Variational principle for a special Cosserat rod

In this paper, we describe the nonlinear models of a rod in three-dimensional space based on the Cosserat theory. Using the pseudo-rigid body method and variational principle, we obtain the motion equations of a Cosserat rod including shear deformations.     

Coupled global dynamics of an axially moving viscoelastic beam

The nonlinear global forced dynamics of an axially moving viscoelastic beam, while both longitudinal and transverse displacements are taken into account, is examined employing a numerical technique. The equations of motion are derived using Newton′s second law of motion, resulting in two partial differential equations for the longitudinal and transverse motions. A two-parameter rheological Kelvin–Voigt energy dissipation mechanism is employed for the viscoelastic structural model, in which the material, not partial, time derivative is used in the viscoelastic constitutive relations; this gives additional terms due to the simultaneous presence of the material damping and the axial speed. The equations of motion for both longitudinal and transverse motions are then discretized via Galerkin’s method, in which the eigenfunctions for the transverse motion of a hinged-hinged linear stationary beam are chosen as the basis functions. The subsequent set of nonlinear ordinary equations is solved numerically by means of the direct time integration via modified Rosenbrock method, resulting in the bifurcation diagrams of Poincaré maps. The results are also presented in the form of time histories, phase-plane portraits, and fast Fourier transform (FFTs) for specific sets of parameters.     

Group theoretic analysis and similarity solution for a nonlinear viscous rod subjected to time dependent velocity impact

Unidirectional wave motion in a nonlinear viscous rod obeying Norton's law in creep, subjected to time dependent velocity impact is considered. From the basic equations of the problem and the four parameter dimensional group of transformations, absolute invariants of the group are constructed to obtain similarity transformations. Similarity representation of the original system of partial differentiation equations is formulated as a system of nonlinear ordinary differential equations with auxiliary conditions. Closed form solutions are obtained for a linear rod, for a nonlinear rod subjected to constant velocity impact and a weekly nonlinear rod. Nonlinear case is solved by a numerical approach based on the quasilinearization method.     

Combination,principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

This study analyses the nonlinear transverse vibration of an axially moving beam subject to two frequency excitation. Focus has been made on simultaneous resonant cases i.e. principal parametric resonance of first mode and combination parametric resonance of additive type involving first two modes in presence of internal resonance. By adopting the direct method of multiple scales, the governing nonlinear integro-partial differential equation for transverse motion is reduced to a set of nonlinear first order ordinary partial differential equations which are solved either by means of continuation algorithm or via direct time integration. Specifically, the frequency response plots and amplitude curves, their stability and bifurcation are obtained using continuation algorithm. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear dynamics of axially moving systems.     

Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam

The three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam is investigated in this paper by means of two numerical techniques. The equations of motion for the longitudinal, transverse, and rotational motions are derived using constitutive relations and via Hamilton’s principle. The Galerkin method is employed to discretize the three partial differential equations of motion, yielding a set of nonlinear ordinary differential equations with coupled terms. This set is solved using the pseudo-arclength continuation technique so as to plot frequency-response curves of the system for different cases. Bifurcation diagrams of Poincaré maps for the system near the first instability are obtained via direct time integration of the discretized equations. Time histories, phase-plane portraits, and fast Fourier transforms are presented for some system parameters.     

