CN106845048A - A kind of inside engaged gear axle decelerator Nonlinear dynamic models method for counting friction and backlash - Google Patents

Abstract

The invention discloses a kind of inside engaged gear axle decelerator Nonlinear dynamic models method for counting friction and backlash, belong to Reliability of Mechanical System field of engineering technology；Concretely comprise the following steps：First, the condition of certain inside engaged gear axle decelerator modeling is set；Then, relative linear displacement of the gear pair along path of contact, static driving error and nonlinear backlash are determined successively；Further, the comprehensive time-variant mesh stiffness of gear pair is approached using Fourier progression forms, the nonlinear dynamical equation of gear pair is set up using Lagrange methods, finally, linear transformation and nondimensionalization treatment are carried out to nonlinear dynamical equation.Model of the invention has preferable versatility, can carry out the inside engaged gear retarder operation state of arbitrary parameter and its numerical simulation of dynamics, and carry out the output simulation of retarder system vibration signal.

Description

A kind of inside engaged gear axle decelerator non-linear dynamic for counting friction and backlash Learn modeling method

Technical field

The invention belongs to Reliability of Mechanical System field of engineering technology, it is related to a kind of friction and the interior of backlash of counting to nibble Close gear shaft decelerator Nonlinear dynamic models method.

Background technology

Inside engaged gear axle decelerator has the advantages that compact conformation, stable performance and efficient work, it is civilian, industrial, Aerospace field is applied widely, therefore the theoretical research to its working mechanism is significant.Reducer gear pair During work, even normal condition, due to the influence of gear tooth friction, backlash and time-variant mesh stiffness, gear vibration table Reveal very strong nonlinear characteristic.

To realize high accuracy, low noise, low vibration and improving the target of gear drive controllability, researcher is in gear Substantial amounts of time and efforts has been put into above the Nonlinear Dynamics Problems of transmission system.

The research of inside engaged gear axle decelerator nonlinear kinetics has to the theory of gear reduction unit and application study Important value.

The content of the invention

Theoretical tool in order to disclose certain type inside engaged gear axle vibration reducer mechanism of the invention, is rubbed by studying the flank of tooth The influence of damping, backlash, time-variant mesh stiffness to its dynamics is wiped, is set up and is included gear tooth friction, non-linear flank The inside engaged gear axle reducer power model of gap function and time-variant mesh stiffness, specifically one kind count friction and flank The inside engaged gear axle decelerator Nonlinear dynamic models method in gap.

Comprise the following steps that：

Step one, the condition being modeled to the decelerator for certain inside engaged gear axle decelerator, setting；

(1) internal gear and external gear of the decelerator are involute spur gear；

(2) two gear gear blanks are considered as rigid body, and the input of decelerator and output shaft are considered as rigid body；The support of two gear shafts is firm Degree is sufficiently large, does not consider the elastic deformation of support；

(3) each parts do not receive axial force in decelerator, and vibration vector has the plane perpendicular to axis；

(4) driving wheel (external gear), the gear teeth of driven pulley (internal gear) do cantilever beam consideration, there are the gear teeth along path of contact With respect to slide displacement；

(5) train middle gear is installed according to reference center distance, and pitch circle overlaps with reference circle；

(6) part's machining errors and alignment error are not counted in.

Step 2, for the inside engaged gear axle decelerator, determine the phase in gear pair path of contact using gear angular displacement To displacement of the lines；

Wherein：X is the relative linear displacement along path of contact in internal gear Meshing Pair；Y is along path of contact in external gear Meshing Pair Relative linear displacement；r₁It is the reference radius of external gear；r₂It is the reference radius of internal gear；θ₁It is the torsion angle of external gear Displacement；θ₂It is the torsional angular displacement of internal gear；T is time variable；E (t) is the static driving error of gear pair, e (t)=e_asin ω_mT, e_aIt is error magnitude, ω_mFor engaging tooth frequently.

Step 3, nonlinear backlash letter is determined according to the relative linear displacement in internal gear Meshing Pair along path of contact Number；

In formula, b is backlash constant.

Step 4, the time-variant mesh stiffness function k that health status lower tooth wheel set is approached using Fourier space_m(t)；

Wherein, k_avIt is the average mesh stiffness of gear pair, k_aIt is stiffness variation amplitude；N rounds numerical value.

