The Two-Dimensional Dynamic Behavior of Conveyor Belts
Ir. G. Lodewijks, Delft University of Technology, The Netherlands
1. SUMMARY 1--------In this paper a new finite element
model of a belt-conveyor system will be introduced. This model has been developed in order to be able to simulate both the longitudinal and transverse dynamic response of the belt during starting and stopping. Application of the model in the design stage of long overland belt-conveyor systems enables the engineer, for example, to design proper belt-conveyor curves by detecting premature lifting of the belt off the idlers. It also enables the design of optimal idler spacing and troughing configuration in order to ensure resonance free belt motion by determining (standing) longitudinal and transverse belt vibrations. Application of feed-back control techniques enables the design of optimal starting and stopping procedures whereas an optimal belt can be selected by taking the dynamic properties of the belt into account. 2. INTRODUCTION 2--------The Netherlands has long been recognised as a country in which transport and transhipment play a major role in the economy. The port of Rotterdam, in particular is known as the gateway to Europe and claims to have the largest harbour system in the world. Besides the large numbers of containers, a large volume of bulk goods also passes through this port. Not all these goods are intended for the Dutch market, many have other destinations and are transhipped in Rotterdam. Good examples of typical bulk goods that are transhipped are coal and iron ore, a significant part of which is intended for the German market. In order to handle the bulk materials a wide range of different mechanical conveyors including belt-conveyors is used. 3--------The length of most belt-conveyor systems erected in the Netherlands is relatively small, since they are mainly used for in-plant movement of bulk materials. The longest belt-conveyor system, which is about 2 km long, is situated on the Maasvlakte, part of the port of Rotterdam, where it is used to transport coal from a bulk terminal to an electricity power station. In addition to domestic projects, an increasing number of Dutch engineering consultancies participates in international projects for the development of large overland belt-conveyor systems. This demands the understanding of typical difficulties encountered during the development of these systems, which are studied in the Department of Transport Technology of the Faculty of Mechanical
Engineering, Delft University of Technology, one of the three Dutch Universities of Technology. 4--------The interaction between the conveyor belt properties, the bulk solids properties, the belt conveyor configuration and the environment all influence the level to which the conveyor-system meets its predefined requirements. Some interactions cause troublesome phenomena so research is initiated into those phenomena which cause practical problems, . One way to classify these problems is to divide them into the category which indicate their underlying causes in relation to the description of belt conveyors. 5---------The two most important dynamic considerations in the description of belt conveyors are the reduction of transient stresses in non-stationary moving belts and the design of belt-conveyor lay-outs for resonance-free operation, . In this paper a new finite element model of a belt-conveyor system will be presented which enables the simulation of the belt's longitudinal and transverse response to starting and stopping procedures and it's motion during steady state operation. It's beyond the scope of this paper to discuss the results of the simulation of a start-up procedure of a belt-conveyor system, therefore an example will be given which show some possibilities of the model。 3. FINITE ELEMENT MODELS OF BELT-CONVEYOR SYSTEMS 6--------If the total power supply, needed to drive a belt-conveyor system, is calculated with design standards like DIN 22101 then 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 response of the belt is called the quasi-static (design) approach. For small belt-conveyor systems this leads to an acceptable design and acceptable operational behavior of the belt. For long belt-conveyor systems, however, this may lead to a poor design, high maintenance costs, short conveyor-component life and well known operational problems like :
? ? ? ? ?
excessive large displacement of the weight of the gravity take-up device premature collapse 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 which can result in spillage of bulk material damage and malfunctioning of (hydrokinetic) drive systems
Many researchers developed models in which the elastic response of the belt is taken into account in order to determine the phenomena responsible for these problems. In most models the belt-conveyor model
consists of finite elements in order to account for the variations of the resistance's and forces exerted on the belt. The global elastic response of the belt is made up by the elastic response of all its elements. These finite element models have been applied in computer software which can be used in the design stage of long belt-conveyor systems. This is called the dynamic (design) approach. Verification of the results of simulation has shown that software programs based on these kind of belt-models are quite successful in predicting the elastic response of the belt during starting and stopping, see for example  and . The finite element models as mentioned above determine only the longitudinal elastic response of the belt. Therefore they fail 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 (shock) stress waves 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 stresses 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 transverse displacements of the belt the development of standing transverse waves the influence of the damping caused by bulk material and by the deformation of the cross- sectional 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 transverse elastic response of the belt is often the cause of breakdowns in long belt-conveyor systems and should therefore be taken into account. The transverse response of a belt can be determined with special models as proposed in  and , but it is more convenient to extend the present finite element models with special elements which take this response into account. 3.1 THE BELT A typical belt-conveyor geometry consisting of a drive pulley, a tail pulley, a vertical gravity take-up, a number of idlers and a plate support is shown in Figure 1. This geometry is taken as an example to illustrate how a finite
element model of a belt conveyor can be developed when only the longitudinal elastic response of the belt is of interest. Since the length of the belt part between the drive pulley and the take-up pulley, Is, is negligible compared to the length of the total belt, L, these pulleys can mathematically be combined to one pulley as long as the mass inertia's of the pulleys of the take-up system are accounted for. Since the resistance forces encountered by the belt during motion vary from place to place depending on the exact local (maintenance) conditions and geometry of the belt conveyor, these forces are distributed along the length of the belt. In order to be able to determine the influence of these distributed forces on the motion of the belt, the belt is divided into a number of finite elements and the forces which act on that specific part of the belt are allocated to the corresponding, element. If the interest is in the longitudinal elastic response of the belt only then the belt is not discredited on those places where it is supported by a pulley which does not force its motion (slip possible). The last step in building, the model is to replace the belt's drive and tensioning system by two forces which represent the drive characteristic and the tension forces. The exact interpretation of the finite elements depends on which resistance's and influences of the interaction between the belt and its supporting structure are taken into account and the mathematical description of the constitutive behavior of the belt material. Depending on this interpretation, the elements can be represented by a system of masses, springs and dashpots as is shown in Figure 1, , where such a system is given for one finite element with nodal points c and c+ 1. The springs K and dashpot H represent the visco-elastic behavior of the belt's tensile member, G represents the belt's variable longitudinal geometric stiffness produced by the vertical acting forces on the belt's cross section between two idlers, V represent the belts velocity dependent resistance's.
