Some analytical and numerical solutions for the safe turn manoeuvres of agricultural aircraft – an overview

2007 ◽  
Vol 111 (1123) ◽  
pp. 593-599 ◽  
Author(s):  
B. Rasuo
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Numerical Results ◽  
System Of Differential Equations ◽  
Bank Angle ◽  
Tangential Load ◽  
Load Factors ◽  
The Mathematical Model

Abstract In this paper, a theoretical study of the turn manoeuvre of an agricultural aircraft is presented. The manoeuvre with changeable altitude is analyzed, together with the, effect of the load factors on the turn manoeuvre characteristics during the field-treating flights. The mathematical model used describes the procedure for the correct climb and descent turn manoeuvre. For a typical agricultural aircraft, the numerical results and limitations of the climb, horizontal and descending turn manoeuvre are given. The problem of turning flight with changeable altitude is described by the system of differential equations which describe the influence of the normal and tangential load factors on velocity, the path angle in the vertical plane and the rate of turn, as a function of the bank angle during turning flight. The system of differential equations of motion was solved on a personal computer with the Runge-Kutta-Merson numerical method. Some analytical and numerical results of this calculation are presented in this paper.

scholarly journals Mathematical Model of Spatial Motion of the Controlled Parachute-Tether System of the Wind Kite Type

2021 ◽  
Vol 24 (4) ◽  
pp. 17-24
Author(s):  
V.M. Churkin ◽  
T.Yu. Churkina ◽  
A.M. Girin
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Solid Body ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Spatial Motion ◽  
Mathematical Task ◽  
Spatial Movement ◽  
Elastic Thread ◽  
The Mathematical Model

Mathematical modeling is created for the mathematical task of spatial motion of the controlled parachute-tether system of the “wind kite” type. The mathematical model parachute-tether system consists of a model of the main parachute and a model of the braking parachute. The parachutes are connected by the tether. The model of the main parachute is supposed to be the solid body. This solid body has two planes of symmetry. The braking parachute is the solid body with axial symmetry. The tether model is an absolutely flexible elastic thread. The tether is connected by ideal hinges with the main parachute and braking parachute. The control of the main parachute is carried out by changing the length of the control slings. Changing the length causes deformation of the dome. This is the reason for the change in its aerodynamics. Maneuvering of the main parachute occurs in the vertical plane, when the length of the control slings changes simultaneously. Maneuvering of the main parachute in space is carried out when the length of the control slings changes, when the slings are given a travel difference. The system of dynamic and kinematic equations is designed for calculating the controlled spatial movement of the main parachute, braking parachute and tether. The option exists when the mass of the tether and the forces applied to the tether cannot be neglected. The motion of the tether is represented by the equations of motion of an absolutely flexible elastic thread in projections on the axis of a natural trihedron. The mathematical model is represented by a system of ordinary differential equations and partial differential equations. The problem is solved using various numerical methods. The solution is possible with the help of an integrated numerical and analytical approach as well.

scholarly journals New Vertically Planed Pendulum Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
A. I. Ismail
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Numerical Procedure ◽  
Second Order ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Motion ◽  
Generalized Coordinates

This article is concerned about the planed rigid body pendulum motion suspended with a spring which is suspended to move on a vertical plane moving uniformly about a horizontal X-axis. This model depends on a system containing three generalized coordinates. The three nonlinear differential equations of motion of the second order are obtained to the elastic string length and the oscillation angles φ 1 and φ 2 which represent the freedom degrees for the pendulum motions. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity ω . The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the approximated fourth-order Runge–Kutta method through programming packages. These solutions are represented graphically to describe and discuss the behavior of the body at any instant for different values of the different physical parameters of the body. The obtained results have been discussed and compared with some previously published works. Some concluding remarks have been presented at the end of this work. The value of this study comes from its wide applications in both civil and military life. The main findings and objectives of the current study are obtaining periodic solutions for the problem and satisfying their accuracy and stabilities through the numerical procedure.

