Curvlinear

10 Key excerpts on "Curvlinear"

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  Engineering Mechanics

  Problems and Solutions

  • Arshad Noor Siddiquee, Zahid A. Khan, Pankul Goel(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  11.1 Introduction When the motion of a body takes place along a curved path, it is called curvilinear motion. The curvilinear motion is known as a two-dimensional motion as it takes place in the X–Y plane. The basic difference between rectilinear and curvilinear motion is that in a rectilinear motion, all particles of a body move parallel to each other along a straight line but in a curvilinear motion all particles move parallel to each other along a curved path in a common plane. Some common examples of curvilinear motion are as follows: a. A vehicle passing over a speed breaker. b. Turning of a vehicle along a curved path. c. Motion of a ball during lofted six in cricket. d. Motion of missile fired from a fighter plane. e. Oscillatory motion of a pendulum. 11.2 Rectangular Coordinates This system is used to analyse the curvilinear motion of a particle moving in X–Y plane. Here the velocity and acceleration are resolved in two perpendicular components along X and Y axis. The resultant value of velocity and acceleration can be determined by using vector approach or combining its components as discussed below: Velocity: Consider a ball during lofted six, it consists two types of motions, one rotary motion and another curvilinear motion. Here the rotary motion is neglected and assumed that all particles of ball will travel with same displacement, velocity and acceleration. Consider a displacement-time graph where a particle is moving along a curved path in X–Y plane. Let at any instant t , particle has position A. If particle moves from point A to B by displacement Δ s during time interval Δ t as shown in Fig. 11.1. Chapter 11 Kinematics: Curvilinear Motion of Particles 530 Engineering Mechanics where Δ s = Δ s x + Δ s y We know that the instantaneous velocity is given by v s t v s s t v s t inst velocity t t x y t x .

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  Engineering Mechanics

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

    (Publisher)

  DOI: 10.1051/978-2-7598-2901-9.c006 © Science Press, EDP Sciences, 2022 6.1 General Curvilinear Motion When a particle moves along a straight line, it undergoes rectilinear motion; whereas a particle moving along a curved path undergoes curvilinear motion. Rectilinear motion has been considered extensively in physics and it will be treated as a special case of curvilinear motion. The kinematics of a particle includes specifying the particle’s position, dis- placement, velocity, and acceleration. Position. A particle is moving along a space curve, figure 6.3. The position of the particle at any instant can be defined by position vector r measured from fixed point O. Apparently, both the magnitude and direction of the vector r will change as the particle moves along its curved path. Then the position vector is a single-valued function of time as follows: r ¼ rðt Þ ð6:1Þ FIG. 6.2 – Plane’s flight attitude. FIG. 6.1 – Plane’s trajectory. 194 Engineering Mechanics Displacement. The displacement of a particle represents the change in its position. Suppose that at a given instant t, the particle is at point P on the curve and position vector is r, as shown in figure 6.3. After a small time interval Δt, the particle moves to new position P 0 and the position vector becomes r 0 . The displacement is then determined by Dr ¼ r 0  r . Velocity. The velocity of a particle is defined as the time rate of change in its position. During the time interval Δt, the average velocity of the particle is v avg ¼ Dr Dt Then the instantaneous velocity is determined by letting Dt ! 0, i.e., v ¼ lim Dt !0 v avg ¼ lim Dt !0 Dr Dt ¼ dr dt ð6:2Þ Since the direction of Dr gradually approaches the tangent to the curved path as Dt ! 0, the direction of dr, also the direction of v, is tangent to the path, as shown in figure 6.3. The magnitude of v, also called the speed, is usually expressed in units of m/s or km/h (kilometer per hour).

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  Conceptual Dynamics

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

    (Publisher)

  For information on vectors and vector operations, see Appendix B. In the discussion of plane curvilinear motion we will examine several coordinate frames. Each coordinate frame may be used to describe the location and motion of a particle in a plane. The choice of which coordinate frame to use will depend on the kind of motion the particle is experiencing and the particular information we are looking for in the problem. One thing is always true no matter what coordinate system is chosen; the velocity vector is always directed tangent to the particle's path of motion as shown in Figure 3.1-1. Even though velocity is always tangent to the path curve, in general, acceleration is not. What is the difference between rectilinear motion and curvilinear motion? 3.1.2) PLANE CURVILINEAR MOTION: CARTESIAN COORDINATES A set of coordinate axes that is frequently used to describe the motion of a particle in a plane is rectangular or Cartesian coordinates. The Cartesian coordinate frame consists of two perpendicular axes: the x-axis and the y-axis as shown in Figure 3.1-2. The Cartesian coordinate system that we will consider in this section is an inertial coordinate frame. This means that the coordinate system does not accelerate. But for our purposes, we can think of an inertial reference frame as one where the origin Conceptual Dynamics Kinematics: Chapter 3 – Kinematics of Particles - Curvilinear Motion 3 - 5 is fixed and does not move and the axes do not rotate. Some coordinate systems, such as the n-t coordinate system, are body-fixed coordinate frames. This means that the coordinate axes are fixed to and move with the body and the axes may rotate. A Cartesian coordinate system is commonly used to solve problems in which the motion proceeds in a straight line, is expressed in terms of the x- and y-components or is non-circular in nature.

