5

In the paper, entitled:

A Closed Form Solution for the Pull-in Voltage of the Micro Bridge

(Link to PDF: https://pdfs.semanticscholar.org/0d31/33707b1243f6b4e3344c4fa19b831b010b8b.pdf)

... the following equation appears:

enter image description here

I really don't know how to solve this for $\eta_{PI}$, even if all constants are known... do you have any idea ?

EDIT: Mathematica Input: ($n_{PI}$ was replaced with a simple $n$)

The nominator:

nom = Integrate[
  b*\[Phi][x]/(g - n*\[Phi][x])^2 + 
   0.265*b^0.25*\[Phi][x]/(g - n*\[Phi][x])^1.25 + 
   0.53*h^0.5*\[Phi][x]/(g - n*\[Phi][x])^1.5, {x, 0, L}]

The denominator:

denom = Integrate[
  2*b*(\[Phi][x])^2/(g - n*\[Phi][x])^3 + 
   0.33125*b^0.25*(\[Phi][x])^2/(g - n*\[Phi][x])^2.25 + 
   0.795*h^0.5*(\[Phi][x])^2/(g - n*\[Phi][x])^2.5, {x, 0, L}]

A typical $ \phi (x)$ function:

\[Phi][x] := a*Sin[x] + b*Cos[x] + c*Sinh[x] + d*Cosh[x]

Some arbitrary constants for numeric solutions to test:

constants = {a->2,b->4,c->7,d->10,g->3,h->2,L->5}

For easier copy-paste:

Clear[\[Phi]]
nom = Integrate[
  b*\[Phi][x]/(g - n*\[Phi][x])^2 + 
   0.265*b^0.25*\[Phi][x]/(g - n*\[Phi][x])^1.25 + 
   0.53*h^0.5*\[Phi][x]/(g - n*\[Phi][x])^1.5, {x, 0, L}]

denom = Integrate[
  2*b*(\[Phi][x])^2/(g - n*\[Phi][x])^3 + 
   0.33125*b^0.25*(\[Phi][x])^2/(g - n*\[Phi][x])^2.25 + 
   0.795*h^0.5*(\[Phi][x])^2/(g - n*\[Phi][x])^2.5, {x, 0, L}]
J. M.'s missing motivation
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james
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    The first thing you could do is to translate the image into Mathematica code. Otherwise, chances are nobody will bother to do it for you, and you won't get any answers. – AccidentalFourierTransform Oct 06 '18 at 19:47
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    Presumably, some of the terms in the integrand have an implicit dependence on x. You will have to work out what this is and make it explicit. – mikado Oct 06 '18 at 20:02
  • @AccidentalFourierTransform Thank you for your comment. Please have a look at my update. – james Oct 07 '18 at 06:44
  • @mikado Thank you for your comment. I added a typical function to my question. Please have a look at the update. – james Oct 07 '18 at 06:44
  • constants = {b->4,b->1} ?,decide. b constant is in meters or micrometers ? – Mariusz Iwaniuk Oct 07 '18 at 09:38
  • @MariuszIwaniuk Sorry. My mistake. I fixed it ! For now, any unit. In the end it will be micrometers. – james Oct 07 '18 at 09:45
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    From paper ϕ[x] is :ϕ[x] == (Cosh[λ x] - Cos[λ x]) - (Cosh[λ L] - Cos[λ L])/(Sinh[λ L] - Sin[λ L])*(Sinh[λ x] - Sin[λ x]) then: λ is ? or from eq (3) ?. Solving eq(3) for λ then: λ =0 ,and then ϕ[x]=0 ? – Mariusz Iwaniuk Oct 07 '18 at 09:57
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    I am mildly amused at the loose use of "closed form" in the paper. – J. M.'s missing motivation Oct 07 '18 at 10:59

1 Answers1

5
b = 50*10^-6;
g = 3*10^-6;
h = 2*10^-6;
L = 250*10^-6;
λ = 10; (* λ = ? .You may change. *)

ϕ[x_?NumericQ] := (Cosh[λ x] - Cos[λ x]) - (Cosh[λ L] - Cos[λ L])/(Sinh[λ L] - Sin[λ L])*(Sinh[λ x] - Sin[λ x])

nom[n_?NumericQ] := NIntegrate[(b ϕ[x])/(g - n ϕ[x])^2 + 53/100*( h^(1/2) ϕ[x])/(g - n ϕ[x])^(3/2) + 
53/200*( b^(1/4) ϕ[x])/(g - n ϕ[x])^(5/4), {x, 0, L}, Method -> "LocalAdaptive"]

denom[n_?NumericQ] := NIntegrate[(2 b ϕ[x]^2)/(g - n ϕ[x])^3 + 159/200*( h^(1/2) ϕ[x]^2)/(g - n ϕ[x])^(5/2) + 
53/160*( b^(1/4) ϕ[x]^2)/(g - n ϕ[x])^(9/4), {x, 0, L}, Method -> "LocalAdaptive"]

Solve for n:

FindRoot[n - nom[n]/denom[n] == 0, {n, 2}, WorkingPrecision -> 20, MaxIterations -> 1000]

(* {n -> 1.2897501610140538697} *)
Mariusz Iwaniuk
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  • Wow ! This is great! Thank you so much !! What doe x_?Numeric do and why do you use it ? – james Oct 07 '18 at 11:16
  • @james .It's defintion of a function that only evaluates when its argument is numeric. More in Documentation, exectute: ?NumericQ. – Mariusz Iwaniuk Oct 07 '18 at 11:31
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    @james Here's a site reference for NumericQ: https://mathematica.stackexchange.com/questions/18393/what-are-the-most-common-pitfalls-awaiting-new-users/26037#26037 – Michael E2 Oct 07 '18 at 12:05
  • @MichaelE2 Thanks for the link to the documentation. – james Oct 07 '18 at 17:38
  • @MariuszIwaniuk Thank you for the explanation ! – james Oct 07 '18 at 17:38
  • @MariuszIwaniuk How do you set the initial point (here n=2) ? What criteria do you use ? It could be anything, right ? I tried with n=100 and it failed. – james Oct 07 '18 at 17:41
  • At the beginning it is some random value. I don't know where is root.I do a plot ListLinePlot[Table[{n, n - nom[n]/denom[n]}, {n, 0, 3, 1}]] and see where is a root.The starting initial point should be as close to the root. Read documentation of FindRoot it's all there or on https://mathematica.stackexchange.com, putt on search Find Root an see question about it. – Mariusz Iwaniuk Oct 07 '18 at 18:11
  • Try: FindRoot[n - nom[n]/denom[n] == 0, {n, -10000}, Method -> {"Newton", "UpdateJacobian" -> 3}] or FindRoot[n - nom[n]/denom[n] == 0, {n, -10000, 1}, Method -> "Secant"] no longer complains . – Mariusz Iwaniuk Oct 07 '18 at 18:22
  • @MariuszIwaniuk Thank you very much for your detailed answer and your suggestion. Very helpful ! Thanks ! – james Oct 08 '18 at 06:30
  • @MariuszIwaniuk I also like to use this function, it seems very robust to find roots: https://mathematica.stackexchange.com/questions/183236/problem-with-nsolve-speed/183240?noredirect=1#comment478213_183240 – james Oct 08 '18 at 07:14
  • Sorry to bother you again. I am trying to calculate nPI for a piece wise defined function... I cannot find my error. I posted a question here: https://mathematica.stackexchange.com/questions/183381/help-for-solving-complicated-function – james Oct 08 '18 at 15:28