A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

Multiple buckled states of rectangular plates

We consider the buckling of a simply supported plate subjected to a constant edge thrust λ. The aspect ratio l is such that the critical thrust (the first bifurcation point of the associated non-linear eigenvalue problem) is of multiplicity two. A study of the non-linear static problem indicates that there are nine possible equilibrium states. One of these corresponds to the unbuckled state while the remaining eight represent buckled states. A linear stability analysis and a calculation of the potential energy of each of the static solutions indicates that four of the solutions are stable and five are unstable.     

Displaced orbits generated by solar sails for the hyperbolic and degenerated cases

Displaced non-Keplerian orbits above planetary bodies can be achieved by orientating the solar sail normal to the sun line. The dynamical systems techniques are employed to analyze the nonlinear dynamics of a displaced orbit and different topologies of equilibria are yielded from the basic configurations of Hill’s region, which have a saddlenode bifurcation point at the degenerated case. The solar sail near hyperbolic or degenerated equilibrium is quite unstable. Therefore, a controller preserving Hamiltonian structure is presented to stabilize the solar sail near hyperbolic or degenerated equilibrium, and to generate the stable Lissajous orbits that stay stable inside the stabilizing region of the controller. The main contribution of this paper is that the controller preserving Hamiltonian structure not only changes the instability of the equilibrium, but also makes the modified elliptic equilibrium become unique for the controlled system. The allocation law of the controller on the sail’s attitude and lightness number is obtained, which verifies that the controller is realizable.     

Dynamic buckling of a 2-DOF imperfect system with symmetric imperfections

Non-linear dynamic buckling of a two-degree-of freedom (2-DOF) imperfect planar system with symmetric imperfections under a step load of infinite duration (autonomous system) is thoroughly discussed using energy and geometric considerations. This system under the same load applied statically exhibits (prior to limit point) an unstable symmetric bifurcation lying on the non-linear primary equilibrium path. With the aid of the total energy-balance equation of the system and the particular geometry (due to symmetric imperfections) of the plane curve corresponding to the zero level total potential energy “surface” exact dynamic buckling loads are obtained without solving the non-linear initial-value problem. The efficiency and the reliability of the technique proposed herein is demonstrated with the aid of various dynamic buckling analyses which are compared with numerical simulation using the Verner-Runge-Kutta scheme, the accuracy of which is checked via the energy-balance equation.     

An investigation of the stability in the large for an autonomous second-order two-degree-of-freedom system

An extension to an algorithm due to Simpson has been developed for the analysis of a second-order two-degree-of-freedom autonomous system. The form of equations considered arises from the study of mechanical systems with a single concentrated non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. For a system possessing a stable equilibrium point and an unstable limit cycle arising from a subcritical Hopf bifurcation, the method has been applied to the problem of predicting the basin of attraction of the equilibrium point. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution. The method was applied to four weakly non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in the cases considered. However, it would be expected that the method would not give such accurate results if the non-linear effect was more significant. This was illustrated for the case of the cubic hardening non-linearity.     

Stabilization of the Furuta Pendulum Based on a Lyapunov Function

We propose a Lyapunov-function-based control for the stabilization of the under-actuated Furuta pendulum. Firstly, by a suitable partial feedback linearization that allows to linearize only the actuated coordinate of the system, we proceed to find a candidate Lyapunov function. Based on this candidate function, we derive a stabilizing controller, in such away that the closed-loop system is locally and asymptotically stable around the unstable vertical equilibrium rest, with a computable domain of attraction.     

New analytical results of energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control solutions for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint. The control objective is to swing the Pendubot up to its unstable equilibrium point (at which two links are in the upright position). We improve the previous energy-based control solutions by analyzing the motion of the Pendubot further. Our main contributions are threefold. First, we provide a bigger control parameter region for achieving the control objective. Specifically, we present a necessary and sufficient condition for avoiding the singular points in the control law. We obtain a necessary and sufficient condition on the control parameter such that the up–down equilibrium point (at which links 1 and 2 are in the upright and downward positions, respectively) is the only undesired closed-loop equilibrium point. Second, we prove that the up–down equilibrium point is a saddle via an elementary proof by using the Routh–Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively. We show that the Pendubot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero provided that these improved conditions on the control parameters are satisfied. Third, we clarify the relationship between the swing-up controller designed via the partial feedback linearization and that designed by the energy-based approach. We present the simulation results for validation of these results.     

