Nonlinear Vibrations Analysis and Dynamic Responses of a Vertical Conveyor System Controlled by a Proportional Derivative Controller

In this paper, we introduce a nonlinear vibrations analysis and dynamic responses of a vertical conveyor system under multi excitation forces. By adding the nonlinear proportional derivative controller (NPD) to the vibrating motion of the vertical vibration conveyor, the energy was transferred between uncontrolled and controlled system. We calculate the approximate solutions of the vibrating system utilizing the method of multiple scales. In addition, we investigate the stability at worst resonance cases using phase plane technique, equations of frequency response, and averaging method. The vertical vibration conveyor behavior was studied numerically at the values of its different parameters. The results exhibit the efficiency of NPD control unit to avoid the oscillations of the vertical conveyor system. Numerical simulations have been carried out using MAPLE and MATLAB software’s to ensure the fidelity of our results. Comparisons are made between analytical and numerical results. Also, findings of the present work are discussed in details and compared with published works.


I. INTRODUCTION
In the field of electronics and mechanics, many researchers have examined oscillations of mechanical systems with periodic loads. Vertical conveyors are considered as efficient samples of control different parameter types for this problem. They have some features such as simple structure, energy used less, and low maintenance cost. The passive control technique is used to investigate the behavior of the system in the presence of multi types of excitation forces [1]- [3]. The dynamic behavior of the inclined cable resonance in the presence of harmonic excitation is discussed by [4]. Others have investigated the nonlinear behavior of the string beam system with multiple excitations at the case 1:1 internal resonance; they showed that there are jump phenomena in the curves of the frequency response [5]. The stability analyses and numerical response of a nonlinear-coupled pitch-roll ship system in the presences of parametric with harmonic excitation The associate editor coordinating the review of this manuscript and approving it for publication was Feiqi Deng . forces were studied [6]. The straight vibrations of the vertical conveyor are checked in different stocking conditions [7]. The nonlinear analysis of the unbalanced mass of a vertical conveyor elevator, dynamic characteristics such as effects of excitations amplitude, nonlinearity, and damping are studied at primary resonance [8]. Shaking conveyers were investigated analytically and numerically applying the method of multiple scales at primary, super-harmonic and sub-harmonic resonances with cubic nonlinear spring and a vibration exciter (ideal and non-ideal). Numerical simulations showed that the important dynamic characteristics of the system and presented a periodic behavior for these conditions [9]- [11]. The mathematical study and nonlinear dynamic analysis of the vertical conveyor oscillations under different excitations were introduced [12]- [15]. The vertical conveyor analysis was investigated using unit control of positive position feedback and negative velocity feedback controls [16], [17]. The vibrations of the vertical conveyor are depressed using the nonlinear saturation controller (NSC), where the system subjected to external excitation [18]. The modified vertical conveyor behavior is studied at different simultaneous resonance cases using multiple scales technique [19]. For a Cartesian manipulator and micro electromechanical system, [20], [21] discussed transfer of energy, the stability, and bifurcation analyses using Poincareì maps and averaging method technique. The horizontally supported Jeffcott-rotor system oscillations was eliminated with nonlinear PD controller at primary resonance [22]. Study of nonlinear damping in large vibration amplitudes for the fractional viscoelastic standard solid model is studied analytically and experimentally [23]. The new vibratory conveyor transport possibilities, changes of direction transporting and velocity are analyzed with the angular velocity changes of the excitation vibrator [24]. The dynamics behavior, the influence of operative parameters, and the frictional properties of the cylindrical parts of the vibratory conveying are discussed [25]. The proportional derivative controller was applied to transfer the energy, and reduce large oscillations in the wind turbine system. In addition, the stability, bifurcation analysis, effect of different parameters were presented numerically in [26]. For more details about the nonlinear vibrations consequence of the geometrically nonlinear stiffness and damping one may refer to the references [27], [28]. Moreover, the analysis detailed of some dynamical systems with different forces was founded in the books [29]- [32]. This paper is organized as follows. In Section.II, the nonlinear vibrations analysis and dynamic responses of a vertical conveyor system under multi excitation forces were introduced. Also, in this section we use multiple method to obtain the approximate solutions of the vibrating system. The stability of the system is analyzed by using averaging method. Section III presents numerically the stability and vertical vibration conveyor behavior at different parameters values of the system. In addition, this section shows the effects of NPD controller on system behaviors and presents a comparison between analytical and numerical results. The key results of the study are presented in Section IV, concluding and important remarks were presented.

