Forward Dynamic Solution and Optimization of a 1-DOF Parallel Mechanism for Vibrating Screen

: In this paper, the ordered single-open-chain (SOC) method in combination with the principle of virtual work is adopted to model and solve the forward dynamics of a single degree-of-freedom (DOF) parallel mechanism (PM), featuring one translation and two rotations (1T2R), which is applied in spatial vibrating screen (i.e., parallel vibrating screen mechanism, PVSM). Afterwards, the dynamic performances of the PVSM is optimized using differential evolution algorithm. Based on the kinematics of the PM, the forward dynamic model is derived and the dynamic response equation is built, of which the coefficient matrix is determined by means of the generalized velocity equation. Moreover, the Euler method was adopted to solve the numerical solution of the differential equation of motion to characterize the motion law and dynamics of the screen surface of the PVSM, which is verified with ADAMS simulation. With the parametric model, the dynamic optimization of PVSM is carried out to maximize the energy transfer efficiency, subject to the constraints on the link mass. The comparison of the dynamic performances of the PVSM with and without optimization reveals the improvement of the PM.


Introduction
Modeling and solving the forward dynamics of the  Ju Li wangju0209@163.com 1 Research Center for Advanced Mechanism Theory, Changzhou University, Changzhou 213016, China 2 Dalian University of Technology, Dalian 116024, China mechanism means to reveal the motion law of the mechanism with the prescribed external wrench (forces and/or moments) and resistance force, considering the mass distributions of the links in motion. Forward dynamics can help to study the movement stability of the mechanism for enhanced control [1], for which the crucial problem lies in the establishment of the dynamic response equation of the mechanism. Compared to the inverse dynamics, the forward dynamics solution and analysis of the mechanism are not so extensively reported in the literature, which can benefits to the design and control of the prototype.
The main approaches to model the mechanism dynamics include Roberson-Wittenburg method Error! Reference source not found., Kane method [3], Lagrange equation [4], Lagrange multiplier method Error! Reference source not found., Hamilton method Error! Reference source not found., and Newton-Euler method [7], etc. The previous methods can make the dynamic equations concise and highly stylized, suitable for computer programming to ease the solving procedure. The principle of virtual work is efficient in solving the inverse dynamics problem, while, it not easy to calculate the reaction forces existing in the joints when solving the forward dynamics. From the perspective of energy, the Lagrange equation can be applied in modeling the dynamics through the computation of the instantaneous kinetic and potential energies of all the moving links, but with the increasing computational burden when the mechanism has multiple links. Similarly, the Newton-Euler method results in high computation cost with the increasing number of links, since it requires the force analysis of each separate link of the mechanism to build the dynamic equation by eliminating the internal forces.
Dynamic modeling and analysis of parallel mechanisms (PMs) have been extensively studied in the literature. For instance, based upon the generalized coordinates, ChenError! Reference source not found. et al. applied the Newton-Euler method to model the dynamic response equation of the 6-UPS 1 spatial parallel mechanism. Zhang [9] et al. used the Lagrange method to model the dynamics of an eccentric cam mechanism. Liu [10] et al. adopted Kane's method to analyze the forward dynamics of a 3-RRRT PM. The foregoing modeling methods does not consider the orderly topological decomposition of the PM, leading to the complex derivation of the differential equation of motion and decreasing efficiency of solving dynamic equation, due to the highly nonlinear sets of equations and coupling of the motion parameters. Alternatively, Yang [1] proposed an ordered single-open-chain method based on the principle of virtual work for dynamic analysis of PMs, which has been applied to analyze the dynamics of a 3-DOF 3-RRR PM and a 4-DOF 2SPS-2RPS PM [11]. This paper will handle the dynamic problem in accordance with this method. With the equations of motion, dynamic performances of the mechanisms can be optimized for improvement. Regarding the optimization methods, particle swarm optimization (PSO), genetic algorithm (GA), ant colony algorithm (ACO), immune algorithm (IA), differential evolution algorithm (DE), and other intelligent algorithms [12] can be adopted to optimize the kinematic and dynamic performances of PMs, with the significant advantages in terms of its practicality and reliability [13]- [16].
In this research, forward dynamic analysis of the one degree-of-freedom (DOF) parallel vibrating screen mechanism (PVSM) is carried out with the integrated principle of virtual work and ordered single-open-chain (SOC) method, wherein the coefficient matrices of the dynamic equation can be readily calculated based on the generalized coordinates. Numerical simulation is carried out to characterize the dynamics of the PVSM. Finally, with the differential evolution algorithm, taking the link mass as the constraints and energy transfer efficiency as the objective function, the dynamic performance of the PVSM is optimized for the structural improvement and further prototype design.

Parallel Vibrating Screen Mechanism
The 1-DOF PVSM with one translation and two rotations (1T2R) is shown in Fig.1 [17] [18]. It is composed of hybrid branches (R1-R2-R3-R4)-R5 and spatial branches S6-S7 , both of which are respectively connected to the 1 Throughout this paper, P, R, U, S and T stand for the prismatic, revolute, universal, spherical and Hooke joints, respectively. moving platform 4 and the base platform 0. Ref. [17] has proved that the PM DOF is equal to 1. The rotating joint R1 is active, and the moving platform 4 can generate translation along the z-axis, rotation  around the x axis, and rotation β around the axis of rotation of joint R5 (i.e., parallel to line R2R3), among which only one of the motion parameters (z, α, β) can be considered as an independent parameter with couple motions of the other two.

