A Special Case of Congruent Centroids of Noncircular Wheels Formed by Arcs of the Logarithmic Spiral

Kresan, Tetiana; Pylypaka, Serhiy; Grischenko, I. Yu.; Babka, Vitaliy

doi:10.32347/0131-579x.2020.98.84-93

Cited by 4 publications

(5 citation statements)

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“…В праці [5] розроблено аналітичну модель установки ґрунтообробних сферичних дисків для визначення геометричних та технологічних характеристик. У працях більш вузького спрямування досліджуються різні аспекти покращення якості обробітку ґрунту такими знаряддями [6][7][8]. Перспективи подальшого удосконалення дискових та інших ґрунтообробних знарядь розглянуто в праці [9].…”

Section: аналіз останніх дослідженьunclassified

“…Для цих даних за формулою (6) знаходимо значення u: u=0,276. Якщо у формулу (6) замість R підставити значення r, то отримаємо: u=0,152. Таким чином, для побудови гелікоїда із обмежувальними зовнішньою і внутрішньою гвинтовими лініями незалежна змінна повинна змінюватися в межах u=0,152…0,276.…”

Section: результати дослідженьunclassified

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Movement of Soil Particles on Surface of Developable Helicoid With Horizontal Axis of Rotation With Given Angle of Attack

Kresan¹

2021

Tehnìka ta energetika

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The article deals with the interaction of a screw cultivator with soil particles. Due to the very wide application in technology, the term "helical surface" is usually understood as the surface of a helical conoid or auger. In this paper, we consider the surface of a deployable helicoid, also linear, but significantly different from the screw surface. The difference lies not only in the geometric shape, but also in the manufacturing technology. If the screw is made by punching or strip rolling with significant deformation of the billet, the unfolded helicoid can be made by simple bending with a minimum of plastic deformation. In terms of theory, if the thickness of the workpiece were zero, there would be no plastic deformation at all when bending it. The working body for soil cultivation consists of a strip of unfolded helical surface, the outer edge of which is sharpened and acts as a blade, and the inner one is rigidly attached to a lattice cylinder. The difference between the radius of the helical line of the blade and the cylinder determines the working depth. The lattice cylinder prevents clogging of the inter-screw space and at the same time performs the additional function of a roller. The body works like a disc tool, that is, the profile of the processed field has protrusions and depressions. At the moment when the moldboard touches the surface of the field, angles of attack and roll arise, similar to the angles of attack and roll of disc guns. The design parameters that provide these angles can be calculated from an analytical description of the surface. The section, that is, the drum with the rotating working surface of the auger, is located so that its axis makes a certain angle with the direction of motion of the unit. This causes an angle of attack and reaction forces that cause the drum to rotate with the surface. From the speed of the aggregate and considering the angle of attack, the angular velocity of the section can be found. The differential equation of motion of the particle after it hits the rotating surface is then generated. The differential equation is drawn in projections on the three axes of the stationary coordinate system. It includes three unknown dependencies: two variables describing the trajectory of the particle sliding on the surface, and the reaction force of the surface. The system is solved numerically. Trajectories of relative and absolute motion of the particle and graphs of changes in its relative and absolute velocities are plotted.

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Section: аналіз останніх дослідженьunclassified

Section: результати дослідженьunclassified

Movement of Soil Particles on Surface of Developable Helicoid With Horizontal Axis of Rotation With Given Angle of Attack

Kresan¹

2021

Tehnìka ta energetika

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“…Rolling along a curvilinear path in a straight line is used in work [8] to construct noncircular wheels, one of which is formed by straight segments, that is, it is a polygon. The formation of congruent noncircular wheels of two symmetrical arcs of the logarithmic spiral is considered in [9]. Paper [10] proposes a geometric model for solving the problem of profiling an uncircular toothed wheel whose centroid consists of interconnected arcs.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…The dependence φ=φ(α) is obtained by integrating expression (9), which ensures equality of curve arcs when turning polar radii ρ and ρ 1 at the corresponding angles:…”

Section: Finding Arc Curves Describing the Profile Of The Teethmentioning

confidence: 99%

“…The advantage of the resulting gear is the absence of friction between the surfaces of the teeth during their work. This is achieved by establishing a dependence (9) between the angles of rotation of the drive and the driven wheels within the operation of one pair of teeth. In this case, there is no friction force, which ultimately prevents the wear of the teeth.…”

