ISO 14839-3:2006

INTERNATIONAL ISO
STANDARD 14839-3
First edition
2006-09-15


Mechanical vibration — Vibration of
rotating machinery equipped with active
magnetic bearings —
Part 3:
Evaluation of stability margin
Vibrations mécaniques — Vibrations de machines rotatives équipées de
paliers magnétiques actifs —
Partie 3: Évaluation de la marge de stabilité




Reference number
ISO 14839-3:2006(E)
©
ISO 2006

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ISO 14839-3:2006(E)
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ISO 14839-3:2006(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Preceding investigation . 1
4 Outline of feedback control systems. 2
5 Measurement procedures . 9
6 Evaluation criteria. 11
Annex A (informative) Case study 1 on evaluation of stability margin . 13
Annex B (informative) Case study 2 on evaluation of stability margin . 25
Annex C (informative) Field data of stability margin . 28
Annex D (informative) Analytical prediction of the system stability. 32
Annex E (informative) Matrix open loop used for a MIMO system. 33
Bibliography . 35

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ISO 14839-3:2006(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 14839-3 was prepared by Technical Committee ISO/TC 108, Mechanical vibration and shock,
Subcommittee SC 2, Measurement and evaluation of mechanical vibration and shock as applied to machines,
vehicles and structures.
ISO 14839 consists of the following parts, under the general title Mechanical vibration — Vibration of rotating
machinery equipped with active magnetic bearings:
⎯ Part 1: Vocabulary
⎯ Part 2: Evaluation of vibration
⎯ Part 3: Evaluation of stability margin
Additional parts are currently in preparation.
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ISO 14839-3:2006(E)
Introduction
While passive bearings, e.g. ball bearings or oil-film bearings, are essentially stable systems, magnetic
bearings are inherently unstable due to the negative stiffness resulting from static magnetic forces. Therefore,
a feedback control is required to provide positive stiffness and positive damping so that the active magnetic
bearing (AMB) operates in a stable equilibrium to maintain the rotor at a centred position. A combination of
electromagnets and a feedback control system is required to constitute an operable AMB system.
In addition to ISO 14839-2 on evaluation of vibration of the AMB rotor systems, evaluation of the stability and
its margin is necessary for safe and reliable operation of the AMB rotor system; this evaluation is specified in
this part of ISO 14839, the objectives of which are as follows:
a) to provide information on the stability margin for mutual understanding between vendors and users,
mechanical engineers and electrical engineers, etc.;
b) to provide an evaluation method for the stability margin that can be useful in simplifying contract concerns,
commission and maintenance;
c) to serve and collect industry consensus on the requirements of system stability as a design and operating
guide for AMB equipped rotors.

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INTERNATIONAL STANDARD ISO 14839-3:2006(E)

Mechanical vibration — Vibration of rotating machinery
equipped with active magnetic bearings —
Part 3:
Evaluation of stability margin
1 Scope
This part of ISO 14839 establishes the stability requirements of rotating machinery equipped with active
magnetic bearings (AMB). It specifies a particular index to evaluate the stability margin and delineates the
measurement of this index.
It is applicable to industrial rotating machines operating at nominal power greater than 15 kW, and not limited
by size or operational rated speed. It covers both rigid AMB rotors and flexible AMB rotors. Small-scale rotors,
such as turbo molecular pumps, spindles, etc., are not addressed.
This part of ISO 14839 concerns the system stability measured during normal steady-state operation in-house
and/or on-site.
The in-house evaluation is an absolute requirement for shipping of the equipment, while the execution of
on-site evaluation depends upon mutual agreement between the purchaser and vendor.
This part of ISO 14839 does not address resonance vibration appearing when passing critical speeds. The
regulation of resonance vibration at critical speeds is established in ISO 10814.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 10814, Mechanical vibration — Susceptibility and sensitivity of machines to unbalance
3 Preceding investigation
The AMB rotor should first be evaluated for damping and stability properties for all relevant operating modes.
There are two parts to this assessment.
First, the run-up behaviour of the system should be evaluated based on modal sensitivities or amplification
factors (Q-factors). This concerns all eigen frequencies that are within the rotational speed range of the rotor.
These eigen frequencies are evaluated by the unbalance response curve around critical speeds measured in
a rotation test.
When the unbalance vibration response is measured as shown in Figure 1, the sharpness of each vibration
peak corresponding to eigen frequencies of the two rigid modes and the first bending mode is evaluated; this
is commonly referred to as Q-factor evaluation. These damping (stability) requirements for an AMB system
during run-up are covered by ISO 10814 (based on Q-factors), and are not the subject of this part of
ISO 14839.
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ISO 14839-3:2006(E)

