Nonlinear dynamics modeling method for electric drive transmission system

Abstract

The invention discloses a nonlinear dynamic modeling method for an electric drive transmission system. The nonlinear dynamic modeling method comprises the following steps: step (1), calculating time-varying meshing stiffness of a secondary helical gear pair based on a potential energy method; (2) introducing a bearing clearance, establishing a bearing mechanical model, analyzing the deformation of a rolling body, and calculating the bearing force of the bearing; (3) introducing a gear backlash nonlinear function, and determining relative displacement and meshing force on a meshing line of the secondary helical gear pair and the differential bevel gear pair; (4) establishing a nonlinear dynamic model of the electric drive transmission system, and listing a system oscillatory differential equation; (5) substituting the parameters, solving, and analyzing nonlinear vibration characteristics of gear pairs of all levels of the system under different external excitation frequencies; the method has the beneficial effects that bearing nonlinear vibration and multistage gear coupling are considered, and a theoretical basis is provided for nonlinear dynamic modeling and analysis of an electric drive transmission system.

Description

Technical Field

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

[0002] As an indispensable core technology in modern transportation and industrial equipment, electric drive transmission systems are widely used in the new energy vehicle industry. Their performance directly impacts the ride comfort, safety, and reliability of new energy vehicles. When the external excitation frequency changes, the system's vibration displacement can exhibit bifurcations and chaos, increasing the probability of failure and safety accidents during mechanical operation. Therefore, analyzing the dynamic characteristics of electric drive transmission systems under different operating conditions, exploring the influence of excitation frequency on the system's nonlinear behavior, and identifying the system's instability region are of great engineering significance.

[0003] In order to solve the above problems, the present invention proposes a nonlinear dynamic modeling method for an electric drive transmission system. The method first calculates the time-varying meshing stiffness of different helical gear pairs in the transmission system based on the potential energy method, introduces the nonlinear function of the tooth side clearance and the comprehensive transmission error, calculates the time-varying meshing force on each meshing line respectively, introduces the bearing clearance, and calculates the bearing support force of each shaft respectively; establishes the nonlinear dynamic equation of the electric drive transmission system, and uses the fourth-order Runge-Kutta method to solve the equation; this method provides a theoretical basis for nonlinear dynamic modeling and analysis of the electric drive transmission system, promotes the development of engineering technology, and can also generate greater social and economic benefits. Summary of the Invention

[0004] In order to overcome the deficiencies of the prior art and improve related technologies, the present invention provides a nonlinear dynamic modeling method for an electric drive transmission system.

[0005] The present invention solves the technical problem by adopting the following technical solution: a nonlinear dynamic modeling method for an electric drive transmission system, characterized by comprising the following steps:

[0006] Step (1): Calculate the time-varying meshing stiffness of the two-stage helical gear pair based on the potential energy method;

[0007] The time-varying mesh stiffness of helical gears is discussed in two cases: one where the base circle radius is smaller than the root circle radius, and the other where the base circle radius is larger than the root circle radius. The former case can simplify the gear teeth into cantilever beams, while the latter case requires consideration of the deformation of the gear teeth between the base circle and the root circle.

[0008] When the base circle radius is smaller than the root circle radius, the gear teeth are simplified as cantilever beams. The Hertzian contact stiffness k of helical gears is hi , bending stiffness k bi , shear stiffness ksi and axial compressive stiffness k ai The calculation formula is as follows:

[0009]

[0010] Among them, α is the pressure angle, α2 is half of the tooth base fillet angle, and α t is the end face pressure angle of the helical gear. α1(t) is the angle between the meshing force F on the contact line and the perpendicular direction of the tooth centerline, which is a function of time t. -α1(t) and α2 change linearly with the contact line, Δy is the infinitesimal value of each tooth slice along the tooth width direction, l i is the length of the contact line of the gear teeth that changes with time during meshing, N is the number of slices, q is the slice number, v is the Poisson's ratio of the gear material, and E is the Young's modulus of the gear material;

[0011] Helical gear base stiffness k fi , the calculation formula is as follows:

[0012]

[0013] Among them, u f is the bending potential energy, L*, M*, P*, Q* can be determined by the following formula:

[0014]

[0015] Among them, h fi A is the ratio of the tooth root radius to the shaft hole radius, i 、B i 、C i 、D i 、E i 、F i is a fixed coefficient;

[0016] When the base circle radius is larger than the root circle radius, considering the deformation of the gear tooth part between the base circle and the root circle, the bending stiffness k of the helical gear is bi , the calculation formula is as follows:

[0017]

[0018] Where α is the pressure angle, α2 is half of the tooth base fillet angle, α1(t) is the angle between the meshing force F on the tooth and the perpendicular direction of the tooth centerline, which is a function of time t. -α1(t) and α2 change linearly with the contact line. Δy is the infinitesimal value of each tooth slice along the tooth width direction, and r b and r f is the base circle radius and tooth root circle radius of the helical gear, N is the number of slices, q is the slice number, x1 is the vertical distance between the tooth root circle radius and the base circle radius, h x1is the distance between a point on the tooth root transition curve and the center line of the gear, d(y) is the displacement of the meshing point and the base circle in the direction of the tooth height, and h(y) is the distance between the meshing point and the center line of the gear in the direction of the tooth height;

