Miniaturizing machines and physical systems is an ongoing effort in human civilization. This effort has been intensified in recent years as market demands for: Intelligent, Robust, Multi-functional and Low cost consumer products has become stronger than ever. The only solution to produce these consumer products is to package many components into the product – making it necessary to miniaturize each individual component. Miniaturization of physical systems is a lot more than just scaling down device components in sizes.

**Types of Scaling Laws**

1) Scaling in Geometry: Scaling of physical size of objects

- Volume (V)and surface (S)are two physical parameters that are frequently involved in machine design.
- Volume leads to the mass and weight of device components.
- Volume relates to both mechanical and thermal inertia. Thermal inertia is a measure on how fast we can heat or cool a solid.
- Surface is related to pressure and the buoyant forces in fluid mechanics. For instance, surface pumping by using piezoelectric means is a practical way for driving fluids flow in capillary conduits.
- When the physical quantity is to be miniaturized, the design engineer must weigh the magnitudes of the possible consequences from the reduction on both the volume and surface of the particular device.
If we let l= linear dimension of a solid, we will have:

2) Scaling of Phenomenological Behavior: Scaling of both size and material characterizations

- Forces are required to make parts to move such as in the case of micro actuators.
- Power is the source for the generation of forces.
- The inertia of solid is related to its mass and the acceleration that is required to initiate or stop the motion of a solid device component.
- In the case of miniaturizing these components, one needs to understand the effect of reduction in the size on the power (P), force (F)or pressure (p), and the time (t)required to deliver the motion.
- Rigid body dynamics is applied in the design of micro actuations and micro sensors, e.g. micro accelerometers (inertia sensors).
- It is important to know how size (scaling) affects the required forces (F), and thus power (P) in the performances of these devices.
- The dynamic force (F) acting on a rigid body in motion with acceleration (a) (or deceleration) can be computed from Newton’s 2ndlaw: F = M.a
- The acceleration (a) in the Newton’s law can be expressed in the following way In scaling:
- Let the displacement of the rigid body, s ∝ (ℓ), in which ℓ= linear scale.
- But velocity, v = s/t, and hence v ∝ (ℓ) t^-1, in which t is the required time.
From particle kinematics, we have:

where vo= the initial velocity. By letting vo= 0, we may express:

Thus, the scaling of dynamic force, F is