CN107341313B - ADAMS-based planetary gear train nonlinear dynamics modeling method - Google Patents
ADAMS-based planetary gear train nonlinear dynamics modeling method Download PDFInfo
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Abstract
The invention discloses an ADAMS-based planetary gear train nonlinear dynamics modeling method, wherein the modeling method includes time-varying engagement stiffness, time-varying engagement damping and related parameter excitation in a model, the model is strong in universality, and dynamics simulation of a planetary gear train under any working condition can be performed; the calculation result of the gear meshing force is more accurate, the influence of the accuracy of the calculation result on the accuracy of the geometric shape of the established model is effectively avoided, and the calculation speed and the accuracy are high. In addition, the gear engagement force equation provided by the invention is suitable for planetary gear trains with different structures, such as a planetary gear train with a fixed gear ring and a differential planetary gear train.
Description
Technical Field
The invention relates to a dynamics modeling method, in particular to an ADAMS-based planetary gear train nonlinear dynamics modeling method.
Background
There are many difficulties in the kinematic and kinetic analysis of complex mechanical systems: the construction of kinetic equations is subject to heavy algebraic and differential calculations and closed analytical solutions are not possible due to the non-linearity of the equations. In this context, the use of computers to solve the analysis and synthesis problems of complex systems has been a major and rapidly evolving research direction in the general field of mechanics in the last two decades. Currently, the multi-body dynamics simulation software (ADAMS) is the most authoritative and widely used mechanical system dynamics software in the world. A user can generate a multi-body dynamics mathematical virtual prototype model of any complex system by using ADAMS software, and can provide comprehensive simulation calculation analysis results from product concept design, scheme demonstration, detailed design, product scheme modification, optimization, test planning and even fault diagnosis stages for the user.
For gear transmission with wide application in mechanical systems, an ADAMS/View provides an impact function for calculating contact force to represent meshing force between gears. The method needs longer time for solving the calculation, and the solving efficiency is low. And the accuracy of the simulation results depends to a large extent on the accuracy of the geometric model. Meanwhile, an actual gear transmission system is a complex dynamic system, and a large number of dynamic excitation factors exist in the meshing process, such as time-varying meshing stiffness, time-varying meshing damping, different tooth-to-tooth meshing phases and the like. Therefore, there is a need for an improvement in the meshing forces between gear teeth.
Disclosure of Invention
The invention provides an ADAMS (adaptive dynamic random access memory) planetary gear train-based nonlinear dynamics modeling method, which considers the time-varying meshing stiffness, the time-varying meshing damping and the tooth-to-tooth phase difference of gears and aims to solve the problem that the meshing acting force between gear teeth is difficult to accurately express in ADAMS software.
In order to achieve the purpose of the invention, the ADAMS-based planetary gear train nonlinear dynamics modeling method provided herein specifically comprises the following steps:
the method comprises the following steps: setting modeling conditions of a planetary gear train;
step two: modeling, introducing the established model into ADAMS, simplifying the model, setting environmental parameters and adding constraints, contact forces and driving functions;
step three: establishing Marker points Angle _ ref at the centers of mass of the sun wheel, the planet wheel, the inner gear ring and the planet carrier, enabling the direction of the Z axis of each Marker point to be the same as that of the Z axis under a global coordinate system, firstly solving the Z axis angular displacement of each gear relative to the Marker points Angle _ ref of the planet carrier, and then multiplying the Z axis angular displacement by the radius of a base circle to obtain a relative displacement equation and a relative linear velocity equation of each gear on a meshing line;
step four: acquiring a gear time-varying meshing stiffness equation by improving a potential energy method;
step five: acquiring a gear time-varying meshing damping equation;
step six: and modifying the gear contact force in the model in ADAMS to ensure that the value of the gear contact force is constant to 0, and defining an engagement force equation according to a relative displacement equation, a relative linear velocity equation, a gear time-varying engagement stiffness equation and a gear time-varying engagement damping equation to complete the nonlinear dynamics modeling of the planetary gear train.
