CN107092750A - A kind of Nonlinear dynamic models method of inside engaged gear axle tooth of reducer damage fault - Google Patents
A kind of Nonlinear dynamic models method of inside engaged gear axle tooth of reducer damage fault Download PDFInfo
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Abstract
The invention discloses a kind of Nonlinear dynamic models method of inside engaged gear axle tooth of reducer damage fault, belong to Reliability of Mechanical System field of engineering technology；Specially：First, the condition of certain inside engaged gear axle decelerator modeling is set；Then, relative linear displacement of the gear pair along path of contact, static driving error and nonlinear backlash are determined successively；Further, the gear teeth time-variant mesh stiffness approached using Fourier space at the time-variant mesh stiffness and single broken teeth failure of health status lower tooth wheel set, and comprehensive time-variant mesh stiffness is calculated, while introducing gear teeth damage factor ε, gear teeth damage fault time-variant mesh stiffness is modeled.Finally, the nonlinear dynamical equation of gear pair is set up, and carries out linear transformation and nondimensionalization processing.The model of the present invention has preferable versatility, by gear teeth damage factor, and the vibration signal output that can slow down to the internal messing of any degree of injury of gear teeth failure carries out numerical simulation, and emulator is high.
Description
Technical field
The invention belongs to Reliability of Mechanical System field of engineering technology, it is related to a kind of inside engaged gear axle tooth of reducer and damages
Hinder the Nonlinear dynamic models method of failure.
Background technology
Inside engaged gear axle decelerator has the advantages that compact conformation, performance be stable and efficient work, it is civilian, industrial,
Aerospace field is applied widely, therefore the theoretical research to its working mechanism is significant.Reducer gear pair
During work, even normal condition, due to the influence of gear tooth friction, backlash and time-variant mesh stiffness, gear vibration table
Reveal very strong nonlinear characteristic.
To realize high accuracy, low noise, low vibration and the target for improving gear drive controllability, researcher is in gear
Substantial amounts of time and efforts has been put into above the Nonlinear Dynamics Problems of transmission system.The ripplings of the gear teeth, peel off and disconnected
Tooth equivalent damage is the most common failure form of gear reduction unit, and the generation development of failure has a strong impact on the reliability index of decelerator,
Serious property and life security loss are even caused, its occurrence and development mechanism is always the weight of scientific and technical research personnel research
Point.
The research of the nonlinear kinetics of inside engaged gear axle tooth of reducer failure is to the theoretical of gear reduction unit and answers
There is important value with research.
The content of the invention
In order to disclose the gear teeth damage fault of inside engaged gear axle decelerator the theory of development mechanism occurs for the present invention, passes through
Gear tooth friction damping, the influence of backlash, time-variant mesh stiffness to its dynamics of decelerator are studied, establishes and includes
Gear tooth friction, non-linear backlash function and time-variant mesh stiffness, gear teeth damage factor and the internal messing for engaging viscous damping
Gear shaft reducer power model, specifically a kind of nonlinear kinetics of inside engaged gear axle tooth of reducer damage fault
Modeling method.
Comprise the following steps that：
Step 1: for some inside engaged gear axle decelerator, setting the condition being modeled to the decelerator；
(1) internal gear and external gear of the decelerator are involute spur gear；
(2) gear teeth failure refers to rippling, peeling and the broken teeth equivalent damage failure mode of reducer gear；
(3) two gear gear blanks are considered as rigid body, and the input of decelerator and output shaft are considered as rigid body；The support stiffness of two gear shafts
It is sufficiently large, do not consider the elastic deformation of support；
(4) each parts are not by axial force in decelerator, and vibration vector has the plane perpendicular to axis；
(5) driving wheel (external gear), the gear teeth of driven pulley (internal gear) do cantilever beam consideration, there are the gear teeth along path of contact
With respect to slide displacement；
(6) train middle gear is installed according to reference center distance, and pitch circle is overlapped with reference circle；
(7) part's machining errors and alignment error are not counted in.
