Exact solution of the differential equation of transverse oscillations of the rod taking into account own weight

The free transverse oscillations of the rod of uniform cross section taking into account its own weight are considered in this work. The appropriate partial differential equation of transverse oscillations of a rod was reduced to two ordinary differential equations for the time function and the amplitude function of deflections. Concurrent with the differential equations for the amplitude state, the equivalent system of differential equation is considered. In total, the exact solution of the initial partial differential equation of transverse oscillations of a rod, expressed in nondimensional fundamental functions and initial parameters is attained. The method of power series was used for the construction of fundamental functions. Due to the exact solution, the formulas in an explicit form for dynamic variables of the state of a rode – deflections, angular displacement, bending moment and transverse force – were defined. The analytical form for equation of free oscillation frequency is defined. That has limited the finding of frequency to definition the unknown non-dimensional parameter through the frequency equation. As a result, the presence of derivative exact solutions provides the possibility to investigate the free oscillations of rod with various types of boundary conditions.


Introduction
It is generally known, that oscillation is the most widespread type of motion. There is no any branch of technology without phenomena where the vibrations take place. At this point, the most important characteristics of oscillatory system are natural frequencies and fundamental mode shapes. Problem of characterization is the actual scientific and practical question, which often reduces to the problem of the solving of appropriate differential equation of motion.
Among others, the important question is the investigation of bending vibration in various buildings and constructions taking into account the effect of axial force. Unfortunately, in the most part of the publication [1][2][3][4][5], where the oscillations of these constructions are considered, an axial force is accepted constant from any height for simplification.
However, in the real structures the axial forces in different cross sections have different values. For example, columns in industrial buildings have different compression ratio at different stories due to concentrated load at floor level. The industrial high-rise buildings like smokestacks, water towers, multipurpose steel towers, used for electric-power transmission line, wind generators supports, antennas of various constructions may be added to this example.
One of the most widespread design models for investigation of transverse oscillations of mentioned buildings is the uniform cross-section rod under the action of building weight in the capacity of variable axial force. The math model of this physical phenomenon is the differential variable-coefficient equation [3,6,7], the exact solution is still unknown. So the investigations are usually made by variational methods. Because of this, the investigations are usually made by variational methods. However, it is evident that the most full and qualitative evaluation of mechanical system can be gained only on the basis of the exact solution of differential equation. As a matter of fact, the focus of this paper is the definition of this solution. ( , ) y x t − the cross motion of the axis point of the rod with coordinate x at time t (dynamic deflection); ( , ) x t ϕ − the dynamic angular displacement; ( , ) M x t − the dynamic bending moment; ( , ) Q x t − the dynamic transverse force; ( , ) f x t − the intensity of inertial forces that appear during oscillation (D'Alembert force).
However, we should note, that all the following formulas are valid for any boundary conditions at the ends of the rod.
It is known [3,6], that the equation of free transverse oscillations taking into account own weight is written as: 4 2 This equation is valid for a model where the longitudinal displacement of cross sections and their twists and shifts are decided to ignore.
Defined dynamic deflection ( , ) y x t from Eq.(1), other dynamic parameters of the rod state are given according to the known formulas [3,6]: .
Using Fourier method, the solution of partial derivative equation (1) is given as where ( ) v x − the amplitude of the transverse displacement, which depends only on variable x ; ( ) T t − the time function t . After the implement (3) in formulas (2), we'll have the similar formulas for the other dynamic parameters: where ( ), ( ), ( ) x M x Q x ϕ − amplitude functions, which are linked by the equalities If we substitute (3) into Eq. (1) and separate the variables there, we will have two differential equations: where 2 p − Fourier method constant. The solution of the Eq. (6), expressed in terms of parameters of initial motion Its analysis shows that motion of mechanical system is oscillatory in nature. The constant is the frequency of free oscillations.
The fundamental mode is the solution of Eq. (7). The main difficulty of the problem is exactly the definition of this solution.
The equivalent system of differential equations is considered along with Eq. (7). The state vector of the rod is used in the function of vector of unknowns. The components of the state vector are amplitude deflections, angular displacement, bending moment and transverse force: Then, writing the collection of formulas (5), (7) in form of matrix we'll have: where ( ) D x − the coefficient matrix of the system, which have the form Transbud-2017

The exact solution of amplitude equation of oscillation. Formulas of state parameters of the rod
For the construction of an exact solution of Eq. (7) we use integration technique proposed and developed in [8].
There are four infinite systems of functions ,0 which are decided continuous ones with their derivatives from the first-order to fourthorder inclusive. With the help of these functions and their derivatives, we make the expansion in powers of parameter 2 P p m = : where ( ) ν -the order of derivate, 1, 2,3, 4 ν = . At the moment we allow, that the ranges (10), (11) are uniform convergent, so it's possible to do term-by-term differentiation.
Following the terminology, accepted in [8], the functions ,0 ( ) For its satisfaction set all the coefficients at powers of parameter P , beginning from zero power: So, the differential equations for primary and generating function definition are attained. Use the power series method for integration this equations, previously accepted next boundary conditions: Integrating both parts (14), one arrives at the equation n k n k j n k j n k n k n k j n k j n k j n k j j j n k n k j n k n k j n k j n k j So the coefficients at the same powers are equated, we'll have: Thus, the coefficients of series (20), (21) are totally definite by formulas (33)−(35) and recurrence formula (32). Disadvantage of these formulas is that they are depending on initial dimensional problem parameters EI and q . This disadvantage can be avoided making the next substitution , , , , , Then the matrix , consisted of these vectors, also will satisfy the system (9). Based on formulas (10), (11) and boundary conditions (17) Therefore taking into account (16) we find: Fundamental matrix, which satisfies the condition (44), is uniquely determined and called matrizant [9].
Thus, allowing for the uniform convergence at the beginning (10), (11), we arrive at a conclusion that the matrix ( ) x Λ , which consists of the sums of these series , is the matrizant of the system (9). On the other hand, in the differential equations theory it is proved that matrizant of equation system with continuous coefficients is always absolutely and uniform convergent matrix series [9]. It follows that because of the uniqueness of matrizant the series (10) and (11) are absolutely and uniform convergent for 1, 2,3 ν = . As for series (11) for 4 ν = , its uniform convergence follows immediately from identity (12). So, previously accepted designations ( ) ( 1,2,3,4) n U ν ν = for series (11) are correct. General solution of differential equation system with the presence of matrizant is given by known [9] formula ( ) ( ) (0) x x Φ =Λ Φ . Writing it in expanded form we get the formulas for amplitude deflections and internal forces: