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

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.

Comparative analysis of soil discarding by spherical disks

The article presents data on the influence of various types of spherical disks on the discarding of soil in the horizontal and vertical planes during its processing. These studies were conducted in order to optimize the selection of working bodies of disk tillage machines in terms of resistance and processing quality. Three types of disks were used in the comparative analysis. Two types of discs with cutouts and one solid spherical disc. On one type of disk, the cutouts are made in the form of circular arcs, and on the other-in the form of arcs of a logarithmic spiral. The conducted studies have shown that within the working surface of the disk, the trajectories of movement of soil particles under the influence of the three types of disks under study differ little from each other. Although it can be noted that the steeper rise of the trajectory in the vertical plane provides a solid disk, and the smallest rise of the trajectory - at the disk with cutouts in the form of a logarithmic spiral. In the horizontal plane, the longitudinal movement of the soil mass is less in disks with cutouts, especially in a disk with cutouts along the arc of a logarithmic spiral. As a result of these studies, it was revealed that the disk working bodies with cutouts on the cutting edge in the form of arcs of a logarithmic spiral showed the best quality and energy indicators.

Rectangles and spirals

Every reader knows about the Golden Rectangle (see [1, pp. 85, 119], [2, 3]), and that it can be subdivided into a square and a smaller copy of itself, and that this process can be continued indefinitely, converging towards the intersection point of diagonals of any two successive rectangles in the sequence. The circumscribed logarithmic spiral passing through the vertices and converging to the same point is also familiar (see [3, 4]), and is analogous to the circumcircle of a regular polygon or a triangle. The approximate logarithmic spiral obtained by drawing a quarter-circle inside each of the squares is equally well known [3, p. 64]. Perhaps slightly less familiar is the inscribed spiral, which is tangential to a side of every rectangle, like the incircle of a triangle or a regular polygon. It does not (quite) coincide with the spiral passing through the point of subdivision of each side, as discussed in [3, pp. 73-77]. The Golden Rectangle, its subdivisions, and the circumscribed and inscribed spirals are illustrated in Figure 1.

Sedimentological evidence for seiching in a swell-dominated headland-bay system: Table Bay, Western Cape, South Africa

Table Bay, South Africa, is a typical headland-bay system with a shoreline that can be described by a logarithmic spiral. A peculiarity and unique feature of Table Bay is the juxtaposition of Robben Island opposite its headland. As a consequence, the bathymetry defines an ellipsoidal basin which was postulated to potentially resonate in the form of long-period standing waves (seiches). One aim of this study, therefore, was to investigate whether any evidence for such resonant oscillations could be detected in the geomorphology and sediment distribution patterns. Indeed, the ellipsoidal shape of the basin can be framed by two converging log-spirals with their centres located opposite each other, one off Robben Island and the other on the Cape Town side of the bay. The so-called apex line, which divides the two spirals into equal parts is aligned SW–NE, i.e. more or less parallel to the direction of ocean wave propagation. The distribution patterns of all sedimentary parameters were found to be characterised by a strikingly similar trend to either side of the apex line. This supports the hypothesis that the basin of Table Bay appears to resonate in the form of a mode 1 standing wave, with the node positioned above the apex line in the centre of the bay. The maximum period of such a standing wave was calculated to be around 37 min. The study demonstrates that large-scale sediment distribution patterns can reveal the existence of specific hydrodynamic processes in coastal embayments. It is recommended that this phenomenon be investigated in greater detail aimed at verifying the existence of resonant oscillations in Table Bay and, in the event, at establishing its precise nature and trigger mechanism.

Uniform Magnetic Field Characteristics Based UHF RFID Tag for Internet of Things Applications

This paper presents a novel inkjet-printed near-field ultra-high-frequency (UHF) radio frequency identification (RFID) tag/sensor design with uniform magnetic field characteristics. The proposed tag is designed using the theory of characteristics mode (TCM). Moreover, the uniformity of current and magnetic field performance is achieved by further optimizing the design using particle swarm optimization (PSO). Compared to traditional electrically small near-field tags, this tag uses the logarithmic spiral as the radiating structure. The benefit of the logarithmic spiral structure lies in its magnetic field receiving area that can be extended to reach a higher reading distance. The combination of TCM and PSO is used to get the uniform magnetic field and desired resonant frequency. Moreover, the PSO was exploited to get a uniform magnetic field in the horizontal plane of the normal phase of the UHF RFID near-field reader antenna. As compared with the frequently-used commercial near field tag (Impinj J41), our design can be readable up to a three times greater read distance. Furthermore, the proposed near-field tag design shows great potential for commercial item-level tagging of expensive jewelry products and sensing applications, such as temperature monitoring of the human body.

A universal power law for modelling the growth and form of teeth, claws, horns, thorns, beaks, and shells

Background A major goal of evolutionary developmental biology is to discover general models and mechanisms that create the phenotypes of organisms. However, universal models of such fundamental growth and form are rare, presumably due to the limited number of physical laws and biological processes that influence growth. One such model is the logarithmic spiral, which has been purported to explain the growth of biological structures such as teeth, claws, horns, and beaks. However, the logarithmic spiral only describes the path of the structure through space, and cannot generate these shapes. Results Here we show a new universal model based on a power law between the radius of the structure and its length, which generates a shape called a ‘power cone’. We describe the underlying ‘power cascade’ model that explains the extreme diversity of tooth shapes in vertebrates, including humans, mammoths, sabre-toothed cats, tyrannosaurs and giant megalodon sharks. This model can be used to predict the age of mammals with ever-growing teeth, including elephants and rodents. We view this as the third general model of tooth development, along with the patterning cascade model for cusp number and spacing, and the inhibitory cascade model that predicts relative tooth size. Beyond the dentition, this new model also describes the growth of claws, horns, antlers and beaks of vertebrates, as well as the fangs and shells of invertebrates, and thorns and prickles of plants. Conclusions The power cone is generated when the radial power growth rate is unequal to the length power growth rate. The power cascade model operates independently of the logarithmic spiral and is present throughout diverse biological systems. The power cascade provides a mechanistic basis for the generation of these pointed structures across the tree of life.