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Neural networks have emerged as a tool for solving differential equations in many branches of engineering and science. But their progress in frequency domain acoustics is limited by the vanishing gradient problem that occurs at higher frequencies. This paper discusses a formulation that can address this issue. The problem of solving the governing differential equation along with the boundary conditions is posed as an unconstrained optimization problem. The acoustic field is approximated to the output of a neural network which is constructed in such a way that it always satisfies the boundary conditions. The applicability of the formulation is demonstrated on popular problems in plane wave acoustic theory. The predicted solution from the neural network formulation is compared with those obtained from the analytical solution. A good agreement is observed between the two solutions. The method of transfer learning to calculate the particle velocity from the existing acoustic pressure field is demonstrated with and without mean flow effects. The sensitivity of the training process to the choice of the activation function and the number of collocation points is studied. • PINNs-based solution for problems in frequency domain acoustics is presented. • Problems associated with the existing Lagrange multiplier method are discussed. • Acoustic field is predicted in the ducts of two different cross-sections. • Mean flow and visco-thermal effects are captured with a maximum error of O (1 0 − 4). • Results show the ability of PINNs to emerge as an acoustic solver in the future. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a numerical method to model wave scattering by local inhomogeneities in elastic open waveguides. The damaged zone of the waveguide and its vicinity are described by a finite-element model. This model is coupled on the cross-section boundaries to a modal representation of the field propagating along the waveguide axis in the undamaged parts, yielding transparent boundary conditions. However, open waveguides are unbounded in the transverse direction, which complicates both the numerical resolution and the physical analysis. Theoretically, the wave fields are described by a discrete sum of trapped modes and two continuous sums on radiation modes. The latter is sometimes approximated by a discrete sum of leaky modes, which grow at infinity in the transverse direction. In this paper, a Perfectly Matched Layer (PML) of finite thickness is introduced to absorb outgoing waves in the transverse direction, yielding three types of discrete modes: trapped modes, leaky modes and PML modes (PML modes oscillate mainly inside the layer and are non-intrinsic to the physics). Two numerical test cases are considered: the scattering at the junction between a closed and an open cylindrical waveguides and the scattering by an axisymmetrical notch. Good agreement with literature results is found. In particular, the influence of PML modes on the scattered field is highlighted through numerical tests. Due to the lack of power orthogonality of leaky modes, it is also shown that the modal cross power can become significant, which complicates the scattering analysis of open waveguides. Finally, the generality of the proposed method is discussed through a three-dimensional test case considering an embedded bar with an oblique break. Graphical abstract Image 1 [ABSTRACT FROM AUTHOR]

Abstract It is well-known that the convective wave equation with a uniform mean flow can be transformed into a standard wave equation without flow by a Prandtl-Glauert-Lorentz type transformation. This paper examines the boundary condition to be used in the transformed coordinates for acoustic scattering by rigid bodies. Recently, based on the energy conservation equation for acoustic waves propagating in a uniform flow, a new solid wall boundary condition, dubbed the Zero Energy Flux (ZEF) solid surface boundary condition, has been proposed. Instead of the commonly used solid surface boundary condition that the normal acoustic velocity be zero, the ZEF condition requires that the acoustic energy flux be zero. In this paper, we point out that when formulated in the acoustic velocity potential and under the ZEF boundary condition, the normal derivative of the velocity potential remains to be zero when computed in the transformed coordinates. As such, numerical methods developed for solving the standard wave equation can be directly applied to the convective wave equation through a use of the Prandtl-Glauert-Lorentz transformation. Utilization of the transformation is particularly beneficial for numerical methods based on the boundary integral formulation because of the simplified kernel function in the transformed coordinates, for both the time domain and the frequency domain approaches. Useful relations for applying the transformation are derived and numerical examples that demonstrate the effectiveness of the transformation and ZEF boundary condition are presented. [ABSTRACT FROM AUTHOR]

This paper presents the theoretical framework of the virtual acoustic volume concept and two related inverse Patch Transfer Functions (iPTF) identification methods (called u-iPTF and m-iPTF depending on the chosen boundary conditions for the virtual volume). They are based on the application of Green׳s identity on an arbitrary closed virtual volume defined around the source. The reconstruction of sound source fields combines discrete acoustic measurements performed at accessible positions around the source with the modal behavior of the chosen virtual acoustic volume. The mode shapes of the virtual volume can be computed by a Finite Element solver to handle the geometrical complexity of the source. As a result, it is possible to identify all the acoustic source fields at the real surface of an irregularly shaped structure and irrespective of its acoustic environment. The m-iPTF method is introduced for the first time in this paper. Conversely to the already published u-iPTF method, the m-iPTF method needs only acoustic pressure and avoids particle velocity measurements. This paper is focused on its validation, both with numerical computations and by experiments on a baffled oil pan. [ABSTRACT FROM AUTHOR]

Health monitoring of rotating structures such as wind turbines and helicopter rotors is generally performed using conventional sensors that provide a limited set of data at discrete locations near or on the hub. These sensors usually provide no data on the blades or inside them where failures might occur. Within this paper, an approach was used to extract the full-field dynamic strain on a wind turbine assembly subject to arbitrary loading conditions. A three-bladed wind turbine having 2.3-m long blades was placed in a semi-built-in boundary condition using a hub, a machining chuck, and a steel block. For three different test cases, the turbine was excited using (1) pluck testing, (2) random impacts on blades with three impact hammers, and (3) random excitation by a mechanical shaker. The response of the structure to the excitations was measured using three-dimensional point tracking. A pair of high-speed cameras was used to measure displacement of optical targets on the structure when the blades were vibrating. The measured displacements at discrete locations were expanded and applied to the finite element model of the structure to extract the full-field dynamic strain. The results of the paper show an excellent correlation between the strain predicted using the proposed approach and the strain measured with strain-gages for each of the three loading conditions. The approach used in this paper to predict the strain showed higher accuracy than the digital image correlation technique. The new expansion approach is able to extract dynamic strain all over the entire structure, even inside the structure beyond the line of sight of the measurement system. Because the method is based on a non-contacting measurement approach, it can be readily applied to a variety of structures having different boundary and operating conditions, including rotating blades. [ABSTRACT FROM AUTHOR]

