Upload
jensen
View
41
Download
0
Embed Size (px)
DESCRIPTION
Dynamic Simulation : Constraint Equations. Objective The objective of this module is to develop the equations for ground, revolute, prismatic, and motion constraints for a planar mechanism. These equations will be developed for a piston-crank assembly in a Boxer style engine. - PowerPoint PPT Presentation
Citation preview
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Dynamic Simulation:Constraint Equations
Objective
The objective of this module is to develop the equations for ground, revolute, prismatic, and motion constraints for a planar mechanism.
These equations will be developed for a piston-crank assembly in a Boxer style engine.
These constraint equations will be used in the next Module (Module 4) to show how position, velocity, and accelerations are computed.
Although the equations developed for this module are for a planar (2D) mechanism, the methods can be generalized to 3D mechanisms.
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Boxer Style Engine
Boxer style engines have a horizontally opposed piston configuration.
This has several advantages Lower center of gravity Lower vertical height Lighter weight Less vibration
Boxer style engines are used by Porsche and Subaru.
Because of their low vertical profile they are often called pancake engines.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 2
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Cross Section View
Cylinder Liner Piston Connecting
Rod
Crank Shaft
Piston Pin
Crank Bearing
Counterweight
Piston Pin
Bearing
Bottom Bearing Cap
Rod Bolt
This module will use the piston-crank portion of this engine to demonstrate how kinematic and motion constraints are developed.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 3
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Planar System
The boxer engine rotating assembly contains four piston assemblies.
Constraint equations will be written for one piston assembly to demonstrate the process.
This single assembly can be represented as a planar mechanism.
A Dynamic Simulation of the complete system will be presented in another module.
Cylinder 1Cylinder 2
Cylinder 3
Cylinder 4
The planar equations will be developed for Cylinder 3.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 4
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Global Coordinate System
The constraint equations will be referenced to the stationary coordinate system shown in the figure.
This reference coordinate system is called the global coordinate system.
Capital letters are used to indicate that a coordinate or vector refers to this coordinate system.
Lower case letters will be used to indicate a coordinate or vector is referred to a body fixed coordinate system associated with a part.
Z
X
Y
X
Cylinder 3
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 5
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Part ID’s
ACylinder Liner
BPiston C
Connecting Rod D
Crank Shaft
ECrank Bearing(Not Visible)
The process of developing the constraint equations is facilitated by identifying each component by a letter.
The five components shown with letters make up the basic system for which the constraint equations will be written.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 6
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Mobility
Gruebler’s equation can be used to establish the mobility of the planar mechanism.
Bodies (B) = 5 Grounded bodies (G) = 2 Revolute joints (R) = 3 Prismatic joints (P) = 1
ACylinder Liner
BPiston C
Connecting Rod D
Crank Shaft
ECrank Bearing(Not Visible)
1)2(3)1(2)3(2)5(3 )(3)(2)(2)(3
GPRBDOF
Mobility
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 7
A mobility of one will require one motion constraint.
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
List of DOF’s
The DOF’s are associated with a set of generalized coordinates.
Each body has 3 DOF and 3 generalized coordinates.
The generalized coordinates for the planar mechanism are listed on the right.
Fifteen constraint equations must be developed that will enable each of the fifteen generalized coordinates to be determined.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 8
E
Ecg
Ecg
D
Dcg
Dcg
C
Ccg
Ccg
B
Bcg
Bcg
A
Acg
Acg
Y
X
Y
X
Y
X
Y
X
Y
X
List of Generalized Coordinates
Format
AcgX
Capital letter indicates that variable is associated with the global coordinate system.
Center of Gravity
Body
X-coordinate of the cg of Body A
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Ground Joints
The cylinder liners are pressed into the engine block and do not move.
The pistons move relative to the cylinder liners and the combination make a prismatic joint.
The cylinder liners must be mathematically grounded or fixed in space.
