Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery Genfeng Lang a, Yuhe Liao a,b,n, Qingcheng Liu a, Jing Lin a,c a Shaanxi Key Laboratory of Product Quality Assurance & Diagnosis, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China b Xi’an Shaangu Power Co., Ltd., Xi’an 710075, PR China c State Key laboratory of Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China

a r t i c l e i n f o

abstract

Article history: Received 27 May 2014 Received in revised form 10 September 2014 Accepted 12 September 2014 Handling Editor: D.J Wagg

The vibration responses of different linear faults all possess some common features, which make fault diagnosis very difficult. Based on the multi-sensor information fusion theory, this paper presents a new qualitative identification method for the diagnosis of linear faults. The excitation–response dynamic equation is constructed and system balancing response with full consideration of system anisotropy is analyzed. Through discussion of the precession orbit shape difference and its dispersive situation, the orbit shape average difference coefficient and the corresponding dispersion term are estimated to obtain the theoretical balancing effect. Finally, the qualitative identification of linear fault can be done according to whether the calculated balancing effect meets the safe operation requirement or not. The dynamic characteristic of the system difference coefficients is verified by a simulation experiment and the case study further testifies the capability and reliability of the proposed method. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction According to the frequency spectral structure of the vibration signal, the faults of rotating machinery can be roughly divided into two broad categories. One is that the frequency spectrum of the fault response is featured with complicated spectral structure and wide spectral bandwidth. Besides the rotational frequency component (1X component), some higher order harmonic and/or sub-harmonic components with apparent amplitudes can also be found in the signal spectrum. Some scholars believe that spectral structure of this kind is an important sign of the presence of nonlinear faults in the rotor– bearing system [1,2]. Study on the mechanism and vibration characteristic of nonlinear faults has been one of the focuses in the rotational machinery fault diagnostic field in recent years. Although there are still many issues pending future investigation, considerable progress has been achieved as the continuous emergence of advanced nonlinear dynamic response analysis methods and intelligent diagnostic methods. The identification and diagnosis capability of nonlinear faults have therefore been significantly improved. The other kind of fault has a much simpler spectral structure than that of its nonlinear counterpart. Although it may not be so accurate, we can still call it linear fault here just for differentiation. No matter what kind of specific linear fault is, the

n Corresponding author at: Shaanxi Key Laboratory of Product Quality Assurance & Diagnosis, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China. Tel./fax: þ86 29 82667938. E-mail address: [email protected] (Y. Liao).

http://dx.doi.org/10.1016/j.jsv.2014.09.018 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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G. Lang et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

