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MECN 4110: Mechanisms Design
Mechanisms Design
MECN 4110
Professor: Dr. Omar E. Meza Castillo
[email protected]
http://facultad.bayamon.inter.edu/omeza
Department of Mechanical Engineering
4110: Mechanisms
MECN
Design- Bayamón
Interamericana
Universidad
MSP21
One thing you learn in science is that there is
no perfect answer, no perfect measure.
A. O. Beckman
Topic 5: Velocity Analysis
Fourbar Linkage
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Chapters Objectives
Up on completion of this chapter, the student will
be able to
Determine velocities of links and points on the fourbar
mechanism by using graphical and analytical
approaches.
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Velocity
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Velocity
Velocity of point P
Multiplying by j rotates the vector by 90°
Velocity is perpendicular to radius of rotation & tangent to path of
motion
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Velocity
Vector r can be written as:
Imaginary
re j rcos j sin
r cos
Multiplying by j gives:
jre j r sin j cos
r
r sin
Multiplying by j rotates a vector 90°
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r cos
r sin
Real
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Velocity
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Velocity
Velocity Difference / Relative Velocity
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Velocity
Methods for Velocity Analysis
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Graphical Approach
Vectors
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Graphical Velocity Analysis
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Graphical Velocity Analysis
Given ω2, find ω3, ω4, VA, VB and VC
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Graphical Velocity Analysis
Given ω2, find ω3, ω4, VA, VB and VC
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Design- Bayamón
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Graphical Velocity Analysis
Given ω2, find ω3, ω4, VA, VB and VC
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Graphical Velocity Analysis
Given ω2, find ω3, ω4, VA, VB and VC
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Analytical Approach
Vector Loop Method
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Analytical Velocity Analysis of a Fourbar Linkage
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Analytical Velocity Analysis of a Fourbar Linkage
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Analytical Velocity Analysis of a Fourbar Linkage
Write the vector loop equation
R2 R3 R4 R1 0
ae j 2 be j3 ce j 4 de j1 0
Position analysis, where: θ1=0o
aCos 2 jaSin 2
bCos 3 jbSin 3
cCos 4 jcSin 4 dCos1 jdSin1 0
aCos 2 jaSin 2
Cos0 1
o
with
bCos 3 jbSin 3
Sin 0 0
o
cCos 4 jcSin 4 d 0
Position Equation
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Analytical Velocity Analysis of a Fourbar Linkage
After solving the position analysis, take the
derivative to get the velocity equation
d 3
d 2
d 2
aSin 2
jaCos 2
bSin 3
dt
dt
dt
d 3
d 4
d 4
jbCos 3
cSin 4
jcCos 4
0
dt
dt
dt
a2 Sin 2 ja2Cos 2 b3 Sin 3
jb3Cos 3 c4 Sin 4 jc4Cos 4 0
Velocity Equation
VA a2 Sin 2 ja2Cos 2 j2 ae j 2
j 3
V
b
Sin
jb
Cos
j
be
V A V BA V B 0
3
3
3
3
3
BA
j 4
V
c
Sin
jc
Cos
j
ce
B
4
4
4
4
4
Where
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Analytical Velocity Analysis of a Fourbar Linkage
Separate Velocity Equation into Real and
Imaginary Parts
a2 Sin 2 b3 Sin3 c4 Sin 4 0
Real Part
a2Cos 2 b3Cos3 c4Cos 4 0
Imaginary Part
Eliminate ω4 and Solve the equations for ω3
b3 Sin3 c4 Sin 4 a2 Sin 2
* Cos 4
b3Cos3 c4Cos 4 a2Cos 2
* Sin4
b3 Sin3Cos 4 c4 Sin 4Cos 4 a2 Sin 2Cos 4
b3Cos3 Sin 4 c4Cos 4 Sin 4 a2Cos 2 Sin 4
