CN107341313A - Planetary gear train Nonlinear dynamic models method based on ADAMS - Google Patents

Planetary gear train Nonlinear dynamic models method based on ADAMS Download PDF

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CN107341313A
CN107341313A CN201710558889.9A CN201710558889A CN107341313A CN 107341313 A CN107341313 A CN 107341313A CN 201710558889 A CN201710558889 A CN 201710558889A CN 107341313 A CN107341313 A CN 107341313A
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msub
mfrac
gear
alpha
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CN107341313B (en
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陈换过
吴建伟
陈文华
李剑敏
陈培
陈特
俞杭
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Zhejiang Sci Tech University ZSTU
Zhejiang University of Science and Technology ZUST
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Abstract

The invention discloses a kind of planetary gear train Nonlinear dynamic models method based on ADAMS, the modeling method has been included in time-variant mesh stiffness, time-varying engagement damping and relevant parameter excitation in a model, institute's established model is versatile, can do the dynamics simulation of the planetary gear train under any operating mode；The result of calculation of gear mesh force is more accurate, and the accuracy that efficiently avoid the geometry due to establishing model has influence on the accuracy of result of calculation, and calculating speed is fast, and precision is high.In addition, built gear mesh force equation provided by the present invention is applied to the planetary gear train of different structure, planetary gear train, differential planetary gear train such as fixed gear ring.

Description

Planetary gear train Nonlinear dynamic models method based on ADAMS
Technical field
It is non-linear more particularly, to a kind of planetary gear train based on ADAMS the present invention relates to a kind of dynamic modeling method Dynamic modeling method.
Background technology
Many difficulties during the kinematics and dynamic analysis of complex mechanical system be present：In tectonodynamics equation Heavy algebraical sum differential calculation is faced, and due to the non-linear analytic solutions for causing that closing can not be tried to achieve of equation.At this Under background, solving analysis of complex system and synthtic price index using computer turns into one of General Mechanics field in the late two decades The main and rapid research direction of progress.At present, Dynamics Simulation software (ADAMS) is most to have authoritative weight in the world, The most wide machinery system dynamics software of use range.User uses ADAMS softwares, can generate more bodies of arbitrarily complicated system Dynamics mathematicization virtual prototype, can provide the user from Product Conceptual Design, demonstration, detailed design, to product Scheme modifying, optimization, experiment planning even fault diagnosis each stage, comprehensive Simulation Analysis result.
For the widely used gear drive in mechanical system, ADAMS/View provides impact functions and calculated Contact force represents the engaging force between gear.This method needs to take the long period when solving and calculating, and asks It is low to solve efficiency.And the accuracy of simulation result is largely dependent upon the accuracy of geometrical model.Meanwhile actual tooth It is a complicated dynamic system to take turns transmission system, a large amount of dynamic exciting factors in engagement process be present, as time-varying engages Rigidity, time-varying engagement damping, different teeth are to mesh phase etc..Therefore, it is necessary to the engaging force between the gear teeth is proposed to change Enter.
The content of the invention
The invention provides one kind to be based on ADAMS planetary gear train Nonlinear dynamic models methods, considers that gear time-varying is nibbled Rigidity, time-varying engagement damping and tooth are closed to phase difference, it is intended to solve to be difficult in ADAMS softwares nibbling between the accurate expression gear teeth Joint force problem.
In order to realize the purpose of the present invention, this provided based on ADAMS planetary gear trains Nonlinear dynamic models side Method, comprise the following steps that：
Step 1：Set the modeling conditions of planetary gear train；
Step 2：Model and import the model built up in ADAMS, model is simplified, set environment parameter simultaneously adds Addition of constraints, contact force and driving function；
Step 3：Marker point Angle_ref are established in the barycenter of sun gear, planetary gear, ring gear, planet carrier, and are made The direction of each Marker points Z axis is identical with the Z axis under global coordinate system, first seeks each gear with respect to planet carrier Marker points Angle_ref Z axis angular displacement, relative displacement equation and relative line of each gear in path of contact are obtained multiplied by with base radius Rate equation；
Step 4：Gear time-variant mesh stiffness equation is obtained by improving potential energy method；
Step 5：Obtain gear time-varying engagement Equation With Damping；
Step 6：Model middle gear contact force is changed in ADAMS, it is 0 to make its value permanent, according to relative displacement equation, phase Engagement force equation is defined to linear velocity equation, gear time-variant mesh stiffness equation and gear time-varying engagement Equation With Damping, completes row Star wheel series Nonlinear dynamic models.
Modeling method provided by the present invention has been included in time-variant mesh stiffness in a model, time-varying engagement damps and relative The parametric excitation such as displacement and speed, institute's established model is versatile, can do the dynamics simulation of the planetary gear train under any operating mode； The result of calculation of gear mesh force is more accurate, and the accuracy that efficiently avoid the geometry due to establishing model has influence on The accuracy of result of calculation, calculating speed is fast, and precision is high.
