CN109472066B - Reflecting surface antenna random error analysis method based on unit central point displacement - Google Patents
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Abstract
The invention discloses a reflecting surface antenna random error analysis method based on unit central point displacement, which comprises the following steps: inputting geometric parameters and electrical parameters of the reflector antenna; calculating a radiation electric field of an ideal antenna far zone; calculating the radiation power of the far zone of the ideal antenna; dividing a triangular mesh of a reflecting surface; calculating first-order and second-order coefficients of the unit of displacement of the electric field relative to the central point of the unit; inputting a surface random error root mean square value; calculating the average value of the radiation power of the antenna far zone; judging whether the electrical property meets the requirement; an output radiated power pattern; and updating the surface random error root mean square value. The invention describes the displacement of the central point of the unit as the random error of the reflector antenna, obtains the influence of the random error of the surface of the reflector antenna on the electrical property based on the second-order approximation formula of the displacement of the central point of the unit, and can guide the processing and manufacturing of the reflector antenna panel.
Description
Technical Field
The invention belongs to the technical field of radar antennas, and particularly relates to a reflecting surface antenna random error analysis method based on unit center point displacement in the field of radar antennas.
Background
Since the reflector antenna is easy to realize characteristics of high gain, narrow beam and the like, it is widely applied to the fields of radio astronomy, radar, communication, detection and the like. During the manufacturing process, the reflector antenna is susceptible to random errors introduced by the manufacturing process and installation, resulting in the deterioration of its electrical performance. The method is used for carrying out relevant research on the influence of the random errors on the surface of the antenna on the electrical performance so as to guide the processing and manufacturing precision of the antenna panel to be a research field in the aspect of reflecting surface antenna error analysis.
An analysis method for analyzing the influence of the random errors on the surface of the antenna on the electrical properties based on a probability method is disclosed in the document "An effective computing method for characterizing the effects of the random surface errors on the average power patterns" (IEEE trans. Antennas and Propagation, vol. 31, no. 1, 1983, 92-98) by Rahmat-Samii. Wang Meng, duan Baoyan, wang Wei, etc., in the document "influence of reflector antenna surface error on average power pattern" (university of west ampere, vol. 41, vol. 6, no. 188-194), a method for calculating an average power pattern in the presence of both surface random error and system error is proposed. Because the surface error is introduced into the electrical property calculation in the form of phase error in the existing methods, the complexity and the time consumption in formula derivation are caused, and the purpose of rapid analysis is difficult to form. Therefore, in order to meet the requirement of improving the analysis efficiency, the invention innovatively describes the antenna radiation electric field as a second-order approximate formula of the displacement of the central point of the triangular unit, and provides a reflecting surface antenna random error analysis method based on the displacement of the central point of the unit.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a reflecting surface antenna random error analysis method based on unit center point displacement. The method describes the displacement of the central point of the unit as the random error of the reflector antenna, and provides an analysis method for analyzing the influence of the random error on the average power directional diagram of the reflector antenna based on a second-order approximation formula of the displacement of the central point of the unit, so that the processing and manufacturing of the reflector antenna panel can be guided.
The technical scheme of the invention is as follows: the reflecting surface antenna random error analysis method based on the unit central point displacement is characterized by comprising the following steps of:
(1) Inputting geometric parameters and electrical parameters of reflector antenna
Inputting the geometric parameters and the electrical parameters of the reflector antenna provided by a user; wherein the geometric parameters comprise radius and focal length; the electrical parameters comprise working wavelength, free space wave constant, aperture field amplitude distribution function, taper pin level, aperture field shape index and electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy;
(2) Calculating far-zone radiation electric field of ideal antenna
Calculating the radiation electric field of the far area of the ideal antenna by using a geometric optics method according to the geometric parameters and the electrical parameters of the reflector antenna;
(3) Calculating the radiation power of the ideal antenna far zone;
(4) Dividing a triangular mesh of a reflecting surface;
(5) Calculating first-order and second-order coefficients of the unit of displacement of the electric field relative to the central point of the unit;
(6) Input surface random error root mean square value
Inputting a random error root mean square value of the surface of the antenna according to the processing and manufacturing error of the reflecting surface antenna panel;
(7) Calculating the average value of the radiation power of the antenna far zone;
(8) Judging whether the electrical property meets the requirement
Judging whether the average value of the radiation power of the antenna far zone meets the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, if so, turning to the step (9), otherwise, turning to the step (10);
(9) Output radiation power pattern
When the average value of the radiation power of the antenna far zone meets the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, outputting a radiation power directional diagram;
(10) Updating surface random error root mean square value
And (5) when the average value of the radiation power of the antenna far area does not meet the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, updating the surface random error root mean square value, and turning to the step (6).
