Dynamic stability and response of fluttered beams subjected to random follower forces

This paper presents a study of non-linear response of a fluttered, cantilevered beam subjected to a random follower force at the free end. The random follower force is characterized as the sum of a post-critical static force and a stationary process with a zero mean. First, the Ritz-Galerkin method is applied to yield a set of discretized system equations. The system equations are then partially uncoupled by a special modal analysis based on normal modes of the corresponding linear, autonomous system at the onset of fluttering. Next, the stochastic averaging method is utilized to get Ito's differential equation governing the amplitude of the fluttered mode. Finally, the probability density function for the amplitude of the fluttered mode is obtained by solving the FPK equation. Numerical results show that the probability density function for the amplitude of the fluttered mode is determined by the sample behavior of the beam near the trivial equilibrium configuration.     

Vibration of thin pre-twisted helical beams

A helical beam has non-zero curvature and tortuosity. When there is a pre-twist, the Frenet triad is not necessary coincident with the principal triad. When an initially straight rod undergoing a large deformation, it will be curved and twisted during each deformation sequence. The purpose of the paper is to establish the governing equations and the associated natural boundary conditions for a pre-twisted helical beam by variational principles and differential geometry. Although circular helix is taken as example, the formulation is valid for inhomogeneous and non-uniform parameters along the centerline that variable curvature and tortuosity can be treated without difficulties. The method can be used to treat helical anisotropy. If the pre-twist rate of a ring is 0.5, we show a thick Möbius ring for the first time. The solution of the natural vibration problem is through Chebyshev discretization and Clenshaw–Curtis integration by means of the Galerkin formulation.     

Dynamic characteristics of multi-span spinning beams with elastic constraints under an axial compressive force

A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed. The displacement and bending angle functions are represented through an improved Fourier series, which ensures the continuity of the derivative at the boundary and enhances the convergence. The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation. The efficiency and ac...     

Dynamic stability for a simply supported beam under periodic axial excitation

This paper studies the dynamic stability for a simply supported straight beam under periodic axial excitation by using the averaging method and the Routh-Hurwitz stability criteria. By considering the first two modes coupled, we discuss the effect of the stability-instability region and the amplitudes of vibration. Furthermore, by studying the principal parametric resonance i.e. subharmonic order $\text{1}\text{2}$, we investigate the effect of the amplitude of the main system by various kinds of non-linearities of the subsystem. Finally, by obtaining the transient results, we describe the beat phenomenon, and harmonic oscillation.     

Dynamic stability of a viscoelastic rotating shaft under parametric random excitation

This paper deals with dynamic stability of a viscoelastic rotating shaft subjected to a parametric random axial compressive thrust, by using moment Lyapunov exponents and the largest Lyapunov exponents as indicators. The equation of motion for the shaft is derived, which is a system of gyroscopic stochastic differential equations. The method of stochastic averaging is used to decouple the governing equations into Itô equations, from which the moment Lyapunov exponent is obtained by using mathematical transformations only. The largest Lyapunov exponent is obtained through its relation with moment Lyapunov exponents. The effects of various parameters on the stochastic dynamic stability are discussed. The approximate analytical results are confirmed by Monte Carlo simulation.     

含脱层损伤复合材料层合梁的冲击动力屈曲

针对含初始缺陷和脱层损伤的复合材料层合梁的轴向冲击动力屈曲问题进行了分析。基于Hamilton原理导出了考虑初始缺陷、轴向和横向惯性、横向剪切变形以及转动惯性影响时含脱层损伤复合材料梁的非线性动力屈曲控制方程;基于B-R准则,采用有限差分方法求解了受轴向冲击载荷作用下含脱层损伤复合材料梁的动力屈曲问题;讨论了冲击速度、初始几何缺陷、铺层角度以及脱层长度等因素对复合材料层合梁动力屈曲的影响。     

Dynamic stability analysis of shear-flexible composite beams

Dynamic stability behavior of the shear-flexible composite beams subjected to the nonconservative force is intensively investigated based on the finite element model using the Hermitian beam elements. For this, a formal engineering approach of the mechanics of the laminated composite beam is presented based on kinematic assumptions consistent with the Timoshenko beam theory, and the shear stiffness of the thin-walled composite beam is explicitly derived from the energy equivalence. An extended Hamilton’s principle is employed to evaluate the mass-, elastic stiffness-, geometric stiffness-, damping-, and load correction stiffness matrices. Evaluation procedures for the critical values of divergence and flutter loads of the nonconservative system with and without damping effects are then briefly introduced. In order to verify the validity and the accuracy of this study, the divergence and flutter loads are presented and compared with the results from other references, and the influence of various parameters on the divergence and flutter behavior of the laminated composite beams is newly addressed: (1) variation of the divergence and flutter loads with or without the effects of shear deformation and rotary inertia with respect to the nonconservativeness parameter and the fiber angle change, (2) influence of the internal and external damping on flutter loads whether to consider the shear deformation or not.     