Specifically, the engagement force equation described in step 6 is：
F in formula (1)spiRatcheting power, k between planetary gear i and sun gearspiFor the time-varying between planetary gear i and sun gear Mesh stiffness, xspiFor planetary gear i and relative displacement of the sun gear in path of contact, cspiBetween planetary gear i and sun gear Time-varying engagement damping,For planetary gear i and relative linear velocity of the sun gear in path of contact；
K in formula (1)spiBy being obtained with next equation：
Or
During formula (3) is monodentate engagement, the time-variant mesh stiffness of a pair of gear vice presidents；Formula (4) is that two pairs of gear teeth are joined simultaneously Total time-variant mesh stiffness during with engaging；Wherein khFor hertz rigidity, kbFor bending stiffness, kaFor radial compression rigidity, ksTo cut Cut rigidity；First pair of gear teeth meshing is represented during i=1；Second pair of gear teeth meshing is represented during i=2；Subscript 1,2 represents driving respectively Gear and driven gear；WithObtained respectively by below equation：
In formula (5)~formula (8)：E is modulus of elasticity, and L is gear axial width, and υ is Poisson's ratio, α1It is meshing point apart from tooth At root d, the line at point of contact and gear centre of the path of contact on basic circle and the angle of gear teeth line of symmetry；α is meshing point apart from tooth At root x, the line at point of contact and gear centre of the path of contact on basic circle and the angle of gear teeth line of symmetry；α2For basic circle and flank profil line The angle that line and gear teeth line of symmetry between intersection point and gear centre are formed；α5Represent when meshing point and root circle distance are 0 When, directed force F and decomposing force FbBetween angle, α in formula5Obtained by below equation group：
R in formula (9)rFor root radius, RbFor base radius, α4Between root circle and flank profil line intersection point and gear centre Line and the angle that is formed of gear teeth line of symmetry；
X in formula (1)spiBy being obtained with next equation：
Or
In formula (10)：Respectively sun gear relative to the angular displacement and planetary gear of planet carrier relative to planet carrier Angular displacement, rbsFor the base radius of sun gear, rbpiFor the base radius of planetary gear,
C in formula (1)spiObtained by below equation：
In formula (12)：ζ is damping ratio, and k is time-variant mesh stiffness, m1、m2Gear mesh middle gear quality is represented respectively；
In formula (1)Obtained by following two equations：
Or
In formula (14)：WithRespectively sun gear with respect to planet carrier linear velocity and planetary gear with respect to planet carrier linear velocity, It can be tried to achieve by corresponding angular velocity and the product of base radius；
F in formula (2)rpiRatcheting power between planetary gear i and ring gear, krpiBetween planetary gear i and ring gear when Become mesh stiffness, xrpiFor planetary gear i and too relative displacement of the ring gear in path of contact, crpiFor planetary gear i and ring gear it Between time-varying engagement damping,For planetary gear i and relative linear velocity of the ring gear in path of contact；
K in formula (2)rpiObtained by equation (3) or (4)；
X in formula (2)rpiObtained by following two equations：
Or
In formula (16)WithFor gear ring relative to planet carrier angular displacement and planetary gear relative to planet carrier angular displacement, rbrFor the base radius of ring gear, rbpiFor the base radius of planetary gear；
C in formula (2)rpiObtained by equation (12)：
In formula (2)By being obtained with next equation：
Or
AZ, WZ are respectively angular displacement function and angular speed function in formula (11), formula (14), formula (16) and formula (18)；Zs、 Zp、ZrThe respectively number of teeth of sun gear, planetary gear and ring gear；α is pressure angle；M is modulus.
Specifically, modeling conditions are described in step 1：
(1) all gears are standard involute spur in train；
(2) two gear gear blanks are considered as rigid body, and the input of train and output shaft are rigid body, do not consider the elastic deformation of support；
(3) train middle gear is installed by reference center distance, and pitch circle overlaps with reference circle；
(4) disregard gear error and tooth surveys gap.
Specifically, the contact force selection impulse function method described in step 2 is calculated.
Specifically, the improvement potential energy method described in step 4 is that the gear teeth in model are reduced into hanging in tooth root institute first Arm beam, the potential energy being stored in meshing gear include Hertz contact potential energy Uh, bowing potential energy Ub, radial compression potential energy UaAnd shearing Potential energy Us, pass through Hertz contact potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy UsCalculate hertz rigidity kh、 Bending stiffness kb, radial compression rigidity kaWith shearing rigidity ks, total mesh stiffness is the cascade of each rigidity；Each energy with just Degree meets following relation：
F is the interaction force of tooth at meshing point in formula (19)~formula (22), and direction is all the time along path of contact direction Intersect with flank profil.
Built gear mesh force equation provided by the present invention is applied to the planetary gear train of different structure, such as fixed gear ring Planetary gear train, differential planetary gear train.
