Method for calculating time-varying meshing stiffness of non-orthogonal asymmetric helical tooth surface gear

Abstract

The invention discloses a non-orthogonal asymmetric helical tooth surface gear time-varying meshing stiffness calculation method. The method comprises the following steps: step (1), constructing a non-orthogonal asymmetric helical tooth surface gear system processing coordinate system variation matrix; (2) establishing a tooth surface equation of the non-orthogonal asymmetric helical tooth surface gear; (3) establishing a time-varying meshing stiffness analytical calculation model of the non-orthogonal asymmetric helical tooth surface gear; (4) solving the time-varying meshing stiffness of the non-orthogonal asymmetric helical tooth surface gear under the combined transmission of different pressure angles and crossed shaft angles; the method has the beneficial effects that the time-varying meshing stiffness of the non-orthogonal asymmetric helical tooth surface gear can be solved, the vacancy of a current non-orthogonal asymmetric helical tooth surface gear system research method is filled, and the method has good universality.

Description

Technical Field

[0001] The present invention relates to the technical field of gear dynamics, and in particular to a method for calculating the time-varying meshing stiffness of non-orthogonal asymmetric helical gears. Background Art

[0002] The non-orthogonal asymmetric helical gear system is a special gear transmission system, which is usually used in high-speed, heavy-load and special transmission scenarios, such as aircraft engines, ship propulsion systems, and rotorcraft. In these application scenarios, the non-orthogonal asymmetric helical gear system can provide better transmission performance and higher transmission efficiency, thereby meeting the high requirements of the equipment for the transmission system. During the transmission process of the non-orthogonal asymmetric helical gear system, the fluctuation of the time-varying mesh stiffness will significantly affect the stability of the transmission system, thereby increasing the risk of mechanical equipment failure. Therefore, it is of great significance to analyze the time-varying mesh stiffness of non-orthogonal asymmetric helical gears and explore the influence mechanism of the pressure angle and shaft angle changes on the system. The existing technology lacks a method for calculating the time-varying mesh stiffness of non-orthogonal asymmetric helical gears.

[0003] In order to solve the above problems, the present invention proposes a method for calculating the time-varying meshing stiffness of non-orthogonal asymmetric helical gears. This method can accurately and effectively study the stiffness changes of non-orthogonal asymmetric helical gears during meshing; it fills the technical gap in the calculation of the time-varying meshing stiffness of non-orthogonal asymmetric helical gears, which not only promotes the development of engineering technology, but also produces greater social and economic benefits. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill a gap in related technologies, this paper provides a method for calculating the time-varying mesh stiffness of non-orthogonal, asymmetric helical gears. This method constructs a transformation matrix for the machining coordinate system of non-orthogonal, asymmetric helical gears, derives the tooth surface equations for the non-orthogonal, asymmetric helical gears, and establishes an analytical calculation model for the time-varying mesh stiffness. Substituting different pressure angles and shaft angles, the method calculates the time-varying mesh stiffness of non-orthogonal, asymmetric helical gears under different pressure angle and shaft angle combinations.

[0005] The technical solution adopted by the present invention to solve the technical problem is as follows: a method for calculating the time-varying meshing stiffness of non-orthogonal asymmetric helical gears, characterized by comprising the following steps:

[0006] Step (1): Construct the machining coordinate system change matrix of the non-orthogonal asymmetric helical gear system; the tooth surface of the non-orthogonal asymmetric helical gear is formed by the tooth surface enveloping of the non-orthogonal asymmetric helical gear shaping cutter, and the coordinate system S1 is the rotation coordinate system of the non-orthogonal asymmetric helical gear. The coordinate system S f is the fixed coordinate system of the non-orthogonal asymmetric helical gear, coordinate system S2 is the rotating coordinate system of the non-orthogonal asymmetric helical gear shaping cutter, and coordinate system Sp It is the fixed coordinate system of the non-orthogonal and asymmetric helical gear shaping cutter;

[0007] The transformation matrix M from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the rotating coordinate system of the non-orthogonal asymmetric helical gear shaping cutter p,2 It can be obtained by the following formula:

[0008]

[0009] Where: ω1 is the angle of rotation of the non-orthogonal asymmetric helical gear shaping cutter;

[0010] The transformation matrix M from the non-orthogonal asymmetric helical gear shaping cutter rotation coordinate system to the non-orthogonal asymmetric helical gear rotation coordinate system 2,1 It can be obtained by the following formula:

