Synthesis and Functional Optimization of a Vibratory Machine with a Parallel Mechanism Structure

Abstract

Vibratory machines are widely used for the separation of granular materials of various densities and characteristics. Vibrations are used in this case to increase the technological speed of separation of these materials. Vibratory machines are limited in functioning to fixed oscillation patterns. Control of the oscillation can offer the possibility of changing the oscillation pattern, adapting it to the type of granular material. This paper proposes a vibratory machine with controllable elastic characteristics and pneumatic actuation. From the perspective of functional generalization of these machines, a parallel mechanism with 3 degrees of mobility, starting from the study of the Stewart platform, was designed. The proposed system has been studied both in terms of structure and kinematics, using instruments of simulation tools available in MATLAB and Simulink 2021. The developed model was validated and optimized using multi-body dynamic software ADAMS 2014, obtaining optimal parameters for one type of granular material.

1. Introduction

Recent developments in engineering sciences are often based on pneumatic devices and vibratory systems, even though they are not always applied in the same context. Some of the research works in these fields are presented below.

In one study [1], an origami-based structure was introduced as a pneumatic device. The material used to fabricate the pneumatic device is made of a hyper-elastic but also soft and non-stretchable material. The most important characteristics of the application of these techniques and materials are that they facilitate an asymmetric geometry with a quadrilateral configuration chamber and an elastic plate. Regarding the materials used in different experiments, some complex structured materials with adjustable parameters were found. The experiments presented in [2] show that these materials respond well to the changes in pressure, with an initial range of frequencies from 29.6 to 145.83 Hz and a five-fold range in the whole domain.

Another subject is tackled by the study published in [3], where the motion of different types of rough particles was measured, which are moving in a pipe, a movement which can be assimilated with a flow in different layers. The results show that the conveying conditions are not as important as the rate of decreasing pressure, at least in the case of stratified flow, sedimentation, and dune formation. In reference [4], in order to control a vibration isolation system to eliminate the possibility of resonance and other negative effects of vibrations, a specific control strategy was proposed based on the H∞ active control theory.

Regarding the control systems [5] of pneumatic devices with disturbances, some researchers studied pneumatic devices with disturbance observers. Reference [6] investigated the use of continuous pneumatic jigging as an efficient technique of solid waste separation, with a focus on the different materials of copper wire and rubber insulator. The research investigated the effect of several parameters on separation efficiency, such as air flow, pulse rate, vibrational force, and bed thickness.

Works [7,8] presented general aspects in studying the vibration of elastic systems. In reference [9], the control of friction-driven oscillations using an impact damper was investigated. An experimental setup of a single-degree-of-freedom friction-driven oscillator with an attached impact damper was designed and fabricated. Reference [10] concerns the formulation and constitutive equations of finite-strain viscoelastic material using multiplicative decomposition in a thermodynamically consistent manner. References [11,12] concerned the above elastic and damping parameters for vibration insulation systems and an evaluation of the level of performance for vibrating screens.

In reference [13], an analysis of the oscillatory conveyor separator was conducted. In order to increase the domain of operation of vibratory machines and also to improve their functional capabilities, the authors in work [14] proposed a new machine structure which uses pneumatic cylinders as actuators while at the same time supporting elements with controllable mechanical characteristics. The scope of the research presented in paper [15] included establishing the parametric performance of the inertia vibrating screens used in sorting the mineral aggregates extracted from water basins (river gravel) or from stone carriers. In reference [16], the influences upon the quality of the mineral aggregates induced by technological vibrations during the sorting process were studied. Reference [17] presents a numerical analytical approach with discrete and continuous parametric variations, from which favorable areas of operation can be established. In this way, the optimization criteria in stabilized harmonic vibration regimes are approached based on an assessment of the vibration amplitude of the force transmitted to the processed material and of the energy dissipated in the system.

Article [18] presented the synthesis of design parameters of a two-frequency inertial vibrator according to the specified power characteristics. Based on the developed mathematical model, the parameters of the variable periodic force are derived for two angular velocities, 157, 314 rad/s, and their ratios, 0.5 and 2. Study [19] developed a new process model for vibrating screens through the combination of discrete element method (DEM) simulation and physics-informed machine learning. In the petroleum exploitation industry, the vibration isolation technology of the traditional vibration screen is difficult to satisfy regarding the security requirement of industrial production; therefore, the anti-resonance system of three exciters is proposed in work [20]. In paper [21], the optimization of the characteristics, the parameters of shock dampers, and dampers for industrial vibrating machines is discussed. Reference [22] discusses the introduction of virtual prototypes since the conceptual design stage of “Virtual Concepts”, in which the coarse models of machinery design variants are simulated to interactively evaluate several solutions and support the best design choices.

