CN103927414A - Vibration response simulation analyzing method for rolling bearing with single-point failures - Google Patents

Abstract

The invention relates to a vibration response simulation analyzing method for a rolling bearing with single-point failures. On the basis of a Hertz contact theory, related knowledge about kinematics and kinetics are utilized to establish a five-degree-of-freedom nonlinear vibration mode of a rolling bearing and kinetic differential equations of the rolling bearing while influencing factors like slipping of rolling balls, oil slick rigidity and the like are taken into consideration, according to that the rolling balls change due to contact deformation is caused when the rolling balls roll through partial failures, a partial failure model of the rolling bearing is introduced into the kinetic differential equations of the rolling bearing, and the single-point failures of the rolling bearing are classified according to length-width ratios of the failures and size ratios between the failures and the rolling balls so as to establish different failure shape functions; the differential equations are solved by an ode solver in MATLAB (matrix laboratory) software, and vibration responses of the rolling bearing are simulated when an inner ring, an outer ring and the rolling balls of the rolling bearing are suffered from the single-point failures. By the aid of the vibration response simulation analyzing method, vibration responses of the bearings with failures in different sizes can be simulated. Compared with traditional methods for acquiring vibration responses of failed bearings, the vibration response simulation analyzing method has the advantages that an experimental period is short and experimental cost is saved. The computation for the vibration responses for the failed rolling bearings can lay the foundation of late failure diagnosis for the rolling bearings.

Description

Bearing vibration response simulating analysis containing Single Point of Faliure

Technical field

The invention belongs to simulation analysis field, be specifically related to a kind of Response Analysis method, particularly a kind of bearing vibration response simulating analysis that contains inner ring, outer ring or rolling body Single Point of Faliure.

Background technology

Rolling bearing is universal component conventional in rotating machinery, and its dynamic behavior affects the dynamic behavior of whole system, when it produces fault, will bring out whole system and produce fault.According to statistics, 30% of rotating machinery fault is caused by bearing fault, therefore, for rolling bearing, carry out condition monitoring and fault diagnosis research very crucial for the normal operation of service equipment, but, due to the mistaken diagnosis that reason causes such as mechanism is unclear, employing technology is improper and feature is not obvious with fail to pinpoint a disease in diagnosis to throw away and happen occasionally.The fault generation of research rolling bearing and the mechanism of transmission and Fault monitoring and diagnosis technology have important engineering and theory value.

The response signal of obtaining bearing is the basic and crucial of bearing failure diagnosis.Traditional method be by artificially arrange fault test obtain vibration signal or etc. the naturally-occurring of middle fault to be produced, experimental period is long, experiment cost is large.In order to promote the diagnosis of variable working condition heavy-duty gear fault in real system and the development of forecasting techniques, the realistic model that foundation produces the specific fault signal of rolling bearing conforming to actual condition is necessary.Set up effective bearing fault Simulation Engineering model, need not wait the naturally-occurring of middle fault to be produced, can directly produce as required variety classes fault, further investigate its failure mechanism and behavioral characteristics and can provide more accurate diagnostic method for the bearing fault detecting under different running statuses.In addition, the signal that fault simulation model produces can also be used to neural network training or is applied to the intelligent diagnostics algorithms such as support vector machine, automatically identify and prediction bearing fault type, reduce the dependence that cannot obtain a large amount of fault samples in actual production process, be conducive to the development of bearing failure diagnosis new method.

For the mechanism of rolling bearing fault, set up the dynamics model of vibration of research object, and to adopt numerical simulation technology to launch further investigation be the method generally adopting in the world at present.In bearing failure diagnosis, the system unit dither frequency that the damage of rolling bearing local surfaces evokes is generally more than 5KHz, the present invention is under non-linear hertz of contact force and radial force effect, combining unit resonator, by adjusting the high frequency proper vibration of the bearing enclose, sensor or other element that are evoked when damage occurs for its rigidity and ratio of damping simulation bearing, consider the factors such as ball slippage, oil film rigidity, set up outer ring, inner ring and the rolling body of rolling bearing

Summary of the invention

The present invention is for more accurately efficiently and emulation rolling bearing Single Point of Faliure vibratory response accurately, a kind of Single Point of Faliure rolling bearing simulating analysis has been proposed, with the method emulation rolling bearing Single Point of Faliure vibration signal, having played can any position of emulation, any large glitch, save the beneficial effect of experimental cost, simultaneously for the emulation of rolling bearing combined failure provides thinking.

For achieving the above object, technical scheme of the present invention is as follows:

A kind of Single Point of Faliure bearing vibration response simulating analysis that contains comprises the following steps:

Step 1 is set up rolling bearing model of nonlinear and dynamic differential equation group

Based on hertz contact theory, use kinematics knowledge relevant with dynamics, consider the influence factors such as ball slippage, oil film rigidity and bearing nonlinear time-varying rigidity and set up 5DOF rolling bearing model of nonlinear.

Step 2 is incorporated into fault in rolling bearing model of nonlinear

According to when rolling bearing has fault, the juxtaposition metamorphose that touches the ball of defect can change, a breakdown switch function is set, fault is incorporated in ball juxtaposition metamorphose formula, and according to Hertz theory, by distortion, solve contact force, contact force is updated in rolling bearing kinetics equation group, complete the introducing of fault.

Step 3 is set up the fault shape function of different faults

All failure conditions that can occur are classified, according to the difference of classification, set up different defect shape functions and represent different defect faults.

Step 4 solves differential equation group and draws bear vibration acceleration responsive figure

Utilize the ode solver in Matlab software, program, solve the differential equation, finally try to achieve outer ring, inner ring, rolling body and contain respectively the bear vibration acceleration responsive curve map of Single Point of Faliure;

The described response of the bearing vibration containing Single Point of Faliure simulating analysis, it is characterized in that: in described step 1), considered the ball slippage of rolling bearing, the factors such as oil film rigidity, and introduce the high frequency proper vibration that unit resonator is simulated the bearing enclose, sensor or other element that are evoked when damage occurs bearing, so more approach bearing actual condition.

The described response of the bearing vibration containing Single Point of Faliure simulating analysis, it is characterized in that: described step 2), defined a breakdown switch function, utilize this function that fault is incorporated in bearing vibration model, completed the mathematical expression of rolling bearing Single Point of Faliure, simple and convenient.

The described response of the bearing vibration containing Single Point of Faliure simulating analysis, it is characterized in that: in described step 3), according to the difference of defect shape, adopt different defect functions to represent defect, being set to a definite value with fault in the past compares, energy emulation, containing the vibratory response of the rolling bearing of various shape defects, makes simulation result more approach operating mode and no longer includes limitation more to approach actual conditions simultaneously like this.

The invention has the beneficial effects as follows: consider the factors such as ball slippage, oil film rigidity, introduce unit resonator, and adopt different defect functions according to the difference of flaw size, defect is incorporated in rolling bearing 5DOF model of nonlinear and dynamic differential equation group by ball juxtaposition metamorphose, finally utilize the ode solver in MATLAB to solve differential equation group, the response of emulation bearing vibration.This lays the foundation for the mechanism research of rolling bearing combined failure, and for rolling bearing fault diagnosis algorithm provides data, has solved traditional cycle that obtains by experiment fault vibration signal long, the problem that cost is high.

Accompanying drawing explanation

Below in conjunction with the drawings and specific embodiments, the invention will be further described.

Fig. 1 is the process flow diagram of the related method of the present invention.

Fig. 2 is 5DOF model of nonlinear of the present invention.

Fig. 3 is that bearing becomes flexibility vibration and load distribution schematic diagram.

Fig. 4 is the schematic diagram of housing washer while there is Single Point of Faliure.

The failure modes situation of Fig. 5 for carrying out according to different fault sizes.

Fig. 6 is fault side schematic view and corresponding fault shape function in different faults situation proposed by the invention.

Fig. 7 is ball schematic diagram and some geometric relationships while touching defect.

Fig. 8 is usingd 1205 bearings as specific embodiment, time-domain diagram and the frequency domain figure of the outer ring fault vibration response that employing the present invention simulates.

Fig. 9 is usingd 1205 bearings as specific embodiment, time-domain diagram and the frequency domain figure of the inner ring fault vibration response that employing the present invention simulates.

Figure 10 is usingd 1205 bearings as specific embodiment, time-domain diagram and the frequency domain figure of the rolling body fault vibration response that employing the present invention simulates.

