1
Theoretical mechanics
2
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
rd
dt
vda
dt
rdvtrr ????
? rectangular coordinates method
)(
)(
)(
3
2
1
tfz
tfy
tfx
?
?
?
zv
yv
xv
z
y
x
?
?
?
?
?
?
za
ya
xa
z
y
x
??
??
??
?
?
?
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
rd
dt
vda
dt
rdvtrr ????
?直角坐标法
)(
)(
)(
3
2
1
tfz
tfy
tfx
?
?
?
zv
yv
xv
z
y
x
?
?
?
?
?
?
za
ya
xa
z
y
x
??
??
??
?
?
?
5
222
zyx vvvv ???
222
zyx aaaa ???
Directions are determined by their related cosines,
? natural coordinates method (when trajectory is known)
dt
dsvtfs ??,)( Along the tangential direction,
2
2
dt
sd
dt
dva ??
?
Along the tangential direction,
?
2v
a n ?
point to the center of curvature,
Resultant acceleration,
n
n a
anaaaa ?
? ??? ),( t g,
22
??a Const,(uniformly accelerated motion),
tavv ??? 0
200
2
1 tatvss
????
)(2 0202 ssavv ??? ?
),,0w h e n( 00 ssvvt ???
6
222
zyx vvvv ???
222
zyx aaaa ???
方向均由相应的方向余弦确定。
?自然法（轨迹已知时）
dt
dsvtfs ??,)( 方向沿切线方向,
2
2
dt
sd
dt
dva ??
?
方向沿切线方向,
?
2v
a n ? 方向指向曲率中心
全加速度,
n
n a
anaaaa ?
? ??? ),( t g,
22
??a 常数（ 匀变速运动 ）,
tavv ??? 0
200
2
1 tatvss
????
)(2 0202 ssavv ??? ?
),0( 00 ssvvt ??? 时
7
?Composite motions of a particle
rea vvv ??
rea aaa ??
（ when the embroiling motion is translation）
krea aaaa ???
（ when the embroiling motion is rotation ）
where ),s i n (2,2
rerekrek vvava ??? ???
?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
dtf ?????? ????
?＝ const,(uniformly
accelerated rotation)
t??? ?? 0
200
2
1 tt ???? ???
)(2 0202 ????? ???
),0w h e n ( 00 ???? ???t
2) Motion of a rigid body
8
?点的合成运动
rea vvv ??
rea aaa ??
（牵连运动为平动时）
krea aaaa ???
（牵连运动为转动时）
其中,),s i n (2,2
rerekrek vvava ??? ???
?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
dtf ?????? ????
?＝常量,
（匀变速转动）
t??? ?? 0
200
2
1 tt ???? ???
)(2 0202 ????? ???
),0( 00 ???? ??? 时t
2．刚体的运动
9
?=const.(uniform rotation),
30,
???? nt ?? Unit of ｎ is rpm
42 ?? ?? Ra
Velocity and acceleration of a point in a rigid body,(Relation between
angular and linear measurement)
?Rv ?
?? Ra ?
2?Ra n ?
Resultant
acceleration,
2),( ?
??natg
Vector
expressions
rv ?? ?
ra ?? ??
va n ?? ?
naaa ?? ?
vr ???? ??
Transmission ratio
of pulley system,
n
n
n
niZ
Z
R
R
n
ni
?
?
?
?
?
?
?
?
?
? 1
3
2
2
11
1
1
2
2
1
2
1
2
1
12,
???????????
?Plane motion ( composition of translation and rotation）
Pole method:(A is the pole)
??,???? ABvvvv BABAAB is the angular velocity,
10
?=常量 （匀速转动）,
30,
???? nt ?? ｎ 的单位,rpm
42 ?? ?? Ra
定轴转动刚体上一点的速度和加速度：（角量与线量的关系）
?Rv ?
?? Ra ?
2?Ra n ?
全加速度,
2),( ?
