CHoPS 2018 9th International Conference on Conveying and Handling of Particulate Solids (10th-14th September 2018) EXPERIMENTAL ANALYSIS OF LATERAL BELT MOVEMENT AND SIMULATION OF MISTRACKING OF CONVEYOR BELTS Hendrik Otto, Chair of Conveying Technology, University of Madeburg, Universitätsplatz 2, 39104 Magdeburg,Germany
[email protected] Andre Katterfeld, Chair of Conveying Technology, University of Madeburg, Universitätsplatz 2, 39104 Magdeburg, Germany
Key Words:
Belt Conveyor, Mistracking, Simulation, Measurements, Belt Training
Mistracking of the conveyor belt is one of the common problems during the operating of a belt conveyor. Reasons and counter-actions for belt mistracking are widely known. However, the cause of mistracking and the effect on the belt can only be predicted based on empirical experience. Known numerical simulation approaches like the Finite Element Method (FEM) cannot be used for a simulation of the belt mistracking because of its high computational effort and the incomplete understanding of the interaction between belt, idlers and the influencing boundary conditions. This paper is divided into two parts. The first part contains experimental investigations of belt mistracking on a new belt conveyor test rig. These tests were carried out to generate a quantitative understanding of belt mistracking and the influencing parameters. The test rig allows a forced mistracking of the belt on a conveyor with a belt width of 650 mm and an axial distance of 15 m. The measured belt movement is a result of disturbance variables within the belt conveyor. The influence of skewed idler stations to the belt tracking can be quantified with this experimental approach. In the second part of the paper a simulation method is presented which can be used for the prediction of the belt deformation and the sideways movement of belt on a conveyor with a tracking problem. Beside the mathematical background of the approach, a comparison of the simulated and the measured results are shown. This validation shows that the simulation method can be used for the prediction of belt mistracking. The developed approach will help to find the most efficient location for counter actions like training idlers or to identify the most influencing mistracking reason for a given belt conveyor design in the future. 1.
INTRODUCTION
Belt mistracking is one of the essential problems in running a belt conveyor. The problem of mistracking has been known since the industrial revolution, where flat belts were first used on drive trains. Belt conveyors were developed from flat belt drive trains. Their increased length increased the problem of tracking as well because a higher number of disturbance values can influence the belt. A sideways movement of a flat belt or a troughed conveyor belt occurs when the belt has a non-orthogonal contact angle on a pulley or an idler. From the angle and the belt velocity the velocity of sideways movement can be calculated. The belt does not move relative to the idler but a different contact point comes into contact while the belt is moving in the conveying direction. Because of the motion of the contact point, it seems that the belt is sliding sideways over the idlers. 2.
MEASUREMENTS ON A BELT CONVEYOR TEST RIG
To analyse the lateral movement of a conveyor belt for a skewed idler station, an experimental conveyor was built that allows a precise assembly of the idler stations on the framework. Head and tail pulley are mounted on a steel frame that introduce the belt tension forces into the rail system on the ground of experimental facility. In Fig. 1 a sketch of the test conveyor is shown. The framework is build out of 45 mm x 45 mm Bosch-Rexroth profiles which allow a modular frame that can easily be adapted for the study of other questions. The idler stations are positioned 0.5 m from each other on the frame. The conveyor can operate on 0 – 2 m/s belt velocity. The 650 mm wide belt is troughed by 30 °. Every 2.5 m a camera system connected to a raspberry pi recognises the belt edge and sends the data to a measuring windows PC. At the head pulley a Laser measuring system works as a reference to validate the signal of the cameras.
CHoPS 2018 9th International Conference on Conveying and Handling of Particulate Solids (10th-14th September 2018) Tensioning forces, power consumption and an incremental sensor for the feed rate are used to record the operational data of the conveyor.