scholarly journals Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1961
Author(s):  
Kanognudge Wuttanachamsri
Keyword(s):  
Mathematical Model ◽  
Exact Solution ◽  
Free Boundary ◽  
Horizontal Plane ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Brinkman Equation ◽  
Average Height ◽  
Free Interface ◽  
The Mathematical Model

Cilia on the surface of ciliated cells in the respiratory system are organelles that beat forward and backward to generate metachronal waves to propel mucus out of lungs. The layer that contains the cilia, coating the interior epithelial surface of the bronchi and bronchiolesis, is called the periciliary layer (PCL). With fluid nourishment, cilia can move efficiently. The fluid in this region is named the PCL fluid and is considered to be an incompressible, viscous, Newtonian fluid. We propose there to be a free boundary at the tips of cilia underlining a gas phase while the cilia are moving forward. The Brinkman equation on a macroscopic scale, in which bundles of cilia are considered rather than individuals, with the Stefan condition was used in the PCL to determine the velocity of the PCL fluid and the height/shape of the free boundary. Regarding the numerical methods, the boundary immobilization technique was applied to immobilize the moving boundaries using coordinate transformation (working with a fixed domain). A finite element method was employed to discretize the mathematical model and a finite difference approach was applied to the Stefan problem to determine the free interface. In this study, an effective stroke is assumed to start when the cilia make a 140∘ angle to the horizontal plane and the velocitiesof cilia increase until the cilia are perpendicular to the horizontal plane. Then, the velocities of the cilia decrease until the cilia make a 40∘ angle with the horizontal plane. From the numerical results, we can see that although the velocities of the cilia increase and then decrease, the free interface at the tips of the cilia continues increasing for the full forward phase. The numerical results are verified and compared with an exact solution and experimental data from the literature. Regarding the fixed boundary, the numerical results converge to the exact solution. Regarding the free interface, the numerical solutions were compared with the average height of the PCL in non-cystic fibrosis (CF) human tissues and were in excellent agreement. This research also proposes possible values of parameters in the mathematical model in order to determine the free interface. Applications of these fluid flows include animal hair, fibers and filter pads, and rice fields.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

Contributions to the Determination of the Equations of Motion for Multidegree of Freedom Systems

1971 ◽  
Vol 93 (1) ◽  
pp. 191-195 ◽  
Author(s):  
Desideriu Maros ◽  
Nicolae Orlandea
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
System Of Differential Equations ◽  
Original Contribution ◽  
Deductive Method ◽  
Method Of Solution ◽  
Further Development ◽  
Three Degrees Of Freedom

This paper is a further development of the kinematic problem presented in our 1967 paper [1] in which we have obtained the transmission functions for different orders of plane systems with many degrees of freedom. This paper establishes the corresponding system of differential equations of motion beginning with these functions. The purpose of this paper is to facilitate computer programming. Our study is based on the work of R. Beyer [2, 3] and is the first original addition to his papers. A second original contribution to Beyer’s theories is the deductive method of solution, from general to particular, which we have, incorporated in our work. Beyer concluded that the cases having two or three degrees of freedom can be considered as particular solutions to the results obtained.

A study of nonlinear biochemical reaction model

2016 ◽  
Vol 09 (05) ◽  
pp. 1650071 ◽  
Author(s):  
Muhammad Asad Iqbal ◽  
Syed Tauseef Mohyud-Din ◽  
Bandar Bin-Mohsin
Keyword(s):  
Differential Equations ◽  
Reaction Model ◽  
Numerical Results ◽  
Picard Iteration ◽  
System Of Differential Equations ◽  
Legendre Wavelets ◽  
Enzyme Substrate ◽  
Runge Kutta Method ◽  
Picard Method ◽  
Order Four

The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method (LWPM). Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge–Kutta method of order four (RK-4). Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.

scholarly journals MATHEMATICAL MODEL OF SPATIAL OSCILLATIONS OF A RAILWAY FOUR-AXLE AUTONOMOUS TRACTION MODULE

2021 ◽  
pp. 55-68
Author(s):  
František Bures
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Mechanical System ◽  
Traffic Safety ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Railway Track ◽  
Theoretical Studies ◽  
System Of Differential Equations ◽  
Spatial Oscillations

