Helical gear transmission system nonlinear dynamic modeling method considering abrasion

Abstract

The invention discloses a nonlinear dynamic modeling method of a helical gear transmission system considering wear. The method comprises the following specific steps: step (1), establishing a time-varying meshing stiffness calculation model of a helical gear pair in a healthy state; (2) establishing a time-varying meshing stiffness calculation model of the wear fault helical gear pair; (3) establishing a helical gear tooth surface friction model under the condition of mixed elastohydrodynamic lubrication; (4) establishing a deformation equation of the tapered roller bearing; (5) establishing a helical gear transmission system model considering wear, and solving the nonlinear vibration characteristics of the helical gear transmission system under different wear degree faults of the helical gear; the method has the beneficial effects that the nonlinear dynamic model of the helical gear transmission system considering wear is established, and technical support is provided for related technologies of fault dynamic modeling of the current helical gear transmission system.

Description

Technical Field

[0001] The present invention relates to the technical field of gear nonlinear dynamics, and in particular to a wear-considered nonlinear dynamics modeling method for a helical gear transmission system. Background Art

[0002] Helical gear transmissions are widely used in various industrial fields, such as automobiles, machine tools, and aviation, due to their advantages such as smooth transmission, strong load-bearing capacity, long service life, suitability for high-speed transmission, and high transmission efficiency. They are especially suitable for high-precision, high-load transmission systems. However, due to the sealing and operation continuity of mechanical transmissions, the degree of wear of helical gears is difficult to monitor in some closed working environments with high requirements. When helical gears have wear failures, they often damage the material and performance of the parts themselves, causing increased vibration and disorder, further leading to operational failure, affecting the efficiency of actual production, and also posing safety hazards. Therefore, it is of great significance to study the nonlinear fault dynamics of helical gear transmission systems of high-speed equipment. It can monitor the degree of wear failure and propose effective vibration monitoring indicators to ensure the safety of production practice. It has high academic value in enriching and expanding the theoretical system of fault dynamics. The existing technology lacks research on the nonlinear fault dynamics of multi-degree-of-freedom helical gear transmission systems.

[0003] To address these issues, this paper proposes a wear-inclusive nonlinear dynamic modeling method for helical gear transmission systems. This method calculates healthy mesh stiffness based on the energy and slicing methods. The mesh stiffness associated with wear failures is then calculated based on the healthy stiffness and the pinion wear model. A friction model for helical gear tooth surfaces under hybrid elastohydrodynamic lubrication is established. A tapered roller bearing is added to couple the gears. Finally, a dynamic model of the helical gear transmission system is established, and the Runge-Kutta method is used to numerically solve the equations. This method can both advance the development of fault dynamics and reduce costs, resulting in significant socioeconomic benefits. Summary of the Invention

[0004] To overcome the shortcomings of the existing technology and fill the gaps in the relevant technology, the present invention provides a nonlinear dynamic modeling method for a helical gear transmission system taking wear into consideration. The technical solutions adopted by the present invention to solve the technical problems are as follows:

[0005] A nonlinear dynamic modeling method for a helical gear transmission system considering wear is characterized by comprising the following steps:

[0006] Step (1): Establish a time-varying meshing stiffness calculation model for healthy helical gear pairs, and use the energy method and the slice method to calculate the meshing stiffness. The potential energy U of the gear consists of four parts: Hertz contact potential energy U h , bending deformation energy U b , shear deformation energy U s , axial compression deformation energy Ua , the corresponding stiffness of the four potential energies can be calculated: Hertz contact stiffness k h , bending stiffness k b , shear stiffness k s , axial compression stiffness k a , the expression is as follows:

[0007]

[0008] Among them, F is the normal contact load on the tooth surface. Slice along the tooth width direction of the helical gear. Each microelement is regarded as a spur gear microelement with a tooth width of dy. Slice along the radial direction on the spur gear microelement. The width of each slice is dx. The base circle radius of the small gear used for the driving wheel is larger than the root circle radius, and the base circle radius of the large gear used for the driven wheel is smaller than the root circle radius. The bending stiffness k of the driving wheel is b1 , shear stiffness k s1 , axial compression deformation stiffness k a1 , Hertz contact stiffness k h for:

[0009]

[0010] Where b is the tooth width of the helical gear, E and ν are the elastic modulus and Poisson's ratio of the gear material, y is the distance from the current driving wheel spur gear infinitesimal to the front face of the gear, x is the distance from the current dx to the tooth root circle plane, α is the pressure angle of the current dx, and R b1 and R f1 are the base circle radius and tooth root circle radius of the driving wheel, L(y) is the distance from the meshing point to the plane formed by the gear on the tooth root circle, h(y) is the distance from the meshing point to the symmetry line of the gear, α1(y) is the pressure angle corresponding to the meshing force on the current driving wheel spur gear infinitesimal element, α2 is half of the corresponding angle of a single tooth of the driving wheel on the base circle, A is the calculation function from the base circle to the tooth top circle in the bending stiffness, and the expressions of α1(y), α2, L(y), h(y), and A are:

[0011]

[0012] Among them, θ1 is the angular displacement of the driving wheel, Z1 and Z2 are the number of teeth of the driving and driven wheels respectively, and α t is the pressure angle of the helical gear end face, h a * is the tooth addendum coefficient, ε β is the axial contact of the helical gear;

[0013] Bending stiffness k of the driven wheel b2 , shear stiffness k s2 , axial compression deformation stiffness k a2 for:

[0014]

[0015] Among them, α3(y) is the pressure angle corresponding to the meshing force on the current driven wheel spur gear infinitesimal element, and α4 is half of the corresponding angle of a single driven wheel tooth on the base circle. The expressions of α3(y) and α4 are:

[0016]

[0017] The comprehensive meshing stiffness k when n pairs of gear teeth are meshing in a healthy state m for:

[0018]

[0019] Among them, k b1i Indicates the bending stiffness of the meshing teeth of the driving wheel in the healthy state, k s1i Indicates the shear stiffness of the meshing teeth of the driving wheel in the healthy state, k a1i Indicates the axial compression deformation stiffness of the i-th pair of meshing teeth of the driving wheel in a healthy state, k b2i represents the bending stiffness of the i-th pair of meshing teeth of the driven wheel, k s2i represents the shear stiffness of the i-th pair of meshing teeth of the driven wheel, k a2i k represents the axial compression deformation stiffness of the i-th pair of meshing teeth of the driven wheel, h is the Hertzian contact stiffness;

[0020] Step (2): Establish a time-varying meshing stiffness calculation model for the helical gear pair with wear faults, and the bending stiffness k of the driving wheel after wear b1 ', shear stiffness k s1 ', axial compression deformation stiffness k a1 ':

[0021]

[0022] Where Ns is the number of gear meshing times, K is the correlation coefficient between wear and the number of gear meshing times Ns, and the distance h'(y) from the meshing point to the gear symmetry line is calculated as follows:

[0023] h'(y)=R b1 [(α1(y)+α2)cosα1(y)-sinα1(y)]-Ns·Kcosα1(y);

[0024] The comprehensive meshing stiffness k of n pairs of gear teeth meshing under wear failure m 'for:

[0025]

[0026] Among them, k b1i' represents the bending stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure, k s1i ' represents the shear stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure, k a1i ' represents the axial compressive deformation stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure;

[0027] Step (3): Establish a friction model for the helical gear tooth surface under mixed elastohydrodynamic lubrication conditions. Use the slicing method to slice the helical gear along the tooth width direction. Each slice is a spur gear infinitesimal element with a tooth width of dy. A1N1B1MB2A2N2 is the meshing line of the front end face of the helical gear, and A1'NN1'NB1'NM'NB2'NA2'NN2' is the meshing line of the rear end face of the gear. The calculation formula for the line segment A1N1 is:

[0028] A1N1=ε β p b +B1M-R1sinα t ;

[0029] Among them, R1 is the driving wheel pitch circle, p b is the helical gear pitch, ε β is the axial contact of the helical gear, and the calculation formula of line segment B1M is:

[0030]

[0031] Among them, R2 is the driven wheel pitch circle, R a2 R is the radius of the driven gear tooth top circle, b2 is the base circle radius of the driven wheel;

[0032] The distance from the front face engagement point to A1 is S(y) = ytanβ b , β b is the base circle helix angle of the helical gear, and the distances S1(y) and S2(y) from the meshing point at the front end face y to N1N1' and N2N2' are calculated as follows:

[0033]

[0034] Get the meshing sliding velocity v of the spur gear infinitesimal element s (y) and entrainment velocity v e The calculation formula of (y):

[0035]

[0036] Among them, the sliding speed of the driving wheel tooth surface is u1(y)=ω1S1(y), the sliding speed of the driven wheel tooth surface is u2(y)=ω2S2(y), ω1 and ω2 are the angular velocities of the driving wheel and the driven wheel respectively, and the friction coefficient function μ(y) and the friction force infinitesimal element df are obtained:

[0037]

[0038] Among them, R avg1 is the surface roughness of the driving wheel, R avg2 is the surface roughness of the driven wheel, F m,i is the meshing force on the i-th pair of teeth, η is the dynamic viscosity coefficient of the tooth contact temperature, v s (y) is the meshing sliding speed of the current spur gear microelement, v e (y) is the entrainment speed of the current spur gear element, A(y) is the signal function of the friction force element df, b(θ1) is the actual meshing width function of the helical gear, and the expressions of A(y) and b(θ1) are:

[0039]