II. MATHEMATICAL ANALYSIS
The model of vertical shaking conveyor with a spring of cubic nonlinearity and linear damping are shown in Figure 1, consist of an elastically cylinder having a helical track, four equal unbalanced masses P give torsional and vertical oscillations for the cylinder, and an electric motor to transfer the directed vibrations along and around the vertical axis leading the load moving upwards along the helix. The vertical and angular (torsional) vibrations in the conveyor are due to the vertical forces components P z and horizontal components P x , respectively.
Proceeding as in Ref. [13], we modified the governing equation of the motion for vertical shaking conveyor as: With the initial conditions x 1 (0) = 0.01,ẋ 1 (0) = 0.01, x 2 (0) = 0.01 andẋ 2 (0) = 0.01, where x 1 and x 2 are the vertical and angular position of the trough of the conveyor system, ε is a small perturbation parameter, µ 1 and µ 2 are the damping coefficients of the vertical and angular springs, α 1 and α 2 are the nonlinear spring coefficients of the vertical and torsional spring, ω 1 , ω 2 , 1 , 2 , and 3 are natural and excitation frequencies of the vertical shaking conveyor respectively, f 1 , f 2 , f 3 , and f 4 are excitation and tuned force amplitudes of the vertical shaking conveyor respectively, p, d, α 3 , and α 4 are linear and nonlinear coefficients of proportional derivative controller.

A. PERTURBATION ANALYSIS
Using the method of multiple scales [31], [32] to obtain the solutions for (1)-(2) in the form: The derivatives can be written in the following: For the first-order approximation, let us introduce the two time scales and the derivatives, where T n = ε n t and D n = ∂ ∂T n , (for n = 0, 1)). We substitute equations (3)-5 into VOLUME 8, 2020 (1) and (2) and equate the coefficients of equal powers of (ε) leads to O ε 0 : O ε 1 : We have written the solutions of equation (6) in the form: where A 1 , A 2 are complex functions in T 1 . Inserting equation (8) into equation (7) with eliminating the secular terms e ±iω 1 T 0 and e ±iω 2 T 0 , the solutions of equation (7) will be in the form: where cc are complex conjugates. The resonance cases are classified into: (A) Primary resonance: 1 (E) Simultaneous resonance: we consider any combination of the above resonance cases as simultaneous resonance.

D. STABILITY ANALYSES AND EQUILIBRIUM SOLUTIONS
The steady-state solution occurs when,ȧ 1 =ȧ 2 =θ 1 =θ 2 = 0 and, the steady state solutions are given by: The stability of the nonlinear solution is analyzed by considering the following: a 1 = a 10 + a 11 , a 2 = a 20 + a 21 , θ 1 = θ 10 + θ 11 , θ 2 = θ 20 + θ 21 (25) where a 10 , a 20 , θ 10 , and θ 20 are the solutions of (19)- (20). Inserting (25) into (19)- (20) and linearizing equations in a 11 , a 21 , θ 11 and θ 21 , we geṫ where l j (j = 1, 2, . . . , 8) are constants. Thus, to investigate the stability of the steady-state solutions, we must find the eigenvalues of the Jacobian matrix, which can be obtained from the following: Expanding this determinant, yields where r 1 , r 2 , r 3 , and r 4 are constants (given in Appendix B). The solution is stable if the real parts of the eigenvalue are negative; otherwise, the solution is unstable. The sufficient conditions for all the roots of (31) to contain negative real parts, we apply the Routh-Hurwitz criterion, so that:

III. RESULTS AND DISCUSSIONS
To investigate the numerical results of system equations (1) and (2), the algorithm of Runge-Kutta of the fourth-order is applied. Also, we studied the stability of the vertical conveyor system with the averaging method and frequency response function, and the different parameters effects on the controlled system behavior were examined. Finally, we investigated the comparison between the analytical results with the numerical ones.

