How to Solve a second order differential equation for variable eta in time domain- with a contained integral over another dummy variable (tau)-

Hello All,

I am new to Matlab and I am trying to solve the attached 'Cummins Equation'. It is an equation of motion in time domain for the variable (eta). All other parameters in the equation M,A,B,C,F,h have been found over a period of interest from time 0 to time t. eta is to be found as function of t.

I have to integrate the contained integral over a dummy tau (convolution of both functions h and etadot) and then solve the resulting ODE. With a known h and an unknown etadot, could I still use conv2 function and continue to solve the ODE?

Will a time marching Runge Kutta method (ode45) work?

I would sincerely appreciate any help in this direction as I have hit a roadblock. Thanks in advance.

Answer 1

[Torsten]

Consider the different numerical methods for Cummins equation explained in

http://www.google.de/imgres?imgurl=http://static-content.springer.com/cover/journal/40722/1/4.jpg&imgrefurl=http://link.springer.com/article/10.1007%252Fs40722-015-0027-1&h=204&w=153&tbnid=3Lgd8TLCtVFH2M:&tbnh=90&tbnw=68&usg=__Iynhr9DcMCdY5Tclz8KraTI1sy0=&docid=xNiunqk6a1OpLM&sa=X&ved=0CE4Q9QEwBmoVChMI_MXNj93pyAIVgvsOCh1eXQHb

In any case, you will have to spend some effort in the solution method. A simple application of a standard tool like ODE45 won't work.

Best wishes

Torsten.

[Vignesh Kumar]

Thanks Torsten. That did accord lot of clarity on how to go about it.

The convolution integral simplifies to this summation. How could I use symbolic math toolbox to maintain the variable zdot through this summation...

and plug the computed I(t) into the resulting first order ODEs? Would the fact that I(t) itself would contain zdot affect the equation?

P.S:I am sorry if I sound silly, but Im using Matlab for the first time and Im forcibly trying to think in terms of Matrices..which is not working very well.in my head.

[Torsten]

Forget the symbolic toolbox.

Use ODE15s with the two equations from above.

To calculate I(t), save the solution for k2(t) up to time t for small time increments deltat during the Integration (e.g. in output_fcn). When ODE15S calls your function routine in which you supply k1' and k2', use this solution for k2(t) up to time t to calculate the convolution integral with the help of the above trapezoidal rule.

Best wishes

Torsten.

[Vignesh Kumar]

Thanks a lot for your time Torsten. Im trying that but my code doesn't seem to work.

[Vignesh Kumar]

Thank you Torsten,

But I have been trying with no result.Could you help?

[Vignesh Kumar]

COULD anyone please help me with this code? I have tried for more than a month but I am unable to crack it.

Answer 2

[ben howey]

Hi, I am trying to do the same thing. I have used a BEM code to calculate the IRF, inf added mass and frequency dependant excitation force and want to calculate the time dependant float position. Have you had any luck in solving this? I dont see how you can solve the convolution integral with the trapz method if z dot is unknown!

Any help or code that I can look through would be much appreciated!!

Thanks

[sivabalan sitti]

There is an open source code available to solve cummins equ,u can try that https://wec-sim.github.io/WEC-Sim/getting_started.html