Unstable Equilibrium Position Research Articles

An approach to account for interfering parametric resonances and anti-resonances applied to examples from rotor dynamics

An initially unstable equilibrium position of a system can be stabilized by introducing a parametric excitation. This is especially of interest for suppressing self-excited vibrations, and the effect is known as parametric anti-resonance which can be observed close to the parametric combination frequencies. For the stability analysis, linearized mechanical systems with an arbitrary number of degrees of freedom and time-periodic damping and stiffness matrices are analyzed. To approximate stability maps analytically, the method of averaging is applied. A state-space representation is outlined which lifts the restriction of symmetric stiffness and damping matrices in the common approaches. The eigenvalues of the slow flow are used to determine stability. Close to a parametric combination frequency, these can change significantly. Restricting the analysis to a single parametric resonance frequency may lead to unsatisfactory and even contradictory stability maps due to the local approximation. Therefore, a novel approach, that is capable to account for the interference between the averaged eigenvalues, is outlined and motivated in an engineering manner. To verify the potential of this approach, two example systems from rotor dynamics are revisited.

Stochastic control stabilizing unstable or chaotic maps

ABSTRACTThis paper considers a stabilizing stochastic control which can be applied to a variety of unstable and even chaotic maps. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps, as well as consider a wider range of parameters for the same maps. This approach allows to stabilize unstable and chaotic maps by noise. The interplay between the map properties and the allowed neighbourhood where a solution can start to be stabilized is explored: as instability of the original map increases, the interval of allowed initial conditions narrows. A directed stochastic control aiming at getting to the target neighbourhood almost sure is combined with a controlling noise. Simulations illustrate that for a variety of problems, an appropriate bounded noise can stabilize an unstable positive equilibrium, without a limitation on the initial value.

Temperature effect on the phonon dispersion stability of zirconium by machine learning driven atomistic simulations

It is well known that conventional harmonic lattice dynamics cannot be applied to energetically unstable crystals at 0 K, such as high temperature body centered cubic (BCC) phase of crystalline Zr. Predicting phonon spectra at finite temperature requires the calculation of force constants to the third, fourth and even higher orders, however, it remains challenging to determine to which order the Taylor expansion of the potential energy surface for different materials should be cut off. Molecular dynamics, on the other hand, intrinsically includes arbitrary orders of phonon anharmonicity, however, its accuracy is severely limited by the empirical potential field used. Using machine learning method, we developed an inter-atomic potential for Zirconium crystals for both the hexagonal closed-packed (HCP) phase and the body centered cubic phase. The developed potential field accurately captures energy-volume relationship, elastic constants and phonon dispersions. The instability of BCC structure is found to originate from the double-well shape of the potential energy surface where the local maxima is located in an unstable equilibrium position. The stabilization of the BCC phase at high temperature is due to the dynamical average of the low-symmetry minima of the double well due to atomic vibrations. Molecular dynamics simulations are then performed to stochastically sample the potential energy surface and to calculate the phonon dispersion at elevated temperature. The phonon renormalization in BCC-Zr is successfully captured by the molecular dynamics simulation at 1188 K.

Analysis and comparison of the main characteristics of linear and nonlinear oscillations

In this paper, the problem of determining the positions of stable and unstable equilibrium positions of a mechanical system of the type of a planetary mechanism with one degree of freedom is considered. Analysis of possible movements of the mechanism is carried out using the phase plane. We investigate the dependence of the number of equilibrium positions on the change in the values of the parameters of the system. It is shown that with changes in the values of the parameters, the nature of the motion of the mechanism can change drastically. Some characteristics of nonlinear oscillations of the mechanism near the positions of stable equilibrium are determined. A comparative analysis of the obtained characteristics with similar characteristics for small (linear) oscillations is carried out.

On the Acoustic Radiation Force Acting Upon a Rigid Spherical Particle Near the Free Liquid Surface

The action of the acoustic radiation force on a rigid spherical particle near the free liquid surface on which a plane sound wave is normally incident is studied. A standing sound pressure wave with displacement node at the free surface results from the interference between the incident and scattered waves. The dependence of the radiation force on the frequency, the radius of the particle, and the distance between it and the liquid surface is established. It is found out that the nodes and antinodes of the standing sound wave are the equilibrium positions of the particle. The antinodes are stable equilibrium positions, while the nodes are unstable equilibrium positions. It is confirmed that the radiation force causes particles to group at the antinodes in the sound field.

