A non-orthogonal bias gear split transmission system dynamics modeling method

Abstract

The application discloses a non-orthogonal bias gear split transmission system dynamics modeling method, which comprises the following steps: (1) establishing a bearing dynamics model; (2) determining the relative displacement and meshing force on the meshing line of the gear pair; (3) establishing a non-orthogonal bias gear split transmission system dynamics equation, and solving the vibration displacement of each part along the meshing line direction by using a numerical method; the non-orthogonal bias gear split transmission system dynamics modeling related technical blank is filled up.

Description

Technical Field

[0001] This invention relates to the field of gear dynamics technology, and in particular to a dynamic modeling method for a non-orthogonal biased gear splitter transmission system. Background Technology

[0002] Face gear pairs, composed of cylindrical gears and conjugate face gears, typically feature compact structure, good flow splitting effect, high load-bearing capacity, and low noise, making them promising for development in the aerospace field. With increasingly stringent requirements for vibration and noise performance, research on the dynamic characteristics of face gear transmission systems is of great significance in determining the mechanisms of vibration and noise generation and providing a theoretical basis for vibration and noise suppression. Furthermore, research on the nonlinear dynamics of face gear systems is also crucial for the normal operation and fault prediction of mechanical equipment. Current technology lacks research on the vibration characteristics of non-orthogonal offset face gear transmission systems.

[0003] To address the aforementioned issues, this invention proposes a dynamic modeling method for a non-orthogonal offset gear splitter transmission system. This method introduces bearings into the system and derives the relative displacement, meshing stiffness, meshing damping, and meshing force of each gear pair along the meshing line. This allows for the establishment of a multi-degree-of-freedom coupled dynamic model of the non-orthogonal offset gear splitter transmission system to analyze its nonlinear dynamic characteristics. This provides technical support for face gear transmission systems, promotes engineering applications, and can generate significant socio-economic benefits. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill the gaps in related technologies, this invention provides a dynamic modeling method for a non-orthogonal offset gear splitter transmission system. This method introduces bearing deformation into the system, derives the relative displacement of each gear pair along the meshing line, and obtains the meshing stiffness, meshing damping, and meshing force of the gear pair, thereby establishing a dynamic model of a non-orthogonal offset gear multi-degree-of-freedom coupled system to analyze its nonlinear dynamic characteristics.

[0005] The technical solution adopted by this invention to solve its technical problem is as follows: A dynamic modeling method for a non-orthogonal biased gear splitter transmission system, characterized by comprising the following steps:

[0006] 1. A dynamic modeling method for a non-orthogonal biased gear splitter transmission system, characterized by comprising the following steps:

[0007] Step (1): Establish a bearing dynamics model; solve for the deformation of the rolling elements after the bearing is subjected to force, and combine the Hertzian contact theory to obtain the interaction force between a single rolling element and the inner and outer rings of the bearing. Then, by superposition, obtain the force generated by the deformation of the rolling elements in the entire bearing.

[0008] Step (2): Determine the relative displacement and meshing force on the gear pair's meshing line:

[0009] ;

[0010] in, This refers to the relative displacement along the meshing line of the gear pair. , Let be the vibration displacement of spur gear 1 along the coordinate axis. The pitch circle radius of the spur gear is 1. The rotational angular displacement of spur gear 1 about its axis of rotation. The perpendicular distance from the axis of rotation of the non-orthogonal offset gear 2 to the midpoint of the tooth width. , , The vibration displacement of the non-orthogonal offset surface gear 2 along the coordinate axis is represented by... The rotational angular displacement of the non-orthogonal offset surface gear 2 about its rotational axis. The offset angle satisfies , For offset distance, The angle between the axes, For pressure angle, This refers to the comprehensive transmission error of non-orthogonal offset surface gear pairs;

[0011] Determine the meshing stiffness of the gear pair Meshing damping Derivation of meshing force expression:

[0012] ;

[0013] in, The backlash function between spur gear 1 and non-orthogonal offset gear 2 is given by decomposing the meshing force into the coordinate system of each gear:

[0014] ;

[0015] in, The meshing force between spur gear 1 and non-orthogonal offset gear 2 is... , Meshing force The y and z axis components acting on spur gear 1 , , Meshing force The x, y, and z axis components acting on the non-orthogonal offset surface gear 2;

[0016] Step (3): Establish the dynamic equations of the non-orthogonal biased gear splitter transmission system, considering the following degrees of freedom:

[0017] ;

[0018] in, , , , , , , , , , , , , , , , , , , , , , The vibration displacements of bearings 11, 12, 21, 22, 31, 32, 41, and 42 along the x, y, and z axes are respectively. , , , , , These are the rotational angular displacements of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 about their respective axes of rotation. , , , , , , , , , , , , , , , , The vibration displacements of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 along the x, y, and z axes are respectively.

