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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

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Presentation Transcript

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

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:

Integration: Integration is commonplace in Science and EngineeringWhy Differentiate, Integrate?

Calculation of Geographic Areas

River ChannelCross Section

Wind-ForceLoading

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

Differentiation: rate of change of a dependent variable with respect to an independent variable.Review: Differentiation

Indefinite Intregral w/ Variable End-PtsIntegral Properties

- Piecewise Property

- Initial/Final Value Formulations

a

c

b

- Linearity → for Constants p & q

Alternative Quotient Rule

- Restate Quotient as rational Exponent, then apply Product rule; to whit:
- Then
- Putting 2nd term over common denom

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

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

- Trapezoidal Form

- Parabolic (Simpson’s) Form

- Higher-Order-Polynomial Tops Lead to increased, but diminishing, accuracy.

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

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

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

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

- The Current can Be integrated Analytically to find v(t), but it’s Painful

- Let’s Tackle The Problem Numerically
- Use the PieceWise Property

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

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

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

Use Trapezoids to approximate the area under the curve:

TrapezoidalRule

n trapezoids

…

a b

Width, Δx=

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

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