1 / 34

Kinematics of a particle

CHAPTER.12. Kinematics of a particle. Mechanics of rigid body. mechanics of rigid body. Galilei. Statics. Dynamics. Statics. Dynamics. Newton. Euler. Lagrange. Equilibrium. Σ F = 0. Kinematics. Kinetics. v=ds/dt a=dv/dt. Σ F = m a. 12.1 Introduction.

gerry
Download Presentation

Kinematics of a particle

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CHAPTER.12 Kinematics of a particle

  2. Mechanics of rigid body mechanics of rigid body Galilei Statics Dynamics Statics Dynamics Newton Euler Lagrange Equilibrium Σ F = 0 Kinematics Kinetics v=ds/dt a=dv/dt Σ F = m a 12.1 Introduction

  3. 12.2 Rectilinear Kinematics: Continuous motion Kinematics Analysis of the geometric aspects of motion. Particle A particle has a mass but negligible size and shape. Rectilinear Kinematics Kinematics of objects moving along straight path and characterized by objects position, velocity and acceleration. Position(1) position vector r A vector used to specify the location of particle P at any instant from origin O.

  4. (2) position coordinate , S An algebraic scalar used to represent the position coordinate of particle P scalar vector r’ r r Ds P P’ o s s’ from O to P. • DisplacementChange in position of a particle , vector (1) Displacement or

  5. (3) Speed speed = magnitude of velocity = | v |Average speed = Total distance/elapsed time = • (2) DistanceTotallength of path traversed by the particle. A positive scalar. • Velocity(1) Average velocity (2) Instantaneous Velocity

  6. Acceleration (1) Average acceleration (2) (Instantaneous) acceleration • Relation involving a , s and vv=ds/dt, dt=ds/va=dv/dt, dt=dv/a • so, ds/v=dv/a vdv=ads

  7. Constants acceleration a = ac

  8. 10. Analysis Procedure • Coordinate System • A. Establish a position coordinate s along the path. • B. Specify the fixed origin and positive direction of the coordinate. (2) Kinematic Equations A. Know the relationship between any two of the four variables a, v, a and t. B. Use the kinematic equations to determine the unknown varaibles

  9. S V a t t t 12.3 Rectilinear Kinematics : Erratic Motion

  10. p 12-4 General Curvilinear Motion 1. Curvilinear motion The particle moves along a curved path. Vector analysis will be used to formulate the particle’s position, velocity and acceleration.

  11. p s p’ r r r ’ o p s r (t) o 2. Position s 3. Displacement = change in position of particle form p to p’

  12. 4. Velocity p v p’ r r r ’ o (1) average velocity 平均 (2) Instantaneous velocity 瞬時 = “tangent” to the curve at Pt .p = “tangent” to the path of motion (3) Speed

  13. Hodograph 5. Acceleration (1) Average acceleration = time rate of change of velocity vectors Hodogragh is a curve of the locus of points for the arrowhead of velocity vector. (2) Instantaneous acceleration which is not tangent to the curve of motion, but tangent to the hodograph.

  14. s path p z r y θ x 12-5 Curvilinear Motion : Rectangular components xyz : fixed rectangular coordinate system

  15. 1. Position vector Here = magnitude of = unit vector = direction of

  16. 2. Velocity 0 0 0 tangent to the path

  17. 3. Acceleration

  18. : initial velocity : Constant downward acceleration : velocity at any instant a= - g j (v0)y v0 (v0)x y v v0 x 12.6 Motion of a projectile

  19. Acceleration Vector a == + = -g = ax + ay V0 = (Vx)o + (Vy)o (known) Position Vector (x,y components) = x + y initial position = xo + yo Velocity Vector = = x + y = Vx + Vy

  20. 1. Horizontal motion, ax=0 Vx = (Vx)0 + axt = (Vx)0 X = X0 + (Vx)0t Same as 1st Eq. One independent eqn X = X0 + (Vx)0t

  21. 2. Vertical motion, ay=-g constant Can be derived from above two Eqs. two independent eqns

  22. 12-7 Curvilinear Motion:Normal and Tangential components. n o’ ρ path o un p t ut Path of motion of a particle is known. 1. Planar motion s Here: t (tangent axis ): axis tangent to the curve at P and positive in the direction of increasing S; ut: unit vector n (normal axis ): axis perpendicular to t axis and directed from P toward to the center of curvature o’; un: unit vector o’ = center of curvature r = radius of curvature p = origin of coordinate system tn

  23. un o’ ρ ut’ dθ un p ut’ dθ p ds dut ut ut (1) Path Function (known) (2) Velocity (3) Acceleration

  24. at: Change in magnitude of velocity an: Change in direction of velocity If the path in y = f ( x )

  25. r:radial coordinate , :transverse coordinate , 12-8 Curvilinear Motion:Cylindrical Components • Polar coordinates (1) coordinates (r,q) q r p r o Reference line (2) Position (3) Velocity

  26. rate of change of the length of the radial coordinate. angular velocity (rad/s)

  27. (4) Acceleration angular acceleration

  28. 3D z y r x 2. Cylindrical coordinates Position vector Velocity Acceleration

  29. A B A B 12.9 Absolute Dependent Motion Analysis of Two Particles • Absolute Dependent MotionThe motion of one particle depends on the corresponding motion of another particle when they are interconnected by inextensible cords which are wrapped around pulleys.

  30. (1)position-coordinate equation A. Specify the location of particles using position coordinates having their origin located at a fixed point or datum line. B. Relate coordinates to the total length of card lT (2)Time Derivatives Take time derivatives of the position-coordinate equation to yield the required velocity and acceleration equations. 2.Analysis procedure

  31. 3. Example Datum Datum B A • position-coordinate equation (2) Time Derivatives

  32. xyz:fixed framex’y’z’:translating frame moving with particle ArA、rB : absolute positions of particle A & BrB/A : relative position of B with respect to A z’ A y’ z rA rB/A x’ o y rB B x 12.10 Relative-Motion Analysis of Two Particles • Translating frames of referenceAframe of reference whose axes do not rotate and are only permitted to translate relative to the fixed frame.

  33. A rA rB/A rA rB rB/A O B rB aB/A:acceleration of B as seen by an observer located at A and translating with x’y’z’ frame. • position vector 3. velocity Vector VB/A : relative velocity observed from the translating frame. 4. acceleration vector

More Related