Planar Kinetics of a Rigid Body

10 Key excerpts on "Planar Kinetics of a Rigid Body"

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  Engineering Mechanics

  • Ping YI, Jun LIU, Feng JIANG(Authors)
  • 2022(Publication Date)
  • EDP Sciences

    (Publisher)

  Chapter 7 Planar Kinematics of Rigid Bodies Objectives  Classify three types of rigid-body planar motion: translation, rotation about a fixed axis and general plane motion.  Investigate rotation about a fixed axis and velocity and acceleration of particles on a rotating body.  Study absolute dependent motion of bodies undergoing planar motion.  Study velocity and acceleration of particles on a body undergoing general plane motion through a relative-motion analysis using a translating reference frame.  Examine the instantaneous center (IC) of rotation and determine velocity of particles on a body undergoing general plane motion by using IC. In machinery, the gears, cams, and links cannot be treated as particles. Their size and shape must be considered and rotation is an important aspect in the analysis of their motions. These objects are treated as rigid bodies. This chapter discusses only planar motion of rigid bodies. Rigid-body planar motion occurs when all the particles of the body move along paths that are equidistant from a fixed plane. For example, when brushing the blackboard, as long as the brush (neglecting its deformation) does not leave the blackboard, the distance from any particle on the brush to the blackboard remains unchanged during the motion. Then, any particle of the brush moves along a path that is equidistant from the fixed blackboard plane and the brush is undergoing planar motion. Many mechanical parts undergo planar motion, such as gears, cams and linkages shown in figure 7.1. DOI: 10.1051/978-2-7598-2901-9.c007 © Science Press, EDP Sciences, 2022 There are three types of rigid-body planar motion and they are defined as follows: Translation. Translation occurs if any line segment inside the body remains parallel to its original orientation during the motion.

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  Engineering Mechanics

  Dynamics

  • L. G. Kraige, J. N. Bolton(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  PART II Dynamics of Rigid Bodies Rigid-body kinematics describes the relationships between the linear and angular motions of bodies without regard to the forces and moments associated with such motions. The designs of gears, cams, connecting links, and many other moving machine parts are largely kinematic problems. Thor Jorgen Udvang/Shutterstock CHAPTER 5 Plane Kinematics of Rigid Bodies 5/1 Introduction In Chapter 2 on particle kinematics, we developed the relationships governing the displacement, velocity, and acceleration of points as they moved along straight or curved paths. In rigid-body kinematics we use these same relationships but must also account for the rotational motion of the body. Thus rigid-body kinematics in- volves both linear and angular displacements, velocities, and accelerations. We need to describe the motion of rigid bodies for two important reasons. First, we frequently need to generate, transmit, or control certain motions by the use of cams, gears, and linkages of various types. Here we must analyze the displacement, velocity, and acceleration of the motion to determine the design geometry of the mechanical parts. Furthermore, as a result of the motion generated, forces may be developed which must be accounted for in the design of the parts. Second, we must often determine the motion of a rigid body caused by the forces applied to it. Calculation of the motion of a rocket under the influence of its thrust and gravitational attraction is an example of such a problem. We need to apply the principles of rigid-body kinematics in both situations. This chapter covers the kinematics of rigid-body motion which may be analyzed as occur- ring in a single plane. In Chapter 7 we will present an introduction to the kinematics of motion in three dimensions.

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  Planar Multibody Dynamics

  Formulation, Programming with MATLAB®, and Applications, Second Edition

  • Parviz E. Nikravesh, Parviz Nikravesh(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  3

  Fundamentals of Planar Kinematics

  This chapter presents a summary of some fundamental concepts of planar kinematics. We first review the kinematics of a single particle, and then we consider the fundamental formulas for the kinematics of a single planar rigid body. Definitions for arrays of coordinates, constraint equations, degrees of freedom, and kinematic joints are also discussed.

  The kinematic definitions and fundamental formulas for kinematics of particles and rigid bodies that are discussed in this chapter will be used extensively throughout this textbook. Therefore, it is important to understand these simple but fundamental formulas at this point. Exercises with MATLAB® are provided to carry out most of the computations, even though most of the examples are very simple and the calculations may be performed on paper. It is our intent to use MATLAB as much as possible and be prepared to solve more complex and realistic problems whenever needed.

