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CN108644342B - Method for establishing non-conical gear antiskid differential mechanism dynamic model - Google Patents

Method for establishing non-conical gear antiskid differential mechanism dynamic model Download PDF

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CN108644342B
CN108644342B CN201810433298.3A CN201810433298A CN108644342B CN 108644342 B CN108644342 B CN 108644342B CN 201810433298 A CN201810433298 A CN 201810433298A CN 108644342 B CN108644342 B CN 108644342B
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gear
differential
limited slip
bevel gear
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韩星会
王兆俊
华林
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Wuhan University of Technology WUT
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H48/00—Differential gearings
    • F16H48/06—Differential gearings with gears having orbital motion
    • F16H48/08—Differential gearings with gears having orbital motion comprising bevel gears
    • G—PHYSICS
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    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/30—Circuit design
    • G06F30/36—Circuit design at the analogue level
    • G06F30/367—Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods

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Abstract

The invention relates to a method for establishing a non-conical gear antiskid differential mechanism dynamic model, which comprises the following steps: s1, establishing a dynamic calculation physical model of the non-bevel gear limited slip differential; s2, selecting a rotating angle of a planet carrier and a rotating angle of a planet gear as generalized coordinates; s3, obtaining the class deviating speed of each component relative to the generalized coordinate; s4, calculating the equivalent moment of inertia and the generalized force of each component according to the class-deviating speed to obtain a basic dynamic model; s5, obtaining an internal friction torque model according to the transmission characteristics of the noncircular bevel gear in the noncircular bevel gear antiskid differential, and bringing the internal friction torque model into a dynamic model; and S6, substituting the locking coefficient into a dynamic equation according to the basic mechanical characteristics of the differential to obtain a dynamic model reflecting the characteristics of the non-bevel gear antiskid differential. The method can accurately obtain the motion rule of the planet gear and the balance position of the antiskid differential in the working process of the noncircular bevel gear antiskid differential, and has the advantages of high calculation precision, high calculation efficiency and the like.

