Optimal Design of Compact Spur Gear Reductions

Abstract The optimal design of compact spur gear reductions includes the selection of bearing and shaft proportions in addition to the gear mesh parameters. Designs for single mesh spur gear reductions are based on optimization of system life, system volume, and system weight including gears, support shafts, and the four bearings. The overall optimization allows component properties to interact, yielding the best composite design. A modified feasible directions search algorithm directs the optimization through a continuous design space. Interpolated polynomials expand the discrete bearing properties and proportions into continuous variables for optimization. After finding the continuous optimum, the designer can analyze near optimal designs for comparison and selection. Design examples show the influence of the bearings on the optimal configurations.

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Optimal Design of Compact Spur Gear Reductions

Journal of Mechanical Design ◽

10.1115/1.2919437 ◽

1994 ◽

Vol 116 (3) ◽

pp. 690-696 ◽

Cited By ~ 6

Author(s):

M. Savage ◽

S. B. Lattime ◽

J. A. Kimmel ◽

H. H. Coe

Keyword(s):

Optimal Design ◽

Search Algorithm ◽

Spur Gear ◽

Optimal Designs ◽

Continuous Variables ◽

Gear Mesh ◽

System Life ◽

Optimal Configurations ◽

System Volume ◽

Selection Of

The optimal design of compact spur gear reductions includes the selection of bearing and shaft proportions in addition to the gear mesh parameters. Designs for single mesh spur gear reductions are based on optimization of system life, system volume, and system weight including gears, support shafts, and the four bearings. The overall optimization allows component properties to interact, yielding the best composite design. A modified feasible directions search algorithm directs the optimization through a continuous design space. Interpolated polynomials expand the discrete bearing properties and proportions into continuous variables for optimization. After finding the continuous optimum, the designer can analyze near optimal designs for comparison and selection. Design examples show the influence of the bearings on the optimal configurations.

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Discrete Optimal Design Formulations: Application to Gear Train Design

18th Design Automation Conference: Volume 1 — Optimum Design, Manufacturing Processes, and Concurrent Engineering ◽

10.1115/detc1992-0132 ◽

1992 ◽

Author(s):

Leonard P. Pomrehn ◽

Panos Y. Papalambros

Keyword(s):

Optimal Design ◽

Spur Gear ◽

Gear Train ◽

Continuous Variables ◽

Discrete Variables ◽

Interesting Aspect ◽

Model Formulation ◽

Design Models ◽

Different Types

Abstract The use of discrete variables in optimal design models offers the opportunity to deal rigorously with an expanded variety of design situations, as opposed to using only continuous variables. However, complexity and solution difficulty increase dramatically and model formulation becomes very important. A particular problem arising from the design of a gear train employing four spur gear pairs is introduced and formulated in several different ways. An interesting aspect of the problem is its exhibition of three different types of discreteness. The problem could serve as a test for a variety of optimization or artificial intellegence techniques. The best known solution is included in this article, while its derivation is given in a sequel article.

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Optimal Tooth Numbers for Compact Standard Spur Gear Sets

Journal of Mechanical Design ◽

10.1115/1.3256424 ◽

1982 ◽

Vol 104 (4) ◽

pp. 749-757 ◽

Cited By ~ 26

Author(s):

M. Savage ◽

J. J. Coy ◽

D. P. Townsend

Keyword(s):

Optimal Design ◽

Objective Function ◽

Design Space ◽

Spur Gear ◽

Gear Ratio ◽

Contact Ratio ◽

Bending Fatigue ◽

Gear Mesh ◽

Compact Design ◽

Center Distance

The design of a standard gear mesh is treated with the objective of minimizing the gear size for a given ratio, pinion torque, and allowable tooth strength. Scoring, pitting fatigue, bending fatigue, and the kinematic limits of contact ratio and interference are considered. A design space is defined in terms of the number of teeth on the pinion and the diametral pitch. This space is then combined with the objective function of minimum center distance to obtain an optimal design region. This region defines the number of pinion teeth for the most compact design. The number is a function of the gear ratio only. A design example illustrating this procedure is also given.

