Showing papers in "Journal of Sound and Vibration in 2009"
TL;DR: In this article, the authors investigated the design and analysis of a novel energy harvesting device that uses magnetic levitation to produce an oscillator with a tunable resonance and derived the governing equations for the mechanical and electrical domains to show the designed system reduces to the form of a Duffing oscillator under both static and dynamic loads.
Abstract: This paper investigates the design and analysis of a novel energy harvesting device that uses magnetic levitation to produce an oscillator with a tunable resonance. The governing equations for the mechanical and electrical domains are derived to show the designed system reduces to the form of a Duffing oscillator under both static and dynamic loads. Thus, nonlinear analyses are required to investigate the energy harvesting potential of this prototypical nonlinear system. Theoretical investigations are followed by a series of experimental tests that validate the response predictions. The motivating hypothesis for the current work was that nonlinear phenomenon could be exploited to improve the effectiveness of energy harvesting devices.
TL;DR: In this paper, an edge-based smoothed finite element method (ES-FEM) was proposed to improve the accuracy of the FEM without much changing to the standard FEM settings.
Abstract: This paper presents an edge-based smoothed finite element method (ES-FEM) to significantly improve the accuracy of the finite element method (FEM) without much changing to the standard FEM settings. The ES-FEM can use different shape of elements but prefers triangular elements that can be much easily generated automatically for complicated domains. In the ES-FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the edges of the triangles. Intensive numerical results demonstrated that the ES-FEM possesses the following excellent properties: (1) the ES-FEM model possesses a close-to-exact stiffness: it is much softer than the “overly-stiff” FEM and much stiffer than the “overly-soft” NS-FEM model; (2) the results are often found superconvergence and ultra-accurate: much more accurate than the linear triangular elements of FEM and even more accurate than those of the FEM using quadrilateral elements with the same sets of nodes; (3) there are no spurious non-zeros energy modes found and hence the method is also temporally stable and works well for vibration analysis and (4) the implementation of the method is straightforward and no penalty parameter is used, and the computational efficiency is better than the FEM using the same sets of nodes. In addition, a novel domain-based selective scheme is proposed leading to a combined ES/NS-FEM model that is immune from volumetric locking and hence works very well for nearly incompressible materials. These properties of the ES-FEM are confirmed using examples of static, free and forced vibration analyses of solids.
TL;DR: In this article, the third-order shear deformation plate theory of Reddy is reformulated using the nonlocal linear elasticity theory of Eringen, which has ability to capture the both small scale effects and quadratic variation of shear strain and consequently shear stress through the plate thickness.
Abstract: The third-order shear deformation plate theory of Reddy [A simple higher-order theory for laminated composite plates, J. Appl. Mech . 51 (1984) 745–752] is reformulated using the nonlocal linear elasticity theory of Eringen. This theory has ability to capture the both small scale effects and quadratic variation of shear strain and consequently shear stress through the plate thickness. Analytical solutions of bending and free vibration of a simply supported rectangular plate are presented using this theory to illustrate the effect of nonlocal theory on deflection and natural frequency of the plates. Finally, the relations between nonlocal third-order, first-order and classical theories are discussed by numerical results.
TL;DR: In this paper, the nonlocal differential constitutive relations of Eringen have been used to solve the governing equations for simply supported boundary conditions for the analysis of double layered nanoplates.
Abstract: Classical plate theory (CLPT) and first-order shear deformation theory (FSDT) of plates are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. Navier's approach has been used to solve the governing equations for simply supported boundary conditions. Analytical solutions for vibration of the nanoplates such as graphene sheets are presented. Nonlocal theories are employed to bring out the effect of the nonlocal parameter on natural frequencies of the nanoplates. The developed theory has been extended to the analysis of double layered nanoplates. Effect of (i) nonlocal parameter, (ii) length, (iii) height, (iv) elastic modulus and (v) stiffness of Winkler foundation of the plate on nondimensional vibration frequencies are investigated. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories of nanoplates and nanoshells.
