topics: [“MEMS”, “ROM”, “modeling”, “mathematics”, “Python”]
In the previous article (MEMS:02),
we visually explored the behavior of a MEMS structure under piezoelectric hysteresis input
through a visualization demo.
In this article, we focus on what lies beneath that behavior:
the mathematical structure and design policy of mems-ana_core.
The purpose of this article is not:
Instead, the goal is to clarify—explicitly and from a designer’s perspective—
Why this mathematical structure was chosen
What is intentionally included in the model, and what is intentionally excluded
This article is the third installment in the mems-ana series.
mems-ana_core corresponds to:
Its defining characteristics are:
Rather than prioritizing raw speed,
the design prioritizes structural readability and interpretability.
The intended targets are typical:
Accordingly, the following are outside the model scope:
If these assumptions do not hold,
mems-ana_core should not be used.
The displacement field $u(x,y,t)$ is expressed as
a product of spatial modes $\phi_n(x,y)$
and time-dependent coefficients $q_n(t)$:
The key design choices here are:
In mems-ana_core, mode selection is guided by:
The result is a model with the minimum number of modes required to capture behavior.
The general ROM-based equation of motion takes the form:
\[M \ddot{\mathbf{q}} + C \dot{\mathbf{q}} + K \mathbf{q} = \mathbf{f}(t)\]where:
Because mems-ana_core primarily targets:
the inertial and damping terms
($\ddot{\mathbf{q}}$, $\dot{\mathbf{q}}$)
are often simplified or omitted.
Piezoelectric actuation is introduced as
an external force term $\mathbf{f}(t)$.
The critical modeling decision is that:
This yields a simplified, design-oriented causal chain:
which preserves interpretability and controllability.
The P–$E_z$ hysteresis used in MEMS:02 is:
Instead, it represents:
Thus, mems-ana_core:
This choice:
Calibration in mems-ana_core can reference:
but the philosophy remains consistent:
Fit numerical values without destroying parameter meaning
In practice, calibration is limited to parameters such as:
Arbitrary parameter fitting is explicitly avoided.
mems-ana_core is not a replacement for FEM.
The intended division of roles is:
FEM results are used as
reference points to reason about “why” behavior occurs.
The implementation of mems-ana_core discussed here
is available in the following GitHub repository:
https://github.com/Samizo-AITL/mems-ana
Please refer to the repository for
the correspondence between equations and code.
Together, these three articles present:
as a single, coherent narrative.
mems-ana_core is designed not as:
A model that produces exact answers
but as:
A model that helps designers think
Rather than minimizing equations,
the priority is not to minimize meaning.
ROMs are approximations—
but good approximations accelerate design thinking.
If this article helps clarify
how and why models are constructed in MEMS design,
it has achieved its purpose.