Chapter 2: Position Analysis of Mechanisms

2.0 Introduction to Position Analysis

Once a tentative design of a mechanism has been synthesized—perhaps through an intuitive process or a first draft based on previously studied kinematic principles—the next fundamental step is to analyze its position. This analysis is the cornerstone of the entire sequence of kinematic and dynamic analyses. Without the ability to determine the position of each link and point in a mechanism at every instant, it is impossible to advance in understanding its behavior.

The primary objective of kinematic analysis is to determine the accelerations of all moving parts of the assembly. This is crucial because, according to Newton’s second law (F = ma), dynamic forces are proportional to acceleration. To calculate the stresses in the mechanism’s components and ensure that the proposed machine will not fail under operating conditions, it is imperative to know the static and dynamic forces acting on its parts. However, to obtain these dynamic forces, accelerations are first required, which are derived from velocities, and these, in turn, are derived from positions. For example, in a four-bar linkage, you might need to calculate the positions, velocities, and accelerations of the links for every two degrees of the input crank, which would involve 180 positions per revolution.

As you will see in this chapter, the position problem can be solved using various methods. Historically, graphical methods were the only practical way to approach this type of analysis. However, the proliferation of inexpensive microcomputers in recent years has revolutionized engineering practice, making computer-solved analytical methods much more efficient and accurate. Once an analytical solution is derived for a specific mechanism, it can be quickly solved (by a computer) for all positions of interest.

In this chapter, we will explore both graphical and analytical approaches for basic types of mechanisms. Position analysis is the first link in the analysis chain; later, in subsequent chapters, we will delve into velocity and acceleration analysis. Thus, here we will start with a given design and analyze how the mechanism will move, and in Chapter 3 (Synthesis of Mechanisms), we will address how to design a mechanism to move according to predefined criteria.