Section: Applying the Proposed Procedures To The Design Of A Pair Of Gear Wheelsmentioning

confidence: 99%

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Designing an outer toothed gear whose wheel teeth are outlined by the logarithmic spiral arcs

Pylypaka

Kresan

Volina

et al. 2021

EEJET

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Toothed gears are the most common mechanical gears in machine building, which are characterized by high reliability and durability, a constant transfer number, and which can transmit high torque. During toothed gear operation, the surfaces of the teeth slide, which gives rise to friction forces and wears their working surfaces. To prevent this, the surfaces of the teeth need constant lubrication. This paper considers the design of a gear tooth engagement, which does not have friction between the surfaces of the teeth since they roll over each other without slipping. The profile of the tooth of such a gear is outlined by congruent arcs, symmetrical relative to the line that connects the center of rotation of the toothed wheel with the top of the tooth. These symmetrical curves at the top of the tooth intersect at the predefined angle. In the depressions of the wheel, adjacent teeth also intersect at the same angle. Such a condition can be ensured by a curve that at all its points crosses the radius-vector emanating from the coordinate origin, also at a stable angle equal to half of the given one. This curve is a logarithmic spiral. If the number of teeth of the drive and driven wheels is the same, then their teeth are congruent. Otherwise, the profiles of the teeth would differ but they could be outlined by congruent arcs of the same logarithmic spiral of the same length taken from different areas of the curve. The minimum possible angle at the top of the teeth is straight. At acute angle, the toothed gear operation is impossible. To build gear wheels with a right angle at the top of the tooth, it would suffice to set the number of teeth of the drive and driven wheels. The center-to-center distance is calculated using the derived formula. The transfer number of such a gear is variable but, with an increase in the number of teeth, the range of its change decreases. The algorithm of wheel construction is given.

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Internal Rolling of Non-Circular Centroids Formed From the Arcs of Logarithmic Spiral

Kresan¹,

Pylypaka²

2020

Tehnìka ta energetika

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In the article the internal rolling of flat centroids on each other with simultaneous rotation around fixed centers is considered. A characteristic feature of the considered centroids is that the profile of each of them is formed by successive connection of identical arcs of the same logarithmic spiral. It is similar to the profile of a gear wheel. As in gears, such centroids can transmit rotational motion. Unlike gears the transmission of rotational motion occurs without sliding of the arcs in the contact area. This is due to the fact that the arc lengths of the tooth profiles are equal. In classical gears, an involute profile is used, which was once proposed by L. Euler [1]. Gears with such a profile are the most common. Other profiles are also known, for example, in Novikov gears, in which the tooth profile is circle or a curve close to a circle. During the operation of these gearing sliding occurs at the point of contact of the teeth, and in the Novikov gear it is less than in gears with involute profile. In these and other gears on both wheels there are circles that roll over each other without slipping. They are are called centroids or splines, whose diameters are the basis for calculating all geometric elements of the gearing. Accordingly and in our case, centroids can serve as the basis for designing a gear with involute or other tooth profile. In the article it is shown that such centroids can be formed with a given number of teeth in the form of a gear, so they can also serve as a gear transmission. The main advantage of such a transmission is the complete absence of sliding, which does not lead to friction of surfaces in the area of contact and their wear. The disadvantage is that the transmission ratio is not constant, it periodically changes periodically. However, for some cases this does not affect significantly on the operation of mechanisms (for example, clock [2] or counting devices). The mathematical description of the profiles of centroids is carried out. The possibility of constructing centroids with an arbitrary permissible number of teeth on each of them. The center distance depends on the number of teeth on each centroid and the angle at the top of the tooth. For the same number of teeth on both centroids they coincide. Pairs of centroids are constructed, and their intermediate positions are shown when one of them is rotated by a given angle. The angle of rotation of the second centroid is determined analytically and is a function of the angle of rotation of the first centroid.

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A Special Case of Congruent Centroids of Noncircular Wheels Formed by Arcs of the Logarithmic Spiral

Cited by 4 publications

References 0 publications

Movement of Soil Particles on Surface of Developable Helicoid With Horizontal Axis of Rotation With Given Angle of Attack

Movement of Soil Particles on Surface of Developable Helicoid With Horizontal Axis of Rotation With Given Angle of Attack

Designing an outer toothed gear whose wheel teeth are outlined by the logarithmic spiral arcs

Internal Rolling of Non-Circular Centroids Formed From the Arcs of Logarithmic Spiral

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