曹青松，向 琴，熊國良

華東交通大學機電工程學院，南昌 330013

1．Introduction

Mechanical fault diagnosis technology，which is usually used to exploit the running states of equipment，includes fault detection，fault identification and fault isolation．Through this diagnosis technology，the performance of equipment could be evaluated as normal or not and the initial cause of failure might be forecast．Since fault characteristics are the basis of fault identification，the prerequisite of fault diagnosis is a kind of in-depth research on fault characteristics．Pedestal looseness is a common fault in a rotor-bearing system and it will reduce holistic stiffness，mechanical damping．All these results could cause violent vibration for the whole system．Therefore，the study of pedestal looseness is very significant for real engineering applications in terms of the safe operation of rotating machinery，the extension of service life，and the improvement of work efficiency．

In the 90’s，Muszynska and Goldman devoted themselves to the research on dynamics of rotating machinery，and made some contribution to this field．They proposed the bilinear model of a rotor system with one loose pedestal．Their numeric results showed that the vibration response included synchronous and subsynchronous fractional components，even leaded to chaos［1］．At present，there are two methods to study the looseness fault characteristics，i．e．，experiment-al method and dynamic modeling method．The former one is limited due to the high cost，so the latter one is adopted widely by domestic and oversea scholars．Dynamic models mainly contain the nonlinear differential equations［2］and finite element model［3］．Nonlinear differential equations could be usually solved by using Matlab software and numerical integration method，and the finite element model could be solved based on the Ansys software．The integration methods such as Runge-Kutta and Newmark-βmethod are usually adopted to solve the problem of looseness fault［4］．Although the available researches have build the foundation for the looseness fault diagnosis，the looseness models are mainly based on the piecewise-linear model that can not describe the real phenomena of looseness due to the unclear fault mechanism．Besides，most of the available results are limited to graphs and seldom achieve the periodic solution of vibration response．This drawback makes it difficult to distinguish the frequency components and the calculation is a time consuming process．

IHB is a kind of method which combines the incremental method and harmonic balance method to solve the nonlinear vibration equations［5］．Since this method has some advantages such as calculation time-saving，easy to converge and appropriate to a strong non-linear system，it is usually applied to solve strong non-linear problems such as piecewiselinear system［6］，Duffing oscillator system［7］and mistuning bladed disks system［8］．

Due to the fact that the vibration equation of rotor system with pedestal looseness and unbalanced force is a strong non-linear and IHB method has excellent performance to solve non-linear equation，this method is introduced into the solution of rotor system with looseness fault to acquire the periodic solution for the vibration response and analyze the fault characteristics of pedestal looseness．The results of present study would provide some theoretical basis for the diagnosis of looseness fault system．

2．Looseness model

In this paper，a sample single-disk Jeffcott system was considered．As shown in Figure1，one end of the shaft is loosed．Assume the looseness clearance is far less than the vibration displacement of the system，and then the imbalance caused by looseness clearance could be ignored．

1)Traditional looseness model．When the vibration amplitude of shaft end is less than or equal to looseness clearance α，stiffness of the system，i．e．，k1，is small．When the vibration amplitude of shaft end is greater than the looseness clearanceα，stiffness of system will be changed to k2due to the impact force from the constrained interface． Therefore，looseness stiffness should be a piecewise-linear function as follows:

2)Continuous looseness model．The looseness model shown in Eq．(1)is relatively idealized．However，the real phenomena of looseness stiffness could not be such step change．It will be continuously and non-linearly changed with the vibration amplitude of shaft end．Since the looseness mechanism was not clear and the complex change of stiffness was hard to describe accurately，a simple continuous model could be proposed．If the vibration amplitude of shaft end is less than or equal to the looseness clearanceα，the stiffness of the system will be linearly changed．However，if the vibration amplitude of shaft end is greater than the looseness clearanceα，the stiffness of system will be a constant value which is equal to the stiffness of shaft．Therefore，the continuous looseness model could be expressed as follows:

Then，the differential equation of motion for Jeffcott rotor system will be written as follows:

Where，m，c，k，k1，k0，F(xiàn)(t)and x denote the total mass of system，viscous damping coefficient，holistic stiffness，stiffness of shaft，looseness stiffness，exciting force of interference outside resulted from eccentric mass and vibration displacement，respectively．

