CN112507485A - Bevel gear time-varying meshing stiffness analysis method based on slice coupling theory - Google Patents
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Abstract
The invention discloses a bevel gear time-varying meshing stiffness analysis method based on a slice coupling theory, and aims to improve the calculation accuracy of the meshing stiffness of a helical cylindrical gear. The method comprises the following implementation steps: outputting basic parameters of the gear system; determining an initial meshing position, and slicing the gear along the tooth width direction; calculating the rigidity of the slicing gear teeth, the coupling rigidity and the rigidity of the gear base; determining the meshing deformation of the gear teeth of the driving gear slice; calculating the meshing force of each slice and the meshing force of the gears; calculating the meshing rigidity of the gear teeth, the rigidity of the gear base and the meshing rigidity of the gear pair; and obtaining the time-varying meshing stiffness of the helical gear system through a period simulation. The method comprehensively considers the coupling effect of the non-contact area and the contact area of the gear teeth, and can accurately simulate the time-varying meshing stiffness characteristic of the simulated helical gear.
Description
Technical Field
The invention belongs to the technical field of machinery, and particularly relates to a bevel gear time-varying meshing stiffness analysis method based on a slice coupling theory.
Background
The helical gear system is widely applied to the fields of aerospace, automobiles, ships and the like, and the performance of the helical gear system determines the performance, service life, safety and reliability of a host to a great extent. Time-varying mesh stiffness is an important internal excitation of a gear system and has a large effect on the vibration characteristics of the gear system.
At present, main research methods for the gear time-varying meshing stiffness characteristic include an analytic method and a finite element method. The finite element method can accurately simulate the actual condition of the gear and has high calculation precision, but has the defects of repeated modeling, low calculation efficiency and the like aiming at different objects. In the analytic method, a gear is equivalent to a variable cross-section cantilever beam, and the rigidity of the variable cross-section cantilever beam is calculated by applying a potential energy method, wherein the rigidity comprises gear tooth bending rigidity, shearing rigidity, radial compression rigidity, contact rigidity and tooth base rigidity. However, in the existing methods, the coupling effect of the deformation of the contact area of the gear teeth and the non-contact area is not considered, and the fact that the non-contact area of the gear teeth resists the deformation of the contact area, so that the meshing rigidity of the helical gear is enhanced.
Disclosure of Invention
Aiming at the problems existing in the existing method, the invention provides a bevel gear time-varying meshing rigidity analysis method based on a slice coupling theory, which deeply considers the influence of the bevel gear slice coupling effect and can accurately simulate the simulation bevel gear time-varying meshing rigidity characteristic.
In order to achieve the purpose, the invention adopts the technical scheme that: a bevel gear time-varying meshing stiffness analysis method based on a slice coupling theory comprises the following steps:
step 1: outputting basic parameters of the gear system, including gear parameters, material parameters and load parameters;
step 2: determining the initial meshing position of the gear, slicing the gear along the tooth width direction, determining the meshing state of each slice, calculating the gear tooth rigidity, the slice coupling rigidity and the tooth base rigidity of each slice, and determining a tooth base rigidity correction factor through a finite element model;
and step 3: make the gear teeth initially engagedThe shape is delta, and the meshing deformation delta of the gear teeth of each driving gear slice in the meshing state and the non-meshing state is further determinedpi；
And 4, step 4: calculating the engaging force F of each sectioniGear mesh force FmComparison of the Gear mesh forces FmWith external load F, if | F-Fm|>FεSetting a new meshing deformation delta and repeating the step 3; if | F-Fm|≤FεObtaining the gear meshing deformation delta under the load F;
and 5: calculating gear tooth meshing rigidity kttFurther consider the tooth base stiffness ktfCalculating the meshing rigidity k of the gear pair;
step 6: and repeating the process at the next meshing position until the simulation of one meshing period is completed, and obtaining the time-varying meshing rigidity of the helical gear system.
Preferably, the step 2 further comprises:
step 2.1: slicing the helical gear along the tooth width direction, wherein each slice is regarded as a slice gear, and each slice gear is regarded as a straight gear;
step 2.2: calculating the gear tooth rigidity of each slice based on a potential energy method;
step 2.3: calculating the rigidity of the tooth base of the slice based on an improved multi-tooth meshing tooth base rigidity calculation method;
step 2.4: calculating slice coupling stiffness based on a slice coupling theory;
step 2.5: and establishing a finite element model of the bevel gear system by adopting finite element analysis software ANSYS, and determining a tooth base rigidity correction factor.
