A Spatial Model of a Vibrating Conveyor Taking Into Consideration The Self-Synchronisation Process of Inertial Vibrators. Analysis of The Effect of The Layout of The vibrators On The Operation of The Conveyor

This article presents a spatial model of a vibrating conveyor supported on steel-elastomer vibration isolators and vibrated by two inertial vibrators as well as the results of analyses of the effect of the layout of vibrators on the operation of the conveyor. The displacement of the line of action of the resultant of the forces of the vibrators beyond the centre of mass of the machine body manifests itself in the departure of the machine body from the desirable rectilinear motion and problems associated with the unevenness of material movement. The model presented in this article takes into account an extremely important coupling between vibrators and the body of the machine, which is responsible for the process of self-synchronisation of the vibrators and the correctness of machine operation. The results of theoretical analyses presented in this paper were veriﬁed by laboratory tests and reference to observa-tions based on the authors’ industrial experience.


Introduction
Vibrating conveyors can be classed as typical transport machines. They are machines of simple construction which are well understood in theoretical and practical terms [13] [2]. They are built in many forms e.g. eccentric, pressure, reaction, and inertial and have many different applications ranging from bulk material handling, through dosing to transport, in which the material is subjected to technological processes such as cooling, heating, drying, wetting, screening, sorting, etc.
The most typical of these is the inertial drive conveyor [3], consisting of a rigid and usually slender trough supported on a flexible suspension and excited to vibration by means of two inertial vibrators. Vibrators rotating in opposite directions synchronise themselves and ensure the generation of a segmental force with a sinusoidal course over time. When the line of action of the force passes through the centre of mass of the conveyor body, the conveyor performs harmonic segmental motion, and when the longitudinal symmetry of the conveyor is ensured, the desirable segmental motion of the trough is obtained to ensure uniform transport of the material. Typical suspension elements are coil springs [11] [14], although recently there has been a trend toward the use of metal-elastomer vibration isolators [5] [16] [17].
The design principles of vibrating conveyors are generally well known, and the description of the movement of a grain as a material point on an oscillating plane is also well understood, yet machine failures are still common in practical applications and the movement of the material is often far from conforming to that assumed. A frequent cause of this state of affairs is improper technology through merging conveyor components leading to residual stresses and cracks in the bodies, and also design errors, in particular failure to maintain the conditions of force and centre of mass. Of significant importance is the omission of body flexural elasticity in the design phase, which can cause structural resonance and disphasing of vibrators. Metal-elastomer vibration isolators, on the other hand, have distinct directional properties in contrast to coil springs, which can affect the synchronisation of vibrators when this occurs in a plane transverse to the plane of operation of the vibration isolators. It is not without significance that the theory of construction of vibrating conveyors is based on models of plane motion, while some of them are clearly designed as machines with spatial motion [20] [15].
In the literature, we can find spatial models of vibrating conveyors. However, these models are simplified to the dynamics of the movement of the trough in which the vibrator action is replaced by a concentrated axial force [12]. Such models cannot reflect the couplings between vibrators and the body [6] and thus the process of self-synchronisation of the vibrators, which is extremely important for the correct operation of the machine [ The ability to model the operation of conveyors, taking into account all their elements, is essential from the point of view of the interaction of machines with their supporting structures. The application of FE models is still very limited due to their very high complexity and resulting long calculation time [7]. The operation method of a conveyor has a very significant impact on the behaviour of the supporting structure. Uncontrolled dynamic action from an improperly operating vibrating conveyor can lead to damage to the structure caused, inter alia, by fatigue cracks [18]. On the other hand, the spatial stiffness and dynamic characteristics of the building structure influence the final movement of the conveyor [23]. The interaction between these two objects has not yet been the subject of detailed theoretical analysis. This article is intended to be the first in a series modelling the motion of vibrating conveyors.

Kinetic energy of the system
The subject of the theoretical analysis was a vibrating conveyor with a structure corresponding to the conveyor shown in Fig.1. The conveyor consists of a body mounted on four metal-elastomer vibration isolators set for vibrations using two inertial vibrators driven from individual drives. Fig. 1 The draw of the vibration conveyor, 1 -trough, 2, 3 -inertial vibrators, 4 -metal-elastomer vibration isolator.
The body of the machine and vibrators perform general motion. A total of 8 coordinates were assigned to the description of the system, 3 of these describing the position of the centre of mass of the machine body, another 3 describing the angular displacement of the body and 2 others the angular displacements of the vibrators. Due to the large angular displacement of the vibrators, it was decided to use Euler angles both to describe the angular position of the vibrators and that of the conveyor body. Fig.2a presents a typical representation of Euler angles, where the mutual positions of two coordinate systems relative to each other (with common coordinate origins) are determined by the angle of precession Ψ, nutation θ and spin ϕ.
In practical applications, especially in the description of vibrating shapes, it is advisable to introduce some modification to the definition of these angles. Namely, they are measured in relation to the new positions of the ξηθ axis, which we obtain after turning the system twice by angles ϕ = π 2 and θ = π 2 , Fig.2b.  By binding the ξηθ system with the central principal axes of inertia of the conveyor body and using the transformation (1) we can associate the coordinates of any point P in the ξηθ system with the coordinates of this point in the xyz system.

