This study deals with a methodology for increasing the efficiency of dynamic process calculations in elastic elements of complex engineering constructions. We studied the complex dynamic processes in a simple engineering construction, a mechanical system of an elastic body–continuous flow of homogeneous medium. The developed methodology is based on the use of a priori information on some of the vibrations forms, the construction of a “simplified” mathematical model of system dynamics, and the obtaining of an analytical relationship that describe the overall range of factors on the elastic vibrations of system. The methodology is used for cases of complex vibrations of elastic bodies, and the obtained results can serve as a basis for choosing the main technological and operational parameters of elastic elements of mechanisms and machines that perform complex vibrations. The results obtained in this work are the basis for calculating the blast effect on the elements of protective structures in order to increase their protective capacity by improving the method of their attachment or by using additional reinforcement, buff load effects on the elements of drilling strings and dynamic processes that occur during surface strengthening by work hardening in order to avoid resonance phenomena, and technological processes of vibration displacement or vibration separation of granular media.
A method for studying the effect of impulse perturbation on the longitudinal oscillations of a homogeneous constant cross-section of the body and the elastic properties of a material which satisfies the essentially nonlinear law of elasticity has been developed. A mathematical model of the process is presented, which is an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of impulse perturbation on the oscillatory process. As for the boundary conditions considered in the work, they are classic of the first, second and third genera. The methodology is based on: the principle of oscillation frequency in nonlinear systems with many degrees of freedom and distributed parameters; basic provisions of asymptotic methods of nonlinear mechanics; the idea of using special periodic Ateb-functions to construct solutions of some classes of nonlinear differential equations; properties of completeness and orthonormality of functions that describe the forms of oscillations of undisturbed motion. In general, the above allowed to obtain relations that describe for the first approximation the defining parameters of the oscillations of an elastic body. Their peculiarity is that even for undisturbed motion, the natural frequency of oscillations depends on the amplitude, and therefore, under the action of a periodic (over time) pulse force on the elastic body, both resonant and nonresonant processes are possible in the latter. It, in contrast to an elastic body with linear or quasilinear elastic properties of the body is determined not only by its basic physical and mechanical properties, but also by the amplitude of oscillations. As a special case, the oscillations of the body under the action of a constant periodic momentum perturbation are considered. It is shown that for the nonresonant case for the first approximation it does not affect the laws of change of amplitude and frequency of the process. As for the resonant is the amplitude of origin through the main resonance significantly depends not only on the speed but also on the points of action of the pulsed perturbation. Moreover, the closer the point of application of the pulsed force to the middle of the elastic body under boundary conditions of the first kind is greater (for boundary conditions of the second kind closer to the end).
The experience of peacekeeping and other military operations shows the growing role of the combat-wheeled vehicles. However, the combat-wheeled vehicles suspension for the base for which a chassis of a serial wheeled vehicle was selected with the armored corps in it does not fully protect the personnel from dynamic overloads while driving with significant irregularities or crosscountry terrain. The armored corps causes a significant increase in the weight of the sprinkled part and so a static deformation of elastic shock absorbers, and also a number of operational features. In order to overcome the above-mentioned disadvantages we modernize the system of sprinkling using the shock absorbers with nonlinear (progressive or regressive) characteristics for such combatwheeled vehicles. However, the influence of the nonlinear force characteristics of the combat-wheeled vehicles' system of sprinkling on the shooting efficiency on the move from stationary mounted small arms are not investigated due to a number of reasons. They relate to the construction and investigation of the solutions to the nonlinear differential equations, which are the mathematical models of the combat-wheeled vehicles' motion. In this paper, we investigate the influence of the power characteristics of the system of sprinkling of combatwheeled vehicles on the transverse oscillations of the sprinkled part, and on the shooting efficiency on move from the stationary mounted small arms. The basis for the determining of these characteristics serve the differential equation of the sprinkled part's perturbed motion. We use the Van der Pol method, adapted to strongly nonlinear differential equations for its integration. It is shown that the magnitude of the dispersion caused by the indicated oscillations of the sprinkled part increases, and in the case of the progressive power characteristic of elastic shock absorbers during the motion along the path with single irregularities the value of the scattering region is greater than that of the regressive one.
A methodology for researching dynamic processes of one-dimensional systems with distributed parameters that are characterized by longitudinal component of motion velocity and are under the effect of periodic impulse forces has been developed. The boundary problem for the generalized non-linear differential Klein–Gordon equation is the mathematical model of dynamics of the systems under study in Euler variables. Its specific feature is that the unexcited analogue does not allow applying the known classical Fourier and D'Alembert methods for building a solution. Non-regularity of the right part for the excited non-linear analogue is another problem. This study shows that the dynamic process of the respective unexcited motion can be treated as overlapping of the direct and reflected waves of different lengths but equal frequencies. Analytical dependencies have been obtained for describing the aforesaid parameters of the waves. They show that the dynamic process in such mechanical systems depends not only on their main physical and mechanical parameters and boundary conditions, but also on the longitudinal motion velocity (relative momentum). As relative momentum increases, the frequency of the process decreases. To describe excited motion, we use the principle of single frequency of oscillations in non-linear systems with concentrated masses and distributed parameters as well as regularization of periodic impulse excitations. The main idea of asymptotic integration of systems with small non-linearity into the class of dynamic systems under study has been generalized. A standard equation for the resonance and non-resonance cases has been obtained. It has been established that for the first approximation, in the non-resonance case, impulse excitation affects only the partial change of the form of oscillations. Resonance processes are possible at a specific relation between the impulse excitation period, the motion velocity of the medium, and physical-mechanical features of the body. The amplitude of transition through resonance becomes higher when impulse actions are applied closer to the middle of the body. As the longitudinal motion velocity increases, it initially increases and then decreases.
The technique of research of dynamic processes of elements of engineering constructions of special purpose from explosive action of projectiles is developed. Elastically reinforced beams with hinged ends were chosen for the physical model of elements of engineering structures. It is assumed that the elastic properties of the latter satisfy the nonlinear technical law of elasticity. A mathematical model of the process of a series of impact actions of projectiles at different points of the element of the protective structure is constructed. The latter is a boundary value problem for a partial differential equation. Its peculiarity is that the external dynamic action is a discrete function of linear and time variables. To determine the dynamic effect of a series of impacts on the object under study, and thus the level of protection of the structure, the basic ideas of perturbation theory methods are extended to new classes of systems. This allowed to obtain an analytical dependence of the deformation of the elastically reinforced element on the basic physical and mechanical characteristics of the material of the protective element, its reinforcement and the characteristics of the external action of the projectiles. It is shown that the most dangerous cases, given the security of the structure, are those when the impact is repeated at equal intervals, in addition, the point of impact is closer to the middle of the protective element. The obtained theoretical results can be the basis for selection at the stage of designing the main physical and mechanical characteristics of the elements of engineering structures and their reinforcement in order to reliably protect personnel and equipment from the maximum possible impact on it of the shock series of projectiles. The reliability of the obtained results is confirmed by: a) generalization of widely tested methods to new classes of dynamical systems; b) obtaining in the limit case the consequences known in scientific sources concerning the linearly elastic characteristics of the elements of protective structures; c) their consistency with the essence of the physical process itself, which is considered in the work.
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