Influence of vibratory conveyor design parameters on the trough motion and the self-synchronization of inertial vibrators

Grzegorz Cieplok

doi:10.1515/eng-2022-0434

Artikel Open Access

Influence of vibratory conveyor design parameters on the trough motion and the self-synchronization of inertial vibrators

Grzegorz Cieplok

Veröffentlicht/Copyright: 20. Januar 2024

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Abstract

A spatial model of a vibratory conveyor supported on steel-elastomer vibration isolators and vibrated by two inertial vibrators is presented in this article. The results of analyses of the effect of the layout of vibrators on the operation of the conveyor are presented. Displacement of the line of action of vibrators resultant force beyond the machine mass center causes the drift of the trough movement from the desirable rectilinear motion and problems associated with the unevenness of material movement. Three main cases were analyzed in the article: the displacement of the center 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 the oblique-angular resonance. The model presented in this article is used to consider coupling between vibrators and the body of the machine, which is responsible for the process of self-synchronization of the vibrators and the correctness of machine operation. The results of the theoretical analyses presented in this article were verified by laboratory tests and references to observations based on the authors’ industrial experience.

Keywords: spatial motion; self-synchronization; conveyor

1 Introduction

Vibratory conveyors can be classified as typical transport machines. They are machines of simple construction that are well understood in theoretical and practical terms. These conveyors are built in many forms, e.g., eccentric, pressure, reaction, and inertial, and have many different applications ranging from bulk material handling to dosing to transport, in which the material is subjected to technological processes such as cooling, heating, drying, wetting, screening, and sorting [1–3].

The most common type of vibrating conveyor is the inertial drive one [4–6], which consists of a rigid and usually slender trough supported on a flexible suspension and excited to vibrations by means of two inertial vibrators. Vibrators rotating in opposite directions synchronize themselves to ensure the generation of a segmental force with a sinusoidal course over time. When the line of action of the force passes through the center of mass of the conveyor body, the conveyor performs harmonic segmental motion, and when the longitudinal symmetry of the conveyor is ensured, the desired segmental motion of the trough is obtained to ensure uniform transport of the material. Typical suspension elements are coil springs [7,8]. However, there has been a recent trend toward the use of metal-elastomer vibration isolators [9–11].

The design principle of vibratory conveyors are generally well known [12–14], and the description of the movement of a grain as a material point or a discrete element on an oscillating plane is also well understood [15,16], yet machine failures are still common in practical applications, and the movement of the material does not often met expectations. Common causes of this phenomena are improper technologies with respect to merging conveyor components, leading to residual stresses and cracks in the bodies, and design errors, in particular the failure to maintain the conditions of force and center of mass. The omission of body flexural elasticity in the design phase is of a significant importance as it can cause structural resonance and disphasing of the vibrators. In contrast, metal-elastomer vibration isolators have distinct directional properties compared to coil springs, which can affect the synchronization of vibrators when this occurs in a plane transverse to the plane of operation of vibration isolators. It is not without significance that the theory of construction of vibratory conveyors is based on models of the plane motion, while some are clearly designed as machines with a spatial motion [17,18].

In the literature, there are cases of spatial models of vibratory conveyors. However, these models are simplified with respect to the dynamics of the trough movement, in which the vibrator action is replaced by a concentrated axial force [19]. Such models cannot be used to reflect the couplings between vibrators and the body [20]; therefore, the process of self-synchronization of the vibrators [21–25] that is very important for the correct operation of the machine cannot be captured.

The aim of this article is to formulate the mathematical model of the vibratory conveyor of a two-vibrator drive in the spatial motion. On the basis of this model, it will be possible to perform simulation investigations of the influence of constructional parameters on the moment synchronizing unbalanced masses of vibrators as well as on the trough motion in a spatial case. This model allows to represent shifting the straight line of a resultant force of inertial vibrators from the mass center of the conveyor as well as to represent the influence of the layout of the main axes of the body inertia versus the vibrators work plane. The model formulated in this study is similar to the one presented in the study by Fang et al. [22], where a case in which the vibrators work plate passes through the mass center of the machine body in accordance with the design rules is provided.

The current study originates from an industrial problem, as the author analyzed the conveyor that was performing strange motions with a significant participation of transverse components and the presence of beats. These unexpected movements disturbed the assumed single-directional material motion versus the trough, causing the conveyor productivity to decrease. An explanation of these malfunctions on the basis of flat model theory was impossible, and building a spatial model, that was able to take into account atypical states, became necessary. The results of these studies are presented in this publication.

2 Model

The dynamic equations of motion of the presented conveyor were derived on the basis of the Lagrange II-kind method of equations. Therefore, subsequent subsections of this publication present stages of the determination of kinetic and potential energies of the system components, taking into account the specificity of large angular displacements of rigid bodies in a general motion. The kinetic and potential energy formulas were applied to determine the Lagrange function. Based on this function, using the wxMaxima program, dynamic equations for each generalized coordinate were determined. These equations were supplemented with generalized forces as a result of the influences of engine driving moments and vibrator resistance to motion. Damping of suspension elements was shown as a function of linear power losses, which was used to determine dissipative influences. Numerical simulations of the conveyor motion on the basis of equations, determined as described earlier, were carried out using the OpenModelica program.