This paper presents a general electromechanical model for predicting the dynamics of thin or moderately thick plates with surface-integrated piezo-patches. Using spectral Tchebychev (ST) technique, the boundary value problem governing the electroelastic dynamics of the two dimensional (2D) plate and piezo-patch structure is developed with Mindlin plate theory assumptions. Mass and stiffness contributions of piezo-patch(es) as well as two-way electromechanical coupling behavior are incorporated in the model for both modal analysis and frequency response calculations. To validate the accuracy of the developed solution technique, the modal analysis results are compared against the existing Rayleigh-Ritz solution from the literature as well as the finite-element simulation results for various piezo-patch sizes on thin and moderately thick host plates; and it is shown that the maximum difference in the predicted natural frequencies between the ST and FE results are below 1%. The electromechanical frequency response functions (FRFs) including the vibration response and electrical output of the system under a transverse point force excitation are obtained using the ST model and the results are shown to match perfectly with the finite element (FE) simulations. Additionally, comparisons of the electromechanical FRFs calculated based on Rayleigh-Ritz method from the literature versus the developed framework is presented to highlight that the exclusion of shear deformation terms in the former model leads to an inaccurate estimation of electroelastic behavior for the case of thicker plates with integrated piezo-patches. Finally, the investigated case studies demonstrate that the computational efficiency of the developed method is significantly higher than that of FE simulations. [ABSTRACT FROM AUTHOR]

Abstract: The three-dimensional transient analysis of functionally graded annular plates with arbitrary boundary conditions is carried out in this paper. The material properties of the FGM plate are assumed to vary smoothly in an exponential law along the thickness direction. The plate is assumed to rest on a two parameter viscoelastic foundation. A semi-analytical method, which integrates the state space method (SSM), Laplace transform and its inversion, as well as the one-dimensional differential quadrature method (DQM), is proposed to obtain the transient response of the plate. The state space method is used to obtain the analytical solution in the thickness direction. The differential quadrature method is employed to approximate the solution in the radial direction. The Laplace transform and the numerical inversion are used to obtain the solution in time domain. Numerical results show a good agreement between the response histories obtained by the present method and finite element method. The effects of the boundary conditions at the edges, the material graded index, the Winkler and shearing layer elastic coefficients, and the damping coefficient are studied. Numerical examples show that the peak response decreases as the material graded index, the Winkler and shearing layer elastic coefficients, and the damping coefficient increase. The results obtained in this paper can serve as benchmark data in further research. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study the free lateral responses of vertically translating media with variable length, velocity and tension, subject to general initial conditions. The translating media are modeled as taut strings with fixed boundaries. The problem can be used as a simple model to describe the lateral vibrations of an elevator cable, for which the length changes linearly in time, or for which the length changes harmonically about a constant mean length. In this paper an initial-boundary value problem for a linear, axially moving string equation is formulated. In the given model a rigid body is attached to the lower end of the string, and the suspension of this rigid body against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to car mass, and for the harmonically length variations small oscillation amplitudes are assumed and it is also assumed that the string mass is small compared to the total mass of the string and the car. A multiple-timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the string. For the linearly varying length analytic approximations of the exact solution are compared with numerical solution. For the harmonically varying length it will be shown that Galerkin׳s truncation method cannot be applied in all cases to obtain approximations valid on long timescales. [Copyright &y& Elsevier]

Abstract: For conservative systems instability can occur only by divergence and the presence of damage can produce both a reduction of the buckling loads and modification of the corresponding mode shapes, depending on the positions and intensities of the damage distribution. For nonconservative systems instability is found to occur by divergence, flutter, or both, characterised by multiple stable and unstable ranges of the loads whose boundary can be altered by the damage distribution. This paper focuses on the stability behaviour of multi-cracked cantilever Euler beam-column subjected to conservative or nonconservative axial loads. The exact flutter and divergence critical loads are obtained by means of the exact closed form solution of the multi-cracked beam-column, derived by the authors in a previous paper. The extensive numerical applications, reported in the paper, aimed at evaluating the influence of several damage scenarios for different values of the degree of nonconservativeness. It is shown how the presence of damage can strongly modify the ranges of divergence and flutter critical loads of the corresponding undamaged cantilever column, which has been the subject of several papers starting from the Pflüger paradoxical results. [Copyright &y& Elsevier]

Abstract: This paper proposes the identification of vibration excitations by an inverse method. The use of the local operator of a structure, for which the vibration field is obtained by measurements, is investigated. This method has already been developed on analytically known structures, but there are as yet few applications. The goal of this paper is to adapt the same technique by using an operator obtained with the Finite Element Method rather than a discretised differential motion equation. The first difficulty is the measurement of rotational degrees of freedom. A concept analogous to an observation matrix enabling only the measurement of displacements is then proposed. The method of extracting the operator on a part of the structure is also discussed. Two variants are proposed: an operator with free boundary conditions and a truncated operator without boundary conditions. The first permits identifying internal forces at the boundaries of the subdomain, whereas the second ignores them. In order to regularise the inverse method, a Tikhonov approach is used. An important point of this paper is the discussion of the effect of this regularisation which implies ambiguities between reconstructed forces and moments in the results. The possibility of an a priori “force only” assumption is then demonstrated and recommended. After illustrations obtained from numerical simulations on beams, an experimental validation is provided for a beam excited by a shaker where several signals are used, giving several signal to noise ratios. The results are discussed and compared to the force measured directly by a piezoelectric sensor. [Copyright &y& Elsevier]

Abstract: The commercial automotive mufflers are generally of a complicated shape with multiply connected parts and complex acoustic elements. The analysis of such complex mufflers has always been a great challenge. In this paper, an Integrated Transfer Matrix method has been developed to analyze complex mufflers. Integrated transfer matrix relates the state variables across the entire cross-section of the muffler shell, as one moves along the axis of the muffler, and can be partitioned appropriately in order to relate the state variables of different tubes constituting the cross-section. The paper presents a generalized one-dimensional (1-D) approach, using the transfer matrices of simple acoustic elements, which are available from the literature. The present approach is robust and flexible owing to its capability to construct an overall matrix of the muffler with the transfer matrices of individual acoustic elements and boundary conditions, which can then be used to evaluate the transmission loss, insertion loss, etc. Results from the present approach have been validated through comparisons with the available experimental and three-dimensional finite element method (FEM) based results. The results show good agreement with both measurements and FEM analysis up to the cut-off frequency. [Copyright &y& Elsevier]

