DYNAMIC ANALYSIS OF LARGE-DIAMETER SAGGED CABLES TAKING INTO ACCOUNT FLEXURAL RIGIDITY

For the purpose of developing a vibration-based tension force evaluation procedure for bridge cables using measured multimode frequencies, an investigation on accurate finite element modelling of large-diameter sagged cables taking into account flexural rigidity and sag extensibility is carried out in this paper. A three-node curved isoparametric finite element is formulated for dynamic analysis of bridge stay cables by regarding the cable as a combination of an “ideal cable element” and a fictitious curved beam element in the variational sense. With the developed finite element formulation, parametric studies are conducted to evaluate the relationship between the modal properties and cable parameters lying in a wide range covering most of the cables in existing cable-supported bridges, and the effect of cable bending stiffness and sag on the natural frequencies. A case study is eventually provided to compare the measured natural frequencies of main cables of the Tsing Ma Bridge and the computed frequencies with and without considering cable bending stiffness. The results show that ignoring bending stiffness gives rise to unacceptable errors in predicting higher order natural frequencies of the cables, and the proposed finite element formulation provides an accurate baseline model for cable tension identification from measured multimode frequencies.     

Single-beam analysis of damaged beams: Comparison using Euler–Bernoulli and Timoshenko beam theory

A new general formulation that is applicable to the damaged, linear elastic structures ‘unified framework’ is used to obtain analytical expressions for natural frequencies and mode shapes. The term mode shapes is used to mean the displacement modes, the section rotation modes, the sectional bending strain modes and sectional shear strain modes. The formulation is applicable to damaged elastic self-adjoint systems. The formulation has two unique aspects: First, the theory is mathematically rigorous since no assumptions are made regarding the physical behavior at a damage location, therefore there is no need to substitute the damage with a hypothetical elastic element such as a spring. Since the beam is not divided at the damage location, rather than an 8 by 8, only a 4 by 4 matrix is solved to obtain the natural frequencies and mode shapes. Second, the inertia effects due to damage which have till now been neglected by researchers are accounted for. The formulation uses a geometric damage model, perturbation of mode shapes and natural frequencies, and a modal superposition technique to obtain and solve the governing differential equation. Timoshenko beam theory is then taken as an example, and its results are compared with results using Euler–Bernoulli beam theory and finite element models. The range of applicability of the two theories is ascertained for damage characteristics such as depth and extent of damage and beam characteristics such as slenderness ratio and Poisson?s ratio. The paper considers rectangular notch like non-propagating damage as an example of the damage.     

Meshless formulation using NURBS basis functions for eigenfrequency changes of beam having multiple open-cracks

This paper presents a meshless formulation using non-uniform rational B-spline (NURBS) basis functions, and its applications to evaluate natural frequencies of a beam having multiple open-cracks. Node-based NURBS basis functions are used to construct the approximation function. The characteristic differentiability of the NURBS basis functions allows it to represent a function having specific degrees of smoothness and/or discontinuity. The discontinuity can be incorporated simply by assigning multiple knots at those locations. Hence, it can yield exact solutions having interior discontinuous derivatives. These advantages of NURBS are well known, and have been used extensively in graphical approximation of geometrical surfaces. However, it is seldom used in other engineering applications. To model the multiple open-cracks in a beam, quartic NURBS basis functions are employed and quadruplicate knots are assigned at the crack locations. Hence, it is capable to model the abrupt changes of slope (the first derivative of displacement) across a crack. In the present applications, additional equivalent massless rotational springs are inserted at the crack locations to represent the local flexibility caused by the cracks. As such, the cracked beam can be treated in the usual manner as a continuous beam. By adopting the meshless Petrov–Galerkin formulation, a generalized stiffness matrix for the cracked beam can be derived. Compared to the conventional finite element method, the present method does not require a finite element mesh for the purposes of interpolation and numerical integration. The advantages and effectiveness of the present method is illustrated in solving the eigenfrequencies of a beam having multiple open-cracks of different depths.     