Determination of Velocity and Acceleration Coefficient Matrix
In accordance with Ref. [18], the angular/linear velocity and acceleration of the center of mass of each link can be expressed as a function of generalized velocity q & and acceleration q && , namely, () Thus, the velocities of the parallel vibrating screen is expressed as Where,

Establishment of the Forward Dynamic Equation
With the principle of virtual work [13]- [16], the sum of the external force/moment) and inertial force/moment of the system on any virtual displacement of the mechanical system should be equal to zero, which yields Where, 1 5 , ,...., 1 5 , ,..., ( ) 0 And the elements in the matrices H , () Pq & ,and Q are given by

Determination of the Coefficients of the Dynamic Response Equation
The key issue for Eq. (6)

Numerical Solution and Verification of Forward Dynamic Equation
By setting the motion parameters to be special values, the elements in the coefficient matrix of the differential equation of motion (6) can be determined. In order to figure out the motion law of each link of the PM, numerical analysis can be adopted to solve the differential equation of motion. The Common methods to solve differential equations include Euler method and RUNGE-KUTTA method [24]. The Euler method features smaller computation burden, which can provide acceptable computation accuracy with smaller increments, which is adopted in this work. The basic principle is depicted as: at the initial condition t=t0, the initial values of the generalized coordinates and generalized speed of the vibrating screen are 0 qq  Sequentially, the generalized speed 1 t q & and generalized coordinates 1 t q at the moment 10 t t t t     can be deduced, which can be treated as the initial conditions for the next moment. Repeating the previous procedures results in the motion law of each link of the PM.
In order to calculate Eq. (6), the parameters of the PM are set in accordance with Ref. [18]. The constant torque acting on the drive joint of the PM in the counterclockwise is T=50(N·mm). The initial angular input position is qt0=1.2 (rad), and the initial velocity is 1 0  t q  (rad/s). Based on Eq.(6), with the step size 10 -4 , the motion profiles of parameters q , q & and q && during the period t=0~0.15s (60°~146°) are calculated, as shown in Fig. 2, whlile the simulation results by ADAMS are shown in Fig. 3. (2) The input angle and speed gradually increase with time, while the input acceleration changes suddenly and sharply after a decrease.
Because the word mainly investigates the motion law of the vibrating screen surface (moving platform is 4), the results obtained are substituted into the displacement, velocity, and acceleration equations of the center of mass of the moving platform 4, and calculated with MATLAB programming, and the results are shown in Fig.4(a)~Fig.8(a).
Then use the ADAMS software for simulation analysis, and get the corresponding curve of displacement, velocity and acceleration of the center of mass of the moving platform 4, as shown in Fig. 4 (b) ~ Fig. 8 (b). (2) Velocity/angular velocity, acceleration/angular acceleration fluctuate greatly in a short period of time, which provides protection for the screening efficiency of the vibrating screen. However the drastic fluctuation of velocity/acceleration in a short period of time can easily cause vibrations in each joint, which results in increased mechanical vibration of the PVSM, which requires optimization of the driving auxiliary input to improve the working stability of the PVSM.

Determine Optimization Parameters and Objective Function
(1) Design variables The previous work Ref. [17] has presented the kinematic optimization of the 1-DOF parallel vibrating screen mechanism (PVSM), such as the dimensions, the trajectory of the screen surface, and the equilibrium of the four-bar linkage. On the other hand, optimization of dynamic performances such as mass parameters and energy transfer efficiency can be considered for further improvement.
In this section, the mass parameters of the links will be optimized based on previously determined link lengths.
Suppose m1, m2, m3, m4, and m5 are the masses of the five links of the PVSM, therefore, the design variables are written as m m m m m  m (16) (2) Optimization constraints According to the original masses of each link, the upper and lower bounds of the design variables are given in Table 1. Then the design variables in the optimization process should meet the overall constraint as follows. For the whole PM, the kinetic energy of the moving platform can be considered as the effective energy of the PM, and the higher the proportion of kinetic energy occupied by the moving platform, the better the energy transmission effect of the PM. In order to evaluate the overall energy transfer performance of the organization, the global energy transfer efficiency index fp is defined as It is calculated that the energy transfer efficiency of the mechanism is 0 p f under the original mass parameters.
From this, the objective function fd for dynamic performance optimization is calculated as

Optimization Analysis and Comparison
After defining the optimization problem, the differential evolution algorithm compiled in MATLAB can be used to optimize the dynamic performance of the PM, where the algorithm parameters are selected according to the Refs. [26] and [27].
(1) Algorithm parameter initialization To reduce the computational cost, the population size and iteration times of the algorithm are reduced The parameter settings shown in Table 2 are obtained from the preliminary estimation. (2) Comparison and analysis of optimization results Figure 9 shows the iterative optimization, from which it can be seen that the objective function begin to converge after 50 iterations, and the final solution is obtained, in comparison with the evaluation index of the PM without optimization, as listed in Table 3.  It can be seen from Table 3 that after optimization, the mass of the driving link and the driven link are significantly reduced, the mass of the moving platform is significantly increased. It is noteworthy that the energy transfer efficiency increases by 57.1%.

Conclusions
(1) This paper presents dynamic analysis of a 1-DOF parallel vibrating screen mechanism using the ordered SOC method and the principle of virtual work, for which the concise equation of motion is derived, wherein the coefficient matrices of the dynamic equation can be readily calculated based on the generalized coordinates.

Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.