Abstract In this paper, we present a novel method for the identification of the local bending stiffness of beams. We use shearography to capture measurements of vibrating beams, so the input data for the identification is the modal slope – the differential of the modal shape. The modal slope is fitted by two Fourier-series functions, one of which is derived from a thin-beam model. The local bending stiffness is identified as the one corresponding with the best match between the measured and the two fitted modal slopes. This identification method, which we call simultaneous Fourier-series fitting, is demonstrated on numerically-generated inputs, as well as on experimental measurements. We use a flat, concave and convex beam, as well as beams with locally varying bending stiffness mimicking local damage to verify the method. It is shown that the method gives accurate results and is robust to noise. Additionally, it has advantageous properties that make it useful and practical: using this method, it is possible to perform the identification from only a sub-region of a beam and even without specifying the boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract An analytical vibration response in the time domain for an axially translating and laterally vibrating string with mixed boundary conditions is considered in this paper. The domain of the string is a constant, dependent upon the general initial conditions. The translating tensioned strings possess different types of mixed boundary conditions, such as fixed_dashpot, fixed_spring-dashpot, fixed_mass-spring-dashpot. An analytical solution using a reflected wave superposition method is presented for a finite translating string. Firstly, the cycle of boundary reflection for strings is provided, which is dependent upon the string length. Each cycle is divided into three time intervals according to the travelling speed and direction of the string. Applying D'Alembert's principle and the reflection properties, expressions for the reflected waves under three different non-classical boundary conditions are derived. Then, the vibrational response of the axially translating string is solved for three time intervals by using a reflected wave superposition method. The accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a Newmark- β method solution. The energy expressions for a travelling string with a fixed_dashpot boundary condition is obtained and the time domain curves for the total energy and the change of energy at the boundaries are given. [ABSTRACT FROM AUTHOR]

Abstract Conventionally, the dynamic modeling of H-type gantry stage(HGS) assumes that the stiffness of mid-beam is extremely high so that its deformation is ignorable. This assumption apparently simplifies the study, but unavoidably compromises the modeling precision as it is not consistent with the real working status especially at high frequency motion. Comparing with similar studies on moving mass problems, the beam vibration of HGS is excited not only by the moving mass and system acceleration but also by the non-synchronized motion of the two moving ends of mid-beam. Therefore, the objective of this paper is to propose an assumed mode method based on simplified mode shapes(AMM-SMS) by simultaneously considering the lateral flexibility and the motion of the ends of mid-beam. The proposed AMM-SMS adopts the mode shapes of a simplified system, which has the same geometry boundary condition(BC) with the investigated system, to discretize the nonlinear kinematic equations of the proposed system. This method improves the modeling precision with low complexity, and can be applied in different BCs of FHGS. The proposed method is verified by various comparisons with the finite element method. Based on the AMM-SMS, the first three natural frequencies and mode shapes are studied. The open-loop responses of the system are obtained to analyze the effects of different BCs, of the motion of end-effector and of the external exciting force on the vibratory behavior. [ABSTRACT FROM AUTHOR]

Abstract The suppression of expansion in thin clamped panels subjected to elevated thermal loading often results in buckling. However, a number of possible post-buckled equilibrium configurations typically exist, and which shape ensues depends on a number of factors including the role of symmetry, boundary conditions, aspect ratio, and the effect of small geometric imperfections associated with the initial shape. It is possible to force the panel to go between different buckled shapes, given a sufficiently large perturbation. Sometimes the panel will spontaneously jump, or snap, when the temperature is gradually increased or decreased (mode jumping). The extent to which these features occur when the thermal loading is applied locally is also investigated. This paper describes some interesting nonlinear (essentially buckling) behavior in thermally-loaded panels from a primarily experimental perspective, with an additional focus on non-uniform heating. The full force of stereo 3D digital image correlation and forward-looking infrared cameras are exploited to provide a relatively complete picture of this behavior. [ABSTRACT FROM AUTHOR]

Abstract: In this paper wave propagation in non-local elastic solids is examined in the framework of the mechanically based non-local elasticity theory established by the author in previous papers. It is shown that such a model coincides with the well-known Kröner–Eringen integral model of non-local elasticity in unbounded domains. The appeal of the proposed model is that the mechanical boundary conditions may easily be imposed because the applied pressure at the boundaries of the solid must be equilibrated by the Cauchy stress. In fact, the long-range forces between different volume elements are modelled, in the body domain, as central body forces applied to the interacting elements. It is shown that the shape change of travelling disturbances coalesces with those predicted by the non-local integral theory of elasticity in unbounded domains, but several differences arise in the case of bounded domains. The wave propagation problem has been formulated by means of the Hamiltonian functional of the proposed mechanically based model of non-local elasticity, introducing an additional term to the elastic potential energy that accounts for elastic long-range interactions. In this way, the wave equation may be obtained in a weak formulation and be further used to provide approximate analytical solutions to the governing equation in the context of standing wave analysis. An equivalent discrete point-spring model, similar to lattice-type networks, has also been introduced to show the mechanical equivalence of the non-local elastic model as well as to provide a mechanical scheme suitable for the numerical treatment of pressure waves travelling in non-local bounded domains. [Copyright &y& Elsevier]

Abstract: This paper presents a dynamic analysis of time-harmonic plane SH-waves propagating in periodically multilayered elastic composites with a strip-like crack. The total wave field in the multilayered elastic structure is described as a sum of incident wave field modeled by the transfer matrix method and the scattered wave field governed by an integral representation containing the crack-opening-displacement. The integral equation derived from the boundary conditions on the crack-faces is solved numerically by a Galerkin method. The paper focuses on resonant and non-resonant regimes of anti-plane wave motion in a stack of elastic layers weakened by a single strip-like crack and wave localization in the vicinity of the crack. The scattered extra displacement induced by the presence of the crack is investigated in detail for both situations of high and low contrast in material properties. Numerical results for the average crack-opening-displacement, the transmission coefficient, the stress intensity factor and the average energy flow are presented and discussed to reveal wave resonance and localization phenomena within the band-gaps and the pass-bands. [Copyright &y& Elsevier]

