Introduction to Scaling Analysis

Scaling analysis is fundamental to predicting the behavior of structures and systems when miniaturized. While each individual application will have restrictions and assumptions based on the proposed function, the approach to scaling analysis will generally follow the generic examples given below.

A Column Supporting a Weight

Assume a square Euler column (slender column) is supporting a load on the centerline of the column. This implies careful and theoretically unobtainable control of the load location. If the dimensions of the column and the load shrink linearly, the resulting analysis shows the strength to weight ratio of the column increases. This is why a small insect can survive a drop onto its legs from a height many times the size of the insect and large animals can not. The analysis shows that the strength to weight ratio goes up linearly as the dimensions decrease. For a 100-fold reduction in size, the structure gets 100-times stronger.

Strength as a Cantilever Beam

A cantilever beam is often found used a structural component in microsystem applications. Assuming the beam has a rigid constraint, and an end-load of a fixed magnitude, then the strength of the beam becomes small very quickly. The stress induced in the beam by the force increases as one over the square of the characteristic dimension. A shrinking of 10 causes a 100-fold increase in the induced stress. Similarly, if the induced stress is to remain the same (a more logical approach), then the force acting on the beam must decrease as the square of the characteristic dimension.

Surface Area to Volume Ratio

Microdevices are often subject to extreme thermal environments or are designed to produce a high temperature with a small response time. For a volume, as the characteristic dimension is decreased the ratio of the surface area to the volume becomes large. This means the ability to transfer (absorb and/or release) heat relative to the ability to store heat, is large. Therefore heat transfer between working fluids is high. Additionally, if the volume is heated internally or externally, the heat will dissipate very quickly giving a rapid response. The large surface area to volume ratio is also important for phenomena such as surface tension, capacitance, and other attributes which are surface, rather than volume, related.

Resistive Power and Time Response

Assume a micro-component is for the purpose of supplying heat by way of resistive heating. The electrical power converted to heat within the component is a function of the material resistivity, the material volume, and the applied voltage. Assuming no losses (which is not a very good assumption given the previous example), then the resistive power will dictate the rate of temperature increase of the component. This rate of temperature increase is proportional to the reciprocal of the square of the characteristic dimension. This results in a rapid response to both heating and cooling.

Many other analyses can be performed for natural frequency of vibration, structural deflection, capacitance, and convection coefficient, for example and these analyses generally follow the same approach shown.