Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration.     

Optimal shape of a twisted and compressed rod

We formulate and solve the problem of determining the shape of an elastic rod stable against buckling and having minimal volume. The rod is loaded by a concentrated force and a couple at its ends. The equilibrium equations are reduced to a single nonlinear second-order equation. The eigenvalues of the linearized version of this equation determine the stability boundary. By using Pontryagin's maximum principle we determine the optimal shape of the rod.     

Buckling of a flexible shaft under torque loads transmitted by Cardan joints

Summary The century-old Greenhill problem of buckling of a flexible shaft, supported at its ends and loaded by external torques, is reconsidered in the case where the loads are transmitted by Cardan joints. It is found that the critical torque is roughly half the Greenhill value and that it depends discontinuously on the relative position of the Cardan joints in the undeformed and unloaded shaft.

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übersicht Das ein Jahrhundert alte Greenhillsche Problem, der Knickung einer auf Drillung beanspruchten Welle wird erneut betrachtet für den Fall, da? das Torsionsmoment von Kardangelenken an den Enden übertragen wird. Das kritische Drillungsmoment ist ganz grob die H?lfte des Greenhillschen Werts und es h?ngt diskontinuierlich ab von der relativen Lage der Kardangelenke in der unbelasteten Welle.

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Dedicated to Professor Dr. Hans Ziegler on the occasion of his 70-th birthday     

Buckling problem for a rod longitudinally compressed by a force smaller than the Euler critical force

It was earlier shown that a rod can buckle under the action of a sudden longitudinal load smaller than the Euler critical load. The buckling mechanism is related to excitation of periodic longitudinal waves generated in the rod by the sudden loading, which in turn lead to transverse parametric resonances. In the linear approximation, the transverse vibration amplitude increases unboundedly, and in the geometrically nonlinear approach, beats with energy exchange from longitudinal to transverse vibrations and back can arise. In this case, the transverse vibration amplitude can be significant. In the present paper, we study how this amplitude responds to the following two factors: the smoothness of application of the longitudinal force and the internal friction forces in the rod material.     

压杆失稳与Liapunov稳定性

根据Kirchhoff理论和Liapunov理论的分析,压杆的平衡状态稳定,而拉杆的平衡状态不稳定. 此结论与压杆失稳的传统理论相悖. 本文解释此现象的产生原因,并说明在应用Liapunov理论讨论静力学中的稳定性问题时,由于时间变量改变为空间变量,运动稳定性理论所反映的物理过程将产生根本改变.     

Elastic deformation of a shaped twisted rod

Kiev Construction Engineering Institute. Translated from Prikladnaya Mekhanika, Vol. 24, No. 8, pp. 85–91, August, 1988.     

The stability of a compressed rotating rod

The classical stability problem of a compressed hinged elastic rod rotating with constant angular velocity about the axis that passes through the hinges is considered. It is assumed that the compressive force is constant and the line of its action coincides with the axis of rotation of the rod. The stability of a solution of the nonlinear problem that describes deformation of the rod under the action of the compressive force and the distributed centrifugal load is studied within the framework of the stability theory of dynamic systems with distributed parameters. The buckling paramcters of the problem are determined. Calculation results are given. Technology Institute, Altai State Technical University, Biisk 659305. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 190–197, July–August, 2000.     

Buckling modes of a compressed plate on an elastic substrate

The buckling modes of a homogeneously compressed elastic plate on a soft elastic substrate are studied. The critical compression is uniquely determined by the bifurcation equation, but this compression is associated with a wide set of buckling modes. It was proved that any solution of the Helmholtz equation satisfies the bifurcation equation. At the same time, in microelectronics, it is required to know which buckling mode is realized. Experimental and theoretical investigations show that the chessboard-like buckling mode should be expected. In what follows, this problem is discussed theoretically. The expected buckling mode can be found by analyzing the energy of the initial postcritical deformation, and the desired mode is determined from the condition of its minimum. The analytic expression of this energy is obtained. Its minimization results in the chessboard-like buckling mode.     

An approximation of the form of a compressed flexible rod

A compact algorithm is proposed for exact calculation of the coordinates of the plane elastic line of an axially compressed flexible rod under any loads. Refined approximate formulas are obtained for calculation of the coordinates of the elastic line with an error not greater than 1% of the rod length even for loads which exceed the critical Euler load by 30%. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 200–203, May–June, 1999.     

Initial transgritical behavior in a compressed rod under creep conditions

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 152–156, March–April, 1993.     

测量压杆临界压力的理论与实验

从细长压杆临界压力的定义出发,分析了实验测定压杆临界压力的困难,阐述了用振动方法测量细长压杆临界压力的理论,介绍了在轴向拉力和压力状态下用振动方法测量压杆临界压力的实验,并与材料力学中通常的压杆稳定实验------测量压溃载荷的实验进行了比较,得出结论：用振动方法可以比较准确地测得细长压杆的临界压力.