Control of the transport direction and velocity of the twoway reversible vibratory conveyor
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Abstract
Vibratory conveyors are very important elements of production lines. Very often there is a need to change the feed transport velocity, especially in the case of feeders, or to alter the transport direction in the case of symmetric production lines. The transport possibilities of the new vibratory conveyor, allowing fast changes of the velocity and direction of transporting, solely by changes of the angular velocity of the excitation vibrator, and patented by the authors, are analysed in this paper. The innovatory control method allowing fast changes of the transport direction and good stabilisation of the given velocity, without the need to pass through its resonance frequencies, is presented. This new solution increases the safety and functionality of machine operations.
Keywords
Vibratory conveyor Resonance zone Direction and velocity control Feed transport1 Introduction
The point of departure for this paper is the authors’ previous research, concerning analysis of the dynamics of vibratory conveyors. This paper constitutes the development of ideas presented in the work [1], where the dynamics of the new conveyor solution patented by the author was analysed. Frahm’s eliminators were applied in this conveyor for damping the trough vibrations in the determined direction. The Frahm’s eliminator, in this case, is not the trough of the conveyor, giving the possibility of changing the direction of transport.
The problem of the control of such a machine is addressed in this paper. In particular, the application of a control system which would enable the feed transport in the reverse mode without the necessity for the system to pass through the resonance zone, related to the eliminator tuned for a lower frequency, was analysed. The influence of the change of the feed mass on the transport velocity and in consequence the reaction of the control system to external disturbances were also examined. Generally, the control law was derived from the specific properties of the dynamic characteristics of the machine.
The new contribution of this work is the development of a control system allowing the rapid change of a transport direction without passing the resonance zone. In addition, the proposed solution is resistant to such disturbances as changes of the feed mass, assuring constant transport velocity.
The feed model utilised in this work—elaborated in the work [2]—was applied in several papers by the authors, among others in [3], which presented the thorough analysis of the feed material transport velocity distribution along the trough, due to rotational vibrations of the whole conveyor caused by noncentral passing of the excitation force by the machine mass centre.
In the paper [4], frequently used conveyors—operating on the basis of the Frahm’s eliminator—in which the trough of the vibrator constitutes the eliminator of frame vibrations causing a significant decrease in forces transmitted to the foundation, were analysed. The strategy of controlling the excitation frequency of such conveyors when they are loaded with a significant feed mass was outlined. The work presented in this paper was further developed in [5], in which control of excitation frequency (initiated by the step change of the feed mass) of two counterrotating vibrators is provided in such way that the force transmitted to the foundations—at loading of the conveyor with the feed—was minimal.
2 Principle of operation of the twoway conveyor
The principle of the conveyor operations is based on the fact that, when the excitation frequency of the vibrator satisfies a condition \(\omega =\sqrt{\frac{k_{f1}}{m_1}}\), the mass \(m_1\) vibrates with the amplitude, which triggers—in the spring \(k_{f1}\)—forces opposite to the excitation force of the vibrator in the direction \(f_1\), causing—in accordance with the Frahm’s eliminator—at small damping in spring \(k_{f1}\), the excitation of the trough vibrations in this direction [7]. This does not change the vibrations in the perpendicular direction \(f_2\), which enables the feed transport to the left. When the vibrator excitation frequency, in the steady state, satisfies the condition \(\omega =\sqrt{\frac{k_{f2}}{m_2}}\), the mass \(m_2\) vibrates with the amplitude, which triggers in the element \(k_2\)—forces opposite to the excitation force of the vibrator in the direction \(f_2\) causing the extinction of the trough vibrations in this direction, but does not change the vibration in the perpendicular direction \(f_1\). This enables the feed transport to the right. This conveyor can be equipped with two or more vibration eliminators, attached to the trough in pairs from each side of the shaft.
3 Simulation of the proposed conveyor structure
The system composed of the model of conveyor, shown in Fig. 1, and the model of feed, shown in Fig. 2, was assumed for the purposes of computer simulation [8]. The model consists of the inertial vibrator of an independent induction drive, described by means of the static characteristics, the machine body performing the flat movement of three degrees of freedom (described by x, y and \(\alpha \)) and supported on a system of vertical springs with dampers of linear characteristic, and five, fourlayer models of the loose feed [2], distributed in various points of the machine work surface [9].

\(F_{j,j1,k}\) and \(T_{j,jj,k}\) are normal and tangent (respectively) components of the pressure of the jth layer on the \(j1\) layer in the kth column,

j—indicator of the material layer (\(j=0\) refers to the machine body),

k—indicator of the column of the material layer.

