Kinematics of Machinery.doc

英汉对照：
Kinematics of Machinery
机械运动学
10.1 Introduction .Because motion is inherent in machinery; kinematic quantities such as velocity and acceleration are of engineering importance in the analysis and design of machine components. Kinematic values in machines have reached extraordinary magnitudes. Rotative speeds, once considered high at 10,000r/min, are approaching 100,000r/min; large rotors of jet engines operate at 10,000
10.1介绍。因为运动是机械所固有的，运动学参数例如速度和加速度等工程参数在分析与设计中的机械零件中起到了很重要的作用。运动的数值在机械中已经起到了特别的重要性。转动速度，一度被视为高于10000转/分钟，接近100000转/分钟，大转子喷气发动机运转速度在10000
to 15,000r/min, and small turbine wheels rotate at 30,000
到15000转/分钟，和小汽轮机的轮子转动速度为30000
to 60,000r/min.
到60000转/分钟。
Size and rotative speed in rotors are related such that such that the smaller the size the greater the allowable rotative speed. A more basic quantity in rotors is peripheral speeds, which depends on rotative speed and size .Peripheral speeds in turbo machinery are reaching
大小和转子旋转速度是相关的，这样，较小的规模能够允许有更大的转动速度。在转子中一个更基本的参数是圆周速度，这取决于转动速度和大小。圆周速度在涡轮机械中能够达到
250 to 500m/s.Peripheral speeds in electric armatures and automotive crankshafts are lower than in aeronautical rotors. Although the rotor, or crank, speeds of linkage mechanisms are low , the trend is toward higher speeds because of the demand for higher rates of productivity from the machines used in printing, paper making, thread spinning, automatic computing, packaging, bottling, automatic machining, and in numerous others applications.
250至500m/s.在电动机的转子和汽车曲轴中圆周速度低于航空转子的速度。尽管转子，或曲柄连杆机构，机械的链接装置速度比较低，但是趋势是走向更高的速度这是因为需求较高的速度在印刷，造纸，螺纹旋转，自动计算，自动灌装，包装，加工，和许多其他应用。
The centripetal acceleration at a rotor periphery depends on the square of the rotative speed and size. In turbines, such accelerations are approaching values of 300,000 to 900,000m/s or about 30,000g to 90,000g, values that may be compared with the acceleration of 10g withstand able by airplane pilots or the 1000g of automotive pistons.
向心加速度在转子的周围取决于转动的速度的平方和大小。在涡轮机中，这种加速度接近于300000至900000m/s²或30000g到90000g，数值可能比得上与10g的飞行员的加速度或者是1000g的汽车活塞产生的加速度
Acceleration is related to force, by Newton’s principle, and in turn related to stress and deformation, which may not be critical in a machine part, depending on the materials used. The speed of a machine is limited ultimately by the properties of the materials of which it consists and the conditions which influence these properties. High temperature arising from friction is a condition in high-speed power machines that influences the strength of the materials. The degree to which the temperature rises also depends on the provisions made for the transmission of heat by coolants such as air, oil, water, or Freon.
加速度与力有关系，由牛顿定律可知，反过来相关的应力和变形，这些也许不是最重要的在机械零件中，而这取决于所用的材料。该机器的速度是有限的是由于其材料的特性，它是由材料的内在和环境所影响的，这些特性。在高速运动下温度过高而引起的摩擦直接影响材料的特性。在一定程度上的温度上升也取决于由于冷却剂所传输的热量由空气，油，水，或氟里昂所提供的。
The successful design of a machine depends on the exploitation of knowledge in the fields of dynamics, stress analysis, thermodynamics, heat transmission, and properties of materials. However, it is the purpose of this chapter to deal solely with kinematic relationships in machines. In subsequent chapters, acceleration and force are discussed in connection with the determination of forces acting on individual links of a mechanism and in connection with machine balance and vibration.
成功设计一个机器取决于在应力分析，动力学，热力学，传热，和材料性能等知识领域的开发应用。然而，这是本章的目的是处理单独的知识领域与机械心中运动学的关系。在随后的章节中，加速度和力的讨论与力在机械装置单独的链接的作用以及机械的平衡和振动的联系。
For bodies rotating about a fixed axis, such as rotors, kinematic values are quickly determined from well-known elementary formulas. However, mechanisms such as the slider crank and its inversions are combinations of links consisting not only of a rotor but of oscillating and reciprocating members as well. Because of the relative velocities and relative accelerations among the several members, together with the many geometric relative positions possible, the kinematic analysis of a linkage is relatively complex compared to that of a rotor. The principles and methods illustrated in this chapter are primarily those for the analysis of linkages consisting of combinations rotors, bars, sliders, cams, gears, and rolling elements.
