NUMERICAL ANALYSIS FOR ENGINEERING
MEAE - 4960 H02
Prof. Ernesto Gutierrez - Miravete
Kinematic Analysis of a Four-Bar Crank
Mechanism with a Redundant Constraint
Roberto A. Alzaga
December 9, 1999
INDEX
Introduction…………………………………………………………………………..3
Theory and Analysis………………………………………………………………..4
General Coordinates…………………………………………………………4
Kinematic Constraint Equations………………………………………..4
Detection of Redundant Constraint…………………………………….5
Degrees of Freedom………………………………………………………….7
Driving Constraint……………………………………………………………7
System Constraint Equations…………………………………………….7
Determinant of System Constraints Jacobian Matrix…………….8
Evaluation of System Constraints Jacobian Matrix
at Singular Point…………….…9
Evaluate Newton's Method for Position………………………………..9
Velocity Approximation………………………………………………….…10
Acceleration Approximation……………………………………………...10
Results………………………………………………………………………………..12
Results of the Double Pendulum Mechanism………………………..12
Results of the Four Bar Linkage Mechanism………………………..16
Four-Bar Linkage Position Error Comparison……………………….19
Conclusions………………………………………………………………………….21
Bibliography…………………………………………………………………………22
Appendix……………………………………………………………………………..23
Gaussian Elimination with Maple Commands and
Inverse of Jacobian………………...23
Newton's Method for Double Pendulum………………………………..28
Newton's Method for Four-Bar Linkage…………………………….….35
2
INTRODUCTION
Mechanical Systems are an integral part of our everyday life. Anything that
is comprised by linkages, cams and gears is defined as a mechanical system. A
mechanical system is defined as a collection of interconnected rigid bodies that
can move relative to one another, consistent with joints that limit relative
motion of pairs of bodies. The motion of a mechanical system may be
prescribed by defining the time history of the position or relative position of
some of its bodies. The motion of the system is then determined by algebraic
kinematics relations or from differential equations of motion and externally
applied forces, in which case the motion of the system is determined by laws of
physics. Kinematics and dynamics of mechanical systems are characterized by
large amplitude motion, which leads to geometric nonlinearity that is reflected
in the algebraic equations of constraint and differential equations of motion.
Three basically different kinds of analysis are employed in the design of
mechanical systems: Kinematic, Dynamic and Inverse Dynamic analysis.
This project is related to a Kinematic analysis of a Four-Bar Crank Mechanism
with a Redundant Constraint. A Kinematic analysis of a mechanical system
concerns the motion of the system independent of forces that produce the
motion. Typically, the time history of position or relative position of one, or
more bodies in the system is prescribed. Time histories of position, velocity,
and acceleration of the remaining bodies are then determined by solving
systems of nonlinear algebraic equations for position and linear algebraic
equations for velocity and acceleration.
The Kinematic analysis of a Four-Bar Crank Mechanism with a Redundant
Constraint will be performed using computerized numerical analysis methods.
This analysis will result in an approximate of the solution of the governing
kinematic equations of this mechanism. The system of equations will be solved
to determine the position, velocity and acceleration of the mechanism for a
given driving condition.
3
THEORY AND ANALYSIS
The Four-Bar Crank Mechanism with a Redundant Constraint to be analyze is
illustrated in the following diagram:
General Coordinates
General Coordinates are defined as any set of variables that uniquely
specifies the position and orientation of all bodies in a mechanism. These
coordinates may by independent (i.e., free to vary arbitrarily) or dependent (i.e.,
required to satisfy equations of constraint). General coordinates are designated
by a column vector q = [q1, q2,…,qnc]T, where nc is the total number of
generalized coordinates used to describe the configuration of the system.
For this Four Bar Linkage mechanism, we have three rigid bodies, each one
defined by their position with respect to x, y and the angle orientation . In
other words q = [x1, y1, 1, x2, y2, 2, x3, y3, 3]T. Since x1, y1, x3 and y3 are
stationary points, x1 = 0, y1 = 0, x3 = 1 and y3 = 0, they are not used in the
analysis to simplify the matrices. The vector of generalized coordinated that
best describes the configuration of this system is:
q = [1, x2, y2, 2, 3]T
Kinematic Constraint Equations
Since joints interconnect rigid bodies that make up a mechanism, there are
equations of constraint that relate the generalized coordinates. Therefore,
Cartesian generalized coordinates are generally dependent. A kinematic
constraint between two bodies imposes conditions on the relative motion
4
between the pair of bodies. When expressed as algebraic equations in terms of
generalized coordinates, they are called holomonic kinematic constraint
equations. This equations are expressed as a vector K(q) = [K1(q),…, Knh(q)].
The equations of constraint must imply the geometry of the joint.
For this Four-Bar Linkage, the equations are develop as follows:
 Position of x2
(1) x2 = cos 1 + cos 2
(2) x2 = 1 + cos 3
 Position of y2
(3) y2 = sin 1 + sin 2
(4) y2 = sin 3
 Length of Segment AB (Redundant Constraint)
(5) (x2 + cos 2 - 2)2 + (y2 + sin 2)2 = 1
Expressed as a vector, we have:
x 2  cos 1  cos  2