[0019] Helical gear shear stiffness k si , the calculation formula is as follows:

[0020]

[0021] Among them, A x1 is the cross-sectional area of the tooth profile corresponding to the meshing force along the base circle to the root circle, G is the shear modulus of the gear material;

[0022] Axial compression stiffness k of helical gear ai , the calculation formula is as follows:

[0023]

[0024] The single tooth meshing stiffness k1 of the helical gear pair is calculated as follows:

[0025]

[0026] Comprehensive meshing stiffness k of helical gear pair m1 , the calculation formula is as follows:

[0027]

[0028] Where m is the number of tooth pairs meshing simultaneously, and n is the number of the meshing tooth pairs;

[0029] Step (2): Introduce bearing clearance, establish bearing mechanical model, analyze rolling element deformation, and calculate bearing support force;

[0030] Angular velocity of the rolling element of the bearing ω bn and position angle θ i (t), the calculation formula is as follows:

[0031]

[0032] Among them, v bi 、v bo Z is the linear velocity at the contact point between the bearing rolling element and the inner and outer rings of the bearing, b is the number of rolling elements, r bi 、r bo is the radius of the inner and outer rings of the bearing, ω bi is the angular velocity of the bearing inner ring around the axis of rotation, and t is the time;

[0033] Bearing rolling element deformation Δ bk , the calculation formula is as follows:

[0034]

[0035] Where x and y are the displacements of the bearing inner ring along the x and y axes, γ is the initial contact angle between the kth bearing rolling element and the bearing inner ring, and Δ θ is the angular deformation of the kth rolling element, θ k is the position angle corresponding to the kth rolling element, r bo is the radius of the bearing outer ring, l is the distance between the bearing outer ring and its center of curvature, b c is the radial clearance of the bearing;

[0036] After the bearing is deformed by force, the contact angle between the bearing rolling element and the bearing inner ring becomes γ', which is calculated as follows:

[0037]

[0038] The force on each rolling element is f bk , the calculation formula is as follows:

[0039]

[0040] Among them, H(Δ bk ) is the Heaviside function, K b is the contact stiffness between the rolling element and the inner ring of the bearing;

[0041] Apply force f to each rolling element bk Decompose it into the x and y directions along the coordinate axis, and sum it to obtain the component force F of the bearing support force along the x and y directions bx and F by , the calculation formula is as follows:

[0042]

[0043] Among them, θ k is the position angle corresponding to the kth rolling element, m b is the number of rolling elements;

[0044] Step (3): Introduce the nonlinear function of tooth side clearance to determine the relative displacement and meshing force on the meshing line between the secondary helical gear pair and the differential bevel gear pair;

[0045] x n1 、x n2 、x n3 is the relative displacement on the meshing line between the first and second stage helical gear pairs and the differential bevel gear pair, and the calculation formula is as follows:

[0046]

[0047] Among them, x ci 、y ci 、z ciis the displacement of the i-th gear along the x, y, and z directions, R ci is the base circle radius of the i-th gear, θ ci is the torsion angle of the i-th gear, i = 1, 2, 3, ..., 6; β1 and β3 are the helical angles of the first and second stage helical gear pairs respectively, δ is the pitch angle of the differential bevel gear pair, α n1 , α n3 , α n5 are the normal pressure angles of the first and second stage helical gear pairs and the pressure angle of the differential bevel gear pair, e n1 (t), e n2 (t), e n3 (t) are the transmission errors of the first and second stage helical gear pairs and the differential bevel gear pair;

[0048] The dynamic meshing force of the first stage helical gear pair is F m1 , F m1 The components of force along the x, y, and z axes are F c1x 、F c1y 、F c1z , the dynamic meshing force of the second-stage helical gear pair is F m2 , F m2 The components of force along the x, y, and z axes are F c2x 、F c2y 、F c2z , the calculation formula is as follows:

[0049]

[0050] Among them, K h1 (t), K h2 (t) is the time-varying meshing stiffness of the first and second stage helical gear pairs, C h1 、C h2 is the meshing damping of the first and second stage helical gear pairs, f(x n1 )、f(x n2 ) is the displacement function on the meshing line of the first and second stage helical gear pairs;

[0051] The dynamic meshing force of the differential bevel gear pair is F m3 , F m3 The components of force along the x, y, and z axes are F c3x 、F c3y 、F c3z , the calculation formula is as follows:

[0052]

[0053] Among them, K h3 (t) is the time-varying meshing stiffness of the differential bevel gear pair, C h3is the meshing damping of the differential bevel gear pair, f(x n3 ) is the displacement function on the meshing line of the differential bevel gear pair;

[0054] Step (4): Establish a nonlinear dynamic model of the electric drive transmission system and list the system vibration differential equations;

[0055] Formulate the differential equations for the vibration of the system with the following degrees of freedom:

