variables3

variables3 - 1.050 Engineering Mechanics I Summary of...

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Unformatted text preview: 1.050 Engineering Mechanics I Summary of variables/concepts Lecture 27 - 37 1 Variable Definition f ( x ) secant f f ( x ) | ( b a ) f ( b ) f ( a ) tangent x x = a Notes & comments Convexity of a function External work Free energy and complementary free energy 1 3 2 N 1 N 2 N 3 1 2 P 3 a b W d * i i x W d = v F r d + v d R r N i = i i * N i i i = N i i * i Complementary free energy Free energy i i N i = i * ( N i ) + i ( i ) i Lectures 27 and 28: Basic concepts: Convexity, external work, free energy, complementary free energy, introduced initially for truss structures (see schematic show in the lower right part). 2 Variable Definition Truss problems * d d ( v R r )= ! v F r com = pot Complementary Potential energy energy com : = pot : = ' ' max ( ( N , R ) ) ' com i i ' ' N S.A. ' ' com ( N i , R ) is equal to pot ( i , i ) min ( ' , ' ) Lower bound i ' K.A. pot i i Upper bound Notes & comments At elastic solution: Potential energy is equal to negative of complementary energy Upper/lower bound At the solution to the elasticity problem, the upper and lower bound coincide Consequence of convexity of elastic potentials , * Lectures 27 and 28: Introduction to potential energy and complementary energy, definition at the elastic solution, upper/lower bound, example of energy bounds for truss structures. The upper/lower bounds of the expressions are a consequence of the convexity of the elastic potentials (see previous slide). 3 Variable Definition Notes & comments Complementary free energy * (1-D) Free energy (1-D) Contributions from external W = F v i d r i W = R r i d r i d W , W * i = 1.. N i = 1.. N work = 1 ( W * + W ) Clapeyrons formulas 2 Significance: Enables one * = 1 ( W * + W ) calculate free energy, 2 complementary free energy, potential energy and pot = 1 ( W * W ) 2 complementary energy directly from the boundary com = 1 ( W W * ) conditions (external work), 2 at the solution (target) !...
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variables3 - 1.050 Engineering Mechanics I Summary of...

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