The discrete element method (DEM) allows for the calculation of the dynamics of the ore. In paper [23], two 2D three-degrees-of-freedom dynamic models for a vibrating screen are tested, using linear and nonlinear approaches for angular displacement. In reference [24], the aim was to simulate the process from a more operational perspective to evaluate process performance and process optimum for different operations. The objective is to model and simulate the discrete phenomena that can cause the process to alter performance and implement it with dynamic process simulations. In reference [25], the combined effects of vibration parameters on the circular screening processes are analyzed based on the Taguchi orthogonal experiment method. In reference [26], a numerical study of the motion of particulates following a circularly vibrating screen deck was performed using the three-dimensional discrete element method (DEM). A deeper understanding of the screening performance of banana screens helps mineral processing engineers to control and optimize them. In article [27], the screening performance of banana screens using a discrete element method (DEM) solver ‘LIGGGHTS’ was investigated.

After analyzing the existing vibrating machines from the point of view of the flexibility of their field of application, the following conclusions were reached [7]:

−

the working body of these machines has a movement imposed by the rigors of the process in which it participates, namely, that of separating or classifying polygranular materials on site; however, this was limited for each type of machine and its structure [7,8].

−

in order to be driven in vibratory motion, the working body of the machine is restrained or suspended with elastic systems with non-adjustable characteristics. This limits the machine’s range of use.

−

the elasticity and damping of the restraint systems by their construction are conditioned by the elasticity of the restraining elements, which changes uncontrollably with the ageing of the restraining material.

Although it is possible to adjust the parameters of the vibration generators (amplitude and frequency), it is not possible to correlate them with the parameters of the system of the resonator. This fact leads to the operation in a different regime from the optimal one [8].

Aiming to solve the above-mentioned problems, a new, flexible solution based on actuators and pneumatic control systems is proposed [10,11].

Three aspects have been considered to ensure the operation of the developed pneumatic system, as follows:

−

technologically correct positioning of the working organ of the machine so that it participates in the process of separation-classification of the polygranular material—positioning function;

−

to support the vibratory motion required by the process by means of the elastic damping effect, without inappropriate changes in position during operation and without changes in the elastic and damping parameters, but with the possibility of being controlled—the elasticity and damping control function;

−

have a simple construction such that, as far as possible, the elastic positioning and restraining element can support, generate, and control, through its construction, the desired vibratory motion—the motion control function.

Research has shown that pneumatic cushioning elements can be successfully used in vibrating machines, and that the pneumatic cushion has adjustable elasticity by means of pressure [11,12,13,14].

A prototype of a vibrating machine containing a controllable tensioning system was realized and tested.

As can be seen, there are some engineering solutions, among other applications, which are based on pneumatic devices but are only used as vibration isolation systems. Research that is focused on vibration generation using pneumatic devices is very hard to find, even if pneumatic devices are broadly used in a large number of industries for other purposes.

2. Description of the Structure of the Mechanism

The Stewart platform is a parallel robot with 6 degrees of mobility, built with six elements connected to each other by a platform. Each element has two spherical couplings and a translation coupling. We can imagine the complex motion of the platform within the 6 degrees of mobility as follows: three translations along the three directions of the Oxyz system and three rotations around the three axes of the same system [15,16,17,18].

The motion of a vibrating platform presents rapidity under the conditions of several motion cycles per second [19,20,21]. Therefore, the mechanical system is rethought with a restricted number of degrees of mobility due to the characteristics of the separation process of the polygranular material. The number of degrees of mobility is reduced to 3—two translations along the Ox and Oy directions in the plane and one rotation around the Oz axis. This restriction is also due to the speed at which the movement is performed.

This new mechanism structure is shown in Figure 1 below.

Four of the translation couplings are locked by the air pressure inside the piston cylinders and are considered to be the elastic restraining elements of the platform. The other two translation couples are pressurized with variable pressures and are considered as actuating systems of the mechanism. Therefore, the mechanism performs a plane-parallel motion in the xOy plane: two translations and one rotation, also with four links, whose stiffness can be variable [21,22].

