Hamiltonian modelling and buckling analysis of a nonlinear flexible beam with actuation at the bottom

ABSTRACT

The use of beams and similar structural elements is finding increasing application in many areas including micro and nanotechnology devices. For the purpose of buckling analysis and control, it is essential to account for nonlinear terms in the strains while modelling these flexible structures. Further, the Poisson’s effect can be accounted in modelling by the use of a two-dimensional stress–strain relationship. This paper studies the buckling effect for a slender, vertical beam (in the clamped-free configuration) with horizontal actuation at the fixed end and a tip-mass at the free end. Including also the inextensibility constraint of the beam, the equations of motion are derived. A preliminary modal analysis of the system has been carried out to describe candidate post-buckling configurations and study the stability properties of these equilibria. The vertical configuration of the beam under the action of gravity is without loss of generality, since the objective is to model a potential field that determines the equilibria. Neglecting the inextensibility constraint, the equations of motion are then casted in port-Hamiltonian form with appropriately defined flows and efforts as a basis for structure-preserving discretization and simulation. Finally, the finite-dimensional model is simulated to obtain the time response of the tip-mass for different loading conditions.     

Fluid-Structure Port-Hamiltonian Model for Incompressible Flows in Tubes with Time Varying Geometries

ABSTRACT

A simple and scalable finite-dimensional model based on the port-Hamiltonian framework is proposed to describe the fluid–structure interaction in tubes with time-varying geometries. For this purpose, the moving tube wall is described by a set of mass-spring-damper systems while the fluid is considered as a one-dimensional incompressible flow described by the average momentum dynamics in a set of incompressible flow sections. To couple these flow sections small compressible volumes are defined to describe the pressure between two adjacent fluid sections. The fluid-structure coupling is done through a power-preserving interconnection between velocities and forces. The resultant model includes external inputs for the fluid and inputs for external forces over the mechanical part that can be used for control or interconnection purposes. Numerical examples show the accordance of this simplified model with finite-element models reported in the literature.     

Topological sensitivity analysis of a multi‐scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi‐scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non‐penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth‐order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi‐scale framework changes when a small circular inclusion is introduced at the micro‐scale level. Copyright © 2009 John Wiley & Sons, Ltd.     

On the determination of natural frequencies and mode shapes of cable-stayed bridges

An accurate analysis of the natural frequencies and mode shapes of a cable-stayed bridge is fundamental to the solution of its dynamic responses due to seismic, wind and traffic loads. In most previous studies, the stay cables have been modelled as single truss elements in conventional finite element analysis. This method is simple but it is inadequate for the accurate dynamic analysis of a cable-stayed bridge because it essentially precludes the transverse cable vibrations. This paper presents a comprehensive study of various modelling schemes for the dynamic analysis of cable-stayed bridges. The modelling schemes studied include the finite element method and the dynamic stiffness method. Both the mesh options of modelling each stay cable as a single truss element with an equivalent modulus and modelling each stay cable by a number of cable elements with the original modulus are studied. Their capability to account for transverse cable vibrations in the overall dynamic analysis as well as their accuracy and efficiency are investigated.     

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.     

The modeling of coalition for environmental protection within co-utility framework

The game theoretic modelling of coalition for environmental protection within the framework of a new concept of co-utility is discussed. The co-utility concept contains mainly two elements. Firstly, agents can increase their payoffs by collaborating with each-other. Secondly, the outcome of collaboration is robust towards internal and external disturbances. The advantages of using of co-utility are two-fold. Primarily, the co-utility concept is broad and can serve as an umbrella concept in all applications where agents have a space for simultaneous improvement of payoffs. Secondly the co-utility concept can be associated with different stability concept such as myopic or farsighted stability. The myopic and farsighted co-utile sets are defined and their element-co-utile outcomes are found.     