Figure 1: Five element composite model .
3.1.1 NON LINEAR TRUSS ELEMENT If only the longitudinal deformation of the belt is of interest then a truss element can be used to model the elastic response of the belt. A truss element as shown in Figure 2 has two nodal points, p and q, and four displacement parameters which determine the component vector x:
xT = [up vp uq vq]
For the in-plane motion of the truss element there are three independent rigid body motions therefore one deformation parameter remains which describes
Figure 2: Definition of the displacements of a truss element
the change of length of the axis of the truss element : ε1 = D1(x) = ∫? o ds?- ds? o 2ds? o dξ
where dso is the length of the undeformed element, ds the length of the deformed element and ξ a dimensionless length coordinate along the axis of the element.
Figure 3: Static sag of a tensioned belt
Although bending, deformations are not included in the truss element, it is possible to take the static influence of small values of the belt sag into account. The static belt sag ratio is defined by (see Figure 3): K1 = δ/1 = q1/8T (3)
where q is the distributed vertical load exerted on the belt by the weight of the belt and the bulk material, 1 the idler space and T the belt tension. The effect of the belt sag on the longitudinal deformation is determined by : εs = 8/3 K?s (4)
which yields the total longitudinal deformation of the non linear truss element:
3.1.2 BEAM ELEMENT
Figure 4: Definition of the nodal point displacements and rotations of a beam element.
If the transverse displacement of the belt is being of interest then the belt can be modelled by a beam element. Also for the in-plane motion of a beam element, which has six displacement parameters, there are three independent rigid body motions. Therefore three deformation parameters remain: the longitudinal deformation parameter, ε1, and two bending deformation parameters, ε2 and ε3.
Figure 5: The bending deformations of a beam element
The bending deformation parameters of the beam element can be defined with the component vector of the beam element (see Figure 4):
xT = [up vp ? uq vq ? p q] (5)
and the deformed configuration as shown in Figure 5:
ε2 = D2(x) = e2p1pq 1o -eq21pq 1o (6)
ε3 = D3(x) =
3.2 THE MOVEMENT OF THE BELT OVER IDLERS AND PULLEYS The movement of a belt is constrained when it moves over an idler or a pulley. In order to account for these constraints, constraint (boundary) conditions have to be added to the finite element description of the belt. This can be done by using multi-body dynamics. The classic description of the dynamics of
multi-body mechanisms is developed for rigid bodies or rigid links which are connected by several constraint conditions. In a finite element description of a (deformable) conveyor belt, where the belt is discretised in a number of finite elements, the links between the elements are deformable. The finite elements are connected by nodal points and therefore share displacement parameters. To determine the movement of the belt, the rigid body modes are eliminated from the deformation modes. If a belt moves over an idler then the length coordinate ξ, which determines the position of the belt on the idler, see Figure 6, is added to the component vector, e.g. (6), thus resulting in a vector of seven displacement parameters.
Figure 6: Belt supported by an idler.
There are two independent rigid body motions for an in-plane supported beam element therefore five deformation parameters remain. Three of them, ε1, ε2 and ε3, determine the deformation of the belt and are already given in 3.1. The remaining two, ε4 and ε5, determine the interaction between the belt and the idler, see Figure 7.
Figure 7: FEM beam element with two constraint conditions.
These deformation parameters can be imagined as springs of infinite stiffness. This implies that: ε4 = D4(x) = (rξ + u ξ)e2 - rid.e2 = 0 ε5 = D5(x) = (r ξ + uξ)e1 - rid.e1 = 0 (7)
If during simulation ε4 > 0 then the belt is lifted off the idler and the constraint conditions are removed from the finite element description of the belt.
