CN111027156B - Method for analyzing reliability of transmission precision of industrial robot speed reducer with crack gear - Google Patents
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Abstract
The invention discloses a method for analyzing the reliability of transmission precision of a speed reducer of an industrial robot with a crack gear, which comprises the steps of firstly analyzing uncertainty factors of the speed reducer of the industrial robot; secondly, analyzing the meshing rigidity of the gear with the cracks by using a finite element simulation method, further solving a kinetic equation, and calculating the transmission error of the gear with the cracks; then, calculating a probability density function of the output error of the industrial robot reducer by using a probability density evolution theory and a Dikela function; and finally, calculating the transmission precision reliability of the industrial robot speed reducer with the crack gear by using a Monte Carlo simulation method. The method applies probability density evolution and a Diecka function to reliability evaluation of mechanical parts, and can realize multi-degree-of-freedom linear and nonlinear mechanical structure dynamic performance response characterization and reliability calculation under the input of high-precision and high-efficiency description uncertainty factors.
Description
Technical Field
The invention belongs to the field of reliability of mechanical parts, and particularly relates to a transmission precision reliability analysis method for a speed reducer of an industrial robot with a crack gear.
Background
Industrial robots are developing in a direction of high speed, high precision, high reliability, multiple degrees of freedom and increased rigidity, and heavy loads or high response speeds are also required in key fields. As one of the core components of an industrial robot, a speed reducer is the core of mechanical transmission, and the transmission precision and reliability of the speed reducer affect the speed and precision of the robot. Due to the influence of random factors such as manufacturing errors, assembly gaps, material performance, use environment and the like, the transmission performance of the industrial robot reducer, namely the RV reducer, has randomness, so that in order to improve the transmission precision of the RV reducer, the reliability of the RV reducer needs to be evaluated by considering the factors, and the design of the RV reducer is further optimized.
The RV reducer can produce the crackle under the continuous effect of variable load, and the crackle can influence the reduction gear rigidity, further influences its transmission precision. Solving highly nonlinear and multi-degree of freedom RV reducer kinetic equations is a complex problem. And the dynamic response and the distribution thereof are more complicated to solve under the influence of uncertain loads and crack factors. The reliability of the complex structure is analyzed based on the probability density evolution theory and the Dikela function, and the method has the following characteristics: (1) the response distribution characteristics of the multi-degree-of-freedom nonlinear structure under complex uncertainty can be calculated, and the reliability of the structure is further calculated with high precision and high efficiency; (2) the Diincar function can greatly simplify the solution of the calculus, and can use the physical quantity convenient for quantification to calculate the dynamic response, thereby reducing the complexity of solving the dynamic response distribution characteristics.
The health condition of the RV reducer directly influences the working condition of the industrial robot, and the analysis and supervision of the health condition of the RV reducer under the influence of various uncertainty factors are necessary. When the gear of the RV reducer is cracked, the transmission precision of the gear is influenced inevitably, and no related effective method is proposed at present to solve the problem of analyzing the reliability of the transmission precision of the gear by considering various uncertain inputs.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a method for analyzing the reliability of the transmission precision of a speed reducer of an industrial robot with a crack gear.
The technical scheme of the invention is as follows: a method for analyzing reliability of transmission precision of a speed reducer of an industrial robot with a crack gear comprises the following steps:
s1, determining uncertainty input of the industrial robot speed reducer;
s2, selecting an input representative point according to the uncertainty input of the industrial robot speed reducer determined in the step S1;
s3, taking the representative point in the step S2 as an input, and carrying out simulation analysis on the meshing rigidity of the gear containing the crack;
s4, substituting the meshing stiffness calculated in the step S3 into a gear dynamic equation, and calculating the transmission error of the gear with the cracks;
s5, analyzing a transmission error probability density function of the gear with the crack;
and S6, carrying out gear transmission precision reliability evaluation on the transmission error probability density function of the gear with the crack established in the step S5.
Further, step S2 specifically uses a point selection method based on number theory to select the input representative point.