Substructure method in high-speed monorail dynamic problems

The study of actions of high-speed moving loads on bridges and elevated tracks remains a topical problem for transport. In the present study, we propose a new method for moving load analysis of elevated tracks (monorail structures or bridges), which permits studying the interaction between two strained objects consisting of rod systems and rigid bodies with viscoelastic links; one of these objects is the moving load (monorail rolling stock), and the other is the carrying structure (monorail elevated track or bridge). The methods for moving load analysis of structures were developed in numerous papers [1–15]. At the first stage, when solving the problem about a beam under the action of the simplest moving load such as a moving weight, two fundamental methods can be used; the same methods are realized for other structures and loads. The first method is based on the use of a generalized coordinate in the expansion of the deflection in the natural shapes of the beam, and the problem is reduced to solving a system of ordinary differential equations with variable coefficients [1–3]. In the second method, after the “beam-weight” system is decomposed, just as in the problem with the weight impact on the beam [4], solving the problem is reduced to solving an integral equation for the dynamic weight reaction [6, 7]. In [1–3], an increase in the number of retained forms leads to an increase in the order of the system of equations; in [6, 7], difficulties arise when solving the integral equations related to the conditional stability of the step procedures. The method proposed in [9, 14] for beams and rod systems combines the above approaches and eliminates their drawbacks, because it permits retaining any necessary number of shapes in the deflection expansion and has a resolving system of equations with an unconditionally stable integration scheme and with a minimum number of unknowns, just as in the method of integral equations [6, 7]. This method is further developed for combined schemes modeling a strained elastic compound moving structure and a monorail elevated track. The problems of development of methods for dynamic analysis of monorails are very topical, especially because of increasing speeds of the rolling stock motion. These structures are studied in [16–18].In the present paper, the above problem is solved by using the method for the moving load analysis and a step procedure of integration with respect to time, which were proposed in [9, 19], respectively. Further, these components are used to enlarge the possibilities of the substructure method in problems of dynamics. In the approach proposed for moving load analysis of structures, for a substructure (having the shape of a boundary element or a superelement) we choose an object moving at a constant speed (a monorail rolling stock); in this case, we use rod boundary elements of large length, which are gathered in a system modeling these objects. In particular, sets of such elements form a model of a monorail rolling stock, namely, carriage hulls, wheeled carts, elements of the wheel spring suspension, models of continuous beams of monorail ways and piers with foundations admitting emergency subsidence and unilateral links. These specialized rigid finite elements with linear and nonlinear links, included into the set of earlier proposed finite elements [14, 19], permit studying unsteady vibrations in the “monorail train-elevated track” (MTET) system taking into account various irregularities on the beam-rail, the pier emergency subsidence, and their elastic support by the basement. In this case, a high degree of the structure spatial digitization is obtained by using rods with distributed parameters in the analysis. The displacements are approximated by linear functions and trigonometric Fourier series, which, as was already noted, permits increasing the number of degrees of freedom of the system under study simultaneously preserving the order of the resolving system of equations.This approach permits studying the stress-strain state in the MTET system and determining accelerations at the desired points of the rolling stock. The proposed numerical procedure permits uniquely solving linear and nonlinear differential equations describing the operation of the model, which replaces the system by a monorail rolling stock consisting of several specialized mutually connected cars and a system of continuous beams on elastic inertial supports.This approach (based on the use of a moving substructure, which is also modeled by a system of boundary rod elements) permits maximally reducing the number of unknowns in the resolving system of equations at each step of its solution [11]. The authors of the preceding investigations of this problem, when studying the simultaneous vibrations of bridges and moving loads, considered only the case in which the rolling stock was represented by sufficiently complicated systems of rigid bodies connected by viscoelastic links [3–18] and the rolling stock motion was described by systems of ordinary differential equations. A specific characteristic of the proposed method is that it is convenient to derive the equations of motion of both the rolling stock and the bridge structure. The method [9, 14] permits obtaining the equations of interaction between the structures as two separate finite-element structures. Hence the researcher need not traditionally write out the system of equations of motion, for example, for the rolling stock (of cars) with finitely many degrees of freedom [3–18].We note several papers where simultaneous vibrations of an elastic moving load and an elastic carrying structure are considered in a rather narrow region and have a specific character. For example, the motion of an elastic rod along an elastic infinite rod on an elastic foundation is studied in [20], and the body of a car moving along a beam is considered as a rod with ten concentrated masses in [21].     

Exact statement of the instability problem for frames and its direct numerical solution

A general approach for the systematic evaluation of the critical buckling load and the determination of the buckling mode is presented. The Navier-Bernoulli beam model is considered, having the possibility of variable cross-section under any type of load (including pressures and thermal loading). With this purpose, the equilibrium equations of each beam element in its deformed configuration under the hypothesis of infinitesimal strains and displacements is considered, resulting in a system of differential equations with variable coefficients for each element. To obtain the nonlinear response of the frame, one should impose the compatibility of displacements and the equilibrium of forces and moments in each beam-end, also in the deformed configuration. The solution is obtained by requiring that the total variation of potential energy is zero at the instant of buckling. The objective of this work is to develop a systematic method to determine the critical buckling load and the bucklingmode of any frame without using the common simplifications usually assumed in matrix analysis or finite element approaches. This way, precise results can be obtained regardless of the discretization done.     