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  Dynamics of Mechanical Systems

  • Carl T. F. Ross(Author)
  • 1997(Publication Date)
  • Woodhead Publishing

    (Publisher)

  CHAPTER TWO Kinematics of Particles 2.1 INTRODUCTION This chapter is divided into two sections, one section is on rectilinear motion and the other section is on plane curvilinear motion. Now rectilinear motion can be described as one dimensional motion or motion in a straight line. The section on rectilinear motion is sub-divided into two sub-sections, namely motion when the acceleration is constant and motion when the acceleration varies with time or distance or velocity. The section on plane curvilinear motion is divided into three sub-sections. One of these sub-sections is on plane motion with reference to the rectangular axes, namely x and y, while another sub-section is on plane motion with reference to the normal and tangential axes of the path of the motion. These axes are often called the not axes, where 'n' is perpendicular to the path of the motion and 't' is tangential to the path of the motion. Thus, if the path of the motion is curvilinear, the directions of 'n' and 't' will vary. The third sub-section on plane motion is on motion in terms of the polar co-ordinates 'r' and '8', where 'r' is radially outwards and '8' is tangential to 'r', in an angular direction. The x-y rectangular co-ordinate system is popular when the motion or its direction is known in terms of the x-y co-ordinates. The not co-ordinate system is popular when the direction of the path that the motion follows is known. This may be in the case of a car or a train travelling around a curve, whose equation is known. Polar co-ordinates are popular for analysing mechanisms and other artefacts, which are rotating about a fixed point. Also considered in this chapter is relative motion of translating axes and the motion of connected particles. 2.2 RECTILINEAR MOTION Rectilinear motion of a particle can be described as motion in a straight line, as shown by Figure 2.1, where the particle 'P' moves to the position 'ph in time'!1t'.

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  Biomechanics For Dummies

  • Steve McCaw(Author)
  • 2014(Publication Date)
  • For Dummies

    (Publisher)

  translation. In linear motion, all points on a body go equally far in the same direction, they travel equally fast, and they move at the same time. All points on the body also speed up and slow down at the same time. There are two forms of linear motion:

  • Rectilinear motion: Motion in a straight line.
  • Curvilinear motion: Motion along a curved line, such as when a body moves through the air as a projectile. Curvilinear motion is simultaneous motion in the up–down (vertical) and forward–backward (horizontal) directions.

  In this chapter, I explain the kinematic descriptors to describe where a body is in space (position) and how far, how fast, and how consistently (speeding up or slowing down) the body moves. I introduce the important quantity of linear momentum because it combines the two critical ideas of the body's current state of motion and the body's resistance, or inertia, to changing its motion. Finally, I demonstrate the use of the three equations of constant acceleration to quantify projectile motion.

  Identifying Position

  Position describes a body's location in space. The location is specified relative to a selected landmark. The landmark is referred to as the origin, because all measures of position are made from, or originate from, this point. Figure 5-1 shows a view from above of a player on a soccer field (or pitch, to be sport-term correct) at a specific instant in time. I've set the origin to a coordinate system (see Chapter 2 ) where the goal line and the sideline meet in the bottom-left corner of the field; the x direction corresponds to the length of the field, and the y

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  Mathematics for the Nonmathematician