\(\mathcal {M}_{\max }\)—breakover torque of the drive motor,

\(\omega _{ss}\)—synchronous frequency of the drive motor,

\(\omega _{\max }\)—frequency of breakover of the drive motor.
4 Results of simulations
4.1 New way of changing the transport direction
It can be seen in Fig. 5 that the highest transport velocities are obtained in the resonance zone, which—in practice—is not technically realisable. On the other hand, the transport direction change can be obtained before the system enters the resonance. Figure 6 presents the dependence of the frequency of the trough horizontal vibrations on vertical ones for the excitation frequency: \(\omega _1=132.9\) rad/s and \(\omega _2=137.9\) rad/s. As can be seen in these figures, the character of vibrations enables the feed transport in the opposite direction, while similar amplitudes of these vibrations, as well as similar excitation frequencies, enable the transport in both directions with similar velocities.
The effect of the change of vibration direction occurs due to the fact that the eliminator, tuned to a lower frequency, ceases fulfilling condition: \(\omega =\sqrt{\frac{k_{f2}}{m_2}}\), and in consequence ceases to eliminate vibrations in its work direction \(f_2\), while the other eliminator, even though it still does not fulfil condition \(\omega =\sqrt{\frac{k_{f1}}{m_1}}\) (since the frequency is too low), is already in a large part—what results from the character of the Frahm’s eliminator operations—eliminating vibrations in the direction \(f_1\), enabling the possibility to obtain nearly rectilinear vibrations in the direction \(f_2\). It should be noted that this is still not the resonance frequency related to the first eliminator, and thus, the vibration amplitudes are at the level of the amplitudes in the operating point.
4.2 Determination of the resonance frequencies of the system
The simulation results confirm the correctness of the above assumption. An example of the amplitude–frequency diagram for eliminators is shown in Fig. 8. Frequencies determined from simplified systems (Fig. 7) on the basis of Eqs. (18, 19) are of a very high accuracy in the case of higher frequencies (\(\omega _4, \omega _5\)) and coincide with the simulated values. Also, frequency \(\omega _3\), related to the whole system rotation—determined from (13)—is the same as the simulated value. However, divergences occur in the case of lower frequencies (\(\omega _1, \omega _2\)). They are the result of the fact that the simulated trough system, together with eliminators, has five degrees of freedom (not two as in the simplified model) and in consequence vibrations in the other direction influence frequencies of the whole system. However, for efficient conveyor operation, the frequencies \(\omega _4\) and \(\omega _5\) are important, and the simplified model provides simple equations, easy for engineering applications. The accurate determination of \(\omega _1\) to \(\omega _3\) values is, in practice, not essential since these frequencies are usually significantly below the machine operation frequency.
4.3 Control of the transport direction and velocity
The control system must ensure the possibility of changing the feed transport direction as well as the stability of its velocity. The velocity of the feed displacement depends on its mass. Under normal working conditions, the feed mass can change in time, which would cause the transport velocity to change (see Fig. 9a). In order to prevent such a situation, the control system must adjust the excitation frequency appropriately, to stabilise the velocity.
A proposition of such a system is presented in Fig. 10. The feedforward controller takes two setpoints as inputs: velocity of conveying \(V_\mathrm{SP}\) and direction of conveying \(\hbox {Dir}_\mathrm{SP}\) (left or right). Based on the setpoints and the disturbance value, the controller computes the control value \(\omega _{\mathrm{CV}}\) (frequency of the drive motor), which provides desired process values: velocity of conveying \(V_\mathrm{PV}\) and direction of conveying \(\hbox {Dir}_\mathrm{PV}\).
The magnified fragment of diagrams showing the dependence of the feed transportation velocity in function of the excitation frequency for various feed masses is shown in Fig. 9b. It is clearly seen that—in this frequency range—the larger the feed mass, the higher the transport velocity. If the required velocity equals e.g. 0.1 m/s (marked in the figure by a dashed line), the coordinates of points of intersection of the velocity trajectory with this line can be read for various feed masses. It is also possible to read the frequency values necessary to ensure the needed displacement velocities for individual masses. In a similar fashion, the displacement in the opposite direction—for the velocity \(\,0.1\) m/s—can be found.
Values of the excitation frequency ensuring the displacement velocity of 0.1 m/s (movement to the right) and of \(\,0.1\) m/s (movement to the left) for individual mass values
Mass (kg)  Frequency (rad/s) (move left)  Frequency (rad/s) (move right) 