Cylinder 1Cylinder 2
Cylinder 3
Cylinder 4
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 9
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Cylinder Liner Ground Equations
The location of the center of gravity and the orientation of the principal axes of inertia are shown in the figures.
The ground constraint equations that fix the position of the c.g. and orientation of the principal axes can be written as
0
0
0 8.156
Acg
Acg
Acg
Y
mmX
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 10
x
y
y
z
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Crank Bearing Ground Joint
The crank bearing is fixed in the engine block and does not move.
The crank shaft rotation relative to the crank bearing can be represented by a revolute joint.
All of the parts in the planar system must lie in the global X-Y plane.
Therefore, a “virtual” crank bearing will be placed at the origin of the global coordinate system so that the planar equations can be developed.
0
0
0
Ecg
Ecg
Ecg
Y
X
Crank Bearing Constraint Equations
X
Y
Z
Virtual Crank Bearing Located at the Origin
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 11
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Summary of Ground Joint Equations
Cylinder Liner Virtual Crank Bearing
0
0
0
Ecg
Ecg
Ecg
Y
X
0
0
0 8.156
Acg
Acg
Acg
Y
mmX
Each of these equations fix one DOF for the respective part in space.
None of the equations are a function of time.
None of the equations involve more than one part.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 12
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
2D Coordinate Transformation Matrix
In subsequent slides it will be necessary to transform the components of a vector from a body fixed coordinate system to the global coordinate system.
This transformation is accomplished with the transformation matrix [T()].
From the figure,
θX
Yx
yθsin x
θsin y
θ cosx
θ
θ cosy
cossinsincos
yxYyxX
Matrix Form
yx
yYX
cossinsincos
cossinsincosy
T
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 13
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Revolute Joint
There are three revolute joints in the piston-crank assembly Between the piston and connecting rod Between the connecting rod and
crankshaft Between the crankshaft and crank
bearing The constraint equations for a
revolute joint will be developed using the two bodies shown in the figure.
Body A and B have the same translational motion at the joint but can have relative rotation.
X
Y
Body A
Body B
xA
yA
A
xB
yB
B
Two bodies connected at a common point that allows relative rotational
motion.
Joint
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 14
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Revolute Joint
X
Y
Body AxA
yA
A
Joint 1
AcgR
1Ar1
cgAR
I
J
ij
The position of Joint 1 on Body A relative to the global coordinate system is given by the equation
The components of are written with respect to the global coordinate system base vectors and the components of are written with respect to the body fixed coordinate system.
AAcg
A rRR 11
jyixr AAA ˆˆ111
JYIXR Acg
Acg
Acg
ˆˆ
AcgR
Ar1
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 15
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Revolute Joint
The components of the body fixed position vector must be transformed to the global coordinate system before the components of the two vectors can be added.
This is accomplished using the transformation matrix introduced earlier.
AAcg
A rRR 11
Position Vector Equation
Component Form
A
A
AA
AA
Acg
Acg
A
A
xx
YX
YX
1
1
1
1
cossinsincos
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 16
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Revolute Joint
The coordinates of Joint 1 on Body A are
Similarly, the coordinates of Joint 1 on Body B are
A
A
AA
AA
Acg
Acg
A
A
yx
YX
YX
1
1
1
1
cossinsincos
B
B
BB
BB
Bcg
Bcg
B
B
yx
YX
YX
1
1
1
1
cossinsincos
In a Revolute Joint the coordinates of the joint must be same for each body.
Thus,
B
B
A
A
YX
YX
1
1
1
1
or
00
cossinsincos
cossinsincos
1
1
1
1
B
B
BB
BB
Bcg
Bcg
A
A
AA
AA
Acg
Acg
yx
YX
yx
YX
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 17
General Form of the Constraint Equations for a Planar Revolute Joint
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Revolute Joint
The general form of the constraint equations for a planar revolute joint is
The specific equations for the three revolute joints in the piston-crank mechanism will now be developed
00
cossinsincos
cossinsincos
1
1
1
1
B
B
BB
BB
Bcg
Bcg
A
A
AA
AA
Acg
Acg
yx
YX
yx
YX
2nd Revolute Joint
Joint 1
Joint 2
Joint 3
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 18
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
1st Revolute Joint
The location of the joint relative to the c.g. is needed to define the parameters &
For the piston,
x
y
Joint 1C.G.
