Determine the equations of motion using Lagrange method The

Determine the equations of motion using Lagrange method. The link of length L is massless. m_1 = 3m.

Solution

Take the kinetic energy and potental energy of the system:

T = 1/2 m1((X\')^{^2} +1/2( mL^² theta \' ^{^2}) = k.e of cart + k.e of inverted pendulum

P.E= 1/2 k X^² +1/2 K(-X)^² + 1/2 K_theta (theta)²= KX^²

Form L=T-V =KE -PE

Lagarange eqns give

d/dt(dL/dX\') -dL/dX =0

and similarly for theta coordinate

result: M₁ X\" =2KX

M L^² (theta)\" - K_theta (Theta) + mgL sin (theta) =0