Unstable Equilibrium Configurations Research Articles

Study of an adiabatic and free rotation of a block around a fixed axis using videoanalysis

This work is a proposal for a basic physics class in which the statics and dynamics of a rigid body are approached simultaneously using a single experimental apparatus. A mechanical system was used to study the rotation of a rigid prismatic block around one of its edges. In the first part of this study (statics), the block is subjected to the action of a variable external force, the intensity of which decreases as the inclination of the block increases. The external force produces a slow rotation (adiabatic rotation) until it is in an unstable equilibrium configuration. The experiment allows data to be collected on the external force as a function of the block’s angle to the support surface. The graph of the theoretical force relative to the angle of the block with the surface is presented together with the experimental points. From the analytical expression of the force, the work is obtained and compared with the work of the weight force. In the graphical representation of work as a function of angle, a symmetry is identified in the curve drawings. In the second part (dynamics), the block is moved to rotate freely. The falling time is calculated considering two types of independent initial conditions. The first condition is a perturbation in angular velocity and the other in the initial angle. The graphs of the falling times as a function of the predicted perturbations are presented and the concordance between the experimental data and the respective theoretical curves can be checked. All equations obtained from the physical modelling are verified experimentally, through laboratory testing, image analysis and videoanalysis. The experimen-tal techniques and results are presented in detail throughout this paper. The practices described in this work are in harmony with active learning in physics teaching.

Simultaneous adjustment of balance maintenance and velocity tracking for a two-wheeled self-balancing vehicle

The purpose of this research is the development a method for simultaneously adjusting the velocity tracking control and the inclination angle stabilization using control techniques for a two-wheeled self-balancing vehicle. The control tasks involve balancing the vehicle around its unstable equilibrium configuration along with steering and velocity tracking. In this study, the mathematical dynamic model of the vehicle is derived using the Lagrange method, under the assumptions of pure rolling and no-slip conditions which are expressed through nonholonomic constraint equations. Along with the mathematical descriptions, a multibody virtual prototype featuring advanced tire-ground interaction modeling has been developed using the MSC Adams software suite. Several classical and modern control strategies are investigated and compared to implement the method. These include Sliding Mode Control (SMC), Proportional Integral Derivative (PID), Feedback Linearization (FL), and Linear Quadratic Regulator Control (LQR) for the under-actuated and unstable subsystem that accounts for the pitch and longitudinal motions. The capabilities of these control strategies are verified and compared not only through Matlab simulation but also using Adams-Matlab co-simulation of the controller and the plant. Although every control technique has its advantages and limitations, the extensive simulation activities conducted for this study suggest that the SMC controller offers superior performances in keeping the system balanced and providing good velocity-tracking responses. Moreover, a Lyapunov-based analysis is used to prove that the sliding mode control achieves finite time convergence to a stable sliding surface. These advantages are counterbalanced by the complexity and the large number of parameters belonging to the designed SMC laws, the scheduling of which can be difficult to implement. For the comparison results another non-linear control strategy, that is, the feedback linearization method, is presented as an alternative. Through the Jacobian linearization approach the mathematical model of the system is linearized, allowing the use of control techniques such as linear quadratic regulation, which are deployed to treat the balancing, steering, and velocity tracking tasks. Finally, the empirical tuning of a PID controller is also demonstrated. The performance and robustness of each controller are evaluated and compared through several driving scenarios both in pure-Matlab and Adams-Matlab co-simulations.