Application of an adaptive bistable power capture mechanism to a point absorber wave energy converter

This work proposes a novel adaptive bistable power capture mechanism, which is applied to a point absorber wave energy converter in regular waves. The adaptive bistable mechanism is realized by two symmetrically oblique main springs (responsible for the bistable characteristics) together with two auxiliary springs (making the bistable system adaptive) and it can adjust the potential function automatically to lower the potential barrier near the unstable equilibrium position. The “adaptive” feature helps solve the low-energy-absorption problem of a conventional bistable wave energy device (whose potential function is invariable in time) caused by the intrawell oscillation in low-amplitude excitation. With a suitable choice of physical and geometric parameters, an adaptive bistable wave energy converter outperforms its linear counterpart while a conventional bistable wave energy device has a poorer performance in low amplitude waves. Even for relatively large waves, an adaptive bistable wave energy converter has a comparable power capture capacity with its conventional bistable counterpart, both of which perform better than a linear wave energy converter especially in the low wave frequency region.

Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting

This paper aims to reveal the nonlinear dynamic mechanism of asymmetric tristable energy harvesters and enhance the energy harvesting performance for different excitations. The general harmonic balance solutions of asymmetric tristable energy harvesters and the corresponding Jacobian matrix for estimating the stability of these analytical solutions are deduced. It is found that asymmetric tristable energy harvesters have seven analytical solutions (four stable solutions) under the appropriate excitation frequency and amplitude, which are verified by using basins of attraction. The influence mechanism of asymmetry of potential wells on energy harvesting performance is analyzed. The results show that the potential barrier is a key factor to determine the orbit height of high-energy interwell oscillations, which influences the amplitude of the output voltage. The influence essence of asymmetry on tristable energy harvesters is to change their potential wells and adjust the distribution of their potential energy. By changing the unstable equilibrium positions, a series of tristable energy harvesters with different dynamic characteristics can be obtained, which will benefit energy harvesting under various excitation conditions.

A Reinforcement Learning Strategy for the Swing-Up of the Double Pendulum on a Cart

The effective control design of a dynamical system traditionally relies on a high level of system understanding, usually expressed in terms of an exact physical model. In contrast to this, reinforcement learning adopts a data-driven approach and constructs an optimal control strategy by interacting with the underlying system. To keep the wear of real-world systems as low as possible, the learning process should be short. In our research, we used the state-of-the-art reinforcement learning method PILCO to design a feedback control strategy for the swing-up of the double pendulum on a cart with remarkably few test iterations at the test bench. PILCO stands for “probabilistic inference for learning control” and requires only few expert knowledge for learning. To achieve the swing-up of a double pendulum on a cart to its upper unstable equilibrium position, we introduce additional state restrictions to PILCO, so that the limited cart distance can be taken into account. Thanks to these measures, we were able to learn the swing up at the real test bench for the first time and in only 27 learning iterations.

Dynamical Analysis of a Tip Balancing Cube

AbstractThis paper discusses some dynamical aspects of a self‐balancing Inertia Wheel Cube (IWC). The acceleration and deceleration of actuated flywheels is used to stabilize the IWC when balancing on one of its corners, which is an unstable equilibrium position. The orientation as well as the rotational velocity of the device are measured by an inertial measurement unit (IMU). The equations of motions are derived with the Projection Equation. Using the rotational velocity of the IWC as part of the generalized velocities leads to well‐structured and interpretable dynamical equations. These equations are the basis for simulation, parameter identification, control synthesis, and stability analysis. The stability of the unstable equilibrium is investigated by means of the linearized motion equations. It can be shown that there are always uncontrollable states by invoking the conservation of momentum. Simulations as well as experimental results are presented. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Balancing a genetic toggle switch by real-time feedback control and periodic forcing

Cybergenetics is a novel field of research aiming at remotely pilot cellular processes in real-time with to leverage the biotechnological potential of synthetic biology. Yet, the control of only a small number of genetic circuits has been tested so far. Here we investigate the control of multistable gene regulatory networks, which are ubiquitously found in nature and play critical roles in cell differentiation and decision-making. Using an in silico feedback control loop, we demonstrate that a bistable genetic toggle switch can be dynamically maintained near its unstable equilibrium position for extended periods of time. Importantly, we show that a direct method based on dual periodic forcing is sufficient to simultaneously maintain many cells in this undecided state. These findings pave the way for the control of more complex cell decision-making systems at both the single cell and the population levels, with vast fundamental and biotechnological applications.

