Cylinder axis element error model under multi-tolerance coupling effect and solving method

Abstract

The invention discloses a cylinder axis element error model and a solving method under a multi-tolerance coupling effect, and the method comprises the steps: 1, describing the geometric element error of a part through the tiny variation of six degrees of freedom; 2, constructing a rectangular coordinate system by taking the center of the rational cylinder as an original point; 3, according to the technical specification, the intersecting line of the straightness tolerance domain and the plane YOZ is a parallel straight line with the interval being TF. 4, the coaxiality not only locks the directions of the geometric elements, but also limits the positions of the geometric elements; the perpendicularity only locks the direction; 5, substituting the variation inequality and the constraint inequality of each error component of the upper and lower boundaries of the straightness tolerance to determine the variation range and the constraint inequality range of each error component; and 6, simulating and sampling error component groups of the upper and lower boundaries by combining the error component probability distribution model. According to the method, whether the error under the design tolerance coupling effect exceeds the required maximum error or not can be accurately verified, the correctness of the design precision grade is verified, and the unnecessary manufacturing cost is reduced.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of tolerance modeling, in particular to a method for establishing and solving a cylindrical axis element error model under the action of multiple tolerance couplings. BACKGROUND

[0002] The machining error of a part is the main source of assembly error and is also a difficult problem that cannot be completely avoided in the manufacturing process. How to accurately predict the machining error at the design stage has always been a key challenge for the industry. Therefore, it is particularly important to establish an accurate geometric element error model and select an appropriate solving method. In actual production, in order to meet the precision requirements of parts, multiple tolerance couplings are often applied to a certain geometric element to reduce the error fluctuation range. By constructing an error model consistent with the working condition and reasonably solving it, the error distribution can be predicted before machining, which can greatly reduce the subsequent manufacturing and repair costs, shorten the debugging time, and effectively improve the overall production efficiency.

[0003] The prior art mainly focuses on the expression of tolerance items and errors, and less on the rationality of the solving method.

[0004] Chinese patent CN113065203B discloses a method for establishing and solving a cylindrical surface element error model under the action of multiple tolerance couplings. The method establishes a cylindrical surface multiple tolerance coupling error model based on the small displacement spinor method, and obtains the error prediction range through two Monte Carlo methods. However, the tolerance items selected in the above research are not representative, and the variation mode of the tolerance band only includes fixed and floating, but does not include translational tolerance band. When determining the tolerance domain range, error model, probability distribution type of different tolerance items and error components, only two Monte Carlo sampling is used to simulate the machining error and component variation interval. This method has three defects: first, the sampling method based on the tolerance domain range cannot predict the machining error, and the obtained sampling values lack practical significance; second, this solving method ignores the comprehensive influence of shape deviation and pose deviation of geometric elements on error; finally, there are operation difficulties in the constraint discrimination process of error component solving results. Therefore, this method has limitations in practicality and reliability. SUMMARY

[0005] The technical problem to be solved by the present application is that the error variation range of the cylindrical axis geometric element under the action of multiple tolerance coupling constraints at the part design stage is uncertain. The present application combines the actual situation and models the error variation of the cylindrical axis under the simultaneous action of coaxiality, straightness and coaxiality tolerance, and based on the part design accuracy grade, linear regression fitting and Monte Carlo method for solving the model, a method for establishing and solving a cylindrical axis element error model under the action of multiple tolerance couplings is proposed.

[0006] The technical scheme of the present application is: a cylindrical axis element error model and solving method under multi-tolerance coupling, comprising the following steps:

[0007] Step one: combine the SDT small displacement torsor method with part tolerance, and use six degrees of freedom of small changes to describe the geometric element error of the part. In SDT, D=(μ, v, w, α, β, γ), wherein μ, v, and w represent the small change vectors along the x, y, and z coordinate axes, and α, β, and γ represent the small change vectors of rotation around the x, y, and z coordinate system. In the SDT expression of different kinds of tolerance constraints, if the translation or rotation sweep trajectory of the geometric element along a certain coordinate axis does not change its shape, then the pose change in this direction has no effect on the shape, and the corresponding component is zero. The error change of the geometric element is described by the non-zero components in SDT, so these non-zero components are called error components.