The modeling method provided by the invention has the advantages that the time-varying meshing stiffness, the time-varying meshing damping, the relative displacement, the speed and other parameter excitations are added into the model, the built model is strong in universality, and the dynamic simulation of the planetary gear train under any working condition can be carried out; the calculation result of the gear meshing force is more accurate, the influence of the accuracy of the calculation result on the accuracy of the geometric shape of the established model is effectively avoided, and the calculation speed and the accuracy are high.
Specifically, the meshing force equation in the sixth step is as follows:
in the formula (1), FspiIs the tooth resultant force, k, between the planet wheel i and the sun wheelspiIs the time-varying meshing stiffness, x, between the planet wheel i and the sun wheelspiIs the relative displacement of the planet gear i and the sun gear on the meshing line, cspiDamping the time-varying mesh between the planet i and the sun,
the relative linear velocity of the planet wheel i and the sun wheel on the meshing line;
in the formula (1), kspiObtained by one of the following equations:
or
The formula (3) is the total time-varying meshing stiffness of the pair of gear pairs during single-tooth meshing; the formula (4) is the total time-varying meshing stiffness when the two pairs of gear teeth participate in meshing simultaneously; wherein k ishIs Hertz stiffness, kbTo bending stiffness, kaTo radial compressive stiffness, ksIs the shear stiffness; when i is 1, the first pair of gear teeth are meshed; when i is 2, the second pair of gear teeth are meshed; subscripts 1, 2 denote a drive gear and a driven gear, respectively;
and
obtained by the following equations, respectively:
in the formulas (5) to (8), E is the elastic modulus, L is the axial thickness of the gear, upsilon is the Poisson ratio, α1α is the angle between the line connecting the tangent point of the meshing line on the base circle and the gear center and the symmetry line of the gear teeth at the position of the meshing point on the base circle away from the tooth root d, and α is the angle between the line connecting the tangent point of the meshing line on the base circle and the gear center and the symmetry line of the gear teeth at the position of the meshing point away from the tooth root xα2α is the included angle formed by the connecting line between the intersection point of the base circle and the tooth profile line and the center of the gear and the symmetric line of the gear teeth5Indicates that the acting force F and the resolving force F are applied when the distance between the meshing point and the tooth root circle is 0bAngle therebetween, formula α5Obtained by the following system of equations:
r in the formula (9)rRoot circle radius, RbRadius of base circle, α4The included angle is formed by a connecting line between the intersection point of the root circle and the tooth profile line and the center of the gear and a symmetric line of the gear teeth;
x in the formula (1)spiObtained by one of the following equations:
or
In formula (10):
the angular displacement of the sun gear relative to the planet carrier and the planet gear relative to the planet carrier, rbsIs the base radius of the sun gear rbpiIs the base radius of the planet wheel,
c in formula (1)spiObtained by the following equation:
in formula (12): zeta is damping ratio, k is time-varying meshing stiffness, m1、m2Respectively representing the mass of the gears in the gear pair;
in the formula (1)
Obtained by the following two equations:
or
In formula (14):
and
the linear speed of the sun wheel relative to the planet carrier and the linear speed of the planet wheel relative to the planet carrier can be obtained by the product of the corresponding angular speed and the radius of the base circle;
in the formula (2), FrpiIs the meshing force, k, between the planet wheel i and the inner gear ringrpiIs the time-varying meshing rigidity, x, between the planet wheel i and the inner gear ringrpiFor the relative displacement of the planet gear i and the solar-internal gear ring on the meshing line, crpiDamping the time-varying meshing between the planet wheel i and the inner gear ring,
the relative linear velocity of the planet gear i and the inner gear ring on the meshing line is obtained;
in the formula (2), krpiObtained by equation (3) or (4);
x in the formula (2)rpiObtained by the following two equations:
or
In formula (16)
And
for angular displacements of the ring gear relative to the planet carrier and of the planet gears relative to the planet carrier, rbrIs the base radius of the inner gear ring, rbpiIs the base radius of the planet wheel;
c in formula (2)rpiObtained by equation (12):
in the formula (2)
Obtained by one of the following equations:
or
AZ and WZ in the formulas (11), (14), (16) and (18) are an angular displacement function and an angular velocity function respectively; zs、Zp、ZrThe numbers of teeth of the sun gear, the planet gear and the inner gear ring are respectively, α is a pressure angle, and m is a modulus.
