包络法的资产负债-螺杆压缩机转子外文文献翻译@中英文翻译@外文翻译

英文原文 A Envelope Method of Gearing Following Stosic 1998, screw compressor rotors are treated here as helical gears with nonparallel and nonintersecting, or crossed axes as presented at Fig. A.1. x01, y01 and x02, y02are the point coordinates at the end rotor section in the coordinate systems fixed to the main and gate rotors, as is presented in Fig. 1.3. is the rotation angle around the X axes. Rotation of the rotor shaft is the natural rotor movement in its bearings. While the main rotor rotates through angle , the gate rotor rotates through angle = r1w/r2w = z2/z1, where r w and z are the pitch circle radii and number of rotor lobes respectively. In addition we define external and internal rotor radii: r1e= r1w+ r1 and r1i= r1w r0. The distance between the rotor axes is C = r1w+ r2w. p is the rotor lead given for unit rotor rotation angle. Indices 1 and 2 relate to the main and gate rotor respectively. Fig. A.1. Coordinate system of helical gears with nonparallel and nonintersecting Axes The procedure starts with a given, or generating surface r1(t, ) for which a meshing, or generated surface is to be determined. A family of such gener-ated surfaces is given in parametric form by: r2(t, , ), where t is a prole parameter while and are motion parameters. r1 =r1(t, )= x1,y1,z1 =x01cos-y01 sin, x01 sin+ y01 cos,p1 (A,.1) 0, 111 tytxtr = 0,c o ss i n,s i nc o s 0101011 tytxtytx (A.2) 0,0, 01010111 xyyxr (A.3) c o ss i n,s i nc o s,),(1111122222 zyzyCxzyxtrr 202020202 ,s ins in,s inco s pyxyx (A.4) 2020202022222 ,s i nc o s,s i ns i n, pyxyxpxyr s i n)(c o s,c o s)(s i n,c o ss i n 121211 CxpCxpyp (A.5) The envelope equation, which determines meshing between the surfaces r1 and r2: 0222 rrtr (A.6) together with equations for these surfaces, completes a system of equations. If a generating surface 1 is dened by the parameter t, the envelope may be used to calculate another parameter , now a function of t, as a meshing condition to define a generated surface 2, now the function of both t and . The cross product in the envelope equation represents a surface normal and r2 is the relative, sliding velocity of two single points on the surfaces 1 and 2 which together form the common tangential point of contact of these two surfaces. Since the equality to zero of a scalar triple product is an invariant property under the applied coordinate system and since the relative velocity may be concurrently represented in both coordinate systems, a convenient form of the meshing condition is dened as: 0211111 rrtrrrtr （ A.7） Insertion of previous expressions into the envelope condition gives: tyytxxppxC 1111211 c o t)( 0)c o t( 12111 txCptypp (A.8) This is applied here to derive the condition of meshing action for crossed helical gears of uniform lead with nonparallel and nonintersecting axes. The method constitutes a gear generation procedure which is generally applicable. It can be used for synthesis purposes of screw compressor rotors, which are electively helical gears with parallel axes. Formed tools for rotor manufacturing are crossed helical gears on non parallel and non intersecting axes with a uniform lead, as in the case of hobbing, or with no lead as in formed milling and grinding. Templates for rotor inspection are the same as planar rotor hobs. In all these cases the tool axes do not intersect the rotor axes. Accordingly the notes present the application of the envelope method to produce a meshing condition for crossed helical gears. The screw rotor gearing is then given as an elementary example of its use while a procedure for forming a hobbing tool is given as a complex case. The shaft angle , centre distance C, and unit leads of two crossed helical gears, p1 and p2 are not interdependent. The meshing of crossed helical gears is still preserved: both gear racks have the same normal cross section prole, and the rack helix angles are related to the shaft angle as = r1+ r2. This is achieved by the implicit shift of the