TL;DR: A free vibration analysis of metal and ceramic functionally graded plates that uses the element-free kp-Ritz method is presented in this paper, where the material properties of the plates are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents.
Abstract: A free vibration analysis of metal and ceramic functionally graded plates that uses the element-free kp-Ritz method is presented. The material properties of the plates are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents. The first-order shear deformation plate theory is employed to account for the transverse shear strain and rotary inertia, and mesh-free kernel particle functions are used to approximate the two-dimensional displacement fields. The eigen-equation is obtained by applying the Ritz procedure to the energy functional of the system. Convergence studies are performed to examine the stability of the proposed method, and comparisons of the solutions derived with those reported in the literature are provided to verify its accuracy. Four types of functionally graded rectangular and skew plates—Al/Al2O3, Al/ZrO2, Ti–6Al–4V/Aluminum oxide, and SUS304/Si3N4—are included in the study, and the effects of the volume fraction, boundary conditions, and length-to-thickness ratio on their frequency characteristics are discussed in detail.
TL;DR: In this article, the authors derived the force transmissibility of a quasi-zero-stiffness isolator, which consists of a vertical spring and two oblique springs that are either linear, linear with pre-stress or softening nonlinear with prestress.
Abstract: In this article the force transmissibility of a quasi-zero-stiffness (QZS) isolator is considered. The isolator comprises a vertical spring and two oblique springs that are either linear, linear with pre-stress or softening nonlinear with pre-stress. The force transmissibility of such a system is derived and compared with that of a linear system. Assuming light damping, simple approximate expressions for the maximum transmissibility and jump-down frequencies are derived. It is shown that there are advantages in having nonlinear and pre-stressed oblique springs compared to the other configurations, and that all the QZS systems can outperform the linear system provided that the system parameters are chosen appropriately.
TL;DR: This paper is the most comprehensive review published to date, of 270 references dealing with different experimental and analytical characterizations of human walking loading, and introduces methods for indirect measurement of time-varying records of walking forces via combination of visual motion tracking (imaging) data and known body mass distribution.
Abstract: Dynamic forces induced by humans walking change simultaneously in time and space, being random in nature and varying considerably not only between different people but also for a single individual who cannot repeat two identical steps. Since these important aspects of walking forces have not been adequately researched in the past, the corresponding lack of knowledge has reflected badly on the quality of their mathematical models used in vibration assessments of pedestrian structures such as footbridges, staircases and floors. To develop better force models which can be used with more confidence in the structural design, an adequate experimental and analytical approach must be taken to account for their complexity. This paper is the most comprehensive review published to date, of 270 references dealing with different experimental and analytical characterizations of human walking loading. The source of dynamic human-induced forces is in fact in the body motion. To date, human motion has attracted a lot of interest in many scientific branches, particularly in medical and sports science, bioengineering, robotics, and space flight programs. Other fields include biologists of various kinds, physiologists, anthropologists, computer scientists (graphics and animation), human factors and ergonomists, etc. It resulted in technologically advanced tools that can help understanding the human movement in more detail. Therefore, in addition to traditional direct force measurements utilizing a force plate and an instrumented treadmill, this review also introduces methods for indirect measurement of time-varying records of walking forces via combination of visual motion tracking (imaging) data and known body mass distribution. The review is therefore an interdisciplinary article that bridges the gaps between biomechanics of human gait and civil engineering dynamics. Finally, the key reason for undertaking this review is the fact that human–structure dynamic interaction and pedestrian synchronization when walking on more or less perceptibly moving structures are increasingly giving serious cause for concern in vibration serviceability design. There is a considerable uncertainty about how excessive structural vibrations modify walking and hence affect pedestrian-induced forces, significantly in many cases. Modelling of this delicate mechanism is one of the challenges that the international civil structural engineering community face nowadays and this review thus provides a step toward understanding better the problem.
TL;DR: In this paper, a simplified mathematical model is proposed to describe the mechanisms leading to modulation sidebands of planetary gear sets, which includes key system parameters such as number of planets, planet position angles, and planet phasing relationships defined by the position angles and the number of teeth of the gears.
