第二模块 动力学基本原理

1、第二模块 动力学基本原理（四）牛顿定律，约束、约束力 1、牛顿定律的基本概念。 2、牛顿定律的使用条件 3、质点的运动微分方程 4、约束和约束反力41 牛顿定律1. Newton's Three Fundamental Laws. (28)Formulated by Sir Isaac Newton in the latter part of the seventeenth century these laws can be stated as follows:(1) Newton's FIRST LAW. If the resultant force acting on a

2、particle is zero, the particle will remain at rest (if originally at rest) or will move with constant speed in a straight line (if originally in motion).(2) Newton's SECOND LAW. If the resultant force acting on a particle is not zero, the particle will have an acceleration proportional to the ma

3、gnitude of the resultant and in the direction of this resultant force.Newtons second law of motion is best understood by imagining the following experiment: A particle is subjected to a force F1 of constant direction and constant magnitude F1. Under the action of that force, the particle is observed

4、 to move in a straight line and in the direction of the force (Fig.4.1.1a). By determining the position of the particle at various instants, we find that its acceleration has a constant magnitude . If the experiment is repeated with forces F2, F3, of different magnitude or direction (Fig.1.1.1b and

5、c), we find each time that the particle moves in the direction of the force acting on it and that the magnitudes , , ,., of the accelerations are proportional to the magnitudes F1, F2, F3,., of the corresponding forces:= constantFig.4.1.1The constant value obtained for the ratio of the magnitudes of

6、 the forces and accelerations is a characteristic of the particle under consideration; it is called the mass of the particle and is denoted by m. When a particle of mass m is acted upon by a force F, the force F and the acceleration a of the particle must therefore satisfy the relationF=ma (4.1.1)Th

7、is relation provides a complete formulation of Newtons second law, it expresses not only that the magnitudes of F and a are proportional but also (since m is a positive scalar) that the vectors F and a have the same direction (Fig.4.1.2). We should note that Eq. (4.1.1) still holds when F is not con

8、stant but varies with time in magnitude or direction. The magnitudes of F and a remain proportional, and the two vectors have the same direction at any given instant. However, they will not, in general, be tangent to the path of the particle.Fig.4.1.2When a particle is subjected simultaneously to se

9、veral forces, Eq. (1.1.1) should be replaced by=ma (4.1.2)where represents the sum, or resultant, of all the forces acting on the particle,(3) Newton's THIRD LAW. The forces of action and reaction between bodies in contact have the same magnitude, same line of action, and opposite sense. For eve

10、ry action, there is an equal and opposite reaction; that is, the forces of interaction between two particles are equal in magnitude and opposite in direction.2、牛顿定律的使用条件（1）在运动学中参考系可以任意选取，但在动力学中则不能任意选取参考系。（2）惯性参考系：与绝对静止空间固连的参考系以及相对其匀速直线平动的参考系。牛顿定律适用于一切惯性参考系。（3）伽利略相对性原理：一切力学方程和定律对所有惯性参考系都是等价的。3、质点的运动微

11、分方程向量式（牛顿第二运动定律） 4、约束和约束反力（五）受力分析、受力图物体的受力分析 受力图 The free-body diagram (FBD) of a body is a sketch of the body showing all forces that act on it. The term free implies that all supports have been removed and replaced by the forces (reactions) that they exert on the body. 在求解静力平衡问题时，必须首先分析物体的受力情况，即进行

12、受力分析。根据问题的已知条件和待求量，从有关结构中恰当选择某一物体（或几个物体组成的系统）作为研究对象。这时，可设想将所选择的对象从与周围的约束（含物体）的接触中分离出来，即解除其所受的约束而代之以相应的约束反力。这一过程称为解除约束。解除约束后的物体，称为分离体，画有分离体及其所受的全部力（包括主动力和约束反力）的简图，称为受力图。The importance of mastering the FBD technique cannot be overemphasized. Free-body diagrams are fundamental to all engineering discip

13、lines that are concerned with the effects that forces have on bodies. The construction of an FBD is the key step that translates a physical problem into a form that can be analyzed mathematically.Forces that act on a body can be divided into two general categoriesreactive forces (or simply reactions