Thermo-mechanical buckling of stepped circular bi-laminates

Thermo-mechanical buckling of stepped circular bi-laminates subjected to a uniform temperature field is investigated analytically. The governing equations are derived via the Principal of Stationary Potential Energy, resulting in self-consistent constitutive equations of the composite structure, as well as the equations of motion and the associated matching and boundary conditions for the composite structure. A loading parameter is identified, and closed form analytical solutions to the non-linear problem are obtained, yielding exact results within the context of the formulation. A stability criterion is established based on the second variation of the total potential energy of the system in order to identify regions of stable and unstable equilibrium configurations, and a critical temperature and a critical membrane force are determined. Simulation results based on the analytical solutions are generated and a critical and characteristic behavior is assessed for both clamped-fixed supports and hinged-fixed supports conditions. The phenomenon of “sling-shot buckling” is observed. Results of parameter studies are also presented, which demonstrate the influence that material and geometric parameters of the system have on critical behavior. In particular, thermal expansion coefficient ratio, modulus ratio and thickness ratio all play a non-trivial role in the temperature carrying capacity of the structure for relatively large patches. Because the loading parameters and geometric parameters appear explicitly in the analytical solution and simulations, the model and results will be useful towards the design of micro-electromechanical systems, smart structures and related configurations.

Mechanics of Compliant Serial Manipulator Composed of Dual-triangle Segments

The paper focuses on the mechanics of a compliant serial manipulator composed of new type of dual-triangle elastic segments. Both the analytical and numerical methods were used to find the manipulator stable and unstable equilibrium configurations, as well as to predict corresponding manipulator shapes. The stiffness analysis was carried on for both loaded and unloaded modes, the stiffness matrices were computed using the Virtual Joint Method (VJM). The results demonstrate that either buckling or quasi-buckling phenomenon may occur under the loading, if the manipulator corresponding initial configuration is straight or non-straight one. Relevant simulation results are presented that confirm the theoretical study.

Extra dimensions’ influence on the equilibrium and radial stability of strange quark stars

We analyze the influence of extra dimensions on the static equilibrium configurations and stability against radial perturbations. For this purpose, we solve stellar structure equations and radial perturbation equations, both modified for a $d$-dimensional spacetime ($d\geq4$) considering that spacetime outside the object is described by a Schwarzschild-Tangherlini metric. These equations are integrated considering a MIT bag model equation of state extended for $d\geq4$. We show that the spacetime dimension influences both the structure and stability of compact objects. For an interval of central energy densities $\rho_{cd}\,G_d$ and total masses $MG_d/(d-3)$, we show that the stars gain more stability when the dimension is increased. In addition, the maximum value of $M{G_d}/(d-3)$ and the zero eigenfrequency of oscillation are found with the same value of $\rho_{cd}\,G_d$; i.e., the peak value of $M{G_d}/(d-3)$ marks the onset of instability. This indicates that the necessary and sufficient conditions to recognize regions constructed by stable and unstable equilibrium configurations against radial perturbations are, respectively, $dM/d\rho_{cd}>0$ and $dM/d\rho_{cd}<0$. We obtain that some physical parameter of the compact object in a $d$-dimensional spacetime, such as the radius and the mass, depend of the normalization. Finally, within the Newtonian framework, the results show that compact objects with adiabatic index $\Gamma_1\geq2(d-2)/(d-1)$ are stable against small radial perturbations.

Optimal reconfigurations between equilibria of two-craft electromagnetic formations along manifolds

As a novel approach to control the relative motion of a spacecraft formation, electromagnetic formation flight (EMFF) has some prominent advantages, such as no propellant consumption and no plume contamination. The relative motion actuated by inter-craft electromagnetic force/torque is characterized by equilibrium states which could be utilized to accomplish particular missions, such as close formation, interferometer, etc. However, to maneuver among and maintain these equilibrium configurations, the control capability would be insufficient with only electromagnetic actuation and the inertial thrust is required, so an optimal method which exploits the invariant manifold theory is investigated in this paper. Firstly, the nonlinear translational dynamic model of two-craft electromagnetic formations is derived, and then relative equilibrium configurations and their stabilities are analytically studied. Secondly, the manifolds displaying characteristics of equilibria for the unstable equilibrium configurations are analyzed for the first time. Based on these manifolds, a novel and generalized method is proposed to formulate and parameterize optimal reconfigurations from one electromagnetic equilibrium configuration to another with the electromagnetic control inputs from the initial value to the target value. Specifically, the system's uncontrolled and discontinuous flows along manifolds compose majority of the transfer trajectories, and then optimal controls are applied to a portion of the discontinuous trajectories to differentially correct them to match continuity. By defining the control histories as a finite set of time independent variables, the optimal control problem is converted to a parameter optimization problem and solved by Particle Swarm Optimization. The method can attain the shape-changing ability and consume as little inertial thrust as possible, in exchange for added electromagnetic control effort. Lastly, numerical simulations are presented to verify the feasibility and optimality of the designed method.

Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation

This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate and fourth order non degenerate parabolic equations, in this work the a priori estimates of the error between the discrete solution and the weak solution to which it converges are obtained for a degenerate fourth order parabolic equation. The theoretical error estimates obtained in the present case state that the norms of the approximation errors, calculated on the support of the solution in the proper functional spaces, are bounded by power laws of the discretization parameters with exponent 1/2, while in the case of the classical Cahn-Hilliard equation with constant mobility the exponent is 1. The estimates are finally succesfully validated by simulation results in one and two space dimensions.

White dwarfs with a surface electrical charge distribution: equilibrium and stability

The equilibrium configuration and the radial stability of white dwarfs composed of charged perfect fluid are investigated. These cases are analyzed through the results obtained from the solution of the hydrostatic equilibrium equation. We regard that the fluid pressure and the fluid energy density follow the relation of a fully degenerate electron gas. For the electric charge distribution in the object, we consider that it is centralized only close to the white dwarfs’ surfaces. We obtain larger and more massive white dwarfs when the total electric charge is increased. To appreciate the effects of the electric charge in the structure of the star, we found that it must be in the order of 10^{20},[mathrm{C}] with which the electric field is about 10^{16},[mathrm{V/cm}]. For white dwarfs with electric fields close to the Schwinger limit, we obtain masses around 2,M_{odot }. We also found that in a system constituted by charged static equilibrium configurations, the maximum mass point found on it marks the onset of the instability. This indicates that the necessary and sufficient conditions to recognize regions constituted by stable and unstable equilibrium configurations against small radial perturbations are respectively dM/drho _c>0 and dM/drho _c<0.

Electron Wave Propagation near a Potential Ridge.

Few-electron atoms possess unstable equilibrium configurations whose many-electron potential surface shows ridges. The diffraction of an electron wave from a potential ridge is shown to deform the potential surface and to create a fictitious force. That force manifests itself as temporary electron-electron attraction. This analysis is based on nonadiabatically modified plane waves in short wavelength approximation.

A Cahn‐Hilliard–type equation with application to tumor growth dynamics

We consider a Cahn‐Hilliard–type equation with degenerate mobility and single‐well potential of Lennard‐Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn‐Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.

Inferring unstable equilibrium configurations from experimental data

This research considers the structural behavior of slender, mechanically buckled beams and panels of the type commonly found in aerospace structures. The specimens were deflected and then clamped in a rigid frame in order to exhibit snap-through. That is, the initial equilibrium and the buckled (snapped-through) equilibrium configurations both co-existed for the given clamped conditions.In order to transit between these two stable equilibrium configurations (for example, under the action of an externally applied load), it is necessary for the structural component to pass through an intermediate unstable equilibrium configuration. A sequence of sudden impacts was imparted to the system, of various strengths and at various locations. The goal of this impact force was to induce relatively intermediate-sized transients that effectively slowed-down in the vicinity of the unstable equilibrium configuration. Thus, monitoring the velocity of the motion, and specifically its slowing down, should give an indication of the presence of an equilibrium configuration, even though it is unstable and not amenable to direct experimental observation. A digital image correlation (DIC) system was used in conjunction with an instrumented impact hammer to track trajectories and statistical methods used to infer the presence of unstable equilibria in both a beam and a panel.

On the experimental identification of unstable static equilibria.

