CN109946077B - Method for establishing fractional order damping rolling bearing fault dynamics gradual model - Google Patents

Method for establishing fractional order damping rolling bearing fault dynamics gradual model Download PDF

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CN109946077B
CN109946077B CN201910116849.8A CN201910116849A CN109946077B CN 109946077 B CN109946077 B CN 109946077B CN 201910116849 A CN201910116849 A CN 201910116849A CN 109946077 B CN109946077 B CN 109946077B
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rolling bearing
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CN109946077A (en
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李志农
李云龙
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Nanchang Hangkong University
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Abstract

The invention discloses a method for establishing a fractional order damping rolling bearing fault dynamics gradual change model, which is characterized in that on the basis of obtaining the equivalent contact rigidity and the equivalent damping of a rolling bearing, the influence of a lubricant and the influence of the deformation of an inner ring when the rolling body breaks down are comprehensively considered, and the rolling bearing gradual change dynamics model containing a single pitting failure is established; applying a fractional order calculus theory to the rolling bearing, deducing a fractional order damping force calculation formula, and respectively establishing an inner ring fault model, an outer ring fault model and a rolling element fault model of the rolling bearing, which take the fractional order damping into consideration; the model provided by the invention has global property and can better reflect the historical dependence process of the system; a good effect can be obtained by using a few parameters; the physical meaning of the expression is clearer, and the expression is simpler; the model takes into account the influence of the lubricating oil between the rolling elements and the inner ring.

Description

Method for establishing fractional order damping rolling bearing fault dynamics gradual model
Technical Field
The invention relates to bearing dynamics and fault diagnosis technology, in particular to a method for establishing a fractional order damping rolling bearing fault dynamics gradual change model.
Background
The rolling bearing is the most important part in rotary machinery of a transmission motion rotor in various machines, has the advantages of small friction force, easy starting, simple lubrication, convenient replacement and the like, and is widely applied to the fields of precision instruments, aerospace, automobiles, machine tools, robots and the like. During the operation of rolling bearings, the bearings may be prematurely damaged due to poor lubrication, fatigue, wear, etc., and statistical data indicate that 30% of rotating machine failures are caused by the bearings. Once a bearing fails, a series of adverse reactions are caused, and a machine in operation or even a production line is shut down, so that huge economic loss is brought to enterprises; the life safety of related workers is seriously threatened. Therefore, in order to ensure safe and smooth high-speed operation of the rotary machine, reduce vibration and noise levels, improve production efficiency, and improve the competitiveness of enterprises in the society, it is necessary to study the failure of the rolling bearing.
In order to effectively diagnose the fault of the rolling bearing, a great deal of research work is carried out on the aspects of the fault mechanism, the dynamic characteristics and the like of the rolling bearing by a plurality of scholars, for example, Walters establishes a motion equation of all dynamic characteristics of the rolling bearing including the displacement, the rotating speed and the relative sliding of each internal element of the rolling body and the retainer for the first time, and establishes a dynamic model of the rolling bearing. Further intensive research is carried out by Harris on the basis of the model of Walters, and the inertia force and the inertia moment generated when the stress and the moment of the rolling body are unbalanced are considered, so that the dynamic model of the rolling bearing is more mature. Gupta improves the dynamic analysis method of the rolling bearing, considers the whole dynamic process of the rolling bearing starting, and establishes a more perfect dynamic analysis model of the rolling bearing. Meeks further develops a dynamic analysis model of the rolling bearing on the basis of Gupta theory, the model treats the contact between the rolling body and the retainer and between the retainer and the ferrule as incomplete elastic contact, and a six-degree-of-freedom dynamic model of the retainer is established. Sopanen proposes a six degree of freedom dynamic model that takes into account hertzian contact deformation and elasto-hydrodynamic, giving a dynamic model that can be calculated from the shape, material properties and radial clearance of the bearing. Cong provides a rolling bearing fault model based on dynamic load analysis, the impact times are considered in the model, and a fault calculation equation is obtained through dynamics and kinematics analysis. Kogan based on classical dynamics and the dynamic equations, proposes a three-dimensional bearing dynamics model that uses hyperbolic tangent function to model friction and Hertz contact spring damping model to represent the interaction between the bearing components.
While some progress has been made in modeling rolling bearing dynamics, there are still some problems: first, the pitting defects are idealized, i.e. the process is not considered a gradual process, provided that the rolling bodies enter and leave the failure zone, releasing and acquiring the full deformation. Secondly, in an actual rotary machine, a failure of a rolling bearing is caused by a change in the condition of a lubricant, and due to the presence of the lubricant, an elastic hydrodynamic oil film having a certain thickness can be formed and maintained between the rolling elements and the cage and the bearing inner ring, and a lubricating oil viscous resistance is generated, but the influence of the lubricant is not considered in the process. Third, they are all based on integer order calculus, with fractional calculus being rarely considered.
