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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 δ 1 δ 2 P δ 3 ξ a b W d ψ * i ψ i x W d = ξ v ⋅ F r d + ξ v d ⋅ R r N i = ∂ ψ i ∂ ψ i * N i ∂ δ i δ i = ∂ N i i ψ * i ψ Complementary free energy Free energy δ i ∑ δ i N i = ψ i * ( N i ) + ψ i ( δ 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 − ( ψ − ξ v ⋅ R r )= ! ψ − ξ v ⋅ F r − ε com = ε pot Complementary Potential energy energy com : ε = pot : ε = ' ' ⎧ max (− ε ( N , R ) ) ⎫ ' com i i ' ' ⎪ ⎪ N S.A. ⎪ ⎪ ' ' − ε com ( N i , R ) ≤ ⎨ is equal to ⎬ ≤ ε pot ( δ i , ξ i ) ⎪ min ε ( δ ' , ξ ' ) ⎪ Lower bound ⎪ ⎩ δ 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 ψ , ψ * 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 ψ * (1-D) ψ Free energy (1-D) Contributions from external W = ∑ F v i d ⋅ ξ r i W = ∑ R r i d ⋅ ξ r i d W , W * i = 1.. N i = 1.. N work ψ = 1 ( W * + W ) Clapeyron’s formulas 2 Significance: Enables one ψ * = 1 ( W * + W ) calculate free energy, 2 complementary free energy, potential energy and ε pot = 1 ( W * − W ) 2 complementary energy directly from the boundary ε 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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