Search

Acoustic black hole (ABH), a thin wedge-shaped structure tapered according to the power-law of power (m) greater than or equal to two, has received much attention to the researchers in wave dynamics due to its potential as a light and effective absorber of elastic waves propagating in beams or plates. In this paper, the Euler-Bernoulli equation for the ABH is reformulated into the form of a generalized hypergeometric differential equation for m > 2. Then, the exact solution is derived in terms of generalized hypergeometric functions (p F q) where p = 0 and q = 3 by dividing the power m into four cases. The derived exact solution is in linearly independent and regular form for arbitrary m. In addition, by using the exact solution, the displacement field of a uniform beam with an ABH and the reflection coefficient from an ABH are calculated to demonstrate the applicability of the present exact solution. This paper aims at providing a mathematical and theoretical basis for the study of the ABHs. [ABSTRACT FROM AUTHOR]

Mathematical models play an increasing role in understanding and predicting machining processes, in particular milling. However, despite the considerable efforts that have been dedicated to this problem, a majority of milling models still rely on simplifying assumptions to calculate the chip thickness. In this paper, the chip thickness is determined without these simplifications, based on a surface function that describes the milled surface and on information about the workpiece boundary. By combining the partial differential equation (PDE) governing the evolution of this surface function with the ordinary differential equations (ODE) governing the tool/machine dynamics, a mixed PDE–ODE formulation is proposed to describe the dynamics of the milling process. The coupled system of differential equations is solved using an algorithm that combines finite difference (ODE) and finite volume (PDE) methods. A case study is presented to compare the proposed approach with the classical delay differential equations (DDE) model formulation for milling processes based on a simplified chip thickness model. The PDE–ODE formulation represents an explicit mathematical model for milling process dynamics; it yields a theoretically exact chip thickness and offers a means to assess the validity of models based on DDE formulation. Moreover, the proposed formulation is capable of simulating transient tool behaviors when the tool is milling the outer region of the workpiece, which is in general neglected by the DDE-based models. • Accurate mathematical description of the milling process. • Evolution of the machined surface around the tool is described. • Chip thickness model affects the accuracy of milling simulation in certain scenarios. • Transient tool behaviors are captured while milling the outer part of the workpiece. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the performance of vehicle suspension system under time-delayed optimal control is investigated. The effect of time delay on control stability of the active suspension system is discussed. The mathematical simulation is used to verify the correctness of the stable interval obtained by differential equation theory for linear systems with constant coefficients and time delay. In order to keep the stability of the system, time-delayed optimal control is designed through the method of state transformation and optimal control theory. The results show that the control strategy could not only guarantee the stability of the system regardless of the variation on control time delay, but also improve the performance of the suspension system. Additionally, influence of the designed active time delay on the amplitude of sprung mass acceleration, suspension deflection, rode holding are analyzed to provide guidance for its further practical engineering application. Vibration control experiments of the active suspension system under harmonic excitation with time-delayed optimal control are figured to compare with simulations. It is seen that the designed time-delayed optimal control strategy has the effectiveness and advantage. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on analysis of a friction-induced mechanical oscillator with cubic nonlinearity and an applied dynamical model of dry friction. The studied model is based on the classical LuGre approach to friction force modelling. Lyapunov exponents spectrum of the frictional oscillator is calculated by means of a recently implemented method of their estimation for non-smooth systems. Results of the numerical research, observations of the nature of the oscillator's response and interaction of the applied dynamic friction model with the oscillator are reported. Highlights • The dynamics of stick-slip oscillator with cubic nonlinearity is demonstrated. • Modified LuGre friction model is applied. • The stability of the system is proven by conditional Lyapunov exponents. • The correlation (synchrony) between oscillator response and dynamics of friction force is confirmed. [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 This paper presents an analytical study on linear and nonlinear free vibration characteristics and dynamic responses of spinning functionally graded (FG) graphene reinforced thin cylindrical shells with various boundary conditions and subjected to a static axial load. The weight fraction of graphene platelet (GPL) nanofillers in each concentric cylindrical layer is constant but follows a layer-wise variation through thickness direction, resulting in position-dependent elastic moduli, mass density and Poisson's ratio along the shell thickness. Based on the Donnell's nonlinear shell theory, the nonlinear partial differential equations of motion for the cylindrical shell are established by using the Hamilton's principle with the effects of centrifugal and Coriolis forces as well as the spin-induced initial hoop tension taken into account. The governing equations for nonlinear vibration of the nanocomposite cylindrical shell with different GPL dispersion patterns are derived from a set of nonlinear ordinary differential equations which are derived by employing the Galerkin approach. Dynamic responses of forward and backward travelling waves are investigated by analyzing the wave form and the frequency spectrum. Special attention is given to the effects of the weight fraction, dispersion patterns and the geometrical size of GPLs, the external axial load and spinning speeds of the cylindrical shell on the linear and nonlinear free vibrations of the nanocomposite cylindrical shell. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we provide a variational derivation of modified Bresse-Timoshenko equations. This process leads to the additional term to so called truncated set of Bresse-Timoshenko equations; this new term is associated with the modified slope inertia. The two sets of truncated versions of the Bresse-Timoshenko equations are contrasted with each other, as well as with the original Bresse-Timoshenko equations on the example of the (a) beam that is simply supported at its both ends, and (b) the cantilever beam with end mass. It is concluded that application of either truncated sets of equations is advantageous over the original Bresse-Timoshenko equations. As far as two truncated sets of equations are concerned, the variationally derived set appears to be preferable. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a nonlinear dynamic differential equations of high-speed cylindrical roller bearing considering the impact of roller dynamic unbalance. The cage vibration characteristics are investigated as a function of roller dynamic unbalance, structure parameters and working conditions. The fast Fourier transformation, cage orbit and vibration acceleration level are used to evaluate the vibration magnitude of the cage. The results show that: (1) Roller dynamic unbalance can affect the state of the cage motion and cause the cage's vibration acceleration level to increase. Once the roller dynamic unbalance exceeds a threshold, the cage will produce a high frequency vibration; (2) The most significant impact on the cage vibration is roller dynamic unbalance in angular direction of roller axis, followed by radial and axial direction; (3) A smaller radial clearance and a reasonable pocket clearance are beneficial to reducing the effect of roller dynamic unbalance on the cage vibration. It is recommended to take a larger cage-race guide clearance within a certain range to lower the cage vibration; (4) The cage vibration due to the roller dynamic unbalance increases with the bearing speed, and is greater than that of the cage with ideal roller. The effect of roller dynamic unbalance on the vibration of cage under high-speed condition is greater than that of the low-speed condition. The increase of the radial load of the bearing, to a certain extent, can reduce the vibration of cage considering the roller dynamic unbalance. [ABSTRACT FROM AUTHOR]