2.1 Kinetic energy of the system

The subject of the theoretical analysis was a vibratory conveyor with a structure corresponding to the conveyor shown in Figure 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.

Figure 1

The draw of the vibratory conveyor. 1: the trough; 2 and 3: the inertial vibrators; and 4: the metal-elastomer vibration isolator. Drawing program: ZWCAD ver. 2018, https://www.zwcad.pl/.

The body of the machine and vibrators perform a general motion. A total of eight coordinates were assigned to the description of the system, three of which describe the position of the center of mass of the machine body, another three describe the angular displacement of the body, and two others describe the angular displacements of the vibrators. Due to the large angular displacement of the vibrators, it was decided to use Euler angles to describe the angular position of both the vibrators and the conveyor body. Figure 2(a) shows 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 .

$Figure 2 Coordinates of the rotation of a rigid body around a fixed point for Euler angles; w w – line of nodes. (a) Euler angles in a traditional system and (b) Euler angle system after revolutions of Ψ = π 2 \Psi =\frac{\pi }{2} and θ = π 2 \theta =\frac{\pi }{2} .$

Figure 2

Coordinates of the rotation of a rigid body around a fixed point for Euler angles; w – line of nodes. (a) Euler angles in a traditional system and (b) Euler angle system after revolutions of Ψ = π 2 and θ = π 2 .

In practical applications, especially in the description of vibrating shapes, it is advisable to introduce some modification to the definition of these angles. Namely, these angles are measured in relation to the new positions of the ξ η θ axis, which were obtained after turning the system twice by angles Ψ = π 2 and θ = π 2 , Figure 2(b).

By binding the ξ η θ system with the central principal axes of inertia of the conveyor body and using the transformation (1), the coordinates of any point P in the ξ η θ system can be associated with the coordinates of this point in the x y z system.

(1) x P y P z P = cos Ψ sin φ sin θ − sin Ψ cos φ cos Ψ cos φ sin θ + sin Ψ sin φ cos Ψ cos θ sin Ψ sin φ sin θ + cos Ψ cos φ sin Ψ cos φ sin θ − cos Ψ sin φ sin Ψ cos θ sin φ cos θ cos φ cos θ − sin θ

Now, it can be simply presented that the coordinates of the vibrators’ centers of mass C 1 and C 2 in a system ξ η ζ , as shown in Figure 3, are as follows:

(2) ξ 1 , 2 = ξ 10 ± e sin ( φ 1 , 2 ) cos ( ν ) ,

(3) η 1 , 2 = η 10 ± e sin ( φ 1 , 2 ) sin ( ν ) ,

(4) ζ 1 , 2 = ± ζ 10 + e cos ( φ 1 , 2 ) ,

where ξ 10 , η 10 , and ζ 10 are the coordinates of the vibrator mounting point.

$Figure 3 Components of the coordinates of the vibrators’ centers of mass C 1 {C}_{1} and C 2 {C}_{2} and the angular velocity vector for the vibrators in the systems ξ 1 , 2 , η 1 , 2 , ζ 1 , 2 {\xi }_{1,2},{\eta }_{1,2},{\zeta }_{1,2} (connected to the body) and rotated by the constant angle ν \nu . ω ( 1 , 2 ) r e l = φ ˙ 1 , 2 {\omega }_{\left(1,2)rel}={\dot{\varphi }}_{1,2} refers to the angular velocity of the vibrators relative to the machine body. (a) Components of vibrator 1 and (b) components of vibrator 2.$

Figure 3

Components of the coordinates of the vibrators’ centers of mass C 1 and C 2 and the angular velocity vector for the vibrators in the systems ξ 1 , 2 , η 1 , 2 , ζ 1 , 2 (connected to the body) and rotated by the constant angle ν . ω ( 1 , 2 ) r e l = φ ˙ 1 , 2 refers to the angular velocity of the vibrators relative to the machine body. (a) Components of vibrator 1 and (b) components of vibrator 2.

Based on (1) and considering small angular shifts of the machine body, it can be written as follows:

(5) x 1 , 2 = ± ζ 10 − Ψ [ ξ 10 ± e cos ( ν ) sin ( φ 1 , 2 ) ] + [ η 10 ± e sin ( ν ) sin ( φ 1 , 2 ) ] θ + e cos ( φ 1 , 2 ) ,

(6) y 1 , 2 = + ξ 10 + Ψ [ ± ζ 10 + e cos ( φ 1 , 2 ) ] − φ [ η 10 ± e sin ( ν ) sin ( φ 1 , 2 ) ] ± e cos ( ν ) sin ( φ 1 , 2 ) ,

(7) z 1 , 2 = + η 10 − θ [ ± ζ 10 + e cos φ ( 1 , 2 ) ] + φ [ ξ 10 ± e cos ( ν ) sin ( φ 1 , 2 ) ] ± e sin ( ν ) sin ( φ 1 , 2 ) .

