Lagrange Formula Research Articles

ALGORITHM OF EULER-LAGRANGE METHOD FOR DEISGNING OF DYNAMIC MODEL

Urgency of the research. Nowadays robotics and mechatronics come to be mainstream. With development in these areas also grow computing fastidiousness. Since there is significant focus on numerical modeling and algorithmization in kinematic and dynamic modeling. Target setting. By automation of whole process of dynamic model design the errors are eliminated as well as the time of designing significantly decreases. Actual scientific researches and issues analysis. Designing of dynamic model by analytical way is very difficult especially in the cases considering high number of DOF. For hyperredundant manipulators it is practically impossible. From this reason whole process is automatized. Uninvestigated parts of general matters defining. The theory of Euler – Lagrange method is automatized by means of robotic view on this issue. The research objective. In the paper, an algorithm for design of dynamic model was introduced. The statement of basic materials. The paper deals with automatic design process of dynamic model for serial kinematic structure mechanisms. In the paper Euler – Lagrange formula is discussed. Analytical way of dynamic modeling should be difficult problem especially for mechanisms with high number of degrees of freedom. From this reason the paper shows the way of automatically designing of dynamic modeling in MATLAB. Our study shows dependence of computing time on increasing DOF. The relation is expressed by function of 3rd order. Subsequently the paper presents automatically generated inverse dynamic model in cooperation with inverse kinematic model as well as trajectory planning task. Conclusions. The paper introduces automatically generated dynamic model for mechanisms with serial kinematic structure. The paper also established the time for designing of dynamic model for several mechanisms with changing DOF.

An Analytical Approach to Modeling of Motion-Response of Floating Structure for Ocean Renewable Energy Conversion System

Ocean renewable energy research has been progressing well. Supporting structures are needed to convert energy from the sea. This paper discusses the response of the floating structure for ocean renewable energy conversion system by providing a simple design of floating structure. Due to its function, the system is limited for the pitching motion. By using the Lagrange formula, the equation of motion of the system can be obtained. In the analysis, there are three variations of wave period to determine the response of floating structure motion. The result shows the trend where the larger wave periods induce larger intersection angle (larger response) of the structure. The floating structure configuration for the ocean energy converter should be determined in such a way that have the most stable motion-response in any condition. The stability of floating structure will affect the current forces in the rotated turbine. It needs a specific design to hold the stability of floating structure.

Generalized Mean-Value Theorem for an Analytic Function and an Algorithm for the Evaluation of the Function of Mean Values

We prove the mean-value theorem for functions analytic in starlike domains, propose an algorithm for finding the function of mean values, and study its analytic continuation. We present a differential equation for the function of mean values and the interpretation of the Lagrange formula for analytic functions in terms of the theory of differential equations. The set of points of the initial values of the function of mean values is analyzed and the upper bound for the radius of expansion of the function of mean values in Taylor’s series is established.

Bend‐bend‐twist vibrations of a wind turbine blade

AbstractDynamics of a wind turbine blade under bend‐bend‐twist coupled vibrations is investigated. The potential and kinetic energy expressions for a straight nonuniform blade are written in terms of beam parameters. Then, the energies are expressed in terms of modal coordinates by using the assumed mode method, and the equations of motion are found by applying Lagrange's formula. The bend‐bend‐twist equations are coupled with each other and have stiffness variations due to centrifugal effects and gravitational parametric terms, which vary cyclicly with the hub angle. To determine the natural frequencies and mode shapes of the system, a modal analysis is applied on the linearized coupled equations of constant angle snapshots of a blade with effects of constant speed rotation. Lower modes of the coupled bend‐bend‐twist model are dominantly in‐plane or out‐of‐plane modes. To investigate the parametric effects, several blade models are analyzed at different angular positions. The stiffness terms involving centrifugal and gravitational effects can be significant for long blades. To further see the effect of blade length on relative parametric stiffness change, the blade models are scaled in size and analyzed at constant rotational speeds, at horizontal and vertical orientations. These studies show that the parametric stiffness effects should be taken into account when designing long blades.

