On the structure of the optimal thrust for the “intermediate” aircraft model

2020 ◽  
Author(s):  
O. Yu. Cherkasov ◽  
N.V. Smirnova
Keyword(s):  
Boundary Value Problem ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Slope Angle ◽  
Vertical Plane ◽  
Material Point ◽  
Process Time ◽  
Brachistochrone Problem ◽  
Thrust Control ◽  
The Pontryagin Maximum Principle

The paper considers a brachistochrone problem modification including in the objective function a fuel consumption penalty apart from the process time. The material point moves in a vertical plane under gravity, viscous nonlinear friction and traction. The trajectory slope angle and thrust are considered as a control variables. The Pontryagin maximum principle allows reducing the optimal control problem to a boundary value problem for a system of two nonlinear differential equations. Qualitative analysis of the resulting system allows studying the key features of extreme trajectories, including their asymptotic behavior. Extreme thrust control is obtained as a function of the velocity and the trajectory slope angle. The structure of extreme thrust is determined, and the number of switches is analytically determined. The results of numerical solving the boundary value problem are presented, illustrating the analytical conclusions.

scholarly journals Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

2021 ◽  
Vol 21 (1) ◽  
pp. 60-75
Author(s):  
Eduard I. Starovoitov ◽  
◽  
Denis V. Leonenko ◽  
Keyword(s):  
Temperature Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Plastic Region ◽  
Local Load ◽  
Natural State ◽  
Elastic Plastic

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

A boundary value problem on the half-line for higher-order nonlinear differential equations

2009 ◽  
Vol 139 (5) ◽  
pp. 1017-1035 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Existence And Uniqueness ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Higher Order ◽  
Half Line ◽  
Nonlinear Ordinary Differential Equations ◽  
Specific Class

This article is devoted to the study of the existence of solutions as well as the existence and uniqueness of solutions to a boundary-value problem on the half-line for higher-order nonlinear ordinary differential equations. An existence result is obtained by the use of the Schauder–Tikhonov theorem. Furthermore, an existence and uniqueness criterion is established using the Banach contraction principle. These two results are applied, in particular, to the specific class of higher-order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of higher-order linear ordinary differential equations, respectively. Moreover, some (general or specific) examples demonstrating the applicability of our results are given.

scholarly journals Approximate method of solving one periodic optimal regulated boundary value problem

2019 ◽  
pp. 66-69
Author(s):  
I. Askerov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Small Parameter ◽  
General Problem ◽  
Asymptotic Method ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Nonlinear Algebraic Equations

In the present work we considered the solution of one periodic optimal regulated boundary value problem by the asymptotic method. For the solution of the problem with extended functional writing, boundary conditions and Euler-Lagrange equations were found. The approach to the solution of the problem depending on a small parameter by seeking a system of nonlinear differential equations and solving Euler-Lagrange equations, the solution of the general problem in the first approach comes down to solving two nonlinear algebraic equations.

On the Computation of Wave Induced Bending and Torsion Moments

1971 ◽  
Vol 15 (03) ◽  
pp. 217-220
Author(s):  
T. Francis Ogilvie
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Vertical Plane ◽  
Bending Moment ◽  
Ship Motions ◽  
Wave Loads ◽  
Cross Member ◽  
Wave Induced ◽  
Incident Waves

In the calculation of wave loads on a ship, one must consider the effects of both the incident waves and the diffraction waves (the latter being caused by the presence of the ship in the incident waves). In the ship-motions problem, Khaskind showed how one can do this without having to solve the diffraction-wave boundary-value problem. Khaskind's procedure is here extended to the calculation of structural loads on a ship. Two examples are discussed: (i) bending moment in the vertical plane of a ship in waves and (ii) torsion in the cross member of a catamaran. Many other applications are possible. In each case, it is necessary to solve a boundary-value problem, but it is generally much simpler than the diffraction problem.

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