BIFURCATION ANALYSIS OF THE STABILITY OF ONE DYNAMICAL SYSTEM WITH A FOLLOWER FORCE

In this paper, for an inverted double pendulum with asymmetric follower force a mathematical model has been constructed in generalized coordinates. The dynamics of the system has been explored through both analytical and numerical approaches. Stability diagrams have been presented. The influence of the follower forces parameters on the dynamical behavior of the pendulum has been analyzed. Numerical simulation has been applied to obtain integral curves and phase portraits. The trajectories of the lower and upper poles of the pendulum have been constructed too.


Introduction
Many researchers have studied the dynamic of pendulum systems. An inverted double pendulum is a combination of two individual pendulums which represents an example of a nonlinear and unstable dynamic system. This model is currently actively used for research and experiments in such fields as automatic aircraft landing system, biped locomotive machines, in robotics, aerospace, marine vehicles, etc. An inverted pendulum with a follower force simulates the structural elements of the props of railway bridges, track-laying machines, and other structures.
An inverted pendulum is a popular benchmark problem in control theory too. The main reason for this is that it constitutes a nonlinear system with unstable and nonminimum-phase behavior and thus reveals many interesting system-theoretic properties Although the pendulum systems have widely used in engineering, the qualitative behavior these systems has not been investigated enough. Bifurcation analysis allows revealing special modes of functioning of the corresponding mechanical systems. The study of the peculiarities of the dynamic behavior of the indicated pendulum makes it possible to optimize its parameters in order to avoid violations of the normal conditions of the functioning of the corresponding mechanical systems and to ensure the strength and stability of its elements.

Analysis of recent research and problem statement
Studying the oscillations of multilink inverted pendulums is a task of current importance in theoretical mechanics. This is a favorite example of a system for demonstration at lectures by students and faculty of physics, dynamics, and theory of

DOI: 10.32703/2617-9040-2019-33-2-4
control [1]. A dynamics of the inverted pendulum are similar to many real-world systems, for instance, in robotics, for modeling of human walking, Segway, biped locomotive machines, flexible space structures, and many more industrial applications [2,3]. For solving the problems of balancing rockets during vertical take-off investigate the stability of the rotary inverted pendulum. It is also a very good model for an automatic aircraft landing system, aircraft stabilization in the turbulent air-flow, stabilization of a cabin in a ship, etc. [4,5].
A double inverted pendulum on a cart is a popular benchmark problem in control theory. This system in control laboratories is widely used as a benchmark for testing control algorithms [6,7]. The main reason for this is that it constitutes a nonlinear system with unstable and nonminimum-phase behavior and thus reveals many interesting systemtheoretic properties [8,9].
In theoretical mechanics, it is important to study the behavior of inverted multilink pendulums with an elastically secured upper end, to which the follower force is applied. Such a model simulates the structural elements of the pillars of railway buildings and bridges. The inverted mathematical pendulum with a follower force is designed too to study the dynamic stability of the cantilever tube that conducts a moving fluid. The first works of this cycle were initiated by the problems of the functioning of oil pipelines [10,11]. Later, Japanese scientists, when examining the Ziegler's double inverted pendulum, supplemented the model with asymmetry of the follower force [1].
In [12,13] author presents a study of the optimal mass distribution in a classical Ziegler's pendulum where local and global extrema can be found explicitly. In the article, proposed to consider an m-link Ziegler-Lobas pendulum loaded by a follower force as a model problem for studying the basic properties of optimization of circulatory systems with respect to stability criteria.
In recent years, there has been particular interest in the bifurcation analysis of nonlinear behavior of inverted pendulums with follower force [14]. Analytical calculations and numerical modeling show that sequential force is the controlling factor in the dynamic behavior of the system. From the standpoint of the qualitative theory of differential equations, this force is a bifurcation parameter on whose values the behavior (stable or unstable) of the model depends. In contrast to conservative systems, the inverted pendulum with a follower force is a non-conservative system that can lose stability both by divergence and by flutter.
In this paper, the dynamic behavior of a double-pendulum system near critical points is investigated both through analytic and numerical approaches. Two critical cases of Lyapunov's theory of stability are considered, which are characterized by one zero eigenvalues and a pair of purely imaginary eigenvalues. In the first case, the divergent bifurcation occurs in the phase space of the pendulum. In the second case, we have a flutter bifurcation.
Bifurcations in the phase space of a double inverted pendulum with a follower force are determined by nonlinear dynamics. Our aim is to understand these dynamics in a qualitative way, with a special interest in stability. All analytical results are verified using the symbolic language of the Maple calculations. It is shown that all numerical solutions agree with the analytic prediction, at least qualitatively.

The purpose and tasks of the study
The purpose of the work is to use the means of the theory of dynamic systems for bifurcation analysis of the peculiarities of the behavior of an inverted double pendulum with a follower force on the upper end.