3.3 THE ROLLING RESISTANCE In order to enable application of a model for the rolling resistance in the finite element model of the belt conveyor an approximate formulation for this resistance has been developed, . Components of the total rolling resistance which is exerted on a belt during motion three parts that account for the major part of the dissipated energy, can be distinguished including: the indentation rolling resistance, the inertia of the idlers (acceleration rolling resistance) and the resistance of the bearings to rotation (bearing resistance). Parameters which determine the rolling resistance factor include the diameter and material of the idlers, belt parameters such as speed, width, material, tension, the ambient temperature, lateral belt load, the idler spacing and trough angle. The total rolling resistance factor that expresses the ratio between the total rolling resistance and the vertical belt load can be defined by: ft = fi + fa + fb (8)
where fi is the indentation rolling resistance factor, fa the acceleration resistance factor and fb the bearings resistance factor. These components are defined by: Fi = CFznzh nhD-nD VbnvK-nk NTnT fa = fb = Mred ??u Fzb Mf Fzbri ?t? (9)
where Fz is distributed vertical belt and bulk material load, h the thickness of the belt cover, D the idler diameter, Vb the belt speed, KN the nominal percent belt load, T the ambient temperature, mired the reduced mass of an idler, b the belt width, u the longitudinal displacement of the belt, Mf the total bearing resistance moment and ri the internal bearing radius. The dynamic and mechanic properties of the belt and belt cover material play an important role in the calculation of the rolling resistance. This enables the selection of belt and belt cover material which minimise the energy dissipated by the rolling resistance. 3.4 THE BELT'S DRIVE SYSTEM To enable the determination of the influence of the rotation of the components of the drive system of a belt conveyor, on the stability of motion of the belt, a model of the drive system is included in the total model of the belt conveyor. The transition elements of the drive system, as for example the reduction box, are modelled with constraint conditions as described in section 3.2. A
reduction box with reduction ratio i can be modelled by a reduction box element with two displacement parameters, ?p and ?q, one rigid body motion (rotation) and therefore one deformation parameter: εred = Dred(x) = i?p + ?q = 0 (10)
To determine the electrical torque of an induction machine, the so-called two axis representation of an electrical machine is adapted. The vector of phase voltages v can be obtained from: v = Ri + ωsGi + L ?i/?t (11)
In eq. (11) i is the vector of phase currents, R the matrix of phase resistance's, C the matrix of inductive phase resistance's, L the matrix of phase inductance's and ωs the electrical angular velocity of the rotor. The electromagnetic torque is equal to: Tc = iTGi (12)
The connection of the motor model and the mechanical components of the drive system is given by the equations of motion of the drive system: Ti = Iij ???j ?t? + Cik ??k ?t Kil? (13)
where T is the torque vector, I the inertia matrix, C the damping matrix, K the stiffness matrix and ? the angle of rotation of the drive component axis's. To simulate a controlled start or stop procedure a feedback routine can be added to the model of the belt's drive system in order to control the drive torque. 3.5 THE EQUATIONS OF MOTION The equations of motion of the total belt conveyor model can be derived with the principle of virtual power which leads to : fk - Mkl ??x1 / ?t? = ζ1Dik (14)
where f is the vector of resistance forces, M the mass matrix and σ the vector of multipliers of Lagrange which may be interpret as the vector of stresses dual to the vector of strains ε. To arrive at the solution for x from this set of equations, integration is necessary. However the results of the integration have to satisfy the constraint conditions. If the zero prescribed strain components of for example e.g. (8) have a residual value then the results of
the integration have to be corrected, also see . It is possible to use the feedback option of the model for example to restrict the vertical movement of the take-up mass. This inverse dynamic problem can be formulated as follows. Given the model of the belt and its drive system, the motion of the take-up system known, determine the motion of the remaining elements in terms of the degrees of freedom of the system and its rates. It is beyond the scope of this paper to discuss all the details of this option. 3.6 EXAMPLE Application of the FEM in the desian stage of long belt conveyor systems enables its proper design. The selected belt strength, for example, can be minimised by minimising, the maximum belt tension using the simulation results of the model. As an example of the features of the finite element model, the transverse vibration of a span of a stationary moving belt between two idler stations will be considered. This should be determined in the design stage of the conveyor in order to ensure resonance free belt support. The effect of the interaction between idlers and a moving belt is important in belt-conveyor design. Geometric imperfections of idlers and pulleys cause the belt on top of these supports to be displaced, yielding a transverse vibration of the belt between the supports. This imposes an alternating axial stress component in the belt. If this component is small compared to the prestress of the belt then the belt will vibrate in it's natural frequency, otherwise the belt's vibration will follow the imposed excitation. The belt can for example be excitated by an eccentricity of the idlers. This kind of vibrations is particularly noticeable on belt conveyor returns. Since the frequency of the imposed excitation depends on the angular speed of the pulleys and idlers, and thus on the belt speed, it is important to determine the influence of the belt speed on the natural frequency of the transverse vibration of the belt between two supports. If the frequency of the imposed excitation approaches the natural frequency of transverse vibration of the belt, resonance phenomena occur. The results of simulation with the finite element model can be used to determine the frequency of transverse vibration of a stationary moving belt span. This frequency is obtained after transformation of the results of the transverse displacement of the belt span from the time domain to the frequency domain using the fast fourier technique. Besides using the finite element model also an analytical approach can be used. The belt can be modelled as a prestressed beam. If the bending stiffness of the belt is neglected, the transverse displacements are small compared to the idler space, Ks << 1, and the increase of the belt length due to the transverse displacement is negligible compared to its initial length, the transverse
vibration of the belt can be approximated by the following linear differential equation, also see Figure 5:
= (c? - C? 2 b)
where v is the transverse displacement of the belt and c2 the wave speed of the transverse waves defined by, : c2 = √g1/8Ks (16)
The first natural transverse frequency of the belt span of Figure 5 can be obtained from eq. (16) if it is assumed that v(O,t)=v(l,t)=0: fb = 1 21 c2 (1 - ?? ) (17)
where ? is the dimensionless speed ratio defined by: ? = Vb / c2 (18)
The frequency fb is different for each individual belt span since the belt tension varies over the length of the conveyor. The excitation frequency of an idler which has a single eccentricity is equal to: fi = Vb / πD (19)
where D is the diameter of the idler. In order to design a resonance free belt support the idler space is subjected to the following condition: L≠ πD 2? (1-?? ) (20)
The results obtained with the linear differential equation (16) however are valid only for low values of the ratio ?. For higher values of ?, as is the case for high-speed conveyors or low belt tensions, the non-linear terms in the full form of e.g. (16) become significant. Therefore numerical simulations using, the FEM model have been made in order to determine the ratio between the linear and the non-linear frequency of transverse vibration of a belt span. These relations have been determined for different values of ? as a function of the sag ratio Ks.