Specifically, the modeling conditions in the first step are as follows:
(1) all gears in the gear train are standard involute straight-tooth cylindrical gears;
(2) the two gear blanks are regarded as rigid bodies, and the input shaft and the output shaft of the gear train are rigid bodies, and the elastic deformation of the support is not considered;
(3) the gears in the gear train are installed according to the standard center distance, and the pitch circle of the gears is superposed with the reference circle;
(4) gear errors and tooth measurement gaps are not counted.
Specifically, the contact force selection impulse function method in the step two is used for calculation.
Specifically, the improved potential energy method described in the fourth step is firstlySimplifying the gear teeth in the model into cantilever beams on a tooth root yard, wherein the potential energy stored in the meshed gears comprises Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsThrough Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsCalculating the Hertz stiffness khBending stiffness kbRadial compression stiffness kaAnd shear stiffness ksThe total meshing stiffness is in a series form of all the stiffnesses; each energy and stiffness satisfy the following relationship:
in the formulas (19) to (22), F is the interaction force of the teeth at the meshing point, the direction is along the meshing line direction, and the direction is always intersected with the tooth profile.
The gear engagement force equation provided by the invention is suitable for planetary gear trains with different structures, such as a planetary gear train with a fixed gear ring and a differential planetary gear train.
The invention has the beneficial effects that: the time-varying meshing stiffness, the time-varying meshing damping and the related parameter excitation are included in the model, the built model is strong in universality and can be used for dynamic simulation of the planetary gear train under any working condition; the calculation result of the gear meshing force is more accurate, the influence of the accuracy of the calculation result on the accuracy of the geometric shape of the established model is effectively avoided, and the calculation speed and the accuracy are high.
Drawings
FIG. 1 is a schematic diagram of a two-stage planetary gear system according to the present invention;
FIG. 2 is a flow chart of a modeling method for ADAMS-based non-linear dynamics of a planetary gear train provided by the invention;
FIG. 3 is a schematic structural diagram 1 of a two-stage planetary gear train nonlinear dynamics model established by the invention;
FIG. 4 is a schematic structural diagram 2 of a two-stage planetary gear train nonlinear dynamics model established by the invention;
FIG. 5 is a first stage sun gear and planet gear meshing stiffness curve;
FIG. 6 is a graph of the meshing stiffness of the first-stage planet gears and the ring gear;
FIG. 7 is a second stage sun gear and planet gear meshing stiffness curve;
FIG. 8 is a curve of the meshing stiffness of the second stage planet gears and the ring gear;
FIG. 9 is a graph of stiffness data introduced into ADAMS;
FIG. 10 is a schematic diagram of gear mesh forces calculated using the mesh force equation provided by the present invention;
FIG. 11 is a schematic diagram of the force applied to the teeth.
Detailed Description
The technical scheme of the invention is further explained by combining the drawings and the specific embodiments.
