Unit 1 General Equilibrium__ Conditions of A System

1 General Equilibrium Conditions of A System 力系的一般平衡条件 In this section we shall consider the conditions that the forces and couples acting upon a body must satisfy in order to be in equilibrium 在这一部分 我们将研究为了使一个物体保持平衡 作用在其上的 力和力偶所必须满足的条件 According to Newton s first law the sum of the forces exerted on a body at rest must be zero Notice however that this law says nothing about the moment or rotational effects of the forces Clearly the total moment must also be zero else the body would rotate 根据牛顿第一定律 施加在一个静止物体上的力系的合力一定为零 然而 请注意这个定律对力矩或力系的转动效应只字未提 显然 合力矩也一定为零 否则物体将会转动 The fundamental problem here is that Newton s first law and second law as originally stated applies only for very small bodies or particles with negligible dimensions and nonzero mass However it can be extended to bodies of finite size as follows 这里的基本问题是原先叙述的牛顿第一定律 和第二定律 只适用 于非常小的物体 或者尺寸可以忽略的非零质量的粒子 然而 它 可以扩展到如下所述的有限尺寸的物体 Consider a system consisting of two particles and let and be the 1f2f forces due to the interaction between them Fig 1 1 These forces are 2 called internal forces since they are due to interactions between bodies within the system Assuming that the internal forces obey Newton s third law we have Suppose that 12ff there are also forces such as and exerted on the 1 2F F3F particles due to interactions with bodies outside the system Such forces are called external forces Clearly and the forces acting upon a particular particle must have the same point of application because a particle has negligible dimensions 考虑一个由两个质点组成的系统 假设 和为它们之间的相互1f2f 作用力 图 1 1 这些力称为内力 因为它们是由于系统内部的物体 之间的相互作用而产生的 假定内力服从牛顿第三定律 我们有 假如还有质点与系统外物体之间的相互作用力施加在质12ff 点上 如 和 这些力称为外力 显然 作用在一个特定粒子1 2F F3F 上的力一定有相同的应用 因为粒子的尺寸可以忽略 We shall say that the system is in equilibrium if each particle within it is in equilibrium In this case by Newton s first law the sum of the forces acting upon each particle must be zero For particle A we have 如果系统内的每一个质点处于平衡 我们就可以说系统是平衡的 3 在本例中 依据牛顿第一定律 作用在每个质点上的力的合力一定 为零 对质点 A 我们有 0 121 fFFFA and for particle B 而对质点 B 有 0 32 FfFB The total force acting upon the system is 作用在系统上的力的总和为 123120ABFFFFFFff Now let us consider the total moment of these forces about some point P Referring to Fig 1 1 we have 现在我们来研究这些力对于同一点 P 的合力矩 由图 1 1 我们有 12 PABMrFrF But so the total moment must also be zero as stated 0ABFF previously 但 所以 如先前所述 合力矩也一定为零 0ABFF Since the forces and have the same line of action the moment 1f2f condition can be rewritten as 由于力 和有相同的作用线 力矩的条件可以改写为1f2f 1121223 0PMrFFffrF But so the conditions on the forces and moments reduce to12ff 但 所以力和力矩的条件简化为12ff 4 1210FFFF And 和 111223 0PMrFrFrF In other words if the system is in equilibrium the sum of the external forces acting upon it is zero and so is the sum of the moments of these forces about an arbitrary point The internal forces need not be considered because their effects cancel out 换句话说 如果系统处于平衡 那么作用在其上的合外力一定为零 而且这些力对于任一点的合力矩也为零 内力不需要考虑 因为它 们的效应相互抵消了 Although we shall not go through the details it should not be too difficult to see that the preceding results hold for a system consisting of any number of particles acted upon by any number of external forces provided the internal forces obey Newton s third law In particular these results apply to bodies of finite extent since such bodies can be thought of as consisting of a large number very small pieces or particles Thus we have the following general equilibrium conditions 尽管我们不去研究细节 也不难发现 倘若内力服从牛顿第三定律 前述结果适用于由许多有多个外力作用其上的质点组成的系统 特 别是 这些结果也适用于有限尺寸的物体 因为这样的物体可认为 是由大量微体或质点组成的 因此 我们得到下列一般平衡条件 5 If a system is in equilibrium then 如果系统处于平衡 那么 and 1 1 0F 0PM where is the sum of the external forces acting upon the system F and is the total moment of these forces about an arbitrary point PM including the moment of any couples which may be acting 这里是作用在系统上的所有外力的总和 而是这些力对F PM 任意点的合力矩 包括系统中可能作用有的力偶的矩 Equations 1 1 are necessary conditions for equilibrium i e if the system is in equilibrium these equations must be satisfied They are not in general sufficient conditions for equilibrium This presents no difficulties however for we shall be dealing only with systems known to be in equilibrium Equations 1 1 are both necessary and sufficient conditions for equilibrium of a rigid body Proof that they are sufficient requires use of Newton s second law and other knowledge beyond the level of this text 方程 1 1 是平衡的必要条件 也就是说 如果系统处于平衡 必 须满足这些方程 一般来说它们不是平衡的充分条件 然而 这并 不会带来任何困难 因为我们的研究仅涉及平衡系统 对于刚体 方程 1 1 既是其平衡的必要条件也是充分条件 检验其充分性需 要应用牛顿第二定律和其它超出本课文的知识 It is important to note that Eqs 1 1 hold for any system in 6 equilibrium regardless of the material of which it is comprised For example they hold for a mass of fluid at rest as well as for solid bodies They also apply to moving systems under certain conditions since Newton s first law upon which they are based applies to particles moving with constant velocity as well as to particles at rest For instance Eqs 1 1 hold for bodies that move in a straight line at constant speed without rotation and for bodies that rotate at a constant rate about a fixed axis through their mass center Typical examples are an airplane in straight level flight at constant speed and the pulley on an electric motor rotating at constant speed However problems involving motion of any kind are usually relegated to texts on dynamics 重要的是要注意到 方程 1 1 适用于任何平衡系统 而不管 组成该系统的物质是什么 例如 他们适用于大量的静止流体和固 体 在某种条件下 它们 指两方程式 也适用于运动系统 因为 它们是建立在牛顿第一定律的基础上 而牛顿第一定律既适用于匀 速运动的质点 也适用于静止的质点 例如 方程 1 1 适用于做 无转动匀速直线运动的物体和以通过质心的固定轴为轴做匀速转动 的物体 典型的例子有做水平匀速直线飞行的飞机和匀速转动的电 动机皮带轮 但是 问题涉及的任何运动一般归类为动力学 When expressed in component Eqs 1 1 yield the six scalar equations 当以分量的形式表示时 方程 1 1 可变形为六个标量方程 0 xF 0yF 0zF 7 1 2 0pxM 0pyM 0pzM These equations can be used in a force analysis of a system to solve for unknown information concerning the external forces and couples acting Since there are six equations we can generally solve for six unknowns If all of unknowns concerning the external forces and couples can be determined from the equilibrium equations the problem is said to be statically determinate If not it is said to be statically indeterminate 利用这些方程对系统进行受力分析 解决就外力和作用力偶而 言的未知问题 由于有 6 个方程 所以我们一般可以解决含六个未 知数的问题 如果通过平衡方程可以解出关于外力和力偶的所有未 知数 我们就说系统是静定的 反之 系统为静不定 When there are more unknowns than e