inertia ratios show that the presence of a concentrated end mass lowers all the natural frequencies, with the fundamental affected most strongly. Rotary inertia is found to have a greater effect than translational inertia, and both of these second order effects are more important in a tightly clamped beam than in a loosely clamped or hinged beam

This paper is concerned with the random vibration of non-linear beams with variable sectional areas subjected to a moving vehicle load. When the track on the beam is assumed to have such irregularities, the dynamic deflections of the beams as well as the moving vehicle vary in a random fashion. The irregularity of the beam in the forward direction of the vehicle is assumed to be a Gaussian random process, and is expressed by a first order linear differential equation with white noise excitation. The longitudinal and transverse deflections of the beam are formulated by using the Galerkin finite element method, and the time evolutions of the means and covariances of deflections of the beam as well as the vertical deflection of the vehicle are computed by using the linearization technique and the implicit direct integration method. Numerical results are presented to compute the dynamic deflections with the forward velocity of the vehicle as a parameter.

Doubly-curved, axisymmetric shell finite elements are used to perform the dynamic analysis of shells of revolution. The modal expansion method is used since it is more efficient than the direct integration method when the axisymmetric shell structures are subjected to certain specific loadings. For the case of complicated loadings such as travelling loads, the modal participation factors are obtained in the form of convolution integrals which can be solved either by Laplace transformation or by numerical integration. Orthogonality conditions between displacement functions and the Fourier expansions of loadings have been used to simplify the consistent loads. It thus becomes possible to perform response calculations, like impulse response, step response, frequency response, travelling loading response, and also static deflections. The effectiveness of the present method is demonstrated through its performance in a series of examples with results compared to alternative solutions with excellent agreements.

Hamilton's principle is used in an extended form to construct a so-called consistent integration operator for the transient response in structural dynamics. The stability and accuracy properties of the method are discussed in detail. Finally, it is shown how the algorithm can be modified to provide it with unconditional stability without significant alteration of its accuracy properties.

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.

The stiffness, mass and gyroscopic matrices of a rotating beam element are developed, a cubic function being used for the transverse displacement. Shear deflection is included by use of end nodal variables of shear strain, along with transverse displacement and cross-section rotation; rotatory inertia effects are included in the energy functional to provide a Timoshenko beam formulation. The gyroscopic effects for small perturbations are linearized as a skew symmetric damping matrix. The formulation is implemented by numerical integration for a linearly tapered circular beam. A technique of reduction of the shear nodal variable prior to global assembly is shown to provide little loss in accuracy with reduced system bandwidth. Numerical comparisons for three previously published beam models are included, with results presented for the case of forward and reverse precession to verify the gyroscopic effects. The utility of the element in a general program for rotor dynamics analysis is identified.

This paper is concerned with the non-linear vibration of immovably supported variable beams, in which the geometric non-linearity due to the axial force generated by stretching the middle surface is taken into account. When the beam is subjected to a moving load, the dynamic deflections of the beam are computed by using a Galerkin finite element formulation, and the time differential terms are integrated by using the implicit direct integration method. Three types of approximations for solving the non-linear problem are considered, in the first of which both the longitudinal deflections and inertia are considered, with only the longitudinal deflections being considered, and in the second neither the longitudinal deflections nor inertia in the third. The results obtained by using these non-linear models are numerically compared with that obtained by the linear model.

The response of a sandwich beam subjected to moving forces (constant as well as pulsating) is analyzed by the use of Fourier and Laplace transforms and compared with the response of an equivalent elastic beam. The results indicate that the critical speed of force on a sandwich beam is always greater than that on an elastic beam of identical mass per unit length and flexural rigidity, and depends on its geometric and shear parameters. For subcritical speeds, the maximum deflection of a sandwich beam is shown to occur earlier than that of an equivalent elastic beam. An increase in the core shear stiffness is shown to be beneficial in reducing the dynamic magnification of the central deflection of the sandwich beam.

The non-stationary random vibration of a beam is investigated. The beam is subjected to a random force with constant mean value which is moving with constant speed along the beam. The statistical characteristics of the first and second order for the deflection and bending moment of the beam are computed by using the correlation method. The numerical results of the coefficient of variation of the deflection at beam span mid-point are given for five basic types of convariances of the force (white noise, constant, exponential cosine, exponential, and cosine wave). The effect of the speed of the movement of the force along the beam as well as the effect of the beam damping is investigated in detail. It is concluded that the resulting beam vibration turns out to be a non-stationary process even though the motion considered is that of a stationary random force.