Bifurcational instability of an atomic lattice

In a recent article N.H. Macmillan and A. Kelly (1972) have confirmed on the basis of a linear eigenvalue analysis that a mechanically stressed perfect crystal can exhibit a bifurcational instability at stresses ranging to 20 per cent below that of the limiting maximum of the primary stress-strain curve. The question thus arises as to whether the branching point is in a non-linear sense either stable or unstable. In the former case, perfect and slightly imperfect crystals would be capable of sustaining stresses over and above the eigenvalue critical stress. In the unstable case, however, this eigenvalue stress would represent the ultimate strength of a perfect solid, while an imperfect crystal would fail at a limiting stress substantially below the eigenvalue.At 20 per cent below the limit point such a branching point is essentially distinct, and the non-linear stability analysis needed to answer this question is provided by a recently established general branching theory for discrete conservative systems. Often, however, the two critical equilibrium states are much nearer than this, and the branching theory is here suitably extended to cover the case of near-compound instabilities.An illustrative study of a close-packed crystal under uniaxial tension is next presented. A kinematically-admissible displacement field is employed and a bifurcation point is located on the primary equilibrium path just before the limiting maximum, the eigenvector being associated with a transverse shearing strain. Under these conditions a corresponding small transverse shearing stress would represent an ‘imperfection’, and the non-linear branching problem is next studied using the new general theory. This shows (in excellent quantitative agreement with an ad hoc numerical solution) that the branching point is non-linearly unstable with a quite severe imperfection-sensitivity which manifests itself as a sharp cusp on the failure-stress locus.     

Solar sail dynamics in the three-body problem: Homoclinic paths of points and orbits

In this paper we consider the orbital dynamics of a solar sail in the Earth-Sun circular restricted three-body problem. The equations of motion of the sail are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the sail. We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described.     

On the global dynamics of path-following control of automated passenger vehicles

The nonlinear dynamics of the path-following control of passenger cars is analyzed in this paper. The effect of specific modeling aspects, such as tire deformation, steering dynamics, feedback delay and controller saturation, is considered. Possible equilibrium points and singularities in the state space are uncovered and analyzed for different vehicle model and controller designs. The equilibrium of stable path following is then analyzed in greater detail: The domains of stabilizing control gains are presented in stability charts and the basin of attraction of the equilibrium along the stable domain is approximated with the help of numerical continuation. Unsafe zones of control gains are highlighted, where the stable equilibrium is surrounded by low-amplitude unstable limit cycles. Finally, it is shown how specific modifications of the control law can remove unwanted equilibrium points and increase the basin of attraction of stable path following, resulting in safer and more reliable control of the vehicle.

     

Global stabilization of 2-DOF underactuated mechanical systems—an equivalent-input-disturbance approach

This paper presents a new method of globally stabilizing a non-linear underactuated mechanical system with two degrees of freedom (DOF). It is based on the idea of equivalent input disturbance (EID), and designing the controller requires only the state variables of position, not velocity. The design procedure has two steps: (1) Use a global homeomorphic coordinate transformation to convert the original system into a new non-linear system. This changes the problem of stabilizing the original system into one of stabilizing the new system. (2) Divide the new system into linear and non-linear parts and take the non-linear part to be an artificial disturbance, thereby enabling use of the EID approach to globally asymptotically stabilize the new system at the origin. The new method was tested through numerical simulations on three well-known 2-DOF underactuated mechanical systems (TORA, beam ball, inertia wheel pendulum). The results demonstrate its validity and its superiority over others.     

Global motion analysis of energy-based control for 3-link planar robot with a single actuator at the first joint

This paper studies nonlinear control of a 3-link planar robot moving in the vertical plane with only the first joint being actuated while the two other revolute joints are passive (called the APP robot below). A nonlinear energy-based controller is proposed, whose objective is to drive the APP robot into an invariant set where the first link is in the upright position and the total mechanical energy converges to its value at the upright equilibrium point (all three links are in the upright position). By presenting and using a new property of the motion of the APP robot, without any condition on its mechanical parameters, this paper proves that if the control gains are larger than specific lower bounds, then only a measure-zero set of initial conditions converges to three strictly unstable equilibrium points instead of converging to the invariant set. This paper presents numerical results for a physical 3-link planar robot to validate the obtained theoretical results and to demonstrate a switch–and–stabilize maneuver in which the energy-based controller is switched to a linear state feedback controller that stabilizes the APP robot at its upright equilibrium point.     

