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...

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 Keywords: inverted pendulum, asymmetric follower force, bifurcation, catastrophe Introduction. The first experimental and theoretical studies on mathematical and physical pendulums (Galileo and Huygens) were oriented toward practical applications such as pendulum clocks. Later pendulums were used as convenient models to study more sophisticated systems, both discrete and continuous. In analyzing the dynamic behavior of pendulum systems in nonconventional force fields, researchers from different countries discovered a number of new interesting effects: synchronization of dynamic systems, stabilization of the upper (unstable) equilibrium position of a pendulum by shifting the point of suspension, etc. The latter effect explains the Indian rope trick described in [14].In the second half of the 20th century, the behavior of a two-link pendulum under the action of a so-called follower force (a force tracking the configuration of a system) seemed paradoxical. Though Lurie [4] supposed that follower forces can specially be introduced into an automatic control system to endow the controlled plant with a certain dynamic property, the first identified follower force had a quite natural origin.The dynamic processes in elastic curvilinear pipes conveying a fluid became of interest in the mid-20th century because of the need for oil pipelining [3]. If r V is the average velocity of particles in the fluid and r is the average radius of curvature of their trajectories, then the normal acceleration isV 2 / r, and a pipe element of length Dx is subject to a transverse centrifugal force m V x ( / ) 2 r D (m = gS is the mass of the fluid per unit length, g is the density of the fluid, S is the cross-sectional area of the pipe).These centrifugal forces [5] distributed along the pipe are equivalent to an end load mV 2 applied at the outlet. For a cantilevered pipe, the force exerted by the fluid flowing out of the free end is tangential to this end and is classed as a follower force. There will be no end load if there is no nozzle. Pflüger [15] was one of the first to pay attention to the unusual phenomena in elastic pipelines. He used Euler's buckling concept to determine the critical follower force that causes instability. The result was paradoxical: no critical load exists. Ziegler [18] found an acceptable theoretical solution. He suggested modeling a continuous rod by a two-link pendulum. The modern concepts of elastic stability and static and dynamic buckling are usually associated with his name [19].Later the Ziegler problem was generalized to inverted pendulums and pendulums with free end elastically fastened [17].Off-center compression in the theory of elastic rods is traditionally associated with the linear eccentricity of the compressive force. Japanese scientists [7,8,13] also introduced angular eccentricity to model structural and design imperfe...

The influence of the pendulum parameters and the follower force on the evolution of equilibrium states is analyzed using a generalized mathematical model of inverted pendulum. Equilibrium curves are plotted using the parameter continuation method. It is shown that the pendulum with certain values of the angular eccentricity has one or three nonvertical equilibrium positions Keywords: mathematical pendulum, asymmetric follower force, equilibrium positions, angular and linear eccentricities Introduction. Pendulums (single-and multilink) have been studied for several centuries now. In the days of Galileo and Huygens, pendulums were regarded as mechanical systems of practical use (pendulum clock). Modern science is interested in pendulum systems because they can model more complicated objects, including continuum systems [1,5]. Mathematical models of pendulums with an arbitrary number of links (n-link pendulums) were developed in [2,10]. The current state of the art in the development of pendulum dynamics is analyzed in detail in [9]. The effect of follower forces on the behavior of pendulums was examined in [2].Recently, many studies have been focused on the effect of the orientation of follower forces (symmetric and asymmetric) [2, 3] and the linear and nonlinear characteristics of springs [7,8] on the dynamic behavior of pendulums. The bifurcations of the equilibrium states and limit cycles of a double pendulum were analyzed in [4,6]. Just a few publications are concerned with the influence of the linear and angular eccentricities, which make the follower forces asymmetric, on the equilibrium states of a pendulum. This paper examines the dependence of the equilibrium states of an inverted simple pendulum on the linear and angular eccentricities. Since one of the equilibrium states is obvious (it is the vertical position), the other equilibrium states can be obtained using the parameter continuation method (in Shinohara's modification [12]). By varying the pendulum parameters, we can analyze their influence on the number and stability type of the singular points of the differential equation of disturbed motion.

