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# Solving second order differential equation?

## Solving second order differential equation?

(OP)
I would like to solve a second order differential equation that represents a spring mass damper system that has an external acceleration applied to it.

I need to know the displacement over time output curve x(t). The input acceleration curve Gz(t)is a data set from experiment.

I have attached a picture (Mathcad Function.jpg) of my Mathcad sheet with the function to solve and a plot of the input acceleration trace.

Is it possible to solve this equation for x(t)?

If so I am not that familiar with Mathcad to know exactly how to do this, any help would be appreciated.

Thanks.

### RE: Solving second order differential equation?

#### Quote:

Is it possible to solve this equation for x(t)?
It is not practical to get a symbolic or exact solution but you can use RK4 to find a numerical solution easily enough.  Look for the Pendulum example in the Mathcad help.  Help->Index->odesolve.

Peter Nachtwey
Delta Computer Systems
http://www.deltamotion.com

### RE: Solving second order differential equation?

(OP)
Hi Peter, I am only after a numerical solution as my input acceleration is numerical. I have followed the pendulum example in help with limited success.

If I use the solve block and instead of submitting Gz(t) as my input acceleration I just use a single value I can obtain a solution for x(t). However If I input the input acceleration as a variable Gz(t) as I need to I cannot obtain a solution.

Accorading to the pendulum help example I believe I need to enter the solve blaock in this fashion Odesolve([vector], x, b, [step]) for my solution to be calcualted by I do not know how to ue this despite my best efforts.

I am not sure what to do with the [vector] part and the help is not particulalry helpful in this regard.

Do you have any examples you may be able to guide me with?

Thanks Peter.

### RE: Solving second order differential equation?

If you only need a numerical solution then this link will show you how to do it using state space methods but you can also use the Runge-Kutta method.

This example is a little more than what you asked for because I show how to control the mass on a spring.  All you are interested in is page 4/24 and this x=A*x+B*u

I made this example to show other how important the second derivative gain is for damping and controlling an under damped system.  I was comparing my PID solution with another persons best PI solution.

Peter Nachtwey
Delta Computer Systems
http://www.deltamotion.com

### RE: Solving second order differential equation?

(OP)
Hi Peter, thanks for this information. From your document I can see you have an extrememly good understanding of Mathcad, far greater than myself.

"the return value of this function must match the problem size"

I am not sure what this means and would be highly greatful if you could offer any further assistance. I have attached my Mathcad file to this post, I was wondering if you could take a quick look at it and give me any advice that you would kindly offer.

Regards,
Rich.

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