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Analytical Velocity Analysis of a Fourbar Linkage
sin(A-B)=sin A cos B - cos A sin B
Summation of two last equations:
b3 Sin 3Cos 4 b3Cos 3 Sin 4
a2 Sin 2Cos 4 a2Cos 2 Sin 4
b3 Sin 3Cos 4 Sin 4Cos 3
a2 Sin 4Cos 2 Sin 2Cos 4
Using Trigonometric Identity:
b3 Sin 3 4 a2 Sin 4 2
a2 Sin ( 4 2 )
3
b Sin ( 3 4 )
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Factorize
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Analytical Velocity Analysis of a Fourbar Linkage
Eliminate ω3 and Solve the equations for ω4
b3 Sin3 c4 Sin 4 a2 Sin 2
* Cos 3
b3Cos3 c4Cos 4 a2Cos 2
* Sin 3
b3 Sin3Cos3 c4 Sin 4Cos3 a2 Sin 2Cos3
b3Cos3 Sin3 c4Cos 4 Sin3 a2Cos 2 Sin3
Summation of two last equations:
c4 Sin 4Cos 3 c4Cos 4 Sin 3
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a2 Sin 2Cos 3 a2Cos 2 Sin 3
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Analytical Velocity Analysis of a Fourbar Linkage
sin(A-B)=sin A cos B - cos A sin B
c4 Sin 4Cos 3 Cos 4 Sin 3
a2 Sin 2Cos 3 Cos 2 Sin 3
Using Trigonometric Identity:
c4 Sin 4 3 a2 Sin 2 3
a2 Sin ( 2 3 )
4
c Sin ( 4 3 )
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Factorize
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Numerical Problem 1
Calculate the angular velocity of link 3 and link 4,
(ω3 & ω4) when θ2=57.3o =1rad and the input
angular velocity, ω2=-1 rad/s
ω3
We have all the requirements except θ3 and θ4
How can we obtain θ3 and θ4?
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Using Position Analysis (Chapter 4) - Mathcad
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Using Working Model (Chapter 4)
Θ3=24.36o
Θ2=57.3o
Θ4=88.01o
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Using Velocity Analysis (Chapter 5) - Mathcad
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a2 Sin ( 4 2 )
b Sin ( 3 4 )
4
a2 Sin ( 2 3 )
c Sin ( 4 3 )
Rad/s
Rad/s
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Using Working Model (Chapter 5)
Θ2=57.3o
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Velocity Analysis of a Inverted Slider-Crank Mechanism
Link 2 (the crank) is the input and link 4 is the
output. Given r1, θ1=0o, r2, θ2, r4, θ4, ω2. Find r4 and
ω4. Remember r4 is a variable so r4≠ 0 in general.
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Velocity Analysis of a Inverted Slider-Crank Mechanism
V4
R 4
V2
r4 , R4
r2 , R2
ω2
ω4
2
r1 , R1 ,1 0o
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Velocity Analysis of a Inverted Slider-Crank Mechanism
Write the vector loop equation
R2 R4 R1 0
r 2 e j 2 r4 e j 4 r1e j1 0
Position analysis, where: θ1=0o
r2Cos 2 jr2 Sin 2 r4Cos 4 jr4 Sin 4
r1Cos1 jr1Sin1 0
with
Cos0 o 1
r2Cos 2 jr2 Sin 2
Sin 0 o 0
r4Cos 4 jr4 Sin 4 r1 0
Position Equation
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Velocity Analysis of a Inverted Slider-Crank Mechanism
After solving the position analysis, take the
derivative to get the velocity equation
d 2
d 2
d 4
r2 Sin 2
jr2Cos 2
r4 Sin 4
dt
dt
dt
d 4
r4Cos 4 jr4Cos 4
jr4 Sin 4 0
dt
r22 Sin 2 jr22Cos 2 r44 Sin 4
r4Cos 4 jr44Cos 4 jr4 Sin 4 0
Velocity Equation
R 4 V4 V2 0
Where
V4
V2
R 4 r4Cos 4 jr4 Sin 4 jr4e j 4
r44 Sin 4 jr44Cos 4 j4 r4 e j 4
r22 Sin 2 jr22Cos 2 j2 r2 e j 2
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Velocity Analysis of a Inverted Slider-Crank Mechanism
Separate Velocity Equation into Real and
Imaginary Parts
r22 Sin2 r44 Sin4 r4Cos4 0
r22Cos2 r44Cos4 r4 Sin4 0
Real Part
Imaginary Part
Eliminate ω4 and Solve the equations for r4
r22 Sin2 r44 Sin4 r4Cos4 0 * Cos 4
r22Cos2 r44Cos4 r4 Sin4 0
* Sin4
r22 Sin 2Cos 4 r44 Sin 4Cos 4 r4Cos 2 4 0
r22Cos 2 Sin 4 r44 Sin 4Cos 4 r4 Sin 2 4 0
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Velocity Analysis of a Inverted Slider-Crank Mechanism