[0040] Among them, ε α is the end face contact of the helical gear, ε λ is the contact ratio of the helical gear, and the total friction force F when n pairs of teeth are meshed is obtained f and the torque M of the friction force on the driving wheel f1 And the torque M of the friction force on the driven wheel f2 The expression is:

[0041]

[0042] Step (4): Establish the deformation equation of the tapered roller bearing and obtain the radial displacement δ of the bearing from the geometric relationship ri and axial displacement δ a The expression is:

[0043]

[0044] Among them, x b 、y b 、z b are the vibration displacements of the bearing in the x, y, and z directions, β i is the rotation angle of the i-th tapered roller, N is the number of tapered rollers, and the contact stiffness K is n Get the support force F of a single tapered roller i for:

[0045] F i =K n (δ ri cosα e +δ a sinα e ) 109 ;

[0046] Among them, α eIt is the angle between the outer contact line of the tapered roller and the axis of the rotating shaft. The support force of a single tapered roller is projected in the x, y, and z directions and summed to obtain the projection F of the bearing support force in the x, y, and z directions. bx 、F by 、F bz for:

[0047]

[0048] Step (5): Establish a helical gear transmission system model considering wear, meshing force F m And the transmission error e(t) is divided into two cases:

[0049] Case 1: Meshing force F in healthy state m Calculate as follows:

[0050]

[0051] Among them, k m is the healthy state of the helical gear, c m is the helical gear meshing damping, f(x n ) is the tooth side clearance function, x n is the displacement of the meshing line, x n The expression is:

[0052] x n =[R b1 θ z1 -R b2 θ z2 +(x1-x2)sinα t +(y1-y2)cosα t +(z1-z2+R b1 θ y1 -R b2 θ y2 )tanβ]cosβ-e(t);

[0053] Among them, θ y1 ,θ z1 is the torsional vibration angular displacement of the driving wheel in the y and z directions, θ y2 ,θ z2 is the torsional vibration angular displacement of the driven wheel in the y and z directions, x1, y1, z1 are the vibration displacements of the driving wheel projected along the x, y, z directions, x2, y2, z2 are the vibration displacements of the driven wheel projected along the x, y, z directions, R b1 and R b2 are the base circle radii of the driving wheel and the driven wheel respectively, β is the helical angle of the helical gear, and e(t) is the transmission error;

[0054] Case 2: Meshing force F under wear failure m Calculate as follows:

[0055]

[0056] Among them, N k m ' is the meshing stiffness of the helical gear wear fault;

[0057] The expression of transmission error e(t) is:

[0058]

[0059] Among them, Nt is time, e0, e r is the mean value and fluctuation amplitude of the transmission error, is the initial phase, ω e is the meshing frequency, e wear is the fluctuation amplitude of wear transmission error, e wear The expression is:

[0060]

[0061] Where n is the number of meshing teeth, h wear,i is the wear amount of the i-th pair of meshing teeth;

[0062] Bearings b1 and b2 are at both ends of the driving wheel. The stiffness and damping of bearing b1 are k b1 、c b1 , the stiffness and damping of bearing b2 are k b2 、c b2 The stiffness and damping between bearings b1 and b2 and the shaft are c1 and k1, and bearings b3 and b4 are at both ends of the driven wheel. The stiffness and damping of bearing b3 are k b3 、c b3 , the stiffness and damping of bearing b4 are k b4 、c b4 , the stiffness and damping between bearings b3, b4 and the shaft are c2 and k2, and the dynamic equations of bearings b1, b2, b3, and b4 are:

[0063]

[0064] Among them, m b1 、m b2 、m b3 、m b4 are the masses of bearings b1, b2, b3, and b4 respectively, and x b1 、y b1 、z b1 is the vibration displacement of bearing b1 projected along the x, y, and z directions, x b2 、y b2 、z b2 is the vibration displacement of bearing b2 projected along the x, y, and z directions, xb3 、y b3 、z b3 is the vibration displacement of bearing b3 projected along the x, y, and z directions, x b4 、y b4 、z b4 is the vibration displacement of bearing b4 projected along the x, y, and z directions, F bx1 、F by1 、F bz1 is the projection of the bearing force of bearing b1 in the x, y, and z directions, F bx2 、F by2 、F bz2 is the projection of the bearing force of bearing b2 in the x, y, and z directions, F bx3 、F by3 、F bz3 is the projection of the support force of bearing b3 in the x, y, and z directions, F bx4 、F by4 、F bz4 is the projection of the support force of bearing b4 in the x, y, and z directions, and g is the acceleration due to gravity;

[0065] The dynamic equations of the driving wheel and the driven wheel are:

[0066]

[0067]