In this paper, the planar maneuver of a flexible satellite with regard to its flexible appendages vibration has been studied. The flexible satellite translates and rotates in a plane; in addition, the flexible appendages can also vibrate in that plane. The system governing equations, which are coupled partial and ordinary differential equations, are obtained based on Hamilton's principle. Then the original system converts to three equivalent subsystems, two of which contains one partial differential equation and one ordinary differential equation along with four boundary conditions, by using change of variables. Employing control forces and one control torque which are applied to the central hub, control objectives, tracking the desired angle and suppressing the satellite and its flexible appendages vibrations, are fulfilled. Furthermore, to eliminate spillover instability phenomenon and to exclude in-domain measurement and actuator usage problem, these control torque and forces are designed based on boundary control method. Lyapunov's direct method is employed to prove the asymptotic stability in absence of any damping effect in modeling the vibrations of flexible appendages. Eventually, in order to demonstrate the effectiveness of the designed boundary control, numerical simulations are presented. [ABSTRACT FROM AUTHOR]

This paper focuses on the performance of the two two-loop implicit sparse matrix numerical integration (TLISMNI) methods as the numerical solution of index-3 differential algebraic equations (DAEs) of motion arising in stiff multibody dynamics with contact and constraint. The original TLISMNI method proposed for the first time by Shabana and Hussein uses the Newmark method or Hilber/Hughes/Taylor (HHT) method as the implicit integration algorithm [TLISMNI (Newmark/HHT) method]. These two integration algorithms consider the accelerations and Lagrange multipliers as basic unknowns and their numerical accuracy is no more than order 2. In order to have a higher integral precision, we propose a new two-loop implicit procedure based on an extended backward differentiation formula scheme [TLISMNI (EBDF) method] which considers coordinates and velocities as basic unknowns. The whole structure of this proposed method is different from the traditional one due to the different unknowns. Furthermore, we apply these two kinds of methods, TLISMNI (Newmark/HHT) method and TLISMNI (BDF/EBDF) method, to the contact and constraint problems of flexible multibody dynamics while few studies of TLISMNI methods have been done on these non-smooth problems. The Baumgarte method and HHT- α method are also used to solve the same contact and constraint problem, and the advantages and disadvantages of all these methods above are compared in this study. Results show that the proposed method only needs to perform less iterations to satisfy the same tolerance of error than the TLISMNI (HHT) method does. In some cases the Baumgarte method and HHT- α method cannot give reasonable results while the TLISMNI methods perform well when the contact and constraint problems are involved. Moreover, the numerical experiment also indicates that the numerical damping property of the HHT method employed in both the TLISMNI method and the GGL formulation (used in the HHT- α method) can behave in a different way to damp out high frequency oscillations induced by impact. [ABSTRACT FROM AUTHOR]

In this paper the axial vibration of a muscle whose fibers are parallel to the direction of muscle compression is investigated. The model is a clamped-free rod with a strongly nonlinear elastic property. Axial vibration is described by a nonlinear partial differential equation. A solution of the equation is constructed for special initial conditions by using the method of separation of variables. The partial differential equation is separated into two uncoupled strongly nonlinear second order differential equations. Both equations, with displacement function and with time function are exactly determined. Exact solutions are given in the form of inverse incomplete and inverse complete Beta function. Using boundary and initial conditions, the frequency of vibration is obtained. It has to be mentioned that the determined frequency represents the exact analytic description for the axially vibrating truly nonlinear clamped-free rod. The procedure suggested in this paper is applied for calculation of the frequency of the longissimus dorsi muscle of a cow. The influence of elasticity order and elasticity coefficient on the frequency property is tested. [ABSTRACT FROM AUTHOR]

This paper presents an alternative pathway in studying the ubiquitous blade/casing rub problem in turbomachinery. Bladed disks interfere with the stationary shroud (casing) for a variety of reasons, such as axial offsets, thermal expansions. Both components being compliant, time-varying interface characteristics, nonlinearities and uncertainties in the rub forces make this dynamics very complex to model and analyze. The main idea in this paper originates from the conjecture that this dynamics is inherently akin to internal machining operation which also deals with compliant cutters (blades) but relatively more rigid workpiece (casing). This analogy directs our attention to the fact that the blade/casing impingement dynamics manifests a ‘regenerative mechanism’ which is impregnated with time delays. The ensuing time-delayed system (TDS) can be stable, which is ideal. If it is unstable, however, the interference amplitudes between the blade and the casing grow, and the nonlinear effects become dominant. If the components survived the exercise, this evolution would reach a limit-cycle behavior. Existing literature indicates that this limit cycle mode is the common state of operation in most modern-day turbomachinery. Consequently, the state-of-the-art research effort is focused on minimizing its amplitude to alleviate the destructive levels of fatigue effect. In this article we consider a different perspective in looking at these problems, by proposing the conditions to achieve stable rub interference. For this, a recent mathematical tool of the authors’ group called the Cluster Treatment of Characteristic Roots (CTCR) is deployed. CTCR declares the complete stability outlook of such time-delayed systems in the space of the operational and design parameters. We show how this new capability can assist the design process of the blade-casing interface. Simulations, relevant stability observations and comparisons with a peer technique are provided for some case studies to demonstrate the capabilities of the approach. [ABSTRACT FROM AUTHOR]

