A calculation method for time-varying mesh stiffness of solid-lubricated spur gears

Abstract

The present invention relates to the field of gear dynamics and provides a method for calculating the time-varying mesh stiffness of solid-lubricated spur gears. The method comprises the following steps: calculating the modified elastic modulus and modified shear modulus of the spur gear under the influence of solid lubrication; calculating the tooth bending stiffness, shear stiffness, radial compression stiffness, Hertzian contact stiffness, matrix flexible deformation stiffness, and inter-tooth structural coupling stiffness; and calculating the time-varying mesh stiffness of the spur gears based on the stiffness series and parallel theory. This method addresses the problem in the prior art of not considering the effect of solid lubricant films on gear stiffness.

Description

Technical Field

[0001] The invention belongs to the field of gear dynamics, and in particular relates to a method for calculating the time-varying meshing stiffness of a solid-lubricated spur gear. Background Art

[0002] Gear transmission systems in space mechanisms are widely used in the transmission joints of space manipulators and solar array actuators. They are core components that enable manipulators to perform critical tasks such as on-orbit assembly of spacecraft, assisting astronauts in extravehicular activities, and orienting solar arrays toward the Sun. Due to the harsh and complex space environment, traditional grease lubricants are inadequate for space missions. Molybdenum disulfide, due to its excellent friction properties in space, has become one of the most effective solid lubricants and is widely used in the gear transmission systems of space mechanisms.

[0003] Currently, most methods for calculating the mesh stiffness of spur gears only consider the effects of the gear's contact stiffness, bending stiffness, shear stiffness, radial compression stiffness, matrix flexible deformation stiffness, and structural coupling stiffness. The total mesh stiffness is then derived through the series and parallel relationships of the stiffness calculations. However, these calculation methods fail to consider the effects of the gear's solid lubricant coating on the gear's stiffness. Coating a solid lubricant coating alters the gear's material properties, and this change directly affects the gear's mesh stiffness. Therefore, mesh stiffness calculated without considering the solid lubricant coating cannot accurately and completely describe the characteristics of the gear transmission system, thus compromising the accuracy of comprehensive analysis of the gear's dynamic performance. Summary of the Invention

[0004] Technical issues to be solved:

[0005] To overcome the shortcomings of existing technologies, this invention provides a method for calculating the time-varying mesh stiffness of solid-lubricated spur gears. By incorporating the effects of the solid lubricant film into the mesh stiffness calculation, this method can more accurately describe the characteristics of the gear transmission system, thereby improving the accuracy of comprehensive analysis of its dynamic performance. This improvement will provide a scientific basis for the design of spacecraft gear transmission systems, optimize the use of solid lubricants, and ensure reliability and stability in the extreme space environment. This invention solves the problem that the existing technology does not consider the impact of the solid lubricant film on gear stiffness.

[0006] The technical solution of the present invention is: a method for calculating the time-varying meshing stiffness of solid-lubricated spur gears, the specific steps of which are as follows:

[0007] Calculation of the modified elastic modulus E of spur gears under the influence of solid lubrication c and the modified shear modulus G c , the formula is as follows:

[0008]

[0009] Where V f is the volume of the solid lubricant film, V s is the gear tooth volume, V t is the total volume, E f is the elastic modulus of the solid lubricating film, E s is the gear tooth elastic modulus, ν is the Poisson’s ratio;

[0010] Calculate the tooth bending stiffness k b , shear stiffness k s , radial compression stiffness k a , Hertz contact stiffness k h , matrix flexible deformation stiffness k nfi and the inter-tooth structural coupling stiffness k nfij ;

[0011] According to the stiffness series and parallel theory, the time-varying meshing stiffness of spur gears is calculated as follows:

[0012]

[0013] Where k hi (i=1,2) is the Hertzian contact stiffness of the i-th pair of teeth; k nti (i=1,2;n=p,g) is the tooth stiffness of the i-th pair of master and driven wheels, including the bending stiffness k b , shear stiffness k s and radial compressive stiffness k a ;k nfi (i=1,2;n=p,g) is the matrix flexible deformation stiffness; k nfij (i=1,2,i≠j;n=p,g) is the inter-tooth structure coupling stiffness, p and g represent the driving wheel and the driven wheel respectively; F i F represents the meshing force between the i-th pair of driving and driven gear teeth, j It represents the meshing force between the jth pair of driving and driven wheel teeth.

[0014] A further technical solution of the present invention is that the gear tooth volume V s The calculation formula is as follows:

[0015]

[0016] Where, d is the gear pitch circle diameter, d f is the gear tooth root circle diameter, B is the tooth width, and z is the number of gear teeth.