The beneficial effects of the invention are as follows：Time-variant mesh stiffness, time-varying engagement damping and related ginseng have been included in a model Number excitation, institute's established model is versatile, can do the dynamics simulation of the planetary gear train under any operating mode；The meter of gear mesh force Calculation result is more accurate, and the accuracy that efficiently avoid the geometry due to establishing model has influence on the accurate of result of calculation Property, calculating speed is fast, and precision is high.
Brief description of the drawings
Fig. 1 is the structure diagram of the gear for two stage planetary gear train system described in the present invention；
Fig. 2 is the flow chart of the planetary gear train Nonlinear dynamic models method provided by the present invention based on ADAMS；
Gear for two stage planetary gear train non-linear dynamic model structural representation Fig. 1 that Fig. 3 is established for the present invention；
Gear for two stage planetary gear train non-linear dynamic model structural representation Fig. 2 that Fig. 4 is established for the present invention；
Fig. 5 is first order sun gear and planetary gear mesh stiffness curve；
Fig. 6 is first order planetary gear and gear ring mesh stiffness curve；
Fig. 7 is second level sun gear and planetary gear mesh stiffness curve；
Fig. 8 is second level planetary gear and gear ring mesh stiffness curve；
Fig. 9 is the rigidity data figure imported in ADAMS；
Figure 10 is the gear engaging force schematic diagram being calculated using engagement force equation provided by the present invention；
Figure 11 is gear teeth meshing stress diagram.
Embodiment
Technical scheme is further described with reference to the drawings and specific embodiments herein.
The application is modeled by taking power dividing type gear for two stage planetary gear train as an example herein, power dividing type gear for two stage planetary gear train Sketch is as shown in Figure 1；The train is made up of gear for two stage planetary gear train, and wherein the first order is common planetary train, and the second level is one Differential planet gear mechanism.Power is inputted by main shaft, power is divided into two parts, a part is delivered to the first order by main shaft On ring gear, another part power is directly connected by main shaft with second level planet carrier, is coupled finally by differential planetary itself Realize that power collaborates on to second level sun gear.Be modeled in ADAMS softwares, and modeling when consider time-variant mesh stiffness, Time-varying engagement damping, modeling procedure is as shown in Fig. 2 be specially：
Step 1：Modeling conditions are set according to planetary gear train, the modeling conditions can according to planetary gear train it is different without Together, modeling conditions are specially used by the application herein：
1st, all gears are standard involute spur in train；
2nd, two gear gear blanks are considered as rigid body, and the input of train and output shaft are rigid body, do not consider the elastic deformation of support；
3rd, train middle gear is installed by reference center distance, and pitch circle overlaps with reference circle；
4th, disregard gear error and tooth surveys gap；
Step 2：Planetary gear train physical model is established by Pro/E, SolidWorks or other 3 d modeling softwares, As shown in Figure 3 and Figure 4；Model is imported in ADAMS after building up, and model is simplified under ADAMS environment, passes through boolean operation Instrument, planetary gear is simplified with planet wheel spindle and is integrated；Set environment parameter, it is specially：The matter of each component is inputted respectively Amount and barycenter mark point are as rotary inertia reference marker point；And various restriction relations are defined, except first order planet carrier is set Outside for fixed joint, other all components are arranged to the revolute around its geometric center, at the same by input shaft respectively with first order tooth Circle, second level planet carrier set fixed joint, and first order sun gear and second level gear ring are set into fixed joint；Can during setting contact force Realized with being calculated using any function in ADAMS, the application is realized using impulse function and calculated herein；
Step 3：Marker point Angle_ref are established in the barycenter of sun gear, planetary gear, ring gear, planet carrier, and are made The direction of each Marker points Z axis is identical with the Z axis under global coordinate system, first seeks each gear with respect to planet carrier Marker points Angle_ref Z axis angular displacement, relative displacement equation and relative line of each gear in path of contact are obtained multiplied by with base radius Rate equation；
Step 4：Gear time-variant mesh stiffness equation is obtained by improving potential energy method；
Step 5：Obtain gear time-varying engagement Equation With Damping；
Step 6：Model middle gear contact force is changed in ADAMS, it is 0 to make its value permanent, according to relative displacement equation, phase Engagement force equation is defined to linear velocity equation, gear time-variant mesh stiffness equation and gear time-varying engagement Equation With Damping, completes row Star wheel series Nonlinear dynamic models.
