机械设计双语版 第3章
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第3页
3.2 Velocity Analysis by the Method of instant Centres
3.2.1 Definition of the Instant Centre Shown in Fig. are two bodies 1 and 2 having relative planar motion. At any instant there exists a pair of coincident points, the absolute velocities of which are the same.
Summary and kernel Problems and Exercises
第2页
பைடு நூலகம்
3.1 Tasks and Methods of Kinematic Analysis
Task: to find positions, velocities and accelerations or angular positions, angular velocities and angular accelerations.
instant centerP12 or P21
第4页
Suppose that the positions of points A and B, the directions of VA2A1 and VB2B1 are known. VA2A1AP. VB2B1BP.
A 2 1
V A2A1 B V B2B1 P12
Chapter 3 Kinematic Analysis of Mechanisms
This chapter mainly introduces the tasks and methods of kinematic analysis. The method of instant centres, graphic and analytical methods are discussed.
第8页
2 1
(3) Higher Pair: Pure-rolling Pair
Instant center—the point of contact
(4)Higher Pair :rolling & sliding Pair Instant center—lies somewhere on the common normal through the point of contact
v3 vP13 1 lP13P14
第13页
Example3 In the cam mechanism as shown in the following fig. the cam2 rotates anti-clockwise at a constant speed ω，Determine the velocity of the follower2 for the position shown.
2) Determine the velocity and angular velocity
(1) Point C： B — Base point，C — Moving point
2 2 E 2
C
vC
v B v CB
⊥AB lABw1
B 1 1 1
3 3 3 D
Direction：⊥CD Magnitude：？
第6页
3.2.2 Number of Instant Centers of a Mechanism
Each pair of links i and j has an instant centre and Pij is identical to Pji. Thus the number N of instant centres of a mechanism with k links is
第15页
3.3 Kinematic Analysis by Graphic Methods
For two points A and B on the same body:
A —— Base point ，B —— moving point
v B v A v BA
n a B a A a BA a A a BA t a BA
2
3
3 B P13
1
第10页
3.2.5 Applications of Instant Centers
Example1: The revolute four-bar linkage. (1) locate all instant centers for the mechanism
(2) find the ratio ω2/ω4
In which： vBA=lAB， vBA Direction：⊥AB，along with ω
anBA=lABw2， anBA Direction：B→A。 atBA=lABa， atBA Direction：⊥AB, along with a
第16页
1、The velocity and acceleration of the points on the same body Given：The length of all links in mechanism, position,
第11页
(2) find the ratio ω2/ω4
P24 : the absolute velocities of which are the same, in both link2 and link 4.
vP24 l P12 P24 2 l P14 P24 4
i24 2 / 4 l P14 P24 / l P12 P24
Position: to determine whether all links will interfere with each other, to find locus Velocity: to calculate the stored kinetic energy or power. Acceleration：To calculate the dynamic forces
V P2 V P1 2 1 P12
At this instant, there is no relative velocity between this pair of coincident points. This pair of coincident points with the same velocities is defined as the instantaneous centre of relative rotation, denoted as P12 or P21.
第12页
Example2:For the following Slider-Crank Mechanism， Given: lAB, lBC,φ，ω1 ①locate all instant centers for the mechanism ②the velocity of follower link3 V3 。 Solution： (1) Find Instant centerP13 (2) Determine V3
P23
(2) Determine V3
v3 l P12 P23 2
第14页
3.2.6 Advantages and Disadvantages of the Method of Instant Centers
Excellent tool in simple mechanisms. Difficult to find Instant Center in a complex mechanism Not be used in acceleration analysis.
Emphases★：
The method of instant centres, graphic method
Difficulties▲: Graphic method
第1页
3.1 Tasks and Methods of Kinematic Analysis
3.2 Velocity Analysis by the Method of instant Centers 3.3 Kinematic Analysis by Graphic Methods 3.4 Kinematic Analysis by Analytical Methods
instant centerP12 One of bodies is static absolute instant center Both of them are moving relative instant center
第5页
Attention: instant center A pair of coincident points, the absolute velocities of which are the same, in both magnitude and direction. Relative velocity is zero.
Methods: graphical method (图解法): 直观，精度不高 Instant(瞬时的) centre method for velocity Vector(矢量) equation method analytical method (解析法) : 精度高，计算烦琐
experimental method (实验法) : 需专门的仪器
(1) Revolute Pair
(2) Sliding Pair Instant center: lies at infinity in either direction perpendicular to the guide-way.
P12
Attention：The common normal may pass through any point !!
k (k 1) k (k 1) N C 2! 2
2 k
k—number of links. The frame is included in the number k.
第7页
3.2.3 The location of the Instant Center of Two Links Connected by a Kinematic Pair
1 , 1
Cal：
2 , 2 ,VC ,VE , aC , a E , 3 , 3
C
Sol： 1) Draw the kinematic diagram of the mechanism
2
Select a length suitable scale ul .
1 A 4
第17页
2 2 B 1 1 E 3 3 3 D (a)
Solution： (1) locate all instant centers 1 1 K N(N 1) 4 (4 1) 6 2 2 Find：P12、P23、P34和P14； Theorem of Three Centres： P12 ， P14 P24 P23 ， P34 P12 ， P23 P14 ， P34 P13
第9页
3.2.4 Theorem of Three Centres (Aronhold-Kennedy Theorem)
Vc 2 Any three links have three instant centers. They must lie on a straight line C
Vc 3
2
A P12
Solution： (1) Find Instant center P23
P23must lie on the straight line connecting P12 and P13. P23 must lie along the common normal n-n through the point of contact C
3.2 Velocity Analysis by the Method of instant Centres
3.2.1 Definition of the Instant Centre Shown in Fig. are two bodies 1 and 2 having relative planar motion. At any instant there exists a pair of coincident points, the absolute velocities of which are the same.