3. Mathematical Model of the Structure of the Mechanism

The vibratory motion of the working organ is characterized by the following equations:

$$x=x_a.\cos (\omega .t+{\varphi }_x); y=y_a.\cos (\omega .t+{\varphi }_y).$$

(1)

where:

$$\begin{array}{l}{x}_a={\displaystyle \frac{F_0.\cos \alpha }{m\sqrt{{\left(p_x^2-{\omega }^2\right)}^2+4n_x^2{\omega }^2}}}; tg{\varphi }_x={\displaystyle \frac{2n_x\omega }{p_x^2-{\omega }^2}};\\ y_a={\displaystyle \frac{F_0.\sin \alpha }{m\sqrt{{\left(p_y^2-{\omega }^2\right)}^2+4n_y^2{\omega }^2}}}; tg{\varphi }_y={\displaystyle \frac{2n_y\omega }{p_y^2-{\omega }^2}}.\end{array}$$

(2)

In which the following is noted:

$$p_x=\sqrt{{\displaystyle \frac{k_x}{m}}}; n_x={\displaystyle \frac{c_x}{2m}}; p_y=\sqrt{{\displaystyle \frac{k_y}{m}}}; n_y={\displaystyle \frac{c_y}{2m}}.$$

(3)

where $p_x,p_y$ are the eigenpulsations in the Ox and Oy directions, respectively, $n_x,n_y$ are the damping factors in the Ox and Oy directions, respectively, $k_x,k_y$ are the stiffness constants in the Ox and Oy directions, respectively, $c_x,{ c}_y$ are the damping coefficients in the directions of Ox and Oy, respectively, $m$ is the mass of the working body, $F_0$ is the amplitude of the exciting force, ${\phi }_x,{\phi }_y$ are the phase shifts of the motion in the directions of Ox and Oy, respectively, and $\omega$ is the pulsation of the exciting force.

The equation of the trajectory of the center of mass (CM) with respect to the center of oscillation (CO), due to the shape of Equation (1), with the same pulsation ω in both directions, is a Lissajous curve of elliptic shape and has the following equation:

$${\displaystyle \frac{x^2}{x_a^2}}+{\displaystyle \frac{y^2}{y_a^2}}-{\displaystyle \frac{2xy}{x_a{y}_a}}\cos ({\varphi }_x-{\varphi }_y)={\sin }^2({\varphi }_x-{\varphi }_y)$$

(4)

Angle of incline ${\alpha }_1$ of the ellipse with reference to the $Ox$ axis is given as follows:

$$tg2{\alpha }_1={\displaystyle \frac{2x_a{y}_a\cos \left({\varphi }_x-{\varphi }_y\right)}{x_a^2-y_a^2}}$$

(5)

Amplitudes of vibration in directions $Ox, Oy$, respectively, are given as follows:

$$a={\displaystyle \frac{x_a{y}_a\left|\sin ({\varphi }_x-{\varphi }_y)\right|}{\sqrt{\frac{1}{2}(x_a^2+y_a^2)-\frac{1}{2}(x_a^2-y_a^2)\cos 2{\alpha }_1-x_a{y}_a\cos ({\varphi }_x-{\varphi }_y).\sin 2{\alpha }_1}}}$$

(6)

$$b={\displaystyle \frac{x_a{y}_a\left|\sin ({\varphi }_x-{\varphi }_y)\right|}{\sqrt{\frac{1}{2}(x_a^2+y_a^2)+\frac{1}{2}(x_a^2-y_a^2).\cos 2{\alpha }_1+x_a{y}_a\cos ({\varphi }_x-{\varphi }_y).\sin 2{\alpha }_1}}}$$

(7)

In Figure 2, the trajectory of the center of mass (CM) of the working element of the mechanical system, the amplitudes of the motions a, b, and the inclination ${\alpha }_1$ relative to the direction of the generating force and the inclination $\gamma$ of the plane of the working element (the plane of a site) are presented [22,23].

What needs to be noted for such mechanical systems is that they operate only in pre-resonant or post-resonant regime so that $\omega$ is smaller or larger than $p_{x,y}$ and the trajectory parameters depend on the intrinsic parameters of the system, as well as the stiffness coefficients k_x,y or the damping coefficients $c_{x,y}$ and, in particular, on their rate, which determines the values for the angle ${\alpha }_1$.

