A distributed-mass approach for dynamic analysis of Timoshenko plane frames

The dynamic stiffness method is the exact method for the dynamic analysis of plane frames using the continuous-coordinate system to consider the effect of mass distribution in beam elements. The dynamic stiffness method may create some null modes where the joints of beam element have null deformation. Unlike the Bernoulli–Euler frames, adding an interior node at the middle of the beam elements cannot normalize all the null modes of flexural vibration in the Timoshenko frames. The floating interior-node scheme is proposed to eliminate the null modes of flexural vibration in the Timoshenko frames. Orthogonal properties of vibration modes in Timoshenko plane frames are theoretically derived, through which the equations of motion in beam elements can be transformed into the decoupled equations of motion in terms of mode amplitudes.     

Coupled bending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effects

The dynamic transfer matrix method for determining natural frequencies and mode shapes of the bending-torsion coupled vibration of axially loaded thin-walled beams with monosymmetrical cross sections is developed by using a general solution of the governing differential equations of motion based on Bernoulli-Euler beam theory. This method takes into account the effect of warping stiffness and gives allowance to the presence of axial force. The dynamic transfer matrix is derived in detail. Two illustrative examples on the application of the present theory are given for bending-torsion coupled beams with thin-walled open cross sections. The effects of axial load and warping stiffness on coupled bending-torsional frequencies are discussed. Compared with those available in the literature, numerical results demonstrate the accuracy and effectiveness of the proposed method.     

Free vibrations of spatial Timoshenko arches

This paper addresses the evaluation of the exact natural frequencies and vibration modes of structures obtained by assemblage of plane circular arched Timoshenko beams. The exact dynamic stiffness matrix of the single circular arch, in which both the in-plane and out-of-plane motions are taken into account, is derived in an useful dimensionless form by revisiting the mathematical approach already adopted by Howson and Jemah (1999 [18]), for the in plane and the out-of-plan natural frequencies of curved Timoshenko beams. The knowledge of the exact dynamic stiffness matrix of the single arch makes the direct evaluation of the exact global dynamic stiffness matrix of spatial arch structures possible. Furthermore, it allows the exact evaluation of the frequencies and the corresponding vibration modes, for the distributed parameter model, through the application of the Wittrick and Williams algorithm. Consistently with the dimensionless form proposed in the derivation of the equations of motion and the dynamic stiffness matrix, an original and extensive parametric analysis on the in-plane and out-of-plane dynamic behaviour of the single arch, for a wide range of structural and geometrical dimensionless parameters, has been performed. Moreover, some numerical applications, relative to the evaluation of exact frequencies and the corresponding mode shapes in spatial arched structures, are reported. The exact solution has been numerically validated by comparing the results with those obtained by a refined finite element simulation.     

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler-Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.     

Free vibration analysis of a cracked beam by finite element method

In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.     

Vibration frequency of a curved blade with weighted edge

The natural vibration frequencies and mode shapes of a curved cylindrical blade with a weighted edge are investigated. A finite element method is used, in which curved cylindrical shell finite elements are utilized to model the blade. The weighted edge is modelled as a beam with its stiffness and mass added into the stiffness and mass of the blade. Vibration frequencies and mode shapes for blades with different boundary conditions and with different radii of curvature are obtained. Finite element results are compared with experimental results.     

Dynamic ground compliance in multistorey buildings

A study is presented of the changes in the characteristics of the natural modes of vibration for multistorey structures which are founded on flexible foundations. First a standard eigenvalue problem is formulated for the proportionally damped case. Then general relationships of changes in natural frequencies and mode shapes are derived for the linear vibration theory. By means of an example problem it is demonstrated, however, that only the first mode obeys the predicted changes of frequencies and mode shapes over a wide range of foundation stiffness. The higher modes are shown to deviate substantially from the linear behaviour. This deviation is ascribed to geometric changes in mode shapes.     

Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects

The paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated.     

The free vibration of compact rotating radial cantilevers

The free vibration of rotating uniform radial cantilever beams of compact cross section is considered, with account taken of centrifugal coupling between motions in the principal elastic planes. For cases other than those in which the principal elastic axes coincide with the equatorial and meridional planes the centrifugal coupling is shown to modify the vibrational behaviour of the compact beam when compared to that of a beam which is infinitely stiff in one principal plane and can result in a considerable reduction in fundamental natural frequency. Results are presented which show how the natural frequencies and mode shapes of the lower modes vary with spin speed for various root offset and cross-sectional configurations.     

Improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme

New approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments. Specifically, the exact eigendata, i.e., the exact natural frequencies and mode shapes, of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration. This is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions. Once the updated matrices for the linear structure are found, the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices. Numerical experiments show that for the same number of elements, the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model. Alternatively, the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model.     

Vibrations of a polygonal plate having orthogonal straight edges by an extended rayleigh-ritz method

An extended Rayleigh-Ritz method is presented for solving vibration problems of a polygonal plate having orthogonal straight edges. The polygonal plate is considered as an assemblage of several rectangular plates. For each element rectangular plate, the transverse displacement is approximated by interpolation functions corresponding to unknown displacements and slopes at the discrete points which are chosen along the edges, and series of trial functions which satisfy homogeneous artificial boundary conditions. By minimizing the energy functional corresponding to the assumed displacement function, the dynamic stiffness matrix of the element rectangular plate, which is similar to that obtained in the finite element method, is derived. The dynamic stiffness matrix of the whole system is obtained by summing up those of the element rectangular plates. Numerical results are presented for the natural frequencies and mode shapes of cantilever L-shaped and T-shaped plates.     