The planetary gear non-linear dynamic model established by above modeling method is included between planetary gear i and sun gear Ratcheting power equation between ratcheting power equation and planetary gear i and ring gear, the planetary gear i and sun gear that the application is established at this Between shown in ratcheting power equation such as formula (1), and shown in such as formula (2) of the ratcheting power equation between planetary gear i and ring gear,
F in formula (1)spiRatcheting power, k between planetary gear i and sun gearspiFor the time-varying between planetary gear i and sun gear Mesh stiffness, xspiFor planetary gear i and relative displacement of the sun gear in path of contact, cspiBetween planetary gear i and sun gear Time-varying engagement damping,For planetary gear i and relative linear velocity of the sun gear in path of contact；
K in formula (1)spiBy being obtained with next equation：
Or
During formula (3) is monodentate engagement, the time-variant mesh stiffness of a pair of gear vice presidents；Formula (4) is that two pairs of gear teeth are joined simultaneously Total time-variant mesh stiffness during with engaging；Wherein khFor hertz rigidity, kbFor bending stiffness, kaFor radial compression rigidity, ksTo cut Cut rigidity；First pair of gear teeth meshing is represented during i=1；Second pair of gear teeth meshing is represented during i=2；Subscript 1,2 represents driving respectively Gear and driven gear；WithObtained respectively by below equation：
In formula (5)~formula (8)：E is modulus of elasticity, and the modulus of elasticity is constant, different according to gear material, modulus of elasticity Difference, if mild steel is 200~210GPa, medium carbon steel 205GPa, steel alloy 210, spheroidal graphite cast-iron is 150~180, and aluminium closes Gold is 71；L is gear axial width, and υ is Poisson's ratio, and the Poisson's ratio is constant, different, such as low-carbon according to gear material difference Steel is 0.24~0.28；Medium carbon steel is 0.24~0.28, and steel alloy is 0.25~0.30, aluminium alloy 0.33.With reference to Figure 11 pairs α in formula (5)~formula (8)1、α、α2And α5Illustrate, wherein α1It is meshing point at tooth root d, path of contact is on basic circle The angle of the line and gear teeth line of symmetry of point of contact and gear centre；α be meshing point at tooth root x, path of contact is on basic circle The angle of the line and gear teeth line of symmetry of point of contact and gear centre；α2For the line between basic circle and flank profil line intersection point and gear centre The angle formed with gear teeth line of symmetry；α5Represent when meshing point and root circle distance are 0, directed force F and decomposing force FbBetween Angle；α in formula5Obtained by below equation group：
R in formula (9)rFor root radius, RbFor base radius, α4Between root circle and flank profil line intersection point and gear centre Line and the angle that is formed of gear teeth line of symmetry, as shown in figure 11；
X in formula (1)sp iBy being obtained with next equation：
Or
In formula (10)：Respectively sun gear relative to the angular displacement and planetary gear of planet carrier relative to planet carrier Angular displacement, rbsFor the base radius of sun gear, rbpiFor the base radius of planetary gear,
C in formula (1)spiObtained by below equation：
In formula (12)：ζ is damping ratio, and k is time-variant mesh stiffness, m1、m2Gear mesh middle gear quality is represented respectively；
In formula (1)Obtained by following two equations：
Or
In formula (14)：WithRespectively sun gear with respect to planet carrier linear velocity and planetary gear with respect to planet carrier linear velocity, It can be tried to achieve by corresponding angular velocity and the product of base radius；
F in formula (2)rpiRatcheting power between planetary gear i and ring gear, krpiBetween planetary gear i and ring gear when Become mesh stiffness, xrpiFor planetary gear i and too relative displacement of the ring gear in path of contact, crpiFor planetary gear i and ring gear it Between time-varying engagement damping,For planetary gear i and relative linear velocity of the ring gear in path of contact；
K in formula (2)rpiObtained by equation (3) or (4)；
X in formula (2)rpiObtained by following two equations：
Or
In formula (16)WithFor gear ring relative to planet carrier angular displacement and planetary gear relative to planet carrier angular displacement, rbrFor the base radius of ring gear, rbpiFor the base radius of planetary gear；
C in formula (2)rpiObtained by equation (12)：
In formula (2)By being obtained with next equation：
Or
AZ, WZ are respectively angular displacement function and angular speed function in formula (11), formula (14), formula (16) and formula (18)；Zs、 Zp、ZrThe respectively number of teeth of sun gear, planetary gear and ring gear；α is pressure angle；M is modulus.In addition formula (11), formula (14), formula (16) and formula (18) is to run the representation under function in ADAMS/View, and wherein function AZ functions are：Return a certain The angle that Marker points rotate around another Marker points Z axis；Wherein function WZ functions are：Return to two Marker point angular velocity vectors Difference is in z-component.
In addition, improvement potential energy method of the present embodiment employed in step 4 is that the gear teeth in model are reduced into tooth first Cantilever beam in root institute, the potential energy being stored in meshing gear include Hertz contact potential energy Uh, bowing potential energy Ub, radial compression gesture Can UaWith shearing potential energy Us, pass through Hertz contact potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy UsCalculate conspicuous Hereby rigidity kh, bending stiffness kb, radial compression rigidity kaWith shearing rigidity ks, total mesh stiffness is the cascade of each rigidity.Respectively Relation between energy and rigidity can obtain corresponding rigidity according to cantilever beam theory according to the energy being stored in the gear teeth, Present embodiment is expressed for the relation between each energy and rigidity by following relation：
F is the interaction force of tooth at meshing point in formula (19)~formula (22), and direction is all the time along path of contact direction Intersect with flank profil；FbAnd FaFor power F two component, as shown in figure 11；D, x and h implication is specifically as shown in figure 11；E is bullet Property modulus, and different, I different according to the material of gearxThe area inertia moment of the part gear teeth between meshing point x and tooth root；G is Modulus of shearing, the modulus of shearing are constant, are obtained according to gear material；AxFor cross-sectional area.