物体绕定轴转动时，例如转子，从众所周知的基本公式运动的数值能够被快速的确定。然而，机械装置例如曲柄滑块及其组合链接不仅包含转子，而且包含振荡和往复运动。因为相对速度和相对加速度之间的几个参数，连同许多几何的相对位置可能，运动学分析联系是比较复杂的比较由于存在一个转子。原则和方法的说明在本章主要是用于分析机构组成的组合转子，连杆，滑块，凸轮，齿轮，和转动元件。
In the following discussions, the individual links of a mechanism are assumed to be rigid bodies in which the distance between two given particles of a moving link remains fixed. Links which undergo large deformations during motion, such as springs, fall in another category and are analyzed as vibrating members.
在下面的讨论中，单独链接的机械装置假设认为是刚性体在给定的两点之间的距离保持不变。链接在运动中的很大变形，例如弹簧，属于另一类，而且被分析成振动的成员。
Most elementary mechanisms are in plane motion or may be analyzed as such. Mechanisms in which all of the particles move in parallel planes are said to be in plane motion. An illustration is a four-bar linkage consisting of two rockers and a connecting rod. This arrangement is often referred to as a double-rocker mechanism.
大部分基本的机械装置是在平面上运动以及在平面上分析分析等。机械装置中所有的零件在平行的平面移动被称为平面运动。有一个例子是一个四杆机构组成的双摇杆和连杆。这种机械装置组成通常称为双摇杆机构。
The motion of a link is expressed in terms of the linear displacements, linear velocities, and linear accelerations of the individual particles which constitute the link. However, the motion of a link may also be expressed in terms of angular displacements, angular velocities, and angular accelerations of lines moving with the rigid link.
运动的一个环节是表示其线性位移，线速度，和线性加速度的单独机械构成的链接。然而，一个链接运动也可以表示其角位移，角速度和角加速度，以及线移动的刚性连接。
In Fig. 10.1, the linear velocity VA and the linear acceleration AA of particle A are shown by the fixed vectors at A. Because of the connecting pin at A, particle A2 on link 2 and particle A3 on link 3 have the same motion, and the vectors shown at A represent the motions of both particles. The angular motions of links 2 and 3 are different as given by the angular velocities W2, W3 and the angular accelerations a2, a3. Usually the angular motion of a driving link is known, or assumed, such as W2 and a2 of Fig. 10.1, and the motions of the connecting and driven links are to be determined.
在图10.1中，线性速度和线性加速度是A点的一部分，可以通过向量从A点出发，因为在A点链接，2的链接在A2，3的链接在A3，它们3个具有相同的运动，两个运动的链接表示在同一个向量。有关于角运动的链接2和3是不同的，角速度为W2，W3，角加速度为a2，a3。通常的角运动的驱动链接是已知的，或假设的，如图10.1，W2和a2是已知的，运动的连接和驱动的联系是确定。
10.2 Linear Motion of a particle. In useful mechanisms, the particles of the links are constrained to move on given
10.2直线运动上的点。在机械装置的应用中，链接的点在很多方向上被限制移动，
paths, many of which, such as circles and straight lines, are obvious. In Fig. 10.1, the particles of links 2 and 4 are constrained to move on circular paths. The particles of link 3, however, are in motion along generally curvilinear paths less simple than circles or straight lines.
路径，其中有许多，如圆与直线，都是是显而易见的。在图10.1中，链接2的点和4的点被限制移动的方向为圆周路径。链接3的点，然而，在沿曲线路径运动比圆或直线都不简单。
A particle in motion on a curvilinear path is said to be in curvilinear translation. The basic kinematic relationships for a particle translating in a plane are well known from the study of mechanics. These are reviewed in the following paragraphs with reference to Fig. 10.2 and were contributed by Professor J.Y. Harrison, University of New South Wales, Australia.
一个点在曲线方向上的运动称为曲线的运动。众所周知的是在机械装置的研究中点在平面上运动是点最基本的运动关系。通过参考图10.2这些都会通过以下的各段再度了解，以及哈里森教授的手稿，新南威尔士大学，澳大利亚。
The linear velocity Vp of a particle, P is the instantaneous rate of change of the position of the particle, or displacement, with respect to time. Referring to Fig.10.2a, in a small interval of time△t, the particle is displaced △S along the curved path from position P to position P’. At the same time, the radius vector of the particle changes from R to R+△R and undergoes an angular displacement △θ. Therefore, the displacement △S is made up of two components: one due to the angular displacement △θ of radius R and the other due to the change in length △R.
点的线性速度为Vp，P是运动上不断变化的点，或位移，相对于时间。如图10.2a，在一个小的时间间隔Δt，点的位移ΔS沿着曲线方向从P点位置运动到P’点位置。同时，点的半径向量的变化为R到R+ΔR，以及经历的角位移为Δθ。因此，点的位移ΔS是由2个部分组成一个部分是由半径R上角位移Δθ组成，另一个部分是由长度ΔR组成。
From Fig. 10.2a
从图10.2a上可知
△S = RΔθp + △Rr
Where p and r are unit vectors perpendicular and parallel to R, respectively. The equation for the velocity of P can be determined as follows:
P和r在单位向量上是垂直的并且平行于R。方程中P的速度可以确定如下：

