A New Calculation Method for Bode and Nyquist Diagrams of Rotor
Startup or Shutdown and Its Application *
First A. Xining Zhang, Second B. Jili Wu , Third C. Jinliang Guo , and Fourth D. Jinjie Xu

Abstract—In order to prepare well for drawing Bode and
Nyquist diagrams of vibration signals collected from rotor
startup or shutdown process, a new Hilbert transform based
calculation method is proposed. Experiment results show that
the proposed method can smooth Bode and Nyquist diagrams,
has more accuracy in amplitude and phase calculation, and also
can not be affected by harmonic components of rotor vibration.
Verifying tests on dynamic balancing experiment reveal that
vibration values of rotor greatly reduce after balancing by using
the information extracted form Bode and Nyquist diagrams.
The vibration signal collected contains multiple frequency
components,only fundamental frequency can reflect rotor
balance, and its harmonic components mainly refelt the other
fault of rotor,such as misalignment and crack. so the Bode and
Nyqusit diagrams mentioned in the paper are just computed
from fundamental component.Theoretically, we assume that
the unbalanced mass will not change during startup and
shutdown process, so the vibration response of fundamental
component is continuous, then Bode and Nyqusit diagrams
will be smooth curves.
I. INTRODUCTION
Generally to draw Bode and Nyquist diagrams, the
vibration signals collected from startup or shutdown process
could be preprocessed through tracking filters to keep
fundamental frequency component and phase-lock-loop
multiplier circuit to collect certain samples in each circle of
rotor. In above processing, key phase signal is adopted in
determining the rotor revolving frequency. Because the
tracking filter and data sampling of current circle mainly
depend on the period of previous circle, these hardware based
methods work very well when the rotor speed changes
smoothly[4-6]. But when the rotor speed changes sharply,
such as the rotor getting through critical speed area, the
performance could be less effective. And these methods which
are highly hardware dependent are comparatively expensive.
In the process of startup or shutdown, rotating machines
experience a non-steady operating condition. Due to the
variational speed , it is difficult to extract unbalancing features
during this process. Only a few imbalance features can be
obtained from stationary vibration signals. In fact, vibration
signals corresponding to startup or shutdown process contain
richer imbalance information, which can be extracted and
utilized for dynamic balance[1].
Bode and Nyquist diagrams are significant monitoring
and diagnosis methods used in rotating machinery[2-3].The
analysis of Bode and Nyquist diagrams is particularly
important for startup or shutdown of rotating machinery. In the
non-stationary startup or shutdown process, the speed of
rotating machinery experiences a increasing or inverse
changing process, meanwhile the frequency of the centrifugal
force produced by imbalance mass also experiences the same
process.The fundamental component of vibration in startup or
shutdown process is exactly corresponding to the vibration
excited by centrifugal force. So Bode and Nyquist diagrams
can well describe the vibration feature excited by centrifugal
force. Therefore, balancing condition of machine can be
monitored and analyzed by Bode and Nyquist diagrams
analysis. This is particularly important for the machine that
can not startup after shutdown.
*Resrach supported by ABC Foundation.
F. A. Xining Zhang is with Mechanical Engineering Department, Xi’an
Jiaotong University, No.28, Xianning West Road, Xi’an, Shaanxi, 710049,
P.R. China ( corresponding author to provide phone: 029-82669053; e-mail:
zhangxining@ mail.xjtu.edu.cn).
S. B. Jili Wu is with Mechanical Engineering Department, Xi ’an
Jiaotong University, No.28, Xianning West Road, Xi’an, Shaanxi, 710049,
P.R. China ( e-mail: wujili@stu.xjtu.edu.cn).
T. C. Jinliang Guo is with Mechanical Engineering Department, Xi’an
Jiaotong University, No.28, Xianning West Road, Xi’an, Shaanxi, 710049,
P.R. China ( e-mail: 871196789@qq.com).
F. D. Jinjie Xu is with Mechanical Engineering Department, Xi’an
Jiaotong University, No.28, Xianning West Road, Xi’an, Shaanxi, 710049,
P.R. China ( e-mail: 362948602@qq.com).