DYNAMIC STABILITY OF AXIALLY MOVING VISCOELASTIC BEAMS WITH PULSATING SPEED

IntroductionThe class of systems with axially moving materials involves power transmission chains,band saw blades and paper sheets during processing. Vibration of such systems is generallyundesirable. The traveling tensioned Euler-Bernoulli beam is the pr…     

Finite element stability analysis of pre-twisted Beck's column

Summary This paper presents a stability and vibration analysis of a pre-twisted cantilever column subjected to vertical and follower loads at its free end. The investigation is carried out using finite element modelling where the effects of transverse shear deformation and rotatory inertia are taken into account in deriving the element matrices. By using the assembly procedure and imposing the boundary conditions, a relationship between the load and vibration frequency is obtained. The critical, divergence and flutter, loads are calculated by evaluating the relationship and shown in figures for various values of non-conservativeness parameter, of twist angle, and those of rigidity ratio.

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Untersuchung der Stabilität des verwundenen Beck-Stabes mit Finiten Elementen

Übersicht Dieser Aufsatz stellt eine Stabilitäts- und Schwingungsuntersuchung des vorverwundenen, einseitig eingespannten Stabes dar, der an seinem freien Ende vertikalen und mitgehenden Lasten ausgesetzt ist. In der Untersuchung wird die Methode der finiten Elemente angewendet, und in der Herleitung der Elementmatrizen werden die Einflüsse der Schubspannungen infolge der Querkraft und der Rotationsträgheit berücksichtigt. Eine Beziehung zwischen der Last und der Schwingungsfrequenz wird dadurch ermittelt, daß man die Gesamtssystemmatrix bildet und die Randbedingungen anwendet. Durch die Auswertung der Beziehung werden die kritischen Divergenz- und Flatterlasten für verschiedene Werte des Nichtkonservativitätsparameters, des Verwindungswinkels und des Steifigkeitsverhältnisses ermittelt und in Bildern dargestellt.

     

含初缺陷裂纹损伤梁的冲击动力屈曲

由Hamilton原理导出考虑初始缺陷及横向剪切变形时裂纹梁的动力屈曲控制方程;应用断裂力学中常用的线弹簧模型将裂纹引入到屈曲控制方程中;基于B-R动力屈曲判断准则,采用数值方法求解了受轴向冲击载荷作用时裂纹梁的动力屈曲;对比讨论了不同冲击速度、初始几何缺陷大小以及分布形式等因素对梁冲击动力屈曲的影响。     

Coupled free,torsional and axial vibration of pre-twisted bars

Summary The present paper deals with the calculation of natural frequencies in a pre-twisted solid with constant cross-section.The axial and torsional displacements of cross-sections are taken into account, as well as the relative inertia forces and couples.The differential equations of the vibration are integrated and their general solution is given.In the paper, by such solution, among the possible cases only the three simplest ones are considered (free-free, fixed-fixed, fixed-free ends) and a numerical example is carried out which enables to evaluate the several orders of magnitude. Other more complex cases will be treated in other papers.

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Sommario Nel presente lavoro viene affrontato il problema della determinazione delle frequenze proprie di vibrazione di un solido svergolato a sezione uniforme.Il problema viene risolto tenendo conto degli spostamenti assiali e torsionali di ogni sezione e delle relative forze e coppie di inerzia.Le equazioni differenziali del moto vengono integrate e ne viene fornita la soluzione generale.Mediante tale soluzione vengono esaminati, fra tutti i diversi casi possibili, i tre casi pratici più semplici (estremità del solido entrambe libere, entrambe incastrate, una libera e l'altra incastrata) e viene inoltre svolto un esempio numerico che consente di ottenere i diversi ordini di grandezza.I casi di maggiore complessità verranno esaminati in successivi lavori.

