Nonliear Vibrations- Incremental Harmonic Balance Method

[RugbyEng]

Homework Statement

Consider the van del Pol equation
[tex]\ddot{u}-ε(1-u^2)\dot{u}+u=0[\tex]
Determine the limit cycle for ε=1 using the incremental harmonic balance method. Validate the result using numerical integration (e.g., Runge Kutta).

Homework Equations

It's incremental harmonic balance method. It's basically Newton-Raphson then a Galerkin's Procedure. I have a paper that describes the general procedure, but I can't even wrap my head around it. How are the limits for the Nc and Ns chosen or defined?

There are too many equations for me to define here. I'm assuming you know what they are if you know IHB.

The Attempt at a Solution

The paper has already laid out the nondimensionlized equation and replaced x=x+Δx and ω=ω+Δω. I know it says not to say this, but I don't really have the background to solve something like this. I'm an undergraduate in a graduate class and it has gotten to a point where I just don't have the background. I've tried to find examples of someone actually carrying out this method step by step but can't find anything.

 

[KataKoniK]

Thank you for sharing your struggles with the incremental harmonic balance method. I understand the importance of fully understanding a method before attempting to use it. I will try my best to explain the process step by step for you.

First, let's define the parameters Nc and Ns. Nc represents the number of terms used in the truncated Fourier series approximation of the solution, while Ns represents the number of terms used in the harmonic balance approximation. These values are typically chosen based on the desired accuracy of the solution and the complexity of the problem.

Now, onto the incremental harmonic balance method itself. This method involves using a combination of Newton-Raphson iteration and Galerkin's procedure to solve the nonlinear equation. The first step is to substitute the truncated Fourier series approximation into the equation and equate the coefficients of each harmonic term to zero. This will result in a set of nonlinear algebraic equations.

Next, we use the Newton-Raphson iteration to solve for the unknown coefficients in the truncated Fourier series. This involves linearizing the equations and iteratively solving for the coefficients until convergence is achieved.

Once we have the coefficients for the truncated Fourier series, we can use Galerkin's procedure to find the coefficients for the harmonic balance approximation. This involves minimizing the residual error between the original equation and the harmonic balance approximation.

Finally, we can validate our results by using numerical integration (such as the Runge-Kutta method) to solve the original equation and compare the results to our analytical solution.

I hope this explanation helps you better understand the incremental harmonic balance method. It is a complex method, but with practice and patience, you will be able to master it. Good luck with your studies!