The modeling is carried out by taking a power split type two-stage planetary gear train as an example, and the power split type two-stage planetary gear train is schematically shown in figure 1; the gear train is composed of two-stage planetary gear trains, wherein the first stage is a common planetary gear train, and the second stage is a differential planetary gear mechanism. The power is input by the main shaft, the power is divided into two parts, one part of the power is transmitted to the first-stage inner gear ring through the main shaft, the other part of the power is directly connected with the second-stage planet carrier through the main shaft, and finally the power is coupled to the second-stage sun gear through the differential planet to realize power confluence. Modeling is carried out on ADAMS software, time-varying meshing stiffness and time-varying meshing damping are considered during modeling, and the modeling steps are shown in FIG. 2 and specifically include:
the method comprises the following steps: setting modeling conditions according to the planetary gear train, wherein the modeling conditions can be different according to different planetary gear trains, and the modeling conditions adopted in the application are specifically as follows:
1. all gears in the gear train are standard involute straight-tooth cylindrical gears;
2. the two gear blanks are regarded as rigid bodies, and the input shaft and the output shaft of the gear train are rigid bodies, and the elastic deformation of the support is not considered;
3. the gears in the gear train are installed according to the standard center distance, and the pitch circle of the gears is superposed with the reference circle;
4. gear errors and tooth measurement gaps are not counted;
step two: establishing a planetary gear train solid model through Pro/E, SolidWorks or other three-dimensional modeling software, as shown in FIGS. 3 and 4; the model is introduced into ADAMS after being built, the model is simplified under the ADAMS environment, and a planetary gear shaft are simplified into a whole through a Boolean operation tool; setting environmental parameters, specifically: respectively inputting the mass and the mass center mark points of each component as rotational inertia reference mark points; defining various constraint relations, except that the first-stage planet carrier is set as a fixed pair, all other components are set as rotating pairs around the geometric center of the fixed pair, meanwhile, the input shaft is respectively provided with the fixed pairs with the first-stage gear ring and the second-stage planet carrier, and the first-stage sun gear and the second-stage gear ring are provided with the fixed pairs; when the contact force is set, any function in ADAMS can be adopted for calculation, and the impulse function is adopted for calculation in the application;
step three: establishing Marker points Angle _ ref at the centers of mass of the sun wheel, the planet wheel, the inner gear ring and the planet carrier, enabling the direction of the Z axis of each Marker point to be the same as that of the Z axis under a global coordinate system, firstly solving the Z axis angular displacement of each gear relative to the Marker points Angle _ ref of the planet carrier, and then multiplying the Z axis angular displacement by the radius of a base circle to obtain a relative displacement equation and a relative linear velocity equation of each gear on a meshing line;
step four: acquiring a gear time-varying meshing stiffness equation by improving a potential energy method;
step five: acquiring a gear time-varying meshing damping equation;
step six: and modifying the gear contact force in the model in ADAMS to ensure that the value of the gear contact force is constant to 0, and defining an engagement force equation according to a relative displacement equation, a relative linear velocity equation, a gear time-varying engagement stiffness equation and a gear time-varying engagement damping equation to complete the nonlinear dynamics modeling of the planetary gear train.
The planetary gear nonlinear dynamical model established by the modeling method comprises a meshing force equation between the planetary gear i and the sun gear and a meshing force equation between the planetary gear i and the inner gear ring, the meshing force equation between the planetary gear i and the sun gear established in the method is shown as a formula (1), the meshing force equation between the planetary gear i and the inner gear ring is shown as a formula (2),
in the formula (1), FspiIs the tooth resultant force, k, between the planet wheel i and the sun wheelspiIs the time-varying meshing stiffness, x, between the planet wheel i and the sun wheelspiIs the relative displacement of the planet gear i and the sun gear on the meshing line, cspiDamping the time-varying mesh between the planet i and the sun,
the relative linear velocity of the planet wheel i and the sun wheel on the meshing line;
