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Dynamic Simulation : Constraint Kinematics. Objective The objective of this module is to show how constraint equations are used to compute the position, velocity, and acceleration of the generalized coordinates . - PowerPoint PPT Presentation
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© 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
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Dynamic Simulation:Constraint Kinematics
Objective
The objective of this module is to show how constraint equations are used to compute the position, velocity, and acceleration of the generalized coordinates.
These equations are kinematic in nature because they do not consider the forces required to cause the motion.
The kinematic and motion constraints developed in the previous module (Module 3) for the piston-crank mechanism are used to demonstrate the mathematics.
© 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
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Notation
The total set of constraint equations needed to define a mechanism includes both kinematic constraints and drive constraints.
There are 15 generalized coordinates and 15 nonlinear constraint equations for the piston-crank assembly used in Module 3.
Since the piston-crank has a mobility of one, only one of the fifteen equations will be a motion constraint that is an explicit function of time.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 2
0,
,
tqq
tqd
k
qk is the set of kinematic constraint equations
tqd , is the set of motion constraint equations
q is the set of generalized coordinates
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Position
Solving the constraint equations will yield the value of each generalized coordinate at a specific instance of time.
The constraint equations are non-linear and the Newton-Raphson method is used as the solution method.
The Newton-Raphson method is iterative and converges when the constraint equations are satisfied.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 3
0,
,
tqq
tqd
k
Constraint Equations
Newton-Raphson Equations
tqqtqqq ii ,,
1
1
where
qtq, is the Jacobian
matrix
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Velocity
The time derivative of the constraint equations is used to determine the velocities of the generalized coordinates.
Since the generalized coordinates are a function of time and the constraint equations are a function of the generalized coordinates and time, the chain rule for partial differentiation must be used.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 4
0,
,
tqq
tqd
k
Constraint Equations
Time Derivative
0,
ttq
qttq
Velocities
tqtq
1
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Acceleration Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 5
The second time derivative of the constraint equations is used to determine the accelerations of the generalized coordinates.
Since the generalized coordinates are a function of time and the constraint equations are a function of the generalized coordinates and time, the chain rule for partial differentiation must be used.
1st Time Derivative of Constraint Equations 0,
ttq
qttq
2nd Time Derivative of Constraint Equations
0
2,
2
2
2
2
2
2
2
ttq
q
tq
tqqq
qqttq
Accelerations
2
22
1
2
2
2tt
qtq
qqqq
qtq
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Summary of Equations
Newton-Raphson Equations
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 6
0,
,
tqq
tqd
k
Constraint Equations
tqqtqqq ii ,,
1
1
Used to determine the position (values of the generalized coordinates) at an instant in time.
Velocities of Generalized Coordinates
tqtq
1
Accelerations of Generalized Coordinates
2
221
2
2
2tt
qtq
qqqqqt
q
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Jacobian
The Jacobian and its inverse is needed to determine the position, velocity, and acceleration of the generalized coordinates.
Each i,j (row,column) term in the Jacobian matrix is given by
q
J Jacobian matrix
j
iji q
J
,
ith constraint equation
jth generalized coordinate
Error messages indicating that the Jacobian is singular are sometimes encountered when running multi-body dynamic programs.
This occurs when there is not a physically realizable solution.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 7
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Piston-Crank Constraint Equations Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 8
The fifteen constraint equations developed for the piston-crank mechanism in Module 3 are given on the right.
Note that only the motion constraint is an explicit function of time.
0)6
0)5
0)4
Ecg
Ecg
Ecg
Y
X
0)3
0)2
08.156)1
Acg
Acg
Acg
Y
X
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
XX
001
cossinsincos
cossinsincos
0110
01)14
BB
BBT
AA
AAT
0314)15 tD
0cossinsincos
0110
01)13
ACG
ACG
BCG
BCG
T
AA
AAT
YX
YX
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Piston-Crank JacobianSection 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 9
000100000000000
000000000coscos
sinsin00
sin
cossin00
0000000000cossinsin
coscossin
010010000000000001001000000000000000sin6.10210cos2810000000000cos6.10201sin2801000000sin4310cos3.4110000000000cos4301sin3.4101000000100000000000000010000000000000001000000000000000000000000100000000000000010000000000000001
2 BA
BA
A
BA
AAAB
cgAcg
ABcg
AcgAA
B
CB
DC
DC
j
i
YY
XX
qJ
The Jacobian of the constraint equations is given below. Although there are many terms, there are a lot of zeros and the derivatives are easily computed.