Step 5, the backlash function using the decelerator, the time-variant mesh stiffness function of conjunction gear wheel set are used Lagrange methods, set up the nonlinear dynamical equation of gear pair；

The nonlinear dynamical equation of gear pair includes the input matrix T of internal gear pair₁With output matrix T₂；

Wherein：I₁It is the rotary inertia of external gear；I₂It is the rotary inertia of internal gear；It is the instantaneous angular acceleration of external gear；It is the instantaneous angular acceleration of internal gear；c_mIt is engagement viscous damping coefficient；l₁It is the arm of force of the engaging friction damping force of external gear； l₂It is the arm of force of the engaging friction damping force of internal gear；λ is direction coefficient, ω₂It is the nominal angular speed of internal gear；It is the instantaneous angular velocity of external gear；It is the instantaneous angular velocity of internal gear；μ is the flank of tooth The coefficient of kinetic friction；

Step 6, the nonlinear dynamical equation to gear pair carry out linear transformation and nondimensionalization treatment, obtain final Count the internal gear pair dimensionless non-linear dynamic model of friction and backlash；

τ=ω_et；ω_eIt is gear pair equivalent intrinsic frequency,m_eIt is gear pair equivalent quality,m₁It is the quality of external gear, m₂It is the quality of internal gear；

e_aIt is the amplitude of gear teeth composition error change, is constant；

The advantage of the invention is that：

(1) a kind of inside engaged gear axle decelerator Nonlinear dynamic models method for counting friction and backlash, base The non-linear dynamic model of internal gear pair is established in Lagrange methods, the versatility of this model very well, can be done The dynamic numerical simulation of the internal gear pair under any operating mode.

(2) a kind of inside engaged gear axle decelerator Nonlinear dynamic models method for counting friction and backlash, mould The excitation of the inner parameters such as frictional damping effect, non-linear backlash function, time-variant mesh stiffness, numerical simulation have been counted in type High precision, the gear reducer system vibration signal emulator of model output is high, and nonlinear dynamic characteristic is obvious.

Brief description of the drawings

Fig. 1 is the structural representation of the inside engaged gear axle decelerator that the present invention is used；

Fig. 2 is inside engaged gear axle decelerator non-linear dynamic model principle schematic of the present invention；

Fig. 3 is the inside engaged gear axle decelerator Nonlinear dynamic models method that the present invention counts friction and backlash Flow chart；

Fig. 4 is the phase mark figure of inside engaged gear axle decelerator steady-state vibration of the present invention；

Fig. 5 is the time-domain diagram of inside engaged gear axle decelerator steady-state vibration of the present invention；

Fig. 6 is the frequency domain figure of inside engaged gear axle decelerator steady-state vibration of the present invention.

Specific embodiment

Below in conjunction with drawings and Examples, the present invention is described in further detail.

The present invention is directed to certain type inside engaged gear axle decelerator, structure as shown in figure 1, being based on Lagrange methods, design It is a kind of comprising being nibbled in the factors such as gear tooth friction, non-linear backlash function, time-variant mesh stiffness and engagement viscous damping Gear shaft decelerator non-linear dynamic model is closed, it is specifically a kind of to count friction and the inside engaged gear axle deceleration of backlash Device Nonlinear dynamic models method.

Schematic diagram as shown in Fig. 2 wherein, N₁N₂Represent the path of contact of internal gear pair；E (t) is the static biography of gear pair Dynamic error；k_mIt is the mesh stiffness of gear pair；c_mIt is engagement viscous damping；α is the gear pair angle of engagement, is also pressure angle； r₁、r₂It is respectively the reference radius of external gear (active) and internal gear (driven)；ω₁、ω₂It is respectively external gear, internal gear Nominal angular speed；l₁、l₂It is respectively the arm of force of Tooth friction force external gear wheel and internal gear；Y is external gear meshing point displacement of the lines； T₁、T₂It is respectively the driving torque and load torque of internal gear pair.