[0019] ,

[0020] ,

[0021] ,

[0022] ,

[0023] ,

[0024] ,

[0025] ,

[0026] ,

[0027] ,

[0028] ,

[0029] ,

[0030] ,

[0031] ,

[0032] ;

[0033] in, , , , , , , , The masses of bearings 11, 12, 21, 22, 31, 32, 41, and 42 are respectively. , , , , , , , , , , , , , , , , , , , , , These are the support damping components of bearings 11, 12, 21, 22, 31, 32, 41, and 42 along the x, y, and z axes, respectively. , , , , , , , , , , , , , , , , , , , , , These are the bearing force components along the x, y, and z axes for bearings 11, 12, 21, 22, 31, 32, 41, and 42, respectively. , , , , , The masses of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 are respectively. , , , , , The moments of inertia are those of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6, respectively. For input torque; , , , These are the torsional stiffness and torsional damping of drive shafts 2 and 3, respectively. This is the load torque; , , , These are the meshing forces between spur gear 1 and non-orthogonal offset surface gear 2, spur gear 1 and non-orthogonal offset surface gear 3, helical gears 4 and 6, and helical gears 5 and 6, respectively. , These are the differences in meshing angles between helical gears 4 and 6, and between helical gears 5 and 6, respectively. The vertical distance from the rotation axis of the non-orthogonal offset gear 3 to the midpoint of the tooth width; , , These are the pitch circle radii of helical gears 4, 5, and 6, respectively.

[0034] The responses of each part of the system can be obtained by solving the above set of vibration differential equations using numerical methods.

[0035] Compared with the prior art, the beneficial effects of this invention are: it proposes a dynamic modeling method that can more accurately study the nonlinear dynamic characteristics of nonorthogonal bias gear splitting transmission systems, improve the gear dynamics theory system, and provide strong support for gear vibration reduction, noise reduction, and other aspects. Attached Figure Description

[0036] Figure 1 Flowchart of dynamic modeling method for non-orthogonal biased gear splitter transmission system;

[0037] Figure 2 Schematic diagram of a non-orthogonal offset surface gear transmission;

[0038] Figure 3 A two-dimensional dynamic model of a non-orthogonal biased gear splitter transmission system;

[0039] Figure 4 A three-dimensional dynamic model of a non-orthogonal biased gear splitter transmission system;

[0040] Figure 5 These are the vibration displacement curves of various parts of the system. Detailed Implementation

[0041] Embodiments of the present invention will be described with reference to the accompanying drawings, which will be further described below. Figure 1 — Figure 5 The specific embodiments of the present invention will be described in detail below.

[0042] like Figure 1 The diagram shows a flowchart of a dynamic modeling method for a non-orthogonal bias gear splitter transmission system, which includes the following steps:

[0043] Step (1): Establish a bearing dynamics model;

[0044] Determining the deformation of rolling elements under bearing load: A rolling bearing generally consists of four parts: rolling elements, inner ring, outer ring, and cage. In this transmission system, it is assumed that the outer ring is fixed to the bearing housing, therefore its linear velocity is zero. It is assumed that the inner ring is rigidly connected to the drive shaft and rotates synchronously with it, so its angular velocity is equal to the angular velocity of the drive shaft. It is assumed that the rolling elements are uniformly distributed in the grooves, and that during operation, the rolling elements and raceways maintain a pure rolling state, neglecting centrifugal force and helical torque. The linear velocities at the contact points between the rolling elements and the inner and outer rings can be expressed as... , where subscript Represents the inner and outer rings of the bearing. These represent the linear velocities at the contact points of the inner and outer rings, respectively. These are the inner and outer radii, respectively. These represent the inner and outer ring angular velocities, with the outer ring angular velocity being zero; the bearing's... The rotation angle of each rolling element at any given moment It can be represented as:

[0045] ;

[0046] in, This refers to the number of rolling elements; vibrations occur during system operation, causing deformation of the bearing rolling elements. The distance between the center of curvature of the inner and outer raceways after the rolling elements deform is [not specified]. and contact angle It can be represented as:

[0047] ;

[0048] in, Let x, y, and z be the center distance of curvature of the inner and outer raceways before deformation, and x, y, and z be the vibration displacements of the rolling element along the coordinate axes. The deformation of the rolling element can be expressed as... , For bearing clearance;

[0049] Using Hertzian contact theory, the interaction forces between a single rolling element and the inner and outer rings of the bearing are obtained:

[0050] ;

[0051] in, It is an axial force. It is a radial force. For the bearing support stiffness, n = 10 / 9. For the Heaviside function;

[0052] The forces generated by the deformation of the rolling elements in the entire bearing are obtained by superposition. Decompose it into the corresponding coordinate system:

[0053] , , ;

[0054] in, The bearing force is along the x-direction. The bearing force is along the y-direction. The bearing force is along the z-direction;

