Modal Analysis for 3 pendulum with springs problem

by Nasser M Abbasi

This small note shows how to use Mathematica to solve symbolically for a problem in vibration to find the natural frequencies that a system of 3 masses will vibrate in. This digram below describes the problem. We use Lagrangian formulation to determine the equation of motions, then use modal analysis to decouple the system and solve it. In this system , the springs are attached at a distant α From the edge. Each pendulum has length L and has masses m1,m2,m3 attached to the end. The initial conditions are θ(0)={Pi/4,Pi/4,Pi/4} and ={0,0,0}

Define a function which accepts the kinetic and potential energy and return back the stiffness and the mass matrix

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Now define the kinetic and potenatial energy

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Now call the above function to generate the stiffness and mass matrix. It also prints the 3 equations of motion

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Now print the STIFFNESS and MASS matrix

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Now that we have the stiffness and mass matrix, we can perform modal analysis. Start by doing the first transformation

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Now find Λ matrix

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Hence the decoupled system of differential equations is

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Now convert the IC from θ (t) space to r (t) space

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Now solve the r (t) system

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Now convert solution from r (t) to θ (t)

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Now plot the solutions

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Created with Wolfram Mathematica 7.0