Embodiment

The present invention is further illustrated with embodiment by reference to the accompanying drawings for lower mask body.

Fig. 1 is the process flow diagram of a kind of response of the bearing vibration containing Single Point of Faliure simulating analysis of the present invention.Specific embodiment of the invention step is as follows:

Step 1 is set up rolling bearing model of nonlinear and dynamic differential equation group

The 5DOF model of nonlinear of model rolling bearing as shown in Figure 2, a unit resonator is introduced in this model lower left, the high frequency proper vibration of simulating the bearing enclose, sensor or other element that are evoked when damage occurs bearing by rigidity and the ratio of damping of adjustment unit resonator.

Based on this model, in rolling bearing, consider after slippage position, the angle φ of j ball _jbe expressed as:

${\mathrm{\φ}}_j=\frac{2\mathrm{\π}(j-1)}{n_b}+{\mathrm{\ω}}_c\mathrm{dt}+{\mathrm{\φ}}_0+{\mathrm{\ξ}}_j\left(0.5\mathrm{rand}\right)\×{\mathrm{\φ}}_{\mathrm{slip}}---\left(1\right)$

Wherein, φ ₀the initial angle position that represents retainer, ω _crepresent retainer angular velocity, ω _sthe angular velocity that represents axle.When rolling body is positioned at supporting region, ξ _jfor+1; When rolling body is positioned at non-bearing district, ξ _jfor-1. Δ ff _m* 100% represents the sudden change percent of average contact frequency, and generally its value is between 1% and 2%, so corresponding φ _slipvalue (0.01rad～0.02rad).

Rolling bearing, in operational process, is affected by rolling body number and Radial Loads power scope, causes the support stiffness cyclical variation of bearing, produce and become flexibility vibration, as shown in Figure 3, the ball in supporting region can come in contact distortion, and total juxtaposition metamorphose of j ball can be expressed as:

δ _j＝(x _s-x _p)cosφ _j+(y _s-y _p)sinφ _j-cj＝1,2,...n _b. （2）

Wherein, c represents bearing clearance, n _brepresent ball number

By hertz contact theory, known, the contact force of j ball and raceway is expressed as:

$f_j=k_b{\mathrm{\δ}}_j^{1.5}---\left(3\right)$

By formula (2) and (3), can be released, bearing can be expressed as in total nonlinear contact power of x and y direction:

$f_x=k_b\mathrm{\Σ}{\mathrm{\γ}}_j{\mathrm{\δ}}_j^{1.5}\cos {\mathrm{\φ}}_j---\left(4\right)$

$f_y=k_b\mathrm{\Σ}{\mathrm{\γ}}_j{\mathrm{\δ}}_j^{1.5}\sin {\mathrm{\φ}}_j---\left(5\right)$

Wherein, γ _jbe a switch function, only have when ball is positioned at supporting region, ball just has distortion, just can produce contact force.Expression formula as follows:

${\mathrm{\γ}}_j=\left\{\begin{array}{cc}1& {\mathrm{\δ}}_j>0\\ 0& \mathrm{otherwise}\end{array}\right.---\left(6\right)$

Finally, according to kinematics and dynamic method, analyze time bearing model of vibration and show that rolling bearing differential equation of motion is:

$\left\{\begin{array}{c}{m}_s{\stackrel{\·\·}{x}}_s+c_s{\stackrel{\·}{x}}_s+k_s{x}_s+f_x=0\\ m_s{\stackrel{\·\·}{y}}_s+c_s{\stackrel{\·}{y}}_s+k_s{y}_s+f_y=F_r\\ m_p{\stackrel{\·\·}{x}}_p+c_p{\stackrel{\·}{x}}_p+k_p{x}_p-f_x=0\\ m_p{\stackrel{\·\·}{y}}_p+(c_p+c_r){\stackrel{\·}{y}}_p+(k_p+k_r)y_p-k_r{y}_b-c_r{\stackrel{\·}{y}}_b-f_y=0\\ m_r{\stackrel{\·\·}{y}}_b+c_r({\stackrel{\·}{y}}_b-{\stackrel{\·}{y}}_p)+k_r(y_b-y_p)=0\end{array}\right.---\left(7\right)$

Step 2 is incorporated into fault in rolling bearing model of nonlinear

When there is local fault in the inside and outside circle of rolling bearing or rolling body, when ball rolls across local fault, can discharge certain deflection, now enter the deflection δ of j rolling body of defect _jbecome

δ _j＝(x _s-x _p)cosφ _j+(y _s-y _p)sinφ _j-c-β _jc _d. （8）

Wherein, β _ja breakdown switch function, when ball is positioned at fault place, β _jvalue is 1, when ball does not touch defect fault, and β _jvalue is 0.

