CN106845048B - Inner gearing gear shaft speed reducer nonlinear dynamics modeling method taking friction and tooth side clearance into account - Google Patents

Inner gearing gear shaft speed reducer nonlinear dynamics modeling method taking friction and tooth side clearance into account Download PDF

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CN106845048B
CN106845048B CN201710269912.2A CN201710269912A CN106845048B CN 106845048 B CN106845048 B CN 106845048B CN 201710269912 A CN201710269912 A CN 201710269912A CN 106845048 B CN106845048 B CN 106845048B
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范磊
王少萍
段海滨
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Beihang University
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Abstract

The invention discloses a nonlinear dynamics modeling method of an inner gearing gear shaft speed reducer with friction and tooth side clearance being counted, belonging to the technical field of mechanical system reliability engineering; the method comprises the following specific steps: firstly, setting a certain modeling condition of the internal gear shaft speed reducer; then, determining the relative linear displacement of the gear pair along the meshing line, the static transmission error and the nonlinear gear backlash in turn; and further, approximating the comprehensive time-varying meshing stiffness of the gear pair by a Fourier series form, establishing a nonlinear kinetic equation of the gear pair by a Lagrange method, and finally performing linear transformation and dimensionless processing on the nonlinear kinetic equation. The model has better universality, can carry out numerical simulation on the running state and the dynamic characteristic of the internal gear reducer with any parameter, and carry out output simulation on the vibration signal of the reducer system.

