! definition and units of angular. ! constant angular acceleration equations. ! relations between linear and angular. !

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1 ANGULAR KINEMATICS Ozkaya and Nodin p Outline! definition and units of angula displacement, angula velocity, and angula acceleation! constant angula acceleation equations! elations between linea and angula kinematic quantities! elative motion! techniques fo measuing and descibing joint angles Stephen Robinovitch, Ph.D. KIN Lectue 10: Linea Kinematics II 2 Most motion involves tanslation and otation (a) = + Angula displacement counte-clockwise otation is positive, clockwise otation is negative +250 o -110 o (b) = + 1 evolution = 360 degees = 2! adians! (adians) = " 180! (degees) (c) 3

2 Angula velocity and acceleation! Angula velocity is denoted with the symbol #, measued in units of ad/s, and defined as! = " = d"! Angula acceleation is denoted with the symbol ", measued in units of ad/s 2, and defined as! = " = # = d# = d 2 " = d $ d" ' & ) % (! If we ae using numeical diffeentiation:! i = " I +1 #" I #1 2$t The impotance of!t in numeical diffeentiation The gaph at left shows the actual vaiation in x(t) and the instantaneous (tangent) velocity dx at time t 1. In the laboatoy, x(t) can be sampled evey!t=(t 1 -t 0 ), o evey!t/2,!t/4, o!t/8. What sampling inteval is equied to obtain a easonable % i =! estimate of dx? I +1 #! I #1 2$t 6!t Kinematic Quantities: Units Angula Kinematics Example quantity linea angula time second (s) second (s) displacement mete (m) adians (ad) velocity acceleation metes pe second (m!s -1 ) adians pe second metes pe second pe second (m!s -2 ) adians pe second pe second (ad!s -1 ) (ad!s -2 ) Fo above pictue, if pointing to 12 epesents an angula displacement of zeo and clockwise otation is positive, angula displacement of minute hand = angula displacement of hou hand = angula velocity of second hand = angula acceleations of all hands = 7 8

3 Constant Angula Acceleation Equations (1)! =! 0 +" 0 t #t2 (2) " = " 0 + #t ( ) (3) " 2 $" 0 2 = 2#! $! 0 9 Relationship between linea and angula quantities s =! "! = s (with! measued in adians) All points on the hamme tavel though the same angula displacement. Howeve, linea displacement (and velocity) inceases linealy with distance fom the axis of otation. s $ d 2 d 1 $ Relationship between linea and angula quantities Any point P on a igid body otating about a fixed axis O has: a velocity that is tangential to the cicula path of motion an acceleation with tangential and nomal (inwadly-diected) components v t = ds = d! = " a t = dv t = d" = # a n = v 2 t = " 2 Angula velocities of limb segments goven linea velocity of ball 12

4 Relative motion: COG velocity also affects linea velocity of ball How is the elease velocity of the ball affected by the hoizontal velocity of the whole-body COG? HAND = COG + HAND/ COG v HAND = v COG + v HAND/ COG a HAND = a COG + a HAND/ COG # am v COG Relative vesus absolute angles! An absolute angle is measued fom an extenal fame of efeence.! A elative angle is the angle fomed between two limb segments. $ el $ abs 13 Relative Angles The pefeed method fo descibing a elative joint position is degees of flexion o extension. Joint position may altenatively be descibed as the angle fomed at the aticulation. (180 " $ el ) $ el Measuing lowe extemity joint angles Diagam at left shows, coodinates (acquied with a motion measuement system) of the geate tochante, lateal femoal epicondyle, and lateal malleolus. Knowing these values, how would we detemine $ knee? ( H, H ) $ knee ( K, K ) ( A, A )

5 Measuing lowe extemity joint angles (contd) tan" = #y #x $" shin = tan -1 k % A k % A " thigh = tan -1 H % K H % K " knee = " thigh %" shin ( H, H ) $ knee ( K, K ) ( A, A ) $ thigh $ shin 17 Instantaneous joint centes of otation! In most biomechanical analyses, we assume that joints have fixed axes of otation! This assumption usually intoduces minimal eo! Howeve, fo many joints, changes in flexion angle cause coesponding changes in the location of the cente of otation (due to olling and sliding between aticulating sufaces) Review Questions! How can the sampling inteval (!t) affect the validity of paametes obtained though numeical diffeentiation?! What is the elationship between linea and angula displacement, velocity, and acceleation?! If a igid body otates with constant angula velocity, what is its acceleation?! Given the (,) positions of the hip, knee, and ankle, how can we detemine knee flexion angle? 19