[0068] Among them, θ x1 and θ x2 are the torsional vibration angular displacements of the driving wheel and the driven wheel in the x direction, m1 and m2 are the masses of the driving wheel and the driven wheel, I1 and I2 are the moments of inertia of the driving wheel and the driven wheel about the rotation axis, k θ1 and c θ1 is the torsional stiffness and torsional damping of the driving wheel, k θ2 and c θ2 are the torsional stiffness and torsional damping of the driven wheel, T1 and T2 are the input and output torques, M f1 and M f2 are the torques of friction on the driving wheel and the driven wheel, F fx 、F fy 、F fz is the projection of friction force in the x, y, and z directions, F x 、F y 、F z is the projection of meshing force in x, y and z directions, F x 、F y 、F z 、F fx 、F fy 、F fzThe expression is:

[0069]

[0070] Among them, F m is the meshing force, F f is the total friction force, α n is the normal pressure angle of the helical gear;

[0071] By solving the dynamic equations, the nonlinear vibration characteristics of the helical gear transmission system under different faults can be obtained, revealing the impact of wear faults on the dynamic response of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 It is a flow chart of the nonlinear dynamic modeling method of the helical gear transmission system considering wear;

[0073] Figure 2 It is a schematic diagram of the driving wheel meshing stiffness slice;

[0074] Figure 3 This is a schematic diagram of the meshing stiffness of the driven gear;

[0075] Figure 4 It is the dynamic model of the helical gear transmission system;

[0076] Figure 5 It is the dynamic response of the helical gear transmission system under wear failure. DETAILED DESCRIPTION

[0077] 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.

[0078] Reference Figure 1 A nonlinear dynamic modeling method for a gear transmission system considering wear is provided, comprising the following steps:

[0079] Step (1): Establish a calculation model for the time-varying meshing stiffness of the healthy helical gear pair, such as Figure 2 and Figure 3 The meshing stiffness models of the driving wheel and the driven wheel are shown respectively. The meshing stiffness is calculated using the energy method and the slice method. The potential energy U of the gear consists of four parts: Hertz contact potential energy U h , bending deformation energy U b , shear deformation energy U s , axial compression deformation energy U a , the corresponding stiffness of the four potential energies can be calculated: Hertz contact stiffness k h , bending stiffness k b , shear stiffness k s , axial compression stiffness ka , the expression is as follows:

[0080]

[0081] Among them, F is the normal contact load on the tooth surface. Slice along the tooth width direction of the helical gear. Each microelement is regarded as a spur gear microelement with a tooth width of dy. Slice along the radial direction on the spur gear microelement. The width of each slice is dx. The base circle radius of the small gear used for the driving wheel is larger than the root circle radius, and the base circle radius of the large gear used for the driven wheel is smaller than the root circle radius. The bending stiffness k of the driving wheel is b1 , shear stiffness k s1 , axial compression deformation stiffness k a1 , Hertz contact stiffness k h for:

[0082]

[0083] Where b is the tooth width of the helical gear, E and ν are the elastic modulus and Poisson's ratio of the gear material, y is the distance from the current driving wheel spur gear infinitesimal to the front face of the gear, x is the distance from the current dx to the tooth root circle plane, α is the pressure angle of the current dx, and R b1 and R f1 are the base circle radius and tooth root circle radius of the driving wheel, L(y) is the distance from the meshing point to the plane formed by the gear on the tooth root circle, h(y) is the distance from the meshing point to the symmetry line of the gear, α1(y) is the pressure angle corresponding to the meshing force on the current driving wheel spur gear infinitesimal element, α2 is half of the corresponding angle of a single tooth of the driving wheel on the base circle, A is the calculation function from the base circle to the tooth top circle in the bending stiffness, and the expressions of α1(y), α2, L(y), h(y), and A are:

[0084]

[0085] Among them, θ1 is the angular displacement of the driving wheel, Z1 and Z2 are the number of teeth of the driving and driven wheels respectively, and α t is the pressure angle of the helical gear end face, h a * is the tooth addendum coefficient, ε β is the axial contact of the helical gear; the bending stiffness k of the driven wheel b2 , shear stiffness k s2 , axial compression deformation stiffness k a2 for:

[0086]

[0087] Among them, α3(y) is the pressure angle corresponding to the meshing force on the current driven wheel spur gear infinitesimal element, and α4 is half of the corresponding angle of a single driven wheel tooth on the base circle. The expressions of α3(y) and α4 are:

[0088]

[0089] The comprehensive meshing stiffness k when n pairs of gear teeth are meshing in a healthy state m for:

[0090]

[0091] Among them, k b1i Indicates the bending stiffness of the meshing teeth of the driving wheel in the healthy state, k s1i Indicates the shear stiffness of the meshing teeth of the driving wheel in a healthy state, k a1i Indicates the axial compression deformation stiffness of the i-th pair of meshing teeth of the driving wheel in a healthy state, k b2i represents the bending stiffness of the i-th pair of meshing teeth of the driven wheel, k s2i represents the shear stiffness of the i-th pair of meshing teeth of the driven wheel, k a2i k represents the axial compression deformation stiffness of the i-th pair of meshing teeth of the driven wheel, h is the Hertzian contact stiffness;