Non-synchronous oscillations in four-dimensional nonlinear semelparous Leslie matrix models

In a certain class of semelparous Leslie matrix models, either a positive equilibrium is stable and an invariant set on the boundary of the nonnegative cone is unstable or vice versa generically if the model dimension is two or three. This dynamic dichotomy is expected to be failed in the four-dimensional case. Our study focuses on a semelparous Leslie matrix model with specific nonlinearities and rigorously proves that the dynamic dichotomy does not hold in the four-dimensional case. This result is derived by showing that the four-dimensional semelparous Leslie matrix model can be uniformly persistent with respect to the boundary of the nonnegative cone even if there exists an unstable positive equilibrium. In such a situation, there are no missing age-classes but population oscillation occurs.

Interaction between an edge dislocation and a bridged crack with surface elasticity

We examine the contribution of crack bridging and surface elasticity to the elastic interaction between a finite crack and an edge dislocation. The surface effect on the crack faces is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. The crack faces are subjected to both normal and shear bridging forces, and the bridging stiffnesses are allowed to vary arbitrarily along the crack. The residual surface tension is taken to be zero in our discussion. The Green’s function method is utilized to reduce the boundary value problem to three first-order Cauchy singular integro-differential equations, which are solved numerically by combining the Chebyshev polynomials and the collocation method. A general formula is derived for calculating the image force acting on the edge dislocation. Our analysis indicates that the stresses exhibit both the weak logarithmic and the strong square root singularities at the crack tips. We note that both crack bridging and surface elasticity influence the magnitude and direction of the image force acting on the edge dislocation. Particularly, the results show that the dislocation may have four stable and two unstable equilibrium positions due to the presence of surface elasticity. In addition, the number and location of the equilibrium positions depend on both surface elasticity and crack bridging.

Periodic nonlinear sliding modes for two uniformly magnetized spheres

A uniformly magnetized sphere slides without friction along the surface of a second, identical sphere that is held fixed in space, subject to the magnetic force and torque of the fixed sphere and the normal force. The free sphere has two stable equilibrium positions and two unstable equilibrium positions. Two small-amplitude oscillatory modes describe the sliding motion of the free sphere near each stable equilibrium, and an unstable oscillatory mode describes the motion near each unstable equilibrium. The three oscillatory modes remain periodic at finite amplitudes, one bifurcating into mixed modes and circumnavigating the free sphere at large energies. For small energies, the free sphere is confined to one of the two discontiguous domains, each surrounding a stable equilibrium position. At large energies, these domains merge and the free sphere may visit both positions. The critical energy at which these domains merge coincides with the cumulation point of an infinite cascade of mixed-mode bifurcations. These findings exploit the equivalence of the force and torque between two uniformly magnetized spheres and the force and torque between two equivalent point dipoles, and offer clues to the rich nonlinear dynamics of this system. Online MagPhyx visualizations illustrate the dynamics.

Flower Pollination for Rotary Inverted Pendulum Stabilization with Delay

Flower pollination is a single objective optimization technique which as a unconstrained optimization method is applied for the stabilization of the rotary inverted pendulum system. It was observed that the flower pollination method gave better sensitivity in control of the pendulum about its upright unstable equilibrium position with less time and definitely indicated that the method is an energy efficient method when compared with other methods like direct pole placement. This method yielded results under the influence of time delay and have proven that the influence of time delay is significantly felt and would cause loss of energy, however the presence of flower pollination for optimization minimizes the loss incurred due to time delay and makes the system significant in terms of sensitivity.