The geometric optical method in the step (2) is a high-frequency approximation method based on aperture field distribution, and the calculation formula is as follows:
wherein E is 0 Representing the far-field radiation electric field of an ideal antenna, ^ representing integral operation, ρ representing the polar coordinate component of a point on a reflecting surface in a caliber plane, Q (ρ) representing the caliber field amplitude distribution function corresponding to the point on the reflecting surface input by a user, exp representing the exponential operation of a natural logarithm, j being an imaginary unit, k being a free space wave constant,
is the position vector of the point on the reflecting surface in the caliber surface>
The unit position vector of the observation point in the far zone, s, and ds, respectively, represent the antenna aperture plane and the integration operation performed in the aperture plane.
The step (3) of calculating the radiation power of the far zone of the ideal antenna is to calculate the radiation power of the far zone of the ideal antenna according to the information of the radiation electric field of the far zone of the ideal antenna by the following formula
Wherein G is 0 Representing the far field radiated power, E, of an ideal antenna 0 The radiation electric field of the far zone of the ideal antenna is shown, the subscript 0 represents the ideal antenna, and the superscript indicates the conjugate operation.
The step (4) of dividing the reflecting surface into triangular meshes is to divide the aperture surface of the reflecting surface into a series of equilateral triangle units according to the geometric parameters and the working wavelength of the antenna provided by the user, wherein the side length of the equilateral triangle unit satisfies the following relational expression
Wherein, λ is the working wavelength, l is the side length of the equilateral triangle of the caliber face.
The step (5) of calculating the first-order and second-order coefficients of the unit of the displacement of the electric field relative to the central point of the unit is to calculate the first-order and second-order coefficients of the unit of the displacement of the electric field relative to the central point of the unit according to the geometric parameters and the electrical parameters of the antenna provided by the user and by combining the triangular unit
Wherein G is i First order coefficient representing displacement of electric field relative to cell center point, H i Representing the second order coefficient of displacement of the electric field relative to the cell center point, subscript i representing the ith triangular cell, ρ i Represents the polar coordinate component of the i-th triangular unit center point in the caliber plane, Q (rho) i ) Represents the aperture field amplitude distribution function input by the user and positioned at the ith triangle central point, exp represents the exponential operation of the natural logarithm, j is an imaginary unit, k is a free space wave constant,
is the position vector of the ith triangle unit center point in the caliber surface>
Unit position vector, xi, of observation point in far zone i And the angular component of the ith triangular unit central point in the feed coordinate system is represented.
The step (7) of calculating the average value of the radiation power of the antenna far zone is to calculate the average value of the radiation power of the antenna far zone by the following formula according to the radiation electric field of the ideal antenna far zone, the radiation power of the far zone, the first order coefficient and the second order coefficient of the unit of the displacement of the electric field relative to the central point of the unit and the surface random error root mean square value
Wherein, the first and the second end of the pipe are connected with each other,
represents the average value of the radiation power of the far zone of the antenna, G 0 Representing the far field radiated power, E, of an ideal antenna 0 Represents the radiation electric field of the far zone of the ideal antenna, subscript 0 represents the ideal antenna, superscript indicates the conjugate operation, M is the total number of the triangular units on the reflecting surface, H i Second order coefficient of cell, G, representing displacement of electric field relative to center point of cell i And a first-order coefficient of the cell representing the displacement of the electric field relative to the center point of the cell, wherein the subscript i represents an ith triangle, the subscript j represents a jth triangle, and sigma represents a random error root mean square value of the antenna surface input by a user.
