INTRODUCTION
The balancing of flexible rotors is one of the pivotal techniques for highspeed rotating machinery in modern industry. In general, high speed balancing procedure must be applied for flexible rotors after lowspeed balancing, because the balance situation of rotor in low speed has been changed in high operating speed. These high speed balancing methods for flexible rotors fall into two categories^{[13]}: modal balancing and influence coefficient balancing. The traditional lowspeed rigidrotor balancing is also applied for flexible rotor, but it is only a prelude to later flexiblerotor balancing. This is done to remove the gross unbalances first before attempting to approach the first critical.
The intention of this study is to propose a new lowspeed balancing method
for highspeed rotating machinery running at the speed between the first and
second critical speeds. It is not rigidrotor balancing, but the extension of
flexiblerotor balancing methods. The first two modal components of unbalance
can be balanced simultaneously at the speed below the first critical speed.
This method is named as LowSpeed HoloBalancing (LSHB), by which the satisfactory
results can be obtained as that at high speed. The rotor needs not run at high
speed or critical speeds to achieve satisfactory balance. In this method, the
threedimensional holospectrum is used to describe the vibration response of
rotor unbalance. It is constructed from vibration information of rotor system
in all bearing sections based on the multiple sensor fusion in data layer. Since
the frequency, amplitude and phase information are fully utilized, the threedimensional
holospectrum can increase the balancing accuracy and efficiency. The effect
of new method was validated by the experiments on a flexible rotor test rig.
HOLOSPECTRUM TECHNIQUE
Description of unbalance responses: In most traditional balancing methods, only the vibration information in one measuring direction is used. This is based on the assumption of equal rigidity in different circumferential directions of rotorbearing system. The errors would occur when the rigidity is different. From the viewpoint of information fusion, it is quite desirable to examine the vibration in a bearing section as a whole, not on individual measuring points. The Initial Phase Point (IPP) on holospectrum effectively fuses information from two sensors in one rotor section and can entirely describe the vibration behavior of rotor in the measuring section^{[4,5]}. Suppose that:

Fig. 1: 
The initialphase point (IPP) on the 1x ellipse 

Fig. 2: 
Configuration of balancing rig x15 
are the synchronous responses of signals picked up from two mutually perpendicular
directions X and Y. Equation (1) can be regarded as the equation
of rotor synchronous rotating orbit or 1X ellipse (Fig. 1).
We define the IPP as the point on first harmonic frequency ellipse, where the
key slot on the rotor locates straightly opposite to the phaser. The initialphase
point (IPP) of holospectrum is defined as
Relationship between initial phase point and trial weights: The experimental rotor rig, which simulates unbalance behavior, is shown in Fig. 2. The numbers 1 to 4 denote the proximity probes measuring the radial vibrations of the rotor in two sections A and B, 5 is the phaser. Both C and D are the two balancing discs. The first critical speed of the rotor is about 2700 rpm and the operating speed of the rotor is about 4600 rpm.
The experiment was held as follows:
1. 
Adding the trial weights on two discs C and D to simulate
the initial force unbalances: T_{C} = T_{D} = 1.0gp0°
. Then, run up to 4600 rpm, measured the rotor original vibrations in both
X and Y directions and drew 3Dholospectra. 
2. 
Changing the mounting angles of trial weights from 0 to 360° with
equal intervals 45°, measured the vibration and got 8 groups of 3Dholospectra.
The phases of IPPs drawn in a plot are shown in Fig. 3a,
which shows the linear relationship between the initial phases and the mounting
peripheral angle of trial weights. 

Fig. 3: 
Relationship between Initial Phase Point and trial weights:
a) Linear relationship between the initial phases of vibrations and the
mounting angles of trail weights; b) Linear relationship between the modulus
of IP vectors and the magnitudes of trail weights 

Fig. 4: 
The decomposition of the 3dimensional holospectra: a) the
original unbalance response under the speeds of 4600 rpm; b) the decomposed
force components at same speeds; c) the decomposed couple components 
3. 
Changing the magnitudes of trial weights from 0.2 to 1.6 g
with equal interval 0.2 g, measured the vibration and got eight groups of
3Dholospectra. The moduli of IP vectors keep the linear relationship with
the trial weights, as shown in Fig. 3b. 
Decomposition of 3Dholospectrum: A 3dimensional holospectrum is composed of 1X ellipses with IPPs, IP vectors and generating lines connecting corresponding sampling points around all 1x ellipses. Such a 3dimensional holospectrum can provide us full information of rotor vibration simultaneously in all bearing sections as a whole. Figure 4a shows a 3dimensional holospectrum, which is composed of 1x ellipses in two bearing sections. The unbalance response expressed as a 3dimensional holospectrum can be decomposed into the force and couple responses^{[6]}. Parallel generating lines indicate force unbalance (Fig. 4b) and the intersecting lines indicate the couple unbalance (Fig. 4c).
VARIATION RULES OF UNBALANCE RESPONSES
Assuming the influence of higher modal components can be neglected equation
and only the first two low modal components of unbalance will be considered,
the unbalance response at the operating speed Ω without damping effects
can be rewritten^{[1,2] }
where, ω_{1 }is the first critical speed, ω_{2} is the second critical speed, ω is the speed of lowspeed balancing below the first critical speed ω_{1}, Ω is the operating speed between ω_{1} and ω_{2}, is force unbalance response, is coupling unbalance response^{[6]}. As above, get The phase change of force unbalance component is
The phase change of coupling unbalance component is
The amplitude change of force unbalance component is
The amplitude change of coupling unbalance component is
When there exists the damping effect, the phase of force unbalance component inverses approximately180° and the phase change of couple unbalance component is approximately equal to 0°, which are consistent with experimental results as shown in Fig. 5.

