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Ball bearings have rolling elements in the form of balls, which in all but the most inexpensive types are held in cages, separators, or retainers, and inner and outer grooved races. 2. Roller bearings have mainly cylindrical, conical, or barrel-shaped rollers instead of balls, but are otherwise quite similar to ball bearings. 3. Needle bearings usually have neither an inner race nor a cage .The needles are retained by integral flanges on the outer race. Transitional types between roller and needle bearings are found in many catalogs. Bearings with rolling contact have no slipstick effect, low starting torque and running friction, and unlike as in journal bearings, the coefficient of friction varies litter with load or speed. Low starting torque is of great advantage in railroad cars, and the railroad industry has given the main impetus for the development of mass-produced roller bearings in the past, mainly for this reason. These bearings may take both radial and axial loads (depending upon the type), and need less space axially but more radially(except needle bearings), and less lubrication and maintenance, than journal bearings .they are also noisier and more expensive, and cannot be repaired easily. Since rolling elements and raceways are theoretically in point or line contact and thus highly stressed in the contact area by typically cyclical loads, all bearings with rolling contact will eventually fail by fatigue when operated at their rated load, metal failure is typically by shear, just below the surface of raceways or rolling elements—a result of the three principal stresses in the x, y and z axis, all compressive .the phenomenon is popularly called spalling and is characterized by the presence of metal flakes in the grease. Even a small bearing load will induce contact stresses above the yield point (up to 500000psi; called hertz stresses after the mathematician who analyzed them). As a result, the contact surfaces are under residual compressive stresses, the phenomenon is popularly called spalling and is characterized by the presence of metal flakes in the grease. Even a small bearing load will induce contact stresses above the yield point (up to 500000psi; called hertz stresses after the mathematician who analyzed them). As a result, the contact surfaces are under residual compressive stresses, balanced by tensile stresses under the surface. Friction between rolling elements and races causes tensile stresses in the surface, believed to cause pitting. The pits, in turn, are the starting point of fatigue failure cracks. Life expectancy The life expectancy of ball bearings is not uniform and is base upon statistics. Usually, rated load values in catalogs give the minimum life (revolutions or hours at a given rpm) for 90% of a group of bearings, the so-called B-10 life . Mean or average life is about three times rated life, while median or 50% survival life is about five times rated life. The considerable spread of the expectancy curve is due to the many possible forms of favorable or unfavorable interaction between the different bearing elements on the basis of dimensional