Key
X rotational speed
Y vibration magnitude
Figure 1 — Q-factor evaluation by unbalance vibration response
The second part, which is covered by this part of ISO 14839, deals with the stability of the system while in
operation at nominal speed from the viewpoint of the AMB control. This analysis is critical since it calls for a
minimum level of robustness with respect to system variations (e.g. gain variations due to sensor drifts caused
by temperature variations) and disturbance forces acting on the rotor (e.g. unbalance forces and higher
harmonic forces). To evaluate the stability margin, several analysis tools are available: gain margin, phase
margin, Nyquist plot criteria, sensitivity function, etc.
4 Outline of feedback control systems
4.1 Open-loop and closed-loop transfer functions
Active magnetic bearings support a rotor without mechanical contact, as shown in Figure 2. AMBs are
typically located near the two ends of the shaft and usually include adjacent displacement sensors and
touch-down bearings. The position control axes are designated x , y at side 1 and x , y at side 2 in the radial
1 1 2 2
directions and z in the thrust (axial) direction. In this manner, five-axis control is usually employed. An example
of a control network for driving the AMB device is shown in Figure 3.

a)  Axial view b)  Rotor system
Key
1 AMB
2 sensor
a
Side 1.
b
Side 2.
Figure 2 — Rotor system equipped with active magnetic bearings
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ISO 14839-3:2006(E)

Key
a
1 mechanical plant rotor E excitation signal Sensor signal.
b
2 position sensor, expressed in V/m F AMB force, expressed in newtons Control signal.
b
c
3 AMB controller, expressed in V/V F disturbance force, expressed in Control current.
d
newtons
4 power amplifier, expressed in A/V
K current stiffness, expressed in newtons
5 electromagnet, expressed in N/A i
per ampere
6 AMB actuator
K negative position stiffness, expressed
s
7 negative position stiffness, expressed in N/m
in newtons per metre
8 AMB
x displacement, expressed in metres
Figure 3 — Block diagram of an AMB system
As shown in these figures, each displacement sensor detects the shaft journal displacement in one radial
direction in the vicinity of the bearing and its signal is fed back to the compensator. The deviation of the rotor
position from the bearing centre is, therefore, reported to the AMB controller. The controller drives the power
amplifiers to supply the coil current and to generate the magnetic force for levitation and vibration control. The
AMB rotor system is generally described by a closed loop in this manner.
The closed loop of Figure 3 is simplified, as shown in Figure 4, using the notation of the transfer function, G ,
r
of the AMB control part and the transfer function, G , of the plant rotor. At a certain point of this closed-loop
p
network, we can inject an excitation, E(s), as harmonic or random signal and measure the response signals,
V and V , directly after and before the injection point, respectively. The ratio of these two signals in the
1 2
frequency domain provides an open-loop transfer function, G , with s = jω, as shown in Equation (1):
o
Vs()
2
Gs()=− (1)
o
Vs()
1
Note that this definition of the open-loop transfer function is very specific. Most AMB systems have multiple
feedback loops (associated with, typically, five axes of control) and testing is typically done with all loops
closed. Consequently, the open-loop transfer function for a given control axis is defined by Equation (1) with
the assumption that all feedback paths are closed when this measurement is made. This definition is different
from the elements of a matrix open-loop transfer function defined with the assumption that all signal paths
from the plant rotor to the controller are broken. See Annex E for a more detailed discussion of this issue.
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ISO 14839-3:2006(E)
The closed-loop transfer function, G , is measured by the ratio as shown in Equation (2):
c
Vs()
2
Gs()=− (2)
c
E()s
The transfer functions of the closed loop, G , and open loop, G , are mutually consistent, as shown in
c o
Equations (3):
G G
o c
G = and G = (3)
c o
1+ G 1− G
o c
The transfer functions, G and G , can typically be obtained using a two-channel FFT analyser.
c o
The measurement of G is shown in Figure 4 a).
o

