Theoretical mechanics
1 Theoretical mechanics
理论力学 篇运动学复习
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means 1. Basic contents. I)Kinematics of a Rectilinear motion Uniform velocity uniformly particle Curvilinear motion accelerated, varying velocity Composite motion: absolute motion, relative motion, embroiling motion 2) Kinematics of a Basic motions translation rigid body 3 Plane motion Rotation about a fixed axis Composite motion: composition of rotations about arallel axes 2. Basic equations: pa 1) Motion of a particle Position vector method r=r(0),I>aA dv d2r a- dt dt rectangular coordinates method x=f1( y=f2() 二=f3(
3 1．Basic contents： 1) Kinematics of a particle Rectilinear motion Curvilinear motion Composite motion：absolute motion, relative motion, embroiling motion Uniform velocity. uniformly accelerated , varying velocity 2) Kinematics of a rigid body Basic motions Plane motion Composite motion：composition of rotations about parallel axes translation Rotation about a fixed axis 2．Basic equations: 1) Motion of a particle Position vector method 2 2 ( ) , , dt d r dt dv a dt dr r =r t v = = = rectangular coordinates method ( ) ( ) ( ) 3 2 1 z f t y f t x f t = = = v z v y v x z y x = = = a z a y a x z y x = = =
远动学 基本内容: 直线运动 1,点的运动学曲线运动匀速匀变速变速 合成运动:绝对运动相对运动牵连运动 基本运动{平动 2刚体运动学平面运动定轴转动 合成运动:绕平行轴转动的合成 基本公式 点的运动 dy d2r 矢量法F=(),卩 dt dt2 直角坐标法 x=f1( as y=f2() 二=f3(
4 一．基本内容： 1.点的运动学 直线运动 曲线运动 合成运动：绝对运动,相对运动,牵连运动 匀速,匀变速,变速 2.刚体运动学 基本运动 平面运动 合成运动：绕平行轴转动的合成 平动 定轴转动 二．基本公式 1．点的运动 矢量法 2 2 ( ) , , dt d r dt dv a dt dr r =r t v = = = 直角坐标法 ( ) ( ) ( ) 3 2 1 z f t y f t x f t = = = v z v y v x z y x = = = a z a y a x z y x = = =
emacs =√vx2+v+v Directions are determined by their related cosines a=、a+an2+a natural coordinates method (when trajectory is known) S=f(1),v= dr along the tangential direction Along the tangential direction, anD2 dt dt point to the center of curvature Resultant acceleration; a=var +an, tg(a,n)= Const (uniformly accelerated motion) V=Vo+art SAtyot+a t2 2 (when t=O,V=VO,S=So v2=v02+2ax(-So)
5 2 2 2 x y z v = v + v + v 2 2 2 a = ax + ay + az Directions are determined by their related cosines. natural coordinates method (when trajectory is known) dt ds s = f (t) , v = Along the tangential direction， 2 2 dt d s dt dv a = = Along the tangential direction， 2 v an = point to the center of curvature. Resultant acceleration： n n a a a a a a n = + , tg( , )= 2 2 a = Const. (uniformly accelerated motion): v v a t = 0 + 2 0 0 2 1 s s v t a t = + + 2 ( ) 0 2 0 2 v =v + a s−s (when 0, , ) 0 0 t = v = v s = s
远动学 v=v +v+v 方向均由相应的方向余弦确定 a=、a+an2+a 自然法(轨迹已知时) s=(),v=方向沿切线方向, dhds方向沿切线方向,方向指向曲率中心 1 dt dt 全加速度:a=√a2+an2,(am)= a=常数(匀变速运动): V=Vo+art SAtyot+a t2 2 (t=0时v=vo,s=S0 v2=v02+2ax(-So)
6 2 2 2 x y z v = v + v + v 2 2 2 a = ax + ay + az 方向均由相应的方向余弦确定。 自然法（轨迹已知时） dt ds s = f (t) , v = 方向沿切线方向， 2 2 dt d s dt dv a = = 方向沿切线方向， 2 v an = 方向指向曲率中心 全加速度： n n a a a a a a n = + , tg( , )= 2 2 a = 常数（匀变速运动）： v v a t = 0 + 2 0 0 2 1 s s v t a t = + + 2 ( ) 0 2 0 2 v =v + a s−s ( 0 , ) 0 0 t= 时v=v s=s