Disclosure of Invention
The invention aims to solve the technical problem of ensuring the safe and stable high-speed running of the rotating machinery, reducing the vibration and noise level and improving the production efficiency by improving the analysis method of the dynamic model of the rolling bearing.
The invention adopts the following technical scheme for solving the technical problems, and the method for establishing the fractional order damping rolling bearing fault dynamics gradual model comprises the following steps:
1) on the basis of obtaining the equivalent contact rigidity and equivalent damping of the rolling bearing, the influence of a lubricant and the influence of the deformation of the inner ring when the rolling body breaks down are comprehensively considered, and a rolling bearing gradual change dynamic model containing a single pitting failure is established;
2) applying a fractional order calculus theory to the rolling bearing, deducing a fractional order damping force calculation formula, and respectively establishing an inner ring fault model, an outer ring fault model and a rolling element fault model of the rolling bearing, which take the fractional order damping into consideration;
a) the specific steps of establishing the fault model of the inner ring of the rolling bearing are as follows:
for a damaged bearing, the contact deformation between the jth rolling element and the raceway is:
δ＝xcosθj+ysinθj-e-βjλ+Hj (1)
Figure GDA0002023669020000021
Figure GDA0002023669020000022
in the formula, X is the radial X-direction displacement of the inner ring; y is the radial Y-direction displacement of the inner ring; thetajIs the position angle of the jth rolling body center; e is the radial gap; lambda is the deformation gradually released when the rolling body rolls over the inner ring fault area; hjIs the oil film thickness; omegacIs the angular velocity of rotation of the cage; z is the number of the rolling bodies; thetalThe starting position angle of the rolling body is numbered 1; omegaiIs the angular velocity of rotation of the rotor; dbIs the diameter of the rolling body; dmIs the pitch circle diameter of the rolling bearing; alpha is a contact angle between the rolling body and the inner ring raceway;
βjis the switching value, and represents the position angle when the rolling body is in the pitting defectWhen the span is within, a certain contact deformation exists; when the rolling body is not in the position angular span of the pitting defect, the deformation amount does not exist any more; it is therefore defined as:
Figure GDA0002023669020000023
Figure GDA0002023669020000028
Figure GDA0002023669020000024
in the formula (I), the compound is shown in the specification,
Figure GDA0002023669020000027
is the position angle of the inner ring pitting failure center; delta phisIs the span angle of the fault; bcHalf the fault width; phispallIs the location angle of the pitting failure center;
when the rolling body passes through the inner ring defect, the released deformation amount is the deformation amount C of the rolling bodydrAnd deformation amount C of inner ringdiSumming; when the rolling body contacts the bottom of the defect, the maximum deformation released is lambdamaxIs equal to the fault depth, i.e.:
λmax＝CdrCdi (7)
Figure GDA0002023669020000025
Figure GDA0002023669020000026
in the formula, riIs the inner circle radius; r isbIs the rolling element radius;
the actual release deflection is:
Figure GDA0002023669020000031
according to the Hertz contact theory, the calculation method of the contact load Q of the point contact between a single rolling body and the inner ring raceway and the outer ring raceway respectively comprises the following steps:
Figure GDA0002023669020000032
in the formula, K is an equivalent contact deformation coefficient which is related to the contact deformation coefficients between the rolling body and the inner and outer ring raceways;
the contact between the rolling body and the inner and outer ring raceways is represented by a group of spring damping models; on the premise that the contact angles of the rolling body and the inner and outer ring raceways are equal, the total equivalent contact deformation coefficient is as follows:
Figure GDA0002023669020000033
Figure GDA0002023669020000034
wherein, the ball bearing n is 1.5; koAnd KiIs the contact deformation coefficient; n isδIs a coefficient related to a principal curvature difference function, and can be obtained by table lookup; Σ ρ is a contact principal curvature sum function, and the calculation method is:
∑ρ＝ρII+ρIII+ρ2I+ρ2II (14)
in the formula, subscripts 1 and 2 represent a rolling element and inner and outer ring raceways, respectively; i represents the axial plane of the rotor; II represents a radial plane perpendicular to I; the specific calculation method of each principal curvature can be obtained by looking up a table;
nδis a calculation coefficient which is related to a main curvature difference function of the rolling bearing and is obtained by looking up a table; the principal curvature difference function is defined as:
Figure GDA0002023669020000035
and projecting all the contact loads Q in the X-axis direction and the Y-axis direction and adding the projection to obtain the total contact load between the inner ring raceway and all the rolling bodies:
Figure GDA0002023669020000036
adopting a Newtonian fluid model, namely considering the lubricant as Newtonian fluid; the tangential friction force borne by the jth rolling element is as follows:
τ＝η*Q (17)
in the formula, eta is the static friction coefficient;
from newton's second law, the kinetic equation of the inner ring of the rolling bearing is:
Figure GDA0002023669020000037
wherein m is the mass of the rolling bearing; c is the equivalent damping of the system; feIs the radial load of the inner ring; frApplying a load to the outer ring;
the damping force is a function of the first derivative of displacement, whose fractional derivative can be expressed as:
Fd＝cx′(t)＝cDαx(t) (19)
in the formula, FdIs a damping force; dαIs a complex variable of the Laplace transform;
the inner ring dynamic model after considering the damping force is as follows:
Figure GDA0002023669020000041
b) the specific steps of establishing the fault model of the outer ring of the rolling bearing are as follows:
the established kinetic equation is as follows:
Figure GDA0002023669020000042
in the formula (I), the compound is shown in the specification,
Figure GDA0002023669020000049
is the included angle between the external load and the X axis;
maximum deflection λmaxAt the rolling body deformation CdrSubtracting the deformation C of the outer ringdoNamely:
λmax＝Cdr-cdo (22)
Figure GDA0002023669020000043
Figure GDA0002023669020000044
in the formula, roIs the outer ring radius;
therefore, the deformation amount calculation method of the actual gradual release is:
Figure GDA0002023669020000045
c) the specific steps of establishing the rolling element fault model of the rolling bearing are as follows:
the dynamic dimensionless model of the rolling element after considering the damping force is as follows:
Figure GDA0002023669020000046
maximum deflection lambda when a rolling element fails and contacts the outer race racewaymaxShould be at the rolling body deformation CdrSubtracting the deformation C of the outer ringdo(ii) a And when the rolling body is in fault and contacts with the inner ring raceway, the maximum deformation amount lambdamaxShould be at the rolling body deformation CdrIn addition, the deformation C of the inner ring is addeddiNamely:
Figure GDA0002023669020000047
therefore, the deformation amount calculation method of the actual gradual release is:
Figure GDA0002023669020000048
the invention considers the fractional order damping characteristic and the lubricating oil action in the dynamic modeling of the rolling bearing, and the proposed model has the following characteristics that (1) the proposed model has global property and can better reflect the historical dependence process of the system; the traditional model has locality and is not suitable for describing a history dependence process; (2) the proposed model overcomes the serious defect that the traditional model theory is not well matched with the experimental result, and a good effect can be obtained by applying a few parameters; (3) compared with the traditional model, the proposed model has clearer physical meaning and simpler expression; (4) the proposed model takes into account the influence of the lubricating oil between the rolling elements and the inner ring, whereas the traditional model ignores the viscous resistance of the lubricating oil, which is obviously not in line with practice.
Drawings
FIG. 1 is an idealized transient model of the present invention;
FIG. 2 is a practical asymptotic model of the present invention;
FIG. 3 is an inner ring single pitting failure model of the present invention;
FIG. 4 is an enlarged schematic view of the local pitting failure of the inner ring of the asymptotic model according to the present invention;
FIG. 5 is a time domain plot of vibration signal displacement for inner ring failure in accordance with the present invention;
FIG. 6 is a time domain plot of vibration signal velocity for an inner ring fault of the present invention;
FIG. 7 is a graph of vibration signal displacement amplitude versus frequency for an inner ring failure of the present invention;
FIG. 8 is a plot of vibration signal velocity amplitude versus frequency for an inner ring fault of the present invention;
FIG. 9 is a bifurcation diagram of the fractional order of the inner race fault of the present invention;
FIG. 10 is a graph of axial trace for an inner race fault of the present invention with a fractional order of 0.6;
FIG. 11 is a graph of the spectrum of the inner ring fault of the present invention with a fractional order of 0.6;
FIG. 12 is an acceleration time domain plot of experimental results of inner ring failure of the present invention;
FIG. 13 is an acceleration amplitude-frequency plot of the experimental results of the inner race fault of the present invention;
FIG. 14 is an outer ring single pitting failure model of the present invention;
FIG. 15 is an enlarged schematic view of the outer ring local pitting failure of the asymptotic model of the present invention;
FIG. 16 is a time domain plot of vibration signal displacement for an outer ring fault of the present invention;
FIG. 17 is a time domain plot of vibration signal velocity for an outer ring fault of the present invention;
FIG. 18 is a graph of vibration signal displacement amplitude versus frequency for a fault in the outer race of the present invention;
FIG. 19 is a plot of vibration signal velocity amplitude versus frequency for a fault in the outer race of the present invention;
FIG. 20 is a bifurcation diagram of the fractional order of the outer ring fault of the present invention;
FIG. 21 is a graph of axial trace for an outer ring fault of the present invention with a fractional order of 0.6;
FIG. 22 is a graph of the spectrum of the outer race fault of the present invention with a fractional order of 0.6;
FIG. 23 is an acceleration time domain plot of the experimental results of the outer ring fault of the present invention;
FIG. 24 is an acceleration amplitude-frequency plot of the experimental results of the outer ring fault of the present invention;
FIG. 25 is a rolling element single pitting failure model of the present invention;
FIG. 26 is a time domain plot of vibration signal displacement for rolling element failure in accordance with the present invention;
FIG. 27 is a time domain plot of vibration signal velocity for rolling element failure in accordance with the present invention;
FIG. 28 is a graph of vibration signal displacement amplitude versus frequency for rolling element failure in accordance with the present invention;
FIG. 29 is a plot of vibration signal velocity amplitude versus frequency for rolling element failure in accordance with the present invention;
FIG. 30 is a bifurcation diagram of the fractional order of rolling element failure of the present invention;
FIG. 31 is a diagram showing the locus of the axis of a rolling element failure of the present invention in a fractional order of 0.6;
FIG. 32 is a graph of the frequency spectrum of a rolling element fault of the present invention with a fractional order of 0.6;
FIG. 33 is a time domain graph of acceleration of experimental results of rolling element failure of the present invention;
fig. 34 is an acceleration amplitude-frequency chart of the experimental result of the rolling element failure of the present invention.