The results for the transverse displacements were transformed to a frequency spectrum using a fast-fourier technique. The frequencies obtained from these spectra were compared to the frequencies obtained from e.g. (18) which yielded the curves as shown in Figure 8. From this figure it follows that for ? smaller that 0.3 the calculation errors are small. For higher values of ? the calculation error made by a linear approximation is more than 10 %. Application of a finite element model of the belt which uses non-linear beam elements therefore enables an accurate determination of the transverse vibrations for high values of ?. For lower values of ? the frequencies of transverse vibration can also be predicted accurate by e.g. (18). However it is not possible to analyse, for example, the interaction between the belt sag and the propagation of longitudinal waves or the lifting of the belt off the idlers as can be done with the finite element model. The determined relation between the belt stress and the frequency of transverse vibrations can also be used in belt tension monitoring systems.
Figure 8: Ratio between the linear and the non-linear frequency of transverse vibration of a belt span supported by two idlers.
4. EXPERIMENTAL VERIFICATION In order to be able to verificate the results of the simulations, experiments have been carried out with the dynamic test facility shown in Figure 9.
Figure 9: Dynamic test facility
With this test facility the transverse vibration of an unloaded flat belt span between two idlers, as for example a return part, can be determined. An acoustic device is used to measure the displacement of the belt. Besides that,
also the tensioning force, belt speed, motor torque, idler rotations and idler space were known during the experiments. 5. EXAMPLE Since the most cost-effective operation conditions of belt conveyors occur in the range of belt widths 0.6 - 1.2 m , the belt's capacity can be varied by varying the belt speed. However before the belt speed is varied the interaction between the belt and the idler should be determined in order to ensure resonance free belt support. To illustrate this the transverse displacement of a stationary moving belt span between two idlers have been measured. The total belt length L was 52.7 m, the idler space I was 3.66 m, the static sag ratio Ks 2.1 %, ? was 0.24 and the belt speed Vb 3.57 m/s. After transformation of this signal by a fast fourier technique the frequency spectrum of Figure 5 was obtained. In Figure 5 three frequencies appear. The first frequency is caused by the passage of the belt splice: fs = Vb/L = 0.067 Hz The second frequency, which appears at 1.94 Hz, is caused by the transverse vibration of the belt.
Figure 10: Frequencies of transverse vibration of a stationary moving belt span supported by two idlers.
The third frequency which appears at 10.5 Hz is caused by the rotation of the idlers. From the numerical simulations Figure 11 was obtained.
Figure 11: Calculated resonance zone's for different idler diameters D. Cross indicates belt speed and idler space during experiment.
Figure 11 shows the zone's where resonance caused by the belt/idler interaction may be expected for three idler diameters. The idlers of the belt conveyor had a diameter of 0.108 m thus resonance phenomena may be expected nearby a belt speed of 0.64 m/s. To check this, the maximum transverse displacement of the belt span has been measured during a start-up of the conveyor.
Figure 12: Measured ratio of the standard deviation of the amplitude of transverse vibration and the static belt sag.
As can be seen in Figure 12 the maximum amplitude of the transverse vibration occur at a belt speed of 0.64 m/s as was predicted by the results of simulation with the finite element model. Therefore the belt speed should not be chosen nearby 0.64 m/s. Although a flat belt is used for the experiments and the theoretical verification, the applied techniques can also be used for troughed belts. 6. CONCLUSIONS Application of beam elements in finite element models of belt conveyors enable the simulation of the transverse displacement of the belt thus enabling the design of resonance free belt supports. The advantage of applying beam elements for small values of ? instead of using a linear differential equation to predict resonance phenomena is that also the interaction between the longitudinal and transverse displacement of the belt and the lifting of the belt off the idlers can be predicted from simulation. 7. REFERENCES 1. Lodewijks, G. (1995), "Present Research at Delft University of Technology, The Netherlands", 1995 5th International Conference on Bulk Material Storage, Handling and transportation, Newcastle, Australia, 10-12 July 1995, The Institution of Engineers, Australia Preprints pp. 381-394. 2. Roberts, A.W. (1994), "Advances in the design of Mechanical Conveyors", Bulk Solids Handling 14, pp. 255-281. 3. Nordell, L.K. and Ciozda, Z.P. (1984), "Transient belt stresses during starting and stopping: Elastic response simulated by finite element methods", Bulk Solids Handling 4, pp. 99-104.