Summary and kernel Problems and Exercises
第2页
பைடு நூலகம்
3.1 Tasks and Methods of Kinematic Analysis
Task: to find positions, velocities and accelerations or angular positions, angular velocities and angular accelerations.
instant centerP12 or P21
第4页
Suppose that the positions of points A and B, the directions of VA2A1 and VB2B1 are known. VA2A1AP. VB2B1BP.
A 2 1
V A2A1 B V B2B1 P12
Chapter 3 Kinematic Analysis of Mechanisms
This chapter mainly introduces the tasks and methods of kinematic analysis. The method of instant centres, graphic and analytical methods are discussed.
第8页
2 1
(3) Higher Pair: Pure-rolling Pair
Instant center—the point of contact
(4)Higher Pair :rolling & sliding Pair Instant center—lies somewhere on the common normal through the point of contact
v3 vP13 1 lP13P14
第13页
Example3 In the cam mechanism as shown in the following fig. the cam2 rotates anti-clockwise at a constant speed ω，Determine the velocity of the follower2 for the position shown.
2) Determine the velocity and angular velocity
(1) Point C： B — Base point，C — Moving point
2 2 E 2
C
vC
v B v CB
⊥AB lABw1
B 1 1 1
3 3 3 D
Direction：⊥CD Magnitude：？
第6页
3.2.2 Number of Instant Centers of a Mechanism
Each pair of links i and j has an instant centre and Pij is identical to Pji. Thus the number N of instant centres of a mechanism with k links is
第15页
3.3 Kinematic Analysis by Graphic Methods
For two points A and B on the same body:
A —— Base point ，B —— moving point
v B v A v BA
n a B a A a BA a A a BA t a BA
2
3
3 B P13
1
第10页
3.2.5 Applications of Instant Centers
Example1: The revolute four-bar linkage. (1) locate all instant centers for the mechanism
(2) find the ratio ω2/ω4
In which： vBA=lAB， vBA Direction：⊥AB，along with ω
anBA=lABw2， anBA Direction：B→A。 atBA=lABa， atBA Direction：⊥AB, along with a
第16页
1、The velocity and acceleration of the points on the same body Given：The length of all links in mechanism, position,
第11页
(2) find the ratio ω2/ω4
P24 : the absolute velocities of which are the same, in both link2 and link 4.
vP24 l P12 P24 2 l P14 P24 4
i24 2 / 4 l P14 P24 / l P12 P24
Position: to determine whether all links will interfere with each other, to find locus Velocity: to calculate the stored kinetic energy or power. Acceleration：To calculate the dynamic forces
V P2 V P1 2 1 P12
At this instant, there is no relative velocity between this pair of coincident points. This pair of coincident points with the same velocities is defined as the instantaneous centre of relative rotation, denoted as P12 or P21.
第12页
Example2:For the following Slider-Crank Mechanism， Given: lAB, lBC,φ，ω1 ①locate all instant centers for the mechanism ②the velocity of follower link3 V3 。 Solution： (1) Find Instant centerP13 (2) Determine V3
P23
(2) Determine V3
v3 l P12 P23 2
第14页
3.2.6 Advantages and Disadvantages of the Method of Instant Centers
Excellent tool in simple mechanisms. Difficult to find Instant Center in a complex mechanism Not be used in acceleration analysis.
Emphases★：
The method of instant centres, graphic method
Difficulties▲: Graphic method
第1页
3.1 Tasks and Methods of Kinematic Analysis
3.2 Velocity Analysis by the Method of instant Centers 3.3 Kinematic Analysis by Graphic Methods 3.4 Kinematic Analysis by Analytical Methods
instant centerP12 One of bodies is static absolute instant center Both of them are moving relative instant center
第5页
Attention: instant center A pair of coincident points, the absolute velocities of which are the same, in both magnitude and direction. Relative velocity is zero.
Methods: graphical method (图解法): 直观，精度不高 Instant(瞬时的) centre method for velocity Vector(矢量) equation method analytical method (解析法) : 精度高，计算烦琐
experimental method (实验法) : 需专门的仪器
(1) Revolute Pair
(2) Sliding Pair Instant center: lies at infinity in either direction perpendicular to the guide-way.
P12
Attention：The common normal may pass through any point !!
k (k 1) k (k 1) N C 2! 2
2 k
k—number of links. The frame is included in the number k.
第7页
3.2.3 The location of the Instant Center of Two Links Connected by a Kinematic Pair
1 , 1
Cal：
2 , 2 ,VC ,VE , aC , a E , 3 , 3
C
Sol： 1) Draw the kinematic diagram of the mechanism
2
Select a length suitable scale ul .
1 A 4
第17页
2 2 B 1 1 E 3 3 3 D (a)
Solution： (1) locate all instant centers 1 1 K N(N 1) 4 (4 1) 6 2 2 Find：P12、P23、P34和P14； Theorem of Three Centres： P12 ， P14 P24 P23 ， P34 P12 ， P23 P14 ， P34 P13
第9页
3.2.4 Theorem of Three Centres (Aronhold-Kennedy Theorem)
Vc 2 Any three links have three instant centers. They must lie on a straight line C
Vc 3
2
A P12
Solution： (1) Find Instant center P23
P23must lie on the straight line connecting P12 and P13. P23 must lie along the common normal n-n through the point of contact C