Description

Method for establishing non-conical gear antiskid differential mechanism dynamic model
Technical Field
The invention relates to the field of non-conical gear dynamics, in particular to a method for establishing a non-conical gear antiskid differential dynamic model.
Background
The noncircular bevel gear can realize variable transmission ratio power and motion transmission between two crossed shafts or staggered shafts, and the noncircular bevel gear differential adopting variable transmission ratio space transmission not only has the advantages of compact structure, stable transmission, high efficiency, long service life and the like, but also has the differential anti-skidding function, and the passing performance on muddy, wet and slippery, ice and snow and other ground is greatly improved compared with the traditional bevel gear differential. In the service process of the noncircular bevel gear antiskid differential, the dynamic performance of the noncircular bevel gear antiskid differential has important influence on the running performance of the whole vehicle.
Because the planet gear of the noncircular bevel gear antiskid differential mechanism does continuous rotary motion and rotates around the planet carrier, the sensor cannot be fixed on the planet carrier to measure the rotation angle and the angular speed of the planet gear. Therefore, at present, qualitative research is mainly carried out on the dynamic performance of the non-bevel gear limited slip differential through kinematic simulation, the kinematic simulation needs complex modeling, the calculation efficiency is low, and the accurate motion rule of the planetary gear and the balance position of the non-bevel gear limited slip differential cannot be obtained.
Disclosure of Invention
The invention aims to provide a method for establishing a non-bevel gear limited slip differential dynamic model with high calculation accuracy and high calculation efficiency.
The technical scheme adopted by the invention for solving the technical problems is as follows: the method for constructing the non-bevel gear antiskid differential mechanism dynamic model comprises the following steps:
s1, establishing a dynamic calculation physical model of the non-bevel gear limited slip differential;
s2, selecting a rotating angle of a planet carrier and a rotating angle of a planet gear as generalized coordinates;
s3, obtaining the eccentric speeds of the components relative to the generalized coordinates according to the variable transmission ratio function between the non-conical gears and the basic speed relation of the non-conical gear antiskid differential gear train;
s4, calculating the equivalent moment of inertia and the generalized force of each component according to the class-deviating speed to obtain a basic dynamic model;
s5, obtaining an internal friction torque model according to the transmission characteristics of the noncircular bevel gear in the noncircular bevel gear antiskid differential, and bringing the internal friction torque model into a dynamic model;
and S6, substituting the locking coefficient into a dynamic equation according to the basic mechanical characteristics of the differential to obtain a dynamic model reflecting the characteristics of the non-bevel gear antiskid differential.
In the above scheme, the variable transmission ratio function between the non-conical gears in the step S3 is
Figure GDA0002518200530000026
And
Figure GDA0002518200530000027
according to the basic speed relationship of the differential gear train, there are:
Figure GDA0002518200530000021
Figure GDA0002518200530000022
integrating the two equations to obtain:
Figure GDA0002518200530000023
calculating the class-biased speed of each component in the non-conical gear antiskid differential system relative to two generalized coordinates according to the formula
Figure GDA0002518200530000024
Figure GDA0002518200530000025
In the formula, ωHSpeed of the planet carrier, ω1Is the absolute angular velocity, ω, of the planet gear2、ω3The angular velocities of the left and right side gears respectively,
Figure GDA0002518200530000031
is the rotating angle of the planet carrier,
Figure GDA0002518200530000032
is the rotation angle of the planet gear, and the planet gear is fixed on the support,
Figure GDA0002518200530000033
the rotation angles of the left and right half axle gears are respectively.
In the above scheme, the equivalent moment of inertia J of the member11、J22、J12Comprises the following steps:
Figure GDA0002518200530000034
JHis the moment of inertia of the planet carrier to its axis of rotation, J1Is the moment of inertia of the planet gear, J2、J3The moment of inertia of the left and right side gears, respectively.
In the scheme, the generalized force Q of the noncircular bevel gear antiskid differential system1、Q2The expression of (a) is:
Figure GDA0002518200530000035
wherein, TrThe calculation expression of (a) is:
Figure GDA0002518200530000036
where Δ ω is the minimum speed change of the moment change, μ is the friction moment coefficient of the train, THTorque transmitted for the engine, T2、T3Torque, T, experienced by the left and right side gears, respectivelyrIs the friction torque generated during the rotation of the gear.
In the above scheme, the whole noncircular bevel gear limited slip differential system has kinetic energy:
Figure GDA0002518200530000037
T2and T3The calculation expression of (a) is:
Figure GDA0002518200530000041
where k represents the locking coefficient.
In the scheme, the dynamic equation of the whole noncircular bevel gear antiskid differential system is as follows:
Figure GDA0002518200530000042
Figure GDA0002518200530000043
in the formula:
Figure GDA0002518200530000044
the method for establishing the non-conical gear antiskid differential mechanism dynamic model has the following beneficial effects:
the method is based on Lagrange's equation and according to the working principle of the non-bevel gear limited slip differential, the key problems of dynamic modeling such as generalized coordinate selection, calculation of partial speed, equivalent moment of inertia, generalized force, internal friction moment and locking coefficient of the non-bevel gear limited slip differential are solved, and an accurate dynamic model of the non-bevel gear limited slip differential system is established. The method can accurately obtain the motion rule of the planet gear and the balance position of the antiskid differential in the working process of the noncircular bevel gear antiskid differential through solving the established kinetic equation, and has the advantages of high calculation precision, high calculation efficiency and the like compared with the kinematics simulation.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a flow chart of a method for establishing a non-bevel-gear limited slip differential dynamic model according to the present invention
FIG. 2 is a non-conical gear limited slip differential dynamics computational physical model;
FIG. 3 is a planetary gear rotation angle variation curve of a non-bevel gear limited slip differential at different initial phase angles;
FIG. 4 is a curve of the change of the rotation angle of the planetary gear of the non-bevel-gear limited slip differential under different friction torque coefficients.
Detailed Description
For a more clear understanding of the technical features, objects and effects of the present invention, embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
