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SCALING LAWS · PART 2 OF 3

Write every force as a power of S.

William Trimmer's trick: instead of solving one force at a time, list how each kind of force scales and work with all of them at once.

F = [ S1   S2   S3   S4 ]
S1 Surface tension: force = surface tension × length of the contact line.
S2 Electrostatic force with the electric field held constant: force ∝ field2 × plate area.
S3 Weight and inertia (they follow mass, so volume), and some magnetic cases.
S4 Magnetic force with the current density held constant.
Read the bracket as four answers in one: whatever you calculate, do it for every column.

THE RULES OF THE GAME

Scaling is just exponent arithmetic.

Multiply Add the exponents: S2 × S1 = S3
Divide Subtract them: S2 ÷ S3 = S−1
Square root Halve it: √(S3) = S1.5
Read the answer Positive exponent: shrinks with the device. Negative: grows as the device shrinks. Zero: does not change.
Plug in S = 0.1 for "10× smaller": S2 becomes 1/100, S−1 becomes 10.

ACCELERATION

Small parts accelerate hard.

Acceleration is force divided by mass, and mass follows volume, [S3]:

a = F / m  ∝  Sf ÷ S3 = Sf − 3
S1 force a ∝ S−2: 10× smaller, 100× more acceleration
S2 force a ∝ S−1: 10× smaller, 10× more acceleration
S3 force a ∝ S0: no change (everything falls at the same rate)
S4 force a ∝ S1: 10× smaller, 10× less acceleration
The force gets weaker, but the mass gets weaker even faster for S1 and S2 forces.

TRAVEL TIME

How long does a move take?

Starting from rest, a part covers distance d in time t when d = ½ a t2, so t = √(2d / a) = √(2d m / F). Distance follows length [S1] and mass follows [S3]:

t ∝ √( S1 × S3 ÷ Sf ) = S(4 − f)/2  →  [ S1.5   S1   S0.5   S0 ]
Every force law except S4 gets faster as you shrink. That is why a MEMS mirror can flip thousands of times a second.

WORKED EXAMPLE

Shrink an electrostatic actuator 10×.

Electrostatic force at constant field is an S2 force. Set S = 0.1 and follow each step:

Force ∝ S2 0.12 = 0.01: a hundred times weaker
Mass ∝ S3 0.13 = 0.001: a thousand times lighter
Acceleration ∝ S−1 0.01 ÷ 0.001 = 10: ten times the acceleration
Time ∝ S1 √(0.1 × 0.001 ÷ 0.01) = 0.1: the move takes a tenth of the time
Weaker force, faster device. When the numbers surprise you, check them with exponents: they never lie.

UNIT 3 STUDY COMPLETE

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Ready for the Fab Challenge?

You've covered Trimmer's Force Scaling.

Based on Prof. Sam Kassegne's ME 585 lecture on scaling laws (San Diego State University) and William Trimmer's work on the scaling of micromechanical devices.