The dynamic response of particles is closely related to screening efficiency. To study the dynamic response of particles, the dynamic equations of a particle on a screening surface are established based on the elastoplastic contact model of spherical particles and are solved for the coal particle. Then, the trajectories of the particles are given with different falling heights and particle radii. The completely different trajectories with slight changes in the falling height and particle radius indicate strong nonlinearity. Second, the nonlinear dynamic behavior under different amplitudes and frequencies is discussed, and the route of transition from quasiperiodic motion to chaotic motion is revealed. Finally, we discuss the average speed along the screening surface considering the frequency, amplitude, friction coefficient, inclination angle, and vibration direction angle. In addition, the convergence conditions of particle motion are proposed, and they are only affected by the inclination angle and friction angle. The results show that in the normal direction of the vibrating screen, the particle motion is quasiperiodic at low frequencies. With increasing frequency, the motion of the particle becomes chaotic, and its Poincaré map becomes petal-shaped. In addition, the number of petals increases at the mutation of the bifurcation diagram. The increase in frequency, amplitude and inclination angle and the decrease in friction coefficient lead to an increase in particle speed along the screen surface. In addition, the particle speed reaches a maximum when the vibration direction angle is 65°. This work provides a theoretical basis for controlling the thickness of granular material flow on a vibrating screen and selecting screening process parameters.

The double-deck vibrating flip-flow screen is taken as the study case, and the dynamic equations, including cubic nonlinear stiffness, are established. The equation in the physical coordinates is transferred to the modal coordinates to study the nonlinear vibration characteristics of the system in resonance, which is analyzed by the multi-scale method. Then, the amplitude-frequency tuning parameter response equation is obtained. The influence of system parameters on the nonlinear system is discussed by solving the amplitude-frequency tuning parameter response equation. It is found that increasing the damping or reducing the harmonic force will suppress the system’s multiple solutions and vibration jumps. Numerical methods are adopted to verify the accuracy of the perturbed solution in both the frequency domain and time domain. Considering that the nonlinear stiffness has little influence in the slightly distant single-valued response resonance region, the displacement response of the double-deck vibrating flip-flow screen in the original physical coordinates system is obtained based on the superposition principle. The amplitude of the main screen frame is 7 mm, and the amplitude of the upper and lower floating frame is 9.4 mm and 5.6 mm, respectively. The phase between the main screen frame and the upper and lower floating frame is 180°, which realizes the stretching and slackening of the deck. This paper guides the design of double-deck vibrating flip-flow screens.

The double-deck vibrating flip-flow screen (DDVFFS) resolves the limitation of the traditional single-deck vibrating flip-flow screen, which cannot perform multi-granularity screening. The nonlinear characteristics of the springs in the DDVFFS lead to issues of unstable vibration and significant deviations between the operational and design amplitudes. Consequently, a dynamic equation incorporating cubic nonlinear stiffness is formulated. These equations are initially expressed in physical coordinates and subsequently transformed into modal coordinates for analyzing the system’s nonlinear vibration characteristics under resonance using the multi-scale method. The study then explores the impact of system parameters on this nonlinear behavior. Increasing damping or reducing harmonic forces proves effective in mitigating multiple solutions and vibration jumps. The operational frequency is chosen within the distant single-valued resonance region to minimize the influence of nonlinear stiffness, ensuring stable operation of the DDVFFS. The calculated amplitudes are as follows: the main frame shows an amplitude of 7 mm, whereas the upper and lower floating frames display amplitudes of 9.4 mm and 5.6 mm, respectively. A 180° phase difference between the main frame and the upper or lower frames facilitates deck stretching and slackening.