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numerical integration - How to solve this equation numerically or analytically


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}]

Answer



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 Ï•[x])/(g - n Ï•[x])^2 + 53/100*( h^(1/2) Ï•[x])/(g - n Ï•[x])^(3/2) + 
53/200*( b^(1/4) Ï•[x])/(g - n Ï•[x])^(5/4), {x, 0, L}, Method -> "LocalAdaptive"]

denom[n_?NumericQ] := NIntegrate[(2 b Ï•[x]^2)/(g - n Ï•[x])^3 + 159/200*( h^(1/2) Ï•[x]^2)/(g - n Ï•[x])^(5/2) + 

53/160*( b^(1/4) Ï•[x]^2)/(g - n Ï•[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} *)

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