Abstract: In this paper, a simplified mathematical model is proposed to describe the mechanisms leading to modulation sidebands of planetary gear sets. The model includes key system parameters such as number of planets, planet position angles, and planet phasing relationships defined by the position angles and the number of teeth of the gears. The model is used to simulate a wide range of gear sets to show that they can be classified in five distinct groups based on their sideband behavior in terms of their frequencies and amplitudes. A special experimental planetary gear set-up is developed and planetary gear sets from of three of these five groups are procured. A methodology is developed to demonstrate modulation sidebands from the ring (internal) gear radial acceleration measurements. For each case, sets of ring gear acceleration measurements at various speed and torque conditions are presented to demonstrate rich sideband activity that agrees well with the model predictions. At the end, based on results of the parametric studies and experiments, general rules on modulation sidebands of planetary gear sets are proposed.
TL;DR: In this paper, the dynamics of conical, cylindrical shells and annular plates were analyzed using the first-order shear deformation theory (FSDT) and the generalized differential quadrature (GDQ) method.
Abstract: This paper focuses on the dynamic behavior of functionally graded conical, cylindrical shells and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The first-order shear deformation theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. The homogeneous isotropic material is inferred as a special case of functionally graded materials (FGM). The governing equations of motion, expressed as functions of five kinematic parameters, are discretized by means of the generalized differential quadrature (GDQ) method. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. For the homogeneous isotropic special case, numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Nastran, Straus, Pro/Mechanica. Very good agreement is observed. Furthermore, the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. Different typologies of non-uniform grid point distributions are considered. Finally, for the functionally graded material case numerical results illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behavior of shell structures.
TL;DR: In this article, the authors studied the effect of damping on power optimality of a piezoelectric vibration-based energy harvester, which utilizes a harvesting circuit employing an inductor and a resistive load.
Abstract: The optimization of power acquired from a piezoelectric vibration-based energy harvester which utilizes a harvesting circuit employing an inductor and a resistive load is described. The optimization problem is formulated as a nonlinear program wherein the Karush–Kuhn–Tucker (KKT) conditions are stated and the resulting cases are treated. In the first part of the manuscript, the case of a purely resistive circuit is analyzed. While this configuration has received considerable attention in the literature, previous efforts have neglected the effect of damping on the optimal parameters. Here, we explore the impact of damping on power optimality and illustrate its quantitative and qualitative effects. Further, we analyze the effect of electromechanical coupling demonstrating that the harvested power decreases beyond an optimal coupling coefficient. This result challenges previous literature suggesting that higher coupling coefficients always culminate in more efficient energy harvesters. In the second part of this work, the effect of adding an inductor to the circuit is examined. It is demonstrated that the addition of the inductor provides substantial improvement to the performance of the energy harvesting device. It is also shown that within realistic values of the coupling coefficient, the optimal harvested power is independent of the coupling coefficient; a result that supports previous findings for the purely resistive circuit.
TL;DR: In this article, an electromechanically coupled finite element (FE) plate model is presented for predicting the electrical power output of piezoelectric energy harvester plates.
Abstract: Vibration-based energy harvesting has been investigated by several researchers over the last decade. The goal in this research field is to power small electronic components by converting the waste vibration energy available in their environment into electrical energy. Recent literature shows that piezoelectric transduction has received the most attention for vibration-to-electricity conversion. In practice, cantilevered beams and plates with piezoceramic layers are employed as piezoelectric energy harvesters. The existing piezoelectric energy harvester models are beam-type lumped parameter, approximate distributed parameter and analytical distributed parameter solutions. However, aspect ratios of piezoelectric energy harvesters in several cases are plate-like and predicting the power output to general (symmetric and asymmetric) excitations requires a plate-type formulation which has not been covered in the energy harvesting literature. In this paper, an electromechanically coupled finite element (FE) plate model is presented for predicting the electrical power output of piezoelectric energy harvester plates. Generalized Hamilton's principle for electroelastic bodies is reviewed and the FE model is derived based on the Kirchhoff plate assumptions as typical piezoelectric energy harvesters are thin structures. Presence of conductive electrodes is taken into account in the FE model. The predictions of the FE model are verified against the analytical solution for a unimorph cantilever and then against the experimental and analytical results of a bimorph cantilever with a tip mass reported in the literature. Finally, an optimization problem is solved where the aluminum wing spar of an unmanned air vehicle (UAV) is modified to obtain a generator spar by embedding piezoceramics for the maximum electrical power without exceeding a prescribed mass addition limit.