Description

Inner gearing gear shaft speed reducer nonlinear dynamics modeling method taking friction and tooth side clearance into account
Technical Field
The invention belongs to the technical field of mechanical system reliability engineering, and relates to a nonlinear dynamics modeling method of an inner gearing gear shaft speed reducer, which takes friction and tooth side clearance into account.
Background
The internal gear shaft speed reducer has the advantages of compact structure, stable performance and high working efficiency, is widely applied in the fields of civil use, industry, aviation and aerospace, and has important significance on theoretical research of the working mechanism of the internal gear shaft speed reducer. When the gear pair of the speed reducer works, even in a normal state, gear vibration shows strong nonlinear characteristics due to the influence of tooth surface friction, tooth backlash and time-varying meshing rigidity.
Researchers have invested a great deal of time and effort in the non-linear dynamics of gear systems in order to achieve the goals of high accuracy, low noise, low vibration, and improved gear controllability.
The research on the nonlinear dynamics of the internal gear shaft speed reducer has important value on the theory and application research of the gear speed reducer.
Disclosure of Invention
The invention discloses a theoretical tool for revealing a vibration mechanism of a certain type of internal gear shaft speed reducer, and establishes an internal gear shaft speed reducer dynamic model comprising tooth surface friction, a nonlinear tooth side clearance function and time-varying meshing rigidity by researching the influence of tooth surface friction damping, tooth side clearance and time-varying meshing rigidity on the dynamic characteristics of the internal gear shaft speed reducer, in particular to a nonlinear dynamic modeling method of the internal gear shaft speed reducer with the friction and the tooth side clearance taken into account.
The method comprises the following specific steps:
step one, aiming at a certain internal gear shaft speed reducer, setting a condition for modeling the speed reducer;
(1) the inner gear and the outer gear of the speed reducer are both involute straight toothed spur gears;
(2) the two gear blanks are regarded as rigid bodies, and the input shaft and the output shaft of the speed reducer are regarded as rigid bodies; the support rigidity of the two gear shafts is enough, and the elastic deformation of the support is not considered;
(3) all parts in the speed reducer are not subjected to axial force, and a vibration vector exists in a plane perpendicular to an axis;
(4) the gear teeth of the driving wheel (external gear) and the driven wheel (internal gear) are considered as cantilever beams, and the relative sliding displacement of the gear teeth along the meshing line exists;
(5) 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;
(6) and the machining error and the installation error of the parts are not counted.
Determining the relative linear displacement on a gear pair meshing line by utilizing the angular displacement of a gear aiming at the internal gear shaft speed reducer;
wherein: x is the relative linear displacement of the internal gear meshing pair along the meshing line; y is the relative linear displacement along the meshing line on the external gear meshing pair; r is1Is the pitch circle radius of the external gear; r is2Is the pitch circle radius of the internal gear; theta1Is the torsional angular displacement of the external gear; theta2Is the torsional angular displacement of the internal gear; t is a time variable; e (t) is the static transmission error of the gear pair, e (t) ═ easinωmt，eaIs the error amplitude, omegamIs the meshing gear frequency.
Thirdly, determining a nonlinear flank clearance function according to the relative linear displacement of the inner gear meshing pair along the meshing line;
Figure BDA0001277073350000022
wherein b is a backlash constant.
Step four, adopting Fourier series to approximate time-varying meshing stiffness function k of the gear pair in a healthy statem(t)；
Figure BDA0001277073350000023
Wherein k isavIs the gear pair average mesh stiffness, kaIs the stiffness variation amplitude; n is an integer value.
Fifthly, establishing a nonlinear kinetic equation of the gear pair by using a gear backlash function of the speed reducer and combining a time-varying meshing stiffness function of the gear pair by adopting a Lagrange method;
the non-linear kinetic equation for a gear pair comprises an input matrix T for the internal gear pair1And an output matrix T2；
Figure BDA0001277073350000024
Wherein: i is1Is the moment of inertia of the outer gear; i is2Is the moment of inertia of the internal gear;
Figure BDA0001277073350000025
is the instantaneous angular acceleration of the outer gear;
Figure BDA0001277073350000026
is the instantaneous angular acceleration of the internal gear; c. CmIs the meshing viscous damping coefficient; l1Is the force arm of the meshing friction damping force of the external gear; l2Is the force arm of the meshing friction damping force of the internal gear; λ is the coefficient of the direction of the friction force,
Figure BDA0001277073350000027
ω2is the nominal angular velocity of the internal gear;
Figure BDA0001277073350000028
is the instantaneous angular velocity of the outer gear;is the instantaneous angular velocity of the internal gear; μ is the tooth flank dynamic friction coefficient;
step six, carrying out linear transformation and dimensionless processing on a nonlinear kinetic equation of the gear pair to obtain a dimensionless nonlinear kinetic model of the internal meshing gear pair, which is finally counted into friction and backlash;
Figure BDA00012770733500000210
Figure BDA0001277073350000031
τ＝ωet；ωeis the equivalent natural frequency of the gear pair,meis the equivalent mass of the gear pair,
Figure BDA0001277073350000033
m1is the mass of the outer gear, m2Is the mass of the internal gear;
Figure BDA0001277073350000034
Figure BDA0001277073350000035
Figure BDA0001277073350000036
Figure BDA0001277073350000037
eais the amplitude of the gear tooth comprehensive error change and is a constant;
Figure BDA0001277073350000038
the invention has the advantages that:
(1) a nonlinear dynamics modeling method of an internal gear shaft speed reducer with friction and tooth side clearance is based on a Lagrange method to establish a nonlinear dynamics model of an internal gear pair, the model has good universality and can be used for dynamic numerical simulation of the internal gear pair under any working condition.
(2) A nonlinear dynamics modeling method of an inner meshing gear shaft speed reducer with friction and tooth side clearance is characterized in that the excitation of internal parameters such as friction damping effect, nonlinear tooth side clearance function, time-varying meshing rigidity and the like is added into a model, the numerical simulation precision is high, the simulation degree of a vibration signal of a gear speed reducer system output by the model is high, and the nonlinear dynamics characteristic is obvious.
Drawings
Fig. 1 is a schematic structural view of an internal gear shaft speed reducer employed in the present invention;
FIG. 2 is a schematic diagram of a nonlinear dynamical model of the internal gear shaft reducer of the present invention;
