KPE 263 Introductory Biomechanics Lecture Notes PDF

KPE 263 Introductory Biomechanics Week 4-Part 1 Angular Kinematics Objectives Distinguish between linear, angular, and general motion Determine and quantify absolute and relative angles Quantify and interpret angular displacement, angular velocity, and angular accel...

KPE 263 Introductory Biomechanics Week 4-Part 1 Angular Kinematics Objectives Distinguish between linear, angular, and general motion Determine and quantify absolute and relative angles Quantify and interpret angular displacement, angular velocity, and angular acceleration 2 What is Biomechanics? BIOMECHANICS RIGID BODIES DEFORMABLE SOLIDS FLUIDS STATICS DYNAMICS Tissue Mechanics STATICS DYNAMICS KINEMATICS KINETICS Displacement/distance Force Velocity/speed Work Acceleration Energy Momentum Linear/Angular Linear/Angular 3 What is Biomechanics? BIOMECHANICS RIGID BODIES DEFORMABLE SOLIDS FLUIDS STATICS DYNAMICS Tissue Mechanics STATICS DYNAMICS KINEMATICS KINETICS Displacement/distance Force Velocity/speed Work Acceleration Energy Momentum Linear/Angular Linear/Angular 4 Angular Kinematics Angular Motion Motion about an axis of rotation All parts of a ridged body move through the same angle but different linear displacement Axis of rotation = line that is perpendicular to the plane in which the rotation occurs Axis of rotation (rotating about a vertical axis) Motion occurring in Motion occurring in sagittal transverse plane plane Axis of rotation (rotating about a vertical axis) 6 Angular Motion Quantifying Angular Motion Angular Motion is the change in position with respect to both spatial (relative to points in space) and temporal (relative to time) frames of reference Angular Displacement (Θ) = change from the initial angular position O ∆𝜃 = 𝜃2 − 𝜃1 Angular Velocity (ω) = change in angular position relative to the time interval it took place 𝜃2 − 𝜃1 O 𝜔= 𝑡2 − 𝑡1 Angular Acceleration (α) = change in angular velocity relative to the interval of that change 𝜔2 − 𝜔1 O 𝛼= 𝑡2 − 𝑡1 7 Angular Motion Representing Angles · Angles (θ) are measured in relation to circles Three different units 𝑠 Radians – most appropriate for angular motion 𝜃(𝑟𝑎𝑑) = a angular 𝑟 for – ratio of arc length to radius use Linearelations I deg3 – unitless 57. 𝑠 Degrees – most common/best understood 𝜃(𝑑𝑒𝑔) = 57.3 𝑟 – 1𝑟𝑎𝑑 = 57.3° (57 3) rad. 