Dynamic modeling method for coupled crack and modification of helical gear

Abstract

The application discloses a helical gear crack and modification coupling damage dynamics modeling method, which comprises the following steps: (1) slicing a helical gear along the tooth width direction to divide the helical gear into a plurality of equal-width lamellas; (2) establishing a tooth root crack model and calculating the time-varying meshing stiffness of the helical gear with cracks based on the cumulative potential energy method; (3) establishing a tooth profile modification model and calculating the time-varying meshing stiffness of the tooth root crack and modification coupling; (4) determining the relative displacement, meshing damping and meshing force on the meshing line of the helical gear pair; and (5) establishing a helical gear pair dynamics equation, substituting the time-varying meshing stiffness calculated by the tooth root crack and tooth profile modification coupling into the dynamics equation to obtain the vibration displacement of the helical gear pair.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of gear dynamics, in particular to a helical gear crack and modification coupling damage dynamics modeling method. BACKGROUND

[0002] The helical gear has the characteristics of stable transmission ratio, compact structure, stable transmission and high transmission efficiency, so it is widely used in complex mechanical equipment in various fields such as rail transportation, ship, aerospace, etc. It is the most common mechanical transmission method. Due to the complex operating state of the gear transmission system, and under the influence of internal dynamic excitation and external load excitation and other factors, the tooth root bears periodic alternating stress, which is easy to cause local crack, seriously affects the dynamic performance and service life of the gear transmission system, and the generation of gear crack will affect the safety and stability of the mechanical equipment operation, and even cause serious safety accidents. Therefore, the related research of gear transmission system dynamics has great significance and engineering application value for the normal operation and safety of mechanical equipment. The prior art lacks research on the crack failure and modification coupling mechanism and vibration characteristics of the helical gear.

[0003] In order to solve the above problems, the present application provides a helical gear crack and modification coupling damage dynamics modeling method, which disperses the helical gear along the tooth width direction into N The same number of thin straight gears with the same width, respectively establishes the tooth width through crack and non-through crack, tooth profile modification model, deduces the time-varying meshing stiffness of the tooth root crack and modification coupling of the thin gear, and then integrates the time-varying meshing stiffness of the helical gear along the tooth width to establish the helical gear pair dynamics model and analyze the dynamics characteristics of the helical gear pair. Provide more accurate technical support for modification gear fault diagnosis, promote the development of engineering technology, and can produce great social and economic benefits. SUMMARY

[0004] In order to overcome the shortcomings of the prior art and fill the gap in related technology, the present application provides a helical gear crack and modification coupling damage dynamics modeling method, which disperses the helical gear along the tooth width direction into N The same number of thin straight gears with the same width, respectively establishes the tooth width through crack and non-through crack, tooth profile modification model, deduces the time-varying meshing stiffness of the tooth root crack and modification coupling of the thin gear, and then integrates the time-varying meshing stiffness of the helical gear along the tooth width to establish the helical gear pair dynamics model and analyze the dynamics characteristics of the helical gear pair.

[0005] The technical solution adopted by the present application to solve its technical problems is as follows: a helical gear crack and modification coupling damage dynamics modeling method, characterized by comprising the following steps:

[0006] Step (1): Calculate the time-varying mesh stiffness of helical gear pair by using the slice method, divide the helical gear into N equal-width slice gears along the tooth width direction; Calculate the time-varying mesh stiffness of the slice spur gears in the meshing state according to the potential energy method, shear potential energy , bending potential energy , radial compression potential energy , and Hertz potential energy ;

[0007] wherein, , is the radial force and tangential force, , represents the force of the meshing tooth pair at the contact point, , , and respectively represent the shear modulus, Young's modulus, tooth width and Poisson's ratio, , is the effective section moment of inertia and the section area at the fixed end of the cantilever beam, so the stiffness of the helical gear can be obtained by integrating the stiffness of each slice spur gear along the tooth width;

[0008] Step (2): Establish a tooth root crack model, the tooth root crack extends along one side of the tooth width and the other side is a through crack: , is the tooth width, is the initial crack depth, is the terminal crack depth;

[0009] and vice versa for the non-through crack: , is the effective crack length along the tooth width direction;

[0010] At this time, the effective section moment of inertia and the section area are respectively:

[0011] , ; and

[0012] wherein, , is the half tooth height at the tooth root, is the crack curve varying with the tooth width, is the crack propagation angle;

[0013] The bending potential energy of the slice gear without cracks can be expressed as:

[0014] ; and

[0015] wherein, This is the distance from the point of engagement to the tooth root. The distance from the meshing point to the center line of the gear. Distance from tooth root x The effective moment of inertia at the section is calculated using the following formula:

[0016] ,

[0017] ,

[0018] ,

[0019] ;

[0020] in, Involute distance from tooth root The distance from the center line of the gear The radius of the base circle;

[0021] The expressions for the bending stiffness, shear stiffness, compressive stiffness, Hertzian contact stiffness, and matrix elastic deformation stiffness of a crack-free helical gear are as follows:

[0022] ,

[0023] ,

[0024] ,

[0025] ,

[0026] ;

[0027] in, , Contact wire length The projection, Number of slices along the tooth width The distance between the point where the contact load passes through the intersection of the tooth centerline and the tooth root arc. , The length of the arc at the tooth root. It is half of the central angle corresponding to the root arc. , For the hub radius, The radius of the tooth root circle;

[0028] Bending stiffness when tooth root cracks are present Shear stiffness and radial compressive stiffness The change will affect the effective section moment of inertia containing the tooth root crack. and cross-sectional area Substitute the potential energy formula to get the changed stiffness of the crack , single tooth engagement stiffness and multi-tooth engagement stiffness Formula:

[0029] ,

[0030] ,

[0031] ;

[0032] Step (3): Establish the tooth profile modification model, and calculate the variable engagement stiffness of the helical gear pair when the tooth root crack and modification are coupled. The formula for calculating the variable engagement stiffness of the crack and modification coupling is wherein is the error of the helical gear pair, is the engagement force;

[0033] Step (4): Determine the relative displacement and engagement damping on the meshing line of the helical gear pair as follows:

[0034] ,

[0035] ;

[0036] The engagement force is , and the engagement damping force is ; is the time-varying engagement stiffness, is the time-varying engagement damping;

[0037] Step (5): Establish the dynamic equation of the helical gear pair when the crack and modification are coupled:

[0038] ;

[0039] wherein , is the mass of the driving wheel and the driven wheel, , is the moment of inertia, , is the input and load torque of the system, respectively, , is the support stiffness, , is the torsional stiffness, , is the support damping; the time-varying engagement stiffness calculated by the tooth root crack and modification coupling is substituted into the dynamic equation to obtain the vibration displacement of the helical gear pair.

[0040] Compared with the prior art, the beneficial effects of the present application are that the proposed dynamic modeling method can more accurately reflect the dynamic characteristics of the gear transmission system under the coupling condition of the tooth root crack and gear modification, perfect the gear dynamics theory system, and provide strong support for gear vibration reduction, noise reduction, gear modification, fault diagnosis and the like. BRIEF DESCRIPTION OF DRAWINGS

[0041] Figure 1 is a helical gear crack and modification coupling damage dynamic modeling method flow chart;

[0042] Figure 2 is a helical gear time-varying meshing stiffness calculation model;

[0043] Figure 3 is a helical gear modification and non-penetrating crack model;

[0044] Figure 4 is a helical gear modification and penetrating crack model;

[0045] Figure 5 is a normal helical gear and a modified helical gear with tooth root crack time-varying meshing stiffness curve;

[0046] Figure 6 is a helical gear transmission system dynamic model;

[0047] Figure 7 is a normal helical gear and a modified helical gear with tooth root crack vibration displacement curve. DETAILED DESCRIPTION

[0048] Embodiments of the present application are described with reference to the accompanying drawings, which are combined below Figure 1 — Figure 7 The specific embodiments of the present application are described in detail.

[0049] As Figure 1 shown is a helical gear crack and modification coupling damage dynamic modeling method flow chart, including the following steps:

[0050] Step (1): the time-varying meshing stiffness of the helical gear pair is calculated by using the slicing method, and the helical gear is divided into N equal-width thin slice gears along the tooth width direction; the thin slice spur gear is simplified as a cantilever beam structure, and the time-varying meshing stiffness of the thin slice spur gear in the meshing state is calculated according to the potential energy method;

[0051] shear potential energy , bending potential energy , radial compression potential energy , and Hertz potential energy ;

[0052] wherein, represents the force of the meshing tooth pair on the contact point; , These are radial force and tangential force. , , and These represent shear modulus, Young's modulus, tooth width, and Poisson's ratio, respectively. , For distance from the fixed end of the cantilever beam The effective moment of inertia and cross-sectional area at the point; the stiffness of the helical gear can be obtained by integrating the stiffness of each thin-plate spur gear along the tooth width;

