A high-speed heavy-load double bevel gear transmission system nonlinear dynamics modeling method

Abstract

The application discloses a high-speed heavy-load double helical gear transmission system nonlinear dynamics modeling method, comprising the following steps: (1) calculating the time-varying meshing stiffness of the helical gear based on the ISO6336-2006 formula; (2) introducing bearing clearance, analyzing rolling body deformation, and calculating bearing support force; (3) introducing a nonlinear function of the tooth side clearance, determining the relative displacement and meshing force on the meshing line of the helical gear pair; (4) establishing a nonlinear dynamics model of the double helical gear system and listing the system dynamics equation, substituting the data into the equation and solving, and drawing displacement bifurcation diagrams on two meshing lines respectively; the beneficial effect is that the nonlinear function of the tooth side clearance, bearing clearance and other factors are introduced into the system, and the bearing vibration is considered, so that the current high-speed heavy-load double helical gear transmission system nonlinear dynamics modeling related technology blank is filled.

Description

Technical Field

[0001] This invention relates to the field of gear dynamics technology, and in particular to a nonlinear dynamics modeling method for a high-speed, heavy-load double helical gear transmission system. Background Technology

[0002] Double helical gears can eliminate the additional axial load during helical gear meshing, increase the contact ratio, and make meshing smoother, thus they are commonly used in the power transmission systems of armed helicopters. However, in the actual meshing process of high-speed, heavy-load double helical gears, nonlinear factors such as tooth backlash and meshing error excitation exist, causing the system motion to exhibit nonlinear characteristics. When the external excitation frequency is changed, the system vibration displacement may exhibit bifurcation and chaos, increasing the probability of malfunctions and safety accidents during mechanical equipment operation. Therefore, analyzing the nonlinear dynamic characteristics of high-speed, heavy-load double helical gear systems, exploring the influence of excitation frequency on the system's nonlinear behavior, and identifying the system's stable operating range are of significant engineering importance. Current technology lacks research on the nonlinear characteristics of high-speed, heavy-load double helical gear transmission systems.

[0003] To address the aforementioned problems, this invention proposes a nonlinear dynamic modeling method for high-speed, heavy-duty double helical gear transmission systems. This method first calculates the time-varying meshing stiffness of the double helical gear pair connected by the intermediate shaft segment, introduces a nonlinear function for tooth flank clearance and a comprehensive transmission error, calculates the time-varying meshing forces on both meshing lines, introduces bearing clearance, and calculates the support forces of the two bearings on the input shaft and the two bearings on the output shaft. Subsequently, the nonlinear dynamic equations of the high-speed, heavy-duty double helical gear transmission system are established, and the equations are solved using the fourth-order Runge-Kutta method. This modeling method can fill a gap in related international technologies, promote engineering technology development, and generate significant socio-economic benefits. Summary of the Invention

[0004] In order to overcome the shortcomings of existing technologies and fill the gaps in related technologies, this invention provides a nonlinear dynamic modeling method for high-speed heavy-load double helical gear transmission systems.

[0005] The technical solution adopted by this invention to solve its technical problem is as follows: A nonlinear dynamic modeling method for a high-speed, heavy-load double helical gear transmission system, characterized by comprising the following steps:

[0006] Step (1): Calculate the time-varying meshing stiffness of the double helical gears connected by the intermediate shaft segment; Step (2): Calculate the meshing force on the two meshing lines of the gears; let x1, y1, z1 be the vibration displacements of gear 1 along the x, y, and z axes respectively; x2, y2, z2 be the vibration displacements of gear 2 along the x, y, and z axes respectively; x3, y3, z3 be the vibration displacements of gear 3 along the x, y, and z axes respectively; x4, y4, z4 be the vibration displacements of gear 4 along the x, y, and z axes respectively; and θ1, θ2, θ3, θ4 be the torsional displacements of gears 1, 2, 3, and 4 respectively; α n β represents the normal pressure angle and helix angle, e represents the static transmission error amplitude, e1(t) and e2(t) represent the static transmission errors on meshing line 1 and meshing line 2, respectively, and R b1 R b2 R b3 R b4 Let the pitch circle radii of gears 1, 2, 3, and 4 be respectively. Then, the displacements on the meshing lines 1 and 2 of the helical gear pair are calculated by the following formula: , With F m1 F m2 F is the meshing force on meshing line 1 and meshing line 2. mx1 F my1 F mz1 F represents the meshing force along the x, y, and z axes on line 1. mx2 F my2 F mz2 b represents the components of the meshing force along the x, y, and z axes on the line of meshing 2. m Assuming half the tooth flank clearance, the meshing force is calculated as follows, with the components along the x, y, and z axes: ; Step (3): Introduce the radial clearance between the rolling elements and the inner and outer raceways of the bearing, and calculate the bearing support force; Step (4): Establish the nonlinear dynamic equations of the system, considering 24 degrees of freedom, namely: {x1,y1,z1,θ1,x2,y2,z2,θ2,x3,y3,z3,θ3,x4,y4,z4,θ4,x A1 ,y A1 ,x A2 ,y A2 ,x B1 ,y B1 ,x B2 ,y B2}; In the system dynamics equations, m1, m2, m3, and m4 represent the masses of gear 1, gear 2, gear 3, and gear 4, respectively, and I1, I2, I3, and I4 represent the moments of inertia of gear 1, gear 2, gear 3, and gear 4, respectively; m A1 m A2 m B1 m B2 The masses of bearings A1, A2, B1, and B2 are respectively; k A1x k A1y k A2x k A2y k B1x k B1y k B2x k B2y The support stiffness of bearings A1, A2, B1, and B2 along the x and y axes are respectively, c. A1x c A1y c A2x c A2y c B1x c B1y c B2x c B2y The bearings A1, A2, B1, and B2 are respectively supported by damping along the x and y axes of the coordinate system; k x1 k y1 k z1 k x2 k y2 k z2 k x3 k y3 k z3 k x4 k y4 k z4 The support stiffness (k) of the connecting shaft segments of gear 1 to bearing A1, gear 2 to bearing B1, gear 3 to bearing A2, and gear 4 to bearing B2 along the x, y, and z axes of the coordinate axes are respectively. θ1 k θ2 k θ3 k θ4 The torsional stiffness of the connecting shaft segments of gear 1 and bearing A1, gear 2 and bearing B1, gear 3 and bearing A2, and gear 4 and bearing B2, respectively; c x1 c x2 c x3 c x4 c y1 c y2 c y3 c y4 c z1 c z2 c z3 c z4The connecting shaft segments of gear 1 to bearing A1, gear 2 to bearing B1, gear 3 to bearing A2, and gear 4 to bearing B2 are respectively supported and damped along the x, y, and z axes of the coordinate axis. θ1 c θ2 c θ3 c θ4 The torsional damping of the connecting shaft segments for gear 1 and bearing A1, gear 2 and bearing B1, gear 3 and bearing A2, and gear 4 and bearing B2 are respectively; k x13 k y13 k z13 k x24 k y24 k z24 The support stiffness of the connecting shaft segments between gears 1 and 3, and between gears 2 and 4 along the x, y, and z axes are respectively given by k. θ13 k θ24 The torsional stiffness of the connecting shaft segments between gears 1 and 3, and between gears 2 and 4, respectively, is c. x13 c y13 c z13 c x24 c y24 c z24 The connecting shafts between gears 1 and 3, and between gears 2 and 4, are supported and damped along the x, y, and z axes, respectively. θ13 c θ24 The torsional damping of the connecting shaft segments between gears 1 and 3, and between gears 2 and 4, is x. A1 y A1 x A2 y A2 x B1 y B1 x B2 y B2 The displacements of bearings A1, A2, B1, and B2 along the x and y axes are respectively represented by F. bxA1 F bxA2 F bxB1 F bxB2 F byA1 F byA2 F byB1 F byB2 These represent the supporting forces of bearings A1, A2, B1, and B2 along the x and y axes, respectively. , , , , , , , ; x n1 x n2 F1 and F2 represent the vibration displacements on meshing lines 1 and 2, respectively, and the amplitudes of the external loads on meshing lines 1 and 2 are respectively. F1 is the ratio of T1 to the radius of gear 1, and F2 is the ratio of T3 to the radius of gear 3. The torsional displacement in the above formula is replaced by the meshing line displacement. ; Attached Figure Description

