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The effect of the type of springs on the equilibrium states of an inverted pendulum is examined. The angular and linear eccentricities of the follower force are taken into account __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 126–137, June 2007.  相似文献   

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This paper studies the vibration and stability of an elastica constrained by a pair of symmetrically placed parallel plane walls. One end of the elastica is fully clamped, while the other end is allowed to slide through a rigid channel under edge thrust. In order to take into account the variation of the contact points between the elastica and the walls during vibration, an Eulerian version of the equations of motion is adopted. It is found that the lowest few natural frequencies approach and remain degenerately zero when point-contact deformations evolve to line-contact patterns. As a consequence, the stability of all line-contact deformations before secondary buckling cannot be determined from the linear vibration analysis. A load-controlled experiment was conducted to find that the elastica jumped from one-point to two-point, and then to three-point contact with the walls without going through any line-contact deformations. These experimental observations are different from the results reported previously by others with different set-ups, in which line-contact deformations did exist. Explanations based on experimental evidences and theoretical analyses are provided to confirm the validity of these previous investigations and clarify the cause of the difference.  相似文献   

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The effect of the linear eccentricity of the follower force on the equilibrium states of an inverted pendulum is examined. Bifurcation points and catastrophes associated with changes in pendulum parameters and type of springs are analyzed. Phase flows are plotted __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 121–129, April 2007.  相似文献   

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This paper studies the snap-through buckling of a hinged elastica subject to a midpoint force. The focus is placed on how different load models affect the deformation and snapping load. Three different load models are considered. In the first model, the point force is fixed onto a midpoint of the elastica. In the second model, the point force is fixed on a central line in space. In the third model, the external force is applied through a rigid bar, which is allowed to slide along the central line in space. When the loaded elastica deforms symmetrically, as in the case when the magnitude of the point force is small, there is no difference between these three load models. However, when the elastica deforms unsymmetrically, the three load models produce different results. Vibration method is used to determine the stability of the equilibrium configurations. It is found that the elastica may snap via either a sub-critical pitchfork bifurcation or a limit-point bifurcation. If the elastica snaps via a sub-critical pitchfork bifurcation, the pre-snapping deformation is symmetric. If the elastica snaps via a limit-point bifurcation, on the other hand, the pre-snapping deformation is unsymmetric. For a specified initial shape of the elastica, different load models predict different snapping loads and pre-snapping deformations. The theoretical predictions are confirmed by experimental observations.  相似文献   

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It is shown that there is a magnitude of the follower force at which two limit cycles, stable and unstable, are born in the phase space of a double simple pendulum  相似文献   

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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。  相似文献   

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The nonlinear static behavior of a linearly elastic cantilever subjected to a nonconservative force of the follower type is formulated and examined. The formulation allows for finite rotations with small strains (the elastica). Exact solutions are found. The investigation is greatly facilitated by means of a phase plane analysis in which the phase plane variables are related to slope angle and bending moment. Some of the interesting and unusual effects occurring in this system are discussed and illustrated with a set of deflection curves for a typical case.Nomenclature x, y coordinates of a point on the deformed elastic axis - slope angle - s arc length - L length of beam (assumed constant) - x L , y L , L values of x, y, at s=L - P applied force - constant angle between P and end tangent - angle between P and the horizontal - EI beam stiffness (assumed constant) - u, dimensionless variables defined by (7) and (8) - c 2 load parameter defined by (10) - k, transformation parameters defined by (21) - F(, k), E(, k) elliptic integrals of the first and second kind - argument of elliptic integrals - 0, 1 values of at u=0 and u=1 - m, n positive integers - N mode number  相似文献   

