Energy-based Stability Research Articles

Gait optimization and energy-based stability for biped locomotion using large-scale programming

AbstractThis paper presents a gait optimization method to generate the locomotion pattern for biped and discuss its stability. The main contribution of this paper is a newly proposed energy-based stability criterion, which permits the dynamic stable walking and could be straight-forwardly generalized to different locomotion scenarios and biped robots. The gait optimization problem is formulated subject to the constraints of the whole-body dynamics and kinematics. The constraints are established based on the modelling of bipedal hybrid dynamical systems. Following the whole-body modelling, the system energy is acquired and then applied to create the stability criterion. The optimization objective is also established on the system energy. The gait optimization is solved by being converted to a large-scale programming problem, where the transcription accuracy is improved via the spectral method. To further reduce the dimensionality of the large-scale problem, the whole-body dynamics is re-constructed. The generalization of the optimized gait is improved by the design of feedback control. The optimization examples demonstrate that the stability criterion naturally leads to a cyclic biped locomotion, though the periodicity was not previously imposed. Two simulation cases, level ground walking and slope walking, verify the generalization of the stability criterion and feedback control. The stability analyses are carried out by investigating the motions of centre of gravity and centre of pressure. It is revealed that if the tracked speed is above 0.3 m/s or the biped accelerates/climbs the slope, the stability criterion accomplishes the dynamic stable walking, where zero moment point criterion is not strictly complied.

Energy-based stability estimates for incompressible media with tensor-nonlinear constitutive relations

Energy-based stability estimates for incompressible media with tensor-nonlinear constitutive relations

Time-variant reliability analysis of underground structures against delayed failure based on the probability density evolution method

The unpredictable delayed failure of underground tunnels may occur after excavation due to the time-dependent behavior of surrounding rock mass, which brings a new consideration to the reliability evaluation of underground structures. In this study, a time-variant reliability analysis procedure via the probability density evolution method (PDEM) is proposed for underground tunnels against delayed failure. The creep-damage theory is introduced to model the time-dependent behavior of rock materials. A new energy-based stability criterion is developed to identify the structural delayed failure. By applying the PDEM with the absorbing boundary condition, the time-varying reliability of underground structures can be obtained. The proposed procedure is applied to a case study involving the deep tunnel with uncertain material parameters. Numerical results demonstrate that the inherent variability of rock properties could result in the large fluctuation of structural delayed responses, which has a significant influence on the time-varying reliability of the tunnels.

Stability Analysis of Microgrid Islanding Transients Based on Interconnected Dissipative Subsystems

To ensure successful islanding of microgrids after a fault has occurred, the transient stability should be analyzed under a set of expected operating conditions during the design and operation of microgrids. Transient stability analysis is conventionally performed with time-domain analysis which is computationally expensive and does not quantify the stability margin. Energy-based methodologies can determine the stability margin, however existing methodologies require significant simplifications to be applied to the microgrid model. The energy-based stability analysis methodology proposed in this paper enables the analysis of high-dimension nonlinear microgrid systems and quantification of the stability margin within reasonable time. The performance of the methodology is validated by analyzing a case study microgrid and comparing the results to time-domain analysis and to a state-of-the-art methodology proposed in the literature. The results indicate that the proposed methodology has a significantly lower computational burden and similar accuracy compared to existing energy-based methodologies. The methodology is able to improve the probability of stable islanding of the case study microgrid from 74% up to 94% when only optimizing the design, and up to 100% when optimizing design and control actions.

Stability-enhanced AP IMEX1-LDG Method: Energy-based Stability and Rigorous AP Property

In our recent work [Z. Peng et al., J. Comput. Phys., 415 (2020), 109485], a family of high-order asymptotic preserving (AP) methods, termed IMEX-LDG methods, are designed to solve some linear kine...

A volatile spray zone model and experimentation in a gas-solid fluidized bed with liquid injection

In a gas-solid fluidized bed, volatile liquid injection results in complex flow behaviors in the spray zone. To characterize the liquid existence forms (droplets and liquid films) and fluidization properties in the volatile spray zone, a mesoscale-theory based volatile spray zone model is developed with the consideration of hydrodynamics, heat transfer, and mass transfer. A new energy-based stability condition is established according to two competition and coordination relationships, consisting of the energy dissipation based on the liquid-solid interface change and suspension and transporting of particles. The modeling results agree well with the experimental bed temperature as well as void fraction data, and show that the variation of liquid existence forms is dependent on the ratio of different energy consumption rates under the stability condition. Moreover, the gas-solid drag coefficient in the volatile spray zone is modified. Simulation results also reasonably describe the bed pressure drop fluctuation and particle velocity distribution.

