帶式輸送機(jī)的機(jī)械傳動(dòng)裝置設(shè)計(jì)【一級(jí)圓柱齒輪減速器】【F=2200Nv=1.5m-s,d=400mm】
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輸送帶的二維動(dòng)態(tài)特性3.1.1非線性梁架(構(gòu)架)元如果只有帶的縱向變形是主要素,那么梁架元就可用于模型的皮帶彈性反應(yīng)。梁架元組成部分有如圖2所示的兩個(gè)結(jié)點(diǎn), P和Q ,四個(gè)位移參數(shù)確定部分載體X:xT = up vp uq vq (1)對(duì)平面運(yùn)動(dòng)的梁架元有三個(gè)獨(dú)立的剛體運(yùn)動(dòng),因此(這公式)仍然是描述一個(gè)變形的參數(shù)。圖2 :梁架元的精確位移梁架元軸的長(zhǎng)度變化, 7 :1 = D1(x) = ods - dsod (2)2dsoDSO是限元未變形的長(zhǎng)度,DS是限元變形的長(zhǎng)度,是沿著有限元軸的無(wú)量綱長(zhǎng)度。圖3 :張帶的靜態(tài)凹陷雖然帶呈彎曲狀態(tài),但梁架元并沒(méi)有變形,這可能考慮到帶小數(shù)值凹陷的靜態(tài)影響。靜態(tài)帶凹陷的比率是有定義的(見(jiàn)圖3 ) :K1 = /1 = q1/8T (3)其中q是暴露在外面帶和散裝物料的重量在豎直方向上分布的荷載, 1是帶輪間距,而T是帶的張力。,帶凹陷的縱向變形影響取決于 7 :s = 8/3 Ks (4)產(chǎn)生了非線性梁架元總的縱向變形。3.1.2梁架元圖4 :節(jié)點(diǎn)的精確位移和旋轉(zhuǎn)的梁架元。如果帶的橫向位移是主要因素,那么梁架元就可以用來(lái)模擬皮帶。同樣對(duì)于擁有六個(gè)位移參數(shù)的梁架元的平面運(yùn)動(dòng)來(lái)說(shuō),相當(dāng)于三個(gè)獨(dú)立的剛體運(yùn)動(dòng)。因此就剩下三個(gè)變形參數(shù)是:縱向變形參數(shù)1 ,兩個(gè)彎曲變形參數(shù)2和3 。圖5 :梁架元的彎曲變形的梁架元彎曲變形的參數(shù)可以定義為梁架元的組成載體(見(jiàn)圖4 ) :xT = up vp p uq vq q (5)和如圖5的變形結(jié)構(gòu)2 = D2(x) =e2p1pq (6)1o3 = D3(x) =-eq21pq1o3.2繞過(guò)托輥及帶輪的帶運(yùn)動(dòng)當(dāng)繞過(guò)托輥或帶輪的時(shí)候,帶運(yùn)動(dòng)是受到約束的。為了說(shuō)明(弄清楚)這些制約因素,影響制約因素(邊界)的條件都必須添加到用來(lái)代模擬帶的有限元中來(lái)。這可以通過(guò)使用多體動(dòng)力學(xué)進(jìn)行描述。多體機(jī)置動(dòng)力學(xué)的經(jīng)典描述,建立起由若干約束條件連接起來(lái)的剛體或剛性鏈接。在(變形)輸送帶的有限元描述里,帶被分離成多個(gè)有限元,有限元之間的聯(lián)系是可變形的。有限元是由節(jié)點(diǎn)連接的,因此分配了位移參數(shù)。要確定帶的運(yùn)動(dòng),排除了剛體模型的變形模式。如果一個(gè)帶繞過(guò)托輥,決定托輥上帶的位置(如見(jiàn)圖6)的帶長(zhǎng)度為,被添加到組件矢量,如:式(6) ,因此產(chǎn)生了7個(gè)位移矢量參數(shù)。圖6 :由托輥支撐的帶梁架元有兩個(gè)獨(dú)立的剛體運(yùn)動(dòng),因此依然有五個(gè)變形參數(shù)存在。其中已經(jīng)在3.1中給出了1 , 2和3 ,確定了帶的變形。剩下4和5 ,確定帶和托輥之間的相互作用,見(jiàn)圖7 。圖7 :兩個(gè)約束條件的梁架元有限元。這些變形參數(shù)可以假設(shè)成無(wú)限剛度的彈性。這意味著:4 = D4(x) = (r + u )e2 - rid.e2 = 0 5 = D5(x) = (r + u)e1 - rid.e1 = 0 (7)如果模擬的是4 0的時(shí)候,那么帶將脫離托輥,而描述帶的有限元上的約束條件也將去除。3.3滾動(dòng)阻力為了使一種模型能應(yīng)用于帶式輸送機(jī)有限元模型的滾動(dòng)阻力,已經(jīng)制定了一種計(jì)算滾動(dòng)阻力的近似公式, 8 。