28 mmBx1By1
0
28
1
1
B
B
y
mmx
Piston Body B
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 19
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
1st Revolute Joint
102.6
Joint 1
Connecting Rod Body C
The location of the joint relative to the c.g. is needed to define the parameters &
From the picture,
Cx1Cy1
0
6.102
1
1
C
C
y
mmx
x
y
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 20
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
1st Revolute Joint
Using the geometry from the piston and connecting rod, the revolute joint constraint equation becomes
0cos6.102sin28
0sin6.102cos28
CC
cgBB
cg
CCcg
BBcg
YY
XX
Joint 1
00
cossinsincos
cossinsincos
1
1
1
1
C
C
CC
CC
Ccg
Ccg
B
B
BB
BB
Bcg
Bcg
yx
YX
yx
YX
0
6.102
1
1
C
C
y
mmx0
28
1
1
B
B
y
mmx
1st Revolute Joint Constraint Equations
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 21
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
2nd Revolute Joint
41.3 mm
00
cossinsincos
cossinsincos
2
2
2
2
D
D
DD
DD
Dcg
Dcg
C
C
CC
CC
Ccg
Ccg
yx
YX
yx
YX
The location of the joint relative to the c.g. is needed to define the parameters &
From the picture,
General Form of Constraint Equation
Body C
Joint 2
Cx2Cy2
0
3.41
2
2
C
C
y
x
x
y
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 22
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
2nd Revolute Joint
43 mm
00
cossinsincos
cossinsincos
2
2
2
2
D
D
DD
DD
Dcg
Dcg
C
C
CC
CC
Ccg
Ccg
yx
YX
yx
YX
The location of the joint relative to the c.g. is needed to define the parameters &
From the picture,
General Form of Constraint Equation
Dx2Dy2
0
43
2
2
D
D
y
x
Joint 2
x
y
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 23
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
2nd Revolute Joint
Using the geometry from the connecting rod and crank shaft, the revolute joint constraint equation becomes
00
cossinsincos
cossinsincos
2
2
2
2
D
D
DD
DD
Dcg
Dcg
C
C
CC
CC
Ccg
Ccg
yx
YX
yx
YX
0
3.41
2
2
C
C
y
x0
43
2
2
D
D
y
x
0cos43sin3.41
0sin43cos3.41
DD
cgCC
cg
DDcg
CCcg
YY
XX
Joint 2
2nd Revolute Joint Constraint Equations
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 24
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
3rd Revolute Joint
00
cossinsincos
cossinsincos
2
2
2
2
E
E
EE
EE
Ecg
Ecg
D
D
DD
DD
Dcg
Dcg
yx
YX
yx
YX
The c.g.’s of both the crank and crank shaft lie at the origin of the global coordinate system.
Therefore, the body fixed coordinates of the joint relative to the c.g. are zero.
General Form of Constraint Equation
00
Ecg
Ecg
Dcg
Dcg
YX
YX
Joint 3
3rd Revolute Joint Constraint Equations
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 25
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Summary of Revolute Joint Equations
2nd Revolute Joint
Joint 1Joint 2
Joint 3
00
Ecg
Ecg
Dcg
Dcg
YX
YX
0cos43sin3.41
0sin43cos3.41
DD
cgCC
cg
DDcg
CCcg
YY
XX
0cos6.102sin28
0sin6.102cos28
CC
cgBB
cg
CCcg
BBcg
YY
XX
Body B Body C Body C
Body D
Body D Body E
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 26
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint
In the planar system the cylindrical joint between the cylinder liner and the piston acts like a prismatic joint.
