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哈爾濱理工大學學士學位論文
附 錄
GEARS
Gears transmit power and motion between moving pasts. Positive transmission of power is accomplished by projections or teeth on the circumference of the gear . There is no slippage as with friction and belt drives , a feature most machinery requires ,because exact speed ratios are essential .Friction drives are used in industry ,where high speeds and light loads are required and where loads subject to impact are transmitted.
When the teeth are built up on the circumference of two rolling disks in contact, recesses must be Provided between the teeth are developed is known as the pitch circle .It is an imaginary circle with the same diameter as a disk that would cause the same relative motion as the gear. All gear design calculations are based on the diameter of the pitch circle. A portion of a gear is shown in Figure 22.13.
Gear Nomenclature
The system of gearing used in the United States is known as the involutes system, because the profile of a gear tooth is principally an involutes curve. An involutes is a curve generated on the circle, the normal of which are all tangent to this circle. The method of generating involutes is shown in Figure 22.14. Assume that a string having a pencil on its end is wrapped around a cylinder. The curve described by pencil as the string is unwound is an involutes, and the cylinder on which it is wound is known as the base circle. The portion of the gear tooth from the base at point a in the figure to the outside diameter at point c is an involutes curve and is the portion that contacts other teeth.. From point b to
point the profile of the base circle on which the involutes is described is inside the pitch circle and is dependent on the angle of thrust of the dear teeth. The relationship existing between the diameter of the pitch circle, D, is Db = Dcosθ where Db = diameter of base circle θ=Angle of thrust between gear tooth.The two common systems have their thrust angles or lines of action at 141/ and .
Figure 22.13 Nomencla ture for Involute spur gear
Other angles are possible, but with larger angles the radial force component tending to force the gears apart becomes greater. If a common tangent is drawn to the pitch circles of two meshing gears. The base circle on which the involutes are drawn are tangent to the line of action.
Most gears transmitting power use the 200, full-deep, involutes tooth form. These gears have the same tooth proportion as the 141/20 full–depth involutes but are stronger at their base because of greater thickness. The 200, fine –pitch involutes gears are-similar to the regular 200 involutes and are made in sizes ranging from 20 to 200 diametral pitch. These gears are used primarily for transmitting motion rather the power. The 200 stub tooth gear has smaller tooth depth than the 200. Full –depth gear and is consequently stronger. Involutes gears fulfill all laws of gearing and have the advantage over some other curves in that the contact action is affected by slight variation of gear center distance.
Figure 22.14 Mothod of genera an Involute tooth surface
The nomenclature of a gear tooth is illustrated in Figure 22.13. the principal definitions and tooth parts for standard 141/20 and 200 involutes gears are discussed here.
The addendum of a tooth is the radial distance from the pitch circle to the outside diameter of addendum circle. Numerically, it is equal to 1 divided by the diametral pitch P.
The addendum is the radial distance from the pitch circle to the root or addendum circle. It is equal to the addendum plus the tooth clearance.
Tooth thickness is the thickness of the tooth measure on the pitch circle. For cut gears the tooth thickness and tooth space are equal. Cast gears are provided with some backlash, the difference between the tooth thickness and tooth space measured on the pitch circle.
The face of a gear tooth is that surface lying between the pitch circle and the addendum circle.
The flank of a gear tooth is that surface lying between the pitch circle and the root circle.
Clearance is a small distance provided so that the top of a meshing tooth will not touch the bottom land of the other gear as it passes the line of centers.
Table 22.2gives the proportions of standard 141/200 involutes gears expressed in term of diametral pitch P and number of teeth N.
Table 22.2 American Gear Manufactures Association Standard for Involute Gearing
Pitch of Gears
The circuit pitch p is the distance from a point on one tooth to the corresponding point on an adjacent tooth, and is measured on the pitch circle. Expressed as an equation.
Metrical gearing is based on the module(mod) instead of the diametral pitch p, as in the English system. The basic metric module formula is mod =D/N=amount of pitch diameter per tooth =millimeters per tooth measured on the pitch diameter. Also, mod=1/p is expressed in millimeters. Also, mod p=25.4.
P = πD/N where D = diameter of the pitch circle
N = number of teeth
The diametral pitch p, often referred to as the pitch of a gear is the ratio of the number of teeth to the pitch diameter. It may be expressed by the following equation: P = N/D
Upon multiplying these two equations the following relationship between circular and diametral pitch results.
Hence,knowing the value of either pitch we may obtail the other by dividing into π.
