Method for calculating stiffness and transient dynamics of an electromechanically coupled variable position gear

Abstract

The application discloses a kind of electromechanical coupling variable position gear stiffness and transient dynamics calculation method, and its specific steps are as follows: (1) establish small modulus variable position gear tooth profile equation;(2) according to tooth profile equation, calculate healthy variable position gear stiffness using material potential energy method;(3) determine crack position and crack form according to the size of dedendum circle and base circle;(4) crack form is substituted into potential energy method calculation formula;(5) establish electromechanical coupling dynamics model under transient operating condition, put healthy gear and crack gear into dynamics, solve vibration displacement;Beneficial effect is to consider the influence mechanism of base circle and graduation circle size on time-varying mesh stiffness, establish small modulus variable position gear time-varying mesh stiffness calculation method, improve the time-varying mesh stiffness calculation precision, improve the accuracy of dynamics analysis.

Description

[0001] The application relates to the field of fault of variable pitch gears, in particular to a method for calculating the stiffness and transient dynamics of an electromechanical coupling variable pitch gear. BACKGROUND

[0002] Gear transmission plays an important role in the mechanical field and has been widely applied to various fields such as automobiles, robots, railway locomotives and wind power generation. In most cases, gears are subjected to impacts such as starting, overloading and speed changing, thereby causing gear failure, and tooth root crack is the most common failure mode. The crack causes the meshing stiffness of the transmission system to be reduced, which is manifested as vibration impact in dynamics. In the joints of service robots, variable pitch gear transmission systems are mostly used, which work under variable speed and variable load conditions. Therefore, the research on variable pitch gear crack and transient dynamics is of great significance. At present, there is a certain gap in the research on the failure mechanism of small modulus variable pitch gears, and most scholars do not consider the vibration displacement of the motor and the vibration displacement under transient working conditions.

[0003] In order to solve the above problems, the application provides a method for calculating the stiffness and transient dynamics of an electromechanical coupling variable pitch gear, which considers the vibration coupling effect of the motor and the small modulus variable pitch gear pair, establishes an electromechanical coupling dynamics model containing crack failure, and analyzes the vibration displacement under transient working conditions: filling the gap in the related technology, promoting the development of engineering technology in the field of precision transmission, and also producing great social and economic benefits. SUMMARY

[0004] In order to overcome the deficiencies of the prior art and fill the gap in the related technology, the application provides a method for calculating the stiffness and transient dynamics of an electromechanical coupling variable pitch gear.

[0005] The technical scheme adopted by the application to solve the technical problems is as follows: a method for calculating the stiffness and transient dynamics of an electromechanical coupling variable pitch gear, characterized by:

[0006] Step (1): for a small modulus variable pitch gear, a tooth profile equation of the variable pitch gear is established according to different variable pitch coefficients, and the formula is: ;

[0007] wherein, is the variable pitch coefficient, is the pressure angle of the reference circle, is the number of teeth, is the matching angle of the current meshing position, is the base circle radius, and different variable pitch profile equations can be obtained for different variable pitch coefficients;

[0008] Step (2): According to the modified tooth profile equation generated in the last step, the coordinates of any point on the tooth profile can be found in the program. The stiffness calculation of the modified gear includes bending stiffness, shear stiffness, compression stiffness and Hertz contact stiffness;

[0009] For transient dynamic study, the contact area of the modified gear pair changes, resulting in changes in its Hertz contact stiffness, which is: ;

[0010] Step (3): Cracks are generated in the modified gear during long-term operation. There are two forms of cracks in the modified gear: single-sided cracks and double-sided cracks. Cracks will affect the meshing stiffness of the modified gear. The position of the root circle and the base circle will affect the bending stiffness, shear stiffness and compression stiffness.

[0011] When the root circle is greater than the base circle, the inverses of the bending stiffness, shear stiffness and compression stiffness are respectively:

[0012] ,

[0013] ,

[0014] ;

[0015] When the root circle is smaller than the base circle, the inverses of the bending stiffness, shear stiffness and compression stiffness are respectively:

[0016] ,

[0017] ,

[0018] ;

[0019] Step (4): Considering that the crack changes along the tooth width side, the crack mainly affects the cross-sectional moment of inertia and the cross-sectional area. According to the integral principle, the total stiffness of the modified gear under the condition of crack is obtained as:

[0020] ;

[0021] wherein, is the Hertz contact stiffness, , is the driving gear and the driven gear, , , are the bending stiffness, shear stiffness and compression stiffness respectively, is the base stiffness;

[0022] Step (5): For brushless motors, the coordinate transformation equation is:

[0023] ;

[0024] The voltage equation, flux linkage equation and electromagnetic torque equation are:

[0025] ,

[0026] ,

[0027] ;