Optimal stability of the Lagrange formula and conditioning of the Newton formula

A pointwise condition number associated to a representation of an interpolation operator is introduced. It is proved that the Lagrange formula is optimal with respect to this conditioning. For other representations of the interpolation operator, an upper bound for the conditioning is derived. A quantitative measure in terms of the Skeel condition number is used to compare the conditioning with the Lagrange representation. The conditioning of the Newton representation is considered for increasing nodes and for nodes in Leja order. For the polynomial Newton formula with n+1 equidistant nodes in increasing order, it is proved that 3n is the best uniform bound of its conditioning and it is attained at the last node. Numerical experiments are included.

Lagrange formula for ordinary continual second-order differential equations

For ordinary continual second-order differential equations, we derive a Lagrange formula and construct their fundamental solutions. We use the Lagrange formula to determine well-posed forms of initial data for these equations and obtain explicit representations of solutions of these initial value problems.

Cross-Layer Optimization and Protocol Analysis for Cognitive Ad Hoc Communications

In the cognitive radio ad hoc networks (CRAHNs), the spectrum availability may change from time to time and hop by hop. Thereby, the performance analysis and optimization for CRAHNs become so intricate that pure divide-and-conquer strategies by layered principles strand. Focusing on the five-layer involved performance analysis of CRAHNs, we set up a cross-layer design framework mathematically and solve it through vertical decomposition approaches. After the convex relaxation, a partial Lagrangian formula, which captures network objective and constraints from the physical layer to application layer, is devised. Then, a cross-layer optimization scheme through a vertical decomposition method (COVD) is proposed, which leads to a novel cross-layer architecture. Through the dual decomposition and primal decomposition, the complex joint optimization problem is decoupled into three subproblems with control parameters flowing back and forth. Although COVD achieves the optimal solution for the joint optimization issue, it incurs high cost in terms of overhead and complexity. Furthermore, we propose a cross-layer optimization design by heuristic algorithm, which reduces the computation complexity by a step-by-step division approach. Finally, simulation results demonstrate the efficiency of the proposed schemes. Complexity analysis and violation of the current protocols are also provided.

Universal curvature identities and Euler–Lagrange formulas for Kähler manifolds

We relate certain universal curvature identities for Kahler manifolds to the Euler–Lagrange equations of the scalar invariants which are defined by pairing characteristic forms with powers of the Kahler form.

Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions

Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.

Improved finite integration method for partial differential equations

Based on the recently developed finite integration method (FIM) for solving one-dimensional partial differential equations by using the trapezoidal rule for numerical quadrature, we improve in this paper the FIM with an alternative extended Simpson׳s rule in which the Cotes and Lagrange formulas are used to determine the first order integral matrix. The improved one-dimensional FIM is then further extended to solve two-dimensional problems. Numerical comparison with the finite difference method and the FIM (Trapezoidal rule) are performed by several one- and two-dimensional real application including the Poisson type differential equations and plate bending problems. It has been shown that the newly revised FIM has made significant improvement in terms of accuracy compare without much sacrifice on the stability and efficiency.

Euler–Lagrange as Pseudo-metric of the RRT algorithm for optimal-time trajectory of flight simulation model in high-density obstacle environment

SUMMARYThis paper introduces a solution to the problem of steering an aerodynamical system, with non-holonomic constraints superimposed on dynamic equations of motion. The proposed approach is a dimensionality reduction of the Optimal Control Problem (OCP) with heavy path constraints to be solved by Rapidly-Exploring Random Tree (RRT) algorithm. In this research, we formulated and solved the OCP with Euler–Lagrange formula in order to find the optimal-time trajectory. The RRT constructs a non-collision path in static, high-dense obstacle environment (i.e. heavy path constraint). Based on a real-world aircraft model, our simulation results found the collision-free path and gave improvements of time and fuel consumption of the optimized Hamiltonian-based model over the original non-optimized model.