14、) and applied forces. Reactions are those forces that are exerted on a body by the supports to which it is attached. Forces acting on a body that are not provided by the supports are called applied forced of course, all forces, both reactive and applied, must be shown on be-body diagrams.The followi

15、ng is the general Procedure for constructing a free-body diagram.1. A sketch of the body is drawn assuming that all supports (surfaces of contact, supporting cables, etc.) have been removed.2. All applied forces are drawn and labeled on the sketch. The weigh of the body is considered to be an applie

16、d force acting at the center of gravity, the center of gravity of a homogeneous body coincides with the centroid of its volume.3. The reactions due to each support are drawn and labeled on the sketch. (If the sense of a reaction is unknown, it should be assumed. The Solution will determine the corre

17、ct sense: A positive result indicates that the assumed sense is correct, whereas a negative result means that the correct sense is opposite to the assumed sense.)4. All relevant angles and dimensions are shown on the sketch.When you have completed this Procedure, you will have a drawing (i.e., a be-

18、body drawing) that contains all of the information necessary for writing the equilibrium equations of the body.The most difficult step to master in the construction of FBDs is the determination of the support reactions. Table 4. l shows the reactions exerted by various coplanar supports; it also lis

19、ts the number of unknowns that are introduced on an FBD by the removal of each support. To be successful at drawing FBDs, you must be completely familiar with the contents of Table 4.l. It is also helpful to understand the physical reasoning that determines the reactions at each support, which are d

20、escribed below.画受力图是求解静力学问题的重要步骤。You should keep the following points in mind when you are drawing free-body diagrams.1. Be neat. Because the equilibrium equations will be derived directly from the free-body diagram, it is essential that the diagram be readable.2. Clearly label all forces, angles, a

21、nd distances with values (if known) or symbols (if the values are not known).3. The support reactions must be consistent with the information presented in Table 4.l.4. Show only forces that are external to the body (this includes support reactions). internal forces occur in equal and opposite pairs

22、and thus will not appear on free-body diagrams.例5一1 重量为G的梯子AB，搁在光滑的水平地面和铅直墙上。在D点用水平绳索与墙相连，如图所示。试画出梯子的受力图。例52 下图所示的结构有杆AC、CD与滑轮B铰接组成。物体重W，用绳子挂在滑轮上。如杆、滑轮及绳子的自重不计，并忽略各处的摩擦，试分别画出滑轮B（包括绳索）、杆AC、CD及整个系统的受力图。在上例中，当取整个系统为研究对象时，AC杆与CD杆在C处不分开，此时这两根杆之间的相互作用力称为内力。一般来说，当取某个系统为研究对象时，该系统内构件之间的内力都不必画出，而只画作用在系统上的外力。需

23、要指出的是，内力和外力的区分不是绝对的，在一定的条件下，内力和外力是可以相互转化的。例如在上例中，如果要画由杆AC、滑轮B、重物和绳子所组成系统的受力图，则杆CD对杆AC的约束反力Fsc（原来整个系统中两杆之间的内力）就成了该系统的外力。Sample Problem 53The homogeneous，250kg triangular plate in rig（a）is supported by a pin at A and a roller at CDrawn the FBD of the plate and determine the number of unknownsSolutionT

24、he FBD of the plate is shown in Fig（b）The pin and roller supports have been removed and replaced by the reactive forces. The forces acting on he plate re described below. W: The Weight of the PlateThe weight of the plate is W = mg = (250)(9.81) = 2453 N. It acts at the centroid G. Only the horizonta

25、l location of G is shown in the figure, because it is sufficient to determine the line of action of W.Ax and Ay: The Components of the Pin Reaction at AFrom Table 4. l, we see that a pin reaction can be shown as two components A. and Ay, which are equivalent to an unknown force acting at an unknown

26、angle. We have shown Ax acting to the right and Ay acting upward. These directions are chosen arbitrarily; the solution of the equilibrium equations will determine the correct sense for each force. Therefore, the free-body diagram would be correct even if A. or Ay were chosen to act in directions opposite to those shown in Fig. (b).NC: The Normal Reaction at CFrom Table 4. l, the force exerted by a roller support is normal to the inclined surface. Therefore, on the FBD we show the force NC at C, inclined at 30o to the vertical. The FBD contai