??natg
用矢量表示为
rv ?? ?
ra ?? ??
va n ?? ?
naaa ?? ?
vr ???? ??
轮系的传动比,
n
n
n
niZ
Z
R
R
n
ni
?
?
?
?
?
?
?
?
?
? 1
3
2
2
11
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）
??,???? ABvvvv BABAAB 为图形角速度
11
?? ?? ABa BA
2??? ABa nBA
??? are the angular velocity and
acceleration respectively
Velocity projection method,? ? ? ?
ABAABB vv ?
Instantaneous velocity center method,
direction determined by ?,
n
BABAAB aaaa ???
?
?e and ? point the
same direction
?e and ? have
opposite directions
,??? PBv B P is the
,PBv B ?
?Composition of rotations around two parallel axes,The composed
motion is rotation about an instantaneous axis,
Absolute angular velocity,
location of the instantaneous axis (across point P) PO PO
r
e
1
2?
?
?rea
??? ??
Instantaneous velocity center,
12
?? ?? ABa BA
2??? ABa nBA
??? 分别为图形的角速度，角加速度
投影法,? ? ? ?
ABAABB vv ?
瞬心法,,??? PBv
B
P点为图形的速度瞬心，,PBv B ? 与 ?一致
n
BABAAB aaaa ???
?
? 绕两平行轴转动的合成 合成结果为绕瞬轴的转动。
绝对角速度,
瞬轴（过 P点）的位置,
PO
PO
r
e
1
2?
?
?
rea ??? ??
?e 与 ?r同向时 ?e 与 ?r反向时
13
3,Steps for problem solving,skills and cautions
1) Analyze the characteristics of moving system and determine the
types of motions of particles or rigid bodies in given problems,
2) Find the known and unknown conditions,
3) Choose suitable methods to establish the relationships among
velocities/accelerations and find their solutions,
For more details about steps,skills and cautions for the available
methods,please refer to summaries in the lessons for problem solving,
14
三．解题步骤．技巧及注意的问题
1.分析题中运动系统的特点及系统中点或刚体的运动形式。
2.弄清已知量和待求量。
3.选择合适的方法建立运动学关系求解。
各种方法的步骤,技巧和使用中注意的问题详见每次习题课中的
总结。
15
4,Examples
[Example 1] A bicycle moves
along a straight road with the
law x=0.1 t２,where t is in
seconds and x is in meters,
R=35cm,l=18cm,the number
of the teeth of the sprocket
wheel,
When t=10,mn is vertical,
Find the absolute velocities of
the pedal m and n,(Assume
that the wheel rolls without
slipping.)
48,18 21 ?? ZZ
16
四．例题
[例 1] 自行车在水平直线道
路上按规律 x=0.1 t２ 行驶,式
中 t以秒计,x以米计, 已知
R=35cm,l=18cm,链轮齿数
。
t=10秒时,mn连线铅垂。
求 此时自行车踏板 m 和 n 的
绝对加速度（设车轮只滚不
滑）。
48,18 21 ?? ZZ
17
Solution,
consider moving point 1, m
moving point 2, n
moving system,bicycle body
static system,road surface
tdtdxvttx xe 20v )cm(10)m(1.0 22 ??????? 20??? dtdvaa xxe
2c m / s 20,c m / s 200 ),s(10 w h e n ?????? exex aavvt
)s/1(743520 ),s/1(74035200 211 ?????? RaRv xx ??
)s/1(14 34818 ),s/1(7154818 21212 ????? ????
18
解, 选取动点 1, m
动点 2, n
动系：自行车身
静系：地面
tdtdxvttx xe 20v )cm(10)m(1.0 22 ??????? 20??? dtdvaa xxe
2c m / s 20,c m / s 200,)s(10 ?????? exex aavvt 时
)s/1(743520 ),s/1(74035200 211 ?????? RaRv xx ??
)s/1(14 34818 ),s/1(7154818 21212 ????? ????
19
Because the embroiling motion is translation
rea aaa ???
?
