Translational Dynamics

7 Key excerpts on "Translational Dynamics"

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  Rigid Body Dynamics

  • Joaquim A. Batlle, Ana Barjau Condomines(Authors)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  This correlation is sometimes described in a simplistic way as a causality relationship: “Dynamics is the study of the motion of bodies caused by the action of forces.” The equations of Newtonian dynamics are differential equations relating the second order derivative of the objects generalized coordinates to the coordinates and their first derivatives, all of them at a same time instant. This simultaneity of motion variables makes it difficult (and formally impossible) to distinguish “causes” and “effects,” as the former should actually precede the latter. That distinction becomes a convention. Principle of Absolute Simultaneity The principle of absolute simultaneity 1 states, in short, that a sequence (order of succession) of events is the same for all observers (or independent from the reference frame). In Newton’s conception of the universe, absolute time goes hand in hand with absolute space: they constitute an immutable stage where physical events occur, they are independent external realities. Other Assumptions Last but not least: as in any scientific theory, Newtonian dynamics have a limited field of application (directly related to the experimental limitations of the seventeenth century!). Outside that field, it yields inaccurate results. The main limitations are:  Low-speed dynamics: objects moving with speeds comparable to the speed of light cannot be treated successfully with this theory.  Medium length scale dynamics: molecular dynamics and long-reach astronomy are also out of scope.  Electromagnetic phenomena are excluded. 1 Batlle, J.A. and Barjau Condomines, A. (2019) Rigid Body Kinematics, Cambridge University Press, chapter 1. 2 Particle Dynamics 1.2 Galilean and Non-Galilean Reference Frames When we enter the field of Newtonian dynamics, we are confronted with a surprising fact: all reference frames are not equivalent for the formulation of the dynamical equations (or of the fundamental laws of Newtonian dynamics).

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  The Cambridge Handbook of Physics Formulas

  • Graham Woan(Author)
  • 2000(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Chapter 3 Dynamics and mechanics 3.1 Introduction Unusually in physics, there is no pithy phrase that sums up the study of dynamics (the way in which forces produce motion), kinematics (the motion of matter), mechanics (the study of the forces and the motion they produce), and statics (the way forces combine to produce equilibrium). We will take the phrase dynamics and mechanics to encompass all the above, although it clearly does not! To some extent this is because the equations governing the motion of matter include some of our oldest insights into the physical world and are consequentially steeped in tradition. One of the more delightful, or for some annoying, facets of this is the occasional use of arcane vocabulary in the description of motion. The epitome must be what Goldstein 1 calls “the jabberwockian sounding statement” the polhode rolls without slipping on the herpolhode lying in the invariable plane , describing “Poinsot’s construction” – a method of visualising the free motion of a spinning rigid body. Despite this, dynamics and mechanics, including fluid mechanics, is arguably the most practically applicable of all the branches of physics. Moreover, and in common with electromagnetism, the study of dynamics and mechanics has spawned a good deal of mathematical apparatus that has found uses in other fields. Most notably, the ideas behind the generalised dynamics of Lagrange and Hamilton lie behind much of quantum mechanics. 1 H. Goldstein, Classical Mechanics , 2nd ed., 1980, Addison-Wesley.