When inside and outside circle exists local fault (Fig. 4 has provided schematic diagram and some geometric relationships of outer ring fault model), failure definition is across angle delta φ _d, fault angle position φ _d, switch function β now _jcan be expressed as:

${\mathrm{\&beta;}}_j=\left\{\begin{array}{cc}1& {\mathrm{\&phi;}}_d<{\mathrm{\&phi;}}_j<{\mathrm{\&phi;}}_d+\mathrm{\&Delta;}{\mathrm{\&phi;}}_d\\ 0& \mathrm{otherwise}\end{array}\right.---\left(9\right)$

When bearing exists outer ring fault, fault occurs in supporting region, and position is fixed, now φ _dit is a definite value.When inner ring exists fault, abort situation is along with inner ring rotates and changes, now φ _da variate, φ _d=ω _st+ φ _d0, φ _d0it is position, primary fault angle.

When rolling body has local defect, the defective rolling body of tool can contact once with inside and outside circle respectively in the process turning around.Because Internal and external cycle raceway radius-of-curvature is different, the fault contacting with Internal and external cycle is across angle delta φ _dwill be different, when ball contacts with Internal and external cycle, the greatest drawback degree of depth touching is also different, so when rolling body has fault, for β _jdefinition adjustment as follows:

${\mathrm{\&beta;}}_j=\left\{\begin{array}{ccc}\begin{array}{c}0\\ 1\\ \frac{c_{\mathrm{dr}}+c_{\mathrm{di}}}{c_{\mathrm{dr}}-c_{\mathrm{do}}}\\ 0\end{array}& \left.\begin{array}{c}0<{\mathrm{\&phi;}}_s<\mathrm{\&Delta;}{\mathrm{\&phi;}}_{\mathrm{do}}\\ \mathrm{\&pi;}<{\mathrm{\&phi;}}_s<\mathrm{\&pi;}+\mathrm{\&Delta;}{\mathrm{\&phi;}}_{\mathrm{di}}\end{array}\right\}& \begin{array}{c}j\&NotEqual;k\\ j=k\\ \mathrm{else}\end{array}\end{array}\right.---\left(10\right)$

Wherein,

(outer ring touches the depth capacity of ball fault) (11)

(inner ring touches the depth capacity of ball fault) (12)

(Maximum Contact loss amount when ball fault contacts with inner ring) (13)

In above-mentioned formula, outer ring radius inner ring radius d _prepresent pitch diameter, D _brepresent ball diameter.

Step 3 is set up the failure function of different faults

When rolling bearing has local defect, when ball rolls across this place, can discharge certain deflection, the definite value that the existing research about local fault bearing system dynamics is is nearly all simply depth of defect by deflection value of setting for of release, and the actual deflection discharging according to defect shape with it different and different from ball size ratio.

The ratio of ball size and flaw size is defined as:

${\mathrm{\&eta;}}_{\mathrm{bd}}=\frac{d}{\min (L,B)}---\left(14\right)$

The length-width-ratio of defect self is defined as:

${\mathrm{\&eta;}}_d=\frac{L}{B}---\left(15\right)$

Wherein, L and B represent the length of defect and wide.

According to the ratio of defect and ball size defectiveness self length-width-ratio also, defect can be divided into 5 kinds of situations shown in Fig. 5: (1) defect is only that a crackle is that ball size is far longer than flaw size, as shown in Fig. 5 (a), η now _bd> > 1; (2) ball size and flaw size are suitable, and defect width is greater than length as shown in Fig. 5 (b), now η _bd> 1and η _d< 1; (3) ball size and flaw size are suitable, and defect width equals length as shown in Fig. 5 (c), now η _bd> 1and η _d=1; (4) ball size and flaw size are suitable, and defect width equals length as shown in Fig. 5 (d), now η _bd> 1and η _d> 1; (5) ball size is far smaller than flaw size as shown in Figure 5 (e) shows, now η _bd≤ 1.