Gears and gear cutters are standardized according to diametral pitch. This pitch can be expressed in even figures or fractions. Circular pitch, being an actual distance, it is expressed in inches and fractions of an inch. A 6-inth gear (6diametral pitch) is one that has 6teeth per inch of pitch diameter . If the pitch diameter is 3 inch, the number of teeth is 3 x 6 or 18.The outside diameter of the gear is equal to the pitch diameter plus twice the addendum distance or 3 in.+2 x 1/6,which is 3.333in.
Any involutes gear of a given diametral pitch will mesh properly with a gear of any other size of the same diametral pitch. However, in cutting gears of various diameters a slight difference in the cutter is necessary to allow for the change in curvature of the involutes as the diameter increases. The extreme case would be a rack tooth ,which would have a straight line as the theoretical tooth profile. For practical reasons the number of teeth in an involutes gear should not be less than 12.
Gear speed
The speeds in rooms ,s and S, of two meshing gears vary inversely with both the pitch diameter and the number of teeth .This may be expressed as follows:
Figure 22.15 Nomenclature for meshing gear and pinon
s/S = D/d =T/t
where Dand d represent pitch diameter as included as indicated in Figure 22.15.T and t represent number of teeth on the gear and pinion.
Center distance : L = (D+d)/2
The speed ratio for a worm gear set depends on the number of teeth on the gear and the lead of the worm. For a single=threaded worm the ratio is
Rpm worm/rpm gear = T/t
Kinds of gears
The gears most commonly used are those that transmission power between two parallel shafts. Such gears having their tooth elements parallel to the ratating shafts are known as spur gears, the smaller of the two being known as a pinion (Figure 22.15).If the elements of the teeth are twisted or helical,as known in figure 22.16B,they are known as helical gears. These gears amay be for connecting shafts that are at an angle in the same or different planes. Helical gears are smooth acting because there is always more than one tooth in contact. Some power is lost because of end thrust, and provision must be made to compensate for this thrust in the bearings. The herringbone gear is equivalent to two helical gears, one having right-hand and the other a left-hand helix.
Figre 22.17 All elements of straight bevel converge at the one opex of the gears
Usually, when two shafts are in the same plane but at an angle with one another, a bevel gear is used. Such a gear is similar in appearance to the frustum of a cone having all the elements of the teeth intersecting at a point, as shown in Figure 22.17. Bevel gears are made with either straight or spiral teeth. When the shafts are at right angles and the two bevel gears are the same size, they are known as miter gears (figure 22.16A). Hypoid gears, an interesting modification of bevel gears shown as Figure 22.16F, have their shaft at right angles by they do not intersect as do the shaft for bevel gears. Correct teeth for these gears are difficult to construct, although a generating process has been developed that produces satisfactory teeth. Zero gears (Figure 22.16D)have curved teeth but have a zero helical angle. They are produced on machines that cut spiral bevels and hypoids. Worm gearing is used where a large speed reduction is desired. The small driving gear is called a worm and the driving gear is called a worm and the driven gear a wheel. The worm resembles a large screw and is set in close to the wheel circumference, the teeth of the wheel being curving to conform to the diameter of the worm. The shafts for such gears are at right angles but not in the same plane. These gears are similar to helical gears in their application, but differ considerably in appearance and method of manufacture. A worm gear set is shown in Figure 22.16C.
Rack gears, which are straight and have no curvature, represent a gear of infinite radius and are used in feeding mechanisms and for reciprocating. They may have either straight or helical teeth. If the rack is bent in the form of a circle, it becomes a bevel gear having a cone apex angle of 180oknown as crown gear. the teeth all converge at the center of the disk and mesh properly with a bevel gear of the same pitch. A gear with internal teeth, known as an annular gear, can be cut to mesh with either a spur or bevel gear, depending on whether the shafts are parallel or intersecting.
Methods of Making Gears
Most gears are produced by some machining process. Accurate machine work is essential for high-speed, long-wearing, quite-operating gears. Die and investment casting of gears has proved satisfactory, but the materials are limited to low-temperature-melting metals and alloys. Consequently, these gears do not have the wearing qualities of heat-treated steel gears. Stamping though reasonably accurate, can be used only in making thin gears from sheet metal.
Commercial methods employed in producing gears are summarized as follows:
A: Casting 1.sand casting 2.Die casting 3.Precision and investment casting
B: Stamping
C: Machining 1.Formed-tooth process a. From cutter in milling machine b. From cutter in broaching machine c. From cutter in shaper 2.Template process 3.Cutter generating process a. cutter gear b. Hobbing c. Rotary cutter d. Reciprocating cutters simulating a rack
D: Power metallurgy
E: Extruding
F: Rolling
G: Grinding
H: Plastic molding
Form Tooth Process
A formed milling cutter, as shown in Figure22.18,is commonly used for cutting a spur gear. Such a cutter used on a milling machine is formed according to the shape of the tooth space to de removed. Theoretically, there should be a different-shape cutter for each size gear of a given pitch as there is a slight change in the curvature of the involutes. However, one cutter can be used for several gears having different numbers of teeth without much sacrifice in their operation. Each pitch cutter is made in eight slightly varying shapes to compensate for this change.
