Equations Of Motion For Beams Research Articles

Impact on the beam of the final length of the moving mass

There have been considered solving direct problem of deforming of isotropic, elastic and hingedly supported beam of finite length. A roller moving at a constant speed along the axis of the beam acts on the beam. The roller has a cylindrical shape of a certain radius and a length that is greater than or equal to the width of the beam. The differential equations of beam motion are analyzed from the point of view of the influence of their components and especially the right-hand parts of the equations on the dynamic behavior of the beam in the case of using certain common materials of the beam and roller, and an option to reduce the equations to a more simplified form is proposed. The unknown functions included in the equations are sought in the form of Fourier series. This allows us to reduce the original equations to ordinary differential equations, which are solved using the Laplace transform. Expressions for coefficients in Fourier series are found using operational calculus and the residue theory. The results of the first numerical experiment on the study of the influence of the roller speed on beam deflections are presented in the form of curves in the figure. For a specific calculated mechanical system in the form of a steel roller, which moves along a steel beam at a constant speed under zero initial conditions, the research results are presented in the form of graphs of beam deflections for different speeds of the roller. The second numerical experiment was carried out to study the propagation of vibrational waves of the beam in the case of motion of the roller at a sufficiently high speed. For this, the figure shows the combined shapes of the beam and the position of the roller at different moments of action of the moving mass. The behavior of the beam at a high speed of movement of the roller was analyzed and a comparison of the deflections of the beam with the deflections of the static model of the beam was made. Further directions for the development of the problem in applied fields of technology and in inverse problems of identifying unknown parameters by indirect manifestations are outlined.

Dynamic Response of PVDF Cantilever Due to Droplet Impact Using an Electromechanical Model.

The dynamic response of a polyvinylidene fluoride (PVDF) cantilever beam under excitation of water droplet impact is investigated by developing an electromechanical model. In the model, the governing equations of beam motion and output voltage are derived in the theoretical way, such that the voltage across the PVDF layer and the cantilever deflection can be predicted. The motion of the beam is described by the multi-mode vibration model through which more accurate results can be obtained. The predicted results of the model are validated by the experiment. Combined with the experiment and the model, the effect of surface wettability on droplet-substrate interaction mechanisms is investigated, which provides an insight into the improvement of mechanical-to-electrical energy conversion efficiency in raindrop energy harvesting (REH) applications. Results show: (1) the droplet splash on a super-hydrophobic beam surface has a positive effect on voltage generation. The splash limit that affects the reaction force of the impacting droplet is experimentally determined and greatly dominant by the Weber number. (2) Small-scaled droplets in splash regime allow generating higher voltage output from a super-hydrophobic beam surface than from an untreated hydrophilic beam surface. (3) Tests of successive droplet impacts also show that a super-hydrophobic surface performs better over a hydrophilic surface by producing constant peak voltage and higher electrical energy harvested. In this case, the voltage measured from the hydrophilic surface decreases gradually as the water layer is accumulated. Overall, the electromechanical behaviors of a super-hydrophobic PVDF cantilever sensor can be well predicted by the model which shows a great potential in energy harvesting by maximizing the inelastic collision upon droplet-substrate interactions.

Dynamics of size-dependant Timoshenko micro beams subjected to moving loads

In this study, dynamic behaviour of size dependant Timoshenko microbeams under the influence of moving loads is modelled and investigated using FEM method. Firstly, the stiffness and mass property matrices of small-scale size-dependant two-node finite element are obtained by using new shape functions derived from the static state of motion equations created using classical Timoshenko beam theory (TBT) and modified couple stress theory (MCST). Subsequently, the load-beam interaction is modelled and integrated with beam motion equations without neglecting the mass inertia effects of the moving load. The effects of different load velocities, mass of the load and material size parameter on the beam dynamics have been investigated in a wide context.