[0055] Step (2): Determine the relative displacement, meshing stiffness, meshing damping, and meshing force of the non-orthogonal offset surface gear pair meshing line and the helical gear pair meshing line:

[0056] ,

[0057] ,

[0058] ,

[0059] ;

[0060] in, The pitch circle radius of the spur gear is 1. The rotational angular displacement of spur gear 1 about its axis of rotation; , These are the perpendicular distances from the rotation axis of non-orthogonal offset gears 2 and 3 to the midpoint of the tooth width, respectively. , , These are the pitch circle radii of helical gears 4, 5, and 6, respectively. , , , , , , , , , , , , , , , , The vibration displacements of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 along the x, y, and z axes are respectively. , , , , , These are the rotational angular displacements of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 about their respective axes of rotation. The offset angle satisfies , This is the offset distance; The angle between the axes; The pressure angle; , These are the differences in meshing angles between helical gears 4 and 6, and between helical gears 5 and 6, respectively. , , , These are the combined transmission errors between spur gear 1 and non-orthogonal offset surface gear 2, spur gear 1 and non-orthogonal offset surface gear 3, helical gears 4 and 6, and helical gears 5 and 6, respectively.

[0061] Determine the meshing stiffness, meshing damping, and meshing force of non-orthogonal offset gear pairs and helical gear pairs:

[0062] ,

[0063] ,

[0064] ,

[0065] ;

[0066] in, The meshing damping ratio; , , , These are the meshing stiffnesses of the non-orthogonal offset surface gear pair 12, the non-orthogonal offset surface gear pair 12, the helical gear pair 46, and the helical gear pair 56, respectively. , , , These are the average meshing stiffnesses of the non-orthogonal offset surface gear pair 12, the helical gear pair 46, and the helical gear pair 56, respectively. , , , These are the meshing stiffness fluctuation amplitudes of non-orthogonal offset surface gear pair 12, non-orthogonal offset surface gear pair 12, helical gear pair 46, and helical gear pair 56, respectively. , , , These are the meshing frequencies of the non-orthogonal offset surface gear pair 12, the non-orthogonal offset surface gear pair 12, the helical gear pair 46, and the helical gear pair 56, respectively. , , , These are the phase differences of non-orthogonal biased surface gear pair 12, non-orthogonal biased surface gear pair 12, helical gear pair 46, and helical gear pair 56, respectively. , , , These are the meshing damping of non-orthogonal offset surface gear pair 12, non-orthogonal offset surface gear pair 12, helical gear pair 46, and helical gear pair 56, respectively. , , , These are the backlash functions for non-orthogonal offset surface gear pair 12, non-orthogonal offset surface gear pair 12, helical gear pair 46, and helical gear pair 56, respectively. , , , These are the meshing forces of the non-orthogonal offset surface gear pair 12, the non-orthogonal offset surface gear pair 12, the helical gear pair 46, and the helical gear pair 56, respectively. , , , , , The moments of inertia of spur gear 1, non-orthogonal offset surface gear 2, non-orthogonal offset surface gear 3, helical gear 4, helical gear 5, and helical gear 6 are respectively.

[0067] Step (3): Establish the dynamic equations of the non-orthogonal biased gear splitter transmission system:

[0068] ,

[0069] ,

[0070] ,

[0071] ,

[0072] ,

[0073] ,

[0074] ,

[0075] ,

[0076] ,

[0077] ,

[0078] ,

[0079] ,

[0080] ,

[0081] ;

[0082] in, , , , , , , , The masses of bearings 11, 12, 21, 22, 31, 32, 41, and 42 are respectively. , , , , , , , , , , , , , , , , , , , , , These are the support damping components of bearings 11, 12, 21, 22, 31, 32, 41, and 42 along the x, y, and z axes, respectively. , , , , , , , , , , , , , , , , , , , , , The vibration displacements of bearings 11, 12, 21, 22, 31, 32, 41, and 42 along the x, y, and z axes are respectively. , , , , , , , , , , , , , , , , , , , , , These are the bearing force components along the x, y, and z axes for bearings 11, 12, 21, 22, 31, 32, 41, and 42, respectively. , , , , , The masses of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 are respectively. For input torque; , , , These are the torsional stiffness and torsional damping of drive shafts 2 and 3, respectively. This is the load torque;

[0083] Solving the above set of vibration differential equations yields the vibration displacements of each part of the system;

[0084] In this example, the gear material parameters are shown in Table 1:

[0085] Table 1 Basic Parameters of Gears

[0086] <![CDATA[Normal modulus m n / mm]]> 4 4 4 4 4 4 Number of teeth z 29 85 85 37 37 85 <![CDATA[Normal pressure angle α n / °]]> 25 25 25 25 25 25 Helix angle β / ° — — — 10 10 10 Gear mass m / kg 1.8 18.2 18.2 3.5 3.5 16.7