[0011]

[0012] Where: γ is the axis intersection angle;

[0013] The transformation matrix M from the rotating coordinate system of the non-orthogonal asymmetric helical gear to the fixed coordinate system of the non-orthogonal asymmetric helical gear 1,f It can be obtained by the following formula:

[0014]

[0015] Where: ω2 is the angle rotated by the non-orthogonal asymmetric helical gear;

[0016] The transformation matrix M from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear face gear p,f It can be obtained by the following formula:

[0017]

[0018] Where: γ is the axis angle, ω1 is the angle of rotation of the non-orthogonal asymmetric helical gear shaping cutter, and ω2 is the angle of rotation of the non-orthogonal asymmetric helical gear face gear;

[0019] Step (2): Establish the tooth surface equation of the non-orthogonal asymmetric helical gear; the position vector r of the active side of the non-orthogonal asymmetric helical gear shaping cutter d (θ d ,e d ) can be obtained by the following formula:

[0020]

[0021] Where: r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, α dis the pressure angle on the active side of the non-orthogonal asymmetric helical gear shaping cutter, θ d is the angle parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter, L lx is the lead of the helix angle, e d is the axial parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter;

[0022] Position vector r of the driven side of the non-orthogonal asymmetric helical gear shaping cutter c (θ c ,e c ) can be obtained from the following formula:

[0023]

[0024] Where: r bc is the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, α c is the pressure angle on the driven side of the non-orthogonal asymmetric helical gear shaping cutter, θ c is the angular parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, L lx is the lead of the helix angle, e c is the axial parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter;

[0025] Meshing equation f of the driving side of non-orthogonal asymmetric helical gear d (e d ,θ d ,ω1) is as follows:

[0026]

[0027] Where: r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter, β is the helix angle, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, N2 is the number of teeth of the non-orthogonal asymmetric helical gear face gear, γ is the axis intersection angle, ω1 is the angle of rotation of the non-orthogonal asymmetric helical gear shaping cutter, α d is the pressure angle on the active side of the non-orthogonal asymmetric helical gear shaping cutter, θ d is the angle parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter, e d is the axial parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter;

[0028] Meshing equation f of the driven side of non-orthogonal asymmetric helical gear c (e c ,θ c ,ω1) is as follows:

[0029]

[0030] Where: r bcis the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, β is the helix angle, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, N2 is the number of teeth of the non-orthogonal asymmetric helical gear face gear, γ is the axis angle, α c is the pressure angle on the driven side of the non-orthogonal asymmetric helical gear shaping cutter, ω1 is the angle through which the non-orthogonal asymmetric helical gear shaping cutter rotates, and θ c is the angular parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, e c is the axial parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter;

[0031] The tooth surface equation of the active side of the non-orthogonal asymmetric helical gear is:

[0032]

[0033] Where: f d (e d ,θ d ,ω1) is the meshing equation of the active side of the non-orthogonal asymmetric helical gear, r d (θ d ,e d ) is the position vector of the active side of the non-orthogonal asymmetric helical gear shaping cutter, M p,f is the transformation matrix from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear face gear, r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter;

[0034] The equation of the driven side tooth surface of the non-orthogonal asymmetric helical gear is:

[0035]

[0036] Where: f c (e c ,θ c ,ω1) is the meshing equation of the driven side of the non-orthogonal asymmetric helical gear, r c (θ c ,e c ) is the position vector of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, M p,f is the transformation matrix from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear face gear, r bc is the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter;

[0037] Step (3): Establish an analytical calculation model for the time-varying meshing stiffness of non-orthogonal asymmetric helical gears; in order to accurately calculate the stiffness of non-orthogonal asymmetric helical gears, the improved potential energy method and slicing method are used to cut the non-orthogonal asymmetric helical gear teeth into several trapezoidal slices along the axial direction. The stiffness of each slice is calculated independently, and the area moment of inertia of each slice is as follows:

[0038]

[0039] Where: I i is the area moment of inertia of the i-th trapezoidal slice, h i is the height of the i-th trapezoidal slice along the tooth profile, a i is the length of the base of the i-th trapezoidal slice, b i is the length of the lower base of the i-th trapezoidal slice;

[0040] Bending stiffness k of gear teeth with non-orthogonal and asymmetric helical tooth surfaces w It is obtained from the following formula:

[0041]

[0042] Where: y f is the axial coordinate of the slice, d is the thickness of the slice projected onto the plane, x t1 and x t2 is the direction coordinate of the boundary point projected onto the plane by the elliptical surface load, F is the total force generated by the load distribution function on the slice, α i is the pressure angle of the i-th trapezoidal slice in the non-orthogonal asymmetric helical gear, x a 、y a are the horizontal and vertical coordinate values of the projection plane, respectively, h i is the height of the i-th trapezoidal slice along the tooth profile, x fc and x fp is the normal vector coordinate of the projection point of the long axis on the projection plane, y fc is the normal vector coordinate of the projection point of the minor axis on the projection plane, E is the Young's modulus of the non-orthogonal asymmetric helical gear, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, a i is the length of the base of the i-th trapezoidal slice, b i is the length of the lower base of the i-th trapezoidal slice;

[0043] Matrix stiffness k of non-orthogonal asymmetric helical gear teeth during single tooth meshing b It is obtained from the following formula:

[0044]

[0045] Where: αi is the pressure angle of the i-th trapezoidal slice in the non-orthogonal asymmetric helical gear, C1, V1, B1, G1 are the polynomial fitting parameters, S is the distance from the front pair of teeth to the back pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear, L f is the distance from the contact point to the base, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, E is the Young's modulus of the non-orthogonal asymmetric helical face gear, d is the thickness of the slice projected onto the plane, and v is the Poisson's ratio;

[0046] The matrix stiffness of the non-orthogonal asymmetric helical gear teeth during double tooth meshing is obtained by the following formula:

[0047]

[0048] Where: k b1 k is the matrix coupling stiffness under the influence of the meshing force of the first pair of teeth on the second pair of teeth during the meshing process of non-orthogonal asymmetric helical gears, b2 is the matrix coupling stiffness under the influence of the meshing force of the rear pair of teeth on the front pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear, α1 is the pressure angle of the front tooth during the meshing process of the non-orthogonal asymmetric helical gear, α2 is the pressure angle of the rear tooth during the meshing process of the non-orthogonal asymmetric helical gear, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, E is the Young's modulus of the non-orthogonal asymmetric helical gear, d is the thickness of the slice projected onto the plane, J2, K2, L2, T2, G2, B2, C2, V2, M2, J3, K3, L3, T3, G3, B3, C3, V3, M3 are the polynomial fitting coefficients, L f1 L is the distance from the contact point of the first pair of teeth to the gear base. f2 is the distance from the contact point of the rear pair of teeth to the base of the gear, and S is the distance from the front pair of teeth to the rear pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear;

[0049] According to Hertz contact theory, the meshing of non-orthogonal asymmetric helical gear transmission system is two isotropic elastic bodies. The tooth contact stiffness k u It can be expressed by the following formula:

[0050]

[0051] Where: F is the total force generated by the load distribution function on the slice, E is the Young's modulus of the non-orthogonal asymmetric helical gear, and d is the thickness of the slice projected onto the plane;

[0052] Single tooth mesh stiffness k of non-orthogonal asymmetric helical gears m1 as follows:

[0053]

[0054] Where: k w is the bending stiffness of the non-orthogonal asymmetric helical gear teeth, k b is the matrix stiffness during single tooth meshing, k u is the gear tooth contact stiffness;

[0055] Double tooth mesh stiffness k of non-orthogonal asymmetric helical gears m2 as follows:

[0056]

[0057] Where: k w is the bending stiffness of the non-orthogonal asymmetric helical gear teeth, k b1 k is the matrix coupling stiffness under the influence of the meshing force of the first pair of teeth on the second pair of teeth during the meshing process of non-orthogonal asymmetric helical gears, b2 k is the matrix coupling stiffness under the influence of the meshing force of the rear pair of teeth on the front pair of teeth during the meshing process of non-orthogonal asymmetric helical gears, u is the gear tooth contact stiffness;

[0058] Step (4): solve the time-varying meshing stiffness of the non-orthogonal asymmetric helical gear under different pressure angle and axis intersection angle combinations; based on the time-varying meshing stiffness analytical calculation model in step (3), solve and analyze the time-varying meshing stiffness calculation equation of the non-orthogonal asymmetric helical gear for different pressure angle and axis intersection angle combinations, and obtain the time-varying meshing stiffness of the non-orthogonal asymmetric helical gear under different pressure angle and axis intersection angle combinations;

[0059] The time-varying meshing stiffness of non-orthogonal asymmetric helical gears under different pressure angles and axis angles was plotted, and the dynamic characteristics of their meshing stiffness were deeply analyzed. Reducing the pressure angle from 30° to 15° and increasing the axis angle from 30° to 90° both played a significant positive role in improving the overall stability of the non-orthogonal asymmetric helical gear transmission system.