It is assumed herein that the major parameter of power dividing type gear for two stage planetary gear train is as shown in table 1.
Table 1：The major parameter of power dividing type gear for two stage planetary gear train
Planetary gear train physical model is established in Pro/E softwares, is then introduced into ADAMS, the model importeding into ADAMS Type shape is as shown in Figure 3 and Figure 4；The good various restriction relations defined in ADAMS, in addition to first order planet carrier is arranged to fixed joint, Other all components are arranged to the revolute around its geometric center, at the same by input shaft respectively with first order gear ring, second level row Carrier sets fixed joint, and first order sun gear and second level gear ring are set into fixed joint；And calculated respectively by improving potential energy method Gear for two stage planetary gear train Gear Meshing Stiffness, and consider mesh phase relation between gear mesh, gear for two stage planetary gear train Gear Meshing Stiffness Result of calculation respectively such as Fig. 5, Fig. 6, as shown in 7 and Fig. 8, wherein Fig. 5 is that first order sun gear and planetary gear mesh stiffness are bent Line, Fig. 6 are first order planetary gear and gear ring mesh stiffness curve, and Fig. 7 is second level sun gear and planetary gear mesh stiffness curve, Fig. 8 is second level planetary gear and gear ring mesh stiffness curve；And import the time-variant mesh stiffness being calculated in ADAMS, when Become mesh stiffness and import influence caused by ADAMS by Fig. 9 expressions, shown wherein (a) is data, (b) is corresponding image Display；The engaging force between the gear teeth finally is described with the operation function of ADAMS offers：Figure 10 is first order planetary gear First pair of engagement force between sun gear and planetary gear and planetary gear and gear ring in system, from simulation result as can be seen that due to time-varying Mesh stiffness acts on, and periodic shock is caused, for first order planetary gear train, in given input speed n=13.9r/min bars Under part, meshing frequency 33Hz, therefore there are 33 Secondary Shocks in the case where simulation time is 1s.Its result is divided for the dynamic of planetary gear train Analysis, Optimal Structure Designing provide effective theoretical foundation.
The general principle and principal character and advantages of the present invention of the present invention has been shown and described above.The technology of the industry Personnel are it should be appreciated that the present invention is not limited to the above embodiments, and the simply explanation described in above-described embodiment and specification is originally The principle of invention, without departing from the spirit and scope of the present invention, various changes and modifications of the present invention are possible, these changes Change and improvement all fall within the protetion scope of the claimed invention.The claimed scope of the invention by appended claims and its Equivalent thereof.

Claims (9)

  1. A kind of 1. planetary gear train Nonlinear dynamic models method based on ADAMS, it is characterised in that：Comprise the following steps that：
    Step 1：Set the modeling conditions of planetary gear train；
    Step 2：Model and import the model built up in ADAMS, model is simplified, set environment parameter is simultaneously added about Beam, contact force and driving function；
    Step 3：Marker point Angle_ref are established in the barycenter of sun gear, planetary gear, ring gear, planet carrier, and are made each The direction of Marker point Z axis is identical with the Z axis under global coordinate system, first seeks each gear with respect to the Angle_ of planet carrier Marker points Ref Z axis angular displacement, relative displacement equation and relative linear velocity side of each gear in path of contact are obtained multiplied by with base radius Journey；
    Step 4：Gear time-variant mesh stiffness equation is obtained by improving potential energy method；
    Step 5：Obtain gear time-varying engagement Equation With Damping；
    Step 6：Model middle gear contact force is changed in ADAMS, it is 0 to make its value permanent, according to relative displacement equation, relative line Rate equation, gear time-variant mesh stiffness equation and gear time-varying engagement Equation With Damping define engagement force equation, complete planetary gear It is Nonlinear dynamic models.
