The Role Of The Kinetic Diagram In The Teaching Of Introductory Rigid Body Dynamics Past, Present, And Future

The introductory engineering dynamics course is widely regarded as one of the most difficult courses that the undergraduate engineering student takes. Further, the rigid-body area of this dynamics course is considered much more difficult than the particle area. One reason for the latter statement is that we have not yet progressed to the best steady-state teaching strategy in the area of rigid-body kinetics. The purpose of this paper is to review the history and current state of affairs in this narrow area and then to advocate a better strategy. Recommendations are made in regard to both diagrams and corresponding equations of motion. Introduction Dynamics did not become a significant issue until the beginning of the machine age. Mechanicists were accustomed to a zero on the right-hand side of the governing equations in statics, so the first direction of particle dynamics was to include a -ma term on the left side of dynamics equations so that the right-hand zero could be retained. Although sometimes referred to as D’Alembert’s Principle, this technique should be called dynamic equilibrium (D’Alembert’s Principle is a virtual-work principle). This -ma term has been called an inertia force, an effective force, a reversed effective force, etc. The technical community eventually took the position that dynamics should not be treated as a special case of statics, but rather the other way around. In other words, we soon placed the ma term on the right side of the equations of motion and included only real (contact and body) forces on the left side. Some textbooks went through a period in which a kinetic diagram (sometimes called a resultant-force diagram) was drawn (in addition to the free-body diagram (FBD)). This diagram merely showed an ma vector (or its components). The usual arrangement was to draw the FBD and then write an equal sign with the kinetic diagram (KD) to the right. Such practice seems to have been largely terminated for particle dynamics. It is the author’s position that, with the conventional teaching of particle dynamics as outlined above, we are in the steady-state, terminal teaching configuration. We note that the teaching of particle statics and particle dynamics is now of identical format. There is the same type of FBD showing only real forces, followed by application of governing equations. The only difference is that, for particle dynamics, the right-hand sides of the governing equations are not zero. P ge 7.182.1 Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition Copyright © 2002, American Society for Engineering Education The story is not yet complete in rigid-body dynamics, however. Again, moving back to the dawn of the machine age, the first attempts at planar rigid-body dynamics were based upon the familiar customs of statics: ma − ( a refers to the mass center here) and Iα − vectors ( I refers to the mass center) were placed directly on the FBD so that they could be treated as an effective force and couple, allowing the use of statics-like equations of motion. As was the case in statics, the educational community eventually rebelled against considering the ma − and Iα − vectors to be the same as real forces and moments in association with the principle of dynamic equilibrium. Rather than eliminating these quantities completely from diagrammed material, however, they were moved to a second diagram (the kinetic diagram) as positive quantities – in effect, they were moved from the left to the right side of the governing equation for rotational motion. The reason this second diagram became “necessary” had to do with the fact that the ma vector might enter the moment equation (depending on what point was chosen as the moment center). Most of the popular introductory dynamics texts use the kinetic-diagram approach outlined above, but only for rigid-body kinetics and not for particle kinetics. Thus, at present, the mainstream teaching strategy consists of one approach for particle and rigid-body statics and particle kinetics, but a different approach (in terms of diagrams) for rigid-body kinetics. The third approach to rigid-body kinetics is to use only a FBD and appropriate governing equations, so that all categories of statics and dynamics are treated in parallel fashions. This philosophy, which was recommended by the author in 1982, will be expanded in the sections that follow. Discussion of the Three Methods In Figure 1, we show generic diagrams associated with the three approaches introduced above, as applied to an arbitrary two-dimensional slab. In the dynamic-equilibrium approach of Figure 1a, we show the applied forces along with ma − and Iα − vectors. The governing equations associated with this approach are F a 0 M α 0 G m I − = − = (1) Figure 1a: The Dynamic-Equilibrium Approach P ge 7.182.2 Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition Copyright © 2002, American Society for Engineering Education One of the disadvantages associated with this method, in addition to treating the kinetic results ( ma and Iα ) the same as if they were causing agents (the applied forces and couples), is that one must decide on a strategy for assuming the senses of a and α. Does one anticipate the correct sense of these two quantities (and then reverse them!), or does one always assume that they are in the positive coordinate directions? In Figure 1b, we show a conventional FBD on the left and on the right is the kinetic diagram indicating the ma and Iα vectors. The governing equations associated with this approach are generally written as F a (or M ρ a) P P m M I mad I m α α = = + = + × (2) Figure 1b: The Kinetic-Diagram Approach The governing rotational equation can be derived in at least two ways. First, and best, one can develop it from first principles. This may be done by considering a general three-dimensional system of particles or by considering a rigid slab as is done in Appendix A. A second derivation is to perform a moment sum about any point in the FBD and equate the results to the moment sum about the same point in the kinetic diagram (KD). The argument is that because the two diagrams are equivalent, performing the same operations in the respective diagrams must also be equivalent. The author finds this second derivation somewhat lacking. But, in fairness, it must be noted that the introductory course does usually not allow sufficient time for the first derivation. And the first derivation does require more effort on the part of the student. An often-cited advantage of the KD approach is the ability to sum moments about a convenient point – that is, a point through which several unknown forces might pass. As we will note below, this advantage is not confined to the KD approach. Furthermore, the method involves the acceleration of the mass center G, which may not be known. As a final comment on the kinetic-diagram method, we note that its use in three-dimensional problems becomes quite labored. A good question for all mechanics educators is “If we do not use the KD for 3-D problems, then why do we introduce it for 2-D problems?” P ge 7.182.3 Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition Copyright © 2002, American Society for Engineering Education In Figure 1c is the pure FBD approach. It is clear that we can always sum moments about the mass center G or about a fixed point O on the body (if one exists). We list as the equation of rotational motion Eq. (A/9) of Appendix A, in order that we might cite its advantages. The author believes that this form is much more powerful than Eq. (A/8), because it gives one the ability to conveniently sum moments about a point whose acceleration is known, which is a frequent occurrence. It of course reduces to the familiar forms if P is the mass center or if P is a nonaccelerating point fixed to the body. F a M α ρ a P P P m I m = = + × (3) Figure 1c: The Pure FBD Approach We now turn to a specific problem: Suppose the 3000-lb car is brought to a halt by skidding all tires. If the coefficient of kinetic friction is 0.8, determine the normal reaction force under the pair of front wheels, that under the pair of rear wheels, and the deceleration during the braking period. Figure 2: A Sample Problem Three solutions to the problem are indicated below.