corresponding vibration signals all possess some common features as follows: the frequency spectrum of the vibration signal has the 1X component as the only or dominant component (although some higher order harmonic and/or subharmonic components could also be found in some cases, the amplitudes of these components are much smaller and can be technically ignored); the amplitude and phase of the 1X component remain stable under the condition of constant rotating speed and load. Two main factors are responsible for the emergence of linear fault. One is related to the excitation forces, which is generated by rotor mass unbalance during the operation of the rotor–bearing system. Actually, rotor mass unbalance is also one of the most representative faults frequently found in industrial rotating machinery, the study on its mechanism and analysis method has been very thorough and systematic. After decades of development, two kinds of methods, the modal balancing method [3] and the influence coefficient method [4], have already been built. Furthermore, aiming at the deficiency of these methods found in practical application, researchers have made some important improvement to these methods. Darlow et al. [5] proposed a unified balancing approach through fusion of the comparative advantage of both the above-mentioned methods. Liu et al. [6] and Liao et al. [7] presented a holobalancing method based on the influence coefficient theory. Han et al. [8] put forward a generalized modal balancing method taking the anisotropic characteristic of a rotor–bearing system into consideration. The development of non-trial balancing methods [9] and active balancing methods [10] further increased the balancing efficiency, and the maintenance cost required has then been obviously reduced. These studies have made important contributions to the development of the rotor dynamic balancing technique. The other factor is mainly linked to the dynamic characteristic of the support system of the rotor–bearing system. Some malfunctions related to the support system (e.g., improper pipe stress or oil temperature, poor bearing installation, to name but a few), are the possible reasons for the degradation of the dynamic characteristic of the support system. In some more complicated situations, the vibration problems could be a combined effect of both factors [11,12]. However, even though the mechanisms of the support malfunction and the rotor mass unbalance are entirely different, they all have identical vibration response features as mentioned above. Here the deficiency of simple spectral structure lies in that it canont provide enough information necessary for the qualitative identification of different linear faults. Due to the theoretical maturity and convenient implementation of the dynamic balancing method, at present field technicians often tend to use balancing correction to solve all linear fault problems without distinction. Unfortunately, this could bring some new problems to the troubleshooting process. The balancing process could be inefficient and in some serious situations it may even be ineffective. Besides the reasons such as miscalculation, disturbance from the system response nonlinearity and incorrect judgment of unbalance mass distribution, another cause that could lead to the failure of the balancing analysis, can be attributed to the faulty application of dynamic balancing to a rotor–bearing system with its actual malfunction unidentified beforehand. Therefore, it is strongly recommended that the state and influence of the overall rotor–bearing system dynamic characteristic be fully considered in practice. In recent years, many scholars have done extensive studies concerning the influence of support characteristic on the vibration of rotor–bearing system. For example, based on the study on a vibration model under the action of system anisotropy and asymmetrical rotor structure of a two-degrees-of-freedom system, Lazarus et al. [13] presented a three-dimensional finite element analysis method for the anisotropic rotor–bearing system. The effect of support stiffness anisotropy on system response is visually reflected in that the behavior of the system varies in different radial directions. The axis precession orbit in this case is actually an ellipse. Since this phenomenon gives more distinct information that vibration signal of any single radial direction can not provide, some scholars then tried to make the fault qualitative analysis utilizing the orbit information. Han et al. [14] decomposed the precession orbit into forward and backward components and proposed a fault identification method using the Shape and Directivity Index (SDI). Qu et al. [15] presented an orbit-based FFT spectrum, i.e., the two-dimensional holospectrum technique, to diagnose the common fault of rotating machinery. Lee [16] and Kim et al. [17] put forward the directional frequency response functions (DFRFs) to analyze the dynamics of an anisotropic rotor–bearing system with asymmetrical rotor structure and applied the DFRFs to diagnose the fault with consideration of the bearing anisotropy. These studies have laid a solid foundation for the exploration of linear fault diagnosis. Actually, these methods are the application of multi-sensor information fusion technique in the fault diagnostic field. The diagnostic methods and information have been greatly enriched since then. However, the vibration information used in these methods is from only one bearing section, which is still insufficient for linear fault diagnosis. Malfunction of the support system usually manifests itself in the degradation of system dynamic characteristics. As to a rotating machine set with multiple bearings, the impact of support system malfunction on system vibration response at each bearing (vibration measuring position) is generally not identical. Both the dynamic properties and the anisotropic characteristics at every bearing location could be different. Consequently, in this case the corresponding rotor axis precession orbits at different axial positions are always of different shapes. Although the dispersion of orbit shape is not related to mass unbalance, it will inevitably interfere with dynamic balancing analysis. This is the main reason why dynamic balancing could be ineffective. Generally, the more serious the orbit shape dispersion situation is, the less effective the dynamic balancing will be. Therefore, the orbit shape and its variation can provide more distinct information that could be useful in the linear fault identification. The diagnosis of linear faults should be implemented from a more macro perspective and take the vibration information of the whole machine set altogether into consideration. The study of this paper is based on the above analysis. The multi-sensor information fusion technique applied in 2D holospectrum is further expanded to all bearing sections. Through analysis of the dispersion of precession orbit shape, a new linear fault qualitative identification method is Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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presented. The excitation–response dynamic equation of an anisotropic rotor–bearing system is constructed and the formation mechanism and dynamic property of the rotor precession orbit are analyzed. The orbit average difference coefficient and deviation term are used to calculate the theoretical balancing effect. Finally, the qualitative identification of linear fault can be done according to whether the balancing effect meets the safe operation requirement or not. The paper is organized as follows: Section 2 gives the dynamic analysis of system response to mass unbalance. The dynamic analysis is further expanded to the axis motion with consideration of system anisotropy in Section 3. Based on that, the basic principle of linear fault qualitative identification approach and the characteristic of the average difference coefficient and the deviation term are discussed in Section 4. In Section 5, a simulation experiment analysis and a case study are presented. Finally, Section 6 gives the conclusion of the proposed approach. 2. Dynamic analysis of system response to mass unbalance No matter how precisely manufactured the rotor is, there is always mass unbalance in it. As the main source of excitation forces, mass unbalance is an important factor related to system vibration that can not be ignored. Therefore, knowing how an anisotropic rotor–bearing system will respond to mass unbalance is a prerequisite. Suppose the rotor–bearing system under consideration is a double-support single-span system and there are two vibration transducers mounted perpendicular to each other on every bearing section. For the convenience of further discussion, a spatial coordinate system is constructed first according to the following rules: the origin of the coordinate system is set at the first bearing position (driving end) and the Z axis of the coordinate system points from the motor along the rotor to the driven end; The X and Y axes are in the same directions of the two vibration transducers, respectively; the relationship of the three axes is defined by the right-hand rule. The rotor–bearing system is discretized in the coordinate system as a lumped mass model with n units, which is shown in Fig. 1. With the torsional deflection of the running rotor being neglected, the force-response equation of the system therefore can be expressed as # " #"   # " #"   # " #"   # " ½F xk  M xx 0 C yx þ ωG C xx K xx K yx p€ p_ p  i þ  i þ  i ¼   : (1) F yk C yy K xy K yy 0 M yy C xy  ωG q€ i q_ i qi Let xi and yi denote the displacements of the ith unit of the rotor in X and Y directions; θxi and θyi denote the deflection angles of the same unit about X and Y axes, respectively; qi and pi in Eq. (1) are the simplified expression of [xi, θyi]T and [yi,  θxi]T. Mxx and Myy are the equivalent inertia parameter matrices of the rotor–bearing system under “perfect balancing state” in X and Y directions, which means the shape center of every rotor unit section coincides with its mass center, Cxx and Cyy are the equivalent damping matrices of the rotor–bearing system in X and Y directions, Cyx and Cxy are the corresponding equivalent cross damping matrices, G is the rotation matrix of the rotor, ω is the rotational speed, Kxx and Kyy are the equivalent stiffness matrices, Kxy and Kyx are the corresponding equivalent cross stiffness matrices and Fxk and Fyk are the generalized excitation forces of the ith unit in X and Y directions, respectively. Consider mass unbalance as the main source of the system excitation force. If the characteristic parameter variation caused by rotor unbalance mass can be neglected, the unbalance-response relationship is 2 "  #3 uk cos ðωt þ αk Þ 6 7 " #"   # " #"   # " #"   # 6 0 M xx 0 C yx þ ωG C xx K xx K yx p€ p_ p k 7 7  i þ  i þ  i  ¼ ω2 F 6 " # (2) 6  7: C yy K xy K yy 0 M yy C xy  ωG q€ i q_ i qi 6 7 sin ð ω t þ α Þ u k k 4 5 0 k F on the right side of Eq. (2) is a transformation matrix. It is determined only by the axial location of the units. uk and αk are the weight and angular position of unbalance mass of the kth unit, respectively. It can be seen that any system shown in Eq. (2) is actually linear time-invariant and the spatial position parameters of every unit are harmonic signals synchronized with the excitation force. Suppose the response of the ith unit is 8   xi ¼ λi cos ωt þ φi > > >  > < θy ¼ σ i cos ωt þ ϑi  i   : (3) > yi ¼ ξi cos ωt þ ψ i > > > :  θx ¼ ν cos ðωt þ υ Þ i

i

i

Fig. 1. Lumped mass model of a double-support single-span rotor–bearing system.

Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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Substituting Eq. (3) into Eq. (2) and through Laplace transformation, we can have the unbalance response equation in complex form as shown in Eq. (4). The fact that the transformation matrix F only depends on the unit axial position (Z coordinate) indicates that there must be an identical mapping relationship between system generalized excitation forces and the system mass unbalance-related excitation forces in either X or Y direction, which can be expressed by F11 in Eq. (4) # #3 # # 3 2 "" 2 "" λi ejðωt þ ϕi Þ uk ejðωt þ αk Þ 7 6 7 " 2 #!  1 " #6 7 6 0 σ ejðωt þ ϑi Þ i 7 ω Mxx þ ωC xx þ K xx ω2 G þ ωC yx þ K yx F 11 0 6 k 6 "" 7 6 "" i 7 2 # # # # (4) ω ¼ 6 7: 6 7 2 2 j ð ω t þ α Þ j ω t þ ψ ð k  ω G þ ω C þK ω M þ ω C þ K 0 F 6 7 6 7 iÞ xy xy yy yy yy 11 e  ju ξ e k i 4 5 4 5 0 νi ejðωt þ υi Þ k i

