Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit

<p style='text-indent:20px;'>We consider the planar double pendulum where its center of mass is attached in an elliptic orbit. We consider the case where the rods of the pendulum have variable length, varying according to the radius vector of the elliptic orbit. We make an Hamiltonian view of the problem, find four linearly stable equilibrium positions and construct the boundary curves of the stability/instability regions in the space of the parameters associated with the pendulum length and the eccentricity of the orbit.</p>

Mathematical models of the gyrostabilizer in different modes of its operation

The article considers the development of a mathematical model of the stabilization and rotation system in the modes of stabilization, targeting, auto-tracking of the target and electrical arrest. The output signals shall be signals proportional to the components of the angular velocities of the line of sight, the angles of pitch and dash of deviation around the axes of the gyrosystem and the angles of inconsistency of the line of sight relative to the optical axis of the homing head. The system of cardan suspension of the stabilization and rotation system is considered, where the actuators are located on the axes of rotation of the outer and inner frames of the cardan suspension. The homing head is mounted on the inner frame. The inner frame is a gyrostabilized platform. Depending on the mode of operation of the stabilization and rotation system: in the stabilization mode, the coordinate system that is stabilized is assumed to be stationary in inertial space; in the auto-tracking mode of the target, the coordinate system that is stabilized by Oxyz is returned according to the change of direction to the target; in the mode of electrical locking, the axes of the coordinate system which is stabilized by Oxyz coincide with the axes of Oxoyozo connected to the body of the main product. To obtain differential equations, the projections of the total vector of the kinetic moment of the inner and outer frames on the axis of the outer frame are taken and written according to the theorem on the change of the kinetic moment of the considered system relative to the axes of suspensions. The total moments of external forces applied to the outer and inner frames around their axes of rotation, which have the following components: moments of actuators, moments of viscous and dry friction, imbalance and other unaccounted for factors around the axes of the outer and inner frames . The moments of the forces of viscous and dry friction are presented in the classical form, taking into account the signs when changing the direction of movement. The mass of the inner frame with all devices mounted on it, and the mass of the entire movable system (outer and inner frames), as well as the radius vector characterizing the displacement of the center of mass, give a static imbalance of the movable system relative to the suspension axis of the i-th frame are components imbalance. The scientific novelty of the work is to obtain a mathematical model for a particular product, as well as the practical feasibility of their application. The result is a differential equation that fully describes this system of stabilization and rotation, takes into account the parameters of actuators, turbulent moments, as well as random effects and can be used depending on the tasks.

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.

Effect of waist ease distribution on aesthetic fit

When a garment is given a certain amount of ease, the body shape characteristics and matching degree of air gap distribution determine the aesthetic fit of garment sculpt. The purpose of this study is to investigate the relationship between the aesthetic fit and air gap distribution of the bust and waist sections. Three factors, the ratio of the horizontal radius vector to the bust and waist sections ( HrvB, HrvW) and the location and proportion of darts ( DL, DP), have been considered in this analysis. Ten female participants with different Hrv (including HrvB and HrvW) were recruited and asked to wear 10 sample blouses with different DL and DP. Samples worn by participants are observed at different angles, and evaluated with scores by 30 experts. The air gap thickness (THA) between the garment and skin are measured by a TC2 scanner and Qualisys three-dimensional motion capture system. The results show that Hrv is the most important factor affecting the aesthetic fit, while DL and DP have significant influence on the formation of garment wrinkles. A round body shape has higher scores when wearing sample garments whose waist dart is close to the front and middle (samples L2, L3) and smaller back middle dart and front waist dart (samples P1, P2), while a flat body shape has higher scores in the opposite condition (sample L5, P5). In terms of air gap distribution, there is a significant positive correlation between HrvB and THA in the front area (132–180°). Here, HrvW was negatively correlated with THA in the back side area. The correlation coefficients are all above 0.6. The DL has a great influence on THA in the side area of the bust section and front side area of the waist section, while DP has a great influence on THA in the front and back areas of the bust section and the back side area of the waist section. In a comparative analysis, this paper proved that the smaller fluctuation of THA and the uniform ease distribution are more likely to present an aesthetic and fit dressing appearance. Although the relationship between THA and the three factors was not significant, the regularity of its distribution is accompanied by a significant change of dressing score.

Some problems of the projectile motion with a square-law resistance

The influence of the force of the quadratic resistance of the medium on the change in some interesting characteristics of the motion of the projectile, which take place when the projectile moves in vacuum, is investigated. Loci are constructed numerically (and partly analytically) that ensures maximization of the arc length of the projectile trajectory and a non-decreasing of the length of the radius-vector. As examples, the motion of a baseball, a tennis ball and a badminton shuttlecock is studied.

Binary and Ternary Fragmentation Analysis of 252Cf Nucleus using Different Nuclear Radii

Pioneering study reveals that a radioactive nucleus may split into two or three fragments and the phenomena are known as binary fission and ternary fission respectively. In order to understand the nuclear stability and related structure aspects, it is of huge interest to explore the fragmentation behavior of a radioactive nucleus in binary and ternary decay modes. In view of this, Binary and ternary fission analysis of 252Cf nucleus is carried out using quantum mechanical fragmentation theory (QMFT). The nuclear potential and Coulomb potential are estimated using different versions of radius vector. The fragmentation structure is found to be independent to the choice of fragment radius for binary as wellas ternary decay paths. The deformation effect is included up to quadrupole (β2) with optimum cold orientations and their influence is explored within binary splitting mode. Moreover, the most probable fission channels explore the role of magic shell effects in binary and ternary fission modes.

Estimation of the Yarkovsky effect on the example of the asteroid 1685 Toro (1948 OA)

On the example of 1685 Toro, secular drifts of orbital elements and the displacement from the unperturbed position were obtained using the analytical solution of the averaged equations of motion of the asteroid in the central gravitational field and additional perturbing acceleration, inversely proportional to the square of the distance to the Sun, in the frame of reference associated with the radius vector. The components of this acceleration are calculated based on the thermophysical characteristics of 1685 Toro within the framework of the Yarkovsky acceleration linear model for spherical asteroids.

Research of the synthesis of radiant intensity indicatrix of multicomponent beam diode module

Widespread use of semiconductor radiation sources in optoelectronic devices for various purposes requires further study of the mechanisms of formation of photometric characteristics of the integrated device in the near illumination zone, where the law of inverted squares is violated. A mathematical model of the multicomponent beam-diode module is proposed. On its basis the analysis of influence of the parameters of separate beam sources on the deformation of the indicatrix of radiant intensity at transition from the far zone to the near one is carried out. It is shown that the determining parameter of the indicatrix change in longitude and polar distance is the distribution in the plane of the modulus of the product of the radiant intensity of a single diode on its polar radius vector. The displacement of the polar angle of the maximum of the vector of the diode radiant intensity is more significant for wide radiation patterns than for concentrated ones. For specific parameters of diodes and geometry of their location the suitability of the proposed model for a priori modeling of beam-diode modules is illustrated.

Implementation of image processing techniques as a tool for form analysis of Romanian folk elements

The present paper proposes a method for an effective analysis of Romanian folk motifs using two basic algorithms – Radius-Vector function (RV) and Principal Component Analysis (PCA). For a better accuracy of the analysis, a combination of these methods has been proposed. The authors selected few Romanian folk motifs (embroidered on blouses, skirts called “poale”, front aprons called “zadii”, sleeveless vest) from Bihor, Arad, and Maramures Counties. The vectorization of traditional motifs was made by GIS software.