A distributed-mass approach for dynamic analysis of Bernoulli-Euler plane frames

The exact dynamic analysis of plane frames should consider the effect of mass distribution in beam elements, which can be achieved by using the dynamic stiffness method. Solving for the natural frequencies and mode shapes from the dynamic stiffness matrix is a nonlinear eigenproblem. The Wittrick-Williams algorithm is a reliable tool to identify the natural frequencies. A deflated matrix method to determine the mode shapes is presented. The dynamic stiffness matrix may create some null modes in which the joints of beam elements have null deformation. Adding an interior node at the middle of beam elements can eliminate the null modes of flexural vibration, but does not eliminate the null modes of axial vibration. A force equilibrium approach to solve for the null modes of axial vibration is presented. Orthogonal conditions of vibration modes in the Bernoulli-Euler plane frames, which are required in solving the transient response, are theoretically derived. The decoupling process for the vibration modes of the same natural frequency is also presented.     

A local flexibility method for vibration-based damage localization and quantification

A method for vibration-based damage localization and quantification, based on quasi-static flexibility, is presented. The experimentally determined flexibility matrix is combined with a virtual load that causes nonzero stresses in a small part of the structure, where a possible local stiffness change is investigated. It is shown that, if the strain–stress relationship for the load is proportional, the ratio of some combination of deformations before and after a stiffness change has occurred, equals the inverse local stiffness ratio. The method is therefore called local flexibility (LF) method. Since the quasi-static flexibility matrix can be composed directly from modal parameters, the LF method allows to determine local stiffness variations directly from measured modal parameters, even if they are determined from output-only data. Although the LF method is in principle generally applicable, the emphasis in this paper is on beam structures. The method is validated with simulation examples of damaged isostatic and hyperstatic beams, and experiments involving a reinforced concrete free–free beam and a three-span prestressed concrete bridge, that are both subjected to a progressive damage test.     

Crack effects on the in-plane static and dynamic stabilities of a curved beam with an edge crack

The effects of a single-edge crack and its locations on the buckling loads, natural frequencies and dynamic stability of circular curved beams are investigated numerically using the finite element method, based on energy approach. This study consists of three stages, namely static stability (buckling) analysis, vibration analysis and dynamic stability analysis. The governing matrix equations are derived from the standard and cracked curved beam elements combined with the local flexibility concept. Approximation for the displacements using coupled interpolations based on the constant-strain, linear-curvature element (SC) has yielded results with reasonable accuracy. The numerical results obtained from the present finite element model are found to be in good agreement with those, both experimental and analytic, of other researchers in the existing literature. Results show that the reductions in buckling load and natural frequency depend not only on the crack depth and crack position, but also on the related mode shape. Analyses also show that the crack effect on the dynamic stability of the considered curved beam is quite limited.     

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.     

On the natural frequencies and mode shapes of a multispan Timoshenko beam carrying a number of various concentrated elements

The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multispan Timoshenko beam carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and spring–mass systems. First, the coefficient matrices for an intermediate pinned support, an intermediate concentrated element, left- and right-end support of a Timoshenko beam are derived. Next, the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method. Finally, the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions, respectively. The effects of distribution of in-span pinned supports and various concentrated elements on the dynamic characteristics of the Timoshenko beam are also studied.     

Free vibration analysis of a uniform multi-span beam carrying multiple spring-mass systems

The literature regarding the free vibration analysis of single-span beams carrying a number of spring-mass systems is plenty, but that of multi-span beams carrying multiple spring-mass systems is fewer. Thus, this paper aims at determining the “exact” solutions for the natural frequencies and mode shapes of a uniform multi-span beam carrying multiple spring-mass systems. Firstly, the coefficient matrices for an intermediate pinned support, an intermediate spring-mass system, left-end support and right-end support of a uniform beam are derived. Next, the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. In this paper, the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made, thus, the last solutions are the exact ones. The effects of attached spring-mass systems on the free vibration characteristics of the 1-4-span beams are studied.     