Divide the coefficient matrix in the right side of Eq. (4) into a 2  2 block matrix " 2 #!  1 " # ω Mxx þ ωC xx þ K xx ω2 G þ ωC yx þK yx F 11 0 2

ω

 ω2 G þ ωC xy þ K xy

ω2 Myy þ ωC yy þ K yy

0

F 11

" ¼

T 11

T 12

T 21

T 22

# :

(5)

Obviously, if and only if block T11 ¼T22 and T21 ¼ –T12, the entire system would be isotropic. In this case the vibration amplitude in any radial direction will be all the same and the phase lag between the two vibration transducers mounted perpendicular to each other will be exactly 901. However, as shown in Eq. (5), the block matrix contains not only the rotor characteristic, but also the dynamic features of the support system. Influenced by the factors of the support system, especially the equivalent stiffness and cross stiffness matrices, the system dynamic characteristic is actually anisotropic. Therefore, both the vibration amplitude and lagging phase in different radial directions will also vary. This situation makes it insufficient to judge the system operating state with vibration signals from any single radial direction. An elementary transformation is implemented to Eq. (4) and the displacement of the ith unit under the action of unbalance excitation force is i 8h   > < λi ejðωt þ ϕi Þ ¼ T x uk ejðωt þ αk Þ h i (6)  : > : ξi ejðωt þ ψ i Þ ¼ T y  juk ejðωt þ αk Þ Tx and Ty are the transfer matrices mainly decided by the system characteristic parameters. Affected by the system anisotropic, Tx and Ty are generally not identical. This further gives the reason why the anisotropic system could respond differently in different radial directions even though the unbalance excitation force is constant. 3. Axis motion analysis with consideration of system anisotropy Since the vibration signal collected by a transducer is the projection of the axis motion in the corresponding radial direction and therefore contains only limited information of the system dynamic characteristic, evaluation of system operating condition with signals of single direction then is inadequate, sometimes even misleading. Considering the fact that the level and characteristic of system anisotropy, which is closely related to the factors like rotor structure, bearing type and installation quality, etc., are reflected through the geometrical parameters of the axis precession orbit, the axis motion is a more complete description of the entire system at certain rotor section compared with single direction vibration. Therefore, using the axis motion as the analysis object is a considerable alternative to implementing the unbalance-response analysis under the condition of system anisotropy. For the convenience of further discussion, we can express the transfer matrices Tx and Ty in complex form as follows:   ( T x ¼ Aik ejγ ik  : (7) T y ¼ χ ik ejςik Aik ejγik

3.1. Orbit analysis of mass unbalance in single unit Let's start our analysis from a simple case first. Suppose all rotor units are in perfect balance state except for the kth unit, and the system response at the ith unit under this condition is investigated. The χ ik ejζik in Eq. (7) is the system difference coefficient, which gives the difference degree of the response at two mutually perpendicular radial directions at the ith unit under constant unbalance force generated by the unbalance mass of the kth unit (uk ejαk ). Therefore, the spatial displacement vector of the ith unit axis then could be expressed as " #"     #" " # # 1 0 cos γ ik þ αk  sin γ ik þ αk cos ðωtÞ λi cos ðωt þ ϕi Þ         ¼ uk Aik : (8) χ ik sin ζ ik χ ik cos ζik cos γ ik þ αk sin γ ik þ αk ξi cos ðωt þ ψ i Þ sin ðωtÞ i

Considering the fact that the rotor–bearing system is anisotropic, χ ik ejζik is generally not equal to 1. Therefore, Eq. (8) is an elliptical motion orbit equation. This elliptical axis precession orbit can be completely described by the following Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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parameters: length of major and minor axes, major axis inclination angle, and position of initial phase point. Other parameters related to axis precession orbit can all be deduced from those mentioned above. With singular value decomposition (SVD) we can further simplify the coefficient matrix on the right side of Eq. (8) as " #"    # 1 0 cos γ ik þ αk  sin γ ik þ αk         uk Aik (9) ¼ U ik ðuk Aik Sik ÞV ik H ik : χ ik sin ζ ik χ ik cos ζ ik cos γ ik þ αk sin γ ik þ αk Among the matrices on the right side of Eq. (9), Hik depends on the phase of the mass unbalance of the kth unit and the corresponding response phase lag of the ith unit. Hik is an orthogonal matrix and Sik is a diagonal matrix, which is the SVD of a lower triangular matrix composed of the system difference coefficient χ ik ejζik . Elements on the diagonal line of Sik contain information about the ratio of the major and minor axes of the precession orbit. Uik is also an orthogonal matrix and is closely related to the major axis inclination angle. Therefore, the shape of the precession orbit, including inclination angle and the ratio of the major and minor axes, is solely determined by the system difference coefficient χ ik ejζik . uk and Aik are the orbit size scale factors which determine the magnitude of the precession orbit. These two factors, together with the system difference coefficient χ ik ejζik , reflect the influence of the kth unit on the axis precession orbit of the ith unit. The total amount of mass unbalance of the kth unit will mainly affect the size of the precession orbit, while the position of mass unbalance determines the position of the initial phase point on the orbit. 3.2. Orbit analysis of mass unbalance in multiple unit Let us expand our analysis to a more general situation. Actually, any real rotor system does have a certain mass unbalance and the unknown unbalance mass is randomly and spatially distributed along the rotor. The most complicated case, therefore, is that every discretized rotor unit has its own unbalance mass. The response at the ith unit is the result of joint action of all these mass unbalances together with system anisotropy. According to the linear superposition and vector synthesis principle, the response of the ith unit can be expressed as the linear superposition of all those displacement vectors caused by each unbalanced unit, respectively. So we have ( " #" " # #     #)" n 1 0 cos γ ik þ αk  sin γ ik þ αk λi cos ðωt þ ϕi Þ cos ðωtÞ         : (10) ¼ ∑ uk Aik χ ik sin ζ ik χ ik cos ζ ik cos γ ik þ αk sin γ ik þ αk ξi cos ðωt þ ψ i Þ sin ðωtÞ k¼1 i