9.73tg
)s/cm(2.84
)
49
4 0 5 0
()
7
27
20(
)()(
)(
7
27
14
3
18
)(
49
4 0 5 0
)
7
15
(18
)(20
1
2
22
22
2
2
2
2
?
?
?
?
???
????
?????
?????
??
?
?
?
?
?
?
?
rmem
n
rm
m
n
rmrmemm
rm
n
rm
em
aa
a
aaaa
la
la
a
?
9.78tg
)s/cm(86
)
49
4050
()
7
27
20(
)()(
)(
7
27
)(
49
4050
)(20
1
2
22
22
?
?
?
?
???
???
???
????
??
?
?
?
??
?
rnen
n
rn
n
n
rnrnenn
rmrn
n
rm
n
rn
en
aa
a
aaaa
aa
aa
a
（ ?n ） （ ?ｍ ）
Moving point m,Moving point n,
20
由于 牵连运动为平动
rea aaa ???
?
9.73tg
)s/cm(2.84
)
49
4 0 5 0
()
7
27
20(
)()(
)(
7
27
14
3
18
)(
49
4 0 5 0
)
7
15
(18
)(20
1
2
22
22
2
2
2
2
?
?
?
?
???
????
?????
?????
??
?
?
?
?
?
?
?
rmem
n
rm
m
n
rmrmemm
rm
n
rm
em
aa
a
aaaa
la
la
a
?
9.78tg
)s/cm(86
)
49
4050
()
7
27
20(
)()(
)(
7
27
)(
49
4050
)(20
1
2
22
22
?
?
?
?
???
???
???
????
??
?
?
?
??
?
rnen
n
rn
n
n
rnrnenn
rmrn
n
rm
n
rn
en
aa
a
aaaa
aa
aa
a
（ ?n ） （ ?ｍ ）
动点 m,动点 n,
21
Solution,OA rotates about a fixed axis; AB,wheel O1 and O2 have plane
motion; rod O1 O2 and plate MN have translation,Consider AB,it is in
instantaneous translation,i.e,?AB=0,,
BA vv ?
2222 m / s 4.0)3060(1.0)30(,m / s 2.030601.030 ?????? ?????????? nOAanOAv AA
)(m / s 2.0 ???? ?AB vv
? ? ? ? 3 22c o s,31s i n,s i nc o s,???? ???? ABABAABB aaaa
)(m / s40.14.022 1tg 22 ?????? ?? AB aa
[Example 2] Crank mechanism makes
the plate MN oscillate,Crank OA= l =
100mm and the rotate speed n=60 rpm,
AB=300mm,wheel O1 and O2do not slid
on the ground surface and the plate MN,
At the moment,OA? O1O2,Find the
velocity and acceleration of the plate MN,
22
解, OA作定轴转动 ; AB,轮 O1,轮 O2均作平面运动 ;杆 O1 O2,平台
MN均作平动 。 研究 AB,图示时作瞬时平动,因此 ?AB=0,
BA vv ?
2222 m / s 4.0)3060(1.0)30(,m / s 2.030601.030 ?????? ?????????? nOAanOAv AA
)(m / s 2.0 ???? ?AB vv
? ? ? ? 3 22c o s,31s i n,s i nc o s,???? ???? ABABAABB aaaa
)(m / s40.14.022 1tg 22 ?????? ?? AB aa
[例 2] 曲柄机构带动平台 MN作
往复运动,曲柄 OA= l = 100mm 转
速 n=60 rpm,AB=300mm,轮 O1,O2
与平台和地面均无相对滑动,图示
时 OA? O1O2 。
求 此时平台的速度与加速度。
23
24
25
Consider rod O1O2,
Take O1 as the pole,n
DODOOD aaaa 111 ???
?
2m / s 80.22
1
1
111
??????? OOODOODx aR
a
Raaaa ?
