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Is this definite integral solvable?

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I have been looking at the eccentricity in induction motor. Through using the Maxwell's equations I have arrived at the equation (1) in the attached file.
However, I cannot find a solution for the equation (2) in a closed form using Mathematica. I believe, that the solution can be greatly simplified as described in the attached file. Am I correct or not?

Attachments:
POSTED BY: Jan Krepela
Answer
18 days ago

By any chance, is ε a mall parameter ε<<1 ?

POSTED BY: Kay Herbert
Answer
18 days ago

Hello Key Herbert, The eccentricity in the induction motor can be all the way to ε=1. Of course it cannot be exactly 1, because it would mean that rotor is rubbing again the startor a disaster for any electric motor. The eccentricity is kind of misterious, because lots of people talk about it, but nobody wants to actually measure it. From my experience, the eccentricity is usually below 0.3. Hence the approximate solution would not give any meaningful solution. Thank you for the response, I have never hoped for something so fast. Regards, jank

POSTED BY: Jan Krepela
Answer
18 days ago

Welcome to Wolfram Community! Please make sure you know the rules: https://wolfr.am/READ-1ST

Please post some code that you tried, this is an essential forum guideline.

POSTED BY: Moderation Team
Answer
18 days ago

If you post actual Mathematica code people are more likely to try it out.

POSTED BY: Daniel Lichtblau
Answer
18 days ago

If I use symmetry take 2 x integral from 0 to Pi and define the constraints on the parameters, Mathematica gives me an analytical answer to the integral:

In[34]:= Integrate[2 Cos[p b]/ (1 + e Cos[b]), {b, 0, Pi}, 
 Assumptions -> e > 0 && p > 0 && p \[Element] Integers && e <= 1]

Out[34]= -((
 2 \[Pi] HypergeometricPFQRegularized[{1/2, 1, 1}, {1 - p, 1 + p}, (
   2 e)/(-1 + e)])/(-1 + e))
POSTED BY: Kay Herbert
Answer
17 days ago

Hello Herbert, At this moment I am kind of overwhelmed. If the function you have arrived at were called sin or arctg I would know what you are talking about. But the Out[34] is something I have never dreamed existed in mathematical language. I have even doubted I have joined the correct community after the e-mail from the moderator. I have a limited knowledge of mathematics, that is why I have bought Mathematica two month ago to help me solve the equations Maxwell's theory led me to. I have no doubt that your solution is absolutely brilliant. However, I would like to bring the solution of the problem to a usable stage. Hence, I would like anybody that works seriously with induction motors to be able to calculate "one directional pull" for a 4 pole motor (p=2) with eccentricity 30%. In the meantime, I have to work hard to learn the Wolfram language to be acceptable. Being just barely tolerated is no fun.

POSTED BY: Jan Krepela
Answer
17 days ago

Hyper geometric functions aren't uncommon. if p=2 it greatly simplifies:

In[19]:= -((
  2 \[Pi] HypergeometricPFQRegularized[{1/2, 1, 1}, {1 - p, 1 + p}, (
    2 e)/(-1 + e)])/(-1 + e)) /. p -> 2

Out[19]= (2 (-2 + 2 Sqrt[(-1 - e)/(-1 + e)] - 
   2 Sqrt[(-1 - e)/(-1 + e)] e + e^2) \[Pi])/(Sqrt[(-1 - e)/(-1 + 
  e)] (-1 + e) e^2)

In[20]:= Simplify[(
 2 (-2 + 2 Sqrt[(-1 - e)/(-1 + e)] - 2 Sqrt[(-1 - e)/(-1 + e)] e + 
    e^2) \[Pi])/(Sqrt[(-1 - e)/(-1 + e)] (-1 + e) e^2)]

Out[20]= -((
 2 (2 - e^2 - 2 Sqrt[(1 + e)/(1 - e)] + 
    2 e Sqrt[(1 + e)/(1 - e)]) \[Pi])/((-1 + e) e^2 Sqrt[(1 + e)/(
  1 - e)]))

enter image description here

POSTED BY: Kay Herbert
Answer
17 days ago

This may be useless information by now, but I looked at the integral and found that there is a relatively simple equation for this integral, at least for integer values of p. See the attached notebook.

Attachments:
POSTED BY: Mike Luntz
Answer
6 days ago

Group Abstract Group Abstract