Step 2: for the inside engaged gear axle decelerator, the phase in gear pair path of contact is determined using gear angular displacement
To displacement of the lines；
Wherein：X is the relative linear displacement along path of contact in internal gear Meshing Pair；Y is along path of contact in external gear Meshing Pair
Relative linear displacement；r1It is the reference radius of external gear；r2It is the reference radius of internal gear；θ1It is the torsion angle of external gear
Displacement；θ2It is the torsional angular displacement of internal gear；T is time variable；E (t) is the static driving error of gear pair, e (t)=easin
ωmT, eaFor error magnitude, ωmFor engaging tooth frequency.
Step 3: determining nonlinear backlash letter according to the relative linear displacement in internal gear Meshing Pair along path of contact
Number；
In formula, b is backlash constant.
Step 4: approaching the time-variant mesh stiffness function k of health status lower tooth wheel set using Fourier spacem(t)；
Wherein, kavIt is the average mesh stiffness of gear pair, kaIt is stiffness variation amplitude；N rounds numerical value.
Step 5: the gear teeth time-variant mesh stiffness function k (t) approached using Fourier space at single broken teeth failure
kafIt is the peak-to-peak value that monodentate mesh stiffness changes at broken teeth, Z1For the number of teeth of external gear that is, driving wheel；ω1It is outer
The nominal angular speed of gear；
Step 6: firm using the time-variant mesh stiffness and the gear teeth time-varying of single broken teeth malfunction of health status gear pair
Degree is made the difference, and combines gear teeth damage factor, and gear teeth damage fault time-variant mesh stiffness is modeled；
Wherein, kmf(t) it is the gear pair time-variant mesh stiffness containing gear teeth damage fault；
ε is gear teeth face damage factor；ε ∈ [0,1], are defined as follows：
Step 7: using the backlash function of the decelerator, with reference to the modeling of gear teeth damage fault time-variant mesh stiffness, adopting
Lagrange methods are used, the nonlinear dynamical equation of gear pair is set up；
The nonlinear dynamical equation of gear pair includes the input matrix T of internal gear pair1With output matrix T2；
Wherein：I1It is the rotary inertia of external gear；I2It is the rotary inertia of internal gear；It is the instantaneous angular acceleration of external gear；It is the instantaneous angular acceleration of internal gear；cmIt is engagement viscous damping coefficient；l1It is the arm of force of the engaging friction damping force of external gear；
l2It is the arm of force of the engaging friction damping force of internal gear；λ is direction coefficient,
ω2It is the nominal angular speed of internal gear；It is the instantaneous angular velocity of external gear；It is the instantaneous angular velocity of internal gear；μ is the flank of tooth
The coefficient of kinetic friction；
Step 8: carrying out linear transformation and nondimensionalization processing to the nonlinear dynamical equation of gear pair, obtain final
It is included in internal gear pair dimensionless non-linear dynamic model under the gear teeth damage fault state of friction and backlash；
Model is as follows：
τ=ωet；ωeIt is gear pair equivalent intrinsic frequency,meIt is gear pair equivalent quality,m1It is the quality of external gear, m2It is the quality of internal gear；
eaIt is the amplitude of gear teeth composition error change, is constant；
The advantage of the invention is that：
(1) a kind of Nonlinear dynamic models method of inside engaged gear axle tooth of reducer damage fault, is based on
Lagrange methods establish the non-linear dynamic model of internal gear pair, and the versatility of this model very well, can appoint
The dynamic numerical simulation of internal gear pair under operating mode of anticipating.
(2) a kind of Nonlinear dynamic models method of inside engaged gear axle tooth of reducer damage fault, introduces the gear teeth
Damage factor ε, the vibration signal output that can slow down to the internal messing of any degree of injury of gear teeth failure carries out numerical simulation.
