CN116702549A - Method and system for calculating fatigue crack growth life of gear under consideration of random dynamic load - Google Patents
Method and system for calculating fatigue crack growth life of gear under consideration of random dynamic load Download PDFInfo
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Abstract
The application provides a method and a system for calculating fatigue crack growth life of a gear under random dynamic load, wherein the method comprises the following steps: establishing a local plane coordinate system at the crack tip to construct a crack propagation equation; calculating a time-varying engagement stiffness characteristic of the root crack; constructing a vibration differential equation of the gear meshing pair and solving to obtain dynamic meshing force of the meshing pair under different crack states of the gear pair; the meshing of the gear pair under different crack states is processed to the dynamic meshing force, and a load spectrum is compiled to obtain variable amplitude load data with multiple periodicity; taking variable amplitude load data as boundary conditions, simulating crack expansion, and reading gear fatigue crack expansion data from a finite element data file; calculating a stress intensity factor based on the gear fatigue crack growth data; fatigue crack growth life is estimated based on the Paris formula and the stress intensity factor. The method can improve the reliability of the gear and the fatigue crack extension life, improve the safety of equipment and reduce the maintenance cost.
Description
Technical Field
The application relates to the field of fatigue detection of mechanical devices, in particular to a method and a system for calculating fatigue crack growth life of a gear under random dynamic load.
Background
The gear transmission has the characteristics of high transmission efficiency, compact structure, reliable work, long service life, stable transmission ratio and the like, and is widely applied to the field of mechanical transmission. Generally, the failure of the gear transmission is mainly the failure of the gear teeth, and the failure modes of the gear teeth are various. Of all failure modes of the gear teeth, the fatigue tooth failure is the largest in proportion, and secondly the surface contact fatigue, so fatigue failure is one of the most dominant modes of gear failure. In order to improve the reliability and service life of gears, it is necessary to study the fatigue life of gears and the root crack simulation method. The fatigue crack initiation and propagation process is characterized by multistage property, uncertainty and the like under the influence of random factors such as environment, materials and the like. In addition, fault excitation generated by fatigue cracks directly affects alternating load amplitude born by gear teeth, and further affects crack propagation speed. Conventional gear fatigue crack models based on static analysis generally assume crack initiation and propagation paths determined under constant cyclic load, but as the gear transmission system is affected by random alternating load in the actual working process, the gear crack propagation characteristics and fatigue life cannot be accurately simulated.
Disclosure of Invention
In order to overcome the defects in the prior art, the application aims to provide a method and a system for calculating the fatigue crack growth life of a gear under random dynamic load.
In order to achieve the above object, the present application provides a method for calculating fatigue crack growth life of a gear under random dynamic load, comprising the steps of:
determining structural and material parameters of the gear mesh pair;
establishing a local plane coordinate system at the crack tip to construct a crack propagation equation;
calculating a time-varying engagement stiffness k containing root crack characteristics m ；
Based on the time-varying engagement stiffness k m Constructing a vibration differential equation of the gear meshing pair according to the lumped parameter model;
solving a vibration differential equation of the gear meshing pair to obtain dynamic meshing force of the meshing pair under different crack states of the gear pair;
the meshing of the gear pair under different crack states is processed to the dynamic meshing force, and a load spectrum is compiled to obtain variable amplitude load data with multiple periodicity;
based on the extended finite element theory, using variable amplitude load data as boundary conditions, simulating a simulated crack extension equation, and reading gear fatigue crack extension data from a finite element data file;
calculating stress intensity factor K based on gear fatigue crack growth data Ⅰ ；
And estimating the fatigue crack growth life N based on a Paris formula and a stress intensity factor.
The optimal scheme of the method for calculating the fatigue crack extension life of the gear under the random dynamic load is as follows: the crack propagation equation isWherein a is the crack length; a, a 0 Is the initial crack length; n is the number of crack propagation sub-steps; a, a n Is crack expansion step length; θ n Is the angle of crack propagation.