As shown in figure 3, comprising the following steps：

Step one, the condition being modeled to the decelerator for certain inside engaged gear axle decelerator, setting；

(1) internal gear and external gear of the decelerator are involute spur gear；

(2) two gear gear blanks are considered as rigid body, and the input of decelerator and output shaft are considered as rigid body；The support of two gear shafts is firm Degree is sufficiently large, does not consider the elastic deformation of support；

(3) each parts do not receive axial force in decelerator, and vibration vector has the plane perpendicular to axis；

(4) driving wheel (external gear), the gear teeth of driven pulley (internal gear) do cantilever beam consideration, there are the gear teeth along path of contact With respect to slide displacement；

(5) train middle gear is installed according to reference center distance, and pitch circle overlaps with reference circle；

(6) part's machining errors and alignment error are not counted in.

Step 2, for the inside engaged gear axle decelerator, determine the phase in gear pair path of contact using gear angular displacement To displacement of the lines；

Wherein：X is the relative linear displacement along path of contact in internal gear Meshing Pair；Y is along path of contact in external gear Meshing Pair Relative linear displacement；r₁It is the reference radius of external gear；r₂It is the reference radius of internal gear；θ₁It is the torsion angle of external gear Displacement；θ₂It is the torsional angular displacement of internal gear；T is time variable；E (t) is the static driving error of gear pair, e (t)=e_asin ω_mT, e_aIt is error magnitude, ω_mFor engaging tooth frequently.

Step 3, nonlinear backlash letter is determined according to the relative linear displacement in internal gear Meshing Pair along path of contact Number；

In formula, b is backlash constant.

Step 4, the time-variant mesh stiffness function k that health status lower tooth wheel set is approached using Fourier space_m(t)；

Wherein, k_avIt is the average mesh stiffness of gear pair, k_aIt is stiffness variation amplitude；N rounds numerical value.

Step 5, the backlash function using the decelerator, the time-variant mesh stiffness function of conjunction gear wheel set are used Lagrange methods, set up the nonlinear dynamical equation of gear pair；

The nonlinear dynamical equation of gear pair includes the input matrix T of internal gear pair₁With output matrix T₂；

Wherein：I₁It is the rotary inertia of external gear；I₂It is the rotary inertia of internal gear；It is the intermittent angle acceleration of external gear Degree；It is the instantaneous angular acceleration of internal gear；c_mIt is engagement viscous damping coefficient；k_mT () is time-variant mesh stiffness；F (x) is tooth Side clearance function；l₁It is the arm of force of the engaging friction damping force of external gear；l₂It is the arm of force of the engaging friction damping force of internal gear；λ It is direction coefficient,ω₂It is the nominal angular speed of internal gear；It is external tooth The instantaneous angular velocity of wheel；It is the instantaneous angular velocity of internal gear；μ is the flank of tooth coefficient of kinetic friction；

Step 6, the nonlinear dynamical equation to gear pair carry out linear transformation and nondimensionalization treatment, obtain final Count the internal gear pair dimensionless non-linear dynamic model of friction and backlash；

The replacement relation wherein used has：

τ=ω_et；ω_eIt is gear pair equivalent intrinsic frequency,m_eIt is gear pair equivalent Quality,m₁It is the quality of external gear, m₂It is the quality of internal gear；

e_aIt is the change of gear teeth composition error Amplitude, is constant；

Embodiment

Modeling and simulating experiment has been carried out to certain type inside engaged gear axle retarder system, the emulation of pure twisting vibration has been obtained As a result, phase mark figure, time-domain diagram and the frequency domain figure of transmission system are obtained；The parameter of simulation example is as shown in the table.

Inside engaged gear decelerator simulation example parameter value table

Numerical simulation solves the phase mark figure of the gear shaft retarder system steady state vibration response for obtaining, as shown in figure 4, time domain Figure is as shown in figure 5, frequency domain figure is as shown in Figure 6.As can be seen from Figure, during normal gear vibration, time-domain curve steadily engages week Phase is 20, there was only meshing frequency 0.048 and its high order frequency in frequency domain figure, does not have sideband, phase mark periodic motion.