Defect according to above five types, the depth of defect that ball can touch when rolling across defect can be divided into following three kinds of situations: (1) is when defect is the first type, outboard profile when ball rolls across defect as shown in Figure 6 (a), the firm contact deficiency of ball has just immediately left defect, now can represent depth of defect with the rectangular function of Fig. 6 (b), depth of defect keeps a definite value always; (2) when defect type is the second and the third defect, side schematic view when ball rolls across defect as shown in Figure 6 (c), ball is in rolling across the process of defect, the depth of defect that ball can touch is along with the rolling of ball slowly increases, depth of defect slowly reduces again after reaching maximal value, now can represent depth of defect with the semisinusoidal function shown in Fig. 6 (d); (3) when defect type is the 4th kind and five defective, schematic diagram when ball rolls across defect as shown in Figure 6 (e), rolling along with ball, the depth of defect that ball can touch slowly increases, after reaching maximal value, remain unchanged a period of time, slowly reduce again, now can represent the depth of defect that ball touches with the piecewise function shown in Fig. 6 (f)

To sum up, the depth of defect function that ball can touch can be expressed as:

$\mathrm{cd}=\left\{\begin{array}{cc}{H}_1& {\mathrm{\&eta;}}_{\mathrm{bd}}>>1\\ H_2& {\mathrm{\&eta;}}_{\mathrm{bd}}>\mathrm{land}{\mathrm{\&eta;}}_d\&le;1\\ H_3& {\mathrm{\&eta;}}_{\mathrm{bd}}>\mathrm{land}{\mathrm{\&eta;}}_d>1\\ H_3& {\mathrm{\&eta;}}_{\mathrm{bd}}\&le;1\end{array}\right.---\left(16\right)$

Wherein, H ₁presentation graphs 6(b) rectangular function shown in

H ₁＝cd' （17）

H ₂presentation graphs 6(d) the semisinusoidal function shown in

H ₃presentation graphs 6(f) piecewise function shown in

Wherein, φ represents that ball enters the angle rolling across after defect, and its scope is [0 Δ φ _d], meanwhile, from the geometric relationship of Fig. 7,

H _d＝0.5d-((0.5d) ²-(0.5B) ²) ^0.5 （20）

Therefore, cd' can be expressed as:

$\mathrm{cd}\text{'}=\left\{\begin{array}{cc}H& H<H_d\\ H_d& H\&GreaterEqual;H_d\end{array}\right.---\left(21\right)$

Step 4 is brought into the contact force after the introducing fault of obtaining by step 2,3 in system of equations (1).Utilize the ode solver in MATLAB software, the numerical solution of Program system of equations (1), is used for the vibratory response of emulation Single Point of Faliure bearing.The present invention adopts 1205 rolling bearings as specific embodiment, the ball number n of 1205 bearings _b=12, ball diameter D _b=7.12mm, pitch diameter D _p=38.5mm.Some physical parameters of 1205 bearings are respectively outer ring, inner ring and rolling body containing the vibration acceleration response curve map of the rolling bearing of Single Point of Faliure in Table 1, Fig. 8-10.

Some physical parameters of table 11205 bearing

Claims (5)

1. containing a bearing vibration response simulating analysis for Single Point of Faliure, it is characterized in that: the method comprises the steps:

Step 1 is set up rolling bearing model of nonlinear and dynamic differential equation group

Based on hertz contact theory, use kinematics knowledge relevant with dynamics, consider ball slippage, oil film rigidity and bearing nonlinear time-varying stiffness effect factor and set up 5DOF rolling bearing model of nonlinear;

Step 2 is incorporated into fault in rolling bearing model of nonlinear

According to when rolling bearing has fault, the juxtaposition metamorphose that touches the ball of defect can change, a breakdown switch function is set, fault is incorporated in ball juxtaposition metamorphose formula, and according to Hertz theory, by distortion, solve contact force, contact force is updated in rolling bearing kinetics equation group, complete the introducing of fault

Step 3 is set up the fault shape function of different faults

All defect situation that can occur are classified, according to the difference of classification, set up different defect shape functions and represent different defect faults.