[0092] Step (2) Establish a time-varying meshing stiffness calculation model for the helical gear pair with wear faults, and the bending stiffness k of the driving wheel after wear b1 ', shear stiffness k s1 ', axial compression deformation stiffness k a1 ':

[0093]

[0094] Where Ns is the number of gear meshing times, K is the correlation coefficient between wear and the number of gear meshing times Ns, and the distance h'(y) from the meshing point to the gear symmetry line is calculated as follows:

[0095] h'(y)=R b1 [(α1(y)+α2)cosα1(y)-sinα1(y)]-Ns·Kcosα1(y);

[0096] The comprehensive meshing stiffness k of n pairs of gear teeth meshing under wear failure m 'for:

[0097]

[0098] Among them, k b1i ' represents the bending stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure, k s1i ' represents the shear stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure, k a1i ' represents the axial compressive deformation stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure;

[0099] Step (3): Establish a friction model for the helical gear tooth surface under mixed elastohydrodynamic lubrication conditions. Use the slicing method to slice the helical gear along the tooth width direction. Each slice is a spur gear infinitesimal element with a tooth width of dy. A1N1B1MB2A2N2 is the meshing line of the front end face of the helical gear, and A1'NN1'NB1'NM'NB2'NA2'NN2' is the meshing line of the rear end face of the gear. The calculation formula for the line segment A1N1 is:

[0100] A1N1=ε β p b +B1M-R1sinα t ;

[0101] Among them, R1 is the driving wheel pitch circle, p b is the helical gear pitch, ε β is the axial contact of the helical gear, and the calculation formula of line segment B1M is:

[0102]

[0103] Among them, R2 is the driven wheel pitch circle, R a2 R is the radius of the driven gear tooth top circle, b2 is the base circle radius of the driven wheel;

[0104] The distance from the front face engagement point to A1 is S(y) = ytanβ b , β b is the helix angle of the base circle of the helical gear, and the distances S1(y) and S2(y) from the meshing point at the front end face y to N1N1' and N2N2' are calculated as follows:

[0105]

[0106] Get the meshing sliding velocity v of the spur gear infinitesimal element s (y) and entrainment velocity v e The calculation formula of (y):

[0107]

[0108] Among them, the sliding speed of the driving wheel tooth surface is u1(y)=ω1S1(y), the sliding speed of the driven wheel tooth surface is u2(y)=ω2S2(y), ω1 and ω2 are the angular velocities of the driving wheel and the driven wheel respectively, and the friction coefficient function μ(y) and the friction force infinitesimal element df are obtained:

[0109]

[0110] Among them, R avg1 is the surface roughness of the driving wheel, R avg2 is the surface roughness of the driven wheel, F m,iis the meshing force on the i-th pair of teeth, η is the dynamic viscosity coefficient of the tooth contact temperature, v s (y) is the meshing sliding speed of the current spur gear microelement, v e (y) is the entrainment speed of the current spur gear element, A(y) is the signal function of the friction force element df, β b is the base circle helix angle of the helical gear, b(θ1) is the actual meshing tooth width function of the helical gear, and the expressions of A(y) and b(θ1) are:

[0111]

[0112] Among them, ε λ is the contact ratio of the helical gear, and the total friction force F when n pairs of teeth are meshed is obtained f and the torque M of the friction force on the driving wheel f1 And the torque M of the friction force on the driven wheel f2 The expression is:

[0113]

[0114] Step (4): Establish the deformation equation of the tapered roller bearing and obtain the radial displacement δ of the bearing from the geometric relationship ri and axial displacement δ a The expression is:

[0115]

[0116] Among them, x b 、y b 、z b are the vibration displacements of the bearing in the x, y, and z directions, β i is the rotation angle of the i-th tapered roller, N is the number of tapered rollers, and the contact stiffness K is n Get the support force F of a single tapered roller i for:

[0117] F i =K n (δ ri cosα e +δ a sinα e ) 109 ;

[0118] Among them, α e It is the angle between the outer contact line of the tapered roller and the axis of the rotating shaft. The support force of a single tapered roller is projected in the x, y, and z directions and summed to obtain the projection F of the bearing support force in the x, y, and z directions. bx 、F by 、F bz for:

[0119]

[0120] Step (5): Figure 4 The model of the helical gear transmission system considering wear is established as shown in the figure. The meshing force F m And the transmission error e(t) is divided into two cases:

[0121] Case 1: Meshing force F in healthy state m Calculate as follows:

[0122]

[0123] Among them, k m is the healthy state of the helical gear, c m is the helical gear meshing damping, f(x n ) is the tooth side clearance function, x n is the displacement of the meshing line, x n The expression is:

