Prediction capabilities of classical and shear deformable beam models excited by a moving mass

In this paper, a comprehensive assessment of design parameters for various beam theories subjected to a moving mass is investigated under different boundary conditions. The design parameters are adopted as the maximum dynamic deflection and bending moment of the beam. To this end, discrete equations of motion for classical Euler-Bernoulli, Timoshenko and higher-order beams under a moving mass are derived based on Hamilton's principle. The reproducing kernel particle method (RKPM) and extended Newmark-β method are utilized for spatial and time discretization of the problem, correspondingly. The design parameter spectra in terms of the beam slenderness, mass weight and velocity of the moving mass are introduced for the mentioned beam theories as well as various boundary conditions. The results indicate the existence of a critical beam slenderness mostly as a function of beam boundary condition, in which, for slenderness lower than this so-called critical one, the application of Euler-Bernoulli or even Timoshenko beam theories would underestimate the real dynamic response of the system. Moreover, there would be a roughly linear relation between the weight of the moving mass and the design parameters for a certain value of the moving mass velocity in most cases of boundary conditions.     

Dynamic contact analysis of a tensioned beam with a moving mass–spring system

The dynamic contact problem of a tensioned beam with clamped-pinned ends is analyzed when the beam contacts a moving mass–spring system. The contact and contact loss conditions are expressed in terms of constraint equations after considering the dynamic contact between the beam and the moving mass. Using these constraints and equations of motion for the beam and moving mass, dynamic contact equations are derived and then discretized using the finite element method, which is based on the Lagrange multiplier method. The time responses for the contact forces are computed from these discretized equations. The contact force variations and contact loss are investigated for the variations of the moving mass velocity, the beam tension, the moving mass, and the stiffness of the moving mass–spring system. In addition, the possibility of contact loss and safe contact conditions between the moving mass and the tensioned beam are also studied.     

THE DYNAMIC ANALYSIS OF A BEAM-MASS SYSTEM DUE TO THE OCCURRENCE OF TWO-COMPONENT PARAMETRIC RESONANCE

The objective of this paper is an analytical and numerical study of the dynamics of a beam--mass system. Special attention is given to the phenomena arising due to the motion of the attached mass and modal interactions produced by the existence of multi-component, specifically two-component, parametric resonance under primary resonance. The two-component parametric resonance occurs when the sums or the differences among internal frequencies are the same, or close, as the dimensionless speed parameter of the moving mass. The effects of two-component parametric resonance post on dynamic condition are investigated. Resonance generated by more than two-component modes are neglected due to its remote probability of occurrence in nature.The mechanics of the problem is Newtonian. Based on the assumption that when the moving mass is set in motion the mass is assumed to be rolling on the beam, the mechanics, including the effects due to friction and convective accelerations, of the interface between the moving mass and the beam are determined.Based on the Bernoulli-Euler beam theory, the coupled non-linear equations of motion of an inextensible beam with an attached moving mass are derived. By employing Galerkin procedure, the partial differential equations which describe the motion of a beam-mass system are reduced to an initial-value problem with finite dimensions. The method of multiple time scales is applied to consider the solutions and the occurrence of internal resonance of the resulting multi-degree-of-freedom beam--mass system with time dependent coefficients.     

Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories

Dynamic analysis of nanotube structures under excitation of a moving nanoparticle is carried out using nonlocal continuum theory of Eringen. To this end, the nanotube structure is modeled by an equivalent continuum structure (ECS) according to the nonlocal Euler-Bernoulli, Timoshenko and higher order beam theories. The nondimensional equations of motion of the nonlocal beams acted upon by a moving nanoparticle are then established. Analytical solutions of the problem are presented for simply supported boundary conditions. The explicit expressions of the critical velocities of the nonlocal beams are derived. Furthermore, the capabilities of various nonlocal beam models in predicting the dynamic deflection of the ECS are examined through various numerical simulations. The role of the scale effect parameter, the slenderness ratio of the ECS and velocity of the moving nanoparticle on the time history of deflection as well as the dynamic amplitude factor of the nonlocal beams are scrutinized in some detail. The results show the importance of using nonlocal shear deformable beam theories, particularly for very stocky nanotube structures acted upon by a moving nanoparticle with low velocity.     

