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Mechanics

Mechanics. Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate. Text: engineering mechanics—DYNAMICS, section 13.2. 2.4 Kinematics in path coordinate.

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Mechanics

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  1. Mechanics Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate Text: engineering mechanics—DYNAMICS, section 13.2 2.4 Kinematics in path(normal-tangential)coordinates

  2. 2.4 Kinematics in path coordinate When a particle moves along a curved path, it is sometimes convenient to use coordinates other than Cartesian(笛卡尔). When the path of motion is known, path coordinate system is often used. • The definition of the path coordinate • The expressions of velocity and acceleration in path coordinate • Motion along a circular path Note: path coordinates can be used to describe three-dimensional particle motion, but we’ll restrict ourself to the plane motion only. 2.4 Kinematics in path(normal-tangential)coordinates

  3. Mechanics 2.4.1 Curvature (曲率) 2.4.2 Definition of path coordinate Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate 2.4.3 Geometric preliminaries 2.4.4 Derivatives of base vector 2.4.5 Velocity in the n-t coordinate system 2.4.6 Acceleration in the n-t coordinate system 2.4.7 Special cases of motion 2.4.8 Sample examples 2.4 Kinematics in path(normal-tangential)coordinates

  4. y B A s +  x Curvature: The amount of degree of bending of a curve.( or the tendency at any point to depart from a tangent drawn to the curve at that point) 2.4.1 Curvature (曲率) curvature at point A If the curve is expressed asy = f(x) Radius of curvature(曲率半径): = 1/K Ifthe curve is a circle of radius R, then K= 1/R 2.4 Kinematics in path(normal-tangential)coordinates

  5. y C  A x 2.4.1 Curvature (曲率) Circle of curvature(曲率圆): the circle that touches the curve (on the concave[凹的] side) and whose radius is the radius of curvature Center of curvature(曲率中心) : the center of the circle of curvature 2.4 Kinematics in path(normal-tangential)coordinates

  6. Mechanics 2.4.1 Curvature (曲率) 2.4.2 Definition of path coordinate Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate 2.4.3 Geometric preliminaries 2.4.4 Derivatives of base vector 2.4.5 Velocity in the n-t coordinate system 2.4.6 Acceleration in the n-t coordinate system 2.4.7 Special cases of motion 2.4.8 Sample examples 2.4 Kinematics in path(normal-tangential)coordinates

  7. 2.4.2 Definition of path coordinate Path coordinate, also known as normal-tangential(n-t) coordinates, describes the motion of a particle in terms of components that are normal and tangent to its path. • The position of the particle is specified by the path coordinates, which is the distance measured along the path from a fixed reference point. normal path o s(t) tangent P • At each position on the path, we can build an-tcoordinate system 2.4 Kinematics in path(normal-tangential)coordinates

  8. et A s en  C 2.4.2 Definition of path coordinate • Theorigin is located on the particle (the origin moves with the particle). • The t-axis is tangent to the path (curve) at the instant considered, positive in the direction of the particle’s motion • The n-axis is perpendicular to the t-axis with the positive direction toward the center of curvature. • The positiven andtdirections are defined by the unit vectorsenandet,respectively. Note:the directions enand et are not fixed, but depends on the location A of the particle 2.4 Kinematics in path(normal-tangential)coordinates

  9. Mechanics 2.4.1 Curvature (曲率) 2.4.2 definition of path coordinate Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate 2.4.3 Geometric preliminaries 2.4.4 Derivatives of base vector 2.4.5 Velocity in the n-t coordinate system 2.4.6 Acceleration in the n-t coordinate system 2.4.7 Special cases of motion 2.4.8 Sample examples 2.4 Kinematics in path(normal-tangential)coordinates