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Velocities
The velocities of the generalized coordinates are computed from the equation
Since the Jacobian is known, this equation can be solved if the array containing the time derivatives of the constraint equations is found.
Only the motion constraint, (15), is an explicit function of time.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 10
tqtq
1
31400000000000000
t
Motion Constraint
Required Array
0314)15 tD
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Acceleration Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 11
The accelerations can be computed if each term is found.
2
221
2
2
2tt
qtq
qqqqqt
q
The time derivative of the Jacobian is zero.
This term is explained on the next slide.
Inverse of the Jacobian
31400000000000000
t
000000000000000
2
2
t
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Acceleration Term
qqqq
This term is evaluated by breaking it down into a series of operations that are easily done on a computer.
Step 1) Multiply the Jacobian by the velocities. This creates a column array.
Step 2) Take the derivative of each row with respect to each generalized coordinate. This operation is similar to finding the Jacobian and results in a matrix.
q
Step 3) Multiply the matrix by the velocities. This results in a column array.
qqqq
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 12
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Redundant Constraints
The Jacobian matrix is an important quantity and enables the position, velocity, and acceleration of the generalized coordinates to be found.
Application of the methods contained in this module requires that the Jacobian have an inverse.
This requires that the determinant of the Jacobian be non-zero or that the rank be equal to the number of generalized coordinates.
The rank of a matrix is equal to the number of independent rows or columns.
Independent rows or columns can not be written as a linear combination of other rows or columns.
If rows or columns of the Jacobian are not independent the Jacobian is singular and the problem does not have a solution.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 13
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Redundant Constraints: Detection
The Dynamic Simulation environment within Autodesk Inventor software assembles the Jacobian and determines its rank as each constraint is added.
The rank gives the number of independent constraints.
The difference between the number of generalized coordinates and the number of independent constraints is equal to the degree of mobility.
The difference between the number of constraints and the number of independent constraints is equal to the degree of redundancy.
nq number of generalized coordinates
nc number of constraintsnic number of independent
constraints
Degree of Mobilitydom= nq – nic
Degree of Redundancydor = nc-nic
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 14
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Redundant Constraints: Reaction Forces
A redundant constraint occurs when the motion associated with a DOF is enforced by too many constraint specifications.
One or more of the constraint specifications can be removed without affecting the mobility of the system.
The joint reactions can not be independently determined when redundant constraints are present.
Although solutions can be obtained they are based on assumptions by the program as to which constraints to use.
Different assumptions will yield different answers.
Joints having friction are particularly effected by redundant constraints.
Friction forces are based on the joint normal forces.
Therefore, the friction forces are incorrect if the joint normal forces are incorrect due to redundant constraints.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 15
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Redundant Constraint: Example
A simple four bar mechanism will have redundant constraints if revolute joints are used at all joints.
The ground link shown in the figure is fixed.
The revolute joint at 1 prevents the drive link from rotating about its long axis and moving normal to the joint plane.
The revolute joint at 2 prevents the coupler from rotating about its long axis and moving normal to the joint plane.
The revolute joint at 4 prevents the rocker from rotating about its long axis and moving normal to the joint plane.
Ground
Drive
Coupler
Rocker
1
2
3
4
A revolute joint at 3 is redundant because neither the rocker or coupler can rotate about their long axis or move normal to the joint plane due to the other revolute joints. These degrees of freedom are already restrained.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 16
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Redundant Constraints: Example
A point-line joint must be used at joint 3.
A point-line joint restricts a point (the center point of the hole at joint 3 on the coupler) to remain on a line (the centerline of the hole at joint 3 on the rocker).
Redundant joints can be confusing and a detailed analysis of what each joint is doing is required to figure out how to remove them.
An example of how to remove redundant constraints is provided in the next module: Module 5.
Ground
Drive
Coupler
Rocker
1
2
3
4Revolute
Revolute
Revolute
Point - Line
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 17
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Module Summary
This module showed how the constraint equations can be used to find the position, velocity, and acceleration of the generalized coordinates.
Kinematic relationships were used during the derivation and no mention of the forces required to impose the motion constraints was made.
The constraint equations for the piston-crank introduced in the previous module (Module 3) were used to demonstrate the mathematical steps.
The Newton-Raphson method is generally used to solve the constraint equations.
The Jacobian is a key component of the overall solution process and the rank of the Jacobian is used to detect redundant constraints.
Section 4 – Dynamic Simulation
Module 4 – Constraint Kinematics
Page 18