The tractability index of memristive circuits: branch‐oriented and tree‐based models

The memory‐resistor or memristor is a new electrical element characterized by a nonlinear charge‐flux relation. This device poses many challenging problems, in particular from the circuit modeling point of view. In this paper, we address the index analysis of certain differential‐algebraic models of memristive circuits; specifically, our attention is focused on so‐called branch‐oriented models, which include in particular tree‐based formulations of the circuit equations. Our approach combines results coming from differential‐algebraic equation theory, matrix analysis and theory of digraphs. This framework should be useful in future studies of dynamical aspects of memristive circuits. Copyright © 2012 John Wiley & Sons, Ltd.     

Passivity analysis of memristive neural networks with different memductance functions

Memristive neural networks have captured the attention of physicists, biologists, ecologists, economists and social scientists. In this paper, we formulate and investigate a class of memristive neural networks with two different types of memductance functions. Some succinct criteria in terms of linear matrix inequalities for the passivity are proposed. Meanwhile, based on the derived criteria, some stability criterion are obtained for the memristive neural networks. These theoretical analysis can characterize the fundamental electrical properties of memristive systems and provide convenience for applications.     

Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses

In this paper, the chaotic behavior of a simplest autonomous memristor-based circuit of fractional order is suppressed by periodic impulses applied to one or several state variables. The circuit consists of two passive linear elements, a capacitor and an inductor, as well as a nonlinear memristive element. It is shown that by applying a sequence of adequate (identical or different) periodic impulses to one or several variables, the chaotic behavior can be suppressed. Impulse values and control timing are determined numerically, based on the bifurcation diagram with impulses as bifurcation parameters. Empirically, the probability to have a reasonably wide range of impulses to suppress chaos is quite large, ensuring that chaos suppression can be implemented, as demonstrated by several examples presented.     

FLIM - A Variant of FEM based on Line Integration

A variant of the finite element method (FEM) for modelling and solving partial differential equations based on triangular and tetrahedral meshes is proposed. While FEM is based on integration over finite elements, the new approach - briefly denoted as FLIM hereafter - uses integration along edges (finite lines). The stiffness matrix, which - for linear triangles and tetrahedra - is identical with the one obtained with FEM, as well as the load vector can solely be obtained by summing up the edge contributions. This new variant requires much lower storage than FEM, especially for three-dimensional problems, but yields the same approximation error and convergence rate as the finite element method. It is shown that its performance, when applied to linear problems, is in close agreement with the performance of the finite element method. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dissipativity analysis of memristive neural networks with time‐varying delays and randomly occurring uncertainties

Dissipativity theory is a very important concept in the field of control system. In this paper, we pay attention to the problem of dissipativity analysis of memristive neural networks with time‐varying delay and randomly occurring uncertainties(ROUs). Under the framework of Filippov solution, differential inclusion theory, by employing a proper Lyapunov functional, and some inequality techniques, the dissipativity criteria are obtained in terms of LMIs. It should be noteworthy that the uncertainty terms as well as the ROUs are separately taken into consideration, in which the uncertainties are norm‐bounded and the ROUs obey certain mutually uncorrelated Bernoulli‐distributed white noise sequences. Finally, the effectiveness of the proposed method will be verified via numerical example. Copyright © 2015 John Wiley & Sons, Ltd.     

Multistability of memristive neural networks with time‐varying delays

The recent discovery of memristive neurodynamic systems holds great promise for realizing large‐scale nanoionic circuits. Development of pattern memory analysis for memristive neurodynamic systems poses several challenges. In this article, it shows that an n‐dimensional memristive neural networks with time‐varying delays can have 2ⁿ locally exponentially stable equilibria in the saturation region. In addition, local exponential stability of delayed memristive neural networks in any designated region is also characterized, which allows the locally exponentially stable equilibria to locate in the designated region. All of these criteria are very easy to be verified. Finally, the effectiveness of the results are illustrated by two numerical examples. © 2014 Wiley Periodicals, Inc. Complexity 21: 177–186, 2015     

An augmented mixed finite element method with Lagrange multipliers: A priori and a posteriori error analyses

In this paper, we provide a priori and a posteriori error analyses of an augmented mixed finite element method with Lagrange multipliers applied to elliptic equations in divergence form with mixed boundary conditions. The augmented scheme is obtained by including the Galerkin least-squares terms arising from the constitutive and equilibrium equations. We use the classical Babuška–Brezzi theory to show that the resulting dual-mixed variational formulation and its Galerkin scheme defined with Raviart–Thomas spaces are well posed, and also to derive the corresponding a priori error estimates and rates of convergence. Then, we develop a reliable and efficient residual-based a posteriori error estimate and a reliable and quasi-efficient Ritz projection-based one, as well. Finally, several numerical results illustrating the performance of the augmented scheme and the associated adaptive algorithms are reported.     

