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The motion of two masses and three springs

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I need some assistance for a 3 spring problem: horizontal configuration, frictionless surface. Spring constant is k for all three springs and mass = m for both weights ie configuration is:

wall| spring - weigh t- spring - weight - spring |wall.

Meed solution using eigenvalues and Lagrangian to obtain equations of motions.

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Solution Summary

The motion of two masses and three springs is discussed. The solution shows how to choose the generalized coordinates, develop the Lagrangian, express the equations of motion and solve them in a matrix form.

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Here is the solution. Of course the eigenvalues are needed to get the characteristic frequencies.
The equations of motion come from the Lagrangian.

First we have to define our generalized coordinates.

These will ...

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