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黄河科技学院本科毕业设计（论文）任务书 工 学院 机械 系 机械设计制造及其自动化 专业 08 级 1 班学 号 080105048 学生 宋珍鹤 指导教师 郭长江 毕业设计（论文）题目 螺旋纸管卷管机设计 毕业设计（论文）工作内容与基本要求（目标、任务、途径、方法，应掌握的原始资料（数据）、参考资料（文献）以及设计技术要求、注意事项等）（纸张不够可加页）一、设计技术要求、原始资料（数据）、参考资料（文献）通过实习调研搜集资料，运用所学知识，并借助CAD等绘图软件，针对化纤用螺旋纸管，设计一种螺旋纸管卷管机。该机必须满足以下要求：1：螺旋纸管卷纸层数6-12层，纸宽60-110mm；2：卷轴直径65-120mm；3：卷管外径80-140mm；4：卷管速度3-8m/min；5：调速方法采用励磁调速电机实现无级调速；在做本课题时，需要查阅机械原理，机械设计，包装纸性能，机械设计手册1-5卷等相关资料。二、设计目标与任务1.查阅文献资料12种以上，外文资料不少于两篇。写出3000字以上的文献综述，单独装订成册。2.翻译外文科技资料，不少于3000汉字，单独装订成册。3.完成开题报告，填写开题报告表。4.拟定设计方案，完成螺旋纸管卷管机结构设计及强度计算等。5.卷管机装配图一张A0；卷管机架零件图一张A0； 卷管机花键轴零件图一张A3；卷管机托板零件图、支撑板零件图各一张A1；6.编写摘要，英中文完全对照，中文不少于300字。7.编写设计说明书，不少于8000字符。三、时间安排1-4 周 完成开题报告、文献翻译、文献综述及总体方案设计5-10 周 完成总体设计、完成部分机构的装配图及部分零件图并撰写说明书10-11 周 修改论文、资格审查等12 周 毕业答辩毕业设计（论文）时间： 2012 年 02 月 13 日至 2012 年 05 月 15 日计 划 答 辩 时 间： 2012 年 05 月 19 日专业（教研室）审批意见：审批人签名：Archive of Applied Mechanics 61 (1991) 523-531 Archive of Applied Mechanics ?9 Springer-Verlag 1991 Dynamic investigations o loads on gear teeth in single gear transmission W. Nadolski, Warsaw Summary: In the paper a discrete-continuous model for the analysis of dynamic loads on gear teeth of a single gear transmission is proposed. In this model a constant equivalent mesh stiffness and ponderable shafts deformed by torsion are taken into account. In the discussion a wave method is applied which utili- zes the wave solution of the equations of motion. Numerical calculations are concentrated on the determina- tion of amplitudes for dynamic loads on gear teeth with respect to frequencies of external excitation in the first and second resonant regions. Untersuchung der dynamischen Zahnbelastung in einstufigen Getrieben Tbersicht: Zur Untersuchung der dynamischen Zahnbeanspruchung in einstufigen Getrieben wird ein )lodell aus diskreten und kontinuierlichen Massen mit einer konstanten, quivalcnten Zahneingriffssteifigkeit und massebehafteten, tordierbaren Wellen vorgeschlagcn. Die Bewegungsgleichungen werden durch einen Wellenfunktionsansatz gel5st. Numerische Berechnungen werden vorwiegend ffir die Amplituden der Zahn- belastnng bei einer nieren Erregung im Bereich der ersten und zweiten Resonanz durchgeffihrt. 1 Introduction In 1, 2 dynamic investigations were performed for the discrete-continuous model of a single gear transmission with rigid gear teeth. In the present paper a similar model is considered, how- ever the mating gears have such profiles of teeth that their equivalent stiffness can be assumed to be constant 3- 6. In the technical literature gear transmissions are mostly analyzed by means of discrete models of one and multi-degrees of freedom 7, 8. In the present paper a discrete-continuous model consists of two ponderable shafts and four rigid bodies with constant mass moments of inertia with respect to the axis of rotation. Those gear transmissions are considered where supporting bearings eliminate deformations due to bending and where the shafts are mainly torsionally deformed. The rigid body at the input is loaded by an external moment which may be arbitrary. Damping is taken into account by means of an equivalent external damping of the viscous type and an equivalent internal damping of the Voigt type. In the considerations a method utilizing the wave solution of the equations of motion is applied. This method enables the determination of dynamics loads on gear teeth, displacements, strains and velocities in steady as well as in transient states. Numerical calculations for selected parameters describing various mechanical properties of the single gear transmission are concen- trated on the determination of amplitudes of dynamic loads on gear teeth with respect to fre- quencies of the external excitation in the first and second resonant regions. 