Vol 87, No 4 (2023)

The work is of a survey analytical nature. The first part of the work presents quaternion and biquaternion methods for describing motion, models of the theory of finite displacements and regular kinematics of a rigid body based on the use of four-dimensional real and dual Euler (Rodrigues–Hamilton) parameters. These models, in contrast to the classical models of kinematics in Euler–Krylov angles and their dual counterparts, do not have division-by-zero features and do not contain trigonometric functions, which increases the efficiency of analytical research and numerical solution of problems in mechanics, inertial navigation, and motion control. The problem of regularization of differential equations of the perturbed spatial two-body problem, which underlies celestial mechanics and space flight mechanics (astrodynamics), is discussed using the Euler parameters, four-dimensional Kustaanheimo–Stiefel variables, and Hamilton quaternions: the problem of eliminating singularities (division by zero), which are generated by the Newtonian gravitational forces acting on a celestial or cosmic body and which complicate the analytical and numerical study of the motion of a body near gravitating bodies or its motion along highly elongated orbits. The history of the regularization problem and the regular Kustaanheim–Stiefel equations, which have found wide application in celestial mechanics and astrodynamics, are presented. We present the quaternion methods of regularization, which have a number of advantages over Kustaanheimo–Stiefel matrix regularization, and various regular quaternion equations of the perturbed spatial two-body problem (for both absolute and relative motion). The results of a comparative study of the accuracy of numerical integration of various forms of regularized equations of celestial mechanics and astrodynamics in Kustaanheimo–Stiefel variables and Newtonian equations in Cartesian coordinates are presented, showing that the accuracy of numerical integration of regularized equations in Kustaanheimo–Stiefel variables is much higher (by several orders of magnitude) than the accuracy of numerical integration Newtonian equations.

The article gives a solution to the problem of possible conditions of regular precession in the motion of a gyrostat around a fixed point under the action of two uniform and one inhomogeneous field. For the known case with equal precession and proper rotation velocities, new solutions are indicated with the precession axis deviated from the symmetry axis of the inhomogeneous field. New cases of regular precession are found, when the ratio of precession and proper rotation velocities is equal to two or one second. Application of the results to precession in orthogonal fields and to precession of a solid are considered.

The problem of controlling the direction of the traction force during the motion of an inertial object is considered. The maximal possible value of the traction force is constant and is determined by the maximal dry friction force. At a finite time interval, the problem of bringing an object to a given rectilinear trajectory with simultaneous velocity maximization in the appropriate direction is considered.

The influence of phase and other constraints on the method of searching for the trajectories of the movement of a civil supersonic aircraft, which are optimal in terms of fuel consumption, is considered. Based on the solutions found by the dynamic programming method, taking into account numerous restrictions on flight altitude, pitch angle, normal high-speed overload, aircraft speed, engine thrust, etc., it is shown that almost all of these conditions can be ignored during the initial stage of calculations, since the optimal solution does not reach them. Therefore, one can first apply the maximum principle, and use the dynamic programming method only in those cases where a substantial part of the constraints turns out to be significant.

The oscillatory process of a rod system of arbitrary shape under shock interaction with a falling load is considered. The system may consist of a large number of rods connected to each other rigidly or pivotally, and striking is assumed to be one of the core elements, thus causing a complex oscillatory process. As an example, vibrations of a rigidly sealed statically indeterminate flat two-post frame experiencing a drop of a load of a given mass and pre-impact velocity are simulated. One of the vertical pillars of the frame has an initial curvature, the presence of which affects the maximum amplitude of transverse vibrations arising from impact. The impact of the load on the frame crossbar is modeled taking into account the deformation in the contact area, which is justified from the point of view of the accuracy of the calculations, because otherwise the magnitude of the impact force will be overestimated. When modeling the impact interaction of the load and the rod system under consideration, it is assumed that the falling load has the shape of a cylinder with a certain length of the generatrix. Linearization of the relationship between force and deformation of cylindrical surfaces is used. The proposed method of modeling the amplitude of transverse oscillations makes it possible to further study the characteristics of the oscillatory process depending on the mass of the falling load and its pre-impact velocity, as well as on the configuration of the rod system. The relevance of the work for calculations of structural elements of various purposes experiencing impact is emphasized, since the presented model can be used for engineering calculations of a wide class of core systems.

The paper develops an analytical method in the classical problem on a flow around a thin rectangular plate of large span. It is shown that with a specific expansion on an orthogonal system of functions with a weight defined by qualitative properties of the solution, the initial 2d integral equation is asymptotically equivalent to a set of independent 1d integral equations. For them, we construct an asymptotic method allied to a boundary layer method, which permits development of analytical representations for basic aerodynamic characteristics. Comparison with the numerical method of discrete vortices shows that precision of the obtained solution is good not only for large but also for medium span of the wing.

The equation for the bending of a long cylindrical shell taking into account the static and dynamic pressures acting on both of its surfaces is given. Particular attention is paid to the role of boundary conditions and pressure, which is average between the static pressures acting on the surfaces. The compression of the wall in thickness is taken into account. The linear bending of a cylindrical panel with an arbitrary opening angle has been studied. The bending of a closed cylindrical shell is considered as a special case.