Time-varying meshing stiffness calculation method for solid lubrication straight toothed spur gear

Abstract

The invention discloses a time-varying meshing stiffness calculation method for a solid lubrication straight spur gear, and belongs to the field of gear dynamics. The method comprises the following steps: calculating a corrected elastic modulus and a corrected shear modulus of the straight toothed spur gear under the influence of solid lubrication; gear tooth bending rigidity, shear rigidity, radial compression rigidity, Hertz contact rigidity, matrix flexible deformation rigidity and inter-tooth structure coupling rigidity are calculated; and calculating to obtain the time-varying meshing stiffness of the straight toothed spur gear according to a stiffness series connection and parallel connection theory. The problem that the influence of a solid lubricating film on the gear rigidity is not considered in the prior art is solved.

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 lubrication spur gear. Background Art

[0002] The gear transmission system of space mechanism is widely used in the transmission joints of space manipulators and solar wing drive devices. It is the core component of the manipulator to complete the on-orbit assembly of spacecraft, assist astronauts in extravehicular activities, and complete the solar wing's orientation to the sun and other key actions. Due to the harsh and complex space environment, traditional grease lubricants are difficult to perform space missions. Molybdenum disulfide has become one of the most effective solid lubricants due to its excellent friction properties in the space environment and is widely used in the gear transmission system of space mechanism.

[0003] At present, when calculating the meshing stiffness of spur gears, most of them only consider the contact stiffness, bending stiffness, shear stiffness, radial compression stiffness, matrix flexible deformation stiffness, and structural coupling stiffness of spur gears, and then obtain the total meshing stiffness through the series and parallel relationship of stiffness calculation. However, these calculation methods do not consider the influence of the solid lubricant coating on the gear stiffness. After the gear is coated with a solid lubricant film, the material properties of the gear change, and the change in material properties will directly affect the meshing stiffness of the gear. Therefore, without considering the meshing stiffness calculated by the solid lubricant film, the characteristics of the gear transmission system cannot be accurately and completely described, thereby affecting the accuracy of the comprehensive analysis of the dynamic performance of the gear. Summary of the invention

[0004] Technical issues to be solved:

[0005] In order to avoid the shortcomings of the prior art, the present invention provides a method for calculating the time-varying meshing stiffness of solid-lubricated spur gears. By incorporating the influence of the solid lubricating film into the calculation of the meshing stiffness, the characteristics of the gear transmission system can be more accurately described, thereby improving the accuracy of the comprehensive analysis of its dynamic performance. This improvement will provide a scientific basis for the design of the gear transmission system of spacecraft, optimize the use of solid lubricants, and ensure reliability and stability in extreme space environments. The present invention solves the problem that the prior art does not consider the influence of the solid lubricating film on the gear stiffness.

[0006] The technical solution of the present invention is: a method for calculating the time-varying meshing stiffness of a solid lubricated spur gear, and the specific steps 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 bending stiffness k of the gear teeth 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, and the formula is as follows:

[0012]

[0013] In the formula, 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 driving 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 represents the meshing force between the i-th pair of driving and driven gear teeth, F 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: 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] In the formula, l s is the involute length of the gear tooth, s a is the tooth thickness at the top circle, h f is the thickness of the solid lubricating film, and B is the tooth width; where,

[0020]

[0021] invλ=tanλ-λ

[0022] In the formula, α 1 is the angle between the line of action of the meshing force and the perpendicular line to the center line of the gear teeth, α 2 is the tooth root 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 tip circle, r is the pitch circle radius, α a is the tooth tip 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] In the formula, α 1 is the angle between the line of action of the meshing force and the perpendicular line to the center line of the gear teeth, 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 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 gear 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 polynomial coefficients respectively.

[0025] A time-varying meshing stiffness calculation system for solid-lubricated spur gears 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 time-varying meshing stiffness calculation method for solid-lubricated spur gears.

[0026] A computer-readable digital storage medium stores computer instructions, and the computer instructions 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 lubricating gears, and finally calculates the time-varying meshing stiffness of the solid lubricating 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 in the present invention is closer to the actual working condition 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 stress and deformation of worn gear teeth.

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

[0033] Figure 4It is a comparison chart of the results of calculating the time-varying meshing stiffness of spur gears by 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 be used to explain the present invention, but should not be construed as limiting the present invention.