[0060] Compared with the prior art, the beneficial effects of the present invention are: a non-orthogonal asymmetric helical gear system machining coordinate system transformation matrix is established, the non-orthogonal asymmetric helical gear tooth surface equation and the time-varying meshing stiffness calculation equation are constructed, and the time-varying meshing stiffness change characteristics of the non-orthogonal asymmetric helical gear under different pressure angle and axis angle combinations are obtained, filling the current gap in the research of non-orthogonal asymmetric helical gear systems, and providing a reference for vibration reduction, noise reduction and performance improvement of non-orthogonal asymmetric helical gear systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1This is a flow chart of the calculation method for the time-varying mesh stiffness of non-orthogonal asymmetric helical gears;

[0062] Figure 2 It is the machining coordinate system of non-orthogonal asymmetric helical gear transmission system;

[0063] Figure 3 It is a calculation model for time-varying mesh stiffness of non-orthogonal asymmetric helical gears;

[0064] Figure 4 This is a time-varying mesh stiffness diagram of a non-orthogonal asymmetric helical gear with an axis angle of 60° and a transmission pressure angle of 15°, 20°, 25°, and 30°.

[0065] Figure 5 This is a time-varying mesh stiffness diagram of a non-orthogonal asymmetric helical gear at a pressure angle of 20° and an axis angle of 30°, 45°, 60°, and 90°. DETAILED DESCRIPTION

[0066] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1-Figure 5 The specific embodiments of the present invention are described in detail.

[0067] Figure 1 The flow chart of the method for calculating the time-varying mesh stiffness of non-orthogonal asymmetric helical gears includes the following steps:

[0068] Step (1): Construct the change matrix of the machining coordinate system of the non-orthogonal asymmetric helical gear system; the tooth surface of the non-orthogonal asymmetric helical gear is formed by the tooth surface envelopment of the non-orthogonal asymmetric helical gear shaping cutter. The machining coordinate system of the non-orthogonal asymmetric helical gear transmission system is as follows: Figure 2 As shown, coordinate system S1 is the rotation coordinate system of the non-orthogonal asymmetric helical gear, and coordinate system S f is the fixed coordinate system of the non-orthogonal asymmetric helical gear, coordinate system S2 is the rotating coordinate system of the non-orthogonal asymmetric helical gear shaping cutter, and coordinate system S p It is the fixed coordinate system of the non-orthogonal and asymmetric helical gear shaping cutter;

[0069] The transformation matrix M from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the rotating coordinate system of the non-orthogonal asymmetric helical gear shaping cutter p,2 It can be obtained by the following formula:

[0070]

[0071] Where: ω1 is the angle of rotation of the non-orthogonal asymmetric helical gear shaping cutter;

[0072] The transformation matrix M from the non-orthogonal asymmetric helical gear shaping cutter rotation coordinate system to the non-orthogonal asymmetric helical gear rotation coordinate system 2,1It can be obtained by the following formula:

[0073]

[0074] Where: γ is the axis intersection angle;

[0075] The transformation matrix M from the rotating coordinate system of the non-orthogonal asymmetric helical gear to the fixed coordinate system of the non-orthogonal asymmetric helical gear 1,f It can be obtained by the following formula:

[0076]

[0077] Where: ω2 is the angle rotated by the non-orthogonal asymmetric helical gear;

[0078] The transformation matrix M from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear face gear p,f It can be obtained by the following formula:

[0079]

[0080] Where: γ is the axis angle, ω1 is the angle of rotation of the non-orthogonal asymmetric helical gear shaping cutter, and ω2 is the angle of rotation of the non-orthogonal asymmetric helical gear face gear;

[0081] Step (2): Establish the tooth surface equation of the non-orthogonal asymmetric helical gear; the position vector r of the active side of the non-orthogonal asymmetric helical gear shaping cutter d (θ d ,e d )as follows:

[0082]

