Analysis of the Mindlin Plate and Identification of the Soil Parameter for Foundation with Gradient Optimizing Method

2012 ◽  
Vol 256-259 ◽  
pp. 3-10
Author(s):  
Tao Hong ◽  
Mei Jun Lu ◽  
Chu Wei Zhou ◽  
Jian Zhang
Keyword(s):  
Differential Equations ◽  
Error Function ◽  
Mindlin Plate ◽  
Mechanical Theory ◽  
Soil Parameter ◽  
True Value ◽  
Initial Soil ◽  
Calculation Results ◽  
Identification Analysis ◽  
Markov Method

Analysis of Mindlin plate is completed. And combined with gradient optimizing method, the Gaussian Markov identification analysis of the soil parameter for the foundation is studied and put forward in detail. After the mechanical theory for the plate on the foundation is introduced, the controlling differential equations of the plate on the foundation which is subjected to vertical loads are deduced. And linear algebra controlling equations for the plate are achieved which leads to solve the original differential equations more easily when the corresponding solutions to the plate on the foundation are gained with Fourier transformative theory. The Gaussian Markov error function for the soil parameter on the plate is established and applied with the gradient optimizing method. The identification steps on the Gaussian Markov error function for the soil parameter are listed. The calculation results verify the conclusions that the soil parameter of the foundation can be efficiently inversed by applying the Gaussian Markov theory. When different initial soil parameter is set, the iterative computations can be convergent to the true value of the soil parameter. And this Gaussian Markov method can also be applied for the problem of inversed analysis of parameters for other foundation models.

Least Square Inversed Analysis of Soil Parameter for Foundation with Two-Order Gradient Theoretic Method

2011 ◽  
Vol 243-249 ◽  
pp. 2294-2299 ◽  
Author(s):  
Yao Feng Xie
Keyword(s):  
Differential Equations ◽  
Error Function ◽  
Least Square Method ◽  
Least Square ◽  
Gradient Theory ◽  
Mechanical Theory ◽  
Soil Parameter ◽  
True Value ◽  
Initial Soil ◽  
Calculation Results

Combined with two-order gradient theory, the least square inversed analysis of the soil parameter for the foundation is studied and put forward in detail. After the mechanical theory for the plate on the foundation is introduced, the controlling differential equations of the plate on the foundation which is subjected to vertical loads are deduced. Through utilizing Fourier transformative theory, the corresponding solutions to the plate on the foundation are gained. Linear algebra controlling equations for the plate are achieved which leads to solve the original differential equations more easily. The least square error function for the soil parameter on the plate is established and applied with the two-order gradient method. The inversed steps on the least square error function for the soil parameter are listed. The calculation results verify the conclusions that the soil parameter of the foundation can be efficiently inversed by applying the least square theory. When different initial soil parameter is set, the iterative computations can be convergent to the true value of the soil parameter. And this least square method can also be applied for the problem of inversed analysis of parameters for other foundation models.

scholarly journals The error function in fractional differential equations

2019 ◽  
Vol 1391 ◽  
pp. 012059
Author(s):  
Jorge Olivares Funes ◽  
Elvis Valero Kari ◽  
Pablo Martin ◽  
Fernando Maass
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Error Function ◽  
Fractional Differential

scholarly journals Some Discussions about the Error Functions on SO(3) and SE(3) for the Guidance of a UAV Using the Screw Algebra Theory

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Yi Zhu ◽  
Xin Chen ◽  
Chuntao Li
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Error Function ◽  
Elliptic Cylinder ◽  
General Situation ◽  
3 Dimensional ◽  
Error Functions ◽  
Aerial Vehicle ◽  
Algebra Theory ◽  
In The Beginning

In this paper a new error function designed on 3-dimensional special Euclidean group SE(3) is proposed for the guidance of a UAV (Unmanned Aerial Vehicle). In the beginning, a detailed 6-DOF (Degree of Freedom) aircraft model is formulated including 12 nonlinear differential equations. Secondly the definitions of the adjoint representations are presented to establish the relationships of the Lie groups SO(3) and SE(3) and their Lie algebras so(3) and se(3). After that the general situation of the differential equations with matrices belonging to SO(3) and SE(3) is presented. According to these equations the features of the error function on SO(3) are discussed. Then an error function on SE(3) is devised which creates a new way of error functions constructing. In the simulation a trajectory tracking example is given with a target trajectory being a curve of elliptic cylinder helix. The result shows that a better tracking performance is obtained with the new devised error function.

Imprecise Solutions of Ordinary Differential Equations for Boundary Value Problems Using Metaheuristic Algorithms

2016 ◽  
pp. 401-421
Author(s):  
Ali Sadollah ◽  
Joong Hoon Kim
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Water Cycle ◽  
Error Function ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Metaheuristic Algorithms ◽  
General Strategy ◽  
Optimization Task

In this chapter, a general strategy is recommended to solve variety of linear and nonlinear ordinary differential equations (ODEs) with boundary value conditions. With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic algorithms, ODEs can be represented as an optimization problem. The purpose is to reduce the weighted residual error (error function) of the ODEs. Boundary values of ODEs are considered as constraints for the optimization model. Inverted generational distance metric is utilized for evaluation and assessment of approximate solutions versus exact solutions. Four ODEs having different orders and features are approximately solved and compared with their exact solutions. The optimization task is carried out using different optimizers including the particle swarm optimization and the water cycle algorithm. The optimization results obtained show that the proposed method equipped with metaheuristic algorithms can be successfully applied for approximate solving of different types of ODEs.