They vary from no.1, which is used to cut gears from 135 teeth to a rack, to no.8, which cuts gears having 12 or 13 teeth. The eight standard involutes cutters are listed in Table 22.3.
Setup of a milling machine to cut spur gears are illustrated in Figure 22.18. A discussion of this process is given the chapter on milling is an accurate process for cutting spur, helical, and worm gears. Although sometimes used for bevel gears, the process is not accurate because of the gradual change in tooth thickness. When used for bevel gears at least two cuts are necessary for each tooth space. The usual practice is to take one center cut of proper depth and about equal to the space at the small end of the tooth. Two shaving cuts are then on each side of the tooth space to give the tooth its proper shape.
Figure 22.18 Setup for cutting a spir gear on a milling machine
Table 22.3 Standard Involute cutters
No.1
135 teeth to a rack
No.2
55 to 134 teeth
No.3
35 to 54 teeth
No.4
26 to 34 teeth
No.5
21 to 25 teeth
No.6
17 to 20 teeth
No.7
14 to 16 teeth
No.8
12 to 13 teeth
The formed-tooth principle may also be utilized in a broaching machine by making the broaching tool conform to the teeth space. Small internal gears can be completely cut in one pass by having a round broaching tool made with the same number of cutters as the gear has teeth. Broaching tool is limited to large-scale production because of the cost of cutters.
齒輪
運動部件之間的能量和運動由齒輪來傳遞。主運動的能量由齒輪四周的凸臺或齒相嚙合來傳遞。由于摩擦和帶傳動，齒輪之間的傳遞無滑移。因為傳遞需要準確的速度，摩擦傳動被廣泛應用于工業(yè)，如高速，輕載以及載荷連續(xù)的地方。
為了保持兩個相嚙合的齒輪以及消除干涉，兩個相嚙合齒輪之間應該留有一點的間隙。向上延伸就是眾所周知的節(jié)圓。節(jié)圓是個假想的圓，載以此為半徑的圓上可以實現(xiàn)齒輪相嚙合。因此所有的齒輪的設計計算建立載節(jié)圓之上的。齒輪的部分如圖 2.2.13所示。
齒輪專業(yè)術(shù)語
由于齒輪的輪廓為漸開線，所以載美國齒輪系統(tǒng)稱之為漸開線系統(tǒng)。漸開線是一條產(chǎn)生于圓且所有的曲線垂直圓的曲線。漸開線的產(chǎn)生方法如圖22.14 所示。假設一個旋轉(zhuǎn)的鉛筆的一端饒在一個圓柱上，隨著鉛筆的旋轉(zhuǎn)，未被破壞的曲線即為漸開線，而被破壞的為基圓。從如圖所示的基圓上的a點到外圓上的c點為漸開線曲線的一部分，在這部分上齒輪相嚙合。從b點到a點以及到根圓的倒角部分為一段射線。漸開線的基圓在節(jié)圓的內(nèi)部，同時基圓的位置決定了齒輪的壓力角。節(jié)圓和基圓直徑之間有如下關(guān)系：