An energy-based stability analysis of misfit dislocations in two-phase electrode particles: A planar model and implications for LiFePO4

It is well known that cycling induced capacity fading in phase transforming intercalation materials is critically linked to the severity of misfit strain between the phases which could result in the emergence of a variety of defects in the material. Here, adopting a planar model and making use of a well-established energy-based framework, we examine stability of a misfit dislocation lying at the phase boundary within a two-phase electrode particle. Within this framework, the stability criterion emerges as a result of competition between the work that must be done against the image forces with the work that is supplied by the phase transformation induced stresses, as the dislocation is pulled from free surface of the solid toward interior. A numerical model accounting for anisotropy of both misfit strain between the phases and elastic properties of the solid is developed. The model is in particular applied to the electrode particles of LiFePO4, a highly promising cathode material for lithium-ion battery applications in which misfit dislocations as a result of phase transformation have also been experimentally observed. The analysis is revealing of size effects on the stability of a misfit dislocation in the electrode particle. Most notably, it is shown that below a critical particle size, no misfit dislocation at the interphase can remain stable within the particle as it becomes energetically more favorable for the dislocation to be driven out of the particle through absorption by the free surfaces. Numerical estimates of the critical size are presented and discussed with reference to available experiments. Through providing a guideline for suppressing misfit dislocations, results of this work could have potentially important implications for disabling a wide range of dislocation-based fatigue mechanisms which could otherwise become active and result in capacity fading upon a large number of cycles.

Enhanced sensitivity‐based decentralised framework for real‐time transient stability assessment in bulk power grids with renewable energy resources

With the increasing trend in the integration and deployment of distributed energy resources, the real-time assessment of power system transient stability is becoming more and more challenging. This study presents an enhanced sensitivity-based decentralised transient stability assessment scheme, which first virtually decomposes the large-scale power network into several smaller areas. Then, a new sensitivity-based scheme is developed to derive and quantify the kinetic energy-based stability assessment index. This index allows taking into account the effects of both AC generators and power electronic-based generator units. In addition, thanks to the new sensitivity-based scheme, the transient stability can be assessed in a decentralised manner using only the boundary buses in each segmented area, which relaxes the need for massive data availability and transfer across the network and enables its applicability for online applications in large-scale power systems. Simulation results on the 48-machine IEEE 140-bus and 544-machine 2000-bus synthetic power grid test systems demonstrate the efficacy of the proposed method.

Power Balancing Approach for Modeling and Stabilization of DC Networks

We propose a set of theoretical and practical tools for the stabilization of dc networks. In order to do so, we propose a new framework to study the dynamics of dc grids—one that permits the combination of power converter models with those impedances—directly in higher-order terms. This mathematical setting includes a convenient modeling specification that allows one to analyze complex high-gain converter topologies and to develop an energy-based stability analysis for dc grids with potential destabilizers such as loads with negative impedance characteristics (e.g., constant power loads). We show that instability problems arise due to a power imbalance between passive and non-passive loads, hence power balancing is a suitable stabilizing mechanism. Moreover, a low-cost hardware solution that ensures stability is provided in the form of a lossless (H-bridge) stabilizing filter. The new theoretical results, combined with the proposed implementation, offer a low-cost alternative with respect to other (battery/super-capacitor) energy-storage-based existing strategies. Experimental results based on a high voltage gain quadratic converter feeding multiple non-passive loads are included.

An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices.

In this work, we revisit a criterion, originally proposed in Friesecke & Pego (Friesecke & Pego 2004 Nonlinearity17, 207-227. (doi:10.1088/0951715/17/1/013)), for the stability of solitary travelling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-travelling frame, as well as at that of the Floquet problem arising when considering the travelling wave as a periodic orbit modulo shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the travelling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Energy shaping for position and speed control of a wheeled inverted pendulum in reduced space

The paper deals with the energy-based stabilization and speed control of a wheeled inverted pendulum, which is an underactuated, unstable mechanical system subject to nonholonomic constraints. We use the method of Controlled Lagrangians for the stabilization of an equilibrium characterized by the length of the driven path, the orientation, and the pitch angle. The approach is systematic and very intuitive, for it is physically motivated. Based on the stabilization results, we design a speed control law. After the presentation of the model under nonholonomic constraints in Lagrangian representation, we provide an elegant solution to the matching equations for kinetic and potential energy shaping for the considered system. Simulations show the applicability of the method, and the comparison with a linear controller emphasizes its performance.

Energy shaping for the robust stabilization of a wheeled inverted pendulum

The paper deals with the robust energy-based stabilization of a wheeled inverted pendulum, which is an underactuated, unstable mechanical system subject to nonholonomic constraints. The equilibrium to be stabilized is characterized by the length of the driven path, the orientation, and the pitch angle. We use the method of Controlled Lagrangians which is applied in a systematic way, and is very intuitive, for it is physically motivated. After a detailed presentation of the model under nonholonomic constraints, we provide an elegant solution of the matching equations for kinetic and potential energy shaping for the considered systems. Simulations show the applicability and robustness of the method.

