THE EXTREMIZATION OF FUNCTIONALS INVOLVING PRODUCTS OF POWERS OF INTEGRALS AND ITS APPLICATION TO AERODYNAMICS
Abstract
The extremization of functionals involving products of powers of n integrals, each of which depends on the same unknown function y(x) is considered. After the necessary conditions for the extremum are formulated, the extremal arc is shown to be governed by a second order differential equation depending on n undand the end-conditions, a system of n + 2 algebraic equations is obtained: it involves n + 2 unknowns, that is, the n undetermined integral parameters and two integration constants. The procedure discussed can be employed in the study of the optimum wings and fuselages for different flow regimes. These problems are generally of the isoperimetric type with variable end-points.termined integral parameters. After the general solution is combined with the definitions of the integral parameters and the end-conditions, a system of n + 2 algebraic equations is obtained: it involves n + 2 unknowns, that is, the n undetermined integral parameters and two integration constants. The procedure discussed can be employed in the study of the optimum wings and fuselages for different flow regimes. These problems are generally of the isoperimetric type with variable end-points. However, they can be reduced to problems involving products of powers of integrals whose end-points are fixed if a simple coordinate transformation is performed. In this connection, a particular example is developed. It refers to the problem of the optimum slender body of revolution in Newtonian flow with zero angle of attack for the case where the diameter and the wetted area are given while the length and the volume are free. The solution minimizing the drag is a cone. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- May 01, 1962
- Accession Number
- AD0277351
Entities
People
- Angelo Miele
Organizations
- Boeing