a)  Measurement of G b)  Measurement of G
o s
Key
G transfer function of the plant rotor
p
G transfer function of the AMB control part
r
E external oscillation signal
G open-loop transfer function
o
G sensitivity function
s
Figure 4 — Two-channel measurement of G and G
o s
4.2 Bode plot of the transfer functions
Once the open-loop transfer function, G , is measured as shown in Figure 5, we can modify it to the
o
closed-loop transfer function, G , as shown in Figure 6. Assuming here that the rated (non-dimensional) speed
c
is N = 8, the peaks of the gain curve at ω = 1, ω = 6 are distributed in the operational speed range so that
1 2
the sharpness, i.e. Q-factor, of these critical speeds are regulated by ISO 10814. This part of ISO 14839
evaluates the stability margin of all of the resulting peaks, noted ω = 1, ω = 6 and ω = 30 in this example.
1 2 3
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ISO 14839-3:2006(E)

Key
X non-dimensional rotational speed
Y1 gain, expressed in decibels. The decibel (dB) scale is a relative measure: − 40 dB = 0,01; − 20 dB = 0,1; 0 dB = 1;
20 dB = 10; 40 dB = 100.
Y2 phase, φ, expressed in degrees
N rated non-dimensional speed
a
Gain.
b
Phase.
Figure 5 — Open-loop transfer function, G
o
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ISO 14839-3:2006(E)

Key
X non-dimensional rotational speed
Y1 gain, expressed in decibels. The decibel (dB) scale is a relative measure: − 40 dB = 0,01; − 20 dB = 0,1; 0 dB = 1;
20 dB = 10; 40 dB = 100.
Y2 phase, φ, expressed in degrees
N rated non-dimensional speed
a
Gain.
b
Phase.
Figure 6 — Closed-loop transfer function, G
c
4.3 Nyquist plot of the open-loop transfer function
Besides the standard display in a Bode plot (see Figure 5), the open-loop transfer function G (jω) can also be
o
displayed on a polar diagram in the form of magnitude ⏐G (jω)⏐and phase of G (jω) as shown in Figure 7
o o
(note the dB polar diagram employed). Such a diagram is called the Nyquist plot of the open-loop transfer
function. Since the characteristic equation is provided by 1 + G (s) = 0, the distance between the Nyquist plot
o
and the critical point A at (− 1, 0) is directly related to the damping of the closed-loop system and its relative
stability. Generally, it can be stated that the larger the curve’s minimum distance from the critical point, the
greater is the system stability.
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ISO 14839-3:2006(E)

a)  ω , ω and ω
1 2 3

b)  ω enlarged
3
a
The decibel (dB) scale is a relative measure: − 40 dB = 0,01; − 20 dB = 0,1; 0 dB = 1; 20 dB = 10; 40 dB = 100.
Figure 7 — Nyquist plot of the open-loop transfer function (dB polar diagram)
The enlargement of this Nyquist plot on a linear polar diagram is drawn in Figure 8, focusing on the critical
point (–1, 0). The shortest distance measured from the critical point is indicated by l = D , where a circle
AB min
of radius D centred at (− 1, 0) is the tangent to the locus. For this example in Figure 8, the gain margin is
min
the distance, l (an intersection between the locus and the real axis), and the phase margin is the angle, φ,
AG
(between the real axis and a line extending from the origin to the intersection, P, between the locus and the
unit circle centred on the origin). In this example, since D < l and D < l , the shortest distance, D ,
min AP min AG min
is a more stringent evaluation criterion compared with the gain margin and the phase margin.
NOTE It can be shown that the shortest distance is always a more conservative measure of stability robustness than
either gain or phase margin.
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ISO 14839-3:2006(E)

Figure 8 — Nyquist plot of the open-loop transfer function (linear polar diagram)
4.4 Sensitivity function
Note that, with the Nyquist plot, interest is focused on the distance between G (jω) and the point (− 1, 0). That
o
is, we want to know how small the value of 1 + G (jω) can be. Alternatively, one can ask how large the inverse
o
of this function can be. If a small minimum value of 1 + G (jω) is undesirable, then so is a large maximum
o
value of 1/[1 + G (jω)]. This latter expression defines the sensitivity function G with s = jω, as shown in
o s
Equation (4):
1
Gs()= (4)
s
1(+Gs)
o
The maximum value of G (jω) is the inverse value of the minimum distance from the point (− 1, 0) to the locus
s
of G (jω) in the Nyquist plot. The corresponding Bode plot of the sensitivity function is shown in Figure 9.
o
The sensitivity function offers two advantages over evaluation of the minimum distance on the Nyquist plot.
First, it is generally easier to construct the maximum magnitude than it is to find the minimum distance. Indeed,
the usual computational method for finding the minimum distance on the Nyquist plot is to find the maximum
value of the sensitivity function and then invert it. Second, measurement of the sensitivity function is relatively
simple. Referring to Figure 4 b), at a certain point of this closed-loop network we can inject an excitation, E(s),
as a harmonic or random signal at an injection point, E, and measure the response signal, V , directly behind
1
the injection point. The ratio of these two signals in the frequency domain provides the sensitivity function, G ,
s
as shown in Equation (5):
Vs()
1
Gs()= (5)
s
E()s
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ISO 14839-3:2006(E)