The invention has the beneficial effects that: firstly, inputting geometric parameters and electrical parameter information of an antenna, and calculating a far-zone radiation electric field and far-zone radiation power of an ideal antenna; secondly, performing triangular mesh division on the reflecting surface, and respectively calculating first-order and second-order coefficients of a unit of displacement of the electric field relative to a central point of the unit; then, calculating the average value of the radiation power of the antenna far zone according to the surface random error root mean square value input by a user; and finally, judging whether the average value of the radiation power meets the electrical property requirement or not, and outputting a radiation power directional diagram so as to guide the processing and manufacturing of the reflecting surface panel.
Compared with the prior art, the invention has the following advantages:
1. according to the method, the average value of the radiation power of the far area of the reflector antenna under the influence of random errors is obtained based on a unit central point displacement second-order approximation formula, on the premise that the calculation accuracy is guaranteed, complex formula derivation is avoided, and the analysis efficiency is improved;
2. the invention obtains the average value of the far-zone radiation power of the reflector antenna from the angle of probability, and the accuracy of analysis is ensured by adopting a second-order approximation formula.
The present invention will be described in further detail below with reference to the accompanying drawings.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a graph comparing the gain averages of the conventional method and the method of the present invention;
fig. 3 is a graph comparing the average power patterns of the conventional method and the method of the present invention.
Detailed Description
The following detailed description of embodiments of the invention is provided in conjunction with the appended drawings:
example 1
The reflecting surface antenna random error analysis method based on the displacement of the central point of the unit comprises the following steps:
step 1, inputting geometric parameters and electrical parameters of reflector antenna
Inputting the geometric parameters and the electrical parameters of the reflector antenna provided by a user; wherein the geometric parameters comprise radius and focal length; the electrical parameters comprise working wavelength, free space wave constant, aperture field amplitude distribution function, taper pin level, aperture field shape index and electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy;
step 2, calculating the far-zone radiation electric field of the ideal antenna
Calculating the radiation electric field of the far area of the ideal antenna by using a geometric optics method according to the geometric parameters and the electrical parameters of the reflector antenna;
step 3, calculating the radiation power of the far zone of the ideal antenna;
step 4, dividing the triangular meshes of the reflecting surfaces;
step 6, inputting surface random error root mean square value
Inputting a random error root mean square value of the surface of the antenna according to the processing and manufacturing error of the reflecting surface antenna panel;
step 7, calculating the average value of the radiation power of the antenna far zone;
step 8, judging whether the electrical property meets the requirement
Judging whether the average value of the radiation power of the antenna far zone meets the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, if so, turning to the step 9, otherwise, turning to the step 10;
step 9, outputting the radiation power directional diagram
When the average value of the radiation power of the antenna far zone meets the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, a radiation power directional diagram is output;
And when the average value of the radiation power of the antenna far area does not meet the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, updating the surface random error root mean square value, and turning to the step 6.
Example 2
As shown in fig. 1, the present invention provides a method for analyzing random errors of a reflector antenna based on displacement of a central point of a unit, comprising the following steps:
step 1, inputting geometric parameters and electrical parameters of reflector antenna
Inputting the geometric parameters and the electrical parameters of the reflector antenna provided by a user; wherein the geometric parameters comprise radius and focal length; the electrical parameters comprise working wavelength, free space wave constant, aperture field amplitude distribution function, taper pin level, aperture field shape index and electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy;
step 2, calculating the far-zone radiation electric field of the ideal antenna
Calculating the radiation electric field of the ideal antenna far zone by using a geometric optics method according to the geometric parameters and the electrical parameters of the reflector antenna; the geometric optical method is a high-frequency approximation method based on aperture field distribution, and the calculation formula is as follows:
wherein E is 0 Representing the far-field radiation electric field of an ideal antenna, ^ representing integral operation, ρ representing the polar coordinate component of a point on a reflecting surface in a caliber plane, Q (ρ) representing the caliber field amplitude distribution function corresponding to the point on the reflecting surface input by a user, exp representing the exponential operation of a natural logarithm, j being an imaginary unit, k being a free space wave constant,
is the position vector of the point on the reflecting surface in the caliber surface>
A unit position vector of a far-zone observation point is obtained, s represents an antenna aperture surface, and ds represents integral operation carried out in the aperture surface;
step 3, calculating the radiation power of the far zone of the ideal antenna
According to the information of the radiation electric field of the far zone of the ideal antenna, the radiation power of the far zone of the ideal antenna is calculated by the following formula
Wherein, G 0 Representing the far field radiated power, E, of an ideal antenna 0 The radiation electric field of the far zone of the ideal antenna is represented, the subscript 0 represents the ideal antenna, and the superscript represents the conjugate operation;
step 4, dividing the triangular mesh of the reflecting surface
Dividing the aperture of the reflecting surface into a series of equilateral triangle units according to the geometric parameters and working wavelength of the antenna provided by the user, wherein the side length of the equilateral triangle unit satisfies the following relational expression
Wherein, lambda is the working wavelength, and l is the side length of the equilateral triangle of the caliber surface.
