Analysis of existing methods of analytical description of complex motion of a point and finding its kinematic characteristics

Abstract

When studying the motion of a point, a certain frame of reference is chosen, in relation to which the motion of the point is considered. Sometimes it is necessary to consider the motion of a point relative to two different reference systems. For example, the movement of a passenger in a train can be analyzed relative to the train and relative to the Earth. The motion of the same point relative to two different reference frames will be different. For example, the rim point of the wheel of a moving railway car will describe a cycloid relative to the Earth, and a circle relative to the car.

When considering the motion of a point relative to two reference frames, the system, which in this problem is conventionally taken as stationary, is called the main frame of reference (fixed), and the other, which moves relative to the main - mobile frame of reference. The motion of a point relative to the main frame of reference is called absolute motion, and its motion relative to the moving frame of reference is called relative motion. A complex motion of a point is a motion of a point in which it simultaneously participates in two or more motions.

The ability to decompose a more complex motion of a point or body into simpler motions by introducing an additional (movable) frame of reference is widely used in kinematic calculations and determines the practical value of the theory of complex motion. In addition, the results of this theory are used in dynamics to study the relative equilibrium and relative motion of bodies under the action of applied forces.

The theory of complex motion of a material point has a clearly completed form and is given in all textbooks on theoretical mechanics. It is based on the fact that the motion of a point is studied simultaneously with respect to two coordinate systems. One of them (the main) is taken as immovable, and the other carries out relative to the immovable relative movement according to the given law. In turn, in a moving coordinate system, the material point carries out the relative motion. The sum of these motions (relative and figurative) is the absolute motion of a point relative to the basic coordinate system. Movements (both figurative and relative) are given by dependencies in time functions.

There is also a natural (natural) way to specify the motion of a material point, in which velocity and acceleration are considered in the projections on the orts of the accompanying trihedron of the trajectory (Frenet trihedron). Thus only simple movement of a point is considered.