A flexible approach to the simulation of hybrid multibody systems

The present works deals with the incorporation of both flexible beam and shell structures into the realm of flexible multibody dynamics. Geometrically exact beam formulations based on classical Simo-Reissner kinematics are suitable for modelling beam-type flexible components in the context of finite-deformation multibody dynamics. So geometrically exact shell formulations are based on Reissner-Mindlin kinematics. In [2], a flexible framework for dealing with flexible structural elements in a multibody context is described. A specific isoparametric finite element discretization of a shell formulation leads to semi-discrete equations of motions assuming the form of differential-algebraic equations (DAEs). A compatible isoparametric formulation of beams has already been developed in [1]. The uniform DAE framework makes possible the incorporation of alternative finite element formulations. In addition to that, various time-stepping schemes such as energy-momentum methods or variational integrators can be applied. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bayesian neural network learning for repeat purchase modelling in direct marketing

We focus on purchase incidence modelling for a European direct mail company. Response models based on statistical and neural network techniques are contrasted. The evidence framework of MacKay is used as an example implementation of Bayesian neural network learning, a method that is fairly robust with respect to problems typically encountered when implementing neural networks. The automatic relevance determination (ARD) method, an integrated feature of this framework, allows us to assess the relative importance of the inputs. The basic response models use operationalisations of the traditionally discussed Recency, Frequency and Monetary (RFM) predictor categories. In a second experiment, the RFM response framework is enriched by the inclusion of other (non-RFM) customer profiling predictors. We contribute to the literature by providing experimental evidence that: (1) Bayesian neural networks offer a viable alternative for purchase incidence modelling; (2) a combined use of all three RFM predictor categories is advocated by the ARD method; (3) the inclusion of non-RFM variables allows to significantly augment the predictive power of the constructed RFM classifiers; (4) this rise is mainly attributed to the inclusion of customer/company interaction variables and a variable measuring whether a customer uses the credit facilities of the direct mailing company.     

Modeling complex multi-component reactive-transport systems: towards a simulation environment based on the concept of a Knowledge Base

A modelling framework within which transport processes in the hydrosphere can be described and interfaced with relevant biogeochemical reactions is presented. Three key elements of this simulation environment are discussed: (1) a numerical engine for solving sets of coupled non-linear process equations; (2) an automated procedure for model code generation (`Automatic Code Generator'); (3) a Web-distributed Knowledge Base (KB) of processes. The Automatic Code Generator translates the information selected in the KB into computer algorithms using the principles defined in the numerical engine. The code CONTRASTE is a first attempt at developing such a modelling framework. It allows one to easily select, adapt and combine a specific set of biogeochemical processes relevant to a user-defined application. The workings of CONTRASTE are described by means of examples which demonstrate how the various components of the simulation environment are coupled and automated. Prospects for future developments towards a fully automated model generation procedure are discussed.     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Standardization and optimum structural design by dynamic programming

The paper is concerned with tools for optimum structural design that can taken into account the industrial production of structural elements. Standardization is considered here as the result of minimizing the cost as a function of both element sizes and numbers of identical elements. The optimum value of the design variables obtained by traditional minimum weight (volume) design is taken as the starting point for the present procedure. Dynamic programming is used for standardization decisions. The method seems to be particularly effective when the cost function for each element size can be regarded as the sum of a constant term (initial investment) and a term that is proportional to the number of identical elements (cost of production).The original idea for the present approach, developed in Section 2, was suggested by Professor N. Distefano. The author will always remember this with gratitude.The author expresses his thanks to Prof. Dr. W. Prager for reviewing the paper.