Dynamics of Longitudinal Impact in the Variable Cross-Section Rods

Dynamics of longitudinal impact in rods of variable cross-section is considered. Rods of various configurations are used as elements of power pulse systems. There is no single method to the construction of a mathematical model of longitudinal impact on rods. The creation of a general method for constructing a mathematical model of longitudinal impact for rods of variable cross-section is the goal of the article. An elastic rod is considered with a cross-sectional area varying in powers of law from the longitudinal coordinate. The solution of the wave equation is obtained using the Fourier method. Special functions are introduced on the basis of recurrence relations for Bessel functions for solving boundary value problems. The expression for the square of the norm is obtained taking into account the orthogonality property of the eigen functions with weight. For example, the impact of an inelastic mass along the wide end of a conical rod is considered. The expressions for the displacements, forces and stresses of the rod sections are obtained for the cases of sudden velocity communication and the application of force. The proposed mathematical model makes it possible to carry out investigations of the stress-strain state in rods of variable and constant cross-section for various conditions of dynamic effects.


Introduction
The research of dynamic processes is carried out using the elastic rod model for a wide range of objects, such as: elements of drilling equipment [1][2][3], buildings and structures [4], power impulse systems [5,6], etc. Especially urgent is the problem of longitudinal impact for power pulse systems, which are used in machines that perform various technological processes: punching, forging, destruction of rocks, concrete coatings, piling, etc. The problem of increasing the productivity of impact machines includes not only increasing the power, but also increasing the efficiency of energy transfer to the process area. The latter is achieved, among other things, on the basis of studying the process of formation of deformation waves by strikers of various geometries and searching for strings structures that create deformation waves with rational parameters [5].
At considering the various models of longitudinal collision of bodies [5,7], the wave model of Saint-Venant's shock is taken as the basis, since it most fully reflects the real dynamic processes in colliding bodies, and for its practical implementation the d'Alembert method is used. The exact solution of the wave model is given by the Fourier method [8].
At solving the equation of longitudinal oscillations of variable cross-section rods, in the case of impact, arise certain mathematical difficulties, for example, the given equation is an equation with variable coefficients, the orthogonality of eigen functions, etc. Therefore, for solving the such 1234567890''"" problems are using various simplifying hypotheses. Thus, in [9] systems with distributed parameters are replaced by single-mass systems (Cox's theory) or various refinements [10], or are using an approximation of the dynamic deformations forms by static [11,12], and in paper [13] is used the method of averaging variable coefficients. In [7,14] are considered the models of longitudinal impact of various shapes rods with the using of their surface approximation by successively conjugate cylindrical sections. Using the operational calculus in work [3], is carried out an analysis of the using effectiveness of application of elements of drilling equipment (picks, drill rods) with different configurations, and for each of the schemes under consideration, a custom mathematical model is being constructed. The longitudinal impact of rods with conical and hyperbolic forms is considered in [15,16], solutions of the wave equation for which are obtained in elementary functions, and in [17] dynamical processes of longitudinal impact in hollow rods of conical shape are studied using Bessel functions.
Thus, based on the fact that in the theory of calculations, at the present time, there is no single approach to constructing a mathematical model of the longitudinal impact of complex configuration rods, the search and development of new solutions to longitudinal impact problems is an actual scientific and practical problem.

Development of a mathematical model
As a mathematical model of the object under consideration, we take an elastic rod of length l, the distributed mass m and the cross-sectional area F, which varies according to the exponent law from the longitudinal coordinate x: when  -material density, 2 F -the cross-sectional area of the larger base of the rod, 1 h и 2 h -the parameters of the cross dimensions of the upper and lower sections are determined by the geometry of the rod cross-section of the rod (the radius of the cross-sections for conical structures), the exponent μ depends on the rod configuration, for example, for conical tubes μ = 1, and for a conic rod of solid cross section μ = 2 (figure 1).

Equation of longitudinal displacements
u , , taking into account the accepted notation, will have the view [17].
Separating the variables, obtain an equation for the eigen functions whose solution has the view [18]. For the convenience of solving problems with different boundary conditions, we introduce the following notation for functions: In point k z  the presented functions take the values: Using the adopted dependencies, arbitrary constants in (3) can be expressed in terms of displacement 0 u and force 0 N in the zero section, then the expressions for displacements and longitudinal forces for an arbitrary waveform can be represented as: On the basis of the reciprocity theorem [19], at the absence of concentrated masses, the eigen functions will be orthogonal with weight       2 1 z z . To find the square of the eigen functions norm, we first proceed in the same way as in [17] for eigen functions with different indices.
Passing to the limit at Having determined the partial derivatives by n  , a square of the eigen functions norm find in the form     .
Now consider the case of forced oscillations, this requires finding a solution   t z u , 2 of non-linear equation The equation solution (7) can be represented as a row by eigen functions As an example, consider the scheme of mass impact M on the upper end of the rod, while the mass is held for some time by the rod, and its lower end is rigidly limited (figure 1), such a circuit can simulate a stress-strain state in waveguides of power pulse systems or in piles in the process of their driving. For the adopted scheme, the movement of the bottom end will be 0 0  u , the boundary condition on the second end of the rod when e(x) -a unit function. The equation for displacements and longitudinal forces for an arbitrary shape is obtained in the form From the boundary condition (10)  Taking into account the fact, that in some approximate methods of dynamic calculation [9,10] is using the value of the first eigen frequency, we estimate the effect of the parameter on the values of the first eigen value of (12) , the frequencies of the rods of constant cross section are determined in the same way [8]. Numerical studies of dependence 1  on the parameter value k for conic hollow roads (ν=0) and solid (ν=-0,5) sections, without taking into account the concentrated mass (   [8] (is indicated by the dot-dash line on the graph figure 2). Given the presence of the concentrated mass on the upper end of the rod (figure 1), the eigen functions will be orthogonal with the weight -Dirac's delta function. Then from the relation (6) the square of the eigen functions norm will be determined by the formula First, we find the movements and forces from the the speed reporting. From the first initial condition (11) it follows that If the second initial condition (11) is satisfied, then in expression (13) the coefficients n  will take the view  . The study showed that different parameters of the scheme do not significantly affect the values of the maximum relative displacements and forces, the stresses in the lower section increase in proportion to the reduction in the cross-sectional area. Those, in systems where the transfer of the pulse is mainly due to the application of the load, and not the speed, in this case the configuration of the rod will not affect the transmission efficiency of the power pulse.

Сonclusions
The proposed mathematical model of the rod longitudinal impact with complex configuration allows both direct investigation of the stress-strain state and, in contrast to numerical methods and means of object modeling, the analytical solution makes it possible to estimate the degree of influence of various model parameters on the unknown quantities. The presented algorithm of calculation can be put in the basis of CAD for imitating modeling of complex mechanical systems. It should also be noted that all the parameters of the model are determined and at the value 5 , 0   that corresponds to the rods of the constant section, so can use the unified approach to the construction of rod models with various configurations