Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: Non-linear model and stability analysis

The stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end is analyzed. The beam is herein modeled as a non-linear Cosserat rod model. Non-linear, partial integro-differential equations of motion are derived expanded up to cubic terms in the transversal displacement and torsional angle of the beam. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Interaction between conservative and non-conservative loads with respect to the stability problem is discussed. The critical spectral properties are derived and the corresponding critical eigenspace is evaluated.     

随从力作用下热弹耦合轴向运动梁的稳定性

研究了切向均布随从力作用下热弹耦合轴向运动梁的稳定性问题。建立了热弹耦合轴向运动梁 在随从力作用下的运动微分方程,采用归一化幂级数法,推导出了2种边界条件下热弹耦合轴向运动梁在随 从力作用下的特征方程。计算了系统的前3阶量纲一复频率,分析了量纲一运动速度、量纲一热弹耦合系数 和量纲一随从力等参数对梁的稳定性的影响。     

Series solution for large deflections of a cantilever beam with variable flexural rigidity

In this paper large deflection and rotation of a nonlinear Bernoulli-Euler beam with variable flexural rigidity and subjected to a static co-planar follower loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytical technique for nonlinear problems, namely the Homotopy Analysis Method (HAM). The present solution can be used for the analysis of a wide range of loads, material/cross section properties and lengths for beams undergoing large deformations. The results obtained from HAM are compared with results reported in previous works. Finally, the load–displacement characteristics of a uniform cantilever beam with different material properties under a follower force applied normal to the deformed beam axis are presented.     

Investigation of the Effect of Axial Loads on the Transverse Vibrations of a Vertical Cantilever with Variable Parameters

The method of influence function is applied to the solution of the boundary-value problem on the free transverse vibrations of a vertical cantilever and a bar subjected to axial loads. To demonstrate the capabilities of the method, a cantilever with the free end under two types of loading — point forces (conservative and follower) and a load distributed along the length (dead load) — is analyzed. A characteristic equation in the general form, which does not depend on the cantilever shape and on the type of axial load, is given. The Cauchy influence function depends on the cantilever shape and the type of axial load. As an example, a tapered cantilever subjected to conservative and follower forces and an elastically supported bar under the dead load are considered in detail. The characteristic equation derived allows one to evaluate the natural frequencies and the Euler critical loads. It is shown that the calculated natural frequencies and critical forces are in a good agreement with the exact values when several terms are retained in the characteristic series. The high accuracy of the method is also confirmed     

Post-buckling of a hinged-fixed beam under uniformly distributed follower forces

Based on geometrically non-linear theory for extensible elastic beams, governing equations of statically post-buckling of a beam with one end hinged and the other fixed, subjected to a uniformly distributed, tangentially compressing follower forces are established. They consist of a boundary-value problem of ordinary differential equations with a strong non-linearity, in which seven unknown functions are contained and the arc length of the deformed axis is considered as one of the basic unknown functions. By using shooting method and in conjunction with analytical continuation, the non-linear governing equations are solved numerically and the equilibrium paths as well as the post-buckled configurations of the deformed beam are presented. A comparison between the results of conservative system and that of the non-conservative systems are given. The results show that the features of the equilibrium paths of the beams under follower loads are evidently different from that under conservative ones.     

ON THE STVENANT FLEXURE FOR CROSS SECTIONS OF A SERFIES OF SYMMETRICAL AIRFOLLS

本文讨论一系列对称翼截面柱体的弯曲问题,柱体空端锁甲外力的方向与主题的对称面垂直.这问题一般可简化为寻找三个平面挑和函数,其中两个函数给出边界值,另一函数给出边界上法向的微商.我们设法将这些调和函数的边界值表达为複变函数的实数或虚数部分,这样所得的复变函数在柱体截面内常会表现有奇异点.本方法中的一个重要内容为：如何削去这些奇异点.本文求得问题的精确普遍解,其中含一个参数,可用来调节截面的厚度.本文最后给出一个例子,阐明求应力函数、扭转厚度及湾区中心的步骤,给出受扭转和弯曲时沿截面边界上剪切的分布情况.     