The absolute velocities (in the X g Y g Z g system) of the centers of the vibrators’ masses can be obtained by differentiation according to time of dependence (5)–(7) and considering the system’s velocity x y z . After the removal components of small higher orders, velocities take forms:

(8) x ˙ 1 , 2 = − Ψ ˙ ξ 10 ± φ ˙ 1 , 2 [ cos ( φ 1 , 2 ) ( e sin ( ν ) θ − Ψ e cos ( ν ) ) ∓ e sin ( φ 1 , 2 ) ] + θ ˙ [ ± e sin ( ν ) sin ( φ 1 , 2 ) + η 10 ] ∓ Ψ ˙ e cos ( ν ) sin ( φ 1 , 2 ) + x ˙ C

(9) y ˙ 1 , 2 = ± Ψ ˙ ζ 10 ± φ ˙ 1 , 2 [ ( e cos ( ν ) − e sin ( ν ) φ ) cos ( φ 1 , 2 ) ∓ Ψ e sin ( φ 1 , 2 ) ] ∓ φ ˙ e sin ( ν ) sin ( φ 1 , 2 ) + Ψ ˙ e cos ( φ 1 , 2 ) − φ ˙ η 10 + y ˙ C

(10) z ˙ 1 , 2 = ∓ θ ˙ ζ 10 + φ ˙ ξ 10 + φ ˙ 1 , 2 [ e sin ( φ 1 , 2 ) θ ± ( e cos ( ν ) φ + e sin ( ν ) ) cos ( φ 1 , 2 ) ] ± ϕ ˙ e cos ( ν ) sin ( φ 1 , 2 ) − θ ˙ e cos ( φ 1 , 2 ) + z ˙ C ,

where x ˙ C , y ˙ C , and z ˙ C are the mass centers of the machine body velocities C in the absolute system.

The kinetic energy of the trough is stated using velocity projections on the axes of the system ξ η ζ and must be expressed by means of generalized coordinates. This transition is made by the following dependency:

(11) ω ξ ω η ω ζ = sin φ cos θ cos φ 0 cos φ cos θ − sin φ 0 − sin θ 0 1 ω Ψ ω θ ω φ .

As a result, for small angular displacements:

(12) ω ξ ≅ θ ˙ + φ Ψ ˙ ,

(13) ω η ≅ Ψ ˙ − φ θ ˙ ,

(14) ω ζ ≅ φ ˙ − θ Ψ ˙ .

Due to the connection of the vibrators with the trough, which is schematically shown in Figure 3, the components of their angular velocities in projections on axes ξ 1 , 2 , η 1 , 2 , and ζ 1 , 2 can be written as follows:

(15) ω ξ 1 , 2 ≅ ω η sin ( ν ) + ω ξ cos ( ν ) ,

(16) ω η 1 , 2 ≅ ± φ ˙ 1 , 2 + ω η cos ( ν ) − ω ξ sin ( ν ) ,

(17) ω ζ 1 , 2 ≅ ω ζ .

Taking into account the axial symmetry of the vibrator, J x princ ≅ J y princ , Figure 4(a), and based on the geometric relationships presented in Figure 4(b), the formula for the kinetic energy of the vibrator rotation can be expressed as follows:

(18) E v i b r 1 r o t = 1 2 J z princ ω 1 z p r i n c 2 + 1 2 J x princ ( ω 1 x princ 2 + ω 1 y princ 2 ) = 1 2 J z princ ω η 1 2 + 1 2 J x princ ( ω ξ 1 2 + ω ζ 1 2 ) ,

(19) E vibr 2 rot = 1 2 J z princ ω 2 z princ 2 + 1 2 J x princ ( ω 2 x princ 2 + ω 2 y princ 2 ) = 1 2 J z princ ω η 2 2 + 1 2 J x princ ( ω ξ 2 2 + ω ζ 2 2 ) .

$Figure 4 Inertial vibrator used in the drive. (a) The vibrator’s principal axes of inertia. Drawing program: FreeCAD ver. 0.19, https://www.freecadweb.org/ (b) Projections of the angular velocity of vibrator 1 onto the vibrator’s principal axes of inertia x p r i n c , y p r i n c , z p r i n c {x}_{princ},{y}_{princ},{z}_{princ} .$

Figure 4

Inertial vibrator used in the drive. (a) The vibrator’s principal axes of inertia. Drawing program: FreeCAD ver. 0.19, https://www.freecadweb.org/ (b) Projections of the angular velocity of vibrator 1 onto the vibrator’s principal axes of inertia x p r i n c , y p r i n c , z p r i n c .

After consideration of the velocities of individual bodies, the energy function (20) for the whole system can be formulated.

