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 ) 中文譯文 A 包絡法的資產負債 螺桿壓縮機轉子 Stosic 1998 年之后，被視為非平行不相交的螺旋齒輪，或在圖的交叉軸。 A.1。 X01， y01 和 x02 之前， y02 是該點的坐標的坐標系統中的固定的主轉子和閘轉子的端部轉子段，如示于圖。 1.3。 是繞 X 軸的旋轉角度。的轉子軸的旋轉，在其軸承是天然的轉子運動。雖然主旋翼旋轉通過角度 ，閘轉子的旋轉通過角度 =r1w / rw = z2/z1 ，其中 rw和 z 是分別的轉子葉片的節距圓的半徑和數量。此外，我們定義外部和內部的轉子半徑： r1e =r1w +r1和 r1i=r1W r0。轉子軸之間的距離是 C =r1W + r2W 。 p 是在給定的單元轉子旋轉角的轉子引線。標 1 和 2 分別涉及的主要 和閘轉子。 圖。 A.1。坐標系與非平行交錯軸斜齒輪 與一個給定的，或產生表面 R1 （ T， ）的嚙合，或產生的表面以確定，該程序開始。一個集合中仍將產生表面參數形式： R2 （ T， ， ） ，其中 t 是一個配置參數， 和是運動參數。 包絡面 r1和 r2之間的嚙合方程，它決定： 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) 包絡方程，它決定了嚙合表面之間的 r1和 r2: 0222 rrtr (A.6) 連同這些表面方程，完成方程系統。如果生成的表面 1 被定義的參數 t ，系統可用于計算另一個參數 ，現在 t 的函數，作為一個嚙合條件來定義一個生成的表面 2，現在， t和的函數的。在包絡方程的交叉乘積表 示的表面法線和 R 2是兩個表面 1和 2 ，它們一起構成了這兩個表面的接觸，共同的切點上的單點的相對滑動速度。由于平等到零的一個標量三重積下施加的坐標系，并是一個不變的屬性，因為相對速度，可以同時在兩個坐標系統的嚙合條件被定義為，以方便的形式表示： 0211111 rrtrrrtr （ A.7） 插入前面的表達式到系統條件給： tyytxxppxC 1111211 c o t)( 0)c o t( 12111 txCptypp (A.8) 這是適用于這里的條件交叉均勻鉛與非平行交錯軸斜齒輪的嚙合動作。的方法構成的齒輪的生成過程，這是普遍適用的。它可用于合成的目的，這是有效地與平行軸的螺旋齒輪的螺桿壓縮機轉子。非平行和非相交軸越過轉子制造的形成工具的螺旋齒輪上具有均勻的引線，在滾齒的情況下，或與如銑削和磨削形成不含鉛。轉子檢查模板平面轉子滾刀一樣。在所有這些情況下，刀具軸不相交的轉子軸。 因此，注意到提出的包絡的方法的應用程序，以產生交叉的螺 旋齒輪的嚙合條件。螺桿轉子齒輪，然后給出作為其使用一個基本例子的，而形成滾齒機工具的過程作為一個復雜的情況下給出。 軸角 ，中心距 C ，和單元信息的兩個交叉的螺旋齒輪， p1和 p2是相互依賴的。交錯軸斜齒輪嚙合仍保存著兩個齒條正截面具有相同的配置文件，并在機架上的螺旋角與軸角 = r1 + r2 。這是通過在 x 方向上的齒條迫使他們相應地調整到適當的機架螺旋角的隱式移位。這當然也包括特殊情況下，這樣的齒輪可以是定向的，使得在軸角的齒輪的螺旋角的總和是等于： = 1+ 2 。此外，中心距離可以 等于齒輪節距半徑的總和：21 rrC 成對的交叉斜齒輪可以與兩個螺旋角相同的符號或每個符號相反，左或右旋的，取決于其鉛和軸角上的組合。 嚙合條件，可以解決只能通過數值方法。對于給定的參數 t ，坐標 X01 ， Y01 和它們的衍生物 所述 X01和 Y01 是已知的。甲猜到參數的值，然后用于計算 X1， Y1 ， T所述 t1和 T1。經修訂的值，然后推導和過程反復進行，直到連續兩個值之間的差異變得足夠小。 對于給定的橫向坐標和齒輪 1 的檔案中的衍生物，可以用來計算 X1， Y1，和 z1坐標其螺旋表面。齒輪 2 的螺旋面的表面，然后可以被計算出來。坐標 z2 的然后，可以使用計算和最后，其橫向的更新點坐標 X2,Y2,可以得到的。 從這樣的分析，可以發現多宗個案。 (i) 當 = 0 ，方程滿足螺桿機轉子和也具有平行軸的螺旋齒輪的嚙合狀態。對于這樣的情況下，齒輪的螺旋角的有相同的值，但符號相反的齒比 i = P2/P1 為負。也可以應用相同的方程的根憂思從齒輪形成的齒條。此外，它描述所形成的平面爐灶，前銑削刀具和模板控制儀器。 (ii) 如果光盤銑削或研磨工具被認為形 成的，它是足夠放置 p2 的 = 0 。這是一個單一的情況下，工具自由轉動時，不影響嚙合過程。因此，反向變換不能直接獲得。 (iii) 全部范圍的嚙合條件是必需生成形成滾齒機工具的檔案。因此，這是最復雜的性態類型的齒輪，它可以從它產生。 B 雷諾運輸定理 繼 Hanjalic ， 1983 年，雷諾運輸定理定義變量在有限的面積 A 的哪個矢量本地法線