According to the geometric parameters and the electrical parameters of the antenna provided by a user, combining the triangular unit, and calculating the first-order and second-order coefficients of the unit of the displacement of the electric field relative to the central point of the unit by the following formula
Wherein, G i First order coefficient, H, representing the displacement of the electric field relative to the center point of the cell i Representing the second order coefficient of displacement of the electric field relative to the cell center point, subscript i representing the ith triangular cell, ρ i Represents the polar coordinate component of the i-th triangular element center point in the caliber plane, Q (rho) i ) At the ith triangle center point representing user inputThe aperture field amplitude distribution function is defined, exp represents the exponential operation of the natural logarithm, j is an imaginary unit, k is a free space wave constant,
is the position vector of the center point of the ith triangular unit in the caliber surface>
Is a unit position vector, xi, of a far-zone observation point i Representing the angle component of the central point of the ith triangular unit under the feed source coordinate system;
step 6, inputting surface random error root mean square value
Inputting a random error root mean square value of the surface of the antenna according to the processing and manufacturing error of the reflecting surface antenna panel;
step 7, calculating the average value of the radiation power of the antenna far zone
Calculating the average value of the radiation power of the far area of the antenna by the following formula according to the radiation electric field of the far area of the ideal antenna, the radiation power of the far area, the first-order and second-order coefficients of the unit of the displacement of the electric field relative to the central point of the unit and the surface random error root mean square value
Wherein, the first and the second end of the pipe are connected with each other,
represents the average value of the radiation power of the far zone of the antenna, G 0 Representing the far field radiation power, E, of an ideal antenna 0 Represents the radiation electric field of the ideal antenna far zone, subscript 0 represents the ideal antenna, superscript represents the conjugate operation, M is the total number of triangular units on the reflecting surface, H i Second order coefficient of cell, G, representing displacement of electric field relative to center point of cell i A first-order coefficient of the cell representing the displacement of the electric field relative to the center point of the cell, wherein a subscript i represents an ith triangle, a subscript j represents a jth triangle, and sigma represents a random error root mean square value of the antenna surface input by a user;
step 8, judging whether the electrical property meets the requirement
Judging whether the average value of the radiation power of the antenna far zone meets the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, if so, turning to the step 9, otherwise, turning to the step 10;
step 9, outputting the radiation power directional diagram
When the average value of the radiation power of the antenna far zone meets the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, outputting a radiation power directional diagram;
And when the average value of the radiation power of the antenna far area does not meet the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, updating the surface random error root mean square value, and turning to the step 6.
The advantages of the present invention can be further illustrated by the following simulation experiments:
1. simulation conditions are as follows:
the aperture of the reflector antenna is 100 lambda, the focal length is 100 lambda, lambda is the working wavelength, and the aperture field amplitude distribution function is
Q is an aperture field amplitude distribution function, a is an aperture plane inner node polar coordinate component, a is an aperture plane radius, P is an aperture field amplitude distribution function control parameter, B + C =1, ET = 20lB, ET is a taper pin of the aperture field amplitude distribution function at the edge of a reflecting surface, ET = -20dB and P =1. And respectively analyzing the calculation results of the antenna power directional diagram with the surface random error root mean square value epsilon of lambda/20-lambda/90.