Key
X non-dimensional rotational speed
Y sensitivity gain, expressed in decibels. The decibel (dB) scale is a relative measure: − 20 dB = 0,1; − 10 dB = 0,315;
0 dB = 1; 10 dB = 3,15.
N rated non-dimensional speed
Figure 9 — Bode plot of the sensitivity function, G
s
5 Measurement procedures
5.1 Transfer functions
In the first step of evaluating the stability margin, one of the transfer functions, G or G , is directly measured
o s
with respect to every closed loop. The generalized controller layout is described in Figure 10, where all
displacement signals of the five axes are fed into a controller to output the command which determines the
magnetic force on the bearings.
NOTE Special controller layouts are shown in Annex A.
In Figure 10, a certain point is selected as the injection point, E, and the open-loop transfer function is
measured according to Equation (1), while all other possible injection points are closed. Once the open-loop
transfer function, G , is measured, the resulting data should be used to form G in accordance with
o s
Equation (4).
The sensitivity function, G , can also be measured directly in accordance with Equation (5).
s
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ISO 14839-3:2006(E)

Key
1 central processing unit (CPU)
Figure 10 — Generalized controller layout
5.2 Stability index
When measuring the open-loop transfer function of one control axis, all of the control axes are closed loops.
By repeating this measurement step by step for each control axis, a set of the open-loop transfer functions is
measured and transferred to sensitivity functions. Otherwise, each sensitivity function is directly measured.
In case of a five-axis control, a total of five sensitivity functions is finally obtained to be evaluated.
The open-loop transfer function or the system’s sensitivity function is measured at rotor standstill and/or
nominal speed but over the maximum frequency range starting from zero.
In general, there is no need to specify an upper frequency limit since the amplitude of the open-loop transfer
function of all technical systems decreases for high frequencies (see Figure 5), rendering both the sensitivity
function and the distance margin 1 (0 dB) for all frequencies above a certain limit (see Figures 7 and 9). This
automatically ensures stability of the AMB system at very high frequencies. However, in practice for the upper
limit, filters are recommended for the signal processing. In this part of ISO 14839, the maximum frequency
f , as given in Equation (6), is set at the larger of
max
a) three times the rated speed, indicated as 3×, or
b) a maximum frequency of 2 kHz:
f=×max(3 , 2 kHz) (6)
max
It is noted that, in digitally controlled AMB systems, it is necessary that this maximum frequency be restricted
to frequencies below the Shannon frequency (half of the sampling frequency).
From these measured sensitivity functions of each axis in the frequency domain for 0uuff , the index to
max
be evaluated is obtained from the following relationship with ω = 2πf, as given in Equation (7):
⎡⎤
GG= max (jω) for 0uuff (7)
max
s,max s max
⎣⎦
i
where i = 1, ., the total number of control axes.
Equation (7) generally states that the system’s overall rating is determined as the worst rating of any of the
transfer functions measured for all five transfer functions individually.
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ISO 14839-3:2006(E)
5.3 Measurement conditions
The rotor system is required to behave stably for normal excitations and expected changes of the operating
conditions. In-house measurements are required for shipping the equipment, while on-site measurements may
be made. The measurements shall be taken under the following conditions:
a) at standstill in-house;
b) at maximum continuous rated speed or nominal power rating in-house or on-site. This location may be
decided by mutual agreement between the purchaser and the vendor.
6 Evaluation criteria
6.1 Criterion I
For evaluation of the stability margin, zone limits are given in Table 1. The definition of each stability zone is
determined by adapting the guidelines of ISO 7919-1.
⎯ Zone A: The sensitivity functions of newly commissioned machines normally fall within this zone.
⎯ Zone B: Machines with the sensitivity functions within this zone are normally considered acceptable
for unrestricted long-term operation.
⎯ Zone C: Machines with the sensitivity functions within this zone are normally considered
unsatisfactory for long-term continuous operation. Generally, the machine may be operated
for a limited period in this condition until a suitable opportunity arises for remedial action.
⎯ Zone D: The sensitivity functions within this zone are normally considered to be sufficiently severe to
cause damage to the machine.
Table 1 — Peak sensitivity at zone limits
Zone Peak sensitivity
Level Factor
A/B 9,5 dB 3
B/C 12 dB 4
C/D 14 dB 5
As an example of this evaluation, the sensitivity function of Figure 9 is redrawn in the Bode plot of Figure 11
with zone limit values indicated by Table 1. As can be seen from Figure 11, all peak values of the sensitivity
function are within zone A.
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ISO 14839-3:2006(E)