2 Assumptions and governing equations Consider the discrete-continuous model of a single gear transmission with parallel axes, Fig. 1. The ponderable shafts 1 and 2 are characterized by the shear modulus G, the polar moment of inertia Ii, the density and the length li (i = 1, 2). The mass moments of inertia of gears 4, 5 and 39* 524 Archive of Applied Mechanics 61 (199i) M(t) J, Rz. u 5 I ! & 2 . Js, Rs Fig. 1. Model of a single gear trans- , I mission 0 I1 II ll +12 ) rigid bodies 3, 6 are Ji (i = 3, 4, 5, 6), respectively. The x-axis is parallel to the shaft axis and its origin coincides with the location of the left end of shaft 1 in an undisturbed state at t = 0. Dis- placements and velocities of shaft cross-sections are assumed to be equal to zero at t -= 0. The ana- lysis does not include numerous effects, e.g., tooth errors from manufacturing, tooth friction, off line of action contact, lubrication between teeth etc. which are usually neglected in dynamic investigations of gear transmissions. With these assumptions the equations governing the motion of a single gear transmission will be linear. It is assumed that during the motion the teeth do not separate, and that shaft displacements in the cross-section x = ll satisfy the relation R4OI (X, t) - RsO( x , t) =t= 0 for x = 11, t 0 (1) where R4, R5 are the centroide radii of gears 4, 5, and Oi(x, t) are angular displacements of shafts 1, 2, respectively. The dynamic loads on gear teeth is given by p = Km(R& - + - -5-i-/j cos x = (2) where Km is the equivalent mesh stiffness, Cm is the coefficient of mesh damping, and is the pressure angle corresponding to the centroide radii R and Rs. It should be pointed out that in 1, 2 the relation (1) is equal to zero, and that the formula (2) describing the dynamic load with- out gear errors is taken from 8. Under the above assumptions the problem of determining the displacements Oi(x, t) and velo- cities Oi/Ot appearing in the (2) is reduced to the solution of the classical wave equations O0i c a20i = 0 (i = 1, 2) (3) t 2 x with the initial conditions 0 Oi(x, O) = - (x, O) = 0 (i = 1, 2) (4) and the boundary conditions M(t)- J3 01 D31 01 ( 201 01 t- T - + GI1 )12 x = 0 forx - O, - DI 01 GI1 D12 + = 0 for x = ll, (5) - W. Nadolski: Dynamic investigations of loads on gear teeth in single gear transmission 525 ( 0 0 0 ( 00 0 t - J- + D-t - GI_Dz- +-x = 0 j 0 00 ( OzOz 0021 - - R et J c for x = l, forx =l +le (5) where M(t) is the external excitation loading the rigid body 3, and Dil (i = 3, 4, 5, 6) and Dis (i = 1, 2) are coefficients of the external and internal damping, respectively 1, 2. The boundary conditions (5) and the comparable conditions in 1, 2 seem to be similar. The difference between them lies in the fact that in the conditions (5) the dynamic load, according to (2), depends on the equivalent mesh stiffness and on the coefficient of the mesh damping. If (1) is equal to zero, then the boundary conditions in 1 and in the present paper are identical. Upon introduction of the non-dimensional quantities - x/ll, i = Ct/I, 0 i = Oi/O0, i i = li/ll, il = Dilll/(Jsc), Dz = Dic/l, M = Mll/(J30o), Ai -= J3/ai, Ki = Iel/Ja, (6) N = 5/R4, K-m = KIn(R, cos c) 2 1/(J3r Cm =- Fro( , cos cr 2 l,/(J3c) the relations (3)- (5) are - - o ( = 1, ), (7) Ot x 00 o(z, o) = -g- (x, o) = 0 ( = l, 2), (8) M(t) 201 D31 C0i ( c201 01 t 0t 2 + K1 