Nonlinear dynamic response and active vibration control of the viscoelastic cable with small sag

IntroductionCablesareveryefficientstructuralmembersandhencehavebeenwidelyusedinmanylong_spanstructures,includingcable_supportbridges,guyedtowersandcable_supportroofs.Sincecablesarelight,veryflexibleandlightlydamped ,structuresutilizingcables,i.e .,cable_structuresystems,usuallyhavevariousdynamicproblems.Theirmodelsarethereforeverimportantinpredictingandcontrollingtheirresponses.Inthelastdecade,thenonlineardynamicvibrationandstabilitybehaviorofcablesandcable_structureshavedrawntheattentionofman…     

Stochastic finite element analysis of non-linear plane trusses

—This study considers the responses of geometrically and materially non-linear plane trusses under random excitations. The stress-strain law in the inelastic range is based on an explicit differential equation model. After a total Lagrangian finite element discretization, the nodal displacements satisfy a system of stochastic non-linear ordinary differential equations with right-hand-sides given by random functions of time. The exact solution of the above stochastic differential equation is generally difficult to obtain. To seek an approximate solution with good accuracy and reasonable computational effort, the stochastic linearization method is used to find the first and second statistical moments (i.e. the mean vector and the one-time covariance matrix) of the nodal displacements. Results of simple structures under Gaussian white-noise excitation indicate that the proposed method has good accuracy (generally underestimates the r.m.s. stationary response by 5–14%) and requires only a small fraction of the computation time of the time-history Monte-Carlo method.     

Nonlinear vibrations in beams and frames: The effect of the deformed equilibrium state

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.     

Constructing an analytical solution for lamb waves using the cosserat continuum approach

The problem of propagation of a Lamb elastic wave in a thin plate is considered using the Cosserat continuum model. The deformed state is characterized by independent displacement and rotation vectors. Solutions of the equations of motion are sought in the form of wave packets specified by a Fourier spectrum of an arbitrary shape for three components of the displacement vector and three components of the rotation vector which depend on time, depth, and the longitudinal coordinate. The initial system of equations is split into two systems, one of which describes a Lamb wave and the second corresponds to a transverse wave whose amplitude depends on depth. Analytical solutions in displacements are obtained for the waves of both types. Unlike the solution for Lamb waves, the solution obtained for the transverse wave has no analogs in classical elasticity theory. The solution for the transverse wave is compared with the solution for the Lamb wave. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 143–150, January–February, 2007. An erratum to this article is available at .     

Instability of sloshing motion in a vessel undergoing pivoted oscillations

Suspending a rectangular vessel partially filled with an inviscid fluid from a single rigid pivoting rod produces an interesting physical model for investigating the dynamic coupling between the fluid and vessel motion. The fluid motion is governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion is given by a modified forced pendulum equation. The fully nonlinear, two-dimensional, equations of motion are derived and linearised for small-amplitude vessel and free-surface motions, and the natural frequencies of the system analysed. It is found that the linear problem exhibits an unstable solution if the rod length is shorter than a critical length which depends on the length of the vessel, the fluid height and the ratio of the fluid and vessel masses. In addition, we identify the existence of 1:1 resonances in the system where the symmetric sloshing modes oscillate with the same frequency as the coupled fluid/vessel motion. The implications of instability and resonance on the nonlinear problem are also briefly discussed.     

微分求积方法对于桩基大变形分析的应用

本文基于大变形的理论,采用弧坐标首先建立了具有初始位移的桩基的非线性数学模型,一组强非线性的微分-积分方程,其中,地基的抗力采用了Winkeler模型;其次,引入变数变换将微分-积分方程转化为一组非线性微分方程,并用微分求积方法离散了方程组,得到一组离散化的非线性代数方程;最后用Newton-Raphson迭代方法对离散化方程进行了求解,得到了桩基变形前后的构形、弯矩和剪力.计算中选取了两种不同类型的初始位移,并考察了它们对桩基大变形力学行为的影响.     