[0087] Using the method described above, the meshing stiffness, meshing damping, and dynamic meshing force of each gear pair are calculated through programming. Then, a dynamic model of the non-orthogonal offset gear splitter transmission system is established, as shown below. Figure 3 and Figure 4 As shown;

[0088] Figure 5 To obtain the vibration displacement curve, where Figure 5 In the middle (a), the displacement is... The vibration displacement curve, Figure 5 (b) represents displacement The vibration displacement curve, Figure 5 (c) represents displacement The vibration displacement curve, Figure 5 In the middle (d), the displacement is... The vibration displacement curve, in which , , , , , , .in, It is half of the tooth flank clearance between spur gear 1 and non-orthogonal offset gear 2. The system's inherent frequency, For equivalent quality.

[0089] The above description is merely a preferred embodiment of the invention and does not constitute any limitation on the invention. 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 method of dynamic modeling of a non-orthogonal bias gear split driveline system, characterized by, Includes the following steps: Step (1): Establish a bearing dynamics model; solve for the deformation of the rolling elements after the bearing is subjected to force, and combine the Hertzian contact theory to obtain the interaction force between a single rolling element and the inner and outer rings of the bearing. Then, by superposition, obtain the force generated by the deformation of the rolling elements in the entire bearing. Step (2): Determine the relative displacement and meshing force on the gear pair's meshing line: ； in, This refers to the relative displacement along the meshing line of the gear pair. , Let be the vibration displacement of spur gear 1 along the coordinate axis. The pitch circle radius of the spur gear is 1. The rotational angular displacement of spur gear 1 about its axis of rotation. The perpendicular distance from the axis of rotation of the non-orthogonal offset gear 2 to the midpoint of the tooth width. , , The vibration displacement of the non-orthogonal offset surface gear 2 along the coordinate axis is represented by... The rotational angular displacement of the non-orthogonal offset surface gear 2 about its rotational axis. The offset angle satisfies , For offset distance, The angle between the axes, For pressure angle, This refers to the comprehensive transmission error of non-orthogonal offset surface gear pairs; Determine the meshing stiffness of the gear pair Meshing damping Derivation of meshing force expression: ； in, The backlash function between spur gear 1 and non-orthogonal offset gear 2 is given by decomposing the meshing force into the coordinate system of each gear: ； in, The meshing force between spur gear 1 and non-orthogonal offset gear 2 is... , Meshing force The y and z axis components acting on spur gear 1 , , Meshing force The x, y, and z axis components acting on the non-orthogonal offset surface gear 2; Step (3): Establish the dynamic equations of the non-orthogonal biased gear splitter transmission system, considering the following degrees of freedom: ； in, , , , These represent the vibration displacements of bearings 11 and 12 along the y and z axes, respectively. , , , , , , , , , , , , , , , , , These are the vibration displacements of bearings 21, 22, 31, 32, 41, and 42 along the x, y, and z axes, respectively. , , , , , These are the rotational angular displacements of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 about their respective axes of rotation. , Let be the vibration displacement of spur gear 1 along the y and z axes. , , , , , , , , , , , , , , The vibration displacements of non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 along the x, y, and z axes are respectively. ， ， ， ， ， ， ， ， ， ， ， ， ， ； in, , , , , , , , The masses of bearings 11, 12, 21, 22, 31, 32, 41, and 42 are respectively. , , , These are the support damping elements for bearings 11 and 12 along the y and z axes, respectively. , , , , , , , , , , , , , , , , , These are the support damping components of bearings 21, 22, 31, 32, 41, and 42 along the x, y, and z axes, respectively. , , , These are the bearing force components of bearings 11 and 12 along the y and z axes, respectively. , , , , , , , , , , , , , , , , , These are the bearing force components along the x, y, and z axes for bearings 21, 22, 31, 32, 41, and 42, respectively. , , , , , The masses of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6 are respectively. , , , , , The moments of inertia are those of spur gear 1, non-orthogonal offset surface gears 2 and 3, and helical gears 4, 5, and 6, respectively. For input torque; , , , These are the torsional stiffness and torsional damping of drive shafts 2 and 3, respectively. This is the load torque; , , , These are the meshing forces between spur gear 1 and non-orthogonal offset surface gear 2, spur gear 1 and non-orthogonal offset surface gear 3, helical gears 4 and 6, and helical gears 5 and 6, respectively. , These are the differences in meshing angles between helical gears 4 and 6, and between helical gears 5 and 6, respectively. The vertical distance from the rotation axis of the non-orthogonal offset gear 3 to the midpoint of the tooth width; , , These are the pitch circle radii of helical gears 4, 5, and 6, respectively. The responses of each part of the system can be obtained by solving the above set of vibration differential equations using numerical methods.

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