  2. 2. the planetary gear train Nonlinear dynamic models method based on ADAMS as claimed in claim 1, it is characterised in that：Step Engagement force equation described in rapid six is：
    <mrow> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>x</mi> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>F</mi> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>x</mi> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    F in formula (1)spiRatcheting power, k between planetary gear i and sun gearspiTime-varying between planetary gear i and sun gear engages Rigidity, xspiFor planetary gear i and relative displacement of the sun gear in path of contact, cspiFor the time-varying between planetary gear i and sun gear Engagement damping,For planetary gear i and relative linear velocity of the sun gear in path of contact；
    K in formula (1)spiBy being obtained with next equation：
    <mrow> <msub> <mi>k</mi> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mi>h</mi> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mn>1</mn> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>b</mi> <mn>2</mn> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    Or
    <mrow> <msub> <mi>k</mi> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>2</mn> </munderover> <mfrac> <mn>1</mn> <mrow> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>h</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>b</mi> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>s</mi> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>b</mi> <mn>2</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>s</mi> <mn>2</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mn>2</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    During formula (3) is monodentate engagement, the time-variant mesh stiffness of a pair of gear vice presidents；Formula (4) is simultaneously participated in for two pairs of gear teeth and nibbled Total time-variant mesh stiffness during conjunction；Wherein khFor hertz rigidity, kbFor bending stiffness, kaFor radial compression rigidity, ksIt is firm to shear Degree；First pair of gear teeth meshing is represented during i=1；Second pair of gear teeth meshing is represented during i=2；Subscript 1,2 represents drive gear respectively With driven gear；WithObtained respectively by below equation：
    <mrow> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mi>h</mi> </msub> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mfrac> <mrow> <mi>&amp;pi;</mi> <mi>E</mi> <mi>L</mi> </mrow> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>&amp;upsi;</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> 1
    <mrow> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mi>b</mi> </msub> </mfrac> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> </mrow> <msub> <mi>&amp;alpha;</mi> <mn>5</mn> </msub> </msubsup> <mfrac> <mrow> <mn>3</mn> <msup> <mrow> <mo>{</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>cos&amp;alpha;</mi> <mn>1</mn> </msub> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;alpha;</mi> <mo>-</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>&amp;rsqb;</mo> <mo>}</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> </mrow> <mrow> <mn>12</mn> <mi>E</mi> <mi>L</mi> <msup> <mrow> <mo>&amp;lsqb;</mo> <mi>sin</mi> <mi>&amp;alpha;</mi> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mi>&amp;alpha;</mi> <mo>&amp;rsqb;</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mi>d</mi> <mi>&amp;alpha;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mi>a</mi> </msub> </mfrac> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> </mrow> <msub> <mi>&amp;alpha;</mi> <mn>5</mn> </msub> </msubsup> <mfrac> <mrow> <msub> <mi>&amp;alpha;</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <msup> <mi>cos&amp;alpha;sin</mi> <mn>2</mn> </msup> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>E</mi> <mi>L</mi> <mo>&amp;lsqb;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;alpha;</mi> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>&amp;rsqb;</mo> </mrow> </mfrac> <mi>d</mi> <mi>&amp;alpha;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mi>s</mi> </msub> </mfrac> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> </mrow> <msub> <mi>&amp;alpha;</mi> <mn>5</mn> </msub> </msubsup> <mfrac> <mrow> <mn>1.2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>&amp;upsi;</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <msup> <mi>cos&amp;alpha;cos</mi> <mn>2</mn> </msup> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> </mrow> <mrow> <mi>E</mi> <mi>L</mi> <mo>&amp;lsqb;</mo> <mi>sin</mi> <mi>&amp;alpha;</mi> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>&amp;rsqb;</mo> </mrow> </mfrac> <mi>d</mi> <mi>&amp;alpha;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    In formula (5)~formula (8)：E is modulus of elasticity, and L is gear axial width, and υ is Poisson's ratio, α1It is meshing point apart from tooth root d Place, the line at point of contact and gear centre of the path of contact on basic circle and the angle of gear teeth line of symmetry；α is meshing point apart from tooth root x Place, the line at point of contact and gear centre of the path of contact on basic circle and the angle of gear teeth line of symmetry；α2Handed over for basic circle and flank profil line The angle that line and gear teeth line of symmetry between point and gear centre are formed；α5Represent when meshing point and root circle distance are 0, Directed force F and decomposing force FbBetween angle, α in formula5Obtained by below equation group：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>R</mi> <mi>r</mi> </msub> <msub> <mi>sin&amp;alpha;</mi> <mn>4</mn> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>b</mi> </msub> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>5</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>cos&amp;alpha;</mi> <mn>5</mn> </msub> <mo>-</mo> <msub> <mi>sin&amp;alpha;</mi> <mn>5</mn> </msub> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>R</mi> <mi>r</mi> </msub> <msub> <mi>cos&amp;alpha;</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>R</mi> <mi>b</mi> </msub> <msub> <mi>cos&amp;alpha;</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>b</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>cos&amp;alpha;</mi> <mn>5</mn> </msub> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>5</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>sin&amp;alpha;</mi> <mn>5</mn> </msub> <mo>-</mo> <msub> <mi>cos&amp;alpha;</mi> <mn>5</mn> </msub> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
