Stable Equilibrium Configurations Research Articles

Minimum energy principles in electrostatics

Abstract We employ the principle of minimum energy of the electrostatic field to derive Laplace's equation for electric potential in charge-free space. We also apply the principle of minimum electrical potential energy of a system of discrete charges to explain the possibility of the existence of their stable equilibrium configuration on the surface of a conductor.

Experimental probing of the configuration space of circular, clamped panels

A thin, post-buckled, circular plate is a classic exhibiter of bi-stability. That is, an elastic structural element that possesses more than a single (stable) equilibrium configuration. The key here is that the plate may be flat, or deflected in a given direction. In the latter case, this is a thoroughly nonlinear phenomenon that typically requires a relatively large perturbation (using a probing force) to deflect the system from one equilibrium configuration to the other, in which we can think of a structure exhibiting an initial configuration, but able to be pushed-through into an inverted, complementary stable configuration, or even into other shapes. The work described here is primarily experimental, exploiting the capabilities of digital image correlation. It points in the direction of identifying nonlinear, global features of the structural behavior, features that tend to have a strong influence on underlying behavior in an operational, dynamic environment. Some of the equilibrium configurations require a degree of interrogation to be revealed, i.e., they are not necessarily observed without some exploratory probing of the configuration space and we present results from a number of experimental studies of clamped, circular panels. The study is preceded by some experiments on initially flat panels, for which there is some access to analytic results and helps to set the stage for nonlinear environment in the post-buckled landscape.

Investigating the equation-of-state, stability and mass–radius relationship of anisotropic and massive neutron stars embedded in f(R,T) modified gravity

In this study, our main focus is to investigate the mass–radius relation and several important properties of massive neutron stars to realize the nature, behavior and evolution of these kinds of compact objects at present time. Also, we want to understand the equation-of-state of the core nuclear matter precisely with their stable equilibrium configuration. We have chosen a few massive binary pulsars and investigated on maximum attainable mass and lowest possible radius of them. We have considered Einstein–Hilbert action as [Formula: see text], where [Formula: see text] is the Ricci scalar and [Formula: see text], the trace of energy–momentum tensor with [Formula: see text] as the coupling parameter and also used modified TOV equations. The interior space-time of the spherical neutron star is matched to the exterior Schwarzschild line element at the surface of the star. The [Formula: see text]–[Formula: see text] curve can predict the maximum achievable mass is about [Formula: see text] with lowest possible radius of around [Formula: see text] km for the massive compact stars under stable equilibrium. We have obtained a clear picture of structural evolution of massive neutron stars through accelerating space-time and can put some constraints on several quantities related to them. It is depicted from our present investigation that all the derived outcomes are compatible with physically adopted regimes which reveal the viability of our current model in the context of [Formula: see text] modified gravity.

Flow Physics of a Rotating Bernoulli Pad: A Numerical Study

Abstract Bernoulli pads are traditionally used for noncontact pick-and-place operations in industry. The normal force produced by a Bernoulli pad allows it to adhere to an object or workpiece. In addition to the normal force, the pad produces shear forces, which allows it to clean a workpiece without contact. A direct relationship between the inlet fluid power and the shear losses motivates us to explore other methods of providing power to the system with the objective of increasing shear forces and thereby improving cleaning efficacy. Here, we numerically investigate a system in which additional mechanical power is added by rotating the Bernoulli pad. The rotating system provides additional fluid forces (normal and shear) for the same inlet fluid power. For a specific pad that we investigated, the maximum wall shear stress increased by ≈15% and the normal force changed from +1.4 N (repulsive) to −6.6 N (attractive) for change in the rotational speed by 60 rad/s. Also, for a given normal attractive force, a stable equilibrium configuration can exist for two mass flow rates, with the higher mass flowrate resulting in a higher stiffness of the flow field.