[0007] Figure 1 This is a flowchart of a nonlinear dynamics modeling method for a high-speed, heavy-load double helical gear transmission system. Figure 2 This is a diagram showing the meshing force analysis of a double helical gear; Figure 3 It is a bearing model for a double helical gear system; Figure 4 It is a multi-degree-of-freedom coupled vibration model of a high-speed, heavy-load double helical gear transmission system; Figure 5 This is the excitation frequency-displacement bifurcation diagram of the meshing line 1 of a double helical gear system; Figure 6 This is a bifurcation diagram of the excitation frequency and displacement of the meshing line of a double helical gear system. Implementation

[0008] Embodiments of the present invention will be described with reference to the accompanying drawings, which will be further described below. Figure 1 — Figure 6 The specific embodiments of the present invention will be described in detail below.

[0009] Reference Figure 1 The diagram shows a flowchart of a nonlinear dynamics modeling method for a high-speed, heavy-load double helical gear transmission system, which includes the following steps: Step (1): Calculate the time-varying meshing stiffness of the two helical gear pairs based on ISO standard 6336-1-2006. The expression is: ; Where: K(t) is the time-varying stiffness of the helical gear meshing, K m For the average meshing stiffness, K i ω represents the amplitude of the stiffness change. m The meshing frequency is equal to the input shaft rotational frequency multiplied by the number of teeth on the input gear; φ0 is the initial phase angle; ε α For the degree of overlap, K c This refers to the meshing stiffness of a single gear pair.