A. SYSTEM BEHAVIOR BEFORE CONTROL
The system behavior was studied numerically at the worst resonance cases by considering the following parameters: In figure 2, we introduce the phase plane and time histories of the two modes of vertical conveyor system before control at primary and combined condition 1 ∼ = ω 1 and 2 − Figure 3 simulates the time histories for the two modes of vertical conveyor system after applying the linear and nonlinear proportional-derivative controller at primary and combined resonance 1 ∼ = ω 1 , 2 − 3 ∼ = ω 2 . In this figure, we suppressed the output steady amplitudes from about 3, and 1 to about 0.14, and 0.1755, respectively and the controller reduced the vibrations of the two modes of the controlled system by about 95.33% and 82.45% from its value before controllers, respectively and, the efficiency of the controllers E a are nearly about 22 for x 1 and 6 for x 2 . Figures 4(a, b) show the transfer of energy between the two modes of the vertical conveyor system at primary and combined resonance case 1 ∼ = ω 1 , 2 − 3 ∼ = ω 2 . These figures show that the energy is transferred from uncontrolled system to the system after applying the PD controller.

D. CURVES OF FREQUENCY RESPONSE FOR THE E. CONTROLLED SYSTEM
This section investigates the stability zone, frequency response curves, and several parameters effect of the controlled system. The solid and dot lines refer to the stable and unstable curves respectively, but the yellow region refers to stability zone regions. The output of the controlled system amplitudes decreased with increasing the values of damping µ 1 , control parameters α 3 , and d as shown in figures 5(a, c, e). Furthermore, for  negative and positive values of α 1 and α 4 , figures 5(b, d) depict the jump phenomena, multi solutions, soft and hard spring due to bent the curves to the right and left respectively. As shown in figure 5(f), increasing the value p shifts the curves of the controlled system to the right, which is useful in the controller performance. In addition, the controlled . Transfer of energy between the two modes of system before and after control at primary and combined resonance case system response increased with increasing the values of the amplitude force f 1 as shown in figures 5(g).
The controlled system behavior decreased with the increase of the damping µ 2 and the control parameter d as shown in figures 6(a, b). Also, with the increased values of the control parameter p, the curves are shifted to the right as showed in figure 6(c). Also, the controlled system amplitudes are directly proportional with increasing of the amplitude force f 4 as shown in figures 6(d).

F. COMPARISON OF ANALYTICAL AND NUMERICAL SIMULATION
In this section, we investigated the validation between the numerical simulation for the system equations (1), (2) with perturbation solution of equations (23) and (24) at different values of system parameters p, d, α 3 and α 4 at primary and combined resonance case 1 ∼ = ω 1 , 2 − 3 ∼ = ω 2 as shown in figure 7. The red line indicates the solution of perturbation, while the blue line refers to numerical simulations. In these figures, we observe a good agreement between the analytical results with the numerical ones.

G. COMPARISON WITH PUBLISHED WORK
This section presents a comparison between our work and previous publish works.  a. Bayıroğlu [8] studied the vertical conveyor vibration without any controller at primary resonance and harmonic excitation force via multiple scales method. b. Bayıroğlu [13] presented a mathematical study and nonlinear dynamic analysis of the vertical conveyor oscillations presented in Ref. [8] at primary, subharmonic, and super-harmonic responses. c. EL-Sayed, and Bauomy [16] [19] investigated the vertical conveyor analysis for Ref. [13] with adding multi parametric excitation forces at two different simultaneous sub-harmonic, and combined resonances without any control. g. In our work, we examined nonlinear vibrations analysis and dynamic responses of a vertical conveyor system for Ref. [13] with adding multi tuned excitation forces using the nonlinear proportional derivative (NPD) controller. Also, we investigate how the energy transfers between uncontrolled and controlled system as shown in figure 4. The stability is analyzed by applying the method of averaging. In the numerical results, the controller reduced the vibrations of the two modes of controlled system by about 95.33% and 82.45% from its value before controllers, respectively and, the