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The effects of a piezoelectric layer on the stability of viscoelastic plates subjected to the follower forces are evaluated. The differential equation of motion of the viscoelastic plate with the piezoelectric layer is formulated using the two-dimensional viscoelastic differential constitutive relation and the thin plate theory. The weak integral form of the differential equations and the force boundary conditions are obtained. Using the element-free Galerkin method, the governing equation of the viscoelastic rectangular plate with elastic dilatation and Kelvin–Voigt distortion is derived, subjected to the follower forces coupled with the piezoelectric effect. A generalized complex eigenvalue problem is solved, and the force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force; this force is used to improve the stability of the non-conservative viscoelastic plate. For the viscoelastic plate with various boundary conditions, the results for the instability type and the critical loads are presented to show the variations in these factors with respect to the location of the piezoelectric layers and the applied voltages. The stability of the viscoelastic plates can be effectively improved by the determination of the optimal location for the piezoelectric layers and the most favorable voltage assignment.  相似文献   

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This paper presents postbuckling behaviors of a variable-arc-length (VAL) circular curved beam subjected to an end follower force. One end of the VAL circular curved beam is hinged while the other end is supported by a frictionless slot, which is fixed horizontally and vertically but is allowed to rotate corresponding to loading direction. When the VAL circular curved beam is deformed, the total arc-length of the circular curved beam varies. Two approaches have been applied for the solution of this problem. The first approach is an elliptic integrals method based on elastica theory, which yields the exact closed-form solution in terms of the first and second kinds of elliptic integrals. For validation of the results, the shooting method is employed for a numerical solution by developing the set of nonlinear governing differential equations together with boundary conditions, and then integrating them by using the fourth-order Runge–Kutta algorithm. The results from both approaches are in very good agreement. From the results, it is found that the VAL circular curved beam subjected to an end follower force can be deformed in many mode shapes. For the first and third modes, the beam exhibits both stable and unstable configurations, whereas for the second mode only an unstable configuration exists. The influences of initial curvature on the critical load and the deformed configurations are highlighted.  相似文献   

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Lorenzo Contri 《Meccanica》1966,1(1-2):61-64
Summary We consider here the problem of the transition between stability and instability of the equilibrium configuration of a linear elastic system subjected to nonconservative follower(1) forces, when their magnitudes are affected by a common factor which is gradually increased.It is known that critical conditions for the stability of equilibrium may occur either by two natural frequencies of the system coinciding or by one natural frequency becoming zero, and that in the second eventuality they can also be recognised by the ordinary static criteria.We give evidence how, as distinct from the case of the conservative forces, the configuration considered can present itself again as stable for values of the load superior to an eventual critical load after the passing of a subsequent characteristic condition of one of the above-mentioned types and in particular of the second type.We indicate the necessity of the completion from this point of view of the treatment of a classic group of problems set by L. Collatz and A. Pflüger.Finally, we give sufficient conditions for the non-occurrence in problems of this kind of characteristic conditions of the second type.This study was presented at the meeting 19th January, 1964 of the Istituto Veneto di Scienze Lettere ed Arti and included in Vol. CXXII pp. 107–119 of the Acts of the same Institute.  相似文献   