4. Funke, A. and K? nneker, F.K. (1988), "Experimental investigations and theory for the design of a long-distance belt-conveyor system", Bulk Solids Handling 8, pp. 567-579. 5. Harrison, A. (1984), "Flexural behaviour of tensioned conveyor belts", Bulk Solids Handling 4 pp. 67-71. 6. Lodewijks, G. (1994), "Transverse vibrations in flexible belt systems", Delft University of Technology, report no. 94.3.TT.4270. 7. Lodewijks, G. (1994), "On the Application of Beam Elements in Finite Element Models of Belt Conveyors, Part 1", Bulk Solids Handling 14, pp. 729-737. 8. Lodewijks, G. (1995), "The Rolling Resistance of Belt Conveyors", Bulk Solids Handling 15, pp. 15-22. 9. Nordell, L.K. and Ciozda, Z.P. (1984), "Transient belt stresses during starting and stopping: Elastic response simulated by finite element methods", Bulk Solid Handling 11, pp. 99-104.
伊. 基. 劳德维加克斯，代尔夫特科技大学，荷兰
本文将介绍一种新的皮带输送系统的有限元模型。 该模型被开发成能用于模 拟皮带在启动和停止时的纵向和横向动态响应。 使工程师能在长距离陆路皮带输 送系统的设计阶段应用该模型， 例如，设计适当的皮带输送机曲线检测元件过早 解除皮带张紧轮。这也能使张紧轮间距和凹槽轮廓的设计最优化，以确保无带运 动的共振和确定纵向和横向带振动。 应用反馈控制技术实现了启动和停止程序的 优化设计，因而计算皮带的动态特性时可以选择最理想的皮带。
荷兰一直以来被认为是一个运输和转运行业在经济中扮演重要角色的国家。 特别是被称为欧洲的门户的鹿特丹港口，声称拥有世界上最大的海港系统。除了 数量庞大的集装箱， 大量的散装货物也都是要通过这个港口的。并非所有这些物 品的目的地都是在荷兰市场， 许多要通往其他目的地的货物转运点都是在鹿特丹 港口。有个很好的例子，典型的散装货物的转运--煤炭和铁矿石，很大一部分， 其目的地是在德国市场。 为了处理大量材料不同地方大范围的转运，使用了机械 运输机，其中就包括带式输送机。 长度最长带式输送系统架设在相对较小的国家--荷兰， 因为它们是主要用于 大量原材料的流动运输。最长的带式输送系统，其长度约为 2 公里长，它位于鹿 特丹港口的一部分--马斯弗拉克特， 它是用来从批发油库运输大量的煤炭到电力 站。 除了国内的工程项目， 越来越多的荷兰工程顾问参与到国际中来开发大型陆 路皮带输送系统。 代尔夫特科技大学是荷兰其中的一个科技大学，而机械工程学 院的交通技术系就是研究在开发这些系统过程中遇到典型的难点。 输送带与散装固体物质之间的相互作用性能，带式输送机结构以及外界环 境都会影响到该输送系统其预定要求达到的合适标准。 有些相互作用造成了一些 令人棘手的现象，因而便开始进入研究这些现象造成的实际问题[ 1 ] 。这些问 题的分类方法之一是，将其根本原因明显涉及到带式输送机的这些问题分为一 类。 非平稳移动皮带的瞬态应力减少和设计皮带输送机时规定空载运作引起的 共振，是描述带式输送机的两个最重要的动态因素[ 2 ] 。本文提出了一种能模 拟程序启动和停止时皮带的纵向和横向响应以及稳定运行时的运动的新的皮带 输送系统有限元模型。 模拟皮带输送系统的启动程序，这超出了本文讨论的结果 范围，因此我们将展示一个比较有可行性的模式的例子。
