slide1
Download
Skip this Video
Download Presentation
Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

Loading in 2 Seconds...

play fullscreen
1 / 33

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] - PowerPoint PPT Presentation


  • 58 Views
  • Uploaded on

Engr/Math/Physics 25. Chp9: Integration & Differentiation. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] Learning Goals. Demonstrate Geometrically the Concepts of Numerical Integ. & Diff. Integrals → Trapezoidal, Simpson’s, and Higher-order rules

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]' - clover


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

Engr/Math/Physics 25

Chp9: Integration

& Differentiation

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

learning goals
Learning Goals
  • Demonstrate Geometrically the Concepts of Numerical Integ. & Diff.
    • Integrals → Trapezoidal, Simpson’s, and Higher-order rules
    • Derivative → Finite Difference Methods
  • Use MATLAB to Numerically Evaluate Math/Data Integrals
  • Use MATLAB to Numerically Evaluate Math/Data Derivatives
why differentiate integrate
We encounter differentiation and integration on a Daily Basis

Differentiation: Many Important Physical processes/phenomena are best Described in Derivative form; Some Examples

Why Differentiate, Integrate?
  • Newton’s 2nd Law:
  • Heat Flux:
  • Drag on a Parachute:
  • Capacitor Current:
why differentiate integrate1
Integration: Integration is commonplace in Science and EngineeringWhy Differentiate, Integrate?

Calculation of Geographic Areas

River ChannelCross Section

Wind-ForceLoading

review integration
Integration: the area under the curve described by the function f(x) with respect to the independent variable x, evaluated between the limits x = a to x = b.Review: Integration
integral properties
Indefinite Intregral w/ Variable End-PtsIntegral Properties
  • Piecewise Property
  • Initial/Final Value Formulations

a

c

b

  • Linearity → for Constants p & q
derivative properties
PRODUCT Rule

Given

Derivative Properties
  • QUOTIENT Rule
    • Given
  • Then
  • Then
alternative quotient rule
Alternative Quotient Rule
  • Restate Quotient as rational Exponent, then apply Product rule; to whit:
  • Then
  • Putting 2nd term over common denom
why numerical methods
Numerical IntegrationWhy Numerical Methods?
  • Very often, the function f(x) to differentiate, or the integrand to integrate, is TOO COMPLEX to yield exact analytical solutions.
  • In most cases in engineering testing, the function f(x) is only available in a TABULATED form with values known only at DISCRETE POINTS
numerical integration
Game Plan: Divide Unknown Area into Strips (or boxes), and Add UpNumerical Integration
  • To Improve Accuracy the TOP of the Strip can Be
    • Slanted Lines
      • Trapezoidal Rule
    • Parabolas
      • Simpson’s Rule
    • Higher Order PolyNomials
strip top effect
Strip-Top Effect
  • Trapezoidal Form
  • Parabolic (Simpson’s) Form
  • Higher-Order-Polynomial Tops Lead to increased, but diminishing, accuracy.
strip count effect
Strip-Count Effect

10 Strips

20 Strips

  • Adaptive Integration → INCREASE the strip-Count in Regions with Large SLOPES
    • More Strips of Constant Width Tends to work just as well
dy dx by finite difference approx

y(x)

y(x-Δx)

y(x)

y(x)

y(x+Δx)

dy/dx by Finite Difference Approx.
  • Derivative at Point-x :
  • Forward Difference

mbkwd

  • Backward Difference

mfwd

dy dx by finite difference approx1

y(x)

y(x-Δx)

y(x)

y(x)

y(x+Δx)

dy/dx by Finite Difference Approx.
  • Central Difference = Average of fwd and bkwd Slopes :

mcent

dy dx by discrete point difference
dy/dx by Discrete-Point Difference
  • From Previous LET
  • The FORWARD Difference Calc
dy dx by discrete point difference1
dy/dx by Discrete-Point Difference
  • The BACKWARD Difference Calc
  • The CENTRAL Difference Calc
finite difference example
Finite Difference Example