This paper shows how the presence of unstable equilibrium configurations of elastic continua is reflected in the behaviour of transients induced by large perturbations. A beam that is axially loaded beyond its critical state typically exhibits two buckled stable equilibrium configurations, separated by one or more unstable equilibria. If the beam is then loaded laterally (effectively like a shallow arch) it may snap-through between these states, including the case in which the loading is applied dynamically and of short duration, i.e. an impact. Such impacts, if applied at random locations and of random strength, will generate an ensemble of transient trajectories that explore the phase space. Given sufficient variety, some of these trajectories will possess initial energy that is close to (just less than or just greater than) the energy required to cause snap-through and will have a tendency to slowdown as they pass close to an unstable configuration: a saddle point in a potential energy surface, for example. Although this close-encounter is relatively straightforward in a system characterized by a single degree of freedom, it is more challenging to identify in a higher order or continuous system, especially in a (necessarily) noisy experimental system. This paper will show how the identification of unstable equilibrium configurations can be achieved using transient dynamics.

Equilibrium and stability of charged strange quark stars

The hydrostatic equilibrium and the stability against radial perturbation of charged strange quark stars composed of a charged perfect fluid are studied. For this purpose, it is considered that the perfect fluid follows the MIT bag model equation of state and the radial charge distribution follows a power-law. The hydrostatic equilibrium and the stability of charged strange stars are investigated through the numerical solutions of the Tolman-Oppenheimer-Volkoff equation and the Chandrasekhar's pulsation equation, being these equations modified from their original form to include the electrical charge. In order to appreciably affect the stellar structure, it is found that the total charge should be of order $10^{20}[\rm C]$, implying an electric field of around $10^{22}[\rm V/m]$. We found the electric charge that produces considerable effect on the structure and stability of the object is close to the star's surface. We obtain that for a range of central energy density the stability of the star decreases with the increment of the total charge and for a range of total mass the electric charge helps to grow the stability of the stars under study. We show that the central energy density used to reach the maximum mass value is the same used to determine the zero eigenfrequency of the fundamental mode when the total charge is fixed, thus indicating that the maximum mass point marks the onset of instability. In other words, when fixing the total charge, the conditions $\frac{dM}{d\rho_c}>0$ and $\frac{dM}{d\rho_c}<0$ are necessary and sufficient to determine the stable and unstable equilibrium configurations regions against radial oscillations.

Snap-Through of Shallow Extensible Arches Under Unilateral Displacement Control

Critical displacements are determined for snap-through of shallow, extensible, and elastic arches that are pushed downward quasi-statically at any point along the span. The initial arch is circular and unstrained, and the ends of the arch are pinned and immovable. When the vertical position at the push-down location reaches a critical value, the arch jumps into an inverted shape (unless the arch is extremely shallow). The critical displacement is given or approximated by an unstable equilibrium configuration of the unloaded arch, for which an analyical formula is derived.

Thermo-mechanical instabilities in patched structures with edge damage

The problem of thermal buckling of patched beam-plates possessing (stationary) symmetric edge detachment is studied. The analytical model is adapted from recent work of the authors and accounts for a (propagating or full) contact zone adjacent to the bonded region. The associated geometrically non-linear problem is recast in a mixed formulation that lends itself to exact analytical solutions in terms of a membrane force, the value of which is determined from an integrability condition. Stability analyses are performed on the equilibrium configurations, revealing salient behavior of the composite structure under thermal loading. Results from numerical simulations indicate the occurrence of sling-shot buckling in the partially detached structure, where the structure is seen to dynamically buckle from one configuration to another configuration (which corresponds to deflection in the opposite sense.) This previously established phenomenon has been shown to be pervasive in intact layered structures under thermal loading. The results of the present study also unveil the occurrence of a, here-to-fore, unreported phenomenon the authors refer to as “buckle-trapping”. During this event, the structure becomes trapped between two unstable equilibrium configurations whose deflections are of opposite sense. This phenomenon is associated with the physical change of the structure through deflection, and its related instabilities. These instabilities, and the associated sequence of events, are seen to be a function of the relative size of the detached region, the support conditions and, consequently, the nature of the contact zone and/or its dissolution.