[0008] Step two: construct a rectangular coordinate system with the center of a cylinder with an ideal radius R and a height of 2l as the origin, the cylinder axis direction coincides with the z-axis of the coordinate system, and the non-zero vectors of SDT are μ, ν, α, and β. Let the coaxiality of the cylinder axis be T P , the perpendicularity be T O , and the straightness be T F , and assume that the coaxiality tolerance domain coincides with the z-axis. Then the equation of the ideal cylinder axis is (taking the YOZ plane as an example):

[0009] y=0

[0010] The equation of the actual cylinder axis geometric element is:

[0011] y=αz+v

[0012] Step three: according to the product geometric technical specification (GPS), the shape variation area of the actual axis of the cylindrical surface should be limited within a cylindrical surface with a diameter of T F , and the intersection line of the straightness tolerance domain and the plane YOZ is a parallel straight line with a spacing of T F , and the relationship equation between the upper and lower boundaries is as follows:

[0013] y t =y b +ΔT

[0014] Therefore, the upper and lower boundary equations of the straightness tolerance domain are:

[0015] y b =α b z+v b

[0016] y t =α t z+v t

[0017] where α b = α t , Since α is very small, cos α = 1, v t = v b + T F . The parameters with subscript b represent the lower boundary parameters of straightness tolerance zone, and the parameters with subscript t represent the upper boundary parameters, where -l≤z≤l.

[0018] Step four: coaxiality locks both the direction and the position of geometric elements; perpendicularity only locks the direction; straightness only restricts the shape without interfering with the direction and position. Thus, the coaxiality tolerance zone remains fixed, the perpendicularity tolerance zone can be translated within the coaxiality zone, and the straightness tolerance zone can be freely floating within the perpendicularity zone. Therefore, there are many complex cases of changes in the coupling constraints of coaxiality, perpendicularity and straightness tolerance bands, but at least one complete axis must exist in the intersection of the three tolerance zones, which is reflected in the two-dimensional plane as at least one complete straight line passing through the upper and lower boundaries of the coaxiality tolerance zone. Selecting the case where the straightness tolerance changes at the maximum value, and taking the YOZ plane as an example, we can get:

[0019]

[0020] According to the coaxiality tolerance zone, we have:

[0021]

[0022] The straightness tolerance lower boundary SDT error component variation inequality and constraint condition are:

[0023]

[0024] The constraint inequality is:

[0025]

[0026] Similarly, the straightness tolerance upper boundary SDT error component variation inequality and constraint inequality are:

[0027]

[0028] Selecting the normal distribution as the probability distribution type of each error component value, denoted as X ~ N(μ, σ 2 ), where μ is the mean and σ 2 is the variance. The probability density function of the normal distribution is as follows:

[0029]

[0030] Step five: According to the accuracy level requirements of coaxiality, perpendicularity and straightness tolerance in the part design stage, the tolerance band size is queried, and the straightness tolerance upper and lower boundary error component variation inequality and constraint inequality are substituted to determine the error component variation range and constraint inequality range

[0031] Step six: According to the error component variation range of the upper and lower boundaries respectively, the error component groups of the upper and lower boundaries are simulated and sampled in combination with the error component probability distribution model. m sampling points are uniformly set in [-1, 1] in the z direction, and according to the known upper and lower boundary equations of the straightness tolerance field, first, the error component groups of the upper and lower boundaries are substituted into the constraint inequality to determine whether the sampled error components meet the constraint. If they do not meet the constraint, the boundary error components need to be resampled. After meeting the constraint, the y value corresponding to each sampling point in the straightness tolerance field range is obtained by random sampling method, and m groups of simulation points in the straightness tolerance field can be obtained. According to these simulation point values, the error component values of the cylindrical axis are solved by linear regression algorithm, and the expression is as follows:

[0032]

[0033] Wherein

[0034] According to the above process, repeat K times. Since different error component variation sequences will affect the final value of the error component, the same number of samples are also extracted according to different sampling sequences, so that 2K actual error samples of each error component are finally retained, and the maximum value is taken as the actual variation range of each error component.