D12- +-x =0 forx = 0, -AKm(01 - NO2) - AC, t - A4K1 DI. + -x / =0 -ANKm(O1 - NO) - ACm2V ( 0011 ( I - AsK2 D22 + Ox =0 002 002 ( Or- + A6D.I - + A6K2 D22 - N 02 t 0201 A4D4t 0-2-1 t Ot 2 Ot for x = 1, (9) N - - +AsD51- ! t Ot z t for x 1, 002 02 1 -57 + x/= for x = 1 + l where 00 is a constant angular displacement, and bars are omitted for convenience. The solution of (7), taking into account the initial conditions (8), is sought in the form OI(X , t) = /l(t - x) - gl(t + x) for 0 _ x _ 1, (lo) 02(x,t) = /2(t - x) + g2(t + x- 2) for l_x_ 1 +12 where the unknown functions i and gi represent, waves caused by the external moment M(t), pro- pagating in the first and the second shaft in a direction consistent and opposite to the x-direction, respectively. In the arguments of/i and 9i it is already taken into account that the first disturb- anee in the first shaft occurs at t = 0 in x = 0 and in the second shaft at t = 1 in z = 1. The functions/i and 9i are continuous and for negative arguments they are identicalto zero. 526 Archive of Applied Mechanics 61 (1991) Differential equations for/i, gi can be obtained by substituting (10) into the boundary con- ditions (9). From the form of the solution (10) it follows that/i, gi become different from zero at different time instants. In the calculations it is convenient to introduce the common argument z for all functions. In 1, 2 where the gear transmission with rigid teeth is studied the largest argument in each equality obtained from the boundary conditions is denoted by z. In the case when the gear teeth are deformable, suitable equations are slightly more complicated because the functions z and glare not independent from each other for actual arguments z and z2 = z + 2, correspondingly. In this case, upon substituting (10) into (9), we have rg(z) + r(z) = rJ.(z - 21) + r,/g(z - 21), r/i() + rd;() = M() + rgl() + rq;(), rg;( + 2) + rl0g;( + 2) + rlg( + 2) + n/;() + r/() (11) = rJ;() + rlJ;() + rlA() + r7() + rly(), ! rlJ2(z) -I- r2o/(z) -b r2/2(z) q- r2291(z q- 2) q- r23gl(z -q- 2) II t ! = r24q2 (z) q- rg(z) + rq(z) q- re7/(z) -q- r:s/(z ) where rl = I + AKeDz, re = A(Ke + DI), rz = AKD.e - 1, r4 =A6(K-D,), r = l + K1D1, r = K1 + Dsl, r7 = KIDI - 1, r s = K1 - Dzl, r = 1 + AK1DI, ro = A(K1 + D41 q- Cm), rll - A4Km, r12 = -ANCm, r3 = -ANKm, rl = AKD - 1, r = A(KI - DI - Cm), r - -rll , (12) r17 : -r12, rlS : -rla, rl 9 : AsK2D22 + 1, r2o = A(K2 + D51 + N2Cm), r21 : AN2K, r2 : -AsNCm, r23 : -AsNKm, r2 : AK2D2 - 1, r5 : A(K2 - D51 - NZCm), r26 : -r21, r27 : -r22, r8 : -r%. The differential equations (11) can be solved by means of the finite difference method. The functions/1,/2, g are determined from (11) for z 0 and the function gl for z : z + 2 0. The functions/i, gi are identical to zero for negative arguments, so from (11)3 it follows that g,(z2) 0 for z2 : z + 2 2. Though the functions/2(z) and gl(z + 2) are not independent, the method of finite differences enables to derive expressions for these functions in dependence on known values of appropriate functions. These expressions are given in the Appendix. 0o = 1 rad, non-dimensional ll = 12 = 1, 3 5Iumerieal results In the numerical calculations the following parameters of the single gear transmission are assumed : dimensional 11 =12 =0.25m, /5 =0.16m, J3- 1.5kgm z, (13) e = 3200m/s, =0.8.104kg/m3; K1 =0.013, K2 - 0.06627, A4 =5, A 6 =0.15, N =4/3, =r/9. The analysis includes the following non-dimensional mesh stiffness Km= 0.005 859, 0.018528, 0.05859, (in dimensions: l0 s 2/m, / 10 s N/m, 109 N/m) for the tooth length equal to 0.10 m, and the coefficient of mesh damping Cm = DI, 9-11. W. Nadolski: Dynamic investigations of loads on gear teeth in single gear transmission 527 01-NO 2 O.,-N2 10 -6 4O 30 20 10 Dil =0.01 0.05 - 0,4r . . . . . . 