  • Morris Kline(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER 14 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION I now propose to set forth those properties which belong to a body whose motion is compounded of two other motions, namely, one uniform and one naturally accelerated; these properties, well worth knowing, I propose to demonstrate in a rigorous manner. GALILEO 14–1  INTRODUCTION We saw in the preceding chapter that simple functions can be used to express physical principles and that by applying algebra to the formulas which express the functions symbolically, we can obtain new physical knowledge. To some extent, then, we have come to recognize the broader significance and usefulness of functions and mathematical processes for science in general. However, we have hardly penetrated as yet the mathematical domain of functions nor have we learned enough applications to sense its real power. In this chapter we shall extend slightly the use of functions. In Chapter 13, we represented the acceleration and speed attained and distance traveled by a falling body by using one formula for each physical quantity. We were enabled thereby to study motion along straight-line paths. We shall now examine motion along curved paths, for example, the motion of an object dropped from a moving plane, or the motion of a projectile shot out from a cannon. It was again Galileo who perceived the basic principle underlying the phenomenon of curvilinear motion. He presented the concept and its mathematical treatment to the world in the Dialogues Concerning Two New Sciences, the very same book in which he treated motion in a straight line. Galileo’s purpose in investigating curvilinear motion was to study the behavior of cannon balls, or projectiles in general. The cannon, introduced in the fourteenth century, had undergone such improvement by Galileo’s time that it could fire a projectile over several miles

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  Practical Stress Analysis in Engineering Design

  • Ronald Huston, Harold Josephs(Authors)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  6 Curvilinear Coordinates 6.1 USE OF CURVILINEAR COORDINATES The formulation of the equilibrium equations for stress and the strain – displacement equations are readily developed in Cartesian coordinates, as in Chapters 4 and 5. In practical stress – strain analyses, however, the geometry often is not rectangular but instead cylindrical, spherical, or of some other curved shape. In these cases, the use of curvilinear coordinates can greatly simplify the analysis. But with curvilinear coordinates, the equilibrium equations and the strain – displacement equations have different and somewhat more complicated forms than those with Cartesian coord-inates. To determine the equation forms in curvilinear coordinates, it is helpful to review some fundamental concepts of curvilinear coordinate analysis. In the following sections, we present a brief review of these concepts. We then apply the resulting equations using cylindrical and spherical coordinates. 6.2 CURVILINEAR COORDINATE SYSTEMS: CYLINDRICAL AND SPHERICAL COORDINATES 6.2.1 C YLINDRICAL C OORDINATES Probably the most familiar and most widely used curvilinear coordinate system is plane polar coordinates and its extension in three dimensions to cylindrical coordinates: consider an XYZ Cartesian system with a point P having coordinates ( x , y , z ) as in Figure 6.1. Next, observe that P may be located relative to the origin O by a position vector OP , or simply p , given by p ¼ x n x þ y n y þ z n z ( 6 : 1 ) where n x , n y , and n z are unit vectors parallel to the X -, Y -, and Z -axes as in Figure 6.2. Suppose now an image of P , say ^ P , is projected onto the X – Y plane as in Figure 6.3. Then we can also locate P relative to O by the vector sum O ^ P þ ^ P P as in Figure 6.4. That is p ¼ OP ¼ O ^ P þ ^ PP ( 6 : 2 ) Let r be the distance from O to ^ P ; z be the distance from ^ P to P ; and u be the angle between O ^ P and the X -axis, as in Figure 6.5.

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  Mathematical Methods for Optical Physics and Engineering

  • Gregory J. Gbur(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  66 Vector calculus in curvilinear coordinate systems x y z x y z x y z (b) (a) Figure 3.2 (a) A plane of constant x, and (b) the intersection of three planes (surfaces) specifying a point. ρ x y z Figure 3.3 A surface of constant ρ. in Fig. 3.3. A system which includes one or more curved coordinates is referred to as a curvilinear coordinate system. An important observation is that the direction of the unit vector associated with a curved surface now depends upon the specific location on this surface; in other words, the unit vectors of a curvilinear coordinate system are position dependent. This will have a tremen- dous influence upon the definition of vector differential quantities such as the gradient, 3.2 General orthogonal coordinate systems 67 divergence and curl. It is to be noted that the coordinates we use, such as φ in cylindri- cal coordinates, need not be physical distances, but may also represent angles or other dimensionless quantities. In general we specify a coordinate system in three dimensions by three coordinates q 1 , q 2 , q 3 which define three surfaces. We will find it convenient to write these coordinates in shorthand as q ≡ (q 1 , q 2 , q 3 ), though q is not typically a vector and does not transform as one under rotations. We may relate these coordinates to the Cartesian coordinates, and vice versa, i.e. general coordinates cylindrical coordinates q 1 , q 2 , q 3 ρ , φ, z x = x(q 1 , q 2 , q 3 ) x = ρ cos φ y = y(q 1 , q 2 , q 3 ) y = ρ sin φ z = z(q 1 , q 2 , q 3 ) z = z (3.1) and q 1 = q 1 (x, y, z) ρ =  x 2 + y 2 q 2 = q 2 (x, y, z) φ = arctan(y/x) q 3 = q 3 (x, y, z) z = z. (3.2) It is to be noted that the specification of φ in Eq. (3.2) above is not quite complete: there is a π ambiguity because of the π -periodicity of arctan. The unit vectors e 1 , e 2 , e 3 of the system coordinates are defined as the positive normal of each of the coordinate surfaces.