[0053] Step (2): Establish a tooth root crack model. The tooth root crack extends along one side of the tooth width, while the other side is a through crack. , For tooth width, This represents the initial crack depth. To terminate the crack depth;

[0054] Conversely, a non-penetrating crack is considered: , The effective crack length along the tooth width direction;

[0055] At this time, the effective cross-sectional moment of inertia and cross-sectional area They are respectively:

[0056] , ;

[0057] in, , It is half the tooth height at the root. The crack curve varies with tooth width. The crack propagation angle;

[0058] Step (3): Derive the calculation method for the time-varying meshing stiffness of crack-free helical gears. The bending potential energy of crack-free thin-plate gears can be expressed as:

[0059] ;

[0060] in, This is the distance from the point of engagement to the tooth root. The distance from the meshing point to the center line of the gear. Distance from tooth root The effective moment of inertia at the section is calculated using the following formula:

[0061] ,

[0062] ,

[0063] ,

[0064] ;

[0065] wherein, is the distance from the involute root to the gear center line, is the distance from the involute root to the gear center line, is the half angle of the corresponding center of the base circle arc, is the meshing angle of any point on the gear tooth profile is the base circle radius, about the driving wheel rotation angle The relationship is:

[0066] ,

[0067] ,

[0068] ;

[0069] wherein, , are the number of teeth of the driving and driven wheels, respectively, is the pressure angle of the helical gear pitch circle;

[0070] The expression of the bending stiffness, shear stiffness, compression stiffness, Hertz contact stiffness, and substrate elastic deformation stiffness of the crack-free helical gear is:

[0071] ,

[0072] ,

[0073] ,

[0074] ,

[0075] ;

[0076] wherein, , , is the projection of the contact line length , is the number of slices along the tooth width, is the distance between the intersection of the contact point load and the tooth center line and the tooth root arc ; is the tooth root arc length, is half the corresponding center angle of the tooth root arc, , is the hub radius, is the tooth root radius, , , , The formula for calculation is:

[0077] ;

[0078] Bending stiffness when tooth root cracks are present Shear stiffness and radial compressive stiffness The change will affect the effective section moment of inertia containing the tooth root crack. and cross-sectional area Substituting into the potential energy formula, we obtain the stiffness change of the crack. Single tooth meshing stiffness and multi-tooth meshing stiffness formula:

[0079] ,

[0080] ,

[0081] ;

[0082] Step (4): Establish the tooth profile modification model, calculate the time-varying meshing stiffness of the helical gear tooth root crack and modification coupling, and the tooth profile modification normalization parameter is: , Among them, the maximum amount of reshaping =0.02 Maximum shaping length =0.6 , The end face module of the helical gear; the formula for calculating the meshing stiffness of the crack and profile coupling is: ,in, For the error of the helical gear pair, ( ; ,in, , (Representing the driving and driven gears respectively, 1 and 2 represent the first and second pairs of teeth respectively.) Tooth profile errors caused by profile modification. It is the meshing force;

[0083] Step (5): Determine the relative displacement and meshing damping on the meshing line of the helical gear pair as follows:

[0084] ;

[0085] in, , , The driven wheel position angle and end face pressure angle are given. The base circle helix angle;

[0086] ;

[0087] The engagement force is , and the engagement damping force is ; is the engagement stiffness, is the engagement damping;

[0088] Step (6): the helical gear pair dynamics equation is:

[0089] ;

[0090] wherein, , is the mass of the driving wheel and the driven wheel, , is the moment of inertia, , is the input and load torque of the system respectively, , is the support stiffness, , is the torsional stiffness, , is the support damping; the time-varying engagement stiffness obtained by coupling the tooth root crack and the tooth profile modification is substituted into the dynamics equation, and the fourth-order method is used to solve the vibration displacement of the helical gear pair.

[0091] In the examples, the material parameters of the helical gear pair are shown in Table 1:

[0092] Table 1 Gear parameters of the helical gear pair

[0093]

[0094] After determining the gear pair parameters, the time-varying engagement stiffness of the helical gear pair is calculated by programming and calculation using the above-mentioned method, and then the dynamics model of the helical gear transmission system is established as shown in Figure 6 .