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Summary The present study is concerned with the theoretical and experimental investigations of the stability of a cantilevered bar subjected to the influence of a fluid jet that acts transversely at the free end. Fluid is carried along the length of the bar by means of a rubber hose placed suitably to avoid asymmetry and an attachment at the free end changes the direction of the fluid flow to achieve the desired result. Thus, in addition to having a transverse follower force at the end, the system is also subjected to the force of Coriolis acceleration.In the first part, using energy considerations, linearized equations of motion are obtained which turn out to involve variable coefficients. Explicit solutions of such equations in forms of known functions do not exist and, therefore, series solutions by using the Frobenius method are constructed. Using a computer, the undetermined coefficients are obtained which numerically converge to any prescribed accuracy. Flutter type instability occurs in the coupled modes of bending and torsion and buckling is completely absent. Stability curves are obtained by varying parameters representing ratio of bending and torsional rigidities, slenderness ratio and Coriolis force. The effect of internal damping is also examined. In the second part experimental results are obtained on models as described before. It is found that the behavior of the models is greatly influenced by the presence of dissipative forces in the material of which the models are made. By suitably accounting for the amount of damping in the system, the correlation between theoretical and experimental results is found to be satisfactory.
Übersicht Es wird die Stabilität eines Kragträgers untersucht, der am freien Ende quer zur Längsachse durch die Reaktionskräfte eines Fluidstrahles belastet wird. Das Fluid wird entlang der Längsachse durch einen Gummischlauch herangeführt, wobei Unsymmetrien sorgfältig vermieden werden. Durch eine Vorrichtung am Ende kann die Richtung des Strahles eingestellt werden. Der Balken wird demnach durch eine Folge-Kraft sowie durch Coriolis-Kräfte beansprucht.Aus Energiebetrachtungen werden im ersten Teil der Arbeit linearisierte Gleichungen abgeleitet, die zeitveränderliche Beiwerte haben. Da sich die Lösungen dieser Gleichung nicht explizit durch bekannte Funktionen ausdrücken lassen, werden Reihenansätze nach dem Verfahren von Frobenius verwendet. Die unbestimmten Beiwerte des Ansatzes werden mit Hilfe eines Computers mit beliebig gewünschter Genauigkeit bestimmt. Eine Instabilität vom Flatter-Typ tritt bei gekoppelter Biegung und Torsion bei Abwesenheit von Knickbeanspruchung auf. Die Stabilitätskurven werden in Abhängigkeit von Biege- und Torsions-Steifigkeit, von Schlankheitsgrad und Coriolis-Kraft ausgerechnet. Außerdem wird der Einfluß der inneren Dämpfung untersucht. Im zweiten Teil wird über Versuchsergebnisse berichtet. Dabei zeigt sich, daß die Ergebnisse stark durch die dissipativen Eigenschaften des Balkenmaterials beeinflußt werden. Unter Berücksichtigung dieser Dämpfung kann die Übereinstimmung von Theorie und Versuch als befriedigend bezeichnet werden.


Formerly Professor of Civil Engineering, The Technological Institute, Northwestern University, Evanston, Illinois.

This research was supported initially under NASA Grants NsG605 and NGR 14-007-067 at Nortwestern University during the stay of one of the authors (S.N.P.) and it was completed at the University of Mississippi with support under Grant GK-3092 from the National Science Foundation.  相似文献   

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An exact solution procedure is formulated for the stability analysis of viscoelastic rectangular plate with two opposite edges simply supported and other two edges clamped as well as the viscoelastic rectangular plate with one edge clamped and other three edges simply supported under the action of tangential follower force. Firstly, by assuming the transverse displacement (W) as independent functions which automatically satisfies the simply supported boundary conditions, the governing partial differential equation is reduced to an ordinary differential equation with variable coefficients. Then, by the normalized power series method and applying the boundary conditions yield the eigenvalue problem of finding the roots of a fourth-order characteristic determinant. The results show that the aspect ratio λ and the dimensionless delay time H have great effects on the types of instability and the critical loads of the viscoelastic plates.  相似文献   

17.
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.  相似文献   

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Equilibrium points, primary and secondary static bifurcation branches, and periodic orbits with their bifurcations of discrete systems under partial follower forces and no initial imperfections are examined. Equilibrium points are computed by solving sets of simultaneous, non-linear algebraic equations, whilst periodic orbits are determined numerically by solving 2- or 4-dimensional non-linear boundary value problems. A specific application is given with Ziegler's 2-DOF cantilever model. Numerous, complicated static bifurcation paths are computed as well as complicated series of periodic orbit bifurcations of relatively large periods. Numerical simulations indicate that chaotic-like transient motions of the system may appear when a forcing parameter increases above the divergence state. At these forcing parameter values, there co-exist numerous branches of bifurcating periodic orbits of the system; it is conjectured that sensitive dependence on initial conditions due to the large number of co-existing periodic orbits causes the chaotic-like transients observed in the numerical simulations.  相似文献   

19.
In solving the problem of the stability of the capillary waves in a drop placed in a variable electric field a drop with an infinite set of capillary waves is treated in accordance with the Rayleigh approach as an oscillatory system with an infinite number of generalized coordinates, in particular the amplitudes of the different modes. In the problem formulation stage the general model used in [7, 8] is employed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 50–55, January–February, 1989.  相似文献   

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