Nondimensionalisation

Objective. Reveal controlling parameter groups.

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Learn: the key idea

Nondimensionalisation rescales variables in a model using characteristic reference values so that equations become dimensionless, revealing the few parameter groups that truly control behaviour. By substituting x = x̃L, t = t̃T for chosen scales L, T into a differential equation and dividing through, the number of independent parameters typically collapses into one or two dimensionless combinations. Example: the damped oscillator mẍ+cẋ+kx=0 nondimensionalised with t̃=t√(k/m) becomes x̃'' + 2ζx̃' + x̃ = 0, where ζ=c/(2√(mk)) is the single damping ratio controlling all qualitative behaviour. This reduces a three-parameter problem to one, exposing the real control knob.

Worked example

The equation mẍ + kx = 0 is nondimensionalised using t̃ = ωt. Find ω (in terms of m and k) that makes the equation become x̃'' + x̃ = 0.

1 $Substitute t̃ = ωt, so d/dt = ω d/dt̃, giving ẍ = ω²x̃''$
2 $Equation becomes mω²x̃'' + kx = 0$
3 $Divide by mω²: x̃'' + (k/(mω²))x̃ = 0$
4 $Require coefficient to equal 1: k/(mω²)=1, so ω = \sqrt(k/m)$

Answer: ω = √(k/m)

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Common mistake. Students nondimensionalise by dividing by arbitrary constants rather than physically meaningful reference scales, producing dimensionless groups that do not correspond to any controlling parameter in the model.

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