[0095] Figure 5 The four time-varying engagement stiffness curves obtained by solving are shown in the following table, wherein the profile modification parameters are =0.6, =0.5. It can be seen that when only modification is performed, the time-varying engagement stiffness curve is reduced and becomes smoother, which is of great significance to the vibration and noise reduction of the gear pair, and the time-varying engagement stiffness curve of the through crack after modification changes more obviously when the crack tooth participates in the engagement; Figure 7 The vibration displacement curves obtained by solving are shown in the following table, wherein Figure 7 (a) is the vibration displacement curve without crack and modification, Figure 7 (b) is the vibration displacement curve of modification without crack,Figure 7 (c) the vibration displacement curve of the modified non-penetrating crack, Figure 7 (d) the vibration displacement curve of the modified penetrating crack, it can be seen that the modification reduces the vibration displacement of the transmission system, the deepening of the crack makes the vibration displacement curve present periodic jumps, the amplitude increases continuously, the vibration impact of the gear transmission system becomes more and more obvious, and the vibration displacement of the penetrating crack is larger than that of the non-penetrating crack, and the vibration impact on the gear transmission system is more prominent.

[0096] The above is only a preferred embodiment of the application, and does not limit the application in any way. Any modification, change and equivalent variation of the above embodiment according to the essence of the application still falls within the protection scope of the application technology.

Claims

1. A method for modeling the coupled damage dynamics of helical gear cracks and profile modification, characterized in that, Includes the following steps: Step (1): The time-varying meshing stiffness of the helical gear pair is calculated using the slicing method. The helical gear is divided along the tooth width direction into... N A thin, uniformly wide gear; The time-varying meshing stiffness and shear potential energy of a thin-plate spur gear in meshing state are calculated using the potential energy method. Bending potential energy Radial compressive potential energy Hertzian potential energy ; in, , These are radial force and tangential force. , This indicates the force exerted by the meshing teeth at the point of contact. , , and These represent shear modulus, Young's modulus, tooth width, and Poisson's ratio, respectively. , For the distance from the fixed end of the cantilever beam Given the effective moment of inertia and cross-sectional area, the stiffness of the helical gear can be obtained by integrating the stiffness of each thin-plate spur gear along the tooth width. Step (2): Establish a tooth root crack model. The tooth root crack extends along one side of the tooth width, while the other side is a through crack. , For tooth width, This represents the initial crack depth. To terminate the crack depth; Conversely, a non-penetrating crack is considered: , The effective crack length along the tooth width direction; At this time, the effective cross-sectional moment of inertia and cross-sectional area They are respectively: ， ； in, , It is half the tooth height at the root. The crack curve varies with tooth width. The crack propagation angle; The bending potential energy of a crack-free thin-plate gear can be expressed as: ； in, This is the distance from the point of engagement to the tooth root. The distance from the meshing point to the center line of the gear. Distance from tooth root x The effective moment of inertia at the section is calculated using the following formula: ， ， ， ； in, Involute distance from tooth root The distance from the center line of the gear The radius of the base circle; The expressions for the bending stiffness, shear stiffness, compressive stiffness, Hertzian contact stiffness, and matrix elastic deformation stiffness of a crack-free helical gear are as follows: ， ， ， ， ； in, , Contact wire length The projection, Number of slices along the tooth width The distance between the point where the contact load passes through the intersection of the tooth centerline and the tooth root arc. , The length of the arc at the tooth root. It is half of the central angle corresponding to the root arc. , For the hub radius, The radius of the tooth root circle; Bending stiffness when tooth root cracks are present Shear stiffness and radial compressive stiffness The change will affect the effective section moment of inertia containing the tooth root crack. and cross-sectional area Substituting into the potential energy formula, we obtain the stiffness change of the crack. Single tooth meshing stiffness and multi-tooth meshing stiffness formula: ， ， ； Step (3): Establish a tooth profile modification model and calculate the time-varying meshing stiffness of the helical gear tooth root crack and modification coupling; the calculation formula for the meshing stiffness of crack and modification coupling is as follows: ,in, For the error of the helical gear pair, It is the meshing force; Step (4): Determine the relative displacement and meshing damping on the meshing line of the helical gear pair as follows: ， ； meshing force is The meshing damping force is ; For time-varying meshing stiffness, For time-varying meshing damping; Step (5): Establish the coupled dynamic equations of crack and profile modification in the helical gear pair: ； in, , Let the masses of the driving wheel and the driven wheel be... , For rotational inertia, , These are the system's input and load torques, respectively. , To support stiffness, , To increase torsional stiffness, , To support damping, the time-varying meshing stiffness obtained by the coupled calculation of tooth root crack and profile modification can be substituted into the dynamic equation to obtain the vibration displacement of the helical gear pair.

Images