𝑠 Revolutions/Rotations – qualitative 𝜃(𝑟𝑒𝑣) = 0.159 𝑟 I revolution : 360 8 Angular Position Angular Position (θ) Absolute Angle pr proximal oximates _ Orientation (angle) of a line segment in space SEGMENT ANGLES Example: Instantaneous angle (θ) of trunk segment 𝑦𝑝𝑟𝑜𝑥 − 𝑦𝑑𝑖𝑠𝑡𝑎𝑙 #gat θ Coordinate 𝑡𝑎𝑛𝜃 = 𝑥𝑝𝑟𝑜𝑥 − 𝑥𝑑𝑖𝑠𝑡𝑎𝑙 (distaly tang = distale 0. 83 - 0 - O = Tan 24 - 0 53. O 0. 37 - 18 =. o tant 29 o 11 Angular Position (θ) Absolute Angle Orientation (angle) of a line segment in space Example: Instantaneous angle (θ) of trunk segment 𝑦𝑝𝑟𝑜𝑥 − 𝑦𝑑𝑖𝑠𝑡𝑎𝑙 𝑡𝑎𝑛𝜃 = 𝑥𝑝𝑟𝑜𝑥 − 𝑥𝑑𝑖𝑠𝑡𝑎𝑙 θ 0.83 − 𝟎. 𝟔𝟏 𝑡𝑎𝑛𝜃 = 0.24 − 𝟎. 𝟓𝟑 0.83 − 𝟎. 𝟔𝟏 𝜃 = 𝑡𝑎𝑛−1 0.24 − 𝟎. 𝟓𝟑 𝜃 = 𝑡𝑎𝑛−1 0.22 𝜃 = −37.18° −0.29 12 Angular Position (θ) Absolute Angle Orientation (angle) of a line segment in space Example: Instantaneous angle (θ) of thigh segment 𝑦𝑝𝑟𝑜𝑥 − 𝑦𝑑𝑖𝑠𝑡𝑎𝑙 𝑡𝑎𝑛𝜃 = 𝑥𝑝𝑟𝑜𝑥 − 𝑥𝑑𝑖𝑠𝑡𝑎𝑙 Fant = 𝟎. 𝟓𝟗 − 0.46 4 ·θ 𝑡𝑎𝑛𝜃 = G :tan- -OSe 0.52 − 0.24 𝟎. 𝟓𝟗 − 0.46 𝜃 = 𝑡𝑎𝑛−1 0 13 90% tant0.52 − 0.24 =24.. 0 o= S 𝜃 = 𝑡𝑎𝑛−1 0.13 𝜃 = 24.90° 0.28 13 Angular Position (θ) If reference axis is fixed, then angular position is absolute e.g., trunk and thigh segment angles on previous slides (horizontal axis is “fixed” to the earth) If reference axis is capable of moving, then angular position is relative e.g., angle of the tibia with respect to the femur when running (knee joint angle) 14 Angular Motion Axis of rotation (rotating about a vertical axis) 15 Angular Position (θ) Absolute vs. Relative Angles 16 Angular Position (θ) Absolute vs. Relative Angles Segment Angles (Absolute) I n Wh flexion E knee angle 17 Angular Position (θ) Relative Angles Angles created by the longitudinal axes of two distinct segments (e.g., knee flexion angle = the angle between the thing and the leg in the sagittal plane) Relative Angles found using the Law of Cosines C b 𝑎2 = 𝑏2 + 𝑐 2 − 2 𝑏𝑐 𝑥 𝑐𝑜𝑠𝐴 A a 𝑏 2 = 𝑎2 + 𝑐 2 − 2 𝑎𝑐 𝑥 𝑐𝑜𝑠𝐵 c 𝑐 2 = 𝑎2 + 𝑏 2 − 2 𝑎𝑏 𝑥 𝑐𝑜𝑠𝐶 B 18  1 Angular Position (θ) 02 Relative Angles: Example 𝑎= 𝑥ℎ𝑖𝑝 − 𝑥𝑎𝑛𝑘𝑙𝑒 2 + 𝑦ℎ𝑖𝑝 − 𝑦𝑎𝑛𝑘𝑙𝑒 2 - ay ---- · 61) 05 a 0. 𝑎a = = + 0.61 − 0.1818) 0.51 − 0.38382 - 0 , 2 + 10. 61 - 0. 2 231 = 20. 069 185 +0. a 𝑎= 0.0169 + 0.185 0. 31m C = ⑤ 45m 0162m 0 b= a =. b C 𝒂 =0.45m b=0.32m c=0.31m A at = b2+ c2 - zbcocosA 𝑎2 = 𝑏2 + 𝑐 2 − 2 𝑏𝑐 𝑥 𝑐𝑜𝑠𝐴 31) 312 (0 32) (0 cost c B · 452-0 32" ,. +o. - 2. a 0 proximal. aS always define 0.452 = 0.322 + 0.312 − 2 relative (0.32)(0.31) 𝑥 𝑐𝑜𝑠𝐴 to A = - 900 dista A = -90° 625 2. 19 Angular Position (θ) Relative Angles: Can also be found if we know absolute angles of two segments 𝜃𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 = 𝜃𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑙 − 𝜃𝑑𝑖𝑠𝑡𝑎𝑙 𝜃𝑘𝑛𝑒𝑒 = 𝜃𝑡ℎ𝑖𝑔ℎ − 𝜃𝑙𝑒𝑔 20