Two Decades Dynamics of Belt Conveyor Systems
Prof.dr.ir. Gabriel Lodewijks
Delft University of Technology, The Netherlands
Summary
The quest for a uful design tool that incorporated the effect of dynamics of conveyor belts on the design of a conveyor system started halfway the 1950's. It was however not until halfway the 1980's that the first uful design tool became available. In the early days of using dynamics of belt conveyor systems the attention was focud on an analysis of both the starts and the stops
of long overland, high tonnage/lift/speed conveyors. With the significant improvement of drive technology over the last twenty years however, it is now possible to start and (operationally) stop a belt conveyor in a very smooth manner. For an analysis of the non-stationary conditions the application of belt conveyor dynamics is not any longer required provided that sufficiently long starting and stopping times are ud. The attention therefore shifted to the analysis of emergency stops and the determination of 'what if' scenarios. It is an illusion to assume that a theoretical analysis gives all the an
swers. A practical verification of the results is of utmost importance to ensure that the assumptions made in the theoretical analysis were right and the advice given to the client correct.
This paper gives an overview of the work done on the mathematical description of dynamics of belt conveyor systems till date and briefly discuss the most important variables that effect belt conveyor dynamics. It will further give some practical recommendations and examples of the application of the belt conveyor dynamics in the design process of conveyor systems, the practical verification of the results and the lessons learned. Finally it will highlight the latest developments in the field and provide answers to frequently asked questions.
1 Introduction
Due to the development of rubber technology, conveyor belts improved significantly after the Second World War and the application of belt conveyor systems for the transportation of bulk materials became widespread. Besides the application for in-plant transportation of bulk materials, the improved belt types enabled application in long overland systems as well. Therefore the capacity as well as the length of belt conveyor systems incread significantly.
To calculate the total power required for driving a belt conveyor system, design standards like
DIN 22101 were, and still are, ud. In the standards the belt is assumed to be an inextensible body. This implies that the forces exerted on the belt during starting and stopping can be derived from Newtonian rigid body dynamics, which yields the belt stress. With this belt stress the maximum extension of the belt can be calculated. This way of determining the elastic respon of the belt is called the quasi-static (design) approach. For low capacity and small belt-conveyor systems this lead to an acceptable design and acceptable operational behavior of the belt. However, up scaling of belt conveyor systems to high capacity or long distance systems introduced operational problems including:
•excessive large displacement of the weight of the gravity take-up device
•premature collap of the belt, mostly due to the failure of the splices
•destruction of the pulleys and major damage of the idlers
•lifting of the belt off the idlers resulting in spillage of bulk material
•damage and malfunctioning of (hydrokinetic) drive systems
枸杞花
The problems, which did not occur in low capacity or short belt conveyor systems, triggered the qu南怀瑾简介
est for design tools that incorporated the effect of dynamics of conveyor belts on the design of
a conveyor system.
2 Modeling belt conveyor dynamics - a historic overview甪直镇
Looking at the development of design tools that incorporate the effect of dynamics of conveyor belts, three periods can be distinguished. The first and perhaps most important period was from 1955 - 1975, the cond period was from 1975 - 1995, and the last period from 1995 - today. The ctions below give a brief overview of each period. For a more extensive overview e [1].
2.1 The first period: 1955 - 1975
To detect the cau of the operational problems mentioned in the introduction, the behavior of the belt and drive system during non-stationary operation were experimentally investigated. Relevant studies include [2]-[5]. The first objective of the experiments was to determine the development of axial (longitudinal) stress waves in the belt and their effect on the belt tension and the drive force. Also the delay effect, caud by the finite propagation speed of the stress waves in the belt, which appears during non-stationary operation of the belt, was of interest.
Oehmen [6] experimentally studied the start-up of a belt conveyor system in detail, accounting for the characteristics of the drive system, the tensioning system and the belt. The results of Oehmen's study were supplemented by Vierling [7] and confirmed by experiments done by others [8]-[10]. With the knowledge gained from the experiments the starting quence of multiple-engine drive systems and the design of automatic tensioning systems could be improved, as has been described by a number of authors [11 ]-[15].
Field tests during starting and stopping of a belt conveyor are not always possible and the number of start-up and stop variations that can be tested is limited. Therefore the need for a mathematical model, which could be ud to obtain detailed insight in the dynamic behavior of a belt during non-stationary operation, incread. Becau of the complexity of the equations of motion that describe the behavior of a conveyor belt during non-stationary operation, the first mathematical models that were implemented for this purpo were the electrical analogue models, e for example [16]-[21]. Parallel to the development of the electrical analogue models, also analytical solutions of the equations of motion were developed. The first rearchers that did an attempt to develop an analytical solution were Havelka [22] and Sobolski [23]. Their simplified model however was not very practical and other rearchers tried to develop more functional models [24]-[29].