(3) a kind of Nonlinear dynamic models method of inside engaged gear axle tooth of reducer damage fault, model is fallen into a trap
The excitation of the inner parameters such as frictional damping effect, non-linear backlash function, time-variant mesh stiffness, numerical simulation precision are entered
Height, the gear reducer system vibration signal emulator of model output is high, and nonlinear dynamic characteristic is obvious.
Brief description of the drawings
Fig. 1 is the structural representation for the inside engaged gear axle decelerator that the present invention is used；
Fig. 2 is inside engaged gear axle tooth of reducer failure non-linear dynamic model principle schematic of the present invention；
Fig. 3 is the gear pair time-variant mesh stiffness waveform diagram that the present invention carries gear teeth damage fault；
Fig. 4 is the Nonlinear dynamic models method flow of inside engaged gear axle tooth of reducer damage fault of the present invention
Figure；
Fig. 5 is the phase mark figure of inside engaged gear axle tooth of reducer health status steady-state vibration of the present invention；
Fig. 6 is the time-domain diagram of invention inside engaged gear axle tooth of reducer health status steady-state vibration；
Fig. 7 is the frequency domain figure of inside engaged gear axle tooth of reducer health status steady-state vibration of the present invention；
Fig. 8 is the phase mark figure of inside engaged gear axle tooth of reducer minor failure state steady-state vibration of the present invention；
Fig. 9 is the time-domain diagram of inside engaged gear axle tooth of reducer minor failure state steady-state vibration of the present invention；
Figure 10 is the frequency domain figure of inside engaged gear axle tooth of reducer minor failure state steady-state vibration of the present invention；
Figure 11 is the phase mark figure of inside engaged gear axle decelerator broken conditions steady-state vibration of the present invention；
Figure 12 is the time-domain diagram of inside engaged gear axle decelerator broken conditions steady-state vibration of the present invention；
Figure 13 is the frequency domain figure of inside engaged gear axle decelerator broken conditions steady-state vibration of the present invention.
Embodiment
Below in conjunction with drawings and examples, the present invention is described in further detail.
The present invention devises a kind of comprising gear tooth friction, non-linear backlash function, gear teeth time-variant mesh stiffness, the gear teeth
The non-linear dynamic model of damage factor and the engagement inside engaged gear axle decelerator of factor such as viscous damping, for object
It is certain type inside engaged gear axle decelerator, based on Lagrange methods, it is proposed that be included in the internal messing tooth of friction and backlash
The Nonlinear dynamic models method of wheel shaft tooth of reducer damage fault.
As shown in figure 1, certain the type inside engaged gear axle reducer structure used for the present embodiment；Set up be included in friction and
The gear teeth damage fault non-linear dynamic model schematic diagram of backlash as shown in Fig. 2
The symbol used in model process is as shown in table 1：
Table 1
Specific modeling procedure is as shown in figure 4, including as follows：
Step 1: for some inside engaged gear axle decelerator, setting the condition being modeled to the decelerator；
(1) internal gear and external gear of the decelerator are involute spur gear；
(2) gear teeth failure refers to rippling, peeling and the broken teeth equivalent damage failure mode of reducer gear；
(3) two gear gear blanks are considered as rigid body, and the input of decelerator and output shaft are considered as rigid body；The support of two gear shafts is firm
Degree is sufficiently large, does not consider the elastic deformation of support；
(4) each parts are not by axial force in decelerator, and vibration vector has the plane perpendicular to axis；
(5) driving wheel (external gear), the gear teeth of driven pulley (internal gear) do cantilever beam consideration, there are the gear teeth along path of contact
With respect to slide displacement；
(6) train middle gear is installed according to reference center distance, and pitch circle is overlapped with reference circle；
(7) part's machining errors and alignment error are not counted in.
