5. Static force analysis

5.1. Force Analysis of Mechanisms

A study of dynamics of machinery cannot be undertaken without a thorough understanding of the basic principles of statics. In this chapter we will review some of these principles of statics and apply these principles to static analysis of machines. We shall discuss static force analysis at length since the same type of analysis will be used in dynamic problems, to be covered later. We shall also discuss the friction forces which are of importance in most moving machinery, and the method of analyzing the static forces in machines by use of the velocity analysis. This latter method is of particular advantage in determining the mechanical advantage of certain machines.

The graphical method of force analysis is a powerful tool and is particularly applicable to the analysis of machinery where the analysis is usually quite complicated. This method will be used in most cases in the analyses to follow. The graphical method has the advantage of being much quicker in most cases, and it usually gives us a better understanding of the forces involved. In dynamic analysis of machinery which we shall consider later on, dynamics problems will be reduced to statics problems by the use of D’Alembert’s principle.

Since most mechanisms which we encounter are plane mechanisms, we can analyze the forces as though they were plane forces. Because of the fact that the links do not always lie in the same plane the forces do not actually lie in the same plane but lie in parallel planes. Therefore such forces can be handled as though they were in the same plane by projecting them all into one plane. The moments caused by the offset, if they are appreciable, can be considered after the plane force analysis is completed.

The force analysis consists of determining all the forces and couples acting on each individual link and the forces acting on all the pairs so that the links and joints will not fail under the imposed forces.

One can set up a general procedure for the force analysis of mechanisms. It is as follows:

1. Draw free-body diagrams of all the links in the mechanisms; for two- and three-force members the directions of the forces can usually be determined.

2. Start the analysis with the links farthest along the mechanism from the links on which the unknown external forces or moments are acting. Usually these will have to be considered simultaneously with an adjoining link, and the analysis can be started with any two links which contain only six unknown quantities in combination, making use of Newton’s law of action and reaction.

3. These two links can then be analyzed by the use of a force polygon.

4. Proceed along the chain from link to link making use of the known forces found in preceding analyses to the link containing the unknown moment or force.

In the present chapter we will consider only the effect of external forces applied to the mechanism; we will not be concerned with the inertial forces. These forces will be discussed later. That is why such an analysis, ignoring the forces concerned with accelerations, is called a static analysis. The weights of the individual links, while usually considered static forces in mechanics, especially in structures, will also be left until we discuss acceleration forces. These weights are actually associated with the inertial forces and their line of action will be covered in our discussion of dynamic forces.

Then, from our viewpoint, a static analysis will be concerned only with the forces associated with the work done by the machine, or the forces associated with the primary functions of the machine. Examples of such forces are the gas pressure on the piston of an internal combustion engine, the forces imposed on the jaws of a rock crusher by a stone being crushed.