英文文献及翻译

1、输送带的二维动态特性3.1.1非线性梁架（构架）元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)对平面运动的梁架元有三个独立的刚体运动，因此（这公式）仍然是描述一个变形的参数。图2 ：梁架元的精确位移梁架元轴的长度变化， 7 ：1 = D1(x) = ods - dsod (2)2dsoDSO是限元未变形的长度，DS是限元变形的长度，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。

2、静态带凹陷的比率是有定义的（见图3 ） ：K1 = /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 ：s = 8/3 Ks (4)产生了非线性梁架元总的纵向变形。3.1.2梁架元图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1 ，两个弯曲变形参数2和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

3、图4 ） ：xT = up vp p uq vq q (5)和如图5的变形结构2 = D2(x) =e2p1pq (6)1o3 = D3(x) =-eq21pq1o3.2绕过托辊及带轮的带运动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。在（变形）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

4、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，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1 , 2和3 ，确定了带的变形。剩下4和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：4 = D4(x) = (r + u )e2 - rid.e2 = 0 5 = D5(x) = (r + u)e1 - rid.e1 = 0 (7)如果模拟的是4 0的时候，那么带将脱离托辊，而描述带

5、的有限元上的约束条件也将去除。3.3滚动阻力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

6、阻力系数。这些组成系数由下面的9确定：Fi = CFznzh nhD-nD VbnvK-nk NTnT(9)fa =Mred uFzb tfb =MfFzbriFZ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。3.4带驱动系统在稳定性的带运动情况下，为了能够测定带式

7、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， p和q ，像一个刚体的（旋转）运动，因此就剩下一个变形参数：red = Dred(x) = ip + 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 + sGi + 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s是电机转子的角速度。电磁转矩等于：Tc =

8、 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：Ti = Iijj+ CikkKil (13)tt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。3.5运动方程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 = 1Dik (14)其中F是阻力矢量，M是模型的质量而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 。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

9、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。3.6实例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

10、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

11、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 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 RESISTANCEIn order to enable application o

12、f a model for the rolling resistance in the finite element model of the belt conveyor an approximate formulation for this resistance has been developed, 8. Components of the total rolling resistance which is exerted on a belt during motion three parts that account for the major part of the dissipate

13、d 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

14、 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

15、 + 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(9)fa =Mred uFzb tfb =MfFzbriwhere Fz is distributed vertical belt and bulk material load,

16、 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, mred 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 beari

17、ng 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 BELTS DRIVE SYSTEM

18、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

19、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:red = Dred(x)

20、= ip + 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 + sGi + L i/t (11)In eq. (11) i is the vector of phase currents, R the matrix of phase r

21、esistances, C the matrix of inductive phase resistances, L the matrix of phase inductances and s 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 equa

22、tions of motion of the drive system:Ti = Iijj+ CikkKil (13)ttwhere 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 axiss.To simulate a controlled start or stop procedure a feedback routine can be added to the

23、 model of the belts drive system in order to control the drive torque.3.5 THE EQUATIONS OF MOTIONThe equations of motion of the total belt conveyor model can be derived with the principle of virtual power which leads to 7:fk - Mkl x1 / t = 1Dik (14)where f is the vector of resistance forces, M the m

24、ass matrix and the vector of multipliers of Lagrange which may be interpret as the vector of stresses dual to the vector of strains . 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

25、. 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 7. 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 dynami

26、c 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 det

27、ails of this option.3.6 EXAMPLEApplication 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 feature

28、s 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

29、 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 componen

30、t is small compared to the prestress of the belt then the belt will vibrate in its natural frequency, otherwise the belts 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 c

31、onveyor 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 th

32、e 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 frequenc

33、y 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 be

34、am. 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 approximat

35、ed by the following linear differential equation, also see Figure 5:v= (c2 - Cb)v- 2Vbv (15)txxtwhere v is the transverse displacement of the belt and c2 the wave speed of the transverse waves defined by, 1:c2 = g1/8Ks(16)The first natural transverse frequency of the belt span of Figure 5 can be obt

36、ained from eq. (16) if it is assumed that v(O,t)=v(l,t)=0:fb =1c2 (1 - ) (17)21where 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 idl

37、er which has a single eccentricity is equal to:fi = Vb / D (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 D(1-) (20)2The results obtained with the linear differential equation (16) however are valid

38、 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 re