Preferably, in step 2.2, the sliced tooth stiffness is calculated by using the following formula:
wherein k isbFor the slice gear tooth bending stiffness, jc (·) represents the integral operation,
indicating the reference circle pressure angle, cos (-) indicates the cosine operation,
which represents the pitch angle of the helical angle,
is the horizontal distance between the point of engagement and the origin, y1,y2Respectively represent the horizontal coordinates of any point on the transition curve and the involute,
is the distance between the meshing point and the centre line of the tooth, sin (·) denotes a sinusoidal operation, E is the modulus of elasticity, Iy1,Iy2,Ay1,Ay2The section inertia moment and the cross-sectional area at any position on the transition curve and the involute are shown, and tau is a displacement angle; k is a radical ofsThe shear stiffness of the gear teeth of the slice is shown, and G is the shear modulus; k is a radical ofaAxial compression stiffness of the slicing gear teeth; k is a radical ofhFor the Hertz contact stiffness of the slice gear, delta l is the thickness of each slice, and ν is the Poisson ratio; k is a radical oftThe slicing gear tooth stiffness.
Preferably, the calculation formula of the multi-tooth meshing tooth base stiffness described in step 2.3 is as follows:
wherein k istfLambda is matrix correction factor and is obtained by finite element analysis, k isfIs the matrix stiffness.
Preferably, the calculation formula of the slice coupling stiffness described in step 2.4 is as follows:
wherein the content of the first and second substances,
represents the stiffness of the ith and (i +1) th slice couplings; ccFor the slice coupling factor, the value is 1 for the bevel gear; k is a radical oftiRepresenting the gear tooth stiffness of the ith slice; and m is the module of the helical gear.
Preferably, in step 3, the gear tooth meshing deformation deltapiCalculated by the following formula:
δt(i+1)＝Γi(i+1)δti
δpi+δgi+δhi＝δi
kpiδpi＝kgiδgi＝khiδhi＝Fi
wherein, deltatiRepresenting a deformation of the slice; ri(i+1)Is the deformation transmission coefficient; k is a radical ofpi、kgi、khiRespectively showing the rigidity of driving gear teeth and driven gear teethStiffness and contact stiffness; deltapi、δgi、δhiRespectively representing the deformation amount; fiFor engagement force, deltaiThe gear pair is meshed and deformed.
Preferably, each slice engaging force F described in step 4iGear mesh force FmThe calculation formula is as follows:
preferably, the gear tooth meshing rigidity k in the step 5ttThe gear pair meshing rigidity k is calculated according to the following formula:
wherein F is the external load.
Compared with the prior art, the invention has the beneficial results that: the method for analyzing the time-varying meshing stiffness of the helical gear based on the slice coupling theory is provided, when the time-varying meshing stiffness of the helical gear is solved, the coupling effect of a non-contact area and a contact area of the gear teeth is comprehensively considered, and the solution accuracy of the time-varying meshing stiffness of the helical gear is greatly improved; the method can accurately simulate the time-varying meshing stiffness characteristic of the simulated helical gear.
Drawings
FIG. 1 is a schematic view of a bevel gear section model;
FIG. 2 is a schematic diagram of a helical gear contact line;
FIG. 3 is a schematic view of a bevel gear slice coupling model provided by the present invention;
FIG. 4 is a schematic view of the relationship between the contact deformation of the driving gear and the driven gear;
FIG. 5 is a flow chart of an implementation of the present invention;
FIG. 6 is a graph comparing the results of the present invention and other methods for determining the time varying meshing stiffness of a helical gear;
FIG. 7 is a helical gear finite element model of an implementation of the present invention.
Detailed Description
The present invention will be described in further detail below with reference to the accompanying drawings.
The steps of the present invention are described in further detail with reference to fig. 5.
