Displacement

4 Key excerpts on "Displacement"

• 

  eBook - ePub

  Mechanical Design for the Stage

  • Alan Hendrickson(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  Assigning a sign of + or − to a position is an arbitrary choice that needs to be made once for a given situation, and then followed consistently from then on. Some choices seem obvious, assume the stage floor is the origin or zero point, and then straight up into the flytower is positive, and down into the trap room is negative, but the opposite assumption is just as valid. In other situations there may be no clear convention. Is stage right + or − relative to the centerline? Ultimately it does not matter what you choose, just pick one and stick with it.

  Displacement

  If an object moves, its position changes. The change of position is called Displacement. To describe Displacements mathematically, they are usually broken up into two distinct components, translational and rotational. Any movement, the rolling of a wheel or the trajectory of a thrown ball, for example, can be described with a sum of translational and rotational components. While accurate and complete, this is far more complex than we need for 99% of stage scenery moves. From here on, through Chapter 6 , we will assume all moves are linear, i.e., motion confined to straight lines; in later chapters all moves will be rotational, or spinning around a fixed axis of rotation. These assumptions are acceptable because flying scenery, wagons, and lifts usually move only in straight lines, and turntables spin around a point that remains stationary. So in summary, and for emphasis:

  Assumption: In Chapters 1 –6 , all moves will be in straight lines.

  Definition of Translational Displacement

  If the position of an object is recorded at two different times, with x1 , a position at an arbitrary time t1 , and x2 the objects position at a later time t2 , then the Displacement of the object is defined as:

  1.1

  Δ x =

  x 2

  −

  x 1

  The Δ is the capital Greek letter delta, and its meaning is “the change of.” So for instance, ΔX is “the change of x,” or “the change of position.” The Δ implies a subtraction of two measurements taken of some variable, with the value of the measurement taken first always subtracted from the value of the one taken second. It is the number subscripts that imply a time sequence. Lower number subscripts always refer to times earlier than higher numbered subscripts. Therefore x1 is a position measurement taken before x2 which likewise refers to a measurement taken before x3

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  eBook - ePub

  Instant Notes in Sport and Exercise Biomechanics

  Second Edition

  • Paul Grimshaw, Michael Cole, Adrian Burden, Neil Fowler(Authors)
  • 2019(Publication Date)
  • Garland Science

    (Publisher)

  Linear motion (or translatory motion) is movement along a line that is either straight or curved and where there is no rotation and all body parts move in the same direction at the same speed. Angular motion involves movement around an axis of rotation. Scalar quantity A quantity that is represented by magnitude (size) only. Vector quantity A quantity that is represented by both magnitude and direction. Distance and Displacement The term distance is classified as a scalar quantity and is expressed with reference to magnitude only (e.g. 140 mi (miles)). Displacement is the vector quantity and is expressed with both magnitude and direction (e.g. 100 m along a track in a straight line from point A to point B). Speed and velocity Speed is the scalar quantity that is used to describe the motion of an object (e.g. 4 m/s). It is calculated as distance divided by time taken. Velocity is the vector quantity and it is also used to describe the motion of an object (e.g. 4.5 m/s from north to south). It is calculated as Displacement divided by time taken. Acceleration Is defined as the change in velocity per unit of time and hence is a vector quantity. It is calculated as velocity divided by time taken. Average and instantaneous Average is the usual term for the arithmetic mean (in this context often over larger periods of time or Displacement). Instantaneous (or tending to instantaneous (towards zero)) refers to smaller increments of time or Displacement in which the velocity or acceleration calculations are made. The smaller the increments of time between successive data points the more the value tends towards an instantaneous value. FURTHER READING The following two resources provide additional reading on the concept of linear kinematics in sports and exercise.

  1    McDonald, C., & Dapena, J. (1991). Linear kinematics of the men’s 110-m and women’s 100-m hurdles races. Medicine and Science in Sports and Exercise , 23 (12), 1382–1391.