In order to solve those problems, two traditional methods
are introduced, which give priority to software processing
instead of hardware. In the first traditional method, vibration
signals are collected only at a number of predetermined speeds
during rotor startup or shutdown process. Then Bode and
Nyquist diagrams are drawn after amplitude and phase
information of fundamental frequency of each vibration signal
record are extracted with Fourier transform. The Bode and
Nyquist diagrams drawn by this method not only suffer from
low quality of smoothness of amplitude curve and phase curve,
but also suffer from the influence of harmonic frequency
components.
In the second traditional method, vibration signals
collected from startup or shutdown process are acquired
continuously. Bode and Nyquist diagrams are drawn after
frequency, amplitude and phase information of vibration
signals corresponding to each rotor revolution are extracted
directly in time domain. Though more smoother Bode and
Nyquist diagrams can be obtained by this method than the
former one, but the huge calculation volume and harmonic
components influence make it infeasible in practice. In order
to eliminate the harmonic components influence, Empirical
Mode Decomposition(EMD) is introduced to extract the
fundamental frequency component from the vibration signal
to draw hologram Bode diagram [7-8]. The method is only
effective for small accelerate range rotating machinery. In case
of large accelerate range, for example, the fundamental
frequency component of startup vibration is often broken into
many small segments distributed in different intrinsic modes.
Complete fundamental frequency component of startup
process still can’t be obtained.
A new approach for complete fundamental frequency
component extracting from vibration signals collected from
startup or shutdown process is presented in the paper. The
application of Bode and Nyquist diagrams drawn by the
proposed method in rotor dynamic balancing is given.
Satisfactory balancing result is achieved, and rotor vibrations
are greatly decreased.
x1 (t )  Ae  nt sin( n2  n 2 t   ).
The particular solution x2 (t ) which reveals steady
vibration can be written as (3). The forced steady vibration
during Startup or shutdown process is mainly considered in
the paper.
x2 (t )  X sin( t   ) .
The amplitude of vibration response
written as follows.
X=
II. THE VIBRATION RESPONSE OF UNBALANCED ROTOR
STARTUP OR SHUTDOWN PROCESS
Bode and Nyquist diagrams are originally proposed to
describe the frequency response characteristics of the system
in automatic control area. Rotor is a typical second-order
vibration system. Once a rotor is under the imbalance state,
forced vibration will be excited by imbalance mass. Therefore,
the fundamental frequency component of forced vibration in
startup or shutdown process is used to describe the balancing
condition of a rotor. And in the forced vibration, the phase of
vibration vector(vibration response) always lags behind the
unbalanced vector(unbalanced mass), when a rotor is at the
speed of critical speed, the lagging phase is up to 90o,
according to this feature, it is very useful to confirm the
unbalanced angle by vibration phase in the Bode diagram at
critical speed. So only the output(vibration response) is
considered, the input(unbalanced mass) is unknown and to be
solved, the output caused by the input can be measured by
sensors. Therefore some features of the output could be
recognized.
(2)
X can be further
Xo
(1  r )  (2 r )2
Xo 
2 2
F meo 2
=
k
k
  arctan
(3)
2 / n
1  ( / n )2
(4)
(5)
(6)
r   / n is frequency ratio,   c 2m
is damping ratio, X o is the displacement due to the static
force F .  is the phase of vibration response.
Where, Where,
X and  are the function of frequency ratio and damping
ratio.When  =0.1 and F / k =1 ,Fig.2 shows characteristics
curves of amplitude-frequency and phase-frequency in Bode
diagram.
Fig.1 shows a rotor system[9]. If only the horizontal
direction vibration is considered, vibration of the rotor system
could be described as(1).
Figure 2. Bode diagram of vibration response of rotating frequency.
Figure 1. Schematic of imbalance rotor.
m  x  c  x  k  x  me 2  sin(  t )
(1)
Where, the mass of rotor is m, e is eccentric distance , c
is damping coefficient, k is rotor stiffness, F is static force.
The solution of (1) includes two parts:
x1 (t ) and x2 (t ) . A
general solution x1 (t ) which reveals damped transient
vibration can be written as
III. THE PRINCIPLE OF HILBERT TRANSFORMATION BASED
BODE AND NYQUIST DIAGRAM CALCULATION
The key of drawing Bode and Nyquist diagrams is to
calculate the amplitude and phase of fundamental vibration
components at each speed in the Startup or shutdown process.
