1Differentiation & Integration• Differentiation – Mathematical process for finding the instantaneous rate of change of a variable (related to finding the slope of a line)– Analytical– Numerical– Graphical• Integration – Mathematical process for finding the total area under a curve (basically a summation process)– Analytical– Numerical– GraphicalCopyright © 2017University of Massachusetts Amherstposition  velocity  accelerationby means of differentiation• 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 timeDerivative is the slope of a lineVel = slope of the pos vs time curveAcc = slope of the vel vs time curvepostitftimes2s1runrise• Basic kinematic formulas give averagevalues onlye.g.,• There will be situations were you need to know the instantaneous velocity and accelerationAverage vs Instantaneous2TimePositiontangentAnalytically (instantaneous)Determine the slope of the tangent linetTimePositionsecantNumerically (very close to instantaneous)Determine the slope of the secant lineti-1 ti ti+1finite difference methodTime (s)Pos (m)0.0 3.200.1 5.800.2 13.600.3 15.400.4 9.800.5 7.60Vel (m/s)??????A numerical technique that provides an approximation to the true instantaneous derivativeFirst finite difference method (3pt)Time (s)Pos (m)0.0 3.200.1 5.800.2 13.600.3 15.400.4 9.800.5 7.60Vel(m/s)?52.048.0-19.0-39.0?Example:Use for points 2 through n-13First forward difference method (2pt)Time (s)Pos (m)0.0 3.200.1 5.800.2 13.600.3 15.400.4 9.800.5 7.60Vel(m/s)26.052.048.0-19.0-39.0?Use for point 1First backward difference method (2pt)Time (s)Pos (m)0.0 3.200.1 5.800.2 13.600.3 15.400.4 9.800.5 7.60Vel(m/s)26.052.048.0-19.0-39.0-22.0Use for point nCalculating accelerationTime (s)Pos (m)0.0 3.200.1 5.800.2 13.600.3 15.400.4 9.800.5 7.60Vel (m/s)26.052.048.0-19.0-39.0-22.0Acc (m/s2)260110-355-435-15170Apply same equations to the velocity dataGraphical Differentiation• 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)4Position (m)Velocity (m/s)Acc (m/s2)Time (s)Position (m)Velocity (m/s)Acc (m/s2)Time (s)Position (m)Velocity (m/s)Acc (m/s2)Time (s)Graphical differentiation1. Assess instantaneous slope at time of interest – draw tangent line2. Find places where slope = 03. Identify regions of (+) and (-) slope4. Find places where slope is steepest5. Fill in by how slope is changing between these pointsIgnore magnitude…. consider slope only5Position (m)Velocity (m/s)Time (s)Position (m)Velocity (m/s)Time (s)Position (m)Velocity (m/s)Time (s)Differentiation - Summary• Analytical - limit formulas (t0)• Numerical – first finite, forward, backward difference methods• Graphical – visual inspection of the slope of a plotted line6acceleration  velocity  positionby 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 producedWhat is integration?• Taking the cumulative area under the curve• Can be thought of as the opposite of differentiation• Think of: how the AREA is ACCUMULATINGVel (m/s)t0t1 t2t3t4t5t0t1 t2t3t4t5Position (m)s4??s1s2s3v1v2v3v4v5Vel (m/s)t0t1 t2t3t4t5t0t1 t2t3t4t5Position (m)s1s2s3s4??v1v2v3v4v57Time (s)acc (m/s2)vel (m/s)disp(m)0.0 -9.81 0.0 0.00.1 -9.810.2 -9.810.3 -9.810.4 -9.810.5 -9.810.6 -9.810.7 -9.810.8 -9.81Time (s)acc (m/s2)vel (m/s)disp(m)0.0 -9.81 0.0 0.00.1 -9.81 -0.9810.2 -9.81 -1.9620.3 -9.81 -2.9430.4 -9.81 -3.9240.5 -9.81 -4.9050.6 -9.81 -5.8860.7 -9.81 -6.8670.8 -9.81 -7.848Time (s)acc (m/s2)vel (m/s)disp (m)0.0 -9.81 0.0 0.00.1 -9.81 -0.981 -0.040.2 -9.81 -1.962 -0.200.3 -9.81 -2.943 -0.440.4 -9.81 -3.924 -0.780.5 -9.81 -4.905 -1.230.6 -9.81 -5.886 -1.770.7 -9.81 -6.867 -2.400.8 -9.81 -7.848 -3.12Graphical 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 accumulating8Graphical example: CM vel/acc in runningGraphical integration1. Assess how the area under the curve is accumulating – draw narrow boxes2. Identify regions of (+) and (-) area3. Find inflection points (minima, maxima)4. Find points where curve crosses zero5. Fill in by: rate of accumulating area = slope between these pointsGraphical integration examplesTime (s)543210Graphical integration examplesTime (s)Graphical integration examplesTime (s)9Which process to use?PosVelAccAccVelPosDifferentiationIntegrationWhich “direction” are you going?Graphically, what to look at?SlopeAccumulating AreaDifferentiation & Integration - SummaryUp nextBack to Linear KinematicsThese 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.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