unit9_notes

unit9_notes - 16.21 Techniques of Structural Analysis and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #9 - Calculus of Variations Raul Radovitzky March 28, 2005 Let u be the actual configuration of a structure or mechanical system. u satisfies the displacement boundary conditions: u = u on S u . Define: u = u + v, where: : scalar v : arbitrary function such that v = 0 on S u We are going to define v as u , the first variation of u : u = v (1) Schematically: u1 u2 u a b u(a) u(b) v 1 As a first property of the first variation : d u du dv = + dx dx dx so we can identify dv dx with the first variation of the derivative of u : du dv = dx dx But: dv dv d dx = dx = dx ( u ) We conclude that: du d = ( u ) dx dx Consider a function of the following form: F = F ( x, u ( x ) , u ( x )) It depends on an independent variable x , another function of x ( u ( x )) and its derivative ( u ( x )). Consider the change in F , when u (therefore u ) changes: F = F ( x, u + u, u + u ) F ( x, u, u ) = F ( x, u + v, u + v ) F ( x, u, u ) expanding in Taylor series: F F 1 2 F 1 2 F F = F + v + v + u 2 ( v ) 2 + ( v )( v ) + F u u 2! 2! uu F F = v + v + h.o.t....
View Full Document

This note was uploaded on 11/07/2011 for the course AERO 16.21 taught by Professor Dffs during the Fall '10 term at MIT.

Page1 / 6

unit9_notes - 16.21 Techniques of Structural Analysis and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online