CN113283025B - Involute tooth profile error modeling method containing system error - Google Patents
Involute tooth profile error modeling method containing system error Download PDFInfo
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- CN113283025B CN113283025B CN202110512824.7A CN202110512824A CN113283025B CN 113283025 B CN113283025 B CN 113283025B CN 202110512824 A CN202110512824 A CN 202110512824A CN 113283025 B CN113283025 B CN 113283025B
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/10—Geometric CAD
- G06F30/17—Mechanical parametric or variational design
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
- F16H—GEARING
- F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
- F16H55/02—Toothed members; Worms
- F16H55/08—Profiling
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
- F16H—GEARING
- F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
- F16H55/02—Toothed members; Worms
- F16H55/17—Toothed wheels
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
- Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation
Abstract
The invention provides an involute tooth profile error modeling method containing a system error, which is characterized by comprising the following steps: firstly, carrying out involute tooth profile error modeling to obtain point cloud data of an actual involute containing a system error; secondly, generating an involute tooth profile error curved surface based on the NURBS numerical model; thirdly, verifying the modeling precision of the curved surface; and fourthly, generating a tooth profile error tooth surface CAD model based on the IGES. The invention can obtain the actual gear geometric model with tooth profile errors.
Description
Technical Field
The invention belongs to the technical field of gear geometric error modeling, and particularly relates to an involute tooth profile error modeling method containing system errors.
Background
The gear system mainly transfers motion and power by means of tooth surface contact engagement, and the tooth surface contact state is greatly influenced by errors of tooth profile surfaces. The tooth profile accuracy is determined by factors in the actual machining of the gear. The machining error causing the tooth profile deviation is divided into a system error and a random error, wherein the system error is changed according to a certain rule in the machining error, is usually far larger than the random error and is dominant, and the random error is irregular and is smaller in proportion. The existing method for expressing the tooth profile error into the random error does not consider the main influence of the system error in the actual processing, and does not really express the duplication of the processing error on the gear geometry.
Disclosure of Invention
In view of the above, the present invention provides an involute tooth profile error modeling method including a system error, which can obtain an actual gear geometric model with a tooth profile error.
The invention is realized by the following technical scheme:
a method for modeling an involute tooth profile error including a system error, comprising the steps of: firstly, carrying out involute tooth profile error modeling to obtain point cloud data on an actual involute containing a system error; secondly, generating an involute tooth profile error curved surface based on the NURBS numerical model; thirdly, verifying the modeling precision of the curved surface; and fourthly, integrating the IGES file obtained based on the NURBS numerical model with the CAD model.
Further, the involute profile error modeling is performed by the following steps: obtaining point cloud data of a theoretical involute according to an ideal equation of an involute tooth profile curve; according to the gear machining rule and the value range of the tooth profile error, expressing the system error which causes the involute tooth profile error into a tiny rotation angle error gamma of a theoretical involute rotating around a tooth root or a tooth top; the random error is added to the involute profile curve as a small disturbance component.
Firstly, point cloud data of a theoretical involute is obtained through the following formula:
wherein r is b Is the radius of the base circle; theta is the rolling angle of any point on the involute, and theta is equal to theta k +α k ；θ k Is the spread angle of k points on the involute, alpha k Is the pressure angle at point k on the involute;
secondly, the value range of the tooth profile error meets the following formula:
ΔF α ≤F α (2)
wherein, Δ F α Is the total deviation of the tooth profile; f α Determining the total tolerance value of the tooth profile according to design requirements;
the systematic errors include: (1) eccentricity errors in workpiece mounting due to Δ x, Δ y; (2) the rotation angle error delta theta; (3) machining tool and (x) 0 ,y 0 ) Error of center distance Δ r b ；
An expression for the systematic error is established by the following equation:
wherein, gamma is a tiny rotation angle error rotating around a tooth root or a tooth top;
thirdly, attaching the random error as a tiny disturbance quantity on the involute tooth profile curve through normal distribution:
Δ∈N(μ,σ) (4)
wherein, mu is the mean value of normal distribution, and sigma is the standard deviation of normal distribution;
fourthly, all errors comprise systematic errors and random errors, and point cloud data of Q' on the actual involute are obtained through the following formula:
wherein Q' is a coordinate value of a point on the actual involute; x is the number of 0 、y 0 、z 0 The coordinate value of the point on the ideal involute;
further, an involute tooth profile error curved surface based on the NURBS numerical model is generated through the following steps: and according to the generated m x n point cloud data on the involute tooth profile curved surface containing the system error, establishing a curved surface model of the actual tooth profile by adopting a NURBS curved surface interpolation reconstruction technology modeling method.