帶運(yùn)動(dòng)中,暴露在帶外面的總滾動(dòng)阻力的組成部分,這三部分是耗能的主要部分,可以區(qū)分為包括:壓痕滾動(dòng)阻力,托輥的慣性(加速滾動(dòng)阻力)和軸承滾動(dòng)阻力(軸承阻力) 。確定滾動(dòng)阻力因素的參數(shù)包括直徑和托輥的材料,以及各種帶參數(shù),如速度,寬度,材料,緊張狀態(tài),環(huán)境溫度,帶橫向負(fù)荷,托輥間距和槽角??倽L動(dòng)阻力的因素,可以表示成總滾動(dòng)阻力和帶垂直負(fù)荷之間的比例,定義為:ft = fi + fa + fb (8)Fi是壓痕滾動(dòng)阻力的系數(shù),F(xiàn)A是加速阻力系數(shù),而FB是軸承阻力系數(shù)。這些組成系數(shù)由下面的9確定:Fi = CFznzh nhD-nD VbnvK-nk NTnT(9)fa =Mred uFzb tfb =MfFzbriFZ是帶垂直方向上分布的負(fù)載和散裝物料的負(fù)載的總和, H是帶的覆蓋厚度,D是托輥的直徑,Vb是帶速,KN是帶負(fù)荷的名義百分之比,T是環(huán)境溫度,Mred是托輥的折算質(zhì)量,B是帶的寬度, U是帶的縱向位移,MF是總的軸承阻力矩和RI是軸承內(nèi)部半徑。在計(jì)算滾動(dòng)阻力中,皮帶的動(dòng)力性能及機(jī)械性能和皮帶上覆蓋的材料發(fā)揮著重要作用。這使得帶的選擇和帶上覆蓋材料,盡量減少由動(dòng)力阻力引起的能源消耗。3.4帶驅(qū)動(dòng)系統(tǒng)在穩(wěn)定性的帶運(yùn)動(dòng)情況下,為了能夠測(cè)定帶式輸送機(jī)驅(qū)動(dòng)系統(tǒng)的旋轉(zhuǎn)組件的影響,這個(gè)帶式輸送機(jī)的總模型必須是含有驅(qū)動(dòng)系統(tǒng)模型。驅(qū)動(dòng)系統(tǒng)的旋轉(zhuǎn)元件,就像一個(gè)減速箱,參照了3.2節(jié)中所述的約束條件。帶有減速比的減速箱,可以用帶兩個(gè)位移參數(shù)的減速元件來(lái)代替, p和q ,像一個(gè)剛體的(旋轉(zhuǎn))運(yùn)動(dòng),因此就剩下一個(gè)變形參數(shù):red = Dred(x) = ip + q = 0 (10)要確定電式扭矩感應(yīng)式電機(jī),是否適應(yīng)所謂的兩軸式電動(dòng)機(jī)。該相電壓的矢量v可從(11)獲得:v = Ri + sGi + L i/t (11)在(11)式中I是相電流矢量,R是模型的相電阻, c是模型的相電感抗,L是模型的相感系數(shù)而s是電機(jī)轉(zhuǎn)子的角速度。電磁轉(zhuǎn)矩等于:Tc = iTGi (12)電機(jī)模型和驅(qū)動(dòng)系統(tǒng)機(jī)械組件是由驅(qū)動(dòng)系統(tǒng)的運(yùn)動(dòng)方程聯(lián)系著的:Ti = Iijj+ CikkKil (13)tt其中T是扭矩矢量,I是模型的慣量,C是模型的阻尼,K是矩陣剛度和是電機(jī)旋轉(zhuǎn)軸的角速度。 模擬啟動(dòng)或停止程序控制反饋的程序可以添加到帶式驅(qū)動(dòng)系統(tǒng)模型中,用來(lái)控制驅(qū)動(dòng)扭矩。3.5運(yùn)動(dòng)方程整個(gè)帶式輸送機(jī)模型的運(yùn)動(dòng)方程可以得出潛在功率的原則, 7 :fk - Mkl x1 / t = 1Dik (14)其中F是阻力矢量,M是模型的質(zhì)量而是拉格朗日乘數(shù)的矢量,可能解釋為雙重壓力矢量to張力矢量 。為了解決帶有X這一組方程,方程一體化是必要的。但是一體化的結(jié)果,必須確保滿足約束條件。如果(8)式中應(yīng)變?yōu)榱悖敲幢仨毤m正一體化結(jié)果,如見(jiàn) 7 ??梢允褂媚P偷姆答佭x擇,例如限制提升物質(zhì)垂直方向上的運(yùn)動(dòng)。這種違逆動(dòng)力學(xué)的問(wèn)題可以用下面公式表示。鑒于帶模型及其驅(qū)動(dòng)系統(tǒng)的提升運(yùn)動(dòng)眾所周知,根據(jù)系統(tǒng)自由度和它的比例(速度)可以確定其他元件的運(yùn)動(dòng)。它超出了本文所討論關(guān)于此項(xiàng)的所有細(xì)節(jié)范圍。