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Effect of Involute Contact Ratio on Spur Gear Dynamics

Journal of Mechanical Design ◽

10.1115/1.2829411 ◽

1999 ◽

Vol 121 (1) ◽

pp. 112-118 ◽

Cited By ~ 102

Author(s):

A. Kahraman ◽

G. W. Blankenship

Keyword(s):

Transmission Error ◽

Spur Gear ◽

Design Guidelines ◽

Forced Response ◽

Contact Ratio ◽

Test Rig ◽

Response Curves ◽

Gear Mesh ◽

Dynamic Transmission Error ◽

Selection Of

The influence of involute contact ratio on the torsional vibration behavior of a spur gear pair is investigated experimentally by measuring the dynamic transmission error of several gear pairs using a specially designed gear test rig. Measured forced response curves are presented, and harmonic amplitudes of dynamic transmission error are compared above and below gear mesh resonances for both unmodified and modified gears having various involute contact ratio values. The influence of involute contact ratio on dynamic transmission error is quantified and a set of generalized, experimentally validated design guidelines for the proper selection of involute contact ratio to achieve quite gear systems is presented. A simplified analytical model is also proposed which accurately describes the effects of involute contact ratio on dynamic transmission error.

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Discrete Optimal Design Formulations With Application to Gear Train Design

Journal of Mechanical Design ◽

10.1115/1.2826695 ◽

1995 ◽

Vol 117 (3) ◽

pp. 419-424 ◽

Cited By ~ 28

Author(s):

L. P. Pomrehn ◽

P. Y. Papalambros

Keyword(s):

Artificial Intelligence ◽

Optimal Design ◽

Spur Gear ◽

Gear Train ◽

Continuous Variables ◽

Discrete Variables ◽

Interesting Aspect ◽

Model Formulation ◽

Design Models ◽

Different Types

The use of discrete variables in optimal design models offers the opportunity to deal rigorously with an expanded variety of design situations, as opposed to using only continuous variables. However, complexity and solution difficulty increase dramatically and model formulation becomes very important. A particular problem arising from the design of a gear train employing four spur gear pairs is introduced and formulated in several different ways. An interesting aspect of the problem is its exhibition of three different types of discreteness. The problem could serve as a test for a variety of optimization or artificial intelligence techniques. The best known solution is included in this article, while its derivation is given in a sequel article.

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Q-optimal experimental designs and close to them experimental designs for polynomial regression on the interval

Industrial laboratory Diagnostics of materials ◽

10.26896/1028-6861-2020-86-5-65-72 ◽

2020 ◽

Vol 86 (5) ◽

pp. 65-72

Author(s):

Yu. D. Grigoriev

Keyword(s):

Optimal Design ◽

Polynomial Regression ◽

Direct Consequence ◽

Experimental Designs ◽

Optimal Designs ◽

Software Systems ◽

General Character ◽

Support Points ◽

Optimal Weights ◽

Universal Expression

The problem of constructing Q-optimal experimental designs for polynomial regression on the interval [–1, 1] is considered. It is shown that well-known Malyutov – Fedorov designs using D-optimal designs (so-called Legendre spectrum) are other than Q-optimal designs. This statement is a direct consequence of Shabados remark which disproved the Erdős hypothesis that the spectrum (support points) of saturated D-optimal designs for polynomial regression on a segment appeared to be support points of saturated Q-optimal designs. We present a saturated exact Q-optimal design for polynomial regression with s = 3 which proves the Shabados notion and then extend this statement to approximate designs. It is shown that when s = 3, 4 the Malyutov – Fedorov theorem on approximate Q-optimal design is also incorrect, though it still stands for s = 1, 2. The Malyutov – Fedorov designs with Legendre spectrum are considered from the standpoint of their proximity to Q-optimal designs. Case studies revealed that they are close enough for small degrees s of polynomial regression. A universal expression for Q-optimal distribution of the weights pi for support points xi for an arbitrary spectrum is derived. The expression is used to tabulate the distribution of weights for Malyutov – Fedorov designs at s = 3, ..., 6. The general character of the obtained expression is noted for Q-optimal weights with A-optimal weight distribution (Pukelsheim distribution) for the same problem statement. In conclusion a brief recommendation on the numerical construction of Q-optimal designs is given. It is noted that in this case in addition to conventional numerical methods some software systems of symbolic computations using methods of resultants and elimination theory can be successfully applied. The examples of Q-optimal designs considered in the paper are constructed using precisely these methods.