The first period was concluded by Funke [30] who developed the first really uful mathematical model. Funke discretid the belt into two homogenous and continuous elements that reprented the carrying and the return strand of the belt. The global elastic respon of the total belt was made up by the elastic respon of the two elements. The motion of the elements was coupled through the motion of the pulleys. His model included time dependent motion resistances and he considered the visco-elastic character of the belt material. With the results obtained from calculations with Funkes model, the insight in the behavior of the belt during non-stationary operation incread considerably. This enabled the improvement of the design of high capacity or long distance belt conveyor systems.
It was then recognid that discretisation of the belt in more than two belt parts would increa the accuracy of the calculations. Instead of using one or two elastic elements with homogenous mass distribution, the belt should be modelled by a number of finite elements to account for the variations of the resistances, mass and forces exerted on the belt. However, the application of this kind of models required advanced computational equipment that was not available at the end
of the sixties [31]. Besides Funke, also Rao [32] and Harrison [33] ud computers to study the development of transient stress during starting and stopping of the belt.
2.2 The cond period: 1975 - 1995
Funkes model [30] basically was a very rudimental finite element model with two elements. Nordell and Ciozda [34] developed the first finite element model of a belt conveyor system that was a significant step forward compared to the model of Funke. Their model included th time dependent drive force, motion resistances and visco-elastic behavior of the be material. Simulation of the dynamics of belt conveyors is one thing, visualisation of the result is another. Morrison [35] illustrated the power of applying computer graphics to visuali the simulation results. Verification of the simulation results has shown that finite element model of belt conveyors are quite successful in predicting the elastic respon of the belt during starting and stopping, e for example [34], [36]-[40]. Even today, the finite element model are still being developed to improve the description of for example the motion resistance
[41].
2.3 The third period: 1995 - today
All the models developed in the period 1975-1995 were made to study the dynamic behavior of a conveyor belt in the axial or longitudinal direction. The models therefore were one dimensional. An i
mportant effect that follows from the cond (vertical or transver) dimension and that has to be accounted for is the effect of the belt sag. Although it is possible to include the effect of the belt sag on the propagation of axial stress waves in a one dimensional model (e [1]) all finite element models as mentioned in ction 2.2 determine only the longitudinal elastic respon of the belt. They therefore fall short in the accurate determination of:
•the motion of the belt over the idlers and the pulleys
•the dynamic drive phenomena
•the bending resistance of the belt
•the development of stress waves with steep gradient of stress rate
•the interaction between the belt sag and the propagation of longitudinal stress waves •the interaction between the idler and the belt
•the influence of the belt speed on the stability of motion of the belt
•the dynamic stress in the belt during passage of the belt over a (driven) pulley
•the influence of parametric resonance of the belt due to the interaction between vibrations of the take up mass or eccentricities of the idlers and the transver
displacements of the belt
•the development of standing transver waves古装穿越电视剧
•the influence of the damping caud by bulk material and by the deformation of the cross-ctional area of the belt and bulk material during passage of an idler •the lifting of the belt off the idlers in convex and concave curves
The transver elastic respon of the belt is also often the cau of breakdowns in long belt-conveyor systems and should therefore be taken into account. The transver respon of a belt can be determined with special (isolated) models as propod in [42]-[43]. It is however more convenient to extend finite element models of belt conveyors with special two-dimensional elements as ud in multibody dynamics that take this respon into account. Lodewijks [1] & [44] developed the first finite element model of a belt conveyor using multibody dynamics. Other rearchers later followed a similar approach using multibody dynamics software packages [45]. Today, finite element models of belt conveyor systems incorporating components of multibody dynamics can be considere
d as state-of-the-art.
2.4 The future
Since today two-dimensional models are state-of-the-art the question is whether or not the future will bring full three-dimensional finite element models of belt conveyor systems. Adding the third dimension will give the opportunity to accurately study the dynamic behavior of conveyor belts in horizontal curves. This is an important issue since the lateral movement of a belt over the idlers determines whether or not the lection of a certain curve radius or idler arrangement is admissible or not. Other important issues are belt tracking and the occurrence of bending belt stress in the curves. The aspects could be studied using a full three dimensional model. One drawback of using a full three dimensional model is that the size of the model, compared to a two-dimensional model, increas with a factor 10 to 20. This implies that even more computational power is required than is necessary for handling a two dimensional model. For comparison: a two-dimensional model has 1000 to 2000 times the number of degrees of freedom (variables) of a one-dimensional model, a three dimensional model will therefore have 10,000 to 40,000 times the degrees of freedom of a one-dimensional model. With the current progress in computational processing speed however, this will eventually not be the bottleneck. A more important bottleneck will be the complexity of the model.