4. Description of Mechanism Operation

The Stewart platform is a parallel robot with 6 degrees of mobility, built with six elements connected to each other by a platform. Each element has two spherical couplings and a translation coupling. We can imagine the complex motion of the platform within the 6 degrees of mobility as follows: three translations along the three directions of the Oxyz system and three rotations around the three axes of the same system.

The motion of a vibrating platform has the characteristic of rapidity under the conditions of several motion cycles per second. Therefore, the mechanical system is rethought with a restricted number of degrees of mobility due to the characteristics of the separation process of the polygranular material. The number of degrees of mobility is reduced to 3:2 translations along the Ox and Oy directions in the plane and one rotation around the Oz axis. This restriction is also due to the speed at which the movement is performed.

This new mechanism structure is shown in the Figure 3 below.

The mechanism performs a plane-parallel motion in the $xOy$ plane: two translations and one rotation, also with four links, whose stiffness can be variable. Two translation couples are pressurized with variable pressures and are considered as actuating systems of the mechanism, while four of the translation couplings are locked by the air pressure inside the piston cylinders and are considered to be the elastic restraining elements of the platform [23,24].

5. Develop the Numerical Model of the Mechanism

The result is the strokes of the pistons of the pneumatic drive motors and the angles of inclination of the drive motors [24,25,26].

Starting from the defined structure of the mechanical system, a MATLAB program was designed to allow the calculation of the strokes of the pistons in pneumatic cylinders and the calculation of their inclination angles, as well as their speeds and accelerations based on the inverse kinematic analysis of the mechanism [27].

MATLAB with Simulink software was used for the inverse kinematics of the machine mechanism. We modeled the pneumatic pistons of the platform with the right segments, with dimensions corresponding to different positions as shown in Figure 4. The input data are represented by the parameters of the trajectory curve of the center of mass of the platform as follows: elliptic Lissajous curve with semi-axes $x_a=2.5 \mathrm{m}\mathrm{m}$ and $y_a=0.5 \mathrm{m}\mathrm{m}$.

The MATLAB program has been designed to render the simplified geometry of the mechanical system as it results from the calculations and graphics presented.

For the input data of the movement of the platform mechanism, we have the following constructive parameters: $l_0$—the initial length of the driving element, $h_0$—the stroke of the driving piston, $p_0$—the position of the base of pistons 2 and 5, $p$—the position of the base of pistons 1, 3, 4, and 6, $H_0$—the height of the initial position of the platform according to Figure 4. Schematic image in MATLAB is presented in Figure 5.

Platform not at rest (initial rest):

$${\left(l_0-h_0\right)}^2={\left(p-p_0\right)}^2+H_0^2$$

(8)

Raised platform (with pistons at 45°):

$$l_0^2={\left(p-p_0\right)}^2+H^2$$

(9)

$$H^2=l_0^2-{\left(p-p_0\right)}^2⟹H=\sqrt{l_0^2-{\left(p-p_0\right)}^2}=\sqrt{H_0^2+h_0\left(2l_0-h_0\right)}$$

(10)

Platform raised and inclined with angle γ:

$$\sin \gamma ={\displaystyle \frac{∆H}{p}}⟹∆H=p·\sin \gamma$$

(11)

(for pistons 4, 5, 6).

Coordinates of the points in Figure 4 are given below:

G₂(−p₀,0); G₁(−p,0); Ω(0,0); G₅(p₀,0); G₄(p,0)

(12)

M₂(−p,H₀); M₁(−p₀,H₀); O₀(0,H₀); M₄(p₀,H₀); M₅(p,H₀)

(13)

P₂(−p,H); P₁(−p₀,H); O(0,H); P₄(p₀,H); P₅(p,H)

(14)

$$I_2(x_{I_2},z_{I_2}); I_1(x_{I_1},z_{I_1}); O(0,H); {I_4\left(x_{I_4},z_{I_4}\right); I}_5(x_{I_5},z_{I_5})$$

(15)

$$x_{I_2}=-p·cos\gamma ; z_{I_2}=H+∆H=H-p·\sin \gamma$$

(16)

$$x_{I_1}={-p}_0·cos\gamma ; z_{I_1}=H+∆H=H-p_0·\sin \gamma$$

(17)

$$x_{I_4}=p_0·cos\gamma ; z_{I_4}=H+∆H=H+p_0·\sin \gamma$$

(18)

$$x_{I_5}=p·cos\gamma ; z_{I_5}=H+∆H=H+p·\sin \gamma$$

(19)