The optimal scheme of the method for calculating the fatigue crack extension life of the gear under the random dynamic load is as follows: the calculation steps of the meshing stiffness of the single tooth and the single pair of gears are as follows:
the time-varying engagement stiffness k m Comprises the meshing rigidity of a single tooth area and the meshing rigidity of a double tooth area;
the calculation steps are as follows:
and calculating the Hertz contact stiffness when the gears are meshed:where E and v are Young's modulus and Poisson's ratio of the tooth material and W is the tooth width;
calculating the matrix stiffness of the gear: wherein ,/> Refer to L * ,M * ,P * and Q*； in the formula ,αm Is the pressure angle; u (u) f Is the tooth root angular distance from the meshing point; s is S f The arc distance between the gear teeth and the gear matrix; l, M, P and Q are X i * The polynomial approximation coefficients referred to; h is a fi Is the ratio of the radius of the tooth root circle to the radius of the hub; θ f Is the included angle between the tooth root and the axle wire of the gear tooth; a is that i 、B i 、C i 、D i and Ei Is a constant coefficient;
constructing a stress model of the gear teeth with initial cracks, integrating in the tooth height direction, and calculating the bending rigidity and the shearing rigidity of the gear teeth in a segmented mode, wherein the bending rigidity, the shearing rigidity and the axial compression rigidity in the gear meshing process are respectively as follows:wherein x is the coordinate in the tooth height direction; d is the meshing point to root crack tip distance; h is the distance from the meshing point of the gear pair to the center line of the gear teeth; g is the modulus of elasticity in shear,
hyis the distance from the crack initiation point to the gear tooth center line; hc is the distance from the crack end point to the gear tooth center line; de is the crack initiation point to meshing point distance; dt is the distance from the crack end point to the tooth root, and Hx is the inertia radius;
meshing stiffness of the single tooth region:
meshing stiffness of the double tooth zone:
wherein g represents a driven wheel; p represents a driving wheel; i represents a single-double tooth meshing zone.
The optimal scheme of the method for calculating the fatigue crack extension life of the gear under the random dynamic load is as follows: the differential equation of vibration of the gear engagement pair is expressed as:
wherein ,/>
Wherein M is a mass matrix; m is m p The quality of the driving wheel is that of the driving wheel; m is m g Is the mass of the driven wheel; j (J) p The rotational inertia of the driving wheel; j (J) g The rotational inertia of the driving wheel; and />Respectively represent x i 、y i and θi Is a vector of acceleration; when i=1, x i Is the displacement vector of the driving wheel in the x direction, y i Is the displacement vector of the driving wheel in the y direction, theta i The angular displacement of the driving wheel gear pair is as follows: when i=2, x i Is a displacement vector of the driven wheel in the x direction; y is i Is a displacement vector of the driven wheel in the y direction; θ i The angular displacement of the driven wheel gear pair is as follows:
in the formula ,/>Respectively represent x i 、y i Is a velocity vector of (2); c px 、c py Representing the support damping of the driving wheel in the x and y directions, k respectively px 、k py Representing the support stiffness of the drive wheel in the x and y directions, c gx 、c gy Representing the support damping of the driven wheel in the x and y directions, k respectively gx 、k gy Representing the supporting rigidity of the driven wheel in the x and y directions respectively; fm represents the meshing pair dynamic meshing force; tp and Tg represent input torques of the driving wheel and the driven wheel respectively; r is R 1 and R2 Representing the base circle radius, alpha, of the driving wheel and the driven wheel respectively p Representing the gear pressure angle.
The optimal scheme of the method for calculating the fatigue crack extension life of the gear under the random dynamic load is as follows: solving a vibration differential equation of the gear meshing pair to obtain a dynamic meshing force F of the meshing pair under different gear tooth crack states m ＝k m f(δ,b t )+c m f 1 (δ,b t), wherein ,km Representing gear mesh stiffness; c m Representing gear mesh damping; f (delta, b) t ) Representing a time-varying tooth flank clearance nonlinear function:
f 1 (δ,b t ) Is f (delta, b) t ) Is a derivative of (2); delta is the gear pair meshing point displacement; b t Is a time-varying flank clearance, 2b t ＝2b 0 +Δb＝2b 0 +2(R 1 +R 2 )(inv(α)-inv(α 0 ) And), wherein b 0 Representing an initial tooth flank clearance; r is R i Representing the radius of the gear base circle; inv (α) represents an involute function, and inv (α) =tan (α) - α, α is a specified angle value.