A functional generalization of the interpolation problem

Given two linearly independent functions f1 and f2, we generalize the interpolating problem to the space πn(f1,f2) spanned by the basis f1n-kf2kk=0n. We show that this problem has a unique solution and represent this solution by a functional analogue of the Lagrange formula. We also give a similar generalization of Hermite interpolation.

On Bivariate Fundamental Polynomials

An n-independent set in two dimensions is a set of nodes admitting (not necessary unique) bivariate interpolation with polynomials of total degree at most n: For an arbitrary n-independent node set X we are interested with the property that each node possesses an n-fundamental polynomial in form of products of linear or quadratic factors. In the present paper we show that each node of X has an n-fundamental polynomial, which is a product of lines, if #X ≤ 2n + 1: Next we prove that each node of X has an n-fundamental polynomial, which is a product of lines or conics, if #X ≤ 2n+[n/2]+1. We bring a counterexample in each case to show that the results are not valid in general if #X ≥ 2n + 2 and #X ≥ 2n + [n/2] + 2; respectively. At the end we bring an algorithm for the construction of above mentioned fundamental polynomials. This, in view of the Lagrange formula, can be used to obtain readily also the interpolation polynomials.

2 자유도 병렬 메니퓰레이터의 동적 모델링

In this paper, two-DOF parallel manipulator has the sliders which execute a linear reciprocating motion depending on parallel guides and the end-effector which can be adjusted arbitrarily. To investigate the dynamic characteristics of the manipulator, the dynamic performance index is used. The index is able to be obtained by the relation between the Jacobian matrix and the inertia matrix. The kinematic and the dynamic analysis find these matrices. Also, the dynamic model of the manipulator is derived from the Lagrange formula. This model represents complicated nonlinear equations of motion. With the simulation results of the dynamic characteristic of the manipulator, we find that the dynamic performance index is based on the selection of the ranges for the continuous movement of the manipulator and the dynamic model derived can be used to the control algorithm development of the manipulator.

Kinematics and dynamics analysis of a three-degree-of-freedom parallel manipulator

Kinematics and dynamics analyses were performed for a spatial 3-revolute joint-revolute joint-clylindric pair (3-RRC) parallel manipulator. This 3-RRC parallel manipulator is composed of a moving platform, a base platform, and three revolute joint-revolute joint-column pair chains which connect the moving platform and the base platform. Firstly, kinematics analysis for 3-RRC parallel manipulator was conducted. Next, on the basis of Lagrange formula, a simply-structured dynamic model of 3-RRC parallel manipulator was derived. Finally, through a calculation example, the variation of motorial parameters of this 3-RRC parallel manipulator, equivalent moment of inertia, driving force/torque and energy consumption was discussed. The research findings have important significance for research and engineering projects such as analyzing dynamic features, mechanism optimization design and control of 3-RRC parallel manipulator.

Development of a tangent linear model (version 1.0) for the High-Order Method Modeling Environment dynamical core

Abstract. We describe development and validation of a tangent linear model for the High-Order Method Modeling Environment, the default dynamical core in the Community Atmosphere Model and the Community Earth System Model that solves a primitive hydrostatic equation using a spectral element method. A tangent linear model is primarily intended to approximate the evolution of perturbations generated by a nonlinear model, provides a computationally efficient way to calculate a nonlinear model trajectory for a short time range, and serves as an intermediate step to write and test adjoint models, as the forward model in the incremental approach to four-dimensional variational data assimilation, and as a tool for stability analysis. Each module in the tangent linear model (version 1.0) is linearized by hands-on derivations, and is validated by the Taylor–Lagrange formula. The linearity checks confirm all modules correctly developed, and the field results of the tangent linear modules converge to the difference field of two nonlinear modules as the magnitude of the initial perturbation is sequentially reduced. Also, experiments for stable integration of the tangent linear model (version 1.0) show that the linear model is also suitable with an extended time step size compared to the time step of the nonlinear model without reducing spatial resolution, or increasing further computational cost. Although the scope of the current implementation leaves room for a set of natural extensions, the results and diagnostic tools presented here should provide guidance for further development of the next generation of the tangent linear model, the corresponding adjoint model, and four-dimensional variational data assimilation, with respect to resolution changes and improvements in linearized physics and dynamics.

