Planar Kinematics of a Rigid Body

12 Key excerpts on "Planar Kinematics of a Rigid Body"

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  eBook - PDF

  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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  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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  Dynamics in Engineering Practice

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

    (Publisher)

  165 4.1 Introduction The derivation of general kinematic equations for bod-ies in plane motion is the central objective of this chap-ter. Planar motion of a body means that all of a body’s motion is confined to a plane, for example, the plane of this page. Figure 4.1 illustrates a rigid body that is trans-lating and rotating in the plane of the figure. The x , y coordinate system is fixed to the rigid body. The origin of the x , y system is point o , which is located in the X , Y system by the vector R o = I R oX + J R oY . The angle θ defines the orientation of both the body and the x , y coordinate system with respect to the X , Y system. Planar kinemat-ics of a rigid body involves keeping track of the position of a specified point on the body (e.g., using R oX and R oY of Figure 4.1 to locate point o ) and the body’s orienta-tion ( θ angle of rotation of Figure 4.1) with respect to a fixed axis. Position, velocity, and acceleration vectors for points were the subjects covered in Chapter 2 on par-ticle kinematics. Formulating planar kinematic problems for rigid bodies requires that we introduce and define angular-velocity and angular-acceleration vectors for the rigid body. Figure 4.1b shows the rigid body at a later time t + Δ t , during which the orientation angle has changed to θ + Δθ . The angular rotation rate of the body is  θ θ θ = ⇒ = lim |( ) . Δ Δ Δ t t d dt 0 / / Using the right-hand rule for an advancing screw, the angular-velocity vector of the body is ω = k  θ . Here, k is the unit vector, which is normal to the plane of the figure and pointing outward along the z -axis of Figure 4.2. Similarly, the angular-acceleration vector for the body is   ω = k θ . Figure 4.2 shows the rigid body and the angular-velocity vector in the X , Y , Z system. The great majority of mechanisms in mechanical engineering are planar; hence, planar kinematics is both important and useful.

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  eBook - ePub

  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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  eBook - ePub

  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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  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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  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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  Engineering Dynamics

  • Cho S.(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  109 C H A P T E R 7 Kinematics of Rigid Bodies 7.1 INTRODUCTION In this chapter the kinematics of rigid bodies are presented. Section 7.1 is concerned with in- troduction to various important definitions and concepts in the kinematics of rigid bodies. The instantaneous center of rotation in plane motion is briefly introduced in Section 7.2. More de- tails presentation and application in motion analysis may be found in textbooks on kinematics and dynamics of machinery, for example in [1–3]. Position vector in a rotating frame of refer- ence is presented in Section 7.3. Rate of change of a vector with respect to a rotating frame of reference is dealt with in Section 7.4. The 3D motion of a point in a rigid body with respect to a rotating frame is considered in Section 7.5 in which three representative examples are included to illustrate the steps in the solution. In a system of particles or rigid bodies, arguably the most fundamental concept is the degrees-of-freedom (dof ) in spaces. By the dof of a system or rigid body one means the minimum number of independent coordinates that are required to completely describe the position of the system. It is independent of the coordinate system adopted in a particular situation. For example, if the system in a Cartesian coordinate system has 6 dof then it has the same number of dof in a spherical coordinate system. For a rigid body in 3D space, there are 3 translational dof, and 3 rotational dof, as shown in Figure 7.1. For a rigid body in a 2D space there are 2 translational dof and 1 rotational dof. A rigid body may, in general, experiences either one or more of the various types of motion to be included in the following. (a) Translation is the motion in which any straight line inside the body maintains the same direction, as shown in Figure 7.2a. Clearly, all particles in the rigid body move in parallel paths. However, if these paths are curved lines such motion is called curvilinear translation, as shown in Figure 7.2b.

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  Conceptual Dynamics

  • Richard C. Hill, Kirstie Plantenberg(Authors)
  • 2013(Publication Date)
  • SDC Publications

    (Publisher)

  Step 2: Analyze the velocities  Start at a position that is fixed or that has known velocity and work outward.  Sometimes two different relationships can be found for the motion of a single point by starting from different points and meeting in the middle.  Apply known information and geometric constraints to eliminate unknowns. Conceptual Dynamics Kinematics: Chapter 4 – Kinematics of Rigid Bodies 4 - 28 Step 3: Analyze the accelerations  Repeat the process presented for the velocity. Typically the expressions used to calculate the accelerations of the system depend on the velocities of the system. Equation Summary Abbreviated variable definition list v A , v B = absolute velocity of points A and B respectively v A/B = velocity of point A relative to point B a A , a B = absolute acceleration of points A and B respectively a A/B = acceleration of point A relative to point B r A/B = position of point A relative to point B  = angular velocity of the rigid body (rad/s)  = angular acceleration of the rigid body (rad/s 2 ) General planar motion / A B A B    v v ω r 2 / / A B A B A B      a a α r r / / ( ) A B A B A B       a a α r ω ω r Conceptual Dynamics Kinematics: Chapter 4 – Kinematics of Rigid Bodies 4 - 29 Example Problem 4.3-4 Consider the following diagram of a piston assembly in an internal combustion engine. The crankshaft OA is rotating at a constant rate of 2000 rpm counter-clockwise. If the piston is at top dead center position ( = 0 o ), determine (a) the velocity and acceleration of piston head B, (b) the angular velocity of connecting rod AB, and (c) the angular acceleration of connecting rod AB. Given: Find: Solution: Conceptual Dynamics Kinematics: Chapter 4 – Kinematics of Rigid Bodies 4 - 30 Conceptual Dynamics Kinematics: Chapter 4 – Kinematics of Rigid Bodies 4 - 31 Example Problem 4.3-5 Consider the four-bar linkage shown.

