轴向运动导电导磁梁的磁弹性振动方程

针对磁场环境中轴向运动导电导磁梁磁弹性耦合振动的理论建模问题进行研究.基于Timoshenko(铁木辛柯)梁理论并考虑几何非线性因素,给出轴向运动弹性梁在横向双向振动下的形变势能、动能计算式以及电磁力和机械力的虚功表达式.应用Hamilton(哈密顿)变分原理,推得磁场中轴向运动Timoshenko梁的非线性磁弹性耦合振动方程,并给出了简化形式的Euler-Bernoulli(欧拉 伯努利)梁磁弹性振动方程.根据电磁理论和相应的电磁本构关系,得到载流导电弹性梁所受电磁力的表达式,基于磁偶极子-电流环路模型给出铁磁弹性梁所受磁体力和磁体力偶的表述形式.通过算例,分析了轴向运动导电弹性梁的奇点分布及其稳定性问题.     

Size-dependent forced vibration of non-uniform bi-directional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass

In this study, general non-uniform material-varying micro-beam models under a moving harmonic load/mass are investigated. Material variation is modeled by combining axial and thickness material grading models using exponential, linear, parabolic and sigmoidal functions. Beam is assumed to be resting on an elastic foundation and in this linear foundation model, foundation modulus is assumed to vary axially with respect to space variable in a non-linear manner ignoring the effect of mass density of foundation on the behavior of micro-beam. Cross-section variation through the length is formulated for both thickness and width variation. Governing equations for such comprehensive beam model is achieved using Hamilton's principle in conjunction with modified couple stress theory to add the scale-effects and solved by discussing explicit and implicit finite element methods with using various-steps and Wilson-theta method. Current methodology is verified using previous studies on simplified problems. A comprehensive parametric study is presented in order to indicate the influence of each design, material and fundamental terms on the forced vibration behavior of such structures under a moving harmonic/constant load/mass. It is shown that by appropriately choosing the material variation in bidirectional functionally graded beams dynamic vibration behavior of such structures could change significantly. Moreover, it is shown that varying cross-section, elastic foundation and type of harmonic moving mass can change the dynamic reaction of the general micro-beam model. From the influence of modified couple stress term on mechanical behavior of such structures it is concluded that this term has crucial effect in varying the dynamic deflections and it is important to acknowledge it in analyzing such structures.     

Vibration and stability of an axially moving Rayleigh beam

In this paper, the vibration and stability of an axially moving beam is investigated. The finite element method with variable-domain elements is used to derive the equations of motion of an axially moving beam based on Rayleigh beam theory. Two kinds of axial motions including constant-speed extension deployment and back-and-forth periodical motion are considered. The vibration and stability of beams with these motions are investigated. For vibration analysis, direct time numerical integration, based on a Runge–Kutta algorithm, is used. For stability analysis of a beam with constant-speed axial extension deployment, eigenvalues of equations of motion are obtained to determine its stability, while Floquet theory is employed to investigate the stability of the beam with back-and-forth periodical axial motion. The effects of oscillation amplitude and frequency of periodical axial movement on the stability of the beam are discussed from the stability chart. Time histories are established to confirm the results from Floquet theory.     

含温度载荷的管板变形的解析解

根据换热器结构形式建立了用于计算管板变形的力学模型,参考ASME Ⅷ 1计算管板的有效弹性常数,并将外筒和换热管束分别等效为会因温度和压力载荷产生轴向变形的弹簧和弹性基础.应用Ritz法建立了管板挠度的解析解,将该解析解与三个不同规模换热器的有限元分析结果进行对比.结果表明,建立的解析解与有限元分析结果吻合良好,验证了推导的管板变形解析解的正确性.研究结果对固定管板式换热器的设计有较大的指导意义.     

Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions

The vibration and stability of a simply supported beam are analyzed when the beam has an axially moving motion as well as a spinning motion. When a beam has spinning and axial motions, rotary inertia plays an important role on the lateral vibration. Compared to previous studies, the present study adopts the Rayleigh beam theory and derives more exact kinetic energy and equations of motion. The rotary inertia terms derived by the present study are compared to those of the previous studies. We investigate the natural frequencies between the present and previous studies. In addition, the critical speed and stability boundary for the spinning and moving speeds are also analyzed. It can be observed from the computed natural frequencies and dynamic responses that the present equations of motion are more reliable than the previous equations because the present equations fully consider the rotary inertia terms.     

