Three identical masses m are connected to 2 identical springs of stiffness k and equilibrium length b, as shown in the figure above. The masses are free to oscillate in one dimension along an axis that runs through all three. They lie on a level, frictionless, horizontal surface. Introduce coordinates, x1, x2 and x3 to measure their displacement from some fixed point on this axis.
Express the Lagrangian of the system in terms of these coordinates and their velocities. Then introduce the coordinates n1, n2, and n3, that measure the displacements of the springs from their equilibrium positions. Write the Lagrangian in terms of these coordinates and their time derivatives. Then find the kinetic and potential energy matrices. What are the frequencies of vibration for the normal modes of this system? What are eigenvectors?
See attached file for diagram.© BrainMass Inc. brainmass.com October 25, 2018, 10:15 am ad1c9bdddf
The solution shows in detail how to derive the Lagrangian for a system of three masses and two springs, obtain the equations of motion, convert it to matrix form and solve for the frequencies and the eigenmodes.
Lagrangian of a Rotating Mass, Spring and Hanging Weight
The mass m1 moves on a smooth horizontal plane, m2 moves vertically under the force of gravity and the spring. Using polar coordinates r, theta for m1, l for m2 and taking b for the total length of the string plus the unstretched length of the spring, find: (diagram attached in file)
a. the Langrangian of the system
b. the equations of motion for mass m1 in terms of the radial coordinate r, and for m2 in terms of l
c. at any given angular velocity, theta, there will be 'equilibrium' values for the positions of m1 and m2. FInd these values r zero and l zero.