A dynamic model of a coupled rotating drum-disk system with elastic support is developed in this paper. By considering the effects of centrifugal and Coriolis forces as well as rotation-induced hoop stress, the governing differential equation of the drum-disk is derived by Donnell's shell theory. The nonlinear amplitude-frequency characteristics of coupled structure are studied. The results indicate that the natural characteristics of the coupling structure are sensitive to the supporting stiffness of the disk, and the sensitive range is affected by rotating speeds. The circumferential wave numbers can affect the characteristics of the drum-disk structure. If the circumferential wave number n = 1 , the vibration response of the drum keeps a stable value under an unbalanced load of the disk, there is no coupling effect if n ≠ 1 . Under the excitation, the nonlinear hardening characteristics of the forward traveling wave are more evident than that of the backward traveling wave. Moreover, because of the coupling effect of the drum and the disk, the supporting stiffness of the disk has certain effect on the nonlinear characteristics of the forward and backward traveling waves. In addition, small length-radius and thickness-radius ratios have a significant effect on the nonlinear characteristics of the coupled structure, which means nonlinear shell theory should be adopted to design rotating drum's parameter for its specific structural parameters. [ABSTRACT FROM AUTHOR]

This paper derives a comprehensive analytical dynamic model of a closed circular beam that has a T shaped cross section. The new model includes in-plane and out-of-plane vibrations derived using continuous media expressions which produces results that have a valid frequency range above those available from traditional lumped parameter models. The web is modeled using two-dimensional elasticity equations for in-plane motion and the classical flexural plate equation for out-of-plane motion. The flange is modeled using two sets of Donnell shell equations: one for the left side of the flange and one for the right side of the flange. The governing differential equations are solved with unknown wave propagation coefficients multiplied by spatial domain and time domain functions which are inserted into equilibrium and continuity equations at the intersection of the web and flange and into boundary conditions at the edges of the system resulting in 24 algebraic equations. These equations are solved to yield the wave propagation coefficients and this produces a solution to the displacement field in all three dimensions. An example problem is formulated and compared to results from finite element analysis. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate temperature effects on the nonlinear vibration behaviors of suspended cables under two-frequency excitation. For this purpose, two combination and simultaneous resonances are chosen and studied in detail. First of all, based on the assumptions of the temperature effects, the partial differential equations of the in-plane and out-of-plane motions with thermal effects under multi-frequency excitations are obtained. The Galerkin method is adopted to discretize the nonlinear dynamic equations, and the single-mode planar discretization is considered. Then, in the absence of the primary and internal resonances, the frequency response equations are obtained by using the multiple scales method. The stability analyses are conducted via investigating the nature of the singular points of equations. After that, temperature effects on nonlinear vibration characteristics of the first symmetric mode are studied. Parametric investigations of temperature effects on corresponding non-dimensional factors and coefficients of linear and nonlinear terms are performed. Numerical results are presented to show the temperature effects via the frequency-response curves and detuning-phase curves of four different sag-to-span ratios. It is found out that effects of temperature variations would lead to significant quantitative and/or qualitative changes of the nonlinear vibration properties, and these effects are closely related to the sag-to-span ratio and the degree of the temperature variation. Specifically, the softening/hardening-type spring behaviors, the response amplitude, the range of the resonance, the intersection and number of branches, the number and phase of the steady-state solutions are all affected by the temperature changes. [ABSTRACT FROM AUTHOR]

In this paper, a closed-form solution of the moving harmonic load problem for continuous Euler–Bernoulli beam systems is presented. The generality of the boundary conditions is taken into account by solving the characteristic equation of the system, obtaining its natural frequencies and mode shapes. The undetermined coefficient method is applied to solve the governing differential equation of motion, determining the base functions of the solution space of the problem. For vertical vibrations, three numerical examples of footbridges are presented. The main contribution of this paper is to provide the closed-form solution of the moving harmonic load problem applied to continuous footbridges including the phase angle in the load definition. In this way, it is possible to find the solution in the time domain of the harmonic component of any load spectra. • A closed-form solution for the moving harmonic load problem is presented. • The solution includes the phase angle in the load definition. • The solution can be applied to a general continuous beam with flexible supports. • The solution can be used in the time domain and for any load spectra. • Great agreement was found between FE-models and the closed-form solution. [ABSTRACT FROM AUTHOR]

In this paper a comprehensive and structured analysis of the stabilizing properties linked to the combination of different chatter suppression techniques in milling was carried out. Specifically, the spindle speed variation, the stiffness variation and the usage of tools with variable pitch and variable helixangles were considered. For this purpose, a unique, general and efficient frequency domain formulation (extended harmonic solution) was developed. It is suitable for dealing with differential delayed equations characterized by multiple and distributed time dependent delays and time periodic dynamics. It was demonstrated that the developed solution converges to the reference cases, which were studied in literature through semi-discretization and its extensions. Moreover, by selecting a proper number of considered harmonics, the computation time was reduced of one order of magnitude, in the face of a slight reduction of accuracy with respect to time domain approaches. The computed stability lobe diagrams demonstrated that the combination of multi-chatter suppression techniques allows exploiting the best features of each methodology. The simultaneous use of the spindle speed variation and the stiffness variation was studied for the first time. The absolute limit of stability increased up to 204%, much more than by adopting the techniques in a separate manner. If the enlargement of the stability region is considered as the reference target, the best results were granted by the combination of all the three strategies. Indeed, the improvement with respect to the use of only two techniques was at least 10%. • Stability of multiple point, distributed, time varying delay differential equation. • Frequency domain formulation for complex milling applications. • Investigation of novel chatter suppression technique combinations. • Improvement of chatter stability with respect to previous results in literature. [ABSTRACT FROM AUTHOR]