(20) ℰ ≅ 1 2 m k ( x ˙ C 2 + y ˙ C 2 + z ˙ C 2 ) + 1 2 J ξ ω ξ 2 + 1 2 J η ω η 2 + 1 2 J ζ ω ζ 2 + 1 2 m 1 ( x ˙ 1 2 + y ˙ 1 2 + z ˙ 1 2 ) + 1 2 J z princ ω η 1 2 + 1 2 J x princ ( ω ξ 1 2 + ω ζ 1 2 ) + 1 2 m 2 ( x ˙ 2 2 + y ˙ 2 2 + z ˙ 2 2 ) + 1 2 J z princ ω η 2 2 + 1 2 J x princ ( ω ξ 2 2 + ω ζ 2 2 ) ,

where m k is the mass of the machine body, m 1 , 2 are the eccentric masses of the vibrators, J ξ , J η , and J ζ are the central principal inertia of the machine body, and J x princ , J y princ , and J z princ are the central principal inertia of the eccentric masses.

2.2 Potential energy of the system

The machine body vibration isolators, as shown in Figure 5, were included in the model as linear elastic elements [9], whose elasticity constants k i j were calculated for the operating point determined by the weight of the machine.

Figure 5

Model of the metal-elastomer vibration isolator. (a) ROSTA AB 15 metal-elastomer vibration isolator and (b) designations of the displacements of the connection plane.

The potential energy of isolator can be represented as a matrix relationship:

(21) V isol = 1 2 Δ x Δ y Δ z Δ φ x Δ φ y Δ φ z T × k 11 k 12 k 13 k 14 k 15 k 16 k 12 k 22 k 23 k 24 k 25 k 26 k 13 k 23 k 33 k 34 k 35 k 36 k 14 k 24 k 34 k 44 k 45 k 46 k 15 k 25 k 35 k 45 k 55 k 56 k 16 k 26 k 36 k 46 k 56 k 66 Δ x Δ y Δ z Δ φ x Δ φ y Δ φ z ,

where Δ x , Δ y , Δ z , Δ φ x , Δ φ y , and Δ φ z denote the displacements of the mounting plane of the vibration isolator to the conveyor body. Assuming small angular displacements Ψ , θ , φ , it can be written as follows:

(22) Δ x = x C − Ψ ξ P + η P θ ,

(23) Δ y = y C + ( − θ ζ P + z C + φ ξ P ) sin ( γ ) + ( Ψ ζ P + y C − η P φ ) cos ( γ ) ,

(24) Δ z = z C + ( − Ψ ζ P − y C + η P φ ) sin ( γ ) + ( − θ ζ P + z C + φ ξ P ) cos ( γ ) ,

(25) Δ φ x = φ ,

(26) Δ φ y = Ψ sin ( γ ) + θ cos ( γ ) ,

(27) Δ φ z = Ψ cos ( γ ) − θ sin ( γ ) ,

and express the potential energy of the vibration isolator as a function of generalized coordinates. Here, ξ P , η P , and ζ P denote the coordinates of the fastening point of the given vibroinsulator to the machine body.

The potential energy of gravity, which is related to the position of the centers of mass of the vibrators and the machine body, can be expressed as a relationship:

(28) V g = m 1 g y ( y 1 + y C ) + m 2 g y ( y 2 + y C ) + m 1 g z ( z 1 + z C ) + m 2 g z ( z 2 + z C ) + m k g y y C + m k g z z C ,

where

(29) g y = − g sin ( γ ) ,

(30) g z = + g cos ( γ ) .

2.3 Suspension damping

The equivalence of the linear damping model (defined by the viscous damping coefficient b ) [26] and the material damping model (defined by the energy dissipation coefficient Ψ ) enables the formulation of the following relationship:

(31) [ B ] = Ψ 2 π ω [ K ] ,

where [ B ] is the damping matrix and ω is the vibration frequency.

Based on the relationship linking the coefficient Ψ and the logarithmic decrement of vibration decay δ :

(32) Ψ = 4 δ ,

the following equations can be formulated:

(33) [ B ] = β [ K ] ,

(34) β = 2 δ π ω 0 ,

where β is the coefficient corresponding to the proportionality coefficient in the Rayleigh damping model. The logarithmic decrement of the vibration decay can be determined from the following relationship:

(35) δ = 1 n ln A i A i + n ,

where A i and A i + n are vibration amplitudes that are apart from each other by n periods of oscillation.

2.4 Dynamic equations of motion

The dynamic equations of the conveyor motion can then be determined on the basis of the Lagrange-Euler equation:

(36) d d t ∂ ℰ ∂ q i ˙ − ∂ ℰ ∂ q i + 1 2 ∂ N ∂ q i ˙ = − ∂ V ∂ q i + Q i ,

where ℰ , V , N , and Q i are the kinetic energy, potential energy, linear loss, and generalized forces of the system, respectively, and q i ∈ { x C , y C , z C , Ψ , θ , φ , φ 1 , φ 2 } .

Due to the elaborate form of the equations, they were limited to only one of them (37), 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.