[0028] The subscript , represents , The axis component, subscript , represents the amount on the stator and rotor; is a differential operator; is the mutual inductance of the stator and rotor, ; , , , , , The voltage, current, resistance, self-inductance, leakage inductance and flux linkage, is the number of magnetic pole pairs, is The angular velocity of the coordinate system, which is equal to the synchronous angular velocity of the stator power supply, is the rotor electric angular velocity;

[0029] The output speed and torque of the motor are associated with the input speed and torque of the gear pair to obtain the electromechanical coupling dynamics equation:

[0030] ;

[0031] The motor electromagnetic torque and rotor speed are used as common variables to transfer data between the motor and the variable tooth gear transmission system in real time, thereby constructing the electromechanical coupling model of the motor-driven multi-stage gear transmission system; according to the coupling model, the multi-degree-of-freedom electromechanical coupling variable tooth gear pair dynamics equation is established, and the bending and torsional deformation of each component, the errors caused by production and installation are considered in the modeling process; the health variable tooth gear stiffness and the crack variable tooth gear stiffness are substituted into the dynamics equation to solve it, and the corresponding vibration displacement can be obtained.

[0032] Compared with the prior art, the present application has the beneficial effects: a time-varying meshing stiffness technical method of the variable tooth gear is established, the stiffness can be calculated in detail according to the sizes of the base circle and the root circle; the time-varying Hertz contact stiffness of the gear contact area under the transient working condition is considered, and the comprehensive stiffness calculation is more accurate; an electromechanical coupling dynamics model is established, the installation and manufacturing errors are considered, and the vibration displacement under the transient condition is analyzed. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 is a flow chart of the stiffness and transient dynamics calculation method of the electromechanical coupling variable tooth gear;

[0034] Figure 2 is a healthy small modulus variable tooth gear;

[0035] Figure 3 is a cracked small modulus variable tooth gear and its crack form;

[0036] Figure 4 is a comparison of the stiffness of single-sided and double-sided through cracks;

[0037] Figure 5 is a comparison of the stiffness of single-sided and double-sided non-through cracks;

[0038] Figure 6 is the transient vibration displacement of a healthy gear pair;

[0039] Figure 7 is the transient vibration displacement of a cracked gear pair. DETAILED DESCRIPTION

[0040] 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.

[0041] The joint of the service robot is limited in volume, and the transmission system adopts a small modulus variable tooth gear. Referring to Figure 1 Fig. 1 is a flow chart of the stiffness and transient dynamics calculation method of the electromechanical coupling variable tooth gear, including the following steps:

[0042] Step (1): a healthy small modulus variable tooth gear, as shown in Figure 2 , the parameters of which are shown in Table 1:

[0043] Table 1

[0044] Gear Teeth Module / mm Variation coefficient / mm Tooth width / mm 1 14 0.2 0.45 3.5 2 82 0.2 0.25 3.5

[0045] The tooth profile equation of the variable tooth gear is established:

[0046]

[0047] wherein,​ is the coefficient of variation, is the pressure angle of the pitch circle, is the number of teeth, is the conjugate angle of the current meshing position, is the base circle radius. The tooth root fillet angle equation is:

[0048] ;

[0049] where, is the pitch circle radius, , , is the tool parameter, whose formula is as follows:

[0050] ;

[0051] Step (2): The force of a single gear is analogous to a cantilever beam structure, and when the gears interact, they are subjected to bending, shear, compression, and Hertz energy, whose expressions are ;

[0052] where ( ) represents the bending, shear, compression, and Hertz contact energy, ( ) represents the bending, shear, compression, and Hertz contact stiffness, is the meshing force at the contact point;

[0053] For transient dynamic research, the contact area is always changing, resulting in a constant change in its Hertz contact stiffness, and its Hertz stiffness is:

[0054] ;

[0055] Step (3): From the parameters in Table 1, it can be found that there are two cases for the tooth root circle and the base circle; the tooth root and base circle positions will affect the bending stiffness, shear stiffness, and compression stiffness; when the tooth root circle is greater than the base circle, the reciprocal of its bending stiffness, shear stiffness, and compression stiffness is respectively:

[0056] ,

[0057] ,

[0058] ;

[0059] When the tooth root circle is smaller than the base circle, the reciprocal of its bending stiffness, shear stiffness, and compression stiffness is respectively:

[0060] ,

[0061] ,

[0062] ;

[0063] wherein, is the elastic modulus; is the contact line tooth width; is the shear modulus; Poisson's ratio, and are the cross-sectional moment of inertia and cross-sectional area at any point of the tooth profile, respectively.

