UMass Amherst KIN 430 - differentiation-integration (1) (9 pages)

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differentiation-integration (1)



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differentiation-integration (1)

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Pages:
9
School:
University of Massachusetts Amherst
Course:
Kin 430 - Biomechanics
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Differentiation Integration Differentiation Mathematical process for finding the instantaneous rate of change of a variable related to finding the slope of a line position velocity acceleration by means of differentiation Analytical Numerical Graphical Integration Mathematical process for finding the total area under a curve basically a summation process The rate of change of one variable pos vel with respect to another time velocity is calculated from position and time acceleration is calculated from vel and time Analytical Numerical Graphical Copyright 2017 University of Massachusetts Amherst Derivative is the slope of a line s2 pos rise Average vs Instantaneous s1 Basic kinematic formulas give average values only run e g tf ti time There will be situations were you need to know the instantaneous velocity and acceleration Vel slope of the pos vs time curve Acc slope of the vel vs time curve 1 Analytically instantaneous Numerically very close to instantaneous Determine the slope of the tangent line Determine the slope of the secant line Position Position tangent t secant ti 1 Time finite difference method Time s Pos m Vel m s 0 0 3 20 A numerical technique that provides an approximation to the true instantaneous derivative ti ti 1 Time First finite difference method 3pt Time s Pos m Vel m s 0 0 3 20 0 1 5 80 52 0 0 2 13 60 48 0 0 3 15 40 19 0 0 1 5 80 0 2 13 60 0 3 15 40 0 4 9 80 0 4 9 80 39 0 0 5 7 60 0 5 7 60 Example Use for points 2 through n 1 2 First forward difference method 2pt First backward difference method 2pt Time s Pos m Vel m s Time s Pos m Vel m s 0 0 3 20 26 0 0 0 3 20 26 0 0 1 5 80 52 0 0 1 5 80 52 0 0 2 13 60 48 0 0 2 13 60 48 0 0 3 15 40 19 0 0 3 15 40 19 0 0 4 9 80 39 0 0 4 9 80 39 0 0 5 7 60 0 5 7 60 22 0 Use for point 1 Use for point n Calculating acceleration Graphical Differentiation Time s Pos m Vel m s Acc m s2 0 0 3 20 26 0 260 0 1 5 80 52 0 110 0 2 13 60 48 0 355 0 3 15 40 19 0 435 0 4 9 80 39 0 15 0 5 7 60 22 0 170 Sometimes it may be necessary and or preferable to evaluate the derivative graphically Can be done by inspecting a position or velocity vs time graph at a few key points Provides a qualitative assessment of the instantaneous velocity or acceleration Apply same equations to the velocity data 3 Position m Acc m s2 Velocity m s Position m Velocity m s Acc m s2 Position m Time s Time s Graphical differentiation Velocity m s 1 Assess instantaneous slope at time of interest draw tangent line 2 Find places where slope 0 3 Identify regions of and slope 4 Find places where slope is steepest 5 Fill in by how slope is changing between these points Acc m s2 Ignore magnitude consider slope only Time s 4 Position m Velocity m s Position m Velocity m s Time s Time s Velocity m s Position m Differentiation Summary Analytical limit formulas t 0 Numerical first finite forward backward difference methods Graphical visual inspection of the slope of a plotted line Time s 5 acceleration velocity position What is integration by means of integration Why Determine velocity and or position from acceleration data such as from an accelerometer Used in biomechanical simulations to predict motion of the body from the forces that are produced v1 Position m t0 t1 t0 t1 v5 v2 t2 t3 t4 t5 s4 s3 s1 v4 s2 t2 t3 v3 Vel m s v3 Vel m s Taking the cumulative area under the curve Can be thought of as the opposite of differentiation Think of how the AREA is ACCUMULATING t4 v1 t0 Position m t5 t1 t0 t1 v5 v2 t2 t3 t4 t5 s4 s3 s1 v4 s2 t2 t3 t4 t5 6 Time s 0 0 0 1 acc m s2 9 81 9 81 Time s 0 0 0 1 acc m s2 9 81 9 81 vel m s 0 0 0 981 0 2 0 3 9 81 9 81 0 2 0 3 9 81 9 81 1 962 2 943 0 4 0 5 9 81 9 81 0 4 0 5 9 81 9 81 3 924 4 905 0 6 0 7 9 81 9 81 0 6 0 7 9 81 9 81 5 886 6 867 0 8 9 81 0 8 9 81 7 848 Time s 0 0 0 1 acc m s2 9 81 9 81 vel m s 0 0 0 981 disp m 0 0 0 04 0 2 9 81 1 962 0 20 0 3 0 4 9 81 9 81 2 943 3 924 0 44 0 78 0 5 0 6 9 81 9 81 4 905 5 886 1 23 1 77 0 7 0 8 9 81 9 81 6 867 7 848 2 40 3 12 vel m s 0 0 disp m 0 0 disp m 0 0 Graphical Integration As with differentiation the process of integration can also be done graphically It involves inspecting an acceleration or velocity vs time graph at key points Remember to keep in mind how the area is accumulating 7 Graphical integration examples Graphical integration Graphical example CM vel acc in running 1 Assess how the area under the curve is accumulating draw narrow boxes 2 Identify regions of and area 3 Find inflection points minima maxima 4 Find points where curve crosses zero 5 Fill in by rate of accumulating area slope between these points 5 4 3 2 1 0 Time s Graphical integration examples Time s Graphical integration examples Time s 8 Differentiation Integration Summary Which direction are you going Which process to use Graphically what to look at Pos Vel Acc Differentiation Slope Acc Vel Pos Integration Accumulating Area Up next Back to Linear Kinematics Acknowledgement The following individuals have contributed substantially to the materials in these slides GE Caldwell PhD University of Massachusetts W McDermott PhD University of Massachusetts Mark Miller PhD University of Massachusetts These notes are to be used only as a study aid in conjunction with the course KIN 430 Biomechanics Kinesiology at the University of Massachusetts Amherst These notes may contain images and or other material that are covered by national and or international copyright law Reproduction and redistribution in any form is forbidden 9


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