• Eigenproperties of a multi-cracked Euler-Bernoulli inextensible circular arch. • Cracks are accounted for in governing equation by means of the distribution theory. • Governing equation defined over a unique integration domain regardless n° cracks. • Integration via Laplace transform is adopted to infer closed form mode shapes. • Both validations and parametric analyses are presented. Despite the numerous explicit solutions of free vibration of arches with regular cross sections, in case of concentrated defects such as cracks, no procedure is available to analyse arch vibrations without sub-division of the integration domain. As a result, curved sub-elements comprised between crack and external constraints, or successive cracks, are considered. In this paper a distributional approach is adopted to provide a formulation of the free vibration differential governing equations of circular inextensible arches over a unique integration domain in presence of multiple concentrated open (non-breathing) cracks. Discontinuities due to the presence of an arbitrary number of cracks are modelled by means of Dirac's deltas. An integration procedure is devised to offer closed form solutions of the relevant vibration modes together with the relevant frequency determinantal equation. Natural frequencies and mode shapes of damaged arches with different damage and restraint configurations have been evaluated and compared with experimental results available in the literature as well as finite element numerical simulations. The presented closed form solutions are also employed for two parametric studies to evaluate the influence of an increasing number of along axis concentrated cracks as well as of the location of cracks along the arch span. [ABSTRACT FROM AUTHOR]

This paper analyzes the mechanical and mathematical models in “Ritto et al. (2013) [1]”. The results are that: (1) the mechanical model is obviously incorrect; (2) the mathematical model is not complete; (3) the differential equation is obviously incorrect; (4) the finite element equation is obviously not discretized from the corresponding mathematical model above, and is obviously incorrect. A mathematical model of dynamics should include the differential equations, the boundary conditions and the initial conditions. [ABSTRACT FROM AUTHOR]

Devices that emit structure-borne sound are commonly decoupled by elastic components to shield the environment from acoustical noise and vibrations. The elastic elements often have a hysteretic behavior that is typically neglected. In order to take hysteretic behavior into account, Bouc developed a differential equation for such materials, especially joints made of rubber or equipped with dampers. In this work, the Bouc model is solved by means of the Gaussian closure technique based on the Kolmogorov equation. Kolmogorov developed a method to derive probability density functions for arbitrary explicit first-order vector differential equations under white noise excitation using a partial differential equation of a multivariate conditional probability distribution. Up to now no analytical solution of the Kolmogorov equation in conjunction with the Bouc model exists. Therefore a wide range of approximate solutions, especially the statistical linearization, were developed. Using the Gaussian closure technique that is an approximation to the Kolmogorov equation assuming a multivariate Gaussian distribution an analytic solution is derived in this paper for the Bouc model. For the stationary case the two methods yield equivalent results, however, in contrast to statistical linearization the presented solution allows to calculate the transient behavior explicitly. Further, stationary case leads to an implicit set of equations that can be solved iteratively with a small number of iterations and without instabilities for specific parameter sets. [ABSTRACT FROM AUTHOR]

Abstract: In this paper a Human–Vehicle–Road (HVR) model, comprising a quarter-car and a biomechanical representation of the driver, is employed for the analysis. Differential equations are provided to describe the motions of various masses under the influence of a harmonic road excitation. These equations are, subsequently, solved to obtain a closed form mathematical expression for the steady-state vertical acceleration measurable at the vehicle–human interface. The solution makes it possible to find optimal parameters for the vehicle suspension system with respect to a specified ride comfort level. The quantitative definition given in the ISO 2631 standard for the ride comfort level is adopted in this paper for the optimization procedure. Numerical examples, based on actually measured road profiles, are presented to prove the validity of the proposed approach and its suitability for the problem at hand. [Copyright &y& Elsevier]

Abstract: In this paper the excited vibrations of a truly nonlinear oscillator are analyzed. The excitation is assumed to be constant and the nonlinearity is pure (without a linear term). The mathematical model is a second-order nonhomogeneous differential equation with strong nonlinear term. Using the first integral, the exact value of period of vibration i.e., angular frequency of oscillator described with a pure nonlinear differential equation with constant excitation is analytically obtained. The closed form solution has the form of gamma function. The period of vibration depends on the value of excitation and of the order and coefficient of the nonlinear term. For the case of pure odd-order-oscillators the approximate solution of differential equation is obtained in the form of trigonometric function. The solution is based on the exact value of period of vibration. For the case when additional small perturbation of the pure oscillator acts, the so called ‘Cveticanin''s averaging method’ for a truly nonlinear oscillator is applied. Two special cases are considered: one, when the additional term is a function of distance, and the second, when damping acts. To prove the correctness of the method the obtained results are compared with those for the linear oscillator. Example of pure cubic oscillator with constant excitation and linear damping is widely discussed. Comparing the analytically obtained results with exact numerical ones it is concluded that they are in a good agreement. The investigations reported in the paper are of special interest for those who are dealing with the problem of vibration reduction in the oscillator with constant excitation and pure nonlinear restoring force the examples of which can be found in various scientific and engineering systems. For example, such mechanical systems are seats in vehicles, supports for machines, cutting machines with periodical motion of the cutting tools, presses, etc. The examples can be find in electronics (electromechanical devices like micro-actuators and micro oscillators), in music instruments (hammers in piano), in human voice producing folds (voice cords), etc. [Copyright &y& Elsevier]