Detailed Description
The invention will be further described with reference to the accompanying drawings.
The dynamic modeling method for the damage fault of the surface of the rolling bearing comprises the following steps:
the current fault model of rolling bearings (as shown in fig. 1) is that the fault is reduced to a rectangular groove, the deformation is caused at the moment when the rolling body is in contact with the fault and is disconnected, and the magnitude of the deformation is equal to the depth of the fault. The actual rolling bearing failure should be (as shown in fig. 2) a circular arc defect, and the rolling element does not deform completely at the moment of contact with the failure, and the maximum deformation is achieved only when the rolling element contacts the bottom of the failure.
1. Fault model of inner ring of rolling bearing:
assume a specific case when there is a local damage failure at a point on the inner ring (as shown in fig. 3). For a damaged bearing, the contact deformation between the jth rolling element and the raceway is:
δ＝xcosθj+ysinθj-e-βjλ+Hj (1)
Figure GDA0002023669020000061
Figure GDA0002023669020000062
in the formula, X is the radial X-direction displacement of the inner ring; y is the radial Y-direction displacement of the inner ring; thetajIs the position angle of the jth rolling body center; e is the radial gap; lambda is the deformation gradually released when the rolling body rolls over the inner ring fault area; hjIs the oil film thickness; omegacIs the angular velocity of rotation of the cage; z is the number of the rolling bodies; theta1The starting position angle of the rolling body is numbered 1; omegaiIs the angular velocity of rotation of the rotor; dbIs the diameter of the rolling body; dmIs the pitch circle diameter of the rolling bearing; alpha is the contact angle between the rolling body and the inner ring raceway.
βjIs a switching value, which indicates that when the rolling body is positioned in the position angular span of the pitting defect, a certain contact deformation exists; when the rolling body is not in the position angle span of the pitting defect, the deformation amount is no longer existed. It is therefore defined as:
Figure GDA0002023669020000063
Figure GDA0002023669020000066
Figure GDA0002023669020000064
in the formula (I), the compound is shown in the specification,
Figure GDA0002023669020000065
is the position angle of the inner ring pitting failure center; delta phisIs the span angle of the fault; bcHalf the fault width; phispallIs the location angle of the pitting failure center, as shown in fig. 1.
An ideal inner ring failure pitting model defines the contact deformation as the failure depth d (as shown in fig. 4). However, the actual inner ring of the rolling bearing is not a rigid body in the complete sense, and the inner ring is also changed when it comes into contact with the rolling elements, so that the amount of deformation released when the rolling elements pass through the inner ring defect should be the amount of deformation C of the rolling elementsdrAnd deformation amount C of inner ringdiSumming; when the rolling body contacts the bottom of the defect, the maximum deformation released ismaxIs equal to the fault depth, i.e.:
λmax＝Cdr+Cdi (7)
Figure GDA0002023669020000071
Figure GDA0002023669020000072
in the formula, riIs the inner circle radius; r isbIs the rolling element radius.
Since the release and recovery of the deformation is an asymptotic process, the contact deformation amount is continuously changed when the rolling body rolls in the inner ring fault area. Therefore, the actual calculation of the released deformation amount should take into account the deformation release of the inner ring and the rolling element and the obtained gradual change process, and then the deformation amount is:
Figure GDA0002023669020000073
according to the Hertz contact theory, the calculation method of the contact load Q of the point contact between a single rolling body and the inner ring raceway and the outer ring raceway respectively comprises the following steps:
Figure GDA0002023669020000074
in the formula, K is an equivalent contact deformation coefficient, and is related to the contact deformation coefficient between the rolling body and the inner and outer ring raceways.