Abstract: The reflection and transmission theories of waves in pyroelectric and piezoelectric medium are studied in this paper. In general in an infinite homogeneous pyroelectric medium there are four bulk wave modes: quasi-longitudinal, two quasi-transversal and temperature waves. In an infinite homogeneous piezoelectric medium there are three bulk wave modes: quasi-longitudinal and two quasi-transversal waves. In the reflection and transmission problem there are five complex boundary conditions in the pyroelectric medium and four complex boundary conditions for the piezoelectric medium. In this paper, we find that the surface waves will be revealed in the reflection and transmission wave problem. The surface waves have the same wave vector component with the incident waves on the interface plane. The two dimensional reflection problem of waves at the interface between the semi-infinite pyroelectric medium and vacuum is researched in greater detail and a numerical example is given. [ABSTRACT FROM AUTHOR]

Abstract A semi-analytical method is proposed for calculating the response of a linear cascade with spanwise mean flow subject to oblique gusts. It is developed based on the boundary value problems defined by Lloyd and Peake (AIAA paper 2008–2840). A gust strength parameter is introduced and the correct three-dimensional (3-D) response is obtained. The classic similarity rules are extended; the approach can be used to extend any 2-D methods to account for oblique gusts and 3-D mean flows. It is validated against analytical approximations for single-airfoil and cascade responses. The method is used to investigate the effect of gust angle α g on the unsteady lift and the sound field. It is found that as α g increases, the 2-D equivalent response varies slightly. However, the 3-D lift is amplified by factor 1 / cos α g , and the spanwise phase variation increases. Cascade effects are also studied. The inter-blade phase angle (IBPA) is important even for very low solidity. As the solidity increases, the chordwise distribution of lift is no longer leading-edge dominant. Cascade effects are small only when the cascade blade count is lower than a limit. A statistics analysis reveals that Mach number is the most important parameter for determining this blade count limit, and frequency is the least important. [ABSTRACT FROM AUTHOR]

This paper presents a new approach, referred to here as the two-dimensional spectral-Tchebychev (2D-ST) technique, to predict the dynamics of thick plates having arbitrary geometries under different boundary conditions. The integral boundary value problem governing the dynamics of plate-like structures is obtained using the Mindlin plate theory and following an energy-based approach. To solve the boundary value problem numerically, a spectral-Tchebychev based solution technique is developed. To simplify the calculation of integral and derivative operations and thus to increase the numerical efficiency of the solution approach, a one-to-one coordinate mapping technique is used to map the arbitrary geometry onto an equivalent rectangular in-plane shape of the plate. The proposed solution technique is applied to various different plate problems to assess the accuracy and show the applicability of the technique. In each case, the convergence of the solution is analyzed, and the predicted (non-dimensional) natural frequencies are compared to those found in the literature or to those found using finite element modeling. It is shown that the calculated natural frequencies converge exponentially with increasing number of Tchebychev polynomials used and are in excellent agreement with those found in the literature and found form a finite elements solution. Therefore, it is concluded that the presented spectral-Tchebychev solution technique can accurately and efficiently capture the dynamics of thick plates having arbitrary geometries. Furthermore, the utility of the 2D-ST is demonstrated by comparing the results obtained using a three-dimensional solution approach. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes and validates a new formulation to investigate the dynamic response of dam–reservoir systems with upstream transmitting boundary conditions (TBCs). The mathematical derivations are provided for the new formulation as well as for exact and various approximate TBCs. The developed analytical equations can be solved numerically to assess the accuracy of a given TBC and determine the associated error independently of FEM or BEM modeling of the reservoir. The method is first validated in the case of semi-infinite reservoirs and an excellent agreement is obtained against classical techniques. The paper presents a fundamental understanding of the behavior of various TBCs and a systematic identification of their influence on the system''s dynamic response, considering: (i) dam flexibility, (ii) water compressibility, (iii) reservoir bottom wave absorption, (iv) reservoir truncation length, and (v) excitation frequency. The new method is used to obtain exact error estimators to evaluate the effects of various TBCs on the dam–reservoir first resonant frequency and hydrodynamic forces acting on the dam upstream face. The proposed formulation can be programmed easily and used efficiently for rigorous assessment of classical or newly developed TBCs for vibrating dam–reservoir systems or similar fluid–structure problems. [Copyright &y& Elsevier]

Abstract: This paper presents a new and efficient method to calculate point mobilities from subcomponents of a full structure. Subcomponent modelling is a commonly used method to gain information on the dynamic behaviour of complex assembly structures using smaller and more efficient models. For instance, point mobility calculations on subcomponent level are used to obtain more accurate input parameters for statistical energy analysis (SEA) models. A full system analysis is often too computationally expensive, so normally only individual subcomponents of the structure are extracted and analysed. This procedure yields a large reduction in computational effort, but also often results in a substantial loss of accuracy. This is due to the use of an approximation of boundary conditions to represent the eliminated remainder part of the structure, i.e. the full structure except the subcomponent at hand. Commonly, these boundary conditions are simplified by assuming clamped, free or simply supported edges. However, this is a huge simplification and may introduce large errors, especially in the low- and mid-frequency ranges. Earlier work has shown that a more accurate description of the boundary conditions can be achieved by describing the interface dynamics by a combination of so-called dynamic waves. In this paper, the method is developed further and a more robust and efficient wave extraction procedure is presented. An industrial body in white BIW is used as a test case and results are presented for three different cases. The results show that the wave-based boundary condition for point impedance calculations from a subcomponent model gives more accurate results than the results obtained with free or clamped boundary conditions. [Copyright &y& Elsevier]

Abstract: The wave propagation approach is used to analyze the free vibration of ring-stiffened cylindrical shell under initial hydrostatic pressure, based on Flügge classical thin shell theory and orthotropic method in this paper. The results obtained are compared with those available in other literature. It has been proved that, free vibration analysis using wave propagation approach with shear diaphragm–shear diaphragm (SD–SD) boundary conditions is absolutely equal to the traditional method. To evaluate the validity and accuracy of the wave propagation approach, an exact solution for arbitrary boundary conditions is also proposed in this paper. The comparisons between these two approaches are carried out for clamped–clamped (C–C) and clamped–shear diaphragm (C–SD) boundary conditions. The results show that wave propagation has high accuracy for long shell and large circumferential numbers. [Copyright &y& Elsevier]