Further, the step S2 specifically includes the following sub-steps:
s21, determining the dimension and the statistical characteristic of the basic random variable X of the structure to be analyzed;
s22, obtaining a generating vector (N, h) of S dimension by means of a Good Lattice Point (GLP) set in a standard independent space1,h2,...,hS) Further, a unit hypercube [ 01 ] is obtained from the formula (1)]SInner point set:
wherein N is the number of point sets to be constructed, hSIs Fibonacci sequence, and the Fibonacci sequence is defined by a recursive method as follows: h isj＝hj-1+hj-2,h0＝h1＝1(n＝1,2,...,S)。
S23, taking the limit of the normalized random variable as L, and carrying out scaling and translation transformation on the point set generated in the step S22 by using the formula (2) to obtain a square [ -L, L]SInner uniformly distributed point set:
θj,k＝2(xj,k-0.5)L,(k＝1,2,…,N,j＝1,2,…,S) (2)
and S24, transforming the sample points of the standard independent space to the sample points corresponding to the original space by utilizing Nataf inverse transformation, wherein the transformed sample points are input representative points.
Further, step S3 specifically adopts a finite element method-based dynamic simulation to analyze the meshing stiffness of the gear with cracks.
Further, the step S3 specifically includes the following sub-steps:
s31, establishing a finite element model of a first-stage speed reducing mechanism, namely a planetary speed reducing mechanism, of the industrial robot speed reducer with the initial cracks according to the representative points determined in the step S2;
s32, solving the finite element model established in S31, analyzing the strain of the planetary reduction mechanism when the gear teeth have cracks, and calculating the gear meshing rigidity K according to the strainpi。
Further, the specific sub-steps of step S4 are as follows:
s41, calculating the gear meshing rigidity K of the planetary reduction mechanism when the gear teeth are cracked according to the specific industrial robot reducer model and the step S32piAnd establishing a corresponding transmission error dynamic model:
wherein F is a reducer load matrix, M is a mass matrix, C is a damping matrix, and K is a stiffness matrix and includes KpiX is a displacement matrix of each part when the load is F and contains the transmission error of the industrial robot speed reducer,
the first order differential of X is represented,
represents the second differential of X;
and S42, solving the transmission error dynamic model by using a Newmark method, and calculating the transmission error of the industrial robot speed reducer in a period when the gear has cracks.
Further, the step S5 specifically adopts a probability density evolution theory and a dickstra function to analyze a transmission error probability density function of the crack-containing gear of the industrial robot.
Further, the step S5 specifically includes the following sub-steps:
s51, obtaining an evolution equation of the system state response joint probability density according to the principle of probability conservation:
in the formula: f (-) represents a probability density function, z is a structural dynamics response vector, h is a first order differential of the response, theta is an uncertainty input vector, and nzNumber of structural dynamic response vectors, zlRepresents the ith structural dynamic response vector, and t is a time variable. The probability density evolution equation under single dynamic response is as shown in formula (5):
wherein:
is the first differential of the kinetic response.
S52 probability density function f of transmission error of industrial robotz(z, t) can be calculated from equation (6) by introducing the Dike function, the probability density function f of the transmission errorz(z, t) calculating the conversion from formula (6) to formula (7):
fz(z,t)＝∫ΩΘfZΘ(z,θ,t)dθ (6)
where δ (·) is the Dikla function, Hi(θ,z0T) is a response uniform discrete point, Pq,iA probability is assigned to each representative point.
S53, calculating the probability density function f of the transmission error through the Gauss Dike-Law approximation functionz(z,t)：
Where σ is a smoothing parameter, and is generally taken as σ ═ Δ Hi。
Further, step S6 specifically adopts a monte carlo-based method to perform gear transmission accuracy reliability evaluation on the transmission error probability density function of the gear with cracks established in step S5.
The invention has the beneficial effects that: according to the method, the relationship between the crack propagation condition of the gear and the transmission performance is analyzed through dynamic simulation, and a transmission error distribution model is established by combining a probability density evolution theory and a Dike function, so that the transmission precision reliability of the crack-containing gear is further evaluated. The method applies probability density evolution and a Diecka function to reliability evaluation of mechanical parts, and can realize multi-degree-of-freedom linear and nonlinear mechanical structure dynamic performance response characterization and reliability calculation under the input of high-precision and high-efficiency description uncertainty factors.