  3.1 A Particle

  The simplest body arising in the study of motion is a particle , or a point mass . Analysis of the behavior of a point mass can lead to the kinematic and dynamic analysis of a body, and eventually to the analysis of a system of bodies. Therefore, it is essential to study the kinematics of particles first. In this textbook, we use the terms particles and points interchangeably.

  3.1.1 Kinematics of a Particle

  The most fundamental step in planar kinematics is to describe the position of a particle or a point in a plane. A plane is described by a nonmoving Cartesian reference frame x–y . The position of a typical particle A in the plane is described by vector

  r →

  A

  as shown in Figure 3.1 . The components of the position vector

  r →

  A

  , or the x–y coordinates of particle A

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  Dynamics of Mechanical Systems

  • Harold Josephs, Ronald Huston(Authors)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  125 5 Planar Motion of Rigid Bodies — Methods of Analysis 5.1 Introduction In this chapter, we consider planar motion — an important special case of the kinematics of rigid bodies. Planar motion characterizes the movement of the vast majority of machine elements and mechanisms. When a body has planar motion, the description of that motion is greatly simplified. Special methods of analysis can be used that provide insight not usually obtained in three-dimensional analyses. We begin our study with a general dis-cussion of coordinates, constraints, and degrees of freedom. We then consider the planar motion of a body and the special methods of analysis that are applicable. 5.2 Coordinates, Constraints, Degrees of Freedom In our discussion, we will use the term coordinate to refer to a parameter locating a particle or to a parameter defining the orientation of a body. In this sense, a coordinate is similar to the measurement used in elementary mathematics to locate a point or to orientate a line. We can bridge the difference between mathematical and physical objects by simply identifying particles with points and bodies with line segments. Coordinates are not unique. For example, a point P in a plane is commonly located either by Cartesian coordinates or polar coordinates as shown in Figure 5.2.1. In Cartesian coordinates, P is located by distances ( x , y ) to coordinate axes. In polar coordinates, P is located by the distance r to the origin (or pole) and by the inclination θ of the line connecting P with the pole. In both coordinate systems, two independent parameters are needed to locate P . If P is free to move in the plane, the values of the coordinates will change as P moves. Because these changes can occur independently for each coordinate, P is said to have two degrees of freedom . If, however, P is restricted in its movement so that it must remain on, say, a curve C , then P is said to be constrained . The coordinates of P are then no longer independent.

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  Dynamics in Engineering Practice

  • Dara W. Childs, Andrew P. Conkey(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  207 5.1 Introduction This chapter revisits all of the principles and objectives covered in Chapter 3 for particles, and now applies them to planar motion of rigid bodies. The objectives include deriving equations of motion using Newtonian and energy approaches, plus developing the principles of conservation of momentum for planar motion of a rigid body. You have presumably just completed Chapter 4 on planar kinematics of rigid bodies and understand the idea of planar motion for a rigid body. One of the attractions of dynamics is that governing equations proceed from a limited number of principles and equations. You enter this chapter knowing (or at least having been exposed to) Newton’s second law of motion for a particle, ∑ = f r m  , where  r is the parti-cle’s acceleration with respect to an inertial coordinate system. The particle kinematics presented in Chapter 2 provides the tools for determining the required  r in Newton’s second law. You will learn in this chapter how to proceed from Newton’s equation for a single particle to derive governing equations of motion for a rigid body , containing an infinite number of particles. Similarly, you worked problems in Chapter 3 involving the kinetic energy for a particle, T = mv 2 /2, where v is the veloc-ity magnitude of the particle with respect to an inertial coordinate system. In this chapter, we will use the par-ticle definition T = mv 2 /2 to develop the defining equa-tion for the kinetic energy of a rigid body. As with Chapter 3, most of the problems in this chapter involve development of models including deriving gov-erning differential equations of motion and kinematic-constraint equations. The derivations will proceed from (1) Newtonian methods using free-body diagrams and (2) work–energy principles. The kinematic-constraint equations of Chapter 4 will frequently be required to develop a complete set of governing equations.

