The results from studies of the dynamic behavior of a double pendulum under the action of a follower force are analyzed. It is pointed out that bifurcations and catastrophes of equilibrium states may occur at some values of the parameters. Differential equations are presented which describe the plane-parallel motion of a pendulum with an arbitrary number of links and angular and linear eccentricities of the follower force whose orientation depends on one parameter. The basic problems of dynamics of pendulums with different number of links are formulated Keywords: multilink pendulum, follower force, eccentricity, bifurcation Engineering Objects Modeled by Series-Connected Pendulums with FollowerForces. The first experimental and theoretical investigations of pendulum dynamics date back to Galileo and Huygens. Later, such investigations intensified because pendulum systems turned out to have many direct applications and to be convenient models of various dynamic systems (physical, hydromechanical, biological, and economic) in which oscillations are of rotational nature.In the 20th century, studies into pendulum systems revealed a number of unusual dynamic phenomena. One of them is the possibility of stabilizing an unstable equilibrium position of the system by a periodic load. This phenomenon has been discovered for an inverted pendulum whose point of suspension vibrates with a small amplitude and certain frequency [19,20]. Though the fact of stabilization has been explained theoretically and illustrated experimentally and is, hence, beyond question, some theoretical aspects of this phenomenon are further researched and developed [4,71,76], including generalization to an n-link pendulum [89]. Recently, pendulum systems have been used as an essential element of dynamic shock absorbers in various crucial engineering devices and structures [23,64].In the first half of the 20th century, unusual and even dangerous oscillatory phenomena were discovered in oil pipelines, in particular the Transarabian pipeline [92]. First attempts to gain insight into these phenomena and to develop adequate theoretical models failed. Pflüger [91] came nearest to resolving this problem, and Ziegler [95] proposed a method of its solution. There are two obvious approaches to the problem: a direct method based on partial differential equations [14,15] and a model method reducing the problem to systems of ordinary differential equations. For the second approach, which reduces a (continuum) hydroelastic problem to a simpler (discrete) problem of theoretical mechanics, the problem of optimal simulation is of primary importance. Certainly, the number of degrees of freedom of a model is very important in this case.Let us discuss, following the monograph [67], dynamic processes in elastic curvilinear pipes conveying a fluid. If r v is the flow velocity and ρ is the local radius of curvature of the pipe, then the particles in the fluid have a normal acceleration of magnitude v 2 / ρ(note that if r v and ρare the cross-section average velocities of p...

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 Introduction. The concepts of bifurcation, limit cycle, catastrophe, attractor, and domain of attraction are widely used in various fields of science. The interest shown in them recently in many countries follows from the fact that the identification of transitions from quantity to quality is the key problem that must be solved to unlock the fundamental secrets of nature and gain a deep insight into its processes and phenomena.The phenomenon of bifurcation (from Latin bifurcus, two-pronged) in the phase (or state) space of a dynamic system represents a qualitative change in the motion of the original mechanical system. There are bifurcations of equilibrium states, where the system is caused to change to another equilibrium state or to periodic motion, bifurcations of limit cycles, etc. Such changes occur at bifurcation points where the evolution path of a nonlinear system splits. At these points, the trajectory branches, with the branch along which further development will occur being only determined by the capabilities of the dynamic system under the current conditions and dependent on the system's parameters and local factors. According to the current concepts, bifurcation theory studies changes in the qualitative behavior of dynamic systems caused by changes in parameter values, and the application of this theory to study sudden reactions of mechanical, physical, chemical, biological, economic, and other systems to smooth variation in the ambient conditions or the properties of the system is called catastrophe theory [1].Results on the bifurcations of equilibrium states of an inverted double pendulum subject to an asymmetric follower force at the upper end are reported in [3][4][5][6][7][8][9][10][11][12][13]. The effect of the nonlinearity of springs on the bifurcations of stationary states of a double pendulum was examined in [12]. In the present paper, we plot bifurcation curves of an inverted single pendulum with an asymmetric follower force and analyze the qualitative changes in the dynamic behavior of the system upon variation in the orientation parameter of this force. We will show that smooth variation in the linear eccentricity of the follower force may cause abrupt transitions (catastrophes) from one stable equilibrium state to another.1. Problem Formulation. The design model of an inverted single pendulum (see [2], Fig. 1) consists of an imponderable rod OA 1 of length l 1 and a material point A 1 of mass m 1 . The lower end O is attached to a viscoelastic hinge (made up, for example, of a spiral spring and a hydraulic damper). Let c be the stiffness of the spiral spring and m 1 be the coefficient of viscosity at the hinge O, which also accounts for external friction. The upper end of the pendulum is attached to a horizontal spring of stiff...

