Static Force Analysis

Introduction

We are now ready for a study of the dynamics of machines and systems. Such a study is usually simplified by starting with the statics of such systems. In our studies of kinematic analysis we limited ourselves to consideration of the geometry of the motions and of the relationships between displacement and time. The forces required to produce those motions or the motion that would result from the application of a given set of forces were not considered.

In the design of a machine, consideration of only those effects that are described by units of length and time is a tremendous simplification. It frees the mind of the complicating influence of many other factors that ultimately enter into the problem, and it permits our attention to be focused on the primary goal, that of designing a mechanism to obtain a desired motion. That was the problem of kinematics, covered in the previous chapters of this book.

The fundamental units in kinematic analysis are length and time; in dynamic analysis they are length, time, and force.

Forces are transmitted between machine members through mating surfaces-that is, from a gear to a shaft or from one gear through meshing teeth to another gear, from a connecting rod through a bearing to a lever, from a V-belt to a pulley, from a cam to a follower, or from a brake drum to a brake shoe. It is necessary to know the magnitudes of these forces for a variety of reasons. The distribution of these forces at the boundaries of mating surfaces must be reasonable, and their intensities must remain within the working limits of the materials composing the surfaces. For example, if the force operating on a sleeve bearing becomes too high, it will squeeze out the oil film and cause metal-to-metal contact, overheating, and rapid failure of the bearing. If the forces between gear teeth are too large, the oil film may be squeezed out from between them. This could result in flaking and spalling of the metal, noise, rough motion, and eventual failure. In our study of dynamics we are interested principally in determining the magnitudes, directions, and locations of the forces, but not in sizing the members on which they act.

Some of the new terms used in this phase of our study are defined as follows.

Force

Our earliest ideas concerning forces arose because of our desire to push, lift, or pull various objects. So force is the action of one body acting on another. Our intuitive concept of force includes such ideas as magnitude, direction, and place of application, and these are called the characteristics of the force.

Matter

Matter is any material substance; if it is completely enclosed, it is called a body.

Mass

Newton defined mass as the quantity of matter of a body as measured by its volume and density. This is not a very satisfactory definition because density is the mass of a unit volume. We can excuse Newton by surmising that perhaps he did not mean it to be a definition. Nevertheless, he recognized the fact that all bodies possess some inherent property that is different than weight. Thus a moon rock has a certain constant amount of substance, even though its moon weight is different from its earth weight. The constant amount of substance, or quantity of matter, is called the mass of the rock.

Inertia

Inertia is the property of mass that causes it to resist any effort to change its motion.

Weight

Weight is the force that results from gravity acting upon a mass. The following quotation is pertinent:

The great advantage of SI units is that there is one, and only one, unit for each physical quantity-the meter for length, the kilogram for mass, the newton for force, the second for time, etc. To be consistent with this unique feature, it follows that a given unit or word should not be used as an accepted technical name for two physical quantities. However, for generations the term "weight" has been used in both technical and nontechnical fields to mean either the force of gravity acting on a body or the mass of the body itself. The reason for the double use of the term for two physical quantities force and mass-is attributed to the dual use of the pound units in our present customary gravitational system in which we often use weight to mean both force and mass.

Particle A particle is a body whose dimensions are so small that they may be neglected. The dimensions are so small that a particle can be considered to be located at a single point; it is not a point, however, in the sense that a particle can consist of matter and can have mass, whereas a point cannot.

Rigid Body All real bodies are either elastic or plastic and will deform, though perhaps only slightly, when acted upon by forces. When the deformation of such a body is small enough to be neglected, such a body is frequently assumed to be rigid-that is, incapable of deformation-in order to simplify the analysis. This assumption of rigidity was the key step that allowed the treatment of kinematics in all previous chapters of this book to be completed without consideration of the forces. Without this simplifying assumption of rigidity, forces and motions are interdependent and kinematic and dynamic analysis require simultaneous solution.

Deformable Body The rigid-body assumption cannot be maintained when internal stresses and strains due to applied forces are to be analysed. If stress is to be found, we must admit to the existence of strain; thus we must consider the body to be capable of deformation, even though small. If the deformations are small enough, in comparison to the gross dimensions and motion of the body, we can then still treat the body as rigid while treating the motion (kinematic) analysis, but must then consider it deformable when stresses are to be found. Using the additional assumption that the forces and stresses remain within the elastic range, such analysis is frequently called elastic-body analysis.

Newton's Laws

As stated in the Principia, Newton's three laws are:

[Law I] Everybody perseveres in its state of rest or uniform motion in a straight line, except in so far as it is compelled to change that state by impressed forces.

[Law 2] Change of motion is proportional to the moving force impressed, and takes place in the direction of the straight line in which such force is impressed.

[Law 3] Reaction is always equal and opposite to action; that is to say, the actions of two bodies upon each other are always equal and directly opposite.

For our purposes, it convenient to restate these laws:

Law 1 If all the forces acting on a particle are balanced, that is, sum to zero, then the particle will either remain at rest or will continue to move in a straight line at a uniform velocity.

Law 2 If the forces acting on a particle are not balanced; the particle will experience an acceleration proportional to the resultant force and in the direction of the resultant force.

Law 3 When two particles interact, a pair of reaction forces come into existence; these forces have the same magnitudes and opposite senses, and they act along the straight line common to the two particles.

Newton's first two laws can be summarized by the equation

which is called the equation of motion for body j. In this equation, AGj is the absolute acceleration of the center of mass of the body j which has mass mj, and this acceleration is produced by the sum of all forces acting on that body-that is, all values of the index i. Both Fij and AGj are vector quantities.