Control of the transport direction and velocity of the two-way reversible vibratory conveyor

It can be noticed that the feed transportation direction in this system changes before it enters the resonance zone (Fig. 5). It should be emphasised that, in the further part of this study, the system of an increased vibrator unbalance \(e=0.086\) was tested, which—as shown in [1]—provides the possibility of high transporting velocities at low excitation frequencies of the two-way conveyor.

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.

To facilitate the analysis of the conveyor operations, its natural frequencies should be determined. The conveyor cannot operate near its natural frequencies, which are mainly dependent on the mass of the trough. Frequencies mainly dependent on the masses of eliminators should be determined since the conveyor operates in their vicinity. It should be mentioned that none of the frequencies—to which eliminators are tuned—are equal to the natural frequency of the system. The resonance frequencies of the system can be analytically determined for this type of the conveyor. However, since this is a system in which five natural frequencies occur—three related to the flat movement of the body and two related to eliminators in two perpendicular directions—the solution of such a system is very difficult. It is much easier to determine the approximate frequencies by the system decomposition. Assuming that masses of eliminators are point masses, the frequency related to the system rotation can be determined from equation (numbering of frequency results from their numerical values—beginning from the lowest)The natural rotational frequency changes insignificantly during the machine operation due to variable distances \(f_1\) and \(f_2\), but since—in practice—values of these displacements are small and masses of eliminators are located near the system centre, the frequency of rotational vibrations can be approximately determined from the equationIn order to determine the remaining frequencies, meaning those which are related to additional degrees of freedom in the form of eliminators, it is possible—for calculation reasons—to simplify the system.

Since the eliminators operate in perpendicular directions, they do not influence each other on the resonance frequencies of the system, and if the suspension of the main mass has the same stiffness coefficients in directions x and y (\(k_x = k_y\)), the diagrams for the determination of resonance frequencies related to eliminators can be simplified as shown in Fig. 7 [4].

Equations of motion of this system are as follows:We expect that the solution will take a form:After insertion of functions (15) and their derivatives into equations of motion (14), the system of homogeneous equations, on account of amplitudes a and b as unknowns, is obtained. With regard to the physical sense of amplitudes, this system should have a nonzero solution. The condition of nonzero solutions of the system of homogeneous equations constitutes the zero value of the determinant formed of coefficients at unknowns. Equating the main determinant of coefficients of amplitudes to zero, it is possible to determine two frequencies of natural vibrations of the system, which, in the whole system, will equal \(\omega _1\) and \(\omega _4\) (due to growing values of successive frequencies). The main determinant of the Eq. (14) will be of the form:This determinant after evolving takes the form:Equation (17) has two positive roots:

In an analogous way, there is a possibility of determining frequencies related to the second eliminator (Fig. 7b), and these frequencies are:Introducing the parameters of the conveyor shown in Nomenclature, the following is obtained:

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.

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.

However, in this case, the classic feedback control in the closed loop would be difficult to apply due to the highly oscillatory character of the machine motion causing its destabilisation. On account of this, the implementation of a simple (and the most popular in industry) PID controller does not find application here. In return, the application of the open loop feed-forward control, in which the feed mass is treated as a measurable disturbance, directly influencing the actual control, seems sensible.

A proposition of such a system is presented in Fig. 10. The feed-forward 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.

The frequency values found in the described way are listed in Tab.1. Based on these measurements, one can find a continuous relationship between the feed mass value and the excitation frequency. This dependence can be approximated by simple linear regression or, for more accurate results, by nonlinear relation. The results of such approximations, using linear regression and nonlinear shape-preserving piecewise cubic Hermite interpolation, are shown in Fig. 11. Such defined dependency can be directly applied as a control law for the feed-forward controller. It should be emphasised that, in the further part of this study, nonlinear approximation was applied for simulations.

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.

The results of the system simulation in the case of changing the feed transport direction from the left to the right are shown in the right column of Fig. 12.

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.

The force in the eliminator 2 suspension and the force originating from the electro-vibrator projected on direction \(f_2\), for the case when the eliminator 2 is operating at partial frequency are presented in Fig. 14. These forces are in the reverse phase, which is due to the character of the eliminator operation. In the case of a small amount of feed or lack thereof, these forces should be equal (Fig. 14b). However, in the case of a large amount of feed (which occurs after 5 s of the simulation—Fig. 14c), the force in the eliminator should be lower due to the energy dissipation in the feed. In order to obtain this, the control system changes the excitation frequency, adjusting it according to the rule described in Sect. 4.3. Diagrams for the other eliminator are analogous.