A sensitivity analysis of the effects of interconnection joint size,flexibility, and inertia on the natural frequencies of Timoshenko frames

The free vibrations of frame structures are influenced by the geometry, stiffness, and inertia of interconnection joints. The effects of generalized joint properties on the natural frequencies and mode shapes are studied for a wide range of natural frequencies by modeling the structure as a Timoshenko continuous system with discretized joints. Dynamic slope-deflection equations are used in the analysis, adapted to the boundary conditions imposed by joints with axial length, axial and rotary stiffness, and inertia. Beam/column axial deformation is also included. Frequency curves are presented for a wide range of beam/column and joint properties to establish the relative importance of model parameters on system free vibrations.     

Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements

This paper employs the numerical assembly method (NAM) to determine the “exact” frequency–response amplitudes of a multiple-span beam carrying a number of various concentrated elements and subjected to a harmonic force, and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force. First, the coefficient matrices for the intermediate concentrated elements, pinned support, applied force, left-end support and right-end support of a beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix. The graph of dynamic response amplitudes versus various exciting frequencies gives the frequency–response curve for any point of a multiple-span beam carrying a number of various concentrated elements. For the case of zero harmonic force, the above-mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained.     

Evaluation of efficiently computed exact vibration characteristics of space platforms assembled from stayed columns

The exact stiffness matrix method computer program BUNVIS finds the natural frequencies and modes of vibration of rigidly jointed three dimensional frames which contain stayed columns very efficiently, by using substructuring and simple substitute columns to compute the stayed column stiffness. BUNVIS is described and applied to a tetrahedral truss which was designed for use in space and which has stayed columns as its members and 21 966 degrees of freedom at its nodes. Locating the first 4978 natural frequencies needed 2 h of VAX-11/780 CPU time and 5860 array locations. These natural frequencies appeared in groups for which the associated modes are discussed.     

Purely axial vibration of thin cylindrical shells with shear-diaphragm boundary conditions

This work presents the free vibration characteristics of a thin walled cylindrical shell at the zeroth axial mode number. The cylindrical shell has shear-diaphragm boundary conditions at each end. The thin shell equations by Flügge are used as these equations of motion lead to more accurate results at low frequencies. The zeroth axial mode number is found to occur at the cut-on of the second class of waves. The mode shapes at these natural frequencies result in a purely axial displacement of the middle surface of the shell. High modal density for the first class of waves occurs before the cutting-on of the second class of waves. Beyond this frequency, the dynamic response is dominated by the latter modes.     

VIBRATIONS OF NON-UNIFORM CONTINUOUS BEAMS UNDER MOVING LOADS

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.     

Dynamic stiffness method for exact inplane free vibration analysis of plates and plate assemblies

The inplane free vibration behaviour of plates is investigated using the dynamic stiffness method. Some distinctive modes which went unnoticed in earlier investigations using the dynamic stiffness method have been addressed by revisiting the problem and focusing on the special set of missing solutions. Results are validated extensively both by published results as well as by numerical studies using NASTRAN and ABAQUS. The accuracy of the finite element method for inplane free vibration analysis is assessed and critically examined through the provision of benchmark solutions. Some representative modes that are missed by well-established dynamic-stiffness-based computer programs are presented. The inplane dynamic stiffness matrix presented is of great importance when combined with the out of plane matrix in order to obtain the closed-form solution for free vibration analysis of structures with complex geometries.     

Free vibration analysis of elastically connected multiple-beams by using the Adomian modified decomposition method

The Adomian modified decomposition method (AMDM) is employed in this paper to investigate the free vibrations of N elastically connected parallel Euler–Bernoulli beams, which are continuously joined by a Winkler-type elastic layer. The proposed AMDM method can be used to analyze the vibration of beam system consisting of an arbitrary number of beams. By using boundary conditions the natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The numerical results for different boundary conditions, beam numbers and the stiffness of the Winkler-type elastic layer are presented. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of multiple-connected beams with arbitrary boundary conditions.     

Mode shapes analysis of a cracked beam and its application for crack detection

In this paper, mode shapes of a cracked beam with a rectangular cross section beam are analysed using finite element method. The 3D beam element is applied for this finite element analysis. The influence of the coupling mechanism between horizontal bending and vertical bending vibrations due to the crack on the mode shapes is investigated. Due to the coupling mechanism the mode shapes of a beam change from plane curves to space curves. Thus, the existence of the crack can be detected based on the mode shapes: when the mode shapes are space curves there is a crack in the beam. Also, when there is a crack, the mode shapes have distortions or sharp changes at the crack position. Thus, the position of the crack can be determined as a position at which the mode shapes exhibit such distortions or sharp changes. While in previous studies using 2D beam element, distortions in the mode shapes caused by a small crack could not be detected, these distortions in the case using the 3D beam element can be amplified and inspected clearly by using the projections of the mode shapes on appropriate planes. The quantitative analysis is also implemented to relate the size and position of the crack with the observed coupled modes. These results can be applied for crack detection of a beam. In this paper, the stiffness matrix of a cracked element obtained from fracture mechanics is presented and numerical simulations of three case studies are provided.     

Free vibration of polar-orthotropic sector plates

The free vibration of ring-shaped polar-orthotropic sector plates is analyzed by the Ritz method using a spline function as an admissible function for the deflection of the plates. For this purpose, the transverse deflection of a sector plate is written in a series of the products of the deflection function of a sectorial beam and that of a circular beam satisfying the boundary conditions. The deflection function of the sectorial beam is approximately expressed by a quintic spline function, which satisfies the equation of flexural vibration of the beam at each point dividing the beam into small elements. The frequency equation of the plate is derived by the conditions for a stationary value of the Lagrangian. The present method is applied to ring-shaped polar-orthotropic sector plates with some combination of boundary conditions, and the natural frequencies and the mode shapes are calculated numerically up to higher modes. This method is very effective for the study of vibration problems of variously shaped anisotropic plates including these sector plates.