Step 1: outputting basic parameters of the gear system, including gear parameters, material parameters and load parameters;
step 2: determining the initial meshing position of the gear, slicing the gear along the tooth width direction as shown in fig. 1, determining the meshing state of each slice, calculating the gear tooth rigidity, slice coupling rigidity and tooth base rigidity of each slice, and determining a tooth base rigidity correction factor through a finite element model;
firstly, slicing the helical gear along the tooth width direction, wherein each slice is regarded as a slice gear, and each slice gear is regarded as a straight gear;
secondly, calculating the gear tooth rigidity of each slice; the slice gear tooth rigidity calculation formula is as follows:
wherein k isbFor the slice gear tooth bending stiffness, jc (·) represents the integral operation,
indicating the reference circle pressure angle, cos (-) indicates the cosine operation,
which represents the pitch angle of the helical angle,
is the horizontal distance between the point of engagement and the origin, y1,y2Respectively represent the horizontal coordinates of any point on the transition curve and the involute,
is the distance between the meshing point and the centre line of the tooth, sin (·) denotes a sinusoidal operation, E is the modulus of elasticity, Iy1,Iy2,Ay1,Ay2The section inertia moment and the cross-sectional area at any position on the transition curve and the involute are shown, and tau is a displacement angle; k is a radical ofsThe shear stiffness of the gear teeth of the slice is shown, and G is the shear modulus; k is a radical ofaAxial compression stiffness of the slicing gear teeth; k is a radical ofhFor the Hertz contact stiffness of the slice gear, delta l is the thickness of each slice, and ν is the Poisson ratio; k is a radical oftThe slicing gear tooth stiffness;
thirdly, calculating the rigidity of the tooth base of the slice based on an improved multi-tooth meshing tooth base rigidity calculation method; the slice tooth base stiffness calculation formula is as follows:
wherein k istfFor the rigidity of the multi-tooth meshing tooth base, lambda is corrected by a base bodyThe coefficients are obtained by finite element analysis, kfIs the matrix stiffness;
fourthly, calculating slice coupling stiffness based on a slice coupling theory;
referring to a schematic diagram of a helical gear contact line shown in fig. 2, a blue solid line in the diagram is the helical gear contact line, the helical gear meshing process starts from the tooth root of one end face of a driving gear and the tooth top of a driven gear, then the contact line is changed from short to long and then from long to short, and finally the tooth top of the other end face of the driving gear and the tooth root of the driven gear are separated in a full-tooth mode; as can be seen from the figure, the meshing of partial slicing gear teeth can occur in the meshing process of the helical gears, and partial slicing gear teeth are separated; the gear teeth of the slicing machine which are meshed generate meshing deformation under the action of meshing force, while the gear teeth of the slicing machine which are separated generate deformation under the action of force transmission, and the force transmission is defined as slicing coupling;
referring to the helical gear slice meshing model shown in fig. 3, the gear tooth may be equivalent to a plurality of slices, wherein each slice is equivalent to a spring; bending deformation, shearing deformation and radial compression deformation of the slicing gear teeth can be transmitted to adjacent slicing gear teeth, and the influence of contact deformation on adjacent teeth can be ignored; the rigidity of each spring is the rigidity k of the slicing gear teethtGear tooth bending stiffness, shear stiffness and radial compression stiffness; the coupling effect of the sliced teeth can also be equivalent to a spring, whose stiffness is expressed as:
wherein the content of the first and second substances,
represents the stiffness of the ith and (i +1) th slice couplings; ccFor the slice coupling factor, the value is 1 for the bevel gear; k is a radical oftiRepresenting the gear tooth stiffness of the ith slice; m is the module of the helical gear;
fifthly, establishing a finite element model of the bevel gear system by using finite element analysis software ANSYS, and determining a tooth base rigidity correction factor as shown in figure 7;
and step 3: referring to the bevel gear slice coupling model of FIG. 3 and the schematic diagram of the contact deformation relationship of the driving gear, the driven gear and FIG. 4, the initial meshing deformation of the gear teeth is delta, and the meshing deformation delta of the gear teeth of each driving gear slice in the meshing state and the non-meshing state is further determinedpi；
The slice gear tooth meshing deformation calculation formula is as follows:
δt(i+1)＝Γi(i+1)δti
δpi+δgi+δhi＝δi
kpiδpi＝kgiδgi＝khiδhi＝Fi
wherein, deltatiRepresenting a deformation of the slice; ri(i+1)Is the deformation transmission coefficient; k is a radical ofpi、kgi、khiRespectively representing the rigidity of the driving gear teeth, the rigidity of the driven gear teeth and the contact rigidity; deltapi、δgi、δhiRespectively representing the deformation amount; fiFor engagement force, deltaiThe gear pair is meshed and deformed;
and 4, step 4: referring to the helical gear slice coupling model of FIG. 3, the meshing force F of each slice is calculatediGear mesh force FmComparison of the Gear mesh forces FmWith external load F, if | F-Fm|>FεSetting a new meshing deformation delta and repeating the step 3; if | F-Fm|≤FεObtaining the gear meshing deformation delta under the load F;
the engaging force F of each sliceiGear mesh force FmThe calculation formula is as follows:
and 5: calculating gear tooth meshing rigidity kttFurther consider the tooth base stiffness ktfCalculating the meshing rigidity k of the gear pair;
the gear tooth meshing rigidity kttThe gear pair meshing rigidity k is calculated according to the following formula:
wherein F is an external load;
step 6: and repeating the process at the next meshing position until the simulation of one meshing period is completed, and obtaining the time-varying meshing rigidity of the helical gear system.
