Research Papers: Design for Manufacture and the Life Cycle

A Novel Region-Division-Based Tolerance Design Method for a Large Number of Discrete Elements Distributed on a Large Surface

[+] Author and Article Information
Guodong Sa

School of Mechanical Engineering,
Zhejiang University,
Hangzhou 310027, China
e-mail: sgd@zju.edu.cn

Zhenyu Liu

State Key Laboratory of CAD&CG,
School of Mechanical Engineering,
Zhejiang University,
Hangzhou 310027, China
e-mail: liuzy@zju.edu.cn

Chan Qiu

State Key Laboratory of CAD&CG,
School of Mechanical Engineering,
Zhejiang University,
Hangzhou 310027, China
e-mail: qc@zju.edu.cn

Jianrong Tan

State Key Laboratory of CAD&CG,
School of Mechanical Engineering,
Zhejiang University,
Hangzhou 310027, China
e-mail: mech@zju.edu.cn

1Corresponding author.

Contributed by the Design for Manufacturing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 3, 2018; final manuscript received September 17, 2018; published online January 11, 2019. Assoc. Editor: Xiaoping Du.

J. Mech. Des 141(4), 041701 (Jan 11, 2019) (18 pages) Paper No: MD-18-1184; doi: 10.1115/1.4041573 History: Received March 03, 2018; Revised September 17, 2018

The array structure is widely used in precise electronic products such as large phased array antennas and large optical telescopes, the main components of which are a large surface base and a large number of high-precision discrete elements mounted on the surface base. The geometric error of discrete elements is inevitable in the manufacturing process and will seriously degrade the product performance. To deal with the tolerance design of discrete elements, a region-division-based tolerance design method is proposed in this paper. The whole array was divided into several regions by our method and the tolerance of discrete elements was correlated with the region importance on the performance. The method specifically includes the following steps: first, the sensitivity of the product performance to geometric errors was analyzed and the statistical relationship between the performance and geometric errors was established. Then, based on the sensitivity matrix, the regional division scheme was developed, and the corresponding tolerance was optimized according to the established relationship function. Finally, the optimal tolerance was selected among the multiple solutions to achieve the best performance. Taking a large phased array as an example, a simulation experiment was performed to verify the effectiveness of the proposed method.

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Fig. 1

(a) The ELT [8] constructed by the European Southern Observatory comprises a 39.3 m diameter primary mirror and a 4.2 m diameter secondary mirror. The left figure shows the model of extremely large telescope, and the right figure illustrates the discrete elements distributed on the secondary mirror. (b) The U.S. active phased array ballistic missile detection radar [9] in Alaska. The two circles sloped at an angle of 20 deg are 32 m high and are the antenna arrays, each composed of 2677 antenna elements mounted on the surface of the building. The left figure shows the whole radar and the right figure shows the part of the discrete elements mounted on the subarray.

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Fig. 2

(a) Actual position error of the array elements measured by photogrammetry. White dots are marks for photogrammetry, and the actual element position error can be acquired by analyzing the coordinates of marks. (b) A planar array antenna with subarray splicing design. Each subarray is the same size and of rectangular structure. (c) Nominal element position and corresponding tolerance zone. For programming convenience in matlab, the subscript labels are indexed in matrix form.

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Fig. 3

Active phased array architecture. The horns on the plate are discrete and the position error of the horn affects the performance seriously.

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Fig. 4

The forward process and backward process in array tolerance design. The former aims to predict or calculate the array pattern, whereas the latter is applied to array element position tolerance design. However, these two processes are separated.

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Fig. 5

The region division-based array element position tolerance design method. The forward process and backward process are both based on sensitivity analysis; the latter depends on the former.

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Fig. 6

Far-field pattern calculation. The coordinate system is fixed in the center of the array for convenience.

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Fig. 7

The flow chart for the regional division-based position tolerance design method

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Fig. 8

The power pattern of both the ideal and realistic array with |Δx|¯. The curve with maximum points at the same level is the ideal power pattern by Eq. (4), the curve whose maximum points are with the largest difference in height represents the realistic power pattern by Eq. (5), and the remaining line represents approximate results by Eq. (10). The simulated array is composed of 20×20 elements with Dolph-Chebyshev weights; the interval between adjacent elements is half wavelength: (a) |Δx|¯=1/30λ, (b) |Δx|¯=1/20λ, and (c) |Δx|¯=1/10λ.

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Fig. 9

(a) The left figure shows the probability density function of the element position when the x coordinate follows the normal distribution, and the right figure illustrates the probability density function of the power degradation ΔP in the direction (θ,ϕ) and (b) the nominal power pattern and the distribution interval (−3σ,+3σ) of degradation ΔP in the direction (θ,ϕ) due to the stochastic position error of the array elements

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Fig. 10

The key angular position in the power pattern

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Fig. 11

The left upper quarter subarray in the whole array is symmetric with the other three subarrays. The region area is limited within interval [Amin, Amax].

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Fig. 12

The amplitude of the tested array

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Fig. 13

(a) Power pattern sensitivity with respect to the x coordinate of the element position, (b) power pattern sensitivity with respect to the y coordinate of the element position, (c) power pattern sensitivity with respect to the z coordinate of the element position, and (d) total power pattern sensitivity with respect to the x, y, and z coordinates of the element position

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Fig. 14

1000 Monte Carlo simulation results. The upper bound is defined by +3τp, the lower bound is calculated by −3τp, and each thin line represents the realistic power pattern with the random elements position error. There are 1000 thin lines, which represent 1000 simulations.

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Fig. 15

(a) The contour line of the average power pattern sensitivity according to Eq. (17); the region enclosed by contour lines gradually changed from triangle to circle; only five contour lines are illustrated here and (b) one possible feasible scheme. Boundary L1 (straight line) is defined by points A and B, boundary L2 (arc) and L3 are defined by radii r1 and r2, and O is the origin of Cartesian coordinates.

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Fig. 16

The optimal region division scheme

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Fig. 17

Power pattern degradation interval with the regional designed position tolerance and uniformly designed position tolerance. The power pattern interval (−3τp,+3τp)optkl with respect to the optimal regional designed tolerance is obviously smaller than the interval (−3τp,+3τp)0kl with respect to the uniformly designed tolerance.

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Fig. 18

Array performance improvement under optimal regional designed position tolerance compared to uniformly designed position tolerance. All the diamond points were calculated by Eq. (22).



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