Influenced by the system anisotropy, each excitation unit will have a system difference coefficient χ ik ejζik at the ith unit. If and only if all these difference coefficients are identical, the comprehensive response of the ith unit will keep its shape of the precession orbit unchanged, as shown in Eq. (11). This situation indicates that the response at the ith unit caused by each unbalance unit, respectively, has identical property. Theoretically, as long as the number and axial location of the balancing planes match the requirement of the modal balancing principle, under such assumptions any arbitrarily selected balancing planes can be used to reduce the overall system vibration to the expected level effectively. " #" " #  #     #"  n  1 0 cos γ  sin γ cos ðωtÞ λi cos ðωt þ ϕi Þ  jðγ ik þ αk Þ          ¼  ∑ uk Aik e : (11)  χ ik cos ζ ik cos γ sin γ ξi cos ðωt þ ψ i Þ sin ðωtÞ k ¼ 1  χ ik sin ζ ik i

However, this is generally not the case. The axial distribution of the system difference coefficients χ ik ejζik depends on the characteristic of system anisotropy and these coefficients are essentially not equal. Besides, some support system malfunctions could further complicate the situation. Therefore, taking all these factors into consideration, the orbit shape of the comprehensive response at the ith unit can be expressed as " ( " #  #     #)" n λi cos ωt þ ϕi cos γ ik þ αk  sin γ ik þ αk cos ðωtÞ   ¼ ∑ uk Aik     : (12) ξi cos ωt þ ψ i χ ik sin γ ik þ αk þ ζ ik χ ik cos γ ik þ αk þ ζ ik sin ðωtÞ k¼1 i

If j∑nk ¼ 1 uk Aik ejðγik þ αk Þ j is not zero, the amplitude (χ i ) and phase (ζ i ) of the response difference coefficient of the ith unit jζ i χ i e are     8  n   n  >  > jðγ ik þ αk þ ζ ik Þ   jðγ ik þ αk Þ  > = χ ¼ u A χ e u A e ∑ ∑     > k ik ik k ik i > k ¼ 1  k ¼ 1  < ! !: (13) n n > > jðγ ik þ αk þ ζ ik Þ jðγ ik þ αk Þ > >  angle ζ ¼ angle u A χ e u A e ∑ ∑ k ik k ik > ik : i k¼1 k¼1 It is obvious that the amplitude and phase of the response difference coefficient χ i ejζi vary along with the change of the mass unbalance distribution and the axial position of the ith unit (vibration measuring unit). The mass unbalance can be seen as a weighting factor in the system difference coefficient χ ik ejζik and will directly affect the orbit shape when the rotor– bearing system is anisotropic. Fig. 2 gives the graphical explanation of this phenomenon. The blue and red ellipses are the axis precession orbits of the ith unit, which are the vibration responses caused by the discrete mass unbalance in two different units, respectively. The green one is the result under the combined effect of these two mass unbalances and it is relatively flat with its major axis almost parallel to the abscissa, as shown in Fig. 2(a). In order to investigate the impact of Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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Fig. 2. Unbalance related orbit shape change. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

mass unbalance on the axis precession orbit, we simulated a balance state change of the unit corresponding to the red orbit, as shown in Fig. 2(b). It can be seen that the size of the red orbit is increased and the initial phase point is shifted to a new position along the orbit. Note that the shape of the red orbit remains constant, which indicates that the difference coefficient of the red orbit is unchanged during this process. It is because this coefficient is solely determined by the system dynamics and is independent of the balance state of the unit. However, the situation could be different when there are multiple outof-balance units in the rotor. The green orbit in Fig. 2(b) is entirely changed compared with the one shown in Fig. 2(a). This kind of phenomenon happens only in anisotropic rotor–bearing system and it shows that it could be insufficient to estimate the system weak stiffness direction only with the shape of axis precession orbit. 4. Linear fault diagnosis Obviously, the system anisotropic has a direct impact on the rotor precession orbit shape, as shown in Fig. 2. This property provides more information and therefore can be utilized in the linear fault diagnosis. However, one thing should be noted that the main difficulty in this process is lack of information about the system dynamic characteristics. Especially to the rotor–bearing system with its dynamic parameters (shown in Eq. (1)) not available, it is not easy to obtain the system running state directly with those theoretical dynamic equations. Here another approach is tried to solve this problem. In most cases, vibration signal is the main source of information that directly reflects the system dynamic characteristic. According to the above discussion, it can be seen that the combined effects of those unknown dynamic parameters on the rotor–bearing system are contained in the transfer matrices Tx and Ty, as shown in Eq. (6). The difference of Tx and Ty, i.e., the system difference coefficient χ ik ejζik , gives the description of the support system dynamic characteristic at the corresponding bearing section. The value of the difference coefficient in essence depends only on the characteristic of the anisotropic support system and has nothing to do with the rotor balance state. Therefore, if we could find a way to obtain the system difference coefficient χ ik ejζik directly from the vibration signals, it could be possible to qualitatively identify the source of linear faults without knowing the exact value of all those system dynamic parameters. 4.1. Basic principle The lumped mass model of a rotor–bearing system is shown in Fig. 1. The vibration response in orbit form at the ith unit is

"



#

λi0 cos ωt þ ϕi0   ξi0 cos ωt þ ψ i0

n

¼ ∑

k¼1

i

( uk0 Aik

"

χ ik

  cos γ ik þ αk0   sin γ ik þ αk0 þ ζ ik

#   #)"  sin γ ik þ αk0 cos ðωt Þ   : χ ik cos γ ik þ αk0 þ ζik sin ðωt Þ

(14)

The orbit shown on the left side of Eq. (14) is a combined effect of both mass unbalance and rotor–bearing system dynamic characteristics, as shown on the right side of the same equation. Since the related parameters are generally unknown, it is difficult to locate the fault source at this stage only with these theoretical formulas. However, we might as well assume that the vibration problem is mainly caused by mass unbalance. The number and axial position of balancing planes are therefore obtained by dynamic balancing analysis to the vibration signals based on this assumption. Suppose there are m balancing planes and the correction mass on the kth balancing plane is denoted by uk ejβk . The system vibration response corresponding only to these balancing masses, i.e., the pure correction mass response, is

3 2 ( " #     #)" λi cos ωt þ ϕi m cos γ ik þ αk  sin γ ik þ αk cos ðωtÞ 4 5     ¼ u A : (15) ∑ k ik   χ ik sin γ ik þ αk þ ζ ik χ ik cos γ ik þ αk þ ζ ik sin ðωtÞ k¼1 ξi cos ωt þ ψ i i

Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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The final balancing effect is the superposition of the left side of Eqs. (14) and (15), which is

3  # "  # 2 " λi cos ωt þ ϕi λi cos ωt þ ϕi λi0 cos ωt þ ϕi0 4 5     ξi cos ωt þ ψ i ¼ ξi0 cos ωt þ ψ i0 þ ξ cos ωt þ ψ  : i i i i

