《理论力学》双语教学 第八章 点的合成运动(117P).ppt

1、1,Chapter 8 Compositive motion of a particle,2,第八章 点的合成运动,3,81 Concept of compositive motion of a particle 81 点的合成运动的概念,82 Theorem of composition of velocities of a particle 82 点的速度合成定理,4,Lesson for problem solving 习题课,83 Theorem of composition of accelerations when the convected motion is translati

2、on 83 牵连运动为平动时点的加速度合成定理,84 Theorem of composition of accelerations when the convected motion is rotation 84 牵连运动为转动时点的加速度合成定理,5,In the previous two chapters, we have considered the motion of a particle or a rigid body with respect to a system of fixed coordinate axes. In practice, however, we often

3、observe the motion with respect to a moving body. For example, we observe the motion of an flying airplane from a moving car, or observe the motion of raindrops from a traveling train. 前两章中我们研究点和刚体的运动，一般都是以地面为参考 体的。然而在实际问题中，还常常要在相对于地面运动着的参 考系上观察和研究物体的运动。例如，从行驶的汽车上观看飞机的运动等，坐在行驶的火车内看下雨的雨点是向后斜落的等。,Why

4、do obtain different results when we observe the motion a object on different coordinate systems or reference objects? It is known that there may exist relationship between two different objects. Next, we are going to study the relationship between the motions of an object and the reference object. F

5、irst of all, we begin with the introduction of some concepts. 为什么在不同的坐标系或参考体上观察物体的运动会有不同的结果呢？我们说事物都是相互联系着的。下面我们就将研究参考体与观察物体运动之间的联系。为了便于研究，下面先来介绍有关的概念。,6,2) Moving coordinate system: A coordinate system fixed to a moving object relative to the earth ground is called moving coordinate system (MCS). Fo

6、r example, a running car. 2.动坐标系：把固结于相对于地面运动物体上的坐标系， 称为动坐标系，简称动系。例如在行驶的汽车。,1) Static coordinate system: A coordinate system fixed to the earth ground is called Static coordinate system (SCS). 1.静坐标系：把固结于地面上的坐标系称为静坐标系,简称静系。,1Coordinate system： 一坐标系：,7,3) Convected motion：motion of the MCS relative to

7、 the SCS. For example, motion of a running care relative to the earth ground. 牵连运动：动系相对于静系的运动 例如：行驶的汽车相对于地面的运动。,3Three kinds of motion and their velocities and accelerations 三三种运动及三种速度与三种加速度。,Absolute motion：motion of the moving point relative to the SCS. 绝对运动：动点对静系的运动。,2) Relative motion：motion of

8、the moving point relative to the MCS. For example, motions of a person in a running bus 相对运动：动点对动系的运动。 例如：人在行驶的汽车里走动。,8,9,10,11,12,13,14,15,16,17,18,19,20,Proof 一证明,21,22,23,24,vrrelative velocity of the moving point; vr动点的相对速度；,veconvected velocity of the moving point, which is the velocity of a po

9、int (convected point) on the MCS; ve动点的牵连速度，是动系上一点(牵连点)的速度,I) When MCS is in translation, all points in it have the same velocity. I) 动系作平动时，动系上各点速度都相等。,25,26,27,Solution：moving point: weight A 解：选取动点: 物块A,28,Absolute motion : curvilinear; 绝对运动: 曲线;,29,absolute velocity va = r direction OA 绝对速度va =

10、r 方向 OA relative velocity vr = ? direction/O1B 相对速度vr = ? 方向/O1B convected velocity ve = ? direction O1B 牵连速度ve = ? 方向O1B,Solution：Choose point A on rod OA杆as the moving point，attach the MCS to rod O1B，and use the earth ground as the SCS. 解：取OA杆上A点为动点，摆杆O1B为动系，基座为静系。,30,31,32,convected velocity ve =

11、OA=2e, direction OA 牵连速度 ve =OA=2e, 方向 OA,absolute velocity va = ? unknown，directin/AB 绝对速度 va = ? 待求，方向/AB,relative velocity vr = ? unknown，directionCA 相对速度 vr = ? 未知，方向CA,33,34,As evidenced from the examples, the general steps should be followed in solving problems about the composition of velocit

