Kinematics 一、运动学的研究对象及任务 1 ．研究对象 Point(particle), Rigid body and System of Rigid Bodies. Point: 不计大小的几何点. 2 ．研究任务 (1) 研究物体的机械运动及运动 的几何性质。 (2) 研究机构传动规律。

例 1 观察轮缘上 M 点的运动轨迹

例 2 观察陀螺的运动特点

例 3 观察机构传动关系及点运动轨迹

二、学习运动学的目的 1 学习动力学的基础 受力分析和运动分析是学习动力学的两大 基础。 2 学习机械原理和设计传动机构的基础。 三、研究方法 不考虑致动原因，只研究运动的几何性质。

Chapter 6 Kinetics of a Particle 6.1 Position, Velocity and Acceleration of a Point: Vector Method Curvilinear Motion Of a Particle:

6.2 Rectangular Coordinates, Curvilinear Coordinates, Polar and Cylindrical Coordinates, etc.. Path: 消去 t 可以得到轨迹方程。 Position: Curvature:

Chapter 7 Kinematics of Rigid Bodies: Simple motions Defination: Translation and rotation about a fixed axis of rigid bodies are called the simple motions of rigid bodies. 特点：刚体的简单运动是刚体运动的最简单形式, 是不可分解的运 动基本形态。刚体的复杂运动均可分解成若干简单运动的合成。 A rigid body is said to be translating if all lines in the body remain parallel to their original positions.Since the shapes of paths for every particle in a translating rigid body are same, the velocities and accelerations of all points of the body are equal. The kinematics of translation reduces to the kinematics of a single point. 关键是平动的判定.

7-1 Rotation about a fixed axis 1. Definition Rotation about a fixed axis is a special case where one line in the body, called the axis of rotation, is fixed in space.

2. Characteristics Since the body is rigid, the path of each point (except points on the axis of rotation) is a circle, which lies in a plane perpendicular to the axis of rotation, with its center on the axis of rotation.

3. Equation of motion

Obviously,  can determine the position of the body. Thus the angular position of a rigid body that is rotating about a fixed axis is specified by a scalar function as following  =  (t) (7.3a) If the unit vector of the rotation axis is donated by k, as right hand rule shown in the Fig. 7.3(c) the angular displacement can also be represented as a vector as  =  (t) k (7.3b) where four fingers of right hand represent the direction of the rotation and the thumb represents the direction of the angular velocity vector.

4. Angular velocity and acceleration Taking the time derivative of Eq. (7.3b) while noting that dk/dt=0 yields  =  k =(d  /dt) k (7.4) where  is the angular velocity of the body with magnitude . Differentiating Eq. (7.4) with respect to time we have  =  k = =(d  /dt) k (7.5) where  is the angular acceleration of the body with magnitude . It should be noted that ,  and  are angular, i.e. integral, position, velocity and acceleration vectors of the body.

In engineering, number of revolution per minute n (rev/min) is usually used as unit of angular velocity. It has the relation with  (rad/s) as following

Two important special cases: (1) Uniformly Rotation  = constant  =  0 +  t (7.7) where  0 is the initial angular position of the body. (2) Uniformly Changeable Velocity Rotation  =constant  =  0 +  t (7.8)  =  0 +  0 t +0.5  t 2 (7.9) where  0 and  0 are the initial angular position and velocity of the body.

Let us consider motion of a point on the rigid body rotating about the fixed axis. 由点的运动学 注意转向与指向的一致性 

The vector form of the velocity and acceleration of B is (1) v is tangent to the path of B; (2) the acceleration component  (  r) is directed toward O, i.e., normal to the path; and (3) the acceleration component  r is tangent to the path. In Fig. 7.3(b) we refer to the acceleration components as a n and a t, because they turn out to be identical to the normal and tangential acceleration components of particle. Recognizing from Fig. 7.3(a) that rsin  = R, the magnitude of the velocity vector is v = |  r | =  rsin  =  R. The magnitudes of the acceleration components are a n = |  (  r)| =  |  r | =  2 rsin  =  2 R and a t,= |  r| =  rsin  =  R.

§7-2 Drive Ratio of Wheel Systems Wheel systems are frequently used to increase or decrease rotating speeds of various machines. In general, wheel systems consist of gears, belt wheels, sprockets, friction wheels and so on. 各轮均作定轴转动 的轮系称为定轴轮系。

1. Definition The ratio of driving wheel angular velocity to driven wheel angular velocity is called the drive ratio, i.e. i 12 =  1   2  n 1  n 2 a. Gears transmission 齿轮传动特点 : ①两轮接触点的速 度大小、方向相同。②两轮接触点的 切向加速度大小、方向相同。

这里用到齿轮的设计原则 : 齿轮的齿数与半径成正比. 因为 : 模数 m 相同. R=mZ b. Drive ratio

c. Belt wheel transmission 皮带轮（链轮）传动适用于两轴距离较远的情况. （开口传动, 交叉传动） 1 ）特点 ①皮带不可伸长 ( 理想化 ) 。 ②设皮带与轮之间无相对 滑动。 ③皮带 ( 链条 ) 上各点 v ， a  大小相同。

二、轮系传动比 若定轴传动轮系由 1~n 个轮组成, 其传动比 : i 1,n =  1 /  n ‘-’ 表示反向传动 ; k: 轮系中外啮合齿轮和交叉皮带传动的次数.

A reducing gear box consists of four gears as shown in Fig. (a). Gear II and gear III are on same shaft. Tooth numbers for these gears are: z 1 =36, z 2 =112, z 3 =32, z 4 =128. Revolution number per minute of driving shaft I is: n=1450 rev/min. Determine revolution number per minute of driven shaft IV. Sample problem 7.1