Combined dynamic stiffness matrix and precise time integration method for transient forced vibration response analysis of beams

A method has been developed for determining the transient response of a beam. The beam is divided into several continuous Timoshenko beam elements. The overall dynamic stiffness matrix is assembled in turn. Using Leung's equation, we derive the overall mass and stiffness matrices which are more suitable for response analysis than the overall dynamic stiffness matrix. The forced vibration of the beam is computed by the precise time integration method. Three illustrative beams are discussed to evaluate the performance of the current method. Solutions calculated by the finite element method and theoretical analysis are also enumerated for comparison. In these examples, we have found that the current method can solve the forced vibration of structures with a higher precision.     

Determination of the timing properties of a pulsed positron lifetime beam by the application of an electron gun and a fast microchannel plate

We show that the timing properties of a pulsed low-energy positron lifetime beam can be conveniently tested by an electron beam. We apply this method to study the time resolution of the beam and electron scattering in flat and ‘sawtooth’ shaped choppers. The results show that (i) time resolution of 160 ps is obtained, (ii) the scattering of the electrons and the secondary electron yield of the flat chopper make the time resolution worse and background poor, and (iii) both these problems can be solved by using a ‘sawtooth’ shaped chopper. We also compare these results to beam simulations.     

High resolution quantitative SIMS analysis of shallow boron implants in silicon using a bevel and image approach

Secondary ion mass spectrometry (SIMS) is frequently used as the preferred tool for dopant profiling due to its sensitivity and depth resolution. However, as dopant profiles become shallower most, if not all of the implant profile lies in the pre-equilibrium or transient region of an SIMS depth profile. In this region sputter yield and ionisation rate vary making accurate quantification of the implant profile very difficult. These problems can be reduced through the use of much lower beam energies or oxygen flooding of the sample. However, most SIMS instruments do not have these capabilities. In this paper an alternative technique for producing an accurate depth profile of a shallow implant, using existing SIMS technology is presented.Through the fabrication of bevels with very small slope angles on a shallow boron implanted silicon via a chemical etch, SIMS ion imaging is performed on the exposed surface. Ion image data is then summed, and in conjunction with accurate measurement of the bevel morphology, a shallow boron implant profile produced. The ‘bevel-image’ profile compares very well with a profile obtained using a 1 keV oxygen beam. To ensure a good dynamic range on the ‘bevel-image’ profile it is important to clean the bevel with a HF etch, prior to imaging.     

Examination of multi-dimensional vibration isolation measures and their correlation to sound radiation over a broad frequency range

This article examines alternate vibration isolation measures for a multi-dimensional system. The isolator and receiver are modelled by the continuous system theory. The source is assumed to be rigid and both force and moment excitations are considered. Our analysis is limited to a linear time-invariant system, and the mobility synthesis method is adopted to describe the overall system behavior. Inverted ‘L’ beam and plate receivers are employed here to incorporate the contribution of their in-plane motions to vibration powers and radiated sound. Multi-dimensional transmissibilities and effectivenesses are comparatively evaluated along with power-based measures for the inverted ‘L’ beam receiver and selected source configurations. Further, sound pressures radiated from the inverted ‘L’ beam receiver are calculated and correlated with power transmitted to the receiver. Interactions within the ‘L’ beam receiver are also analyzed and measures that could identify dominant transfer paths within a system are examined. Sound measurements and predictions for the inverted ‘L’ plate receiver demonstrate that a rank order based on free field sound pressures, at one or more locations, may be regarded as a measure of isolation performance. Measured insertion losses for sound pressure match well with those based on computed results although further study is needed in relation to some discrepancies shown in the results. Finally, several emerging research topics are identified.     

Vibration characteristics of a cantilever plate with attached spring-mass system

Coupled free vibration analysis has been performed on a cantilever thin plate carrying a spring-mass system attached on an arbitrary point by using Rayleigh-Ritz method. Influence of an attached ‘spring-mass’ system, i.e., attached position, relative values of mass and spring constant, on the coupled vibration characteristics of the system has been clarified comparing with those of uncoupled ones. Optimal attached position to maximize coupled plate natural frequency is also investigated and shown in contour diagrams. The influence of an attached mass has also been investigated, as the limiting case whereby the spring stiffness of the ‘spring-mass’ system approaches infinity.     