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  Advanced Dynamics

  • Donald T. Greenwood(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 Introduction to particle dynamics In the study of dynamics at an advanced level, it is important to consider many approaches and points of view in order that one may attain a broad theoretical perspective of the subject. As we proceed we shall emphasize those methods which are particularly effective in the analysis of relatively difficult problems in dynamics. At this point, however, it is well to review some of the basic principles in the dynamical analysis of systems of particles. In the process, the kinematics of particle motion will be reviewed, and many of the notational conventions will be established. 1.1 Particle motion The laws of motion for a particle Let us consider Newton’s three laws of motion which were published in 1687 in his Prin-cipia . They can be stated as follows: I. Every body continues in its state of rest, or of uniform motion in a straight line, unless compelled to change that state by forces acting upon it. II. The time rate of change of linear momentum of a body is proportional to the force acting upon it and occurs in the direction in which the force acts. III. To every action there is an equal and opposite reaction; that is, the mutual forces of two bodies acting upon each other are equal in magnitude and opposite in direction. In the dynamical analysis of a system of particles using Newton’s laws, we can interpret the word “body” to mean a particle, that is, a certain fixed mass concentrated at a point. The first two of Newton’s laws, as applied to a particle, can be summarized by the law of motion : F = m a (1.1) Here F is the total force applied to the particle of mass m and it includes both direct contact forces and field forces such as gravity or electromagnetic forces. The acceleration a of the particle must be measured relative to an inertial or Newtonian frame of reference. An example of an inertial frame is an xyz set of axes which is not rotating relative to the “fixed”

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  Mechanical Engineering Systems

  • Richard Gentle, Peter Edwards, William Bolton(Authors)
  • 2001(Publication Date)
  • Newnes

    (Publisher)

  4

  Dynamics

  Summary

  This chapter deals with movement. In the first part the movement is considered without taking into account any forces. This is a subject called kinematics and it is important for analysing the motion of vehicles, missiles and engineering components which move backwards and forwards, by dealing with displacement, speed, velocity and acceleration. These quantities are defined when we look at uniform motion in a straight line. This subject is extended to look at the particular case of motion under the action of gravity, including trajectories. This chapter also looks at how the equations of motion in a straight line can be adapted to angular motion. Finally in the first half the subject of relative velocity is covered as this is very useful in understanding the movement of the individual components in rotating machinery.

  In the second part of this chapter we consider the situation where there is a resultant force or moment on a body and so it starts to move or rotate. This topic is known as dynamics and the situation is described by Newton’s laws of motion. Once moving forces are involved, we need to look at the mechanical work that is being performed and so the chapter goes on to describe work, power and efficiency. Newton’s original work in this area of dynamics was concerned with something called momentum and so this idea is also pursued here, covering the principle of conservation of momentum. The chapter extends Newton’s laws and the principle of conservation of momentum to rotary motion, and includes a brief description of d’Alembert’s principle which allows a dynamic problem to be converted into a static problem.

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  Mechanics and Strength of Materials

  • Bogdan Skalmierski(Author)
  • 2013(Publication Date)
  • Elsevier

    (Publisher)

  CHAPTER 2 The Dynamics of a Particle 2.1 Fundamental definitions and theorems In this chapter we shall discuss the dynamic aspects of a particle in motion. We begin with the basic laws of dynamics. Axioi 1 (Newton's second law). If a force P acts on a particle, the acceleration thus produced is proportional to that force, which can be written as follows: P= m a, (1) where m is the mass of the particle. We shall treat mass as a primary concept. The force P should be regarded as the resultant of the forces acting on the particle, that is, n P = R j . (la) f= 1 The cited law brings into association three basic concepts (force, mass and motion) of mechanics. It is valid in inertial systems (see Chapter 5). Under the SI system, the unit of force is the newton: 1 kgms -2 = 11. For a unit of force we can also take the force with which the earth attracts 1 kg of mass: 1 kgf = 1 kg • g, where g = 9.80665 m s — 2 and is the normal value of acceleration of gravity. The equality sign is valid between inert and heavy mass. Axioi 2 (Newton's third law). If a particle A acts on another particle B with force P AD , then simultaneously B acts on A with force PBA of equal absolute value but with an opposite sense, i.e. R A B + PBa = = 0 . (2) (4) a V P i = 70 THE DYNAMICS OF A PARTICLE Ch. 2 This is known as the law of action and reaction. We shall now introduce the definition of work. By work one should understand a process in which resistance is being overcome along a certain route. This definition is, however, imprecise, and for that reason it is better to define work in concise mathematical notation: Work analytically formulated is a curvilinear integral df B W = P • ds . (3) A If under the integral (3) a total differential occurs, then the forces doing the work are said to have potential V. A decrease in potential is tantamount to an increase in work: — ~ V = d W. Therefore Potential forces, as it will easily be seen, act in the direction of the maximum drop of the potential.