gear racks in the x direction forcing them to adjust accordingly to the appropriate rack helix angles. This certainly includes special cases, like that of gears which may be orientated so that the shaft angle is equal to the sum of the gear helix angles: = 1+ 2. Furthermore a centre distance may be equal to the sum of the gear pitch radii :C = r1+ r2. Pairs of crossed helical gears may be with either both helix angles of the same sign or each of opposite sign, left or right handed, depending on the combination of their lead and shaft angle . The meshing condition can be solved only by numerical methods. For the given parameter t, the coordinates x01 and y01 and their derivatives x01t and y01t are known. A guessed value of parameter is then used to calculate x1, y1, x1 t and y1t. A revised value of is then derived and the procedure repeated until the difference between two consecutive values becomes sufficiently small. For given transverse coordinates and derivatives of gear 1 prole, can be used to calculate the x1, y1, and z1 coordinates of its helicoid surfaces. The gear 2 helicoid surfaces may then be calculated. Coordinate z2 can then be used to calculate and nally, its transverse prole point coordinates x2, y2 can be obtained. A number of cases can be identied from this analysis. (i) When = 0, the equation meets the meshing condition of screw machine rotors and also helical gears with parallel axes. For such a case, the gear helix angles have the same value, but opposite sign and the gear ratio i = p2/p1 is negative. The same equation may also be applied for the gen-eration of a rack formed from gears. Additionally it describes the formed planar hob, front milling tool and the template control instrument.122 A Envelope Method of Gearing (ii) If a disc formed milling or grinding tool is considered, it is suffcient to place p2= 0. This is a singular case when tool free rotation does not affect the meshing process. Therefore, a reverse transformation cannot be obtained directly. (iii) The full scope of the meshing condition is required for the generation of the prole of a formed hobbing tool. This is therefore the most compli-cated type of gear which can be generated from it. B Reynolds Transport Theorem Following Hanjalic, 1983, Reynolds Transport Theorem denes a change of variable in a control volume V limited by area A of which vector the local normal is dA and which travels at local speed v. This control volume may, but need not necessarily coincide with an engineering or physical material system. The rate of change of variable in time within the volume is: vVdVtt (B.1) Therefore, it may be concluded that the change of variable in the volume V is caused by: change of the specic variable m/ in time within the volume because of sources (and sinks) in the volume, t dV which is called a local change and movement of the control volume which takes a new space with variable in it and leaves its old space, causing a change in time of for v.dA and which is called convective change The rst contribution may be represented by a volume integral:. dVtV (B.2) while the second contribution may be represented by a surface integral: A dAV (B.3) Therefore: AV VVdAVdVtdVdtdt ( B.4) which is a mathematical representation of Reynolds Transport Theorem. Applied to a material system contained within the control volume V m which has surface A m and velocity v which is identical to the fluid velocity w, Reynolds Transport Theorem reads: dAWdVtddtdt AmVm VmVm V (B.5) If that control volume is chosen at one instant to coincide with the control volume V , the volume integrals are identical for V and Vm and the surface integrals are identical for A and Am , however, the time derivatives of these integrals are different, because the control volumes will not coincide in the next time interval. However, there is a term which is identical for the both times intervals: dVtdVt VmV (B.6) therefore, AVAmVm dAvtdAwt (B.7) or: dAvwttAVVm (B.8) If the control volume is xed in