Platform edges circle of the points M₁ and M₄:

$$x^2+{\left(z-H_0\right)}^2=p^2$$

(20)

Platform inner circle of the points I₁ and I₄:

$$x^2+{\left(z-H\right)}^2=p_0^2$$

(21)

intersected with the platform line:

$$z-H=0$$

(22)

and then with the line ${\displaystyle \frac{x}{a}}={\displaystyle \frac{z}{b}}$, which must pass through the following points:

Positioning and repositioning of piston-cylinder 5

$$G_5{I}_5=\sqrt{{\left(x_{I_5}-x_{G_5}\right)}^2+{\left(z_{I_5}-z_{G_5}\right)}^2}=\sqrt{{\left(p·cos\gamma -p_0\right)}^2+{\left(H+p·\sin \gamma \right)}^2}$$

(23)

$$G_5{P}_5=\sqrt{{\left(x_{P_5}-x_{G_5}\right)}^2+{\left(z_{P_5}-z_{G_5}\right)}^2}=\sqrt{{\left(p-p_0\right)}^2+H^2}$$

(24)

$$G_5{M}_5=\sqrt{{\left(x_{M_5}-x_{G_5}\right)}^2+{\left(z_{M_5}-z_{G_5}\right)}^2}=\sqrt{{\left(p-p_0\right)}^2+H_0^2}$$

(25)

Positioning and repositioning of piston-cylinders 4 and 6

$$G_4{I}_4=\sqrt{{\left(x_{I_4}-x_{G_4}\right)}^2+{\left(z_{I_4}-z_{G_4}\right)}^2}=\sqrt{{\left(p_0·cos\gamma -p\right)}^2+{\left(H+p_0·\sin \gamma \right)}^2}$$

(26)

$$G_4{P}_4=\sqrt{{\left(x_{P_4}-x_{G_4}\right)}^2+{\left(z_{P_4}-z_{G_4}\right)}^2}=\sqrt{{\left(p_0-p\right)}^2+H^2}$$

(27)

$$G_4{M}_4=\sqrt{{\left(x_{M_4}-x_{G_4}\right)}^2+{\left(z_{M_4}-z_{G_4}\right)}^2}=\sqrt{{\left(p_0-p\right)}^2+H_0^2}$$

(28)

Positioning and repositioning of piston-cylinders 1 and 3

$$G_1{I}_1=\sqrt{{\left(x_{I_1}-x_{G_1}\right)}^2+{\left(z_{I_1}-z_{G_1}\right)}^2}=\sqrt{{\left({-p}_0·cos\gamma +p\right)}^2+{\left(H-p_0·\sin \gamma \right)}^2}$$

(29)

$$G_1{P}_1=\sqrt{{\left(x_{P_1}-x_{G_1}\right)}^2+{\left(z_{P_1}-z_{G_1}\right)}^2}=\sqrt{{\left({-p}_0+p\right)}^2+H^2}$$

(30)

$$G_1{M}_1=\sqrt{{\left(x_{M_1}-x_{G_1}\right)}^2+{\left(z_{M_1}-z_{G_1}\right)}^2}=\sqrt{{\left({-p}_0+p\right)}^2+H_0^2}$$

(31)

Positioning and repositioning of piston-cylinder 2

$$G_2{I}_2=\sqrt{{\left(x_{I_2}-x_{G_2}\right)}^2+{\left(z_{I_2}-z_{G_2}\right)}^2}=\sqrt{{\left(-p·cos\gamma +p_0\right)}^2+{\left(H-p·\sin \gamma \right)}^2}$$

(32)

$$G_2{P}_2=\sqrt{{\left(x_{P_2}-x_{G_2}\right)}^2+{\left(z_{P_2}-z_{G_2}\right)}^2}=\sqrt{{\left(-p+p_0\right)}^2+H^2}$$

(33)

$$G_2{M}_2=\sqrt{{\left(x_{M_2}-x_{G_2}\right)}^2+{\left(z_{M_2}-z_{G_2}\right)}^2}=\sqrt{{\left(-p+p_0\right)}^2+H_0^2}$$

(34)