FIG. 3 is a flow chart of a method of modeling the non-linear dynamics of an internal gear shaft reducer incorporating friction and backlash according to the present invention;
FIG. 4 is a trace plot of steady state vibration of the ring gear shaft reducer of the present invention;
FIG. 5 is a time domain plot of steady state vibration of the ring gear shaft reducer of the present invention;
fig. 6 is a frequency domain diagram of steady state vibration of the ring gear shaft speed reducer of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
The invention relates to a nonlinear dynamics model of an inner gear shaft speed reducer, which aims at a certain type of inner gear shaft speed reducer, has a structure shown in figure 1, and designs a nonlinear dynamics model of the inner gear shaft speed reducer containing factors such as tooth surface friction, a nonlinear tooth side clearance function, time-varying meshing rigidity, meshing viscous damping and the like based on a Lagrange method, in particular to a nonlinear dynamics modeling method of the inner gear shaft speed reducer with friction and tooth side clearance taken into account.
The schematic diagram is shown in FIG. 2, wherein N1N2A meshing line representing a pair of internal gear wheels; e (t) is the static transmission error of the gear pair; k is a radical ofmIs the meshing stiffness of the gear pair; c. CmIs the viscous damping of the meshing, α is the gear pair meshing angle, also the gear pressure angle, r1、r2The reference circle radii of the external gear (driving) and the internal gear (driven) respectively; omega1、ω2Nominal angular velocities of the external gear and the internal gear respectively; l1、l2Force arms of the tooth surface friction force to the outer gear and the inner gear respectively; y is the external gear meshing point linear displacement; t is1、T2The drive torque and the load torque of the ring gear pair are respectively.
As shown in fig. 3, the method comprises the following steps:
step one, aiming at a certain internal gear shaft speed reducer, setting a condition for modeling the speed reducer;
(1) the inner gear and the outer gear of the speed reducer are both involute straight toothed spur gears;
(2) the two gear blanks are regarded as rigid bodies, and the input shaft and the output shaft of the speed reducer are regarded as rigid bodies; the support rigidity of the two gear shafts is enough, and the elastic deformation of the support is not considered;
(3) all parts in the speed reducer are not subjected to axial force, and a vibration vector exists in a plane perpendicular to an axis;
(4) the gear teeth of the driving wheel (external gear) and the driven wheel (internal gear) are considered as cantilever beams, and the relative sliding displacement of the gear teeth along the meshing line exists;
(5) 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;
(6) and the machining error and the installation error of the parts are not counted.
Determining the relative linear displacement on a gear pair meshing line by utilizing the angular displacement of a gear aiming at the internal gear shaft speed reducer;
Figure BDA0001277073350000041
wherein: x is the relative linear displacement of the internal gear meshing pair along the meshing line; y is the relative linear displacement along the meshing line on the external gear meshing pair; r is1Is the pitch circle radius of the external gear; r is2Is the pitch circle radius of the internal gear; theta1Is the torsional angular displacement of the external gear; theta2Is the torsional angular displacement of the internal gear; t is a time variable; e (t) is the static transmission error of the gear pair, e (t) ═ easinωmt，eaIs the error amplitude, omegamIs the meshing gear frequency.
Thirdly, determining a nonlinear flank clearance function according to the relative linear displacement of the inner gear meshing pair along the meshing line;
Figure BDA0001277073350000042
wherein b is a backlash constant.
Step four, adopting Fourier series to approximate time-varying meshing stiffness function k of the gear pair in a healthy statem(t)；
Figure BDA0001277073350000051
Wherein k isavIs the gear pair average mesh stiffness, kaIs the stiffness variation amplitude; n is an integer value.
Fifthly, establishing a nonlinear kinetic equation of the gear pair by using a gear backlash function of the speed reducer and combining a time-varying meshing stiffness function of the gear pair by adopting a Lagrange method;
the non-linear kinetic equation for a gear pair comprises an input matrix T for the internal gear pair1And an output matrix T2；
Figure BDA0001277073350000052
Wherein: i is1Is the moment of inertia of the outer gear; i is2Is the moment of inertia of the internal gear;
Figure BDA0001277073350000053
is the instantaneous angular acceleration of the outer gear;
Figure BDA0001277073350000054
is the instantaneous angular acceleration of the internal gear; c. CmIs the meshing viscous damping coefficient; k is a radical ofm(t) is the time-varying mesh stiffness; (x) is a flank clearance function; l1Is the force arm of the meshing friction damping force of the external gear; l2Is the force arm of the meshing friction damping force of the internal gear; λ is the coefficient of the direction of the friction force,
Figure BDA0001277073350000055
ω2is the nominal angular velocity of the internal gear;
Figure BDA0001277073350000056
is the instantaneous angular velocity of the outer gear;
Figure BDA0001277073350000057
is the instantaneous angular velocity of the internal gear; μ is the tooth flank dynamic friction coefficient;
step six, carrying out linear transformation and dimensionless processing on a nonlinear kinetic equation of the gear pair to obtain a dimensionless nonlinear kinetic model of the internal meshing gear pair, which is finally counted into friction and backlash;
Figure BDA0001277073350000058
the substitution relations used therein are:
Figure BDA0001277073350000059
τ＝ωet；ωeis the equivalent natural frequency of the gear pair,
Figure BDA00012770733500000510
meis the equivalent mass of the gear pair,
Figure BDA00012770733500000511
m1is the mass of the outer gear, m2Is the mass of the internal gear;
Figure BDA00012770733500000512
Figure BDA00012770733500000513
Figure BDA00012770733500000514
eais the amplitude of the gear tooth comprehensive error change and is a constant;
Figure BDA0001277073350000061
examples
Carrying out modeling simulation test on a certain type of internal gear shaft speed reducer system to obtain a simulation result of pure torsional vibration and obtain a phase trace diagram, a time domain diagram and a frequency domain diagram of a transmission system; the parameters of the simulation examples are shown in the following table.
Inside engaged gear reducer simulation example parameter value-taking table
Figure BDA0001277073350000062
The trace diagram of the steady-state response of the vibration of the gear shaft speed reducer system obtained by numerical simulation is shown in fig. 4, the time domain diagram is shown in fig. 5, and the frequency domain diagram is shown in fig. 6. It can be seen from the figure that when the normal gear vibrates, the stable meshing period of the time domain curve is 20, and the frequency domain graph only has the meshing frequency of 0.048 and the higher frequency multiplication thereof, and has no side frequency band and the trace moves periodically.

