Numerical integration of rotation with geometry propagators

2016 ◽  
Vol 07 (03) ◽  
pp. 1650017
Author(s):  
Jianhua Tang ◽  
Linfang Qian ◽  
Qiang Yin
Keyword(s):  
Numerical Integration ◽  
Fourth Order ◽  
Integration Method ◽  
Dynamic Equations ◽  
Reconstruction Method ◽  
Order Accuracy ◽  
Runge Kutta ◽  
Multi Body ◽  
Different Order ◽  
Body Dynamic

Commutator free method is an effective method for solving rotating integration. Numerical examples show that the use of the proposed combining method can achieve the same order accuracy with less computation than other geometry integration method. However, it is difficult to be directly applied to mechanic dynamics solutions. In this paper, commutator free method which is often applied to rotation integration and classical Runge–Kutta (RK) method which is usually operated in Linear space are combined to solve the multi-body dynamic equations. The explicit Runge–Kutta coefficients are reconstructed to meet different order accuracy integration methods. The reconstruction method is discussed and coefficients are given. With this method, the dynamic equations can be solved accurately and economically without much modification on the classical numerical integration. Moreover, CG method and CF method can also be combined with adaptive RK method without many changes. Finally, the results of the examples show that with less computation, fourth-order combining method is as accurate as fourth-order Crouch–Grossman algorithm.

scholarly journals Multi Body Dynamic Equations of Belt Conveyor and the Reasonable Starting Mode

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1489
Author(s):  
Yongbo Guo ◽  
Fansheng Wang
Keyword(s):  
Finite Element ◽  
Recursive Formula ◽  
Conservative System ◽  
Conveyor Belt ◽  
Dynamic Equations ◽  
Lagrange’S Equation ◽  
Rigid Finite Element Method ◽  
Potential Capacity ◽  
Multi Body ◽  
Body Dynamic

Based on the rigid finite element method and multibody dynamics, a discrete model of a flexible conveyor belt considering the material viscoelasticity is established. RFE (rigid finite element) and SDE (spring damping element) are used to describe the rigidity and flexibility of a conveyor belt. The dynamic differential equations of the RFE are derived by using Lagrange’s equation of the second kind of the non-conservative system. The generalized elastic potential capacity and generalized dissipation force of the SDE are considered. The forward recursive formula is used to construct the conveyor belt model. The validity of dynamic equations of conveyor belt is verified by field test. The starting mode of the conveyor is simulated by the model.

A New Efficient Ode Solver For Solving Initial Value Problem

1998 ◽  
pp. 47-56
Author(s):  
Nazeeruddin Yaacob ◽  
Bahrom Sanugi
Keyword(s):  
Stability Analysis ◽  
Explicit Formula ◽  
Fourth Order ◽  
Initial Value Problem ◽  
Harmonic Mean ◽  
Kutta Method ◽  
Initial Value ◽  
Order Accuracy ◽  
Runge Kutta Method ◽  
Runge Kutta

In this paper we develop a new three-stage,fourth order explicit formula of Runge-Kutta type based on Arithmetic and Harmonic means.The error and stability analyses of this method indicate that the method is stable and efficient for nonstiff problems.Two examples are given which illustrate the fcurth order accuracy of the method. Keywords: Runge-Kutta method, Harmonic Mean, three-stage, fourth-order, covergence and stability analysis.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

Continuous multi-body dynamic analysis of float-over deck installation with rapid load transfer technique in open waters

2021 ◽  
Vol 224 ◽  
pp. 108729
Author(s):  
Shujie Zhao ◽  
Xun Meng ◽  
Huajun Li ◽  
Dejiang Li ◽  
Qiang Fu
Keyword(s):  
Dynamic Analysis ◽  
Load Transfer ◽  
Multi Body ◽  
Body Dynamic

Numerical Integration Method for Stability Analysis of Milling With Variable Spindle Speeds

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Ye Ding ◽  
Jinbo Niu ◽  
LiMin Zhu ◽  
Han Ding
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Numerical Integration ◽  
Integration Method ◽  
Transcendental Equation ◽  
Spindle Speed ◽  
Machining Parameters ◽  
Numerical Integration Method ◽  
Stability Diagrams ◽  
Time Period

A semi-analytical method is presented in this paper for stability analysis of milling with a variable spindle speed (VSS), periodically modulated around a nominal spindle speed. Taking the regenerative effect into account, the dynamics of the VSS milling is governed by a delay-differential equation (DDE) with time-periodic coefficients and a time-varying delay. By reformulating the original DDE in an integral-equation form, one time period is divided into a series of subintervals. With the aid of numerical integrations, the transition matrix over one time period is then obtained to determine the milling stability by using Floquet theory. On this basis, the stability lobes consisting of critical machining parameters can be calculated. Unlike the constant spindle speed (CSS) milling, the time delay for the VSS is determined by an integral transcendental equation which is accurately calculated with an ordinary differential equation (ODE) based method instead of the formerly adopted approximation expressions. The proposed numerical integration method is verified with high computational efficiency and accuracy by comparing with other methods via a two-degree-of-freedom milling example. With the proposed method, this paper details the influence of modulation parameters on stability diagrams for the VSS milling.

The Numerical Methods of Peridynamics Theory Used in Failure Analysis of Materials

2014 ◽  
Vol 1030-1032 ◽  
pp. 223-227
Author(s):  
Lin Fan ◽  
Song Rong Qian ◽  
Teng Fei Ma
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Numerical Integration ◽  
Integration Method ◽  
Equation Of Motion ◽  
Numerical Integration Method ◽  
Volume Integral ◽  
Volume Integral Equation ◽  
Method Of Molecular Dynamics ◽  
Classic Theory

In order to analysis the force situation of the material which is discontinuity,we can used the new theory called peridynamics to slove it.Peridynamics theory is a new method of molecular dynamics that develops very quickly.Peridynamics theory used the volume integral equation to constructed the model,used the volume integral equation to calculated the PD force in the horizon.So It doesn’t need to assumed the material’s continuity which must assumed that use partial differential equation to formulates the equation of motion. Destruction and the expend of crack which have been included in the peridynamics’ equation of motion.Do not need other additional conditions.In this paper,we introduce the peridynamics theory modeling method and introduce the relations between peridynamics and classic theory of mechanics.We also introduce the numerical integration method of peridynamics.Finally implementation the numerical integration in prototype microelastic brittle material.Through these work to show the advantage of peridynamics to analysis the force situation of the material.

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