Statics 2. The law of equivalent law of a force couple in space A force couple acting on a rigid body can be replaced by any other force couple of the same moment and turning direction lying in a parallel plane without altering the external effect on that body R Proo:①Daw∥I,cd∥ab 2Draw a couple of equilibrium E forces R and R'at point e, (R=R'should be satisfied) 3 Composing Fi andR we get R F2 at the point d composing FIand r we get F2 at point, R-FIF2,R-FI=F2 4 The force couple(Fi. fi) in plane I is replaced by(F,. F2 )in p lane ll 21
21 Proof: ①Draw II//Ⅰ, cd // ab ②Draw a couple of equilibrium forces R and R' at point E, (-R=R' should be satisfied) ③ Composing F1 and R we get F2 at the point d composing F1 'and R' we get F2 ' at point c , R-F1=F2 , R'-F1 '= F2 ' . ④ The force couple (F1，F1 ') in plane I is replaced by (F2， F2 ') in plane II. 2. The law of equivalent law of a force couple in space A force couple acting on a rigid body can be replaced by any other force couple of the same moment and turning direction lying in a parallel plane without altering the external effect on that body
学 二、空间力偶的等效定理 作用在同一刚体的两平行平面的两个力偶,若它们的转向相 同,力偶矩的大小相等,则两个力偶等效 证①作I∥I,cd∥ab R ②作一对平衡力RR(在E点且 F F2 使-R=R ③由反向平行力合成得 F1与R合成得F2,作用在d点 R F1与R合成得F2,作用在c点 且R-F1=F2,RF1′F2 ④在内的力偶(F1,F1)等效变成内的(F2,F2) 22
22 [证] ①作II//Ⅰ，cd // ab ②作一对平衡力R, R' (在E点,且 使-R=R') ③由反向平行力合成得： F1与R合成得F2，作用在d点 F1 '与R'合成得F2 '，作用在c点 且R-F1=F2 ，R'- F1 '= F2 ' ④在I内的力偶（F1，F1 '）等效变成II内的（ F2， F2 ' ） 二、空间力偶的等效定理 作用在同一刚体的两平行平面的两个力偶，若它们的转向相 同，力偶矩的大小相等，则两个力偶等效
Statics Because the force couple in space is a free vector it has three elements Othe magnitude of the moment of the force couple m 2the direction of the the moment of the force couple is the normal of the action plane of the force couple 3the turning direction, given by the right-hand rule 3. Composition and equilibrium of a system of force couple in space Because the moment of a force couple in space is a free vector,it can be translated to any point without changing its direction Especially, it can be translated to the concurrent point Because the moment are vectors they can be composed according to usual vector rules. The resultant moment of force couple is the sum of the moments of the component force couples
23 Because the force couple in space is a free vector it has three elements ①the magnitude of the moment of the force couple ②the direction of the the moment of the force couple is the normal of the action plane of the force couple. ③the turning direction, given by the right-hand rule. m 3. Composition and equilibrium of a system of force couple in space Because the moment of a force couple in space is a free vector, it can be translated to any point without changing its direction. Especially, it can be translated to the concurrent point. Because the moment are vectors they can be composed according to usual vector rules. The resultant moment of force couple is the sum of the moments of the component force couples
学 由此可得出,空间力偶矩是自由矢量,它有三个要素: ①力偶矩的大小= ②力偶矩的方向—与力偶作用面法线方向相同 ③转向—遵循右手螺旋规则 三、空间力偶系的合成与平衡 由于空间力偶系是自由矢量,只要方向不变,可移至任意 一点,故可使其滑至汇交于某点,由于是矢量,它的合成符合 矢量运算法则。合力偶矩=分力偶矩的矢量和 24
24 由此可得出，空间力偶矩是自由矢量，它有三个要素： ①力偶矩的大小= ②力偶矩的方向——与力偶作用面法线方向相同 ③转向——遵循右手螺旋规则。 m 三、空间力偶系的合成与平衡 由于空间力偶系是自由矢量，只要方向不变，可移至任意 一点，故可使其滑至汇交于某点，由于是矢量，它的合成符合 矢量运算法则。 合力偶矩 = 分力偶矩的矢量和
Statics m=m1+m2+m3+…+mn=∑m m= m,+ma+m, cosa=., cos B=", cosy=m Obviously the equilibrium condition of the force couple system in space Is m=∑m=0 The projections are =0 ∑u3=0 2=0
25 = = + + + + = n i m m m m mn mi 1 1 2 3 The projections are mx =0 my =0 mz =0 m m m m m m m m m m x y z = x + y + z ;cos= ,cos = ,cosg = 2 2 2 = =0 m mi Obviously the equilibrium condition of the force couple system in space is