Step 4 solves differential equation group and draws bear vibration acceleration responsive figure

Utilize the ode solver in Matlab software, program, solve the differential equation, finally try to achieve outer ring, inner ring, rolling body and contain respectively the bear vibration acceleration responsive curve map of Single Point of Faliure.

2. the bearing vibration containing Single Point of Faliure according to claim 1 responds simulating analysis, it is characterized in that: above-mentioned steps 1), considered the ball slippage of rolling bearing, the factors such as oil film rigidity, and introduce the high frequency proper vibration that unit resonator is simulated the bearing enclose, sensor or other element that are evoked when damage occurs bearing, so more approach bearing actual condition.

3. the bearing vibration containing Single Point of Faliure according to claim 1 responds simulating analysis, it is characterized in that: above-mentioned steps 2), defined a breakdown switch function, utilize this function that fault is incorporated in bearing vibration model, completed the mathematical expression of rolling bearing Single Point of Faliure, simple and convenient.

4. the bearing vibration containing Single Point of Faliure according to claim 1 responds simulating analysis, it is characterized in that: above-mentioned steps 3), according to the difference of defect shape, adopt different defect functions to represent defect, being set to a definite value with fault in the past compares, energy emulation, containing the vibratory response of the rolling bearing of various shape defects, makes simulation result more approach operating mode and no longer includes limitation more to approach actual conditions simultaneously like this.

5. the response of the bearing vibration containing Single Point of Faliure simulating analysis according to claim 1, is characterized in that: a kind of bearing vibration containing Single Point of Faliure of the present invention responds simulating analysis, and its concrete implementation step is as follows,

Step 1 is set up rolling bearing model of nonlinear and dynamic differential equation group

The 5DOF model of nonlinear of model rolling bearing, a unit resonator is introduced in this model lower left, the high frequency proper vibration of simulating the bearing enclose, sensor or other element that are evoked when damage occurs bearing by rigidity and the ratio of damping of adjustment unit resonator;

Based on this model, in rolling bearing, consider after slippage position, the angle φ of j ball _jbe expressed as:

${\mathrm{\&phi;}}_j=\frac{2\mathrm{\&pi;}(j-1)}{n_b}+{\mathrm{\&omega;}}_c\mathrm{dt}+{\mathrm{\&phi;}}_0+{\mathrm{\&xi;}}_j\left(0.5\mathrm{rand}\right)\&times;{\mathrm{\&phi;}}_{\mathrm{slip}}---\left(1\right)$

Wherein, φ ₀the initial angle position that represents retainer, ω _crepresent retainer angular velocity, ω _sthe angular velocity that represents axle; When rolling body is positioned at supporting region, ξ _jfor+1; When rolling body is positioned at non-bearing district, ξ _jfor-1; Δ ff _m* 100% represents the sudden change percent of average contact frequency, and generally its value is between 1% and 2%, so corresponding φ _slipvalue (0.01rad～0.02rad);

Rolling bearing is in operational process, affected by rolling body number and Radial Loads power scope, cause the support stiffness cyclical variation of bearing, produce and become flexibility vibration, ball in supporting region can come in contact distortion, and total juxtaposition metamorphose of j ball can be expressed as:

δ _j＝(x _s-x _p)cosφ _j+(y _s-y _p)sinφ _j-c j＝1,2,...n _b. （2）

Wherein, c represents bearing clearance, n _brepresent ball number

By hertz contact theory, known, the contact force of j ball and raceway is expressed as:

$f_j=k_b{\mathrm{\&delta;}}_j^{1.5}---\left(3\right)$

By formula (2) and (3), can be released, bearing can be expressed as in total nonlinear contact power of x and y direction:

$f_x=k_b\mathrm{\&Sigma;}{\mathrm{\&gamma;}}_j{\mathrm{\&delta;}}_j^{1.5}\cos {\mathrm{\&phi;}}_j---\left(4\right)$

$f_y=k_b\mathrm{\&Sigma;}{\mathrm{\&gamma;}}_j{\mathrm{\&delta;}}_j^{1.5}\sin {\mathrm{\&phi;}}_j---\left(5\right)$

Wherein, γ _jbe a switch function, only have when ball is positioned at supporting region, ball just has distortion, just can produce contact force; Expression formula as follows:

