**Introduction:**

The Complex or Double pendulum exhibits one of the most chaotic motion characteristics known to us. It is considered to be unpredictable in its movement. However, we can solve the motion of such systems using the equations of motion. But if you attempt to solve the movement of a double pendulum using its equation of motion analytically, you will understand that it is very challenging. Hence, in this project, we will be solving the motion of a double pendulum numerically.

**Steps to be done:**

- Understand the theory
- Set up simulation
- Analyse the results:
- Trace the path of the pendulum movement
- Plot the reaction forces acting on the frame at various points when the pendulum is in motion.

**Theory:**

Any mechanical system can be solved using two methods: Energy method and Force method. In this project, we are going to adapt the energy method to calculate the reaction forces on the pendulum at different points. We will begin this by deriving the Lagrangian equation of this system (which is the difference between the kinetic and potential energy in the system). From the Lagrangian of this system, the Combined Euler Lagrange Differential Equation for θ1 and θ2 can be derived as

Upon solving this PDE, we can arrive at the solution of the system. But doing this analytically is complex. Hence this same system will be simulated in HyperWorks.

**Setting up the simulation: **

### Software used: MotionSolve from HyperWorks

For the simulation, we will consider the system with an initial configuration as follows. One of the things about this PDE is that it is extremely sensitive to initial conditions. When you solve a PDE numerically, it provides numerous solutions. Hence it is important that you specify the boundary conditions to correctly extract the results you need.

From the scheme above the simulation can now be setup in HyperWorks,

The mains steps involved in the simulation setup is as follows,

- Creating hardpoints for the joint locations
- Creating geometries for the rods and associating them with bodies
- Creating joints for the Compound Pendulum and connecting them with the bodies
- Creating the outputs to measure trace and reaction forces
- Making sure gravity is switched on – else the equation will be unsolvable
- Setting the time and running the simulation

**Results:**

Upon the completion of the simulation, we get the following results,

**Result for Trace:**

Trace is defined as the trajectory followed by a point in space. In our case, we trace the path of the bob at the end of the pendulum. The trace result appeared as follows,

**Results for the reaction forces at ‘P0’ of the system:**

When a pendulum oscillates, it imparts a reactive force to the ground/frame it is attached to. In our case, the points P1 and P2 impart a reactive force on point P0.

There are 3 components to this reaction force,

- X component
- Y component
- Z component

but since the entire motion takes place in 2D we can neglect the 3^{rd} component (Z component)

Upon plotting the reaction forces of X and Y component, here is the result we obtained:

**Graph for X component reaction forces:**

**Graph for Y component reaction forces:**

**Graph for Net magnitude of reaction forces:**

You can see the animation of the simulated pendulum here:

**Conclusion: **

Thus, we have obtained the trajectory/range of the pendulum and the reaction forces at various points via simulation.

Using the obtained data, we can calculate various factors like the load capacity, the maximum force that the pendulum can withstand, the force on the joints/frame etc. Hence, we can solve a double pendulum system numerically rather than using analytical methods.

Project submitted by,

Thanuj Singaravelan