Db = Dcosθ(Db為基圓的直徑，θ為齒輪的壓力角)?，F(xiàn)在廣泛運用的兩個壓力角（或作用線）為140/2o和20o，其他的角度也是可能的，但是跟隨角度的變大。如果對于兩個相嚙合的節(jié)圓的線相切，壓力角（或作用線）以140/2o為好，這時在基圓上所有的漸開線與作用線相切。
大多數(shù)傳遞動力的齒輪使用壓力角為20o，全切深的漸開線齒輪。20o漸開線齒輪和140/2o的漸開線齒輪具有相同的齒部分，但由于20o的基圓上有較厚的齒，因而強度更高。如同標準的20o齒輪一樣，20o精切節(jié)圓漸開線齒輪有從20到200大小不等直徑的節(jié)圓的齒輪。這些齒輪主要用于傳遞運動而非能量。與20o全切深的齒輪相比較，20o的輕型的齒輪有著較淺的切屑深度，但是強度更高。漸開線齒輪滿足齒輪設計的所有的原理，因為作用線不受齒輪中心距的變化而受到影響，因為必其他的曲線輪廓更加的有利于齒輪的嚙合。
如圖22.13所示，對齒輪的相關(guān)的術(shù)語進行了闡述。同時液討論了標準的20o和141/20o齒輪漸開線齒輪的規(guī)律定義以及齒的各部分進行了說明。定義如下：
齒頂高為節(jié)圓到齒頂圓之間的距離，數(shù)值上等于1除以節(jié)圓直徑P。
齒根高為節(jié)圓到齒根圓（或齒根高）之間的距離，其等于齒頂高乘以齒間隙。齒厚為在節(jié)圓上測量得到的齒的厚度。對于切削齒輪，齒厚和齒間隙是相等的。然而對于鑄造齒輪，由于提供了一定的緩沖，因而對于在節(jié)圓上齒厚和齒間隙的測量有一定的區(qū)別。
齒輪的前刀面為位于節(jié)圓和齒頂圓之間的面。
齒輪的后刀面為位于節(jié)圓和齒根圓之間的面。
間隙為以各小的距離，他的提供是為了在經(jīng)過中心線時，兩個相嚙合的齒輪不發(fā)生干涉（嚙合齒輪的頂部不與令一個齒輪的根部相撞）。
表22.2根據(jù)節(jié)圓直徑P和齒數(shù)N給出了141/20o和20o標準的漸開線齒輪的部分的數(shù)據(jù)。
齒輪節(jié)圓
齒距P為從齒輪的一側(cè)的一點刀另外一個相鄰點之間的距離，通常在節(jié)圓上測量大小。方程式為：
P = πD/N（D等于節(jié)圓的直徑，N等于齒的數(shù)目）
節(jié)圓直徑P，通常指的是在節(jié)圓上，齒輪的齒數(shù)除以節(jié)圓的直徑。方程式如下：
P = N/D
如果將上式相乘可以得到齒距和節(jié)圓直徑之間的如下關(guān)系：P=π。因此，如果我們知道了兩者之間的任何一個值，就可以通過除以π得到令一個值。
齒輪和齒輪的加工根據(jù)不同的節(jié)圓直徑可以實現(xiàn)標準化。節(jié)圓可以通過數(shù)字或者分數(shù)來表示。由于齒距是一個實際的距離，因此可以以英寸或者是幾英寸表示。一個6節(jié)圓齒輪為在節(jié)圓直徑上每英寸6個齒的齒輪。如果節(jié)圓的直徑為3英寸，則齒數(shù)為三乘以六為十八。外圓直徑等于節(jié)圓直徑乘以2倍的齒頂高，或者3+1x1/6=3.333英寸。
任意給定節(jié)圓直徑的漸開線齒輪將與同節(jié)圓直徑的任意尺寸的齒輪正確的嚙合。然而，在加工不同直徑的切削齒輪時，在切削過程中，隨著直徑的增加，漸開線的曲率發(fā)生輕微的變化時必須的允許的。但是齒條時一個特例，如果為理論的齒形，齒條應該為以條直線。但是由于實際的原因，一個漸開線齒輪的齒數(shù)則必須的大于12。
齒輪速度
對于s和S兩個齒輪來說，兩個相嚙合的齒輪的速度（rpms）與節(jié)圓直徑和齒數(shù)成反比。表達方程式為：s/S=D/d=T/t。如圖22.15所示，D、d代表節(jié)圓的直徑，T、t代表大齒輪和小齒輪的齒數(shù)，中心距：L=(D+d)/2
對于蝸輪蝸桿的速度由齒輪齒數(shù)和蝸桿的頭數(shù)決定。對于單線蝸桿比率如下所示：(rpms orm)/(rpm gear) = T/1
齒輪種類