The contact between the rolling bodies and the inner and outer ring raceways can be represented by a set of spring damping models. On the premise that the contact angles of the rolling body and the inner and outer ring raceways are equal, the total equivalent contact deformation coefficient is as follows:
Figure GDA0002023669020000075
Figure GDA0002023669020000076
in the formula, the ball bearing n is 1.5. KoAnd KiIs the contact deformation coefficient; n isδIs a coefficient related to a principal curvature difference function, and can be obtained by table lookup; sigmaρIs a contact principal curvature sum function, and the calculation method is as follows:
∑ρ＝ρII+ρ1II+ρ2I+ρ2II (14)
in the formula, subscripts 1 and 2 represent a rolling element and inner and outer ring raceways, respectively; i represents the axial plane of the rotor; II represents a radial plane perpendicular to I. The specific calculation method of each principal curvature can be obtained by looking up a table.
nδIs a calculation coefficient which is related to a main curvature difference function of the rolling bearing and is obtained by looking up a table. The principal curvature difference function is defined as:
Figure GDA0002023669020000077
and projecting all the contact loads Q in the X-axis direction and the Y-axis direction and adding the projection to obtain the total contact load between the inner ring raceway and all the rolling bodies:
Figure GDA0002023669020000078
because the internal geometry and the working environment of the bearing are relatively complex, the establishment of a real rolling body frictional resistance model is relatively difficult. Therefore, most rolling bearing dynamic models adopt empirical and semi-empirical models to deal with the oil film frictional resistance of the rolling elements with the inner ring and the outer ring. The newtonian fluid model is used herein, i.e., the lubricant is considered to be a newtonian fluid. Considering that the rolling bodies both roll and slide along a plane, the tangential friction force can be divided into a pure rolling friction force and a sliding friction force. Considering the viscosity of the lubricating oil, and the elastic hydrodynamic lubricating oil film generated between the rolling element and the raceway is thin, generally 0.3-1 μm, so the tangential friction force borne by the jth rolling element is as follows:
τ＝η*Q (17)
wherein η is the coefficient of static friction.
From newton's second law, the kinetic equation of the inner ring of the rolling bearing is:
Figure GDA0002023669020000081
wherein m is the mass of the rolling bearing; c is the equivalent damping of the system; feIs the radial load of the inner ring; frAnd applying load to the outer ring.
Taking the 6220 type bearing as an example, the main parameters of the bearing are as follows: the diameter of the rolling body is 25.4mm, the diameter of the pitch circle is 140mm, the diameter of the outer ring is 190.8mm, the diameter of the inner ring is 89.2mm, the number of the rolling bodies is 10, the rotating speed is 719.57rad/min, and the size of a simulation fault is 0.3mm multiplied by 0.3 mm. The initial conditions in simulation were: the initial displacement x ═ y ═ 0, the initial speed x ═ y ═ 0, the time is from 0.2s to 0.3s, the equivalent damping c ═ 200N × s/m, and the rotor speed is 1772 r/min.
Fig. 5 and 6 are a vibration signal displacement time domain graph and a velocity time domain graph of an inner ring fault, respectively. It can be derived from fig. 6 that when there is a pitting failure of the inner ring of the rolling bearing, there are some peaks, but most of the peaks are not particularly prominent; in the displacement time domain waveform diagram of fig. 5, if there are about 4-5 quasi-periodic signals in 0.4s, the frequency is 10-12.5 Hz, and the characteristic frequency is preliminarily estimated as the rotor frequency (11.99 Hz).
Fig. 7 and 8 are a graph of amplitude-frequency displacement and a graph of amplitude-frequency velocity of the vibration signal of the inner ring fault, respectively. The main frequencies involved are basically the rotor frequency (12.21Hz), the frequency at which the rolling elements fail through the inner ring and its second and third harmonics (70.84Hz, 141.7Hz and 212.5Hz) and the side frequencies caused by the rotor speed (83.05Hz, 153.9Hz, etc.), but in fig. 8 the inner ring failure characteristic frequencies quadruple and quintupled (288.2Hz and 354.2Hz) occur.
The damping force is a function of the first derivative of displacement, whose fractional derivative can be expressed as:
Fd＝cx′(t)＝cDax(t) (19)
in the formula, FdIs a damping force; dαIs a complex variable of the lagrange transform.
The inner ring dynamic model after considering the fractional order damping force is as follows:
Figure GDA0002023669020000082
fig. 9 is a bifurcation diagram of the inner ring single point failure system. It can be seen from fig. 9 that changing the order of the rolling bearing fractional order has a large effect on the motion characteristics of the system. When the fractional order alpha is less than or equal to 0.2, the rolling bearing system is in a chaotic state; when alpha is more than 0.2, the rolling bearing system enters a periodic motion state.