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  Tensor Analysis

  • Heinz Schade, Klaus Neemann, Andrea Dziubek, Edmond Rusjan, Andrea Dziubek, Edmond Rusjan(Authors)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  4 Tensor Analysis in Curvilinear Coordinates 4.1 Curvilinear Coordinates 4.1.1 Curvilinear Coordinate Systems 1. Until now, we described a point P in space using a Cartesian coordinate system. Such a coordinate system consists of an origin O and an orthonormal basis e i at this origin, and we described the point P by the three coordinates x i of the position vector x = 󳨀󳨀→ OP = x i e i . Let us consider a set of transformation equations (one-to-one, except at singular points), u i = u i ( x j ) , x i = x i ( u j ) . (4.1) Then every point in space is uniquely described by the three variables u i . Thus we can also call the u i point coordinates, i. e. each set of transformation equations (4.1), together with a Cartesian coordinate system, defines again a coordinate system. All these coordinate systems are called curvilinear coordinate systems. 1 As an example we consider cylindrical coordinates. They are described by the transformation equations x 1 = u 1 cos u 2 , x 2 = u 1 sin u 2 , x 3 = u 3 . We obtain conversely u 1 = √ x 2 1 + x 2 2 , u 2 = arctan x 2 x 1 , u 3 = x 3 or in the usual notation x 1 = x , x 2 = y , u 1 = R , u 2 = φ , x 3 = u 3 = z x = R cos φ , y = R sin φ , z = z , R = √ x 2 + y 2 , φ = arctan y x , z = z . (4.2) 2. From d x i = (𝜕 x i /𝜕 u j ) d u j it follows that 𝜕 x i /𝜕 x k = δ ik = (𝜕 x i /𝜕 u j ) (𝜕 u j /𝜕 x k ) , and analogously from d u i = (𝜕 u i /𝜕 x j ) d x j it follows that 𝜕 u i /𝜕 u k = δ ik = (𝜕 u i /𝜕 x j )(𝜕 x j /𝜕 u k ) , i. e. the partial derivatives of the transformation equations satisfy the orthogonality relations 𝜕 x i 𝜕 u j 𝜕 u j 𝜕 x k = δ ik , 𝜕 u i 𝜕 x j 𝜕 x j 𝜕 u k = δ ik . (4.3) 1 Since the transformation equations (2.9) between two Cartesian coordinate systems are special cases of (4.1), Cartesian coordinate systems are special cases of curvilinear coordinate systems. https://doi.org/10.1515/9783110404265-004 160 | 4 Tensor Analysis in Curvilinear Coordinates 3.

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  Mathematical Physics

  Applied Mathematics for Scientists and Engineers

  • Bruce R. Kusse, Erik A. Westwig(Authors)
  • 2010(Publication Date)
  • Wiley-VCH

    (Publisher)

  This chapter generalizes the concepts of the previous chapters to include curvi- linear coordinate systems. The two most common systems, spherical and cylindrical, are described first, in order to provide a framework for the more abstract discussion of generalized curvilinear coordinates that follows. 3.1 THE POSITION VECTOR The position vector F(P) associated with a point P describes the offset of P from the origin of the coordinate system. It has a magnitude equal to the distance from the origin to P, and a direction that points from the origin to P. 44 THE CYLINDRICAL SYSTEM 45 Figure 3.1 The Position Vector It seems natural to draw the position vector between the origin and P , as shown in Figure 3.l(a). While this is fine for Cartesian coordinate systems, it can lead to difficulties in curvilinear systems. The problems arise because of the position dependence of the curvilinear basis vectors. When we draw any vector, we must always be careful to specify where it is located. If we did not do this, it would not be clear how to expand the vector in terms of the basis vectors. To get around this difficulty, both the vector and the basis vector should be drawn emanating from the point in question. The curvilinear vector components are then easily obtained by projecting the vector onto the basis vectors at that point. Consequently, to determine the components of the position vector, it is better to draw it, as well as the basis vectors, emanating from P . This is shown in Figure 3.l(b). There are situations, however, when it is better to draw the position vector emanating from the origin. For example, line integrals, such as the one shown in Figure 2.2, are best described in this way, because then the tip of the position vector follows the path of the integration. We will place the position vector as shown in Figure 3.l(a) or (b), depending upon which is most appropriate for the given situation.

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