Non-conservative stability of multi-step non-uniform columns

The non-conservative stability of non-uniform columns under the combined action of concentrated and variably distributed forces is solved analytically. Two types of follower force system are considered: (i) concentrated follower forces and variably distributed follower forces, (ii) concentrated follower forces and variably distributed conservative forces. The exact solutions for stability of four kinds of one-step non-uniform columns subjected to the two types of follower force system are derived for the first time. Then a new exact approach, which combines the exact solutions of one-step columns and the transfer matrix method, is presented for the non-conservative stability analysis of multi-step non-uniform columns. The advantage of the proposed method is that the resulting eigenvalue equation for a multi-step non-uniform column with any kinds of two end support configurations, an arbitrary number of spring supports and concentrated masses can be conveniently determined from a second order determinant. The decrease in the determinant order, as compared with previously developed procedures, leads to significant savings in the computational effort. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM.     

Cantilever beam equilibrium configurations

This article investigates a method for obtaining all equilibrium configurations of a cantilever beam subjected to an end load with a constant angle of inclination. The formulation is based on plane finite-strain beam theory in the elastic domain. An example of a cantilever beam subjected to a horizontal pressure force is discussed in detail.     

非保守功能梯度弹性组合曲梁的精确模型及其数值解

基于直法线假设,采用可伸长梁的几何非线性理论,建立了功能梯度材料弹性组合曲梁受切线均布随从力作用下的静态大变形数学模型。该模型不仅计及了轴线伸长,同时也精确地考虑了梁的初始曲率对变形的影响以及轴向变形与弯曲变形之间的耦合效应。用打靶法数值求解了由金属和陶瓷两相材料所构成的一种FGM组合曲梁在沿轴线均布切向随动载荷作用下的非线性平面弯曲问题,给出了不同梯度指标下FGM弹性曲梁随载荷参数大范围变化的平衡路径,并与金属和陶瓷两种单相材料曲梁的相应特性进行了比较。     

Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions

The paper provides an exact analytical solution for the equilibrium configurations of a cantilever rod subject to inclined force and tip moment acting on its free end. The solution is given in terms of Jacobi’s elliptical functions and illustrated by several numerical examples and several graphical presentations of shapes of deformed cantilevers. Possible forms of the underlying elastica of a cantilever are discussed in detail, and various simple formulas are given for calculating the characteristic dimensions of the elastica. For the case when a cantilever is subject only to applied force, three load conditions are discussed: the follower load problem, the load determination problem, and the conservative load problem. For all cases, either a formula or an effective procedure for determining the solution is provided. In particular, a new efficient procedure is given to determine all possible equilibrium shapes in the case of the conservative load problem.     

Die Wirkung äu\erer und innerer Dämpfung auf die Stabilität des Prandtlschen Kragträgers unter mitgehender Momentenbelastung

übersicht über das Prinzip von Hamilton wird das nichtlineare Randwertproblem eines kippenden Kragträgers hergeleitet, der durch ein mitgehendes, zeitunabhängiges Endmoment belastet wird. Die Stabilität des sich einstellenden Gleichgewichtszustandes wird untersucht. Der Einflu\ sowohl äu\erer als auch innerer Dämpfung auf die kritische Last wird eingehend diskutiert.

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Effects of external and internal damping on the stability of Prandtl's cantilever subjected to a follower bending moment

Summary Using Hamilton's principle the non-linear boundary value problem for lateral buckling of a cantilever beam subjected to a follower time-independent bending moment is derived. The stability of the resulting state of equilibrium is studied. The influence of external and internal damping on the critical load is discussed in detail.

     

STATIC STUDY OF CANTILEVER BEAM STICTION UNDER ELECTROSTATIC FORCE INFLUENCE

I. INTRODUCTION For capacitor-like microelectromechanical systems (MEMS) structure[1??6], the voltage betweenthe structure and substrate causes attractive force. The sources of the voltage can be an arti?ciallymounted device[2 , 7??9] or the temporar…     

Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory

This two-part contribution presents a beam theory with a non-uniform warping including the effects of torsion and shear forces, and valid for any homogeneous cross-section made of isotropic elastic material. Part I is devoted to the theoretical developments and part II discusses analytical and numerical results obtained for torsion and shear-bending of cantilever beams made of different kinds of cross-section. The theory is based on a kinematics assuming that the cross-section maintains its shape and including three independent warping parameters associated to the three warping functions corresponding to torsion and shear forces. Starting from this displacement model and using the principle of virtual work, the corresponding beam theory is derived. For this theory, closed-form results are obtained for the cross-sectional constants and the three-dimensional expressions of the normal and shear stresses. Comparison with classical beam theories is carried out and additional effects due to the non-uniformity of the warping are highlighted. In particular, the contributions of primary and secondary internal forces and the effect of the non-symmetry of the cross-section on the structural behavior of the beam are specified. Simplified versions of this theory, wherein the number of degrees of freedom is reduced, are also presented. The analytical and numerical analyzes presented in part II give responses on the quality of this non-uniform beam theory and indicate also when its simplified versions could be applied.     

