Lecture 09v


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Lecture 09v

  1. 1. Angular Kinematics Objectives: • Introduce the angular concepts of absolute and relative angles, displacement, distance, velocity, and speed • Learn how to compute angular displacement, velocity, and speed • Learn to compute and estimate instantaneous angular velocity Angular KinematicsKinematics• The form, pattern, or sequencing of movement with respect to time• Forces causing the motion are not consideredAngular Motion (Rotation)• All points in an object or system move in a circle about a single axis of rotation. All points move through the same angle in the same timeAngular Kinematics• The kinematics of particles, objects, or systems undergoing angular motion 1
  2. 2. Angular Kinematics & Motion• Most volitional movement is performed through rotation SHOULDER of the body segments NECK• The body is often analyzed ELBOW as a collection of rigid, LUMBAR rotating segments linked at HIP the joint centers• This is a rough KNEE approximation ANKLE Measuring AnglesDegrees: Radians: 90 π/2 57.3° 1 radian180 0, 360 π 0, 2 π 270 3π/21 radian = 57.3° π = 3.141591 revolution = 360° = 2π radians Note: Excel usesθ(degrees) = (180/π)× θ(radians) radians! 2
  3. 3. Positive vs. Negative Angles By convention, when describing angular kinematics: • positive angles counterclockwise rotation • negative angles clockwise rotation Positive: Negative: +90° -270° +57.3°+180° 0,+360° -180° 0,-360° -57.3° +270° -90° Absolute Angle (or Angle of Inclination) • Angular orientation of a line segment with respect to a fixed line of reference • Use the same reference for all absolute angles θ θ Trunk angle Trunk angle from horizontal from vertical 3
  4. 4. Angular Displacement • Change in the angular position or orientation of a line segment • Doesn’t depend on the path between orientations • Has angular units (e.g. degrees, radians) axis of rotation initial final orientation orientation angular displacement Computing Angular Displacement• Compute angular displacement (∆θ) by subtraction of angular positions: ∆θ = θfinal – θinitial ∆θ final orientation θfinal initial orientation θinitial axis of rotation 4
  5. 5. Angular Distance• Sum of the magnitude of all angular changes undergone by a rotating body• Has angular units of length (e.g. degrees, radians)• Distance ≥ (Magnitude of displacement) Angular final orientation Distance = 225° -90°intermediate Angular orientation 135° Displacement = 45° axis of rotation initial orientation Example Problem #1 A figure skater spins 10.5 revolutions in a clockwise direction, pauses, then spins 60° counterclockwise before skating away. What were the total angular distance and angular displacement? 5
  6. 6. Relative Angle• Angle between two line segments• Compute relative angle by subtraction of absolute angular positions: θ(1→2) = θ2 – θ1 θ(1→2) segment 1 segment 2 θ2 θ1 axis of rotation Joint Angles• Joint angles are relative angles between longitudinal axes of adjacent segments (or between anterior- posterior axes for internal rotation) θelbow θshoulder θhip θknee Use a consistent sign convention for joint angles θankle (e.g. + = flexion) 6
  7. 7. Computing Joint Angles• Involves subtracting absolute angles of segments• Exact formula and order of subtraction depends on the joint and the convention chosen θknee = θleg– θthigh θknee = 180° + θthigh– θleg HIP θthigh HIP θthigh θknee θleg θleg KNEE ANKLE KNEE θknee ANKLE Angular Velocity• The rate of change in the absolute or relative angular position or orientation of a line segment change in angular angular angular position displacement velocity = = change in time change in time• Shorthand notation: θfinal – θinitial ∆θ ω= = tfinal – tinitial ∆t• Has units of (angular units)/time (e.g. radians/s, °/s) 7
  8. 8. Angular Speed• The angular distance traveled divided by the time taken to cover it• Equal to the average magnitude of the instantaneous angular velocity over that time. angular distance angular speed = change in time• Has units of (angular units)/time (e.g. radians/s, °/s) Angular Speed vs. Velocity Angular end of follow- Distance = 225° through -90° end of Angular backswing 135° Displacement = 45° tennis player racquet at start Assume tennis stroke shown takes 0.75 s: 225° +45° Speed = Velocity = 0.75 s 0.75 s = 300°/s = +60°/s 8
  9. 9. Example Problem #2The figure skater of Problem #1 completes the first (clockwise) spin in 3 s, pauses for 1 s, then completes the second (counterclockwise) spin in 0.3 s.What were her average angular velocity and average angular speed during the first spin?What were her average angular velocity and average angular speed for the skill as a whole? Example Problem #3A person is performing a squat exercise. She starts from a standing (i.e. anatomical) position.At her lowest point, 2 seconds later, her knees are flexed to 60° and her hips are flexed to 90° from the anatomical position.1 second later, she has risen back to the standing position and completed the exercise.What were the average knee and hip angular velocities during each phase of the exercise? for the exercise as a whole? 9
  10. 10. Average vs. Instantaneous Velocity • Previous formulas give us the average velocity between an initial time (t1) and a final time (t2) • Instantaneous angular velocity is the angular velocity at a single instant in time • Can estimate instantaneous angular velocity using the central difference method: θ (at t1 + ∆t) – θ (at t1 – ∆t) ω (at t1) = 2 ∆t where ∆t is a very small change in time Angular Velocity as a Slope• Graph of angular position vs. time slope = instantaneous ω at t1 θ (degrees) slope = average ω from t1 to t2 ∆θ(1→2) ∆t(1→2) ∆t t1 t2 time (s) 10
  11. 11. Estimating Angular Velocity Identify points with zero slope = pointsθ (deg) with zero velocity Portions of the curve with positive slope time (s) have positive velocity (i.e. velocity in theω (deg/s) + direction) Portions of the curve with negative slope 0 time (s) have negative velocity (i.e. velocity in the – direction) Example Problem #4 A gymnast swings back and forth from the high bar as shown below. Sketch her angular velocity. 80 60 40 angle (deg) 20 0 0 2 4 6 8 10 -20 θ -40 -60 time (s) 11