Consider plate MN,since there is no slipping between the plate
and wheel O1 and O2,we have
)(m / s26.1 ??? DMN vv
)(m / s80.2 2 ??? DxMN aa
Consider wheel O1,P1 is its instantaneous
velocity center
)(m / s26.14.02 1 ???? ?OD vv
R
v
R
a OO 11
11,?? ??
)(m / s 2.0
)(m / s40.1
21
21
2
????
????
?BOO
BOO
vvv
aaa
26
研究杆 O1O2,
)(m / s 2.0
)(m / s40.1
21
21
2
????
????
?BOO
BOO
vvv
aaa
以 O1为基点,n
DODOOD aaaa 111 ???
?
2m / s 80.22
1
1
111
??????? OOODOODx aR
a
Raaaa ?
研究平台,由于平台与轮 O1,O2接触处无相对滑动，所以
)(m / s26.1 ??? Dvv 平
)(m / s80.2 2 ??? Dxaa 平
研究轮 O1,P1为其速度瞬心
)(m / s26.14.02 1 ???? ?OD vv
R
v
R
a OO 11
11,?? ??
27
[Example 3] Determine the instantaneous velocity centers for all
rigid bodies having plane motions and draw the direction of each
angular velocity,(wheels roll without slipping.)
?wheel O has plane motion,
and P1 is the instantaneous
velocity center,?O
? Rod AB has plane motion,
and P2 is the instantaneous
velocity center,?AB
? wheel C has plane motion,
and P１ is the instantaneous
velocity center,?C
? BD has plane motion,
P2 is the instantaneous
velocity center,?BD
? AB has plane motion,
and P3 is the instantaneous
velocity center,?AB
28
[例 3] 画出图示作平面运动构件的速度瞬心的位置以及角速度
转向（各轮子均为纯滚动）,
? 轮 O作平面运动,
P１ 为其速度瞬心,?O
? 杆 AB作平面运动
P2为速度瞬心,?AB
? 轮 C作平面运动,
P1为速度瞬心,?C
? BD作平面运动,
P2为速度瞬心,?BD
? AB作平面运动,
P3为速度瞬心,?AB
29
30
Theoretical mechanics
2
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
rd
dt
vda
dt
rdvtrr ????
? rectangular coordinates method
)(
)(
)(
3
2
1
tfz
tfy
tfx
?
?
?
zv
yv
xv
z
y
x
?
?
?
?
?
?
za
ya
xa
z
y
x
??
??
??
?
?
?
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
rd
dt
vda
dt
rdvtrr ????
?直角坐标法
)(
)(
)(
3
2
1
tfz
tfy
tfx
?
?
?
zv
yv
xv
z
y
x
?
?
?
?
?
?
za
ya
xa
z
y
x
??
??
??
?
?
?
5
222
zyx vvvv ???
222
zyx aaaa ???
Directions are determined by their related cosines,
? natural coordinates method (when trajectory is known)
dt
dsvtfs ??,)( Along the tangential direction,
2
2
dt
sd
dt
dva ??
?
Along the tangential direction,
?
2v
a n ?
point to the center of curvature,
Resultant acceleration,
n
n a
anaaaa ?
? ??? ),( t g,
22
??a Const,(uniformly accelerated motion),
tavv ??? 0
200
2
1 tatvss
????
)(2 0202 ssavv ??? ?
),,0w h e n( 00 ssvvt ???
6
222
zyx vvvv ???
222
zyx aaaa ???
方向均由相应的方向余弦确定。
?自然法（轨迹已知时）
dt
dsvtfs ??,)( 方向沿切线方向,
2
2
dt
sd
dt
dva ??
?
方向沿切线方向,
?
2v
a n ? 方向指向曲率中心
全加速度,
n
n a
anaaaa ?
? ??? ),( t g,
22
??a 常数（ 匀变速运动 ）,
tavv ??? 0
200
2
1 tatvss
????
)(2 0202 ssavv ??? ?
),0( 00 ssvvt ??? 时
7
?Composite motions of a particle
rea vvv ??
rea aaa ??
（ when the embroiling motion is translation）
krea aaaa ???