Step 2: for the inside engaged gear axle decelerator, the phase in gear pair path of contact is determined using gear angular displacement
To displacement of the lines；
Represent the N in the path of contact such as Fig. 2 of internal gear pair1N2；
Wherein：X is the relative linear displacement along path of contact in internal gear Meshing Pair；Y is along path of contact in external gear Meshing Pair
Relative linear displacement；r1It is the reference radius of external gear, is constant value；r2It is the reference radius of internal gear, is constant value；θ1It is
The instantaneous angular displacement of external gear, is the function on t；θ2It is the instantaneous angular displacement of internal gear, is the function on t；T is the time
Variable；E (t) is the static driving error of gear pair, e (t)=easinωmT, eaFor error magnitude, ωmFor engaging tooth frequency.
Step 3: determining nonlinear backlash letter according to the relative linear displacement in internal gear Meshing Pair along path of contact
Number；
In formula, b is backlash constant.
Step 4: approaching the time-variant mesh stiffness function k of health status lower tooth wheel set using Fourier spacem(t)；
Wherein, kavIt is the average mesh stiffness of gear pair, kaIt is stiffness variation amplitude；N rounds numerical value.
As shown in Figure 3 a, normal condition Gear Meshing Stiffness, mesh cycle T0Under, the peak-to-peak value k of mesh stiffness changea；Nibble
The average value for closing stiffness variation is kav；
Step 5: the gear teeth time-variant mesh stiffness function k (t) approached using Fourier space at single broken teeth failure
kafIt is the peak-to-peak value that monodentate mesh stiffness changes at broken teeth, Z1It is also the number of teeth of driving wheel for external gear, is constant value；
ω1It is the nominal angular speed of external gear, is constant value；
As shown in Figure 3 b, mesh stiffness during the single broken teeth failure of the flank of tooth, swing circle TzUnder, Z1For the tooth of driving gear
Number；The average value of broken teeth mesh stiffness change is kav1；
Step 6: utilizing health status gear pair time-variant mesh stiffness and the gear teeth time-varying rigidity of single broken teeth malfunction
Make the difference, and combine gear teeth damage factor, gear teeth damage fault time-variant mesh stiffness is modeled；
Wherein, kmf(t) it is the gear pair time-variant mesh stiffness containing gear teeth damage fault；
ε is gear teeth face damage factor；ε ∈ [0,1], are defined as follows：
As shown in Figure 3 c, the peak-to-peak value k that monodentate mesh stiffness changes at broken teethavf=kav-kav1。
Step 7: using the backlash function of the decelerator, with reference to the modeling of gear teeth damage fault time-variant mesh stiffness, adopting
Lagrange methods are used, the nonlinear dynamical equation of gear pair is set up；
The nonlinear dynamical equation of gear pair includes the input matrix T of internal gear pair1With output matrix T2；Input
Matrix is also that driving torque is used；Output matrix is also that load torque is used；
Wherein：I1It is the rotary inertia of external gear, is constant value；I2It is the rotary inertia of internal gear, is constant value；It is external tooth
The instantaneous angular acceleration of wheel；It is the instantaneous angular acceleration of internal gear；cmIt is engagement viscous damping coefficient；l1It is nibbling for external gear
The arm of force of friction damping force is closed, is constant value；l2It is the arm of force of the engaging friction damping force of internal gear, is constant value；λ is frictional force side
To coefficient,ω2It is the nominal angular speed of internal gear；It is the intermittent angle of external gear
Speed, is t function；It is the instantaneous angular velocity of internal gear, is t function；μ is the flank of tooth coefficient of kinetic friction；
Step 8: carrying out linear transformation and nondimensionalization processing to the nonlinear dynamical equation of gear pair, obtain final
It is included in internal gear pair dimensionless non-linear dynamic model under the gear teeth damage fault state of friction and backlash；
The commonly used Algebraic Expression of nondimensionalization process includesτ、ζ1、ζ2、k2、Fav、Fe；Symbol and parameter
Between transformational relation it is as shown in table 2；
Table 2
Model is as follows：
τ=ωet；ωeIt is gear pair equivalent intrinsic frequency,meIt is gear pair equivalent
Quality,m1It is the quality of external gear, m2It is the quality of internal gear；
eaIt is the amplitude of gear teeth composition error change, is constant；
Embodiment
Modeling and simulating experiment has been carried out to inside engaged gear axle decelerator of certain type with gear teeth damage fault, it has been obtained
The simulation result of pure twisting vibration, obtains phase mark figure, time-domain diagram and the frequency domain figure of transmission system, and accordingly have studied friction and
Influence of the mesh stiffness to system vibration.According to twisting vibration curve, the characteristic value of failure is extracted and has depicted characteristic value
Curve.The parameter of simulation example is as shown in the table.