Firstly, generating m x n point cloud data with tooth profile errors through a formula (1), a formula (2), a formula (3), a formula (4) and a formula (5);
secondly, according to the generated m x n point cloud data, a NURBS curved surface interpolation reconstruction technology modeling method is adopted to obtain a curved surface model with the characteristics of the irregular geometric shape of the actual tooth profile surface;
and further, verifying the surface modeling precision through the distance from the point set on the actual involute tooth profile curved surface to the point set on the ideal involute tooth profile curved surface.
Firstly, verifying that the geometric error of the obtained tooth profile is within the total tolerance range of the tooth profile by using the following formula:
F′ α ＝|d Max |+|d Min |≤F α (7)
wherein (x) i ,y i ) Is a point on the actual involute profile curve, (x) 0i ,y 0i ) Is a point on the surface of the ideal involute profile, F α Is the tooth profile total tolerance value, F' α Is the actual tooth profile error.
Further, an IGES-based tooth profile error tooth surface CAD model is generated by the following steps: according to an IGES file generated by a modeling method based on NURBS curved surface interpolation reconstruction technology, integrating the IGES file with a CAD model to obtain an actual tooth profile curved surface with tooth profile errors; and obtaining a gear geometric model with tooth profile errors by using a flexible modeling and materialized modeling method.
Has the advantages that: the invention provides a modeling method of involute tooth profile errors including system errors, which is characterized in that a NURBS curved surface interpolation reconstruction technology modeling method is adopted to establish a curved surface model with irregular geometric shape characteristics of an actual tooth profile surface, and an actual gear geometric model with tooth profile errors can be obtained.
Drawings
FIG. 1 is a schematic view of an involute profile;
FIG. 2 is an involute profile curve and base radius error;
FIG. 3 is a development of an actual involute profile curve;
FIG. 4 is a schematic view of the actual involute profile curve;
FIG. 5 is a diagram of involute surface point cloud data and an interpolated three-dimensional surface;
FIG. 6 is an actual tooth profile surface with a tooth profile error;
FIG. 7 is a geometric model of a gear with tooth profile error
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
Example 1:
the embodiment provides an involute tooth profile error modeling method containing a system error, which comprises the following steps:
as shown in fig. 1, the variation range of the tooth profile error is shown, and the taken system error and the random error are within the variation range of the tooth profile error.
ΔF α ≤F α (1)
Wherein, Δ F α Is the total deviation of the tooth profile; f α Determining the total tolerance value of the tooth profile according to design requirements;
referring to fig. 2 for an ideal involute profile curve and base radius error, the systematic error that determines the involute profile error can be represented by a slight rotational angle error γ around the addendum or dedendum.
First, referring to fig. 2(a), point cloud data on a theoretical involute is obtained by the following formula:
wherein r is b Is the radius of the base circle; theta is the rolling angle of any point on the involute, and theta is equal to theta k +α k ；θ k Is the spread angle of k points on the involute; alpha is alpha k Is the pressure angle at point k on the involute;
second, referring to FIG. 3, the involute profile curve is expressed in two dimensions, and the systematic error is expressed as a small rotational angle error γ around the z-axis:
thirdly, the random error is expressed by normal distribution:
Δ∈N(μ,σ) (4)
wherein mu is the mean value of normal distribution, and sigma is the standard deviation of normal distribution;
fourthly, referring to fig. 3, since all errors include a systematic error and a random error, a point Q' on the actual involute is obtained by combining equation (2), equation (3) and equation (4):
wherein x is 0 、y 0 、z 0 The coordinate value of the point on the ideal involute;
fifthly, referring to fig. 4(a), m × n point cloud data with tooth profile errors are generated through a formula (1), a formula (2), a formula (3), a formula (4) and a formula (5), wherein fig. 4(a) only shows a local comparison between one actual involute tooth profile curve and an ideal involute tooth profile curve;
referring to fig. 4(b), there are 76 points along the involute direction and 48 involutes along the tooth width direction for the number of points used to construct the actual involute tooth profile curved surface in the calculation example;
referring to fig. 5(a), the point cloud data of the actual involute profile curved surface;
sixthly, referring to the attached figure 5(b), establishing an actual involute tooth profile curved surface model by using a NURBS curved surface interpolation reconstruction technology modeling method;
seventhly, verifying the curved surface modeling precision: verifying the modeling precision of the curved surface through the distance d from the point set on the actual involute tooth profile curved surface to the point set on the ideal involute tooth profile curved surface;
