Multidimensional Planar Monoreactive Oscillations

The relevance of the study is determined by the fact that fluctuations of inertial masses are found everywhere. In the field of construction and use of aviation and rocket technology, this topic is of particular importance. Like a three-dimensional plane coordinate system in coordinate plane Z, a multidimensional system with \(n\) axes \(0{{x}_{{z1}}},0{{x}_{{z2}}},...,0{{x}_{{zn}}}\) turned relative to each other through angles \({{2\pi } \mathord{\left/ {\vphantom {{2\pi } n}} \right. \kern-0em} n}\) can be considered. There is an arbitrary vector \({\mathbf{R}}\) emanating from the origin 0, \({\mathbf{R}} \subset Z\). It is proved that points \({{x}_{1}},{{x}_{2}},...,{{x}_{n}}\), which are the coordinates of the end of vector \({\mathbf{R}}\) in the coordinate system \(0{{x}_{{z1}}},0{{x}_{{z2}}},...,0{{x}_{{zn}}}\), are the vertices of a regular polygon. The shape and dimensions of the polygon are not related to the coordinates of vector \({\mathbf{R}}\), i.e., are unchanged. The center of a regular polygon in all cases coincides with the middle of vector \({\mathbf{R}}\). In the considered (idealized) case, the polygon, at the vertices of which there are oscillating loads of masses m, lies in the Z plane (multipiston mechanism). In the considered multidimensional planar monoreactive oscillator, free harmonic linear oscillations of loads can occur. In this case, only kinetic energy is involved in the energy exchange. There is no need for elastic elements. The oscillator has no fixed natural oscillation frequency. The frequency depends on the initial speeds and positions of the loads. A regular polygon \({{x}_{1}},{{x}_{2}},...,{{x}_{n}}\) performs a double rotation (around point 0 and around point r). At the same time, the loads carry out linear harmonic oscillations with amplitude \(R\). The use of a crank-slider or crank-and-rod mechanism will allow organizing the parallel movement of goods.