A modeling technique for active control design studies with application to spacecraft microvibrations

Microvibrations, at frequencies between 1 and 1000 Hz, generated by on board equipment, can propagate throughout a spacecraft structure and affect the performance of sensitive payloads. To investigate strategies to reduce these dynamic disturbances by means of active control systems, realistic yet simple structural models are necessary to represent the dynamics of the electromechanical system. In this paper a modeling technique which meets this requirement is presented, and the resulting mathematical model is used to develop some initial results on active control strategies. Attention is focused on a mass loaded panel subjected to point excitation sources, the objective being to minimize the displacement at an arbitrary output location. Piezoelectric patches acting as sensors and actuators are employed. The equations of motion are derived by using Lagrange's equation with vibration mode shapes as the Ritz functions. The number of sensors/actuators and their location is variable. The set of equations obtained is then transformed into state variables and some initial controller design studies are undertaken. These are based on standard linear systems optimal control theory where the resulting controller is implemented by a state observer. It is demonstrated that the proposed modeling technique is a feasible realistic basis for in-depth controller design/evaluation studies.     

DYNAMIC RESPONSE OF A ROTATING FLEXIBLE ARM CARRYING A MOVING MASS

A clamped-free flexible arm rotating in a horizontal plane and carrying a moving mass is studied in this paper. The arm is modelled by the Euler-Bernoulli beam theory in which rotatory inertia and shear deformation effects are ignored. The assumed mode method in conjunction with Hamilton's principle is used to derive the equation of motion of the system which takes into account the effect of centrifugal stiffening due to the rotation of the beam. The eigenfunctions of a cantilever beam which satisfy the prescribed geometric boundary conditions are used as basis functions in the assumed mode method. The equation of motion is expressed in non-dimensional matrix form. Pre-designed transformed cosine profiles are used as trajectory inputs for the hub angle and the moving mass. The equation of motion is solved numerically using the fourth order Runge-Kutta method. Graphical results are presented to show the influence of centrifugal stiffening effect, moving mass values, mass travelling time, hub angle and mass trajectory profile on the deflection of the beam.     

Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations

The formulation of three-dimensional dynamic behavior of a Beam On Elastic Foundation (BOEF) under moving loads and a moving mass is considered. The weight of the vehicle is modeled as a moving point load, however the effect of the lateral excitation is considered by modeling: (case 1) a lateral moving load with random intensity for wind excitation and (case 2) a moving mass just in lateral direction of the beam for earthquake excitation. A Dirac-delta function is used to describe the position of the moving load and the moving mass along the beam. The beam foundations are considered as elastic Winkler-type in two perpendicular transverse directions. This model is proposed to investigate the bending response of the rails under the effect of traveling vehicle weight while a random excitation such as earthquake or wind takes place. The results showed the importance of considering the effect of earthquake/wind actions as in bending stress of the beam on elastic foundations. The effect of different regions (different support stiffness) and different velocities of the vehicle on the response of the beam are investigated in mentioned directions. At the end, a linear optimal control algorithm with displacement–velocity feedback is proposed as a solution to suppress the response of BOEFs. By the method of modal analyses and taking into account enough number of vibration modes, state-space equation is obtained, then sufficient number of actuators was chosen for each direction. Stochastic analyses were performed in lateral direction in order to illustrate a comprehensive view for the response of the beam under the random moving load in both controlled and uncontrolled systems. Furthermore, the efficiency of control algorithm on critical velocities is verified by parametric analyses in the vertical direction with the constant moving load for different regions.     

On the dynamics of interaction between a moving mass and an infinite one-dimensional elastic structure at the stability limit

