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TECHNICAL PAPERS

# An Evaluation of Stability Indices Using Sensitivity Functions for Active Magnetic Bearing Supported High-Speed Rotor

[+] Author and Article Information
Naohiko Takahashi

Tsuchiura Research Laboratory, Research & Development Group, Hitachi Plant Technologies, Ltd., 603 Kandatsu–machi, Tsuchiura–shi, Ibaraki–ken 300–0013, Japannaohiko.takahashi.qb@hitachi-pt.com

Hiroyuki Fujiwara

Department of Mechanical Engineering, National Defense Academy, 1–10–20 Hashirimizu, Yokosuka–shi, Kanagawa–ken 239–8686, Japanhiroyuki@nda.ac.jp

Osami Matsushita

Department of Mechanical Engineering, National Defense Academy, 1–10–20 Hashirimizu, Yokosuka–shi, Kanagawa–ken 239–8686, Japanosami@nda.ac.jp

Makoto Ito

Development Department, Shinkawa Sensor Technology, Inc., 4–22 Yoshikawa Kogyodanchi, Higashihiroshima–shi, Hiroshima–ken 739–0153, Japanitom@sst.shinkawa.co.jp

Yasuo Fukushima

Tsuchiura Works, Hitachi Plant Technologies, Ltd., 603 Kandatsu–machi, Tsuchiura–shi, Ibaraki–ken 300–0013, Japanyasuo.fukushima.hj@hitachi-pt.com

J. Vib. Acoust 129(2), 230-238 (Oct 05, 2006) (9 pages) doi:10.1115/1.2424979 History: Received November 02, 2005; Revised October 05, 2006

## Abstract

In active magnetic bearing (AMB) systems, stability is the most important factor for reliable operation. Rotor positions in radial direction are regulated by four-axis control in AMB, i.e., a radial system is to be treated as a multi-input multioutput (MIMO) system. One of the general indices representing the stability of a MIMO system is “maximum singular value” of a sensitivity function matrix, which needs full matrix elements for calculation. On the other hand, ISO 14839-3 employs “maximum gain” of the diagonal elements. In this concept, each control axis is considered as an independent single-input single-output (SISO) system and thus the stability indices can be determined with just four sensitivity functions. This paper discusses the stability indices using sensitivity functions as SISO systems with parallel/conical mode treatment and/or side-by-side treatment, and as a MIMO system with using maximum singular value; the paper also highlights the differences among these approaches. In addition, a conversion from usual $x∕y$ axis form to forward/backward form is proposed, and the stability is evaluated in its converted form. For experimental demonstration, a test rig diverted from a high-speed compressor was used. The transfer functions were measured by exciting the control circuits with swept signals at rotor standstill and at its $30,000$ revolutions/min rotational speed. For stability limit evaluation, the control loop gains were increased in one case, and in another case phase lags were inserted in the controller to lead the system close to unstable intentionally. In this experiment, the side-by-side assessment, which conforms to the ISO standard, indicates the least sensitive results, but the difference from the other assessments are not so great as to lead to inadequate evaluations. Converting the transfer functions to the forward/backward form decouples the mixed peaks due to gyroscopic effect in bode plot at rotation and gives much closer assessment to maximum singular value assessment. If large phase lags are inserted into the controller, the second bending mode is destabilized, but the sensitivity functions do not catch this instability. The ISO standard can be used practically in determining the stability of the AMB system, nevertheless it must be borne in mind that the sensitivity functions do not always highlight the instability in bending modes.

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## Figures

Figure 1

Test rig for AMB stability evaluation

Figure 2

Free–free mode shapes

Figure 3

Control system block diagram

Figure 4

Open-loop transfer function of x-axis parallel system, −xp∕xp′, at standstill

Figure 5

Open-loop transfer function of x-axis conical system, −xc∕xc′, at standstill

Figure 9

Maximum singular value plot of sensitivity function matrix at standstill (x axis)

Figure 10

Maximum singular value of only off-diagonal element matrix in parallel/conical evaluation

Figure 11

Open-loop transfer function of x-axis parallel system, −xp∕xp′, at 30,000revolutions∕min

Figure 17

Sensitivity function in critical gain case at 30,000revolutions∕min (x axis, parallel/conical evaluation)

Figure 6

Nyquist plot at standstill: (a) parallel system −xp∕xp′; (b) conical system −xc∕xc′

Figure 7

Sensitivity function at standstill (x axis, parallel/conical evaluation)

Figure 8

Sensitivity function at standstill (x axis, side-by-side evaluation)

Figure 18

Sensitivity function in critical phase case at 30,000revolutions∕min (x axis, parallel/conical evaluation)

Figure 12

Open-loop transfer function of x-axis conical system, −xc∕xc′, at 30,000revolutions∕min

Figure 13

Open-loop transfer function of parallel forward system at 30,000revolutions∕min

Figure 14

Open-loop transfer function of parallel backward system at 30,000revolutions∕min

Figure 15

Open-loop transfer function of conical forward system at 30,000revolutions∕min

Figure 16

Open-loop transfer function of conical backward system at 30,000revolutions∕min

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