Key
X non-dimensional rotational speed
Y sensitivity gain and zone limits, expressed in decibels
A, B, C, D stability zones
N rated non-dimensional speed
Figure 11 — Evaluation of the stability margin of G (jω)
s
6.2 Criterion II
This criterion provides an assessment of the change in the stability margins measured periodically from the
average values. A significant change in magnitudes of the stability margin can occur that would require
remedial action even though zone C of Criterion I has not been reached. Such changes can be progressive
with time or instantaneous and can point to incipient damage or some other irregularity.
Criterion II is specified on the basis of the change in magnitude of the stability margin occurring under steady-
state operating conditions. When criterion II is applied, it is essential that the measurements being compared
are taken under approximately the same machine operating conditions. Significant changes from the normal
magnitudes should be regulated to less than 25 % of the upper boundary value of zone B, as defined in
Table 1, because a potentially serious fault can be indicated. When a change in the magnitudes is beyond this
regulation, the reason for the change shall be determined, and appropriate remediation steps must be planned.
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ISO 14839-3:2006(E)
Annex A
(informative)

Case study 1 on evaluation of stability margin
A.1 Test rotor
A test rig and the corresponding rotor is shown in Figures A.1 and A.2. The rotor specification is listed in
Table A.1.
The eigen modes and eigen frequencies under free-free conditions are shown in Figure A.3.

Key
1 thrust AMB
2 radial AMB
3 vacuum chamber
4 flexible rotor
5 motor
Figure A.1 — Test rig
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ISO 14839-3:2006(E)
Dimensions in millimetres

Key
1 thrust AMB rotor
2 radial AMB rotor
3 motor rotor
a
Nodal points.
b
Shaft element length, L.
c
Shaft element diameter, D .
0
Figure A.2 — Structure of the flexible rotor
Table A.1 — Specification of the rotor
Mass 31,4 kg
Shaft diameter 37 mm
Total length 1316 mm
Rated speed 250 rev/s
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ISO 14839-3:2006(E)

Key
1 plane of AMB
2 plane of sensor
Figure A.3 — Eigen modes and eigen frequencies, N , of the flexible rotor
Ci
As shown in the critical speed map in Figure A.4, the intersection between the eigen frequency curve and the
AMB stiffness curve indicates the critical speeds of this rotor, noted as N . In the range of operational speeds
Ci
up to 250 rev/s, four critical speeds are laid out, designated as N and N indicating the rigid modes and
C1 C2
N and N corresponding to the first two bending modes.
C3 C4

Key
X stiffness, expressed in newtons per metre
Y natural frequency, hertz
a c
Free-free. AMB.
b d
Pinned-pinned. Rated speed.
Figure A.4 — Critical speed map of the flexible rotor
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ISO 14839-3:2006(E)
A.2 AMB controllers
Two type of controllers are prepared; “PID” means proportional, integral and differential actions.
a) Type 1: Controller transfer function Gs=×PID PBF (phase bump filter):
( )
r
The controller layout is decentralized as shown in Figure A.5. Bode plots for the evaluation of the stability
margin are shown in Figure A.6.

Figure A.5 — Decentralized control
The Bode plot of the open-loop transfer function of the rotor at standstill, G , was measured as shown in
o
Figure A.6 a) by using a 2-channel FFT analyser. This open-loop transfer function is described in the Nyquist
in dB polar diagram in Figure A.6 b), which in Figure A.6 c) shows the behaviour of the shaded range around
the frequency 645 Hz corresponding to the N mode. The orbit is very close to the critical point (− 1, 0), so
C6
that this AMB system is not so stable.
These measured data were rearranged t
...