The equilibrium states of an inverted two-link simple pendulum with an asymmetric follower force are classified depending on the characteristics of the springs (hard, soft, or linear) at the upper end and at the hinges. Phase portraits are plotted. The bifurcation points on the equilibrium curves are identified. Emphasis is on fold and cusp catastrophes Keywords: two-link inverted simple pendulum, asymmetric follower force, spring characteristics of different typesIntroduction. In the mid-20th century, increased speeds of oil piping occasioned dangerous oscillatory phenomena in the pipe-fluid system. In particular, the unexpected vibrations of the Transarabian oil pipeline discovered in 1950 were described by Paidoussis and Issid in 1974 [23, 26]. Experts associated them with the buckling of a compressed slender column. Pflüger [22] tried to find the critical compressive force based on Euler's concept of buckling for a slender column compressed at the ends by longitudinal forces (active one and subgrade reaction). The result appeared paradoxical: no critical force existed in spite of what intuition obviously suggested. Ziegler [27] was able to explain the paradox. He modeled a slender column by a discrete system-a two-link pendulum. Certainly, the disagreement between Pflüger's and Ziegler's results is not due to the discrete approximation of a continuous system. This, in particular, follows from stability analyses of elastic pipelines where their continuous nature is taken into account [3,10]. It is after Ziegler [9, 28] that static and dynamic stability began to be distinguished in the theory of stability of elastic systems [1,2,9].A distinctive feature of the problem solved by Pflüger [22] and Ziegler [27] is the presence of a so-called follower force that remains tangential to the curved (curvilinear) axis of the column. The specific behavior of mechanical systems with follower forces is due to the fact that these forces (like the forces of radial correction in gyroscopy) are represented by a skew-symmetric matrix in the linearized equations of perturbed motion. In arbitrary elastic systems, such forces follow any changes in their configuration. By this definition, the cornering forces acting on tires can also be attributed to follower forces.Koiter [15] doubted whether follower forces are realistic. Sugiyama, Langthjem, and Ryu [25] held the opposite opinion. The philosophical sense and physical feasibility of follower forces were discussed by Elishakoff in the review [11]. Japanese scientists [13,14,20] introduced angular eccentricity to model manufacturing imperfections of pipes and inaccuracy in the application of compressive loads. Linear eccentricity is possible too [4][5][6][7]. It was shown in [8] that as the angular eccentricity of the follower force is smoothly varied, a two-link pendulum may jump from one stable equilibrium state to another, even if its springs are linear. The statement of dynamic problems for chain systems consisting of series-connected simple pendulums was extended in [19] to in...

The effect of the concurrent use of springs with characteristics of different types (hard, soft, or linear) on the equilibrium of an inverted simple pendulum is studied Keywords: inverted pendulum, springs of different types, bifurcations, catastrophes Introduction. Pendulum systems have long attracted the attention of scientists in many countries. The development of dynamics of such systems, dating back to Galileo and Huygens, is reviewed in [6, 16]. Of the numerous modern approaches to the research of pendulum systems, the following are worthy of mention: local bifurcations via normal forms [9, 18], coupled flutter and divergent bifurcations [15, 22], nonlinear damping [8], stabilization of the unstable upper equilibrium position by displacing the point of suspension [1, 17], deterministic chaos [2, 7, 21], etc. Especially noteworthy are the series of studies into the effect of follower forces on the motion of single-link and multilink pendulums [3-5, 10-15, 19] because such forces make the force field nonpotential, positional and introduce new features into the dynamics of mechanical systems. Such studies were stimulated by Pflüger and Ziegler [3] who examined constant forces directed along the bent axis of a compressed rod. Japanese authors [10,11] generalized this problem formulation to forces tracking the configuration of the deformed rod and being located asymmetrically about its axis.The bifurcations of the equilibrium states of an inverted simple pendulum with angular and linear eccentricities of the follower force were analyzed in [4,5]. The springs of the pendulum had characteristics of the same type: hard, soft, or linear. In what follows, we will analyze how the difference in the types of the spring characteristics at the upper end and at the point of suspension of an inverted simple pendulum affects the bifurcations and stability of its equilibrium states.