Fig. 1: Sketch of the test rig at the University of Magdeburg During operation of the test conveyor the the 5th idler station is skewed in steps of 2° from the perpendicular position to 10°. The lateral movement is measured. In Fig. 2 it is shown that the belt has an eigen deformation because of fabrication derivations. These lead to a periodic signal which is known as the “Belt Tracking Signature” [6]. The eigen-deformation is mainly driven by an angle offset in the belt connection but also uneven pretensions in a steel cord belt or an uneven layer of fabric in a fabric belt can have an enormous influence [7]. The periodicity depends on the velocity of the belt because it repeats after one revolution of the belt. From this point of view it is better to refer the belt movement not to the time but to the distance of feed. In Fig. 2 three graphs are presented. They show the raw date of camera 4 (in the top), the extracted belt tracking signature (in the middle) and a clean signal as the difference of the first two signals (in the bottom). After three revolutions of the belt respectively every 150 m of belt travel, the idler station no. 5 is skewed by -2 ° until a maximum angle of -6 ° is reached in this measurement. From the lower graph in Fig. 2 it can be concluded that the relative amount of mistracking reduces with the increasing skewing angle. After three revolutions of the belt the lateral position reaches a steady state.
Fig. 2: Raw signal of the mistracking caused by an idler with an skewing angle of up to – 2 ° to - 6 ° in steps The results of the measurement are presented in Fig. 3. For the analysis the measurement is perform five times. The presented deformations and positions are determined by the average of the steady state at every camera. In the left figure the absolute position of the belt is shown. The maximum side travel is reached by an idler skew of 6 °. The amount of mistracking is asymptotically increasing up to a maximum of 5 mm if the value of 5.5 mm for 10 ° is ignored. The right figure shows the deformation of the belt at long the conveyor. The deformation also increases asymptotically with the angle of the skewing of the idler station. The deformation is limited by the maximum of the forces between belt and idler station which are limited to the maximum of the friction between belt and idler surface.
CHoPS 2018 9th International Conference on Conveying and Handling of Particulate Solids (10th-14th September 2018)
Fig. 3: Mistracking of the conveyor belt over the length of the conveyor between 2 ° and 10 ° of skewing angle (left) compared to the deformation of the belt (right) The maximum of deformation of the measurements appears between 2.5 m and 5 m behind the point with the skewed idler station. It is difficult to determine the exact defomation of the belt because on the one hand the lateral position can only be measured at the cameras and on the other hand the belt tracking signature can only be eliminated by the complex data filtering. When comparing the belt tracking signature with more than 8 mm with the deformation due to the skewed idler station with 1.5 – 2 mm, it is clear that the incluence of the belt tracking signure can have an importet influence to the maximum of force that the frictional connection between belt and idler can handle. 3.
SIMULATION MODEL FOR BELT MISTRACKING
SIMULATION MODEL The misalignment of flat belts can be described as a geometric effect [4]. The sideways movement of the belt does not depend on the forces in contact, but to geometric boundaries. For this a frictional contact between the belt and the idlers is assumed anytime during the simulation. This simplification is only valid if the forces for misalignment are small enough. The mathematical approach for the computation is a cantilever beam, which has the bending stiffness of a conveyor belt around the z-axis. This beam is hyperstatic because it has a contact support at each idler station. The solution can be calculated analytically and numerically. In this case the numeric approach should be used, because it is more efficient to adapt it to different boundary conditions.
Fig. 4: Definition of the contact angle and resulting velocities In Fig. 4 the determination for the misalignment angle α is shown. α is the angle between the normal of the idler and the tangent of the belt in the point of contact and it is given by: Equation 1 α=γ-β The point of first contact is the point of the deformed conveyor belt, where it comes into contact with the idler. Ideally this would be exact at the centre of the idler. Due to the deformation of a weight loaded belt, the point of
CHoPS 2018 9th International Conference on Conveying and Handling of Particulate Solids (10th-14th September 2018) first contact moves backwards on the conveyor. If α is equal to zero, the derivation of the beams contact point has the same value as the normal of the idler. So the belt will not move from its contact point sideways on the idler. For the computation of a whole belt, the belt is reduced to its centre axis. The belt is modelled as a linear beam with small deformation compared to its length. Generally the formulation of a linear Finite Element Model (FEM) can be described in one equation. The force vector 𝒇 equals the stiffness matrix 𝑲 multiplied with the deformation vector 𝒖 Equation 2 𝒇 = 𝑲𝒖. In this simulation both, force and displacement vector are partially known. So it is not possible to rearrange the Eq. 2 to its unknown vector. For the solution of the system, the system has to be divided into a part for fixed degrees of freedom and a part for free degrees of freedom. The evaluation of the elements of the displacement vector 𝒖 is done under the assumption that the belt conveying velocity is 𝑥̇ . The sideways movement 𝑥𝑡 in the reference system of an idler given by the contact angle and an arbitrary time interval 𝑡 Equation 3 𝑥𝑡 = − ∫ sin 𝛼 𝑥̇ 𝑑𝑡. 0
Following the deflection has to be transformed into the global coordinate system: 𝑥𝑔 = 𝑥𝑡 cos 𝛽.