The optimal scheme of the method for calculating the fatigue crack extension life of the gear under the random dynamic load is as follows: based on the rain flow counting principle, the meshing of the gear pair under different crack states is processed for dynamic meshing force, firstly, the obtained dynamic meshing force of the gear pair is filtered, small amplitude is removed, peak-valley value is reserved, then, discretization is carried out on data, finally, rain flow cycle counting is carried out, a load spectrum is compiled, and amplitude-variable load data with multiple periodicity are obtained.
The optimal scheme of the method for calculating the fatigue crack extension life of the gear under the random dynamic load is as follows: the expression of the fatigue crack growth data of the gear is:
wherein I is the total number of grids; n (N) i A shape function representing a standard finite element mode; u (u) i Node displacement representing a standard finite element mode; h (x) represents the displacement jump characteristic of both sides of the crack; a, a i Representing degrees of freedom of expansion nodes associated with the jump discontinuous expansion function; f (x) represents a tip gradient field function; b i Representing the node degree of freedom for crack tip expansion.
The optimal scheme of the method for calculating the fatigue crack extension life of the gear under the random dynamic load is as follows: the specific steps for calculating the stress intensity factor based on the fatigue crack propagation data of the gear are as follows:
selecting an integral region on a fatigue crack surface of the gear, and calculating an energy release rate J of a crack tip in the integral region:
wherein: Γ represents an integral integrating path; w represents strain energy density; sigma represents the stress vector on the integral path boundary; ds represents the displacement on the integrating path; x, y represent the local coordinates of the crack; u represents a displacement vector on the integral path;
stress intensity factor under elastic deformationWherein E represents an elastic modulus; v represents poisson's ratio.
The optimal scheme of the method for calculating the fatigue crack extension life of the gear under the random dynamic load is as follows: the Paris formula for calculating fatigue crack growth life is:
wherein C and m are material constants, deltaK represents stress intensity factor range, and K is stress intensity factor of adjacent crack length Ⅰ And the difference between them, N, represents fatigue crack growth life.
The application also provides a system for analyzing the fatigue crack extension life of the gear under the consideration of random dynamic loading, which comprises a data acquisition module, a processing module and a storage module, wherein the data acquisition module is in communication connection with the processing module, the data acquisition module acquires the structure and material parameters of the gear meshing pair and sends the structure and material parameters to the processing module, the processing module is in communication connection with the storage module, and the storage module is used for storing at least one executable instruction, and the executable instruction enables the processing module to execute the method for calculating the fatigue crack extension life of the gear under the consideration of random dynamic loading.
The beneficial effects of the application are as follows:
according to the application, a vibration differential equation of the gear meshing pair comprising crack characteristics and time-varying tooth gaps is used, so that dynamic meshing force under different crack lengths is obtained, the dynamic meshing force is processed into a dynamic load spectrum by utilizing rain flow counting, the dynamic load spectrum is used as a load boundary condition and substituted into a finite element model, and the fatigue crack extension life of the gear is analyzed and calculated to obtain a conclusion, so that the gear fatigue design has the following several meanings.
(1) The reliability and the service life of the gear are improved: by researching the fatigue crack extension life of the gear, the reason of fatigue failure of the gear can be found out, so that the design and materials of the gear are optimized, the reliability and the life of the gear are improved, and the influence of gear faults on the operation of equipment is reduced.
(2) The safety of equipment is improved: gears are an important component of many mechanical devices and may cause equipment failure or accidents if the gears experience fatigue cracks during use. By researching the fatigue crack extension life of the gear, the safety of equipment can be improved, and accidents caused by gear faults can be reduced.