[0017] A further technical solution of the present invention is that the volume V of the solid lubricating film is f The calculation formula is as follows:

[0018] V f =(2l s +s a )h f B

[0019] Where, l s is the involute length of the gear teeth, s a is the tooth top circle thickness, h f is the thickness of the solid lubricating film, B is the tooth width; where,

[0020]

[0021] invλ=tanλ-λ

[0022] Where α1 is the angle between the meshing force line and the vertical line of the gear tooth center line, α2 is the tooth root circle half angle, r b is the base circle radius of the gear, m is the gear module, r a is the radius of the tooth top circle, r is the pitch circle radius, α a is the tooth top circle pressure angle, α is the pitch circle pressure angle, and inv is the involute angle formula.

[0023] A further technical solution of the present invention is: the bending stiffness k b , shear stiffness k s , radial compression stiffness k a , Hertz contact stiffness k h , matrix flexible deformation stiffness k nfi and the inter-tooth structural coupling stiffness k nfij The calculation expressions are:

[0024] Where α1 is the angle between the meshing force line and the perpendicular line of the gear tooth centerline, α2 is the tooth root circle half angle, θ1 and θ2 are the meshing angles of gear tooth 1 and gear tooth 2 respectively, and r b is the base circle radius of the gear, r f is the root circle radius of the gear, ∑h β is the cumulative wear at any pressure angle β on the involute, h x is the half tooth thickness at the horizontal distance x from the tooth root, B is the tooth width, F is the gear tooth meshing force, u i S is the distance from the intersection of the meshing line and the gear tooth centerline to the tooth root circle. f is the length of the tooth root arc, E c , G c are the modified elastic modulus and shear modulus, L * 、M * 、P * , Q * and L i 、M i 、Pi , Q i 、R i 、S i 、T i 、U i 、V i Represent the coefficients of each polynomial respectively.

[0025] A system for calculating the time-varying mesh stiffness of a solid-lubricated spur gear comprises at least one processor and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, and the computer program is executed by the at least one processor to enable the at least one processor to execute the method for calculating the time-varying mesh stiffness of the solid-lubricated spur gear.

[0026] A computer-readable digital storage medium stores computer instructions, which are used to enable a processor to implement the method for calculating the time-varying meshing stiffness of a solid-lubricated spur gear when the computer instructions are executed.

[0027] Beneficial effects

[0028] The beneficial effects of the present invention are as follows: when calculating the time-varying meshing stiffness of spur gears, the present invention takes into account the influence of the solid lubricating film on the surface of the gear pair, calculates the volume of the solid lubricating film and the gear teeth based on the geometric characteristics of the gears, corrects the elastic modulus and shear modulus of the solid-lubricated gears, and finally calculates the time-varying meshing stiffness of the solid-lubricated gear pair based on the stiffness series-parallel theory, combining bending stiffness, shear stiffness, radial compression stiffness, Hertz contact stiffness, matrix flexible deformation stiffness and inter-tooth structure coupling stiffness.

[0029] After verification, the results show that the calculation method of the time-varying meshing stiffness of spur gears considering the influence of solid lubrication is closer to the actual working conditions of solid lubrication gear transmission. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 It is a schematic diagram of the solid lubrication film on gear teeth.

[0031] Figure 2 It is a schematic diagram of the calculation of the force and deformation of worn gear teeth.

[0032] Figure 3 It is a schematic diagram of the coupling effect of inter-tooth structure.

[0033] Figure 4 This is a comparison chart of the results of calculating the time-varying meshing stiffness of spur gears using the present invention and the standard method. DETAILED DESCRIPTION

[0034] The embodiments described below with reference to the accompanying drawings are exemplary and are intended to explain the present invention, but should not be construed as limiting the present invention.

[0035] The current calculation of the mesh stiffness of spur gears fails to consider the mesh stiffness derived from the solid lubricant film, resulting in an inaccurate and complete description of the gear transmission system's characteristics, which in turn affects the accuracy of comprehensive analysis of the gear's dynamic performance. This paper proposes a method for calculating the time-varying mesh stiffness of solid-lubricated spur gears. By incorporating the effects of the solid lubricant film into the mesh stiffness calculation, it can more accurately describe the gear transmission system's characteristics, thereby improving the accuracy of comprehensive analysis of its dynamic performance. This improvement will provide a scientific basis for the design of spacecraft gear transmission systems, optimize the use of solid lubricants, and ensure reliability and stability in the extreme space environment.