Controlling a Variable-Length Exoskeleton Link

Vestnik MEI ◽  
2021 ◽  
Vol 3 (3) ◽  
pp. 110-120
Author(s):  
Andrey V. Borisov ◽  
◽  
Konstantin D. Filippenkov ◽  
Keyword(s):  
Differential Equations ◽  
Musculoskeletal System ◽  
Biomechanical Properties ◽  
Variable Length ◽  
Electromechanical Model ◽  
Inertial Parameters ◽  
Wide Range ◽  
Calculation Results ◽  
New Generation ◽  
Support Motion

The aim of the study is to develop a spatial electromechanical model of a variable-length link for use in telescopic manipulators, anthropomorphic robots, exoskeletons, and in studying the human musculoskeletal system. The proposed link model has two massive absolutely solid sections at the ends and a weightless section of variable length located between them. The study was carried out using the methods of theoretical mechanics, electromechanics, mathematical modeling, engineering design, numerical methods for solving systems of ordinary differential equations, control theory, nonlinear dynamics, experimental methods, and empirical data on the biomechanical properties of the human musculoskeletal system. The reliability of the obtained results is substantiated by a rigorous use of the above-mentioned methods. As a result of the study, a system of Lagrange-Maxwell differential equations was written, and an electromechanical model of an anthropomorphic system was developed in the Matlab Simulink software package. With the specified geometric and inertial parameters of a variable-length link corresponding to an average person's leg lower part and the time corresponding to the single-support motion phase, the electric motors and reducing gears implementing the human musculoskeletal system link's biomechanical motion fragment are selected. All of the selected motors have a sufficient operating parameters margin. The trajectories of all generalized coordinates along which the anthropomorphic system performs its necessary motion are determined. The mechanism load diagrams are obtained. The control system for the motors is synthesized, and the positioning error is evaluated. The novelty of the approach is that the newly developed electromechanical models of controlled variable-length links have a wide range of applying the obtained results and can be used in designing anthropomorphic robots and comfortable new-generation exoskeletons. Thus, the electromechanical model of a variable-length link with the parameters corresponding to the average person's leg lower part has been developed. The electric drives and transmissions able to implement a motion close to the anthropomorphic one have been selected; its implementation has been demonstrated, and the numerical calculation results are given.

scholarly journals Comparison of Artificial Neural Network Architecture in Solving Ordinary Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Susmita Mall ◽  
S. Chakraverty
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Network Architecture ◽  
Initial Conditions ◽  
Error Function ◽  
Back Propagation ◽  
Feed Forward Neural Network ◽  
Error Back Propagation ◽  
Hidden Layer

This paper investigates the solution of Ordinary Differential Equations (ODEs) with initial conditions using Regression Based Algorithm (RBA) and compares the results with arbitrary- and regression-based initial weights for different numbers of nodes in hidden layer. Here, we have used feed forward neural network and error back propagation method for minimizing the error function and for the modification of the parameters (weights and biases). Initial weights are taken as combination of random as well as by the proposed regression based model. We present the method for solving a variety of problems and the results are compared. Here, the number of nodes in hidden layer has been fixed according to the degree of polynomial in the regression fitting. For this, the input and output data are fitted first with various degree polynomials using regression analysis and the coefficients involved are taken as initial weights to start with the neural training. Fixing of the hidden nodes depends upon the degree of the polynomial. For the example problems, the analytical results have been compared with neural results with arbitrary and regression based weights with four, five, and six nodes in hidden layer and are found to be in good agreement.

Free Bending Vibrations of Integrally-Stiffened and/or Stepped-Thickness Rectangular Plates or Panels With a Non-Central Plate Stiffener

2007 ◽  
Author(s):  
U. Yuceoglu ◽  
N. Gemalmayan ◽  
O. Sunar
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Orthotropic Materials ◽  
Mindlin Plate ◽  
Al Alloy ◽  
Bending Vibrations ◽  
Central Plate ◽  
Plate Elements ◽  
Stress Resultant ◽  
Free Bending

The present study is primarily concerned with the “Free Bending Vibrations of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with a Non-Central Plate Stiffener”. The general theoretical formulation is based on the “Mindlin Plate Theory”. The plate elements of the system are considered to be made of dissimilar orthotropic materials with unequal thicknesses. The transverse shear deformations and the transverse and the rotary moments of inertia of plate elements are included in the analysis. The damping effects, however, are neglected. The dynamic equations of the orthotropic “Mindlin Plates” in combination with the stress resultant-displacement expressions are algebraically manipulated. They are eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in the “state vectors” form. The resulting differential equations system is numerically integrated by making use of the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”. The mode shapes with their dimensionless natural frequencies are presented for various support conditions in the “isotropic” Al-Alloy and in the “orthotropic” composite cases. Additionally, the effect of some of the important parameters such as (“Stiffener Position Ratio”, “Thickness Ratio”, “Stiffener Length (or Width) Ratio)” on the dimensionless natural frequencies are investigated and plotted. Based on the numerical results, some brief but important conclusions are presented.

scholarly journals An application of the finite difference method for solving the mass spring system equation

2020 ◽  
Vol 16 (3) ◽  
pp. 404
Author(s):  
Gabariela Purnama Ningsi ◽  
Fransiskus Nendi ◽  
Lana Sugiarti
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
System Equation ◽  
Limit Value ◽  
Calculation Results ◽  
Spring System ◽  
The Finite Difference Method ◽  
Mass Spring

The numerical method is one method that can be used to solve differential equations, both differential equations that are easy or difficult to solve analytically. The solution obtained from the calculation results is an approximate solution or a solution that approaches an analytic solution, not an analytic solution. That is, in solving differential equations numerically, there is always an error. In this paper, an analytical solution is described and described and the application of different methods in solving a damped mass spring system with a known limit value. The error between the analytic and numerical solutions obtained is very small.

Sign in / Sign up

Export Citation Format

Share Document