Simulation of radial oscillations of a spring-loaded roll in a roll compactor

Carried out simulation of oscillations of a spring-loaded roll in a roll compactor when interacting the powder being compacted with the rolls. Considering the separation of the feed and compaction areas in the contact area of the roll with the material being compacted, we obtain the dependence of the force acting on the roll on the gap size between the rolls. It is shown that this dependence is non-linear, and it can be described with a sufficiently high accuracy degree by an exponential function with a negative exponent in the working range. The given numerical solution of the equation of free nonlinear oscillations of the spring-loaded roll has shown that considering the deformation of the material being compacted leads to a reduction of the natural frequency of the system by 20–25 % compared to the case, where the pressure force of the powder on the roll is assumed to be independent of the gap size. The nonlinearity of the dependence of the pressure force on the gap also leads to the increase by 10 % in the calculated values of the maximum displacements. The developed approach to the calculation of oscillations of the spring-loaded roll in the roll compactor enables to take into account the peculiarities of deformation of the powder being compacted during its interaction with the rolls. In addition, it allows estimating the frequencies and oscillation amplitudes and setting the optimum range of spring rate values, at which the occurrence of resonance in the machine is not possible.

Nonlinear oscillations in a constant gravitational field

Exploring non-linear oscillations is a challenging task since the related differential equations cannot be directly solved in terms of elementary functions. Thus, complicated mathematical or numerical methods are usually employed to find accurate or approximate expressions that describe the behavior of the system with respect to time. In this paper, the vertical oscillations of an object under the influence of its weight and an opposite force with magnitude F=cyn, where n>0 are being explored. Accurate and approximate simple solutions regarding the object’s position with respect to time are presented and the dependence of the oscillation’s period from the oscillation’s range of displacements and the exponent n is revealed. In addition, the special case in which n=3/2 (which describes the oscillation of a rigid sphere on an elastic half space) is also highlighted. Lastly, it is shown that similar cases (such as the case of a force with magnitude F=kx+λx2) can be also treated using the same approach.

Nonlinear oscillations of a sandwich plate with a 3D-printed honeycomb core

A three-layer sandwich plate with a FDM-printed honeycomb core made of polycarbonate is considered. The upper and lower faces of the sandwich are made of a carbon fiber-reinforced composite. To study the response of the sandwich plate, the honeycomb core is replaced with a homogeneous layer with appropriate mechanical properties. To verify the honeycomb core model, a finite-element simulation of the representative volume of the core was performed using the ANSYS software package. A modification of the high-order shear theory is used to describe the structure dynamics. The assumed-mode method is used to simulate nonlinear forced oscillations of the plate. The Rayleigh–Ritz method is used to calculate the eigenfrequencies and eigenmodes of the plate, in which the displacement of the plate points during nonlinear oscillations are expanded. This technique allows one to obtain a finite-degree-of-freedom nonlinear dynamic system, which describes the oscillations of the plate. The frequency response of the system is calculated using the continuation approach applied to a two-point boundary value problem for nonlinear ordinary differential equations and the Floquet multiplier method, which allows one to determine the stability and bifurcations of periodic solutions. The resonance behavior of the system is analyzed using its frequency response. The proposed technique is used to analyze the forced oscillations of a square three-layer plate clamped along the contour. The results of the analysis of the free oscillations of the plate are compared with those of ANSYS finite-element simulation, and the convergence of the results with increasing number of basis functions is analyzed. The comparison shows that the results are in close agreement. The analysis of the forced oscillations shows that the plate executes essentially nonlinear oscillations with two saddle-node bifurcations in the frequency response curve, in which the periodic motion stability of the system changes. The nonlinear oscillations of the plate near the first fundamental resonance are mostly monoharmonic. They may be calculated using the describing function method.

Analysis of subharmonic oscillations in multi-phase ferroresonance circuits using a mathematical model

The article about solution of system of nonlinear differential equations that are almost impossible to solve by analytical methods by constructing a mathematical model of nonlinear oscillations occurring in three-phase ferroresonance circuits. A system of nonlinear differential equations was formed by approximating the volt-ampere characteristics of a ferromagnetic element in a ferroresonance circuit. Mathematical models for solving technical problems characterizing subharmonic oscillation processes in three-phase ferroresonance circuits and systems using the finite-difference method are expressed in the form of differential equations without appropriate initial conditions. Amathematical model of a system of equations representing subharmonic oscillations in ferroresonance connections depending on the value of the selected parameters in the field of change of any variable is considered.

Optimization of Nonlinear Unbalanced Flexible Rotating Shaft Passing Through Critical Speeds

This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for this work. The shaft is modeled as a beam and the Euler–Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6 degrees of freedom. In order to solve these equations numerically, the finite element method (FEM) is used. Furthermore, for different bearing properties, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. Then the optimal values of bearing stiffness and damping are calculated to achieve the minimum vibration amplitude, which causes to pass easier through critical speeds. It is concluded that the value of damping and stiffness of bearing change the rotor critical speeds and also significantly affect the dynamic behavior of the rotor system. These effects are also presented graphically and discussed.

Physics of nonlinear oscillations with nonlocal variables

In cases where physical processes cannot be described by linear equations, and nonlinear equations are difficult to solve mathematically, we have to use approximate solutions to such problems. One such example is the description of the Kapitsa pendulum, which is a pendulum with a vibrating suspension point. In contrast to the previously known methods of describing such a problem, in this paper we propose to use additional variables in the form of higher derivatives, which allows us to obtain corrections that give a more detailed contribution to the description of this problem.