531.011A problem formulation and differential equations are given to describe the plane-parallel motion of an inverted multilink pendulum with an asymmetric follower force acting at the elastically restrained upper end. The physical nonlinearities of the springs are taken into account. The possible mechanisms of energy dissipation are described Keywords: inverted multilink pendulum, asymmetric follower force, elastically restrained upper end, physical nonlinearityIntroduction. Engineering Objects Modeled by Pendulums with Follower Forces. Unusual and even dangerous oscillatory phenomena in oil pipelines were discovered in the first half of the 20th century. Pflüger [9] was the first to try to gain an insight into them. He modeled the Transarabian pipeline by an elastic cantilever rod with a follower force acting at the free end. Despite the incorrect final result [6], Pflüger's work gave rise to numerous follow-ups. The problem was of theoretical interest because of the presence of nonpotential positional forces generated by the follower force. Ziegler [12] proposed to model the cantilever rod by a simple double-link pendulum.In [11] the rod was modeled by an inverted pendulum with elastically restrained end. The influence of an asymmetric follower force on the equilibrium states of a double pendulum was studied in [7], and the influence of the force orientation on the stability and instability regions in [5]. The evolution of limit cycles in the asymptotic-stability domain of the vertical equilibrium state of a pendulum with a varying follower force was examined in [2]. Pitchfork bifurcations (triple equilibrium) caused by an asymmetric follower force were analyzed in [3][4]. Methods of bifurcation theory were also used in [10].The above-cited studies considered double-link pendulums with linear springs. A double pendulum with nonlinear helical springs at the upper end and in joints was mathematically described in [8]. It was shown there that the equilibrium deflection angles of the links are single-valued functions of the angular eccentricity of the follower force when the springs are hard and the corresponding curves have bifurcation points when the springs are linear or soft. In the latter case, the equilibrium state of the pendulum may lose stability either colliding with another equilibrium state or abruptly changing into another equilibrium state during smooth variation in the pendulum parameters (catastrophe).A double-link pendulum is obviously not a perfect discrete model of an elastic rod, and the optimal number of its links is yet to be determined. As a first step, we will develop here a mathematical model of an inverted simple pendulum with an arbitrary number of links and nonlinear viscoelastic elements (helical springs at the upper end and in the joints).