(16)

i

The purpose of dynamic balancing is to find a proper balancing scheme to minimize the left side of Eq. (16). This is a more complicated situation if we take the anisotropic nature of the support system into consideration. Since the balancing effect is the superposition of two elliptical orbits with different shapes, as shown on the left side of Eqs. (14) and (15), it could be directly affected by the system dynamic property. Unlike the normal balancing analysis under isotropic assumption, the shape similarity of the two orbits, which is determined by the difference coefficient and the size scale factor on the right side of Eqs. (14) and (15), has a significant impact on the balancing effect shown in Eq. (16). Obviously, the closer the shape of the two elliptical orbits is, the better the balancing effect will be. In other words, whether the dynamic balancing correction is suitable to solve the vibration problem depends on the dispersion degree of the difference coefficients in Eqs. (14) and (15). Furthermore, another factor should be mentioned. As we have discussed earlier in the above subsection, the mass unbalance distribution in any rotor is a continuous random spatial curve, while at the same time the rotor dynamic balancing can only be done with several discrete correction masses. Both the number and the axial positions of the correction masses are limited. It is impossible, also unnecessary, to correct the mass unbalance on this curve point-by-point in practice. Consequently, the balancing effect obtained is only an approximation to the ideal balance state in any circumstances. Once the number and axial position of the balancing planes are determined, theoretically there must be a corresponding best balancing effect. No matter which specific balancing scheme is applied, the actual balancing effect obtained can only gradually approach this theoretical state and can never exceed it. One favorable property of this theoretical balancing effect is that it is unrelated to specific balancing scheme, so no trial weight test is needed. As long as the system difference coefficients are available, it can be directly calculated with vibration signals without knowing the exact system dynamic characteristic parameters. The importance of this theoretical value lies in that it can be used as a criterion for the qualitative identification of linear faults. This is because it depends largely on the dispersion degree of the difference coefficients, i.e., the shape difference of the rotor precession orbit. Therefore, if the theoretical balancing effect can meet the safe and stable operation requirement of the rotor–bearing system, then mass unbalance should be the main source of the vibration problem and it is feasible to solve the problem with dynamic balancing correction. Otherwise, if the value obtained is too high and unacceptable, it suggests that there might be some malfunctions originating from the support system besides rotor mass unbalance. In this case dynamic balancing could be ineffective and our attention should turn to checking the operating condition of the support system first. By doing so, the qualitative identification of linear faults can be realized.

4.2. Average difference coefficient and the deviation term estimation It is now clear that system anisotropy and its dispersion level could directly affect the theoretical balancing effect, according to the above analysis. Especially when there are some support system malfunctions, system anisotropic situation could be more dispersed and complicated than normal condition at different measuring positions. The difference coefficients corresponding to different axial unbalanced units are generally not equal, as shown in Fig. 2, which indicates that the difference coefficients could actually disperse to a certain extent. In order to evaluate the theoretical balancing effect under this condition, an average difference coefficient χ ejζ is introduced here to represent the overall state of the system. The vibration response of the ith unit shown in Eq. (14) can be modified as " (  # ( " #"     #) )" # n λi0 cos ωt þ ϕi0 cos γ ik þ αk0  sin γ ik þ αk0 1 0 cos ðωt Þ   ¼ ∑ uk0 Aik     þ Ri ; (17) ξi0 cos ωt þ ψ i0 cos γ ik þ αk0 sin γ ik þ αk0 χ sin ζ χ cos ζ sin ðωt Þ k¼1 i

where Ri is the deviation term. Eq. (17) shows that the orbit formed vibration response at the ith unit consists of two parts. One is related to the average difference coefficient and the other is the deviation term. Putting the right side of Eq. (14) onto the left side of Eq. (17), we have a clearer expression of Ri as "    # " # " # 1 0 cos ϕi0 sin ϕi0 1 0   ξi0   λ     ¼ λi0 ξi0 Ri : (18) π π i0 cos ϕi0  sin ϕi0 χ sin ζ χ cos ζ λi0 sin ψ i0 þ 2  ϕi0 λi0 cos ψ i0 þ 2  ϕi0 It can be seen that the orbit shape of the deviation term is determined by those of the original vibration and average difference coefficients-related precession orbit. Similarly, the deviation term of the ith unit corresponding to average difference coefficient can be expressed as ( " " #) # 1 0 1 0     ξi RESi ¼ λi ξi sin ψ þ π  ϕ : (19)  τi π χ sin ζ χ cos ζ i i 2 λ λ cos ψ i þ 2  ϕi i

i

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Note that there is a slight difference between Eqs. (18) and (19). Since the orbit size scale factor τi, which is the product of uk and Aik, is related to the correction mass, it could not be exactly equal to the original unbalance mass. In order to avoid confusion, we use τi here in Eq. (19) instead of λi0. For any industry rotating machinery, there could be several measuring points along the shaft train. Generally, the dynamic characteristics at these points are not identical and the vibration responses at different points also vary even under stable operating conditions. The difference coefficients and the precession orbits corresponding to these points therefore are also different. Considering the fact that the system dynamic characteristics necessary for the calculation of the system difference coefficient at all measuring points are generally unknown, one possible solution to this problem is to use the average difference coefficient χ ejζ to represent the overall system. Therefore, if the χ ejζ can be obtained directly with system vibration response, the theoretical balancing effect can be calculated. Specifically, the average difference coefficient χ ejζ can be obtained by minimizing the sum of the square of the length of the major and minor axes of the residue precession orbits RESi at all measuring points, i.e., the trace of the matrix product of RESi and its transpose matrix. Suppose the whole rotor– bearing system has m measuring points, the sum of the trace is 



i 2 h

i2 m m   h π π τi  λi0 2 þ τi χ sin ζ  ξi0 sin ψ i0 þ  ϕi0 þ τi χ cos ζ  ξi0 cos ψ i0 þ  ϕi0 : (20) ∑ tr RESi RESi T ¼ ∑ 2 2 i¼1 i¼1 The minimization of Eq. (20) needs the following partial differential equation group to be held:





8 8 8 T T T <∂∑m <∂∑m <∂∑m i ¼ 1 tr RESi RESi i ¼ 1 tr RESi RESi i ¼ 1 tr RESi RESi     ¼ 0; ¼ 0: ¼ 0; : : : δτi δ χ sin ζ δ χ cos ζ

(21)