[0064] Step (4): The gear transmission system in the robot joint needs to be frequently reversed, which leads to fatigue cracks in the gear too quickly, and the cracks often appear on both sides of a tooth, as shown in Figure 3 ;

[0065] When the crack appears on one side, the formula is:

[0066] , ;

[0067] wherein, is the effective tooth thickness on the crack side, is the crack depth at different tooth width positions, is the crack propagation angle;

[0068] When the crack appears on both sides:

[0069] , ; and

[0070] The crack gear exists in two forms of penetrating along the tooth width and not penetrating along the tooth width, when penetrating:

[0071] ;

[0072] When the crack does not penetrate along the tooth width direction, the parabolic equation is:

[0073] ;

[0074] wherein: is the crack depth on the other side end surface when the crack penetrates along the tooth width, is the effective crack length along the tooth width direction, is the crack depth on the crack starting end surface;

[0075] According to the micro-element method, the stiffness under the thickness of each tooth can be obtained, and the total stiffness under the crack condition is ;

[0076] wherein, is the Hertz contact stiffness, , drive gear, driven gear, , , bending stiffness, shear stiffness, compression stiffness, base stiffness;

[0077] Step (5): According to the stiffness calculation, the stiffness is obtained as shown in Figure 4 , 5 The gear pair dynamics equation is:

[0078] ;

[0079] wherein, mass, moment of inertia, time-varying meshing damping, time-varying meshing stiffness, support damping, support stiffness, relative displacement along the meshing line direction, comprehensive error, motor torque, load torque;

[0080] The output speed and torque of the motor are associated with the input speed and torque of the gear pair to obtain the electromechanical coupling dynamics equation:

[0081] ;

[0082] The health gear stiffness and the crack gear stiffness are substituted into the dynamics, and the Runge-Kutta is used to solve it, so as to obtain the corresponding vibration displacement as shown in Figure 6 , 7 Compared with the previous calculation method, this method can better reflect the crack vibration characteristics under transient working conditions, and the vibration displacement at the crack is more affected by the transient state.

[0083] The above is only a preferred embodiment of the application, not any limitation on the application, any modification, change and equivalent change of the above embodiment according to the essence of the application are still within the protection scope of the application technology.

Claims

1.A method for calculating the stiffness and transient dynamics of an electromechanical coupling variable displacement gear, characterized by the following steps: Step (2): According to the variable displacement tooth profile equation generated in the previous step, the coordinates of any point on the tooth profile can be found in the program. The variable displacement gear stiffness calculation includes bending stiffness, shear stiffness, compression stiffness, and Hertz contact stiffness; Step (3): The variable displacement gear generates cracks during long-term operation. There are two forms of cracks: single-sided cracks and double-sided cracks. The cracks will affect the meshing stiffness of the variable displacement gear. The bending stiffness, shear stiffness, and compression stiffness are affected by the position of the tooth root and the base circle; When the tooth root circle is greater than the base circle, the reciprocals of the bending stiffness, shear stiffness, and compression stiffness are respectively: When the tooth root circle is smaller than the base circle, the reciprocals of the bending stiffness, shear stiffness, and compression stiffness are respectively: Step (4): Considering that the crack changes along the tooth width side, the crack mainly affects the cross-sectional moment of inertia and the cross-sectional area. According to the integral principle, the total stiffness of the variable displacement gear under the condition of crack is obtained as: Step (5): For a brushless motor, the coordinate transformation equation after Park transformation is: Therefore, the voltage equation, flux linkage equation, and electromagnetic torque equation are: The output speed and torque of the motor are related to the input speed and torque of the gear pair, and the electromechanical coupling dynamics equation is obtained: The motor electromagnetic torque and rotor speed are used as common variables to transfer data between the motor and the variable displacement gear transmission system in real time, thereby constructing the electromechanical coupling model of the motor-driven multi-stage gear transmission system. According to the coupling model, the multi-degree-of-freedom electromechanical coupling variable displacement gear pair dynamics equation is established. In the modeling process, the bending and torsional deformation of each component, the errors caused by production and manufacturing, and the errors caused by installation are considered. The healthy variable displacement gear stiffness and the crack variable displacement gear stiffness are substituted into the dynamics equation to solve it, and the corresponding vibration displacement can be obtained. Step (1): for small modulus variable displacement gear, according to different variable displacement coefficient, the tooth profile equation of variable displacement gear is established, and the formula is: ; wherein, is the displacement coefficient, is the pressure angle of the reference circle, is the number of teeth, is the engagement angle of the current position, is the base circle radius, different displacement profile equations can be obtained for different displacement coefficients; ​ For the transient dynamics study, the contact area of the variable-dimension gear pair changes, resulting in the change of its Hertz contact stiffness, which is: ; ​ ​ ， ， ； ​ ， ， ； ​ ； wherein is the hertz contact stiffness, , is the drive gear, the driven gear, , , are the bending stiffness, the shear stiffness, the compression stiffness, respectively, is the base stiffness; ​ ； ​ ， ， ； subscript , represent , axial component, subscript , represent quantities on the stator and rotor; is the differential operator; is the mutual inductance of stator and rotor, ; , , , , L l , are the voltage, current, resistance, self-inductance, leakage inductance and flux linkage, respectively, n e is the number of pole pairs, is the angular velocity of the coordinate system, which has the value of the synchronous angular velocity of the stator power supply, is the rotor electric angular velocity; ​ ； ​

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