Theory of inverted pendulum with follower force revisited

Abstract

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...

Influence of concurrent use of springs with characteristics of different types on the equilibrium of an inverted pendulum

“…Following [4,5], we assume, unless otherwise sated, that m 1 N×m, e = 0. The angular eccentricity dof the follower force is assumed to vary continuously.…”

“…The angular eccentricity dof the follower force is assumed to vary continuously. As in [5], the values of the angle j 1 can be found as the roots of the equation…”

“…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.…”

“…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.…”

“…These are the following combinations: hs, hl, sl, sh, lh, ls. Note that the combinations hh, ss, and ll were examined in [5]. This paper suggests that the presence of a hard spring should double the number of combinations because "small" and "large" limit travels of springs should be distinguished.…”