Differentiation Integration Differentiation Mathematical process for Differentiation Mathematical process for finding the instantaneous rate of change of a finding the instantaneous rate of change of a variable related to finding the slope of a line variable related to finding the slope of a line Integration Mathematical process for finding Integration Mathematical process for finding the total area under a curve basically a the total area under a curve basically a summation process summation process Methods Methods Analytical Analytical Numerical Numerical Graphical Graphical Differentiation position velocity acceleration The rate of change of one variable e g position velocity with respect to another variable time Velocity is the rate of change of position with respect to time Acceleration is the rate of change of velocity with respect to time 1 2 1 Derivative Slope of a Line rise s2s1 position or velocity Velocity slopeof the position vs timecurve Acceleration slopeof the velocity vs timecurve run time rise run slope t2 t1 s t Derivative Slope of a Line rise rise s3 High velocity s2s1 Low velocity position or velocity Velocity slopeof the position vs timecurve Acceleration slopeof the velocity vs timecurve run run time rise run slope t2 t1 s t 3 4 2 Derivative Slope of a Line rise rise v3 High acceleration velocity v2v1 Low acceleration Velocity slopeof the position vs timecurve Acceleration slopeof the velocity vs timecurve run run time rise run slope s t t2 t1 Analytical Differentiation Determine the slope of the tangentline to find the instantaneous velocity or acceleration tangent t 0 v lim t s t 0 n o i t i s o P t Time 5 6 3 Gait Analysis Tutorial You Force platform High speed Infrared camera 7 8 4 Determine the slope of the secantline close to instantaneous Numerical Differentiation n o i t i s o TimeP ti 1 ti ti 1 secant v i s i 1 s i 1 2 t Finite Difference Method Numerical Differentiation Time s 0 0 0 1 0 2 0 3 0 4 0 5 Pos m 3 20 5 80 13 60 15 40 9 80 7 60 Vel m s A numerical technique that provides an approximation to the true instantaneous derivative 9 10 5 s i s i i 1 v 1 2 t Example cid 2870 cid 3404 13 60 cid 3398 3 20 2 cid 4666 0 1 cid 4667 cid 2870 cid 3404 10 40 0 2 cid 3404 Use for points 2 through n 1 Numerical Differentiation First Finite Difference requires 3 data points cid 3404 0 1 Vel m s Time s 0 0 0 1 0 2 0 3 0 4 0 5 Pos m 3 20 5 80 13 60 15 40 9 80 7 60 Time s 0 0 0 1 0 2 0 3 0 4 0 5 Pos m 3 20 5 80 13 60 15 40 9 80 7 60 52 0 48 0 19 0 39 0 Vel m s 26 0 52 0 48 0 19 0 39 0 Use for point 1 Numerical Differentiation First Forward Difference requires 2 data points cid 3404 0 1 v 1 s 1 s 2 t cid 2869 cid 3404 5 80 cid 3398 3 20 0 1 cid 2869 cid 3404 2 60 0 1 cid 3404 11 12 6 First Backward Difference requires 2 data points Numerical Differentiation Time s 0 0 0 1 0 2 0 3 0 4 0 5 Pos m 3 20 5 80 13 60 15 40 9 80 7 60 cid 3404 0 1 v n s n s 1 n t cid 3041 cid 3404 7 60 cid 3398 9 80 0 1 cid 3041 cid 3404 2 60 0 1 cid 3404 cid 3398 Vel m s 26 0 52 0 48 0 19 0 39 0 22 0Use for point n 13 14 Calculating Acceleration Numerical Differentiation Time s 0 0 0 1 0 2 0 3 0 4 0 5 Pos m 3 20 5 80 13 60 15 40 9 80 7 60 Vel m s 26 0 52 0 48 0 19 0 39 0 22 0 Acc m s2 a 1 v 1 v 2 t a i v i v 1 2 t i 1 Apply same difference method equations to the velocity data a n n v v 1 n t 7 Calculating Acceleration Numerical Differentiation Time s 0 0 0 1 0 2 0 3 0 4 0 5 Pos m 3 20 5 80 13 60 15 40 9 80 7 60 Vel m s 26 0 52 0 48 0 19 0 39 0 22 0 Acc m s2 260 110 355 435 15 170 a 1 v 1 v 2 t a i v i v 1 2 t i 1 Apply same difference method equations to the velocity data a n n v v 1 n t Graphical 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 qualitativeassessment of the instantaneous velocity or acceleration 15 16 8 0 2 4 6 8 10 Time s m n o i t i s o P s m y t i c o l e V 2 s m c c A 20 15 10 5 0 100 80 60 40 20 0 20 100 80 60 40 20 0 20 m n o i t i s o P s m y t i c o e V l 2 s m c c A 100 80 60 40 20 0 20 15 10 5 0 100 80 60 40 20 0 20 17 18 0 2 4 6 8 10 Time s 9 Graphical Differentiation 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 Ignore magnitude Consider slope only 19 21 m n o i t i s o P s m y t i c o e V l 2 s m c c A 1 5 0 5 1 0 Time s 6 4 2 0 2 4 6 40 20 0 Tasks 20 1 Draw the velocity curve 40 2 Draw the acceleration curve 200 3 Determine the velocity magnitude 100 4 Determine the acceleration magnitude 0 100 200 Magnitudes of curves will NOT be asked for during exams 0 0 1 5 1 0 0 5 10 m n o i t i s o P s m y t i c o e V l m n o i t i s o P s m y t i c o e V l 40 20 0 20 40 Time s Time s 23 24 11 6 4 2 0 2 4 6 40 20 0 20 40 m n o i t i s o P s m y t i c o l e V …