Through derivation of Eq. (20) according to Eq. (21), we get the theoretical balancing effect related parameter estimation equations as 8

o m n π > > χ sin ζ ¼ ∑m 1 τi 2 ∑ τi ξi0 sin ψ i0 þ  ϕi0 > > > i ¼ 1 2 > i¼1 > <

o m n π 1 (22) χ cos ζ ¼ ∑m τi 2 ∑ τi ξi0 cos ψ i0 þ  ϕi0 : > i ¼ 1 2 > i¼1 > >

  > > > τi ¼ 1 2 λi þ χξi0 cos ψ i0 þ π  ϕi0  ζ : 2 ð1 þ χ Þ These are nonlinear equations and can be solved with an iteration process. Put the parameters obtained by Eq. (22) into Eq. (19), the deviation term corresponding to average difference coefficient at every measuring unit can be calculated. If the difference coefficients at all measuring units are identical and equal to the average difference coefficient, the deviation term can be seen as an approximation of the theoretical balancing effect. Based on this prerequisite, the theoretical balancing effect can therefore be calculated with Eqs. (22) and (19). If the balancing planes used are inconsistent with the rotor mass unbalance distribution, or there are some possible malfunctions with the support system, the system difference coefficients then will be dispersed and not equal to the average difference coefficient. The deviation term Ri in Eqs. (17) and (18) in this situation is therefore not negligible. The deviation term obtained with Eq. (19) could be too conservative and not suitable for the estimation of theoretical balancing effect. In this case, the estimation process should take this into consideration to ensure a reliable result. Therefore, Eq. (19) should be adjusted with a weighting factor, as follows: ( " " #) # 1 0 1 0     ^ i ¼ λ i ξi     ξ RES τ : (23)  π π i i εχ sin ζ þ κ εχ cos ζ þ κ λi sin ψ i þ 2  ϕi λi cos ψ i þ 2  ϕi The theoretical balancing effect can then be calculated. The qualitative identification of linear faults can therefore be made according to whether the balancing effect meets the safe operation requirement or not.

5. Simulation analysis and case study 5.1. Simulation analysis of difference coefficient characteristic The support system anisotropy is ubiquitous. Especially when there are some support system malfunctions, the supporting characteristic will be more complicated and will inevitably affect the system response. The transfer matrices Tx and Ty in Eqs. (6) and (7) show that the anisotropic characteristic could affect the system response in one bearing section. In this subsection, a simulation analysis will be implemented to further discuss the characteristic of the system difference coefficient χ ik ejζik . The lumped mass model, which is shown in Fig. 1, is divided into 13 units and units marked by m1 and mn are the two bearings. The first three natural frequencies are about 13 Hz, 38 Hz and 68 Hz. Parameters related to bearing stiffness and Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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damping are denoted by k1, kn þ 1, c1 and cn þ 1, respectively. Since the effect of damping factors is comparatively small and can be neglected, only the supporting stiffness is considered in this simulation analysis. Furthermore, in order to simplify the system model, the effect of rotation matrix G is also ignored and the system dynamic parameters, including equivalent mass, stiffness and damping matrices, of all units are assumed to be constant. There are two measuring points set on both units m1 and mn, which are 901 apart and are denoted by X and Y, respectively. Consider the most frequently seen situation. The dynamic characteristics of the two bearings are both anisotropic. Neither the weak stiffness direction nor the stiffness difference level between the two-probe mounting directions is identical. This means the bearing stiffness meets the following requirement: ( k1;x  k1;y 4 0; kn þ 1;x  kn þ 1;y o0 : (24) jk1;x  k1;y ja jkn þ 1;x kn þ 1;y j Suppose initially the simulated rotor–bearing system model is in perfect running state. The excitation unit is successfully shifted from unit 1 to unit 13 and the frequency responses for every situation at rotational speed range [0 Hz, 80 Hz] are therefore simulated. The purpose of this simulation analysis is to investigate the effect of both non-uniform axial stiffness and the anisotropic bearing stiffness on system response. Figs. 3 and 4 give the amplitude–frequency response and phase– frequency response in X direction at both bearings. The amplitude–frequency response and phase–frequency response of the Y direction vibration are similar to those of the X direction, so only the response of the X direction is considered here. Three clear resonance peaks (the deep red area) and phase reversals can be seen in Figs. 3 and 4 at 13 Hz, 38 Hz and 68 Hz, respectively, which are the first three-order system natural frequencies. It is obvious that the system responses at the two bearings are of significant differences. Note that the amplitude–frequency responses at both m1 and mn might be insensitive (i.e., low amplitude) to the excitation force located at some specific units. The response at unit m1 is insensitive to the excitation force located close to m1 at low speed range, as shown in Fig. 3(a). Similarly, the same situation can also be found in Fig. 4(a), where the amplitude–frequency response at unit mn is also insensitive to the excitation force located close to mn at low speed range. With the increase of the rotational speed, the vibration insensitive related excitation force position gradually moves away to

Fig. 3. The amplitude–frequency response (a) and phase–frequency response (b) of the X direction vibration at bearing m1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. The amplitude–frequency response (a) and phase–frequency response (b) of the X direction vibration at bearing mn. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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the other end of the rotor, as shown in Figs. 3(a) and 4(a). The curve (in light green) traced out during this process intersects the 2nd order resonance peak (the red zone at 38 Hz) in the middle of the rotor (around unit 7), which is the node of the 2nd order mode shape. When the rotational speed is above 50 Hz, another insensitive excitation force unit begins to appear. These two curves (both in light green) intersect the 3rd order resonance peak (the red zone at 68 Hz) close to units 4 and 10, respectively, which are the two nodes of the 3rd order mode shape. Knowing this phenomenon could be very helpful in dynamic balancing analysis, balancing planes should be deliberately selected according to the rotational speed and rotor mode shape to avoid using those that are insensitive to system response. Figs. 5 and 6 give the variation trends of the response difference coefficients of both bearings in the same process. We can see that the amplitude and phase of response difference coefficients vary with the variation of rotational speed and excitation force position. This indicates that the rotor precession orbit will change its shape (mainly refers to the orbit eccentricity and inclination angle) as the variation of rotational speed and rotor balance state. This phenomenon occurs only when the support system is anisotropic, which has been verified in Section 3.2. Another noteworthy phenomenon is that there would be a drastic variation in difference coefficients when the excitation force happens to be located in the insensitive excitation force area. For the convenience of discussion, detailed figures of this situation are shown in Fig. 7, which are slices of Figs. 5(a) and 6(a) at 30 Hz. This indicates that small changes of excitation force position in this area could cause significant variation of orbit shape. It is also noted that the system vibration response amplitude corresponding to these insensitive excitation force areas is very small, as shown in Fig. 8. This indicates that, no matter how drastic a change the difference coefficient will be in the vicinity of this area, its impact on system response is negligible. While corresponding to the other system units, even though the differences coefficients are still not constant, the changes are relatively slow compared with this insensitive area. Therefore, it is feasible to use average difference coefficients instead of the actual value to evaluate the characteristic difference level and evaluate the system running state, as long as the rotational speed is far away from the critical speeds. This further verifies the assumption we made in Section 4.2.