(3) The maintenance cost is reduced: gear failure requires repair or replacement, which results in increased downtime and maintenance costs. By researching the fatigue crack extension life of the gear, the time of gear faults can be predicted, the gear can be maintained in advance, and the downtime and maintenance cost are reduced.
Additional aspects and advantages of the application will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the application.
Drawings
The foregoing and/or additional aspects and advantages of the application will become apparent and may be better understood from the following description of embodiments taken in conjunction with the accompanying drawings in which:
FIG. 1 is a six degree of freedom gear dynamics lumped parameter model;
FIG. 2 is a tooth force model with initial cracks;
FIG. 3 is a flow chart of a live load process;
FIG. 4 is a schematic diagram of a two-dimensional integration zone;
FIG. 5 is a gear tooth extension finite element model;
FIG. 6 is gear dynamic mesh forces during dynamic crack propagation;
FIG. 7 is a gear dynamic mesh force rain flow histogram during crack propagation;
FIG. 8 is a graph of fatigue crack path comparison for gears under dynamic and static loading;
FIG. 9 is a graph comparing fatigue crack growth life under dynamic and static loading of a gear.
Detailed Description
Embodiments of the present application are described in detail below, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to like or similar elements or elements having like or similar functions throughout. The embodiments described below by referring to the drawings are illustrative only and are not to be construed as limiting the application.
In the description of the present application, unless otherwise specified and defined, it should be noted that the terms "mounted," "connected," and "coupled" are to be construed broadly, and may be, for example, mechanical or electrical, or may be in communication with each other between two elements, directly or indirectly through intermediaries, as would be understood by those skilled in the art, in view of the specific meaning of the terms described above.
As shown in fig. 1, the application provides a method for calculating fatigue crack growth life of a gear under random dynamic load, which comprises the following steps:
the structural and material parameters of the gear mesh pair are determined.
The local plane coordinate system established at the crack tip is a coordinate system with the x-axis as the initial crack propagation direction and the y-axis as the coordinate system perpendicular to the initial crack direction, and the crack propagation equation is thatWherein a is the crack length; a, a 0 Is the initial crack length; a, a i Representing crack propagation step length; θ n Representing the angle of crack propagation.
Calculating a time-varying engagement stiffness k containing root crack characteristics m 。
Specifically, the time-varying engagement stiffness k m Comprises the meshing rigidity of a single tooth area and the meshing rigidity of a double tooth area;
in this embodiment, the meshing stiffness of the single tooth region:
meshing stiffness of the double tooth zone:
wherein g represents a driven wheel, and p represents a driving wheel; i represents a single-double tooth meshing zone, when i is 1, a single tooth zone, when i is 2, a double tooth zone, k h The calculation formula of the Hertz contact rigidity is thatE and v are Young's modulus and Poisson's ratio of the tooth material, and W is the tooth width. k (k) f The matrix rigidity of the gear is calculated as wherein ,
refer to L * ,M * ,P * and Q* ，α m Is the pressure angle; u (u) f Is the tooth root angular distance from the meshing point; s is S f The arc distance between the gear teeth and the gear matrix; l, M, P and Q are X i * The polynomial approximation coefficients referred to; h is a fi Is the ratio of the radius of the tooth root circle to the radius of the hub; θ f Is the included angle between the tooth root and the axle wire of the gear tooth; a is that i 、B i 、C i 、D i and Ei Is an empirical coefficient and is a constant.
k b K is bending stiffness during gear engagement s In order to achieve a shear stiffness, the material,k a the calculation steps of the axial compression rigidity are as follows: constructing a stress model of the gear teeth with initial cracks, integrating in the tooth height direction, and calculating the bending rigidity and the shearing rigidity of the gear teeth in a segmented mode, wherein the bending rigidity, the shearing rigidity and the axial compression rigidity in the gear meshing process are respectively as follows:wherein x is the coordinate in the tooth height direction, and h is the distance from the meshing point of the gear pair to the center line of the gear tooth; g is the shear modulus of elasticity, ">
d is the meshing point to root crack tip distance; h is a y 、h c 、d e and dt As shown in fig. 2, hy is the distance from the crack initiation point to the tooth centerline; hc is the distance from the crack end point to the gear tooth center line; de is the crack initiation point to meshing point distance; dt is the crack end to root angular distance and Hx is the radius of inertia.