Drawings
FIG. 1 is a schematic flow chart of the method for analyzing the transmission precision reliability of the industrial robot speed reducer with the crack gear based on probability density evolution theory and Dikela function;
FIG. 2 is a schematic view of a gear crack model of a planetary reduction mechanism of an industrial robot speed reducer according to an embodiment of the invention;
FIG. 3 is a schematic diagram of a finite element simulation model of a crack-containing gear of the planetary reduction mechanism of the industrial robot reducer in the embodiment of the invention;
FIG. 4 is a schematic diagram of an "equivalent model" of a transmission error kinetic equation in an embodiment of the present invention.
FIG. 5 is a schematic diagram of the probability density distribution and cumulative distribution result of the transmission errors of the speed reducer in the embodiment of the invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
According to the method, the relationship between the crack propagation condition of the gear and the transmission performance is analyzed through dynamic simulation, and a transmission error distribution model is established by combining a probability density evolution theory and a Dike function, so that the transmission precision reliability of the crack-containing gear is further evaluated. The method applies probability density evolution and a Diecka function to reliability evaluation of mechanical parts, and can realize multi-degree-of-freedom linear and nonlinear mechanical structure dynamic performance response characterization and reliability calculation under the input of high-precision and high-efficiency description uncertainty factors.
As shown in FIG. 1, the method for analyzing the reliability of the transmission precision of the industrial robot speed reducer with the crack gear based on the probability density evolution theory and the Dikela function comprises the following steps:
s1, determining uncertainty input of the industrial robot speed reducer;
s2, selecting an input representative point based on a number theory point selection method according to the uncertainty input of the industrial robot speed reducer determined in the step S1;
s3, taking the representative point in the step S2 as input, and analyzing the time-varying meshing rigidity of the gear with the crack based on finite element method dynamic simulation;
s4, substituting the meshing stiffness calculated in the step S3 into a gear dynamic equation, and calculating the transmission error of the gear with the cracks;
s5, analyzing a transmission error probability density function of the gear with the crack;
and S6, carrying out gear transmission precision reliability evaluation on the transmission error probability density function of the gear with the crack established in the step S5.
In step S1, the present invention takes an RV-20E type industrial robot decelerator as an example, and determines its random uncertainty input. Specifically, the uncertainty input for analyzing the industrial robot reducer is as follows: load F and initial crack length l. The distribution information of the determined load F and the initial crack length l according to the rated working table of the RV-20E type industrial robot speed reducer is shown in the table 1:
TABLE 1 distribution information of random variables and random parameters
In step S2, selecting an input representative point based on a number theory point selection method according to the uncertainty input of the industrial robot speed reducer determined in step S1; the method specifically comprises the following steps:
and S21, determining the dimension and the statistical characteristics of the basic random variable X of the structure to be analyzed, such as the mean, the standard deviation and the covariance matrix.
S22, obtaining a generating vector (N, h) of an S dimension by virtue of a GLP set in a standard independent space1,h2.…,hs) Further, a unit hypercube [ 01 ] is obtained from the formula (15)]SInner point set:
in the formula, N is the number of point sets to be constructed, hsIs Fibonacci sequence, and the Fibonacci sequence is defined by a recursive method as follows: h isj＝hj-1+hj-2,h0＝h1＝1(n＝1,2,...,S)。
S23, taking the limit of the normalized random variable as L, and using the formula (16) to carry out scaling and translation transformation on the point set generated in the step S22 to obtain a square [ -L, L]SUniformly distributed point sets within.
θj,k＝2(xj,k-0.5)L,(k＝1,2,…,N,j＝1,2,…,S) (16)
And S24, transforming the sample points of the standard independent space to the corresponding sample points of the original space by utilizing Nataf inverse transformation.
In step 3, the representative point in step S2 is used as input, and the meshing rigidity of the gear containing the cracks is analyzed on the basis of finite element method dynamic simulation; the method specifically comprises the following steps:
s31, establishing a finite element model of a first-stage speed reducing mechanism, namely a planetary speed reducing mechanism, of the industrial robot speed reducer with the initial crack according to the representative point determined in the step S2, wherein the crack length is l, and the load is F.
S32, solving a finite element model of the neutralization potential in S31, analyzing the strain delta of the planetary reduction mechanism when the gear teeth have cracks (as shown in figure 3), and calculating the gear mesh stiffness K according to the strainpi。
In the formula, FcIs the tooth flank force calculated from the retarder load F.