tolerances, surface finish and varying structural properties, as well as conditions of service. For plain ball bearing, radial load ratings in pounds (as given in catalogs) are based upon a given rpm and a given number of hours of life with 90% survival. The product rpm 60 hr life is then equal to . Other manufactures give rated loads on the basis of revolutions instead of hours of life (again, 90%survival). In that case, the number of hours of life can be found by dividing by rpmx60. it will be clear from the foregoing that ,for a given load, changes in rpm(within the design limits as given by the manufacturer)affect life in hours inversely; thus where n=rpm and H=life(hr.).changes in load, on the other hand, affect life in revolutions exponentially; thus = where F=load(lb) and B=life(revolutions).values of k range between 3 and 4 (3 for ball bearings;3.3for most roller and needle bearings).note that for k=3 ,doubling the load means reducing the life in revolutions by a factor of 8. The relation between B(life in revolutions)and H(life in hours),is as follows: B=Hrpm60 Other factors affecting bearing life The rated life of any bearing with rolling contact is based upon proper application conditions, such as adequate lubrication, good alignment, and adherence to recommended interference fit values for the races, if applicable. Any negative variance from these conditions will affect fatigue life unfavorably. On the other hand, life can be extend considerably by materials, the manufacture can produce a bearing with great life expectancy by increasing the accuracy of manufacture i.e. by reducing manufacture tolerances Static and dynamic load In tables giving permissible load data for ball dynamic loads, i.e, with the bearing stationary or rotating. The permissible static load is smaller than the permissible dynamic load due to the possibility of permanent indentation (flattening of the contact areas when the bearing is at rest). This phenomenon is called Brinelling .Some permanent indentation is almost unavoidable due to the high contact stresses, however, it has been found that permanent deformations smaller than times the element diameter usually do not unfavorably affect the bearingˊs operation. Rotation factor When the outer race rotates and the inner race is stationary (the reversal of common practice), the rated load may be increased by a factor of 1.2,since a ball rolls further per revolution on the outer race when the outer race rotates (the ball rotates in the same sense as the race) than on the inner race (where ball rotation is opposite to that of the race), and life is limited by ball revolutions. Equivalent dynamic load A plain ball bearing is designed to take radial loads. However, due to the curvature of the tracks in an axial plane of both inner and outer races, some axial load may be imposed on a radial ball bearing as well. Note that an axial load is taken by all balls , at least in theory, while a radial load is carried by less than half their number. Manufacturesˊcatalogs give simple conversion methods to and axial loads occur simultaneously in a rotating ball bearing. One manufacture uses a load conversion facture Fc which, when multiplied with the radial component R, gives the equivalent radial load P, which is then used to select a properly size bearing. These load conversion factors depend upon the ratio of the thrust load and the radial load(T/R)and vary for different types of bearings。 