[0083] Where: r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, α d is the pressure angle on the active side of the non-orthogonal asymmetric helical gear shaping cutter, θ d is the angle parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter, L lx is the lead of the helix angle, e d is the axial parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter;

[0084] Position vector r of the driven side of the non-orthogonal asymmetric helical gear shaping cutter c (θ c ,e c )as follows:

[0085]

[0086] Where: r bcis the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, α c is the pressure angle on the driven side of the non-orthogonal asymmetric helical gear shaping cutter, θ c is the angular parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, L lx is the lead of the helix angle, e c is the axial parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter;

[0087] Meshing equation f of the driving side of non-orthogonal asymmetric helical gear d (e d ,θ d ,ω1) is obtained by the following formula:

[0088]

[0089] Where: r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter, β is the helix angle, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, N2 is the number of teeth of the non-orthogonal asymmetric helical gear face gear, γ is the axis intersection angle, ω1 is the angle of rotation of the non-orthogonal asymmetric helical gear shaping cutter, α d is the pressure angle on the active side of the non-orthogonal asymmetric helical gear shaping cutter, θ d is the angle parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter, e d is the axial parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter;

[0090] Meshing equation f of the driven side of non-orthogonal asymmetric helical gear c (e c ,θ c ,ω1) is obtained by the following formula:

[0091]

[0092] Where: r bc is the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, β is the helix angle, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, N2 is the number of teeth of the non-orthogonal asymmetric helical gear face gear, γ is the axis angle, α c is the pressure angle on the driven side of the non-orthogonal asymmetric helical gear shaping cutter, ω1 is the angle through which the non-orthogonal asymmetric helical gear shaping cutter rotates, and θ c is the angular parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, e c is the axial parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter;

[0093] The tooth surface equation of the active side of the non-orthogonal asymmetric helical gear is:

[0094]

[0095] Where: f d (e d ,θ d ,ω1) is the meshing equation of the active side of the non-orthogonal asymmetric helical gear, r d (θ d ,e d ) is the position vector of the active side of the non-orthogonal asymmetric helical gear shaping cutter, M p,f is the transformation matrix from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear face gear, r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter;

[0096] The equation of the driven side tooth surface of the non-orthogonal asymmetric helical gear is:

[0097]

[0098] Where: f c (e c ,θ c ,ω1) is the meshing equation of the driven side of the non-orthogonal asymmetric helical gear, r c (θ c ,e c ) is the position vector of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, M p,f is the transformation matrix from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear face gear, r bc is the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter;

[0099] Step (3): Establish an analytical calculation model for the time-varying meshing stiffness of non-orthogonal asymmetric helical gears; In order to accurately calculate the stiffness of non-orthogonal asymmetric helical gears, the improved potential energy method and slicing method are used, such as Figure 3 As shown in the calculation model of time-varying mesh stiffness of non-orthogonal asymmetric helical gears, the non-orthogonal asymmetric helical gear teeth are divided into several trapezoidal slices along the axial direction. The stiffness of each slice is calculated independently, and the area moment of inertia of each slice is:

[0100]

[0101] Where: I i is the area moment of inertia of the i-th trapezoidal slice, h i is the height of the i-th trapezoidal slice along the tooth profile, a i is the length of the base of the i-th trapezoidal slice, b i is the length of the lower base of the i-th trapezoidal slice;

[0102] Bending stiffness k of gear teeth with non-orthogonal and asymmetric helical tooth surfaces w for:

[0103]

[0104] Where: y f is the axial coordinate of the slice, d is the thickness of the slice projected onto the plane, x t1 and x t2 is the direction coordinate of the boundary point projected onto the plane by the elliptical surface load, F is the total force generated by the load distribution function on the slice, α i is the pressure angle of the i-th trapezoidal slice in the non-orthogonal asymmetric helical gear, x a 、y a are the horizontal and vertical coordinate values of the projection plane, respectively, h i is the height of the i-th trapezoidal slice along the tooth profile, x fc and x fp is the normal vector coordinate of the projection point of the long axis on the projection plane, y fc is the normal vector coordinate of the projection point of the minor axis on the projection plane, E is the Young's modulus of the non-orthogonal asymmetric helical gear, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, a i is the length of the base of the i-th trapezoidal slice, b i is the length of the lower base of the i-th trapezoidal slice;

