Transverse vibration spectrum of a part of a moving rod under a longitudinal load

We study the natural transverse vibrations of a constant-length part of a rectilinear thin rod moving along a neutral line in unstrained state. The displacement occurs between two fixed coaxial rod guides (clamps) and the distance between them is equal to the length of the vibration part of the rod. Moreover, it is assumed that a constant longitudinal force acts along the neutral line, and two significantly different cases are distinguished: the force extends the rod, and the force compresses the rod. The vibration process is described by a nonself-adjoint boundary value problem. For arbitrary values of the rod displacement velocity and longitudinal forces, the natural frequencies of possible vibration modes are determined and analyzed by a numerical-analytical procedure with a prescribed accuracy. The global properties of the spectrum depending on the velocity, the longitudinal force, and mode number are determined. For higher-order modes, the domains are discovered where the frequencies nonuniquely depend on the rod motion velocity and the value and direction of the longitudinal force and where the lower vibration frequencies can be absent as the rod motion velocity increases. It is shown that the picture of partial vibrations from the standpoint of an immovable observer cardinally differs from the commonly known picture for the immovable rod. The obtained results are interesting for studying vibrations of various elements of moving elastic media, including the case of systems with moving boundaries. These results can also have technical applications in problems of dynamics and strength of devices, machines, and mechanisms in textile industry in the filament and rope manufacturing and in metallurgy, in particular, in rolling of metallic rods and strips, in wire drawing, and in production of plastic articles and paper rolls. The developed technique for calculating the natural frequencies and shape modes can be used to analyze the transverse vibrations of parts of transport pipelines with rapidly flowing liquids.