Investigations on the dynamic snap-through of MFC bonded self-resetting bistable laminates

Unsymmetric composite laminates having two stable equilibrium configurations have been studied extensively in the recent past due to their potential applications in morphing structures. Surface bonded Macro Fiber Composites (MFC) actuators have been considered as a viable solution to trigger the snap-through transition in bistable laminates. Although MFC bonded bistable laminates are widely used in morphing applications, they might require considerably high voltage inputs to achieve the required levels of actuation control during the shape transition. As a solution, other possible energy sources can be combined with active MFC patches to reduce the snap-through energy required from the single source of MFC actuation. In this work, we examine the dynamic behavior of bistable composite plates actuated using MFC actuators, where external vibration energy has been used to assist with the MFC-controlled actuation between stable states. A refined semi-analytical model based on the Rayleigh–Ritz formulation has been proposed, where the membrane energy and the bending energy are separately evaluated. Bending components are directly evaluated using the approximated transverse displacement functions, whereas the membrane components are evaluated separately by combining compatibility conditions and equilibrium equations. Results from the proposed semi-analytical framework are compared with a full geometrically nonlinear finite element framework and necessary experimental observations. The results show a significant reduction in the snap-through energy demand on MFC layers where external dynamic excitation assists the snap-through process. Additionally, a parametric study is performed using variable stiffness (VS) fiber orientation parameters, achieving bistable laminate-MFC configurations that lower snap-through requirements through the proposed morphing strategy. Thus, the study offers to aid a multi-efficient snap-through strategy for the morphing of multistable composite structures.

Enhancing the Design Space of Bistable Laminates by Tailoring the Attachment Boundary Conditions

Abstract Bistable and multistable laminates are structural elements with more than one stable equilibrium configuration. The bistability makes them very interesting for the design of compliant mechanisms. However, these laminates are extremely sensitive to boundary conditions and attachment methods. It has been shown prior in the literature that restrictive boundary conditions can lead to the loss of bistability or unwanted deformation modes. This article develops attachment concepts that leverage the structural behavior changes caused by boundary conditions to expand the design space of bistable elements for structural applications. A systematic means of using the boundary conditions to improve the stability margins and load introduction of bistable tape springs is demonstrated. The stability margins and the achievable angles have been quantified using an analytical model previously developed by Guest and Pellegrino. Subsequently, high-fidelity finite element (FE) simulations are done in abaqus™ to determine the limits of the proposed method in terms of localization effects. Based on the analytical model and simulation outcomes, layup, size, and attachment points are determined, and two prototypes are fabricated, illustrating the effectiveness of the proposed method in expanding the design space of traditional bistable tape springs.

Dynamic response of piezoelectrically actuated bistable cross-ply laminates under oscillating impulse voltages

Multistable composite laminates are ideal candidates for morphing applications due to their ability to switch between multiple stable states with proper actuation mechanisms. Surface-bonded macro fiber composite (MFC) actuators to trigger snap-through between one stable shape to another by applying external voltages have attracted considerable attention recently. In this work, dynamic snap-through of bistable unsymmetric cross-ply laminate using MFC actuators under oscillating impulse voltages has been investigated. Both unimorph and bimorph laminate-MFC configurations are considered for the analysis. The stable equilibrium configurations of the MFC-actuated bistable laminate, considering the effects of MFC bondage, are discussed using the proposed finite element (FE) framework. Under the action of oscillating impulse voltages, the single-well and cross-well oscillations of the active bistable plate are analyzed in detail through time responses and phase portraits. The FE results show the advantage of actuating MFC layers through the oscillating impulse voltages as it leads to a reduction in the maximum snap-through voltage requirements.

Model-independent traversable wormholes from baryon acoustic oscillations

In this paper, we investigate the model-independent traversable wormholes from baryon acoustic oscillations. Firstly, we place the statistical constraints on the average dark energy equation of state ωav by only using BAO data. Subsequently, two specific wormhole solutions are obtained, i.e, the cases of the constant redshift function and a special choice for the shape function. For the first case, we analyze the traversabilities of the wormhole configuration, and for the second case, we find that one can construct theoretically a traversable wormhole with infinitesimal amounts of average null energy condition violating phantom fluid. Furthermore, we perform the stability analysis for the first case, and find that the stable equilibrium configurations may increase for increasing values of the throat radius of the wormhole in the cases of a positive and a negative surface energy density. It is worth noting that the obtained wormhole solutions are static and spherically symmetrical metric, and that we assume ωav to be a constant between different redshifts when placing constraints, hence, these wormhole solutions can be interpreted as stable and static phantom wormholes configurations at some certain redshift which lies in the range [0.32, 2.34].