Inside engaged gear decelerator simulation example parameter value table
During actual emulation, gear teeth damage factor ε difference values 0/0.2/1, simulation gear teeth normal condition/flank of tooth is light
Micro-damage/break of gear tooth totally 3 kinds of health/malfunctions；Obtained inside engaged gear vibration reducer under three state phase mark,
Vibration characteristics under time domain and frequency curve, each state preferably reflected.
1) normal gear vibration during ε=0
During ε=0, gear teeth health status is good, phase of the gear pair without tooth surface damage failure, now gear vibration steady-state response
Mark is as shown in figure 5, it can be seen from the stable state curve map of health status during normal gear vibration, as shown in fig. 6, time domain is bent
Line steady mesh cycle is 20, frequency domain figure as shown in fig. 7, only meshing frequency 0.048 and its high order frequency, without sideband,
Phase mark periodic motion.
2) flank of tooth slight damage of ε=0.2
During ε=0.2, that is, the minor failures such as gear teeth infant cracking, spot corrosion and peeling are represented, now the numerical value of gear train is asked
Gear teeth face minor failure steady-state response is shown in solution curve, its phase mark is as shown in Figure 8.Stable state under minor failure state
When curve it can be seen from the figure that gear pitting corrosion vibrates, time-domain diagram is as shown in figure 9, low-frequency fluctuation occurs in time-domain curve, and frequency domain figure is such as
Shown in Figure 10, have and occur slight sideband at meshing frequency 0.111 and its high order frequency, meshing frequency, sideband is at intervals of turning
Frequently 0.008, phase mark periodic motion.
3) the broken teeth failure of ε=1
During ε=1, indicate there is broken teeth failure in gear train.Now showed in the numerical solution curve of gear train
The steady-state response gone out under broken teeth malfunction.As shown in figure 11, the steady-state response curve under broken conditions can be seen its phase mark
Go out, when gear tooth breakage vibrates, as shown in figure 12, substantially impact modulation phenomenon occurs to time-domain curve in time-domain curve, and the cycle is rotation
Turn-week phase, frequency curve as shown in figure 13, has at meshing frequency 0.111 and its high order frequency, meshing frequency in frequency domain figure and occurred
Obvious sideband, sideband is at intervals of frequency 0.2586 is turned, and phase mark periodic motion is not obvious.
