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Figure 24- Tension Diagram
Figure 25
Analysis and Simulation
Many will argue the major reason for our ability to build complex conveyors as described above is advancements in the analysis and simulation tools available to the designer. A component manufacturer can usually test his product to insure it meets the specification; however the system engineer can seldom test the finished system until it is completed on site. Therefore computational methods and tools are absolutely critical to simulate the interactions of various perse disciplines and components.
Dynamic Starting and Stopping
When performing starting and stopping calculations per CEMA or DIN 22101 (static analysis), it is assumed all masses are accelerated at the same time and rate; in other words the belt is a rigid body (non-elastic). In reality, drive torque transmitted to the belt via the drive pulley creates a stress wave which starts the belt moving gradually as the wave propagates along the belt. Stress variations along the belt (and therefore elastic stretch of the belt) are caused by these longitudinal waves dampened by resistances to motion as described above. 7
Many publications since 1959 have documented that neglecting belt elasticity in high capacity and/or long length conveyors during stopping and starting can lead to incorrect selection of the belting, drives, take-up, etc. Failure to include transient response to elasticity can result in inaccurate prediction of:
Maximum belt stresses
Maximum forces on pulleys
Minimum belt stresses and material spillage
Take-up force requirements
Take-up travel and speed requirements
Drive slip
Breakaway torque
Holdback torque
Load sharing between multiple drives
Material stability on an incline
It is, therefore, important a mathematical model of the belt conveyor that takes belt elasticity into account during stopping and starting be considered in these critical, long applications.
A model of the complete conveyor system can be achieved by piding the conveyor into a series of finite elements. Each element has a mass and rheological spring as illustrated in Figure 26.
Figure 26
Many methods of analyzing a belt’s physical behavior as a rheological spring have been studied and various techniques have been used. An appropriate model needs to address:
1. Elastic modulus of the belt longitudinal tensile member
2. Resistances to motion which are velocity dependent (i.e. idlers)
3. Viscoelastic losses due to rubber-idler indentation
4. Apparent belt modulus changes due to belt sag between idlers
Since the mathematics necessary to solve these dynamic problems are very complex, it is not the goal of this presentation to detail the theoretical basis of dynamic analysis. Rather, the purpose is to stress that as belt lengths increase and as horizontal curves and distributed power becomes more common, the importance of dynamic analysis taking belt elasticity into account is vital to properly develop control algorithms during both stopping and starting.
Using the 8.5 km conveyor in Figure 23 as an example, two simulations of starting were performed to compare control algorithms. With a 2x1000 kW drive installed at the head end, a 2x1000 kW drive at a midpoint carry side location and a 1x1000kW drive at the tail, extreme care must be taken to insure proper coordination of all drives is maintained.
Figure 27 illustrates a 90 second start with very poor coordination and severe oscillations in torque with corresponding oscillations in velocity and belt tensions. The T1/T2 slip ratio indicates drive slip could occur. Figure 28 shows the corresponding charts from a relatively good 180 second start coordinated to safely and
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