Bifurcations and Stability Analysis of Elastic Slender Structures Using Static Discrete Elastic Rods Method

Abstract Discrete elastic rods (DER) method provides a computationally efficient means of simulating the nonlinear dynamics of one-dimensional slender structures. However, this dynamic-based framework can only provide first-order stable equilibrium configuration when combined with the dynamic relaxation method, while the unstable equilibria and potential critical points (i.e., bifurcation and fold point) cannot be obtained, which are important for understanding the bifurcation and stability landscape of slender bodies. Our approach modifies the existing DER technique from dynamic simulation to a static framework and computes eigenvalues and eigenvectors of the tangential stiffness matrix after each load incremental step for bifurcation and stability analysis. This treatment can capture both stable and unstable equilibrium modes, critical points, and trace solution curves. Three representative types of structures—beams, strips, and gridshells—are used as demonstrations to show the effectiveness of the modified numerical framework, which provides a robust tool for unveiling the bifurcation and multistable behaviors of slender structures.

On the elastic snapping of structural elements

A bistable structural component possesses more than one stable equilibrium configuration. In terms of the strain energy stored in bending, this can be thought of as a system with not only an initial equilibrium configuration represented by an isolated minimum, but also a remote minimum that might be accessed given a sufficient disturbance. Whether the system is able to stay in this new position, or revert back to the initial state when the disturbance is removed is an important practical issue. This distinction is largely determined by geometry. Continuous elastic structures of the type produced using a 3D-printer are necessarily of relatively high-order in a dimensional sense, and most previous studies have used nonlinear finite element analysis to determine parameter sensitivity. However, there is a role to be played by discrete low-order models in which the same type of essential qualitative behavior can be captured, but where the parameter dependency (in this case whether a system stays in its inverted, snapped, configuration, or not) can be assessed more directly than numerical simulation. This short note develops a discrete model specifically designed to address this issue, and presents the outcomes of some tests on 3D-printed elements.

Click mechanism for racing car self-levelling flap

This paper presents a simulation and experimental study on a passive “click” mechanism, which is designed to adapt the angle of incidence of a racing car flap along a circuit to have high down force when the car undertakes low speed corners and low drag force when the car sprints along the straights. The mechanism is composed by four rigid linkages connected via flexible pivot junctions. The paper first provides a parametric study that shows how the stiffness of the flexible pivot junctions, the length of the linkages and the rest angles of the linkages influence the typical N-shaped resistant moment–joint rotation function, which is at the basis of the clicking effect. The parametric study is then used to design a mechanism, which is characterised by a stable high pitch equilibrium configuration only and maintains the N-shaped moment-rotation feature such that the flap snaps from high-to-low pitch and vice versa at given critical speeds during acceleration and deceleration of the car. Finally, the dynamic response of a flap mounted on the proposed monostable clicking joint is analysed in detail and contrasted with that of a flap fixed on classical passive joints built with a stiff spring or a soft pre-loaded spring.

Neutrally stable double-curved shells by inflection point propagation

Elastic structures that can deflect without springback, known as neutrally stable structures, form a remarkable group within their field, since they require the energetic state to remain unchanged during elastic deformation. Several examples in the literature obtain this state of neutral stability by the application of pre-stress, either as a result of manufacturing processes or the application of imposed boundary conditions. In this paper, we present a new class of neutrally stable structure that exhibits neutral stability as part of a continuous deformation process, while also allowing a stress-free configuration to exist. The transition of a double-curved compliant shell from its stress-free stable equilibrium towards its second stable equilibrium, through a range of neutrally stable equilibrium configurations forms the basis of this investigation. To design this neutrally stable shell, an optimization is employed to obtain an ideal set of variables that defines a varying thickness profile. Numerical analysis of the resulting optimized shell structure predicts a substantial region of near-constant energy and associated near-zero loads within this unique deformation mode. Additively manufactured prototypes demonstrate the validity of the modeled results by featuring a continuous equilibrium within the range of motion. These results lay the foundation for compliant beam elements with a neutrally stable bending degree of freedom.

