Page 298 - The Mechatronics Handbook

0066_frame_C14.fm Page 12 Wednesday, January 9, 2002 1:49 PM

• making accurate assumptions, simplify the problem to make the studied MEMS mathematically
tractable (mathematical models, which are the idealization of physical phenomena, are never
absolutely accurate, and comprehensive mathematical models simplify the reality to allow the
designer to perform a thorough analysis and make accurate predictions of the system performance).
The second step is to derive equations that relate the variables and events:
• define and specify the basic laws (Kirchhoff, Lagrange, Maxwell, Newton, and others) to be used
to obtain the equations of motion. Mathematical models of electromagnetic, electronic, and
mechanical microscale subsystems can be found and augmented to derive mathematical models
of MEMS using defined variables and events;
• derive mathematical models;
The third step is the simulation, analysis, and validation:

• identify the numerical and analytic methods to be used in analysis and simulations;
• analytically and/or numerically solve the mathematical equations (e.g., differential or difference
equations, nonlinear equations, etc.);
• using information variables (measured or observed) and events, synthesize the fitting and mis-
match functionals;
• verify the results through the comprehensive comparison of the solution (model input-state-output-
event mapping sets) with the experimental data (experimental input-state-output-event mapping
sets);
• calculate the fitting and mismatch functionals;
• examine the analytical and numerical data against new experimental data and evidence.

If the matching with the desired accuracy is not guaranteed, the mathematical model of MEMS must be
refined, and the designer must start the cycle again.
Electromagnetic theory and classical mechanics form the basis for the development of mathematical
models of MEMS. It was illustrated that MEMS can be modeled using Maxwell’s equations and torsional-
mechanical equations of motion. However, from modeling, analysis, design, control, and simulation
perspectives, the mathematical models as given by ordinary differential equations can be derived and used.
Consider the rotational microstructure (bar magnet, current loop, and microsolenoid) in a uniform
magnetic field, see Fig. 14.5. The microstructure rotates if the electromagnetic torque is developed. The
electromagnetic field must be studied to find the electromagnetic torque.
B,
The torque tends to align the magnetic moment with and
m
T = m × B

FIGURE 14.5 Clockwise rotation of the motion microstructure.

©2002 CRC Press LLC