Claims (3)
1. a kind of Nonlinear dynamic models method of inside engaged gear axle tooth of reducer damage fault, it is characterised in that tool
Body step is as follows：
Step 1: for some inside engaged gear axle decelerator, setting the condition being modeled to the decelerator；
Step 2: for the inside engaged gear axle decelerator, the relative line in gear pair path of contact is determined using gear angular displacement
Displacement；
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Wherein：X is the relative linear displacement along path of contact in internal gear Meshing Pair；Y is the phase along path of contact in external gear Meshing Pair
To displacement of the lines；r1It is the reference radius of external gear；r2It is the reference radius of internal gear；θ1It is the torsional angular displacement of external gear；
θ2It is the torsional angular displacement of internal gear；T is time variable；E (t) is the static driving error of gear pair, e (t)=easinωmT,
eaFor error magnitude, ωmFor engaging tooth frequency；
Step 3: determining nonlinear backlash function according to the relative linear displacement in internal gear Meshing Pair along path of contact；
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In formula, b is backlash constant；
Step 4: approaching the time-variant mesh stiffness function k of health status lower tooth wheel set using Fourier spacem(t)；
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Step 5: the gear teeth time-variant mesh stiffness function k (t) approached using Fourier space at single broken teeth failure
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kafIt is the peak-to-peak value that monodentate mesh stiffness changes at broken teeth, Z1For the number of teeth of external gear that is, driving wheel；ω1It is external gear
Nominal angular speed；
Step 6: being done using the time-variant mesh stiffness and the gear teeth time-varying rigidity of single broken teeth malfunction of health status gear pair
Difference, and gear teeth damage factor is combined, gear teeth damage fault time-variant mesh stiffness is modeled；
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<mi>k</mi>
<mi>m</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>&epsiv;</mi>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<msub>
<mi>k</mi>
<mi>a</mi>
</msub>
</mrow>
<mrow>
<msub>
<mi>n&pi;k</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</mfrac>
<munderover>
<mi>&Sigma;</mi>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mi>sin</mi>
<mi> </mi>
<msub>
<mi>n&omega;</mi>
<mi>m</mi>
</msub>
<mi>t</mi>
<mo>,</mo>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mn>5</mn>
<mo>,</mo>
<mn>...</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mo>-</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&epsiv;k</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
<mrow>
<mo>{</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<msub>
<mi>k</mi>
<mrow>
<mi>a</mi>
<mi>f</mi>
</mrow>
</msub>
</mrow>
<mrow>
<msub>
<mi>Z</mi>
<mn>1</mn>
</msub>
<msub>
<mi>k</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</mfrac>
<mo>+</mo>
<munderover>
<mi>&Sigma;</mi>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mrow>
<mo>{</mo>
<mrow>
<mfrac>
<msub>
<mi>k</mi>
<mrow>
<mi>a</mi>
<mi>f</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>n&pi;k</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</mfrac>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>n</mi>
<mi>&pi;</mi>
</mrow>
<msub>
<mi>Z</mi>
<mn>1</mn>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mi>cos</mi>
<mi> </mi>
<msub>
<mi>n&omega;</mi>
<mn>1</mn>
</msub>
<mi>t</mi>
<mo>+</mo>
<mfrac>
<msub>
<mi>k</mi>
<mrow>
<mi>a</mi>
<mi>f</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>n&pi;k</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</mfrac>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>-</mo>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>n</mi>
<mi>&pi;</mi>
</mrow>
<msub>
<mi>Z</mi>
<mn>1</mn>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mi>sin</mi>
<mi> </mi>
<msub>
<mi>n&omega;</mi>
<mn>1</mn>
</msub>
<mi>t</mi>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mi>n</mi>
<mi>&pi;</mi>
</mrow>
</mfrac>
<munderover>
<mi>&Sigma;</mi>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mi>sin</mi>
<mi> </mi>
<msub>
<mi>n&omega;</mi>
<mi>m</mi>
</msub>
<mi>t</mi>
<mo>,</mo>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mn>5</mn>
<mo>,</mo>
<mn>...</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mo>-</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&epsiv;k</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