Intrawell and interwell oscillations for bi-stable cross-ply laminates actuated by MFC under oscillating impulse voltages

The snap-through is a typical geometric nonlinear problem for the bi-stable laminate structures. Using the Macro Fiber Composite (MFC) actuators to actuate the snap-through behavior of the morphing structures has attracted considerable attention. In this paper, the intrawell oscillations and interwell oscillations are studied for the asymmetric four layers cross-ply bi-stable rectangular laminate actuated by the d33 type MFC actuator. It is assumed that the four edges of the bi-stable laminate are free while the geometry center of it is fixed. On the basis of the classical laminated plate theory and the Hamilton’s principle, the equations of motion of the bi-stable laminate which are expressed by curvatures are presented. The stable equilibrium configurations of the bi-stable laminate considering the effects of MFC are given. Under the action of oscillating impulse voltages, the intrawell oscillations and interwell oscillations of the system are analyzed in detail through the frequency-response curves, time response and the phase portraits. The numerical results show that the snap-through behavior is easier to realize for the bi-stable laminate actuated by the oscillating impulse voltages than that actuated by the static voltages. Furthermore, snap-through behavior can be better controlled by changing the amplitude and frequency of the applied voltages in a certain range.

Vibration of an Incompressible viscoelastic shell of rate type

We study the oscillations of a spherical shell of a rate-type viscoelastic solid subject to a pressure difference between the inner and the outer surface. The stable equilibrium configurations, in the class of spherically symmetric deformations, correspond to the minima of the elastic energy function. Numerical simulations indicate that the way in which the equilibrium state is reached, strongly depends on the material parameters.

Identifying Friction in a Nonlinear Chaotic System Using a Universal Adaptive Stabilizer

This paper proposes a friction model parameter identification routine that can work with highly nonlinear and chaotic systems. The chosen system for this study is a passively-actuated tilted Furuta pendulum, which is known to have a highly non linear and coupled model. The pendulum is tilted to ensure the existence of a stable equilibrium configuration for all its degrees of freedom, and the link weights are the only external forces applied to the system. A nonlinear analytical model of the pendulum is derived, and a continuous friction model considering static friction, dynamic friction, viscous friction, and the stribeck effect is selected from the literature. A high-gain Universal Adaptive Stabilizer (UAS) observer is designed to identify friction model parameters using joint angle measurements. The methodology is tested in simulation and validated on an experimental setup. Despite the high nonlinearity of the system, the methodology is proven to converge to the exact parameter values, in simulation, and to yield qualitative parameter magnitudes in experiments where the goodness of fit was around 85\% on average. The discrepancy between the simulation and the experimental results is attributed to the limitations of the friction model. The main advantage of the proposed method is the significant reduction in computational needs and the time required relative to conventional optimization-based identification routines. The proposed approach yielded more than 99\% reduction in the estimation time while being considerably more accurate than the optimization approach in every test performed. One more advantage is that the approach can be easily adapted to fit other models to experimental data.

Equilibrium Configuration Analysis and Equilibrium-Based Trajectory Generation Method for Under-Constrained Cable-Driven Parallel Robot

Under-constrained cable-driven parallel robots (UCCDPRs) manipulate the end-effector (EE) by employing fewer number of cables than the degree of freedom of the EE, which causes unwanted swaying motions and oscillations. These unwanted vibrations can be suppressed by using the regenerated EE trajectory through the input-shaper. However, in the case of the UCCDPR systems, generating the feasible EE trajectory and deriving the natural frequency for designing the input-shaper is not straightforward since finding the stable equilibrium configuration is challenging. This paper proposes a novel methodology to find the stable equilibrium configuration of the general UCCDPRs in the assigned EE position. With the proposed method, the EE trajectory can be reproduced as a vibration suppression trajectory through an input shaper designed based on the analysis of the natural frequencies, as well as generating feasible EE trajectories based on the equilibrium configuration. Also, even if the orientation trajectory is not given and only the position trajectory is given, the cable length to follow the given position trajectory of the EE can be obtained based on the equilibrium configuration corresponding to the given position trajectory. Computer simulations and hardware experiments were conducted to verify the effectiveness and performance of the proposed method.