01-N0 0.1 0.01 /X J, o.o 0.2 0.4 0.6 p Fig. 2. Amplitude-frequency. curves of the functions 01 - NO2 and 01 - JT02 The function of the external moment M(t) can be arbitrary, i.e. irregular or regular, periodic or nonperiodie. Here it is assumed in the form M(t) = a sin (pt) (1r where a = 10 6, and p is a non-dimensional external frequency. The considerations focus on the determination of amplitudes of dynamic loads on gear teeth with respect to frequencies of external excitation in the steady states for the first and second resonant regions. The dynamic load P expressed by (2) depends on the relative displacements 01 (1, t) - N02(1, t), the relative velocities 01(1, t) - N02(1, t), and on the coefficients Kin, Urn. The effect of damping on 01 - 2V02, 01 - N02, and the effect of Km and Cm on the load P is shown in Figs. 2, 3 and 4. The amplitude-frequency curves for the relative displacements 01(1, t) -N02(1, t) (conti- nuous lines) and for the relative velocities 01(1, t) - N02(1, t) (dotted lines) presented in Fig. 2 are obtained using (11) with the parameters (13), and for the additional parameters Km =0.018528, C = 0, Dil =0,0.01,0.05,0.1 (i =3,4,5,6), (15) Di = 0, 0.01, 0.05 (i = 1, 2). It appeared that the effect of external damping on the studied functions was appreciable, but the effect of internal damping was rather insignificant. Each curve in Fig. 2 for fixed Dil corresponds to the three values of the coefficients Di2, so an effect of internal damping is not observed. Fur- ther numerical calculations are performed for coefficients of internal damping DI2 = D22 = 0.01, and for coefficients of external damping Dil = 0.05 (i = 3, 4, 5, 6). From Fig. 3 where the dia- grams of amplitudes PA of dynamic loads for the equivalent mesh stiffness Km= 0.005 859, 0.018528, 0.05859 and C = 0 are plotted it follows that the curves are regular, namely, in the first resonant region the maximum amplitude of the load increases with increase of K. The am- plitude-frequency curves for the dynamic loads shown in Figs. 4, 5, 6, 7 and 9 are obtained only for Km= 0.018528. 528 & 1.5 1.0 0.5 Km=0.05859 0.018528 0.005859 0 0.2 0.4 0.6 p Fig. 3. Amplitudes of dynamic loads for some K m Archive of Applied lechanics 61 (1991) 10 -5 1.5 Gin=0 O.Ol o.o5 1.0 oJo 0.5 i D 0 0.2 0.4 0.6 p Fig. 4. Amplitudes of dynamic loads for some C m The effect of the coefficient of mesh damping Cm on the dynamic loads is investigated for C, = 0, 0.01, 0.05, 0.1. From Fig. 4 it follows that the effect of C, is rather insignificant, except for the neighbourhood of the first resonance. The single gear transmission is characterized by many parameters (13). For instance, the effect of the parameters K1, K2 and A 8 on the amplitude-frequency curves for dynamic loads P is presented in Figs. 5, 6 and 7. From Fig. 6 it follows that the maximum load P inereases with increase of the parameter K in the first resonant region. Such a regularity is not observed in Figs. 5 and 7, showing the amplitude-frequency curves for P with various K1 and As, respectively. *l 10 .6 1.5 K1=0.0130 12 J 0.06558 o . 