In step 4, the meshing stiffness calculated in step S3 is substituted into a gear dynamic equation, and the transmission error of the gear with the cracks is calculated; the method specifically comprises the following steps:
s41, calculating the gear meshing rigidity K of the planetary reduction mechanism when the gear teeth are cracked according to the specific industrial robot reducer model and the calculated gear meshing rigidity K in S32piAnd establishing a corresponding transmission error dynamic model. The invention uses RV-20E type industrial robot reducer as an example, and uses a 'grade model' as a transmission error dynamic model of the industrial robot reducer. The "equivalent model" method is a method in which each component in a mechanism is a rigid body, a play between the components is replaced with a spring, and a machining error, a clearance, a mounting error, and a minute displacement caused by a deviation from a theoretical position of each component are replaced with a spring variation between the components. And (4) deriving the stress of each part according to the equivalent error, and establishing a kinematic equation of the part. The micro displacement of each part can be solved by simultaneously solving a kinematic equation set, so that the rotation angle deviation of the output shaft, namely the transmission error, can be calculated.
The RV-20E type speed reducer provided by the invention mainly comprises an input shaft, 2 planet wheels, 2 crankshafts, 2 cycloidal gears, 40 pin gears, 1 planet carrier, 1 pin gear shell and the like. Fig. 3 is a schematic diagram of an equivalent model of the RV reducer, and it can be seen from the diagram that there are 17 displacement variables of the equivalent model, so the model is a 17-degree-of-freedom planar model.
And establishing a kinetic equation set according to the model:
the relevant parameters in the equation are shown in fig. 4, and equation (18) is written in matrix form:
wherein F is the reducer load matrix, M is the mass matrix, C is the damping matrix, K is the stiffness matrix and includes KpiAnd X is a displacement matrix of each part when the load is F and comprises:
XT＝[Xs,Ys,Xp1,Xp2,Yp1,Yp2,θp1,θp2,θd1,θd2,θdo1,θdo2,η1,η2,Xca,Yca,θca]
wherein, thetacaFor actual output of angle of rotation theta of speed reducer of industrial robotcThe theoretical output corner of industrial robot reduction gear, industrial robot reduction gear transmission error e can be solved by following mode:
e＝θca-θc (20)
and taking the maximum value of the amplitude of the transmission error of the industrial robot in the period to represent the level of the transmission error under the current input.
S42, solving the dynamic model established in the step S32 by using a Newmark method to simulate the transmission error, solving X in a kinematic equation set, and calculating the transmission error of the industrial robot speed reducer in a period when the gear has cracks.
In step S5, analyzing a transmission error probability density function of the crack-containing gear of the industrial robot based on probability density evolution theory and a dickstra function, wherein step S5 specifically includes the following sub-steps:
s51, obtaining an evolution equation of the system state response joint probability density according to the principle of probability conservation:
in the formula: f (-) represents the probability density function, z is the structural dynamics response vector, h is the first differential of the response, theta is the uncertainty input vector, and t is time. The probability density evolution equation under single dynamic response is as shown in formula (5):
in the formula:
is the first differential of the kinetic response.
S52 probability density function f of transmission error of industrial robotz(z, t) can be calculated from equation (6) by introducing the Dike function, the probability density function f of the transmission errorz(z, t) calculating the conversion from formula (6) to formula (7):
fz(z,t)＝∫ΩΘfZΘ(z,θ,t)dθ (23)
where δ (·) is the Dikla function, Hi(θ,z0T) is a response uniform discrete point, Pq,iA probability is assigned to each representative point.
S53, calculating the probability density function f of the transmission error through the Gauss Dike-Law approximation functionz(z,t)：
Where σ is a smoothing parameter, and is generally taken as σ ═ Δ Hi。
In step 6, a monte carlo-based method is specifically adopted to evaluate the reliability of the gear transmission precision of the transmission error probability density function of the gear with the crack established in step S5.
According to the method, the relationship between the crack propagation condition of the gear and the transmission performance is analyzed through dynamic simulation, and a transmission error distribution model is established by combining a probability density evolution theory and a Dike function, so that the transmission precision reliability of the crack-containing gear is further evaluated. The method applies probability density evolution and a Diecka function to reliability evaluation of mechanical parts, and can realize multi-degree-of-freedom linear and nonlinear mechanical structure dynamic performance response characterization and reliability calculation under the input of high-precision and high-efficiency description uncertainty factors.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.