（ when the embroiling motion is rotation ）
where ),s i n (2,2
rerekrek vvava ??? ???
?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
dtf ?????? ????
?＝ const,(uniformly
accelerated rotation)
t??? ?? 0
200
2
1 tt ???? ???
)(2 0202 ????? ???
),0w h e n ( 00 ???? ???t
2) Motion of a rigid body
8
?点的合成运动
rea vvv ??
rea aaa ??
（牵连运动为平动时）
krea aaaa ???
（牵连运动为转动时）
其中,),s i n (2,2
rerekrek vvava ??? ???
?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
dtf ?????? ????
?＝常量,
（匀变速转动）
t??? ?? 0
200
2
1 tt ???? ???
)(2 0202 ????? ???
),0( 00 ???? ??? 时t
2．刚体的运动
9
?=const.(uniform rotation),
30,
???? nt ?? Unit of ｎ is rpm
42 ?? ?? Ra
Velocity and acceleration of a point in a rigid body,(Relation between
angular and linear measurement)
?Rv ?
?? Ra ?
2?Ra n ?
Resultant
acceleration,
2),( ?
??natg
Vector
expressions
rv ?? ?
ra ?? ??
va n ?? ?
naaa ?? ?
vr ???? ??
Transmission ratio
of pulley system,
n
n
n
niZ
Z
R
R
n
ni
?
?
?
?
?
?
?
?
?
? 1
3
2
2
11
1
1
2
2
1
2
1
2
1
12,
???????????
?Plane motion ( composition of translation and rotation）
Pole method:(A is the pole)
??,???? ABvvvv BABAAB is the angular velocity,
10
?=常量 （匀速转动）,
30,
???? nt ?? ｎ 的单位,rpm
42 ?? ?? Ra
定轴转动刚体上一点的速度和加速度：（角量与线量的关系）
?Rv ?
?? Ra ?
2?Ra n ?
全加速度,
2),( ?
??natg
用矢量表示为
rv ?? ?
ra ?? ??
va n ?? ?
naaa ?? ?
vr ???? ??
轮系的传动比,
n
n
n
niZ
Z
R
R
n
ni
?
?
?
?
?
?
?
?
?
? 1
3
2
2
11
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）
??,???? ABvvvv BABAAB 为图形角速度
11
?? ?? ABa BA
2??? ABa nBA
??? are the angular velocity and
acceleration respectively
Velocity projection method,? ? ? ?
ABAABB vv ?
Instantaneous velocity center method,
direction determined by ?,
n
BABAAB aaaa ???
?
?e and ? point the
same direction
?e and ? have
opposite directions
,??? PBv B P is the
,PBv B ?
?Composition of rotations around two parallel axes,The composed
motion is rotation about an instantaneous axis,
Absolute angular velocity,
location of the instantaneous axis (across point P) PO PO
r
e
1
2?
?
?rea
??? ??
Instantaneous velocity center,
12
?? ?? ABa BA
2??? ABa nBA
??? 分别为图形的角速度，角加速度
投影法,? ? ? ?
ABAABB vv ?
瞬心法,,??? PBv
B
P点为图形的速度瞬心，,PBv B ? 与 ?一致
n
BABAAB aaaa ???
?
? 绕两平行轴转动的合成 合成结果为绕瞬轴的转动。
绝对角速度,
瞬轴（过 P点）的位置,
PO
PO
r
e
1
2?
?
?
rea ??? ??