Claims (2)

1. A nonlinear dynamics modeling method of an inner meshing gear shaft speed reducer with friction and tooth side clearance, which is characterized by comprising the following specific steps:
step one, aiming at a certain internal gear shaft speed reducer, setting a condition for modeling the speed reducer;
determining the relative linear displacement on a gear pair meshing line by utilizing the angular displacement of a gear aiming at the internal gear shaft speed reducer;
Figure FDA0002187491350000011
wherein: x is the relative linear displacement of the internal gear meshing pair along the meshing line; y is the relative linear displacement along the meshing line on the external gear meshing pair; r is1Is the pitch circle radius of the external gear; r is2Is the pitch circle radius of the internal gear; theta1Is the torsional angular displacement of the external gear; theta2Is the torsional angular displacement of the internal gear; t is a time variable; e (t) is the static transmission error of the gear pair, e (t) ═ easinωmt，eaIs the error amplitude, omegamIs the meshing tooth frequency;
thirdly, determining a nonlinear flank clearance function according to the relative linear displacement of the inner gear meshing pair along the meshing line;
Figure FDA0002187491350000012
wherein b is a backlash constant;
step four, adopting Fourier series to approximate time-varying meshing stiffness function k of the gear pair in a healthy statem(t)；
Figure FDA0002187491350000013
Wherein k isavIs gear pair averagingEngagement stiffness, kaIs the stiffness variation amplitude; n is an integer value;
fifthly, establishing a nonlinear kinetic equation of the gear pair by using a gear backlash function of the speed reducer and combining a time-varying meshing stiffness function of the gear pair by adopting a Lagrange method;
the non-linear kinetic equation for a gear pair comprises an input matrix T for the internal gear pair1And an output matrix T2；
Figure FDA0002187491350000014
Wherein: i is1Is the moment of inertia of the outer gear; i is2Is the moment of inertia of the internal gear;
Figure FDA0002187491350000015
is the instantaneous angular acceleration of the outer gear;
Figure FDA0002187491350000016
is the instantaneous angular acceleration of the internal gear; c. CmIs the meshing viscous damping coefficient; l1Is the force arm of the meshing friction damping force of the external gear; l2Is the force arm of the meshing friction damping force of the internal gear; λ is the coefficient of the direction of the friction force,ω2is the nominal angular velocity of the internal gear;
Figure FDA0002187491350000018
is the instantaneous angular velocity of the outer gear;
Figure FDA0002187491350000019
is the instantaneous angular velocity of the internal gear; μ is the tooth flank dynamic friction coefficient;
Figure FDA00021874913500000110
is the relative linear speed of the internal gear meshing pair along the meshing lineDegree;
step six, carrying out linear transformation and dimensionless processing on a nonlinear kinetic equation of the gear pair to obtain a dimensionless nonlinear kinetic model of the internal meshing gear pair, which is finally counted into friction and backlash;
Figure FDA0002187491350000021
Figure FDA0002187491350000022
representing a dimensionless nonlinear dynamic model of the internal gear pair; τ ═ ωet；ωeIs the equivalent natural frequency of the gear pair,
Figure FDA0002187491350000023
meis the equivalent mass of the gear pair,
Figure FDA0002187491350000024
m1is the mass of the outer gear, m2Is the mass of the internal gear;
Figure FDA0002187491350000025
Figure FDA0002187491350000026
Figure FDA0002187491350000028
eais the amplitude of the gear tooth comprehensive error change and is a constant;
representing a dimensionless nonlinear dynamic model of the external gear pair;
Figure FDA00021874913500000210
Figure FDA00021874913500000211
2. the method for modeling the nonlinear dynamics of a ring gear shaft reducer taking into account friction and backlash as set forth in claim 1, wherein said first step is specifically:
(1) the inner gear and the outer gear of the speed reducer are both involute straight toothed spur gears;
(2) the two gear blanks are regarded as rigid bodies, and the input shaft and the output shaft of the speed reducer are regarded as rigid bodies; the support rigidity of the two gear shafts is enough, and the elastic deformation of the support is not considered;
(3) all parts in the speed reducer are not subjected to axial force, and a vibration vector exists in a plane perpendicular to an axis;
(4) the gear teeth of the outer gear and the inner gear are considered as cantilever beams, and the relative sliding displacement of the gear teeth along the meshing line exists;
(5) 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;
(6) and the machining error and the installation error of the parts are not counted.
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