Abstract: Frame structures with viscoelastic dampers mounted on them are considered in this paper. Viscoelastic (VE) dampers are modelled using two, three-parameter, fractional rheological models. The structures are treated as elastic linear systems. The equation of motion of the whole system (structure with dampers) is written in terms of state-space variables. The resulting matrix equation of motion is the fractional differential equation. The proposed state space formulation is new and does not require matrices with huge dimensions. The paper is devoted to determine the dynamic properties of the considered structures. The nonlinear eigenvalue problem is formulated from which the dynamic parameters of the system can be determined. The continuation method is used to solve the nonlinear eigenvalue problem. Moreover, results of typical calculations are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the dynamic response of a double-string system traversed by a constant or a harmonically oscillating moving force is considered. The force is moving with a constant velocity on the top string. The strings are identical, parallel, one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a double-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case a part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations. [ABSTRACT FROM AUTHOR]

Abstract: The lateral–torsional vibrations of composite beams are investigated in this paper, based on a variational approach. Composite beams studied in this paper are classified as composite beams with interlayer slip (composite beams with partial interaction or layered wood beams) or three-layer sandwich beams. It will be shown in this paper that both structural systems are governed by the same nature of differential equations. The theoretical framework of lateral–torsional vibrations of composite beams is given, and some engineering results are presented for the pinned–pinned strip composite beam. A simple closed-form solution is achieved for the lateral–torsional natural frequencies. The results are analogous to the ones obtained for the in-plane vibrations of composite beams (sandwich beam or composite beams with partial interaction) where the natural frequencies increase with the stiffness of the connection. Extension of these results to some more complex loading cases is envisaged, and a numerical procedure will be probably needed in the general case. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to show that the image source method (ISM) can be used both for analytically calculating Green''s functions of particular simply supported convex polygonal plates and for predicting medium and high frequency vibrations of arbitrarily shaped simply supported convex polygonal plates with controllable precision. In the first part of the paper, the method for obtaining the Green''s function of a polygonal plate by ISM is developed. Examples for plates of different geometries (rectangle, isosceles right triangle, half-equilateral triangle and equilateral triangle) are given. In the second part of the paper, the pertinence of ISM for predicting medium and high frequency vibrations of arbitrarily shaped simply supported convex polygonal plates is investigated. An approximation based on the exclusion of image sources beyond a certain distance from the receiver is used in order to take advantage of the dissipation of vibrational energy through wave propagation. We investigate the influence of structural damping and truncation distance on the accuracy of such approximation. The computed responses are in good agreement with reference solutions, which are analytically known or obtained by the finite element method. [Copyright &y& Elsevier]

In this paper a new analytical method for solving the differential equations which describe the motion of the oscillator with fraction order elastic force is introduced. Using the first integral of motion the exact period of vibration in the form of the Euler beta function is obtained. Based on that value the approximate solution of the strong nonlinear fraction order differential equation is formed. The suggested procedure is applied for various fraction values, but also for the pure quadratic, cubic and quintic oscillators. The advantage of the suggested method is that is valid for all fraction values , the accuracy of the approximate solution is very high as the period of vibration is exactly analytically determined and is independent on the time. In the paper the fraction order differential equation with small linear and nonlinear terms is also considered. The Krylov–Bogolubov method is extended due to the fact that frequency of vibration depends on the amplitude of vibration. The approximate solution is with time variable amplitude, phase and frequency. The linear damped fraction order oscillator is widely discussed. The approximate solution of the differential equation which describes the vibrations of such oscillator is compared with exact numerical one and shows a good agreement. [Copyright &y& Elsevier]

The paper deals with the vibration of suspended bridges subjected to the simultaneous action of moving loads and vertical support motions due to earthquake. The basic partial integro-differential equation is applied to the vertical vibration of a suspended beam. The dynamic actions of traffic loads are modelled as a row of equidistant moving forces, while the earthquake is considered by vertical motions of supports. The governing equation is solved first analytically to receive an ordinary differential equation and next numerically. Moreover, the designed world's largest suspended bridge—Messina Bridge—is investigated (central span of length 3.3km). The paper studies the effect of various lags of the earthquake arrival because the earthquake may appear at any time when the train moves along a large-span bridge. The modified Kobe earthquake records have been applied to calculations. The results indicate that the interaction of both the moving and seismic forces may substantially amplify the response of long-span suspended bridges in the vicinity of the supports and increase with the rising speed of trains. [Copyright &y& Elsevier]

The propagation of sound waves in a horizontally stratified environment, a classic problem in ocean acoustics, has traditionally been calculated using normal modes. Most programs based on the normal mode model are discretized using the finite difference method (FDM). In this paper, a Legendre collocation method (LCM) based on domain decomposition is proposed to solve this problem. A set of collocation points cannot penetrate multiple layers of media, thus necessitating domain decomposition and the use of multiple sets of collocation points. The solution process of this method proceeds entirely in physical space, requiring that the original differential equation be strictly established at the collocation points; thus, a dense matrix eigenvalue system is formed, from which the solution for the horizontal wavenumbers and modes can be directly obtained. Numerical experiments are presented to demonstrate the validity and applicability of this method. A comparison with other methods shows that the LCM proposed in this article is more accurate than the FDM and offers approximately the same accuracy as, but a faster calculation speed than, other types of spectral methods. • This paper proposes a new method to calculate acoustic propagation in horizontally stratified marine environment. • The accuracy of the new method is greater than that of the classical finite difference method. • The new method is far faster than the Legendre-Galerkin and Chebyshev-Tau spectral methods. [ABSTRACT FROM AUTHOR]

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]

