Finding Center of Mass and Moment of Inertia

Lagrange’s Method to find EOM 1. Identify number of generalized coordinates and any generalized forces 2. Choose generalized coordinates ξ1, ξ2, . . . 3. Find kinetic energy T and potential energy V in terms of these generalized coordinates 4. Assemble Lagrangian L = T − V 5. Express generalized forces in terms of the generalized coordinates δW = Fxδx δW = Ξjδξj • When finding generalized forces which require solving δx in terms of δξ, sometimes it is easiest to find velocities, then cancel dt and make dx into δx and dξ into δξ. • When breaking force F into components Fx, don’t forget the sign • Measure springs deflections from static equilibrium and gravity won’t appear in equations of motion • When finding kinetic energy of rigid bodies, place coordinate system at CG and such that it is a set of principal axes, then the moment of inertia is about the CG not the physical point of rotation. 6. Evaluate ∂L ∂ξ̇j and ∂L ∂ξj 7. Use the formula d dt ( ∂L ∂ξ̇j ) − ∂L ∂ξj = Ξj