Figure 1．Schematic diagram of a Jeffcott rotor system with looseness fault

3．IHB method solution

IHB method was adopted to solve the Eq．(3)．If kx=k0x+f(x)，f(x)can be written as follows:

For convenience，we introduce a phase angleτ given by

and then the independent variable could be changed from t to τ．Therefore，Eq．(3)could be transformed into the following equation．

Assume that the current state of vibration corresponding to the excitation parametersω0and F0is denoted by x0(τ)，a neighboring state could be expressed by adding the corresponding increments as follows:

For a small incrementΔx，the function f(x)may be written approximately as follows:

Where，f'( x0) represents the value of the first derivative of f(x)with respect to x at x0which could be expressed by using the step function．Put the Eq．(7)and Eq．(8)into Eq．(6)，and consider the parametersΔω andΔF are equal to zero and ignore the small value of higher order，then a linearized incremental equation could be obtained as the following form:

Where，

R is the correction term which will equal to zero if the values of ω0、F0and x0are the precise solutions of Eq．(6)．

The approximate periodic solution x0may be expressed as

Accordingly，the increment is as follows:

Where，

and N is the number of harmonic terms．

The Galerkin procedure is carried out with the Δan’s and Δbn’s as the generalized coordinate

Eq．(11)and Eq．(12)are substituted into Eq．(16)，one could obtain Eq．(17)after the rearrangement

Where，

The explicit expressions for Kij，Ri，Piand ΔQi(i=1，2;j=1，2)could be worked out as follows:

①Elements of the matrix K:

Where，

②Elements of the matrix R:

The nonlinear part of the elements of K and R are expressed by

4．Simulation analysis

The initial state of system A0andω0are given to iterate in the program with the following equation

The updated matrices K(i+1)，R(i+1)are then evaluated from Eqs．(18) ～ (25)in terms of the(i+1)th amplitude vector A(i+1)for the next iteration．In order to obtain the next point of solution，the iteration process will be proceeded until the criteria(R)≤ε is satisfied，whereεis the permissible error according to the given accuracy．A will be obtained when the iteration process is stopped，which is the coefficient of the harmonic terms．

Parameters of the Jeffcott rotor system are as follows:total mass of the system m=3 kg，eccentric mass Δm=5 g，eccentric displacement e=30．8 mm，system span L=630 mm，diameter of shaft d=15．4 mm，and modulus of elasticity E=206 GPa．The stiffness of shaft k2is 1．081 ×105N/m when the system is normal．

Since there exist only one unbalanced force from eccentric rotor，one pedestal looseness was considered．In this paper，the fault characteristics of rotor system with looseness was studied only in the vertical direction since looseness fault has directivity．Assume the rotate speed ω0is 1 400 r/min or 147 rad/s，looseness clearance α is 5 μm and the number of harmonic N is 12．

The periodic solution of the rotor system without looseness fault could be obtained by the IHB method，the time domain waveform，amplitude spectrogram，phase diagram and Poincare sectional drawing were shown in Figure 2．Figure 2(a)is the time domain waveform，which is smooth and the vibration displacement is approximate 34μm．From Figure 2(b)one could understand that the component of the vibration frequency is one fold and equal to the rotation frequency of rotor．The rotor motion is periodic since the phase diagram is a closed curve and there is only one point on the Poincare sectional drawing．

Figure 2．The time domain waveform，amplitude spectrogram，phase diagram and Poincare sectional drawing of the system without looseness

If the looseness stiffness k0is 3．6 ×104N/m when the fault happened，the periodic solution of the rotor system with looseness fault could be obtained by using the IHB method，the time domain waveform，amplitude spectrogram，phase diagram and Poincare sectional drawing were shown in Figure 3．In this figure，the time domain waveform became abnormal and the maximum vibration displacement was increased to 42 μm．Furthermore，the vibration response has complicated frequency components which consist of 1/3X，3/3X，5/3X，7/3X(X indicates the rotation frequency of rotor)and so on．The phase diagram is still a closed curve but more complicated．The rotor motion was judged to be period-three from the three points on the Poincare sectional drawing．