Suspension damping
The equivalence of the linear damping model (defined by the viscous damping coefficient b) and the material damping model (defined by the energy dissipation coefficient Ψ) enables the formulation of the following relationship: where: [B] -damping matrix, ω -vibration frequency. Based on the relationship linking the coefficient Ψ and the logarithmic decrement of vibration decay δ: one can formulate the equation: where β is the coefficient corresponding to the proportionality coefficient in the Rayleigh damping model. The logarithmic decrement of vibration decay can be determined from the relationship: where: A i , A i+n are vibration amplitudes distant from each other by n periods of oscillation.

Dynamic equations of motion
Dynamic equations of the conveyor motion can then be determined on the basis of the Lagrange-Euler equation: in which: E, V , N , Q i -kinetic energy, potential energy, power of the system linear lost and generalized forces, Due to the elaborate form of the equations, we are limited to presenting only one of them (36), i.e., the equation determined for the coordinate y C . This equation does not include the components of damping forces which, because of their similarity to the components of elastic forces, can easily be completed by the reader.

Verification of the theoretical model
Computer simulations were carried out based on the mathematical model. They were based on the physical parameters of a laboratory conveyor built specifically for the purpose of analysing the effect of structural asymmetry on the movement properties of the conveyor. A photograph of the test stand and a list of its physical parameters are presented in Fig. 6 and Table 1, respectively. The logarithmic decrement of vibration δ and the damping coefficient β for the suspension system were calculated based on the recorded natural vibrations of the conveyor body, Fig. 6b.
Slightly higher values were obtained for the horizontal vibration direction: δ = 0.32 i β = 0.01. In order to evaluate the degree of agreement between the model and the real system, the results of the simulation studies were compared with the measured results. Two points were selected for comparison, the first P1 lying in the central section of the trough surface and the second P2 lying at the height of the centre of mass of the conveyor body. In the first case, the vertical component of the vibrations was measured, see Fig. 7a, in the second case -vibrations in the direction of desired work (on an axis at an inclination of 30 0 to the horizontal plane), see Fig. 8a.
As can be observed, good agreement was obtained between the measured and simulated courses, Fig. 7b, 8bparticularly in the range of steady state operation and machine coastdown. The largest differences occurred in the startup phase. It was in this respect, however, that the mathematical model of the system differed most from the real one due to the description of the drive motors. Those in the model were described using Klos' static mechanical characteristics, which do not capture the transients of the motor and may differ significantly from the mechanical characteristics of the motor used in the experiment.
However, the startup and coastdown phases will not be of fundamental importance in analyses of the influence of design parameters on the operating properties of the conveyor since these are determined on the basis of the machine's steady state operation.       Figs. 9 and 10 show the trajectories of four points located in places where vibration isolators are coupled with the conveyor trough; these points permit the unambiguous determination of the spatial movement of the trough. Fig. 9 shows the case in which the line of action of the resultant force of the vibrators was adjusted to pass through the centre of mass of the body. In this case, one can observe almost congruent movement of all points of the trough, both in the horizontal plane of the trough and in its vertical cross-section.
A slight rotation of the trajectory of horizontal motion can be observed (Fig. 9a). However, it is caused by the presence of resistance to conveyor movement which, in the form of moments of a force, must be overcome by the drive motors mounted to the conveyor body. This rotation should, therefore, similarly reveal itself in all subsequent simulations in which no additional resistances were introduced in the system. Fig. 10 shows analogous courses, but in the case in which the line of action of the resultant force of vibrators was displaced by 0.132 m, and the angles γ and ν assumed the values, respectively, of 7 0 and −23 0 . This is a situation corresponding to the laboratory conveyor presented in Chapter 3. In this case, one can clearly observe how the movement of the trough varied in the vertical cross-section. The points on the side of the drive units, labelled 1, 2, vibrated much less intensely than those on the opposite side of the trough, denoted 3 and 4. The vibration angle of the trough on the side of the drive units clearly decreased with respect to the case presented in Fig. 9 and clearly increased on the opposite side of the trough. The resulting change in throw factors for the trough manifested itself in a change in the velocity of movement of the transported material -less on the drive unit side, greater on the material discharge side. This effect was clearly observed in experimental studies.      Increasing the angle of disphasing of the vibrators can entail a rotation of their resultant force and an increase in the amplitude of transverse vibrations (in the x direction) of the conveyor body. Apart from emergency situations, such as the seizure of bearings on one of the vibrators, there is no reason for this to occur in practice.
However, there are cases in which a clear increase in the amplitude of transverse vibrations, movement of the material oblique with respect to the trough axis, and rumbling of conveyor body vibrations are observed. This situation was observed by authors on a long industrial conveyor. The explanation for this behaviour of the conveyor can be provided by the structural resonance of its body. A long and flaccid body may have a relatively low first harmonic of natural vibrations that can enter into resonance with the forcing of the vibrators. Working near by the resonance causes the vibrators to become severely disphased and can lead to breaking of the synchronising bond.
Structural resonance cannot be described by the model presented in this paper. However, the oblique-angular resonance of the body can be artificially modelled by increasing the elastic coefficients of two selected vibration isolators located diagonally opposite to one another. The results shown in Fig. 11 were obtained in this manner. There is a clear change in the trajectory of the conveyor trough with a significant contribution from the transverse component, Fig. 11a, and pronounced disphasing of the vibrators, Fig. 11b -in the case analysed reaching approx. 19 0 .   in paper [4] was used to analyse the synchronising moment. This refers to the ideal case in which, due to the assumed symmetry, the equations of plane motion of the conveyor in the plane of operation were obtained, and its analytical form could be determined. where: According to formula (43), the maximum value of the synchronising torque for the physical parameters in Table 1 was 0.5 Nm. From simulation studies based on the model thus derived, a value of 0.59 Nm was obtained. The effect of the arrangement of angles γ and ν, Fig. 13, on the value of the synchronising torque is presented in Table 2. The following assumptions were made in the simulations: the inertial parameters of the conveyor, the condition of the centre of mass and a trough vibration angle equal to 30 0 with respect to the horizontal level. As can be seen, the arrangement of angles significantly influenced the maximum value of the synchronising torque. The smallest value was obtained in the situation in which the working plane of the vibrators coincided with the plane of the main axes of the Cζη conveyor and the highest was obtained when these planes were rotated maximally with respect to each other. The difference was 0.23 Nm, which, with respect to the reference value of 0.59 N, represented as much as 39%.  (Table 1), where the centre of mass was situated 0.132 m above the resultant action line, this moment increased to 1.2 Nm. In general, it can be said that an increase in the asymmetry of the system, as was the case in the research presented in the paper cited [4], increases the synchronising torque of the vibrators.