the coordinate system, i.e. if it does not move, v = 0 and consequently: dVtt VV (B.9) therefore: AVVmdAwdVtt (B.10) Finally application of Gauss theorem leads to the common form: dVwdVtt VVVm (B.11) As stated before, a change of variable is caused by the sources q within the volume V and influences outside the volume. These effects may be proportional to the system mass or volume or they may act at the system surface. The rst effect is given by a volume integral and the second effect is given by a surface integral. VVAVm AmAVm qdVdVqqvdAqq v d Vt (B.12) q can be scalar, vector or tensor. The combination of the two last equations gives: A VVq d VdAwdVt Or: 0 dVqwtV (B.13) Omitting integral signs gives: 0 qwt (B.14) This is the well known conservation law form of variable . Since for = 1, this becomes the continuity equation: 0 wt nally it is: 0 qwwt Or: qwtdtD (B.15) dtD / is the material or substantial derivative of variable . This equation is very convenient for the derivation of particular conservation laws. As previously mentioned = 1 leads to the continuity equation, = u to the momentum equation, = e, where e is specic internal energy, leads to the energy equation, = s, to the entropy equation and so on. If the surfaces, where the fluid carrying variable enters or leaves the control volume, can be identied, a convective change may conveniently be written: o u tino u tinA mmmddAw )()( (B.16) where the over scores indicate the variable average at entry/exit surface sections. This leads to the macroscopic form of the conservation law: QmmQdtddtdoutinoutinVV )()( (B.17) which states in words: (rate of change of ) = (inflow ) (outflow ) +(source of ) 中文译文 A 包络法的资产负债 螺杆压缩机转子 Stosic 1998 年之后，被视为非平行不相交的螺旋齿轮，或在图的交叉轴。 A.1。 X01， y01 和 x02 之前， y02 是该点的坐标的坐标系统中的固定的主转子和闸转子的端部转子段，如示于图。 1.3。 是绕 X 轴的旋转角度。的转子轴的旋转，在其轴承是天然的转子运动。虽然主旋翼旋转通过角度 ，闸转子的旋转通过角度 =r1w / rw = z2/z1 ，其中 rw和 z 是分别的转子叶片的节距圆的半径和数量。此外，我们定义外部和内部的转子半径： r1e =r1w +r1和 r1i=r1W r0。转子轴之间的距离是 C =r1W + r2W 。 p 是在给定的单元转子旋转角的转子引线。标 1 和 2 分别涉及的主要 和闸转子。 图。 A.1。坐标系与非平行交错轴斜齿轮 与一个给定的，或产生表面 R1 （ T， ）的啮合，或产生的表面以确定，该程序开始。一个集合中仍将产生表面参数形式： R2 （ T， ， ） ，其中 t 是一个配置参数， 和是运动参数。 包络面 r1和 r2之间的啮合方程，它决定： r1 =r1(t, )= x1,y1,z1 =x01cos -y01 sin , x01 sin + y01 cos ,p1 (A,.1) 0, 111 tytxtr = 0,c o ss i n,s i nc o s 0101011 tytxtytx (A.2) 0,0, 01010111 xyyxr (A.3) c o ss i n,s i nc o s,),( 1111122222 zyzyCxzyxtrr 202020202 ,s ins in,s inco s pyxyx (A.4) 2020202022222 ,s i nc o s,s i ns i n, pyxyxpxyr s i n)(c o s,c o s)(s i n,c o ss i n 121211 CxpCxpyp (A.5) 包络方程，它决定了啮合表面之间的 r1和 r2: 0222 rrtr (A.6) 连同这些表面方程，完成方程系统。如果生成的表面 1 被定义的参数 t ，系统可用于计算另一个参数 ，现在 t 的函数，作为一个啮合条件来定义一个生成的表面 2，现在， t和的函数的。在包络方程的交叉乘积表 示的表面法线和 R 2是两个表面 1和 2 ，它们一起构成了这两个表面的接触，共同的切点上的单点的相对滑动速度。由于平等到零的一个标量三重积下施加的坐标系，并是一个不变的属性，因为相对速度，可以同时在两个坐标系统的啮合条件被定义为，以方便的形式表示： 0211111 rrtrrrtr （ A.7） 插入前面的表达式到系统条件给： tyytxxppxC 1111211 c o t)( 0)c o t( 12111 txCptypp (A.8) 这是适用于这里的条件交叉均匀铅与非平行交错轴斜齿轮的啮合动作。的方法构成的齿轮的生成过程，这是普遍适用的。它可用于合成的目的，这是有效地与平行轴的螺旋齿轮的螺杆压缩机转子。非平行和非相交轴越过转子制造的形成工具的螺旋齿轮上具有均匀的引线，在滚齿的情况下，或与如铣削和磨削形成不含铅。转子检查模板平面转子滚刀一样。在所有这些情况下，刀具轴不相交的转子轴。 因此，注意到提出的包络的方法的应用程序，以产生交叉的螺 旋齿轮的啮合条件。螺杆转子齿轮，然后给出作为其使用一个基本例子的，而形成滚齿机工具的过程作为一个复杂的情况下给出。 轴角 ，中心距 C ，和单元信息的两个交叉的螺旋齿轮， p1和 p2是相互依赖的。交错轴斜齿轮啮合仍保存着两个齿条正截面具有相同的配置文件，并在机架上的螺旋角与轴角 = r1 + r2 。这是通过在 x 方向上的齿条迫使他们相应地调整到适当的机架螺旋角的隐式移位。这当然也包括特殊情况下，这样的齿轮可以是定向的，使得在轴角的齿轮的螺旋角的总和是等于： = 1+ 2 。此外，中心距离可以 等于齿轮节距半径的总和：21 rrC 成对的交叉斜齿轮可以与两个螺旋角相同的符号或每个符号相反，左或右旋的，取决于其铅和轴角上的组合。 啮合条件，可以解决只能通过数值方法。对于给定的参数 t ，坐标 X01 ， Y01 和它们的衍生物 所述 X01和 Y01 是已知的。甲猜到参数的值，然后用于计算 X1， Y1 ， T所述 t1和 T1。经修订的值，然后推导和过程反复进行，直到连续两个值之间的差异变得足够小。 对于给定的横向坐标和齿轮 1 的档案中的衍生物，可以用来计算 X1， Y1，和 z1坐标其螺旋表面。齿轮 2 的螺旋面的表面，然后可以被计算出来。坐标 z2 的然后，可以使用计算和最后，其横向的更新点坐标 X2,Y2,可以得到的。 从这样的分析，可以发现多宗个案。 (i) 当 = 0 ，方程满足螺杆机转子和也具有平行轴的螺旋齿轮的啮合状态。对于这样的情况下，齿轮的螺旋角的有相同的值，但符号相反的齿比 i = P2/P1 为负。也可以应用相同的方程的根忧思从齿轮形成的齿条。此外，它描述所形成的平面炉灶，前铣削刀具和模板控制仪器。 (ii) 如果光盘铣削或研磨工具被认为形 成的，它是足够放置 p2 的 = 0 。这是一个单一的情况下，工具自由转动时，不影响啮合过程。因此，反向变换不能直接获得。 (iii) 全部范围的啮合条件是必需生成形成滚齿机工具的档案。因此，这是最复杂的性态类型的齿轮，它可以从它产生。 B 雷诺运输定理 继 Hanjalic ， 1983 年，雷诺运输定理定义变量在有限的面积 A 的哪个矢量本地法线