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  eBook - ePub

  Kinematics and Dynamics of Mechanical Systems, Second Edition

  Implementation in MATLAB® and SimMechanics®

  • Kevin Russell, Qiong Shen, Rajpal S. Sodhi(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  4

  Kinematic Analysis of Planar Mechanisms

  CONCEPT OVERVIEW

  In this chapter, the reader will gain a central understanding regarding

  1. Link velocity and acceleration components in planar space
  2. The Newton–Raphson method for a set of two simultaneous equations
  3. Vector-loop-based displacement, velocity, and acceleration equation formulation and solution
  4. Kinematics of mechanism link locations of interest
  5. Instant centers in relative planar motion
  6. Instant center generation and application in velocity analysis
  7. Centrode generation and application in coupler motion replication
  8. Configurations of closed-loop mechanisms
  9. Relationships between general angular velocity and time
  10. Cognate construction and application

  4.1 Introduction

  In a kinematic analysis, positions, displacements, velocities, and accelerations of mechanism links are determined either qualitatively or quantitatively . In a quantitative kinematic analysis, equations that fully describe the motion of the mechanism links are used. Qualitative methods include constructing and measuring mechanism schematics and polygons to determine the positions, velocities, and accelerations of mechanism links. As intended by the authors, the kinematic analysis methods presented in this textbook are all quantitative. Kinematic equations for the planar four-bar, slider-crank, geared five-bar, Watt II, and Stephenson III mechanisms are formulated in this chapter. Displacement equations are formulated by taking the sum of the closed vector loop(s) in each mechanism (as introduced in Section 2.2.2 ) [1 ]. Taking the first and second derivatives of the vector-loop displacement equations introduces mechanism link velocity and acceleration variables, respectively, and, ultimately, produces mechanism velocity and acceleration equations, respectively.

  As noted in Section 3.1.4 , the Watt II and Stephenson III mechanisms (Figure 3.6 ) are the planar multiloop mechanisms of choice for analysis in this textbook. Both mechanisms have two links that exhibit complex motion. As illustrated in Figure 3.6 , Watt II and Stephenson III mechanisms are comprised of a planar four-bar mechanism and an additional dyad.*

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  eBook - ePub

  Advanced Engineering Dynamics

  • H. Harrison, T. Nettleton(Authors)
  • 1997(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  4

  Rigid Body Motion in Three Dimensions

  4.1 Introduction

  A rigid body is an idealization of a solid object for which no change in volume or shape is permissible. This means that the separation between any two particles of the body remains constant.

  If we know the positions of three non-colinear points, i, j and k , then the position of the body in space is defined. However, there are three equations of constraint of the form |

  ri −rj

  | = constant so the number of degrees of freedom is 3 × 3 − 3 = 6.

  4.2 Rotation

  If the line joining any two points changes its orientation in space then the body has suffered a rotation. If no rotation is taking place then all particles will be moving along parallel paths. If the paths are straight then the motion is described as rectilinear translation and if not the motion is curvilinear translation. From the definitions it is clear that a body can move along a circular path but there need be no rotation of the body.

  It follows that for any pure translational motion there is no relative motion between individual particles. Conversely any relative motion must be due to some rotation.

  The rotation of a rigid body can be described in terms of the motion of points on a sphere of radius a centred on some arbitrary reference point, say i . The body, shown in Fig. 4.1 , is now reorientated so that the points j and k are moved, by any means, to positions j′ and k′ . The arc of the great circle joining j and k will be the same length as the arc joining j′ and k′, by definition of a rigid body. Next we construct the great circle through points j and j′ and another through the points k and k′ . We now draw great circles which are the perpendicular bisectors of arcs jj′ and kk′ . These two circles intersect at point N. The figure is now completed by drawing the four great circles through N and the points j, k, j′ and k′ respectively.

  Fig. 4.1

  By the definition of the perpendicular bisector arc Nj = arc Nj′ and arc Nk = arc Nk′ . Also arc jk = arc j′k′ and thus it follows that the spherical triangle k Nj is congruent with k′ Nj′ . Now the angle k Nj = k′ Nj′ and the angle k Nj′ is common; therefore angle k Nk′ = j Nj′

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  eBook - ePub

  Multi-Body Kinematics and Dynamics with Lie Groups

  • Dominique Paul Chevallier, Jean Lerbet(Authors)
  • 2017(Publication Date)
  • ISTE Press - Elsevier

    (Publisher)

  [BOU 08] .

  In the present chapter we show that only the mathematical properties of the displacement group are actually involved in the kinematics of rigid body and rigid body systems. It is only in section 4.4 , when we shall aim at showing the link with the usual exposition of mechanics, that we shall refer to a model of body as a set of particles; at the end, we shall notice that the mathematical structure of kinematics (and further of dynamics) will retain nothing from this model.

  4.3 The position space of a rigid body

  Let S be the space depicting in a mathematical language the positions of a rigid body seen by an observer. To be clear, any observer uses some devices to locate the positions of objects in space, it can perform some measures and uses some set of parameters to locate the body (as coordinates of the position of a point of the body and three angles defining its orientation in space). After all, the process amounts to measure the displacements of the observed body with respect to a reference body fixed in the “laboratory”. But there exists an infinity of ways to do this and the points of S describe the positions leaving aside the choice of the parameters.

  The main property of S is the following: there is a left action of the Euclidean displacement group on S and this action is free and transitive . In other words, there is an operation

  D × S → S :

  g s

  ↦ g . s

  such that, for all

  s ∈ S

  and for all g ,

  h ∈ D

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