Nonlinear Dynamics of Temperature-Dependent FG-GRC Laminated Beams Resting on Visco-Pasternak Foundations

This paper studies the nonlinear dynamic responses of graphene-reinforced composite (GRC) beams in a thermal environment. It is assumed that a laminated beam rests on a Pasternak foundation with viscosity and consists of GRC layers with various volume fractions of graphene reinforcement to construct a functionally graded (FG) pattern along the transverse direction of the beam. An extended Halpin–Tsai model which is calibrated against the results from molecular dynamics (MD) simulations is used to evaluate the material properties of GRC layers. The mechanical model of the beam is on the establishment of a third-order shear deformation beam theory and includes the von-Kármán nonlinearity effect. The model also considers the foundation support and the temperature variation. The two-step perturbation technique is first applied to solve the beam motion equations and to derive the nonlinear dynamic load–deflection equation of the beam. Then a Runge–Kutta numerical method is applied and the solutions for this nonlinear equation are obtained. The influence of FG patterns, visco-elastic foundation, ambient temperature and applied load on transient response behaviors of simply supported FG-GRC laminated beams is revealed and examined in detail.

An assessment and extension of geometrically nonlinear beam theories

Abstract Equations of motion for beams assuming only longitudinal, normal stress are derived for large-amplitude beam deflection. The formulation includes various physical extensibility conditions without expansion or truncation of their nonlinear nature. The Lagrange multiplier for an inextensible beam model is explicitly written. The limits of the applicability of various beams models with purely longitudinal, normal stress are explored, and the validity of a pure transverse displacement for a beam with fixed ends is quantified.

Mitigating the sound of a flapping airfoil using optimal structural properties distributions

We study the effects of structural properties distributions on the sound radiation of a thin elastic airfoil. Focusing on flapping-flight in a high-Reynolds and low-Mach regime, we seek optimal conditions to mitigate flapping-sound while maintaining aerodynamic efficiency. The airfoil is immersed in a two-dimensional potential flow field, and is subjected to leading-edge actuation in the form of a small-amplitude periodic heaving motion. The near-field dynamics is governed by Euler-Bernoulli's beam equation of motion and analyzed using thin airfoil theory in conjunction with a discrete-vortex wake model. The near-field results are introduced as an effective dipole-type source term to the right-hand-side of the Powell-Howe acoustic wave equation, and the associated far-field sound is calculated using Green's function formulation. Considering the elastic flapping configuration, we seek optimal material properties and a linear thickness distribution to reduce the sound of an otherwise rigid airfoil while retaining the lift amplitude of the rigid configuration. To this end, the aeroacoustic model is introduced into an optimization scheme, where minimal sound amplitude is sought in the lift direction, subjected to the aforementioned lift force constraint with a 33% assigned tolerance value. The relatively wide tolerance produces in turn an effective Pareto front, reflecting the trade-off between the competing objectives of lower sound levels and aerodynamic efficiency. Compared with the rigid heaving airfoil, over 10 [dB] sound reduction was obtained for the optimal flexible configuration producing the same lift amplitude. The Pareto front also displays a linear relation between sound reduction ratio and lift amplitude ratio, indicating a further substantial sound reduction of up to 30 [dB] for smaller lift ratio values. The effect of optimal structure thickness distribution on the equi-lift sound reduction mechanism is two-fold: The motion and wake dipoles are shifted to an antiphased mode thus reducing the total sound signal, and the motion dipole which represents the unsteady forces, exerted by the airfoil on the fluid while transversing, is fixed in magnitude, thus retaining the lift amplitude value. The system dynamics leading to the opposing sound dipoles is evoked by phase-locking the airfoil motion and circulation, and following Kelvin's theorem, antiphase-locking the airfoil motion and wake circulation. Implications on flapping flags and streamers is briefly discussed as well.