The present invention will be described in further detail with reference to examples.
The parameters and material properties of the selected helical gears are shown in table 1.
TABLE 1 helical gear parameters
After the helical gear parameters are selected, a finite element model of the helical gear pair is established to verify the current model, and the advancement and the effectiveness of the current model are determined by analyzing the helical gear time-varying meshing stiffness considering the slice coupling and the time-varying meshing stiffness not considering the slice coupling.
(1) Bevel gear time-varying meshing stiffness considering slice coupling
Considering the coupling of the helical gear slices, establishing a time-varying meshing stiffness model of the helical gear pair by using the method, and enabling the gear to be equivalent to a straight gear slice along the tooth width direction; setting the number of the slices as 100 to calculate the gear tooth rigidity, the slice coupling rigidity and the tooth base rigidity of each slice, wherein the slice coupling factor C c1, the tooth base stiffness correction factor lambda is equal to 1.09, and simulation of 50 points of one period is carried out, so that the time varying meshing stiffness of the helical gear is obtained and is shown as a black solid line in figure 6.
(2) Helical gear time-varying meshing stiffness without considering slice coupling
Let slice coupling factor C when slice coupling is not consideredcEqual to 0, and the other parameters are the same as when slice coupling is considered, and calculating the helical gear time varying meshing stiffness is shown in the red band implementation in fig. 6.
(3) Finite element method for calculating time-varying meshing stiffness of helical gear
Based on the helical gear parameters in the table 1, a finite element analysis software ANSYS is adopted to establish a helical gear system finite element model, as shown in FIG. 7, wherein the gear adopts Solid185 units, the surface of the gear is refined, and the gear meshing adopts Conta173 and Targe170 units; the surface of a gear shaft hole is rigidly connected to the central point of a gear, and the driving gear is constrained by other degrees of freedom of the central point except the degree of freedom of rotation around the direction of the gear; each degree of freedom of the central point of the driven gear is restricted, external load torque is applied to the rotation direction of the driving gear, time-varying meshing rigidity of the helical gear at the moment is obtained through finite element contact analysis, and one meshing period is divided into 20 points to be simulated, so that the time-varying meshing rigidity of the helical gear is obtained, as shown by a black dotted line in fig. 6.
In this example, the results of comparing the time-varying meshing stiffness of the helical gears calculated by the three methods are shown in fig. 6. In the present invention, the helical gear meshing zone is divided into a single-tooth meshing zone and a double-tooth meshing zone, and the double-tooth meshing zone is composed of a transition zone and a full double-tooth meshing zone. As can be seen from the figure, the trend of the time-varying meshing stiffness calculation method of the bevel gear considering slice coupling is completely consistent with that of the finite element method in a single-tooth meshing area, a transition area and a complete double-tooth area, and the maximum relative error is about 0.5%. Compared with a finite element method and a slice coupling method, the slice decoupling-free method has basically the same result in a single-tooth meshing area and a complete double-tooth area, but has lower rigidity than the other two methods in a transition area, mainly because the coupling effect of a non-contact area of the gear teeth and a contact area of the gear teeth is not considered. The slice coupling method can consider the coupling effect of the non-contact area and the contact area of the gear teeth, and simulate the time-varying meshing rigidity of the helical gear more accurately. In summary, the slice coupling method can accurately simulate the meshing characteristics of helical gears.
Claims (8)
1. The bevel gear time-varying meshing stiffness analysis method based on the slice coupling theory is characterized by comprising the following steps of:
step 1: outputting basic parameters of the gear system, including gear parameters, material parameters and load parameters;
step 2: determining the initial meshing position of the gear, slicing the gear along the tooth width direction, determining the meshing state of each slice, calculating the gear tooth rigidity, the slice coupling rigidity and the tooth base rigidity of each slice, and determining a tooth base rigidity correction factor through a finite element model;
and step 3: the initial meshing deformation of the gear teeth is delta, and the meshing deformation delta of the gear teeth of each driving gear slice in the meshing state and the non-meshing state is further determinedpi；
And 4, step 4: calculating the engaging force F of each sectioniGear mesh force FmComparison of the Gear mesh forces FmWith external load F, if | F-Fm|>FεSetting a new meshing deformation delta and repeating the step 3; if | F-Fm|≤FεObtaining the gear meshing deformation delta under the load F;
and 5: calculating gear tooth meshing rigidity kttFurther consider the tooth base stiffness ktfCalculating the meshing rigidity k of the gear pair;
step 6: and repeating the process at the next meshing position until the simulation of one meshing period is completed, and obtaining the time-varying meshing rigidity of the helical gear system.