最普通使用的齒輪為傳遞動力的平行軸齒輪。其中齒輪軸平行于旋轉(zhuǎn)軸的稱為直齒輪，如圖 22.15 所示，兩個齒輪中較小的稱為小齒輪。如圖 22.16 所示，如果齒輪示扭曲或斜的，，則稱為斜齒輪，所有齒輪的命名與在同一平面或不在同一平面成什么樣的角度相聯(lián)系。斜齒輪由于總是超過一個齒輪在嚙合，所以應該采取一些措施來補償根切的影響。人字形齒輪等價于兩個斜齒輪，其中一個為右手系，另一個為左手系齒輪。
通常，當兩個軸在同一平面內(nèi)，同時彼此之間有一定的角度時，使用斜齒輪。這樣的齒輪表面與所有齒輪的齒形延長線相交于一點的圓錐一樣，圖形如22.17所示。通常斜齒輪的形狀為直齒或者是螺旋形齒。當兩軸之間的角度為90o時，同時這兩個斜齒輪有相同的尺寸，我們稱之為等徑傘齒輪（如圖22.16A所示）。如圖22.16F所示的偏軸傘齒輪，是對斜齒輪有趣的修改，雖然斜齒輪之間的角度為90o，但是并不相交。雖然對于齒輪，使人滿意的加工方法不斷的發(fā)展，但是制造準確的齒形仍然使困難的。零度齒輪（如圖22.16D所示）有曲線的齒，斜齒的角度為零度，它通過在機器上加工螺旋形斜齒和偏軸傘齒輪得到所要求的齒輪。對于蝸輪蝸桿，常被用于快速降速的地方。小的驅(qū)動的齒輪為蝸桿，被驅(qū)動的齒輪為蝸輪蝸桿好象一個大的螺栓，緊緊地束縛在蝸輪的四周上，被切削的蝸輪齒與螺桿的直徑相對應。蝸輪蝸桿的軸之間呈90o，但不在同一平面內(nèi)。它的工作原理和斜齒輪的工作原理非常類似，但是表面和加工的方法有著明顯的區(qū)別。蝸輪蝸桿的形狀如圖22.16C所示。齒輪齒條為直的、無曲率、且半徑無窮大的齒輪，被廣泛的應用于機械的進給運動和往復運動之中。他們可以是直齒或者是斜齒。如果齒條彎成一個圓，則它就成為眾所周知的冠齒輪，冠齒輪的圓錐的最高角度可以是180o，所有的齒的延長線匯集于中點，能夠和相同節(jié)圓的斜齒輪進行正確的嚙合。對于內(nèi)齒輪來說，是一個內(nèi)齒的齒輪，根據(jù)軸是平行還是相交的不同，可以加工出與直齒輪或斜齒輪嚙合的齒輪。
齒輪的加工的方法
大多數(shù)的齒輪的加工由普通的機床進行加工。為了獲得高速，耐磨，穩(wěn)定的齒輪，用精密的機床進行加工也是很有必要的。因此，模板成型和銷蝕模使齒輪的加工發(fā)展到了令人滿意的程度，但是其加工的材料僅限于低熔融的金屬和合金。同時，沒有熱加工鋼鐵齒輪耐磨的性質(zhì)。雖然模鍛的精度很高，但是僅能制造板金類的精細齒輪。商用的加工齒輪的方法如下所示：
A 鑄造
1 砂型鑄造 2 壓模鑄造 3 精密和銷蝕模鑄造
B 沖壓
C 機械加工
1 成型加工 a 銑床成型加工 b 拉床成型加工 c 牛頭刨床成型加工
2 模板加工
3 切削加工 a 刨床切削加工 b 滾齒加工 c 旋轉(zhuǎn)切削加工 d 往復切削加工齒條
D 冶金加工
E 擠壓加工
F 磨削成型
G 磨削成型
H 塑料模具成型
成型齒輪刀具加工
如圖22.18所示，通常一個成型的銑削用于直齒輪的加工。而使用的機床的型號由被切削的齒槽的形狀來決定的。理論上來說，由于漸開線的曲率存在著輕微的變化，所以給定的節(jié)圓不同，則有不同的切削形狀的切削刀具。然而，一種切削的方式就可以在沒有任何缺陷的操作中加工出很多具有不同齒數(shù)的齒輪。每個切削的節(jié)圓根據(jù)形狀的變化等分為八份，來補償形狀發(fā)生的變化。如表22.3所示，列出了8種標準的漸開線切削方式，變化的范圍從NO.1的135齒到齒條，到NO.8的12齒或13齒。
如圖22.18所示，闡明了切削直齒輪的銑床的安裝，并詳細介紹了關(guān)于銑床加工的過程。成型銑對于直齒輪和斜齒輪以及蝸輪蝸桿的加工十分的準確。雖然有時候也用于傘齒輪的加工，但是由于齒厚的不斷變化，加工過程不是那么的準確。當用于傘齒輪的加工時，對于每個槽的加工，需要加工兩次。通常實際中，先確定中心切削的正確的尺寸，其尺寸為齒輪小端的距離。對于齒槽的兩邊的加工，則通過兩個剃齒刀的切削來得正確的齒形。
我們通過使用拉刀加工齒輪的齒向間隙可以看出成型刀具的加工的原理對于拉床也是適用的。小內(nèi)齒輪可以根據(jù)有相同切削的齒數(shù)的齒輪進行多次的走刀而加工出來。但是由于切削的成本較高，拉削僅用于大規(guī)模的生產(chǎn)加工中。
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