Fig. 10 and 11 are an inner ring single-point failure axial center locus diagram and a frequency spectrum diagram of the rolling bearing when the fractional order α is 0.6, respectively. As can be seen from fig. 10, the axle center locus diagram, although being more complex, exhibits a motion characteristic of a distinct 8-shaped figure, which indicates that the motion state of the rolling bearing at this time is a periodic motion, which corresponds to the conclusion of the order bifurcation diagram. And the abscissa range of the axle center track diagram is [ -8, 8 [ -8]10-7m, range of ordinate [ -10, 10]10-7And m, obtaining that the axis locus of the single-point fault of the inner ring of the rolling bearing is elliptical. It can be observed from the system spectrum diagram of fig. 11 that the system exhibits 4 × frequency multiplication components. But the dominant component is the 2 x multiplied frequency component and the 6 x multiplied frequency component is also present in the spectrum.
Fig. 12 and 13 are an acceleration time domain graph and an amplitude frequency graph of the experimental result of the inner ring fault, respectively. No explicit useful information can be seen from the time domain waveform diagram of fig. 12. From the spectrogram of fig. 13, the rotational frequency (13Hz), the inner ring fault characteristic frequency and its higher harmonics (73Hz, 150Hz, 199Hz, 285Hz and 341Hz) and the side frequencies (65Hz and 162Hz, etc.) caused by the rotor frequency conversion can be derived. Even a 1.5 multiple frequency (106Hz) of the inner ring fault signature frequency appears in the figure.
2. Fault model of outer ring of rolling bearing:
because the outer ring and the frame are generally in a tight transition fit or interference fit, the speed of the outer ring is zero. The outer ring fault model is shown in fig. 14:
as shown in fig. 15, the maximum deformation amount λmaxAt the rolling body deformation CdrSubtracting the deformation C of the outer ringdoNamely:
λmax＝Cdr-Cdo (21)
Figure GDA0002023669020000091
Figure GDA0002023669020000092
in the formula, roIs the outer ring radius.
Therefore, the deformation amount calculation method of the actual gradual release is:
Figure GDA0002023669020000093
the established kinetic equation is as follows:
Figure GDA0002023669020000094
in the formula (I), the compound is shown in the specification,
Figure GDA0002023669020000095
is the angle between the applied load and the X-axis.
Fig. 16 and 17 are a vibration signal displacement time domain diagram and a velocity time domain diagram of an outer ring fault, respectively. FIG. 17 shows that when the outer ring of the rolling bearing has pitting failure, the failure characteristic frequency of the outer ring (19-20 peaks in 0.4 s) can be seen; and the displacement time domain waveform diagram of FIG. 16 has about 4-5 quasi-periodic signals (frequency is 10-12.5 Hz) in 0.4s, and also has 4-5 peak values in each periodic signal, which is just the fault characteristic frequency of the outer ring.
Fig. 18 and 19 are a displacement amplitude-frequency diagram and a velocity amplitude-frequency diagram, respectively, of the vibration signal in the case of an outer ring fault. The main frequencies included are basically the rotor frequency (12.21Hz), the frequency when the rolling bodies pass through the outer ring to fail and the second and third harmonics thereof (48.85Hz, 97.7Hz and 146.6Hz) and the side frequency caused by the rotor speed (87.93Hz, etc.), and the outer ring failure characteristic frequency quadruple and quintupling harmonics and higher frequency-doubled harmonics (195.4Hz, 244.3Hz, 293.1Hz and 344.44Hz, etc.) also appear.
Fig. 20 is a bifurcation diagram of the outer ring single point fault system. As can be seen from fig. 20, changing the order of the rolling bearing fractional order has a large influence on the motion characteristics of the system. When the fractional order alpha is less than or equal to 0.1, the rolling bearing system is in a chaotic state; when alpha is more than 0.1, the rolling bearing system enters a periodic motion state.
Fig. 21 and 22 are an outer ring single-point failure axial center locus diagram and a frequency spectrum diagram of the rolling bearing when the fractional order α is 0.6, respectively. As can be seen from fig. 21, the axial locus diagram, although being more complex, exhibits a motion characteristic of a distinct inner 8 shape, which indicates that the motion state of the rolling bearing at this time is a periodic motion, which corresponds to the conclusion of the order bifurcation diagram. And the range of the abscissa of the axle center track diagram is [ -1.5, 1.5]10-4m, the range of the ordinate is [ -1, 1]10-4m, and the axle center track of the single-point fault of the inner ring of the rolling bearing is approximately circular. As can be observed from the system spectrum diagram of fig. 22, the system is mainly composed of 2 × frequency multiplication components, and frequency multiplication components such as 1 × frequency multiplication components and 3 × frequency multiplication components appear in the spectrum.
Fig. 23 and 24 are an acceleration time domain graph and an amplitude frequency graph of the experimental result of the outer ring fault, respectively. No explicit useful information can be seen from the time domain waveform diagram of fig. 23. The rotational frequency (12Hz), the outer ring fault characteristic frequency and its higher harmonics (52Hz, 99Hz, 199Hz, 155Hz and 207Hz) can be derived from the spectrogram of fig. 24.