A simple one-dimensional finite element model is relatively easy to make, to u, and the results are relatively easy to understand. A three-dimensional model on the other hand is much more complicated and certainly (today and in the near future) not fit as a day-to-day engineering tool. Also, with increasing accuracy of the finite element model of the belt itlf, the accuracy of models of other conveyor components like the motor, fluid coupling etc. becomes more important as well. It therefore remains to be en whether or not three-dimensional finite element model will be applied to model belt conveyor systems.
An important development that will affect the future of belt conveyor systems is the application of dynamic simulation techniques in their control systems. Today, control systems of belt conveyors are rigid: they can control a start or stop procedure according to fixed procedures. Even in ca of the application of a proportional brake system the braking procedure is rigid, although the brake tting can be changed. Current control procedures do not account for the circumstances under which a conveyor stopped, in ca of a start, or started, in ca of a stop. Today, for example most starts are either velocity or torque controlled. No difference is made between the start of a fully loaded belt or an empty belt, where as far as power and belt tension is concerned, there is an important difference. The same holds for example for the conditions under which a conveyor stopped. As far as the contro
l procedures are concerned, no difference is made between a start after an emergency stop and a start after an operational stop. As far as belt tension distribution and nsitivity to the starting procedure is concerned there is an important difference. This implies that engineers are faced with the fact that they have to design control procedures that work more or less under all operational circumstances. The procedures however are far from optimal, they put the conveyor system in most (all normal) operational conditions under a relatively high strain just to make sure that they also work in the worst ca conditions.
In the future, large belt conveyor systems will have intelligent control systems that for example keep track of the load on the belt and the operational status of the system. Both the status and load conditions can be recognid and the optimum operational procedures can be determined using the results of dynamic simulations that are performed on the spot. Feedback from the conveyor system using nsors in for example the belt, the pulley lagging, the tensioning system and the drive system ensure that the input parameters for the dynamic calculations are kept up to date. The result will be that on one hand safety factors of the conveyor's components can be (further) decread and that on the other hand the overall system safety and reliability will increa. The rearch group of the author is already working on the smart control systems.
3 Important design parameters
For the description of the dynamic behavior of conveyor belts it is important to know at what speed stress (or tension) waves travel through a belt. Two wave speeds can be distinguished: c1
and c 2. Wave speed c 1 is the speed at which axial stress waves travel through a belt. Wave speed c 2 is the axial propagation speed of transver waves that determines the frequency of transver vibrations of a belt supported by two idlers (belt flap frequency). Their definition is as follows:
___ E b c 1 = √ ρb
___ ____
σ T c 2 = √ ρb = √ ρb A b
(1)In the above given equations is E b the Young's modulus of the belt, ρb the density of the belt, σ the belt stress, A b the cross ctional area of the belt and T the belt tension. Typically the wave propagation speed c 1 ranges from about 1,000 m/s for fabric belts to about 2,500 m/s for steelcord belts. Wave speed c 2 typically varies from about 25 till 100 m/s. The wave speeds
however, depend not only on the density of the belt material but on the weight of the bulk material on the belt and the load of the (reduced) mass of the supporting idlers as well.
If the two wave speeds are rewritten to:
活动预算方案___ ______ E b E b A b
c 1 = √ ρb = √ m'g _____ ___
T (2) T c 2 = √ ρb A b = √ m'g退休基金
where m'g is the belt weight per unit of length, then the effective wave speeds for an unloaded belt can be written as follows:
_______ _______ E b A b m'g c 1,effU = √ m'g + m'r = √ m'g + m'r
人设什么意思c 1 = C U c 1 (3)c 2,effU = c 2 where m'r is the reduce
d mass of th
e idlers per unit o
公司国庆放假通知
f length, and for a loaded, supported belt:
_____________ _____________ E b A b m'g c 1,effL = √ m'g + m'l + m'r = √ m'g + m'l + m'r c 1 = C L1c 1 ________ ________
T m'g c 2,effL = √ m'g + m'l = √ m'g + m'l c 2 = C L2c 2 (4)where m'l is the weight of bulk material on the belt. In practice the equations (4) reprent the lower limit of the speed calculation, and the equations (3) reprent the upper speed limit. When
