Scaling Laws

Micro Intuition

• Linear extrapolation is easy but we are at a loss when considering the implications that shrinking of length has on surface area to volume ratios and on the relative strength of external forces

• Micro intuition can be misleading.

• Our aim is to develop a systematic approach about the likely behavior of downsized systems so we do not need to rely on micro intuition alone.

Scaling Laws

• They allow us to determine whether physical phenomena will scale more favorably or will scale poorly.

• Generally, smaller things are less effected by volume dependent phenomena such as mass and inertia, and are more effected by surface area dependent phenomena such as contact forces or heat transfer.

As you decrease the size

• Friction > inertia

• Heat dissipation > Heat storage

• Electrostatic force > Magnetic Force

Surface Area to Volume Ratio

Surface Area : Volume Ratio

For Example

What are the implications of this?

• Volume relates, for example, to both mechanical and thermal inertia. Thermal inertia is a measure on how fast we can heat or cool a solid. It is an important parameter in the design of a thermally actuated devices.

Thermal Scaling

• Small object will loose heat rapidly, the dissipation of waste heat is not problematic in many cases.

Mathematical Approach

• Mathematically, a scaling law is a law that describes the variations of physical quantities with the size of the system.

• Use of dimensional analysis.

Scaling effects on spring constant (k)

• Consider a beam, length L, width w, Thickness t, and Youngs Modulus E.

Stress in a rod connected to a mass experiencing a

constant acceleration

Resistance

Resistance

Capacitance

Given a parallel plate capacitor with plate area wL=A and plate separation d, the capacitance is given as

C = ε A / d

where ε = permittivity of gap insulator material

Electrostatic Forces

Electrostatic Forces

Electromagnetism

• Faraday’s law governs the induced force (or a motion) in the wire under the influence of a magnetic field.

• The scaling of electromagnetic force follows: F ∝ S4.

• For electromagnets, as S decreases, these forces decrease because it is difficult to generate large magnetic fields with small coils of wire.

• However permanent magnets maintain their strength as they are scaled down in size, and it is often advantageous to design magnetic systems that use the interaction between an electromagnet and a permanent magnet.

Magnet Scaling

Fluid Mechanics

Use of Matrix Formalization

• To design micromechanical actuators, it is helpful to understand how forces scale. Use of a matrix formalism notation is very handy to describe how different forces scale into the small (and large) domain. It is called the Trimmer’s vertical bracket approach.

• Formulated by William Trimmer in 1986.

Use of Matrix Formalization

• The top element in this notation refers to the case where the force scales as S1. The next one down refers to a case where the force scales as S2, etc.

• If the system becomes one-tenth its original size, all the dimensions decrease by a tenth. The mass of a system, m, scales as (S3) and, as systems become smaller, the scaling of the force also determines the acceleration (a), transit time (t), and the amount of power per unit volume (PV-1)

Scaling of forces

• The force due to surface tension scales as S1

• The force due to electrostatics with constant field scales as S2

• The force due to certain magnetic forces scales as S3

• Gravitational forces scale as S4

• Summarizing The Trimmer Notation

Order Force Scale, F Acceleration, a Time, t Power Density, P/V

1 1 -2 1.5 -2.5

2 2 -1 1 -1

3 3 0 0.5 0.5

4 4 1 0 2

• List of Physical Phenomena and their scaling

Benefits Of Scaling

• Speed (Frequency increase, Thermal Time constraints reduce)

• Power Consumption (actuation energy reduce, heating power reduces)

• Robustness (g-force resilience increases)

• Economy (batch fabrication)