Transcript Introduction Fundamentals of Analytical Analysis 2 Fundamentals of An Analytical Method The vector-loop method is a classical procedure that provides a set of.

Introduction
Fundamentals of Analytical Analysis 2
Fundamentals of An Analytical Method
The vector-loop method is a classical procedure that provides a set of vector
equations that can be solved either graphically for the kinematics of a planar
mechanism. This lesson reviews the basic ideas and rules in constructing the
vectors, and consequently the corresponding vector loop, for different
mechanisms.
When vector loop equations are transformed to algebraic equations, they can
be solved analytically or numerically to determine the kinematics of a system.
The analytical formulations will be discussed in the upcoming lessons.
P. Nikravesh, AME, U of A
Vector loop
Fundamentals of Analytical Analysis 2
Vector Loop
Position vectors in a mechanism creates one or more vector loops around the
linkage. If we move around a loop, the vectors in that loop take us from one link
through a joint, to another link, and another joint, and so on until we return to the
same link that we started from.
In the following examples we see how vector loops are constructed for some
commonly used mechanisms.
Vector loops lead to algebraic equations that can be solved for the kinematics of a
mechanism.
In order for a vector loop to yield a solvable set of algebraic equations, some
fundamental rules must be followed when the vectors are defined.
P. Nikravesh, AME, U of A
Vector loop
Fundamentals of Analytical Analysis 2
Vector Loop For A Fourbar
The ground link, the three moving links, and
the four pin joints (A, B, O2, and O4) form a
closed chain.
The following four vectors form a loop:
P
►
RBA
The four vectors form the following vectorloop equation:
RAO2 + RBA - RBO4 - RO4O2 = 0
The ground link can be represented by
vector RO4O2 or it can be presented as the
sum of two vectors, one horizontal and one
vertical: ►
The vector loop equation can be revised as:
RAO2 + RBA - RBO4 - RO4Q - RQO2 = 0
Either set of ground vectors could be used
for kinematic analysis.
P. Nikravesh, AME, U of A
B
A
RBO4
RAO2
RO4O2
O4
RO4Q
O2
RQO2
Q
Vector loop
Fundamentals of Analytical Analysis 2
Vector Loop For A Slider-Crank
The ground link, the three moving links,
the three pin joints (A, B, and O2), and the
sliding joint form a closed chain (loop).
A
The following three vectors form a loop: ►
The three vectors form the following
vector-loop equation:
RAO2 + RBA - RBO2 = 0
Note that vector RBO2 will have a variable
magnitude when the links move. The
magnitude of this vector represents the
distance of the slider block, point B, from
the ground reference point; i.e., point O2.
RBA
RAO2
B
O2
RBO2
►
►
Rule: When there is a sliding joint in a mechanism, we must define a variable-length
vector along or parallel to the axis of the joint.
P. Nikravesh, AME, U of A
Vector loop
Fundamentals of Analytical Analysis 2
Vector Loop For An Offset Slider-Crank
The ground link, the three moving links, the
three pin joints (A, B, and O2), and the sliding
joint form a closed chain.
The following four vectors form a loop:
• Two fixed-length vectors:
A
RAO2
RO2Q
►
• One variable-length and one fixed vector: ►
RBA
O2
B
RBO2
Q
RBQ
The four vectors form the following vector-loop
equation:
RAO2 + RBA - RBQ + RO2Q = 0
Can we replace vectors RBQ and RO2Q with a
vector from O2 to B ► which yields the
following vector-loop equation?
RAO2 + RBA - RBO2 = 0
Answer: NO! Remember the rule!
Useless!
► ►
RAO2 + RBA - RBQ + RO2Q = 0
P. Nikravesh, AME, U of A
Rule: When there is a sliding
joint in a mechanism, we must
define a variable-length vector
along or parallel to the axis of
the joint.
Vector loop
Fundamentals of Analytical Analysis 2
Vector Loop For An Inverted Slider-Crank
The ground link, the three moving links,
the three pin joints (A, O2, and O4), and
the sliding joint form a closed chain.
The following vectors form a loop: ►
A
• Two fixed-length vectors
RAB
• One variable-length and variable-angle
vector
RAO2
• Two fixed vectors (or one vector) for the
ground
O2
RO2Q
The vectors form the following vector-loop
equation:
Q
RAO2 - RAB - RBO4 - RO4Q + RO2Q = 0
Can we replace vectors RAB and RBO4
with a vector from O4 to A
► which
yields the following vector-loop equation?
RAO2 - RAO4 - RO4Q + RO2Q = 0
Answer: NO! Remember the rule!
P. Nikravesh, AME, U of A
Useless!
►
B
RAO4
RBO4
RO4Q
O4