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Characteristic frequencies and modes of a system of springs

Consider a system of 3 springs (A, B and C) connected in a line. The left hand end of spring A is anchored to a wall as is the right hand end of spring C. Between springs A and B is mass M and between springs B and C is mass N.

Consider the following 2 situations:

1. mass M = N and the spring constants are spring A = k. B=2k, C=k

2. mass M = 2N and the spring constants are A=4k, B=k, C = k

Find the characteristic frequencies and the characteristic modes of vibration for the 2 situations.

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There is some technical Classical Mechanics stuff involved in this problem which I tried to explain. But it may not be explained adequately. If you need more detailed explanations, then you can submit a question restricted to me for 1 point tomorrow. I will then be able to send a personal message free of charge to you addressing that question. I don't think that the Brainmass staff work on weekends, so emailing them directly may not work.

Let's denote the displacement of the left hand mass by x1, so that positive x1 means a displacement to the right. And similarly, we denote by x2 the displacement of the right hand mass, also such that positive x2 means a displacement to the right. The potential energy of the system can be found as follows. Recall that if a spring with spring constant k is streched by an amount x the potential energy of that spring is 1/2 k x^2. So, we need to find out by how much each of the three springs is stretched if the diplacements of the masses are x1 and x2.

Clearly the spring on the right will be streched by x1,
the middle spring will be streched by x2 - x1,
the right hand spring will be streched by -x2

So, the potential energy function is:

V(x1,x2) = 1/2 A x1^2 + 1/2 B (x1 - x2)^2 + 1/2 C x2^2 (1)

Basically this is all you need to know, because the force on mass one is simply minus the derivative of V w.r.t. x1 and you can equate that to the mass of M times the second derivative of x1 w.r.t. the time. The force on mass 2 is minus the derivative of V w.r.t. x2 and you must equate that to the mass N times the second derivative of x2 w.r.t. time. You then get two linear differential equations for two functions which you can solve. This is, however, not the most efficient way to solve this ...

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