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# unit9_notes - 16.21 Techniques of Structural Analysis and...

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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....
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## 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.

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unit9_notes - 16.21 Techniques of Structural Analysis and...

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