In order to prevent the teeth from deforming during engagement due to load and heat propagation, a certain backlash is left during manufacturing. During the operation of a gear, the backlash is a time-varying parameter due to temperature variations, manufacturing errors, wear of the tooth flanks, etc. In the embodiment, the influence of the change of the backlash on the dynamic excitation of the gear is considered, the time-varying backlash bt is calculated, and the expression of the time-varying backlash bt is 2b t ＝2b 0 +Δb＝2b 0 +2(R 1 +R 2 )(inv(α)-inv(α 0 ) And), wherein b 0 Representing an initial tooth flank clearance; r is R i Representing the radius of the gear base circle; inv (α) represents the involute opening function, inv (α) =tan (α) - α, α being the specified angle value, typically taken as 20 °.
And (3) establishing a six-degree-of-freedom gear dynamics lumped parameter model shown in fig. 1, and constructing a vibration differential equation of the gear meshing pair according to the lumped parameter model. The method for constructing the lumped parameter model is the prior art and will not be described herein.
The differential equation of vibration of the gear mesh pair is expressed here as:
wherein ,/>M is a mass matrix; m is m p The quality of the driving wheel is that of the driving wheel; m is m g Is the mass of the driven wheel; j (J) p The rotational inertia of the driving wheel; j (J) g The rotational inertia of the driving wheel; /> and />Respectively represent x i 、y i and θi When i=1, x i Is the displacement vector of the driving wheel in the x direction, y i Is the displacement vector of the driving wheel in the y direction, theta i The angular displacement of the driving wheel gear pair is as follows: when i=2, x i Is a displacement vector of the driven wheel in the x direction; y is i Is a displacement vector of the driven wheel in the y direction; θ i Is the angular displacement of the driven wheel gear pair.
in the formula ,/>Respectively represent x i 、y i Is a velocity vector of (2); c px 、c py Representing the support damping of the driving wheel in the x and y directions, k respectively px 、k py Representing the support stiffness of the drive wheel in the x and y directions, c gx 、c gy Representing the support damping of the driven wheel in the x and y directions, k respectively gx 、k gy Representing the supporting rigidity of the driven wheel in the x and y directions respectively; tp and Tg represent input torques of the driving wheel and the driven wheel respectively; r is R 1 and R2 Representing the base circle radius, alpha, of the driving wheel and the driven wheel respectively p Representing the gear pressure angle, these parameters are known parameters, and Fm represents the mesh versus dynamic mesh force.
Solving a vibration differential equation of the gear meshing pair, in the embodiment, preferably but not limited to adopting ODE45 to solve the vibration differential equation of the gear meshing pair, obtaining dynamic response of the gear meshing pair, and obtaining dynamic meshing force of the meshing pair under different gear tooth crack states; in this embodiment, the dynamic engagement force of the engagement pair is F m ＝k m f(δ,b t )+c m f 1 (δ,b t), wherein ,km Representing gear mesh stiffness; c m Representing gear engagement damping, determined by an empirical formula; f (delta, b) t ) Representing a time-varying tooth flank clearance nonlinear function; f (f) 1 (δ,b t ) Is f (delta, b) t ) Is used as a derivative of the function of (c),here δ is the gear pair meshing point displacement; b t Is a time-varying flank gap.
Based on the rain flow counting principle, the meshing of the gear pair in different crack states is processed to the dynamic meshing force, and a load spectrum is compiled to obtain amplitude-variable load data with multiple periodicity.
Specifically, on the basis of a rain flow counting algorithm, the obtained dynamic meshing force of the gear pair is filtered, small amplitude is removed, peak-valley values are reserved, then discretization is carried out on data, finally the rain flow cycle counting is carried out, corresponding calculation is carried out, a load spectrum is compiled, and amplitude-variable load data with multiple periodicity are obtained.