The paper herein deals with the study of the dynamic behaviour generated by the instability of the vibration of a loaded mass, uniformly moving along an Euler-Bernoulli beam on a viscoelastic foundation, induced by the anomalous Doppler waves excited in the beam. This issue is relevant for the case of modern trains travelling along a track with soft soil when the trains speed exceeds the phase velocity of the waves induced in the track. The model corresponds to a railway vehicle reduced to a loaded wheel running along a (half) track. The beam takes account of the bending stiffness of the rail and the mass of the track, including the mass of the rail, semi-sleepers and half of the ballast layer, where the viscoelastic foundation represents the subgrade. The model includes the wheel/rail Hertzian contact and it allows the simulation of the possibility of contact loss. The nonlinear equations of motion are integrated using a numerical approach based on the Green’s function method. When the vibration becomes unstable, the system evolution is a limit cycle characterised by a succession of shocks, due to the action of two opposite factors: the anomalous Doppler waves that pump energy at the interface between the moving mass and the beam, thus forcing the mass to take off, and the static load that push the mass downwards. The frequency of the shocks increases at higher velocity and the magnitude of the impact force decreases; the most dangerous velocity is the critical one, which represents the stability limit of the linear approximation of the motion equations. The transient behaviour that precedes the limit cycle appearance is being analysed. The Hertzian contact influences the time history of the limit cycle and the magnitude of the impact force and, therefore, it is essential to be included in the model. To the authors’ knowledge, this problem has never been dealt with.     

非惯性系下考虑剪切变形的柔性梁的动力学建模

研究非惯性坐标系下考虑剪切变形的柔性梁的动力学建模. 首先借鉴Euler-Bernoulli梁的几何非线性变形模式,考虑了Timoshenko梁弯曲以及剪切变形产生的几何非线性效应对纵向、横向变形位移的影响,在考虑两个方向的变形耦合项后,利用有限元法对柔性梁进行了离散,采用Lagrange方程建立了柔性梁的动力学模型,首次建立了包含变形二次耦合量的Timoshenko梁的动力学方程. 关键词： 非惯性坐标系 剪切变形 柔性梁 动力学建模     

On the nonlinear primary resonances of a piezoelectric laminated micro system under electrostatic control voltage

In this article, a comprehensive nonlinear analysis for a piezoelectric laminated micro system around its static deflection is presented. This static deflection is created by an electrostatic DC control voltage through an electrode plate. The micro system beam is assumed as an elastic Euler-Bernoulli beam with clamped-free end conditions. The dynamic equations of this model have been derived by using the Hamilton method and considering the nonlinear inertia, curvature, piezoelectric and electrostatic terms. The static and dynamic solutions have been achieved by using the Galerkin method and the multiple-scales perturbation approach, respectively. The results are compared with numerical and other existing experimental results. By studying the primary resonance excitation, the effects of different parameters such as geometry, material and excitations voltage on the system?s softening and hardening behaviors are evaluated. In a piezoelectrically actuated micro system it was showed that because of existence of curvature and inertia nonlinear terms a small change in excitation amplitude can lead to the formation and expansion of nonlinear response. In this paper, it is demonstrated that by applying an electrostatic DC control voltage, these nonlinearities can be controlled and altered to a linear domain. This model can be used to design a nano or micro-scale smart device.     

Electro-mechanical response of functionally graded beams with imperfectly integrated surface piezoelectric layers

The time-dependent behavior of a simply-supported functionally graded beam bonded with piezoelectric sensors and actuators is studied using the state-space method. The creep behavior of bonding adhesives between piezoelectric layers and beam is characterized by a Kelvin-Voigt viscoelastic model, which is practical in a high temperature circumstance. Both the host elastic functionally graded beam and the piezoelectric layers are orthotropic and in a state of plane stress, with the former being inhomogeneous along the thickness direction. A laminate model is employed to approximate the host beam. Moreover, the coupling effect between the elastic deformation and electric field in piezoelectric layers is considered. Results indicate that the viscoelastic property of interfacial adhesives has a significant effect on the function of bonded actuators and sensors with time elapsing.     

DYNAMIC RESPONSE OF A BEAM WITH A CRACK SUBJECT TO A MOVING MASS

An iterative modal analysis approach is developed to determine the effect of transverse cracks on the dynamic behavior of simply supported undamped Bernoulli-Euler beams subject to a moving mass. The presence of crack results in higher deflections and alters the beam response patterns. In particular, the largest deflection in the beam for a given speed takes longer to build up, and a discontinuity appears in the slope of the beam deflected shape at the crack location. Crack effects become more noticeable as crack depth increases. The effect of the inertia force due to the moving mass is, in general, qualitatively similar and additive to the effect of the crack. The exact effect of crack and mass depends on the speed, time, crack size, crack location, and the moving mass level. Other approximate methods, namely a stationary mass model and a single iteration technique, are also evaluated. The stationary mass approach is useful for light moving masses (<20% of beam mass) and cracks at mid-span. For other cases, the errors can be unacceptably large. The results of the single-iteration approximation are quite close to the iterative modal analysis approach, which indicates that this approximate solution is an excellent tool for the analysis of the moving mass problem.     