Interaction between a screw dislocation and a circular nano-inhomogeneity with a bimaterial interface

The problem of a screw dislocation interacting with a circular nano-inhomogeneity near a bimaterial interface is investigated. The stress boundary condition at the interface between the inhomogeneity and the matrix is modified by incorporating surface/interface stress. The analytical solutions to the problem in explicit series are obtained by an efficient complex variable method associated with the conformal mapping function. The image force exerted on the screw dislocation is also derived using the generalized Peach–Koehler formula. The results indicate that the elastic interference of the screw dislocation and the nano-inhomogeneity is strongly affected by a combination of material elastic dissimilarity, the radius of the inclusion, the distance from the center of inclusion to the bimaterial interface, and the surface/interface stress between the inclusion and the matrix. Additionally, it is found that when the inclusion and Material 3 are both harder than the matrix (μ1 > μ2 and μ3 > μ2), a new stable equilibrium position for the screw dislocation in the matrix appears near the bimaterial interface; when the inclusion and Material 3 are both softer than the matrix (μ1 < μ2 and μ3 < μ2), a new unstable equilibrium position exists close to the bimaterial interface.

Optimal Synthesis in the Control Problem of an n-Link Inverted Pendulum with a Moving Base

In this paper, we consider the problem of stabilization of an n-link inverted pendulum on a movable base (cart). A cart is allowed to move along the horizontal axis. A force applied to the cart is considered as a control. The problem is to minimize the mean square deviation of the pendulum from the vertical line. For the linearized model, we show that, for small deviations from the upper unstable equilibrium position, the optimal regime contains trajectories with more and more frequent switchings. Namely, the optimal trajectories with infinite number of switchings are shown to attain, in finite time, the singular surface and then continue these motion with singular control over the singular surface, approaching the origin in an infinite time. It is shown that the costructed solutions are globally optimal.

Bistable Mechanisms for Space Applications.

Compliant bistable mechanisms are monolithic devices with two stable equilibrium positions separated by an unstable equilibrium position. They show promise in space applications as nonexplosive release mechanisms in deployment systems, thereby eliminating friction and improving the reliability and precision of those mechanical devices. This paper presents both analytical and numerical models that are used to predict bistable behavior and can be used to create bistable mechanisms in materials not previously feasible for compliant mechanisms. Materials compatible with space applications are evaluated for use as bistable mechanisms and prototypes are fabricated in three different materials. Pin-puller and cutter release mechanisms are proposed as potential space applications.

On the dynamics of Armbruster Guckenheimer Kim galactic potential in a rotating reference frame

In this article, we are interested in studying some dynamics aspects for the Armbruster Guckenheimer Kim galactic potential in a rotating reference frame. We introduce a non-integrability condition for this problem using Painleve analysis. The equilibrium positions are given and their stability is studied. Furthermore, we prove the force resulting from the rotation of the reference frame can be used to stabilize the unstable maximum equilibrium positions. The periodic solutions near the equilibrium positions are constructed by applying Lyapunov method. The permitted region of motion is determined.

Optimal conditions with chattering in the inverted two-link pendulum control problem

The plane motion of a two-link inverted mathematical pendulum, attached by a hinge to a moving trolley, is studied. The pendulum is controlled by a bounded force applied to the trolley. The problem of the minimization of the mean square deviation of the pendulum from an unstable equilibrium position is considered. Pontryagin's maximum principle is used. An optimal feedback control, containing special second order trajectories and trajectories with chattering is constructed for a linearized model. It is proved that, before emerging onto a special manifold, the optimal trajectories experience a chattering after a finite period of time and then reach the unstable equilibrium after an infinite time by a specific mode. The global optimality of the solution constructed is proved.

Virtual bottom for ships sailing in restricted waterways (unsteady squat)

The focus of this paper is to update the classic criterion for the determination of the safe keel clearance for ships sailing in restricted waterways. Ship׳s dynamics is taken into account by integrating the notion of the unsteady squat. This paper shows the existence of unstable equilibrium position of the ship during heave motion. Furthermore, it proposes a new mathematical expression to evaluate it as a function of canal and ship parameters. It is shown that this unstable equilibrium position can be considered as a virtual bottom for ship. It should not be reached during ship motion in order to avoid grounding. The importance of this result, confirmed both by an analytical model as well as by a numerical model with Finite Elements Method, lies in the reduction of the safety margin actually allowed for pilots. So this result points out the knowledge of a virtual bottom that could lie above the real nautical bottom.