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I would like to transform :

This expression of the transmittance of a 1Dof mechanical system :

T = (k + I c ω)/(k + I c ω - m ω^2)

TO

enter image description here

with 

ω0^2 = k/m

and λ = c / (2*m*ω0)

Can you help me to conduct these subsitutions with Mathematica ?

I failed with /. for the moment.

Thank you for your help

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  • $\begingroup$ T //. {ω0 -> k/m, λ -> c / (2*m*ω0)}? $\endgroup$
    – march
    Aug 24, 2017 at 21:58
  • $\begingroup$ @march it doesn't work yet since ω0 is not yet defined in the initial definition of T and the same for λ $\endgroup$
    – Bendesarts
    Aug 24, 2017 at 22:00
  • $\begingroup$ Oh, perhaps you meant T /. {k -> m*ω0, c -> λ*(2*m*ω0)} // Simplify? $\endgroup$
    – march
    Aug 24, 2017 at 22:02

1 Answer 1

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We can get close, but not exactly, by considering the transmittance squared and using a variable r for the ratio ω/ω0.

T = (k + I c ω)/(k + I c ω - m ω^2)
trns2 = T (T /. I -> -I) // Simplify
trns2 = trns2 /. k -> m ω0^2 /. 
    c -> 2 m ω0 λ /. ω -> r ω0 // Simplify
num = Numerator[trns2]
den = Collect[Denominator[trns2], λ, Simplify]
trns = Sqrt[num]/Sqrt[den] /. r -> ω/ω0
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