[0056]

[0057] Among them, x ci 、y ci 、z ci ,θ ci is the displacement and torsion angle of gear i along the x, y, and z directions, where the subscript i is the gear number, i = 1, 2, 3, ..., 6; x bj 、y bj is the vibration displacement of bearing j along the x and y directions, where subscript j is the bearing number, j = 1, 2, 3, …, 6;

[0058] The vibration differential equation of the first-stage helical gear pair is:

[0059]

[0060] Among them, m c1 、m c2 、m e1 is the equivalent mass of gear 1, gear 2 and the first-stage helical gear pair, K c1x , K c1y , K c1z is the support stiffness of the input shaft along the x, y, and z directions, C c1x 、C c1y 、C c1z is the support damping of the input shaft in the x, y, and z directions, K c2x , K c2y , K c2z is the support stiffness of the intermediate shaft along the x, y, and z directions, C c2x 、C c2y 、C c2z is the support damping of the intermediate shaft along the x, y, and z directions, F m1 is the dynamic meshing force of the first-stage helical gear pair, F c1x 、F c1y 、F c1z is the component force of the dynamic meshing force of the first-stage helical gear pair along the x, y, and z directions, α n1 is the normal pressure angle of the first-stage helical gear pair, β1 is the helical angle of the first-stage helical gear pair, and x c1 、y c1 、z c1is the displacement of gear 1 along the x, y, and z directions, x c2 、y c2 、z c2 is the displacement of gear 2 along the x, y, and z directions, e n1 (t) is the transmission error of the first-stage helical gear pair;

[0061] The vibration differential equation of the second-stage helical gear pair is:

[0062]

[0063] Among them, m c3 、m c4 、m e2 is the equivalent mass of gear 3, gear 4 and the second-stage helical gear pair, K c3x , K c3y , K c3z is the support stiffness of the output shaft along the x, y, and z directions, C c3x 、C c3y 、C c3z is the support damping of the output shaft in the x, y, and z directions, F m2 is the dynamic meshing force of the second-stage helical gear pair, F c2x 、F c2y 、F c2z is the component force of the dynamic meshing force of the second-stage helical gear pair along the x, y, and z directions, α n3 is the normal pressure angle of the second-stage helical gear pair, β3 is the helical angle of the second-stage helical gear pair, and x c3 、y c3 、z c3 is the displacement of gear 3 along the x, y, and z directions, x c4 、y c4 、z c4 is the displacement of gear 4 along the x, y, and z directions, e n2 (t) is the transmission error of the second-stage helical gear pair;

[0064] Vibration differential equation of bearing I:

[0065]

[0066] Among them, m b1 、m b2 is the mass of bearings 1 and 2, K b1x , K b1y , K b2x , K b2y is the support stiffness of bearings 1 and 2 along the x and y directions, C b1x 、C b1y 、C b2x 、C b2yis the support damping of bearings 1 and 2 along the x and y directions, F b1x 、F b1y 、F b2x 、F b2y is the component force of the support force of bearings 1 and 2 along the x and y directions, x b1 、y b1 、x b2 、y b2 is the displacement of bearings 1 and 2 along the x and y directions;

[0067] Vibration differential equation of bearing II:

[0068]

[0069] Among them, m b3 、m b4 is the mass of bearings 3 and 4, K b3x , K b3y , K b4x , K b4y is the support stiffness of bearings 3 and 4 along the x and y directions, C b3x 、C b3y 、C b4x 、C b4y is the support damping of bearings 3 and 4 along the x and y directions, F b3x 、F b3y 、F b4x 、F b4y is the component of the support force of bearings 3 and 4 along the x and y directions, x b3 、y b3 、x b4 、y b4 is the displacement of bearings 3 and 4 along the x and y directions;

[0070] Vibration differential equation of bearing III:

[0071]

[0072] Among them, m b5 、m b6 is the mass of bearings 5 and 6, K b5x , K b5y , K b6x , K b6y is the support stiffness of bearings 5 and 6 along the x and y directions, C b5x 、C b5y 、C b6x 、C b6y is the support damping of bearings 5 and 6 along the x and y directions, F b5x 、F b5y 、F b6x 、F b6yis the component of the support force of bearings 5 and 6 along the x and y directions, x b5 、y b5 、x b6 、y b6 is the displacement of bearings 5 and 6 along the x and y directions;

[0073] Vibration differential equation of differential bevel gear pair:

[0074]

[0075] Among them, m c5 、m c6 、m e3 is the equivalent mass of gear 5, gear 6 and the differential bevel gear pair, K c4x , K c4y , K c4z is the support stiffness of the output shaft along the x, y, and z directions, C c4x 、C c4y 、C c4z is the support damping of the output shaft in the x, y, and z directions, F m3 is the dynamic meshing force of the differential bevel gear pair, F c3x 、F c3y 、F c3z is the component force of the dynamic meshing force of the differential bevel gear pair along the x, y, and z directions, α n5 is the normal pressure angle of the differential bevel gear pair, δ is the pitch cone angle of the differential bevel gear pair, x c5 、y c5 、z c5 is the displacement of gear 5 along the x, y, and z directions, x c6 、y c6 、z c6 is the displacement of gear 6 along the x, y, and z directions, e n3 (t) is the transmission error of the differential bevel gear pair;