[0010] Step (2): Calculate the meshing force on the two meshing lines of the gears; let x1, y1, and z1 be the vibration displacements of gear 1 along the x, y, and z axes, respectively; let x2, y2, and z2 be the vibration displacements of gear 2 along the x, y, and z axes, respectively; let x3, y3, and z3 be the vibration displacements of gear 3 along the x, y, and z axes, respectively; let x4, y4, and z4 be the vibration displacements of gear 4 along the x, y, and z axes, respectively; and let θ1, θ2, θ3, and θ4 be the torsional displacements of gears 1, 2, 3, and 4, respectively; α n β represents the normal pressure angle and helix angle, e represents the static transmission error amplitude, e1(t) and e2(t) represent the static transmission errors on meshing line 1 and meshing line 2, respectively, and R b1 R b2 R b3 R b4 Let the pitch circle radii of gears 1, 2, 3, and 4 be respectively. Then, the displacements on the meshing lines 1 and 2 of the helical gear pair are calculated by the following formula: , With C m For the meshing damping of the helical gear pair, F m1 F m2 F is the meshing force on meshing line 1 and meshing line 2. mx1 F my1 F mz1 F represents the meshing force along the x, y, and z axes on line 1. mx2 F my2 F mz2 b represents the components of the meshing force along the x, y, and z axes on the line of meshing 2. m It is half of the tooth flank clearance, f(x) n1 f(x) n2 The nonlinear functions of tooth flank clearance for meshing lines 1 and 2 are respectively given. The meshing force and its components in each direction of the coordinate axis are calculated as follows: , ; Step (3): Introduce the radial clearance between the rolling element and the inner and outer raceways of the bearing, and calculate the bearing support force; the linear velocity at the contact point between the rolling element and the inner and outer rings is v. bi v bo , with Z b r is the number of rolling elements. bi r bo ω represents the inner and outer ring radii of the bearing. bi ω bo The angular velocities of the inner and outer rings about the axis of rotation are based on... Figure 3 The relationship shown in the bearing model is the angular velocity ω of the rolling element's revolution. bnand position angle θ i The calculation method for (t) is as follows: ; In the initial state, l is the center distance between the outer ring curvature centers, and the initial contact angle between the rolling element and both the inner and outer rings is γ. i After being deformed by force, the contact angle between the rolling element and the inner and outer rings becomes γ. ’ i The center distance between the outer ring curvature centers becomes l', δ is the rolling element deformation, c is the bearing radial clearance, x, y, and z are the bearing displacements along the x, y, and z axes, respectively, and F bx F by F bz These represent the x, y, and z components of the bearing support force, respectively. Let H(δ) denote the Heaviside function, and K... b Calculate the forces acting on the bearing in each direction of the coordinate system to determine the contact stiffness between the rolling elements and the raceway. , ; Step (4): Establish the nonlinear dynamic equations of the system, considering 24 degrees of freedom, namely: {x1,y1,z1,θ1,x2,y2,z2,θ2,x3,y3,z3,θ3,x4,y4,z4,θ4,x A1 ,y A1 ,x A2 ,y A2 ,x B1 ,y B1 ,x B2 ,y B2}; First, taking into account factors such as bearing nonlinear vibration, tooth backlash nonlinearity, and meshing error excitation, a dynamic model of the high-speed heavy-load double helical gear transmission system is established, such as... Figure 4 As shown; Figure 4 In the equation, m1, m2, m3, and m4 represent the masses of gear 1, gear 2, gear 3, and gear 4, respectively; I1, I2, I3, and I4 represent the moments of inertia of gear 1, gear 2, gear 3, and gear 4, respectively. A1 