Equation 4
For simulation of mistracking linear deformation of the belt is assumed. The body will be modelled as an isotropic two dimensional beam. Another method would be the extraction of the stiffness matrix from a commercial FEM program and the order reduction according to [3]. In this approach each node of the beam will have one rotational degree of freedom for the bending around the vertical axis and a translational degree orthogonal to the conveying direction. From Eq. 4 the displacement of nodes on idlers are known. These displacements are given for the computation of the system in Eq. 2. Therefore the equation must be parted into two pieces [2]. Those degrees of freedom which are used for the fixation of the beam on the idlers are collected in a mounting vector. The vector 𝒖 contains both, the mounted 𝒖𝑙 and the free degrees of freedom 𝒖𝑓 . 𝒖𝑓 Equation 5 𝒖 = [𝒖 ] 𝑙 The different sorting of displacements leads to a different sorting of stiffness matrix from Eq. (2) as well. 𝒇𝑙 are the forces in the contact zone, 𝒇𝑓 are the forces acting on the free nodes of the FEM system. 𝑲𝑓𝑓 𝑲𝑙𝑓 𝒖𝑓 𝒇𝑓 Equation 6 [ ][ ] = [ ] 𝑲𝑓𝑙 𝑲𝑙𝑙 𝒖𝑙 𝒇𝑙 For given boundary conditions the system in Eq. 6 can be solved for the deflection of the free nodes. −1 Equation 7 𝒖𝑓 = 𝑲𝑓𝑓 (𝒇𝑓 − 𝑲𝑙𝑓 𝒖𝑙 ) After that, the forces on the mounted nodes can be computed. 𝒇𝑓 = 𝑲𝑓𝑙 𝒖𝑓 − 𝑲𝑙𝑙 𝒖𝑙
Equation 8
Now all forces and deformations are known. This computation must be executed for each time step in a simulation of discrete time steps.
VERIFICATION OF SIMULATION RESULTS To use this described simulation model in practice, it has to be verified. For this purpose, an experiment was set up with a flat belt running over three idlers. The simulation model must show a good correlation between the movement of the flat belt and the simulation. If the correlation is not satisfying, it is not to be expected that the simulation model can be used for the complex problem of the troughed conveyor belt.
CHoPS 2018 9th International Conference on Conveying and Handling of Particulate Solids (10th-14th September 2018)
Fig. 5: Sketch of the measuring system of a flat belt test rig for the validation of the simulation model (left) and the occurring buckling (right) Fig. 5 shows the basic test setup of the experiment. The central idler (2) at the measurement point is rotated in the defined angle around the z-axis. The belt has a length of 4200 mm and a width of 50 mm. Before starting the measurement, the belt is perpendicular aligned to all idlers. The middle idler is rotated before the data acquisition is started. The experiment uses the same video camera that is also used on the troughed conveyor. The y-location of the belt is filmed and automatically analysed. For this purpose, an image processing algorithm was developed, which uses edge detection by contrast amplification. The positioning of the belt edge can be measured with an accuracy of 0.1 - 0.3 mm. One of the important measurement errors arise from lens distortions, but it can be eliminated by the constant width of the belt. The belt movement and the simulation results are compared in figure Fig. 6. In the simulation it is important to consider the buckling of the belt like it is shown in Fig. 5 (right). The buckling can be simulated as a local reduction of the stiffness. If buckling at the head and tail pulley is considered in the simulation, the experiment and the simulation show very good congruence.