${\mathrm{\&gamma;}}_j=\left\{\begin{array}{cc}1& {\mathrm{\&delta;}}_j>0\\ 0& \mathrm{otherwise}\end{array}\right.---\left(6\right)$

Finally, according to kinematics and dynamic method, analyze time bearing model of vibration and show that rolling bearing differential equation of motion is:

$\left\{\begin{array}{c}{m}_s{\stackrel{\&CenterDot;\&CenterDot;}{x}}_s+c_s{\stackrel{\&CenterDot;}{x}}_s+k_s{x}_s+f_x=0\\ m_s{\stackrel{\&CenterDot;\&CenterDot;}{y}}_s+c_s{\stackrel{\&CenterDot;}{y}}_s+k_s{y}_s+f_y=F_r\\ m_p{\stackrel{\&CenterDot;\&CenterDot;}{x}}_p+c_p{\stackrel{\&CenterDot;}{x}}_p+k_p{x}_p-f_x=0\\ m_p{\stackrel{\&CenterDot;\&CenterDot;}{y}}_p+(c_p+c_r){\stackrel{\&CenterDot;}{y}}_p+(k_p+k_r)y_p-k_r{y}_b-c_r{\stackrel{\&CenterDot;}{y}}_b-f_y=0\\ m_r{\stackrel{\&CenterDot;\&CenterDot;}{y}}_b+c_r({\stackrel{\&CenterDot;}{y}}_b-{\stackrel{\&CenterDot;}{y}}_p)+k_r(y_b-y_p)=0\end{array}\right.---\left(7\right)$

Step 2 is incorporated into fault in rolling bearing model of nonlinear

When there is local fault in the inside and outside circle of rolling bearing or rolling body, when ball rolls across local fault, can discharge certain deflection, now enter the deflection δ of j rolling body of defect _jbecome

δ _j＝(x _s-x _p)cosφ _j+(y _s-y _p)sinφ _j-c-β _jc _d. （8）

Wherein, β _ja breakdown switch function, when ball is positioned at fault place, β _jvalue is 1, when ball does not touch defect fault, and β _jvalue is 0;

When inside and outside circle exists local fault (Fig. 4 has provided schematic diagram and some geometric relationships of outer ring fault model), failure definition is across angle delta φ _d, fault angle position φ _d, switch function β now _jcan be expressed as:

${\mathrm{\&beta;}}_j=\left\{\begin{array}{cc}1& {\mathrm{\&phi;}}_d<{\mathrm{\&phi;}}_j<{\mathrm{\&phi;}}_d+\mathrm{\&Delta;}{\mathrm{\&phi;}}_d\\ 0& \mathrm{otherwise}\end{array}\right.---\left(9\right)$

When bearing exists outer ring fault, fault occurs in supporting region, and position is fixed, now φ _dit is a definite value; When inner ring exists fault, abort situation is along with inner ring rotates and changes, now φ _da variate, φ _d=ω _st+ φ _d0, φ _d0it is position, primary fault angle;

When rolling body has local defect, the defective rolling body of tool can contact once with inside and outside circle respectively in the process turning around; Because Internal and external cycle raceway radius-of-curvature is different, the fault contacting with Internal and external cycle is across angle delta φ _dwill be different, when ball contacts with Internal and external cycle, the greatest drawback degree of depth touching is also different, so when rolling body has fault, for β _jdefinition adjustment as follows:

${\mathrm{\&beta;}}_j=\left\{\begin{array}{ccc}\begin{array}{c}0\\ 1\\ \frac{c_{\mathrm{dr}}+c_{\mathrm{di}}}{c_{\mathrm{dr}}-c_{\mathrm{do}}}\\ 0\end{array}& \left.\begin{array}{c}0<{\mathrm{\&phi;}}_s<\mathrm{\&Delta;}{\mathrm{\&phi;}}_{\mathrm{do}}\\ \mathrm{\&pi;}<{\mathrm{\&phi;}}_s<\mathrm{\&pi;}+\mathrm{\&Delta;}{\mathrm{\&phi;}}_{\mathrm{di}}\end{array}\right\}& \begin{array}{c}j\&NotEqual;k\\ j=k\\ \mathrm{else}\end{array}\end{array}\right.---\left(10\right)$