m A2 m B1 m B2 The masses of bearings A1, A2, B1, and B2 are respectively; k A1x k A2x k B1x k B2x k A1y k A2y k B1y k B2yFor the inner rings of bearings A1, A2, B1, and B2, along the x and y axes of the coordinate system, c A1x c A2x c B1x c B2x c A1y c A2y c B1y c B2y Damping is provided for the inner rings of each bearing along the x and y axes of the coordinate system; k x1 k x2 k x3 k x4 k y1 k y2 k y3 k y4 k z1 k z2 k z3 k z4 k represents the support stiffness of the shaft segment connecting the gear and bearing along the x, y, and z axes of the coordinate axis. θ1 k θ2 k θ3 k θ4 For the torsional stiffness of the shaft section connecting the gear and bearing; c x1 c x2 c x3 c x4 c y1 c y2 c y3 c y4 c z1 c z2 c z3 c z4 For the gear and bearing connecting shaft section, damping is provided along the x, y, and z axes of the coordinate system. θ1 c θ2 c θ3 c θ4 For the torsional damping of the shaft section connecting the gear and bearing; k x13 k x24 k y13 k y24 k z13 k z24 k represents the support stiffness of the intermediate shaft segment along the x, y, and z axes. θ13 k θ24 For the torsional stiffness of the intermediate shaft section, c x13 c x24 c y13 c y24 c z13 c z24 For the dimensionless support damping of the intermediate shaft segment along the x, y, and z axes, cθ13 c θ24 For the dimensionless torsional damping of the intermediate shaft segment, x A1 y A1 x A2 y A2 x B1 y B1 x B2 y B2 These represent the displacements of each bearing along the x and y axes of the coordinate system, respectively, F. bxA1 F bxA2 F bxB1 F bxB2 F byA1 F byA2 F byB1 F byB2 The dimensionless support forces of each bearing along the x and y axes are given; subsequently, the nonlinear dynamic equations of the high-speed heavy-duty double helical gear transmission system are listed: , , , , , , , ; x n1 x n2 F1 and F2 represent the vibration displacements on meshing lines 1 and 2, respectively, and the amplitudes of the external loads on meshing lines 1 and 2 are respectively. F1 is the ratio of the input torque T1 on gear 1 to the radius of gear 1, and F2 is the ratio of the input torque T3 on gear 3 to the radius of gear 3. The torsional displacement in the above formula is replaced by the meshing line displacement. ; The parameters of the double helical gear used in establishing the nonlinear dynamic model of high-speed heavy-load double helical gear are as follows: Table 1 Basic Parameters of Gears Number of teeth 22 41 22 41 Module (mm) 5 5 5 5 Pressure angle (°) 20 20 20 20 Helix angle (°) 22 22 22 22 Tooth width (mm) 60 60 60 60 At different external excitation frequencies ω e The fourth-order Runge-Kutta method is used to solve the equation system, and the solution is used to draw the bifurcation diagram of the external excitation frequency-vibration displacement of the system. Figure 5 This is a bifurcation diagram of the external excitation frequency and vibration displacement of the system's meshing line 1. Figure 6The diagram shows the bifurcation of external excitation frequency and vibration displacement along meshing line 1 of the system. As can be seen from the diagram, when other system parameters remain unchanged, the nonlinear response of the system exhibits richness when the external excitation frequency is changed. When the external excitation increases slowly, the nonlinear motion on meshing lines 1 and 2 of the system follows the following trend: from single-period motion to period-doubling bifurcation motion leading to chaos, then from chaotic motion state to quasi-periodic motion state, and finally returning to stability. The above description is merely a preferred embodiment of the invention and is not intended to limit the invention in any way. Any modifications, alterations, or equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the protection scope of the invention.