Gearing A friction drive consists of two cylinders rolling together under some pressure. When no slipping occurs, the tangential velocity V at the line of contact of the two cylinders is, of course, the same for each. From physics we know that V=w*r., in which V is the tangential velocity (in./sec), W is the rotational velocity (rad /sec), and r is the radius of rotation. Since V=w1r1 and also V=w2r2, we can equate both equations. Thus, dividing both sides by we get where is the speed ratio (Mw). When w is divided by 2π and multiplied by 60,we obtain rpm(n).Therefore, Mw= That is, the speed ratio of two cylinders in rolling contact is equal to the inverse ratio of their radii or diameters, in consistent units. The drawback inherent to a friction drive, such as that described above , is that it is liable to slip when power of any consequence is transmitted. It can be used only for very small torque applications, such as phonograph turntable drives and the like. The positive prevention of slippage in the transmission of large quantities of power requires the use of teeth, penetrating into the surface of each cylinder of the friction drive. Mating cylinders provided with teeth are called gears. The diameter of each of the original rolling cylinders is called the pitch diameter, and the sectional outline is called the pitch circle. The curved shape of the tooth outline must be such that no change in speed ratio occurs during the passing contact of each tooth with its mating tooth on the other gear. This is a basic requirement for all gearing. Curves that satisfy this requirement are called conjugate curves. While several of such curves exist, the one almost universally used at this time is the involute. This curve is that described by a point on a string as it is being unwound from a cylinder. In spur gears, the simplest of all gears, the teeth are straight lengthwise and parallel to the axis of rotation. The spur gear nomenclature which is mostly applicable to all other types of gears as well. Nomenclature and definitions The pitch point P is the point of the pitch circle. The pitch diameter Dp is the diameter of the pitch circle, and is equal to twice the pitch radius. The addendum (i.e., that which is added) is the radial distance from the pitch circle to the top of tooth (the crest) .The dedendum (i.e, that which is deducted ) is the radial distance from the pitch circle to the bottom of the groove between adjacent teeth (the root). Clearance is the difference between addendum and dedendum in mating gears. Clearance prevents binding caused by any possible eccentricity. The circle pitch Pc is the distance between corresponding sides of neighboring teeth, measured along the pitch circle. The diametral pitch Pd is the number of teeth of a gear for each inch of pitch diameter.