Wherein,

(outer ring touches the depth capacity of ball fault) (11)

(inner ring touches the depth capacity of ball fault) (12)

(Maximum Contact loss amount when ball fault contacts with inner ring) (13)

In above-mentioned formula, outer ring radius inner ring radius d _prepresent pitch diameter, D _brepresent ball diameter;

Step 3 is set up the failure function of different faults

When rolling bearing has local defect, when ball rolls across this place, can discharge certain deflection, the definite value that the existing research about local fault bearing system dynamics is is nearly all simply depth of defect by deflection value of setting for of release, and the actual deflection discharging according to defect shape with it different and different from ball size ratio;

The ratio of ball size and flaw size is defined as:

${\mathrm{\&eta;}}_{\mathrm{bd}}=\frac{d}{\min (L,B)}---\left(14\right)$

The length-width-ratio of defect self is defined as:

${\mathrm{\&eta;}}_d=\frac{L}{B}---\left(15\right)$

Wherein, L and B represent the length of defect and wide;

According to the ratio of defect and ball size defectiveness self length-width-ratio also, defect can be divided into 5 kinds of situations: (1) defect is only that a crackle is that ball size is far longer than flaw size, now η _bd> > 1; (2) ball size and flaw size are suitable, and defect width is greater than shown in length, now η _bd> 1and η _d< 1; (3) ball size and flaw size are suitable, and defect width equals length, now η _bd> 1and η _d=1; (4) ball size and flaw size are suitable, and defect width equals length, now η _bd> 1and η _d> 1; (5) ball size is far smaller than flaw size, now η _bd≤ 1;

Defect according to above five types, the depth of defect that ball can touch when rolling across defect can be divided into following three kinds of situations: (1) is when defect is the first type, side when ball rolls across defect, the firm contact deficiency of ball has just immediately left defect, now can represent depth of defect with rectangular function, depth of defect keeps a definite value always; (2) when defect type is the second and the third defect, side schematic view when ball rolls across defect, ball is in rolling across the process of defect, the depth of defect that ball can touch is along with the rolling of ball slowly increases, depth of defect slowly reduces again after reaching maximal value, now can represent depth of defect with semisinusoidal function; (3) when defect type is the 4th kind and five defective, schematic diagram when ball rolls across defect, rolling along with ball, the depth of defect that ball can touch slowly increases, after reaching maximal value, remain unchanged a period of time, slowly reduce again, now can with piecewise function represent the depth of defect that ball touches;

To sum up, the depth of defect function that ball can touch can be expressed as:

$\mathrm{cd}=\left\{\begin{array}{cc}{H}_1& {\mathrm{\&eta;}}_{\mathrm{bd}}>>1\\ H_2& {\mathrm{\&eta;}}_{\mathrm{bd}}>\mathrm{land}{\mathrm{\&eta;}}_d\&le;1\\ H_3& {\mathrm{\&eta;}}_{\mathrm{bd}}>\mathrm{land}{\mathrm{\&eta;}}_d>1\\ H_3& {\mathrm{\&eta;}}_{\mathrm{bd}}\&le;1\end{array}\right.---\left(16\right)$

Wherein, H ₁presentation graphs 6(b) rectangular function shown in

H ₁＝cd' （17）

H ₂presentation graphs 6(d) the semisinusoidal function shown in

H ₃the piecewise function representing

Wherein, φ represents that ball enters the angle rolling across after defect, and its scope is [0 Δ φ _d], meanwhile, from geometric relationship,

H _d＝0.5d-((0.5d) ²-(0.5B) ²) ^0.5 （20）

Therefore, cd' can be expressed as:

$\mathrm{cd}\text{'}=\left\{\begin{array}{cc}H& H<H_d\\ H_d& H\&GreaterEqual;H_d\end{array}\right.---\left(21\right)$

Step 4 is brought into the contact force after the introducing fault of obtaining by step 2,3 in system of equations (1); Utilize the ode solver in MATLAB software, the numerical solution of Program system of equations (1), is used for the vibratory response of emulation Single Point of Faliure bearing; The present invention adopts 1205 rolling bearings as specific embodiment, the ball number n of 1205 bearings _b=12, ball diameter D _b=7.12mm, pitch diameter D _p=38.5mm.