[0076] Step (5): Substitute the parameters and solve to analyze the nonlinear vibration characteristics of the gear pairs at all levels of the system under different external excitation frequencies;

[0077] A three-dimensional phase diagram of the relationship between the dimensionless displacement, velocity and external excitation frequency of each gear pair is drawn. Through analysis, it can be seen that when the external excitation frequency is changed, the nonlinear response of the system is rich, showing chaotic, period-doubling and single-period vibration states. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] Figure 1 It is a flow chart of the nonlinear dynamic modeling method of the electric drive transmission system;

[0079] Figure 2 is the time-varying mesh stiffness model of helical gears;

[0080] Figure 3 It is a multi-degree-of-freedom coupled vibration model of the electric drive transmission system;

[0081] Figure 4 It is a three-dimensional phase diagram of the electric drive transmission system at different external excitation frequencies at each level. DETAILED DESCRIPTION

[0082] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1 —4 The specific implementation methods of the present invention are described in detail.

[0083] like Figure 1 The figure shows a flow chart of the nonlinear dynamic modeling method of the electric drive transmission system, which includes the following steps:

[0084] Step (1): Calculate the time-varying meshing stiffness of the two-stage helical gear pair based on the potential energy method;

[0085] Build as Figure 2 The time-varying mesh stiffness model of helical gears is shown. The time-varying mesh stiffness of helical gears is discussed in two cases: one where the base circle radius is smaller than the root circle radius, and the other where the base circle radius is larger than the root circle radius. The former simplifies the gear teeth into cantilever beams, while the latter requires consideration of the deformation of the gear teeth between the base circle and the root circle.

[0086] When the base circle radius is smaller than the root circle radius, the gear teeth are simplified as cantilever beams. The Hertzian contact stiffness k of helical gears is hi , bending stiffness k bi , shear stiffness k si and axial compressive stiffness k ai The calculation formula is as follows:

[0087]

[0088]

[0089] Among them, α is the pressure angle, α2 is half of the tooth base fillet angle, and α t is the end face pressure angle of the helical gear. α1(t) is the angle between the meshing force F on the contact line and the perpendicular direction of the tooth centerline, which is a function of time t. -α1(t) and α2 change linearly with the contact line, Δy is the infinitesimal value of each tooth slice along the tooth width direction, l i is the length of the contact line of the gear teeth that changes with time during meshing, N is the number of slices, q is the slice number, v is the Poisson's ratio of the gear material, and E is the Young's modulus of the gear material;

[0090] Helical gear base stiffness k fi , the calculation formula is as follows:

[0091]

[0092] Among them, u f is the bending potential energy, L*, M*, P*, Q* can be determined by the following formula:

[0093]

[0094] Among them, h fi A is the ratio of the tooth root radius to the shaft hole radius, i 、B i 、C i 、D i 、E i 、F i is a fixed coefficient, which can be determined by looking up Table 1;

[0095] Table 1A i 、B i 、C i 、D i 、E i 、F i The coefficient value of

[0096]

[0097] When the base circle radius is larger than the root circle radius, considering the deformation of the gear tooth part between the base circle and the root circle, the bending stiffness k of the helical gear is bi , the calculation formula is as follows:

[0098]

[0099] Where α is the pressure angle, α2 is half of the tooth base fillet angle, α1(t) is the angle between the meshing force F on the tooth and the perpendicular direction of the tooth centerline, which is a function of time t. -α1(t) and α2 change linearly with the contact line. Δy is the infinitesimal value of each tooth slice along the tooth width direction, and r b and r f is the base circle radius and tooth root circle radius of the helical gear, N is the number of slices, q is the slice number, x1 is the vertical distance between the tooth root circle radius and the base circle radius, h x1 is the distance between a point on the tooth root transition curve and the center line of the gear, d(y) is the displacement between the meshing point and the base circle in the direction of the tooth height, and h(y) is the distance between the meshing point and the center line of the gear in the direction of the tooth height;

[0100] Helical gear shear stiffness k si , the calculation formula is as follows:

[0101]

[0102] Among them, A x1is the cross-sectional area of the tooth profile corresponding to the meshing force along the base circle to the root circle, G is the shear modulus of the gear material;

[0103] Axial compression stiffness k of helical gear ai , the calculation formula is as follows:

[0104]

[0105] The single tooth meshing stiffness k1 of the helical gear pair is calculated as follows:

[0106]

[0107] Comprehensive meshing stiffness k of helical gear pair m1 , the calculation formula is as follows:

[0108]

[0109] Where m is the number of tooth pairs meshing simultaneously, and n is the number of the meshing tooth pairs;

[0110] Step (2): Introduce bearing clearance, establish bearing mechanical model, analyze rolling element deformation, and calculate bearing support force;