( m k + 2 m w ) y ¨ C + m w e [ − φ ˙ 2 2 sin ( ν ) ϕ + φ ¨ sin ( ν ) + φ ˙ 2 2 cos ( ν ) − ( Ψ φ ¨ 2 + 2 Ψ ˙ φ ˙ 2 ) ] sin ( φ 2 ) + m w e [ φ ¨ 2 sin ( ν ) ϕ + 2 φ ˙ φ ˙ 2 sin ( ν ) − φ ¨ 2 cos ( ν ) + ( Ψ ¨ − Ψ φ ˙ 2 2 ) ] cos ( φ 2 ) + m w e [ φ ˙ 1 2 sin ( ν ) ϕ − φ ¨ sin ( ν ) − φ ˙ 1 2 cos ( ν ) − ( Ψ φ ¨ 1 + 2 Ψ ˙ φ ˙ 1 ) ] sin ( φ 1 ) − m w e [ φ ¨ 1 sin ( ν ) ϕ + 2 φ ˙ φ ˙ 1 sin ( ν ) + φ ¨ 1 cos ( ν ) + ( Ψ ¨ − Ψ φ ˙ 1 2 ) ] cos ( φ 1 ) − 2 φ ¨ η 10 m w = 4 [ k 13 sin ( γ ) − k 12 cos ( γ ) ] x C + 4 [ − k 33 sin ( γ ) 2 + 2 k 23 cos ( γ ) sin ( γ ) − k 22 cos ( γ ) 2 ] y C + 4 [ k 23 sin ( γ ) 2 + ( k 33 − k 22 ) cos ( γ ) sin ( γ ) − k 23 cos ( γ ) 2 ] z C

+ − k 33 ∑ i = 1 4 ζ i + 4 k 35 sin ( γ ) 2 + 2 k 23 ∑ i = 1 4 ζ i + 4 ( k 36 − k 25 ) cos ( γ ) − k 13 ∑ i = 1 4 ζ i sin ( γ ) − k 22 ∑ i = 1 4 ζ i + 4 k 26 cos ( γ ) 2 + k 12 ∑ i = 1 4 ξ i cos ( γ ) Ψ + ( k 22 − k 33 ) ∑ i = 1 4 ζ i + 4 k 35 + 4 k 26 cos ( γ ) + k 13 ∑ i = 1 4 η i sin ( γ ) − k 23 ∑ i = 1 4 ζ i + 4 k 36 sin ( γ ) 2 + k 23 ∑ i = 1 4 ζ i − 4 k 25 cos ( γ ) 2 − k 12 ∑ i = 1 4 η i cos ( γ ) θ

(37) + − k 23 ∑ i = 1 4 ξ i + k 22 ∑ i = 1 4 η i cos ( γ ) 2 − 4 k 24 cos ( γ ) + k 23 ∑ i = 1 4 ξ i + k 33 ∑ i = 1 4 η i sin ( γ ) 2 + ( k 33 − k 22 ) ∑ i = 1 4 ξ i − 2 k 23 ∑ i = 1 4 η i cos ( γ ) + 4 k 34 sin ( γ ) φ + 2 m w g sin ( γ ) y C = Q y C .

In the equations of motion, all generalized forces Q i take a value of zero except:

(38) Q φ 1 = M el 1 − M load 1 ,

(39) Q φ 2 = M el 2 − M load 2 ,

(40) Q Ψ ≅ ( M el 2 − M el 1 + M load 1 − M load 2 ) cos ( ν ) ,

(41) Q θ ≅ ( M el 1 − M el 2 − M load 1 + M load 2 ) sin ( ν ) ,

where M el 1 , 2 is the electromagnetic torque of the drive motors and M load 1 , 2 are the moments of resistance of the vibrators.

3 Verification of the theoretical model

Computer simulations were carried out based on the mathematical model. These simulations are based on the physical parameters of a laboratory conveyor built specifically for the purpose of analyzing 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 shown in Figure 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, as shown in Figure 6(b).

Figure 6

Test stand. (a) Photograph of a laboratory conveyor. 1. Measurement point P1; and 2. Measurement point P2. (b) The acceleration of natural vibrations in the vertical axis.

Table 1

Conveyor parameters

Par.	Value	Unit	Par.	Value	Unit
m k	50.92	kg	ξ 10	0.151	m
m w	4.0	kg	η 10	‒ 0.132	m
e	0.0211	m	ζ 10	0.099	m
J ξ	1.77	kgm²	ξ 1	0.458	m
J η	5.76	kgm²	η 1	0.199	m
J ζ	6.26	kgm²	ζ 1	0.198	m
J w	0.0035	kgm²	ξ 2	0.458	m
J w x y	0.0006	kgm²	η 2	0.199	m
k 11	9.97 × 1 0 3	N/m	ζ 2	‒ 0.198	m
k 12	27.19	N/m	ξ 3	‒ 0.351	m
k 13	‒ 91.10	N/m	η 3	0.101	m
k 14	7.16	N/rad	ζ 3	‒ 0.198	m
k 15	‒ 833.02	N/rad	ξ 4	‒ 0.351	m
k 16	278.76	N/rad	η 4	0.101	m
k 22	1.88 × 1 0 3	N/m	ζ 4	0.198	m
k 23	‒ 132.51	N/m	k 24	164.23	N/rad
k 25	‒ 5.06	N/rad	k 26	6.63	N/rad
k 33	5.94 × 1 0 3	N/m	k 34	‒ 172.65	N/rad
k 35	19.37	N/rad	k 36	‒ 1.71	N/rad
k 45	‒ 1.05	Nm/rad	k 46	0.796	Nm/rad
k 55	120.50	Nm/rad	k 56	‒ 23.72	Nm/rad
k 66	120.50	Nm/rad	M load 1 , 2	0.0087	Nm
k 44	21.42	Nm/rad