Fig. 5: 
Bode diagram of unbalance response in the A plane: a) Force
component; b) couple component 
LOWSPEED HOLOBALANCING OF A FLEXIBLE ROTOR
As discussed above, it can be seen that when δ_{f}, δ_{c}, r_{f} and r_{c }are all known, the unbalance response at high operating speed can be deduced from that at low speed below the first critical speed, according to equation (47). Thus, the rotor balancing does not have to be operated at high operating speed. Based on it, a new balancing method called as ISHB, is proposed for balancing flexible rotor. In this section, we will discuss the implementation process and validity of the new method in details through a balancing case.
The configuration of experimental rotor rig is shown in Fig. 2. The operating speed of rotor is Ω = 4600 rpm and the balancing speed is ω = 1900 rpm.
The balancing steps are as follows:
1. 
At one rundown stage, the vibrations caused by the original
unbalance are measured at speeds both Ω and ω. From equation (2),
get R^{Ω}(s_{A}), R^{Ω}(s_{B}),
R^{ω}(s_{A}) and R^{ω}(s_{B}). 
2. 
The trial weights are added on two balancing discs to simulate the force
unbalance, T^{1}_{C} = T^{1}_{D} = 1.0gp270°.
Rotor vibrations including R^{ω}_{1} (s_{A})
and R^{ω}_{1} (s_{B}) are measured at speed
ω and then the test weights T^{1}_{C} and T^{1}_{c}
are taken off. 
3. 
The trial weights are added on two balancing discs to simulate the couple
unbalance in the second run: T^{2}_{C} = 1.0 gp45°,
T^{2}_{D} = 1.0 gp225°. Rotor vibration in R^{ω}_{2}
(s_{A}) and R^{ω}_{2} (s_{B}) are measured
at speed ω and then the test weights T^{2}_{C} and
T^{2}_{C} are taken off. 
4. 
Calculating δ_{f} and r_{f}: through the decomposition
of 3Dholospectrum, R^{Ω}(s), R^{ω}(s) can be
decomposed as equation (3). Inserting the decomposed results into equation
(3) and (6) gives δ_{f} = 145.7° and r_{f} = 1.47. 
5. 
Calculating δ_{c} and r_{c}: δ_{c} =
9.6° and r_{c} = 5.0. 
6. 
Calculating the vibration response of the unit force unbalance weights
at the speed ω 
7. 
Calculating the vibration response of the unit couple unbalance
weights at the speed ω 
8. 
The correcting weights on two balancing discs for force unbalance
can be calculated 
Similar to equations (5.4), the correcting weights for couple unbalance can be calculated

Fig. 6: 
Results of Lowspeed holobalancing 
Table 1: 
Effects of Lowspeed holobalancing 

The total correcting weights can be calculated, P_{c} = 1.19gp104.0° and P_{D} = 0.57gp280.9. Adding the correcting weights P_{C }and P_{D}, then running up to 4600 rpm and measuring the rotor residual vibrations R’(s_{A}) and R’(s_{B}). The measuring results in detail are listed in Table 1. The vibration responses expressed by the 3Dholospectra, before and after correction by the Lowspeed holobalancing method, are shown in Fig. 6. The thin lines represent the original vibrations and the bold lines represent the residual vibrations. Using this Lowspeed holobalancing technique, vibration levels of an experimental rotor rig were successfully reduced to as much as 50% of their original levels. It is evidential that the correction weights are reasonable and the new balancing method is effective.
CONCLUSION
This study presents a new balancing method, named lowspeed holobalancing. It is an extension of flexible rotor balancing method, but avoids test runs at high operating speed or critical speeds in the balancing process. The key features of new method are that the vibration responses of rotor are described by threedimensional holospectrum and the decomposition of holospectrum are employed to investigate the variation rules of first two modal components at the runup or rundown stages. The first two modal components can be canceled simultaneously at the speed below the first critical speed of flexible rotors. Experimental results showed that this new method can reduce the rotor vibration due to unbalance effectively, safely and conveniently.
ACKNOWLEDGMENT
This study was supported by the National Natural Science Foundation of China under grant number 59335033.