Fig. 5. The amplitude (a) and phase (b) of the response difference coefficient at unit m1.

Fig. 6. The amplitude (a) and phase (b) of the response difference coefficient at unit mn.

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Fig. 7. The slices of Fig. 5(a) (a) and Fig. 6(a) (b) at 30 Hz.

Fig. 8. The slices of Fig. 3(a) (a) and Fig. 4(a) (b) at 30 Hz.

5.2. Case study Fig. 9 is a 50 MW steam turbine generator set of an oil refinery plant in central China. This machinery is composed of a turbine and a generator. The two sub-units are connected via a rigid coupling. Fig. 10 gives the sketch map of the rotor set structure together with the arrangement of vibration transducers. There are four bearings along the rotor set (marked successively from the driving end to the free end) and two coplanar and mutually perpendicular probes are mounted on each bearing section. The rated speed is 3000 rev/min. The vibration alarm level is 80 mm (peak-to-peak value) and the danger level is 120 mm. To ensure the long term safe and stable operation of the machine set, it requires that the vibration level at all measuring points should be below 60 mm. In this case, the vibrations of the faulty machine set at most measuring points exceeded the alarm level. The largest one (4_X) was even over 200 mm. Figs. 11–14 give the frequency spectra of the vibration signals collected from every measuring point of the faulty machine set. It is clear that, even though some low amplitude higher order harmonic components can be found, all spectra have the synchronous component (1X component) as the dominant component. Vibration surveillance shows that the amplitude and phase of the 1X components at all measuring points are stable at working speed. Due to insufficient data acquisition equipment, the coast down process data at only three measuring points, including 2_X, 3_Y and 4_X, were fully recorded. Fig. 15 gives the waterfall plots of these processes. All vibration components other than the synchronous one are of relatively low amplitude. The presence of prominent 1X component resonance peaks, which is one of the most important characteristics of mass unbalance, seemingly indicates that mass unbalance might possibly be the source of the vibration problem. This case is a typical linear fault and therefore only the 1X component related parameters will be considered in the following analysis. Table 1 gives the detailed information about the 1X component of the faulty machine set at working speed (3000 rev/min). Analysis of the vibration signals made by field technicians shows that the fault response characteristic is consistent with that of the mass unbalance, dynamic balancing correction is implemented. Two balancing planes, which are on the free end and coupling end of the generator rotor, respectively, are deliberately selected and two trial weights, both 287 g∠1091, Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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Fig. 9. The faulty 50 MW steam turbine generator set.

Fig. 10. The sketch map of the machine set and the transducers arrangement.

Fig. 11. The frequency spectra at measuring point 1_X (a) and 1_Y (b).

Fig. 12. The frequency spectra at measuring point 2_X (a) and 2_Y (b).

Fig. 13. The frequency spectra at measuring point 3_X (a) and 3_Y (b).

Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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Fig. 14. The frequency spectra at measuring point 4_X (a) and 4_Y (b).

Fig. 15. The waterfall plot of 2_X (a), 3_Y (b) and 4_X (c). Table 1 The vibration data of the faulty machine set (1X component only) (mm∠1). 1# Bearing

2# Bearing

3# Bearing

4# Bearing

1_x

1_y

2_x

3_y

3_x

3_y

4_x

4_y

141∠237

75∠74

107∠288

43∠42

94∠325

121∠106

209∠302

68∠82

are applied successively on these two planes to get the trial weight response and the corresponding influence coefficients. Table 2 gives the trial weight response data.

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Table 2 The trial weight response data (1X component) (mm∠1). Trial weight position

Probe direction

1# Bearing

2# Bearing

3# Bearing

4# Bearing

Free end of generator rotor

X Y

103∠229 73∠76

111∠286 51∠44

41∠322 67∠94

222∠276 77∠45

Coupling end of generator rotor

X Y

141∠237 85∠75

134∠278 59∠46

108∠309 119∠88

197∠290 106∠37

Table 3 Possible balancing effect. Bearing sequence

Original fault response Possible residual vibration

Length of precession orbit major axis (mm) 1# Bearing (mm)

2# Bearing (mm)

3# Bearing (mm)

4# Bearing (mm)

158 118

109 75

145 92

216 108

Fig. 16. The 3D holospectrum of the fault response.

Based on the fault response and trial weight response, the balancing scheme (426 g∠1541 on the free end and 374 g∠2301 on the coupling end of the generator rotor, respectively) and its possible balancing effects (residual vibration) are obtained with the weighted least square influence coefficient balancing approach. Table 3 compares the possible residual vibration of the balancing scheme with the original fault response. In order to better describe the overall vibration situation, here the length of the precession orbit major axis is used instead of vibration amplitude of one radial direction. Even though the possible residual vibrations at all bearing sections can be reduced to a certain extent with this balancing scheme, it still can not meet the safe operation requirement. The possible residual vibration is much worse than we expected and this balancing scheme is abandoned. The vibration data obtained during this process is then analyzed with the method of this paper. With the data listed in Table 1, we have the 3D holospectrum of the fault response, as shown in Fig. 16. The 3D holospectrum shown in Fig. 16 gives us a complete description of the overall vibration situation of the faulty machine set. All precession orbits are of different shapes, which makes the generation lines connecting adjacent orbits twisted. The 3D holospectrum in such a structure indicates that the faulty machine set is running in a rather complicated situation. This can be further explained with the system dynamic parameters, which are calculated with the fault response data and are listed in Table 4. It can be seen that the response difference coefficients χ i ejζi are of relatively large dispersion. Therefore, the average difference coefficient χ ejζ used to calculate the theoretical balancing effect must be modified with Eq. (23) to have this situation considered, as we have discussed in Section 4.2. The estimation result is listed in Table 5. The theoretical balancing effect listed in Table 5 basically matches with the possible residual vibration obtained with trial test, which is listed in Table 3. This verifies the capability of this approach and it is an important indication that, besides mass unbalance, there must be some abnormalities related to the support system. It could then be very difficult to solve this vibration problem with dynamic balancing method only. Since the system responses of the whole balancing process are all available, more detailed analysis can be done to further clarify this situation. With the trial weights and the corresponding trial response, accurate system dynamic parameters can be calculated. Table 6 lists the detail. The system difference coefficients corresponding to the two trial weights at all four bearing sections range from 0.07 to 1.50 (when the balancing plane is on the free end of the rotor) and from 0.55 to 4 (when the balancing plane is on the coupling end of the rotor), respectively. Consider the variation of the difference coefficient at one bearing section (for example, the bearing section no. 1), the change of trial weight position (from the free end to the coupling end of the generator rotor) dramatically increases the difference coefficient amplitude from 0.07 to 4. The dispersion situation of the Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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Table 4 The dynamic parameters of the faulty machine set. Dynamic parameters