如果用来驱动皮带输送系统的总电源， 是用德国工业标准 22101 来计算设计 的， 然后带假设成一个不可拓展的机构。这意味在带启动和停止时施加在带上的 压力， 可从牛顿刚体动力学的理论中推导出来。带最大的延长可以用带应力计算 出来的。这种通过确定皮带弹性反应的方式被称为准静态（设计）的方法。对于 小型皮带输送系统，这就使得了一个带的设计和运行状态合格。然而，对于长距 离皮带输送系统，这可能变成一个有缺陷的设计，导致维修费用高，缩短运输机 零件的寿命和众所周知的工作问题，如： ?机器的重量牵引位移过大 ?带的过早崩裂，最主要地引起绞接头的破损 ?破坏托辊和造成皮带张紧轮的重大损害 ?使皮带脱离皮带张紧轮，这可能导致散装原材料的溢出 ?造成（液压动力的）驱动系统的损坏和失灵 在许多研究人员开发出的模型中， 皮带的弹性反应是被用来计算以确定这种 现象引起的问题。在大多数模型中，包括皮带输送机的有限元模型，也是为了用 来计算在皮带上阻力和压力的变化。 皮带的全局弹性反应是由所有零件的弹性响 应组成。 这种有限元模型已经应用在计算机软件，它可以用在长距离皮带输送系 统的设计阶段。这就是所谓的动态（设计）的方法。模拟结果验证表明，基于这 种带模型的软件程序，预测（系统）启动和停止时带的弹性反应是相当成功，例 如见[ 3 ] 和[ 4 ] 。 上述的有限元模型确定的只是皮带的纵向弹性反应。因此，他们不能准确地 确定出： ?托辊和张紧轮上皮带的运动 ?动力驱动的状态 ?带的阻力弯曲 ?（震动）应力波的演变 ?带凹陷与应力波的纵向传播之间的相互作用 ?皮带和托辊之间的相互作用 ?皮带稳定运动时带速的影响 ?通过托辊（驱动）的皮带上的动态应力。 ?皮带共振的参数对于提升物品时候或由托辊的偏心率引起的振动和皮带的横向 位移的相互关系的影响 ?竖直横向波的发展 ?由大量散装材料以及在皮带横截面面积的变形所引起的阻力的影响 ?脱离托辊的带产生的凸.凹曲线 皮带的横向弹性反应往往是导致长距离皮带输送系统故障的原因， 因此应当 加以考虑。需要有中提到的特殊模型，才能确定带的横向响应，但是要是 考虑到特殊因素的（横向）响应，就能更方便地扩展现存的有限元模型。
一个典型的皮带输送机结构组成包括驱动滚筒，尾部托辊，一个垂直向上提 升的带轮， 一些托辊和一底盘如图 1 所示。这个结构为例来说明如何有限元模型 的输送带被开发只有带的纵向弹性响应成为主体。 由于驱动滚筒和提升带轮之间部分皮带的长度 Ls，与皮带的总长度 L 相比 是可忽略不计的， 只要考虑到提升系统中带轮的质量惯性，这些带轮可以数理性 地看成为一个带轮。 由于带从一点到另一点的运动变化中所遇到的阻力，根据当 地精确（维护）的条件和带式输送机的结构，沿着带的长度分布。为了能够确定 带运动中分布应力的影响， 皮带被划分为多个不同的有限元素，带上应力被具体 地分配到相对应的元素。 如果关心的只是皮带的纵向弹性反应，由带轮无力量驱 动的运动（有滑移的可能），带就会这些地方起不了作用。设计的最后一步，该 模型可以由两股带有驱动特征和张力特性的力量取代带的驱动系统和张力系统。 确切的说， 有限元取决于哪些阻力以及在带和其支撑结构之间的相互作用影 响， 考虑到这些问题可能与数学描述皮带材料的基本特性有关系。 根据这一解释， 其要素可以由一个系统块代表，如图 1所示的是弹簧和阻尼，这样的系统给 出了一个有限元与节点 C 和 C + 1 。弹簧 K 和阻尼 H 代表带拉伸的粘弹性状，G 代表皮带的可变纵向的结构刚度， 是由作用在两个带轮交错的横截面上垂直的力 的所产生，V 代表皮带速度取决于阻力的。
图 1 ：五限元综合模型[ 9 ] 。
如果只有带的纵向变形是主要素， 那么梁架元就可用于模型的皮带弹性反应。 梁架元组成部分有如图 2 所示的两个结点， P 和 Q ，四个位移参数确定部分载 体 X： xT = [up vp uq vq] (1)
图 2 ：梁架元的精确位移 梁架元轴的长度变化， [ 7 ] ： ε1 = D1(x) = ∫? o ds?- ds? o 2ds? o dξ
DSO 是限元未变形的长度，DS 是限元变形的长度，ξ 是沿着有限元轴的无量纲长 度。
图 3 ：张带的静态凹陷 虽然带呈弯曲状态， 但梁架元并没有变形，这可能考虑到带小数值凹陷的静态影 响。静态带凹陷的比率是有定义的（见图 3 ） ： K1 = δ/1 = q1/8T (3)
其中 q 是暴露在外面带和散装物料的重量在竖直方向上分布的荷载， 1 是带轮 间距，而 T 是带的张力。，带凹陷的纵向变形影响取决于[ 7 ] ： εs = 8/3 K?s 产生了非线性梁架元总的纵向变形。 (4)
图 4 ：节点的精确位移和旋转的梁架元。
如果带的横向位移是主要因素， 那么梁架元就可以用来模拟皮带。同样对于拥有 六个位移参数的梁架元的平面运动来说，相当于三个独立的刚体运动。因此就剩 下三个变形参数是：纵向变形参数ε 1 ，两个弯曲变形参数ε 2 和ε 3 。
图 5 ：梁架元的弯曲变形的 梁架元弯曲变形的参数可以定义为梁架元的组成载体（见图 4 ） ：
xT = [up vp ?p uq vq ?q] (5)
和如图 5 的变形结构
ε2 = D2(x) = e2p1pq 1o (6) ε3 = D3(x) = -eq21pq 1o
当绕过托辊或带轮的时候，带运动是受到约束的。为了说明（弄清楚）这些 制约因素，影响制约因素（边界）的条件都必须添加到用来代模拟带的有限元中 来。这可以通过使用多体动力学进行描述。多体机置动力学的经典描述，建立起 由若干约束条件连接起来的刚体或刚性链接。 （变形） 在 输送带的有限元描述里， 带被分离成多个有限元， 有限元之间的联系是可变形的。 有限元是由节点连接的， 因此分配了位移参数。要确定带的运动，排除了刚体模型的变形模式。如果一个 带绕过托辊，，决定托辊上带的位置（如见图 6）的带长度为ξ ，被添加到组件 矢量，如：式（6） ，因此产生了 7 个位移矢量参数。