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.     

To analyse the performance of tapered and MMI assisted splitter on the basis of geographical parameters

A literal model using geometric examination was formulated to calculate the optimum width and length of multimode interference structure used in the 1 × 2 power splitter and plot electric field intensity for input and each output signal. Designing and simulation of the wavelength response of 1 × 2 splitter using Tapered and MMI assisted structure. Mechanism on the critical geographical parameters like length and width of MMI structure, separation distance between output ports ‘S’, arm angle ‘α’ which give the optimum power and electric field intensity for the considered necessary wavelength. Finite difference beam propagation methods have been used to simulate the conventional and tapered MMI device behaviour.     

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.     

Structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one flexible plate

The nonlinear natural frequency of a rectangular box, which consists of one flexible plate and five rigid plates, is studied in this paper. The flexible plate is assumed to vibrate like a simple piston. The behavior of the structural-acoustic coupling between the flexible plate and the air cavity is analyzed by using the proposed finite element modal method. The system finite element equation is reduced and expressed in terms of the modal coordinates with small degrees of freedom by using the proposed reduction method. The system nonlinear stiffness matrix representing the large amplitude vibration can be transformed to be a constant modal matrix. The natural frequencies are determined by using the harmonic balance method to solve the eigenvalue equations of the structural-acoustic system. The effect of the cavity depth on the natural frequencies and convergence studies are discussed in detail.     

EXPRESSIONS FOR PREDICTING FUNDAMENTAL NATURAL FREQUENCIES OF NON-CYLINDRICAL HELICAL SPRINGS

Numerical and analytical studies are performed for the free vibration analysis of non-cylindrical (conical, barrel and hyperboloidal types) helical springs. The stiffness matrix method is used in the numerical analysis. A total of 12 degrees of freedom (six displacements and six rotations) is described for an element. The exact element stiffness matrix and the exact concentrated element inertia matrix are used in the formulation. The rotary inertia, the shear and extensional deformation effects are considered in the analysis. Comparison of the numerical results with the reported results obtained numerically and experimentally gives satisfactory values. After verification of the numerical frequencies, the non-dimensional fundamental frequencies of fixed-fixed non-cylindrical helical springs with circular section are expressed in a simple formula with a maximum absolute relative error of 5% using those numerical values for the constant helix pitch angles (5°, 10°, and 15°). These expressions restricted to the fundamental frequencies are also verified with ANSYS results.     

Vibration modelling of helical springs with non-uniform ends

Helical springs constitute an integral part of many mechanical systems. Usually, a helical spring is modelled as a massless, frequency independent stiffness element. For a typical suspension spring, these assumptions are only valid in the quasi-static case or at low frequencies. At higher frequencies, the influence of the internal resonances of the spring grows and thus a detailed model is required. In some cases, such as when the spring is uniform, analytical models can be developed. However, in typical springs, only the central turns are uniform; the ends are often not (for example, having a varying helix angle or cross-section). Thus, obtaining analytical models in this case can be very difficult if at all possible. In this paper, the modelling of such non-uniform springs are considered. The uniform (central) part of helical springs is modelled using the wave and finite element (WFE) method since a helical spring can be regarded as a curved waveguide. The WFE model is obtained by post-processing the finite element (FE) model of a single straight or curved beam element using periodic structure theory. This yields the wave characteristics which can be used to find the dynamic stiffness matrix of the central turns of the spring. As for the non-uniform ends, they are modelled using the standard finite element (FE) method. The dynamic stiffness matrices of the ends and the central turns can be assembled as in standard FE yielding a FE/WFE model whose size is much smaller than a full FE model of the spring. This can be used to predict the stiffness of the spring and the force transmissibility. Numerical examples are presented.