Fig. 6: Results of measurement and simulation of belt mistracking for a skewing angle of 1 ° Simulations and measurements start with an undeformed belt. The in this case one at 1 ° skewed idler station in the middle between head and tail pulley drives the belt sideways. The lateral movement reaches a steady state where the derivation of the lateral position is constant. To reach this steady state the 4.2 m long belt needs one to two revolution. At the beginning of the first revolution the undeformed belt moves much faster because the contact angles are not influenced by the deformation of the belt in the contact area. If the buckling is consider in the simulation the deformation of the belt is not evenly distributed which allows a different contact angle at the middle idler [5]. With the comparison of the simulation of the flat belt system with the experiments, the simulation model can be seen as validated. 4.
SIMULATION OF A TROUGHED BELT CONVEYOR
The simulation of a troughed conveyor is performed by the same simulation model which is presented above. In contrast to this simulation the belt conveyor has more than three idlers. The conveyor is troughed by three idlers with an angle of 30 °. This is the main difference to a flat belt system. The trough of the belt acts as a self-centring effect. If the belt moves mistracked through the idler station the gravitational forces bends the belt sideways and
CHoPS 2018 9th International Conference on Conveying and Handling of Particulate Solids (10th-14th September 2018) leads to a contact angle smaller than when using a flat belt. A second self-centring effect is established by a crowned head pulley. Both of these effects limit the lateral movement of the belt to a maximum. This maximum is also influenced by the maximum of friction in the contact between the idler and the belt. The maximum friction force in this simulation model is set to an estimated value of 40 N. In Fig. 7 a direct comparison between the results of a skewing angle of 8 ° are shown.
Fig. 7: Results of measurement and simulation of a troughed belt conveyor for a skewing angle of 8 ° If the results are comparted, it is obviously that the total amount a lateral movement match in good congruence. The deformation in the simulation seems to be more uneven than the measurements. This has two reasons. The first reason is the distance of 2.5 m from camera to camera and a coarse resolution of the deformation in the measurement. The second reason lies in the extrapolation of the signal. At the tail pulley no camera is installed, so the measurement of camera 5 is connected to the measurement of the laser beam. This neglects deformations in the low run of the belt, which can have an influence that is not measureable, yet. 5.
CONCLUSION
Measurements and simulations are performed on flat a troughed belts. On flat belt systems the comparison match very well. On troughed belt conveyors the belt tracking signature, frictional contact between idlers and belt, the self-centring effects, as well as a low resolution of measurements points along the conveyor make it difficult to parameterize the simulation model and compare it against measurements. However, the presented measurements show good congruence in deformation and total lateral traveling distance. As a conclusion the simulation model are validated and reliable for flat belt systems and even for troughed conveyor belts. The parameters of the simulation like the maximum of lateral force will need further investigations. 6.
REFERENCES 1. P. Koutsovasilis, Model Order Reduction in Structural Mechanics, Dissertation, Technische Universität Dresden, 2009. 2. K.J. Bathe and P. Zimmermann, Finite-Elemente-Methoden, Springer, Berlin, Heidelberg, New York, 2. Edition, 2002. 3. U. F Meißner; A. Maurial.: Die Methode der finiten Elemente. 2. Berlin Heidelberg: Springer, 2000. 4. M. Egger, Laufverhalten von Flachbändern: Lenkmechanismen beim seitlichen Führen von Bändern, Stahleisen Communications, Verlag Stahleisen, Düsseldorf, 2008. 5. H. Otto; A. Katterfeld; Prediction and simulation of the mistracking of conveyors belts In: 8th International Conference on Conveying and Handling of Particulate Solids (CHOPS 2015). - Tel-Aviv, 2015. 6. M.P. Evans.: Conveyor Belt Engineering for the Coal and Mineral Mining Industries, Society for Mining, Metallurgy, and Exploration, Pennsylvania State University, 1993 7. S. Tappeiner: Häufige Ursachen für Qualitätsmängel an Stahlseilfördergurten. In: Fachtagung Schüttgutfördertechnik 20; Magdeburg, 2015.