[0036] The above technical solution is further described below with reference to the accompanying drawings and examples:

[0037] This embodiment provides a method for calculating the time-varying mesh stiffness of solid lubricated spur gears. Figure 1 、 Figure 2 and Figure 3 , the calculation method includes the following steps:

[0038] Step 1: Consider the influence of the solid lubricant coating on the gear pair surface and calculate the corrected material properties under the influence of solid lubrication;

[0039] Specifically, the modified elastic modulus E of the spur gear considering the influence of solid lubrication is calculated. c and the modified shear modulus G c , the modified elastic modulus E c and the modified shear modulus G c The calculation method is:

[0040]

[0041] Where V f is the volume of the solid lubricant film, V s is the gear tooth volume, V t is the total volume, E f is the elastic modulus of the solid lubricating film, E s is the gear tooth elastic modulus, ν is the Poisson's ratio, and the volume of the solid lubricating film and the gear tooth is calculated by the following method:

[0042]

[0043] V f =(2l s +s a )h f B

[0044] Where, d is the gear pitch circle diameter, d f is the gear tooth root diameter, B is the tooth width, z is the number of gear teeth, l s is the involute length of the gear teeth, s a is the tooth top circle thickness, h f is the thickness of the solid lubricant film, B is the tooth width, and the calculation method of the involute length of the gear tooth and the tooth thickness of the tooth top circle is:

[0045]

[0046] Where m is the gear module, r a is the radius of the tooth top circle, r is the pitch circle radius, α a is the tooth top circle pressure angle, α is the pitch circle pressure angle, and inv is the involute angle formula.

[0047] Step 2: Calculate the time-varying meshing stiffness of the solid lubricated gear teeth during the meshing process of a single pair of teeth;

[0048] Specifically, according to the potential energy method, the tooth bending stiffness k is calculated b , which is calculated as follows:

[0049]

[0050] Calculate the tooth shear stiffness k s , which is calculated as follows:

[0051]

[0052] Calculate the radial compressive stiffness k of the gear teeth a , which is calculated as follows:

[0053]

[0054] Where α1 is the angle between the meshing force line and the vertical line of the gear tooth center line, α2 is the tooth root circle half angle, r b is the base circle radius of the gear, r f is the root circle radius of the gear, ∑h β is the cumulative wear at any pressure angle β on the involute, h x is the half tooth thickness at the horizontal distance x from the tooth root.

[0055] According to Hertz contact theory, calculate the contact stiffness k of the gear teeth h :

[0056]

[0057] Calculate the matrix flexible deformation stiffness k of the gear tooth nfi and the inter-tooth structural coupling stiffness knfij :

[0058]

[0059] Where θ1 and θ2 are the meshing angles of gear teeth 1 and 2, respectively; F is the meshing force of the gear teeth; u i S is the distance from the intersection of the meshing line and the gear tooth centerline to the tooth root circle. f is the length of the tooth root arc, E c , G c are the modified elastic modulus and shear modulus, L * 、M * 、P * , Q * and L i 、M i 、P i , Q i 、R i 、S i 、T i 、U i 、V i Represent the coefficients of each polynomial respectively.

[0060] Step 3: Calculate the time-varying mesh stiffness of the solid lubricated spur gear based on the stiffness series and parallel theory;

[0061] According to the above steps, the bending stiffness k of the solid lubrication gear teeth is calculated. b , shear stiffness k s , radial compression stiffness k a , Hertz contact stiffness k h , matrix flexible deformation stiffness k nfi and the inter-tooth structural coupling stiffness k nfij , then the total time-varying meshing stiffness k of a pair of meshing gears is c for:

[0062]

[0063] Among them, k hi (i=1,2) is the Hertzian contact stiffness of the i-th pair of teeth; k nti (i=1,2;n=p,g) is the tooth stiffness of the i-th pair of master and driven wheels, including the bending stiffness k b , shear stiffness k s and radial compressive stiffness k a ;k nfi (i=1,2;n=p,g) is the matrix flexible deformation stiffness; k nfij (i=1,2,i≠j;n=p,g) is the inter-tooth structural coupling stiffness, p and g represent the driving wheel and the driven wheel respectively.

[0064] A time-varying mesh stiffness calculation system for solid-lubricated spur gears, characterized by comprising at least one processor and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, and the computer program is executed by the at least one processor so that the at least one processor can execute the method for calculating the time-varying mesh stiffness of solid-lubricated spur gears.

[0065] A computer-readable digital storage medium is characterized in that: the computer-readable digital storage medium stores computer instructions, and the computer instructions are used to enable a processor to implement the time-varying mesh stiffness calculation method of solid-lubricated spur gears when executed.

[0066] Based on the above steps, combined with the experimental example parameter table shown in Table 1, Figure 1 Schematic diagram of the solid lubricating film on the gear teeth of the experimental example shown, Figure 2 The schematic diagram of the calculation of the stress and deformation of the worn gear teeth in the experimental example shown Figure 3 Schematic diagram of the inter-tooth structural coupling effect of the experimental example shown, and the time-varying mesh stiffness of the spur gear considering the solid lubricant film is calculated.