     

Dynamic stability analysis and DQM for beams with variable cross-section

The present paper deals with the dynamic behaviour of a clamped beam subjected to a sub-tangential follower force at the free end. The aim of this work is to obtain the frequency–axial load relationship for a beam with a variable circular cross-section. In this way, one can identify both divergence critical loads – where the frequency goes to zero – and the flutter critical load – in correspondence with two frequencies coalescence. The numerical approach adopted for solving the partial differential equation of motion is the differential quadrature method (henceforth DQM). This method was proposed by Bellmann and Casti [Bellmann, R.E., Casti, J., 1971. Differential quadrature and long-term integration. J. Math. Anal. 34, 235–238] and has been employed recently in the solution of solid mechanics problems by Bert and Malik [Bert, C.W., Malik, M., 1996. Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev., ASME, 49 (1), 1–28] and Chen et al. [Chen, W., Stritz, A.G., Bert, C.W., 1997. A new approach to the differential quadrature method for fourth-order equations. Int. J. Numer. Method Eng. 40, 1941–1956]. More precisely, a modified version of this method has been used, as proposed by De Rosa and Franciosi [De Rosa, M.A., Franciosi, C., 1998a. On natural boundary conditions and DQM. Mech. Res. Commun. 25 (3), 279–286; De Rosa, M.A., Franciosi, C., 1998b. Non classical boundary conditions and DQM. J. Sound Vibrat. 212(4), 743–748] to satisfy all the boundary conditions.Some frequencies–axial loads relationships are reported in order to show the influence of tapering on the critical loads.     

Analysis of bending and buckling of pre-twisted beams: A bioinspired study

Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanical properties of elas- tic beams by using the Timoshenko beam model. Particular attention is paid to the transverse bending and axial buckling of a pre-twisted rectangular beam. The analytical solution is first derived for the deflection of a clamped-free beam under a uniformly or periodically distributed transverse force. The critical buckling condition of the beam subjected to its self- weight and an axial compressive force is further solved. The results show that the twisting morphology can significantly improve the resistance of beams to both transverse bending and axial buckling. This study helps understand some phenomena associated with twisting chirality in nature and provides inspirations for the design of novel devices and structures.     

Dynamic response of beams under moving loads with finite deformation

Nonlinear Dynamics - A novel material parameter-dependent model is proposed in this work to investigate the nonlinear vibration of a beam under a moving load within a finite deformation framework....     

Bernoulli–Euler beams with random field properties under random field loads: fractal and Hurst effects

Responses of Bernoulli–Euler beams with random field properties and also possibly under random field forcing are studied for random fields with linear, Matérn, Cauchy, and Dagum covariances. The latter two allow decoupling of the fractal dimension and Hurst effect. We find second-order characteristics of the beam displacement under various boundary conditions. In a number of cases, the results may be obtained in explicit analytical (albeit lengthy) forms, but as Cauchy and Dagum models are being introduced, one has to resort to numerics.     

Dynamic stability of electrostatically actuated initially curved shallow micro beams

Micro and nano devices incorporating bistable structural elements have functional advantages including the existence of several stable configurations at the same actuation force, extended working range, and tunable resonant frequencies. In this work, after a short review of operational principles of bistable micro devices, results of a theoretical and numerical investigation of the transient dynamics of an initially curved, shallow, double-clamped micro beam, actuated by distributed electrostatic and inertial forces are presented. Due to the unique combination of mechanical and electrostatic nonlinearities, typically not encountered in large scale structures, the device exhibits sequential snap-through and electrostatic (pull-in) instabilities. A phase plane analysis, performed using a consistently derived lumped model along with the numerical results, indicate that critical voltages corresponding to the dynamic snap-through and pull-in instabilities are lower than their static counterparts, while the minimal curvature required for the appearance of the dynamic snap-through is higher than in the static case. The boundaries of the bistability region of a quasi-statically loaded beam are found in terms of the geometrical and loading parameters and are shown to be bounded from above by the dynamic pull-in instability. Some of the post-buckling states cannot be reached under suddenly applied or quasi-statically increasing voltages: specially tailored loading schemes are suggested for realization of these configurations often beneficial in applications.     

压杆的动态稳定性

在动力学范畴内讨论压杆稳定性问题. 证明压杆的一次近似动态Lyapunov稳定性要求静态Lyapunov稳定性与欧拉稳定性条件同时得到满足.