Combination of circular motions in machines and mechanisms

In technical systems, including aviation and space technology, and in particular, in aircraft transmissions, bearings, orbital systems, helicopter mechanisms, and many others, the combined rotational movements are widespread, and when designing, it is important to understand the nature of joint motion. The paper aimed at the generalization of the principle of the combination of motions in circular movements. The author considered the x'0'y' coordinate system that rotates in the x0y coordinate system without angular acceleration with the velocity ω. The radius of rotation is equal to ρ₁. Wherein 0x || 0'x', 0y || 0'y'. An object a rotates in the x'0'y' coordinate system without angular acceleration with the velocity ±ω. The radius of rotation is equal to ρ₂. The study identified that at reverse rotations, the trajectory of joint motion represents an ellipse. The author determined all standard ellipse characteristics relating to the case under the study and identified the elliptical trajectory inclination. The study shows that if the joint motion trajectory is elliptical and the semi-axes are equal (ρ₁+ρ₂) and |ρ₁−ρ₂|, then an object a undergoes circular motion in the x'0'y' coordinate system without angular acceleration with the velocity −ω. Just as the result of the superposition of two non-accelerated straight movements is also non-accelerated, i.e. a uniform and rectilinear movement, at the one-way rotations, the joint motion trajectory represents a circle. At circular motions with multiple speeds, the joint motion trajectory represents a snail. The practical aspect of the study is determined by the fact that the resulting formulas can be directly used in the CAD system when performing design works.