ASE 365 - Lecture 23

ASE 365 - Lecture 23 - Principle of virtual work example R...

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Unformatted text preview: Principle of virtual work example R 1 m1g T R 2 T m2g Newtonian FBDs Eliminate T, R1, R2. θ θ α θ δθ θ α δθ δ sin ) sin( sin ) ( ) sin( ) ( 2 1 2 1 m m r g m r g m W =- ⇒ =-- = : approach PVW m 1 m 2 θeq R L r g m 1 δθ r θ α - δθ r θ g m 2 ∑ = ∂ ∂ = ∂ ∂ + + ∂ ∂ + ∂ ∂ = = + + = = = = n j j j i n n i i i i n i i i i i n i i n i i n i i q q q q q q q q q q q z y x q q q z z q q q y y q q q x x 1 2 2 1 1 2 1 2 1 2 1 2 1 ) , , ( ) ( , ) , , ( ), , , ( ), , , ( δ δ δ δ δ r r r r r r k j i r becomes nt displaceme virtual the so , have we since s, coordinate d generalize of terms In . a is where becomes work virtual of principle The force d generalize j n j j j n j j j i N i i N i n j j j i i N i i i Q q Q q q q q W 1 1 1 1 1 1 = = ∂ ∂ ⋅ = ∂ ∂ ⋅ = ⋅ = ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = δ δ δ δ δ r F r F r F PVW extension—d’Alembert’s principle ( 29 ( 29 ) ( ) ( 1 1 = ⋅- + = ⋅- + + = ∑ ∑ = = N i i i i i N i i i i i i m m ' W r r F r r f F δ δ δ : case dynamic the to extended be can work virtual of principle The force". inertia " an is Here : law 2nd s Newton' Rewrite ) ( ) ( i i i i i i i i i i m m m r r f F r f F - =- + + ⇒ = + motion. of equations derive to used be can which equations, s Lagrange' for foundation the lays This ( 29 ( 29 ) ( ) ( 1 1 = ⋅- + = ⋅- + + = ∑ ∑ = = N i i i i i N i i i i i i m m ' W r r F r r f F δ δ δ ) , ( 2 1 1 1 n i i i i N i i q q m T r r r r = ⋅ = ∑ = , : energy Kinetic . Recall . so , So ∑ ∑ = = ∂ ∂ = ∂ ∂ = ∂ ∂ ∂ ∂ = n j j j i i j i j i n j j j i i q q q q q q 1 1 δ δ r r r r r r . and Now j i i N i i j i i N i i j j i i N i i j q m q m q T q m q T ∂ ∂ ⋅ = ∂ ∂ ⋅ = ∂ ∂ ∂ ∂ ⋅ = ∂ ∂ ∑ ∑ ∑ = = = r r r r r r 1 1 1 . So j j i i N i i j i i N i i j i i N i i j q T q m q m q m q T dt d ∂ ∂ + ∂ ∂ ⋅ = ∂ ∂ ⋅ + ∂ ∂ ⋅ = ∂ ∂ ∑ ∑ ∑ = = = r r r r r r 1 1 1 ( 29 ( 29 ) ( ) ( 1 1 1 = ∂ ∂ ⋅- + = ⋅- + = ∑ ∑ ∑ = = = j n j j i N i i i i N i i i i i q q m m ' W δ δ δ r r F r r F . work virtual and energy kinetic Need j n j j q Q W T δ δ ∑ = = 1 j j i i N i i j q T q m q T dt d ∂ ∂...
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This note was uploaded on 09/18/2011 for the course ASE 365 taught by Professor Staff during the Spring '10 term at University of Texas.

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ASE 365 - Lecture 23 - Principle of virtual work example R...

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