[0124] x n =[R b1 θ z1 -R b2 θ z2 +(x1-x2)sinα t +(y1-y2)cosα t +(z1-z2+R b1 θ y1 -R b2 θ y2 )tanβ]cosβ-e(t);

[0125] Among them, θ y1 ,θ z1 is the torsional vibration angular displacement of the driving wheel in the y and z directions, θ y2 ,θ z2 is the torsional vibration angular displacement of the driven wheel in the y and z directions, x1, y1, z1 are the vibration displacements of the driving wheel projected along the x, y, z directions, x2, y2, z2 are the vibration displacements of the driven wheel projected along the x, y, z directions, R b1 and R b2 are the base circle radii of the driving wheel and the driven wheel respectively, β is the helical angle of the helical gear, and e(t) is the transmission error;

[0126] Case 2: Meshing force F under wear failure m Calculate as follows:

[0127]

[0128] Among them, N k m ' is the meshing stiffness of the helical gear wear fault;

[0129] The expression of transmission error e(t) is:

[0130]

[0131] Among them, t is time, e0, e r is the mean value and fluctuation amplitude of the transmission error, is the initial phase, ω e is the meshing frequency, e wear is the fluctuation amplitude of wear transmission error, e wear The expression is:

[0132]

[0133] Where n is the number of meshing teeth, h wear,i is the wear amount of the i-th pair of meshing teeth;

[0134] Bearings b1 and b2 are at both ends of the driving wheel. The stiffness and damping of bearing b1 are k b1 、c b1 , the stiffness and damping of bearing b2 are k b2 、c b2 The stiffness and damping between bearings b1 and b2 and the shaft are c1 and k1, and bearings b3 and b4 are at both ends of the driven wheel. The stiffness and damping of bearing b3 are k b3 、c b3 , the stiffness and damping of bearing b4 are k b4 、c b4 , the stiffness and damping between bearings b3, b4 and the shaft are c2 and k2, and the dynamic equations of bearings b1, b2, b3, and b4 are:

[0135]

[0136] Among them, m b1 、m b2 、m b3 、m b4 are the masses of bearings b1, b2, b3, and b4 respectively, and x b1 、y b1 、z b1 is the vibration displacement of bearing b1 projected along the x, y, and z directions, x b2 、y b2 、z b2 is the vibration displacement of bearing b2 projected along the x, y, and z directions, x b3 、y b3 、z b3 is the vibration displacement of bearing b3 projected along the x, y, and z directions, x b4 、y b4 、z b4is the vibration displacement of bearing b4 projected along the x, y, and z directions, F bx1 、F by1 、F bz1 is the projection of the bearing force of bearing b1 in the x, y, and z directions, F bx2 、F by2 、F bz2 is the projection of the bearing force of bearing b2 in the x, y, and z directions, F bx3 、F by3 、F bz3 is the projection of the support force of bearing b3 in the x, y, and z directions, F bx4 、F by4 、F bz4 is the projection of the support force of bearing b4 in the x, y, and z directions, and g is the acceleration due to gravity;

[0137] The dynamic equations of the driving wheel and the driven wheel are:

[0138]

[0139]

[0140] Among them, θ x1 and θ x2 are the torsional vibration angular displacements of the driving wheel and the driven wheel in the x direction, m1 and m2 are the masses of the driving wheel and the driven wheel, I1 and I2 are the moments of inertia of the driving wheel and the driven wheel about the rotation axis, k θ1 and c θ1 is the torsional stiffness and torsional damping of the driving wheel, k θ2 and c θ2 are the torsional stiffness and torsional damping of the driven wheel, T1 and T2 are the input and output torques, M f1 and M f2 are the torques of friction on the driving wheel and the driven wheel, F fx 、F fy 、F fz is the projection of friction force in the x, y, and z directions, F x 、F y 、F z is the projection of meshing force in x, y and z directions, F x 、F y 、F z 、F fx 、F fy 、F fz The expression is:

[0141]

[0142] Among them, F m is the meshing force, F fis the total friction force, α n is the normal pressure angle of the helical gear;

[0143] By solving the dynamic equations, the nonlinear vibration characteristics of the helical gear transmission system under different faults can be obtained, revealing the impact of wear faults on the dynamic response of the system.

[0144] In this example, the gear parameters of the helical gear transmission system are shown in Table 1, and the bearing parameters are shown in Table 2:

[0145] Table 1 Gear parameters of helical gear transmission system

[0146]

[0147] Table 2 Bearing parameters of helical gear transmission system

[0148]

[0149] Substitute the transmission related parameters and use the Runge-Taku method to solve the dynamic equations. Figure 5 The nonlinear dynamic response characteristic diagram of the wear fault helical gear transmission system. The system has single-cycle, multi-cycle and chaotic nonlinear vibration characteristics. By adjusting the speed, the working stability of the faulty system can be improved and the system service life can be extended.

