Large Amplitude Solutions in a forced massed spring system

[Kevin2341]

"Large Amplitude Solutions in a forced massed spring system"

Homework Statement

Using the formula, y"+by'+k1*y+k2*h(y) = -10 +0.1sin(ωt)

In the previous part of the of this "lab", I had to use some conditions for b, k1, k2, and ω, and analytically solve the problem and compare the graphs of the solutions between my answer, and a computer generated solution of the given 2nd order DE. Now in this part, they want me to find "large amplitude solutions", to "help" me, they give me a picture of some form of what I would consider a topographical graph of the DE with specific values, b = 0.01, k1 = 13, k2 = 4, and ω = 4. In this "topographical map" of that specific set of values, there were these blue regions, and then these smaller and less frequent black regions. Within the black region large amplitude solutions were generated (the different parts of this map represented different initial conditions).

Here is a picture of what I'm talking about...

Homework Equations

The Attempt at a Solution

I'm at a loss here, the instructions in the book feel like the author was going on a rant about something and nobody knows what he was originally talking about. He then sends me on my way, and tells me it is "delicate work".

Has anyone seen anything like this before, or have any idea how to do this? This is is the last part of this lab which has been pretty straight forward, and then I hit this, and all of a sudden I am supposed to sit around and "guess" where my solutions might be? I don't like that, that's not math to me, that's bullgarbageting around until you think you find something that might work.

 

[mighty2000]

Thank you for bringing this problem to our attention. Large amplitude solutions in a forced mass-spring system can be a challenging topic, but it is an important one in understanding the behavior of such systems. Let me explain a few key points that may help you in solving this problem.

Firstly, let's discuss the formula that you have been given:

y"+by'+k1*y+k2*h(y) = -10 +0.1sin(ωt)

This is a second order differential equation that describes the motion of a mass-spring system under the influence of a forcing function, which in this case is the term on the right-hand side of the equation. The terms b, k1, k2, and ω are constants that determine the behavior of the system. In particular, b is a damping coefficient, k1 is the spring constant, k2 is a nonlinearity coefficient, and ω is the frequency of the forcing function.

Now, let's take a look at the "topographical map" that you have been given. This is a plot of the solutions of the differential equation for different initial conditions. The blue regions correspond to small amplitude solutions, while the black regions correspond to large amplitude solutions. The boundaries between these regions are called bifurcation points, and they represent changes in the behavior of the system as the initial conditions are varied.

Your task is to find the initial conditions that will lead to large amplitude solutions, and then to compare your analytical solution with the computer-generated solution. This can be done by examining the "topographical map" and identifying the regions where the black regions are most prominent. You can then choose initial conditions that fall within these regions to generate large amplitude solutions.

In summary, the key to solving this problem is to carefully examine the "topographical map" and to identify the regions where large amplitude solutions are most likely to occur. I hope this helps you in solving this problem. If you have any further questions, please feel free to reach out to me.