TL;DR: In this paper, a solution strategy is presented that allows for the evaluation of the second-order statistics of the response due to dynamic excitation based on the power spectral density function of the track unevenness.
Abstract: In predictions of railway-induced vibrations, a distinction is generally made between the quasi-static and dynamic excitation. The quasi-static excitation is related to the static component of the axle loads. The dynamic excitation is due to dynamic train–track interaction, which is generated by a large number of excitation mechanisms, such as the spatial variation of the support stiffness and the wheel and track unevenness. In the present paper, the quasi-static excitation and the dynamic excitation due to random track unevenness are evaluated by means of numerical predictions. A solution strategy is presented that allows for the evaluation of the second-order statistics of the response due to dynamic excitation based on the power spectral density function of the track unevenness. Due to the motion of the train, the second-order statistics of the response at a fixed point in the free field are non-stationary and an appropriate solution procedure is required. The quasi-static and dynamic contribution to the track and free-field response are analysed for the case of InterCity and high-speed trains running at a subcritical train speed. It is shown how the train speed affects the quasi-static and dynamic contribution. Finally, results of numerical predictions for different train speeds are compared with field measurements that have been performed at a site along the high-speed line L2 Brussels–Koln within the frame of homologation tests.
TL;DR: In this paper, a new concept of energy-harvesting, the flutter-mill, is proposed in which these flutter motions are utilized to generate electrical power, based on the energy analysis of the fluid-structure interaction system.
Abstract: Cantilevered flexible plates in axial flow lose stability at sufficiently high flow velocity. Once the instability threshold is exceeded, flutter takes place, and energy is continuously pumped into the plate from the surrounding fluid flow, sustaining the flutter motion. This kind of self-induced, self-sustained vibration can be utilized to extract energy from the fluid flow. This paper studies the energy transfer between the fluid flow and the plate. Then, based on the energy analysis of the fluid–structure interaction system, a new concept of energy-harvesting, the flutter-mill, is proposed in which these flutter motions are utilized to generate electrical power.
TL;DR: In this article, an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations is presented. But the main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the harmonic balance method.
Abstract: Combining the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time-domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called asymptotic numerical method (ANM) continuation technique. Several classical examples are presented to illustrate the use of this numerical approach.
TL;DR: In this article, a non-dimensional analysis of the magnetic support is considered and it is shown analytically that for cubical magnets the ratio of force to displacement is directly proportional to face area.
Abstract: This paper presents an analysis of a magnetic levitation system for vibration isolation. A non-dimensional analysis of the magnetic support is considered and it is shown analytically that for cubical magnets the ratio of force to displacement is directly proportional to face area. The arrangement of magnets examined uses a negative stiffness element to reduce the natural frequency of the suspension. Design criteria are imposed on the system to satisfy mass loading, bandwidth of the required isolation, expected magnitude of the vibration disturbance and required robustness of the system. The vibration response of a system designed to satisfy these requirements is compared to an equivalent linear system and is shown to become increasingly nonlinear as the system moves towards instability.
TL;DR: In this article, a model based technique for fault diagnosis of rotor bearing system is described using residual generation technique, residual vibrations are generated from experimental results for the rotor bearing systems subject to misalignment and unbalance, and then the residual forces due to presence of faults are calculated.