Abstract: In this paper an initial boundary value problem for a linear equation describing an axially moving stretched beam will be considered. The velocity of the beam is assumed to be time varying. Since the order of magnitude of the bending stiffness terms depends on the vibration modes and the frequencies involved a combination of two simplified models (that is, a string equation for the lower frequencies and a beam with string effect equation for the higher frequencies) will be used to describe the transversal vibrations of the system accurately. Based on the calculations of the natural frequencies the regions of applicability of these sub-models will be determined. A two time-scales perturbation method will be used to construct formal asymptotic approximations of the solutions. Non-resonant and some resonant cases will be studied for four different values of the relative errors. An important implication of the earlier results in the literature is that for these types of axially moving continua problems the use of only string-like models is not appropriate. To describe the dynamics of these types of axially moving continua problems correctly one has to include (small) bending stiffness in the model. In this paper it is explicitly shown how to work with a combined model that is a string model at the low frequencies and a tensioned beam model at the higher frequencies. [Copyright &y& Elsevier]

Abstract: This paper extends the analysis of the self-excitated vibrations of a drilling structure presented in an earlier paper [T. Richard, C. Germay, E. Detournay, A simplified model to explore the root cause of stick-slip vibrations in drilling systems with drag bits, Journal of Sound and Vibration 305 (3) (2007) 432–456] by basing the formulation of the model on a continuum representation of the drillstring rather than on a characterization of the drilling structure by a 2 degree of freedom system. The particular boundary conditions at the bit–rock interface, which according to this model are responsible for the self-excited vibrations, account for both cutting and frictional contact processes. The cutting process combined with the quasi-helical motion of the bit leads to a regenerative effect that introduces a coupling between the axial and torsional modes of vibrations and a state-dependent delay in the governing equations, while the frictional contact process is associated with discontinuities in the boundary conditions when the bit sticks in its axial and angular motion. The dynamic response of the drilling structure is computed using the finite element method. While the general tendencies of the system response predicted by the discrete model are confirmed by this computational model (for example that the occurrence of stick-slip vibrations as well as the risk of bit bouncing are enhanced with an increase of the weight-on-bit or a decrease of the rotational speed), new features in the self-excited response of the drillstring can be detected. In particular, stick-slip vibrations are predicted to occur at natural frequencies of the drillstring different from the fundamental one (as sometimes observed in field operations), depending on the operating parameters. [Copyright &y& Elsevier]

This paper is concerned with active wave control of a distributed parameter structure. It is the purpose of this paper to present the active wave control method of a rectangular panel and to clarify fundamental properties of the control system. Firstly, a transfer matrix method for a rectangular panel is introduced to describe the wave dynamics of the structure. This is followed by the derivation of feedforward control laws for absorbing reflected waves or eliminating transmitted waves. In the proposed method, the control laws are including the modal actuation scheme for uncontrolled direction. Then, from a viewpoint of numerical analyses, basic properties of the proposed method are verified. It is found that the reflected wave absorbing control enables the inactivation of all vibration modes in the controlled direction and the transmitted wave eliminating control enables the generation of an almost vibration-free state. It is also found that some phenomena different from the case of a beam-like structure appear under certain boundary conditions. [Copyright &y& Elsevier]

This paper deals with the free transverse vibration of a non-homogeneous tapered beam subjected to general axial forces, with arbitrarily located internal hinge and elastics supports, and ends elastically restrained against rotation and translation. A rigorous and complete development is presented. First, a brief description of several papers previously published is included. Second, the Hamilton principle is rigorously stated by defining the domain D of the action integral and the space Da of admissible directions. The differential equations, boundary conditions, and particularly the transitions conditions, are obtained. Third, the transition conditions are analysed for several sets of restraints conditions. Fourth, the existence and uniqueness of the weak solutions of the boundary value problem and the eigenvalue problem which, respectively, govern the statical and dynamical behaviour of the mentioned beam is treated. Finally, the method of separation of variables is used for the determination of the exact frequencies and mode shapes and a modern application of the Ritz method to obtain approximate eigenvalues. In order to obtain an indication of the accuracy of the developed mathematical model, some cases available in the literature have been considered. New results are presented for different boundary conditions and restraint conditions in the internal hinge. [Copyright &y& Elsevier]

In the present paper, the Mindlin plate theory is used to study buckling of in-plane loaded isotropic rectangular plates with different boundary conditions. The novelty of the paper is that the analytical closed-form solution is developed without any use of approximation for a combination of six different boundary conditions; specifically, two opposite edges are simply supported and any of the other two edges can be simply supported, clamped or free. Monoaxial in-plane compressive loads on both directions are considered, as well as equal biaxial loads. The present analytical solution can be obtained with any required accuracy and can be used as benchmark. Dimensionless critical buckling loads and mode shapes are given for the six cases analyzed. The effect of boundary conditions, loading conditions, variations of aspect ratios and thickness ratios are examined and discussed in detail. Finally, based on comparison with previously published results, the accuracy of the results is shown. [Copyright &y& Elsevier]

Abstract: In this paper the wind-induced, horizontal vibrations of a weakly damped vertical Euler–Bernoulli beam with and without a tip-mass will be studied. The damping is assumed to be boundary damping and global Kelvin–Voigt damping. The boundary damping is assumed to be proportional to the velocity of the beam at the top. The horizontal vibrations of the beam can be described by an initial-boundary value problem. In this paper, the multiple-timescales perturbation method will be applied to construct approximations of the solutions of the problem. Also it will be shown that a combination of boundary damping and Kelvin–Voigt damping can be used to damp the wind-induced vibrations of a vertical beam with tip-mass uniformly. [Copyright &y& Elsevier]

Abstract: The frequency spectra of the higher-order model, for laminated arches, are identified through a novel process, proposed in this paper. Besides the basic spectra corresponding to axial, flexural and shear, presence of higher-order spectra is established. Coupled spectra of first-order and higher-order degrees of freedom, in various combinations, are identified. Based on this spectra identification framework, the frequencies of sandwich and composite arches, for various boundary conditions are classified and presented. Also, the role of curvature and aspect ratio on the fundamental frequency of arches with various end-conditions is investigated, in the second part of this paper. [Copyright &y& Elsevier]

This paper presents active boundary control—ABC—of an Euler–Bernoulli beam, which enables one to generate a desired boundary condition at any designated position of a target beam structure, thereby permitting the structure to possess desired properties characterized by the boundary condition. Furthermore, ABC has potential to create a completely vibration-free state in the designated area of a beam. This paper begins by presenting the principle of ABC using a transfer matrix method, the optimal control law of the ABC system being derived. It is found that, in addition to conventional four classical boundary conditions: free, pinned, clamped and sliding support, ABC can generate two more boundary conditions that may not be observed in real systems but realized by ABC. It is also found that as a result of applying ABC to a specific location, including a current conventional boundary of a beam, a completely vibration-free state in the target region of a beam can be realized. Finally, an experiment using an adaptive feedforward control was conducted, demonstrating that ABC enables the generation of a desired boundary condition at the designated location of a target beam, and of a completely vibration-free state of a beam. [Copyright &y& Elsevier]

