Constant Angular Acceleration

74 Pages·2013·2.44 MB·English

Chapter 16
Planar Kinematics of a
Rigid Body
APPLICATIONS
Passengers on this amusement ride are subjected to
curvilinear translation since the vehicle moves in a circular
path but always remains upright.
If the angular motion of the rotating arms is known, how
can we determine the velocity and acceleration experienced
by the passengers?
Does each passenger feel the same acceleration?
15-2
PLANAR RIGID BODY MOTION
平
移
Translation: Translation occurs if every line segment on
the body remains parallel to its original direction during the
motion. When all points move along straight lines, the
motion is called rectilinear translation. When the paths of
motion are curved lines, the motion is called curvilinear
translation.
15-3
PLANAR RIGID BODY MOTION
(continued)
Rotation about a fixed axis: In this case, all
the particles of the body, except those on
the axis of rotation, move along circular
paths in planes perpendicular to the axis of
rotation.
繞固定軸旋轉
General plane motion: In this case, the
body undergoes both translation and
rotation. Translation occurs within a
plane and rotation occurs about an axis
perpendicular to this plane.
一般平面運動
15-4
15-5
16.2 Translation (平移)
r = r + r (平移時，相關位置不變∴r = const.)
B A B/ A B/ A
⇒一次導數： v = v
B A
⇒二次導數： a = a
B A
平移時，所有的點有相 同的速度和加速度
15-6
16.3 Rotation About a Fixed Axis
θ: angular position
dθ: angular displacement
方向以 "右手定則 " 決定
dθ
ω: 角速度，θ 對時間的變化率，即 ω=
dt
α: 角加速度，ω 對時間的變化率
dω d2θ
即 α= =
dt dt2
dω dθ
又 α= ⋅
dt dθ
dθ dω
= ⋅
dt dθ
dω
=ω
dθ
⇒ α dθ=ω dω (compare ads = vdv)
15-7
Motion of Point P --Velocity
P 以 O 為心，r 為半徑，沿圓周運動
由極座標：v = r
&
r
v = rθ&
θ
r = const. ∴r = 0
Q &
⇒ v = v = rθ&
θ
又ω=θ&
∴v = rω
由上圖的右手定則，決定 v 的方向
⇒ v =ω×r (=ω×r)
P
(r 為旋轉軸上任一點到 P 的位置向量 )
P
v =ω (r sinφ) =ω (r)
P
15-8
Motion of Point P --Acceleration
dv v2
a = , a =
t dt n r
dω
由 v =ωr , α=
dt
d(ωr) dω
得 a = = r =αr
t dt dt
(ωr)2
a = =ω2r
n r
vector
dv dω dr
a = = × r +ω× P
P
dt dt dt
=α× r +ω× (ω× r )
P P
a =αr sinφ=αr
Q
t P
⇒ a = a + a
t n
=α× r −ω2r
15-9
Constant Angular Acceleration
(1) α=α = 0 ⇒ uniform rotation
C
⇒ω= const.
即 θ=θ +ωt
O
(2) α=α ≠ 0 ⇒ uniformly acceleration rotation
C
⎧ ω=ω +αt
O C
⎪
⎪ 1
θ=θ +ωt + αt2
⎨
O O 2 C
⎪
⎪ 消去 t ⇒ω2 =ω2 + 2α (θ-θ )
⎩
O C O
與直線運動的關係相似
v2 = v 2 + 2a (s − s )
O C O
15-10