16.61 Aerospace Dynamics Spring 2003 Derivation of Lagrange's equation Two approaches (A) Start with energy expressions Formulation Lagrange's Equations (Greenwood, 6-6) Interpretation Newton's laws (B)Start with Newtons Laws Formulation Lagrange's equations (Wells, Chapters 3&4) Interpretation Energy Expressions (A)Replicated the application of Lagranges equations in solving problems (B)Provides more insight and feel for the physics Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Derivation of Lagrangeís Equation • Two approaches (A) Start with energy expressions Formulation Lagrangeís Equations (Greenwood, 6-6) Interpretation Newtonís Laws (B) Start with Newtonís Laws Formulation Lagrangeís Equations (Wells, Chapters 3&4) Interpretation Energy Expressions (A) Replicated the application of Lagrangeís equations in solving problems (B) Provides more insight and feel for the physics Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 11
16.61 Aerospace Dynamics Spring 2003 Our pr 1. Start with Newton 2. apply virtual work 3. Introduce generalized coordinates 4. Eliminate constraints 5. Using definition of derivatives, eliminate explicit use of acceleration Start with a single particle with a single constraint, e. g Marble rolling on a frictionless sphere Conical pendulum m Free body Diagram X Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Our process 1. Start with Newton 2. Apply virtual work 3. Introduce generalized coordinates 4. Eliminate constraints 5. Using definition of derivatives, eliminate explicit use of acceleration • Start with a single particle with a single constraint, e.g. o Marble rolling on a frictionless sphere, o Conical pendulum θ φ r = L x y z m F Free body θ Diagram φ r = L x y z m F Free body Diagram Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 12