Claims

1. A nonlinear dynamic modeling method for a high-speed, heavy-load double helical gear transmission system, characterized in that, Includes the following steps: Step (1): Calculate the time-varying meshing stiffness of the double helical gears connected by the intermediate shaft segment; Step (2): Calculate the meshing force on the two meshing lines of the gear; x 1 、y 1 、z 1 represents gear 1 along the coordinate axis x Towards, y Xianghe z Towards vibration displacement, x 2 、y 2 、z 2 represents gear 2 along the coordinate axis x Towards, y Xianghe z Towards vibration displacement, x 3 、y 3 、z 3 represents gear 3 along the coordinate axis x Towards, y Xianghe z Towards vibration displacement, x 4 、y 4 、z 4 represents gear 4 along the coordinate axis x Towards, y Xianghe z Towards vibration displacement, θ 1. θ 2. θ 3. θ 4 represents the torsional displacement of gear 1, gear 2, gear 3, and gear 4, respectively. α n , β For the normal pressure angle and helix angle, e For static propagation error amplitude, e 1( t ), e 2( t The static transmission errors on meshing lines 1 and 2 are respectively. R b1 , R b2 , R b3 , R b4 Let the pitch circle radii of gears 1, 2, 3, and 4 be respectively. Then, the displacements on the meshing lines 1 and 2 of the helical gear pair are calculated by the following formula: by F m1 , F m2 The meshing force on meshing line 1 and meshing line 2, F mx1 , F my1 , F mz1 The meshing force along the coordinate axis on line of engagement 1 x Towards, y Xianghe z Component force, F mx2 , F my2 , F mz2 The meshing force along the coordinate axis on line of engagement 2 x Towards, y Xianghe z Component force, meshing force along the coordinate axis x Towards, y Xianghe z The calculation method for the axial force is as follows: Step (3): Introduce the radial clearance between the rolling elements and the inner and outer raceways of the bearing, and calculate the bearing support force; Step (4): Establish the nonlinear dynamic equations of the system, considering 24 degrees of freedom, namely: { x 1 ,y 1 ,z 1 ,θ 1 ,x 2 ,y 2 ,z 2 ,θ 2 ,x 3 ,y 3 ,z 3 ,θ 3 ,x 4 ,y 4 ,z 4 ,θ 4 ,x A1 ,y A1 ,x A2 ,y A2 ,x B1 ,y B1 ,x B2 ,y B2}； In the system dynamics equations, m 1. m 2. m 3. m 4 are gear 1, gear 2, gear 3, and gear 4 respectively. Gear 4 mass, I 1. I 2. I 3. I 4 represent the moments of inertia of gear 1, gear 2, gear 3, and gear 4, respectively. m A1 , m A2 , m B1 , m B2 Bearings A 1. A 2. B 1. B 2. Quality; k A1x 、k A1y 、k A2x 、k A2y 、k B1x 、k B1y 、k B2x 、k B2y Bearings A 1. A 2. B 1. B 2. Along the coordinate axes x Towards, y Towards support stiffness, c A1x 、c A1y 、c A2x 、c A2y 、c B1x 、c B1y 、c B2x 、c B2y Bearings A 1. A 2. B 1. B 2. Along the coordinate axes x Towards, y Towards support damping; k x1 、k y1 、k z1 、k x2 、k y2 、k z2 、k x3 、k y3 、k z3 、k x4 、k y4 、k z4 Gear 1 and bearing respectively A 1. Gear 2 and bearing B 1. Gear 3 and bearing A 2. Gear 4 and bearing B 2 Connecting shaft segments along the coordinate axes x Towards, y towards and z Towards support stiffness, k t1 、k t2 、k t3 、k t4 Gear 1 and bearing respectively A 1. Gear 2 and bearing B 1. Gear 3 and bearing A 2. Gear 4 and bearing B 2. Torsional stiffness of the connecting shaft segment; c x1 、c x2 、c x3 、c x4 、c y1 、c y2 、c y3 、c y4 、c z1 、c z2 、c z3 、c z4 Gear 1 and bearing respectively A 1. Gear 2 and bearing B 1. Gear 3 and bearing A 2. Gear 4 and bearing B 2 Connecting shaft segments along the coordinate axes x Towards, y towards and z Towards support damping, c t1 、 c t2 、c t3 、c t4 Gear 1 and bearing respectively A 1. Gear 2 and bearing B 1. Gear 3 and bearing A 2. Gear 4 and bearing B 2. Torsional damping of the connecting shaft section; k x13 、k y13 、k z13 、k x24 、k y24 、k z24 The connecting shaft segments between gears 1 and 3, and between gears 2 and 4, are located along the coordinate axes. x Towards, y Xianghe z Towards support stiffness, k t13 、k t24 These refer to the torsional stiffness of the connecting shaft segments between gears 1 and 3, and between gears 2 and 4. c x13 、c y13 、c z13 、c x24 、c y24 、c z24 The connecting shaft segments between gears 1 and 3, and between gears 2 and 4, are located along the coordinate axes. x Towards, y Xianghe z Towards support damping, c t13 、c t24 These are the torsional damping components of the connecting shafts between gears 1 and 3, and between gears 2 and 4. x A1 , y A1 , x A2 , y A2 , x B1 , y B1 , x B2 , y B2 Bearings A 1. A 2. B 1. B 2. Along the coordinate axes x Towards, y Displacement, F bxA1 、F bxA2 、F bxB1 、F bxB2 、F byA1 、F byA2 、F byB1 、F byB2 Bearings A 1. A 2. B 1. B 2. Along the coordinate axes x Towards, y Towards supporting force; x n1 , x n2 These represent the vibration displacements on meshing lines 1 and 2, respectively. F 1. F 2 represents the amplitude of the external load on meshing line 1 and meshing line 2, respectively. F 1 is a value T The ratio of the radius of gear 1 to the radius of gear 1. F 2 values T The ratio of 3 to the radius of gear 3 is used to replace the torsional displacement in the above formula with the displacement of the meshing line: 。

Images