Using stability analysis of discrete elastic systems to study the buckling of nanostructures

Stubility/instability criteria of discrete elastic systems are used to study the buckling of nanostructures. The deformation of nanostructures is simulated by solving the nonlinear equations of molecular mechanics. The external forces applied to the nanostructure are assumed to be dead (that is the directions of their action remain constant throughout nanostructure deformation). We note that the positive-definiteness property of the tangential stiffness matrix of a nanostructure is a universal sufficient stability criterion for both equilibrium states and quasi-static/dynamic motions of the nanostructure. The equilibrium configurations are stable in Lyapunov's sense, and quasi-static/dynamic motions are stable in a finite time interval t isin (0, T cr ) in which the positive-definiteness property of this matrix is preserved. For dynamic motions of nanostructures, the stability property in this time interval follows from Lee’s criterion of quasi-bifurcation of solutions of second order ODEs. The non-positive definiteness of the tangential stiffness matrix of nanostructures at a time t > T cr corresponds to both unstable equilibrium configurations and unstable dynamic motions. Computer procedures for determining the critical time and buckling mode(s) are developed using this criterion and are implemented in the PIONER FE code. This code is used to obtain new solutions for the deformation and buckling of twisted (10, 10) armchair and (10, 0) zigzag single-walled carbon nanotubes

Numerical continuation of bistable composite cylindrical shells

Multi-stability of thin composite structures has shown potential for morphing applications. This paper, focusing on the bistability of composite cylindrical shells, aims to gather the analytical understanding to design bistability into shells and to define and study the parameters constraining their design envelope. A classic Rayleigh–Ritz method is used in conjunction with recently developed shell models taking into account curvature effects. A path-following method is used to solve the resulting non-linear set of equilibrium equations and to explore the design space. The method, which allows the capture of multiple stable and unstable equilibrium configurations, is benchmarked against finite-element computations.

Dynamics of an electrostatically charged elastic rod in fluid

We investigate the effects of electrostatic and steric repulsion on the dynamics of a pre-twisted charged elastic rod immersed in a viscous incompressible fluid. Equations of motion of the rod include the fluid–structure interaction, rod elasticity and a combination of two interactions that prevent self-contact, namely the electrostatic interaction and hard-core repulsion. The governing equations are solved using the generalized immersed-boundary method. We find that after perturbation, a pre-twisted minicircle collapses into a compact supercoiled configuration. The collapse proceeds along a complex trajectory that may pass near several unstable equilibrium configurations, before it settles in a locally stable equilibrium. The dwell time near an unstable equilibrium can be up to several microseconds. Both the final configuration and the transition path are sensitive to the initial excess link, ionic strength of the solvent and the initial perturbation.

Rayleigh instability of the inverted one-cell amphibian embryo

The one-cell amphibian embryo is modeled as a rigid spherical shell containing equal volumes of two immiscible fluids with different densities and viscosities and a surface tension between them. The fluids represent denser yolk in the bottom hemisphere and clearer cytoplasm and the germinal vesicle in the top hemisphere. The unstable equilibrium configuration of the inverted system (the heavier fluid on top) depends on the value of the contact angle. The theoretically calculated normal modes of perturbation and the instability of each mode are in agreement with the results from ComFlo computational fluid dynamic simulations of the same system. The two dominant types of modes of perturbation give rise to axisymmetric and asymmetric sloshing of the cytoplasm of the inverted embryos, respectively. This work quantifies our hypothesis that the axisymmetric mode corresponds to failure of development, and the asymmetric sloshing mode corresponds to development proceeding normally, but with reversed pigmentation, for inverted embryos.

Variable core model and the Peierls stress for the mixed (screw-edge) dislocation

A variable core model of a moving crystal dislocation is proposed and used to derive an expression for the Peierls stress. The dislocation width varies periodically as a dislocation moves through the lattice, which leads to an expression for the Peierls stress in terms of the difference of the total energies in the crystal corresponding to stable and unstable equilibrium configurations of the dislocation, rather than the difference in the misfit energies alone. Results for both edge and mixed dislocations are given and proposed to be used in conjunction with ab initio calculations.