Non-standard finite difference methods (NSFDM) introduced by Mickens [Non-standard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994] are interesting alternatives to the traditional finite difference and finite volume methods. When applied to linear hyperbolic conservation laws, these methods reproduce exact solutions. In this paper, the NSFDM is first extended to hyperbolic systems of conservation laws, by a novel utilization of the decoupled equations using characteristic variables. In the second part of this paper, the NSFDM is studied for its efficacy in application to nonlinear scalar hyperbolic conservation laws. The original NSFDMs introduced by Mickens (1994) were not in conservation form, which is an important feature in capturing discontinuities at the right locations. Mickens [Construction and analysis of a non-standard finite difference scheme for the Burgers–Fisher equations, Journal of Sound and Vibration 257 (4) (2002) 791–797] recently introduced a NSFDM in conservative form. This method captures the shock waves exactly, without any numerical dissipation. In this paper, this algorithm is tested for the case of expansion waves with sonic points and is found to generate unphysical expansion shocks. As a remedy to this defect, we use the strategy of composite schemes [R. Liska, B. Wendroff, Composite schemes for conservation laws, SIAM Journal of Numerical Analysis 35 (6) (1998) 2250–2271] in which the accurate NSFDM is used as the basic scheme and localized relaxation NSFDM is used as the supporting scheme which acts like a filter. Relaxation schemes introduced by Jin and Xin [The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications in Pure and Applied Mathematics 48 (1995) 235–276] are based on relaxation systems which replace the nonlinear hyperbolic conservation laws by a semi-linear system with a stiff relaxation term. The relaxation parameter () is chosen locally on the three point stencil of grid which makes the proposed method more efficient. This composite scheme overcomes the problem of unphysical expansion shocks and captures the shock waves with an accuracy better than the upwind relaxation scheme, as demonstrated by the test cases, together with comparisons with popular numerical methods like Roe scheme and ENO schemes. [Copyright &y& Elsevier]

From the viewpoint of differential eigenvalue problem of Hamiltonian system, a linear finite-time H ∞ filtering problem can be addressed by computing eigenvalue and solving Riccati differential equation of an associated linear Hamiltonian system. This paper shows how a problem of determining the minimum induced norm of the H ∞ filter is formulated as a Hamiltonian differential eigenvalue problem. The H ∞ filters concerned here include central filter, perturbed filter and decentralised filter. The methods presented in the paper are based on characteristics of eigensolution of the corresponding Hamiltonian system and Riccati differential equation. With eigensolution of the Hamiltonian system arising from the central H ∞ filtering problem, variational methods are proposed to compute eigenvalues of perturbed Hamiltonian systems and large-scale Hamiltonian systems derived from perturbed H ∞ filters and decentralised H ∞ filters, respectively. Then, eigenvalues can be obtained by calculating stationary values of corresponding extended Rayleigh''s quotient with dual argument functions, which is essentially different from the well-known Rayleigh''s quotient of Lagrange systems with only one independent argument function. Numerical examples are also presented to illustrate the variational approaches presented in this paper. [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: In this paper, the dynamic behavior of a block-on-belt system subjected to simultaneous parametric and external excitations is studied, in which the parametric excitation arises from the variable stiffness of a spring in the system while the external excitation consists of a harmonic force. In addition, the system is damped by dry friction that follows the classical Coulomb''s law. The focus of the paper is on analyzing the influence of the parametric excitation on the qualitative features of system dynamics, under various frequency ratios between the external and parametric excitations. Numerical simulations are carried out, whose results are visualized by means of bifurcation diagrams, Poincaré sections and Lyapunov exponents. The results show that the system possesses rich dynamics characterized by periodic, quasi-periodic and chaotic attractors. Furthermore, it is found that the parametric excitation amplitude and the frequency ratio between the excitations have a significant impact on the dynamical behavior of the system. [Copyright &y& Elsevier]

Abstract: In a companion paper (Input torque balancing using a cam-based centrifugal pendulum: design procedure and example, J. Sound Vib.), the cam-based centrifugal pendulum (CBCP) was introduced as a simple, cam-based, input torque balancing mechanism. The differential equation that governs the CBCP cam design was derived and a methodology for solving it was developed. Furthermore, in a design example, the CBCP was applied to balance the input torque of a high-speed cam-follower mechanism, driving the sley of a weaving loom. The present paper firstly shows how the design parameters for this particular design example can be optimized, so as to obtain a compact and technologically feasible mechanism. The formulation of the optimization problem is based on a parameterization of the CBCP rotor and coupler shape. Because of its nonconvex nature, the optimization problem is solved using a multi-start sequential quadratic programming (SQP) approach. A design chart, based on an exhaustive analysis, is introduced which (i) allows the designer to perform the design optimization in a quick and approximative way, and (ii) gives considerable insight into the behavior of the SQP-algorithm. Secondly, the CBCP is applied to an industrial case study, that is, a weaving loom. The robustness of the CBCP is illustrated by showing that input torque balancing solely the sley movement enhances the overall dynamic machine behavior, despite the presence of the non-balanced shed motion. A particular contribution of this part is the determination of the weaving loom regime behavior in the frequency domain, an approach which is believed to be novel in mechanism literature. [Copyright &y& Elsevier]

The purpose of this paper is to analyze triply coupled vibrations of thin-walled beams with arbitrary open cross-section. Starting from the Vlasov''s theory, the governing differential equations for coupled bending and torsional vibrations were performed using the principle of virtual displacements. In the case of a simply supported thin-walled beam, a closed-form solution for the natural frequencies of free harmonic vibrations was derived. The significance of neglecting cross-sectional warping and rotary inertia on the accuracy of results was analyzed. A recent paper on the same subject is discussed, with a critical review of it. [Copyright &y& Elsevier]