If the looseness stiffness k0is 0．6 ×104N/m when the fault happened，the same work could be done using the same method．Time domain waveform，amplitude spectrogram，phase diagram and Poincare sectional drawing were shown in Figure 4，respectively．The time domain waveform is messier than the previous one and the vibration displacement got increased apparently．The frequency components of vibration response became more adjacent which consist of 1/3X，2/3X，3/3X，4/3X and so on．It proved the importance of the Poincare sectional drawing for the phase diagram as shown in Figure 4(c)was too disordered to distinguish the motion state of rotor．The rotor motion is period-three，as shown in Figure 4(d)．

Figure 3．The time domain waveform，amplitude spectrogram，phase diagram and the Poincare sectional drawing of the system when the looseness stiffness k0=3．6 ×104 N/m

Figure 4．The time domain waveform，amplitude spectrogram，phase diagram and the Poincare sectional drawing of the system when the looseness stiffness k0=0．6 ×104 N/m

The simulation results showed that not only the vibration amplitude of the failing system is greater than that of the normal one，but also the frequency components become more complex．The phase diagram and the Poincare sectional drawing became more disperse with the decrease of looseness stiffness．All these results almost correspond to the result of reference［4］．It also could explain that the IHB method is capable to solve this issue like rotor system with looseness fault，and it also proves the validity of the continuous looseness model．

5．Conclusion

In this paper，the nonlinear dynamics and fault characteristics of a Jeffcott rotor system with pedestal looseness was studied．IHB method was exploited to solve a dynamics model with a continuous changed stiffness．Some conclusions could be drawn as follows:

1)The time domain waveform of vibration response for rotor system is smooth when there was no looseness fault，the vibration amplitude is relatively small，and there exists only 1X amplitude spectrogram，the rotor motion is period-one．

2)When the looseness fault happened，the time domain waveform of vibration response became messy，the vibration amplitude gets increased，and the degree of mess depends on the reduction of the looseness stiffness．The rotor motion became period-three．

3)The results also confirmed the feasibility that the IHB method could be applied to solve the issue of looseness system．The new continuous model built some foundation for nonlinear continuous looseness model which could describe the looseness mechanism best．And this research work could provide a theoretical basis for the diagnosis of looseness fault．

［1］ Muszynska A，Goldman P．Chaotic responses of unbalanced rotor/bearing/stator systems with looseness or rubs［J］．Chaos，Solitons ＆ Fractals，1995，5(9):1683-1704．

［2］ Goldman P，Muszynska A．Dynamic effects in mechanical structures with gaps and impacting:order and chaos［J］．Journal of Vibration and Acoustics，1994，116(4):541-547．

［3］ LIU Xiandong，HE Tian，LI Qihan．Dynamic model of rotor system with support loosening and its diagnosis method［J］．Journal of Aerospace Power，2005，20(1):54-59．

［4］ MA Hui，SUN Wei，Xuejun Wang，et al．Characteristic analysis of looseness fault in rotor system［J］．Journal of Northeastern University，2009，30(3):400-404．

［5］ LIU Yuanfeng，ZHAO Mei，ZHU Houjun．Chaotic Behavior of a rotor under rub and looseness［J］．Journal of Vibration and Shock，2002，21(4):36-38，21．

［6］ YANG Yongfeng，REN Xingmin，QIN Wei-yang．Nonlinear Dynamics Behaviors of Rotor Systems with Crack and Pedestal Looseness Faults［J］．Mechanical Science and Technology，2005，24(8):985-987．

［7］ CHEN Shuhui．Quantitative analysis method of strongly nonlinear vibration system［M］．Beijing:Science Press，2006．

［8］ Lau SL，ZHANGWS．Nonlinear vibrations of piecewise linear systems by incremental harmonic balance method，Journal of Applied Mechanics，1992，59:153-160．

［9］ Leung A Y T，CHUI SK．Nonlinear vibration of coupled duffing oscillators by an improved incremental harmonic balance method［J］．Journal of Sound and Vibration，1995，181(4):619-633．

［10］ WANG Ailun，LONG Qing．Application of IHB method to study the response characteristics of mistuning bladed disks with strong non-linear friction［J］．Journal of Mechanical Strength，2012，34(2):159-164．