图 6 ：由托辊支撑的带 梁架元有两个独立的刚体运动，因此依然有五个变形参数存在。其中已经在 3.1 中给出了ε 1 , ε 2 和ε 3 ，确定了带的变形。剩下ε 4 和ε 5 ，确定带和 托辊之间的相互作用，见图 7 。
图 7 ：两个约束条件的梁架元有限元。 这些变形参数可以假设成无限刚度的弹性。这意味着： ε4 = D4(x) = (rξ + u ξ)e2 - rid.e2 = 0 ε5 = D5(x) = (r ξ + uξ)e1 - rid.e1 = 0 (7)
如果模拟的是ε 4 > 0 的时候，那么带将脱离托辊，而描述带的有限元上的约束 条件也将去除。
为了使一种模型能应用于带式输送机有限元模型的滚动阻力，已经制定了 一种计算滚动阻力的近似公式， [ 8 ] 。带运动中，暴露在带外面的总滚动阻 力的组成部分，这三部分是耗能的主要部分，可以区分为包括：压痕滚动阻力， 托辊的惯性（加速滚动阻力）和轴承滚动阻力（轴承阻力） 。确定滚动阻力因 素的参数包括直径和托辊的材料，以及各种带参数，如速度，宽度，材料，紧张 状态，环境温度，带横向负荷，托辊间距和槽角。总滚动阻力的因素，可以表示 成总滚动阻力和带垂直负荷之间的比例，定义为： ft = fi + fa + fb (8)
Fi 是压痕滚动阻力的系数，FA 是加速阻力系数，而 FB 是轴承阻力系数。这些组 成系数由下面的确定： Fi = CFznzh nhD-nD VbnvK-nk NTnT fa = fb = Mred ??u Fzb Mf Fzbri ?t? (9)
FZ 是带垂直方向上分布的负载和散装物料的负载的总和， H 是带的覆盖厚度，D 是托辊的直径，Vb 是带速，KN 是带负荷的名义百分之比，T 是环境温度，Mred 是托辊的折算质量，B 是带的宽度， U 是带的纵向位移，MF 是总的轴承阻力矩 和 RI 是轴承内部半径。在计算滚动阻力中，皮带的动力性能及机械性能和皮带 上覆盖的材料发挥着重要作用。 这使得带的选择和带上覆盖材料，尽量减少由动 力阻力引起的能源消耗。
在稳定性的带运动情况下，为了能够测定带式输送机驱动系统的旋转组件 的影响， 这个带式输送机的总模型必须是含有驱动系统模型。驱动系统的旋转元 件，就像一个减速箱，参照了 3.2 节中所述的约束条件。带有减速比的减速箱， 可以用带两个位移参数的减速元件来代替， μ p 和μ q ，像一个刚体的（旋转） 运动，因此就剩下一个变形参数： εred = Dred(x) = i?p + ?q = 0 (10)
要确定电式扭矩感应式电机，是否适应所谓的两轴式电动机。该相电压的矢量 v 可从（11）获得： v = Ri + ωsGi + L ?i/?t (11)
在（11）式中 I 是相电流矢量，R 是模型的相电阻， c 是模型的相电感抗，L 是 模型的相感系数而ω s 是电机转子的角速度。电磁转矩等于： Tc = iTGi (12)
电机模型和驱动系统机械组件是由驱动系统的运动方程联系着的： ???j ??k Ti = Iij + Cik Kil? (13) ?t? ?t 其中 T 是扭矩矢量，I 是模型的惯量，C 是模型的阻尼，K 是矩阵刚度和 ? 是电 机旋转轴的角速度。 模拟启动或停止程序控制反馈的程序可以添加到带式驱动系统模型中， 用来 控制驱动扭矩。
整个带式输送机模型的运动方程可以得出潜在功率的原则， [ 7 ] ： fk - Mkl ??x1 / ?t? = ζ1Dik (14)
其中 F 是阻力矢量，M 是模型的质量而σ 是拉格朗日乘数的矢量，可能解释为双 重压力矢量 to 张力矢量ε 。 为了解决带有 X 这一组方程， 方程一体化是必要的。 但是一体化的结果，必须确保满足约束条件。如果(8)式中应变为零，那么必须 纠正一体化结果，如见[ 7 ] 。可以使用模型的反馈选择，例如限制提升物质垂 直方向上的运动。 这种违逆动力学的问题可以用下面公式表示。鉴于带模型及其 驱动系统的提升运动众所周知，根据系统自由度和它的比例（速度）可以确定其 他元件的运动。它超出了本文所讨论关于此项的所有细节范围。
为了在长距离带式输送机系统设计阶段能够正确设计，应用了有限元法。例 如带强度的选择， 可以减少的尽量减少，使用模型模拟的结果确定传送带的最大 张力。 以有限元模型的功能作为例子，应该考虑到在两个托辊位置范围之间稳定 移动带的横向振动。 在运输机的设计阶段这必须被确定， 才得以确保空带的共振。 对于皮带输送机的设计来说，托辊和移动带间相互作用影响是很重要的。托 辊的及带轮的几何不完善性， 导致带脱离托辊和带轮能支撑的位置，在带和支撑 带轮之间产生一种横向振动。 这对带施加了一部分的交互轴向应力。如果这部分 力是比皮带的预应力小，那么带将在它的固有频率中振动，否则带将被迫振动。 皮带是会受迫振动的，例如受托辊的偏心率影响。在输送带返程中，这种振动特 别值得注意。 