y 2  sin 1  cos  2


 K (q )  
x 2  cos  3  1
0


y 2  sin  3


( x 2  cos  2  2) 2  ( y 2  sin  2 ) 2  1
Detection of Redundant Constraint
This matrix includes one redundant constraint for the system, in other
words, a constraint that is already satisfied by one or a combination of the
other constraints already established. As is geometrically clear, 1 = 3 and 2 =
0, thus, from the kinematic constraints (1) and (3), x2 = 1 + cos 1 and y2 = sin
1. Substituting these results into kinematic constraint (5) shows that it is
identically satisfied, in other words, kinematic constraint (5) is a redundant
constraint.
The Constraint Jacobian row rank criterion identifies this redundancy. The
Jacobian Kq (q) of the Kinematic Constraint Vector is as follows:
 qK (q) 
   i 


q  q j 
nxk
5
 sin 1
  cos 
1

K
 q (q )   0

 0
 0
sin  2

0
1
 cos  2
0 

1
0
0
sin 3 

0
1
0
 cos 3 
2( x2  cos  2  2) 2( y 2  sin  2 )  2 sin  2 ( x2  cos  2  2)  2 cos  2 ( y 2  sin  2 )
0 
1
0
0
The row rank (column rank) of a matrix is defined as the largest number of
linearly independent rows (columns) in the matrix. A square matrix with
linearly independent rows (columns) is said to have full rank or nonsingular,
meaning that for such matrix, there is an inverse A-1 such that AA-1 = A-1A = I.
This property is necessary for the solution of the kinematic analysis. The rank
of a matrix can be established using Gaussian Elimination.
The Jacobian of the Kinematic Constraints is evaluated at the assembled
configuration with 1 = 3 = /3 (60), 2 = 0, from kinematic constraints (1) and
(2), x2 = 1.5 and y2 = 0.8660 respectively.
0.8660254037

 0.5

K
0
 q (q )  
0

0


0
1
0
1
0
1


1
1
0

0
0
0.8660254037

1
0
 0.5


1.73205087 1.73205087
0
0
0
0
Using Gaussian Elimination Command in Maple for this matrix, the Jacobian is
reduced to
0.8660254037

0

 qK (q 0 )  
0

0


0
1
1
0
0
0

0
0
0.8660254037 

1.732050807 1.732050807  0.8660254037

0
0
0.11


0
0
0
0
0
0
The reduced Jacobian is rank deficient and the last constraint is identified as
the redundant constraint and is removed from the Kinematic Constraint Matrix.
Even if a small change is made in 10, and hence 30, the same conclusion
follows, so the redundancy is not an isolated singular point. An isolated
singular point occurs at a configuration of the mechanism where a unique
solution of the equations of kinematics cannot be determined.
6
Degrees of Freedom
If the kinematic constraints are consistent and independent as evaluated by
detecting the redundant constraint, the system is said to have nc (# of
generalized coordinates) - nh (# of independent kinematic constraints) degrees
of freedom. In other words, the degrees of freedom identify the number of
drivers needed to give movement to the mechanism. Again, nc is equal to 3
times (x, y ) the number of bodies in the mechanism.
DOF = 3(nb) - nh
where:
 nb = number of bodies is 3 neglecting ground and
 nh = number of holonomic kinematic constraint equations is 8, which are not
reflected in the Kinematic Constraint Equation Matrix because we did
not include x1 = 0, y1 = 0, x3 = 1, y3 = 0 to simplify the analysis due to
the fact that these are static values
The degrees of freedom for this mechanism is
DOF = 3(3) - 8 = 1
This means that we need only one Driving Constraint to be able to move the
mechanism.
Driving Constraint
The Driving Constraint selected for this mechanism is:
1 = t/180 for 0 ≤ t ≤ 180
System Constraint Equations
When we combine the Kinematic Constraint Equations with the Driving
Constraint we obtain the System Constraint Equations
  K ( q) 
 KD (q, t )  D
0
 ( q, t )
7
 x 2  cos 1  cos  2 
 y  sin   cos  
1
2
 2
 KD (q, t )   x 2  cos  3  1   0