Therefore,
因此，

Or
或
(10.1)
The acceleration of P is given by
P的加速度被给出为
(10.2)
The acceleration of P therefore consists of two components, one of magnitude 2wr (dR/dt) + Rwr in the direction of the unit vector p, the other of magnitude (dR/dt)-Rwr in the direction of the unit r. The equation for Ap can also be expressed as
因此P的加速度由2部分构成，一部分的大小为2wr （dR/dt）+Rwr在直线单位向量P上，另一部分的大小为（d²R/dt²）-Rwr²在直线单位向量r上。Ap的公式也能够表达为
(10.3)
When the origin of the coordinate system coincides with the center of curvature, dR/dθ and dR/dθ are zero so that dR/dt and dR/dt are also zero. Under this condition Eq. 10.1 can be simplified to give
当在坐标系的原点与曲率中心是相同的，dR/dθ和d²R/d²θ为零因此dR/dt和d²R/dt²也为零在这种情况下，公式10.1可以简化为
Vp = Wr × R

And
与
| Vp | = Rwr

From Eq. 10.3
由10.3式
（10.5）
The term wr *( wr*R) is the normal component of the acceleration with direction from point P toward the center of curvature, and ar * R is the tangential component with direction tangent to the curve at point P. Equation 10.5 may, therefore, be written as
Wr×(Wr×R)是P点的加速度方向沿着曲率中心的正常的组成部分，ɑr×R点P在曲线切线方向上的组成部分。方程10.5也许，因此，被写为