Relation between the coordinates of the piston-cylinder in an inclined position

$$G_5{I}_5=\sqrt{{\left(x_{I_5}-x_{G_5}\right)}^2+{\left(z_{I_5}-z_{G_5}\right)}^2}=\sqrt{{\left(p·cos\gamma -p_0\right)}^2+{\left(H+p·\sin \gamma \right)}^2}$$

(35)

$$G_4{I}_4=\sqrt{{\left(x_{I_4}-x_{G_4}\right)}^2+{\left(z_{I_4}-z_{G_4}\right)}^2}=\sqrt{{\left(p-p_0·cos\gamma \right)}^2+{\left(H+p_0·\sin \gamma \right)}^2}$$

(36)

$$G_1{I}_1=\sqrt{{\left(x_{I_1}-x_{G_1}\right)}^2+{\left(z_{I_1}-z_{G_1}\right)}^2}=\sqrt{{\left({p-p}_0·cos\gamma \right)}^2+{\left(H-p_0·\sin \gamma \right)}^2}$$

(37)

$$G_2{I}_2=\sqrt{{\left(x_{I_2}-x_{G_2}\right)}^2+{\left(z_{I_2}-z_{G_2}\right)}^2}=\sqrt{{\left(p·cos\gamma -p_0\right)}^2+{\left(H-p·\sin \gamma \right)}^2}$$

(38)

Relation between the coordinates of the piston-cylinder in horizontal position

$$G_5{P}_5=\sqrt{{\left(x_{P_5}-x_{G_5}\right)}^2+{\left(z_{P_5}-z_{G_5}\right)}^2}=\sqrt{{\left(p-p_0\right)}^2+H^2}$$

(39)

$$G_4{P}_4=\sqrt{{\left(x_{P_4}-x_{G_4}\right)}^2+{\left(z_{P_4}-z_{G_4}\right)}^2}=\sqrt{{\left(p_0-p\right)}^2+H^2}$$

(40)

$$G_1{P}_1=\sqrt{{\left(x_{P_1}-x_{G_1}\right)}^2+{\left(z_{P_1}-z_{G_1}\right)}^2}=\sqrt{{\left({p-p}_0\right)}^2+H^2}$$

(41)

$$G_2{P}_2=\sqrt{{\left(x_{P_2}-x_{G_2}\right)}^2+{\left(z_{P_2}-z_{G_2}\right)}^2}=\sqrt{{\left(p-p_0\right)}^2+H^2}$$

(42)

Relation between the coordinates of the piston-cylinder in the repause position

$$G_5{M}_5=\sqrt{{\left(x_{M_5}-x_{G_5}\right)}^2+{\left(z_{M_5}-z_{G_5}\right)}^2}=\sqrt{{\left(p-p_0\right)}^2+H_0^2}$$

(43)

$$G_4{M}_4=\sqrt{{\left(x_{M_4}-x_{G_4}\right)}^2+{\left(z_{M_4}-z_{G_4}\right)}^2}=\sqrt{{\left({p-p}_0\right)}^2+H_0^2}$$

(44)

$$G_1{M}_1=\sqrt{{\left(x_{M_1}-x_{G_1}\right)}^2+{\left(z_{M_1}-z_{G_1}\right)}^2}=\sqrt{{\left({p-p}_0\right)}^2+H_0^2}$$

(45)

$$G_2{M}_2=\sqrt{{\left(x_{M_2}-x_{G_2}\right)}^2+{\left(z_{M_2}-z_{G_2}\right)}^2}=\sqrt{{\left(p-p_0\right)}^2+H_0^2}$$

(46)

A new mechanical system has been modeled in order to be used to cover several usable cases of motion to create an optimal granular material separation process.

This new type of system replaces the bearings with six pneumatic piston-cylinder subassemblies, totaling 12 mechanism elements and 6 cylindrical couplings. These subassemblies are assembled in a structure containing 11 rotating couplings, 1 spherical coupling, and a platform, which is also the working element of the system.

The purpose of four of the piston-cylinder subassemblies is because the air cushion-type bearings have variable elastic and damping properties capable of replacing the restraining elements with constant properties.

The other two piston-cylinder subassemblies are controlled actuators, generating forces in order to achieve the desired regime of motion.

