• 호남수학학술지
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• 제40권2호
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• pp.251-264
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• 2018

In the present study, we consider the curves in Galilean 4-space ${\mathbb{G}}_4$. We find out the involute curves of order k (k = 1, 2, 3) of a given curve. We get the relationships between the Frenet apparatus of a given curve and its involute curves of order k.

• 한국정밀공학회지
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• 제15권5호
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• pp.85-92
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• 1998

This present paper describes a mathematical model of profile shifted elliptical gears, and this model is based on the concepts of envelope theory and conjugate geometry between the blank and the straight-sided rack cutter. The geometric model of the rack cutter includes working regions generating involute curves and fillets for trocoidal curves, and furthermore the addendum modified coeff. is considered for avoiding undercutting. The addendum modified coeff. is changed linearly along with pitch curves and must be the same absolute value at both major semi-axis and minor semi-axis. If undercutting is at all pronounced, the undercut tooth not only are weakened in strength, but lose a small portion of the involute adjacent to the base circle, then this loss of involute may cause a serious reduction in the length of contact. A very effective method of avoiding undercutting is to use the so-called profile shifted gearing. Non-undercutting condition is examined with the change of eccentricity and addendum modified coeff. in elliptical gears and then the minimum number of tooth is proposed not to gernerate undercutting phenomenon.

• 한국정밀공학회:학술대회논문집
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• 한국정밀공학회 1997년도 춘계학술대회 논문집
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• pp.572-577
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• 1997

This present paper describes a mathematical model of profile elliptical gears, and this model is based on the concepts of envelop theory and conjugate geometry between the blank and the straight-sided rack cutter. The geometric model of the rack cutter includes working regions generating involute curves andd fillets for trocoidal curves, and furthermore the addendum modified coeff,is considered for avoiding undercutting. The addendum modified coeff, is changed linearly along with pitch curves and must be the must be the same absolute value at both major semi-axis and minor semi-axis. If undercutting is at all pronounced, the undercut tooth not only are weakened in strength, but lose a small portion of the involute adjacent to the base circle, then this loss of involute may ncause a serios reduction in the length of contact. A very effective method of avoiding undercutting is to use the so-called profile shifted gearing. Non-undercutting conditon is examined with the change of eccentricity and addendum modefied coeff. in elliptical gears and then the minimum number of tooth is proposed not to gernerate undercutting phenomenon.

• 대한수학회논문집
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• 제38권2호
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• pp.589-598
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• 2023

Magnetic curves are the trajectories of charged particals which are influenced by magnetic fields and they satisfy the Lorentz equation. It is important to find relationships between magnetic curves and other special curves. This paper is a study of magnetic curves and this kind of relationships. We give the relationship between β-magnetic curves and Mannheim, Bertrand, involute-evolute curves and we give some geometric properties about them. Then, we study this subject for γ-magnetic curves. Finally, we give an evaluation of what we did.

• 한국소성가공학회:학술대회논문집
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• 한국소성가공학회 2003년도 추계학술대회논문집
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• pp.34-38
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• 2003

In metal working, cold forging that has profit to satisfy dimension accuracy is using in various manufacturing products. Recently, most of the interest thing is precision forging of gear. Gear forging product is more strength than broaching gear, and it has many advantages with reduction of factory expenses. The reason of difficulty to improve accuracy of gear dimension compare to another products is the dimension accuracy is very high, approximately 10$\mu\textrm{m}$, and because die of involute teeth and elastic strain of forged tool differ from standard curve. This paper represent quantitative analysis of die and teeth of forged tool, namely difference of curves, with experiments and analyze the factor of dimension gap, finally, will design compensated involute curve.

• 한국소성가공학회:학술대회논문집
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• 한국소성가공학회 2003년도 춘계학술대회논문집
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• pp.44-48
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• 2003

In metal working, cold forging that has profit to satisfy dimension accuracy is using in various manufacturing products. Recently, most of the interest thing is precision forging of gear, Gear forging product is more strength than broaching gear, and it has many advantages with reduction of factory expenses. The reason of difficulty to improve accuracy of gear dimension compare to another products is the dimension accuracy is very high, approximately 10$\mu\textrm{m}$, and because die of involute teeth and elastic strain of forged tool differ from standard curve. This paper represent quantitative analysis of die and teeth of forged tool, namely difference of curves, with experiments and analyze the factor of dimension gap, finally, will design compensated involute curve.

• 한국정밀공학회지
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• 제13권4호
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• pp.129-135
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• 1996

The area of gear vibration and noise, has recently been the focus of many studies. The proper kinematic and geometric design of gears, the mathematical modeling of gear system are essential for a good design. This work present a gear disign for reducing noise, and practical approaches used for machinery noise reduction slong with the summary of methods available for predicting gear noise in terms of the transmis- sion error, and show a comparative study with other methods. A new tooth profile modification is proposed for reducing vibration and noise of involute gears. The method is based on the use of cubic spline curves. The tooth profile is constrained to assume an involute shape during the loaded operation. Thus the new gear profile assures conjugate motion at all points along the line of action. The new profile is found to result in a more uniform static transmission error compared to not only standard involute profile but also modificated profile therby contributing to the improvement of vibration and noise characteristics of the gear.

• Tribology and Lubricants
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• 제24권5호
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• pp.221-227
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• 2008

This paper presents a step-by-step design of noncircular gears. From the diagram of angular velocity ratio of a noncircular gear pair, the pitch curves of the two mating gears are determined, and the perimeter of the pitch curve has been divided into equal-length segments by the number of teeth. A master tooth profile, which is a composite curve of circular arcs that represents involute, has been introduced. A noncircular gear pair has been designed by imposing the master tooth on the divided points of the pitch curve, and a full fillet has been achieved between neighbour teeth. Thus, the whole profile of the noncircular gear is a composite curve of arcs only, and consequently NC codes for wire EDM can be easily generated.

• Kyungpook Mathematical Journal
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• 제58권1호
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• pp.117-135
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• 2018

The orthogonal trajectories of the first tangents of the curve are called the involutes of x. The hyperspheres which have higher order contact with a curve x are known osculating hyperspheres of x. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve x in n-dimensional Euclidean space ${\mathbb{E}}^n$. In the present study, we give a characterization of involute curves of order k (resp. evolute curves) of the given curve x in n-dimensional Euclidean space ${\mathbb{E}}^n$. Further, we obtain some results on these type of curves in ${\mathbb{E}}^3$ and ${\mathbb{E}}^4$, respectively.

• 한국수학사학회지
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• 제37권4호
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• pp.93-105
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• 2024

Research on the caustic, the envelope of light rays, began with the geometric optics studies of Huygens and others in the 17th century. One of the most important problems in optics in the 17th century was focusing the sun's rays. This was a problem that had to be solved in order to manufacture various practical optical instruments at the time. Beginning with research on the caustic during this period, the concept of envelope became generalized and expanded to various fields until the 19th century. This paper examines the mathematical history involved in the study of envelope curves. We compare several methods of defining the envelope and provide an example of calculating the envelope accordingly.