A description of the original mathematical model of spatial oscillations of a four-axle autonomous traction module during its movement along straight and curved sections of the railway track is proposed. In this case, the design of a four-axle autonomous traction module is presented as a complex mechanical system, and the track is considered as an elastic-viscous inertial system. The equations of motion were compiled using the Lagrange method of the ІІ kind. For each of the solids, the kinetic energy is determined by the Koenig theorem. The potential energy component is obtained by the Clapeyron theorem, as the sum of the energies accumulated in the elastic elements of the system during their deformations. The dissipative energy was also taken into account when compiling the equations of motion. Generalized forces that have no potential, in this case, include the forces of interaction between wheels and rails, which are determined using the creep hypothesis. It is important to note that the model takes into account the forces in the bonds between the bodies of the system and the spatial displacements of the rigid bodies of the mechanical system, taking into account possible restrictions. Moreover, the mathematical model developed by the author is a system of differential equations of the 100th order. This mathematical model is the base for further theoretical studies of the dynamics of railway four-axle autonomous traction modules in single motion or when moving as part of a train. To solve the resulting system of differential equations, the author develops special software that allows for complex theoretical studies of spatial oscillations of a four-axle autonomous tractionmodule to determine the indicators of its dynamic loading and traffic safety. 

Lagrangian descriptions of dissipative systems: a review

2020 ◽  
pp. 108128652097183
Author(s):  
Alberto Maria Bersani ◽  
Paolo Caressa
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dissipative Systems ◽  
Lagrangian Formalism ◽  
Lagrangian Description ◽  
System Of Differential Equations ◽  
Helmholtz Conditions ◽  
Necessary And Sufficient

In this paper, we review classical and recent results on the Lagrangian description of dissipative systems. After having recalled Rayleigh extension of Lagrangian formalism to equations of motion with dissipative forces, we describe Helmholtz conditions, which represent necessary and sufficient conditions for the existence of a Lagrangian function for a system of differential equations. These conditions are presented in different formalisms, some of them published in the last decades. In particular, we state the necessary and sufficient conditions in terms of multiplier factors, discussing the conditions for the existence of equivalent Lagrangians for the same system of differential equations. Some examples are discussed, to show the application of the techniques described in the theorems stated in this paper.

Numerical Simulations of Bio-Convection in the Stream-Wise and Cross-Flow Directions Comprising Nanofluid Conveying Motile Microorganism: Analysis of Multiple Solutions

2021 ◽  
pp. 2150058
Author(s):  
Umair Khan ◽  
A. zaib ◽  
A. Ishak ◽  
S. Abu Bakar ◽  
Taseer Muhammad
Keyword(s):  
Differential Equations ◽  
Radiation Effects ◽  
Numerical Solutions ◽  
Cross Flow ◽  
Nanofluid Flow ◽  
Nusselt Numbers ◽  
Opposite Behavior ◽  
Flow Directions ◽  
New Variables ◽  
The Mathematical Model

This work tackles the phenomenon of motile microorganisms and nanoliquid flow in the esteem of cross-flow (CF) and stream-wise (SW) direction. The analysis exposed to viscous dissipation, Brownian motion, thermal radiation, magnetic function, and thermophoresis impacts is encountered. The mathematical model consists of the partial differential equations (PDEs) switched into nonlinear ordinary differential equations (ODEs) through proper transformations of new variables. The multiple outcomes of the flow problem are achieved through the Lobatto IIIA formula. The features of controlling constraints are sketched for the motile organism, temperature, velocities (CF and SW), and concentration fields. Also, the Sherwood and the Nusselt numbers along with motile density and friction factor are sketched. One imperative numerical outcome of this research is the existence of dual numerical solutions for the nanofluid flow. The upshots indicate that the profiles of microorganisms decelerate due to bio-convection Schmidt and Péclet numbers. The magnetic function decelerates the velocity in the directions of SW and CF in the branch of the first solution and upsurges in the branch of the second solution. The concentration profile uplifts due to [Formula: see text] in both solutions while the opposite behavior is observed for different values of [Formula: see text] in both solutions. The temperature uplifts due to magnetic and radiation effects in both solutions.

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