    R in formula (9)rFor root radius, RbFor base radius, α4For the company between root circle and flank profil line intersection point and gear centre The angle that line and gear teeth line of symmetry are formed；
    X in formula (1)spiBy being obtained with next equation：
    <mrow> <msub> <mi>x</mi> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>&amp;theta;</mi> <mi>s</mi> <mi>c</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>&amp;theta;</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> <mi>c</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
    Or
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>Z</mi> <mi>s</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>A</mi> <mi>Z</mi> <mrow> <mo>(</mo> <mi>S</mi> <mi>u</mi> <mi>n</mi> <mi>g</mi> <mi>e</mi> <mi>a</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>,</mo> <mi>C</mi> <mi>a</mi> <mi>r</mi> <mi>r</mi> <mi>i</mi> <mi>e</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>Z</mi> <mi>p</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>A</mi> <mi>Z</mi> <mrow> <mo>(</mo> <mi>P</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mi>e</mi> <mi>t</mi> <mo>,</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>,</mo> <mi>C</mi> <mi>a</mi> <mi>r</mi> <mi>r</mi> <mi>i</mi> <mi>e</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
    In formula (10)：Respectively sun gear relative to planet carrier angular displacement and planetary gear relative to planet carrier angle position Move, rbsFor the base radius of sun gear, rbpiFor the base radius of planetary gear,
    C in formula (1)spiObtained by below equation：
    <mrow> <mi>c</mi> <mo>=</mo> <mn>2</mn> <mi>&amp;zeta;</mi> <msqrt> <mrow> <mi>k</mi> <mfrac> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </mfrac> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
    In formula (12)：ζ is damping ratio, and k is time-variant mesh stiffness, m1、m2Gear mesh middle gear quality is represented respectively；
    In formula (1)Obtained by following two equations：
    <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> <mi>c</mi> </msubsup> <mo>+</mo> <msubsup> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>p</mi> <mi>i</mi> </mrow> <mi>c</mi> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
    Or
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>Z</mi> <mi>s</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>W</mi> <mi>Z</mi> <mrow> <mo>(</mo> <mi>S</mi> <mi>u</mi> <mi>n</mi> <mi>g</mi> <mi>e</mi> <mi>a</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>,</mo> <mi>C</mi> <mi>a</mi> <mi>r</mi> <mi>r</mi> <mi>i</mi> <mi>e</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>Z</mi> <mi>p</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>W</mi> <mi>Z</mi> <mrow> <mo>(</mo> <mi>P</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mi>e</mi> <mi>t</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>,</mo> <mi>C</mi> <mi>a</mi> <mi>r</mi> <mi>r</mi> <mi>i</mi> <mi>e</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
    In formula (14)：WithRespectively sun gear with respect to planet carrier linear velocity and planetary gear with respect to planet carrier linear velocity, can be with Tried to achieve by corresponding angular velocity and the product of base radius；
    F in formula (2)rpiRatcheting power between planetary gear i and ring gear, krpiTime-varying between planetary gear i and ring gear is nibbled Close rigidity, xrpiFor planetary gear i and too relative displacement of the ring gear in path of contact, crpiBetween planetary gear i and ring gear Time-varying engagement damping,For planetary gear i and relative linear velocity of the ring gear in path of contact；
    K in formula (2)rpiObtained by equation (3) or (4)；
    X in formula (2)rpiObtained by following two equations：
    <mrow> <msub> <mi>x</mi> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>&amp;theta;</mi> <mi>r</mi> <mi>c</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>-</mo> <msubsup> <mi>&amp;theta;</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> <mi>c</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
    Or
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>Z</mi> <mi>r</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>A</mi> <mi>Z</mi> <mrow> <mo>(</mo> <mi>R</mi> <mi>i</mi> <mi>n</mi> <mi>g</mi> <mi>g</mi> <mi>e</mi> <mi>a</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>,</mo> <mi>C</mi> <mi>a</mi> <mi>r</mi> <mi>r</mi> <mi>i</mi> <mi>e</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>Z</mi> <mi>p</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>A</mi> <mi>Z</mi> <mrow> <mo>(</mo> <mi>P</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mi>e</mi> <mi>t</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>,</mo> <mi>C</mi> <mi>a</mi> <mi>r</mi> <mi>r</mi> <mi>i</mi> <mi>e</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
    In formula (16)WithFor gear ring relative to the angular displacement and planetary gear of planet carrier relative to the angular displacement of planet carrier, rbrFor The base radius of ring gear, rbpiFor the base radius of planetary gear；
    C in formula (2)rpiObtained by equation (12)：
    In formula (2)By being obtained with next equation：
    <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>r</mi> <mi>c</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>p</mi> <mi>i</mi> </mrow> <mi>c</mi> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
    Or
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>Z</mi> <mi>r</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>W</mi> <mi>Z</mi> <mrow> <mo>(</mo> <mi>R</mi> <mi>i</mi> <mi>n</mi> <mi>g</mi> <mi>g</mi> <mi>e</mi> <mi>a</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>,</mo> <mi>C</mi> <mi>a</mi> <mi>r</mi> <mi>r</mi> <mi>i</mi> <mi>e</mi> <mi>r</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>Z</mi> <mi>p</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>W</mi> <mi>Z</mi> <mrow> <mo>(</mo> <mi>P</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mi>e</mi> <mi>t</mi> <mo>.</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>,</mo> <mi>C</mi> <mi>a</mi> <mi>r</mi> <mi>r</mi> <mi>i</mi> <mi>e</mi> <mi>r</mi> <mo>,</mo> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mi>l</mi> <mi>e</mi> <mo>_</mo> <mi>r</mi> <mi>e</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
    AZ, WZ are respectively angular displacement function and angular speed function in formula (11), formula (14), formula (16) and formula (18)；Zs、Zp、Zr The respectively number of teeth of sun gear, planetary gear and ring gear；α is pressure angle；M is modulus.