A stable second-order scheme for fluid–structure interaction with strong added-mass effects

In this paper, we present a stable second-order time accurate scheme for solving fluid–structure interaction problems. The scheme uses so-called Combined Field with Explicit Interface (CFEI) advancing formulation based on the Arbitrary Lagrangian–Eulerian approach with finite element procedure. Although loosely-coupled partitioned schemes are often popular choices for simulating FSI problems, these schemes may suffer from inherent instability at low structure to fluid density ratios. We show that our second-order scheme is stable for any mass density ratio and hence is able to handle strong added-mass effects. Energy-based stability proof relies heavily on the connections among extrapolation formula, trapezoidal scheme for second-order equation, and backward difference method for first-order equation.Numerical accuracy and stability of the scheme is assessed with the aid of two-dimensional fluid–structure interaction problems of increasing complexity. We confirm second-order temporal accuracy by numerical experiments on an elastic semi-circular cylinder problem. We verify the accuracy of coupled solutions with respect to the benchmark solutions of a cylinder-elastic bar and the Navier–Stokes flow system. To study the stability of the proposed scheme for strong added-mass effects, we present new results using the combined field formulation for flexible flapping motion of a thin-membrane structure with low mass ratio and strong added-mass effects in a uniform axial flow. Using a systematic series of fluid–structure simulations, a detailed analysis of the coupled response as a function of mass ratio for the case of very low bending rigidity has been presented.

Localization of flow structures using -norm optimization

AbstractStability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density ${E}_{1} (t)= (1/ {V}_{\Omega } )\int \nolimits _{\Omega } e(\boldsymbol{x}, t)\hspace{0.167em} \mathrm{d} \Omega $ (where $e(\boldsymbol{x}, t)= \vert \boldsymbol{u}{\vert }^{2} / 2$) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain $\Omega $ of volume ${V}_{\Omega } $. This paper explores the use of $p$-norms in determining optimal perturbations for ‘energy’ growth over prescribed time intervals of length $T$. For $p= 1$ the traditional energy-based stability analysis is recovered, while for large $p\gg 1$, localization of the optimal perturbations is observed which identifies confined regions, or ‘hotspots’, in the domain where significant energy growth can be expected. In addition, the $p$-norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the $\infty $-norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the $\infty $-norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed $p$-norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth.

Fabrication of metallic nanoparticles by spinodal dewetting of thin films: A high-throughput approach

Metal nanoparticles on structured supports are used in a variety of technological applications including biosensing, energy harvesting, and electronics. In every case, the functions and properties of the metallic nanostructures depend on both their composition and structure (i.e. size, shape, and spatial distribution). Among the challenges to the development of metal nanoparticles for these applications is the characterization of relationships between their structure and their functional properties over multiple structural degrees of freedom spanning a large range of values. In this work, a method for creating a morphological gradient of metal nanoparticles on a substrate is described. The approach, suited for high-throughput fabrication and characterization, is based on spinodal dewetting of a metallic thin film from its substrate. Through control of initial film thickness, anneal temperature, and anneal time, spinodal dewetting results in supported nanoparticles with well-defined and controlled structure. The approach is demonstrated through its application to preparation of Pd nanoparticles on a silicon nitride substrate. The morphologies of the particles were characterized by scanning electron and atomic force microscopies. Free energy-based stability and topological analyses were used to confirm the dewetting mechanism. In addition, the stability theory provides a connection to the thermophysical properties of the resulting nanoparticle array. The dewetting approach is general to any metal/support system and provides an alternative, inexpensive, and robust means to rapidly create metal nanostructures with control of morphology. It shows promise for large scale production of metal nanoparticles structures, as well as understanding basic stability properties of thin metal films.

