Professor! Today's topic is about magnetic bearing analysis, right? What is it?

It's non-contact support using electromagnets. It's essential for the high-speed operation of rotating machinery, controlled via PID for current stiffness and displacement stiffness.

What kind of equations are involved in magnetic bearing analysis?

The analysis involves Maxwell's equations for electromagnetism and the equations of motion for the rotor.

How do you actually solve these equations using a computer?

We use spatial discretization with the Finite Element Method (FEM). This involves assembling element stiffness matrices to construct the global stiffness equation.

What specifically is meant by matrix solution algorithms?

It refers to solving the system of equations using direct methods (like LU decomposition, Cholesky decomposition) or iterative methods (like the Conjugate Gradient method, GMRES). For large-scale problems, preconditioned iterative methods are particularly effective.

磁気軸受解析

Category: 解析 | Integrated 2026-04-06

Magnetic bearing electromagnetic force vs air gap and current characteristic curves with nonlinear reluctance model — 磁気軸受の電磁力特性：電流-力曲線（左）とギャップ-力非線形特性（右）。緑点が動作点、オレンジ破線が線形化接線。

Overview

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先生！今日は磁気軸受解析の話なんですよね？どんなものなんですか？

Theory and Physics

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Non-contact support using electromagnets. Essential for high-speed rotation machinery. PID control of current stiffness and displacement stiffness.

Governing Equations

$$ F = \frac{B^2 A}{2\mu_0} $$

$$ F_{net} = k_i i + k_x x $$

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Professor, your explanation is easy to understand! The haze around describing magnetic bearing analysis has cleared up.

Discretization Method

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How do you actually solve these equations on a computer?

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We use spatial discretization by the Finite Element Method (FEM). We assemble the element stiffness matrix and construct the global stiffness equation.

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We perform transformation to the weak form (variational form) and use formulation by the Galerkin method using test functions and shape functions. The choice of element type (low-order elements vs. high-order elements, full integration vs. reduced integration) is directly linked to the trade-off between solution accuracy and computational cost.

Matrix Solution Algorithms

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What exactly do you mean by matrix solution algorithms?

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We solve the simultaneous equations using direct methods (LU decomposition, Cholesky decomposition) or iterative methods (CG method, GMRES method). For large-scale problems, preconditioned iterative methods are effective.

Solution Method	Classification	Memory Usage	Applicable Scale
LU decomposition	Direct Method	O(n²)	Small to Medium Scale
Cholesky decomposition	Direct Method (Symmetric Positive Definite)	O(n²)	Small to Medium Scale
PCG Method	Iterative Method	O(n)	Large Scale
GMRES method	Iterative Method	O(n·m)	Large Scale / Non-symmetric
AMG Preconditioner	Preprocessing	O(n)	Very Large Scale

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So, if you cut corners on the finite element method part, you'll pay for it later. I'll keep that in mind!

Implementation in Commercial Tools

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So, what software can be used to do magnetic bearing analysis?

Tool Name	Developer/Current	Main File Format
COMSOL Multiphysics	COMSOL AB	.mph
JMAG-Designer	JSOL Corporation	.jmag, .jproj
Ansys Mechanical (formerly ANSYS Structural)	Ansys Inc.	.cdb, .rst, .db, .ans, .mac
Abaqus FEA (SIMULIA)	Dassault Systèmes SIMULIA	.inp, .odb, .cae, .sta, .msg

Vendor Lineage and Product Integration History

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Is the background of each software quite dramatic?

COMSOL Multiphysics

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Tell me about "COMSOL Multiphysics"!

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Founded in Sweden in 1986. Started as FEMLAB with MATLAB integration, later renamed to COMSOL. Strong in multiphysics.

Current Affiliation: COMSOL AB

JMAG-Designer

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What exactly is JMAG?

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Developed by Japan's JSOL Corporation. An electromagnetic field analysis tool specialized for electrical equipment design.

Current Affiliation: JSOL Corporation

Ansys Mechanical (formerly ANSYS Structural)

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Tell me about "Ansys Mechanical"!

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Developed in 1970 by Swanson Analysis Systems Inc. (SASI). Based on APDL (Ansys Parametric Design Language).

Current Affiliation: Ansys Inc.

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Ah, I see! So that's how it was established in Sweden in 1986.

File Formats and Interoperability

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Are there any points to note when transferring data between different software?

Format	Extension	Type	Overview
STEP	.stp/.step	Neutral CAD	ISO 10303 compliant 3D CAD data exchange format. Supports geometry + PMI.
IGES	.igs/.iges	Neutral CAD	Early CAD data exchange standard. Has issues with surface data compatibility. Transition to STEP is progressing.
VTK	.vtk/.vtu	Visualization	Visualization Toolkit format. Used by ParaView, etc.

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When converting models between different solvers, attention must be paid to the correspondence of element types, compatibility of material models, and differences in the representation of loads and boundary conditions. Particularly, high-order elements and special elements (cohesive elements, user-defined elements, etc.) often cannot be directly converted between solvers.

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I see... Formats seem simple at first glance, but they're actually quite deep, aren't they?

Practical Considerations

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Are there things like "field wisdom" that aren't covered in textbooks?

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Verifying mesh convergence, validating the appropriateness of boundary conditions, and performing sensitivity analysis of material parameters are extremely important.