Claims (2)
1. A method for analyzing reliability of transmission precision of a speed reducer of an industrial robot with a crack gear comprises the following steps:
s1, determining uncertainty input of the industrial robot speed reducer;
s2, selecting an input representative point according to the uncertainty input of the industrial robot speed reducer determined in the step S1;
the method specifically comprises the following steps:
s21, determining the dimension and the statistical characteristic of the basic random variable X of the structure to be analyzed;
s22, obtaining a generated vector (h) of S dimension by the aid of grid point set in standard independent space1,h2,...,hS) Further, a unit hypercube [ 01 ] is obtained from the formula (1)]SInner point set:
wherein N is the number of point sets to be constructed, hSIs Fibonacci sequence, and the Fibonacci sequence is defined by a recursive method as follows: h isj＝hj-1+hj-2,h0＝h1＝1(j＝1,2,...,S)；
S23, taking the limit of the normalized random variable as L, and carrying out scaling and translation transformation on the point set generated in the step S22 by using the formula (2) to obtain a square [ -L, L]SInner uniformly distributed point set:
θj,k＝2(xj,k-0.5)L,(k＝1,2,…,N,j＝1,2,…,S) (2)
s24, transforming the sample points of the standard independent space to the sample points corresponding to the original space by utilizing Nataf inverse transformation, wherein the sample points obtained by transformation are input representative points;
s3, taking the representative point in the step S2 as an input, and carrying out simulation analysis on the meshing rigidity of the gear containing the crack;
the method specifically comprises the following steps:
s31, establishing a finite element model of a first-stage speed reducing mechanism, namely a planetary speed reducing mechanism, of the industrial robot speed reducer with the initial cracks according to the representative points determined in the step S2;
s32, solving the finite element model established in the step S31, analyzing the strain of the planetary reduction mechanism when the gear teeth have cracks, and calculating the gear meshing rigidity K according to the strainpi；
S4, substituting the meshing stiffness calculated in the step S3 into a gear dynamics equation, and calculating the transmission error of the gear with the cracks;
the method comprises the following specific steps:
s41, calculating the gear meshing rigidity K of the planetary reduction mechanism when the gear has cracks according to the specific industrial robot reducer model and the step S32piAnd establishing a corresponding transmission error dynamic model:
wherein F is a reducer load matrix, M is a mass matrix, C is a damping matrix, and K is a stiffness matrix and includes a gear mesh stiffness KpiX is a displacement matrix of each part when the load is F and contains the transmission error of the industrial robot speed reducer,
the first derivative of X is represented by the equation,
represents the second derivative of X;
s42, solving the transmission error dynamic model by using a Newmark method, and calculating the transmission error of the industrial robot speed reducer in a period when the gear has cracks;
s5, analyzing a transmission error probability density function of the gear with the crack;
the method specifically comprises the following steps:
s51, obtaining an evolution equation of the system state response joint probability density according to the principle of probability conservation:
wherein f (-) represents a probability density function, z is a structural dynamics response vector, h is a first order differential of the response, θ is an uncertainty input vector, nzNumber of structural dynamic response vectors, zlThe I structural dynamics response vector is expressed, t is a time variable, and the probability density evolution equation under single dynamic response is as shown in the formula (5):
wherein the content of the first and second substances,
is the first differential of the kinetic response;
s52 probability density function f of transmission error of industrial robotz(z, t) can be calculated from equation (6) by introducing the Dikela function, the probability density function f of the transmission error of the industrial robotz(z, t) calculating the conversion from formula (6) to formula (7):
fz(z,t)＝∫ΩΘfZΘ(z,θ,t)dθ (6)
where δ (·) is the Dikla function, Hi(θ,z0T) is a response uniform discrete point, Pq,iAssigning a probability to each representative point;
s53, calculating probability density function f of transmission error of industrial robot through Gauss Dike Laval functionz(z,t)：
Wherein σ is a smoothing parameter;
and S6, carrying out gear transmission precision reliability evaluation on the transmission error probability density function of the gear with the crack established in the step S5.
2. The method for analyzing the transmission precision reliability of the industrial robot speed reducer with the crack gear as claimed in claim 1, wherein the step S6 is implemented by performing the gear transmission precision reliability evaluation on the transmission error probability density function of the crack gear established in the step S5 by using a monte carlo-based method.
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