[0035] The beneficial effects of the present application are that the multi-tolerance coupling cylindrical axis element error model establishment and solving method has an important role in predicting the variation of the cylindrical axis element error under the constraint of multi-tolerance coupling in actual engineering. The axis geometric element exists in the shaft hole type part, determines the positioning accuracy and fitting quality of the part when the hole and the shaft are assembled, and plays an important role in the transmission system. Through the characteristic analysis of the coaxiality tolerance, perpendicularity tolerance and straightness tolerance, the cylindrical axis element error model under the constraint of multi-tolerance coupling is established, the error component variation inequality and constraint inequality are expressed through mathematical analysis method, the precision grade requirements of each tolerance in the part design stage are combined, the linear regression fitting solving and Monte Carlo simulation sampling method are combined, and the actual machining error of the cylindrical axis geometric element is effectively predicted. Compared with the translation and rotation existing in the cylindrical axis error, the shape change of the axis is also considered, and is added to the SDT expression of the error through linear regression fitting processing, the variation range of the cylindrical axis element error under the constraint of multi-tolerance coupling is more accurately predicted, so that whether the error under the designed tolerance coupling action exceeds the required maximum error can be more accurately verified, the correctness of the design precision grade is verified, and unnecessary manufacturing cost is reduced. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 The cylindrical axis error variation graph under the coupling action of the coaxiality tolerance, perpendicularity tolerance and straightness tolerance in the embodiment of the present application.

[0037] Figure 2 The two-dimensional upper and lower boundary graph of the straightness tolerance domain in the embodiment of the present application.

[0038] Figure 3 The graph of the straightness tolerance translation amount in the extreme case in the embodiment of the present application.

[0039] Figure 4 The graph of the straightness tolerance rotation amount in the extreme case in the embodiment of the present application

[0040] Figure 5 The sampling solving process flow chart (taking the order of α b ,ν b as an example) in the embodiment of the present application.

[0041] Figure 6 The model construction and solving flow chart in the embodiment of the present application. DETAILED DESCRIPTION

[0042] The technical scheme of the present application is further described below

[0043] Referring to Figures 1-6 , the specific steps of the multi-tolerance coupling cylindrical axis element error model establishment and solving method of the present application are as follows

[0044] Step one: Combine the SDT small displacement torsor method with part tolerance, and use six degrees of freedom of small changes to describe the geometric element error of the part. In SDT, D=(μ,ν,w,α,β,γ), where μ,ν,w represent the small change vectors along the x, y, z coordinate axes, and α,β,γ represent the small change vectors around the x, y, z coordinate system, respectively. In the SDT expression of different types of tolerance constraints, when the geometric element moves or rotates along a certain coordinate axis, if the trajectory swept does not change compared with its own shape feature, it means that the pose change of the geometric element in the corresponding direction has no effect on the shape feature of the geometric element, and the corresponding change component is zero. Since the geometric element error change is expressed by the non-zero component of SDT, the non-zero component of the geometric element SDT is called the error component.

[0045] Step two: According to the product geometric technical specification (GPS) and actual functional requirements, the shape tolerance is usually more stringent than the position tolerance in the design. In the common shaft-hole assembly, due to the priority of function, the consideration of manufacturing and detection cost, the coaxiality tolerance band is usually larger than the perpendicularity tolerance band. A rectangular coordinate system is constructed with the center of the ideal cylinder with radius R and height 2l as the origin, the cylinder axis direction coincides with the z-axis of the coordinate system, and the SDT non-zero vectors are μ,ν,α and β. Let the coaxiality of the cylinder axis be T P , the perpendicularity be T O , and the straightness be T F , and assume that the coaxiality tolerance domain axis coincides with the z-axis. The specific case of multi-tolerance coupling of the cylinder axis geometric element is shown in Figure 1 . Then the equation of the ideal cylinder axis is (take the YOZ plane as an example):

[0046] y=0

[0047] The equation of the actual cylinder axis geometric element is:

[0048] y=αz+v

[0049] Step three: According to the product geometric technical specification (GPS), the shape change region of the actual axis of the cylindrical surface should be limited within the cylindrical surface with diameter T F , the intersection line of the straightness tolerance domain and the plane YOZ is a parallel straight line with a spacing of T F , and the relationship equation between the upper and lower boundaries is shown in Figure 2 ,

[0050] y t =y b +ΔT

[0051] where The parameters with subscript b represent the lower boundary parameters of straightness tolerance zone, and the parameters with subscript t represent the upper boundary parameters, where -l≤z≤l.