Vibration of axially moving hyperelastic beam with finite deformation

In this paper, we study the vibration of an axially moving hyperelastic beam under simply supported condition. The kinematic of the axially moving beam have been described by Eulerian-Lagrangian formulation. In continuum mechanics frame, the finite deformation formula and a higher order shear deformation beam theory are applied to describe the deformation of the axially moving hyperelastic beam. In these formulas the material parameter, shear deformation and the geometric non-linearity have been taken into account. Through the Hamilton principle, the governing equations of nonlinear vibration are obtained, where the transverse vibration is coupled with the longitudinal vibration. When the velocity is a constant, the critical speed and natural frequencies are determined by solving the corresponding linear equations. Meantime, effects of the geometrical and material parameters on the critical speed and natural frequencies have been investigated. Comparisons among the critical velocities of the hyperelastic and Euler linear beam are also made. The results show that the critical velocity of hyperelastic beam is larger than that of linear Euler–Bernoulli beam. For the natural frequencies, we have the same conclusions. Lastly, by the multiple scales method, the leading order analytical solutions of the equilibrium state of axially moving hyperelastic beam in the supercritical regime are obtained. Furthermore the amplitudes of analytical solutions of the hyperelastic beam have been compared with that of linear Euler–Bernoulli beam. The effects of the material and geometrical parameters on the asymptotic solutions and the amplitude has been analyzed.     

Vibrations of a beam-string complex system under a moving force

In this paper, the dynamic response of an Euler-Bernoulli beam and string system traversed by a constant moving force is considered. The force is moving with a constant velocity on the top beam. The complex system is finite, simply supported, parallel one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a beam-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case the aperiodic part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations. The dynamic influence lines of complex systems may be used for the analysis the complex models of moving load. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic response of clamped axially moving beams: Integral transform solution

The generalized integral transform technique (GITT) is employed to obtain a hybrid analytical-numerical solution for dynamic response of clamped axially moving beams. The use of the GITT approach in the analysis of the transverse vibration equation leads to a coupled system of second order differential equations in the dimensionless temporal variable. The resulting transformed ODE system is then solved numerically with automatic global accuracy control by using the subroutine DIVPAG from IMSL Library. Excellent convergence behavior is shown by comparing the vibration displacement of different points along the beam length. Numerical results are presented for different values of axial translation velocity and flexural stiffness. A set of reference results for the transverse vibration displacement of axially moving beam is provided for future co-validation purposes.     

Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity

Nonlinearly parametric resonances of axially accelerating moving viscoelastic sandwich beams with time-dependent tension are investigated in this paper. Based on the Kelvin differential constitutive equation, the controlling equation of the transverse vibration of a beam with large deflection is established. The system has been subjected to a time varying velocity and a harmonic axial tension. Here the governing equation of motion contains linear parametric terms and two frequencies, one is the frequency of axially moving velocity and the other one is the frequency of varying tension. The method of multiple scales is applied directly to the governing equation to obtain the complex eigenfunctions and natural frequencies of the system. The elimination of secular terms leads to the steady-state response and amplitude of vibrations. The influence of various parameters such as initial tension on natural frequencies and the amplitude of axial fluctuation, the phase angle between the two frequencies on response curves has been investigated for two different resonance conditions. With the help of numerical results, it has been shown that by using suitable initial tension, the amplitude of axial fluctuation, the phase angle, the vibration of the sandwich beam can be significantly controlled.     

The steady vibrations and resistance of a railway track to the uniform motion of an unbalanced wheel

A model of a railway track, in the form of an infinite Timoshenko beam resting on equally spaced massive visco-elastic supports, is considered. Steady vertical vibrations of the track due to a harmonic force moving along it at a constant velocity are investigated. The vertical displacement of the track is represented in a moving system of coordinates by a generalized Fourier series. The steady vertical vibrations of a massive rigid wheel rolling along the track at a constant velocity and loaded by a vertical harmonic force are investigated. The track-wheel interaction force is expressed as a generalized Fourier series whose coefficients are determined using an equality relating the vertical displacements of the wheel and the track. Vibrations of the wheel due to centrifugal force and periodic changes in the track parameters are considered. Parametric vibrations of a wheel moving at a constant velocity under a static load due to periodic variation in the stiffness of the track are investigated. The force with which the track resists the uniform motion of an unbalanced wheel is computed.     