Structure and stability of the finite-area von Kármán street

By using a recently developed numerical method, we explore in detail the possible inviscid equilibrium flows for a Kármán street comprising uniform, large-area vortices. In order to determine stability, we make use of an energy-based stability argument (originally proposed by Lord Kelvin), whose previous implementation had been unsuccessful in determining stability for the Kármán street [P. G. Saffman and J. C. Schatzman, “Stability of a vortex street of finite vortices,” J. Fluid Mech. 117, 171–186 (1982)10.1017/S0022112082001578]. We discuss in detail the issues affecting this interpretation of Kelvin's ideas, and show that this energy-based argument cannot detect subharmonic instabilities. To find superharmonic instabilities, we employ a recently introduced approach, which constitutes a reliable implementation of Kelvin's stability ideas [P. Luzzatto-Fegiz and C. H. K. Williamson, “Stability of conservative flows and new steady fluid solutions from bifurcation diagrams exploiting a variational argument,” Phys. Rev. Lett. 104, 044504 (2010)10.1103/PhysRevLett.104.044504]. For periodic flows, this leads us to organize solutions into families with fixed impulse I, and to construct diagrams involving the flow energy E and horizontal spacing (i.e., wavelength) L. Families of large-I vortex streets exhibit a turning point in L, and terminate with “cat's eyes” vortices (as also suggested by previous investigators). However, for low-I streets, the solution families display a multitude of turning points (leading to multiple possible streets, for given L), and terminate with teardrop-shaped vortices. This is radically different from previous suggestions in the literature. These two qualitatively different limiting states are connected by a special street, whereby vortices from opposite rows touch, such that each vortex boundary exhibits three corners. Furthermore, by following the family of I = 0 streets to small L, we gain access to a large, hitherto unexplored flow regime, involving streets with L significantly smaller than previously believed possible. To elucidate in detail the possible solution regimes, we introduce a map of spacing L, versus impulse I, which we construct by numerically computing a large number of steady vortex configurations. For each constant-impulse family of steady vortices, our stability approach also reveals a single superharmonic bifurcation, leading to new families of vortex streets, which exhibit lower symmetry.

Displacement-velocity correction schemes for incompressible fluid–structure interaction

We propose a new class of time-marching schemes for the explicit coupling of an incompressible fluid and an elastic solid (not necessarily thin). We state a general energy-based stability result and illustrate the accuracy of the different variants in a numerical benchmark.

A Virtual Model of Spring Reverberation

The digital emulation of analog audio effects and synthesis components, through the simulation of lumped circuit components has seen a large amount of activity in recent years; electromechanical effects have seen rather less, primarily because they employ distributed mechanical components, which are not easily dealt with in a rigorous manner using typical audio processing constructs such as delay lines and digital filters. Spring reverberation is an example of such a system-a spring exhibits complex, highly dispersive behavior, including coupling between different types of wave propagation (longitudinal and transverse). Standard numerical techniques, such as finite difference schemes are a good match to such a problem, but require specialized design and analysis techniques in the context of audio processing. A model of helical spring vibration is introduced, along with a family of finite difference schemes suitable for time domain simulation. Various topics are covered, including numerical stability conditions, tuning of the scheme to the response of the model system, numerical boundary conditions and connection to an excitation and readout, implementation details, as well as computational requirements. Simulation results are presented, and full energy-based stability analysis appears in an ????? .

Elastic buckling of cold-formed steel columns and beams with holes

Simplified methods for approximating the global, distortional, and local critical elastic buckling loads of cold-formed steel columns and beams with holes are developed and summarized. These methods provide engineering approximations appropriate for design, but are intended to be general enough to accommodate the range of hole shapes, locations, and spacings common in industry. The simplified methods are developed as a convenient alternative to shell finite element eigen-buckling analysis, which require laborious and subjective visual identification methods as well as commercial software not generally accessible to the engineering community. Global buckling of cold-formed steel beams and columns, including the influence of holes, is predicted with approximate “weighted average” cross-section properties formulated from classical energy-based stability solutions. Distortional and local buckling of a cold-formed steel member with holes are determined with the semi-analytical finite strip method, considering appropriate modifications to the element thickness and choice of buckling half-wavelength. The proposed methods are verified with shell finite element eigen-buckling studies. Unambiguous, simple methods for elastic buckling prediction of members with holes is central to the extension of the Direct Strength Method (DSM) for cold-formed steel member ultimate strength determination.

Analytical Modeling of Impact Resistance and Damage Tolerance of Laminated Composite Plates

The present study is concerned with a fully analytical model for predicting impact resistance and damage tolerance characteristics of laminated fiber reinforced composite plates. The energy balance for damage development during the impact process is established with the help of a localized deformation field, in which various effects such as the anisotropy of the delaminated sublaminates, elastic property degradation due to matrix cracking, multiple delaminations, and membrane deformation within the damage zone are taken into account. This results in upper and lower bounds of the delamination threshold load and a deflection-dependent critical load for stable delamination propagation. On the basis of the conservation of energy principle, simple relationships are found between the resulting damage area and the impact energy as well as the peak impact load. The global buckling load of damaged rectangular plates is derived from the energy-based stability criterion, and the residual ultimate strength is calculated using the von Karman postbuckling strength theory. The analytical solutions obtained turn out to be capable of providing highly accurate predictions compared with the standard impact and the compression-after-impact experiments of carbon-fiber-reinforced polyetherimide and aromatic polymer composites.