Introduction

1. 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.

2. 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.

3. 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: The four vectors form the following vector-loop 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 ► B RBA A RBO4 RAO2 O4 RO4O2 RO4Q ► Q O2 RQO2

4. 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). 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. A ► 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.

5. 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: • One variable-length and one fixed vector: • 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! • RAO2 + RBA - RBQ + RO2Q = 0 A RAO2 RBA O2 B RO2Q RBO2 ► ► Q RBQ ► 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. Useless! ► ►

6. 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: • Two fixed-length vectors • One variable-length and variable-angle vector • Two fixed vectors (or one vector) for the ground • The vectors form the following vector-loop equation: • 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! ► A RAB RAO2 O2 B RAO4 RO2Q RBO4 O4 Q RO4Q ► Useless! ►