1730_帶式輸送機(jī)的機(jī)械傳動裝置,輸送,機(jī)械傳動,裝置 The Two-Dimensional Dynamic Behavior of Conveyor Belts3.1.1 NON LINEAR TRUSS ELEMENTIf 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] (1)For the in-plane motion of the truss element there are three independent rigid body motions therefore one deformation parameter remains which describesFigure 2: Definition of the displacements of a truss elementthe change of length of the axis of the truss element [7]:ds2 - ds2oε1 = D1(x) = ∫1 o2ds2odξ (2)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 beltAlthough 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 [7]:εs = 8/3 K2s (4)which yields the total longitudinal deformation of the non linear truss element:3.1.2 BEAM ELEMENTFigure 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 elementThe bending deformation parameters of the beam element can be defined with the component vector of the beam element (see Figure 4):xT = [up vp μp uq vq μq] (5)and the deformed configuration as shown in Figure 5:e2p1pqε2 = D2(x) =1o-eq21pqε3 = D3(x) =1o(6)3.2 THE MOVEMENT OF THE BELT OVER IDLERS AND PULLEYSThe 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 = 0ε5 = 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 RESISTANCEIn 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, [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 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 NTnTMred ?2ufa =Fzb ?t2Mf fb =Fzbri(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, 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 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 SYSTEMTo 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, μp and μq, one rigid body motion (rotation) and therefore one deformation parameter:εred = Dred(x) = iμp + μq = 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:?2?j ??kTi = Iij?t2+ Cik?tKil? (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 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 ?2x1 / ?t2 = σ1Dik (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 [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 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 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 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:?2v ?2v ?2v?t2= (c22 - C2b)?x2- 2Vb?x?t(15)where 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 obtained from eq. (16) if it is assumed that v(O,t)=v(l,t)=0:1 fb =21c2 (1 - ?2) (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:πDL ≠2?(1-?2) (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 VERIFICATIONIn 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 facilityWith 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. EXAMPLESince the most cost-effective operation conditions of belt conveyors occur in the range of belt widths 0.6 - 1.2 m [2], 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 HzThe 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. CONCLUSIONSApplication 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.