[0150] 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 nonlinear dynamic modeling method for helical gear transmission system considering wear, characterized by The following steps are involved: Step (1): Establish a calculation model for the time-varying meshing stiffness of a healthy helical gear pair. Slice along the tooth width direction of the helical gear. Each microelement is regarded as a spur gear microelement with a tooth width of dy. Slice along the radial direction of the spur gear microelement. Each slice has a width of dx. The driving gear uses a small gear with a base circle radius larger than the root circle radius. The driven gear uses a large gear with a base circle radius smaller than the root circle radius. Bending stiffness k of the driving wheel b1 , shear stiffness k s1 , axial compression deformation stiffness k a1 for: Where b is the tooth width of the helical gear, E and ν are the elastic modulus and Poisson's ratio of the gear material, y is the distance from the current driving wheel spur gear infinitesimal to the front face of the gear, x is the distance from the current dx to the tooth root circle plane, α is the pressure angle of the current dx, and R b1 and R f1 are the base circle radius and tooth root circle radius of the driving wheel, L(y) is the distance from the meshing point to the plane formed by the gear on the tooth root circle, h(y) is the distance from the meshing point to the symmetry line of the gear, α1(y) is the pressure angle corresponding to the meshing force on the current driving wheel spur gear infinitesimal element, α2 is half of the corresponding angle of a single tooth of the driving wheel on the base circle, A is the calculation function from the base circle to the tooth top circle in the bending stiffness, and the expressions of α1(y), α2, L(y), h(y), and A are: Among them, θ1 is the angular displacement of the driving wheel, Z1 and Z2 are the number of teeth of the driving and driven wheels respectively, and α t is the pressure angle of the helical gear end face, h a * is the tooth addendum coefficient, ε β is the axial contact of the helical gear; Bending stiffness k of the driven wheel b2 , shear stiffness k s2 , axial compression deformation stiffness k a2 for: Among them, α3(y) is the pressure angle corresponding to the meshing force on the current driven wheel spur gear infinitesimal element, and α4 is half of the corresponding angle of a single driven wheel tooth on the base circle. The expressions of α3(y) and α4 are: The comprehensive meshing stiffness k when n pairs of gear teeth are meshing in a healthy state m for: Among them, k b1i Indicates the bending stiffness of the meshing teeth of the driving wheel in the healthy state, k s1i Indicates the shear stiffness of the meshing teeth of the driving wheel in the healthy state, k a1i Indicates the axial compression deformation stiffness of the i-th pair of meshing teeth of the driving wheel in a healthy state, k b2i represents the bending stiffness of the i-th pair of meshing teeth of the driven wheel, k s2i represents the shear stiffness of the i-th pair of meshing teeth of the driven wheel, k a2i k represents the axial compression deformation stiffness of the i-th pair of meshing teeth of the driven wheel, h is the Hertzian contact stiffness; Step (2): Establish a time-varying meshing stiffness calculation model for the helical gear pair with wear faults, and obtain the bending stiffness k of the driving wheel after wear. b1 ', shear stiffness k s1 ', axial compression deformation stiffness k a1 'for: Where Ns is the number of gear meshing times, K is the correlation coefficient between wear and the number of gear meshing times Ns, and the distance h'(y) from the meshing point to the gear symmetry line is calculated as follows: h'(y)=R b1 [(α1(y)+α2)cosα1(y)-sinα1(y)]-Ns·Kcosα1(y); The comprehensive meshing stiffness k of n pairs of gear teeth meshing under wear failure m 'for: Among them, k b1i ' represents the bending stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure, k s1i ' represents the shear stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure, k a1i ' represents the axial compressive deformation stiffness of the i-th pair of meshing teeth of the driving wheel under wear failure; Step (3): Establish a friction model for the helical gear tooth surface under mixed elastohydrodynamic lubrication conditions. Use the slicing method to slice the helical gear along the tooth width direction to obtain the friction coefficient function μ(y) and the friction force element df: Among them, R avg1 is the surface roughness of the driving wheel, R avg2 is the surface roughness of the driven wheel, F m,i is the meshing force on the i-th pair of teeth, η is the dynamic viscosity coefficient of the tooth contact temperature, v s (y) is the meshing sliding speed of the current spur gear microelement, v e (y) is the entrainment speed of the current spur gear element, A(y) is the signal function of the friction element df, β b is the base circle helix angle of the helical gear, b(θ1) is the actual meshing tooth width function of the helical gear, and the expression of b(θ1) is: Among them, p b is the helical gear pitch, ε λ and ε