3. Fault model of rolling element of rolling bearing:
because a plurality of rolling bodies exist in the rolling bearing, and the rolling bodies not only revolve but also rotate during the rotation process. When the rolling body has a fault, the rolling body can be respectively contacted with the inner ring raceway and the outer ring raceway, so that a fault damage model of the rolling body is complex.
It is assumed that when there is a local single pitting failure on the rolling elements, the specific case thereof is as shown in fig. 25.
Maximum deflection lambda when a rolling element fails and contacts the outer race racewaymaxShould be at the rolling body deformation CdrSubtracting the deformation C of the outer ringdo(ii) a And when the rolling body is in fault and contacts with the inner ring raceway, the maximum deformation amount lambdamaxShould be at the rolling body deformation CdrIn addition, the deformation C of the inner ring is addeddiNamely:
Figure GDA0002023669020000101
therefore, the deformation amount calculation method of the actual gradual release is:
Figure GDA0002023669020000102
the dynamic dimensionless model of the rolling element after considering the damping force is as follows:
Figure GDA0002023669020000103
fig. 26 and 27 are a time domain graph of vibration signal displacement and a time domain graph of velocity of a rolling element failure, respectively. From fig. 26 and 27, it can be obtained that when the inner ring of the rolling bearing has the pitting failure, the speed time domain waveform diagram and the displacement waveform diagram have some same peak values, and if about 4-5 peak values exist within 0.4s, the frequency is 10-12.5 Hz, and the characteristic frequency is preliminarily estimated to be the rotor frequency (11.99 Hz).
Fig. 28 and 29 are a displacement amplitude-frequency diagram and a velocity amplitude-frequency diagram, respectively, of the vibration signal of the rolling element failure. Only 7.33Hz (approximately equal to the cage rotation frequency) and 62.27Hz (approximately equal to the characteristic frequency of rolling element failure) can be seen from the figure.
Fig. 30 is a branch diagram of the rolling element single point failure system. As can be seen from fig. 30, changing the order of the rolling bearing fractional order has a large influence on the motion characteristics of the system. When the fractional order alpha is less than or equal to 0.2, the rolling bearing system is in a chaotic state; when α >0.2, the rolling bearing system enters a periodic motion state.
Fig. 31 and 32 are a rolling element single point failure axial center locus diagram and a frequency spectrum diagram of the rolling element rolling bearing when the fractional order α is 0.6, respectively. As can be seen from fig. 31, the axle center locus diagram, although being more complex, exhibits a motion characteristic of a distinct ellipse, which indicates that the motion state of the rolling bearing at this time is a periodic motion, which corresponds to the conclusion of the order bifurcation diagram. And the abscissa range of the axle center track diagram is [ -1.5, 1.5]10-7m, the ordinate range is [ -4, 4]10-7m, and the axle center track of the rolling element single-point fault of the rolling bearing can be obtained to be elliptical. As can be observed from the system spectrum diagram of fig. 31, the system is dominated by 2 × frequency multiplication components.
Fig. 33 and 34 are an acceleration time domain diagram and an amplitude frequency diagram of the experimental results of the rolling element failure, respectively. It is difficult to find any information from the time domain waveform diagram of fig. 33. In the low-band frequency domain diagram of fig. 34, the main characteristic frequency of 145.7Hz is included, and some side bands, such as 121.4Hz, appear.

Claims (1)

1. The method for establishing the fractional order damping rolling bearing fault dynamics gradual model is characterized by comprising the following steps of:
1) on the basis of obtaining the equivalent contact rigidity and equivalent damping of the rolling bearing, the influence of a lubricant and the influence of the deformation of the inner ring when the rolling body breaks down are comprehensively considered, and a rolling bearing gradual change dynamic model containing a single pitting failure is established;
2) applying a fractional order calculus theory to the rolling bearing, deducing a fractional order damping force calculation formula, and respectively establishing an inner ring fault model, an outer ring fault model and a rolling element fault model of the rolling bearing, which take the fractional order damping into consideration;
a) the specific steps of establishing the fault model of the inner ring of the rolling bearing are as follows:
for a damaged bearing, the contact deformation δ between the jth rolling element and the raceway is:
δ＝xcosθj+ysinθj-e-βjλ+Hj (1)
Figure FDA0002973520050000011
Figure FDA0002973520050000012
in the formula, X is the radial X-direction displacement of the inner ring; y is the radial Y-direction displacement of the inner ring; thetajIs the position angle of the jth rolling body center; e is the radial gap; lambda is deformation of gradual release; hjIs the oil film thickness; omegacIs the angular velocity of rotation of the cage; z is the number of the rolling bodies; theta1The starting position angle of the rolling body is numbered 1; omegaiIs the angular velocity of rotation of the rotor; dbIs the diameter of the rolling body; dmIs the pitch circle diameter of the rolling bearing; alpha is a contact angle between the rolling body and the inner ring raceway;