ForwardDifference

Analytical

finite difference fence post errors
Finite Difference Fence-Post Errors
  • If we have data vectors for x & f(x) we can calc m = df(x)/dx by the Fwd, Bkwd or Central Difference methods
  • If there are 1 to n Data points then can NOT calc
    • mfwd for pt-n (cannot extend fwd beyond n-1)
    • mbk for pt-1 (cannot extend bkwd beyond 1)
    • mcnt for pt-1 and pt-n (cannot extend bk beyond 1, cannot extend fwd beyond n)

Time For

Live Demo

cap charging
Cap Charging
  • The Current can Be integrated Analytically to find v(t), but it’s Painful
  • Let’s Tackle The Problem Numerically
  • Use the PieceWise Property
digression
Digression
  • For More Info on
  • See pages 333-335 from
cap chrg piecewise integration
Cap ChrgPieceWise Integration
  • Game Plan
    • Make Function for i(t)/C
    • Divide 300 mS interval into 1 mS pieces
    • Use 1-300 FOR Loop to collect
      • Vector for Time-Plot
      • Use ΔV summation to Create a V-Plotting Vector

Time For

Live Demo

  • File List
    • Fcn → iOverC_CapCharge.m
    • Calc& Plot → Cap_Charge_Soln_1111.m
file codes

function [Cap_Charge] = iOverC_CapCharge(time)

Cap_Charge = (1/0.001)*(10 + 300*exp(-5*time).*sin(25*pi*time))/1000;

% Cap Charge for Prob for Chp9 in COULOMBS

File Codes

% B. Mayer 08Nov11

% Cap Charging: Piecewise Ingegration

% Cap_Charge_Soln_1111.m

%

% use 500 pts using LinSpace

% => Ask user for max time

tmax = input(\'Enter Max Time in Sec = \')

tmin = 0; n = 500;

t = linspace(tmin,tmax,n); % in Sec

TimePts =length(t) % 2X check number of time points

%

% Initalize the Vminus1 & Plotting Vectors

Vminus1 = 0;

Vplot = 0;

tplot = 0;

%

% Use FOR Loop with Lobratto Integrating quadl function on Cap Charge

% Function

for k = 1:n-1

tplot(k) = t(k);

del_v(k) = quadl(\'iOverC_CapCharge\', t(k), t(k+1));

% The Incremental Area Under the Curve; can be + or -

Vplot(k) = Vminus1 + del_v(k);

Vminus1 = Vplot(k);

end

plot(1000*tplot, del_v), xlabel(\'time (mS)\'), ylabel(\'DelV (V)\'),...

title(\'Capacitor Voltage PieceWise Integral\'), grid

disp(\'Showing del_v PLOT - hit any key to show V(t) plot\')

pause

plot(1000*tplot, Vplot), xlabel(\'time (mS)\'), ylabel(\'Cap Potential (V)\'),...

title(\'Capacitor Voltage\'), grid

units analysis
Examine the Integrand fromUnits Analysis
  • A → A (a base unit)
  • S → S (a base unit)
  • F → m−2•kg−1•S4•A2
  • V → m2•kg•S−3•A−1
  • The Integrand Units
  • Or
  • Recall From ENGR10 A, S, & F in SI Base Units
  • But

all done for today
All Done for Today

Use Trapezoids to approximate the area under the curve:

TrapezoidalRule

n trapezoids

a b

Width, Δx=

slide32

Engr/Math/Physics 25

Appendix

Time For

Live Demo

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

dy dx example
dy/dx example

x = [1.215994, 2.263081, 3.031708, 4.061534, 5.122477, 6.12396, 7.099754, 8.070701, 9.215382, 10.04629, 11.16794, 12.22816, 13.02504, 14.13544, 15.20385, 16.01526]

y = [0.381713355 1.350058777 1.537968679 2.093069052 1.002924647 1.123878013 7.781303297 14.2596343 13.96413795 4.352973409 51.45863097 22.85918559 100.8729773 106.5041434 34.15277499 134.2488143]

plot(x,y),xlabel(\'x\'), ylabel(\'y\'), grid

ad