y 2  sin  3


t


1 


180
We now again, evaluate the Jacobian Matrix of the System Constraint
Equations, to verify for Singular Points in the movement of the mechanism, and
also, the Jacobian will be used in the calculation of the Velocity and
Acceleration of the Four Bar System.
 sin 1 1 0 sin  2
 cos  0 1  cos 
1
2

 q ( q, t )   0
1 0
0

0 1
0
 0
 1
0 0
0

0 

sin 3 

 cos 3 
0 
0
Determinant of System Constraints Jacobian Matrix
By calculating at which conditions the determinant of the Jacobian is equal
to zero (singular points, at which no solution exists), we can determine the
intervals at which the mechanism can be analyze,
|q| = -(cos 2*sin 3 + sin 2*cos 3)
Since this Four Bar Linkage has 2 = 0 always because of its configuration (r1 =
r2 = r3), the determinant is reduced to
|q| = -sin 3
The mechanism encounters singular points when 1 = k, where k = 0,1,2,…,n.
At those points the Jacobian is singular or rank deficient. The Implicit Function
Theorem summarizes the importance of this finding. Let q0 be a solution of the
System Constraint Matrix KD(q) at t = t0 and let the function (q,t) be twice
continuously differentiable with respect to its arguments. Then if, the Jacobian
is nonsingular at (q0, t0) there exists a unique solution q = f(t) in some interval
of time about t0, such that f(t0) = q0. Furthermore, the solution of q = f(t) is
8
twice continuously differentiable with respect to time, that is, velocity q' = f'(t)
and acceleration q'' = f''(t) are continuous.
Evaluation of System Constraints Jacobian Matrix at Singular Point
To demonstrate one of the singular points in this mechanism, lets take k = 0,
we have 1 = 3 = 0, 2 = 0, from kinematic constraints (1) and (2), x2 = 2 and y2
= 0 respectively.
0
 1

 q (q 0 , t )   0

0
 1
1 0
0 1
1 0
0 1
0 0
0
1 0 

0
0

0  1
0
0 
0
Using Gaussian Elimination Command in Maple for this matrix, the Jacobian is
reduced to:
 1
0

 q (q 0 , t )   0

0
 0
0 1 1
1 0
0 1
0 0
0 0
0
0
0

0  1

1 1 
0
0 
The reduced Jacobian is rank deficient, which might suggest that there might
be another redundant constraint in the System Constraint Equations.
However, an arbitrarily small perturbation of 10, and hence, 30, yields a matrix
with full rank. Thus, proving that the mechanism has a singular point at 1 =
0, specifically a lock-up configuration. This configuration indicates that there is
a solution at one or both sides of an isolated singular point. With this
evaluation, we can conclude that the range of movement for this Four Bar
Linkage occurs from 0 < 1 < .
Evaluate Newton's Method for Position
To approximate the Position of the Four- Bar Linkage, the Newton's Method
for Nonlinear Systems was used. This is an iterative technique that begins
with an estimate q(0) of a configuration that satisfies the System Constraint
9
Equations KD(q) at time t. At a typical iteration k, the following equation is
solved for a correction q(k):
q(q(k),t) q(k) = -(q(k),t)
which is then added to the estimate q(k) to obtain an improved estimate
q(k+1) = q(k) + q(k), for k = 0,1…
Iteration is continued until an error tolerance in satisfied. This method has the
property that it is quadratically convergent if a right initial estimate is establish.
For the Four-Bar Linkage analyzed, the initial approximation used was
q(0) = [ /180, 1 + cos (/180), sin (/180), 0, /180]T
after that, the solution of q(t-1) was used as an initial estimate for the solution
process of q(t).
Velocity Approximation
Presuming that the Jacobian Matrix is nonsingular and that a solution of the
kinematic equations for position has been obtained numerically. If we
differentiate the System Constraint Equations with respect to time:
qq' = - t = 
where
 q is the Jacobian of the System Constraint Equations
 q' is the velocity components of the general coordinates
q' = [ 1', x2', y2', 2', 3' ]T
 t is the derivative of the System Constraint Equations with respect to time
- t = [0, 0, 0, 0, /180]T
Here is where the property of the nonsingular Jacobian Matrix comes into
account. The velocity equations can be re-arrange to
q' = q-1(- t)
Acceleration Approximation
Just as the velocity equations were obtained by differentiating the System
Constraint Equations, differentiating both sides of the velocity equations with
respect to time:
qq'' = - ( qq' )q - 2qtq' - tt = 
where
10
 q is the Jacobian of the System Constraint Equations
 q'' is the acceleration components of the general coordinates
q'' = [ 1'', x2'', y2'', 2'', 3'' ]T
 qt is the Jacobian Matrix differentiated with respect to time
  2i 
 qt  
0

