Longitudinal vibrations of rods

. Elastic structures of buildings and machines, the dynamic behavior mathematical model of which is the problem of longitudinal vibrations of rods, are widespread in modern technology. In this regard, the study of issues related to the longitudinal vibrations of the rods is also an urgent problem currently. In terms of solving such problems, we considered the problem formulation of longitudinal (free and forced) vibration of rods, obtained the spectra of natural frequencies  n and own forms  n (x) vibration, u(x, t) – the function of cross-sections displacement in the longitudinal direction of the rod has


Introduction
Let us consider all these points in more detail. Let us draw a design scheme (Fig. 1) and consider natural motions. It should be noted that they are described in detail in the following literature sources [1][2][3][4][5].

Fig. 1. Homogeneous rod and selected element dx
Let us consider a homogeneous rod (Fig. 1) with a total mass m =   A (material density, Across-sectional area) vibrating in the longitudinal direction [6,7]. In this case, we admit the use of the flat sections and the d'Alembert principle hypothesis. Longitudinal sections' displacements are characterized by the function u(x, t). The relative deformation is  = x u   . Then, according to Hooke's law, it is possible to be written as: Let us consider the equilibrium of the selected element dx. The inertia force of (d'Alembert force) is equal to dx t u m dI 2 2    All forces applied to the selected element to the main axis of the rod are projected in the following way: and taking into account (1), we have: After some transformations, we finally obtain the equation of transverse vibration of a homogeneous rod of constant cross section, which is a homogeneous differential equation in partial derivatives of hyperbolic type.
Note that the equation (2) is identical to the classical equation of a single-span homogeneous string natural motion.
Further, the boundary conditions are added to the problem (2), which can be very diverse depending on the conditions for fixing the ends of the rod [8][9][10]. Let us consider a particular case of rigid attachment at both ends of the rod. In this case, the boundary conditions will have the form: The solution to the problem (2), (3) is found using the method of variables separation as the product Substitution by the method of variables separation in (2) gives: And substitution in boundary conditions (3) Next, we obtain the spectrum of natural frequencies of vibrations n and own forms n(x) l n a 2 For the input parameters a=1, l=1 the formula (4) gives the data for Table 1 of the first 4 natural frequencies The corresponding eigenmodes were found using the MATLAB programming system. The graphs are shown in Fig. 2.

Forced vibrations from kinematic disturbances
For kinematic perturbations, the following formula is obtained It gives the amplitude of stresses and its properties , In both cases, from cos k l = 0    (x) A σ , since in this case the frequency of disturbances ω coincides with natural frequencies n ω . This means that resonance leads to infinite values of the voltage amplitude.
Here are the graphs of the H (x) amplitudes along the rod. The resonance frequency graphs are not shown, since they strive for .

Stresses in the cross-sections of the rod at natural motions (n).
In the process of vibrations, not only the deflections of the rod u(x, t), but also normal stresses σ(x, t) and longitudinal force N(x, t) change.
Normal stresses according to Hooke's law will be These formulas determine the stresses and longitudinal force during natural motions.
Based on the calculations results, it is possible to build the stress diagrams which will give an opportunity to carry out the calculations on strength.  Table 4. Values of natural motions spectra

Conclusion
The obtained spectra of natural frequencies n and own forms n(x) vibration, the function of cross-sections displacement in the longitudinal direction of the rod u(x, t) has been found.