Question

Planar double pendulum, Setup Consider a planar double pendulum (in the r-y plane) in the Earth's gravitational field in the y direction as shown above. The length of both strings is ( and both masses are equal, and denoted by m

1. How many DOFs are there in this case? Write them in terms of angles.
2. Find the Lagrangian.
3. Write the EOMs (but do not solve them)

Answer 1

The planar double pendulum has two degrees of freedom represented by two angles. The Lagrangian of the system can be derived from the kinetic and potential energies of both masses.

Step-by-step explanation:

The planar double pendulum has two degrees of freedom, which are represented by the two angles in the system. Let's call them θ1 and θ2.

The Lagrangian of the system can be derived by considering the kinetic and potential energies of both masses. It is given by:

L = (1/2)m1l1^2θ1'² + (1/2)m2(l1^2θ1'² + l2^2θ2'² + 2l1l2θ1'θ2'cos(θ1 - θ2))) - m1gl1cos(θ1) - m2gl2cos(θ2)

The equations of motion (EOMs) can be found by taking the partial derivatives of the Lagrangian with respect to the angles and their derivatives. They are given by:

(m1 + m2)l1θ1'' + m2l2θ2''cos(θ1 - θ2) = -(m2l2θ2'²sin(θ1 - θ2)) - (g/l1)(m1 + m2)sin(θ1)

m2l2θ2'' + m2l1θ1''cos(θ1 - θ2) = -m2l1θ1'²sin(θ1 - θ2) - (g/l2)m2sin(θ2)