A prismatic joint allows two bodies to translate relative to each other along a common axis.
The two bodies cannot rotate independent of each other.
The equations for a planar prismatic joint are based on the geometry shown in the figure.
X
Y Common Axis
Body A
Body B
xA
yA
A
xA
yA
B
Two bodies A & B that translate relative to one another along a
common axis.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 27
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint Constraint Equations
The points P and Q in Body A lie on the common axis and are connected by the vector PQ.
The points R and S in Body B lie on the common axis and are connected by the vector RS.
The vector PR also lies on the common axis and connects the points P and R.
The three vectors must be parallel.
Alternatively, vectors PR and RS must be perpendicular to .
X
Y Common Axis
P
Q
Body A
Body B
xA
yA
A
xA
yA
B
R S
Two bodies A & B that translate relative to one another along a
common axis.
PQ
PQ
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 28
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint Constraint Equations
The vector PQ with components written with respect to the body fixed coordinate system of Body A are
The components of the vector PQ with respect to the global coordinate system are
X
Y
P
Q
Body A
Body B
xA
yA
A
xA
yA
B
R S
Two bodies A & B that translate relative to one another along a
common axis.
PQjyixQP A
PQAPQ
ˆˆ
APQ
APQ
AA
AA
APQ
APQ
yx
YX
cossinsincos
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 29
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint Constraint Equations
The vector RS with components written with respect to the body fixed coordinate system of Body B are
The components of the vector RS with respect to the global coordinate system are
X
Y
P
Q
Body A
Body B
xA
yA
A
xA
yA
B
R S
Two bodies A & B that translate relative to one another along a
common axis.
PQjyixSR B
RSBRS
ˆˆ
BRS
BRS
BB
BB
BRS
BRS
yx
YX
cossinsincos
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 30
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint Constraint Equations
The third vector is directed from point P to point R.
Point P has the coordinates
Point R has the coordinates
The vector has componentsX
Y
P
Q
Body A
Body B
xA
yA
A
xA
yA
B
R S
Two bodies A & B that translate relative to one another along a
common axis.
PQ
Ap
Ap
AA
AA
ACG
ACG
AP
AP
yx
YX
YX
cossinsincos
BR
BR
BB
BB
BCG
BCG
BR
BR
yx
YX
YX
cossinsincos
AP
AP
AA
AA
ACG
ACG
BR
BR
BB
BB
BCG
BCG
PR
PR
yx
YX
yx
YX
YX
cossinsincos
cossinsincos
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 31
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint Constraint Equations
The vector perpendicular to PQ has components
The dot product of two vectors that are perpendicular to each other is zero.
X
Y
P
Q
Body A
Body B
xA
yA
A
xA
yA
B
R S
Two bodies A & B that translate relative to one another along a
common axis.
PQ
APQ
APQ
AA
AA
APQ
APQ
yx
YX
0110
cossinsincos
0
PR
PRAPQ
APQ Y
XXXRPQP
First Constraint Eq.
0
BRS
BRSA
PQAPQ Y
XXXRPQP
Second Constraint Eq.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 32
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint Constraint Equations
Substituting the vector components from the previous slides into the first constraint equation yields
Substituting the vector components from the previous slides into the second constraint equation yields
0
PR
PRAPQ
APQ Y
XXXRPQP
0cossinsincos
cossinsincos
cossinsincos
0110
AP
AP
AA
AA
ACG
ACG
BR
BR
BB
BB
BCG
BCG
T
AA
AATAPQ
APQ y
xYX
yx
YX
yx
0
BRS
BRSA
PQAPQ Y
XXXRPQP
0cossinsincos
cossinsincos
0110
BRS
BRS
BB
BBT
AA
AATAPQ
APQ y
xyx
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 33
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Summary of Prismatic Constraint Equations
The two constraint equations for a planar prismatic joint are
0cossinsincos
cossinsincos
cossinsincos
0110
AP
AP
AA
AA
ACG
ACG
BR
BR
BB
BB
BCG
BCG
T
AA
AATAPQ
APQ y
xYX
yx
YX
yx
0cossinsincos
cossinsincos
0110
BRS
BRS
BB
BBT
AA
AATAPQ
APQ y
xyx
1st Constraint Equation
2nd Constraint Equation
The vector components at the beginning and end of each equation are based on the body fixed coordinate systems and are constant. The only variables are the generalized coordinates of Body A and B.
