Presentation on theme: "Angular Variables Linear Angular Position m s deg. or rad. q Velocity"— Presentation transcript:
1Angular Variables Linear Angular Position m s deg. or rad. q Velocity rad/swAcceleration2a
2Radians What is a radian? q = 1 rad = 57.3 360 = 2 p rad orr360o= 2pradqrWhat is a radian?a unitless measure of anglesthe SI unit for angular measurement1 radian is the angular distance covered when the arclength equals the radius
4Measuring AnglesRelative Angles (joint angles) The angle between the longitudinal axis of two adjacent segments.Absolute Angles(segment angles) The angle between a segment and the right horizontal of the distal end.Should be measuredconsistently on same sidejointstraight fully extendedposition is generallydefined as 0 degreesShould be consistentlymeasured in the samedirection from a singlereference - eitherhorizontal or vertical
5Measuring AnglesThe typical data that we have to work with in biomechanics are the x and y locations of the segment endpoints. These are digitized from video or film.(x2,y2)(x3,y3)(x4,y4)(x5,y5)(0,0)YX(x1,y1)Frame 1
6Tools for Measuring Body Angles goniometerselectrogoniometers (aka Elgon)potentiometersLeighton Flexometergravity based assessment of absolute angleICR - Instantaneous Center of Rotationoften have translation of the bones as wellas rotation so the exact axis moves within jt
7Calculating Absolute Angles Absolute angles can be calculated from the endpoint coordinates by using the arctangent (inverse tangent) function.qoppadj(x1,y1)(x2,y2)opp = y2-y1adj = x2-x1
8Calculating Relative Angles Relative angles can be calculated in one of two ways:1) Law of Cosines (useful if you have the segment lengths)q(x1,y1)(x2,y2)abc(x3,y3)c2 = a2 + b2 - 2ab(cosq)
9Calculating Relative Angles 2) Calculated from two absolute angles. (useful if you have the absolute angles)q3 = q1 + (180 - q2)q1q3q2
10CSB Gait Standards qtrunk qthigh qleg qfoot qhip qknee qankle CanadianSociety ofBiomechanicsqtrunkqthighqlegqfootqhipqkneeqankleAnatomical position is zero degrees.RIGHT sagittal viewsegment anglesjoint angles
11CSB Gait Standards qtrunk qthigh qleg qfoot qhip qknee qankle CanadianSociety ofBiomechanicsqtrunkqthighqlegqfootqhipqkneeqankleAnatomical position is zero degrees.LEFT sagittal viewsegment anglesjoint angles
12CSB Gait Standards (joint angles) RH-reference frame only! qhip = qthigh - qtrunkqhip> 0: flexed position qhip< 0: (hyper-)extended positionslope of qhip v. t > 0 flexingslope of qhip v. t < 0 extendingqknee = qthigh - qlegqknee> 0: flexed position qknee< 0: (hyper-)extended positionslope of qknee v. t > 0 flexingslope of qknee v. t < 0 extendingqankle = qfoot - qleg - 90odorsiflexed plantar flexed -dorsiflexing (slope +) plantar flexing (slope -)
13Angle ExampleThe following coordinates were digitized from the right lower extremity of a person walking. Calculate the thigh, leg and knee angles from these coordinates.HIP (4,10)KNEE (6,4)ANKLE (5,0)
23Angular Motion Vectors The representation of the angular motion vector is complicated by the fact that the motion is circular while vectors are represented by straight lines.
24Angular Motion Vectors Right Hand Rule: the vector is represented by an arrow drawn so that if curled fingers of the right hand point in the direction of the rotation, the direction of thevector coincideswith the directionof the extendedthumb.
25Angular Motion Vectors A segment rotating counterclockwise (CCW) has a positive value and is represented by a vector pointing out of the page.+-A segment rotating clockwise (CW) has a negative value and is represented by a vector pointing into the page.
26Angular Distance vs. Displacement analogous to linear distance and displacementangular distancelength of the angular path taken along a pathangular displacementfinal angular position relative to initial position q = qf - qiAngular Kinematics
27Angular Distance vs. Displacement Angular Displacement
28Angular PositionExample - Arm Curls231,4Consider 4 points in motion1. Start2. Top3. Horiz on way down4. End
29Position 1: -90Position 2: +75Position 3: 0Position 4: -90NOTE: startingpoint is NOT 01,432
30Distance and Displacement Computing AngularDistance and Displacement1,432f q1 to2 to3 to1 to 2 to1 to 2 to 3 to
33Angular Velocity (w)Angular velocity is the rate of change of angular position.It indicates how fast the angle is changing.Positive values indicate a counter clockwise rotation while negative values indicate a clockwise rotation.units: rad/s or degrees/s
34Angular Acceleration (a) Angular acceleration is the rate of change of angular velocity.It indicates how fast the angular velocity is changing.The sign of the acceleration vector is independent of the direction of rotation.units: rad/s2 or degrees/s2
35Equations of Constantly Accelerated Angular Motion Eqn 1:Eqn 2:Eqn 3:
36Angular to Linear consider an arm rotating about the shoulder Point B on the arm moves through a greater distance than point A, but the time of movement is the same. Therefore, the linear velocity (Dp/Dt) of point B is greater than point A.The magnitude of this linear velocity is related to the distance from the axis of rotation (r).
37Angular to LinearThe following formula convert angular parameters to linear parameters:s = qrv = wrat = arac = w2r or v2/rNote: the angles must be measured in radians NOT degrees
38q to s (s = qr)rqrThe right horizontal is 0o and positive angles proceed counter-clockwise. example: r = 1m, q = 100o, What is s?s = 100*1 = 100 mNO!!! q must be in radianss = (100 deg* 1rad/57.3 deg)*1m = 1.75 m
39w to v (v = wr) hip tangential velocity radial axis ankle The direction of the velocity vector (v) is perpendicular to the radial axis and in the direction of the motion. This velocity is called the tangential velocity. example: r = 1m, w = 4 rad/sec, What is the magnitude of v?v = 4rad/s*1m = 4 m/s
40Bowling example vt = tangential velocity w = angular velocity r = radiusGiven w = 720 deg/s at releaser = 0.9 mCalculate vtEquation: vt = wrwrvtvtFirst convert deg/s to rad/s: 720deg*1rad/57.3deg = rad/s
41Batting example vt = wr choosing the right bat Things to consider when you want to use a longer bat:1) What is most important in swing?- contact velocity2) If you have a longer bat that doesn’t inhibit angular velocity then it is good - WHY?3) If you are not strong enough to handle the longer bat then what happens to angular velocity? Contact velocity?
42a to at (at = ar) Increasing angular speed ccw: positive a. Decreasing angular speed ccw: negative a.Increasing angular speed cw: negative a.Decreasing angular speed cw: positive a.There is a tangential acceleration whenever the angular speed is changing.
43Centripetal Acceleration TDCw is constantBy examining the components of the velocity it is clear that there is acceleration even when the angular velocity is constant.
44a to ac (ac = w2r or ac = v2/r) Even if the velocity vector is not changing magnitude, the direction of the vector is constantly changing during angular motion.There is an acceleration toward the axis of rotation that accounts for this change in direction of the velocity vector.This acceleration is called centripetal, axial, radial or normal acceleration.
45Resultant Linear Acceleration Since the tangential acceleration and the centripetal acceleration are orthogonal (perpendicular), the magnitude of the resultant linear acceleration can be found using the Pythagorean Theorem:acat