1# Bearing

2# Bearing

3# Bearing

4# Bearing

Size scale factor τi χ i ejζi

141 0.53∠2531

107 0.4∠3351

94 1.29∠3091

209 0.32∠3101

Table 5 The average difference coefficient and theoretical balancing effect. Bearing sequence

Length of precession orbit major axis (mm)

Original fault response Theoretical balancing effect RESi εχejðζ þ κÞ

1# Bearing

2# Bearing

3# Bearing

4# Bearing

158 112 0.92∠3151

109 66

145 72

216 116

Table 6 System dynamic parameters calculated with trial weight response. Trial weight position

Dynamic parameters

1# Bearing

2# Bearing

3# Bearing

4# Bearing

Free end of generator rotor

Size scale factor τi χ ik ejζik

140 0.07∠3361

20 1.50∠2921

190 1.05∠2971

340 0.47∠3111

Coupling end of generator rotor

Size scale factor τi χ ik ejζik

10 4∠3251

110 0.55∠2831

110 1.18∠3411

150 1.73∠2821

Table 7 The influence coefficients. Trial weight position

Probe direction

Influence coefficients 1# Bearing

2# Bearing

3# Bearing

4# Bearing

Free end of generator rotor

X Y

0.14∠2541 0.01∠1381

0.02∠981 0.03∠2861

0.18∠1961 0.20∠411

0.34∠1351 0.16∠3571

Coupling end of generator rotor

X Y

0.00∠01 0.04∠2591

0.12∠961 0.06∠2841

0.10∠871 0.13∠3371

0.15∠1511 0.26∠3441

system difference coefficients is almost the same as that of the fault response and less affected by the change of rotor balance state. However, since the dynamic balancing analysis is completely based on system response, the balancing analysis could be severely affected by the support system running state in such a complicated situation and may cause a misleading result. This is because the abnormality of the support system will directly change the system stiffness distribution and this in turn will lead to a distorted unbalance response. For more details, Table 7 lists the influence coefficients obtained during the balancing process. Fig. 17 gives the comparison of the fault response and the two pure trial weight system responses. Fig. 17 shows that the pure trial responses generated by nearly 300 g trial weight are far less than the fault response. Especially in some bearing sections, there is not any response at all. This indicates that the system response is not a reflection of mass unbalance only, some malfunctions of the support system could be coexisting in the system. Besides, the backward precession direction at bearing section no. 1 is another important proof of the presence of support system malfunction. Based on the above analysis, it can be concluded that the vibration problem in this case is actually a combined result of both mass unbalance and support system malfunction. If we try to solve this vibration problem with dynamic balancing approach only, the correction weights required could be unreasonably large and actually inapplicable. A more reasonable approach should be to check the state of the support system and eliminate its possible malfunctions first of all and apply field balancing thereafter if necessary. According to our suggestion, an overhaul is made to the faulty machine set. Special attention is paid to check the support system installation conditions. It is found that the bearing parameters at bearing no. 2 and no. 3 are not consistent with Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i

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Fig. 17. The comparison of the fault response and the two pure trial responses. (a) The pure trial weight response of 287 g∠1091 in the free end balancing plane (bold line). (b) The pure trial weight response of 287 g∠1091 in the coupling end balancing plane (bold line).

installation standard, which degraded the alignment of the shaft train. The system is readjusted and after that field balancing is applied. The problem has been resolved successfully with the vibration amplitudes at all measuring points lower than 60 mm. 6. Conclusions Although the mechanisms of linear faults may be different, the corresponding vibration responses generally have similar frequency spectral structure. One common characteristic of these spectra is that they all have 1X component as the only or dominant component. Since simple spectral structure can not provide enough feature information necessary for fault diagnosis, it is then very difficult to identify different linear faults solely with vibration response spectra of single radial direction. In this paper a linear fault diagnosis approach is presented using rotor axis precession orbit as the object of analysis. Dynamical analysis is implemented on a rotor–bearing system with consideration of the combined effect of mass unbalance and system anisotropy. The orbit difference coefficient is adopted to describe the shape variation of the precession orbit and the dispersion situation of the difference coefficients is investigated. On this basis, the average difference coefficient of the faulty machine set is estimated to represent the system dynamic characteristic. Therefore, the theoretical balancing effect, i. e., the lowest possible residual vibration, is obtained. Since this balancing effect has nothing to do with the specific balancing scheme and is only decided by the system dynamic characteristic, it can be used as a criterion for us to qualitatively identify the source of linear fault. The characteristic of the difference coefficient is analyzed through a simulation analysis. The capability and feasibility of this approach is verified by a real case application.

Acknowledgment The work is supported by the Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20120201120043), the National Science and Technology Major Project (Grant no. 2014ZX04001191), the National Natural Science Foundation of China (Grant no. 51125022) and the Fundamental Research Funds for the Central Universities of China (Grant no. CXTD2014001). The authors would also like to thank the anonymous reviewers and Dr. Myles d’Airelle for their constructive suggestions in improving the clarity of the original manuscript. References [1] Y. Ishida, Nonlinear vibrations and chaos in rotordynamics, JSME International Journal, Series C 37 (2) (1994) 237–245. [2] G. Adiletta, A.R. Guido, C. Rossi, Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part I: Theoretical Analysis. Nonlinear Dynamics 14 (1) (1997) 57–87. [3] R.E.D. Bishop, The vibration of rotating shafts, Journal of Mechanical Engineering Science 1 (1) (1959) 50–65. [4] T.P. Goodman, A least-squares method for computing balance corrections, Journal of Engineering for Industry, Trans of the ASME, Series B 86 (3) (1964) 273–279.

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Please cite this article as: G. Lang, et al., Study on the precession orbit shape analysis-based linear fault qualitative identification method for rotating machinery, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j. jsv.2014.09.018i