These equations are easily evaluated in a computer program.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 34
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint
The prismatic joint formed by the cylinder liner and the piston lies along the global X-axis.
Point P is chosen to lie at the c.g. of the cylinder liner.
Point Q is chosen to lie 1 mm to the right on the x-axis.
Point R is chosen to lie at the c.g. of the piston.
Point S is chosen to lie 1 mm to the right on the x-axis.
xy
xy
P Q R S
0
1
APQ
APQ
y
x
Vector components
of PQ
0
1
BRS
BRS
y
x
Vector components
of RS
Point Coordinates
0
0
AP
AP
y
x0
0
BR
BR
y
x
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 35
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Prismatic Joint
Substitution of the vector components and point coordinates into the two prismatic joint equations yields
000
cossinsincos
00
cossinsincos
cossinsincos
0110
01
AA
AA
ACG
ACG
BB
BB
BCG
BCG
T
AA
AAT
YX
YX
001
cossinsincos
cossinsincos
0110
01
BB
BBT
AA
AAT
1st Constraint Equation
which reduces to
0cossin
sincos0110
01
ACG
ACG
BCG
BCG
T
AA
AAT
YX
YX
2nd Constraint Equation
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 36
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Motion Constraint
One motion constraint is required to make the mechanism stable.
The rotation of the crankshaft (Body D) will be given an angular speed of 3,000 rpm.
A 3,000 rpm engine speed is equal to 314 rad/sec.
Although all fifteen generalized coordinates are a function of time, this is the only constraint equation that explicitly contains time as a variable.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 37
0314 tD
Motion Constraint
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Summary of Constraint Equations
There are five planar bodies each having three DOF giving a total of fifteen DOF. Fifteen unknowns requires fifteen equations.
0)6
0)5
0)4
Ecg
Ecg
Ecg
Y
X
0)3
0)2
08.156)1
Acg
Acg
Acg
Y
X
Ground Constraint 1
Ground Constraint 2
0cos6.102sin28)10
0sin6.102cos28)9
CC
cgBB
cg
CCcg
BBcg
YY
XX
0cos43sin3.41)8
0sin43cos3.41)7
DD
cgCC
cg
DDcg
CCcg
YY
XX
0)12
0)11
Ecg
Dcg
Ecg
Dcg
YY
XXRevolute Joint 1
Revolute Joint 2
Revolute Joint 3
001
cossinsincos
cossinsincos
0110
01)14
BB
BBT
AA
AAT
Prismatic Joint
Motion Constraint0314)15 tD
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 38
0cossin
sincos0110
01)13
ACG
ACG
BCG
BCG
T
AA
AAT
YX
YX
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Summary of Constraint Equations
Only one of the constraint equations is time dependent (Motion Constraint).
Most of the constraint equations are non-linear. All of the constraints are algebraic equations and none are
differential equations. Geometric quantities (dimensions and distances) contained in
the constraint equations can be found from information in a 3D CAD model.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 39
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Module Summary
The constraint equations for ground, revolute, and prismatic joints have been developed for a planar mechanism.
The constraint equation for a rotational motion constraint has been developed for a planar mechanism.
These equations were used to determine the fifteen equations necessary for a piston-crank assembly taken from a Boxer engine model.
In some cases the constraint equations are very simple and in other cases they are complex.
Only the motion constraint is an explicit function of time. All of the constraint equations are algebraic. These equations will be applied in the next module: Module 4.
Section 4 – Dynamic Simulation
Module 3 – Constraint Equations
Page 40