Abstract: Vibration monitoring is one of the primary techniques of condition monitoring of rotating machines. Shaft misalignment and rotor unbalance are the main sources of vibration in rotating machines. In this study a model based technique for fault diagnosis of rotor–bearing system is described. Using the residual generation technique, residual vibrations are generated from experimental results for the rotor bearing system subject to misalignment and unbalance, and then the residual forces due to presence of faults are calculated. These residual forces are compared with the equivalent theoretical forces due to faults. The fault condition and location of faults are successfully detected by this model based technique.
TL;DR: In this paper, the displacement solution is expressed as a two-dimensional Fourier series supplemented with several one-dimensional series, which is capable of representing any function (including the exact displacement solution) whose third-order partial derivatives are (required to be) continuous over the area of the plate.
Abstract: An analytical method is developed for the vibration analysis of rectangular plates with elastically restrained edges The displacement solution is expressed as a two-dimensional Fourier series supplemented with several one-dimensional Fourier series Mathematically, such a series expansion is capable of representing any function (including the exact displacement solution) whose third-order partial derivatives are (required to be) continuous over the area of the plate Since the discontinuities (or jumps) potentially related to the partial derivatives at the edges (when they are periodically extended onto the entire x – y plane as implied by a two-dimensional Fourier series expansion) have been explicitly “absorbed” by the supplementary terms, all the series expansions for up to the fourth-order derivatives can be directly obtained through term-by-term differentiations of the displacement series Thus, an exact solution can be obtained by letting the series simultaneously satisfy the governing differential equation and the boundary conditions on a point-wise basis Because the series solution has to be truncated numerically, the “exact solution” should be understood as a solution with arbitrary precision Several numerical examples are presented to illustrate the excellent accuracy of the current solution The proposed method can be directly extended to other more complicated boundary conditions involving non-uniform elastic restraints, point supports, partial supports, and their combinations
TL;DR: In this paper, the authors presented a thermal-mechanical vibration analysis of functionally graded (FG) beams and functionally graded sandwich (FGSW) beams using modified differential quadrature method (MDQM) and modified weighting coefficient matrix (MWCM).
Abstract: Thermo-mechanical vibration analysis of functionally graded (FG) beams and functionally graded sandwich (FGSW) beams are presented. The functionally graded material (FGM) beams are considered to be resting on variable (i) Winkler foundation and (ii) two-parameter elastic foundation. The material properties of these beams are assumed to be varying in the thickness direction. The governing differential equations for beam vibration are being solved using the modified differential quadrature method (MDQM). The applied kinematic boundary conditions are implemented using the modified weighting coefficient matrix (MWCM). The weighting coefficients are generated from the Chebyshev basis function. Present results for the vibration of isotropic beam with variable Winkler foundation are in good agreement with those reported in the literature. Parametric study on the vibration response of FG beams and FGSW beams are being investigated. These parameters include (i) temperature distributions, (ii) power-law index, (iii) variable Winkler foundation modulus, (iv) two-parameter elastic foundation modulus and (v) normalized core thickness of FGSW beams.
TL;DR: In this article, the authors proposed an indirect approach for extracting bridge frequencies from a passing vehicle, which works mainly for the first frequency and can be used to extract bridge frequencies of higher modes.
Abstract: The indirect approach previously proposed for extracting the bridge frequencies from a passing vehicle works mainly for the first frequency In order to extract bridge frequencies of higher modes, the vehicle response will first be processed by the empirical mode decomposition (EMD) to generate the intrinsic mode functions (IMFs), and then by the fast Fourier transform One feature with the EMD technique is that frequencies of higher modes can be made more visible by the sifting process To verify the feasibility of such an approach, the vehicle response generated by the finite element simulations will first be analyzed, with results compared with the analytical ones Then the same procedure will be adopted to extract bridge frequencies from the recorded response of a passing vehicle, with results compared with those from an ambient vibration test It was demonstrated that using the IMFs computed from the vehicle response, rather than based on the original data, bridge frequencies of higher modes can be successfully extracted Future directions of research are highlighted
TL;DR: In this paper, the authors studied the nonlinear vibration of beams made of functionally graded materials (FGMs) containing an open edge crack based on Timoshenko beam theory and von Karman geometric nonlinearity.