In this paper, a non-linear strategy, based on the centre manifold, the rational approximants and the alternating frequency/time domain method has been developed, in order to study the non-linear dynamical behaviour of a system in the neighbourhood of a critical steady state equilibrium point. The stability analysis and the non-linear dynamics of a complex braking system with a non-linear rotor/stator contact are presented. Moreover, one of the most important steps of this paper is the determination of the non-linear behaviour and the limit cycle amplitudes of this complex system. In order to conduct this study, the dynamic response is evaluated by using applying the centre manifold, the rational approximants and the alternating frequency/time domain method, that permit to obtain rapidly and efficiently the non-linear behaviour of the system. The dynamic response obtained by applying this method is compared with that evaluated through numerical integration. [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 investigation, a new formulation for the wheel/rail contact problem based on the elastic force approach is presented. Crucial to the success of any elastic force formulation for the wheel/rail contact problem is the accurate prediction of the location of the contact points. To this end, features of multibody formulations that allow introducing additional differential equations are exploited in this investigation in order to obtain a good estimate of the rail arc length travelled by the wheel set. In the formulation presented in this paper, four parameters are used to describe the wheel and the rail surfaces. In order to determine the location of the points of contact between the wheel and the rail, a first order differential equation for the rail arc length is introduced and is integrated simultaneously with the multibody equations of motion of the wheel/rail system. The method presented in this paper allows for multiple points of contact between the wheel and the rail by using an optimized search for all possible contact points. The normal contact forces are calculated and used with non-linear expressions for the creepages to determine the creep forces. The paper also discusses two different procedures for the analysis of the two-point contact in the wheel/rail interaction. Numerical results obtained using the elastic force model are presented and compared with the results obtained using the constraint approach. [Copyright &y& Elsevier]

A previously developed perturbation method is generalized for computing the simplest normal form (at each level of computation, the minimum number of terms are retained) of general n-dimensional differential equations. This “direct” approach, combining the normal form theory with center manifold theory in one unified procedure, can be used to systematically compute the simplest (or unique) normal form. Two particular singularities of the Jacobian of the system are considered in this paper: the first one is associated with one pair of purely imaginary eigenvalues (Hopf-type singularity), and the other corresponds to a simple zero and a pair of purely imaginary eigenvalues (Hopf-zero-type singularity). The approach can be easily formulated and implemented using a computer algebra system. Maple programs have been developed in this paper which can be “automatically” executed by a user without the knowledge of computer algebra. A physical oscillator model is studied in detail to show the computational efficiency of the “direct” method, and the advantage of using the simplest normal form, which greatly simplifies the analysis on dynamical systems, in particular, for bifurcations and stability. [Copyright &y& Elsevier]

This paper considers an accepted model of the metal cutting process dynamics in the context of an approximate analysis of the resulting non-linear differential equations of motion. The process model is based upon the established mechanics of orthogonal cutting and results in a pair of non-linear ordinary differential equations which are then restated in a form suitable for approximate analytical solution. The chosen solution technique is the perturbation method of multiple time scales and approximate closed-form solutions are generated for the most important non-resonant case. Numerical data are then substituted into the analytical solutions and key results are obtained and presented. Some comparisons between the exact numerical calculations for the forces involved and their reduced and simplified analytical counterparts are given. It is shown that there is almost no discernible difference between the two thus confirming the validity of the excitation functions adopted in the analysis for the data sets used, these being chosen to represent a real orthogonal cutting process. In an attempt to provide guidance for the selection of technological parameters for the avoidance of primary chatter, this paper determines for the first time the stability regions in terms of the depth of cut and the cutting speed co-ordinates. [Copyright &y& Elsevier]

This paper investigates moment stability of the plate with viscoelasticity, and determinates the expressions of p th moment Lyapunov exponents, for a finite value of p. Although, both are the significant indexes to study stochastic stability, in comparison with maximum Lyapunov exponent, moment Lyapunov exponent is more considerable and is more difficult to evaluate. In this paper, we take a non-Gaussian colored noise as the excitation of the system. Applying stochastic averaging method and the technique proposed by Larionov, the coupled equations governing the motion of the system are converted to the Itô differential equations, through which the p th moment Lyapunov exponents are conveniently evaluated, and then, the largest Lyapunov exponents are given by its relationship with the moment Lyapunov exponents. The stability boundaries, which indicate the transition state, and stability indexes of the system, which represents the probability for an almost surely stable system exceeds a threshold, are then discussed in detail. Via the numerical simulation, it is shown that the analytical expressions of moment Lyapunov exponents agree well with numerical results. At last, the effects of both noise parameter and system ones on moment Lyapunov exponents are studied, from which quantitative analysis for the practical systems with viscoelasticity can be supplied. [ABSTRACT FROM AUTHOR]

The transverse vibrations of axially moving strings with time dependent velocity and its instability are examined in this paper. The velocity function is assumed to be a harmonically varying function about a constant low mean speed. Approximate solutions are sought using the two timescales perturbation method in conjunction with the method of characteristic coordinates. Moreover, it will turn out that infinitely many resonances in the system can arise, when the velocity fluctuation frequency is equal (or close) to an odd multiple of the lowest natural frequency of the constant velocity system. The dynamic displacement-response and the energy of the belt system obtained using the method of characteristic coordinates are compared numerically. It is shown that the energy of the belt system can become large in a resonance case. In addition, the amplitude of vibration is also found to be growing on long timescales, that is, on a timescale of order ε −1 . [ABSTRACT FROM AUTHOR]