Traction problem in electro-elasticity and bifurcation theory

This paper is the first of a series of two. It will deal with the problem of static traction problem with minor deformations for a material which is governed by the electrostriction phenomenon. Two approaches to this problem will be described. We can consider either the equilibrium equations which are naturally non-linear, or the equations after linearization. The linearization of equations must be done near a natural state. Locally, under some conditions, we can establish the existence and the uniqueness of the solutions. We use the local theorem of implicit functions. The problem can be approached more globally. If we consider the non-linear equations, we can use a natural principle of these equations: the independence of the choice of the observer, also known as objectivity property. This property makes it possible for us to take into account an action of the rotations group of the Euclidean space, and consequently to take into account all the trivial solutions. It is then possible to prove within the space of all configurations the existence of the non-linear equations solutions and to find their number.This work presents a thorough and detailed approach to a non-linear theory, the geometric arguments of which make it possible for us to prove the existence of all the solutions and to study their stability in the aggregate; this last aspect will be developed in the second paper. Not only can this theory anticipate the eventual existence of a stable solution, it can also anticipate that an unstable solution in terms of the elasticity can, thanks to the effect of an electric field, become stable in terms of the electro-elasticity.     

Non-linear in-plane analysis and buckling of pinned–fixed shallow arches subjected to a central concentrated load

This paper is concerned with an analytical study of the non-linear elastic in-plane behaviour and buckling of pinned–fixed shallow circular arches that are subjected to a central concentrated radial load. Because the boundary conditions provided by the pinned support and fixed support of a pinned–fixed arch are quite different from those of a pinned–pinned or a fixed–fixed arch, the non-linear behaviour of a pinned–fixed arch is more complicated than that of its pinned–pinned or fixed–fixed counterpart. Analytical solutions for the non-linear equilibrium path for shallow pinned–fixed circular arches are derived. The non-linear equilibrium path for a pinned–fixed arch may have one or three unstable equilibrium paths and may include two or four limit points. This is different from pinned–pinned and fixed–fixed arches that have only two limit points. The number of limit points in the non-linear equilibrium path of a pinned–fixed arch depends on the slenderness and the included angle of the arch. The switches in terms of an arch geometry parameter, which is introduced in the paper, are derived for distinguishing between arches with two limit points and those with four limit points and for distinguishing between a pinned–fixed arch and a beam curved in-elevation. It is also shown that a pinned–fixed arch under a central concentrated load can buckle in a limit point mode, but cannot buckle in a bifurcation mode. This contrasts with the buckling behaviour of pinned–pinned or fixed–fixed arches under a central concentrated load, which may buckle both in a bifurcation mode and in a limit point mode. An analytical solution for the limit point buckling load of shallow pinned–fixed circular arches is also derived. Comparisons with finite element results show that the analytical solutions can accurately predict the non-linear buckling and postbuckling behaviour of shallow pinned–fixed arches. Although the solutions are derived for shallow pinned–fixed arches, comparisons with the finite element results demonstrate that they can also provide reasonable predictions for the buckling load of deep pinned–fixed arches under a central concentrated load.     

Exceedance probability criterion based stochastic optimal polynomial control of Duffing oscillators

An optimal polynomial control strategy is developed in the context of the physical stochastic optimal control scheme of structures that is well-adapted to randomly-driven non-linear dynamical systems. A class of Duffing oscillators with polynomial active tendons subjected to random ground motions is investigated for illustrative purposes. Numerical studies reveal that using an exceedance probability criterion with the minimum of the failure probability of system quantities in energy trade-off sense, a linear control with the 1st-order controller suffices even for strongly non-linear systems. This bypasses the need to utilize non-linear controls with the higher-order controller which may be associated with dynamical instabilities due to time delay and computational dynamics. The statistical variability, meanwhile, of system responses gains an obvious reduction, and the system performance is significantly improved. The 1st-order controller, however, does not have the same control effect to the higher-order controller when control criteria currently in used are employed, e.g. system second-order statistics evaluation and Lyapunov asymptotic stability condition, as indicated in the comparative studies of the exceedance probability criterion against the two control criteria. Besides, the proposed optimal polynomial control is insensitive to the non-linearity strength of the class of base-excited non-linear oscillators whereby a robust control of systems can be implemented, while the LQG control in conjunction with the statistical linearization technique, using a band-limited white noise input, does not have this advantage.     

