Uploaded on
10 May 2020
Skill-Lync
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.
Any mechanical system can be solved using two methods: The energy method and the 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.
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,
Upon the completion of the simulation, we get the following results,
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,
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,
but since the entire motion takes place in 2D we can neglect the 3rd component (Z component)
Upon plotting the reaction forces of X and Y component, here is the result we obtained:
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.
If you want to work on multi-body dynamics projects like the one mentioned here, you can enroll in the course and in no time, work on your own ideas.
Author
MehulMukesh Shah
Author
Skill-Lync
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