12、ies: 由上述例题可看出，求解合成运动的速度问题的一般步骤为： Select moving point, MCS and SCS; 选取动点，动系和静系。 Analysis of the related three kinds of motions; 三种运动的分析。,35, Find unknowns from the parallelogram of velocities; Proper selection of the moving point, MCS and SCS is the key point in solving such problems. 根据速度平行四边形，求出未知量

13、。 恰当地选择动点、动系和静系是求解合成运动问题的关键。, Analysis of three kinds of velocities; 三种速度的分析。,36,Guidelines for the selection of moving point, MCS and SCS 动点、动系和静系的选择原则 The moving point, MCS and SCS must be in three different rigid bodies. Otherwise, one motion out of the absolute, relative and convected motion wil

14、l be absent (always resting indeed)，and consequently it will not suitable for the application of the composition of motions. 动点、动系和静系必须分别属于三个不同的物体，否则绝对、相对和牵连运动中就缺少一种运动，不能成为合成运动,37, The relative path of the moving point relative to the MCS must be easy to determine. (except to find the convected moti

15、on when the absolute and relative motions are given. ) 动点相对动系的相对运动轨迹易于直观判断（已知绝对运动和牵连运动求解相对运动的问题除外）。,38,39,40,Absolute motion: rectilinear motion, 绝对运动: 直线运动,Absolute velocity: 绝对速度:,41,42,Noting the convected motion is translation，we have 由于牵连运动为平动，故,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,What

16、 is the correct relationship among these accelerations？ What is the additional term 2 vr why does it appear? Next, we are going to seek the answers to these questions and derive the theorem of composition of accelerations when the convected motion is rotation. 那么他们之间的关系是什么呢？ 2 vr 又是怎样出现的呢？它是什么呢？下面我们

17、就来讨论这些问题，推证牵连运动为转动时点的加速度合成定理。,58,59,convected velocity 牵连速度 Relative velocity 相对速度 Absolute velocity 绝对速度,60,Analysis of the velocity change in an interval Dt relative velocity： draw the triangle composed of After cutting a length of in the vector , can be decomposed into and .,Dt 时间间隔内的速度变化分析 相对速度：

18、由作速度矢量三角形， 在 矢量上截取 长度后， 分解为 和,61,62,where represents the direction change of the convected velocity caused by the convected rotation in an interval Dt, which is independent on the relative motion. represents the magnitude change of the convected velocity caused by the relative motion in an interval

19、Dt, which is dependent on the relative velocity .,其中: 表示Dt内由于牵连转动而引起的牵连速度方向的改 变量，与相对运动无关。 表示Dt内动点的牵连速度，由于相对运动而引起的 大小改变量，与相对速度 有关。,63,64,65,66,67,68,69,70,71,72,73,74,Example 3 A rectangular plate ABCD rotates about axis z at an constant angular velocity of . Point M1 and M2 move along the diagonal B

20、D and side CD respectively，and their velocities relative to the plate are and respectively at this moment. Calculate the Coriolis accelerations at point M1 and M2 and determine their directions.,75,76,77,78,1Concepts and formulae 一概念及公式 1). One point, two systems and three motions 1. 一点、二系、三运动 The a

21、bsolute motion of a point is the composition its relative and convected motions Composition of motions 点的绝对运动为点的相对运动与牵连运动的合成,79,80,81,82,83,For Guiding rod-block mechanisms, a typical way is to attach the MCS to the guiding rod and take the sliding block as the moving point; 导杆滑块机构：典型方法是动系固结于导杆，取滑块为

22、动点。,84,85,Acceleration-related problems. Since more than three vectors are involved, analytical (projection) method should be applied to find the unknowns and the axes to be projected on should be chosen to make most of the projections be easily calculated. 加速度问题, 往往超过三个矢量, 一般采用解析（投影）法求 解，投影轴的选取依解题简便的要求而定。,2) Velocity-related problems. It is simple to find the unknow