 q j t  nhxnc
 tt is the double derivative of the System Constraint Equations with respect
tt = 0
to time
 2 
 tt   2 i 
0

t

 nhx1
 The term ( qq' )q is the result of the chain rule of differentiation, this term is
calculated as follows:
 q'
q
q
 nc  2  i

 
q' k 
 ncxnc
 k 1 q j qk
 1 ' sin 1  x2 ' 2 ' sin  2 
   ' cos   y ' ' cos  
1
2
2
2
 1
x 2 ' 3 ' sin  3

(1)  q q'  


y 2 ' 3 ' cos  3




1 '
1 ' cos 1
 ' sin 
1
 1
(2) ( q q' ) q  
0

 0
 0
0 0  2 ' cos 12
0 0
 2 ' sin  2
0 0
0
0 0
0
0 0
0


0

3 ' cos 3 
3 ' sin 3 

0
0
  (1 ' 2 cos 1   2 ' 2 cos  2 )


2
2
  (1 ' sin 1   2 ' sin  2 ) 

(3) ( q q' ) q q'  
 3 ' 2 cos 3


2
 3 ' sin 3




0
By substituting in the acceleration equations and using the nonsingular
property of the Jacobian
q'' = q-1 [-( qq' )q]
11
RESULTS
The programming tool used to model the Four-Bar Linkage Mechanism was
Maple. Although, the solution for the position of this specific Four-Bar Linkage
is close form (because 2 = 0) and can be easily compared to approximated value
obtained with the Newton's Method, actual velocity and accelerations solutions
were not available. I did find a solution for position, velocity and acceleration of
a Double Pendulum Mechanism. To certify the Maple Program created for the
Four-Bar Linkage, I first ran a case for the Double Pendulum at only one
configuration and compared it to the textbook solution found.
Results of the Double Pendulum Mechanism
The Double Pendulum Mechanism is illustrated in the following diagram:
Please refer to the Theory and Analysis section if you have any questions on
how the following equations were derived:
 Cartesian Generalized Coordinates
q = [x1, y1, 1, x2, y2, 2]T
 Kinematic Constraints
Position of x1
(1) x1 = cos 1
Position of y1
(2) y1 = sin 1
Position of x2
(3) x2 = cos 1 + cos 2
Position of y2
(4) y2 = sin 1 + sin 2
12
(5) y2 = -1
x1  cos 1




y1  sin 1


 K (q )   x 2  cos 1  cos  2   0


 y 2  sin 1  sin  2 


y2  1
 Degrees of Freedom
DOF = 3(nb) - nh = 3(2) - 5 = 1
This means that one driving condition is needed for movement of the
mechanism.
 Driving Constraint
1 = 5/3 + t/6, 0 ≤ t ≤ 3/2
 System Constraint Equations
x1  cos 1




y1  sin 1


x

cos


cos


1
2
 KD (q )   2
0
y

sin


sin

2
1
2




y2  1


 1  5 / 3   / 6 
 Jacobian Matrix
1
0

0
 q ( q, t )  
0
0

0
0
sin 1
0 0
1  cos 1 0 0
0
sin 1
1 0
0  cos 1 0 1
0
0
0 1
0
1
0 0

0 

sin  2 

 cos  2 
0 

0 
0
 Determinant of Jacobian
|q| = -cos 2
Singular when 2 = /2 + k, for k integer.
 Evaluate Newton's Method for Position
The solution of this mechanism was only calculated for t = 0 and compared
to the textbook results to validate the Four Bar Linkage Model Program.
The initial approximation to solve the System Constraint Equations is
q(0) = [ 0.5, -0.866, 5/3, 1.5, -1.0, 6.14]T
13
 Velocity Approximation
qq' = - t = 
where
q is the Jacobian of the System Constraint Equations
q' is the velocity components of the general coordinates
q' = [x2', y2', 1', x2', y2', 2']T
t is the derivative of the System Constraint Equations with respect to time
- t = [0, 0, 0, 0, /6]T
 Acceleration Approximation
qq'' = - ( qq' )q - 2qtq' - tt = 
where
q is the Jacobian of the System Constraint Equations
q'' is the acceleration components of the general coordinates
q'' = [x2'', y2'', 1'', x2'', y2'', 2'']T
qt is the Jacobian Matrix differentiated with respect to time
qt = 0
tt is the double derivative of the System Constraint Equations with respect
tt = 0
to time
The term ( qq' )q is the result of the chain rule of differentiation