[0067] Table 1

[0068]

[0069] The maximum value of the single tooth meshing stiffness obtained by the calculation method of the present invention is about 2.56×10 8 N / m, and the average time-varying meshing stiffness is 3.74×10 8 N / m. The maximum value of the single tooth meshing stiffness calculated by the ISO standard method is 2.68×10 8 N / m, and the average time-varying meshing stiffness is 3.97×10 8 N / m.

[0070] The ISO standard method does not take into account the solid lubricating film of the gears, so the meshing stiffness calculated theoretically is too high. Compared with the results obtained by the calculation method of the present invention, the maximum error of the single tooth meshing stiffness is about 4.7%, and the average error of the time-varying meshing stiffness is about 6.1%. Figure 4 shown.

[0071] Obviously, the calculation method of time-varying mesh stiffness of spur gears considering solid lubrication proposed in the present invention improves the calculation accuracy of time-varying mesh stiffness and is closer to the actual working condition of spur gear transmission.

[0072] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention without departing from the principles and purpose of the present invention.

Claims

1. A method for calculating the time-varying mesh stiffness of solid lubricated spur gears, characterized in that The specific steps are as follows: Calculation of the Modified Elastic Modulus of Spur Gears Under the Influence of Solid Lubrication E c and modified shear modulus G c , the formula is as follows: Where, V f is the volume of the solid lubricating film, V s is the volume of the gear teeth, V t is the total volume, E f is the elastic modulus of the solid lubricating film, E s is the gear tooth elastic modulus, ν is Poisson's ratio; Calculating the bending stiffness of gear teeth k b , shear stiffness k s , radial compression stiffness k a , Hertz contact stiffness k h , matrix flexible deformation stiffness k nfi and inter-tooth structural coupling stiffness k nfij ; According to the stiffness series and parallel theory, the time-varying meshing stiffness of spur gears is calculated as follows: Where, k hi ( i =1, 2) is the i Hertzian contact stiffness of teeth; k nti ( i =1, 2; n =p, g) is the i Tooth stiffness of the master and driven wheels, including bending stiffness k b , shear stiffness k s and radial compressive stiffness k a ; k nfi ( i =1, 2; n =p, g) is the matrix flexible deformation stiffness; k nfij ( i =1, 2, i ≠ j ; n =p, g) is the coupling stiffness of the inter-tooth structure, p, g represent the driving wheel and the driven wheel respectively; F i Indicates the i The meshing force between the master and driven gear teeth, F j Indicates the j The meshing force between the master and driven wheel teeth; The volume of the solid lubricating film V f The calculation formula is as follows: Where, l s is the involute length of the gear teeth, s a is the tooth top circle thickness, h f is the solid lubricant film thickness, B is the tooth width; Where, α 1 is the angle between the line of action of the meshing force and the vertical line of the gear tooth center line, α 2 is the tooth root half angle, r b is the gear base circle radius, m is the gear module, r a is the radius of the tooth tip circle, r is the pitch circle radius, α a is the tooth tip pressure angle, α is the pitch circle pressure angle, inv is the formula for the involute angle.

2. The method for calculating the time-varying mesh stiffness of a solid-lubricated spur gear according to claim 1, characterized in that: The gear tooth volume V s The calculation formula is as follows: Where, d is the gear pitch circle diameter, d f is the gear tooth root diameter, B is the tooth width, z is the number of gear teeth.

3. The method for calculating the time-varying mesh stiffness of a solid-lubricated spur gear according to claim 1, wherein: The bending stiffness k b , shear stiffness k s , radial compression stiffness k a , Hertz contact stiffness k h , matrix flexible deformation stiffness k nfi and inter-tooth structural coupling stiffness k nfij The calculation expressions are: Where, α 1 is the angle between the line of action of the meshing force and the vertical line of the gear tooth center line, α 2 is the tooth root half angle, θ 1. θ 2 are the meshing angles of gear teeth 1 and 2, r b is the gear base circle radius, r f is the root circle radius of the gear, ∑ h β is any pressure angle on the involute β The accumulated wear at h x Horizontal distance from tooth root x Half the tooth thickness, B is the tooth width, F is the gear tooth meshing force, u i is the distance from the intersection of the meshing line and the gear tooth centerline to the tooth root circle, S f is the length of the tooth root arc, E c , G c are the modified elastic modulus and shear modulus, as well as L i 、 M i 、 P i 、 Q i 、 R i 、 S i 、 T i 、 U i 、 V i Represent the coefficients of each polynomial respectively.

4. A time-varying mesh stiffness calculation system for solid-lubricated spur gears, characterized by: The invention comprises at least one processor and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, and the computer program is executed by the at least one processor so that the at least one processor can execute the method for calculating the time-varying meshing stiffness of a solid-lubricated spur gear according to any one of claims 1 to 3.

5. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores computer instructions, and the computer instructions are used to enable a processor to implement the time-varying mesh stiffness calculation method of a solid-lubricated spur gear according to any one of claims 1 to 3 when executed.

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