A novel composite multi-layer piezoelectric energy harvester

A typical linear piezoelectric energy harvester (PEH) is represented by a unimorph or bimorph cantilever beam. To improve the efficiency of linear PEHs, classical strategies involve the increase of the beam length, tapering or adding additional cantilever beams to the free end. In this work we discuss the design of novel type of composite linear multi-layer piezoelectric energy harvester (MPEH). MPEHs here consist of carbon fibre laminates used as conducting layers, and glass fibre laminas as insulating components. We develop first a electromechanical model of the MPEH with parallel connection of PZT layers based on Euler-Bernoulli beam theory. The voltage and beam motion equations are obtained for harmonic excitations at arbitrary frequencies, and the coupling effect can be obtained from the response of the system. A direct comparison between MPEH and PEH configurations is performed both from the simulation (analytical and numerical) and experimental point of views. The experiments agree well with the model developed, and show that a MPEH configuration with the same flexural stiffness of a PEH can generate up to 1.98–2.5 times higher voltage output than a typical piezoelectric energy harvester with the same load resistance.

Development of an Active Curved Beam Model—Part II: Kinetics and Internal Activation

Utilizing the kinematics, presented in the Part I, an active large deformation beam model for slender, flexible, or soft robots is developed from the d'Alembert's principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross sections of a deforming beam. In the derivation of the beam model, Élie Cartan's moving frame method is utilized. The resulting large deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko beam model is derived by linearizing the nonlinear planar equations. For an active, simply supported Timoshenko beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active C1 beam elements in a nonlinear finite element (FE) code.

A novel inertial energy harvester using magnetic shape memory alloy

This paper studies the output voltage from a novel inertial energy harvester using magnetic shape memory alloys (MSMAs). The MSMA elements are attached to the root of a cantilever beam by means of two steps. In order to get electrical voltage, two coils are wound around the MSMAs and a shock load is applied to a tip mass at the end of the beam to have vibration in it. The beam vibration causes strain in the MSMAs along their longitudinal directions and as a result the magnetic flux alters in the coils. The change of magnetic flux in the surrounding coil produces an AC voltage. In order to predict the output voltage, the nonlinear governing equations of beam motion based on Euler–Bernoulli model and von Kármán theory are derived. A thermodynamics-based constitutive model is used to predict the nonlinear strain and magnetization response of the MSMAs. Also, the induced voltage during martensite variant reorientation in MSMAs is investigated with the help of Faraday’s law of induction. Finally, the effect of different parameters including bias magnetic field, pre-strain and number of MSMA elements are investigated in details. The results show that this novel energy harvester has the capability of using as an alternative to the current piezoelectric and magnetostrictive ones for harvesting energy from ambient vibration.

Dynamic Stability Analysis of Beams with Shear Deformation and Rotary Inertia Subjected to Periodic Axial Forces by Using Discrete Singular Convolution Method

AbstractThis paper presents dynamic stability analysis of beams with shear deformation and rotary inertia subjected to periodic axial forces by employing a discrete singular convolution (DSC) algorithm with regularized Shannon kernel. The shear effect has been taken into account based on Engesser’s and Haringx’s theories, respectively. A modified algorithm is proposed to solve the governing equations of beam motion. The influence of rotary inertia, shear deformation, and damping on dynamic instability regions has been investigated. The obtained numerical results are compared with those of the existing method. Numerical results indicate that the modified algorithm is a reliable approach for the solutions of this kind of problem in this investigation. The differences between dynamic instability regions of beams based on Engesser’s and Haringx’s theories are also presented.

Discrete Singular Convolution Method for Dynamic Stability Analysis of Beams under Periodic Axial Forces

AbstractIn the present study, the discrete singular convolution (DSC) method is used to analyze the dynamic stability of beams subjected to different boundary conditions under periodic axial forces. A unified DSC algorithm is established to solve the governing equations of beam motion under different boundary conditions. The regularized Shannon’s delta kernel is selected as singular convolution to illustrate the present algorithm. The matched interface and boundary method is applied to the treatment of elastic restraint boundary conditions. The effects of a constant term in the periodic axial force and damping on dynamic instability regions are also investigated. The obtained numerical results are compared with those of Bolotin’s method. Numerical results indicate that the DSC method is a reliable method for dynamic stability analysis of beams.

