PMP & LMDFE Systems: Kinetics

Transcription

1 . 2 PMP & LMDFE Systems: Kinetics 2 1

2 Chapter 2: PMP & LMDFE SYSTEMS: KINETICS 2 2 TABLE OF CONTENTS Page 2.1. Introduction Newton Laws of Motion First Law Second Law Third Law Forces Concept of Force Force Decomposition Action-Reaction (AR) Force Decomposition Internal-External Force Decomposition Particle Forces External and Internal Forces Resultant Force and Moment

3 NEWTON LAWS OF MOTION 2.1. Introduction This Chapter continues the study of point-mass particle (PMP) systems by going over their kinetics. This is the study of the effect of forces upon the motion. Modeling such effects using Newton s laws results in the equations of motion (EOM) of the system. The concept of work is briefly introduced in this Chapter, since it is necessary to support some criterion for force decomposition. The concept and applications of work and energy principles are elaborated in following Chapters Newton Laws of Motion The laws of motion enuntiated by Newton in his Principia [269] will be stated in axiomatic form, as derivable from observation First Law Original Statement. Lex I: Corpus omne perseverare in status suo quiescendivei movendi uniformiter in directum, nisi quatenus a viribus impressis cogito statum illum mutare. Literal English Translation. An object at rest will remain at rest unless acted upon by an external and unbalanced force. An object in motion will remain in motion unless acted upon by an external and unbalanced force. Modern Version. A body remains in a state of rest or of uniform rectilinear motion unless compelled to change its state by acting forces. This is essentially Galileo s law of inertia, enunciated by him long before Newton: A body moving on a level surface will continue in the same direction at a constant speed unless disturbed. The concept was further developed by Descartes, whereas the term inertia was introduced by Kepler. However Newton recognized the fundamental important of the concept and stated it as the First Law of Motion. Inadefinition preceding the statement of this law, Newton introduces the body quantity of motion as the measure of the same, arising from the velocity and the quantity of matter conjunctly [347, p. 3]. By quantity of matter Newton meant the mass of the body. The product of mass and velocity is what we now call momentum. Since the mass is a scalar whereas velocity is a vector, the momentum is also a vector: p = m v = m u. (2.1) Of course vectors have not been invented at Newton s time. 1 Using the modern notation (2.1) we can restate the First Law succintly as: if no forces act on the body, the momentum p is conserved: in which C is independent of time. p = C, (2.2) 1 Vector analysis was simultaneously created by the American mathematician J. W. Gibbs and the English electrical engineer O. Heaviside in the late XIX Century; cf. [76]. The notation of Gibbs is that primarily in use today. Vector analysis and matrix algebra were not linked until the appearance of digital computers in the 1950w. This synergistic marriage was delayed by the unseemly preoccupation of algebrists with determinants for over a century. 2 3

4 Chapter 2: PMP & LMDFE SYSTEMS: KINETICS Second Law Original Statement. Lex II: Mutationem motus proportionalem vi motrici impressate, et fieri secumdum lineam rectam qua vis illa imprimatur. Literal English Translation. The alteration of motion is ever proportional to the motive force impressed, and is made in the right line in which the force is impressed. Modern Version. The change in motion is proportional to the acting force, and takes place in the direction of the straight line along which the force acts. Using again the modern concept of momentum (2.1), Newton s change in motion can be briefly stated as: the change with time of the momentum: ṗ = dp/dt is proportional to the acting force on the body, denoted by f. Thus the second law can be written ṗ = f. (2.3) This may be labeled as the law of momentum. Evidently if f = 0, p is constant in time, thus recovering the conservation law (2.2). If the mass m is constant in time, (2.3) reduces to the more familiar mass times acceleration expression m v = m ü = f. (2.4) This form is called Newton s law of acceleration in the literature. Although more restrictive, (2.4) has a wide application range, since time-varying mass problems are rare in Classical Mechanics Third Law Original Statement. Lex III. Actionem contrariam semper et æqualem esse reactionem: sive corporus duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi. Literal English Translation. All forces occur in pairs, and these two forces are equal in magnitude and of opposite directions. Modern Version. For every action there is an equal and opposite reaction. This is the principle of action and reaction. It says that forces are always paired in nature. Example from [347, p. 6]: the falling stone attracts the Earth just as strongly as the Earth attracts the stone. This law makes it possible to move from the mechanics of single mass points to that of systems, whether at rest (statics) or moving (dynamics) Forces The term force appears in each of the three laws introduced above. In [384, p. 527], Truesdell cites Hamel as writing in the concept of force lies the chief difficulty in the whole of mechanics. These difficulties motivated Hertz [190] to construct a Mechanics without the force concept. His attempt, however, was not successful within the framework of Classical Mechanics. So what is force? The dictionary is no help. In fact, definitions pertinent to mechanical forces given by the American Heritage Dictionary are: 2 They are more important in relativistic mechanics, because particle mass depends on its velocity relative to light speed. 2 4