A generalized mathematical theory of a double mathematical pendulum with follower force is used to analyze the stability of the vertical equilibrium position of the pendulum with both linear and nonlinear (hard and soft) elastic elements in the critical case of one zero root of the characteristic equation. The influence of the parameters of these elements on the safe and dangerous sections of the stability boundary is demonstrated Keywords: double mathematical pendulum, follower force, critical case, vertical position of pendulum, linear and nonlinear (hard and soft) springsIntroduction. An inverted double-link pendulum with a follower force acting at the elastically restrained upper end models an elastic rod compressed at the ends (by an external force and support reaction) [6]. Euler [4] formulated and partially solved the problem on stability of the vertical position of an elastic straight rod under vertical load. He has determined the critical load as the minimum force under which the rod is in equilibrium not only in the initial rectilinear configuration but also in an infinitesimally close (neighboring) curved configuration [7].Tracing further the evolution of approaches to the dynamic problem for a double pendulum with follower force, we should mention the unusual and even dangerous oscillatory phenomena in oil pipelines discovered in the first half of the 20th century. First attempts to gain insight into these phenomena and develop adequate theoretical models failed. Pflüger [9] came nearest to resolving this problem in 1950. He studied the static stability of a cantilever rod subjected at the free end to a force that follows its configuration. Such forces have been termed follower forces, in contrast to dead forces. These forces failed to be described by Euler's static method [10]. Ziegler (1952) gave a clue to the problem. It turned out that a mechanical system consisting of a cantilevered elastic pipe and a fluid flowing inside it can be simulated by a simple double-link pendulum with follower force at the upper end. In the present paper, we use such a model to study the influence of the nonlinearity of the horizontal spring at the upper end and two spiral springs in the hinges of the pendulum on the stability of its vertical equilibrium position in the critical case of one zero root of the characteristic equation.1. Problem Formulation. Let us consider the generalized pendulum model developed in [2]. The model is called generalized because it encompasses all possible types of the restoring force and moments: nonlinear (hard and soft) and linear.Two material points A 1 and A 2 of masses m 1 and m 2 , respectively, are pivotally connected with imponderable rigid bars OA 1 and A 1 A 2 of lengths l 1 and l 2 , respectively. The bars can deviate from the vertical by angles ϕ 1 and ϕ 2 , respectively [2].The upper end of the link A 1 A 2 is connected to a horizontal spring. The joints O and A 1 are viscoelastic, their elasticity being due to spiral springs and viscosity due to hydraulic dampers (dashpots).W...

This paper formulates a problem for and gives the differential equations of plane-parallel motion of an inverted n-link pendulum subject to an asymmetric follower force applied at the upper end via a spring. The physical nonlinearities of springs are taken into account. The possible mechanisms of energy dissipation are described Keywords: inverted multilink pendulum, asymmetric follower force, physical nonlinearity Introduction. Unusual and even dangerous oscillatory phenomena in oil pipelines were discovered in the first half of the 20th century. They were believed to be due to the loss of stability of the stationary solution after the oil velocity exceeds a certain (critical) level. To comprehend and explain the phenomenon, Pflüger [17] used Euler's buckling concept developed to determine the critical compressive force on a vertical elastic column. This concept, formulated more than two centuries ago, worked flawlessly for quite a while. However, Pflüger's problem included a follower force, a factor not previously encountered. It is a force that tracks the varying configuration of the system (the compressive force in Pflüger's problem remains tangent to the deformed elastic column).For this very reason the use of Euler's concept produced a paradoxical result: no critical load in Pflüger's problem in spite of intuitive considerations. Ziegler [18] explained the paradox by using a simple double-link pendulum to model the column under a follower force. He showed that besides static instability there may occur dynamic instability. This is how types of instability of elastic systems are now called [1,5,19,20]. There are also other terms: divergent and flutter (oscillatory) instability.The compressive force in the Ziegler pendulum is directed along the last link. It can be resolved into two components: vertical (present in Euler's problem) and horizontal (absent in Euler's problem). The magnitudes of both components depend on the angle of deviation of the last link from the vertical. The horizontal component is responsible for the follower force's being nonpotentional [2]. Japanese authors [7,8,15] have introduced asymmetric follower forces, the asymmetry caused by angular eccentricity (parameters d and k). The effect of this asymmetry on the bifurcations, stability, and catastrophes of a two-link pendulum are studied in [4,9,[11][12][13][14].Besides modeling an elastic column by a double-link pendulum, it is possible to directly study the column as a continuum system [3,6].Though the two-link approximation allowed Ziegler to explain the contradictions in Pflüger's problem, it raised new questions. One is the number of links. It is clear that approximating a curve with a chord (one-link pendulum) is not the best choice. However, the double-link approximation is insufficient too.Thus, we need to study the plane-parallel motion of an n-link pendulum for n = 3, 4, ... and then to compare the results obtained. It would be ideal to solve the problem for an arbitrary n and then to pass to the limit as n ® + ¥. This is how ...