( do not confuse Pd with Dp). The circle pitch and the diametral pitch are related as follows: The line of centers is a line passing through the centers of two mating gears. The center distance C (measured along the line of centers) equals the sum of the pitch radii of pinion and gear () or half the sum of the pitch diameters: Tooth width is the width of the tooth, measured along the Space width is the distance between facing sides of adjacent teeth, measured along the pitch circle. Tooth width plus space width equals the circular pitch. Face width measures tooth width in an axial direction. The circle from which the involute is generated is called the base circle. Backlash is the space width minus the tooth. It is necessitated by the tolerance of the manufacturing processes used. The face of the tooth is the active surface of the tooth outside the pitch cylinder. The flank of the tooth is the active surface inside the pitch cylinder. The fillet is the rounded corner at the base of the tooth. The working depth is the sum of the addendum of a gear and the addendum of its mating gearing The base pitch is similar to the circular pitch but is measured along the base circle instead of along the pitch circle. It can easily be seen that the base radius equals the pitch radius times the cosine of the pressure angle. Since, for a given angle, the ratio between any subtended arc and its radius is constant, it is also true that the base pitch equals the circle pitch times the cosine of the pressure angleΨ.the pressure angleΨ is the angle between the common tangent to the base circles, and the common tangent to the pitch circles at the pitch point. At present, the preferred pressure angle for spur angle gears is . In newer designs this angle replace the value of formerly used. The AGMA (American gear manufactures association) has establish the following proportions for pressure angle standard spur gears: Addendum= Dedendum= Minimum clearance=dedendum-addendum= In order to mate properly, gears running together must have :(1) the same pitch, (2) the same pressure angle, and (3) the same addendum and dedendum .The last requirement is valid for standard only. Since the number of teeth of each of two mating gears is proportional to its respective pitch diameter (prove), it is easier and thus customary to express the speed ratio in tooth numbers rather than pitch diameters; thus, Mw= Where subscript p=pinion, and g=gear. Dpg=pitch diameter of gear, and Dpp=pitch diameter of pinion; n=rpm, and N=tooth number. Recall that the involute is the curve almost universally used at present for shaping the outline of gear teeth, let us now analyze the action of a pair of mating involute teeth. 中文譯文 機 械 設 計 基 礎 滾動接觸軸承 在滾動接觸軸承中，軸直接或間接的由滾動體支撐著。如球滾子，圓柱滾子和圓錐滾子，還有滾針（一種長徑比較大的圓柱滾子）。一些偶然間遇到的 名詞“減摩軸承”表明這種類型（軸承）只有很少（或者沒有）摩擦。其實這種認識是錯誤的，因為摩擦不僅僅是滑動軸承的通性。在滾動接觸軸承中，摩擦損耗是由于滾動接觸中的 表面塑性變形、滾動體與保持架、擋圈、密封圈或者它們相互間的滑動摩擦、以及潤滑劑的剪切引起的。 特征 1.球軸承有球形滾動體，這種幾乎是最便宜的滾動體有由保持架、擋圈和內外滾道固定。 2.滾動軸承主要有圓柱滾子、圓錐滾子或水桶形滾子以替代滾球，但在其他方面與球軸承非常的相似 3.滾針軸承通常既無內圈也無保持架。滾針是由整個法蘭固定在外圓上。滾子和滾針軸承之間的過渡型軸承在許多種類中廣泛應用。 與滑動軸承不同，滾動軸承無滑動面粘附現(xiàn)象的影響，其啟動力矩低，運轉摩擦小，摩擦系數(shù)幾乎不隨外載和轉速變化。低啟動轉矩是鐵路機車的主要優(yōu)點之一。在過去由于大批量生產的滾子軸承的發(fā)展促進了鐵路機車的發(fā)展，主要就是這個原因。 這些軸承能承受徑向或軸向載荷（主要取決于軸承的類型）。與滑動軸承相比，滾動軸承占用很小的空間，但其徑向空間很大。滾動軸承也僅需要很小的潤滑和維護。