[0105] Matrix stiffness k of non-orthogonal asymmetric helical gears during single tooth meshing b for:

[0106]

[0107] Where: α i is the pressure angle of the i-th trapezoidal slice in the non-orthogonal asymmetric helical gear, C1, V1, B1, G1 are the polynomial fitting parameters, S is the distance from the front pair of teeth to the back pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear, L f is the distance from the contact point to the base, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, E is the Young's modulus of the non-orthogonal asymmetric helical face gear, d is the thickness of the slice projected onto the plane, and v is the Poisson's ratio;

[0108] The matrix stiffness of the non-orthogonal asymmetric helical gear during double tooth meshing is:

[0109]

[0110] Where: k b1k is the matrix coupling stiffness under the influence of the meshing force of the first pair of teeth on the second pair of teeth during the meshing process of non-orthogonal asymmetric helical gears, b2 is the matrix coupling stiffness under the influence of the meshing force of the rear pair of teeth on the front pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear, α1 is the pressure angle of the front tooth during the meshing process of the non-orthogonal asymmetric helical gear, α2 is the pressure angle of the rear tooth during the meshing process of the non-orthogonal asymmetric helical gear, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, E is the Young's modulus of the non-orthogonal asymmetric helical gear, d is the thickness of the slice projected onto the plane, J2, K2, L2, T2, G2, B2, C2, V2, M2, J3, K3, L3, T3, G3, B3, C3, V3, M3 are the polynomial fitting coefficients, L f1 L is the distance from the contact point of the first pair of teeth to the gear base. f2 is the distance from the contact point of the rear pair of teeth to the base of the gear, and S is the distance from the front pair of teeth to the rear pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear;

[0111] According to Hertz contact theory, the meshing of non-orthogonal asymmetric helical gear transmission system is two isotropic elastic bodies. The tooth contact stiffness k u It can be expressed by the following formula:

[0112]

[0113] Where: F is the total force generated by the load distribution function on the slice, E is the Young's modulus of the non-orthogonal asymmetric helical gear, and d is the thickness of the slice projected onto the plane;

[0114] Single tooth mesh stiffness k of non-orthogonal asymmetric helical gears m1 for:

[0115]

[0116] Where: k w is the bending stiffness of the non-orthogonal asymmetric helical gear teeth, k b is the matrix stiffness during single tooth meshing, k u is the gear tooth contact stiffness;

[0117] Double tooth mesh stiffness k of non-orthogonal asymmetric helical gears m2 for:

[0118]

[0119] Where: k w is the bending stiffness of the non-orthogonal asymmetric helical gear teeth, k b1k is the matrix coupling stiffness under the influence of the meshing force of the first pair of teeth on the second pair of teeth during the meshing process of non-orthogonal asymmetric helical gears; b2 k is the matrix coupling stiffness under the influence of the meshing force of the rear pair of teeth on the front pair of teeth during the meshing process of non-orthogonal asymmetric helical gears; u is the gear tooth contact stiffness;

[0120] Step (4): solve the time-varying meshing stiffness of the non-orthogonal asymmetric helical gear under different pressure angle and axis intersection angle combinations; based on the time-varying meshing stiffness analytical calculation model in step (3), solve and analyze the time-varying meshing stiffness calculation equation of the non-orthogonal asymmetric helical gear for different pressure angle and axis intersection angle combinations, and obtain the time-varying meshing stiffness of the non-orthogonal asymmetric helical gear under different pressure angle and axis intersection angle combinations;

[0121] The time-varying meshing stiffness of non-orthogonal asymmetric helical gears under different pressure angles and axis angles was plotted, and the dynamic characteristics of their meshing stiffness were deeply analyzed. The reduction of the pressure angle from 30° to 15° and the increase of the axis angle from 30° to 90° both played a significant positive role in improving the overall stability of the non-orthogonal asymmetric helical gear transmission system.

[0122] In the example, the parameters selected by the system are shown in Table 1. Using the above method, the time-varying meshing stiffness of the non-orthogonal asymmetric helical gear under different pressure angles and axis angles is calculated by MATLAB programming. The results are as follows: Figure 4 and Figure 5 shown.