An EMQ model with price and time dependent demand under the effect of reliability and inflation

The paper deals with an economic manufacturing quantity (EMQ) model for the selling price and the time dependent demand pattern in an imperfect production process. Due to long-run, machine breakdown may occur, as a result, the system may shift to out-of-control state from in-control state and production systems begin to produce imperfect quality items. The production of imperfect items increases with time. All imperfect quality items are reworked at a fixed cost to restore these to its original quality. To reduce the production of the imperfect quality items, we consider reliability as a decision variable along with the development cost and the production cost as a function of reliability. The profit function is maximized by Euler–Lagrange formula. The numerical example, sensitively analysis, and graphical illustrations are given to illustrate the model.

Geometrical prediction of maximum power point for photovoltaics

It is important to drive solar photovoltaic (PV) system to its utmost capacity using maximum power point (MPP) tracking algorithms. This paper presents a direct MPP prediction method for a PV system considering the geometry of the I–V characteristic of a solar cell and a module. In the first step, known as parallelogram exploration (PGE), the MPP is determined from a parallelogram constructed using the open circuit (OC) and the short circuit (SC) points of the I–V characteristic and Lagrangian interpolation. In the second step, accurate values of voltage and power at the MPP, defined as Vmp and Pmp respectively, are decided by the Lagrangian interpolation formula, known as the Lagrangian interpolation exploration (LIE). Specifically, this method works with a few (V, I) data points instead most of the MPP algorithms work with (P, V) data points. The performance of the method is examined by several PV technologies including silicon, copper indium gallium selenide (CIGS), copper zinc tin sulphide selenide (CZTSSe), organic, dye sensitized solar cell (DSSC) and organic tandem cells’ data previously reported in literatures. The effectiveness of the method is tested experimentally for a few silicon cells’ I–V characteristics considering variation in the light intensity and the temperature. At last, the method is also employed for a 10W silicon module tested in the field. To testify the preciseness of the method, an absolute value of the derivative of power (P) with respect to voltage (V) defined as (dP/dV) is evaluated and plotted against V. The method estimates the MPP parameters with high accuracy for any kind of PV technologies with different environmental conditions. In future, this method proposes a guide line to construct control scheme for real-time MPPT tracking in the PV system.

Reliability analysis of stiffened tank-roof stability with multiple random variables using minimum distance and Lagrange methods

Based on the reliability and buckling analysis of the steel tanks with a stiffened dome-roof, a mathematical model with multiple random variables was established and a new method for calculating and analysing the reliability and stability of the tank-roof by minimum distance and Lagrange formulae is provided. The analysis result of the tank as an engineering example by the proposed method shows that the thickness of both dome-roof-plates and stiffened ribs would affect the reliability of the tank-roof under different external loads. It is demonstrated that decisive factors affecting tank-roof reliability may be analysed such as: material thickness and external load. A reliability assessment is also needed performed to prove the correctness of both model and method. The assessment results were consistent with those from the engineering example. A baseline, reference method, for tank design and operation is thus proposed.

An economic manufacturing quantity model with probabilistic deterioration in a production system

This paper develops an economic manufacturing quantity (EMQ) model with deterioration and exponential demand in a production system over a finite time horizon under the effect of inflation and time value of money. The production rate is a dynamic variable (varying with time) in a production system. Due to a long run process, the machinery system is converted from in-control state to out-of-control state which results the production of improper items. The improper items are reworked at a fixed cost to make it as proper. With the increasing value of time, the production of improper item also increases. To reduce the production of the improper items, the systems have to be more reliable and with less amount of failure. In this direction, the model considers that the development cost, production cost, and material cost are dependent on the reliability parameter. The deterioration of the product is considered probabilistic to make the research a more realistic one. By considering the reliability parameter as a decision variable, we try to obtain the associated profit of the system which we have to maximize. To derive the maximization procedure, we use Euler–Lagrange formula from control theory. We outline some numerical examples along with graphical representations and sensitivity analysis to illustrate the model.