Construction of a physical-mathematical model of oscillations of the unbalanced vibrator of the pneumatic sorting table

Abstract

A physical-mathematical model of oscillations of unbalanced vibrators of a pneumatic sorting table as non-stationary oscillations of an impulse-loaded round plate with various options for fixing its contour has been built. The axisymmetric non-stationary oscillations of a round plate supported by a one-sided round base were considered in two ways of fixing its contour, namely, when it is tightly clamped and freely supported. It was assumed that the linearly elastic base resists only compression and does not perceive stretching. It is shown that for certain durations of the transverse force pulse in time, the amplitude of the deflection of the middle of the plate in the direction of action of the external pulse can be smaller than the amplitude of the deflection in the opposite direction. At the same time, in the second case, there is no contact of the plate with the base. It has been proven that this dynamic effect of the asymmetry of the elastic characteristic of the system also applies to bending moments and is more clearly manifested when the contour is freely supported than when it is tightly clamped. For rectangular and sinusoidal pulses, closed-loop solutions of the equations of motion of the plate during its contact with the base and after separation from the base were constructed. Compact formulas were derived for calculating the amplitudes of positive and negative deflections in both directions from the zero position of static equilibrium. Formulas have been obtained for calculating the time it takes for the plate to obtain extreme deflection values, which is achieved due to the selection of a special axisymmetric distribution of dynamic pressure on the plate. Under such a load, the plate simultaneously detaches from the base at all points, except for the contour points, which reduces the nonlinear boundary value problem to a sequence of two linear problems. Numerical integration of the differential equation was carried out to check the reliability of the constructed analytical solutions. Adequacy of the model was proven for the following values of initial parameters: modulus of elasticity, 2.1·1011 Pa; the Poisson ratio of the plate material, 0.25; plate thickness, 7...10 mm; the maximum pressure on the plate, 4·103 Pa; the bending stiffness of the plate, 6402.6667 N·m