Claims (2)

1. a kind of to count the inside engaged gear axle decelerator Nonlinear dynamic models method rubbed with backlash, its feature exists In comprising the following steps that：

Step one, the condition being modeled to the decelerator for certain inside engaged gear axle decelerator, setting；

Step 2, for the inside engaged gear axle decelerator, the relative line in gear pair path of contact is determined using gear angular displacement Displacement；

$\left\{\begin{array}{c}x=r_1{\mathrm{\θ}}_1-r_2{\mathrm{\θ}}_2-e\left(t\right)\\ y=r_1{\mathrm{\θ}}_1\end{array}\right.$

Wherein：X is the relative linear displacement along path of contact in internal gear Meshing Pair；Y is the phase along path of contact in external gear Meshing Pair To displacement of the lines；r₁It is the reference radius of external gear；r₂It is the reference radius of internal gear；θ₁It is the torsional angular displacement of external gear； θ₂It is the torsional angular displacement of internal gear；T is time variable；E (t) is the static driving error of gear pair, e (t)=e_a sinω_mT, e_aIt is error magnitude, ω_mFor engaging tooth frequently；

Step 3, nonlinear backlash function is determined according to the relative linear displacement in internal gear Meshing Pair along path of contact；

$f\left(x\right)=\left\{\begin{array}{cc}x-b,& x>b\\ 0,& \left|x\right|\&le;b\\ x+b,& x<-b\end{array}\right.$

In formula, b is backlash constant；

Step 4, the time-variant mesh stiffness function k that health status lower tooth wheel set is approached using Fourier space_m(t)；

$k_m\left(t\right)=k_{av}+\frac{2k_a}{n\mathrm{\&pi;}}\underset{n=1}{\overset{\mathrm{\&infin;}}{\&Sigma;}}\sin {\mathrm{n\&omega;}}_{m}t,n=1,3,5,...$

Wherein, k_avIt is the average mesh stiffness of gear pair, k_aIt is stiffness variation amplitude；N rounds numerical value；

Step 5, the backlash function using the decelerator, the time-variant mesh stiffness function of conjunction gear wheel set are used Lagrange methods, set up the nonlinear dynamical equation of gear pair；

The nonlinear dynamical equation of gear pair includes the input matrix T of internal gear pair₁With output matrix T₂；

$\left\{\begin{array}{c}{I}_1{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_1+r_1\&lsqb;c_m\stackrel{\&CenterDot;}{x}+k_m\left(t\right)f\left(x\right)\&rsqb;+l_1\mathrm{\&lambda;}\mathrm{\&mu;}\&lsqb;c_m\stackrel{\&CenterDot;}{x}+k_m\left(t\right)f\left(x\right)\&rsqb;=T_1\\ I_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_2-r_2\&lsqb;c_m\stackrel{\&CenterDot;}{x}+k_m\left(t\right)f\left(x\right)\&rsqb;-l_2\mathrm{\&lambda;}\mathrm{\&mu;}\&lsqb;c_m\stackrel{\&CenterDot;}{x}+k_m\left(t\right)f\left(x\right)\&rsqb;=T_2\end{array}\right.$

Wherein：I₁It is the rotary inertia of external gear；I₂It is the rotary inertia of internal gear；It is the instantaneous angular acceleration of external gear； It is the instantaneous angular acceleration of internal gear；c_mIt is engagement viscous damping coefficient；l₁It is the arm of force of the engaging friction damping force of external gear；l₂ It is the arm of force of the engaging friction damping force of internal gear；λ is direction coefficient, ω₂It is the nominal angular speed of internal gear；It is the instantaneous angular velocity of external gear；It is the instantaneous angular velocity of internal gear；μ is the flank of tooth The coefficient of kinetic friction；

Step 6, the nonlinear dynamical equation to gear pair carry out linear transformation and nondimensionalization treatment, are finally counted Friction and the internal gear pair dimensionless non-linear dynamic model of backlash；

