16.61 Aerospace Dynamics Spring 2003 Generalized forces revisited Derived Lagrange s equation from d'Alembert's equation ∑m(8x+16y+22)=∑(Fx+F+F。=) Define virtual displacements sx Substitute in and noting the independence of the 8q,, for each DOf we get one Lagrange equation oa +F +F Applying lots of calculus on LHS and noting independence of the Sq, for each dof we get a Lagrange equation d a ar ∑ +F +F Further, we moved the conservative forces(those derivable from a potential function to the lhs aLaL ∑ az +F +F Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)16.61 Aerospace Dynamics Spring 2003 Generalized Forces Revisited • Derived Lagrange’s Equation from D’Alembert’s equation: ( ) ( ) 1 1 i i i p p i i i i i i i x i y i z i i i m x δ x y δ δ y z z F δ x F δ y F δ z = = ∑ ∑ && + + && && = + + • Define virtual displacements 1 N i i j j j x x q = q   ∂ =    ∂    δ ∑ δ • Substitute in and noting the independence of the j δ q , for each DOF we get one Lagrange equation: 1 1 i i i p p i i i i i i i i i r x y z i i r r r r r r x y z x y z m x y z q F F F q = = q q q q q q   ∂ ∂ ∂  ∂ ∂ ∂   + + =  + + ∂ ∂ ∂ ∂ ∂ ∂    ∑ ∑ && && && r    δ δ • Applying lots of calculus on LHS and noting independence of the i δ q , for each DOF we get a Lagrange equation: 1 i i i p i i x y z r r i r r x i r d T T y z F F F dt q q = q q q   ∂ ∂  ∂ ∂ ∂    − =  + + ∂ ∂ ∂ ∂ ∂    & ∑   • Further, we “moved” the conservative forces (those derivable from a potential function to the LHS: 1 i i i p i i x y z r r i r r x i r d L L y z F F F dt q q = q q q   ∂ ∂  ∂ ∂ ∂    − =  + + ∂ ∂ ∂ ∂ ∂    & ∑   Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 2