in the formula (1), kspiObtained by one of the following equations:
or
The formula (3) is the total time-varying meshing stiffness of the pair of gear pairs during single-tooth meshing; the formula (4) is the total time-varying meshing stiffness when the two pairs of gear teeth participate in meshing simultaneously; wherein k ishIs Hertz stiffness, kbTo bending stiffness, kaTo radial compressive stiffness, ksIs the shear stiffness; when i is 1, the first pair of gear teeth are meshed; when i is 2, the second pair of gear teeth are meshed; subscripts 1, 2 denote a drive gear and a driven gear, respectively;
and
obtained by the following equations, respectively:
in the formulas (5) to (8), E is an elastic modulus which is a constant and is different according to gear materials, such as 200-210 GPa for low-carbon steel, 205GPa for medium-carbon steel, 210 for alloy steel, 150-180 for ductile cast iron, 71 for aluminum alloy, L is the axial thickness of the gear, upsilon is a Poisson ratio which is a constant and is different according to gear materials, such as 0.24-0.28 for low-carbon steel, 0.24-0.28 for medium-carbon steel, 0.25-0.30 for alloy steel and 0.33 for aluminum alloy, and α in the formulas (5) to (8) is compared with FIG. 111、α、α2And α5To explain, wherein α1The included angle between the connecting line of the tangent point of the meshing line on the base circle and the gear center and the gear tooth symmetry line is d, the included angle between the connecting line of the tangent point of the meshing line on the base circle and the gear tooth symmetry line is α, the included angle between the connecting line of the tangent point of the meshing line on the base circle and the gear center and the gear tooth symmetry line is α2α is the included angle formed by the connecting line between the intersection point of the base circle and the tooth profile line and the center of the gear and the symmetric line of the gear teeth5Indicates that the acting force F and the resolving force F are applied when the distance between the meshing point and the tooth root circle is 0bAngle therebetween, formula α5Obtained by the following system of equations:
r in the formula (9)rRoot circle radius, RbRadius of base circle, α4Is an included angle formed by a connecting line between the intersection point of the root circle and the tooth profile line and the center of the gear and a symmetric line of the gear teeth, as shown in fig. 11;
x in the formula (1)sp iObtained by one of the following equations:
or
In formula (10):
the angular displacement of the sun gear relative to the planet carrier and the planet gear relative to the planet carrier, rbsIs the base radius of the sun gear rbpiIs the base radius of the planet wheel,
c in formula (1)spiObtained by the following equation:
in formula (12): zeta is damping ratio, k is time-varying meshing stiffness, m1、m2Respectively representing the mass of the gears in the gear pair;
in the formula (1)
Obtained by the following two equations:
or
In formula (14):
and
the linear speed of the sun wheel relative to the planet carrier and the linear speed of the planet wheel relative to the planet carrier can be obtained by the product of the corresponding angular speed and the radius of the base circle;
in the formula (2), FrpiIs the meshing force, k, between the planet wheel i and the inner gear ringrpiIs the time-varying meshing rigidity, x, between the planet wheel i and the inner gear ringrpiFor the relative displacement of the planet gear i and the solar-internal gear ring on the meshing line, crpiDamping the time-varying meshing between the planet wheel i and the inner gear ring,
the relative linear velocity of the planet gear i and the inner gear ring on the meshing line is obtained;
in the formula (2), krpiObtained by equation (3) or (4);
x in the formula (2)rpiObtained by the following two equations:
or
In formula (16)
And
for angular displacements of the ring gear relative to the planet carrier and of the planet gears relative to the planet carrier, rbrIs the base radius of the inner gear ring, rbpiIs the base radius of the planet wheel;
c in formula (2)rpiObtained by equation (12):
in the formula (2)
Obtained by one of the following equations:
or
AZ and WZ in the formulas (11), (14), (16) and (18) are an angular displacement function and an angular velocity function respectively; zs、Zp、ZrThe self-adaptive automatic transmission system comprises a sun gear, a planet gear and an inner gear ring, wherein the sun gear, the planet gear and the inner gear ring are respectively provided with tooth numbers, α is a pressure angle, m is a modulus, and formulas (11), (14), (16) and (18) are expressed under an ADAMS/View running function, wherein a function AZ is used for returning the angle of a certain Marker point rotating around a Z axis of another Marker point, and a function WZ is used for returning the angular velocity vector difference of the two Marker points on the component of the Z axis.