For A 1 = 25.61 m/s 2 , A 11 = 1.46 m/s 2 , t 1 = 0.353 s , t 11 = 3.142 s , and n = 10 , the following values were obtained:

(42) δ = 0.286 ,

(43) β = 0.008 .

Slightly higher values were obtained for the horizontal vibration direction: δ = 0.32 and β = 0.01 .

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 that lies in the central section of the trough surface, and the second P2 that lies at the height of the center mass of the conveyor body. In the first case, the vertical component of the vibrations was measured; see Figure 7(a). In the second case, vibrations in the direction of the desired work (on an axis at an inclination of 3 0 ∘ to the horizontal plane) were measured; see Figure 8(a).

Figure 7

Comparison of vertical vibrations. (a) Course of point P1 obtained from the measurement along the vertical axis over the center of mass. (b) Course of point P1 obtained from simulation studies along the vertical axis over the center of mass.

Figure 8

Comparison of vibrations in the travel direction. (a) Course of point P2 obtained from the measurement along the axis of travel. (b) Course of point P2 obtained from simulation studies along the axis of travel.

The measurement system consisting of a four-channel module of the data acquisition VIBDAQ 4+, piezoelectric accelerometers PCB 353B33 type, and a portable computer (measurement data storage) was utilized to record the time history in experimental studies. Integration of the vibration acceleration signals to obtain displacement pathways was performed in the MATLAB program using the Signal Analyzer Toolbox module. In addition, the physical parameters of the conveyor, i.e., the masses and the mass moments of inertia, were determined using the Salome Meca program on the basis of machine technical drawings and material parameters of its components. The total mass of the machine, determined from the theoretical calculations, was verified by weighing. The difference was equal to 0.4 kg, which means that it was within the weight measuring error.

Good agreement was obtained between the measured and simulated courses, as shown in Figures 7(b) and 8(b), particularly in the range of the steady-state operation and machine coastdown. The largest differences occurred in the start-up 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 the 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.

4 Results of simulations

Based on the analytical model of the conveyor, simulation studies were conducted on the effect of the selected design parameters of the conveyor on its operation. Investigations were carried out on the following:

the effect of the displacement of the conveyor’s center of mass from the line of action of the resultant force of the inertial vibrators,
the effect of the angular shift of the main central inertia axes of the conveyor in relation to the plane of operation of the vibrators, and
the effect of angular resonance of the body-suspension system.

Each of these effects were studied on the trajectory of the movement of the trough and the value of the synchronizing torque.

Figures 9 and 10 show the trajectories of four points located at places where vibration isolators were coupled with the conveyor trough; these points allow for the unambiguous determination of the spatial movement of the trough. Figure 9 shows the case in which the line of action of the resultant force of the vibrators was adjusted to pass through the center 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.

$Figure 9 Motion trajectories of the mounting points of metal–elastomer vibration isolators to the trough body in the global coordinate system. η 10 = 0 {\eta }_{10}=0 . (a) x y xy trajectory and (b) y z yz trajectory.$

Figure 9

Motion trajectories of the mounting points of metal–elastomer vibration isolators to the trough body in the global coordinate system. η 10 = 0 . (a) x y trajectory and (b) y z trajectory.

$Figure 10 Motion trajectories of the mounting points of metal–elastomer vibration isolators to the trough body in the global coordinate system. η 10 = − 0.132 m {\eta }_{10}=-0.132\hspace{0.33em}m . (a) x y xy trajectory and (b) y z yz trajectory.$

Figure 10

Motion trajectories of the mounting points of metal–elastomer vibration isolators to the trough body in the global coordinate system. η 10 = − 0.132 m . (a) x y trajectory and (b) y z trajectory.

Figure 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 of 7 ∘ and − 2 3 ∘ , respectively. This is a situation corresponding to the laboratory conveyor presented in Section 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, labeled 1 and 2, vibrated much less intensely than those on the opposite side of the trough, labeled 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 Figure 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, as shown in Figure 11.

Figure 11

A photograph of the material on the conveyor trough.