Based on the extended finite element theory, amplitude variation load data is used as boundary conditions, crack extension is simulated, and gear fatigue crack extension data are read from a finite element data file.
In particular, the fatigue crack growth is simulated by adopting an expansion finite element method,the extended finite element is based on the conventional finite element and the unit decomposition method, and the advantages of the conventional finite element are maintained. When the expanding finite element analysis fracture problem is applied, the position of a crack surface is not considered, grids are directly divided on the gear fatigue crack surface, a unit decomposition method is applied, an additional function is added in the finite element analysis method to improve the degree of freedom, generalized Heaviside functions are used for strengthening unit nodes of a crack penetration area, progressive displacement field functions of the crack tip are used for strengthening unit nodes containing the crack tip to reflect the local characteristics of the crack tip area, and thus the existence of discontinuity can be indirectly reflected by the additional function, and the existence of strong discontinuity of the crack is mutually independent with the grid of the finite element when the grids are divided, so that the difficulty brought by high-density grid division in a high-stress area such as the crack tip and the deformation concentration area is overcome. Obtaining fatigue crack propagation data of the gear, wherein the expression isWherein I is the total number of grids; n (N) i A shape function representing a standard finite element mode, u i Node displacement representing a standard finite element mode; h (x) represents the displacement jump characteristic of both sides of the crack, essentially considered as an extension of the sign distance function, a i Representing the degree of freedom of the extended node in relation to the jump discontinuous extended function, F (x) representing the tip gradient field function, b i The node degrees of freedom representing crack tip expansion are all known parameters, which can be read from a finite metadata file.
Calculating a stress intensity factor based on the gear fatigue crack growth data:
in the embodiment, part of the grids are selected as an integral area on the fatigue crack surface of the gear, the energy release rate J of the crack tip is calculated in the integral area, and then the stress intensity factor is indirectly obtained by utilizing a corresponding formula. Here, the energy release rate of the crack tip is obtained by means of an interactive integral solutionWherein: Γ represents an integral integrating path; w represents strain energyThe density and sigma represent stress vectors on the boundary of the integral path, which can be obtained by finite element simulation and are known parameters; ds represents the displacement on the integrating path; x, y represent the local coordinates of the crack; u represents the displacement vector on the integral path.
In the case of elastic deformation, the energy release rate J and the stress intensity factor K Ⅰ Has the following relationWherein E represents an elastic modulus; v represents poisson's ratio.
Fatigue crack growth life is estimated based on the Paris formula and the stress intensity factor.
The Paris formula for calculating fatigue crack growth life is:where a is a crack propagation equation, C and m are material constants, ΔK represents a stress intensity factor range, and K is a stress intensity factor of adjacent crack lengths Ⅰ And the difference value is N, wherein N represents the fatigue crack extension life, and the gear fatigue crack extension life can be obtained by solving N.
The application also provides a gear fatigue crack extension life calculation system under the consideration of random dynamic loading, which comprises a data acquisition module, a processing module and a storage module, wherein the data acquisition module is in communication connection with the processing module, the data acquisition module acquires the structure and material parameters of the gear meshing pair and sends the structure and material parameters to the processing module, the processing module is in communication connection with the storage module, and the storage module is used for storing at least one executable instruction, and the executable instruction enables the processing module to execute the gear fatigue crack extension life calculation method under the consideration of random dynamic loading to estimate the gear fatigue crack extension life.
In the description of the present specification, a description referring to terms "one embodiment," "some embodiments," "examples," "specific examples," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present application. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiments or examples. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the present application have been shown and described, it will be understood by those of ordinary skill in the art that: many changes, modifications, substitutions and variations may be made to the embodiments without departing from the spirit and principles of the application, the scope of which is defined by the claims and their equivalents.
