P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 2Introduction 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 Fundamentals of Analytical Analysis 2 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. Vector loop Vector Loop

P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 2 The ground link, the three moving links, and the four pin joints (A, B, O 2, and O 4 ) form a closed chain. The following four vectors form a loop: The four vectors form the following vector- loop equation: R AO 2 + R BA - R BO 4 - R O 4 O 2 = 0 The ground link can be represented by vector R O 4 O 2 or it can be presented as the sum of two vectors, one horizontal and one vertical: The vector loop equation can be revised as: R AO 2 + R BA - R BO 4 - R O 4 Q - R QO 2 = 0 Either set of ground vectors could be used for kinematic analysis. Vector loop ► Vector Loop For A Fourbar A O4O4 B R O4O2 R AO2 O2O2 R BO4 R BA ► Q R QO2 R O4Q P

P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 2 The ground link, the three moving links, the three pin joints (A, B, and O 2 ), 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: R AO 2 + R BA - R BO 2 = 0 Note that vector R BO 2 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 O 2. Vector loop ► Vector Loop For A Slider-Crank A B R BO2 R AO2 O2O2 R BA ► 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 Fundamentals of Analytical Analysis 2 The ground link, the three moving links, the three pin joints (A, B, and O 2 ), 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: R AO 2 + R BA - R BQ + R O 2Q = 0 Can we replace vectors R BQ and R O 2Q with a vector from O 2 to B which yields the following vector-loop equation? R AO 2 + R BA - R BO 2 = 0 Answer: NO! Remember the rule! R AO 2 + R BA - R BQ + R O 2Q = 0 Vector loop ► Vector Loop For An Offset Slider-Crank A B R BQ R AO2 O2O2 R BA ► Q R O2Q ► R BO2 ► 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!

P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 2 The ground link, the three moving links, the three pin joints (A, O 2, and O 4 ), 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: R AO 2 - R AB - R BO 4 - R O 4 Q + R O 2 Q = 0 Can we replace vectors R AB and R BO 4 with a vector from O 4 to A which yields the following vector-loop equation? R AO 2 - R AO4 - R O 4 Q + R O 2 Q = 0 Answer: NO! Remember the rule! Vector loop ► Vector Loop For An Inverted Slider-Crank A R O2Q R AO2 O2O2 R AB Q R O4Q O4O4 B R BO4 ► R AO4 ► Useless!