IV. CONCLUSION
Vibrating conveyors are widely used in elevators, iron and steel industry, metallurgy industry, chemical plants, feedstock, small parts for processing equipment and production lines to transport a wide range of bulk materials and particles. The nonlinear vibrations analysis and dynamic responses of a vertical conveyor system under multi excitation forces were studied. The approximate solutions and resonance cases of vibrating system were calculated utilizing the method of multiple scales, the approximate solutions of higher orders are very complicated due to the nonlinearity and its degree and therefore software's algorithms should be used. The stability, vertical vibration conveyor behavior was achieved numerically at different parameter values of the system. From the overall study, we concluded: 1. The output steady amplitudes for the two modes of controlled system are nearly about 95.33% and 82.45% from its value before controllers, respectively. 2. The efficiency of the controller E a for the two modes of controlled system is about 22 and 6. 3. The output steady amplitudes of controlled system are monotonous decreasing function in the damping µ 1 , µ 2 , the controller parameters α 3 , d and monotonous increasing function in the amplitude force f 1 , f 4 . 4. The Hopf bifurcations, saddle-node, and jump phenomena were appeared for varying the controller nonlinearity α 1 and α 4 . 5. For the best performance of the controller, the natural frequency ω 1 must be adjusted to the measured 1 value and the natural frequency ω 2 must be adjusted to the measured 2 − 3 value for the two modes of controlled system respectively. 6. Increasing the values µ 1 , µ 2 , d, and α 3 have tightened the energy channel between the controller and the system modes specifically which is useful in the controller performance. 7. Increasing the controller value p, shifts the curves to the right which is a problem for the controller performance. 8. The analytical results are well agreement with the numerical simulations. In future work, we can study the proposed system via multi controller such as, modified positive position feedback (MPPF), nonlinear saturation controller (NSC) and nonlinear proportional integral derivative (NPID) controller. Moreover, the suggested controlled system encourages the experimental studies to present a design algorithm, analysis, and the computational complexity of the controller design.

APPENDIX A
As in Ref. [13]: Apply subsequently Lagrange's equation to obtain action equations for vertical conveyor: where q i and Q i are respectively the coordinates and forces. The kinetic energy is denoted by T which is given as: where m and I z are respectively the trough mass and inertia moment of the conveyor. The potential energy (V ) is given by: where the vertical and angular positions are donated by z and ψ respectively. The nonlinear constants of the vertical and angular springs are given by k 1 , k 2 , k t1 , k t2 . The Rayleigh dissipation function D is where the vertical and angular springs damping constants are denoted by c and c t , respectively. By using the Lagrange's equation for two coordinates q 1 = z and q 2 = ψ, we obtain the following equations of motion as: mz + cż + k 1 z + k 2 z 3 = P z , (A.5) VOLUME 8, 2020 HAMMAD ALOTAIBI received the Ph.D. degree in mathematics from Adelaide University, Australia, in 2017. His areas of research interests include modeling of complex multi scale dynamical systems, atomic simulation, and the numerical and analytical computational methods for solving differential equations.
E. R. EL-ZAHAR received the M.Sc. and Ph.D. degrees in engineering mathematics from Menofia University, Egypt, in 2004 and 2008, respectively. He is currently a Full Professor of engineering mathematics with the Department of Mathematics, Prince Sattam Bin Abdulaziz University. He is the author of more than 70 scientific articles and two textbooks in refereed journals and international conferences. He has served as an editorial board member and a referee for many reputed international journals. His research interests include theory of differential equations and its application, numerical analysis, modeling, numerical, semi-analytical, and computational methods for solving differential equations and engineering systems. VOLUME 8, 2020