Про динамічний ефект несиметрії силової характеристики коливальної системи при імпульсному навантаженні

Abstract

Описано динамічний ефект несиметрії силової характеристики коливальної механічної системи з одним ступенем вільності при короткочасному силовому імпульсному навантаженні. Він полягає в тому, що максимальні переміщення системи у напрямі дії прикладеного імпульсу менші, ніж протилежні за напрямом максимальні переміщення розвантаженої системи, при її вільних коливаннях. Показано, що умови прояву ефекту пов'язані не тільки з власними параметрами коливальної системи, а й з тривалістю імпульса. Розглянуто дію двох варіантів силових імпульсів. Перший прямокутний, а другий – у вигляді однієї півхвилі синусоїди.The motion of a nonlinear mechanical system with one degree of freedom under force impulse loading is considered. The elastic characteristic of the system is approximated by segments of two straight lines that have a common point in the equilibrium position with zero displacement. The dynamic effect of asymmetry of the power characteristic of an oscillating mechanical system is described. It consists in the fact that the maximum displacement of the system in the direction of the applied external pulse is less than the amplitude deviation of the unloaded system in the opposite direction with its free oscillations. Conditions are defined when this inequality holds. To this end, analytical solutions of the differential equations of motion were constructed in the regions of positive (with greater rigidity) and negative (with less rigidity) displacements, and then the solutions were padded. Compact formulas are derived for calculating the maxima of displacements and the time of their attainment. It is shown that the expressions for the maxima can also be derived from the energy relationships, without solving the differential equations of motion. The action of two variants of pulses of finite duration is considered. The first is rectangular, with a constant instantaneous applied force, which also instantly disappears after some time, and the second is sinusoidal - in the form of one half-wave of the sinusoid. It is established that the height of these pulses, that is, the maximum value of the applied force, does not enter into the conditions for the manifestation of the dynamic effect for the adopted kusano linear elasticity characteristic of the system. But the conditions for the manifestation of the dynamic effect are related to the proper parameters of the vibrational system (mass and rigidity), and also depend on the duration of the pulse in time. For relatively long duration of the action of pulses with respect to the period of free oscillations of the oscillator, this effect is not manifested. Numerical examples are given that confirm the adequacy of the derived calculation formulas and illustrate the possibilities of the theory presented. To compare the results, we numerically integrate the differential equation of motion on a computer. It is shown that the results obtained using analytical and numerical solutions of the Cauchy problem are in good agreement, and the dynamic effect of asymmetry of the power characteristic of the oscillating system is manifested for both forms of the pulses considered.