Macaulay functions in MUPad

In general the Macaulay function (<x-a>^n) is evaluated as 0 for x<a and x^n for x>a.

The integral is defined as int(<x-a>^n)=(<x-a>^(n+1))/(n+1)

I would like to implement functions of this type in MuPad to allow solving of beam deflection problems symbolically. However, I am not able to find a function that behaves as the Macaulay functions do. Any suggestions would be appreciated.

[Nathan Hardenberg]

Interesting to see someone from 6 years ago having the same Idea as me. Im trying it with Matlab instead of Mupad, but have not found a way to implement it. Especially if it should be able to solve as symbolic. Maybe this would be possible with an addon.

lg

[alejandro buitrago lopez]

were any of you able to solve this, i' currently trying to program this so i can print the diagrams in matlab.

[Nathan Hardenberg]

I did not solve the deflection problem, but I wrote a symbolic solver for shear force and bending moment. It is pretty well documented in the ReadMe file and you obviously can get some inspiration from the code.

https://github.com/icegoogles/Symbolic-Beam-Calculator

I also found a Python implementation that should be able to symbolically calculate deflection, shear force etc. Having said this, I did not use it yet.

https://pypi.org/project/symbeam/

Answer 1

[Nathan Hardenberg]

Not quite the solution you are looking for, but it is certainly possible to use the heaviside function. The heaviside function is also integratable and differentiable, so it is possible to use it instead.

<x - a>^0 is heaviside(x-a)

For functions with a higher exponent you have to write something like this

x^2 * heaviside(x-a)

this "switches" the function x^2 on (after a). Before it is 0

heaviside function: https://de.mathworks.com/help/symbolic/heaviside.html

[Walter Roberson]

<x - a>^0 is heaviside(x-a)

@Nathan Hardenberg

No, the ^0 form is 1 everywhere, including at -inf and +inf and 0

a = 2;
x = linspace(-5,5);
y1 = (x-a).^0;
y2 = heaviside(x-a);
stairs(x, y1, '-*');
hold on
stairs(x, y2, ':+')
hold off
legend({'\^0', 'heaviside'}, 'Location', 'best')
ylim([-0.1 1.1])

[Nathan Hardenberg]

You are right in the fact, that is not !

But here < and > are meant as macaulay brackets. With this notation my statement is correct. See the equation here.

It's probably better written as equation:

But note that the term is not defined at . In MATLAB is defined as at this position. It is sometimes also defined as 1.