Mathematical modelling of an inverted pendulum system (in generalized coordinates)
The double inverted pendulum system consists of two rigid weightless links of length l 1 and l 2 that carry two concentrated masses m 1 and m 2 , respectively. The pendulum is attached to a firm basement by a viscoelastic revolute joint with the stiffness coefficient c 1 and the damping coefficient μ 1 . Another viscoelastic revolute joint with the stiffness coefficient c 2 and the damping coefficient μ 2 connects the two rods.
The positions of the pendulum's lower and upper poles respectively are given by: where φ 1 and φ 2 are the generalized coordinates (the angular displacements measured from the vertical).
A follower force P , generally asymmetric, is applied to the upper pole of the pendulum. Asymmetry of this force determine the angle eccentricity δ = const and orientation parameter k = const. The direction of action of the force P varies depending on the change in the configuration of the mechanical system, caused by this force. The presence of a follower force makes the dynamical system under study non-conservative. To achieve a mathematical description, system behavior is described by Lagrange's equations: where   , T kinetic energy of the system, 12 , The method begins by writing down the kinetic energy T of the system. This energy consists of a two component (for the lower and upper poles of the pendulum): According to formulas (1) the total energy of the system in generalized coordinates is: Then Lagrange equations (2) for the inverted pendulum can be written in a more compact matrix form: Here M is a matrix of mass: Therefore, the equation (3) is the dynamic equations for a mechanical system with 2 degrees of freedom.
Next, we use the parameters of the lower link (m 1 , l 1 , c 1 ) as the base and introduce the dimensionless time: Dimensionless quantities of the pendulum marking with overbars: P is a dimensionless module of the follower force, c is dimensionless stiffness of the horizontal spring at the upper pole of the pendulum. We consider these two parameters as the essential parameters of the studied system. At different values of essential parameters, we observe various patterns of motion of the studied dynamic system.
The linear model of the triple link inverted pendulum is represented in state-space form as follows:  the vector-column of the variables state of system. The eigenvalues associated with matrix A of the linearized differential equations (4) determine the stability of the position of the equilibrium of the pendulum. We will denote the dimensionless value of the follower force through P d if one eigenvalue of the matrix A is zero. If the pair of eigenvalues is purely imaginary, then we denote the module of the force P f . The symbols are associated with divergent and flutter bifurcations. For values P d and P f we have critical cases of the Lyapunov's stability theory.
When the image point of the system passes through the P d boundary, the divergent bifurcation occurs. In the phase space of the pendulum, the point 0 x  loses its stability, but another stable point appears. The pendulum has a non-vertical equilibrium position.
When the image point of the system passes through the P f boundary, the Hopf bifurcation occurs. The vertical equilibrium position of the pendulum ( 1 0   , 2 0   ) loses its stability, the periodic solution of the equations (3) appears. In the phase space of the pendulum, for this periodic solution corresponds to the limit cycle (stable or unstable). For the pendulum we have oscillatory motion.

Simulation results
After obtaining the mathematical model of the system, we need to analyze the stability of the system in order to further understand the features of the system. Here we carried out some numerical simulations using the Maple system. Numerical simulation is carried out for an inverted double pendulum with the parameters specified in Table 1.     Table 2.   Table 3. . After this, the point again passes through the boundary of P d and falls into domain ( 2,2) D .
In Fig. 2 influence of the follower forces orientation parameter k on the system's stability is represented. As the magnitude of k increases, the domain ( 2,2) D of the flutter instability extends. The curve P f , which corresponds to a pair of purely imaginary eigenvalues of matrix A, limits this domain.
The starting point of the integration of the system of the differential equations (4) in Fig. 3 is denoted by S. Over time, the image point of the system goes away from the origin and goes to a stable boundary cycle. A phase trajectory constructed in space 2 2 ,, t      more clearly demonstrates the behavior of the system (Fig. 3, b). Curves on Fig. 3 D(2,2) The asymmetry of the follower force has a strong influence on the stability and the bifurcation of the system. Fig. 4 shows how the angular eccentricity δ of the force P affects the angles of deviation of the pendulum links from the vertical.  To fully analyze the behavior of the pendulum, it is still necessary to identify safe and dangerous areas of the boundary of the stability domain. A task of current importance in the qualitative analysis of a double pendulum is to study the behavior of the solutions near the asymptotic stability boundaries in the first approximation because this behavior determines the mechanisms whereby the pendulum loses stability in equilibrium positions. As termed in [14,15], the stability boundary of an equilibrium state is safe if the stability is asymptotic in a critical case and dangerous if not asymptotic. Physically, a safe asymptotic-stability boundary means soft loss of stability. In this case, stable oscillations (limit cycles) occur instead of asymptotically stable equilibrium. If a dangerous stability boundary is approached by a representative point of the system moving within the stability domain, the equilibrium state will practically lose stability, though equilibrium state is asymptotically stable in the sense of Lyapunov from the formal-mathematical point of view.

Conclusions
In the present study, using Lyapunov's stability theory, features of the dynamical behavior of an inverted pendulum with follower force were investigated. By means of the theory of dynamical systems, a bifurcation analysis of the pendulum's behavior peculiarities was made when the essential parameters of the system were changed. Numerical methods are used to study bifurcation cases. A specially developed Mapleprogram for determining Lyapunov values allows you to analyze the behavior of a dynamic system near the boundaries of stability domains. The study of the features of the dynamic behavior of the indicated pendulum makes it possible to optimize its parameters in order to avoid violations of the normal conditions of the functioning of the corresponding mechanical systems.