Summary
This article presents the construction of a mathematical model of a vibrating conveyor in general motion driven by two inertial vibrators. The model allows one to describe the phenomenon of self-synchronisation of the vibrators commonly used in drives of vibrating machines. The steel-elastomer vibration isolators, which clearly exhibit directional properties due to their structure, were fully modelled. The model was used to analyse the movement of the conveyor in the steady state as well as transient states associated with the start-up and coastdown phases for the machine. In all three cases, very good agreement was obtained between the computer simulation results and the experimental results (less so in the start-up phase due to the highly simplified model of the drive motors). The high utility value of the model enabled a study to be conducted to analyse the features of conveyors that are a typical or deviate from the accepted rules for designing vibrating conveyors. Three such cases are analysed herein: displacement of the centre of mass of the machine body from the straight line of action of the resultant for the vibrators, rotation of the main axes of inertia of the conveyor body in the working plane of the conveyor, and oblique-angular resonance. The displacement of the centre of mass in relation to the resultant force from the vibrators has a very clear effect on the trajectory of motion of the conveyor trough plane, clearly differentiating the motion of the feed section and the discharge section. This is an undesirable feature, increasing the unevenness of material movement in the trough. However, this characteristic increases the value of the synchronising moment of the vibrators, which, in the case of conveyors with a weak synchronising bond, could have a positive effect. However, in the case of counter-rotation vibrator drives, this characteristic should not be of great importance, as in this case the synchronising bond is relatively high. The rotation of the main axes of inertia of the conveyor body in the vertical plane of the conveyor also resulted in significant changes in the synchronising torque. An advantage was proved in increasing the angle between the working plane of the vibrators and the main axes of the conveyor inertia. In general, it may be concluded that, in the case of the conveyor structure presented in this article, the most beneficial solution in terms of uniformity of material movement in the conveyor trough and the highest value of torque synchronising the vibrators is the structure maintaining the condition of the centre of mass and the maximum rotation angle of the vibrators' plane and the main axes of the conveyor body's inertia. Interesting results were obtained for conveyor operation near by obliqueangular resonance. Resonance had a great effect on the synchronisation of the vibrators. In the case of a displacement of the centre of mass with respect to the line of action of the resultant, an increase in the disphasing of the vibrators and a corresponding increase in vibration in the direction perpendicular to the plane of work of the conveyor was observed.
In the typical case in which the line of action passed through the centre of mass, the synchronising bond was broken and beat phenomena of the body in all directions of conveyor movement was observed. It should be noted that similar behaviour was observed on industrial conveyors suspected of operating on the verge of the structural resonance of the conveyor body.

Declarations
Funding This work was supported by: AGH University of Science and Technology in Krakow under the grant no. 16.16.130.942 and Cracow University of Technology.

Conflicts of interest
Not applicable. Availability of data and material All data generated or analysed during this study are included in this published article.

Code availability
Computer simulations were based on own code implemented in OpenModelica software. Other graphic materials were also prepared with the use of free software, such as FreeCad, InkSpace without breaking the license rights. The code generated during the current study is available from the corresponding authors on reasonable request.