Due to the difficulty in distinguishing the velocity components of the material particles in the photograph, the problem of revealing the influence of the machine motion on the movement of the material was performed in an intermediate way by applying the LIGGGHTS software. In this program, the motion of the machine was defined on the basis of the experimental measurement (2.87 mm for the vibration amplitude in the work direction, 0.001 rad for the angular vibration amplitude according to a phase of the vibration of the vertical component, 25 Hz for the frequency of vibrations, and 3 0 ∘ for the angle of vibrations). The simulation results are presented in Figure 12. It can be noticed that as a material nears the trough end, its average vertical component grows, and the velocities of individual particles in this direction differentiate. It manifests itself by an increased dynamic of the particle movement that is easily visible to the naked eye (compare Figure 11). Despite the increased amplitude of the velocity in the vertical direction, no essential change in the material movement in the transport direction is observed.

Figure 12

Results of the computer simulation of the material movement on the conveyor trough using the LIGGGHTS program. (a) Visualization of the material particle motion using the LIGGGHTS program. The group of 1,000 representative particles is labeled in purple and (b) courses of average velocities of the group of representative particles with distinct components in directions x , y , z . This view shows the average value as a line, while coloring the area between q1–q3 (upper/lower quartile) ranges.

This behavior agrees with the kinematics of the trough movement. That is, the horizontal component of the trough movement, which originates from the angular movement, has a similar value throughout its length, while the vertical component increases in proportion to the increase in distance from the center of mass C . The schematic presentation of the fraction of individual components is presented in Figure 13. On the basis of this figure, it can be stated that in case of the direction of the vibrators resulting in a force that passes below the mass center of the machine body C , the horizontal component Δ y subtracks from the horizontal component of the translatory motion, which causes the material transportation process to slow down. When the direction of the vibrator resulting force passes above the mass center C , the velocity of the transportation of the material increases.

$Figure 13 Fractions of components in the conveyor trough movement. Translatory component: A transl {A}_{{\rm{transl}}} ; angular components: Δ y P A = Δ y P B = Δ y P D ≅ − φ ⋅ h \Delta {y}_{{P}_{A}}=\Delta {y}_{{P}_{B}}=\Delta {y}_{{P}_{D}}\cong -\varphi \cdot h , Δ z P A ≅ − φ ⋅ a 1 \Delta {z}_{{P}_{A}}\cong -\varphi \cdot {a}_{1} , Δ z P B ≅ 0 \Delta {z}_{{P}_{B}}\cong 0 , Δ z P D ≅ φ ⋅ a 2 \Delta {z}_{{P}_{D}}\cong \varphi \cdot {a}_{2} , and R R is the resulting force of the vibrators.$

Figure 13

Fractions of components in the conveyor trough movement. Translatory component: A transl ; angular components: Δ y P A = Δ y P B = Δ y P D ≅ − φ ⋅ h , Δ z P A ≅ − φ ⋅ a 1 , Δ z P B ≅ 0 , Δ z P D ≅ φ ⋅ a 2 , and R is the resulting force of the vibrators.

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 phenomenon to occur in practice.

However, there are cases where there is a clear increase in the amplitude of transverse vibrations; as a result, the movement of the material oblique with respect to the trough axis and rumbling of the conveyor body vibrations are observed. This situation was observed by the author on a long industrial conveyor. The explanation for this behavior 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 resonance with the forcing of the vibrators. Working near the resonance causes the vibrators to become severely disphased, which can lead to breaking of the synchronizing bond.

The structural resonance cannot be described by the model presented in this article. However, the oblique-angular resonance of the body can be artificially modeled by increasing the elastic coefficients of two selected vibration isolators located diagonally opposite to one another. The results shown in Figure 14 are obtained in this manner. There is a clear change in the trajectory of the conveyor trough with a significant contribution from the transverse component, Figure 14a, and pronounced disphasing of the vibrators, Figure 14b, in the case analyzed, reaching approximately 1 9 ∘ .

Figure 14

Oblique-angular resonance. (a) x y trajectory and (b) vibrators disphasing.

The situation in which the vibration beat phenomena appears is presented in Figure 15. This result was obtained by returning to the case in which the resultant of the vibrator forces passes through the center of conveyor mass. The beat phenomena is obtained in this case as a result of breaking of the synchronizing bond between the vibrators, which is clearly shown in Figure 15(b).

Figure 15

Oblique-angular resonance and beat phenomena. (a) Motion trajectory of the body center mass and (b) disphasing of the vibrators.

The formula (44) developed in the study by Cieplok and Wójcik [27] was used to analyze the synchronization moment, which refers to the ideal case in which, due to assumed symmetry, the equations of plane motion of the conveyor were obtained in the plane of operation and its analytical form could be determined.

(44) M syncMAX ≈ m w 2 e 2 L 2 ω 2 m k * 2 + 4 m w ( m w − m k * ) cos ( β ) 2 m k * ( J η * m k * − 4 L 2 m w 2 cos ( β ) 2 ) ,

where

(45) m k * = m k + 2 m w ,

(46) J η * = J η + 2 m w ( ζ 10 2 + ξ 10 2 + e 2 ) ,

(47) ξ 10 = L cos β ,

(48) ζ 10 = L sin β .