[0123] Figure 4 This is a time-varying mesh stiffness diagram of a non-orthogonal asymmetric helical gear with an axis angle of 60° and a transmission pressure angle of 15°, 20°, 25°, and 30°. Figure 5 Figure 2 shows the time-varying mesh stiffness of a non-orthogonal, asymmetric helical gear at a 20° pressure angle and at shaft angles of 30°, 45°, 60°, and 90°. Within a certain range, decreasing the pressure angle and increasing the shaft angle can improve system performance, reduce noise, and increase the service life of the structure.

[0124] Table 1 Basic parameters of non-orthogonal asymmetric helical gear transmission system

[0125]

[0126] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modifications, changes and equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the scope of protection of the technology of the invention.

Claims

1. A method for calculating the time-varying mesh stiffness of non-orthogonal asymmetric helical gears, characterized in that: The following steps are involved: Step (1): Construct the change matrix of the machining coordinate system of the non-orthogonal asymmetric helical gear system; the tooth surface of the non-orthogonal asymmetric helical gear is formed by the tooth surface of the non-orthogonal asymmetric helical gear shaping cutter, and the change matrix M from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear is p,f It can be obtained by the following formula: Where: γ is the axis angle, ω1 is the angle of rotation of the non-orthogonal asymmetric helical gear shaping cutter, and ω2 is the angle of rotation of the non-orthogonal asymmetric helical gear face gear; Step (2): Establish the tooth surface equation of the non-orthogonal asymmetric helical gear; the position vector r of the active side of the non-orthogonal asymmetric helical gear shaping cutter d (θ d ,e d ) can be obtained by the following formula: Where: r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, α d is the pressure angle on the active side of the non-orthogonal asymmetric helical gear shaping cutter, θ d is the angle parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter, L lx is the lead of the helix angle, e d is the axial parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter; Position vector r of the driven side of the non-orthogonal asymmetric helical gear shaping cutter c (θ c ,e c ) can be obtained from the following formula: Where: r bc is the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, α c is the pressure angle on the driven side of the non-orthogonal asymmetric helical gear shaping cutter, θ c is the angular parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, L lx is the lead of the helix angle, e c is the axial parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter; Meshing equation f of the driving side of non-orthogonal asymmetric helical gear d (e d ,θ d ,ω1) is as follows: Where: r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter, β is the helix angle, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, N2 is the number of teeth of the non-orthogonal asymmetric helical gear face gear, γ is the axis intersection angle, ω1 is the angle of rotation of the non-orthogonal asymmetric helical gear shaping cutter, α d is the pressure angle on the active side of the non-orthogonal asymmetric helical gear shaping cutter, θ d is the angle parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter, e d is the axial parameter of the active side of the non-orthogonal asymmetric helical gear shaping cutter; Meshing equation f of the driven side of non-orthogonal asymmetric helical gear c (e c ,θ c ,ω1) is as follows: Where: r bc is the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, β is the helix angle, N1 is the number of teeth of the non-orthogonal asymmetric helical gear shaping cutter, N2 is the number of teeth of the non-orthogonal asymmetric helical gear face gear, γ is the axis angle, α c is the pressure angle on the driven side of the non-orthogonal asymmetric helical gear shaping cutter, ω1 is the angle through which the non-orthogonal asymmetric helical gear shaping cutter rotates, and θ c is the angular parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, e c is the axial parameter of the driven side of the non-orthogonal asymmetric helical gear shaping cutter; The tooth surface equation of the active side of the non-orthogonal asymmetric helical gear is: Where: f d (e d ,θ d ,ω1) is the meshing equation of the active side of the non-orthogonal asymmetric helical gear, r d (θ d ,e d ) is the position vector of the active side of the non-orthogonal asymmetric helical gear shaping cutter, M p,f is the transformation matrix from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear face gear, r bd is the base circle radius of the active side of the non-orthogonal asymmetric helical gear shaping cutter; The equation of the driven side tooth surface of the non-orthogonal asymmetric helical gear is: Where: f c (e c ,θ c ,ω1) is the meshing equation of the driven side of the non-orthogonal asymmetric helical gear, r c (θ c ,e c ) is the position vector of the driven side of the non-orthogonal asymmetric helical gear shaping cutter, M p,f is the transformation matrix from the fixed coordinate system of the non-orthogonal asymmetric helical gear shaping cutter to the fixed coordinate system of the non-orthogonal asymmetric helical gear