Where
在哪儿

and
与

If the condition should arise where the origin of the coordinate system is on the normal to the curve through the point, dR/dθ and, therefore, dR/dt will be zero and Eqs.10.1 and 10.3 can be modified accordingly.
如果这种情况应该发生在坐标系的原点垂直于曲线上的点，dR/dθ，因此，dR/dt也将为零以及方程10.1和10.3可以被修改。
Figure 10.2b shows the instantaneous directional orientation with respect to the tangent and normal to be path for the velocity vector Vp and the component vectors of acceleration Ap and Ap ; the radius of curvature C of path. The direction of Ap is tangent to the path and its sense is for increasing velocity. The resultant acceleration Ap is the vector sum of Ap and Ap as shown.
如图10.2b所示，瞬时速度向量Vp分为切线方向和垂直方向以及向量加速度的组成分为Ap和Ap；曲线沿半径方向是连续的，指出Ap在垂直于P的方向以及指向曲率中心C是很重要的，Ap的方向为P的切线方向也就是增加速度。结论就是加速度Ap是向量Ap和Ap的总和。
Equation 10,4a, 10.6a, and 10.7a are used to calculate only the magnitudes of the vectors describing the linear motion of a particle, and they appear repeatedly in the development of kinematic relationships of particles in mechanisms were the origin of the coordinate system coincides with the center of curvature.
方程10,4a，10.6a，和10.7a是只用来计算向量的大小是用来描述点的直线运动，点在机械装置运动发展的关系中，它们的坐标系的原点和曲率中心看来是重复的。
10.3 Angular Motion. Angular velocity and angular acceleration are the first and second derivatives respectively of the angular displacement θ of a line with respect to time t. In machine analysis, the angular motion of a link is expressed by the angular motion of any line visualized fixed to the link. In Fig.10.3, line AB is in angular motion because of its angular displacement with respect to time. Lines BC and AC undergo the same angular displacements with respect to time as line AB because triangle ABC is fixed in position with link 3 as a rigid body. Since all lines of link 3 have the same angular motion, the angular velocity and angular acceleration of these lines are w3 and a3 of the link, with the subscript denoting the link number.
10.3角运动。角速度和角加速度分别为角位移的θ相对于时间t的一介导数和二介导数。在分析机械装置的运动中，连接处的角运动的可以通过固定连接出线性角运动来表示。如图10.3所示，由于角位移随着时间的变化直线AB可以用角运动表示。因为三个点ABC固定在杆3的刚体上，直线BC和AC所具有的角位移随着时间的变化与直线AB相同。因为刚体杆3上所有的线具有相同的角位移、角速度、角加速度，它们可以表示为w3和a3，可以用下角标标注数字表示。
Angular motion of a link may be the same or different from the angular motions of the radii of curvature of the paths of the individual particles of the link. In Fig.10.3, since all particles of link 2 are moving on circular paths having a common center of curvature at the fixed center O2, it is obvious that wr and ar of the radii of curvature of the paths of all particles are equal to the respective angular velocity and angular acceleration w2 and a2 of the link. In the case of the connecting link 3 in Fig.10.3, which is not rotating about a fixed center, wr and ar of the radius of curvature of the path of given particle are not the same as w3 and a3 of link 3.
角运动的一个链接可能是相同或不同的角运动的曲率半径的单个质点单独部分的链接。在图10.3中，因为杆2上所有的部分在圆周方向的运动有一个共同的曲率中心固定在O2点，很明显，所有曲率半径的所有质点的wr和ar都等于各自的角速度和角加速度即w2和a2。在图10.3的杆3的链接上，并不是绕固定点的旋转运动，所有曲率半径的所有给定质点的wr和ar和杆3的w3和a3不同。
It is an important concept in mechanics that a particle, which has the infinitely small size of a point, may have only linear motion. Angular motion is that of a line, and since a particle is a point, not a line, it is not considered to be in angular motion. This concept must be fully understood the relative motion among particles. For example, the velocity of the particle on link 2 at O2 in Fig.10.3 relative to the velocity of any particle on the fixed link 1 is zero. Linear velocity is implied, and it is incorrect to hold that, by virtue of the angular motion of link 2, the particle O2 has the angular velocity of the link.
在机械装置中，质点具有无限小的尺寸是一个重要的概念，可能只有直线运动。角运动是一个线运动，因为质点是一个点，不是线，它是不被认为是角运动。这种观念必须要充分理解各质点之间的相对运动。例如，在图10.3中杆2上的质点的速度相对于固定在O2上的杆1为零。线性速度是隐含的，它不是正确的由于杆2的角运动以及点O2在杆2上的角速度。
10.4 Relative Motion. As will be shown in a later section, the relative motion between particles is very important in the kinematic analysis of mechanisms. In Fig.10.4a, P and Q are particles moving relative to a fixed reference plane at the respective velocities of Vp and VQ, and it is necessary to determine the relative velocity VpQ between the two particles. In determining VpQ use will be made of the fact that the addition of equal velocities to each particle does not change the relative velocity of the two particles. Therefore, if P and Q are each given a velocity equal and opposite to VQ, the particle Q becomes stationary in the fixed plane and P acquires an additional velocity component –VQ relative to the fixed plane. The new absolute velocity of P (Vp-VQ), therefore, becomes the relative velocity VpQ because Q is now fixed relative to the reference plane. This is shown by the vector diagram of Fig.10.4b from which the equation for VpQ becomes
10.4相对运动。在以后的章节中，两点之间的相对运动在机械运动分析中有着很大的作用。如图.10.4a，P点和Q点是质点的运动在相对固定的参考平面各自的速度为VP和VQ，确定两点之间的相对速度VPQ是必须的。在确定VPQ时，VPQ是由另一质点的相等速度的大小但方向相反以及本身质点的速度所组成的。因此，如果P点和Q点分别给定速度相等方向相反VQ，Q点固定在平面上不动，P点相对于平面获得另一个质点的速度-VQ。P点的绝对速度为VP-VQ，因此，它成为相对速度VPQ因为Q点是在相对固定的参考平面上。这是所表示出的矢量图10.4b，从该方程得出VPQ
VPQ = VP - VQ (10.5)

In a similar manner VQp can be obtained by the addition of –Vp to each particle. This is shown in Fig.10.4c, and VQp is given by the equation
以类似的方式得知VQP包含另一质点的速度-VP。如图10.4c所示，从该方程得出VQP
VQP = VQ - VP

The vector equation for the acceleration of particle P relative to particle Q is similar in form to Eq.10.5.
类似的P点相对于Q点