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  Stress, Strain, and Structural Dynamics

  An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes

  • Bingen Yang(Author)
  • 2005(Publication Date)
  • Academic Press

    (Publisher)

  Section 9.6 .

  9.2 Dynamics of Particles

  9.2.1 Preliminaries

  Units

  Some commonly used quantities in engineering mechanics are listed in Table 9.2.1 , where units in both the Standard International (SI) and U.S. Customary Systems are shown.

  TABLE 9.2.1 Units

  Newton’s Laws

  Classical mechanics was established based upon the following three laws of motion that were first stated by Isaac Newton in 1687:

  First Law: A particle remains at rest or moves with a constant velocity along a straight line if the sum of the forces acting on it is zero.

  Second Law: The rate of change of the linear momentum of a particle is equal to the sum of the forces acting on it.

  Third Law: The forces exerted by two particles upon each other, which are called the action and reaction forces, are equal in magnitude, opposite in direction, and collinear.

  Mathematically, Newton’s second law is expressed by

  (9.1)

  where m and ν are the mass and velocity of the particle, respectively; F is the resultant or sum of all forces acting on the particle; and mv is the linear momentum of the particle. If the mass m is constant, Eq. (9.1) becomes

  (9.2)

  where is the acceleration of the particle. Equation (9.2) is a commonly used expression of Newton’s second law, which indicates that the acceleration of a particle of constant mass is proportional to the resultant force.

  Besides Newton’s laws of motion, Newton’s law of gravitation is also important in dynamics, especially in orbital dynamics; see Sec. 9.2.6 .

  Force

  The motion of a particle or body is caused by forces. A force is an action of pull or push applied by one body to another. A force is characterized by its magnitude, direction of action, and point of application. Thus, forces can be expressed by vectors, and superposition of forces follows the parallelogram rule of vector summation. For instance, the sum of forces F 1 and F 2 in Fig. 9.2.1(a) is obtained by the diagonal of the parallelogram formed by the forces, as shown in Fig. 9.2.1(b)

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  Intermediate Dynamics

  • Patrick Hamill(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  Once the acceleration of a body is known, the laws of kinematics determine its velocity and position at any later time. This means that if you know the net force acting on a body, you can calculate its position at any future time. The ability to predict the motion is the power of the second law. Recall that dynamics is the study of how a force affects the motion of a body. As you might suspect, dynamics usually involves accelerating bodies. However, in some situations there may be forces acting on a body but nevertheless the acceleration is zero. Consider, for example, a body acted upon by two equal and opposing forces. The effects of these forces cancel out and the body does not accelerate. Zero acceleration is an important special case in dynamics and is called statics. Statics is of particular interest to civil engineers who want to make sure the structures they design, such as bridges and skyscrapers, will have zero acceleration. Statics was treated briefly in Section 1.6 and will be dealt with in greater detail in Chapter 18. The principle of superposition states that if two or more forces act on a particle, the net effect is the same as that of a single force equal to the vector sum of all the forces. You will be exposed to the principle of superposition in other areas of physics. For example, the net electric field at a point is the vector sum of all the electric fields acting at that point. 8 7 If you are interested in some of the philosophical implications of mass, force, inertia and Newton’s laws, you might enjoy the three articles by Franck Wilczek entitled, “Whence the Force of F = ma?”. These articles were published in Physics Today in the issues of December 2004, July 2005, and October 2005. The article “Drop Test” by Adrian Cho (Science, 6 March, 2015, p. 1096) describes three experiments to measure the equivalence of inertial mass and gravitational mass to better than one part in ten trillion.

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