Abstract: Based on the Flügge thin shell theory, this paper presents exact solutions for the vibration of circular cylindrical shells with step-wise thickness variations in the axial direction. The shell is sub-divided into multiple segments at the locations of thickness variations. The state-space technique is adopted to derive the homogenous differential equations for a shell segment and domain decomposition method is employed to impose the equilibrium and compatibility requirements along the interfaces of the shell segments. To ensure the correctness of the present results, comparisons are made with one paper available in the open literature based on the Donnell–Mushtari theory. Shells with various combinations of end boundary conditions can be analyzed by the proposed method. Furthermore, the influences of the shell thickness ratios, locations of step-wise thickness variations and step thickness ratios on the natural frequencies and mode shapes are examined. The exact vibration results can serve as important benchmark values for researchers to validate their numerical methods for such circular cylindrical shells. [Copyright &y& Elsevier]

Abstract: This paper proposes a hybrid method for the prediction of vibrational and acoustic responses of low-damping system in the medium-to-high frequency ranges by using the power flow analysis (PFA) algorithm and statistical energy analysis (SEA) coupling concepts. The main part of this method is the application of the coupling loss factor (CLF) of SEA to the boundary condition of PFA in reverberant system. First, for hybrid PFA, the hybrid boundary conditions on 1-D and 2-D cases were derived in the general form. To verify the derived boundary conditions, numerical analyses for each case were performed. The hybrid PFA solutions using derived boundary conditions were compared with the classical PFA solutions with various reverberance factors including the effects of the characteristic length, excitation frequency and group velocity besides damping loss factor of the subsystem. Additionally, the hybrid PFA on 3-D case and the hybrid power flow finite element method (PFFEM) for hybrid PFA of built-up structures are described in the other companion paper. [Copyright &y& Elsevier]

Abstract: This paper introduces an original, non-destructive diagnosis tool using modal data only. Our objective is twofold: to identify damage in structural elements (beams) and to establish a classification of the connections between elements using no a priori information except the modal data. This paper introduces enhanced wavelet analysis as a tool for damage detection based on mode shape data, using an analytical mode shape fit as the undamaged state. Then we demonstrate that damage severity can be identified using a supervised neural networks approach. To supplement the local diagnosis, boundary conditions are characterised using spatial frequency. Finally, a damaged beam of a wooden portal is analysed using a high-resolution mode shape acquired with a laser Doppler vibrometer. The experimental results confirm the accuracy of the method for detecting damages and classifying boundary conditions. [Copyright &y& Elsevier]

Abstract: In this paper, the first nine frequency parameters of circular and annular plates with variable thickness and combined boundary conditions are computed for different thickness to radius ratios. Several combined boundary conditions are considered for inner and outer edges. These are free, soft and hard simply supported, and clamped boundary conditions. Three-dimensional elasticity theory is used. Results of this paper are compared with works of other authors and the results of finite element method analysis which are in agreement. Data for thick circular and annular plates with linear and parabolic thickness variation and different combinations are presented for the first time. [Copyright &y& Elsevier]

Abstract: The literature regarding the “exact” solutions of natural frequencies and mode shapes of a uniform beam carrying multiple two-degree-of-freedom (2-dof) spring–mass systems is rare, thus, this paper aims at studying this problem using the numerical assembly method (NAM). First of all, the equivalent springs for replacing the effect of a 2-dof spring–mass system are determined. Next, the coefficient matrix for a 2-dof spring–mass system attached to the uniform beam is derived based on the compatibility of deformations and equilibrium of forces (including moments). The coefficient matrices for the left end and right end of the beam are also derived based on the various boundary conditions of the beam. Combining the coefficient matrices for all the 2-dof spring–mass systems attached to the beam and the coefficient matrices for the boundary conditions of the beam, one obtains the overall coefficient matrix of the constrained beam (i.e., the beam carrying any number of 2-dof spring–mass systems). The product of the overall coefficient matrix and the vector for all the integration constants yields a set of simultaneous equations. Let the coefficient determinant of the last simultaneous equations equal to zero, one obtains the frequency equation. The roots of the last frequency equation denote the natural frequencies of the constrained beam. Substituting the roots of the frequency equation into the set of simultaneous equations one may determine the associated mode shapes of the constrained beam. In this paper, the “exact” solution refers to the one obtained from the “continuous” model instead of the “discrete” mode, besides, the accuracy of the analytical-and-numerical combined method (ANCM) given by the existing literature is dependent on the total number of vibration modes considered, but this is not true for the accuracy of the NAM adopted here. To confirm the reliability of the presented theory, all the numerical results obtained from NAM are compared with the corresponding ones obtained from the conventional finite element method (FEM) and good agreement is achieved. [Copyright &y& Elsevier]

Abstract: The paper presents a new, efficient and accurate method for experimentally determining structural damping properties of stiff materials in flexural vibration. The estimation method seeks a wavenumber that forces data at all measurement points to conform with the general forced-vibration solution for a beam, and, thus, to have the same wave coefficients. The method does not depend on beam boundary conditions, making it relatively simple to implement in any laboratory setting. The paper shows results of using the method to estimate the complex modulus of aluminum and of polymethyl methacrylate (PMMA) beams in the frequency range from 30 to 800Hz.Results compare very well with those obtained by more conventional methods and with those previously published over most of the frequencies of interest. Discrepancies in results for frequencies less than 80Hz are attributed to the difficulty in measuring differences of velocity between points that are near each other at low frequency, and a technique for avoiding such discrepancies is outlined. Evidence shows that the error in the results near 350Hz is due to a resonance in the scanning laser vibrometer that was used to take the experimental measurements. [Copyright &y& Elsevier]