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  Engineering Dynamics

  A Comprehensive Introduction

  • N. Jeremy Kasdin, Derek A. Paley(Authors)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  PART THREE Relative Motion and Rigid-Body Dynamics in Two Dimensions CHAPTER EIGHT Relative Motion in a Rotating Frame This chapter is devoted to relative motion in a rotating reference frame. Of course, you have already been introduced to some rotating frames, namely, the polar and path frames. These frames were enormously useful for solving certain problems, simplifying the kinematics, and providing new insights into particle motion. This chapter examines problems where using multiple rotating frames leads to important new formulas for dynamical systems. In particular, we see how the kinematics of certain planar rigid bodies are simplified by attaching a reference frame to them and then applying our previous results. We then complete the study of planar kinematics by examining the relative motion of a particle in a translating and rotating frame of reference. This expands our treatment in Chapter 3, where we restricted the discussion of relative motion to a translating frame. This chapter introduces one of the most important formulas of the book and the precursor to our subsequent treatment of rigid-body dynamics—the transport equation. 8.1 Rotational Motion of a Planar Rigid Body This section focuses on the kinematics of a rotating and translating rigid body. We have alluded to rigid bodies before without rigorously defining them. For now, you simply need to recognize that a rigid body is a collection of particles constrained so as not to move relative to one another. In many cases we consider a continuous collection that forms a solid body, such as a disk, rod, or sphere. Alternatively, we may consider a finite collection of particles connected by massless rigid links. What is important for our study of kinematics is that to every rigid body we can attach a body frame. Recall Definition 3.1: a reference frame is equivalent to a rigid body. Since all points of a rigid body are fixed with respect to one another, we can use them to define a reference frame.

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  Research Methods in Biomechanics

  • D. Gordon E. Robertson, Graham E. Caldwell, Joseph Hamill, Gary Kamen, Saunders Whittlesey(Authors)
  • 2013(Publication Date)
  • Human Kinetics

    (Publisher)

  appendix C ) and mathematical principles (appendixes D and E) that are required for data collection and analysis in kinematics. Note that text in boldface is a concept described in the glossary at the end of the book.

  Passage contains an image

  Chapter 1

  Planar Kinematics

  D. Gordon E. Robertson and Graham E. Caldwell

  K inematics is the study of bodies in motion without regard to the causes of the motion. It is concerned with describing and quantifying both the linear and angular positions of bodies and their time derivatives. In this chapter and the next, we

          examine how to describe a body’s position;

          describe how to determine the number of independent quantities (called degrees of freedom) necessary to describe a point or a body in space;

          explain how to measure and calculate changes in linear position (displacement) and the time derivatives velocity and acceleration;

          define how to measure and calculate changes in angular position (angular displacement) and the time derivatives angular velocity and angular acceleration;

          describe how to present the results of a kinematic analysis; and

          explain how to directly measure position, velocity, and acceleration by using motion capture systems or transducers.

  Examples showing how kinematic measurements are used in biomechanics research and, in particular, methods for processing kinematic variables for planar (two-dimensional; 2-D) analyses are presented in this chapter. In chapter 2 , additional concepts for collecting and analyzing spatial (three-dimensional; 3-D) kinematics are introduced.

  Kinematics is the preferred analytical tool for researchers interested in questions such as these: Who is faster? What is the range of motion of a joint? How do two motion patterns differ? An important application of kinematic data is their use as input values for inverse dynamics analyses performed to estimate the forces and moments acting across the joints of a linked system of rigid bodies (see chapters 5 , 6 , and 7

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  Engineering Mechanics

  Dynamics

  • Benson H. Tongue(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  O B J E C T I V E S Upon completion of this chapter you will be able to: ◆ Analyze the time-dependent behavior of a particle: where it is and how it is moving ◆ Describe a particle’s motion in a variety of coordinate systems ◆ Switch from one description to another ◆ Deal with relative motion (one particle that’s moving with respect to another) as well as constrained motion A t last we’ve reached the point we’ve all been waiting for—the kinetics of two-dimensional rigid bodies. Of course, three-dimensional motion is even more fun (you can’t have gyroscopic motion in two dimensions), but it’s also a topic that’s probably beyond what your class covers. So, for all intents and purposes, this is the main event of your course. After finishing this chapter, you’ll be fully equipped to handle a very reasonable range of problems. What sort of problems? Anything involving two-dimensional motion of rigid bodies. You will be able to model a car, for instance, and include the engine, drivetrain, and wheels. (But planar motion only. If you want to model a thrown football’s wobble, then you’ll need a three-dimensional analysis.) Robotic arms that move in-plane are on the menu, as are automatic door openers, rudders on airplanes, and bicycle gearchains. Benson Tongue (cont.) 383 (cont.) All we’ll need to solve the two-dimensional problems of this chapter are three scalar equations (along with some constraints). The first two equations come from a force balance, ∑ F = m a . By breaking this vector relationship into components along two orthogonal directions (e r -e θ , i- j, e t -e n , and so forth), we will get two scalar equations. The third equation is our moment balance, and it tells us how the rotational acceleration of the body is related to the applied forces and moments.