细长结构几何非线性分析的子结构方法

将细长结构沿长度方向划分为多个子结构,并在每个子结构上建立一个随结构一起运动的连体基,则结构内任意点的位移可分解为连体基的转动和相对于连体基的小位移。利用细长结构这样的变形特征,本文详细讨论了连体基的转动,给出了与连体基选择方式相协调的节点位移及其虚变分表达式,并将子结构内部位移凝聚到了边界节点上。在此基础上,提出了一种细长结构几何非线性分析的子结构方法,可在不损失计算精度的前提下大幅度降低求解规模,从而提高了计算效率。数值算例验证了所提方法的有效性。     

弹性薄板分析的条形传递函数方法

提出一种用于矩形弹性薄板变形分析的条形传递函数方法．一个矩形区域首先沿某一个方向被剖分成若干个条形子域,分割这些子域的直线称为结线,在结线上定义位移函数,它是结线坐标的一维函数,结线的两个端点称为结点．为适应复杂边界条件,在边界结线上定义若干结点,该结线的位移函数用结点位移参数插值表示．每个条形子域的变形用结线位移函数和适当的插值函数（形函数）表示．结线位移函数和结点位移参数满足的平衡微分方程及代数方程由变分原理给出     

Non-linear global dynamics of an axially moving plate

In the present study, the geometrically non-linear dynamics of an axially moving plate is examined by constructing the bifurcation diagrams of Poincaré maps for the system in the sub and supercritical regimes. The von Kármán plate theory is employed to model the system by retaining in-plane displacements and inertia. The governing equations of motion of this gyroscopic system are obtained based on an energy method by means of the Lagrange equations which yields a set of second-order non-linear ordinary differential equations with coupled terms. A change of variables is employed to transform this set into a set of first-order non-linear ordinary differential equations. The resulting equations are solved using direct time integration, yielding time-varying generalized coordinates for the in-plane and out-of-plane motions. From these time histories, the bifurcation diagrams of Poincaré maps, phase-plane portraits, and Poincaré sections are constructed at points of interest in the parameter space for both the axial speed regimes.     

Nonlinear vibrations of FGM rectangular plates in thermal environments

Geometrically nonlinear vibrations of FGM rectangular plates in thermal environments are investigated via multi-modal energy approach. Both nonlinear first-order shear deformation theory and von Karman theory are used to model simply supported FGM plates with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. A pseudo-arclength continuation and collocation scheme is used and it is revealed that, in order to obtain the accurate natural frequency in thermal environments, an analysis based on the full nonlinear model is unavoidable since the plate loses its original flat configuration due to thermal loads. The effect of temperature variations as well as volume fraction exponent is discussed and it is illustrated that thermally deformed FGM plates have stronger hardening behaviour; on the other hand, the effect of volume fraction exponent is not significant, but modal interactions may rise in thermally deformed FGM plates that could not be seen in their undeformed isotropic counterparts. Moreover, a bifurcation analysis is carried out using Gear’s backward differentiation formula (BDF); bifurcation diagrams of Poincaré maps and maximum Lyapunov exponents are obtained in order to detect and classify bifurcations and complex nonlinear dynamics.     

Bending analyses of 1D orthorhombic quasicrystal plates

The meshless Petrov–Galerkin method (MLPG) is applied to plate bending analysis in 1D orthorhombic quasicrystals (QCs) under static and transient dynamic loads. The Bak and elasto-hydrodynamic models are applied for phason governing equation in the elastodynamic case. The phason displacement for the orthorhombic QC in the first-order shear deformation plate theory depends only on the in-plane coordinates on the mean plate surface. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding this node. The coupled governing partial differential equations are satisfied in a weak-form on small fictitious subdomains. The spatial variations of the phonon and phason displacements are approximated by the moving least-squares (MLS) scheme. After performing the spatial MLS approximation, a system of ordinary differential equations (ODEs) for nodal unknowns is obtained. The system of the ODEs of the second order is solved by the Houbolt finite-difference scheme. Our numerical examples demonstrate clearly the effect of the coupling parameter on both static and dynamic phonon/phason deflections.