Abstract: In this paper the Network Simulation Method (NSM) is proposed as a tool for solving the transverse-motion of thin beams resulting from a bending action. The reliability and robustness of the proposed technique is demonstrated with two examples, in which a pinned–pinned beam and a clamped-mass load beam are, respectively, submitted to different flexural forces. The NSM is a numerical technique for the solution of linear and non-linear problems, which are defined by a mathematical model. Assuming that the electrical variables of voltage and current are equivalent, respectively, to the displacement and the spatial variation of the displacement, a network model for each volume element is designed such that electrical equations are formally equivalent to the spatial discretised equations of the mechanical model. The whole network model, including the devices associated with the boundary conditions, is solved by the numerical computer code PSPICE. The numerical results have been compared with those obtained by the mode-superposition method: the agreement between them is impressive. The temporal behaviour of each quantity involved in the problem is obtained without transform complications, since the whole study has been conducted in the time-domain, with the possibility of obtaining the corresponding spectral content at the end of calculation. The main contribution of the present paper, therefore, is to employ the NSM numerical method in order to arrive at solutions, in the time-domain, of dynamic mechanical problems. One must also emphasize that NSM has not previously been used by other authors to solve physical problems like the ones presented here. [Copyright &y& Elsevier]

In this paper, the aim is to find the non-linear mode shapes and natural frequencies for a class of one-dimensional continuous damped systems with weak cubic inertia, damping and stiffness non-linearities. This paper presents general formulations for natural frequencies and mode shapes with all non-linearity effects. Initially the non-linear system with general boundary conditions is discretized, and using a two-dimensional manifold, the model of cubic non-linearities is constructed and the general equation of motion which governs non-linear system is derived. The method of multiple scales is then used to extend the non-linear mode shapes and natural frequencies. During this analysis, it is realized that when the natural frequencies of the linear system become equal to the natural frequencies of the non-linear system a one-to-one internal resonance will appear. Also, there is a three-to-one internal resonance which is not dependent on the damping of the system. Finally, general formulations of amplitude for vibrations, natural frequencies and mode shapes of the non-linear system are obtained in parametric forms. Thus, a non-linear problem with some simple integration can be solved. The formulations are capable of handling any non-linearities in inertia, damping, stiffness, or any combination of them under any arbitrary boundary conditions. [Copyright &y& Elsevier]

In this paper, initial–boundary-value problems for a beam equation (with string effect) are considered. These problems can be used as simple models to describe the vertical vibrations of a conveyor belt, for which the velocity is small with respect to the wave speed. In this paper, the belt is assumed to move with a time-varying velocity V(t)=ϵ(V0+α sin(Ωt)). Formal asymptotic approximations of the solutions are constructed to show the complicated dynamical behaviour of the belt. Complicated dynamical behaviour of the belt system occurs when the frequency Ω is the sum or difference of any two natural frequencies of the system with zero belt velocity. For special values of the belt parameters these sum type and difference type of internal resonances coincide giving rise to even more complicated dynamical behaviour. Some examples (including detuning cases) have been studied in detail. [Copyright &y& Elsevier]

This paper examines the motions of reference systems linked to deformable bodies under simultaneously vibration and large translations and rotations. These motions depend on the particular type of linkage between the moving reference system and the deformable body, which is defined by the so-called reference conditions. When using the Rayleigh–Ritz method, the reference conditions also dictate the boundary conditions to be fulfilled by the shape functions used to describe the body''s elasticity. This paper analyses three different types of reference conditions, namely: free linkage, rigid linkage and two-point linkage. It is shown that, moving reference frames only evolve at a constant velocity in the absence of external forces when the free linkage is used. The reference velocities for systems with a free linkage are designated rigid body equivalent velocities for the deformable body here. Such velocities can also be calculated under other types of reference conditions and are usually functions of the elastic and reference co-ordinates, and also of their derivatives. Rigid body equivalent velocities are useful for purposes such as estimating the trajectory of deformable bodies moving freely in space without the need to examine the deformations they undergo. Also, their calculation is required with a view to determining the kinematic restitution coefficient for deformable body collisions, which is dealt within Part II of this series. [Copyright &y& Elsevier]

In this paper, a convected frequency-domain approach for acoustic scattering prediction is suggested, for sources and scattering surfaces in a uniform constant flow. Frequency-domain moving-medium formulations are used for prediction of the incident acoustic pressure and acoustic pressure gradient. The latter is required for evaluation of the hardwall boundary condition in a moving medium and allows a convected definition of the equivalent sources. The scattering approach is validated by the analytical case of scattering of a pulsating monopole source by a rigid sphere. The applicability of the methodology to moving-medium problems is demonstrated for rotating and pulsating monopole point sources in a uniform flow, located near an infinite flat scattering surface. The suggested approach allows frequency-domain scattering predictions for sources in a uniform constant flow of any subsonic velocity, enabling direct inclusion of convection effects on incident and scattered acoustics. The hardwall boundary condition is thus evaluated directly in a moving medium. The need for a Lorentz transform is obviated, overcoming the complexity it introduces and the limitations it imposes. [ABSTRACT FROM AUTHOR]

The coupled plates are commonly seen in flight vehicle structures, which are always subjected to a severe combination of thermal, aerodynamic and mechanical loads during the cruise. In this paper, a unified solution is derived for the aero-thermo-elastic flutter problems of the coupled plate structure with general boundary conditions, in which the classical and elastic boundary conditions can be dealt with. The Mindlin plate theory in conjunction with the supersonic piston theory to consider the supersonic flow effect is adopted to formulate the theoretical model of the heated coupled plate structure subjected to supersonic flow. Each displacement component of the coupled plate is constructed as a two-dimensional Fourier series supplemented with auxiliary functions in order to satisfy the possible boundary conditions. The flutter analysis of a single flat plate, to which great efforts have been devoted, can be considered as a special case in the proposed model by setting the coupling angle of the coupled plate equal to zero. A considerable number of numerical cases are presented to show the accuracy and efficiency of the proposed method. The effects of the aerodynamic pressure, boundary condition, coupling spring and temperature change on the flutter characteristics of the coupled plate are also discussed in detail. The present formulation permits to analyze the flutter problems of the coupled plate having arbitrary coupling angles and boundary conditions. [ABSTRACT FROM AUTHOR]