2. And (3) simulation results:
the method is adopted to calculate the radiation power directional diagram in the presence of surface random errors, and is compared with the traditional method. Fig. 2 is a curve of the mean gain value of the antenna obtained by the conventional method and the method of the present invention according to the variation of the root mean square value of the random error of the surface. FIG. 3 is a graph of the average power pattern of the antenna using the conventional method and the method of the present invention when the mean square error ε of the surface random errors is λ/30. It can be seen that when the mean square value of the random errors on the surface is less than lambda/30, the method of the invention has better coincidence with the traditional method on the antenna gain, and simultaneously the coincidence of the main lobe and the near-side lobe region is also very good.
In summary, the invention firstly inputs the geometric parameters and electrical parameter information of the antenna, and calculates the far-zone radiation electric field and far-zone radiation power of the ideal antenna; secondly, performing triangular mesh division on the reflecting surface, and respectively calculating first-order and second-order coefficients of a unit of displacement of the electric field relative to a central point of the unit; then, calculating the average value of the radiation power of the antenna far zone according to the surface random error root mean square value input by a user; and finally, judging whether the average value of the radiation power meets the electrical property requirement or not, and outputting a radiation power directional diagram so as to guide the processing and manufacturing of the reflecting surface panel.
Compared with the prior art, the invention has the following advantages:
1. according to the method, the average value of the radiation power of the far area of the reflector antenna under the influence of random errors is obtained based on a unit central point displacement second-order approximation formula, on the premise that the calculation accuracy is guaranteed, complex formula derivation is avoided, and the analysis efficiency is improved;
2. the invention obtains the average value of the far-zone radiation power of the reflector antenna from the angle of probability, and the accuracy of analysis is ensured by adopting a second-order approximation formula.
The parts of the present embodiment not described in detail are common means known in the art, and are not described here. The above examples are merely illustrative of the present invention and should not be construed as limiting the scope of the invention, which is intended to be covered by the claims and any design similar or equivalent to the scope of the invention.
Claims (5)
1. The reflecting surface antenna random error analysis method based on the unit central point displacement is characterized by comprising the following steps of:
(1) Inputting geometric parameters and electrical parameters of reflector antenna
Inputting the geometric parameters and the electrical parameters of the reflector antenna provided by a user; wherein the geometric parameters comprise radius and focal length; the electrical parameters comprise working wavelength, free space wave constant, aperture field amplitude distribution function, taper pin level, aperture field shape index and electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy;
(2) Calculating far-zone radiation electric field of ideal antenna
Calculating the radiation electric field of the ideal antenna far zone by using a geometric optics method according to the geometric parameters and the electrical parameters of the reflector antenna;
(3) Calculating the radiation power of the ideal antenna far zone;
(4) Dividing a triangular mesh of a reflecting surface;
(5) Calculating first-order and second-order coefficients of the unit of displacement of the electric field relative to the central point of the unit;
(6) Input surface random error root mean square value
Inputting a random error root mean square value of the surface of the antenna according to the processing and manufacturing error of the reflecting surface antenna panel;
(7) Calculating the average value of the radiation power of the antenna far zone;
(8) Judging whether the electrical property meets the requirement
Judging whether the average value of the radiation power of the antenna far zone meets the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, if so, turning to the step (9), otherwise, turning to the step (10);
(9) Output radiation power pattern
When the average value of the radiation power of the antenna far zone meets the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, outputting a radiation power directional diagram;
(10) Updating surface random error root mean square value
When the average value of the radiation power of the antenna far area does not meet the electrical performance requirements including antenna gain, lobe width, side lobe level and pointing accuracy, updating the surface random error root mean square value, and turning to the step (6);
the step (7) of calculating the average value of the radiation power of the antenna far area is to calculate the average value of the radiation power of the antenna far area by the following formula according to the radiation electric field of the ideal antenna far area, the radiation power of the far area, the first order coefficient and the second order coefficient of the unit of the displacement of the electric field relative to the central point of the unit and the surface random error root mean square value
Wherein the content of the first and second substances,
represents the average value of the radiation power of the far zone of the antenna, G 0 Representing the far field radiation power, E, of an ideal antenna 0 Represents the radiation electric field of the ideal antenna far zone, subscript 0 represents the ideal antenna, superscript represents the conjugate operation, M is the total number of triangular units on the reflecting surface, H i Second order coefficient of cell, G, representing displacement of electric field relative to center point of cell i And a first-order coefficient of the cell representing the displacement of the electric field relative to the center point of the cell, wherein the subscript i represents an ith triangle, the subscript j represents a jth triangle, and sigma represents a random error root mean square value of the antenna surface input by a user.