βjis a switching value, which indicates that when the rolling body is positioned in the position angular span of the pitting defect, a certain contact deformation exists; when the rolling body is not in the position angular span of the pitting defect, the deformation amount does not exist any more; thus it is fixedMeaning as follows:
Figure FDA0002973520050000013
Figure FDA0002973520050000019
Figure FDA0002973520050000014
in the formula (I), the compound is shown in the specification,
Figure FDA0002973520050000015
is the position angle of the inner ring pitting failure center; delta phisIs the span angle of the fault; bcHalf the fault width; phispallIs the initial position angle of the pitting failure center; r isiIs the inner circle radius;
when the rolling body passes through the inner ring fault region, the released deformation amount is the deformation amount C of the rolling bodydrAnd deformation amount C of inner ringdiSumming; when the rolling body contacts the bottom of the defect, the maximum deformation released is lambdamax1Is equal to the fault depth, i.e.:
λmax1＝Cdr+Cdi (7)
Figure FDA0002973520050000016
Figure FDA0002973520050000017
in the formula, rbIs the rolling element radius;
the deformation amount of gradual release is as follows:
Figure FDA0002973520050000018
according to the Hertz contact theory, the method for calculating the contact load Q of the point contact between a single rolling body and the inner and outer ring raceways comprises the following steps:
Q＝K×δ3/2 (11)
wherein K is the total equivalent contact deformation coefficient, which is related to the contact deformation coefficient between the rolling body and the inner and outer ring raceways;
the contact between the rolling body and the inner and outer ring raceways is represented by a group of spring damping models; on the premise that the contact angles of the rolling body and the inner and outer ring raceways are equal, the total equivalent contact deformation coefficient is as follows:
Figure FDA0002973520050000021
Figure FDA0002973520050000022
wherein, the ball bearing n is 1.5; koAnd KiIs the contact deformation coefficient; n isδIs a coefficient related to a principal curvature difference function, and can be obtained by table lookup; Σ ρ is a contact principal curvature sum function, and the calculation method is:
∑ρ＝ρ1Ⅰ+ρ1Ⅱ+ρ2Ⅰ+ρ2Ⅱ (14)
in the formula, subscripts 1 and 2 represent a rolling element and inner and outer ring raceways, respectively; i represents the axial plane of the rotor; II represents a radial plane perpendicular to I; the specific calculation method of each principal curvature can be obtained by looking up a table;
nδis a calculation coefficient which is related to a main curvature difference function of the rolling bearing and is obtained by looking up a table; the principal curvature difference function is defined as:
Figure FDA0002973520050000023
and projecting all the contact loads Q in the X-axis direction and the Y-axis direction and adding the projection to obtain the total contact loads between the inner ring raceway and all the rolling bodies in the X-axis direction and the Y-axis direction:
Figure FDA0002973520050000024
Figure FDA0002973520050000025
adopting a Newtonian fluid model, namely considering the lubricant as Newtonian fluid; the tangential friction force borne by the jth rolling element is as follows:
τ＝η*Q (18)
in the formula, eta is the static friction coefficient;
from newton's second law, the kinetic equation of the inner ring of the rolling bearing is:
Figure FDA0002973520050000026
wherein m is the mass of the rolling bearing; c is the equivalent damping of the system; feIs the radial load of the inner ring; frApplying a load to the outer ring;
the damping force is a function of the first derivative of displacement, whose fractional derivative can be expressed as:
Fd＝cx′＝cDαx (20)
in the formula, FdIs a damping force; dαIs a complex variable of the Laplace transform;
the inner ring dynamic model after considering the damping force is as follows:
Figure FDA0002973520050000031
b) the specific steps of establishing the fault model of the outer ring of the rolling bearing are as follows:
the established kinetic equation is as follows:
Figure FDA0002973520050000032
in the formula (I), the compound is shown in the specification,
Figure FDA0002973520050000033
is the included angle between the external load and the X axis;
maximum deflection λmax2At the rolling body deformation CdrSubtracting the deformation C of the outer ringdoNamely:
λmax2＝Cdr-Cdo (23)
Figure FDA0002973520050000034
Figure FDA0002973520050000035
in the formula, roIs the outer ring radius;
therefore, the deformation amount calculation method of gradual release is:
Figure FDA0002973520050000036
c) the specific steps of establishing the rolling element fault model of the rolling bearing are as follows:
the dynamic dimensionless model of the rolling element after considering the damping force is as follows:
Figure FDA0002973520050000037
when the rolling body fails andmaximum deformation lambda when the outer ring raceways are in contactmax3Should be at the rolling body deformation CdrSubtracting the deformation C of the outer ringdo(ii) a And when the rolling body is in fault and contacts with the inner ring raceway, the maximum deformation amount lambdamax3Should be at the rolling body deformation CdrIn addition, the deformation C of the inner ring is addeddiNamely:
Figure FDA0002973520050000038
therefore, the deformation amount calculation method of gradual release is:
Figure FDA0002973520050000039
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