  10. ds = d  et= dr/ds dr = dset  During an infinitesimal time intervaldt, the particle moves from A to B along its path. 2.4.3 Geometric preliminaries Arc length:dsinfinitesimal Displacement: dr, |dr| = ds Sincedsis infinitesimal, we can approximately consider the arc AB as a part of the circle of curvature at A. ds B s dr y A Whereis measured in radians. is the radius of curvature at A  d  Sincedris tangent to the path at A C x o 2.4 Kinematics in path(normal-tangential)coordinates

  11. Mechanics 2.4.1 Curvature (曲率) 2.4.2 definition of path coordinate Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate 2.4.3 Geometric preliminaries 2.4.4 Derivatives of base vector 2.4.5 Velocity in the n-t coordinate system 2.4.6 Acceleration in the n-t coordinate system 2.4.7 Special cases of motion 2.4.8 Sample examples 2.4 Kinematics in path(normal-tangential)coordinates

  12. 2.4.4 Derivatives of base vector Since the directions ofenandet,vary with the position of the particle, theirtime derivatives are not zero. • Express the base vectors in terms of rectangular components. en= -cosi -sinj et= -sini +cosj • Differentiating with respect to time  2.4 Kinematics in path(normal-tangential)coordinates

  13. Mechanics 2.4.1 Curvature (曲率) 2.4.2 definition of path coordinate Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate 2.4.3 Geometric preliminaries 2.4.4 Derivatives of base vector 2.4.5 Velocity in the n-t coordinate system 2.4.6 Acceleration in the n-t coordinate system 2.4.7 Special cases of motion 2.4.8 Sample examples 2.4 Kinematics in path(normal-tangential)coordinates

  14. 2.4.5 Velocity in the n-t coordinate system The velocity vector is alwaystangentto the path of motion (t-direction). The magnitude is determined by taking the time derivative of the path function,s(t). Herev defines the magnitude of the velocity (speed) and etdefines the direction of the velocity vector. 2.4 Kinematics in path(normal-tangential)coordinates

  15. Mechanics 2.4.1 Curvature (曲率) 2.4.2 definition of path coordinate Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate 2.4.3 Geometric preliminaries 2.4.4 Derivatives of base vector 2.4.5 Velocity in the n-t coordinate system 2.4.6 Acceleration in the n-t coordinate system 2.4.7 Special cases of motion 2.4.8 Sample examples 2.4 Kinematics in path(normal-tangential)coordinates

  16. 2.4.6 Acceleration in the n-t coordinate system Acceleration is the time rate of change of velocity: 2.4 Kinematics in path(normal-tangential)coordinates

  17. 2.4.6 Acceleration in the n-t coordinate system There are two components to the acceleration vector: a = at et + an en • The tangential component is tangent to the curve or (using chain rule!) atis caused by a change in the speed of the particle • If the speed is increasing,athas the same direction as the velocity. • If the speed is decreasing,atand the velocity have opposite directions 2.4 Kinematics in path(normal-tangential)coordinates

  18. 2.4.6 Acceleration in the n-t coordinate system • The normal or centripetal(向心的) component is always directed toward the center of curvature of the curve. anis due to a change in the direction of the velocity. • The magnitude of the acceleration vector is • The direction of the acceleration vector is is the angle betweenaandv 2.4 Kinematics in path(normal-tangential)coordinates

  19. Mechanics 2.4.1 Curvature (曲率) 2.4.2 definition of path coordinate Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate 2.4.3 Geometric preliminaries 2.4.4 Derivatives of base vector 2.4.5 Velocity in the n-t coordinate system 2.4.6 Acceleration in the n-t coordinate system 2.4.7 Special cases of motion 2.4.8 Sample examples 2.4 Kinematics in path(normal-tangential)coordinates

  20. 2.4.7 Special cases of motion There are some special cases of motion to consider. • The particle moves along a straight line. In this case, only the tangential component is nonzero and represents the time rate of change of the magnitude of the velocity vector. 2.4 Kinematics in path(normal-tangential)coordinates