The Structure and Stability of Massive Hot White Dwarfs

Abstract We investigate the structure and stability against radial oscillations, pycnonuclear reactions, and inverse β-decay of hot white dwarfs. We consider the fluid matter to be made up of nucleons and electrons confined in a Wigner–Seitz cell surrounded by free photons. It is considered that the temperature depends on the mass density considering the presence of an isothermal core. We find that the temperature produces remarkable effects on the equilibrium and radial stability of white dwarfs. The stable equilibrium configuration results are compared with those for white dwarfs estimated from the Extreme Ultraviolet Explorer survey and the Sloan Digital Sky Survey. We derive masses, radii, and central temperatures for the most massive white dwarfs according to the surface gravity and effective temperature reported by the surveys. We note that these massive stars are in the mass region where general relativity effects are important. These stars are near the threshold of instabilities due to radial oscillations, pycnonuclear reactions, and inverse β-decay. Regarding the radial stability of these stars as a function of the temperature, we find that it decreases with the increment of central temperature. We also find that the maximum-mass point and the zero eigenfrequencies of the fundamental mode are determined at the same central energy density. Regarding low-temperature stars, pycnonuclear reactions occur in similar central energy densities, and the central energy density threshold for inverse β-decay is not modified. For T c ≥ 1.0 × 108 [K], the onset of radial instability is attained before pycnonuclear reaction and inverse β-decay.

Advanced nonlinear buckling analysis of a compressed single layer graphene sheet using the molecular mechanics method

The standard molecular mechanics (MM) method with the DREIDING force field (see Mayo et al. The Journal of Physical Chemistry, 1990, 94: 8897–8909) and the molecular structural mechanics (MSM) method with Bernoulli–Euler beam elements are used to study the quasi-static nonlinear buckling and post-buckling behavior of a compressed nearly square single layer graphene sheet (SLGS) for different types of boundary conditions. The novelty of this study is the finding that well-calibrated parameter sets provide similar values of buckling forces/modes and similar post-buckling deformations for stable equilibrium configurations of SLGSs in simulations using the standard MM and MSM methods. In addition, the effect of accounting for non-bonded van der Waals (vdW) forces on the advanced post-buckling deformation modes of compressed SLGSs was studied for the first time. It is shown that the column-like post-buckling deformation modes of compressed SLGSs obtained without the inclusion of vdW interatomic forces are similar to the well-known ones for the planar Euler elastica, and the advanced post-buckling modes obtained with the inclusion of interatomic vdW forces are qualitatively different from the corresponding modes for the planar Euler elastica. In addition, the study shows the theoretical possibility of the existence of post-buckling out-of-plane equilibrium configurations whose stability is provided by attractive vdW forces.

Atomistic modelling of near-crack-tip plasticity * *This work was supported by EPSRC, Grant No. EP/S028870/1.

An atomistic model of near-crack-tip plasticity on a square lattice under anti-plane shear kinematics is formulated and studied. The model is based upon a new geometric and functional framework of a lattice manifold complex, which ensures that the crack surface is fully taken into account, while preserving the crucial notion of duality. As a result, existence of locally stable equilibrium configurations containing both a crack opening and dislocations is established. Notably, with the boundary in the form of a crack surface accounted for, no minimum separation between a dislocation core and the crack surface or the crack tip is required. The work presented here constitutes a foundation for several further studies aiming to put the phenomenon of near-crack-tip plasticity on a rigorous footing.

Bayesian Optimization of Equilibrium States in Elastomeric Beams

Abstract Architected elastomeric beam networks have great potential for energy absorption, multi-resonant vibration isolation, and multi-bandgap elastic wave control, due to the reconfigurability and programmability of their mechanical buckling instabilities. However, navigating this design space is challenging due to bifurcations between mono- and bistable beam designs, inherent geometric nonlinearities, and the strong dependence of buckling properties on beam geometry. To investigate these challenges, we developed a Bayesian optimization framework to control the equilibrium states of an inclined elastomeric beam, while also tuning the energy to transition between these configurations. Leveraging symmetry to reduce the design space, the beam shape is parameterized using a Fourier series representation. A penalty method is developed to include monostable designs in objective functions with dependencies on bistable features, enabling monostable results to still be incorporated in the Gaussian process surrogate and contribute to the optimization process. Two objectives are optimized in this study, including the position of the second stable equilibrium configuration and the ratio of output to input energy between the two stable states. A scalarized multi-objective optimization is also carried out to study the trade-off between equilibrium position and the energetics of transition between the stable states. The predicted designs are qualitatively verified through experimental testing. Collectively, the study explores a new parameter space for beam buckling, introduces a penalty method to regularize between mono- and bistable domains, and provides a library of beams as building blocks to assemble and analyze in future studies.