Plastic dynamic stability of a column under nonconservative forces

I.Introduction'Whenadeformableobjectmovinginamedium,thedirectionsofinteractiveforcessuchasfrictionalforceshouldvarywiththedeformationoftheobject.Thisisakindofnonconservativeforces.Forexample,thedirectionofgasfrictionalforceactingontherocketsandthefossiles…     

A consistent equilibrium in a cross-section of an elastic–plastic beam

A phenomenon of inequality of equilibrium and constitutive internal forces in a cross-section of elastic–plastic beams is common to many finite element formulations. It is here discussed in a rate-independent, elastic–plastic beam context, and a possible treatment is presented. The starting point of our discussion is Reissners finite-strain beam theory, and its finite element implementation. The questions of the consistency of interpolations for displacements and rotations, and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem. Approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method. The novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross-section. An advantage of the present approach is that the locations, where the balance of equilibrium and constitutive moments should be satisfied, can be prescribed in advance. In order to minimize the error, explicit analytical expressions are used for the constitutive forces; for a rectangular cross-section and bilinear constitutive law, they are given in Appendix A. The comparison between the results of the two finite element formulations, the one using consistent, and the other inconsistent equilibrium in a cross-section, is shown for a cantilever beam subjected to a point load. The problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method, in which the coordinate system is transformed such to follow high gradients of curvature.     

非保守力作用下FGM矩形板的稳定性分析

对受均布随从力作用的功能梯度材料(FGM)矩形板,引入应力函数,得到了以应力函数和挠度函数表示的耦合运动微分方程组。用Fourier级数法研究了四边简支FGM非保守矩形板的稳定性,给出了不同边长比和不同梯度指标下频率和发散载荷的变化曲线,以及梯度指标变化对频率和发散载荷的影响。     

Dynamic edge angles of wetting upon spreading of a drop over a solid surface

The hydrodynamic free-boundary problem of the axisymmetric spreading of a viscous-fluid drop over the smooth surface of a solid under the action of capillary forces and under the conditions of weak gravitation is considered. For finite inclination angles of the free surface and small capillary numbers, the problem is reduced to the simpler hydrodynamic problem in a region with known boundary by the asymptotic method. An expression for the dynamic edge angle of the drop is obtained. It is shown that in addition to the local inclination angle of the boundary near the contact line of three phases, one drop has several dynamic edge angles. These angles are calculated for small Reynolds and Bond numbers. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences, Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 101–107, January–February, 1999.     

FLUTTER INSTABILITY OF SUPPORTED PIPES CONVEYING FLUID SUBJECTED TO DISTRIBUTED FOLLOWER FORCES

In the past decades,it has been reported that divergence is the expected form of instability for fluid-conveying pipes with both ends supported.In this paper,the form of instability of supported pipes ...     

Bending of Bars under a Follower Load

Exact solutions of the problem of nonlinear bending of a thin bar under a point follower load are given. The problem is studied for an arbitrary follower angle and the particular cases of axial and transverse follower forces are considered. The solutions are written in unified parametric form and expressed in terms of Jacobi elliptic functions.     

On indentation of a pair of rigid punches connected by an elastic beam into an elastic half-plane with regard to the friction and adhesion forces in the contact region

The contact problem of indentation of a pair of rigid punches with plane bases connected by an elastic beam into the boundary of an elastic half-plane is considered under the conditions of plane strain state. The external load is generated by lumped forces applied to the punches and a uniformly distributed normal load acting on the beam.It is assumed that the contact between the punch and the elastic half-plane can be described by L. A. Galin’s statement, i.e., it is assumed that the adhesion acts in the interior part of each of the contact regions and the tangential stresses obeying the Coulomb law act on their boundaries.With the symmetry taken into account, the problem is stated only for a single punch, and solving this problem is reduced to a system of four singular integral equations for the tangential and normal stresses in the adhesion region and the contact pressure in the sliding zones. The solution of the constitutive system together with three conditions of equilibrium of the system of punches connected by a beam is constructed by direct numerical integration by the method of mechanical quadratures.As a result of the numerical analysis, the contact stress distribution functions were constructed and the values of the sliding zones and the punch rotation angle were determined for various values of the geometric, elastic, and force characteristics.