Response of infinite length bars and beams with periodically varying area

This talk develops a solution method for the longitudinal motion of a rod or the flexural motion of a beam of infinite length whose area varies periodically. The conventional rod or beam equation of motion is used with the area and moment of inertia expressed using analytical functions of the longitudinal (horizontal) spatial variable. The displacement field is written as a series expansion using a periodic form for the horizontal wavenumber. The area and moment of inertia expressions are each expanded into a Fourier series. These are inserted into the differential equations of motion and the resulting algebraic equations are orthogonalized to produce a matrix equation whose solution provides the unknown wave propagation coefficients, thus yielding the displacement of the system. An example problem of both a rod and beam are analyzed for three different geometrical shapes. The solutions to both problems are compared to results from finite element analysis for validation. Dispersion curves of the systems are shown graphically. Convergence of the series solutions is illustrated and discussed.

Response of infinite length rods and beams with periodically varying area

This paper develops a solution method for the longitudinal motion of a rod or the flexural motion of a beam of infinite length whose area varies periodically. The conventional rod or beam equation of motion is used with the area and moment of inertia expressed using analytical functions of the longitudinal (horizontal) spatial variable. The displacement field is written as a series expansion using a periodic form for the horizontal wavenumber. The area and moment of inertia expressions are each expanded into a Fourier series. These are inserted into the differential equations of motion and the resulting algebraic equations are orthogonalized to produce a matrix equation whose solution provides the unknown wave propagation coefficients, thus yielding the displacement of the system. An example problem of both a rod and beam are analyzed for three different geometrical shapes. The solutions to both problems are compared to results from finite element analysis for validation. Dispersion curves of the systems are shown graphically. Convergence of the series solutions is illustrated and discussed.

Repair of a cracked Timoshenko beam subjected to a moving mass using piezoelectric patches

This paper presents an analytical method for the application of piezoelectric patches for the repair of cracked beams subjected to a moving mass. The beam equations of motion are obtained based on the Timoshenko beam theory by including the dynamic effect of a moving mass traveling along a vibrating path. The criterion used for the repair is altering the first natural frequency of the cracked beam towards that of the healthy beam using a piezoelectric patch. Conceptually, an external voltage is applied to actuate a piezoelectric patch bonded on the beam. This affects the closure of the crack so that the singularity induced by the crack tip will be decreased. The equations of motion are discretized by using the assumed modes method. The cracked beam is modeled as number of segments connected by two massless springs at the crack locations (one, extensional and the other, rotational). The relationships between any two spans can be obtained by considering the compatibility requirements on the crack section and on the ends of the piezoelectric patch. Using the analytical transfer matrix method, eigensolutions of the system can be calculated explicitly. Finally, numerical simulations are performed with respect to different conditions such as the moving load velocity. It is seen that when the piezoelectric patch is used, the maximum deflection of the cracked beam approaches maximum deflection of the healthy beam.

Connection design for two-floating beam system for minimum hydroelastic response

This paper is concerned with the connection design for a two-floating beam system for minimum hydroelastic response. The frequency domain approach is used for the hydroelastic analysis. The fluid is modelled as an ideal fluid, and the floating beams are modelled by the Euler–Bernoulli beam theory. The boundary element method (BEM) and the finite element method (FEM) are applied to solve the governing equation of the fluid motion and the beam equation of motion, respectively. The study aims to investigate the optimum location and rotational stiffness of the connection for the two-floating beam system with the view to minimize the compliance. The study also investigates the effects of relative beam stiffnesses on the hydroelastic response of the two-floating beam system.

The application of topological models in the problem of the dynamic interaction of a rigid bodies system

The differential equations of beam motion considering of moving, elastic and dissipative forces are presented on types of oscillations and integrate longitudinal, torsional, bending as well as shearing vibrations. The basic topological structure is figured in the form of the weighed and dual graphs for determination of force and kinematic parameters. The representing algebraic system is noted as tensor relations.