6. Dynamic Model of the Mechanism

The peculiarity of this system is that the generating forces act on the working element (platform) of the vibrating mechanical system, not on the center of mass (CM), but at its extreme points along the Ox axis in the couplings Joint_2pl and Joint_5pl. The diagram of the generating forces $F_2\left(t\right), F_5\left(t\right)$ and the reaction forces $F_1, F_3, F_4, F_6$ of the piston-cylinder subassemblies are shown in Figure 6.

The structure components are shown in Table 1. The pneumatic piston-cylinder subassemblies are considered to be translating torques. Four of the six piston-cylinder subassemblies (Φ32) are pressurized with compressed air and considered as elastic spring elements with variably controlled stiffness. The other two piston-cylinder subassemblies (Φ40) have the functionality of pneumatic actuators.

Table 1. Components in Figure 7.

Nr.	Component Name	Pcs
1	Platform plate 850 × 400 × 12.5 mm	1
2	Platform-actuator fixed element Φ32	4
3	Platform-actuator mobile element Φ32	4
4	Platform-actuator fixed element Φ40	2
16	Platform-actuator mobile element Φ40	2
17	Piston actuator Φ32	4
9	Actuator cylinder Φ32	4
10	Piston actuator Φ40	2
18	Actuator cylinder Φ40	2
15	Base plate 850 × 400 × 50 mm	1

Table 1. Components in Figure 7.

Nr.	Component Name	Pcs
1	Platform plate 850 × 400 × 12.5 mm	1
2	Platform-actuator fixed element Φ32	4
3	Platform-actuator mobile element Φ32	4
4	Platform-actuator fixed element Φ40	2
16	Platform-actuator mobile element Φ40	2
17	Piston actuator Φ32	4
9	Actuator cylinder Φ32	4
10	Piston actuator Φ40	2
18	Actuator cylinder Φ40	2
15	Base plate 850 × 400 × 50 mm	1

7. Functional Optimization of the Vibrating System

For the functional optimization of the mechanical vibrating system, the MD-ADAMS model was developed (Figure 8).

Based on the constructive parameters determined through inverse kinematics performed in MATLAB and MD-ADAMS, a model was developed. The modeled mechanical system has the following characteristics:

Total mass of the moving part is as follows: m = 12,078 kg, angles ${\beta }_{\mathrm{2,5}}=41.42367°$, ${\beta }_{\mathrm{1,4}}=43.667781°$, stiffness constants for each cylinder-piston subassembly $k_{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}}=100 \mathrm{N}/\mathrm{m}\mathrm{m}$, damping constants for each cylinder-piston subassembly $c_{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}}=1.5 \mathrm{N}/\mathrm{m}\mathrm{m}$.

The actuating forces of cylinder-piston subassemblies 2 and 5 are as follows:

$$F_{2,5}(t)=F_0.(\sin (\omega .t\mp {\varphi }_f)+1)=600.(\sin (2.\pi .16.t\mp 0.5397.\pi )+1)$$

(47)

ADAMS model is simulated in dynamic mode under the action of actuating forces (47), and the menu of ADAMS software permits the monitoring of different kinematic parameters, such as the position of the center of mass (CM) of the platform: Platform_Xpozition and Platform_Ypozition. ADAMS software has a function of optimization, which implies defining an objective function and establishing the variables for design that influence the objective function. The objective function Maxim Platform_Xpozition was defined to ensure sufficient time for granular material to separate, which results in an elliptical trajectory with the major semi-axis maximal.

The established design variables, with their names in ADAMS, in order to achieve optimization, are as follows:

• .model_1.DV_1—amplitude of force $F_0$,
• .model_1.DV_2—the offset between the actuating forces $F_2$ and $F_5$,
• .model_1.DV_3—stiffness constants for each cylinder-piston sub-assembly k_1,2,3,4,5,6,

The default values for the variables are as follows:

• .model_1.DV_1 = 650 [N],
• .model_1.DV_2 = 0.5*pi [rad],
• .model_1.DV_3 = 100 [N/mm],

During the simulation, the values of the three design variables vary around the nominal values with a relative percentage range from −30 to +30 relative to the nominal values. As a consequence of the optimization process, ADAMS indicates the values of the variables for which the objective function is maximized, which means that Platform_Xpozition is the maximum (

Table 2. Design of experiment summary.

Trial	Objective Function	DV_1	DV_2	DV_3
118	1.4948	850.00	1.6997	100.00

).

The results are represented graphically in Figure 9, Figure 10, Figure 11 and Figure 12 and in Table 2.