[0035] Based on the fact that the meshing stiffness of spur gears is currently calculated without considering the meshing stiffness calculated by the solid lubricating film, the characteristics of the gear transmission system cannot be accurately and completely described, thereby affecting the accuracy of the comprehensive analysis of the dynamic performance of the gears, and the like, the present invention provides a method for calculating the time-varying meshing stiffness of solid lubricating spur gears, which can more accurately describe the characteristics of the gear transmission system by incorporating the influence of the solid lubricating film into the calculation of the meshing stiffness, thereby improving the accuracy of the comprehensive analysis of its dynamic performance. This improvement will provide a scientific basis for the design of the gear transmission system of spacecraft, optimize the use of solid lubricants, and ensure reliability and stability in extreme space environments.

[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 meshing 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 surface of the gear pair 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 circle diameter, B is the tooth width, z is the number of gear teeth, l s is the involute length of the gear tooth, s a is the tooth thickness at the top circle, h f is the thickness of the solid lubricating 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] In the formula, m is the gear module, r is a is the radius of the tooth tip circle, r is the pitch circle radius, α a is the tooth tip circle pressure angle, α is the pitch circle pressure angle, and inv is the involute angle formula.

[0047] Step 2, calculating the time-varying meshing stiffness of the solid lubrication gear teeth during the meshing process of a single pair of teeth;

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

[0049]

[0050] Calculation of tooth shear stiffness k s , which is calculated as:

[0051]

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

[0053]

[0054] In the formula, α 1 is the angle between the line of action of the meshing force and the perpendicular line to the center line of the gear teeth, α 2 is the tooth root 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 a 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 k nfij :

[0058]

[0059] In the formula, θ 1 ,θ 2 are the meshing angles of gear teeth 1 and 2, F is the meshing force of 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 gear 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 polynomial coefficients respectively.

[0060] Step 3: Calculate the time-varying meshing stiffness of the solid lubricated spur gear according to 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 ;knfi (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 and g represent the driving wheel and the driven wheel respectively.

[0064] A time-varying meshing stiffness calculation system for solid-lubricated spur gears, characterized in that it includes 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 time-varying meshing stiffness calculation method for solid-lubricated spur gears.

[0065] A computer-readable digital storage medium, 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 meshing 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 in FIG. Figure 2 The schematic diagram of the calculation of the stress and deformation of the worn gear teeth in the experimental example shown in the figure and 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 lubricating 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 single tooth meshing stiffness calculated by 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 calculation results of meshing stiffness calculated by the ISO standard method are too large in theory because the solid lubricating film of the gear is not taken into account. Compared with the results obtained by the calculation method of the present invention, the maximum error of single tooth meshing stiffness is about 4.7%, and the average error of time-varying meshing stiffness is about 6.1%. Figure 4 shown.

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

[0072] Although the embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary 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 intent 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 E of spur gears under the influence of solid lubrication c and the modified shear modulus G c , the formula is as follows: 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; Calculate the bending stiffness k of the gear teeth 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 ; According to the stiffness series and parallel theory, the time-varying meshing stiffness of spur gears is calculated, and the formula is as follows: In the formula, 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 driving 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 represents the meshing force between the i-th pair of driving and driven gear teeth, F j It represents the meshing force between the jth pair of driving and driven wheel teeth.

2. The method for calculating the time-varying meshing stiffness of a solid lubrication 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 circle diameter, B is the tooth width, and z is the number of gear teeth.

3. The method for calculating the time-varying meshing stiffness of a solid lubrication spur gear according to claim 2, characterized in that: The volume V of the solid lubricating film f The calculation formula is as follows: V f =(2l s +s a )h f B In the formula, l s is the involute length of the gear tooth, s a is the tooth thickness at the top circle, h f is the thickness of the solid lubricating film, and B is the tooth width; where, invλ=tanλ-λ 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, r b is the base circle radius of the gear, 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 circle pressure angle, α is the pitch circle pressure angle, and inv is the involute angle formula.

4. The method for calculating the time-varying meshing stiffness of a solid lubrication spur gear according to claim 3, characterized in that: 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: 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 gear 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 polynomial coefficients respectively.

5. A time-varying meshing stiffness calculation system for solid lubrication spur gears, characterized in that: It 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 time-varying meshing stiffness calculation method of the solid-lubricated spur gear according to any one of claims 1 to 4.

6. 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 method for calculating the time-varying mesh stiffness of a solid-lubricated spur gear as described in any one of claims 1 to 4 when executed.

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