Aeroelastic Analysis of Bearingless Rotors Using Large Deflection Beam Theory

The aeroelastic response and stability of bearingless rotors are investigated using large deflection beam theory. The bearingless rotor configuration consists of a single flexbeam with a wraparound-type torque tube and pitch links located at the leading edge and trailing edge of the torque tube. The outboard main blade, flexbeam, and torque tube are all assumed to be an elastic beam undergoing arbitrary large displacements and rotations, which are discretized into beam finite elements. The finite element equations of motion for beams are obtained from Hamilton's principle. Two-dimensional quasi-steady strip theory is used to evaluate aerodynamic forces in both hover and forward flight. For the analysis of hover flight, the nonlinear equations of motion are solved for an equilibrium position through an iterative procedure. The modal approach method based on coupled rotating natural modes is used for the stability analysis. For the analysis of forward flight, the nonlinear periodic blade steady response is obtained by integrating the full finite element equation in time through a coupled trim procedure with a vehicle trim. After the coupled trim response is computed, the aeroelastic response is calculated through a time-marching solution procedure under small perturbations assumption. A stability analysis is then performed using a moving block analysis. The results of the full finite element analysis using the large deflection beam theory are quite different from those of a previously published modal analysis using the moderate deflection-type beam theory.

Space charge calculations for sub-three-dimensional particle-in-cell code

A novel approach for modeling high-current, charged particle beams in a three-dimensional manner is introduced. While the integration of beam motion equations is done as in completely 3D particle-in-cell codes, the space charge forces are found by an approximately self-consistent inclusion of the transverse and longitudinal fields. The algorithm is dramatically faster than fully 3D algorithms with computational times comparable to 2D field solvers. In addition, a sparser spatial grid and fewer required macroparticles provide significantly reduced memory demands. The proposed sub-3D technique has been verified with good agreement with other independent algorithms.

BEAM EQUATIONS OF MOTION DERIVED FROM THE DISPLACEMENT FIELD OF LAMB PLATE THEORY

Lamb の板の波動伝播の弾性論解に含まれる変位場を基にして, 充実長方形断面はりの運動方程式を誘導し, 位相速度曲線の一致を確認した. この式中に含まれる動的断面2次モーメントや断面積などの断面諸量と周波数の関係を調べた. 無限長のはりあるいは板などの媒質を伝わる弾性波の第1次モードは, 周波数無限大において等しく Rayleigh 表面波に収束する. この事実から, 半無限体に対しても Rayleigh 表面波の動的断面2次半径を定義できることを示し, Poisson 材料において, それが静的な断面2次半径の7割弱ほどの値に低減する事を証明した. さらに第2次以上のモードではこの動的断面2次半径は周波数無限大の極限で零となり, 高次モードが媒質から消失する物理的事実とも対応することを証明した.

Aeroelastic Analysis of a Hingeless Rotor Blade in Forward Flight

The aeroelastic response and stability of isotropic and composite rotor blades are investigated using a large deflection-type beam theory. The finite element equations of motion for beams undergoing arbitrary large displacements and rotations, but small strains, are obtained from Hamilton's principle. The sectional elastic constants of a composite box beam including warping deformations are determined from the refined cross-sectional finite element method. The analysis is performed for a soft-in-plane hingeless rotor in free flight propulsive trim. The nonlinear periodic blade steady response is obtained by integrating the full finite element equation in time through a coupled trim procedure with a vehicle trim. After the coupled trim response is computed, the aeroelastic response is calculated through a time-marching solution procedure under small perturbations assumption, and then the stability analysis is performed by using a moving block analysis. Numerical results of rotating natural frequencies, blade response, and aeroelastic stability are presented. The results of the full finite element analysis using the large deflection-type beam theory are quite different from those of a previously published modal analysis using the moderate deflection-type beam theory.