Abstract: Nonlinear vibration of beams made of functionally graded materials (FGMs) containing an open edge crack is studied in this paper based on Timoshenko beam theory and von Karman geometric nonlinearity. The cracked section is modeled by a massless elastic rotational spring. It is assumed that material properties follow exponential distributions through beam thickness. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of cracked FGM beams with different end supports. A detailed parametric study is conducted to study the influences of crack depth, crack location, material property gradient, slenderness ratio, and end supports on the nonlinear free vibration characteristics of cracked FGM beams. It is found that unlike isotropic homogeneous beams, both intact and cracked FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of bending-extension coupling in FGM beams.
TL;DR: In this article, an analytical model is proposed to study the nonlinear dynamic behavior of rolling element bearing systems including surface defects, and the results were obtained in the form of time series, frequency responses and phase trajectories.
Abstract: In this paper an analytical model is proposed to study the nonlinear dynamic behavior of rolling element bearing systems including surface defects. Various surface defects due to local imperfections on raceways and rolling elements are introduced to the proposed model. The contact force of each rolling element described according to nonlinear Hertzian contact deformation and the effect of internal radial clearance has been taken into account. Mathematical expressions were derived for inner race, outer race and rolling element local defects. To overcome the strong nonlinearity of the governing equations of motion, a modified Newmark time integration technique was used to solve the equations of motion numerically. The results were obtained in the form of time series, frequency responses and phase trajectories. The validity of the proposed model verified by comparison of frequency components of the system response with those obtained from experiments. The classical Floquet theory has been applied to the proposed model to investigate the linear stability of the defective bearing rotor systems as the parameters of the system changes. The peak-to-peak frequency response of the system for each case is obtained and the basic routes to periodic, quasi-periodic and chaotic motions for different internal radial clearances are determined. The current study provides a powerful tool for design and health monitoring of machine systems.
TL;DR: In this paper, a tuned passive control is used to damp unstable combustion systems, with particular emphasis on systems which exhibit multiple unstable modes, and two algorithms are developed, one for identifying the characteristics of all modes present in real time, and another for tuning the neck areas of the Helmholtz resonators.
Abstract: In this work, tuned passive control is used to damp unstable combustion systems, with particular emphasis on systems which exhibit multiple unstable modes. Helmholtz resonators are used as passive dampers. The frequency at which they offer maximum damping is varied by altering their geometry; in this work, geometry changes are achieved by varying the area of the Helmholtz resonator neck. For each unstable mode exhibited by the combustion system, a separate Helmholtz resonator has its neck area tuned. Two algorithms are developed, one for identifying the characteristics of all modes present in real time, and another for tuning the neck areas of the Helmholtz resonators. These algorithms are successfully implemented in numerical simulations of a longitudinal combustor exhibiting two unstable modes. The algorithms result in both modes being stabilised as long as two Helmholtz resonators are used. Experiments are then conducted on a Rijke tube with its upper part split into two branches of differing lengths, shaped like a ‘Y’. The differing lengths give rise to two unstable modes at different frequencies. A Helmholtz resonator is attached to each branch; the neck area of both can be varied by means of an ‘iris’ valve, which opens and closes like a camera lens. On implementing the procedure for tuning the neck areas, both unstable modes are stabilised, and stability is maintained for large changes in operating condition. This confirms that the procedure developed is sufficiently robust for use in real combustion systems exhibiting multiple unstable modes.
TL;DR: The wave and finite element (WFE) method is a numerical approach to the calculation of the wave properties of structures of arbitrary complexity as discussed by the authors, which is prone to numerical difficulties.