In addition, the improved potential energy method adopted in the fourth step of the embodiment is to firstly simplify the gear teeth in the model into cantilever beams on a tooth root yard, and the potential energy stored in the meshed gears comprises Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsThrough Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsCalculating the Hertz stiffness khBending stiffness kbRadial compression stiffness kaAnd shear stiffness ksThe total meshing stiffness is in the form of a series of individual stiffnesses. The relationship between each energy and the stiffness can be obtained according to the cantilever theory and the energy stored in the gear teeth, and the embodiment is expressed by the following relationship for the relationship between each energy and the stiffness:
f in the formulas (19) to (22) is the interaction force of the teeth at the meshing point, the direction is along the meshing line direction, and the interaction force is always intersected with the tooth profile; fbAnd FaTwo components of force F, as shown in fig. 11; d. the meanings of x, and h are specifically shown in FIG. 11; e is the elastic modulus, which varies depending on the material of the gear, IxThe area moment of inertia of part of gear teeth between the meshing point x and the tooth root; g is a shear modulus which is a constant and is obtained according to a gear material; a. thexIs the cross-sectional area.
It is assumed here that the main parameters of the power split two-stage planetary gear train are as shown in table 1.
Table 1: main parameters of power split type two-stage planetary gear train
Establishing a planetary gear train entity model in Pro/E software, then introducing into ADAMS, wherein the model shape introduced into ADAMS is shown in figures 3 and 4; various constraint relations are defined in ADAMS, except that a first-stage planet carrier is set as a fixed pair, all other components are set as rotating pairs around the geometric center of the first-stage planet carrier, an input shaft and a first-stage gear ring and a second-stage planet carrier are respectively provided with fixed pairs, and a first-stage sun gear and a second-stage gear ring are provided with fixed pairs; the meshing stiffness of the gears of the two-stage planetary gear train is respectively calculated by an improved potential energy method, and the meshing phase relationship between the gear pairs is considered, and the calculation results of the meshing stiffness of the gears of the two-stage planetary gear train are respectively shown in fig. 5, fig. 6, fig. 7 and fig. 8, wherein fig. 5 is a first-stage sun gear and planetary gear meshing stiffness curve, fig. 6 is a first-stage planetary gear and gear ring meshing stiffness curve, fig. 7 is a second-stage sun gear and planetary gear meshing stiffness curve, and fig. 8 is a second-stage planetary gear and gear ring meshing stiffness curve; introducing the calculated time-varying meshing stiffness into the ADAMS, wherein the influence of the time-varying meshing stiffness introduced into the ADAMS is represented by fig. 9, in which (a) is data display and (b) is corresponding image display; finally, the meshing force between the gear teeth is described by using an operation function provided by ADAMS: fig. 10 shows the meshing forces between the first pair of sun gears and planetary gears and between the planetary gears and the ring gear in the first-stage planetary gear train, and it can be seen from the simulation result that periodic impacts are caused due to the time-varying meshing stiffness effect, and for the first-stage planetary gear train, under the condition that the given input rotation speed n is 13.9r/min, the meshing frequency is 33Hz, so that 33 impacts are caused under the condition that the simulation time is 1 s. The result provides effective theoretical basis for the dynamic analysis and the structure optimization design of the planetary gear train.
The foregoing shows and describes the general principles and broad features of the present invention and advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (8)
1. An ADAMS-based planetary gear train nonlinear dynamics modeling method is characterized in that: the method comprises the following specific steps:
the method comprises the following steps: setting modeling conditions of a planetary gear train;
step two: modeling, introducing the established model into ADAMS, simplifying the model, setting environmental parameters and adding constraints, contact forces and driving functions;
step three: establishing Marker points Angle _ ref at the centers of mass of the sun wheel, the planet wheel, the inner gear ring and the planet carrier, enabling the direction of the Z axis of each Marker point to be the same as that of the Z axis under a global coordinate system, firstly solving the Z axis angular displacement of each gear relative to the Marker points Angle _ ref of the planet carrier, and then multiplying the Z axis angular displacement by the radius of a base circle to obtain a relative displacement equation and a relative linear velocity equation of each gear on a meshing line;