?e 与 ?r同向时 ?e 与 ?r反向时
13
3,Steps for problem solving,skills and cautions
1) Analyze the characteristics of moving system and determine the
types of motions of particles or rigid bodies in given problems,
2) Find the known and unknown conditions,
3) Choose suitable methods to establish the relationships among
velocities/accelerations and find their solutions,
For more details about steps,skills and cautions for the available
methods,please refer to summaries in the lessons for problem solving,
14
三．解题步骤．技巧及注意的问题
1.分析题中运动系统的特点及系统中点或刚体的运动形式。
2.弄清已知量和待求量。
3.选择合适的方法建立运动学关系求解。
各种方法的步骤,技巧和使用中注意的问题详见每次习题课中的
总结。
15
4,Examples
[Example 1] A bicycle moves
along a straight road with the
law x=0.1 t２,where t is in
seconds and x is in meters,
R=35cm,l=18cm,the number
of the teeth of the sprocket
wheel,
When t=10,mn is vertical,
Find the absolute velocities of
the pedal m and n,(Assume
that the wheel rolls without
slipping.)
48,18 21 ?? ZZ
16
四．例题
[例 1] 自行车在水平直线道
路上按规律 x=0.1 t２ 行驶,式
中 t以秒计,x以米计, 已知
R=35cm,l=18cm,链轮齿数
。
t=10秒时,mn连线铅垂。
求 此时自行车踏板 m 和 n 的
绝对加速度（设车轮只滚不
滑）。
48,18 21 ?? ZZ
17
Solution,
consider moving point 1, m
moving point 2, n
moving system,bicycle body
static system,road surface
tdtdxvttx xe 20v )cm(10)m(1.0 22 ??????? 20??? dtdvaa xxe
2c m / s 20,c m / s 200 ),s(10 w h e n ?????? exex aavvt
)s/1(743520 ),s/1(74035200 211 ?????? RaRv xx ??
)s/1(14 34818 ),s/1(7154818 21212 ????? ????
18
解, 选取动点 1, m
动点 2, n
动系：自行车身
静系：地面
tdtdxvttx xe 20v )cm(10)m(1.0 22 ??????? 20??? dtdvaa xxe
2c m / s 20,c m / s 200,)s(10 ?????? exex aavvt 时
)s/1(743520 ),s/1(74035200 211 ?????? RaRv xx ??
)s/1(14 34818 ),s/1(7154818 21212 ????? ????
19
Because the embroiling motion is translation
rea aaa ???
?
9.73tg
)s/cm(2.84
)
49
4 0 5 0
()
7
27
20(
)()(
)(
7
27
14
3
18
)(
49
4 0 5 0
)
7
15
(18
)(20
1
2
22
22
2
2
2
2
?
?
?
?
???
????
?????
?????
??
?
?
?
?
?
?
?
rmem
n
rm
m
n
rmrmemm
rm
n
rm
em
aa
a
aaaa
la
la
a
?
9.78tg
)s/cm(86
)
49
4050
()
7
27
20(
)()(
)(
7
27
)(
49
4050
)(20
1
2
22
22
?
?
?
?
???
???
???
????
??
?
?
?
??
?
rnen
n
rn
n
n
rnrnenn
rmrn
n
rm
n
rn
en
aa
a
aaaa
aa
aa
a
（ ?n ） （ ?ｍ ）
Moving point m,Moving point n,
20
由于 牵连运动为平动
rea aaa ???
?
9.73tg
)s/cm(2.84
)
49
4 0 5 0
()
7
27
20(
)()(
)(
7
27
14
3
18
)(
49
4 0 5 0
)
7
15
(18
)(20
1
2
22
22
2
2
2
2
?
?
?
?
???
????
?????
?????
??
?
?
?
?
?
?
?
rmem
n
rm
m
n
rmrmemm
rm
n
rm
em
aa
a
aaaa
la
la
a
?
9.78tg
)s/cm(86
)
49
4050
()
7
27
20(
)()(
)(
7
27
)(
49
4050
)(20
1
2
22
22
?
?
?
?
???
???
???
????
??
?
?
?
??
?
rnen
n
rn
n
n
rnrnenn
rmrn
n
rm
n
rn
en
aa
a
aaaa
aa
aa
a
（ ?n ） （ ?ｍ ）
动点 m,动点 n,
21
Solution,OA rotates about a fixed axis; AB,wheel O1 and O2 have plane
motion; rod O1 O2 and plate MN have translation,Consider AB,it is in
instantaneous translation,i.e,?AB=0,,
BA vv ?