5 .L DID 0 0,2 0.4 0.6 p Fig. ,5. Amplitudes of dynamic loads for some K 1 I/ /.176 0.5 0 0.2 0.4 0.6 p Fig. 6. Amplitudes of dynamic loads for some K s W. Nadolski: Dynamic investigations of loads on gear teeth in single gear transmission 529 1.5 1.0 0.5 0 0.6 I I I I 0.2 0.4 Fig. 7. Amplitudes of dynamic loads for A : 0,03, 0.15, 0.75 , 10 - 1.5 1.0 0.5 m=0.05859 0,018528 0.005859 5 10 6 1.5 1.0 0,5 K = 0,30648 0.06627 0.01690 0.00858 f - - I I I i l I I I I I I 0.1 0.3 0.5 0.7 AG 0 0.2 0.4 0.6 A6 Fig. S. The effect of K m on resonant amplitudes of dynamic loads Fig. 9. The effect of K1 and K 2 on resonant ampli- tudes of dynamic loads From Figs. 2- 7 it follows that the maximum amplitudes for the dynamic loads P occur in the first resonance, i.e., when the frequency of external excitation is equal to oi - the first frequency of free vibration. For this reason in the further numerical calculations resonant amplitudes are discussed, i.e., the amplitudes for p = coi. For instance, they are determined for selected values of K m and/1, K2 in the dependence on A G. The first and next frequencies of free vibration are obtained by seeking the solution of (7) using the method of separation of variables and neglected damping and the external moment in the boundary conditions (9). The effect of parameters Km and/1, K2 on the resonant amplitudes for dynamic loads P is presented in Figs. 8 and 9 for A s C (0.03, 0.77). Suitable diagrams for Km = 0.005 859, 0.018528, 530 Archive of Applied Mechanics 61 (1991) 0.05859 are plotted in Fig. 8. From Fig. 8 ib follows that each curve of the resonant amplitudes for the dynamic loads increases monotonically, reaches the maximum value, and then decreases monotonically with the increase of the parameter A s. For fixed A s the resonant amplitudes increase with increase of Km. Similar conclusions can be drawn from the diagrams in Fig. 9 for the resonant amplitudes obtained for the indicated parameter pairs K, Ks. The values of the constant/1 are determined for dl = 0.06 m, 0.10 m . ,0.22 m, and the values of the constant Ks for ds = 0.09 m, 0.15 m . , 0.33 m where d and ds are the diameters of shafts 1 and 2, re- spectively. 4 Final remarks The considerations presented in this paper have a theoretical character. They concern the dis- crete-continuous model of a single gear transmission with constant equivalent mesh stiffness and torsionally deformed ponderable shafts. In the analysis the wave method is applied. Numerical results give information on the effect of selected parameters, characterizing the single gear trans- mission, on the amplitudes of dynanfie loads on the gear teeth. In practice, the obtained informa- tion can be useful in designing single gear transmissions in which the equivalent mesh stiffness can be considered to be constant. Appendix: Determination of functions gl(z2) and/2(z) The functions g(z2) and/2(z) are determined from the third and fourth equations of set (12). The use of the method of finite differences gives g,(z2) - 291(z2 - Az) + gl(z2 - 2Az) % (Az)S + rio gl(z2) - g(zs - Az) Az h(z) - h(z - 3z) + r1191(z2) + rt2 + rlJ2(z) Az t! t ! = q4/, (z) + flail(z) + r,Jl(Z) + r1792(z) + rlsg2(z), h(z) - 2h(z - Az) + h(z - dz) h(z) - h(z - ) + ro + rj(z) rl (zjz) 2 zlz g(z2) - g,(z. - dz) + rs + r3gl(z2) Az Defining tt ! t - r2492 (z) -t- r2592(z) - r26.q2(z) + r27/1(2) - r28/l(Z)* $1 = r, + qoAZ + rlt(Az) , Se = (rl + rl#Jz) Az, $3 = (2% 4- qoAz) gl(z2 - Az) - rggl(zs - 2Az) + rlAz/2(z - Az) $4 = (r22 + r2aAz) Az, S = r,g + r2sAz + r21(Az) 2, S s = (2r19 + r2oAZ) A(z - dz) - rt9A(z - 2Az) 4- r2gt(z2 - 2z) Az + (a)2 rg(z) + rssg() + 2sqs(z) + ./;(z) + 8A(z), gl(z2) and/2(z) are expressed as follows $385 -SsS6 $3S4 -S2Se el(z) - h(z) - $1S5 -SsS4 SsS -$1S5 W. Nadolski: Dynamic investigations
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