Said-Ball Cubic Transition Curve And Its Application To Spur Gear Design
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Date
2015-09
Authors
Yahaya, Saifudin Hafiz
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Abstract
Undoubtedly, gears are some of the most widely used elements in consumer
and industrial machineries. Past studies have shown that an involute curve is the
most common profile used in designing the spur gear tooth, developed based on the
approximation theories such as Chebyshev approximation and the tracing points
method. However, these employed methods are not accurate (or inexact) and only
focusing on the approximation concepts. Gear noise reduction and tooth strength are
continually being the focus of exploration and experimentation particularly the
modification of the tooth shape or tooth profile. Therefore, this study is to design the
S and C-shaped transition curves using Said-Ball cubic curve based on the third and
fifth cases of circle to circle templates with some mathematical proofs. One of the
objectives is to investigate the capability of this proposed S and C-shaped model in
reducing sound or noise level through an acoustic experiment. In this study, spur gear
is chosen as a case model due to its fundamental gear and the fact that it is simple to
construct and manufacture. Based on the conducted experiment and simulation,
results show that by using Said-Ball cubic curve, the theories of S and C-shaped
transition curves have successfully developed. These curves have been
mathematically proven, by using the concavity and second derivative tests and also
Kneserās theorem. It is also revealed that S and C-shaped transition curves can be
applied successfully in designing spur gear tooth. This proves that the solid model of
spur gear can also be developed through the integrated use of mathematical and CAD software. When measured through linear static analysis, fatigue analysis and DE, the
applicability of the proposed design and the material, AISI 304 shows that First-order
Newton interpolating polynomial can be employed as a fatigue predictor for all
design models. The new teeth design methods, S and C-shaped curves are the
acceptable methods in designing the spur gear teeth where both methods have
presented DE greater than 85% of the design effectiveness. All models have also
been successfully measured via dynamic and acoustic response analyses. C-shaped
model has been proven to have the lowest displacement when compared to S-shaped
(transition) and EM models. By utilizing this model, it is proven that gear noise or
sound can be reduced consistently. C-shaped model is more reliable than other
models in accordance to PS. It is proven that tooth profile modification is the main
factor in reducing sound or noise in a very significant and consistent way. The
contribution of this study will be beneficial to the designers or manufacturers in
designing the spur gear profiles where the above-mentioned curves can be applied as
an alternate method of these profiles. For future research, the capability of C
transition curve can be further explored in designing aerodynamic models for
example, car, high-speed train, bullet etc. Study on gear design can also be further
explored on other type of gears such as helical herringbone gear.
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Mathematics