Abstract: The wave and finite element (WFE) method is a numerical approach to the calculation of the wave properties of structures of arbitrary complexity. The method starts from a finite element (FE) model of only a short segment of the structure, typically by using existing element libraries and commercial FE packages. The dynamic stiffness matrix of the segment is obtained, a periodicity condition applied and an eigenvalue problem formed whose solution gives the dispersion equations and wave mode shapes. These define a wave basis from which the forced response can be found straightforwardly. Although straightforward in application, the WFE method is prone to numerical difficulties. These are discussed in this paper and methods to avoid or remove them described. Attention is focused on 1-dimensional waveguide structures, for which numerical problems are most severe. Three ways of phrasing the eigenvalue problem for free wave propagation are presented and a method based on singular value decomposition is proposed to determine eigenvectors at low frequencies. Discretisation errors are seen to occur if the segment is too large, while round-off errors occur if the segment is too small. This can be overcome by forming a super-segment from the concatenation of two or more segments. The forced response is then considered. The use of a reduced wave basis removes many problems. Direct calculation of the waves excited by a point force is very prone to poor numerical conditioning but can be circumvented by exploiting the orthogonality of the left and right eigenvectors. Numerical examples are presented.
TL;DR: In this article, an analytical technique for the treatment of certain class of nonlinear damping functions is developed, and the case of piecewise-quadratic damping is investigated in a system comprising of a linear oscillator and a nonlinear energy sink.
Abstract: In this work, response regimes are investigated in a system comprising of a linear oscillator (subject to harmonic excitation) and a nonlinear energy sink (NES) with nonlinear damping characteristics. An analytical technique for the treatment of certain class of nonlinear damping functions is developed. Special attention is paid to the case of piecewise-quadratic damping, motivated by possible applications. It is demonstrated that the NES with a properly tuned piecewise-quadratic damping element allows complete elimination of undesirable periodic regimes. In this way, an efficient system of vibration absorption is obtained, and its performance can overcome that of a tuned mass damper (TMD). Numerical results agree satisfactorily with the analytical predictions.
TL;DR: In this paper, exact closed-form expressions for the vibration modes of the Euler-Bernoulli beam in the presence of multiple concentrated cracks are presented explicitly as functions of four integration constants only, to be determined by the standard boundary conditions.
Abstract: In this study, exact closed-form expressions for the vibration modes of the Euler–Bernoulli beam in the presence of multiple concentrated cracks are presented. The proposed expressions are provided explicitly as functions of four integration constants only, to be determined by the standard boundary conditions. The enforcement of the boundary conditions leads to explicit expressions of the natural frequency equations. Besides the evaluation of the natural frequencies, neither computational work nor recurrence expressions for the vibration modes are required. The cracks, that are not subjected to the closing phenomenon, are modelled as a sequence of Dirac's delta generalised functions in the flexural stiffness. The Eigen-mode governing equation is formulated over the entire domain of the beam without enforcement of any continuity conditions, which are already accounted for in the adopted flexural stiffness model. The vibration modes of beams with different numbers of cracks under different boundary conditions have been analysed by means of the proposed closed-form expressions in order to show their efficiency.
TL;DR: In this paper, the entropy noise mechanism was investigated both experimentally and numerically on a generic test case, where electrical heating was used to generate non-isentropic perturbations in a spatially varying average flow field set by the geometrical boundary conditions of an axisymmetric convergent-divergent nozzle.
Abstract: The entropy noise mechanism was investigated both experimentally and numerically on a generic test case. The model experiment features electrical heating to generate non-isentropic perturbations in a spatially varying average flow field. This flow field is set by the geometrical boundary conditions of an axisymmetric convergent–divergent nozzle in an otherwise straight tube. The considered flow conditions range from low subsonic to a transonic chocked base flow. The general response of the system to an abrupt heating pulse is studied experimentally and numerically. Furthermore, for two specific cases a detailed investigation of the pressure response in the outlet section is provided. Comprehensive experimental data are provided for the validation of numerical methods with respect to entropy noise. The numerical investigations use a commercial available computational fluid dynamics (CFD) method with partially and non-reflective boundary conditions for unsteady compressible simulations on one hand and a high-order CAA method with a time-domain impedance model on the other hand. It is found that the determination of reflections from the downstream and/or upstream open ends of the test configuration are necessary for the correct prediction of the experiment. The results of both methods are analyzed for the presence of acoustic sources considering the source term of an acoustic analogy and the acoustic intensity, respectively. Strong sources are found in the convergent/divergent nozzle by both methods. These sources show a much larger source strength than the direct sources due to the unsteady heat input. A saturation of the peak pressure response with a increasing Mach number in the nozzle throat above 0.8 is attributed to a phase shift of the source contributions between nozzle and diffuser. The presented results enable a deep understanding of the entropy noise phenomenon especially due to the combination of experiments and two fairly different numerical approaches. However, even in spite of the simplified model case the investigated entropy noise mechanism still appears in a comprehensive complexity. Therefore, and because of the increasing relevance as an aero-engine noise source, further research on entropy noise, also under application of the presented reference cases, should be performed.