10  138.0102  133.2192 
20  138.0416  133.1250 
30  138.1358  132.6537 
40  138.1515  132.3082 
50  138.2615  131.7427 
5 Results of the simulation of the control system
5.1 Change of the transport direction
The results of the system simulation in the case of changing the conveying direction (of feed mass 20 kg) from the right to the left are presented in the left column of Fig. 12. In the fourth second of the simulation, the control changes linearly from the initial value of \(\omega _1=133.1\) rad/s to \(\omega _2 = 138.04\) rad/s in 1 s (Fig. 12b). The control change is accompanied by the change of the conveying direction (seen in Fig. 12b). Before the direction change, the average feed velocity was app. 0.1 m/s. After this change, the average velocity oscillated around \(\,0.1\) m/s value (Fig. 12a). The average velocity values were obtained by means of the first order recursive filter.
In the plot of the trough velocity (Fig. 12c), the pathway of the first half of the simulation time (i.e. from 0 to 5 s)—before the transport direction changes—was drawn in blue, and the pathway of the second half of the simulation time (i.e. from 5 to 10 s)—after the transport direction changes—in orange. Due to that, both vibration directions are clearly visible.
5.2 Stabilisation of the transport velocity
A significantly different control problem is the stabilisation of the transport velocity in the event of outside disturbances, which in this case are mainly changes of the feed mass load.
Simulation experiments based on the step change of the mass of the feed during the system operation were performed. In the first case, the influence of the step change of the feed mass (from 10 to 50 kg) on its velocity without the control system was shown. The excitation frequency was constant (133.22 rad/s). A drift of the average conveying velocity from the given value (0.1 m/s) starting in the fifth second of the simulation, i.e. in the moment of the step change, is shown in Fig. 13a (left column).
In the second case, the control system, by changing the excitation frequency (based on the dependence between the feed mass values and the excitation frequency found in the previous section), stabilises the velocity at the given level. It is seen in Fig. 13a (right column) that, due to the controller intervention (i.e. step change of the excitation frequency from 133.22 to 131.74 rad/s in the fifth second of the simulation), the average feed velocity is maintained at the given velocity 0.1 m/s, regardless of the mass step change.
6 Conclusions
In the classic constructions of vibratory conveyors, the change of the transport direction is related to the necessity of stopping the vibrator and starting it in the opposite direction. Such operation requires passing the whole system through the resonance zone and takes a lot of time. It is also very disadvantageous, since it is always related to the occurrence of vibrations of an amplitude several times higher than in normal operation.
The concept of the authors’ vibratory conveyor of a simple and reliable construction, presented in this paper, offers a solution to this problem. In this conveyor, the fast change of the transportation direction of the material is possible only by the change of the excitation frequency of the vibrator without the need of changing the direction of its rotation. Moreover, during the change of the direction there is no need to pass through any resonance zone, which improves the safety and reliability of the machine operation and significantly accelerates its reaction time for the change of direction.
A separate problem is the influence of the feed mass on its transport velocity. In the classic conveyor, the mass increase raises the transport velocity, which—from the point of view of industrial applications—is highly disadvantageous. On account of this, the control system and the stabilisation of the feed transport velocity are realised by the simple feedforward controller utilising nonlinear dependency between the feed mass and excitation frequency, as proposed in this work. This controller—on the basis of continuously measuring the feed mass—adjusts the excitation frequency to assure the constant transport velocity of the material in both directions. For industrial use, the digital filtration of the mass measurements should be applied, and relevant dead zones of the controller insensitiveness should be introduced to avoid its excessive reaction to measurement disturbances.
Finally, it seems worth adding that, for the correct operation of the controller, continuous and reliable measuring of the feed mass is necessary, which in some industrial applications can be difficult. However, instead of measuring the mass, measurement of the force transmitted to the foundation by the machine can be carried out, on the basis of which the mass can be calculated, which in some cases can be simpler.
7 Nomenclature
\(l_1=l_2=0.5\) (m)  Distances from the mass centre to the flexible suspension 
\(h=0\) (m)  Distance from centre of main mass to trough surface 
\(H=0\) (m)  Distance from centre of main mass to the suspension hitch 
\(b_x=b_y=50\) (Ns/m)  Coefficients of suspension damping in X and Y directions 
\(b_{f1}=60\) (Ns/m)  Coefficient of suspension damping in eliminator 1 
\(b_{f2}=50\) (Ns/m)  Coefficient of suspension damping in eliminator 2 
\(k_x=k_y=75{,}000\) (N/m)  Coefficients of suspension stiffness in X and Y directions 
\(k_{f1}=444{,}132\) (N/m)  Coefficient of suspension stiffness of eliminator 1 on its operation direction 
\(k_{f2}=320885\) (N/m)  Coefficient of suspension stiffness of eliminator 2 on its operation direction 
\(m_1=m_2=18\) (kg)  Masses of eliminators 
\(m_r=120\) (kg)  Mass of the conveyor trough 
\(m_w=5\) (kg)  Mass of the inertial vibrator 
\(J_r=25\) (kg/m\(^2\))  Mass moment of the trough inertia 
\(J_w=0.1\) (kg/m\(^2\))  Mass moment of the vibrator inertia 
\(\phi =variable\)  Rotation angle of inertial vibrator 
\(\alpha =variable\)  Rotation angle of the main mass 
\(\beta _1=45^{\circ }\)  Angle at which eliminator 1 is suspended 
\(\beta _2=135^{\circ }\)  Angle at which eliminator 2 is suspended 
\(R=0.05\)  Restitution rate 
\(\mu =0.4\)  Friction coefficient between feed elements and the trough 
\(p=1\)  Hertz–Stajerman coefficient for flat surfaces 
\(k_\mathrm{s}=10^8\) (N/m)  Feed stiffness coefficient 
\(e=0.086\) (m)  Eccentricity of the inertial vibrator 
\(\omega _{ss}=variable\) (rad/s)  Synchronous frequency of the drive motor 
\(\omega _{\max }=variable\) (rad/s)  Frequency of breakover of the drive motor 
\(M_{\max }=49.4\) (Nm)  Breakover torque of the drive motor 
Notes
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