As shown in FIG. 1, the method for establishing the non-bevel-gear limited slip differential dynamic model comprises the following steps:
s1, establishing a dynamic calculation physical model of the non-bevel gear limited slip differential;
s2, selecting a rotating angle of a planet carrier and a rotating angle of a planet gear as generalized coordinates;
s3, obtaining the eccentric speeds of the components relative to the generalized coordinates according to the variable transmission ratio function between the non-conical gears and the basic speed relation of the non-conical gear antiskid differential gear train;
s4, calculating the equivalent moment of inertia and the generalized force of each component according to the class-deviating speed to obtain a basic dynamic model;
s5, obtaining an internal friction torque model according to the transmission characteristics of the noncircular bevel gear in the noncircular bevel gear antiskid differential, and bringing the internal friction torque model into a dynamic model;
and S6, substituting the locking coefficient into a dynamic equation according to the basic mechanical characteristics of the differential to obtain a dynamic model reflecting the characteristics of the non-bevel gear antiskid differential.
The following detailed description of the embodiments is made with reference to fig. 2-4:
the noncircular gear limited slip differential is a differential gear train with two degrees of freedom, and a physical model for dynamically calculating the noncircular gear limited slip differential is established according to the working principle of the noncircular gear limited slip differential, as shown in fig. 2.
In FIG. 2,. omega.HThe speed of the planet carrier (i.e. the speed of the gearwheel at the input of the differential), THTorque delivered to the engine, JHIs the moment of inertia of the planet carrier (including the planet wheels) to its axis of rotation. Omega1Is the absolute angular velocity of the planet gear, J1Is the moment of inertia of the planet gear, omega2、ω3Angular velocities, T, of left and right side gears, respectively2、T3The torque to which the left and right side gears are subjected (i.e., the torque acting on the wheels), J, respectively2、J3The moment of inertia of the left and right side gears (including the wheels) respectively. T isrIs the friction torque generated during the rotation of the gear.
Because the engine has very hard mechanical property, the speed can be kept constant in a certain torque range, and the speeds of the side gears 2 and 3 are both based on the planetary gear 1, the rotating angle of the planetary gear is selected in the process of establishing a dynamic model
Figure GDA0002518200530000061
Angle of rotation with respect to the planet carrier
Figure GDA0002518200530000062
Is a generalized coordinate.
In the noncircular bevel gear limited slip differential, the gear ratio functions of noncircular bevel gears 1 and 2 are as follows:
Figure GDA0002518200530000063
the transmission ratio function between the non-conical gears 1, 3 is:
Figure GDA0002518200530000064
when the engine runs at a constant speed, the differential shell moves at a constant speed, the rotating speed of the engine is 36deg/s, and the rotating angle of the differential shell is taken
Figure GDA0002518200530000065
And angular velocity
Figure GDA0002518200530000066
Satisfy the relation:
Figure GDA0002518200530000067
the non-bevel gear antiskid differential system is a differential gear train with two degrees of freedom, and according to the basic speed relationship of the differential gear train, the non-bevel gear antiskid differential system comprises the following components:
Figure GDA0002518200530000068
Figure GDA0002518200530000069
integrating the two equations to obtain:
Figure GDA00025182005300000610
according to the formula (1), the class deviation speeds of all members in the noncircular bevel gear antiskid differential system relative to two generalized coordinates are obtained
Figure GDA00025182005300000611
Comprises the following steps:
Figure GDA0002518200530000071
obtaining the equivalent moment of inertia J of each component according to the class-deviating speed of each component relative to the generalized coordinate in the formula (2)11、J22、J12Comprises the following steps:
Figure GDA0002518200530000072
in the formula, J1、J2、J3、JHThe moment of inertia of the planet gear, the left and right half axle gears and the planet carrier, J2＝J3Taking values according to parameters of the non-conical gear in the dynamic model solving process, and taking J1＝3×10-3kg.m2，J2＝J3＝6×10-3kg.m2，JH＝1.5×10-2kg.m2。
System generalized force Q1、Q2The expression of (a) is:
Figure GDA0002518200530000073
wherein, TrThe calculation expression of (a) is:
Figure GDA0002518200530000074
in the process of solving, T is takenH＝200N·m，Δω＝0.01。
The whole noncircular bevel gear antiskid differential system has the kinetic energy as follows:
Figure GDA0002518200530000081
the torque transmitted from the engine is distributed to all the left and right half-shaft gears, and therefore: t isH＝T2+T3
The locking coefficient k is used forThe antiskid performance and the torque distribution characteristic of the antiskid differential mechanism are measured and equal to the torque ratio of driving wheels at the two sides of the speed of the vehicle, so that:
Figure GDA0002518200530000082
T2for the torque, T, distributed on the half-shaft on the side of greater ground adhesion coefficient3The distributed torque on the half shaft on the side with smaller adhesion coefficient.
It is possible to obtain:
Figure GDA0002518200530000083
substituting expressions in formula (2), formula (3), formula (4), formula (5) and formula (6) into the Lagrange equation, the dynamic model of the whole noncircular bevel gear antiskid differential system can be obtained as follows:
Figure GDA0002518200530000084
Figure GDA0002518200530000085
in the formula:
Figure GDA0002518200530000086
the established non-conical gear antiskid differential dynamic model contains an initial phase angle
Figure GDA0002518200530000087
Three parameters of a locking coefficient k and a friction torque coefficient mu of the wheel train,
Figure GDA0002518200530000091
k and mu are main parameters of the non-bevel gear limited slip differential dynamics analysis, and different constant values can be taken to be brought into a dynamics model to carry out solving calculation according to the requirements. Programming and solving the obtained dynamic model in MATLAB by a Runge-Kutta method to obtain a change curve of the rotation angle of the planetary gear,the dynamic performance of the non-bevel gear antiskid differential mechanism under different working conditions can be theoretically analyzed.
The friction torque coefficient mu is 0, the locking coefficient k is 1, and the initial phase angle
Figure GDA0002518200530000092
The change curve of the rotation angle of the planetary gear obtained by solving the kinetic equation at-45 degrees and-60 degrees is shown in FIG. 3.
Friction torque coefficient mu is 0.05, 0.06 and 0.08, locking coefficient k is 2, initial phase angle
Figure GDA0002518200530000093
The change curve of the rotation angle of the planetary gear obtained by solving the kinetic equation when 0 is taken is shown in fig. 4.
While the present invention has been described with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, which are illustrative and not restrictive, and it will be apparent to those skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (6)