sin(A-B)=sin A cos B - cos A sin B
Summation of two last equations:
r22 Sin 2Cos 4 r22Cos 2 Sin 4
r4Cos 2 4 r4 Sin 2 4 0
r22 Sin 4Cos 2 Cos 4 Sin 2
r4 Cos 4 Sin 4 0
2
2
Using Trigonometric Identity:
r22 Sin4 2 r4 0
r4 r22 Sin4 2
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Factorize
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Velocity Analysis of a Inverted Slider-Crank Mechanism
Eliminate for r4 and Solve the equations for ω4
r22 Sin2 r44 Sin4 r4Cos4 0 * Sin4
r22Cos2 r44Cos4 r4 Sin4 0 * Cos4
r22 Sin 2 Sin 4 r44 Sin 2 4 r4Cos 4 Sin 4 0
r22Cos 2Cos 4 r44Cos 2 4 r4Cos 4 Sin 4 0
Summation of two last equations:
r22 Sin 2 Sin 4 r22Cos 2Cos 4
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r44 Sin 2 4 r44Cos 2 4 0
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Velocity Analysis of a Inverted Slider-Crank Mechanism
cos(A-B)=sin A sin B + cos A cos B
r44 Sin 2 4 Cos 2 4
r22 Sin 4 Sin 2 Cos 4Cos 2
Using Trigonometric Identity:
r44 r22Cos4 2
r22
4
Cos( 4 2 )
r4
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Factorize
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Numerical Problem 2
Calculate the angular velocity and linear velocity
of link 4, (ω4 & r4) when r1=0.2m, θ1=0o, r2=0.1m,
θ2=70o, and the input angular velocity,
ω2=25rad/s
We have all the requirements except r4 and θ4
How can we obtain r4 and θ4?
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Using Position Analysis (Chapter 4) - Mathcad
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Using Velocity Analysis (Chapter 5) - Mathcad
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r4 r22 Sin4 2
m/s
r22
Cos( 4 2 )
r4
Rad/s
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Velocity Analysis of Offset Slider-Crank
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Velocity Analysis of Offset Slider-Crank
Remember d is a variable so d≠ 0 in general.
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Velocity Analysis of Offset Slider-Crank
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Velocity Analysis of Offset Slider-Crank
Write the vector loop equation
R2 R3 R4 R1 0
ae j 2 be j 3 ce j 4 de j1 0
Position analysis, where: θ1=0o,
and θ4=90o
aCos 2 jaSin 2 bCos 3 jbSin 3
cCos 4 jcSin 4 dCos1 jdSin1 0
aCos 2 jaSin 2 bCos 3
Cos0o 1, Sin 0o 0
jbSin 3 jc d 0
with Cos90o 0, Sin90o 1
Position Equation
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Velocity Analysis of Offset Slider-Crank
After solving the position analysis, take the
derivative to get the velocity equation
d 3
d 2
d 2
aSin 2
jaCos 2
bSin 3
dt
dt
dt
d 3 d d
jbCos 3
0
dt
dt
a2 Sin 2 ja2Cos 2 b3 Sin 3
jb3Cos 3 d 0
Velocity Equation
VA a2 Sin 2 ja2Cos 2 j2 ae j 2
j 3
V
b
Sin
jb
Cos
j
be
V A V BA V B 0
BA
3
3
3
3
3
V
d
B
Where
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Velocity Analysis of Offset Slider-Crank
Separate Velocity Equation into Real and
Imaginary Parts
a2 Sin 2 b3 Sin3 d 0
a2Cos 2 b3Cos 3 0
From imaginary part:
b3Cos3 a2Cos 2
aCos 2
3
2
bCos 3
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Real Part
Imaginary Part
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Velocity Analysis of Offset Slider-Crank
From real part:
a2 Sin 2 b3 Sin3 d
d a2 Sin 2 b3 Sin3
d dot is horizontal speed of slider relative to ground
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Velocity Analysis of a Inverted Crank Slider
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Velocity Analysis of a Inverted Crank Slider
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Homework 5 http://facultad. bayamon.inter.edu/omeza/
Omar E. Meza Castillo Ph.D.
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¿Preguntas?
Comentarios
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GRACIAS
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