TL;DR: In this paper, the authors developed models for predicting particle dampers (PDs) behavior using the Discrete Element Method (DEM), where individual particles are typically represented as elements with mass and rotational inertia.
Abstract: This paper presents initial work on developing models for predicting particle dampers (PDs) behaviour using the Discrete Element Method (DEM). In the DEM approach, individual particles are typically represented as elements with mass and rotational inertia. Contacts between particles and with walls are represented using springs, dampers and sliding friction interfaces. In order to use DEM to predict damper behaviour adequately, it is important to identify representative models of the contact conditions. It is particularly important to get the appropriate trade-off between accuracy and computational efficiency as PDs have so many individual elements. In order to understand appropriate models, experimental work was carried out to understand interactions between the typically small (∼1.5–3 mm diameter) particles used. Measurements were made of coefficient of restitution and interface friction. These were used to give an indication of the level of uncertainty that the simplest (linear) models might assume. These data were used to predict energy dissipation in a PD via a DEM simulation. The results were compared with that of an experiment.
TL;DR: In this paper, a comprehensive model of an electrostatically actuated microcantilever beam separated from the ground plane by relatively larger gap is formulated accounting for the nonlinearities of the system arising out of electric forces, geometry of the deflected beam and the inertial terms.
Abstract: A comprehensive model of an electrostatically actuated microcantilever beam separated from the ground plane by relatively larger gap is formulated accounting for the nonlinearities of the system arising out of electric forces, geometry of the deflected beam and the inertial terms. Since the gap is relatively large, the electrostatic model is formulated incorporating higher order correction of electrostatic forces. First static analysis is carried out to match the results obtained from the proposed model with the results provided by other researchers. It is observed that reduced order model exhibits good convergence when five or more number of modes is considered for the analysis. Dynamic analysis of the model is performed with five modes. The study indicates that although electrostatic forces cause softening characteristics whereas geometric nonlinearity produces stiffening effect on the microstructure, the nonlinearities play a significant role when pull-in occurs. The consideration of slope and curvature of deformable electrode for modelling the electrostatic forces for large gap separations predicts more accurate results. For applications in and around pull-in zone, the large deflection model needs to be considered for effective design.
TL;DR: In this paper, the performance of a nonlinear tuned mass damper (NTMD) was investigated using the numerical continuation software AUTO and the optimization of periodic solutions and parameter sweeps.
Abstract: We explore the performance of a nonlinear tuned mass damper (NTMD), which is modeled as a two degree of freedom system with a cubic nonlinearity. This nonlinearity is physically derived from a geometric configuration of two pairs of springs. The springs in one pair rotate as they extend, which results in a hardening spring stiffness. The other pair provides a linear stiffness term. We perform an extensive numerical study of periodic responses of the NTMD using the numerical continuation software AUTO. In our search for optimal design parameters we mainly employ two techniques, the optimization of periodic solutions and parameter sweeps. During our investigation we discovered a family of detached resonance curves for vanishing linear spring stiffness, a feature that was missed in an earlier study. These detached resonance response curves seem to be a weakness of the NTMD when used as a passive device, because they essentially restore a main resonance peak. However, since this family is detached from the low-amplitude responses there is an opportunity for designing a semi-active device.
TL;DR: In this article, the authors extended the analysis of the self-excited vibrations of a drilling structure presented in an earlier paper 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.
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.