[0111] Angular velocity of the rolling element of the bearing ω bn and position angle θ i (t), the calculation formula is as follows:

[0112]

[0113] Among them, v bi 、v bo Z is the linear velocity at the contact point between the bearing rolling element and the inner and outer rings of the bearing, b is the number of rolling elements, r bi 、r bo is the radius of the inner and outer rings of the bearing, ω bi is the angular velocity of the bearing inner ring around the axis of rotation, and t is the time;

[0114] Bearing rolling element deformation Δ bk , the calculation formula is as follows:

[0115]

[0116] Where x and y are the displacements of the bearing inner ring along the x and y axes, γ is the initial contact angle between the kth bearing rolling element and the bearing inner ring, and Δ θ is the angular deformation of the kth rolling element, θ k is the position angle corresponding to the kth rolling element, r bo is the radius of the bearing outer ring, l is the distance between the bearing outer ring and its center of curvature, b c is the radial clearance of the bearing;

[0117] After the bearing is deformed by force, the contact angle between the bearing rolling element and the bearing inner ring becomes γ', which is calculated as follows:

[0118]

[0119] The force on each rolling element is f bk , the calculation formula is as follows:

[0120]

[0121] Among them, H(Δ bk ) is the Heaviside function, K b is the contact stiffness between the rolling element and the inner ring of the bearing;

[0122] Apply force f to each rolling element bk Decompose it into the x and y directions along the coordinate axis, and sum it to obtain the component force F of the bearing support force along the x and y directions bx and F by , the calculation formula is as follows:

[0123]

[0124] Among them, θ k is the position angle corresponding to the kth rolling element, m b is the number of rolling elements;

[0125] Step (3): Introduce the nonlinear function of tooth side clearance to determine the relative displacement and meshing force on the meshing line between the secondary helical gear pair and the differential bevel gear pair;

[0126] x n1 、x n2 、x n3 is the relative displacement on the meshing line between the first and second stage helical gear pairs and the differential bevel gear pair, and the calculation formula is as follows:

[0127]

[0128] Among them, x ci 、y ci 、z ci is the displacement of the i-th gear along the x, y, and z directions, R ci is the base circle radius of the i-th gear, θ ci is the torsion angle of the i-th gear, i = 1, 2, 3, ..., 6; β1 and β3 are the helical angles of the first and second stage helical gear pairs respectively, δ is the pitch angle of the differential bevel gear pair, α n1 , α n3 , α n5 are the normal pressure angles of the first and second stage helical gear pairs and the pressure angle of the differential bevel gear pair, en1 (t), e n2 (t), e n3 (t) are the transmission errors of the first and second stage helical gear pairs and the differential bevel gear pair;

[0129] The dynamic meshing force of the first stage helical gear pair is F m1 , F m1 The components of force along the x, y, and z axes are F c1x 、F c1y 、F c1z , the dynamic meshing force of the second-stage helical gear pair is F m2 , F m2 The components of force along the x, y, and z axes are F c2x 、F c2y 、F c2z , the calculation formula is as follows:

[0130]

[0131] Among them, K h1 (t), K h2 (t) is the time-varying meshing stiffness of the first and second stage helical gear pairs, C h1 、C h2 is the meshing damping of the first and second stage helical gear pairs, f(x n1 )、f(x n2 ) is the displacement function on the meshing line of the first and second stage helical gear pairs;

[0132] The dynamic meshing force of the differential bevel gear pair is F m3 , F m3 The components of force along the x, y, and z axes are F c3x 、F c3y 、F c3z , the calculation formula is as follows:

[0133]

[0134] Among them, K h3 (t) is the time-varying meshing stiffness of the differential bevel gear pair, C h3 is the meshing damping of the differential bevel gear pair, f(x n3 ) is the displacement function on the meshing line of the differential bevel gear pair;

[0135] Step (4): Establish a nonlinear dynamic model of the electric drive transmission system and list the system vibration differential equations;

[0136] The dynamic model of the electric drive transmission system, such as Figure 3 As shown, the system vibration differential equation including the following degrees of freedom is established:

[0137]

[0138] Among them, x ci 、y ci 、z ci ,θ ci is the displacement and torsion angle of gear i along the x, y, and z directions, where the subscript i is the gear number, i = 1, 2, 3, ..., 6; x bj 、y bj is the vibration displacement of bearing j along the x and y directions, where subscript j is the bearing number, j = 1, 2, 3, …, 6;

[0139] The vibration differential equation of the first-stage helical gear pair is:

[0140]

