help with symbolic int result

I ran into a problem, symbolic int results include int, adding a second variable, theta + psi, when using symbolic int.

Attached the "Code" and "Result" below.

Rather than randomly implementing all permutations of "assume()" and "simplify()" would someone please guide me to generating a result with the symsbolic "int" term?

Thank you

Code:

syms theta psi No Lo
N1          = (sym(1)/sym(2))*(No/sym(1))*sign(sin(sym(1)*theta + psi));
assume(theta>0 & theta<2*pi)
assume(theta, "real")
L11         = Lo*int((subs(N1,theta, theta + psi))^2, theta, 0, 2*pi)    
simplify(L11, Steps=10)

Result:

(Lo*No^2*int(sign(sin(theta + 2*psi))^2, theta, 0, 2*pi))/4

[Torsten]

The result should be No^2*pi/2.

But although I tried to restrict psi and theta to real values, MATLAB was not able to integrate

(subs(N1,theta, theta + psi))^2

[CD]

Difficult to explain.

If you look at the attached code, L11 is derived from N1, integrated over theta, and L22 is derived from N2, integrated over theta.

L12 is a derived from L11 and L22, intergrated over theta.

At the end, Matlab returns a single value (L12 = 68).

I would like to add a phase offset, psi, which exists in the final equation of L12.

This way I can find L12 (single value) given say L11 (or L22) is offset from L22 by a desired offset, psi.

No idea what to call this.

Hope this helps.

Thank you,

Example that returns a single value (after a bunch of steps):

syms theta No Lo
N1          = (sym(1)/sym(2))*(No/sym(1))*sign(sin(sym(1)*theta));
L11         = Lo*int(N1^2, theta, 0, 2*pi)
N2          = piecewise((0  < theta) & (theta < pi),   -(1/pi)*theta+0.5, ...
                        (pi < theta) & (theta < 2*pi),  (1/pi)*theta-1.5);
L22         = Lo*int(N2^2,theta,0,2*pi)
L12         = Lo*int(L11*L22,0, 2*pi) 
Lo = 1;
L12         = vpa(subs(L12), 4)

% L12 = 68.0 [units]

[Torsten]

And what is your question ?

[CD]

Torsten,

Walter Roberson is working this in a direction that I like.

Answer 1

[Walter Roberson]

syms theta psi No Lo real
assumeAlso(theta>0 & theta<2*pi)

N1          = (sym(1)/sym(2))*(No/sym(1))*sign(sin(sym(1)*theta + psi));
L11         = Lo*int((subs(N1,theta, theta + psi))^2, theta, 0, 2*pi)    

L11 = 

simplify(L11, Steps=10)

ans = 

The sign of psi is not defined, so this is the best you can do.

[Walter Roberson]

Whereas if you constrain psi instead of theta

syms theta psi No Lo real
assumeAlso(psi>0 & psi<2*pi)

N1          = (sym(1)/sym(2))*(No/sym(1))*sign(sin(sym(1)*theta + psi));
L11         = Lo*int((subs(N1,theta, theta + psi))^2, theta, 0, 2*pi)    

L11 = 

simplify(L11, Steps=10)

ans = 

[CD]

Ok, the output appears to be piecewise (view of live editor, not as obvious with script).

So I have seen this via script so at the time I did not know what to make of it.

I understand that this results essentually defines L11 for what may be all values of psi (assuming with 0 to 2pi).

How would I continue on with this L11 result?

How may I set or fix psi with in L11 then continue???

For example (there is a question in the middle of the example):

clc
clear all
syms theta psi No Lo real
assumeAlso(psi>0 & psi<2*pi)
N1          = (sym(1)/sym(2))*(No/sym(1))*sign(sin(sym(1)*theta + psi));
L11         = Lo*int((subs(N1,theta, theta + psi))^2, theta, 0, 2*pi)    

L11 = 

At this point L11 could be / is a function (piecewise) of psi.

How might I set psi (within L11) to some value/offset here then continue on with the following code???

%simplify(L11, Steps=10)
N2          = piecewise((0  < theta) & (theta < pi),   -(1/pi)*theta+0.5, ...
    (pi < theta) & (theta < 2*pi),  (1/pi)*theta-1.5);
L22         = Lo*int(N2^2,theta,0,2*pi)

L22 = 

L12         = Lo*int(L11*L22,0, 2*pi)

L12 = 

Lo = 1, No = 1;

Lo = 1

L12         = vpa(subs(L12), 4)                     % L2 = xx [units]

L12 = 

5.168

[Torsten]

I don't understand the piecewise result for L11. sign(...)^2 is always 1 (except maybe for one or two values in the interval 0<theta<2*pi depending on psi), thus the integral over sign(...)^2 will always be 2*pi.

[CD]

Perhaps better question:

Given my latest code list ...

Should I stop at the ouput of L11 then rebuild the derivation of L11 with psi removed given the piecewise results with psi then continue to derive L12?

Or is there a smarter way of getting to L12 with psi part of L11?

[Walter Roberson]

How might I set psi (within L11) to some value/offset here then continue on with the following code???

newL11  = subs(L11, psi, DesiredValue)