Theoretical Mechanics Chapter 2: Special cases of force systems in a plane
1 Theoretical Mechanics
理论力学 AL 2
2
Statics Introduction Force system: coplanar force system or three-dimensional x无法显 force system C planar (coplanar system of concurrent forces force 2coplanar system of parallel forces(coplanar system: system of force couples is a special cast general case of a force system in a plane Special kast of force system in a plane: coplanar system bf concurrent forces, coplanar system of force couples and coplanar system df parallel forces Coplanar system of concurrent forces: Example: the Forces whose lines of action intersect at one hook ofla crane point are called concurrent, if all the concurrent forces acting on a body lie in one plane - the form a coplanar system of concurrent forces The methods of study: the graphical method 72 and the analytical method 3
3 Coplanar system of concurrent forces: Forces whose lines of action intersect at one point are called concurrent, if all the concurrent forces acting on a body lie in one plane, they form a coplanar system of concurrent forces. Introduction ①coplanar system of concurrent forces ②coplanar system of parallel forces (coplanar system of force couples is a special cast) ③general case of a force system in a plane The methods of study: the graphical method and the analytical method. Example: the hook of a crane Force system: coplanar force system or three-dimensional force system Coplanar force system: Special cast of force system in a plane: coplanar system of concurrent forces, coplanar system of force couples and coplanar system of parallel forces
学 引言 力系为面力系、空间力系 ①平面汇交力系 平面力系②平面平行力系(平面力偶系是其中的特殊情况) (③平面一般力系平面任意力系) 平面特殊力系:指的是平面汇交力系、平面力偶系和平面平 行力系 平面汁交力系: 各力的作用线都在同一平面内且例:起重机的挂钩 汇交升一点的力系。 T 研究方法:几何法,解析法。 12
4 平面汇交力系： 各力的作用线都在同一平面内且 汇交于一点的力系。 引 言 ①平面汇交力系 平面力系 ②平面平行力系(平面力偶系是其中的特殊情况 ) ③平面一般力系(平面任意力系) 研究方法：几何法，解析法。 例：起重机的挂钩。 力系分为：平面力系、空间力系 平面特殊力系：指的是平面汇交力系、平面力偶系和平面平 行力系
Chapter 2: Special cases of force systems in a plane 82-1 The graphical method of composition and the equilibrium of a coplanar system of concurrent forces 82-2 The analytical method of composition and the equilibrium of a coplanar system of concurrent forces 82-3 The concepts and the character of a moment and a force couple 82-4 The composition and the equilibrium of a coplanar system of force couples 82-5 The composition and the equilibrium of a coplanar system of parallel forces
5 §2–1 The graphical method of composition and the equilibrium of a coplanar system of concurrent forces §2–2 The analytical method of composition and the equilibrium of a coplanar system of concurrent forces §2–3 The concepts and the character of a moment and a force couple §2–4 The composition and the equilibrium of a coplanar system of force couples §2–5 The composition and the equilibrium of a coplanar system of parallel forces Chapter 2: Special cases of force systems in a plane
第二章平面特殊力系 §2-1平面汇交力系合成和平衡的几何法 §2-2平面汇交力系合成和平衡的解析法 §2-3力矩、力偶的概念及其性质 §2-4平面力偶系的合成与平衡 §2.5平面平行力系的合成与平衡
6 §2–1 平面汇交力系合成和平衡的几何法 §2–2 平面汇交力系合成和平衡的解析法 §2–3 力矩 、力偶的概念及其性质 §2–4 平面力偶系的合成与平衡 §2–5 平面平行力系的合成与平衡 第二章 平面特殊力系
Statics 82-1 The graphical method of composition and the equilibrium of a coplanar system of concurrent forces I The graphical method of the composition of forces 1. The composition of two concurrent forces 2. The composition of any coplanar concurrent forces 180- Fi F R R Fi F2 A A∠ A A F2 cos(1809-a)=-cosa F3 F By the parallelogram rule or constructing a force triangle According to the law of cosines The fo F orce R VF+F+2F F coSa polygon Fi According to the law of cosines F R R the direction of the resultant force is sin sin( 180=0
7 §2–1 The graphical method of composition and the equilibrium of a coplanar system of concurrent forces Ⅰ The graphical method of the composition of forces 2 1 2 cos 2 2 2 R = F1 + F + F F 2. The composition of any coplanar concurrent forces The force polygon: 1.The composition of two concurrent forces According to the law of cosines, the direction of the resultant force is: According to the law of cosines: cos(180−)=−cos By the parallelogram rule or constructing a force triangle. sin sin(180 ) 1 − = F R
学 §2-1平面汇交力系合成与平衡的几何法 合成的几何法 1.两个共点力的合成 2任意个共点力的合成 F 180°a R R Fi Fi A∠q F2 F2 F2 A● F cos(80°a)=cos Fa 由力的平行四边形法则作, F 也可用力的三角形来作。 F2 F 由余弦定理: 为力多边形 R=√F2+F2+2 f, F coSa F 合力方向由正弦定理: sinsin( 180-c) R
8 §2-1 平面汇交力系合成与平衡的几何法 一、合成的几何法 2 1 2 cos 2 2 2 R = F1 + F + F F sin sin(180 ) 1 − = F R 2. 任意个共点力的合成 为力多边形 1.两个共点力的合成 合力方向由正弦定理： 由余弦定理： cos(180−)=−cos 由力的平行四边形法则作， 也可用力的三角形来作
Statics Conclusion: R=F+F+F,+E In general: R-EF The resultant of a coplanar system of concurrent forces is equal to the geometrical sum of these forces and it applies to the point of intersection of these forces. II The graphical condition of equilibrium The necessary and sufficient condition is R=∑F=0 If the resultant force is zero the force F2 F polygon draw with these forces is closed So the graphical condition of equilibrium of a coplanar system of concurrent forces is F5 R The force polygon is closed or The geometrical sum of all forces is zero
9 Conclusion: In general: The resultant of a coplanar system of concurrent forces is equal to the geometrical sum of these forces and it applies to the point of intersection of these forces. Ⅱ The graphical condition of equilibrium R=F R F1 F2 F3 F4 = + + + If the resultant force is zero, the force polygon draw with these forces is closed. So the graphical condition of equilibrium of a coplanar system of concurrent forces is: The necessary and sufficient condition is: R=F =0 The force polygon is closed or The geometrical sum of all forces is zero
学 结论:R=F+F+F+F 即:R=∑F 即:平面汇交力系的合力等于各分力的矢量和,合力的作用 线通过各力的汇交点。 二、平面汇交力系平衡的几何条件 平面汇交力系平衡的充要条件是:R=∑F=0 在上面几何法求力系的合力中,合力为 F2 F 零意味着力多边形自行封闭。所以平面 汇交力系平衡的必要与充分的几何条件 F5 R 是或 力多边形自行封闭 力系中各力的矢量和等于零 10
10 结论： 即： 即：平面汇交力系的合力等于各分力的矢量和，合力的作用 线通过各力的汇交点。 二、平面汇交力系平衡的几何条件 R=F R F1 F2 F3 F4 = + + + 在上面几何法求力系的合力中，合力为 零意味着力多边形自行封闭。所以平面 汇交力系平衡的必要与充分的几何条件 是： 平面汇交力系平衡的充要条件是： R=F =0 力多边形自行封闭 或 力系中各力的矢量和等于零