The equations of an inverted pendulum with an arbitrary number of links and an asymmetric follower force

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 __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 106–114, May 2007.     

Equilibrium states of a pendulum with an asymmetric follower force

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.     

Bifurcation analysis of the motion of an asymmetric double pendulum subjected to a follower force: Codimension three problem

Local and global bifurcations in the motion of a double pendulum subjected to a follower force have been studied when the follower force and the springs at the joints have structural asymmetries. The bifurcations of the system are examined in the neighborhood of double zero eigenvalues. Applying the center manifold and the normal form theorem to a four-dimensional governing equation, we finally obtain a two-dimensional equation with three unfolding parameters. The local bifurcation boundaries can be obtained for the criteria for the pitchfork and the Hopf bifurcation. The Melnikov theorem is used to find the global bifurcation boundaries for appearance of a homoclinic orbit and coalescence of two limit cycles. Numerical simulation was performed using the original four-dimensional equation to confirm the analytical prediction.     

Evolution of the equilibrium states of an inverted pendulum

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 __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 122–131, March 2007.     

Influence of material and geometrical nonlinearities on the bifurcations of equilibrium states of a two-link pendulum

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 __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 115–128, August 2007.     

Double impact periodic orbits for an inverted pendulum

There exist many types of possible periodic orbits that impact at the walls for the inverted pendulum impacting between two rigid walls. Previous studies only focused on single impact periodic orbits and symmetric periodic orbits that bounce back and forth between the two walls. They respectively correspond to Types I and II orbits in the Chow, Shaw and Rand classification. In this paper we discuss two types of double impact periodic orbits that have not been studied before. The equations need to be solved for double impact orbits are transcendental and it is very hard to see the structure of the solutions. Consequently the analysis of double impact orbits is much more difficult than that of Types I and II orbits. A combination of analytical and numerical methods is employed to investigate the existence, stability and bifurcations of these orbits. Grazing bifurcations, which do not present for Types I and II orbits, are also observed.     

Generalized Mathematical Model of an Inverted Multilink Pendulum with Follower Force

A 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__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 5, pp. 124–130, May 2005.     

Steering performance of an inverted pendulum vehicle with pedals as a personal mobility vehicle

From the viewpoints of environmental protection, support for the aged and ensuring the right to mobility, there is a need to develop a new type of mobility vehicle that provides more effective transportation. The authors propose an inverted pendulum vehicle with pedals as one of the forms of personal mobility vehicles (PMVs). In this paper, the steering performance of the inverted pendulum vehicle with pedals is discussed based on experiments on a prototype. From the experimental results, it was confirmed that the errors from the five subjects for the target trajectory and the five-grade evaluation of the maneuverability were similar. Finally, we created an inverted pendulum vehicle with pedals to which was added a reaction actuator for the steering system. From the experimental results, it was found that setting appropriate feedback gains for the handle steering angle and its rate of rotation, which control the right and left wheel driving torques, resulted in greatly improved maneuverability.     

Divergent Bifurcations of a Double Pendulum under the Action of an Asymmetric Follower Force

The paper studies the influence of the orientation parameter of the follower force on the behavior of the pendulum with a simple zero eigenvalue in the linearization matrix     

Dynamic Behavior of Multilink Pendulums under Follower Forces

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 __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 3–35, June 2005.     

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

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 __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 107–113, July 2007.     

Stability domains of the vertical equilibrium state of a triple simple pendulum

The stability boundaries for the vertical equilibrium position of a triple simple pendulum subject to asymmetric follower force are analyzed. The effect of the upper spring and the magnitude of the follower force on the roots of the characteristic equation is studied Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 122–134, October 2008.     

Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.     

随从力作用下热弹耦合轴向运动梁的稳定性

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。     

Asynchronous free oscillations of linear mechanical systems: A general appraisal and a digression on a column with a follower force

The concept of asynchronous oscillatory response of mechanical systems in free vibrations is addressed as an introduction to a definition of asynchronous modes of vibration. Connection of the asynchronous modes with the localization and other dynamical phenomena is discussed. Then, conditions for the occurrence of asynchronous modes are investigated in depth. It is seen that a non-conservative system, with an asymmetric stiffness matrix – caused, for instance, by circulatory loads, may have an asynchronous mode. Yet, it is also seen that non-conservativeness is not actually mandatory. Two- to four-degree-of-freedom models of a column with a follower force are addressed, allowing for a mathematical modelling and phenomenological discussion.     

Limit cycles of a double pendulum subject to a follower force

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     

Bifurcations in the dynamics of an orthogonal double pendulum

The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1:1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.     

An analogy between a gravity modulated porous layer heated from below and the inverted pendulum with an oscillating pivot point

The governing equations for the inverted pendulum is developed and shown to be analogous to the gravity modulated porous layer heated from below. In particular the temperature in a gravity modulated porous layer heated from below (R>0-unstable) is likened to the motion of an inverted pendulum (unstable). As gravity modulation stabilizes convection, Transport Porous Med. 57 (2004), 113, it is found that oscillating the pivot point of an unstable, inverted pendulum stabilizes the motion.     

Group delay induced instabilities and Hopf bifurcations, of a controlled double pendulum

Digital filters, frequently used in active control of mechanical systems, enable one to improve the signal-to-noise ratio and the control performance, but introduce group delays into the control loops simultaneously. In order to gain an insight into the effects of a digital filter on a controlled mechanical system, this paper presents the stability switches and the corresponding Hopf bifurcations of a double pendulum system with the linear quadratic control having a digital filter via theoretical analysis, numerical simulations and experiments. In this study, the digital filters are used to remove the undesired noise of high frequency, which is embedded in the control signal, and are modeled as the components of pure time delay during the theoretical analysis and numerical simulations. The study shows that a digital filter with moderate specifications can not only improve the vibration reduction effectively, but also save the energy consumption of the servo-motor remarkably. However, over demanding specifications will make the group delay of the filter exceed a critical value and cause either a divergent motion or a self-excited vibration through a Hopf bifurcation, the occurrence of which depends on both the stability and the size of the basin of attraction of the bifurcating periodic motion. The experimental results well coincide with the theoretical and numerical ones, and strongly support the simplification of the digital filters as the components of pure time delay. Finally, some suggestions are made to avoid the group delay induced instability.     

Attitude stabilization of the rotational axis of a carrying body by pendulum dampers

The paper addresses attitude stabilization of the rotational axis of an asymmetric carrying body by pendulum dampers. Steady motions in which the kinetic energy of the system takes stationary values are identified. Whether these motions are stable is established __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 120–128, October 2007.