F′ α ＝|d Max |+|d Min |≤F α (7)
wherein (x) i ,y i ) Is a point on the actual involute profile curve, (x) 0i ,y 0i ) Is a point on the surface of the ideal involute profile curve, F α Is the tooth profile total tolerance value, F' α Is the actual tooth profile error;
eighthly, referring to the attached figure 6, integrating an IGES file obtained by a NURBS curved surface interpolation reconstruction technology modeling method with a CAD model to obtain a curved surface model of an actual tooth profile error surface; establishing an actual gear geometric model with tooth profile errors by using flexible modeling and materialization technologies;
see fig. 6(a), for actual tooth profile error surface;
see fig. 6(b), for an enlargement of the tooth profile error surface;
referring to fig. 7, a geometric model of a gear with tooth profile error is shown, wherein fig. 7(a) shows an actual single tooth and fig. 7(b) shows an actual geometric model of a gear;
the involute tooth profile error modeling method can obtain the actual involute tooth profile curved surface formed in the gear machining process, and can provide a gear geometric model with errors for further researching the tooth surface contact problem of the gear. The method can be widely applied to the evaluation of the tooth profile error in the gear design.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (3)
1. A method of modeling involute tooth profile errors including system errors, comprising:
the method comprises the following steps of firstly, carrying out involute tooth profile error modeling to obtain point cloud data on an actual involute containing a system error, and specifically realizing that:
obtaining point cloud data on an ideal involute according to an ideal equation of the involute tooth profile curve;
expressing the system error forming the involute tooth profile error into a tiny rotation angle error gamma rotating around a tooth root or a tooth top according to the rule of the system error in machining;
since all errors include both systematic and random errors, the random error is added as a small perturbation to the involute profile curve:
the ideal equation of the involute tooth profile curve is as follows:
wherein r is b Is the base circle radius; theta is the rolling angle of any point on the involute, and theta is equal to theta k +α k ；θ k Is the spread angle of k points on the involute, alpha k Is the pressure angle at point k on the involute;
the systematic errors include: (1) eccentricity errors in workpiece mounting due to Δ x, Δ y; (2) the rotation angle error delta theta; (3) machining tool and (x) 0 ,y 0 ) Error of center distance Δ r b ；
According to the forming reason of the involute tooth profile error in the gear machining process, a mathematical model of the system error is established through the following formula:
wherein, gamma is a tiny rotation angle error rotating around a tooth root or a tooth top;
the random error is represented by a normal distribution:
Δ∈N(μ,σ) (3)
wherein mu is the mean value of normal distribution, and sigma is the standard deviation of normal distribution;
combining the formula (1), the formula (2) and the formula (3), obtaining point cloud data of Q' on the actual involute by the following formula:
wherein x is 0 、y 0 、z 0 The coordinate value of the point on the ideal involute;
secondly, generating an involute tooth profile error curved surface based on the NURBS numerical model;
thirdly, verifying the curved surface modeling precision, which is specifically realized as follows:
verifying the modeling precision of the curved surface through the distance d from the point set on the actual involute tooth profile curved surface to the point set on the ideal involute tooth profile curved surface;
F′ α ＝|d Max |+|d Min |≤F α (6)
wherein (x) i ,y i ) Is the point on the actual involute profile curve, (x) 0i ,y 0i ) Is a point on the surface of the ideal involute profile, F α Is the total tooth profile tolerance value, F' α Is the actual tooth profile error;
and fourthly, generating a tooth profile error tooth surface CAD model based on IGES.
2. The method of claim 1 wherein generating an involute tooth profile error surface based on a NURBS numerical model by;
generating m & ltn & gt point cloud data of a tooth profile error curved surface according to the involute tooth profile error mathematical model;
and according to the generated point cloud data, establishing a curved surface model of the irregular geometric shape characteristics of the actual tooth profile surface by adopting a NURBS curved surface interpolation reconstruction technology modeling method.
3. The method of claim 1 wherein generating an IGES-based tooth profile error flank CAD model by generating an IGES-based tooth profile error flank CAD model;
integrating an IGES file generated by a modeling method based on a NURBS curved surface interpolation reconstruction technology with a CAD model to obtain an actual tooth profile curved surface with a tooth profile error;
and obtaining a gear geometric model with tooth profile errors by using a flexible modeling and materialized modeling method.
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