A strong-form boundary collocation method, the singular boundary method (SBM), is developed in this paper for the wave propagation analysis at low and moderate wavenumbers in periodic structures. The SBM is of several advantages including mathematically simple, easy-to-program, meshless with the application of the concept of origin intensity factors in order to eliminate the singularity of the fundamental solutions and avoid the numerical evaluation of the singular integrals in the boundary element method. Due to the periodic behaviors of the structures, the SBM coefficient matrix can be represented as a block Toeplitz matrix. By employing three different fast Toeplitz-matrix solvers, the computational time and storage requirements are significantly reduced in the proposed SBM analysis. To demonstrate the effectiveness of the proposed SBM formulation for wave propagation analysis in periodic structures, several benchmark examples are presented and discussed The proposed SBM results are compared with the analytical solutions, the reference results and the COMSOL software. [ABSTRACT FROM AUTHOR]

Free and forced nonlinear vibrations of a traveling belt with non-homogeneous boundary conditions are studied. The axially moving materials in operation are always externally excited and produce strong vibrations. The moving materials with the homogeneous boundary condition are usually considered. In this paper, the non-homogeneous boundaries are introduced by the support wheels. Equilibrium deformation of the belt is produced by the non-homogeneous boundaries. In order to solve the equilibrium deformation, the differential and integral quadrature methods (DIQMs) are utilized to develop an iterative scheme. The influence of the equilibrium deformation on free and forced nonlinear vibrations of the belt is explored. The DIQMs are applied to solve the natural frequencies and forced resonance responses of transverse vibration around the equilibrium deformation. The Galerkin truncation method (GTM) is utilized to confirm the DIQMs' results. The numerical results demonstrate that the non-homogeneous boundary conditions cause the transverse vibration to deviate from the straight equilibrium, increase the natural frequencies, and lead to coexistence of square nonlinear terms and cubic nonlinear terms. Moreover, the influence of non-homogeneous boundaries can be exacerbated by the axial speed. Therefore, non-homogeneous boundary conditions of axially moving materials especially should be taken into account. [ABSTRACT FROM AUTHOR]

In this paper, it is shown how characteristic coordinates, or equivalently how the well-known formula of d’Alembert, can be used to solve initial-boundary value problems for wave equations on fixed, bounded intervals involving Robin type of boundary conditions with time-dependent coefficients. A Robin boundary condition is a condition that specifies a linear combination of the dependent variable and its first order space-derivative on a boundary of the interval. Analytical methods, such as the method of separation of variables (SOV) or the Laplace transform method, are not applicable to those types of problems. The obtained analytical results by applying the proposed method, are in complete agreement with those obtained by using the numerical, finite difference method. For problems with time-independent coefficients in the Robin boundary condition(s), the results of the proposed method also completely agree with those as for instance obtained by the method of separation of variables, or by the finite difference method. [ABSTRACT FROM AUTHOR]

In this paper, an analytical method is presented to study an initial-boundary value problem describing the transverse displacements of a vertically moving beam under boundary excitation. The length of the beam is linearly varying in time, i.e., the axial, vertical velocity of the beam is assumed to be constant. The bending stiffness of the beam is assumed to be small. This problem may be regarded as a model describing the lateral vibrations of an elevator cable excited at its boundaries by the wind-induced building sway. Slow variation of the cable length leads to a singular perturbation problem which is expressed in slowly changing, time-dependent coefficients in the governing differential equation. By providing an interior layer analysis, infinitely many resonance manifolds are detected. Further, the initial-boundary value problem is studied in detail using a three-timescales perturbation method. The constructed formal approximations of the solutions are in agreement with the numerical results. [ABSTRACT FROM AUTHOR]

This paper presents a three-dimensional analytical model for the prediction of ground vibrations from two parallel tunnels embedded in a full-space. The two tunnels are modelled as cylindrical shells of infinite length, and the surrounding soil is modelled as a full-space with two cylindrical cavities. A virtual interface is introduced to divide the soil into the right layer and the left layer. By transforming the cylindrical waves into the plane waves, the solution of wave propagation in the full-space with two cylindrical cavities is obtained. The transformations from the plane waves to cylindrical waves are then used to satisfy the boundary conditions on the tunnel-soil interfaces. The proposed model provides a highly efficient tool to predict the ground vibration induced by the underground railway, which accounts for the dynamic interaction between neighbouring tunnels. Analysis of the vibration fields produced over a range of frequencies and soil properties is conducted. When the distance between the two tunnels is smaller than three times the tunnel diameter, the interaction between neighbouring tunnels is highly significant, at times in the order of 20 dB. It is necessary to consider the interaction between neighbouring tunnels for the prediction of ground vibrations induced underground railways. [ABSTRACT FROM AUTHOR]

In this paper, we propose the boundary control for undesired vibrations suppression with output constraint of the flapping-wing robotic aircraft (FWRA). We also present the dynamics of the flexible wing of FWRA with governing equations and boundary conditions, which are partial differential equations (PDEs) and ordinary differential equations (ODEs), respectively. An energy-based barrier Lyapunov function is introduced to analyze the system stability and prevent violation of output constraint. With the effect of the proposed boundary controller, distributed states of the system remain in the constrained spaces. Then the IBLF-based boundary controls are proposed to assess the stability of the FWRA in the presence of output constraint. [ABSTRACT FROM AUTHOR]

Engineers and designers are often faced with the task of selecting materials that minimizes structural weight whilst meeting the required strength and stiffness. In many cases fibre reinforced composites (FRCs) are the materials of choice since they possess a combination of high strength and low density. Depending on the application, composites are frequently constructed to form long slender beam-like structures or flat thin plate-like structures. Such structures when subjected to random excitation have the potential to excite higher order vibratory modes which can contribute significantly to structure-borne sound. Statistical Energy Analysis (SEA) is a framework for modeling the high frequency vibration of structures. The modal density, which is typically defined as the number of modes per unit Hertz in a frequency band, is a fundamental parameter when applying SEA. This study derives formulas for the modal density of a fibre reinforced composite beam coupled in bending and torsion. The effects of shear deformation and rotary inertia are accounted for in the formulation. The modal density is shown to be insensitive to boundary conditions. Numerical analyses were carried out to investigate the variation of modal density with fibre orientation including and excluding the effects of shear deformation and rotary inertia. It was observed that neglecting such effects leads to underestimating the mode count in a particular frequency band. In each frequency band there exists a fibre orientation for which the modal density is minimized. This angular orientation is shown to be dependent on the shear rigidity as well as the bending, torsional and coupling rigidities. The foregoing observation becomes more pronounced with increasing frequency. The paper also addresses the modal density beyond the wave-mode transition frequency where the beam supports three propagating waves. [ABSTRACT FROM AUTHOR]