The paper investigates free vibrations of an absolutely rigid body, supported by a set of linearly elastic springs and performing a plane-parallel motion. The proposed system has two degrees of freedom, which makes it elementary to determine the frequencies and modes of its natural oscillations by using exact analytical expressions. However, these expressions are rather cumbersome, which makes it difficult to study the behavior of frequencies and modes when the characteristics of the model change. Therefore, the aim of the work was to find out the qualitative properties of the modes of free vibrations depending on the elastic, inertial and geometric characteristics of the system, as well as to study the effect of changing the position of elastic supports on its natural frequencies. The main qualitative characteristic of the mode of natural vibrations of the system in consideration is the position of its node – a point that remains stationary during natural vibrations. For the practically important case of a system with two supports, it has been established in the work that, in the general case, of two modes corresponding to two different natural frequencies, one has a node located inside the gap between the supports, and the other – outside this gap. Analytical conditions are found that must be satisfied by the inertial and geometric characteristics of the system, which make it possible to determine which of the two modes corresponds to the internal position of the node. It is noted that these conditions do not depend on the stiffness of the supports. Analytical results were also obtained, allowing to determine a more accurate qualitative localization of the node. To clarify the behavior of natural frequencies when the position of the supports changes, an explicit expression is obtained for the derivative of the square of the natural frequency of the system with respect to the coordinate defining the position of the support. This expression can be used to solve a variety of problems related to the control and optimization of the operating modes of engineering structures subjected to dynamic, in particular periodic, effects. The results of the work were obtained using qualitative methods of the mathematical theory of oscillations. In particular, the theorem on the effect of imposing constraints on the natural frequencies of an elastic system is systematically used.
Natural vibrations of a cylindrical shell and the foundation of shell theory.