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Effect of Contact Ratio on Spur Gear Dynamic Load With No Tooth Profile Modifications

Journal of Mechanical Design ◽

10.1115/1.2826905 ◽

1996 ◽

Vol 118 (3) ◽

pp. 439-443 ◽

Cited By ~ 34

Author(s):

Chuen-Huei Liou ◽

Hsiang Hsi Lin ◽

F. B. Oswald ◽

D. P. Townsend

Keyword(s):

Dynamic Load ◽

Spur Gear ◽

Tooth Size ◽

Contact Ratio ◽

Gear Dynamics ◽

Wide Range ◽

Gear Contact ◽

Center Distance ◽

Selection Of ◽

Better Than

This paper presents a computer simulation showing how the gear contact ratio affects the dynamic load on a spur gear transmission. The contact ratio can be affected by the tooth addendum, the pressure angle, the tooth size (diametral pitch), and the center distance. The analysis presented in this paper was performed by using the NASA gear dynamics code DANST. In the analysis, the contact ratio was varied over the range 1.20 to 2.40 by changing the length of the tooth addendum. In order to simplify the analysis, other parameters related to contact ratio were held constant. The contact ratio was found to have a significant influence on gear dynamics. Over a wide range of operating speeds, a contact ratio close to 2.0 minimized dynamic load. For low-contact-ratio gears (contact ratio less than two), increasing the contact ratio reduced gear dynamic load. For high-contact-ratio gears (contact ratio equal to or greater than 2.0), the selection of contact ratio should take into consideration the intended operating speeds. In general, high-contact-ratio gears minimized dynamic load better than low-contact-ratio gears.

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Optimal Selection of Conductors in Three-Phase Distribution Networks Using a Discrete Version of the Vortex Search Algorithm

Computation ◽

10.3390/computation9070080 ◽

2021 ◽

Vol 9 (7) ◽

pp. 80

Author(s):

John Fernando Martínez-Gil ◽

Nicolas Alejandro Moyano-García ◽

Oscar Danilo Montoya ◽

Jorge Alexander Alarcon-Villamil

Keyword(s):

Objective Function ◽

Power Flow ◽

Phase Distribution ◽

Search Algorithm ◽

Distribution Networks ◽

Discrete Version ◽

Optimal Selection ◽

Flow Method ◽

Three Phase ◽

Selection Of

In this study, a new methodology is proposed to perform optimal selection of conductors in three-phase distribution networks through a discrete version of the metaheuristic method of vortex search. To represent the problem, a single-objective mathematical model with a mixed-integer nonlinear programming (MINLP) structure is used. As an objective function, minimization of the investment costs in conductors together with the technical losses of the network for a study period of one year is considered. Additionally, the model will be implemented in balanced and unbalanced test systems and with variations in the connection of their loads, i.e., Δ− and Y−connections. To evaluate the costs of the energy losses, a classical backward/forward three-phase power-flow method is implemented. Two test systems used in the specialized literature were employed, which comprise 8 and 27 nodes with radial structures in medium voltage levels. All computational implementations were developed in the MATLAB programming environment, and all results were evaluated in DigSILENT software to verify the effectiveness and the proposed three-phase unbalanced power-flow method. Comparative analyses with classical and Chu & Beasley genetic algorithms, tabu search algorithm, and exact MINLP approaches demonstrate the efficiency of the proposed optimization approach regarding the final value of the objective function.

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