Generalized Mathematical Model of an Inverted Multilink Pendulum with Follower Force

Abstract

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).

Dynamics of a rope modeled as a multi-body system with elastic joints

“…This has led to the non-linear characteristic of spiral springs placed in the joints. Such an elastic characteristic is typical for hard springs [5] and close to the intuitive view on behaviour of folded ropes or cables. The Lagrange equations of motion have been derived for the general case of the system with rheonomic constraints.…”

“…The work [4], in turn, is devoted to dynamics of a flyline. Since the present paper develops a multi-body system, the reader is referred especially to [5,6], where authors discuss models of inverted multiple pendulums with various elastic characteristics of springs in joints.…”

Investigation of the Mathematical Model of a Single Pendulum Under the Action of the Follower Force

“…An analysis of the dynamics development of pendulum systems is given in [10]. The research results of bifurcations of equilibrium states of an inverted double pendulum loaded at the upper end by an asymmetric follower force are presented in [1][2][3][4][5][6][7][8][9][10]. The equilibrium states of a single inverted pendulum are considered in this paper.…”

Dynamics of theN-link pendulum: a fractional perspective

“…Known for their rich dynamics, pendular systems (PS) have been widely used for modelling and as benchmark systems in many areas, namely long-chain molecular physics (Mezić, 2006;Vesely, 2013), control theory (Boubaker, 2013;Kurdekar & Borkar, 2013;Larcombe, 1992), robotics (Ali, Motoi, Heerden, & Kawamura, 2013;Brisilla & Sankaranarayanan, 2015), biomechanics (Barin, 1992;Schiehlen, 2014), building structures (Housner, 1963;Zayas, Low, & Mahin, 1990), chaos and nonlinear dynamics (Gmiterko & Grossman, 2010;Hedrih, 2008;Lobas, 2005;Yu & Bi, 1998), among others (Baker & Blackburn, 2005;Jadlovská, Sarnovskỳ, Vojtek, & Vošček, 2015).…”