 1 ' 2 cos 1


 1 ' 2 sin 1


 (1 ' 2 cos 1   2 ' 2 cos  2 )
( q q' ) q q'  

2
2
  (1 ' sin 1   2 ' sin  2 ) 
0




0


 Results and Comparison
The following table summarizes the results for position, velocity and
acceleration obtained with the Maple Program for the Double Pendulum
Mechanism at t = 0.
14
Position Results Comparison
X1
Y1
Phi1
X2
Y2
Phi2
(IN)
(IN)
(RAD)
(IN)
(IN)
(RAD)
Textbook Results
0.5000
-0.8660
5.2360
1.4910
-1.0000
6.1488
Program Results
0.5000
-0.8660
5.2360
1.4910
-1.0000
6.1488
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
X1'
Y1'
Phi2'
X2'
Y2'
Phi2'
(IN/SEC)
(IN/SEC)
(RAD/SEC)
(IN/SEC)
(IN/SEC)
(RAD/SEC)
Textbook Results
0.4534
0.2618
0.5236
0.4180
0.0000
-0.2642
Program Results
0.4534
0.2618
0.5236
0.4181
0.0000
-0.2642
0.00%
0.00%
0.00%
0.01%
0.00%
0.00%
Phi2''
X2''
Y2''
Phi2''
Actual Error
Velocity Results Comparison
Actual Error
Acceleration Results Comparison
X1''
Y1''
(IN/SEC^2)
(IN/SEC^2)
Textbook Results
-0.1371
0.2374
0.0000
-0.2397
0.0000
-0.2490
Program Results
-0.1371
0.2374
0.0000
-0.2396
0.0000
-0.2490
0.00%
0.00%
0.00%
0.01%
0.00%
0.00%
Actual Error
(RAD/SEC^2) (IN/SEC^2) (RAD/SEC^2) (RAD/SEC^2)
These results show zero error in almost every parameter from position to
acceleration, validating the Maple Program of Newton's Method developed
specifically for the Four Bar Linkage Mechanism discussed.
15
Results of the Four-Bar Linkage Mechanism
The Four-Bar Linkage Mechanism was analyze for a range of movement
driven by the time variable in the driving constraint of 1 = t/180 for
t = 1, 2,…179. The results were tabulated and graphed as follows:
q = [1, x2, y2, 2, 3]T
 General Coordinate 1
Position, Velocity & Acceleration vs. Time for Phi1
3.5
3
2.5
2
1.5
1
0.5
0
0
20
40
60
80
100
120
140
160
180
Time (sec.)
Angle
Velocity
Acceleration
Angle 1 is the one driven by the input. It has a constant velocity of /180
and acceleration is zero due to the contant velocity . Since 3 = 1 because of
the mechanism configuration, the same solution is expected for 3.
16
 General Coordinate x2
Position, Velocity & Acceleration vs. Time for x2
2.5
2
1.5
1
0.5
0
0
20
40
60
80
100
120
140
160
180
-0.5
Time (sec)
Position
Velocity
Acceleration
The position of x2 goes from 2 when t =1 and rotates counter clockwise until
it reaches almost the origin of the global Cartesian coordinates. Velocity and
acceleration numbers are close to zero following the derivatives of the
position graph.
17
 General Coordinate y2
Position, Velocity & Acceleration vs. Time for y2
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
120
140
160
180
-0.2
Time (sec)
Position
Velocity
Acceleration
The position of y2 starts close to zero at t =1 and reaches a maximum when
t = 90 or 1 = /2.
 General Coordinate 2
Basically since all the length are equal to 1, 2, its velocity and acceleration
will always be 0, . This is part of the inherent restriction due to the
configuration of the Four-Bar Linkage.
 General Coordinate 3
Since 3 = 1, the graph for the position, velocity and acceleration are the
same as for 1.
18
Four-Bar Linkage Position Error Comparison
Since the solution of the position vector for the Four-Bar Linkage System
Constraints is of close form, due to the fact that for this particular mechanism
 2 = 0, the actual calculated positions and angles can be compared to the
approximated results from Newton's Method. We will compare the results for
t = 30, 60, 90, 120 and 150 or 1 = /6, /3, /2, 2/3 and 5/6.
From the System Constraint Equations
 x 2  cos 1  cos  2 
 y  sin   cos  
1
2
 2
 KD (q, t )   x 2  cos  3  1   0