2. The method for analyzing the time-varying meshing stiffness of the helical gear based on the slice coupling theory as claimed in claim 1, wherein the step 2 further comprises:
step 2.1: slicing the helical gear along the tooth width direction, wherein each slice is regarded as a slice gear, and each slice gear is regarded as a straight gear;
step 2.2: calculating the gear tooth rigidity of each slice based on a potential energy method;
step 2.3: calculating the rigidity of the tooth base of the slice based on an improved multi-tooth meshing tooth base rigidity calculation method;
step 2.4: calculating slice coupling stiffness based on a slice coupling theory;
step 2.5: and establishing a finite element model of the bevel gear system by adopting finite element analysis software ANSYS, and determining a tooth base rigidity correction factor.
3. The bevel gear time-varying meshing stiffness analysis method based on the slice coupling theory as claimed in claim 2, wherein the slice gear tooth stiffness calculation formula of step 2.2 is as follows:
wherein k isbFor the slice gear tooth bending stiffness, jc (·) represents the integral operation,
indicating the reference circle pressure angle, cos (-) indicates the cosine operation,
which represents the pitch angle of the helical angle,
is the horizontal distance between the point of engagement and the origin, y1,y2Respectively represent the horizontal coordinates of any point on the transition curve and the involute,
is the distance between the meshing point and the centre line of the tooth, sin (·) denotes a sinusoidal operation, E is the modulus of elasticity, Iy1,Iy2,Ay1,Ay2The section inertia moment and the cross-sectional area at any position on the transition curve and the involute are shown, and tau is a displacement angle; k is a radical ofsThe shear stiffness of the gear teeth of the slice is shown, and G is the shear modulus; k is a radical ofaAxial compression stiffness of the slicing gear teeth; k is a radical ofhFor the Hertz contact stiffness of the slice gear, delta l is the thickness of each slice, and ν is the Poisson ratio; k is a radical oftThe slicing gear tooth stiffness.
4. The method for analyzing the time-varying meshing stiffness of the helical gear based on the slice coupling theory as claimed in claim 2, wherein the calculation formula of the multi-tooth meshing tooth base stiffness in the step 2.3 is as follows:
wherein k istfA correction system using lambda as matrix for the rigidity of the multi-tooth meshing tooth baseThe numbers are obtained by finite element analysis, kfIs the matrix stiffness.
5. The bevel gear time-varying meshing stiffness analysis method based on the slice coupling theory as claimed in claim 2, wherein the slice coupling stiffness calculation formula of step 2.4 is:
wherein the content of the first and second substances,
represents the stiffness of the ith and (i +1) th slice couplings; ccFor the slice coupling factor, the value is 1 for the bevel gear; k is a radical oftiRepresenting the gear tooth stiffness of the ith slice; and m is the module of the helical gear.
6. The method for analyzing time-varying meshing stiffness of helical gear based on slice coupling theory as claimed in claim 1, wherein the gear meshing deformation δ in the step 3piThe calculation formula is as follows:
δt(i+1)＝Γi(i+1)δti
δpi+δgi+δhi＝δi
kpiδpi＝kgiδgi＝khiδhi＝Fi
wherein, deltatiRepresenting a deformation of the slice; ri(i+1)Is the deformation transmission coefficient; k is a radical ofpi、kgi、khiRespectively representing the rigidity of the driving gear teeth, the rigidity of the driven gear teeth and the contact rigidity; deltapi、δgi、δhiRespectively representing the deformation amount; fiFor engagement force, deltaiThe gear pair is meshed and deformed.
7. The method for analyzing time-varying meshing stiffness of helical gear based on slice coupling theory as claimed in claim 1, wherein the meshing force F of each slice in the step 4iGear mesh force FmThe calculation formula is as follows:
8. the bevel gear time-varying meshing stiffness analysis method based on the slice coupling theory as claimed in claim 1, wherein the gear tooth meshing stiffness k in the step 5ttThe calculation formula of the meshing rigidity k of the gear pair is as follows:
wherein F is the external load.
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|---|---|---|---|---|
| CN116992702A (en) * | 2023-09-29 | 2023-11-03 | 季华实验室 | Method, device, equipment and medium for determining tooth back side meshing stiffness of internal gear pair |
| CN116992702B (en) * | 2023-09-29 | 2023-12-22 | 季华实验室 | Method, device, equipment and medium for determining tooth back side meshing stiffness of internal gear pair |
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