What is the expression for a series of sine equations in a spring-mass system?

[Brian4455]

Homework Statement

I'm trying to figure out what the expression of a series of sine equations would be. The problem deals with a series of masses attached by springs. In an equation describing the energy at a specific mass in the structure there is an expression that looks like this:

- B/m * (2pi/c) * sin ( 2pi/c * x_k)

x_k is a function not a partial derivative. It is the displacement function for the kth element in the spring mass system. The function x_k is dependent on two variables, one time and one displacement.

So I'm trying to take the above expression which is for the kth element and apply the expression to all the elements in the system.

Homework Equations

The equation the above expression comes from is the following:

x_k" = d/m(x_(k+1) - 2x_k + x_(k-1)) - B/m *(2pi/c)*sin(2pi/c*x_k)

when the expression I am having trouble with is left out the equation becomes the wave equation when applied to the whole system. So it would be:

d^2x/dt^2 = K/u*(d^2x/dz^2)

The Attempt at a Solution

I think the expression goes to 0 when applied to the whole system. Sine ranges in value from -1 to 1 and all of these terms ranging in value from -1 to 1 would cancel each other out. I might be thinking of a perfect situation though. 2pi/c*x_k might continually create angle values that make sine return all positive values or all negative values. I've considered using Power Series but I think that doesn't apply to this problem. The key is to account for the values that x_k will give.

 

[Brian4455]

I guess my latest attempt would be to add the expression:

+ F*B/m*(2pi/c)

to the wave equation when taking into account all of the weights. F is the new constant and it could be positive or negative. So I guess I see the equation resulting being the wave equation with a constant added on the end of it. I was thinking that the expression made the wave equation quasilinear but now I'm thinking the expression is constant and not dependent on the funciton x.