2. The reflecting surface antenna random error analysis method based on unit center point displacement according to claim 1, characterized in that: the geometric optical method in the step (2) is a high-frequency approximation method based on aperture field distribution, and the calculation formula is as follows:
wherein E is 0 Representing the far-field radiation electric field of an ideal antenna, ^ representing integral operation, ρ representing the polar coordinate component of a point on a reflecting surface in a caliber plane, Q (ρ) representing the caliber field amplitude distribution function corresponding to the point on the reflecting surface input by a user, exp representing the exponential operation of a natural logarithm, j being an imaginary unit, k being a free space wave constant,
is the position vector of the point on the reflecting surface in the caliber surface>
The unit position vector of the observation point in the far zone, s, and ds, respectively, represent the antenna aperture plane and the integration operation performed in the aperture plane.
3. The reflecting surface antenna random error analysis method based on unit center point displacement according to claim 1, characterized in that: the step (3) of calculating the radiation power of the far zone of the ideal antenna is to calculate the radiation power of the far zone of the ideal antenna according to the information of the radiation electric field of the far zone of the ideal antenna by the following formula
Wherein, G 0 Representing the far field radiated power, E, of an ideal antenna 0 The radiation electric field of the far zone of the ideal antenna is shown, the subscript 0 represents the ideal antenna, and the superscript indicates the conjugation operation.
4. The reflecting surface antenna random error analysis method based on unit center point displacement according to claim 1, characterized in that: the step (4) of dividing the reflecting surface into triangular meshes is to divide the aperture surface of the reflecting surface into a series of equilateral triangle units according to the geometric parameters and the working wavelength of the antenna provided by the user, wherein the side length of the equilateral triangle unit satisfies the following relational expression
Wherein, λ is the working wavelength, l is the side length of the equilateral triangle of the caliber face.
5. The reflecting surface antenna random error analysis method based on unit center point displacement according to claim 1, characterized in that: the step (5) of calculating the first-order and second-order coefficients of the unit of the displacement of the electric field relative to the central point of the unit is to calculate the first-order and second-order coefficients of the unit of the displacement of the electric field relative to the central point of the unit according to the geometric parameters and the electrical parameters of the antenna provided by the user and by combining a triangular unit through the following formula
Wherein G is i First order coefficient representing displacement of electric field relative to cell center point, H i Representing the second order coefficient of displacement of the electric field relative to the cell center point, subscript i representing the ith triangular cell, ρ i Represents the polar coordinate component of the i-th triangular unit center point in the caliber plane, Q (rho) i ) Represents the aperture field amplitude distribution function input by the user and positioned at the ith triangle central point, exp represents the exponential operation of the natural logarithm, j is an imaginary unit, k is a free space wave constant,
is the position vector of the ith triangle unit center point in the caliber surface>
Unit position vector, xi, of observation point in far zone i And the angular component of the ith triangular unit central point in the feed coordinate system is represented. />
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| CN107169174A (en) * | 2017-04-24 | 2017-09-15 | 西安电子科技大学 | Umbrella antenna electromechanics Integrated Optimal Design method under optimal focal length |
| CN107679336A (en) * | 2017-10-20 | 2018-02-09 | 西安电子科技大学 | Reflector antenna surface random error analysis method based on Two-order approximation formula |
| CN108533715A (en) * | 2018-06-28 | 2018-09-14 | 西安交通大学 | A kind of two-way conjugate tooth profile design method for Harmonic Gears |
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