  21. 2.4.7 Special cases of motion • The particle moves along a curve at constant speed. In this case, only the normal component is nonzero and represents the time rate of change of the direction of the velocity vector. 2.4 Kinematics in path(normal-tangential)coordinates

  22. 2.4.7 Special cases of motion • The tangential component of ais constant, at= (at)c. • In this case, Heresoand voare the initial position and velocity of the particle att = 0. 2.4 Kinematics in path(normal-tangential)coordinates

  23. 2.4.7 Special cases of motion a • Motion along a circular path  If the path is a circle of radius R  R Where = d/dtis called the angular velocity Tangential acceleration Normal (or centripetal) acceleration.anis always directed toward the center of the circle 2.4 Kinematics in path(normal-tangential)coordinates

  24. Mechanics 2.4.1 Curvature (曲率) 2.4.2 definition of path coordinate Chapter 2 Kinematics of particle 2.4 Kinematics in path coordinate 2.4.3 Geometric preliminaries 2.4.4 Derivatives of base vector 2.4.5 Velocity in the n-t coordinate system 2.4.6 Acceleration in the n-t coordinate system 2.4.7 Special cases of motion 2.4.8 Sample examples 2.4 Kinematics in path(normal-tangential)coordinates

  25. 2.4.8 Sample examples Sample example 2.4-1 Given:Starting from rest, a motorboat travels around a circular path ofr = 50 m at a speed that increases with time v = (0.2 t2) m/s. Find: The magnitudes of the boat’s velocity and acceleration at the instant t = 3 s. 2.4 Kinematics in path(normal-tangential)coordinates

  26. Plan:The boat starts from rest (v = 0 when t = 0). 1) Calculate the velocity att = 3s usingv(t). 2) Calculate the tangential and normal components of acceleration and then the magnitude of the acceleration vector. 2.4.8 Sample examples Solution: 1) The velocity vector isv = vet, where the magnitude is given byv = (0.2t2) m/s. Att = 3s: v = 0.2t2 = 0.2(3)2 = 1.8 m/s 2) The acceleration vector is 2.4 Kinematics in path(normal-tangential)coordinates

  27. 2.4.8 Sample examples Tangential component: At t = 3s: at = 0.4t = 0.4(3) = 1.2 m/s2 Normal component: At t = 3s: an = [(0.2)(32)]2/(50) = 0.0648 m/s2 The magnitude of the acceleration is a = [(at)2 + (an)2]0.5 = [(1.2)2 + (0.0648)2]0.5 = 1.20 m/s2 2.4 Kinematics in path(normal-tangential)coordinates

  28. 2.4.8 Sample examples Sample example 2.4-2 Given:A jet plane travels along a vertical parabolic path defined by the equationy = 0.4x2. At point A, the jet has a speed of 200m/s, which is increasing at the rate of 0.8 m/s2. Find: The magnitude of the plane’s acceleration when it is at point A. 2.4 Kinematics in path(normal-tangential)coordinates

  29. 2.4.8 Sample examples Plan:1) The change in the speed of the plane (0.8 m/s2) is the tangential component of the total acceleration. 2) Calculate the radius of curvature of the path at A. 3) Calculate the normal component of acceleration. 4) Determine the magnitude of the acceleration vector. Solution: 1) Thetangential componentof acceleration is the rate of increase of the plane’s speed, soat = dv/dt = 0.8 m/s2. 2.4 Kinematics in path(normal-tangential)coordinates

  30. 2.4.8 Sample examples 2) Determine theradius of curvatureat point A (x = 5 km): At x =5 km, dy/dx = 0.8(5) = 4, d2y/dx2 = 0.8 3) Thenormal componentof acceleration is an = v2/r = (200)2/(87.62 x 103) = 0.457 m/s2 4) Themagnitudeof the acceleration vector is a = [(at)2 + (an)2]0.5 = [(0.8)2 + (0.457)2]0.5 = 0.921 m/s2 2.4 Kinematics in path(normal-tangential)coordinates

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