[0141] Among them, m c1 、m c2 、m e1 is the equivalent mass of gear 1, gear 2 and the first-stage helical gear pair, K c1x , K c1y , K c1z is the support stiffness of the input shaft along the x, y, and z directions, C c1x 、C c1y 、C c1z is the support damping of the input shaft in the x, y, and z directions, K c2x , K c2y , K c2z is the support stiffness of the intermediate shaft along the x, y, and z directions, C c2x 、C c2y 、C c2z is the support damping of the intermediate shaft along the x, y, and z directions, F m1 is the dynamic meshing force of the first-stage helical gear pair, F c1x 、F c1y 、F c1z is the component force of the dynamic meshing force of the first-stage helical gear pair along the x, y, and z directions, α n1 is the normal pressure angle of the first-stage helical gear pair, β1 is the helical angle of the first-stage helical gear pair, and x c1 、y c1 、z c1 is the displacement of gear 1 along the x, y, and z directions, x c2 、y c2 、z c2 is the displacement of gear 2 along the x, y, and z directions, e n1 (t) is the transmission error of the first-stage helical gear pair;

[0142] The vibration differential equation of the second-stage helical gear pair is:

[0143]

[0144] Among them, m c3 、m c4 、m e2 is the equivalent mass of gear 3, gear 4 and the second-stage helical gear pair, K c3x , K c3y , K c3z is the support stiffness of the output shaft along the x, y, and z directions, C c3x 、C c3y 、C c3z is the support damping of the output shaft in the x, y, and z directions, F m2 is the dynamic meshing force of the second-stage helical gear pair, F c2x 、F c2y 、F c2z is the component force of the dynamic meshing force of the second-stage helical gear pair along the x, y, and z directions, α n3 is the normal pressure angle of the second-stage helical gear pair, β3 is the helical angle of the second-stage helical gear pair, and x c3 、y c3 、z c3 is the displacement of gear 3 along the x, y, and z directions, x c4 、y c4 、z c4 is the displacement of gear 4 along the x, y, and z directions, e n2 (t) is the transmission error of the second-stage helical gear pair;

[0145] Vibration differential equation of bearing I:

[0146]

[0147] Among them, m b1 、m b2 is the mass of bearings 1 and 2, K b1x , K b1y , K b2x , K b2y is the support stiffness of bearings 1 and 2 along the x and y directions, C b1x 、C b1y 、C b2x 、C b2y is the support damping of bearings 1 and 2 along the x and y directions, F b1x 、F b1y 、F b2x 、F b2y is the component force of the support force of bearings 1 and 2 along the x and y directions, x b1 、y b1 、x b2 、y b2 is the displacement of bearings 1 and 2 along the x and y directions;

[0148] Vibration differential equation of bearing II:

[0149]

[0150] Among them, m b3 、m b4 is the mass of bearings 3 and 4, K b3x , K b3y , K b4x , K b4y is the support stiffness of bearings 3 and 4 along the x and y directions, C b3x 、C b3y 、C b4x 、C b4y is the support damping of bearings 3 and 4 along the x and y directions, F b3x 、F b3y 、F b4x 、F b4y is the component of the support force of bearings 3 and 4 along the x and y directions, x b3 、y b3 、x b4 、y b4 is the displacement of bearings 3 and 4 along the x and y directions;

[0151] Vibration differential equation of bearing III:

[0152]

[0153] Among them, m b5 、m b6 is the mass of bearings 5 and 6, K b5x , K b5y , K b6x , K b6y is the support stiffness of bearings 5 and 6 along the x and y directions, C b5x 、C b5y 、C b6x 、C b6y is the support damping of bearings 5 and 6 along the x and y directions, F b5x 、F b5y 、F b6x 、F b6y is the component of the support force of bearings 5 and 6 along the x and y directions, x b5 、y b5 、x b6 、y b6 is the displacement of bearings 5 and 6 along the x and y directions;

[0154] Vibration differential equation of differential bevel gear pair:

[0155]

[0156] Among them, m c5 、m c6 、m e3 is the equivalent mass of gear 5, gear 6 and the differential bevel gear pair, K c4x , K c4y , K c4z is the support stiffness of the output shaft along the x, y, and z directions, C c4x 、C c4y 、C c4z is the support damping of the output shaft in the x, y, and z directions, F m3 is the dynamic meshing force of the differential bevel gear pair, F c3x 、F c3y 、F c3z is the component force of the dynamic meshing force of the differential bevel gear pair along the x, y, and z directions, α n5 is the normal pressure angle of the differential bevel gear pair, δ is the pitch cone angle of the differential bevel gear pair, x c5 、y c5 、z c5 is the displacement of gear 5 along the x, y, and z directions, x c6 、y c6 、z c6 is the displacement of gear 6 along the x, y, and z directions, e n3 (t) is the transmission error of the differential bevel gear pair;

[0157] During the establishment of the nonlinear dynamic model of the electric drive transmission system, the parameters of each gear are shown in Table 2:

[0158] Table 2 Basic parameters of gears

[0159]

[0160] Step (5): Substitute the parameters and solve to analyze the nonlinear vibration characteristics of the gear pairs at all levels of the system under different external excitation frequencies;

[0161] Draw a three-dimensional phase diagram of the relationship between the dimensionless displacement, speed and external excitation frequency of each gear pair, such as Figure 4 The following are the three-dimensional phase diagrams of the first and second stage helical gear pairs and the differential bevel gear pair. Analysis shows that when the external excitation frequency is changed, the nonlinear response of the system is rich, showing chaotic, period-doubling, and single-period vibration states.