Many wave propagation problems (in acoustics or in railway catenary or cable car dynamics, for example) can be solved with high efficiency if the computational domain can be truncated to a small region of interest with appropriate absorbing boundary conditions. In this paper, highly absorbing and stable boundary conditions for linear partial differential equations discretized by finite difference schemes are directly designed using a flexible, optimization-based formulation. The proposed optimization approach to the computation of the absorbing boundary conditions is capable of optimizing the accuracy (the absorbing quality of the boundary condition) while guaranteeing stability of the discretized partial differential equations with the absorbing boundary conditions in place. Penalty functions are proposed that explicitly quantify errors introduced by the boundary condition on the solution of the bounded domain compared to the solution of the unbounded domain problem. Together with the stability condition the described approach can be applied on various types of linear partial differential equations and is thus applicable for generic wave propagation problems. Its flexibility and efficiency is demonstrated for two engineering problems: The Euler–Bernoulli beam under axial load, which can be used to model cables as well as catenary flexural dynamics, and a two-dimensional wave as commonly encountered in acoustics. The accuracy of the absorbing boundary conditions obtained by the proposed concept is compared to analytical absorbing boundary conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a method for reducing the residual vibration of a flexible beam deployed from a translating hub. Whereas previous studies have discussed reducing vibration in translating constant-length beams, this study investigates a vibration reduction method for translating beams of variable length. The partial differential equation of motion for a translating beam is derived and transformed into a variational equation. Based on the discretized equations from the variational equation, the dynamic responses of the flexible beam under translation are analyzed. A vibration reduction method is proposed that is effective for both constant- and variable-length deploying translating beams. [Copyright &y& Elsevier]

• On the stability boundary the mass-beam-foundation system is in the form of a steady state periodic motion. • The infinite beam is divided into three segments, in each of which the differential equation is solved analytically. • The point mass model overestimates the stability of the mass-beam-foundation system. The stability of a distributed mass of finite length travelling uniformly on an infinite beam resting on a visco-elastic foundation is considered in this paper. The focus is placed on the effects of the length of the distributed mass on the stability of the mass-beam-foundation system. It is argued that on the stability boundary, the mass-beam-foundation system is in the form of a steady state periodic motion. To seek the periodic solution, the infinite beam is divided into three segments, in each of which the differential equation is solved analytically. By comparing the stability boundary curves of the distributed mass of finite length with the one of infinite length, it is found that the stability boundary curves of finite length do not converge to the one of infinite length. It is believed that the course of the failure to converge towards the infinite length model is the differences in the boundary conditions at infinity. On the other hand, the stability boundary curves of finite length do converge to the point mass model when the mass length approaches zero. Compared to the point mass model, the distributed mass model is less stable. More precisely, for a specified system damping and total mass, the critical speed is in general much smaller if distributed model is adopted. The longer the mass length, the smaller the critical speed is. In short, the conventional point mass model overestimates the stability of the mass-beam-foundation system. [ABSTRACT FROM AUTHOR]

Abstract: Vibration suppression and chaotification are the key issues in the study of the concealment capability of underwater vehicles. Time delay control is superior in chaotification, but the involved dynamics are sensitive and complex. This paper presents an analysis to obtain an analytical solution of the nonlinear delay differential equations and determine the effect of delay control on the vibration amplitude. Besides, by checking the stability of the analytical solution, dependence of chaotification upon the time delay control parameters is examined. Based on the theoretical derivation, the effects of different configurations of system parameters and delay control parameters on vibration amplitude are demonstrated in numerical simulation. What׳s more, the outcome of our results shows the significant role the time delay control plays in vibration suppression and chaotification. According to the analytical solution and stability analysis, not only can the appropriate delay be found to reduce the vibration amplitude, but also the suitable delay control setting can be selected for chaotification. [Copyright &y& Elsevier]

Abstract: This paper introduces a formalism which extends that of “Green's function” and that of “the Volterra series”. These formalisms are typically used to solve, respectively, linear inhomogeneous space–time differential equations in physics and weakly nonlinear time-differential input-to-output systems in automatic control. While Green's function is a space–time integral kernel which fully characterizes a linear problem, the Volterra series expansions involve a sequence of multi-variate time integral kernels (of convolution type for time-invariant systems). The extension proposed here consists in combining the two approaches, by introducing a series expansion based on multi-variate space–time integral kernels. This series allows the representation of the space–time solution of weakly nonlinear boundary problems excited by an “input” which depends on space and time. This formalism is introduced on and applied to a nonlinear model of a damped string that is excited by a transverse mass force . The Green–Volterra kernels that solve the transverse displacement dynamics are computed. The first-order kernel exactly corresponds to Green's function of the linearized problem. The higher order kernels satisfy a sequence of linear boundary problems that lead to (both) analytic closed-form solutions and modal decompositions. These results lead to an efficient simulation structure, which proves to be as simple as the one based on the Volterra series, that has been obtained in a previous work for excitation forces with separated variables . Numerical results are presented. [Copyright &y& Elsevier]

Abstract: Energy Flow Analysis (EFA) has been developed to predict the vibrational energy density of the system structures in the medium-to-high frequency range. The elementary longitudinal wave theory is often used to describe the longitudinal vibration of a slender rod. However, for relatively large diameter rods or high frequency ranges, the elementary longitudinal wave theory is inaccurate because the lateral motions are not taken into account. In this paper, vibrational energy flow models are developed to analyze the longitudinally vibrating Rayleigh–Love rod considering the effect of lateral inertia, and the Rayleigh–Bishop rod considering the effect not only of the lateral inertia but also of the shear stiffness. The derived energy governing equations are second-order differential equations which predict the time and space averaged energy density and active intensity distributions in a rod. To verify the accuracy of the developed energy flow models, various numerical analyses are performed for a rod and coupled rods. Also, the EFA results for the Rayleigh–Love and Rayleigh–Bishop rods are compared with the analytical solutions for these models, the traditional energy flow solutions, and the analytical solutions for the classical rod. [Copyright &y& Elsevier]