step four: acquiring a gear time-varying meshing stiffness equation by improving a potential energy method;
step five: acquiring a gear time-varying meshing damping equation;
step six: modifying the gear contact force in the model in ADAMS to make the value constant to 0, defining an engagement force equation according to a relative displacement equation, a relative linear velocity equation, a gear time-varying engagement stiffness equation and a gear time-varying engagement damping equation, and completing the nonlinear dynamics modeling of the planetary gear train; the meshing force equation is as follows:
in the formula (1), FspiIs the tooth resultant force, k, between the planet wheel i and the sun wheelspiIs the time-varying meshing stiffness, x, between the planet wheel i and the sun wheelspiIs the relative displacement of the planet gear i and the sun gear on the meshing line, cspiDamping the time-varying mesh between the planet i and the sun,
the relative linear velocity of the planet wheel i and the sun wheel on the meshing line;
in the formula (1), kspiObtained by one of the following equations:
or
The formula (3) is the total time-varying meshing stiffness of the pair of gear pairs during single-tooth meshing; the formula (4) is the total time-varying meshing stiffness when the two pairs of gear teeth participate in meshing simultaneously; wherein k ishIs Hertz stiffness, kbTo bending stiffness, kaTo radial compressive stiffness, ksIs the shear stiffness; when i is 1, the first pair of gear teeth are meshed; when i is 2, the second pair of gear teeth are meshed; subscripts 1, 2 denote a drive gear and a driven gear, respectively;
and
obtained by the following equations, respectively:
in the formulas (5) to (8), E is the elastic modulus, L is the axial thickness of the gear, upsilon is the Poisson ratio, α1The included angle between the connecting line of the tangent point of the meshing line on the base circle and the gear center and the gear tooth symmetry line is d, the included angle between the connecting line of the tangent point of the meshing line on the base circle and the gear tooth symmetry line is α, the included angle between the connecting line of the tangent point of the meshing line on the base circle and the gear center and the gear tooth symmetry line is α2α is the included angle formed by the connecting line between the intersection point of the base circle and the tooth profile line and the center of the gear and the symmetric line of the gear teeth5Indicates that the acting force F and the resolving force F are applied when the distance between the meshing point and the tooth root circle is 0bAngle therebetween, formula α5Obtained by the following system of equations:
r in the formula (9)rRoot circle radius, RbRadius of base circle, α4The included angle is formed by a connecting line between the intersection point of the root circle and the tooth profile line and the center of the gear and a symmetric line of the gear teeth;
x in the formula (1)spiObtained by one of the following equations:
or
In formula (10):
the angular displacement of the sun gear relative to the planet carrier and the planet gear relative to the planet carrier, rbsIs the base radius of the sun gear rbpiIs the base radius of the planet wheel,
c in formula (1)spiObtained by the following equation:
in formula (12): zeta is damping ratio, k is time-varying meshing stiffness, m1、m2Respectively representing the mass of the gears in the gear pair;
in the formula (1)
Obtained by the following two equations:
or
In formula (14):
and
the linear speed of the sun wheel relative to the planet carrier and the linear speed of the planet wheel relative to the planet carrier can be obtained by the product of the corresponding angular speed and the radius of the base circle;
in the formula (2), FrpiIs the meshing force, k, between the planet wheel i and the inner gear ringrpiIs the time-varying meshing rigidity, x, between the planet wheel i and the inner gear ringrpiFor the relative displacement of the planet gear i and the solar-internal gear ring on the meshing line, crpiDamping the time-varying meshing between the planet wheel i and the inner gear ring,
the relative linear velocity of the planet gear i and the inner gear ring on the meshing line is obtained;
in the formula (2), krpiObtained by equation (3) or (4);
x in the formula (2)rpiBy the following twoEquation acquisition:
or
In formula (16)
And
for angular displacements of the ring gear relative to the planet carrier and of the planet gears relative to the planet carrier, rbrIs the base radius of the inner gear ring, rbpiIs the base radius of the planet wheel;
c in formula (2)rpiObtained by equation (12):
in the formula (2)
Obtained by one of the following equations:
or
AZ and WZ in the formulas (11), (14), (16) and (18) are an angular displacement function and an angular velocity function respectively; zs、Zp、ZrThe numbers of teeth of the sun gear, the planet gear and the inner gear ring are respectively, α is a pressure angle, and m is a modulus.