face gear, r bc is the base circle radius of the driven side of the non-orthogonal asymmetric helical gear shaping cutter; Step (3): Establish an analytical calculation model for the time-varying meshing stiffness of non-orthogonal asymmetric helical gears; in order to accurately calculate the stiffness of non-orthogonal asymmetric helical gears, the improved potential energy method and slicing method are used to cut the non-orthogonal asymmetric helical gear teeth into several trapezoidal slices along the axial direction. The stiffness of each slice is calculated independently. The bending stiffness k of the non-orthogonal asymmetric helical gear teeth is w It is obtained from the following formula: Where: y f is the axial coordinate of the slice, d is the thickness of the slice projected onto the plane, x t1 and x t2 is the direction coordinate of the boundary point projected onto the plane by the elliptical surface load, F is the total force generated by the load distribution function on the slice, α i is the pressure angle of the i-th trapezoidal slice in the non-orthogonal asymmetric helical gear, x a 、y a are the horizontal and vertical coordinate values of the projection plane, respectively, h i is the height of the i-th trapezoidal slice along the tooth profile, x fc and x fp is the normal vector coordinate of the projection point of the long axis on the projection plane, y fc is the normal vector coordinate of the projection point of the minor axis on the projection plane, E is the Young's modulus of the non-orthogonal asymmetric helical gear, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, a i is the length of the base of the i-th trapezoidal slice, b i The length of the lower base of the i-th trapezoidal slice; Matrix stiffness k of non-orthogonal asymmetric helical gear teeth during single tooth meshing b It is obtained from the following formula: Where: α i is the pressure angle of the i-th trapezoidal slice in the non-orthogonal asymmetric helical gear, C1, V1, B1, G1 are the polynomial fitting parameters, S is the distance from the front pair of teeth to the back pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear, L f is the distance from the contact point to the base, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, E is the Young's modulus of the non-orthogonal asymmetric helical face gear, d is the thickness of the slice projected onto the plane, and v is the Poisson's ratio; The matrix stiffness of the non-orthogonal asymmetric helical gear teeth during double tooth meshing is obtained by the following formula: Where: k b1 k is the matrix coupling stiffness under the influence of the meshing force of the first pair of teeth on the second pair of teeth during the meshing process of non-orthogonal asymmetric helical gears, b2 is the matrix coupling stiffness under the influence of the meshing force of the rear pair of teeth on the front pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear, α1 is the pressure angle of the front tooth during the meshing process of the non-orthogonal asymmetric helical gear, α2 is the pressure angle of the rear tooth during the meshing process of the non-orthogonal asymmetric helical gear, I i is the area moment of inertia of the i-th trapezoidal slice along the tooth profile, E is the Young's modulus of the non-orthogonal asymmetric helical gear, d is the thickness of the slice projected onto the plane, J2, K2, L2, T2, G2, B2, C2, V2, M2, J3, K3, L3, T3, G3, B3, C3, V3, M3 are the polynomial fitting coefficients, L f1 L is the distance from the contact point of the first pair of teeth to the gear base. f2 is the distance from the contact point of the rear pair of teeth to the base of the gear, and S is the distance from the front pair of teeth to the rear pair of teeth during the meshing process of the non-orthogonal asymmetric helical gear; Single tooth mesh stiffness k of non-orthogonal asymmetric helical gears m1 as follows: Where: k w is the bending stiffness of the non-orthogonal asymmetric helical gear teeth, k b is the matrix stiffness during single tooth meshing, k u is the gear tooth contact stiffness; Double tooth mesh stiffness k of non-orthogonal asymmetric helical gears m2 as follows: Where: k w is the bending stiffness of the non-orthogonal asymmetric helical gear teeth, k b1 k is the matrix coupling stiffness under the influence of the meshing force of the first pair of teeth on the second pair of teeth during the meshing process of non-orthogonal asymmetric helical gears, b2 k is the matrix coupling stiffness under the influence of the meshing force of the rear pair of teeth on the front pair of teeth during the meshing process of non-orthogonal asymmetric helical gears, u is the gear tooth contact stiffness; Step (4): Solve the time-varying meshing stiffness of the non-orthogonal asymmetric helical gear under different pressure angle and axis angle combinations; based on the time-varying meshing stiffness analytical calculation model in step (3), complete the solution and analysis of the time-varying meshing stiffness calculation equation of the non-orthogonal asymmetric helical gear for different pressure angle and axis angle combinations, and obtain the time-varying meshing stiffness of the non-orthogonal asymmetric helical gear under different pressure angle and axis angle combinations.

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