y 2  sin  3


t


1 
180


it is seen, that because 2=0 throughout the movement of the mechanism,
direct calculation of the position vector can be obtained:
1 = *t/180
x2 = cos 1 + cos (0)
y2 = sin 1 + sin (0)
3 = 1 = sin-1(y2)
Calculated Close Form
Time
(SEC)
Phi1
(RAD)
X2
(IN)
Y2
(IN)
Phi2
(RAD)
Phi3
(RAD)
30
0.5236
1.8660
0.5000
0.0000
0.5236
60
1.0472
1.5000
0.8660
0.0000
1.0472
90
1.5708
1.0000
1.0000
0.0000
1.5708
120
2.0944
0.5000
0.8660
0.0000
2.0944
150
2.6180
0.1340
0.5000
0.0000
2.6180
19
Newton's Method
Time
(SEC)
Phi1
(RAD)
X2
(IN)
Y2
(IN)
Phi2
(RAD)
Phi3
(RAD)
30
0.5236
1.8660
0.5000
0.0000
0.5236
60
1.0472
1.5000
0.8660
0.0000
1.0472
90
1.5708
1.0000
1.0000
0.0000
1.5708
120
2.0944
0.5000
0.8660
0.0000
2.0944
150
2.6180
0.1340
0.5000
0.0000
2.6180
Time
(SEC)
Phi1
(RAD)
X2
(IN)
Y2
(IN)
Phi2
(RAD)
Phi3
(RAD)
30
0.00%
0.00%
0.00%
0.00%
0.00%
60
0.00%
0.00%
0.00%
0.00%
0.00%
90
0.00%
0.00%
0.00%
0.00%
0.00%
120
0.00%
0.00%
0.00%
0.00%
0.00%
150
0.00%
0.00%
0.00%
0.00%
0.00%
Actual Error
No actual errors were found for the position vector.
20
CONCLUSIONS
The use of computer aided kinematic analysis is a very powerful tool when it
comes to new designs. It enables the engineer to iterate quit rapidly the
configuration of the mechanism without consuming too much time. The key to
the ease of the use of a computer is that time has to be spent to make sure that
the program is correctly modeling the real mechanism by the incorporation of
the correct Kinematic Constraints. All the basic steps to put together a simple
model for computer analysis were discussed in this project.
First, we had to define the General Coordinates of the Four-Bar Linkage
mechanism presented. Then, the Kinematic Constraints that relate the General
Coordinates were established. To verify that we did not establish a redundant
constraint, the Jacobian of the Kinematic Constraints was evaluated at the
assembled configuration. After performing Gaussian Elimination to this matrix,
we determined that it was rank deficient related to a redundant constraint
already satisfied by the other Kinematic Constraints established. After
eliminating the redundant constraint and calculating the Degrees of Freedom,
one Driving Condition was incorporated to be able to give movement to the
mechanism.
The Jacobian of the now System Constraint Equations was calculated. When
the determinant of this matrix is zero, singularity occurs and the mechanism
cannot be defined at that specific point. For this Four Bar Linkage, we
demonstrated that isolated singular points occur at 1 = k for k integer.
Furthermore, the mechanism analyzed can only move between 0 < 1 < .
Finally the matrix equations for position, velocity and acceleration were
established and used as an input to the Newton Method for Nonlinear System.
Since no data was on hand to compare errors, another mechanism was used in
the same process just described and compared to known values at only one
point, yielding errors from 0% to 0.01%.
21
BIBLIOGRAPHY
(1) Burden, R., and Faires, J.D., Numerical Analysis,
Brooks/Cole Publishing Company, Pacific Groove, CA, 1997.
(2) Haug, Eduard J., Computer-Aided Kinematics and Dynamics of Mechanical
Systems Volume 1: Basic Methods,
Allyn and Bacon, Needham Heights, MA, 1989.
(3) Analytical Tool Used
Maple V Relese 5
Waterloo Maple Inc.
22
APPENDIX
23