Claims (10)
1. The method for calculating the fatigue crack growth life of the gear under the consideration of random dynamic load is characterized by comprising the following steps of:
determining structural and material parameters of the gear mesh pair;
establishing a local plane coordinate system at the crack tip to construct a crack propagation equation;
calculating a time-varying engagement stiffness k containing root crack characteristics m ；
Based on the time-varying engagement stiffness k m Constructing a vibration differential equation of the gear meshing pair according to the lumped parameter model;
solving a vibration differential equation of the gear meshing pair to obtain dynamic meshing force of the meshing pair under different crack states of the gear pair;
the meshing of the gear pair under different crack states is processed to the dynamic meshing force, and a load spectrum is compiled to obtain variable amplitude load data with multiple periodicity;
based on the extended finite element theory, using variable amplitude load data as boundary conditions, simulating a simulated crack extension equation, and reading gear fatigue crack extension data from a finite element data file;
calculating stress intensity factor K based on gear fatigue crack growth data Ⅰ ；
And estimating the fatigue crack growth life N based on a Paris formula and a stress intensity factor.
2. The method for calculating fatigue crack growth life of a gear under random dynamic load according to claim 1, wherein the crack growth equation isWherein a is the crack length; a, a 0 Is the initial crack length; n is the number of crack propagation sub-steps; a, a n Is crack expansion step length; θ n Is the angle of crack propagation.
3. The method for calculating fatigue crack growth life of a gear under random dynamic load according to claim 1, wherein the time-varying engagement stiffness k m Comprises the meshing rigidity of a single tooth area and the meshing rigidity of a double tooth area;
the calculation steps are as follows:
and calculating the Hertz contact stiffness when the gears are meshed:where E and v are Young's modulus and Poisson's ratio of the tooth material and W is the tooth width;
calculating the matrix stiffness of the gear: wherein , refer to L * ,M * ,P * and Q*； in the formula ,αm Is the pressure angle; u (u) f Is the tooth root angular distance from the meshing point; s is S f The arc distance between the gear teeth and the gear matrix; l, M, P and Q are X i * The polynomial approximation coefficients referred to; h is a fi Is the ratio of the radius of the tooth root circle to the radius of the hub; θ f Is the included angle between the tooth root and the axle wire of the gear tooth; a is that i 、B i 、C i 、D i and Ei Is a constant coefficient;
constructing a stress model of the gear teeth with initial cracks, integrating in the tooth height direction, and calculating the bending rigidity and the shearing rigidity of the gear teeth in a segmented mode, wherein the bending rigidity, the shearing rigidity and the axial compression rigidity in the gear meshing process are respectively as follows:wherein x is the coordinate in the tooth height direction; d is the meshing point to root crack tip distance; h is the distance from the meshing point of the gear pair to the center line of the gear teeth; g is the modulus of elasticity in shear,
hyis the distance from the crack initiation point to the gear tooth center line; hc is the distance from the crack end point to the gear tooth center line; de is the crack initiation point to meshing point distance; dt is the distance from the crack end point to the tooth root, and Hx is the inertia radius;
meshing stiffness of the single tooth region:
meshing stiffness of the double tooth zone:
wherein g represents a driven wheel; p represents a driving wheel; i represents a single-double tooth meshing zone.
4. The method for calculating fatigue crack growth life of a gear under random dynamic load according to claim 1, wherein the vibration differential equation of the gear meshing pair is expressed as:
wherein ,/>
Wherein M is a mass matrix; m is m p The quality of the driving wheel is that of the driving wheel; m is m g Is the mass of the driven wheel; j (J) p The rotational inertia of the driving wheel; j (J) g The rotational inertia of the driving wheel; and />Respectively represent x i 、y i and θi Is a vector of acceleration; when i=1, x i Is the displacement vector of the driving wheel in the x direction, y i Is the displacement vector of the driving wheel in the y direction, theta i The angular displacement of the driving wheel gear pair is as follows: when i=2, x i Is a displacement vector of the driven wheel in the x direction; y is i Is a displacement vector of the driven wheel in the y direction; θ i The angular displacement of the driven wheel gear pair is as follows:
in the formula ,/>Respectively represent x i 、y i Is a velocity vector of (2); c px 、c py Representing the support damping of the driving wheel in the x and y directions, k respectively px 、k py Representing the support stiffness of the drive wheel in the x and y directions, c gx 、c gy Representing the support damping of the driven wheel in the x and y directions, k respectively gx 、k gy Representing the supporting rigidity of the driven wheel in the x and y directions respectively; fm represents the meshing pair dynamic meshing force; tp and Tg are respectively substitutedInput torque of the driving wheel and the driven wheel is shown; r is R 1 and R2 Representing the base circle radius, alpha, of the driving wheel and the driven wheel respectively p Representing the gear pressure angle.