Qualitative criteria in non-linear dynamic buckling and stability of autonomous dissipative systems

A general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented. Attention is focused on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path. Using the total energy equation, the theory of point and periodic attractors of the basin of attraction of a stable equilibrium point, of local and global bifurcations, of the inset and outset manifolds of a saddle and of the geometry of the channel of motion, the stability of the fundamental equilibrium path and the mechanism of dynamic buckling are thoroughly discussed. This allows us to establish useful qualitative criteria leading to exact, approximate and upper/lower bound buckling estimates without integrating the highly non-linear initial-value problem. The individual and coupling effect of geometric and material non-linearities of damping and mass distribution on the dynamic buckling load are also examined. A comparison of the results of the above qualitative analysis with those obtained via numerical simulation is performed on several two- and three-degree-of-freedom models of engineering importance.     

Lyapunov-Based Controller for the Inverted Pendulum Cart System

A nonlinear control force is presented to stabilize the under-actuated inverted pendulum mounted on a cart. The control strategy is based on partial feedback linearization, in a first stage, to linearize only the actuated coordinate of the inverted pendulum, and then, a suitable Lyapunov function is formed to obtain a stabilizing feedback controller. The obtained closed-loop system is locally asymptotically stable around its unstable equilibrium point. Additionally, it has a very large attraction domain.Contributed by Prof. F. Pfeiffer.     

About Linear Stability for Multiple Gas Balls

The results presented in this paper are generalizations of earlier work on the linear stability of non-rotating round gas balls in equilibrium, with respect to perturbations with zero angular momentum. Here we allow a more general barotropic equation of state for the gas, a non-zero angular momentum of the equilibrium state, and we are considering arbitrary numbers of gas balls, intending to use the result later to prove non-linear stability. The result requires an energy stability condition, which we verify for a single, slowly rotating gas ball, and the restricted class of equations of state used in earlier papers.     

Bifurcating self-excited vibrations of a horizontally rotating viscoelastic shaft

Summary Self-excited postcritical vibrations of a rotating geometrically non-linear shaft caused by internal friction are analysed in this paper using the Hopf bifurcation theory. Stable periodic vibrations bifurcate from the non-trivial equilibrium which becomes unstable itself. Ordinary differential equations of motion are obtained by means of Galerkin's method. Bifurcating periodic solution is constructed in a parametric form due to Iooss and Joseph.

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Verzweigende selbsterregte Schwingungen eines horizontal gelagerten viskoelastischen Rotors

Übersicht Die von der inneren Reibung abhängigen selbsterregten Schwingungen einer drehenden geometrisch nichtlinearen Welle werden in dieser Arbeit mit Hilfe der Hopfschen Bifurkationstheorie analysiert. Die stabilen periodischen Schwingungen verzweigen sich ausgehend von der Gleichgewichtslage, die selbst instabil wird. Die Bewegungsgleichungen werden mit Hilfe der Galerkinschen Methode ausgewertet. Verzweigungslösungen werden in parametrischer Form nach Iooss und Joseph konstruiert.

     

Active flow control of laminar boundary layers for variable flow conditions

This work investigates the stability of a fxLMS controller for active wave cancelation of broad-band Tollmien–Schlichting disturbances in a flat plate boundary-layer with a single DBD plasma actuator. In particular the influence of a changing free stream velocity and the resulting off-design operation of the control algorithm is analyzed up to an unstable behavior. As the main reason for unstable controller operation in the off-design case the difference between actual and predicted phase angle of the disturbances at the position of the error sensor is identified. A method for an online adjustment of the secondary-path model to different free-stream velocities is presented. Finally a wall-bounded method based on the disturbances phase speed is developed that can cope with changes of the physical secondary path not only due to changes of the free-stream velocity but also due to changes of the pressure distribution. This method enables the extension of the stable operation range of the control system significantly.