由于受迫振动的频率取决于带轮和托辊的角速度，因此对于带的速 度，确定在带轮和托辊之间，带在自然频率状况下，横向振动中带速影响，这个 是很重要的。 如果受迫振动的频率接近于皮带横向振动的固有频率，将发生共振 现象。 有限元模型的模拟结果可用于确定稳定移动的带的横向振动频率范围。 该频 率是利用快速傅立叶技术从时域范围到频域范围，带横向位移变换后得到的结 果。除了使用有限元模型外也可以运用近似分析法。 皮带可以模拟成一个预应力梁。如果皮带的弯曲硬度可以被忽略，横向位移 比托辊间距还小，Ks << 1 ，并且带增加的长度相对于横向位移的原始长度来说 是微不足道，带的横向振动可近似为下列线性微分方程，如见图 15 ： ??v ?t? ??v ?x? ??v ?x?t
= (c? - C? 2 b)
其中 V 是皮带的横向位移和 C2 是横向波的波速度，由（16）式定义： c2 = √g1/8Ks (16)
首先，图 5 中带的横向固有频率范围可从公式（16）获得，如果假定 v(O,t)=v(l,t)=0： 1 fb = c2 (1 - ?? ) (17) 21 ? 是无量纲的速比，由（18）式确定： ? = Vb / c2 (18)
FB 是不同带的各自独立的频率范围，由于输送带长度方向上带张力变化。托辊 的受迫振动频率，使托辊产生了一个偏心率等于： fi = Vb / πD (19)
其中 D 是托辊的直径。 为了设计一个在托辊间距中无支撑的共振，这受到以下条
由线性微分方程（16）所取得的成果不过是只适用于小数值的速比 ?。对于 大数值的速比 ? 来说，如高速运输机或低的带张力，在（16）式中所有非线性条 件就显得重要的。因此，数值模拟的运用，有限元模型的开发，都是为了确定带 横向振动线性和非线性频率之间的比例范围。 这些关系已被确定适合不同的数值 的 ?，例如说一个功能凹陷的比率 Ks。 使用快速傅里叶技术将横向位移结果的转化为频谱。 从这些频谱中获得的频 率与公式 （18） 获得的频率相比， 其产生了图 8 所显示的曲线。 从这一数字可见， 对小于 0.3 的 ? 来说，计算误差很小。对于大数值的 ? 来说，运用线性近似值法 产生的计算误差达到 10 ％以上。运用了皮带采用非线性梁架元的有限元模型， 因此可以准确地确定大数值 ? 的横向振动。 对于小数值 ? 的横向振动的频率也可以用公式（18）准确地预测。然而，它 不能分析， 例如带凹陷和纵向波的传播之间的相互作用，或者同样可以看成有限 元模型的脱离托辊的皮带。 这决定带应力和横向振动频率之间的关系可以用于皮带张力监测系统。
图 8 ：由两个托辊支撑的带的横向振动线性和非线性频率之间的比例。
为了使模拟的结果能够得到验证， 实验中使用了动态试验设备， 如图 9 所示。
图 9 ：动态试验设施 使用这试验设施能够确定的两个托辊的间距和卸荷扁带的横向振动， 例如返 程部分的。声音装置是用来测量皮带的位移。此外，还有在试验中为我们所知的 张紧力，带速，电机转矩，托辊转子与托辊的距离。
由于最具有成本效益带式输送机的操作条件中出现了宽度范围为 0.6m1.2m[ 2 ] 的各种皮带 ，可通过变换不同的带速改变带的输送能力， 。然而在带 速度被改变之前，应确定带和托辊之间的相互作用，以确保无支撑的带的共振。 为了说明稳定移动的带的横向位移这一点，测量了两个托辊的间隔。带的总长度 L 是 52.7m，托辊间距 I 是 3.66m，静态凹陷的比例常数是 2.1 ％ ，? 为 0.24 而带速 Vb 为 3.57m/ s。 这个信号的后期转化由如图 5 所示的快速傅里叶技术频谱获得。 在图 5 中 出 现了 3 个频率。第一频率是由带结合处所引起的： fs = Vb/L = 0.067 Hz 第二个频率，出现在 1.94 赫兹，是由皮带的横向振动所造成的。
图 10 ：带稳定移动时横向振动频率
第三个频率出现在 10.5Hz，是由托辊的旋转所造成的，从图 11 所示的数值 模拟获得。
图 11 ：计算共振区的不同托辊的直径 D. 贯穿实验表明皮带速度和托辊间距。
图 11 显示的是拖过带与托辊互动引起的共振区可以预测三个托辊的直径。该带 式输送机的托辊直径为 0.108M，从而可以预测皮带速度邻近 0.64M/S 的共振现 象。为了验证结果，在启动运输机的时候测量了带的最大横向位移跨度。
图 12 ：测量横向振动和带静态凹陷幅度的标准差的比例。 在图 12 中，可以看出横向振动的最大振幅发生在带速为 0.64M/S 处，正如 有限元模型模拟预测的结果一样。因此，带速度不应选择临近 0.64 米/ s 的。 虽然是用扁带进行实验和理论的验证的，但是这种应用技术也可运用于槽型带 中。
带式输送机有限元模型中梁架元的应用，带横向位移的模拟，从而使能够设 计出带无支撑的共振。 对于小数值的 ? 来说， 采用梁架元代替线性微分方程预测 共振现象的优势是同样可以预测到皮带纵向和横向位移的之间的相互作用以及 从模拟中预见皮带脱离托辊。
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