5. The method for calculating fatigue crack growth life of gear under random dynamic load according to claim 4, wherein the vibration differential equation of the gear meshing pair is solved to obtain the dynamic meshing force F of the meshing pair under different gear tooth crack states m ＝k m f(δ,b t )+c m f 1 (δ,b t), wherein ,km Representing gear mesh stiffness; c m Representing gear mesh damping; f (delta, b) t ) Representing a time-varying tooth flank clearance nonlinear function:
f 1 (δ,b t ) Is f (delta, b) t ) Is a derivative of (2); delta is the gear pair meshing point displacement; b t Is a time-varying flank clearance, 2b t ＝2b 0 +Δb＝2b 0 +2(R 1 +R 2 )(inv(α)-inv(α 0 ) And), wherein b 0 Representing an initial tooth flank clearance; r is R i Representing the radius of the gear base circle; inv (α) represents an involute function, and inv (α) =tan (α) - α, α is a specified angle value.
6. The method for calculating the fatigue crack growth life of the gear under random dynamic load considered as claimed in claim 1, wherein the meshing of the gear pair under different crack states is processed based on a rain flow counting principle, the obtained dynamic meshing force of the gear pair is filtered firstly, small amplitude is removed, peak valley values are reserved, then data are discretized, finally rain flow cycle counting is carried out, and a load spectrum is compiled, so that amplitude variable load data with multiple periodicity are obtained.
7. The method for calculating fatigue crack growth life of a gear under random dynamic load according to claim 1, wherein,
the expression of the fatigue crack growth data of the gear is:
wherein I is the total number of grids; n (N) i A shape function representing a standard finite element mode; u (u) i Node displacement representing a standard finite element mode; h (x) represents the displacement jump characteristic of both sides of the crack; a, a i Representing degrees of freedom of expansion nodes associated with the jump discontinuous expansion function; f (x) represents a tip gradient field function; b i Representing the node degree of freedom for crack tip expansion.
8. The method for calculating the fatigue crack growth life of the gear under random dynamic load according to claim 1, wherein the specific step of calculating the stress intensity factor based on the fatigue crack growth data of the gear is as follows:
selecting an integral region on a fatigue crack surface of the gear, and calculating an energy release rate J of a crack tip in the integral region:
wherein: Γ represents an integral integrating path; w represents strain energy density; sigma represents the stress vector on the integral path boundary; ds represents the displacement on the integrating path; x, y represent the local coordinates of the crack; u represents a displacement vector on the integral path;
stress intensity factor under elastic deformationWherein E represents an elastic modulus; v represents poisson's ratio.
9. The method for calculating fatigue crack growth life of a gear under random dynamic load according to claim 1, wherein the Paris formula for calculating fatigue crack growth life is:
wherein C and m are material constants, deltaK represents stress intensity factor range, and K is stress intensity factor of adjacent crack length Ⅰ And the difference between them, N, represents fatigue crack growth life.
10. The system for analyzing the fatigue crack growth life of the gear under the random dynamic load is characterized by comprising a data acquisition module, a processing module and a storage module, wherein the data acquisition module is in communication connection with the processing module, the data acquisition module acquires the structure and material parameters of a gear meshing pair and sends the structure and material parameters to the processing module, the processing module is in communication connection with the storage module, and the storage module is used for storing at least one executable instruction, and the executable instruction enables the processing module to execute the method for calculating the fatigue crack growth life of the gear under the random dynamic load according to any one of claims 1 to 9.
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