A particle filtering approach for structural system identification in vehicle-structure interaction problems

The problem of identification of parameters of a beam-moving oscillator system based on measurement of time histories of beam strains and displacements is considered. The governing equations of motion here have time varying coefficients. The parameters to be identified are however time invariant and consist of mass, stiffness and damping characteristics of the beam and oscillator subsystems. A strategy based on dynamic state estimation method, that employs particle filtering algorithms, is proposed to tackle the identification problem. The method can take into account measurement noise, guideway unevenness, spatially incomplete measurements, finite element models for supporting structure and moving vehicle, and imperfections in the formulation of the mathematical models. Numerical illustrations based on synthetic data on beam-oscillator system are presented to demonstrate the satisfactory performance of the proposed procedure.     

An efficient model of an equipment loaded panel for active control design studies

An effective investigation of alternative control strategies for the reduction of vibration levels in satellite structures requires realistic, yet efficient, structural models to simulate the dynamics of the system. These models should include the effects of the sources, receivers, supporting structure, sensors, and actuators. In this paper, a modeling technique which meets these requirements is developed and some active control strategies are briefly investigated. The particular subject of investigation is an equipment-loaded panel and the equations of motion are derived using the Lagrange-Rayleigh-Ritz (LRR) approach. The various pieces of equipment on the panel are mounted on active or passive suspensions, and resonators are used to represent the internal dynamics of the mounted equipment. Control of the panel, which transmits vibrations from sources to receivers, is by means of piezoelectric patches and the excitation consists of dynamic loads acting on the equipment enclosures and/or directly on the panel. The control objective is to minimize the displacement at an arbitrary output location. The LRR model developed is verified against one produced by using the finite-element method. Finally, some initial controller design studies are undertaken to investigate and compare the effectiveness of different control strategies (e.g., minimization at the source, along the vibration path, or at the receiver).     

Closed-form solution for free vibration of piezoelectric coupled annular plates using Levinson plate theory

Free vibration analysis of annular moderately thick plates integrated with piezoelectric layers is investigated in this study for different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Levinson plate theory (LPT). The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study the closed-form solution for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. To demonstrate the accuracy of the present solution, comparison studies is first carried out with the available data in the literature and then natural frequencies of the piezoelectric coupled annular plate are presented for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric and boundary conditions. Present analytical model provides design reference for piezoelectric material application, such as sensors, actuators and ultrasonic motors.     

On utilizing delayed feedback for active-multimode vibration control of cantilever beams

We present a single-input single-output multimode delayed-feedback control methodology to mitigate the free vibrations of a flexible cantilever beam. For the purpose of controller design and stability analysis, we consider a reduced-order model consisting of the first n vibration modes. The temporal variation of these modes is represented by a set of nonlinearly coupled ordinary-differential equations that capture the evolving dynamics of the beam. Considering a linearized version of these equations, we derive a set of analytical conditions that are solved numerically to assess the stability of the closed-loop system. To verify these conditions, we characterize the stability boundaries using the first two vibration modes and compare them to damping contours obtained by long-time integration of the full nonlinear equations of motion. Simulations show excellent agreement between both approaches. We analyze the effect of the size and location of the piezoelectric patch and the location of the sensor on the stability of the response. We show that the stability boundaries are highly dependent on these parameters. Finally, we implement the controller on a cantilever beam for different controller gain-delay combinations and assess the performance using time histories of the beam response. Numerical simulations clearly demonstrate the controller ability to mitigate vibrations emanating from multiple modes simultaneously.     

Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load

The vibration of beams on foundations under moving loads has many applications in several fields, such as pavements in highways or rails in railways. However, most of the current studies only consider the energy dissipation mechanism of the foundation through viscous behavior; this assumption is unrealistic for soils. The shear rigidity and radius of gyration of the beam are also usually excluded. Therefore, this study investigates the vibration of an infinite Timoshenko beam resting on a hysteretically damped elastic foundation under a moving load with constant or harmonic amplitude. The governing differential equations of motion are formulated on the basis of the Hamilton principle and Timoshenko beam theory, and are then transformed into two algebraic equations through a double Fourier transform with respect to moving space and time. Beam deflection is obtained by inverse fast Fourier transform. The solution is verified through comparison with the closed-form solution of an Euler-Bernoulli beam on a Winkler foundation. Numerical examples are used to investigate:(a) the effect of the spatial distribution of the load, and(b) the effects of the beam properties on the deflected shape, maximum displacement, critical frequency, and critical velocity. These findings can serve as references for the performance and safety assessment of railway and highway structures.     

Nonlinear vibration of micromachined asymmetric resonators

In this paper, the nonlinear dynamics of a beam-type resonant structure due to stretching of the beam is addressed. The resonant beam is excited by attached electrostatic comb-drive actuators. This structure is modeled as a thin beam-lumped mass system, in which an initial axial force is exerted to the beam. This axial force may have different origins, e.g., residual stress due to micro-machining. The governing equations of motion are derived using the mode summation method, generalized orthogonality condition, and multiple scales method for both free and forced vibrations. The effects of the initial axial force, modal damping of the beam, the location, mass, and rotary inertia of the lumped mass on the free and forced vibration of the resonator are investigated. For the case of the forced vibration, the primary resonance of the first mode is investigated. It has been shown that there are certain combinations of the model parameters depicting a remarkable dynamic behavior, in which the second to first resonance frequencies ratio is close to three. These particular cases result in the internal resonance between the first and second modes. This phenomenon is investigated in detail.     

Nonlocal vibration of SWBNNT embedded in bundle of CNTs under a moving nanoparticle

In this study, an analytical method of the small scale parameter on the vibration of single-walled Boron Nitride nanotube (SWBNNT) under a moving nanoparticle is presented. SWBNNT is embedded in bundle of carbon nanotubes (CNTs) which is simulated as Pasternak foundation. Using Euler–Bernoulli beam (EBB) model, Hamilton's principle and nonlocal piezoelasticity theory, the higher order governing equation is derived. The effects of electric field, elastic medium, slenderness ratio and small scale parameter are investigated on the vibration behavior of SWBNNT under a moving nanoparticle. Results indicate the importance of using surrounding elastic medium in decrease of normalized dynamic deflection. Indeed, the normalized dynamic deflection decreases with the increase of the elastic medium stiffness values. The electric field has significant role on the nondimensional fundamental frequencies, as a smart controller. The results of this work is hoped to be of use in design and manufacturing of smart nano-electro-mechanical devices in advanced medical applications such as drug delivery systems with great applications in biomechanics.     

Theoretical modeling and experimental realization of dynamically magnified thermoacoustic-piezoelectric energy harvesters

Conventional thermoacoustic-piezoelectric (TAP) harvesters convert thermal energy, such as solar or waste heat energy, directly into electrical energy without the need for any moving components. The input thermal energy generates a steep temperature gradient along a porous medium. At a critical threshold of the temperature gradient, self-sustained acoustic waves are developed inside an acoustic resonator. The associated pressure fluctuations impinge on a piezoelectric diaphragm, placed at the end of the resonator. In this study, the TAP harvester is coupled with an auxiliary elastic structure in the form of a simple spring–mass system to amplify the strain experienced by the piezoelectric element. The auxiliary structure is referred to as a dynamic magnifier and has been shown in different areas to significantly amplify the deflection of vibrating structures. A comprehensive model of the dynamically magnified thermoacoustic-piezoelectric (DMTAP) harvester has been developed that includes equations of motions of the system?s mechanical components, the harvested voltage, the mechanical impedance of the coupled structure at the resonator end and the equations necessary to compute the self-excited frequencies of oscillations inside the acoustic resonator. Theoretical results confirmed that significant amplification of the harvested power is feasible if the magnifier?s parameters are properly chosen. The performance characteristics of experimental prototypes of a thermoacoustic-piezoelectric resonator with and without the magnifier are examined. The obtained experimental findings are validated against the theoretical results. Dynamic magnifiers serve as a novel approach to enhance the effectiveness of thermoacoustic energy harvested from waste heat by increasing the efficiency of their harvesting components.