$\left\{\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\stackrel{\&OverBar;}{x}}\left(\mathrm{\&tau;}\right)+2{\mathrm{\&zeta;}}_1\&lsqb;1+g_1\left(\mathrm{\&tau;}\right)\mathrm{\&mu;}\&rsqb;\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{x}}\left(\mathrm{\&tau;}\right)+\&lsqb;1+g_1\left(\mathrm{\&tau;}\right)\mathrm{\&mu;}\&rsqb;\&lsqb;1+k_1\sin \left(\stackrel{\&OverBar;}{\mathrm{\&omega;}}\mathrm{\&tau;}\right)\&rsqb;f\left(\stackrel{\&OverBar;}{x}\right)=\frac{m_2-m_1}{m_1+m_2}{F}_{av}\\ +F_e{\stackrel{\&OverBar;}{\mathrm{\&omega;}}}^2\sin \stackrel{\&OverBar;}{\mathrm{\&omega;}}\mathrm{\&tau;}\\ \stackrel{\&CenterDot;\&CenterDot;}{\stackrel{\&OverBar;}{y}}\left(\mathrm{\&tau;}\right)+2{\mathrm{\&zeta;}}_2\&lsqb;1+g_2\left(\mathrm{\&tau;}\right)\mathrm{\&mu;}\&rsqb;\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{x}}\left(\mathrm{\&tau;}\right)+\&lsqb;1+g_2\left(\mathrm{\&tau;}\right)\mathrm{\&mu;}\&rsqb;k_2\left(1+k_1\sin \stackrel{\&OverBar;}{\mathrm{\&omega;}}\mathrm{\&tau;}\right)f\left(\stackrel{\&OverBar;}{x}\right)=\frac{m_e{F}_{av}}{m_1}\end{array}\right.$

τ=ω_et；ω_eIt is gear pair equivalent intrinsic frequency,m_eIt is gear pair equivalent quality,m₁It is the quality of external gear, m₂It is the quality of internal gear；

${\mathrm{\&zeta;}}_1=\frac{c_m}{\left(2m_e{\mathrm{\&omega;}}_e\right)};g_1\left(\mathrm{\&tau;}\right)=(\frac{l_1}{m_1{r}_1}+\frac{l_2}{m_2{r}_2})m_e\mathrm{\&lambda;};\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{x}}\left(\mathrm{\&tau;}\right)=\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{x}}\left(t\right)\&times;\frac{1}{{\mathrm{\&omega;}}_e};$

$k_1=\frac{k_a}{k_{av}};\stackrel{\&OverBar;}{\mathrm{\&omega;}}=\frac{{\mathrm{\&omega;}}_m}{{\mathrm{\&omega;}}_e};$

$f\left(\stackrel{\&OverBar;}{x}\right)=\left\{\begin{array}{cc}\stackrel{\&OverBar;}{x}-1,& \stackrel{\&OverBar;}{x}>1\\ 0,& \left|\stackrel{\&OverBar;}{x}\right|\&le;1\\ \stackrel{\&OverBar;}{x}+1,& \stackrel{\&OverBar;}{x}<-1\end{array}\right.,\stackrel{\&OverBar;}{x}=\frac{x}{b};$

e_aIt is the amplitude of gear teeth composition error change, is constant；

$\stackrel{\&CenterDot;\&CenterDot;}{\stackrel{\&OverBar;}{y}}\left(\mathrm{\&tau;}\right)=\frac{\stackrel{\&CenterDot;\&CenterDot;}{y}\left(t\right)}{{\mathrm{\&omega;}}_e^2};{\mathrm{\&zeta;}}_2=\frac{c_m}{\left(2m_1{\mathrm{\&omega;}}_e\right)};g_2\left(\mathrm{\&tau;}\right)=\frac{l_1\mathrm{\&lambda;}}{r_1};k_2=\frac{{\mathrm{\&omega;}}_1^2}{{\mathrm{\&omega;}}_e^2}.$

2. it is a kind of as claimed in claim 1 to count the inside engaged gear axle decelerator nonlinear kinetics rubbed with backlash Modeling method, it is characterised in that described step one is specially：

(1) internal gear and external gear of the decelerator are involute spur gear；

(2) two gear gear blanks are considered as rigid body, and the input of decelerator and output shaft are considered as rigid body；The support stiffness foot of two gear shafts It is enough big, do not consider the elastic deformation of support；

(3) each parts do not receive axial force in decelerator, and vibration vector has the plane perpendicular to axis；

(4) driving wheel (external gear), the gear teeth of driven pulley (internal gear) do cantilever beam consideration, there are the gear teeth along the relative of path of contact Slide displacement；

(5) train middle gear is installed according to reference center distance, and pitch circle overlaps with reference circle；

(6) part's machining errors and alignment error are not counted in.