但是滾動軸承噪聲大且非常的昂貴，而且修理也不容易。 由于滾子部件在理論上是點接觸或是線接觸的，因而在接觸區(qū)域由于循環(huán)載荷而產生很高的壓力。所有的滾動接觸軸承最終會由于施加在它們上的額定載荷產生疲勞而最終損壞。在滾道或滾動體的表面下由于剪切力發(fā)生金屬損傷這是因為它受到x、y、三個方向的主應力。這種現(xiàn)象通常稱為剝落。其特征是在潤滑脂中出現(xiàn)金屬薄片。即使是一個很小的軸承載荷也會導致超過屈服點的接觸應力，因此，接觸表面受殘余壓應力，而在表面下則承受拉應力。滾動體與軸承內外圈之間的摩擦產生拉應力。由此而產生凹坑，這凹坑，就是疲勞損傷裂紋的起始點。 預期壽命 所有的球軸承的預期壽命是不一樣的。而是統(tǒng)計出來的。通常產品樣本的額定載荷值給出了一組軸承內90％的軸承所能達到的最小壽命（轉數(shù)或額定轉速下的小時數(shù)）即所謂的B-10壽命。平均壽命大約是額定壽命的3倍。而50％的 軸承所達到的壽命通常是額定壽命的5倍。 對普通球軸承，徑向載荷（以磅作單位）實基于以給定轉速和一給定的軸承所達到壽命的90％的小時數(shù)的。產品的每分鐘轉速（rpm）60hour（小時）然后換算到。 其它生產商定的額定載荷是基于轉而不是小時數(shù)。壽命（以小時計）可以由除以(rpm60)得到。通過前述，我們可以清楚的知道，對于一給定載荷，壽命轉化為rpm影響與小時計的壽命是成反比關系的。即 . （ 這里n表示rpm，H表示壽命）另一方面，轉化成載荷影響以轉速計的壽命是指數(shù)關系的，即(F代表載荷（磅），B代表壽命(轉速),K的值在3－4之間取，球軸承取3，大多數(shù)的滾子和滾針軸承取3.3)假定K=3,把載荷2，這樣就意味著以轉速計的壽命將為原來的1/8。 B和H之間的關系如下所示：B=Hrpm60 其它影響軸承壽命的因素 所有的球接觸軸承的額定壽命基于在合適的應用條件下的。例如足夠的潤滑，良好的調整以及與所薦的過盈配合值相匹配。任何不符合上述使用條件的都會對疲勞壽命產生不利的影響。 從另一個方面說，如果能夠給予良好的潤滑，軸承壽命還是能夠延長的。同時對于同一尺寸同一材料，制造商如果能夠提高制造精度，如減小制造誤差，就能制造出預期壽命更長的軸承。 靜態(tài)載荷和動態(tài)載荷 在表中，對各種球軸承施加容許的載荷。靜態(tài)載荷和動態(tài)載荷產生的差異是明顯的。例如：穩(wěn)定的軸承或旋轉的軸承，由于永久壓痕的可能性，容許的靜態(tài)載荷一般都小于容許的動態(tài)載荷。這現(xiàn)象稱之為布里涅耳現(xiàn)象。由于接觸壓力高，一些永久壓痕幾乎是不可避免的。然而，已證實那些小于滾動體直徑的永久壓痕通常不大影響軸承的運作。 旋轉因素 當軸承外圈旋轉而內圈靜止時（與通常的情況相反）其額定載荷將增加到1.2倍，因為當外圈旋轉時，球滾動體每轉對外圈的作用力比對內作用力更甚。其壽命也由于球滾動體的旋轉而受到限制。 當量動載荷 普通的求軸承是設計成承受徑向載荷的。然而由于在軸承內圈外圈軸向面上滾道曲率影響。徑向求軸承也承受了一些軸向力，軸向載荷被所有的滾球承受（至少在理論上，而徑向載荷則只有少于半數(shù)的滾動球來承擔。 如果一旋轉軸承同時承受了徑向載荷和軸向載荷，品目錄手冊給了一個簡單方法去計算單量徑向載荷。某一廠家采用了載荷轉換因子FC。當它與徑向分力R相乘，就得到當量徑向載荷P,以此為標準去選取大小合適的軸承。這個載荷轉化因子只取決于進推力與徑向載荷的比，而且隨不同類型軸承取值也不一樣。 齒輪 摩擦驅動是在兩個旋轉圓柱體之間的壓力下產生的。當未產生滑移時，兩個圓柱齒輪嚙合處的切向速度v對每個齒輪來說是相同的。從物理學中我們知道速度v=wr,在此式子中，v指切向速度，w指角速度，r指旋轉半徑。因為v=,同時=,所以我們得＝。兩邊都除以得，稱為速度比（Mw）。 當w除以2，再乘以60，我們就得到了轉速（n）因此Mw＝。也就是說在單位一致的情況下，兩圓柱在旋轉接觸面上的速度比與它們的半徑或直徑成反比。 摩擦驅動的固有的缺點，如上述所描述的，在于它在傳動功率時易產生滑移。因此它只能適用于一些小力矩方面。例如拍照的驅動等等。 在大功率傳動中，在滾動體上加工齒形能有效的防止滑動。兩嚙合齒輪中的較小的齒輪稱為小齒輪。大著稱為大齒輪。每個原滾動體的直徑稱為節(jié)園直徑。圓柱的橫截面輪廓線稱為節(jié)園。輪齒的外廓線應當滿足以下要求：在兩個嚙合齒輪嚙合所經過的接觸面?zhèn)鲃颖炔话l(fā)生變化。這是對所有齒輪的一個基本要求。能滿足這個要求的曲線都稱為共扼曲線。 在已有的這樣的曲線中，現(xiàn)在用的最廣泛的是漸開線。這樣的曲線可以如下描述：一直線繞一圓周純滾動時直線上某點的軌跡。 在直齒圓柱齒輪（最簡單的一類齒輪）中。所有的齒都是直線縱向的平行于其旋轉軸的，直齒圓柱齒輪所用的一些術語大多數(shù)也適用于其它所有類型的齒輪。 術語和定義 節(jié)點P就是節(jié)園的相切點。節(jié)園直徑是節(jié)園的直徑，也等于節(jié)園半徑的兩倍。齒頂高是從節(jié)園到齒頂?shù)膹较蚋叨?。齒根高是節(jié)園與相鄰近齒槽底部的徑向高度。頂隙是一對嚙合齒輪中，一齒輪的齒頂圓與另一齒輪的齒根圓之間的間隙。頂隙可避免各種可能的偏心所引起的相互干涉。 周節(jié)是指沿著分度圓上測量的相鄰兩齒同測齒廓漸的距離。徑節(jié)是指節(jié)園直徑單位英尺長度上齒輪的齒數(shù)。（不要混淆和）周節(jié)與徑節(jié)之間的關系如下所示: =. 中心線是指穿過兩嚙合齒輪中心的連線。中心距C（沿中西線測量所得）等于小齒輪的節(jié)園半徑與大齒輪節(jié)園半徑之和，也等于兩節(jié)園直徑和的一半。 輪齒寬度是指在節(jié)園上測得的齒的寬度。齒寬也可稱為齒厚。槽寬是指在節(jié)園上相鄰兩齒同側齒廓間的弧長。齒厚與齒槽寬之和等于周節(jié)。沿軸向測得的齒寬即表面寬度。形成漸開線的圓稱為基圓。側隙是齒槽寬與齒厚的之差。齒側間隙一般在制造過程中由公差保證。齒頂面是節(jié)圓柱外的嚙合表面。齒根面是節(jié)圓柱內的嚙合表面。倒角是在齒輪基部的圓角。 工作深度是齒頂高與其相嚙合的另一齒輪的齒頂高（例如在標準齒輪中是2倍的齒頂高）。 基圓節(jié)距與周節(jié)相似，只是前者是在基圓上測得而不是在節(jié)圓上。我們可以知道;基圓半徑等于節(jié)圓搬進乘以壓力角的余弦。因為對于一個給定的齒輪，任何對應的弧長與其半徑的比值都是一個常數(shù)。所以我們也可以說基圓節(jié)距也等于周節(jié)乘以壓力角的余弦。壓力角是指兩基圓的公切線在節(jié)點處的夾角?，F(xiàn)在，圓柱齒輪的參照壓力角是。在最新的設計中，這個壓力角已取代了以前常用的。 AGMA(美國齒輪制造協(xié)會)已經為壓力角為的標準圓柱齒輪制定了如下的比率; 齒頂高＝ 齒根高= 最小頂隙=齒根高-齒頂高= 為了能夠正確的嚙合，齒輪傳動必須具備下列3個條件： （1）相同的節(jié)距 （2）相同的壓力角。 （3）相同的齒頂高和齒根高（適用于標準齒輪。） 因為兩嚙合齒輪的齒數(shù)是與他們各自的節(jié)圓直徑成比例的，因此我們習慣上易用齒數(shù)而不是用節(jié)圓直徑來表示速度比。即 Mw= 這里的下角p表示小齒輪，g表示大齒輪，表示大齒輪的節(jié)圓直徑，是小齒輪的節(jié)圓直徑。n是轉速rpm，N是齒數(shù)。