As the amplitude and the corresponding phase of transient
vibration signal vary with time in the process. So Hilbert
transformation is introduced to acquire them. For the sake of
convenience, the amplitude curve and the phase curve against
time, which are calculated by Hilbert transformation, are
named as Hilbert amplitude curve and Hilbert phase curve of
original vibration signal.Averaging the upper and lower
envelop curves could weaken the effects of harmonic
frequency, a another better method to eliminate the effects of
harmonic frequency is the fractional fourier transform and a
narrowband filtering in the fractional fourier domain[10].
Then the average curve of the upper and lower envelope
curves is adopted as corrected Hilbert amplitude curve.
Hilbert phase curve of vibration signal changes as a
function of time, While the phase in Bode and Nyquist
diagram is relative to the phase of keyphasor signal. So where
needs a keyphase signal to realize the phase conversion.
Specifically, on Hilbert phase curve, Phase values extracted
relative to the key pulse of keyphasor signal are used to get a
corrected change curve of phases by curve interpolation, the
phases on the curve is relative to initial phase value of
keyphasor signal at each sampling time. So far, the calculation
steps of Bode and Nyquist diagrams can be summarized as
follows:
and 2. Horizontal and vertical vibrations at section B are
measured by current eddy sensors 3 and 4. Two plates on the
rotor shaft which are used for balance discs are marked as C
and D. There are 24 evenly distributed screwed holes on each
plate, which are used for adding balancing weight. Keyphasor
signal is collected by eddy current sensor 5. Sensors 1 and 3
are defined as the direction X, while direction of sensors 2 and
4 are defined as Y. Acc/Dec rate is controlled by speed
controller according to the setting value. The vibration and key
phase signals during Startup or shutdown process is collected
by tape recorder of Sony PC208AX.
1) Collection of vibration signal and keyphasor signal
during Startup or shutdown process of investigated rotor.
2) Determine the times of each phase pulse on keyphasor
signal.
3) Performing Hilbert transform of the vibration
signal x (t ) , and constructing analytic function z (t ) .
z(t )  x(t )  jy(t )  a(t )e j(t ) .
(7)
4) Calculation of Hilbert amplitude curve a (t ) and Hilbert
phase curve  (t ) .
a(t )  x(t ) 2  y (t ) 2 .
(t )  arctan
y (t )
.
x(t )
(8)
(9)
5) Performing the correction of Hilbert amplitude curve by
averaging the upper and lower envelope curves of Hilbert
amplitude curve obtained in step 4.
6) Performing the correction of Hilbert phase curve by
extracting the phase values on Hilbert phase curve
corresponding to the times of key phase pulse on keyphasor
signal, and phase interpolating at each sampling times.
7) Draw Bode and Nyquist diagrams with corrected
Hilbert amplitude curve and Hilbert phase curve.
IV. EXPERIMENTS AND DATA ANALYSIS
The Startup or shutdown experiments are carried out on
the rotor test rigs of RK4, which is manufactured by Bently
Corporation. The structure of RK4 is shown in Fig.3. A
measurement section A is at motor end of rotor, while
measurement B is at free end of rotor. Horizontal and vertical
vibrations of section A are measured by eddy current sensors 1
Figure 3. Structure of Bently RK4.
In the experiment, the rotor speed-up range is set between
300(r/min) and 4500(r/min). The vibration signals and
keyphase signal are collected during the speed-up process in
experiment. Signal sampling frequency of each channel is
24000(Hz).
The vibration signal and keyphase signal coming from
sensors 1 and sensor 5 respectively are selected and analyzed
with the purposed method. The Bode and Nyquist diagram of
fundamental frequency component of vibration signal are
shown in fig.4. For the convenience of comparison, fig.5 gives
Bode and Nyquist diagrams drawn by the conventional
method with signals collected discontinuously at certain
predetermined speeds. It is easy to see that Bode and Nyquist
diagram drawn by the proposed method is much smoother.
