Approximate Solution of Differential Equations Governing the Orientation of a Rigid Body in a Reference Frame

1987 ◽  
Vol 54 (1) ◽  
pp. 232-234 ◽  
Author(s):  
T. R. Kane ◽  
D. A. Levinson
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Rigid Body ◽  
Reference Frame

Solution of Kinematical Differential Equations for a Rigid Body

1973 ◽  
Vol 40 (1) ◽  
pp. 109-113 ◽  
Author(s):  
T. R. Kane
Keyword(s):  
Approximate Solution ◽  
Angular Velocity ◽  
Differential Equations ◽  
Rigid Body ◽  
Digital Computer ◽  
Special Form ◽  
Unsolved Problem ◽  
Computer Solution ◽  
Constant Magnitude ◽  
Matrix Exponentials

This paper deals with the following problem: Given the orientation of a rigid body B at time t0 and the angular velocity of B for t ≥ t0, find the orientation of B for t ≥ t0. An approximate solution is found by employing an “averaging” technique and a lemma (proved in the paper) concerned with a special form of matrix exponentials. As a test, the heretofore unsolved problem of an axisymmetric rigid body subjected to the action of a body-fixed transverse torque of constant magnitude is worked out, and results are compared with those of a digital computer solution.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

A New Numerical Method for Solving Nonlinear Fractional Fokker–Planck Differential Equations

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
BeiBei Guo ◽  
Wei Jiang ◽  
ChiPing Zhang
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Simple Algorithm ◽  
High Accuracy ◽  
Computational Work ◽  
Reproducing Kernel Theory ◽  
Transport Problems ◽  
Fokker Planck

The nonlinear fractional-order Fokker–Planck differential equations have been used in many physical transport problems which take place under the influence of an external force filed. Therefore, high-accuracy numerical solutions are always needed. In this article, reproducing kernel theory is used to solve a class of nonlinear fractional Fokker–Planck differential equations. The main characteristic of this approach is that it induces a simple algorithm to get the approximate solution of the equation. At the same time, an effective method for obtaining the approximate solution is established. In addition, some numerical examples are given to demonstrate that our method has lesser computational work and higher precision.

A Friendly Iterative Technique for Solving Nonlinear Integro-Differential and Systems of Nonlinear Integro-Differential Equations

2018 ◽  
Vol 15 (03) ◽  
pp. 1850016 ◽  
Author(s):  
A. A. Hemeda
Keyword(s):  
Approximate Solution ◽  
Differential Operator ◽  
Differential Equations ◽  
The Other ◽  
Iterative Technique ◽  
Restrictive Assumption ◽  
Numerical Examples ◽  
Linear And Nonlinear ◽  
Computational Procedures ◽  
Adomian’S Polynomials

In this work, a simple new iterative technique based on the integral operator, the inverse of the differential operator in the problem under consideration, is introduced to solve nonlinear integro-differential and systems of nonlinear integro-differential equations (IDEs). The introduced technique is simpler and shorter in its computational procedures and time than the other methods. In addition, it does not require discretization, linearization or any restrictive assumption of any form in providing analytical or approximate solution to linear and nonlinear equations. Also, this technique does not require calculating Adomian’s polynomials, Lagrange’s multiplier values or equating the terms of equal powers of the impeding parameter which need more computational procedures and time. These advantages make it reliable and its efficiency is demonstrated with numerical examples.

Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

scholarly journals Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
S. Narayanamoorthy ◽  
T. L. Yookesh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Approximate Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Delay Differential

We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

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