<mrow>
<mo>{</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<msub>
<mi>k</mi>
<mn>3</mn>
</msub>
</mrow>
<msub>
<mi>Z</mi>
<mn>1</mn>
</msub>
</mfrac>
<mo>+</mo>
<munderover>
<mi>&Sigma;</mi>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mrow>
<mo>{</mo>
<mrow>
<mfrac>
<msub>
<mi>k</mi>
<mn>3</mn>
</msub>
<mrow>
<mi>n</mi>
<mi>&pi;</mi>
</mrow>
</mfrac>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>n</mi>
<mi>&pi;</mi>
</mrow>
<msub>
<mi>Z</mi>
<mn>1</mn>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mi>cos</mi>
<mi> </mi>
<msub>
<mi>n&omega;</mi>
<mn>1</mn>
</msub>
<mi>t</mi>
<mo>+</mo>
<mfrac>
<msub>
<mi>k</mi>
<mn>3</mn>
</msub>
<mrow>
<mi>n</mi>
<mi>&pi;</mi>
</mrow>
</mfrac>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>-</mo>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>n</mi>
<mi>&pi;</mi>
</mrow>
<msub>
<mi>Z</mi>
<mn>1</mn>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mi>sin</mi>
<mi> </mi>
<msub>
<mi>n&omega;</mi>
<mn>1</mn>
</msub>
<mi>t</mi>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
1
Wherein, kmf(t) it is the gear pair time-variant mesh stiffness containing gear teeth damage fault；
ε is gear teeth face damage factor；
Step 7: using the backlash function of the decelerator, with reference to the modeling of gear teeth damage fault time-variant mesh stiffness, using
Lagrange methods, set up the nonlinear dynamical equation of gear pair；
The nonlinear dynamical equation of gear pair includes the input matrix T of internal gear pair1With output matrix T2；
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
<msub>
<mover>
<mi>&theta;</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>&lsqb;</mo>
<msub>
<mi>c</mi>
<mi>m</mi>
</msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>m</mi>
<mi>f</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>+</mo>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
<mi>&lambda;</mi>
<mi>&mu;</mi>
<mo>&lsqb;</mo>
<msub>
<mi>c</mi>
<mi>m</mi>
</msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>m</mi>
<mi>f</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>=</mo>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>I</mi>
<mn>2</mn>
</msub>
<msub>
<mover>
<mi>&theta;</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>-</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
<mo>&lsqb;</mo>
<msub>
<mi>c</mi>
<mi>m</mi>
</msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>m</mi>
<mi>f</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>-</mo>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
<mi>&lambda;</mi>
<mi>&mu;</mi>
<mo>&lsqb;</mo>
<msub>
<mi>c</mi>
<mi>m</mi>
</msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>m</mi>
<mi>f</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>=</mo>
<msub>
<mi>T</mi>
<mn>2</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Wherein：I1It is the rotary inertia of external gear；I2It is the rotary inertia of internal gear；It is the instantaneous angular acceleration of external gear；
It is the instantaneous angular acceleration of internal gear；cmIt is engagement viscous damping coefficient；l1It is the arm of force of the engaging friction damping force of external gear；l2
It is the arm of force of the engaging friction damping force of internal gear；λ is direction coefficient,
ω2It is the nominal angular speed of internal gear；It is the instantaneous angular velocity of external gear；It is the instantaneous angular velocity of internal gear；μ is the flank of tooth
The coefficient of kinetic friction；
Step 8: carrying out linear transformation and nondimensionalization processing to the nonlinear dynamical equation of gear pair, finally it is included in
Internal gear pair dimensionless non-linear dynamic model under the gear teeth damage fault state of friction and backlash；
Model is as follows：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mover>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
<msub>
<mi>&zeta;</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>g</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mi>&mu;</mi>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mover>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>g</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mi>&mu;</mi>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>k</mi>
<mn>11</mn>
</msub>
<mo>-</mo>
<msub>
<mi>&epsiv;k</mi>
<mn>12</mn>
</msub>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>m</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>m</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
<msub>
<mi>F</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