5 FORCES 1. Capacity to do work or cause physical change; strength; power. 2. Power made operative against resistance; exertion. 3. The use of such power or exertion. These three statements confuse power and force, a serious mistake (as defined later, power is work done in unit time). The capacity to do work is energy, which has dimensions of force times motion. Hence the foregoing definitions are technically useless Concept of Force As suggested by Sommerfeld [347], in Classical Mechanics force can be concretely defined as the right hand side of the law of momentum (2.3). It is true that the left side still contains the mass, which so far has only been defined qualitately as quantity of matter. But some properties of forces can now be stated. First, since the momentum defined in (2.1) is a vector, so is its time rate ṗ. Consequently force is a vector: it has direction and magnitude. Second, the rule of composition of forces appears as a corollary to the laws of motion: two forces applied to the same mass point compound to act like the diagonal of the parallelogram formed by their vectors. If the forces are denoted by f 1 and f 2, that parallelogram law states that the law of momentum (2.3) is simply ṗ = f 1 + f 2 = f. (2.5) Thus forces add as vectors to provide a resultant force. The above equation expresses succintly the principle of superposition of forces Force Decomposition The three Newtonian laws of motion, plus the principle (2.5), are sufficient to develop all of Classical Mechanics for PMP systems. Because of the importance of the force superposition principle, some authors elevate its status to that of a Fourth Law of Motion. Without taking sides in this issue, the crucial role of (2.5) lies in the possibility of decomposing forces. Force decomposition is crucial to the analysis of general PMP systems. The total force acting on each particle is in general a composition of effects that originate from different sources. For example, some forces may depend on position, some on displacements, others on velocities, and others on accelerations. Some forces may be prescribed as data, while others, called reactions, depend on kinematic constraints 4 and are unknown until the response is obtained. In summary, force decomposition is essential to the construction of the correct equations of motion, their boundary and initial conditions, as well as their solution. It turns out that for complex dynamic systems there are several possible decompositions. Obvious question: which one should be used? Quick generic answer: whatever it works. More precisely, a decomposition that allows solvable equations of motion (EOM) to be formulated and solved in a 3 This is a precise restatement in vector notation of Galileo s fuzzier Principle of Superposition: If a body is subjected to two physical influences that are independent of each other, it responds to each without modifying its response to the other. Galileo used this principle to study the motion of projectiles by superposing vertical and horizontal motions. 4 Kinematic constraints were introduced in

6 Chapter 2: PMP & LMDFE SYSTEMS: KINETICS 2 6 systematic way. The qualifier systematic is important. Often ad-hoc decompositions allow a case by case investigation of specific problems, but are not extendible beyond those. Decompositions based on physical intuition do exist. A common one in older textbooks is: contact forces (e.g. collision) versus action-at-a-distance forces (e.g. gravity). The distinction is easy to understand and explain but largely useless aside of some special problems. Several proven top-level force decompositions are introduced in the following sections Action-Reaction (AR) Force Decomposition A useful top-level force decomposition givide forces into action and reaction: { Applied forces Action forces Interaction forces Forces { (2.6) E-reactions Reaction forces C-reactions Reaction forces are those associated with kinematic constraints. They are the forces required to exactly enforce the constraint if the kinematic devices on which it depends are removed. All remaining forces are action forces. If the dynamic system is unconstrained, all forces are of action type. Reaction forces are subclassified into E-reactions and C-reactions. E-reactions are associated with the environment of the dynamic systemwhile C-reactions are associated with constraints internal to the system They are also called external reactions and internal reactions, respectively. Reaction forces are in subclassified into applied forces and interaction forces. Applied forces are those specified directly on a particle from physical effects external to the dynamic system. For example: gravity, fluid pressure, electromagnetic fields. Interaction forces are particle-to-particle effects, which typically depend on relative displacements, velocities or accelerations of the linked particles. Why is this decomposition useful? It helps to select unknowns. If all or part of the reaction forces are carried along in the EOM, the result is the Lagrange multiplier method, which is more fully described in the next Chapter. (TBC) 2 6