Dynamics of a Pendulum 8 Define the lagrangian as the difference between the kinetic and potential energies L=T-V=,m()+(re d)I-mgri-cosd 8 Nonlinear dynamic equations found using Lagranges Equation d al aI 0 where q= generalized DOF dt(as丿ac 8 Results in the following equations P: mr's-m(re) sin o cos o+mgr sin o=0 m(rsin￠)(+2mr6sin￠cosp=0Dynamics of a Pendulum Dynamics of a Pendulum F Define the Lagrangian as the difference between the kinetic and potential energies: F Nonlinear dynamic equations found using Lagrange’s Equation: F Results in the following equations L = T − V = 1 2m ( rφ ś ) 2 + (rθ śsin φ ) 2 [ ] − mgr [1 − cos φ ] d dt ∂ L ∂q ś ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ∂ L ∂ q = 0 where q = generalized DOF [ ]φ : m r 2 φśś − m( r θś ) 2 sin φcosφ + mgrsin φ = 0 [ ]θ : m ( rsin φ ) 2θ śś + 2m r 2θ śφ śsin φcosφ = 0