Kinematics Composite motions of a particle V=v+卫 aa=ae+a.( when the embroiling motion is translation a =a ta +a c when the embroiling motion is rotation where ak=2O×,=2o,si(a 2) Motion of a rigid body Translation( can be simplified to the motion of a particle) At any moment, the trajectories, velocities and accelerations at all points in the body are identical Rotation about a fixed axis p=f(t),@ do d o 8= 0=00+ E=const: uniformly accelerated rotation) 0=90+00t+ar2 ( when t=00=00,9=90) O2=002+2(-() 7
7 Composite motions of a particle a e r v = v + v aa = ae + ar （when the embroiling motion is translation） aa = ae + ar + ak （ when the embroiling motion is rotation ） where 2 , 2 sin( , ) k e r k e r e r a = v a = v v Translation ( can be simplified to the motion of a particle） At any moment, the trajectories, velocities and accelerations at all points in the body are identical. Rotation about a fixed axis 2 2 ( ) , , dt d dt d dt d f t = = = = ＝const: (uniformly accelerated rotation) = +t 0 2 0 0 2 1 = + t+ t 2 ( ) 0 2 0 2 = + − (when 0 , ) = =0 =0 t 2) Motion of a rigid body
远动学 点的合成运动 Va=v+y an=a2+a,(牵连运动为平动时) dn=a2+an+ak(牵连运动为转动时) 其中,a=2可×,aA=2msn(可2,形) 2.刚体的运动 平动(可简化为一点的运动) 任一瞬时,各点的轨迹形状相同,各点的速度和加速度均相等 定轴转动 =f(0),o=a0 do d dt dt dt2 E=常量: 0=00+at (匀变速转动) 0=90+0(+2a2(=0时o=on;o=9) Q2=00+26(-0)
8 点的合成运动 a e r v = v + v aa = ae + ar （牵连运动为平动时） aa = ae + ar + ak （牵连运动为转动时） 其中， 2 , 2 sin( , ) k e r k e r e r a = v a = v v 平动（可简化为一点的运动） 任一瞬时, 各点的轨迹形状相同, 各点的速度和加速度均相等 定轴转动 2 2 ( ) , , dt d dt d dt d f t = = = = ＝常量： （匀变速转动） = +t 0 2 0 0 2 1 = + t+ t 2 ( ) 0 2 0 2 = + − ( 0 , ) = 时 =0 =0 t 2．刚体的运动
Kinematics o=const. (uniform rotation): p=ot,o=20 Unit of nis rpm Velocity and acceleration of a point in a rigid body: (Relation between angular and linear measurement v=R a= re a=E×F Resultant acceleration: a R@ Vector expressions a,=O×v arve+04 a=a2×n tgla, n =E×1+O×1 Transmission ratio of pulley system: 2 0,n, R 是=会 n Plane motion( composition of translation and rotation Pole method: (A is the pole) VB=VA+BA VBA=,o is the angular velocity
9 =const.(uniform rotation): 30 , n = t = Unit of ｎis rpm 2 4 a = R + Velocity and acceleration of a point in a rigid body: (Relation between angular and linear measurement) v = R a = R 2 an = R Resultant acceleration： 2 ( , ) tg a n = Vector expressions v = r a = r a v n = a = a an = r + v Transmission ratio of pulley system: n n n n i Z Z R R n n i 1 3 2 2 1 1 1 1 2 2 1 2 1 2 1 12 , − = = = = = = Plane motion ( composition of translation and rotation） Pole method:(A is the pole) vB =vA +vBA vBA=AB , is the angular velocity
远动学 0=常量(匀速转动): 30 n的单位:rpm 定轴转动刚体上一点的速度和加速度:(角量与线量的关系) v=RO V=⑦×F a= Ra =E×F d= Ro 用矢量表示为a=D×p 全加速度:a=R√E2+o4 a=a× tgla, n)= =E×F+D×下 轮系的传动比:42= r o2 R2 00 平面运动(平动和转动的合成) 基点法:(A为基点) vB=VA+ BA VBA=AB,O为图形角速度 10
10 =常量（匀速转动）： 30 , n = t = ｎ的单位：rpm 2 4 a = R + 定轴转动刚体上一点的速度和加速度：（角量与线量的关系） v = R a = R 2 an = R 全加速度： 2 ( , ) tg a n = 用矢量表示为 v = r a = r a v n = a = a an = r + v 轮系的传动比： n n n n i Z Z R R n n i 1 3 2 2 1 1 1 1 2 2 1 2 1 2 1 12 , − = = = = = = 平面运动（平动和转动的合成） 基点法：（A为基点） vB =vA +vBA vBA=AB , 为图形角速度