Especially, the resolution gets much higher through the critical
speed. Besides, Bode and Nyquist diagram calculated with the
proposed method is free of amplitude error and phase error
caused by the average effect of Fourier transform in
computation process.
(a) Bode diagram.
(a) Bode diagram.
(b) Nyquist diagram.
Figure 4. Bode and Nyquist diagrams drawn by the proposed method.
Another comparison of Bode and Nyquist diagrams drawn
by the proposed method and traditional method is illustrated in
fig.6 and fig.7. The Bode and Nyquist diagrams shown in fig.6
are drawn by the proposed method. Bode and Nyquist
diagrams shown in figure 7 are drawn by the second
traditional method. In fig.7(b), it is obvious that it exists a
protruding point caused by the interference of second
harmonic frequency component. The above interference does
not appeared in fig.6(b).
(b) Nyquist diagram.
Figure 5. Bode and Nyquist diagrams drawn by the first traditional method
with signals at certain speeds.
Experimental analysis indicates that the proposed Hilbert
based Bode and Nyquist diagrams calculation method is low
in computation volume comparing with the traditional method,
it can effectively extract the information from transient
vibration signal, errors both from average effect of Fourier
transform in computation and the interference of second
harmonic frequency component are suppressed, and accurate
Bode and Nyquist diagrams are obtained.
(a) Bode diagram.
V. APPLICATION OF BODE AND NYQUIST DIAGRAMS IN
DYNAMIC BALANCING
As Bode and Nyquist diagrams directly show the
balancing condition of rotor, the information on Bode and
Nyquist diagrams can be utilized for rotor balancing. The
following experiments are carried out on RK4. Sensor
installation and data acquisition system in the experiment are
same as those in part 3 of the paper.
(b) Nyquist diagram.
Figure 6. Bode and Nyquist diagrams drawn by the proposed method.
Unbalancing mass EC  1.2 g90() and ED  1.2 g90()
are artificially introduced on disk C and disk D to simulate
unbalance in experiment. Bode diagram of the Startup process
is drawn with the proposed method. Low-speed balancing
method for flexible rotor is employed, and the balancing
weights calculated on disks C and D are EPC  1.63g271()
and EPD  1.54 g273() respectively. Obviously, the calculated
balancing weights is only a little bigger than the artificially
introduced weights in magnitude. The angles of balancing
weights are almost right in the opposite positions of the
artificially introduced weights. Vibrations are collected after
balancing added weights. Fig.8 is the comparison between
original vibration and vibration after balancing. It is obvious
that rotor vibration reduces greatly after balancing with
weights calculated according to Bode and Nyquist diagrams.
Figure 8. Comparison of vibration before and after balancing.
VI. CONCLUSION
(a) Bode diagram.
The paper propose a Hilbert transform based Bode and
Nyquist diagram calculating method, which is used to process
vibration signals of rotor Startup or shutdown. The principle
and the calculation steps of the method are presented in the
paper. The performance of the method is verified by
experiment. At last, one balancing example is given, in which
Bode diagram calculated by the proposed method is employed
in balancing weights calculation. Then main conclusions can
be summarized as follows.
1) The proposed method can extract the amplitude curve
and phase curve fundamental frequency component from
vibration signals, and It can be used to calculate Bode and
Nyquist diagrams with the vibrations in Startup or shutdown
process.
(b) Nyquist diagram.
Figure 7.
Bode and Nyquist diagrams drawn by the second traditional
method.
2) The results of principle study and experiment analysis
indicate that the proposed method is insusceptible to Errors
both from average effect of Fourier transform in computation
and the interference of second harmonic frequency component.
So accurate Bode and Nyquist diagrams can be obtained.
3) The proposed method has the advantageous of low
computation volume comparing with the traditional method.
4) Application of Bode and Nyquist diagrams in rotor
balancing suggests that Bode and Nyquist diagrams calculated
by the proposed method are correct and can be used in rotor
dynamic balancing and balance monitoring.
ACKNOWLEDGMENT
The authors gratefully acknowledge the contribution of
State Key Laboratory for Manufacturing Systems Engineering,
the project of National Key Basic Research Program
(2009CB724405), the project of National Natural Science
Foundation of China(51275379) and reviewers’ comments.
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