2. An ADAMS-based non-linear dynamics modeling method of planetary gear trains according to claim 1, characterized in that:
in the first step, the modeling conditions are as follows:
(1) all gears in the gear train are standard involute straight-tooth cylindrical gears;
(2) the two gear blanks are regarded as rigid bodies, and the input shaft and the output shaft of the gear train are rigid bodies, and the elastic deformation of the support is not considered;
(3) the gears in the gear train are installed according to the standard center distance, and the pitch circle of the gears is superposed with the reference circle;
(4) gear errors and tooth measurement gaps are not counted.
3. An ADAMS-based non-linear dynamics modeling method of planetary gear trains according to claim 1, characterized in that:
and step two, the contact force selection impulse function method is used for calculation.
4. An ADAMS-based non-linear dynamics modeling method of planetary gear trains according to claim 2, characterized in that: and step two, the contact force selection impulse function method is used for calculation.
5. An ADAMS-based non-linear dynamics modeling method of planetary gear trains according to claim 1, characterized in that: the potential energy improvement method described in the fourth step is to simplify the gear teeth in the model into cantilever beams on the root yard, and the potential energy stored in the meshed gears includes Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsThrough Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsCalculating the Hertz stiffness khBending stiffness kbRadial compression stiffness kaAnd shear stiffness ksThe total meshing stiffness is in a series form of all the stiffnesses; each energy and stiffness satisfy the following relationship:
in the formulas (19) to (22), F is the interaction force of the teeth at the meshing point, the direction is along the meshing line direction, and the direction is always intersected with the tooth profile.
6. An ADAMS-based non-linear dynamics modeling method of planetary gear trains according to claim 2, characterized in that: the potential energy improvement method described in the fourth step is to simplify the gear teeth in the model into cantilever beams on the root yard, and the potential energy stored in the meshed gears includes Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsThrough Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsCalculating the Hertz stiffness khBending stiffness kbRadial compression stiffness kaAnd shear stiffness ksThe total meshing stiffness is in a series form of all the stiffnesses; each energy and stiffness satisfy the following relationship:
in the formulas (19) to (22), F is the interaction force of the teeth at the meshing point, the direction is along the meshing line direction, and the direction is always intersected with the tooth profile.
7. An ADAMS-based non-linear dynamics modeling method of planetary gear trains according to claim 3, characterized in that: the potential energy improvement method described in the fourth step is to simplify the gear teeth in the model into cantilever beams on the root yard, and the potential energy stored in the meshed gears includes Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsThrough Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsCalculating the Hertz stiffness khBending stiffness kbRadial compression stiffness kaAnd shear stiffness ksThe total meshing stiffness is in a series form of all the stiffnesses; each energy and stiffness satisfy the following relationship:
in the formulas (19) to (22), F is the interaction force of the teeth at the meshing point, the direction is along the meshing line direction, and the direction is always intersected with the tooth profile.
8. The method as claimed in claim 4An ADAMS (adaptive dynamic analysis of moving objects) planetary gear train nonlinear dynamics modeling method is characterized by comprising the following steps: the potential energy improvement method described in the fourth step is to simplify the gear teeth in the model into cantilever beams on the root yard, and the potential energy stored in the meshed gears includes Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsThrough Hertz contact potential energy UhBending potential energy UbRadial compression potential energy UaAnd shear potential energy UsCalculating the Hertz stiffness khBending stiffness kbRadial compression stiffness kaAnd shear stiffness ksThe total meshing stiffness is in a series form of all the stiffnesses; each energy and stiffness satisfy the following relationship:
in the formulas (19) to (22), F is the interaction force of the teeth at the meshing point, the direction is along the meshing line direction, and the direction is always intersected with the tooth profile.
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| CN113609609A (en) * | 2021-07-23 | 2021-11-05 | 南京航空航天大学 | Method for analyzing dynamic characteristics of multi-stage planetary gear structure |
| CN116341105A (en) * | 2023-03-13 | 2023-06-27 | 江苏省金象传动设备股份有限公司 | Modeling method for herringbone tooth planetary transmission system dynamics under multi-source excitation |
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