α The total friction force F when n pairs of teeth are meshed is obtained as f The expression is: Step (4): Establish the deformation equation of the tapered roller bearing and obtain the radial displacement δ of the bearing from the geometric relationship ri and axial displacement δ a The expression is: Among them, x b 、y b 、z b are the vibration displacements of the bearing in the x, y, and z directions, β i is the rotation angle of the i-th tapered roller, N is the number of tapered rollers, and the contact stiffness K is n Get the support force F of a single tapered roller i for: F i =K n (d ri thing e +d a Sinai e ) 109 ; Among them, α e It is the angle between the outer contact line of the tapered roller and the axis of the rotating shaft. The support force of a single tapered roller is projected in the x, y, and z directions and summed to obtain the projection F of the bearing support force in the x, y, and z directions. bx 、F by 、F bz for: Step (5): Establish a helical gear transmission system model considering wear, meshing force F m And the transmission error e(t) is divided into two cases: Case 1: Meshing force F in healthy state m Calculate as follows: Among them, k m is the healthy state of the helical gear, c m is the helical gear meshing damping, f(x n ) is the tooth side clearance function, x n is the displacement of the meshing line, x n The expression is: x n =[R b1 i z1 -R b2 i z2 +(x1-x2)sine t +(y1-y2)cosα t +(z1-z2+R b1 i y1 -R b2 i y2 )tanβ]cosβ-e(t); Among them, θ y1 ,θ z1 is the torsional vibration angular displacement of the driving wheel in the y and z directions, θ y2 ,θ z2 is the torsional vibration angular displacement of the driven wheel in the y and z directions, x1, y1, z1 are the vibration displacements of the driving wheel projected along the x, y, z directions, x2, y2, z2 are the vibration displacements of the driven wheel projected along the x, y, z directions, R b1 and R b2 are the base circle radii of the driving wheel and the driven wheel respectively, β is the helical angle of the helical gear, and e(t) is the transmission error; Case 2: Meshing force F under wear failure m Calculate as follows: Among them, k m ' is the meshing stiffness of the helical gear wear fault; The expression of transmission error e(t) is: Among them, t is time, e0, e r is the mean value and fluctuation amplitude of the transmission error, is the initial phase, ω e is the meshing frequency, e wear is the fluctuation amplitude of wear transmission error, e wear The expression is: Where n is the number of meshing teeth, h wear,i is the wear amount of the i-th pair of meshing teeth; Bearings b1 and b2 are at both ends of the driving wheel. The stiffness and damping of bearing b1 are k b1 、c b1 , the stiffness and damping of bearing b2 are k b2 、c b2 The stiffness and damping between bearings b1 and b2 and the shaft are c1 and k1, and bearings b3 and b4 are at both ends of the driven wheel. The stiffness and damping of bearing b3 are k b3 、c b3 , the stiffness and damping of bearing b4 are k b4 、c b4 , the stiffness and damping between bearings b3, b4 and the shaft are c2 and k2, and the dynamic equations of bearings b1, b2, b3, and b4 are: Among them, m b1 、m b2 、m b3 、m b4 are the masses of bearings b1, b2, b3, and b4 respectively, and x b1 、y b1 、z b1 is the vibration displacement of bearing b1 projected along the x, y, and z directions, x b2 、y b2 、z b2 is the vibration displacement of bearing b2 projected along the x, y, and z directions, x b3 、y b3 、z b3 is the vibration displacement of bearing b3 projected along the x, y, and z directions, x b4 、y b4 、z b4 is the vibration displacement of bearing b4 projected along the x, y, and z directions, F bx1 、F by1 、F bz1 is the projection of the bearing force of bearing b1 in the x, y, and z directions, F bx2 、F by2 、F bz2 is the projection of the bearing force of bearing b2 in the x, y, and z directions, F bx3 、F by3 、F bz3 is the projection of the support force of bearing b3 in the x, y, and z directions, F bx4 、F by4 、F bz4 is the projection of the support force of bearing b4 in the x, y, and z directions, and g is the acceleration due to gravity; The dynamic equations of the driving wheel and the driven wheel are: Among them, θ x1 and θ x2 are the torsional vibration angular displacements of the driving wheel and the driven wheel in the x direction, m1 and m2 are the masses of the driving wheel and the driven wheel, I1 and I2 are the moments of inertia of the driving wheel and the driven wheel about the rotation axis, k θ1 and c θ1 is the torsional stiffness and torsional damping of the driving wheel, k θ2 and c θ2 are the torsional stiffness and torsional damping of the driven wheel, T1 and T2 are the input and output torques, M f1 and M f2 are the torques of friction on the driving wheel and the driven wheel, F fx 、F fy 、F fz is the projection of friction force in the x, y, and z directions, F x 、F y 、F z is the projection of meshing force in x, y and z directions, F x 、F y 、F z 、F fx 、F fy 、F fz The expression is: Among them, F m is the meshing force, F f is the total friction force, α n is the normal pressure angle of the helical gear.

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