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

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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( 0 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
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= = + + + = = + + = = = = 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 0 1 1 1 1 1 1 = = = = = ∑ ∑ = = = = = = δ δ δ δ δ r F r F r F
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PVW extension—d’Alembert’s principle ( 29 ( 29 0 ) ( ) ( 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 0 r f F r f F - = - + + = + motion. of equations derive to used be can which equations, s Lagrange' for foundation the lays This
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( 29 ( 29 0 ) ( ) ( 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
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( 29 ( 29 0 ) ( ) ( 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 + = = r r 1 0 1 1 1 = - = = = = j n j j i N i i i N i j i i q q m q δ r r r F 0 1 = - - = = j n j j j j q q T q T dt d Q δ n j Q q T q T dt d j
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