[0052] Step four: Since the coaxiality tolerance simultaneously constrains the direction and position of geometric elements, the perpendicularity tolerance constrains the direction of geometric elements, and the straightness tolerance only constrains the shape of geometric elements, not the direction and position of geometric elements, so the variation of the three tolerance zones is as follows: the coaxiality tolerance zone is fixed, the perpendicularity tolerance zone can move horizontally in the coaxiality tolerance zone, and the straightness tolerance zone floats in the perpendicularity tolerance zone. Therefore, the variation of the coaxiality, perpendicularity and straightness tolerance zone coupling constraints has many complex cases, but it must satisfy that there is at least one complete axis in the intersection part of the three tolerance zones, which is reflected in the two-dimensional plane as at least one complete straight line passing through the upper and lower boundaries of the coaxiality tolerance zone. Select the straightness tolerance variation at the maximum value as shown in Figure 3 、 Figure 4 According to the geometric relationship, we have:

[0053]

[0054] According to the coaxiality tolerance zone, we have:

[0055]

[0056] The straightness tolerance lower boundary SDT error component variation inequality and constraint inequality are:

[0057]

[0058] Similarly, the straightness tolerance upper boundary SDT error component variation inequality and constraint inequality are:

[0059]

[0060] Select normal distribution as the probability distribution type of each error component value, denoted as X~N(μ,σ 2 ), where μ is the mean, σ 2 is the variance, and the probability density function of the normal distribution is as follows:

[0061]

[0062] Step five: According to the precision grade requirements of coaxiality, perpendicularity and straightness tolerance in the part design stage, the tolerance zone width is queried, and the straightness tolerance upper and lower boundary error component variation inequality and constraint inequality are substituted to determine the error component variation range and constraint inequality

[0063] Step six: Use the Monte Carlo method to simulate sampling of the error component group of the lower boundary according to the probability distribution type of the error component, and obtain αb and v b , and then the upper boundary error component group of the straightness tolerance domain is obtained through the relationship between the upper and lower boundaries

[0064] α b = α t

[0065]

[0066] wherein, since α is small, cos α = 1, v t = v b + T F .

[0067] Then the upper and lower boundary equations of the straightness error need to be determined,

[0068] m sampling points are uniformly set in the [-l, l] in the z direction, and the z coordinate values of the sampling points are substituted into the known upper and lower boundary equations of the straightness tolerance domain

[0069]

[0070] Then, whether the obtained boundary error component meets the constraint is determined in combination with the constraint inequality of the upper and lower boundaries. Then, whether the obtained error component meets the constraint is determined in combination with the error component group of the upper and lower boundaries. If the constraint is not met, the boundary error needs to be resampled. After the constraint is met, the y value corresponding to each sampling point in the straightness tolerance domain range is obtained through random sampling, and m groups of simulation points in the straightness tolerance domain can be obtained.

[0071] Taking the sampling sequence of α b and v b as an example, α b is first sampled, and then v b is sampled. At this time, whether the constraint is met is determined by substituting the two into the constraint inequality. If the constraint is not met, v b is resampled. In addition, an upper limit N1 is set to prevent the cycle from being trapped in the resampling of v b . When the sampling of v b reaches the set upper limit, the sampling of α b is started again. If the constraint is met, the current α b and v b values are retained, and α t and v t are calculated. The y coordinates of the m simulation points are obtained through the Monte Carlo method again according to the straightness tolerance domain boundary equation. At this time, the curve composed of these simulation points represents the prediction of the actual machining error of the geometric elements of the cylindrical axis.