1. A method for establishing a non-bevel gear limited slip differential dynamic model is characterized by comprising the following steps:
s1, establishing a dynamic calculation physical model of the non-bevel gear limited slip differential;
s2, selecting a rotating angle of a planet carrier and a rotating angle of a planet gear as generalized coordinates;
s3, obtaining the eccentric speeds of the components relative to the generalized coordinates according to the variable transmission ratio function between the non-conical gears and the basic speed relation of the non-conical gear antiskid differential gear train;
s4, calculating the equivalent moment of inertia and the generalized force of each component according to the class-deviating speed to obtain a basic dynamic model;
s5, obtaining an internal friction torque model according to the transmission characteristics of the noncircular bevel gear in the noncircular bevel gear antiskid differential, and bringing the internal friction torque model into a dynamic model;
and S6, substituting the locking coefficient into a dynamic equation according to the basic mechanical characteristics of the differential to obtain a dynamic model reflecting the characteristics of the non-bevel gear antiskid differential.
2. The method for modeling the dynamics of a non-bevel-gear limited slip differential according to claim 1, wherein the variable transmission ratio function between the non-bevel gears in step S3 is
Figure FDA0002518200520000011
And
Figure FDA0002518200520000012
according to the basic speed relationship of the differential gear train, there are:
Figure FDA0002518200520000013
Figure FDA0002518200520000014
integrating the two equations to obtain:
Figure FDA0002518200520000015
calculating the class-biased speed of each component in the non-conical gear antiskid differential system relative to two generalized coordinates according to the formula
Figure FDA0002518200520000016
Figure FDA0002518200520000021
In the formula, ωHSpeed of the planet carrier, ω1Is the absolute angular velocity, ω, of the planet gear2、ω3The angular velocities of the left and right side gears respectively,
Figure FDA0002518200520000022
is the rotating angle of the planet carrier,
Figure FDA0002518200520000023
is the rotation angle of the planet gear, and the planet gear is fixed on the support,
Figure FDA0002518200520000024
the rotation angles of the left and right half axle gears are respectively.
3. The method of modeling a non-conical gear limited slip differential kinetic model according to claim 2 wherein the equivalent moment of inertia J of the member11、J22、J12Comprises the following steps:
Figure FDA0002518200520000025
JHis the moment of inertia of the planet carrier to its axis of rotation, J1Is the moment of inertia of the planet gear, J2、J3The moment of inertia of the left and right side gears, respectively.
4. The method of modeling a non-conical-gear limited slip differential kinetic model according to claim 3 wherein the non-conical-gear limited slip differential system generalized force Q1、Q2The expression of (a) is:
Figure FDA0002518200520000026
wherein, TrThe calculation expression of (a) is:
Figure FDA0002518200520000031
where Δ ω is the minimum speed change of the moment change and μ is the trainCoefficient of friction torque, THTorque transmitted for the engine, T2、T3Torque, T, experienced by the left and right side gears, respectivelyrIs the friction torque generated during the rotation of the gear.
5. The method of building a non-conical-gear limited slip differential kinetic model according to claim 4, wherein the entire non-conical-gear limited slip differential system has kinetic energy of:
Figure FDA0002518200520000032
T2and T3The calculation expression of (a) is:
Figure FDA0002518200520000033
where k represents the locking coefficient.
6. The method for building a non-bevel-gear limited slip differential kinetic model according to claim 5, wherein the kinetic equation of the entire non-bevel-gear limited slip differential system is:
Figure FDA0002518200520000034
Figure FDA0002518200520000035
in the formula:
Figure FDA0002518200520000041
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