<mo>+</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>F</mi>
<mi>e</mi>
</msub>
<msup>
<mover>
<mi>&omega;</mi>
<mo>&OverBar;</mo>
</mover>
<mn>2</mn>
</msup>
<mi>sin</mi>
<mover>
<mi>&omega;</mi>
<mo>&OverBar;</mo>
</mover>
<mi>&tau;</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mover>
<mover>
<mi>y</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
<msub>
<mi>&zeta;</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>g</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mi>&mu;</mi>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mover>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>g</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mi>&mu;</mi>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>k</mi>
<mn>11</mn>
</msub>
<mo>-</mo>
<msub>
<mi>&epsiv;k</mi>
<mn>12</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<msub>
<mi>m</mi>
<mi>e</mi>
</msub>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
</mfrac>
<msub>
<mi>F</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
τ=ωet；ωeIt is gear pair equivalent intrinsic frequency,meIt is gear pair equivalent quality,m1It is the quality of external gear, m2It is the quality of internal gear；
<mrow>
<msub>
<mi>&zeta;</mi>
<mn>1</mn>
</msub>
<mo>=</mo>
<mfrac>
<msub>
<mi>c</mi>
<mi>m</mi>
</msub>
<mrow>
<mo>(</mo>
<mn>2</mn>
<msub>
<mi>m</mi>
<mi>e</mi>
</msub>
<msub>
<mi>&omega;</mi>
<mi>e</mi>
</msub>
<mo>)</mo>
</mrow>
</mfrac>
<mo>;</mo>
<msub>
<mi>g</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mfrac>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
<mrow>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
<mrow>
<msub>
<mi>m</mi>
<mn>2</mn>
</msub>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
<mo>)</mo>
</mrow>
<msub>
<mi>m</mi>
<mi>e</mi>
</msub>
<mi>&lambda;</mi>
<mo>;</mo>
<mover>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mover>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>&times;</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>&omega;</mi>
<mi>e</mi>
</msub>
</mfrac>
<mo>;</mo>
</mrow>
<mrow>
<msub>
<mi>k</mi>
<mn>11</mn>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mi>n</mi>
<mi>&pi;</mi>
</mrow>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
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2. a kind of Nonlinear dynamic models side of inside engaged gear axle tooth of reducer damage fault as claimed in claim 1
Method, it is characterised in that described step one is specially：
(1) internal gear and external gear of the decelerator are involute spur gear；
(2) gear teeth failure refers to the damage fault form of rippling, peeling and the broken teeth of reducer gear；
(3) two gear gear blanks are considered as rigid body, and the input of decelerator and output shaft are considered as rigid body；The support stiffness of two gear shafts is enough
Greatly, the elastic deformation of support is not considered；
(4) each parts are not by axial force in decelerator, and vibration vector has the plane perpendicular to axis；
(5) driving wheel (external gear), the gear teeth of driven pulley (internal gear) do cantilever beam consideration, there are the gear teeth along the relative of path of contact
Slide displacement；
(6) train middle gear is installed according to reference center distance, and pitch circle is overlapped with reference circle；
(7) part's machining errors and alignment error are not counted in.
3. a kind of Nonlinear dynamic models side of inside engaged gear axle tooth of reducer damage fault as claimed in claim 1
Method, it is characterised in that described gear teeth face damage factor ε ∈ [0,1], is defined as follows：
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| CN107729626A (en) * | 2017-09-26 | 2018-02-23 | 西安交通大学 | A kind of gear pitting corrosion model modelling approach based on probability distribution |
| CN107677438A (en) * | 2017-10-12 | 2018-02-09 | 电子科技大学 | Epicyclic gearbox sun gear gradual change spot corrosion time-variant mesh stiffness analysis method |
| CN108694302A (en) * | 2018-08-03 | 2018-10-23 | 宝鸡文理学院 | A kind of method for diagnosing faults based on gear train assembly nonlinear stability |
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| CN110362858A (en) * | 2019-06-05 | 2019-10-22 | 徐州圣邦机械有限公司 | A kind of reliability estimation method of high pressure crescent gear pump gear pair |
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| CN114354187B (en) * | 2022-01-05 | 2022-12-06 | 上海交通大学 | Gear fault classification detection method and system based on identification of meshing stiffness |
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