  2    Murphy, A. J., Lockie, R. G., & Coutts, A. J. (2003). Kinematic determinants of early acceleration in field sport athletes. Journal of Sports Science & Medicine , 2

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  eBook - ePub

  Basic Mechanics with Engineering Applications

  • J Jones, J Burdess, J Fawcett(Authors)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  OY axis is superfluous.

  Position and Displacement

  The position of a point at P may be described by its distance x along the line OX, as shown in Fig. 1.7 . In this case the position vector of the point is r = x. If the point now moves to P' such that its new position vector is x + δx, the change in position δx is called the Displacement of the point.

  FIG . 1.7

  The number of degrees of freedom of a point is always equal to the number of independent co-ordinates needed to specify its Displacement. In this case the Displacement can be expressed in terms of a change in the single coordinate x, so that a point moving on a straight line has one degree of freedom. Since the direction of the Displacement vector is specified by the straight line, all vectors must lie along OX. Hence when adding these vectors, the vector triangle of Fig. 1.3 will reduce to a straight line. This is equivalent to simple scalar addition, taking account of sign.

  The recommended SI units for position and Displacement are metres, m, or millimetres, mm.

  Velocity

  Consider the point P with position x at time t. Let the point be moving such that a small time δt later it has moved to a new position x + δx. The Displacement δx is the change in position, and is assumed to be positive in the same sense as x, i.e. along OX. The velocity ν of the point is defined as its rate of change of position and is given by

  This limit defines the velocity of the point at position x. The derivative dx/dt is often written as , where the dot denotes differentiation with respect to time.

  Velocity, like position, is a vector quantity and in the case of straight line motion its direction is defined by its sign, which is the same as that of the incremental Displacement δx. The recommended SI unit of time is the second, s, so that the SI unit of velocity will be m s−1

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  eBook - ePub

  Measurement and Instrumentation in Engineering

  Principles and Basic Laboratory Experiments

  • Francis S. Tse, Ivan E. Morse(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  7 Displacement, Motion, Force, Torque, and Pressure Measurements 7-1.  INTRODUCTION

  Length and time are basic measurements. In this chapter we describe common methods for measuring Displacement, velocity, acceleration, force, torque, and pressure. These topics are Displacement related, and flexures are often used as primary sensors for their measurements.

  The descriptions cannot be inclusive, since numerous usual and unusual methods of measurement are described in the literature. For example, 40 methods for thickness measurements are presented in a NBS publication [1 ]. Basic principles are described, but the engineer should be well aware of details, inherent errors, capabilities, limitations, and sources of loading in applications.

  Length is a distance between two points when one of the points is the datum. It may describe the dimension or the size of a machine part. Displacement implies a motion. A linear Displacement describes the distance traversed between two points when some other point is often the datum. Velocity and acceleration refer to the motion of an object, assuming rigid body motion.

  Velocity and acceleration are the first and second time derivatives of Displacement, as expressed in Eq. 7-1 for rectilinear motions. A motion may be rectilinear, angular, periodic, transitory, one-dimensional, or a vectorial quantity in three-dimensional space.

  Displacement: x = x 2 − x 1 Velocity: x ˙ = d x d t Acceleration: x ¨ = d 2 x d t 2

  (7-1)

  7-2.  DIMENSION AND Displacement MEASUREMENTS

  A dimension refers to a static measurement. It is as essential as a motion or a dynamic measurement in engineering. Without precision dimensional measurements, or metrology, parts would not be interchangeable and machines could not be mass produced.

  Although somewhat intuitive, dimensional measurements should be exercised with care. First, the problem must be clearly defined. Before measuring the length of a bar, it should be asked: “Are the ends plane?” Before measuring the diameter of a cylinder, the “diameter” must be defined. It should be asked; “Is the cylinder round?” This will be illustrated presently. Furthermore, temperature and extraneous inputs must be considered in a more precise measurement. Some skill or feel is required in using simple tools, such as a go/no-go plug gage.

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