2222 m / s 4.0)3060(1.0)30(,m / s 2.030601.030 ?????? ?????????? nOAanOAv AA
)(m / s 2.0 ???? ?AB vv
? ? ? ? 3 22c o s,31s i n,s i nc o s,???? ???? ABABAABB aaaa
)(m / s40.14.022 1tg 22 ?????? ?? AB aa
[Example 2] Crank mechanism makes
the plate MN oscillate,Crank OA= l =
100mm and the rotate speed n=60 rpm,
AB=300mm,wheel O1 and O2do not slid
on the ground surface and the plate MN,
At the moment,OA? O1O2,Find the
velocity and acceleration of the plate MN,
22
解, OA作定轴转动 ; AB,轮 O1,轮 O2均作平面运动 ;杆 O1 O2,平台
MN均作平动 。 研究 AB,图示时作瞬时平动,因此 ?AB=0,
BA vv ?
2222 m / s 4.0)3060(1.0)30(,m / s 2.030601.030 ?????? ?????????? nOAanOAv AA
)(m / s 2.0 ???? ?AB vv
? ? ? ? 3 22c o s,31s i n,s i nc o s,???? ???? ABABAABB aaaa
)(m / s40.14.022 1tg 22 ?????? ?? AB aa
[例 2] 曲柄机构带动平台 MN作
往复运动,曲柄 OA= l = 100mm 转
速 n=60 rpm,AB=300mm,轮 O1,O2
与平台和地面均无相对滑动,图示
时 OA? O1O2 。
求 此时平台的速度与加速度。
23
24
25
Consider rod O1O2,
Take O1 as the pole,n
DODOOD aaaa 111 ???
?
2m / s 80.22
1
1
111
??????? OOODOODx aR
a
Raaaa ?
Consider plate MN,since there is no slipping between the plate
and wheel O1 and O2,we have
)(m / s26.1 ??? DMN vv
)(m / s80.2 2 ??? DxMN aa
Consider wheel O1,P1 is its instantaneous
velocity center
)(m / s26.14.02 1 ???? ?OD vv
R
v
R
a OO 11
11,?? ??
)(m / s 2.0
)(m / s40.1
21
21
2
????
????
?BOO
BOO
vvv
aaa
26
研究杆 O1O2,
)(m / s 2.0
)(m / s40.1
21
21
2
????
????
?BOO
BOO
vvv
aaa
以 O1为基点,n
DODOOD aaaa 111 ???
?
2m / s 80.22
1
1
111
??????? OOODOODx aR
a
Raaaa ?
研究平台,由于平台与轮 O1,O2接触处无相对滑动，所以
)(m / s26.1 ??? Dvv 平
)(m / s80.2 2 ??? Dxaa 平
研究轮 O1,P1为其速度瞬心
)(m / s26.14.02 1 ???? ?OD vv
R
v
R
a OO 11
11,?? ??
27
[Example 3] Determine the instantaneous velocity centers for all
rigid bodies having plane motions and draw the direction of each
angular velocity,(wheels roll without slipping.)
?wheel O has plane motion,
and P1 is the instantaneous
velocity center,?O
? Rod AB has plane motion,
and P2 is the instantaneous
velocity center,?AB
? wheel C has plane motion,
and P１ is the instantaneous
velocity center,?C
? BD has plane motion,
P2 is the instantaneous
velocity center,?BD
? AB has plane motion,
and P3 is the instantaneous
velocity center,?AB
28
[例 3] 画出图示作平面运动构件的速度瞬心的位置以及角速度
转向（各轮子均为纯滚动）,
? 轮 O作平面运动,
P１ 为其速度瞬心,?O
? 杆 AB作平面运动
P2为速度瞬心,?AB
? 轮 C作平面运动,
P1为速度瞬心,?C
? BD作平面运动,
P2为速度瞬心,?BD
? AB作平面运动,
P3为速度瞬心,?AB
29
30