[0072] According to these simulation point values, a linear regression algorithm is used to fit and solve the error component values at this time, and the expression is as follows:

[0073]

[0074] wherein

[0075] The above process needs to be repeated K times to form the total sample of the error component. The simulation sampling process of each error component is as shown in Figure 5

[0076] Since different error component variation sequences will affect the final value of the error component, the same number of samples are also extracted according to different sampling sequences, so that a total of 2K actual error samples are retained for each error component, and the maximum value is taken as the actual variation range of each error component.​

Claims

1. A method for solving cylindric axis element error model under multi-tolerance coupling, characterized in that: Comprising the following steps: Step one: combine the SDT small displacement torsor method with part tolerance, use six degrees of freedom of the small change to describe the geometric element error of the part; in SDT, D=(μ, v, w, α, β, γ), where μ, v, w represent the small change vector along the x, y, z three coordinate axes, and α, β, γ represent the small change vector rotating around the x, y, z three coordinate system respectively; Step two: take the center of the cylinder with ideal radius R and height 2l as the origin to construct a rectangular coordinate system, the cylinder axis direction coincides with the z axis of the coordinate system, and the non-zero vectors of SDT are μ, ν, α and β; let the coaxiality of the cylinder axis be T P , the perpendicularity be T O , and the straightness be T F , and the axis of the coaxiality tolerance field coincides with the z axis; Step three: According to the product geometry technical specification GPS, the shape variation area of the actual axis of the cylindrical surface should be limited within the cylindrical surface with diameter T F , the intersection line of the straightness tolerance field and the plane YOZ is a parallel straight line with a distance of T F , and the relationship equation between the upper and lower boundaries is as follows: y t = y b + ΔT The upper and lower boundary equations of straightness tolerance domain are: y b = a b z + v b y t = a t z + v t where α b = α t , Since α is small, cos α = 1, v t = v b + T F ; the parameters with subscript b represent the lower boundary parameters of the straightness tolerance zone, and the parameters with subscript t are the upper boundary parameters, where -l ≤ z ≤ l. Step four: coaxiality not only locks the direction of geometric elements, but also limits its position; perpendicularity only locks the direction; straightness only constrains the shape without interfering with the direction and position; Step five: according to the accuracy level requirements of coaxiality, perpendicularity and straightness tolerance in the design stage of the part, query the tolerance band size, and substitute it into the straightness tolerance upper and lower boundary error component variation inequality and constraint inequality to determine the error component variation range and constraint inequality range; Step six: according to the error component variation range of the upper and lower boundaries, respectively, combined with the error component probability distribution model, the error component groups of the upper and lower boundaries are simulated and sampled; m sampling points are uniformly set in the z direction [-l, l], according to the known straightness tolerance domain upper and lower boundary equation, first combine the error component groups of the upper and lower boundaries to substitute into the constraint inequality to determine whether the sampled error components meet the constraint, if not, re-sample the boundary error components, and if they meet the constraint, then obtain the corresponding y value of each sampling point in the straightness tolerance domain range by random sampling method, and get m groups of simulated points in the straightness tolerance domain; according to the simulated points, the error component value of the cylindrical axis is solved by linear regression algorithm, and the expression is as follows: wherein According to the above process, repeat K times, finally keep 2K actual error samples for each error component, and take the maximum value as the actual variation range of each error component.

2. The method of claim 1, wherein: In step one, in the SDT expression of different kinds of tolerance constraints, if the translation or rotation scanning track of the geometric element along a certain coordinate axis does not change its shape, the pose variation in this direction has no effect on the shape, and the corresponding component is zero; the error variation of the geometric element is described by the non-zero components in SDT, so these non-zero components are called error components.

3. The method of claim 1, wherein: In step two, the equation of the ideal cylindrical axis is: y＝0 The equation of the actual cylindrical axis geometric element is: y = αz + v.

4. The method of claim 1, wherein: In step four, select the straightness tolerance variation in YOZ plane under the condition of maximum value: According to the coaxiality tolerance domain, we have: Then the straightness tolerance lower boundary SDT error component variation inequality and constraint condition are: The constraint inequality is: The straightness tolerance upper boundary SDT error component variation inequality and constraint inequality are: The normal distribution is selected as the probability distribution type of each error component value, denoted as X ~ N(μ, σ 2 ), where μ is the mean value and σ 2 is the variance. The probability density function of the normal distribution is as follows:

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