[0162] 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 an electric drive transmission system, characterized in that: The following steps are involved: Step (1): Calculate the time-varying meshing stiffness of the two-stage helical gear pair based on the potential energy method; When the base circle radius is larger than the root circle radius, considering the deformation of the gear teeth between the base circle and the root circle, the bending stiffness k of the helical gear is bi , the calculation formula is as follows: Where α is the pressure angle, α2 is half of the tooth base fillet angle, α1(t) is the angle between the meshing force F on the tooth and the perpendicular direction of the tooth centerline, which is a function of time t. -α1(t) and α2 change linearly with the contact line. Δy is the infinitesimal value of each tooth slice along the tooth width direction. E is the Young's modulus of the gear, and r is the linear function of the tooth width. b and r f is the base circle radius and tooth root circle radius of the helical gear, N is the number of slices, q is the slice number, x1 is the vertical distance between the tooth root circle radius and the base circle radius, h x1 is the distance between any point on the tooth root transition curve and the gear centerline, d(y) is the displacement between the meshing point and the base circle in the tooth height direction, and h(y) is the distance between the meshing point and the gear centerline in the tooth height direction; Helical gear shear stiffness k si , the calculation formula is as follows: Among them, A x1 is the cross-sectional area of the tooth profile corresponding to the meshing force along the base circle to the root circle, G is the shear modulus of the gear, and v is the Poisson's ratio of the material used for the gear; Axial compression stiffness k of helical gear ai , the calculation formula is as follows: Step (2): Introduce bearing clearance, establish bearing mechanical model, analyze rolling element deformation, and calculate bearing support force; Bearing rolling element deformation Δ bk , the calculation formula is as follows: Where x and y are the displacements of the bearing inner ring along the coordinate axis in the x and y directions, γ is the initial contact angle between the kth rolling element of the bearing and the bearing inner ring, Δ θ is the angular deformation of the kth rolling element, θ k is the position angle corresponding to the kth rolling element, r bo is the radius of the bearing outer ring, l is the distance between the bearing outer ring and its center of curvature, b c is the radial clearance of the bearing; After the bearing is deformed by force, the contact angle between the bearing rolling element and the bearing inner ring becomes γ', which is calculated as follows: The force on each rolling element is f bk , the calculation formula is as follows: Among them, H(Δ bk ) is the Heaviside function, K b is the contact stiffness between the rolling element and the inner ring of the bearing; Apply force f to each rolling element bk Decompose it into the x and y directions along the coordinate axis, and sum it to get the component force F of the bearing support force along the x and y directions bx and F by , the calculation formula is as follows: Among them, θ k is the position angle corresponding to the kth rolling element, m b is the number of rolling elements; Step (3): Introduce the nonlinear function of tooth side clearance to determine the relative displacement and meshing force on the meshing line between the secondary helical gear pair and the differential bevel gear pair; x n1 、x n2 、x n3 is the relative displacement on the meshing line between the first and second stage helical gear pairs and the differential bevel gear pair, and the calculation formula is as follows: Among them, x ci 、y ci 、z ci is the displacement of the i-th gear along the x, y, and z directions, R ci is the base circle radius of the i-th gear, θ ci is the torsion angle of the i-th gear, i = 1, 2, 3, ..., 6; β1 and β3 are the helical angles of the first and second stage helical gear pairs respectively, δ is the pitch angle of the differential bevel gear pair, α n1 , α n3 , α n5 are the normal pressure angles of the first and second stage helical gear pairs and the pressure angle of the differential bevel gear pair, e n1 (t), e n2 (t), e n3 (t) are the transmission errors of the first and second stage helical gear pairs and the differential bevel gear pair; The dynamic meshing force of the first stage helical gear pair is F m1 , F m1 The components of force along the x, y, and z axes are F c1x 、F c1y 、F c1z , the dynamic meshing force of the second-stage helical gear pair is F m2 , F m2 The components of force along the x, y, and z axes are F c2x 、F c2y 、F c2z , the calculation formula is as follows: Among them, K h1 (t), K h2 (t) is the time-varying meshing stiffness of the first and second stage helical gear pairs, C h1 、C h2 is the meshing damping of the first and second stage helical gear pairs, f(x n1 )、f(x n2 ) is the displacement function on the meshing line of the first and second stage helical gear pairs; The dynamic meshing force of the differential bevel gear pair is F m3 , F m3 The components of force along the x, y, and z axes are F c3x 、F c3y 、F c3z , the calculation formula is as follows: Among them, K h3 (t) is the time-varying meshing stiffness of the differential bevel gear pair, C h3 is the meshing damping of the differential bevel gear pair, f(x n3 ) is the displacement function on the meshing line of the differential bevel gear pair; Step (4): Establish a nonlinear dynamic model of the electric drive transmission system and list the system vibration differential equations; Formulate the differential equations for the vibration of the system with the following degrees of freedom: Among them, x ci 、y ci 、z ci ,θ ci is the displacement and torsion angle of gear i along the x, y, and z directions, where the subscript i is the gear number, i = 1, 2, 3, ..., 6; x bj 、y bj is the vibration displacement of bearing j along the x and y directions, where subscript j is the bearing number, j = 1, 2, 3, …, 6; The vibration differential equation of the first-stage helical gear pair is: Among them, m c1 、m c2 、m e1 is the equivalent mass of gear 1, gear 2 and the first-stage helical gear pair, K c1x , K c1y , K c1z is the support stiffness of the input shaft along the x, y, and z directions, C c1x 、C c1y 、C c1z is the support damping of the input shaft in the x, y, and z directions, K c2x , K c2y , K c2z is the support stiffness of the intermediate shaft along the x, y, and z directions, C c2x 、C c2y 、C c2z is the support damping of the intermediate shaft along the x, y, and z directions, F m1 is the dynamic meshing force of the first-stage helical gear pair, F c1x 、F c1y 、F c1z is the component force of the dynamic meshing force of the first-stage helical gear pair along the x, y, and z directions, α n1 is the normal pressure angle of the first-stage helical gear pair, β1 is the helical angle of the first-stage helical gear pair, and x c1 、y c1 、z c1 is the displacement of gear 1 along the x, y, and z directions, x c2 、y c2 、z c2 is the displacement of gear 2 along the x, y, and z directions, e n1 (t) is the transmission error of the first-stage helical gear pair; The vibration differential equation of the second-stage helical gear pair is: Among them, m c3 、m c4 、m e2 is the equivalent mass of gear 3, gear 4 and the second-stage helical gear pair, K c3x , K c3y , K c3z is the support stiffness of the output shaft along the x, y, and z directions, C c3x 、C c3y 、C c3z is the support damping of the output shaft in the x, y, and z directions, F m2 is the dynamic meshing force of the second-stage helical gear pair, F c2x 、F c2y 、F c2z is the component force of the dynamic meshing force of the second-stage helical gear pair along the x, y, and z directions, α n3 is the normal pressure angle of the second-stage helical gear pair, β3 is the helical angle of the second-stage helical gear pair, and x c3 、y c3 、z c3 is the displacement of gear 3 along the x, y, and z directions, x c4 、y c4 、z c4 is the displacement of gear 4 along the x, y, and z directions, e n2 (t) is the transmission error of the second-stage helical gear pair; Vibration differential equation of bearing I: Among them, m b1 、m b2 is the mass of bearings 1 and 2, K b1x , K b1y , K b2x , K b2y is the support stiffness of bearings 1 and 2 along the x and y directions, C b1x 、C b1y 、C b2x 、C b2y is the support damping of bearings 1 and 2 along the x and y directions, F b1x 、F b1y 、F b2x 、F b2y is the component force of the support force of bearings 1 and 2 along the x and y directions, x b1 、y b1 、x b2 、y b2 is the displacement of bearings 1 and 2 along the x and y directions; Vibration differential equation of bearing II: Among them, m b3 、m b4 is the mass of bearings 3 and 4, K b3x , K b3y , K b4x , K b4y is the support stiffness of bearings 3 and 4 along the x and y directions, C b3x 、C b3y 、C b4x 、C b4y is the support damping of bearings 3 and 4 along the x and y directions, F b3x 、F b3y 、F b4x 、F b4y is the component of the support force of bearings 3 and 4 along the x and y directions, x b3 、y b3 、x b4 、y b4 is the displacement of bearings 3 and 4 along the x and y directions; Vibration differential equation of bearing III: Among them, m b5 、m b6 is the mass of bearings 5 and 6, K b5x , K b5y , K b6x , K b6y is the support stiffness of bearings 5 and 6 along the x and y directions, C b5x 、C b5y 、C b6x 、C b6y is the support damping of bearings 5 and 6 along the x and y directions, F b5x 、F b5y 、F b6x 、F b6y is the component of the support force of bearings 5 and 6 along the x and y directions, x b5 、y b5 、x b6 、y b6 is the displacement of bearings 5 and 6 along the x and y directions; Vibration differential equation of differential bevel gear pair: Among them, m c5 、m c6 、m e3 is the equivalent mass of gear 5, gear 6 and the differential bevel gear pair, K c4x , K c4y , K c4z is the support stiffness of the output shaft along the x, y, and z directions, C c4x 、C c4y 、C c4z is the support damping of the output shaft in the x, y, and z directions, F m3 is the dynamic meshing force of the differential bevel gear pair, F c3x 、F c3y 、F c3z is the component force of the dynamic meshing force of the differential bevel gear pair along the x, y, and z directions, α n5 is the normal pressure angle of the differential bevel gear pair, δ is the pitch cone angle of the differential bevel gear pair, x c5 、y c5 、z c5 is the displacement of gear 5 along the x, y, and z directions, x c6 、y c6 、z c6 is the displacement of gear 6 along the x, y, and z directions, e n3 (t) is the transmission error of the differential bevel gear pair; Step (5): Substitute the parameters and solve to analyze the nonlinear vibration characteristics of the gear pairs at all levels of the system under different external excitation frequencies.

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