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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)

  A rigid body is an object in which the distance between any two points remains the same. By definition, a rigid body does not deform, or change shape. In this section, the two-dimensional motion of slablike rigid bodies is considered.

  Translation

  Translation is a motion in which any straight line in a body keeps the same direction during the motion; see Fig. 9.3.1 . In translation, all points of the body have the same velocity and the same acceleration, i.e.,

  Rotation about a Point

  The rotation of a body is shown in Fig. 9.3.2 , which is a motion about an axis that is perpendicular to plane Oxy and passes reference point O . The velocity of any point P is

  FIGURE 9.3.1 A rigid body in translation.

  FIGURE 9.3.2 The rotation of a rigid body about a point.

  (9.73)

  and the acceleration at P is

  (9.74a)

  with the tangential and normal acceleration components

  (9.74b)

  Here ω is the angular velocity, ω and α are the magnitudes of the angular velocity and acceleration, and k is a unit vector that is perpendicular to plane Oxy (determined by the right-hand rule). The magnitudes of the velocity and acceleration are given by

  (9.75)

  where The angular velocity and acceleration can be expressed as

  (9.76)

  where θ is the rotational displacement of the body. The unit of θ is radians (rad), the unit of ω is rad/s, and the unit of α is rad/s2 .

  General Plane Motion

  A general plane motion can be decomposed into a translation defined by the motion of any reference point and a simultaneous rotation about that point. For instance, let A be a reference point of the body in Fig. 9.3.3 . The velocity νB of point B is the sum of the velocity νA of the translation with A and the velocity νB/A relative to A :

  (9.77)

  The

  v B/A

  is a relative velocity associated with rotation about A

  (9.78)

  The angular velocity ω is independent of the reference point. In other words, any two points of a rigid body have the same angular velocity. The acceleration of point B

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  Meriam's Engineering Mechanics

  Dynamics

  • L. G. Kraige, J. N. Bolton(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  206 CHAPTER 6 Plane Kinetics of Rigid Bodies done by the other force, so that the net work done is dU = F ( b d 𝜃 ) = M d 𝜃 due to the rotational part of the motion. If the couple acts in the sense opposite to the rotation, the work done is negative. During a finite rotation, the work done by a couple M whose plane is parallel to the plane of motion is, therefore, U = ∙ M d u1D703 Kinetic Energy We now use the familiar expression for the kinetic energy of a particle to develop expressions for the kinetic energy of a rigid body for each of the three classes of rigid-body plane motion illustrated in Fig. 6 ∕ 12 . (a) Translation. The translating rigid body of Fig. 6 ∕ 12 a has a mass m and all of its particles have a common velocity v . The kinetic energy of any particle of mass m i of the body is T i = 1 2 m i v 2 , so for the entire body T = Σ 1 2 m i v 2 = 1 2 v 2 Σ m i or T = 1 2 mv 2 (6 ∕ 7) This expression holds for both rectilinear and curvilinear translation. (b) Fixed-axis rotation. The rigid body in Fig. 6 ∕ 12 b rotates with an angular velocity 𝜔 about the fixed axis through O . The kinetic energy of a representative particle of mass m i is T i = 1 2 m i ( r i u1D714 ) 2 . Thus, for the entire body T = 1 2 u1D714 2 Σ m i r i uni00A0 2 . But the moment of inertia of the body about O is I O = Σ m i r i uni00A0 2 , so T = 1 2 I O u1D714 2 (6 ∕ 8) Note the similarity in the forms of the kinetic energy expressions for translation and rotation. You should verify that the dimensions of the two expressions are identical. (c) General plane motion. The rigid body in Fig. 6 ∕ 12 c executes plane motion where, at the instant considered, the velocity of its mass center G is v and its angu-lar velocity is 𝜔 . The velocity v i of a representative particle of mass m i may be ex-pressed in terms of the mass-center velocity v and the velocity 𝜌 i 𝜔 relative to the mass center as shown.

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