3.6實(shí)例為了在長(zhǎng)距離帶式輸送機(jī)系統(tǒng)設(shè)計(jì)階段能夠正確設(shè)計(jì),應(yīng)用了有限元法。例如帶強(qiáng)度的選擇,可以減少的盡量減少,使用模型模擬的結(jié)果確定傳送帶的最大張力。以有限元模型的功能作為例子,應(yīng)該考慮到在兩個(gè)托輥位置范圍之間穩(wěn)定移動(dòng)帶的橫向振動(dòng)。在運(yùn)輸機(jī)的設(shè)計(jì)階段這必須被確定,才得以確保空帶的共振。 對(duì)于皮帶輸送機(jī)的設(shè)計(jì)來(lái)說(shuō),托輥和移動(dòng)帶間相互作用影響是很重要的。托輥的及帶輪的幾何不完善性,導(dǎo)致帶脫離托輥和帶輪能支撐的位置,在帶和支撐帶輪之間產(chǎn)生一種橫向振動(dòng)。這對(duì)帶施加了一部分的交互軸向應(yīng)力。如果這部分力是比皮帶的預(yù)應(yīng)力小,那么帶將在它的固有頻率中振動(dòng),否則帶將被迫振動(dòng)。皮帶是會(huì)受迫振動(dòng)的,例如受托輥的偏心率影響。在輸送帶返程中,這種振動(dòng)特別值得注意。由于受迫振動(dòng)的頻率取決于帶輪和托輥的角速度,因此對(duì)于帶的速度,確定在帶輪和托輥之間,帶在自然頻率狀況下,橫向振動(dòng)中帶速影響,這個(gè)是很重要的。如果受迫振動(dòng)的頻率接近于皮帶橫向振動(dòng)的固有頻率,將發(fā)生共振現(xiàn)象。 有限元模型的模擬結(jié)果可用于確定穩(wěn)定移動(dòng)的帶的橫向振動(dòng)頻率范圍。該頻率是利用快速傅立葉技術(shù)從時(shí)域范圍到頻域范圍,帶橫向位移變換后得到的結(jié)果。除了使用有限元模型外也可以運(yùn)用近似分析法。皮帶可以模擬成一個(gè)預(yù)應(yīng)力梁。如果皮帶的彎曲硬度可以被忽略,橫向位移比托輥間距還小,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 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 NTnT(9)fa =Mred uFzb tfb =MfFzbriwhere 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 BELTS 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, p and q, one rigid body motion (rotation) and therefore one deformation parameter:red = Dred(x) = 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 resistances, 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 equations 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 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 mass 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. 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 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 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: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 obtained 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 idler 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 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
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