As a consequence of the optimization, the optimal values for the design values (model.DV_1, model.DV_2, model.DV_3) for which the major axis of the ellipse, Platform Xpozition, is the maximum. With these new values, a simulation of the ADAMS model was realized, and the optimal trajectory of the center of mass (CM) of the platform is presented in Figure 13.

8. Discussion

In the field of vibrating machines, two approaches are distinguished: one approach is about machine dynamics [18,28,29,30], and the other is about the dynamics of granular material [31,32].

The structure of the new vibratory mechanical system is similar to the parallel structure of a Stewart platform, permitting 3 degrees of mobility, which ensures a motion in a vertical plane along a Lissajous-type elliptic trajectory due to the equality of pulsation between horizontal motion and vertical motion.

Based on the constructive parameters determined through inverse kinematics performed in MATLAB and MD-ADAMS, a model was developed. The modeled mechanical system has the following characteristics:

Total mass of the moving part is: m = 12,078 kg, angles ${\beta }_{\mathrm{2,5}}=41.42367°$, ${\beta }_{\mathrm{1,4}}=43.667781°$, stiffness constants for each cylinder-piston subassembly $k_{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}}=100 \mathrm{N}/\mathrm{m}\mathrm{m}$, damping constants for each cylinder-piston subassembly $c_{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}}=1.5 \mathrm{N}/\mathrm{m}\mathrm{m}$.

The actuating forces of cylinder-piston subassemblies have been concluded to be in Equation (47).

The possibility of the adjustment of the elastic characteristics of the elastic supporting system, with the help of the four cylinder-piston subsystems, assures the flexibility of adjustment of the vibratory regime by adjusting the pneumatic pressure inside the chambers of the cylinders. This possibility eliminates the situation in which the elastic characteristics are depreciating in time and also allows the possibility of positioning the platform at an angle specific for granular material separation, for which the system is used.

The pneumatic actuating system, assured by the two-cylinder-piston subsystems in the middle, also provided by a variable regime of pressure, allows the possibility to adjust the vibratory motion.

Theoretical and practical observations revealed that the positioning of the six-cylinder-piston subsystems has to be adjusted at 45° angles or close to this value.

In recent years, and in many applications in different industries, pneumatic devices equipped with control systems have been used to achieve the positioning and controlled speed of mechanical parts. Such devices are not used to produce controlled vibratory or oscillatory motions. This is the reason why we consider that the solutions described in this paper represent a new, viable approach. A prototype of the described vibrating mechanical system has been constructed and laboratory tested, and will be presented in a future paper together with the experimental results.

9. Conclusions

A mechanism for a vibrating platform with only 3 degrees of mobility has been synthesized based on the analysis of a Stewart platform.

The mechanism has elastic and damping elements with variable characteristics and a pneumatic actuation system.

This structure allows the characteristics of the spring elements to be adjusted in such a way as to result in the optimum platform trajectory.

In the structural and kinematical synthesis of the vibratory machine, we started from the elliptical trajectory of the center of mass, considered to be optimal (as shape) for the separation of the granular material. This is the hypothesis from which we started to generate the structural synthesis of the machine. The elliptical trajectory has also been presented in previous works and papers. It has the following stages:

• Structural and kinematic synthesis of the machine using software MATLAB with Simulink.
• Generating the dynamic model of the machine using the software ADAMS. Simulation of the dynamical model established the elliptical shape trajectory for the center of mass of the machine platform.
• Functional optimization of the vibratory machine was performed with ADAMS software. In this process, three optimization variables were selected: DV_1—amplitude of force $F_0$, DV_2—the offset between the actuating forces $F_2$ and $F_5$, DV_3—stiffness constants for each cylinder-piston sub-assembly k_1,2,3,4,5,6. As a consequence of the optimization process, the optimal values of the three variables were achieved, which ensure the elliptical trajectory of the center of mass with a maximal major axis.

The structure with adjusted pneumatic pressure in cylinder-piston subassemblies allows combating the depreciation over time of the elastic characteristics.

For granular materials whose density is different from case to case, the kinematics and dynamics of the platform can be adapted by modifying the elastic characteristics of supports and the actuation force.

Future research directions will be focused on a 3D study for the separation process of granular material and, implicitly, on the structural and kinematic synthesis of the process, with a configuration similar to a Stewart platform, still using pneumatic actuating systems.