According to formula (44), the maximum value of the synchronizing torque for the physical parameters in Table 1 was 0.5 Nm. From simulation studies based on the derived model, a value of 0.59 Nm was obtained.

The effect of the arrangement of angles γ and ν , Figure 16, on the value of the synchronizing torque is presented in Table 2. The arrangement of angles significantly influenced the maximum value of the synchronizing 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 C ζ η 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% of the value.

Figure 16

Axis system.

Table 2

Effect of the rotations of the main inertia central axes of the conveyor in relation to the plane of the vibrators’ work

γ ( ∘ )	ν ( ∘ )	ξ 10 (m)	η 10 (m)	Δ M syncMAX (Nm)
30.0	0.0	0.151	0.0	0.59
25.0	‒ 5.0	0.150	‒ 0.013	0.59
20.0	‒ 10.0	0.149	‒ 0.026	0.61
15.0	‒ 15.0	0.146	‒ 0.039	0.64
10.0	‒ 20.0	0.142	‒ 0.052	0.69
7.0	‒ 23.0	0.139	‒ 0.059	0.72
5.0	‒ 25.0	0.137	‒ 0.064	0.75
0.0	‒ 30.0	0.138	‒ 0.076	0.82

Simulations of the system, in which an additional displacement of the body’s center of mass occurred beyond the straight line of action of the resultant vibrators, also indicated a significant increase. Based on the parameters corresponding to the laboratory bench (Table 1), where the center of mass was 0.132 m above the resulting action line, this moment increased to 1.2 Nm.

In general, it can be said that an increase in system asymmetry, as was the case in the research presented in the cited article [27], increases the synchronizing torque of the vibrators.

5 Summary

The construction of a mathematical model of a vibratory conveyor in general motion driven by two inertial vibrators was presented in this article. This model was used to describe the phenomena of self-synchronization of the vibrators commonly used in the drives of vibrating machines. The steel-elastomer vibration isolators, which clearly exhibited directional properties due to their structure, were fully modeled. The model was also used to analyze the movement of the conveyor in the steady state as well as transient states associated with the start and stop phases of the machine. In all three cases, very good agreement was obtained between the computer simulation results and the experimental results (there was less agreements in the start-up phase because of the highly simplified model of the drive motors). The high utility value of the model allowed a study to be conducted to analyze the characteristics of the conveyors that are typical or deviate from the accepted rules for designing vibratory conveyors. Three such cases were analyzed: the displacement of the center 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 center of mass in relation to the resultant force of the vibrators had a very clear effect on the trajectory of motion of the conveyor trough, clearly differentiating the motion of the feed section and the discharge section. This undesirable feature increased the unevenness of material movement in the trough. However, this characteristic increased the value of the synchronizing moment of the vibrators, which, in the case of conveyors with a weak synchronizing bond, could have a positive effect. However, in the case of counterrotation vibrator drives, this characteristic should not be of great importance, as in this case, the synchronizing bond was 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 synchronizing torque. An advantage was proven by increasing the angle between the vibrator working plane 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 synchronizing the vibrators was the structure maintaining the condition of the center of mass and as big as possible the rotation angle of the vibrators’ working plane versus the main axes of the conveyor body’s inertia.

Interesting results were obtained for the conveyor operation near the oblique-angular resonance. The resonance had a great effect on the synchronization of the vibrators. In the case of a displacement of the center 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 was observed in the direction perpendicular to the plane of the conveyor work. In the typical case, in which the line of action passed through the center of mass, the synchronizing bond was broken, and beat phenomena of the body in all directions of the conveyor movement were observed. It should be noted that similar behavior was observed on industrial conveyors suspected of operating on the verge of the structural resonance of the conveyor body.

The issue of the influence of structural resonances of the machine body on the synchronization of inertial vibrators is particularly interesting in the field of the construction of slender bodies. You can meet them, for example, in the case of long conveyors or vibrating screens. However, the analysis of this issue is much more demanding in relation to the presented work because it requires modeling machine parts as deformable bodies, taking into account the problem of large rotation of vibrators. Due to the practical importance of this issue, it will be the goal of further research by the author related to the issues of constructing vibrating machines.

Acknowledgements

The author would like to acknowledge Rafal Sieńko for translating the article and preparing Figure No. 1 used in this work.

Funding information: This work was supported by AGH University of Science and Technology in Krakow (Grant No. 16.16.130.942).
Author contributions: Conceptualization, formal analysis and investigation, funding acquisition, supervision: Grzegorz Cieplok.
Conflict of interest: The author states that they not conflicts of interest.
Code availability: Computer simulations were based on own code implemented in OpenModelica, wxMaxima, and LIGGGHTS software. Other graphic materials were also prepared with the use of free software, such as FreeCad, InkSpace, and Paraview without breaking the license rights. The code generated during the current study is available from the corresponding author on reasonable request.
Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2023-01-24

Revised: 2023-03-17

Accepted: 2023-03-28

Published Online: 2024-01-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/eng-2022-0434

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spatial motion; self-synchronization; conveyor

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