  3. 3. the planetary gear train Nonlinear dynamic models method based on ADAMS as claimed in claim 1 or 2, it is characterised in that： Modeling conditions are described in step 1：
    (1) all gears are standard involute spur in train；
    (2) two gear gear blanks are considered as rigid body, and the input of train and output shaft are rigid body, do not consider the elastic deformation of support；
    (3) train middle gear is installed by reference center distance, and pitch circle overlaps with reference circle；
    (4) disregard gear error and tooth surveys gap.
  4. 4. the planetary gear train Nonlinear dynamic models method based on ADAMS as claimed in claim 1 or 2, it is characterised in that： Contact force selection impulse function method described in step 2 is calculated.
  5. 5. the planetary gear train Nonlinear dynamic models method based on ADAMS as claimed in claim 3, it is characterised in that：Step Contact force selection impulse function method described in rapid two is calculated.
  6. 6. the planetary gear train Nonlinear dynamic models method based on ADAMS as claimed in claim 1 or 2, it is characterised in that： Improvement potential energy method described in step 4 is the cantilever beam being reduced to the gear teeth in model in tooth root institute first, is stored in engagement Potential energy in gear includes Hertz contact potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy Us, pass through hertz Contact potential Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy UsCalculate hertz rigidity kh, bending stiffness kb, radially Compression stiffness kaWith shearing rigidity ks, total mesh stiffness is the cascade of each rigidity；Each energy meets following relation with rigidity：
    <mrow> <msub> <mi>U</mi> <mi>h</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>h</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>b</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>b</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow> 3
    <mrow> <msub> <mi>U</mi> <mi>a</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>a</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>s</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
    In formula (19)~formula (22) F be meshing point at tooth interaction force, direction along path of contact direction, and be all the time with tooth Exterior feature is intersecting.
  7. 7. the planetary gear train Nonlinear dynamic models method based on ADAMS as claimed in claim 3, it is characterised in that：Step Improvement potential energy method described in rapid four is the cantilever beam being reduced to the gear teeth in model in tooth root institute first, is stored in engaging tooth Potential energy in wheel includes Hertz contact potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy Us, connect by hertz Touch potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy UsCalculate hertz rigidity kh, bending stiffness kb, radially press Contracting rigidity kaWith shearing rigidity ks, total mesh stiffness is the cascade of each rigidity；Each energy meets following relation with rigidity：
    <mrow> <msub> <mi>U</mi> <mi>h</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>h</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>b</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>b</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>a</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>a</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>s</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
    In formula (19)~formula (22) F be meshing point at tooth interaction force, direction along path of contact direction, and be all the time with tooth Exterior feature is intersecting.
  8. 8. the planetary gear train Nonlinear dynamic models method based on ADAMS as claimed in claim 4, it is characterised in that：Step Improvement potential energy method described in rapid four is the cantilever beam being reduced to the gear teeth in model in tooth root institute first, is stored in engaging tooth Potential energy in wheel includes Hertz contact potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy Us, connect by hertz Touch potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy UsCalculate hertz rigidity kh, bending stiffness kb, radially press Contracting rigidity kaWith shearing rigidity ks, total mesh stiffness is the cascade of each rigidity；Each energy meets following relation with rigidity：
    <mrow> <msub> <mi>U</mi> <mi>h</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>h</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>b</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>b</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>a</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>a</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>s</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
    In formula (19)~formula (22) F be meshing point at tooth interaction force, direction along path of contact direction, and be all the time with tooth Exterior feature is intersecting.
  9. 9. the planetary gear train Nonlinear dynamic models method based on ADAMS as claimed in claim 5, it is characterised in that：Step Improvement potential energy method described in rapid four is the cantilever beam being reduced to the gear teeth in model in tooth root institute first, is stored in engaging tooth Potential energy in wheel includes Hertz contact potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy Us, connect by hertz Touch potential energy Uh, bowing potential energy Ub, radial compression potential energy UaWith shearing potential energy UsCalculate hertz rigidity kh, bending stiffness kb, radially press Contracting rigidity kaWith shearing rigidity ks, total mesh stiffness is the cascade of each rigidity；Each energy meets following relation with rigidity：
    <mrow> <msub> <mi>U</mi> <mi>h</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>h</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>b</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>b</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>a</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>a</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>U</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <msup> <mi>F</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mi>s</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
    In formula (19)~formula (22) F be meshing point at tooth interaction force, direction along path of contact direction, and be all the time with tooth Exterior feature is intersecting.
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