Vibration of a single-walled carbon nanotube embedded in an elastic medium under a moving internal nanoparticle

Vibration of single-walled carbon nanotube embedded in an elastic medium under excitation of a moving nanoparticle is analyzed in this paper. Based on the Winkler spring model and the Euler–Bernoulli beam model, the time-domain responses of the single-walled carbon nanotube subjected to the moving transverse load with three different boundary conditions are computed by using the Newmark method. The effects of velocity and the excitation frequency of the moving internal nanoparticle and boundary conditions on the dynamic deflections of the single-walled carbon nanotube are discussed.     

The problem of the equilibrium of a helical spring in the non-linear three-dimensional theory of elasticity

The problem of the loading of a helical spring by an axial force and a torque is considered using the three-dimensional equations of the non-linear theory of elasticity. The problem is reduced to a two-dimensional boundary-value problem for a plane region in the form of the transverse cross section of the coil of the spring. The solution of the two-dimensional problem obtained enables the equations of equilibrium in the volume of the body and the boundary conditions on the side surface to be satisfied exactly. The boundary conditions at the ends of the spring are satisfied in the integral Saint-Venant sense. The problem of the equivalent prismatic beam in the theory of springs is discussed from the position of the solution of the non-linear Saint-Venant problem obtained. The results can be used for accurate calculations of springs in the non-linear strain region, and also when developing applied non-linear theories of elastic rods with curvature and twisting.     

Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: Non-transforming spectral element analysis

Engineering systems, such as rolled steel beams, chain and belt drives and high-speed paper, can be modeled as axially translating beams. This article scrutinizes vibration and stability of an axially translating viscoelastic Timoshenko beam constrained by simple supports and subjected to axial pretension. The viscoelastic form of general rheological model is adopted to constitute the material of the beam. The partial differential equations governing transverse motion of the beam are derived from the extended form of Hamilton's principle. The non-transforming spectral element method (NTSEM) is applied to transform the governing equations into a set of ordinary differential equations. The formulation is similar to conventional FFT-based spectral element model except that Daubechies wavelet basis functions are used for temporal discretization. Influences of translating velocities, axial tensile force, viscoelastic parameter, shear deformation, beam model and boundary condition types are investigated on the underlying dynamic response and stability via the NTSEM and demonstrated via numerical simulations.     

Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams

Non-linear parametric vibration and stability of an axially moving Timoshenko beam are considered for two dynamic models; the first one, with considering only the transverse displacement and the second one, with considering both longitudinal and transverse displacements. The set of non-linear partial-differential equations of both models are derived using an energy approach. The method of multiple scales is applied directly to both models, and using the equation order one, the mode shape equations and natural frequencies are obtained. Then, for the equation order epsilon, the solvability conditions are considered for the resonance case and the stability boundaries are formulated analytically via Routh–Hurwitz criterion. Eventually, some numerical examples are provided to show the differences in the behavior of the above-mentioned non-linear models.     

Well-posedness and exponential stability of a thermoelastic Joint–Leg–Beam system with Robin boundary conditions

An important class of proposed large space structures features a triangular truss backbone. In this paper we study thermomechanical behavior of a truss component; namely, a triangular frame consisting of two thin-walled circular beams connected through a joint. Transverse and axial mechanical motions of the beams are coupled though a mechanical joint. The nature of the external solar load suggests a decomposition of the temperature fields in the beams leading to two heat equations for each beam. One of these fields models the circumferential average temperature and is coupled to axial motions of the beam, while the second field accounts for a temperature gradient across the beam and is coupled to beam bending. The resulting system of partial and ordinary differential equations formally describes the coupled thermomechanical behavior of the joint–beam system. The main work is in developing an appropriate state-space form and then using semigroup theory to establish well-posedness and exponential stability.     

Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation

This paper investigates bifurcation and chaos in transverse motion of axially accelerating viscoelastic beams. The Kelvin model is used to describe the viscoelastic property of the beam material, and the Lagrangian strain is used to account for geometric nonlinearity due to small but finite stretching of the beam. The transverse motion is governed by a nonlinear partial-differential equation. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. When the Galerkin truncation is based on the eigenfunctions of a linear non-translating beam subjected to the same boundary constraints, a computation technique is proposed by regrouping nonlinear terms. The scheme can be easily implemented in practical computations. When the transport speed is assumed to be a constant mean speed with small harmonic variations, the Poincaré map is numerically calculated based on 4-term Galerkin truncation to identify dynamical behaviors. The bifurcation diagrams are present for varying one of the following parameter: the axial speed fluctuation amplitude, the mean axial speed and the beam viscosity coefficient, while other parameters are unchanged.     

轴向运动梁的横向振动分析

论述了轴向运动梁横向振动问题以及研究轴向运动梁横向振动问题的方法,指出对轴向运动梁横向振动问题研究中存在的一些错误并进行了更正.针对一端可看作固定边界条件的轴向运动悬臂梁,基于连续体的模态叠加法,推导出含自重效应的轴向运动梁动力响应的计算公式,进行实例计算,并对计算结果进行了详细的讨论,得出影响轴向运动梁振动响应的因素主要有速度和运动方向.     

Control and exponential stabilization for the equation of an axially moving viscoelastic strip

The aim through this work is to suppress the transverse vibrations of an axially moving viscoelastic strip. A controller mechanism (dynamic actuator) is attached at the right boundary to control the undesirable vibrations. The moving strip is modeled as a moving beam pulled at a constant speed through 2 eyelets. The left eyelet is fixed in the sense that there is no transverse displacement (see Figure 1 ). The mathematical model of this system consists of an integro‐partial differential equation describing the dynamic of the strip and an integro‐differential equation describing the dynamic of the actuator. The multiplier method is used to design a boundary control law ensuring an exponential stabilization result.     

Analytical study on dynamic responses of a curved beam subjected to three-directional moving loads

Considering the warping resistance, inertia force and moving three-directional loads, a more comprehensive set of governing equations for vertical, torsional, radial and axial motions of the curved beam are derived. The analytical solutions for vertical, torsional, radial and axial responses of the curved beam subjected to three-directional moving loads are obtained, using the Galerkin method to discretize the partial differential equations and the modal superposition method to decouple the ordinary differential equations. The analytical results are compared with the numerical integration and a published work to verify the validity of the proposed solutions. Effects of Galerkin truncation terms and damping ratio on solution convergence are also discussed. Considering first-mode and higher-mode truncation respectively, the conditions of resonance and cancellation are analyzed for vertical, torsional, radial and axial motions of the curved beam. Taking a curved bridge under passage of a vehicle as an example, the influences of system parameters, such as vehicle speed, braking acceleration, bridge curve radius, bridge span and bridge deck elastic modulus, on bridge midpoint vibration are explored. The proposed approach and results may be beneficial to enhance understanding the three-directional vehicle-induced dynamic responses of curved bridges. It is shown that when the axial motion, or the multiple moving loads are involved, the first-order truncation are not accurate enough and one should use higher-mode truncation to study the responses of curved beams. In addition, it is necessary to consider damping in the vibration study of curved beams.     

Moving Cracks in Composite Materials with Initial Stresses

The dynamic problems of fracture mechanics for composite materials with initial stresses are considered in the case of cracks moving at a constant rate along a straight line. In the continuum approximation, composite materials are modeled by orthotropic nonlinearly elastic bodies with an arbitrary form of the elastic potential. A three-dimensional linearized theory of elasticity is used. The complex potentials of plane and antiplane problems of the linearized theory are used for dynamic problems. Exact solutions for Modes I, II, and III in the case of moving cracks are obtained using the Keldysh-Sedov methods. Asymptotic formulas for stresses and displacements near the crack tip for Modes I, II, and III are presented. The basic mechanical effects are analyzed with respect to the problems considered.