FRECUENCIAS DE COLOCACIONES LÉXICAS
3.2. Procedimiento de elaboración del listado
3.2.1. Selección de la base
Suppose an experimental data set produced a plot as shown in Figure 8.3 and it was desired to obtain an analytical expression that comes close to fitting the data. Let us assume that the –L ≤ x ≤ L. If not, make it so by shifting the origin. Actually the original abscissa data were for 0≤ ≤t 10 5. seconds The data were shifted by letting . x t= - 5 25. .
The t domain was subdivided into 70 equal spaces, with Dt =10 5 70 0 15. / = . second. Thus, xi+1 – xi is uniform over the entire domain. An attempt to fit a poly-nomial approximating curve to these data would not be successful. However, the use of a Fourier series could give a reasonable analytical expression approximating the data. If uc is the approximating curve, then by a Fourier series,
uc x a a m x
where
a L u x dx
L L
0 1
=2
∫
- ( )a L u x m x
L dx
m L
= L
-
1
∫
( )cos πb L u x m x
L dx
m L
= L
-
1
∫
( )sin πUsing 30 terms in the series and Simpson’s rule on integration an approximat-ing curve as shown in Figure 8.4 was obtained.
The a m x
L b m x
mcos π msin Lπ
+
terms can be put into the following form by the trigonomic identity acosb+bsinb=csin(b φ- ),where c represents the ampli-tude. The amplitude, c, is given by
c= (a2+b2) A plot of amplitude vs. m
Lπ is shown in Figure 8.5.
4.5 ×104 u vs. x
4 3.5 3
u 2.5
2 1.5 1 0.5
0–6 –4 –2 0
x 2 4 6
Figure 8.3 Experimental data of u vs. x.
6 ×104 Fourier Series Fit 5
4
3
u, uc (m/s) 2 1
0
–1–6 –4 –2 0
t 2 4 6
Red–uc Green–u
Figure 8.4 Fourier series fit of the data. (See color insert following page 334.)
Amp. vs. n*pi/L 7000
6000
5000
4000
Amp. 3000
2000
1000
00 2 4 6 8 10
n*pi/L
12 14 16 18
Figure 8.5 Fourier series coefficient amplitudes vs. n L
π.
Projects
Project 8.1
A formula describing the fluid level, heq, in a tank, as a function of time, as the fluid discharges through a small orifice, is
h h C A
A g t
eq eq o d
T
= , - 0
2 2 (P8.1a)
where
Cd = the discharge coefficient.
heq,o = the fluid level in the tank at time, t = 0.
A0 = the area of the orifice.
AT = the cross-sectional area of the tank.
An experiment consisting of a cylindrical tank with a small orifice was used to determine Cd for that particular orifice and cylinder. The tank walls were transpar-ent and a ruler was pasted to the wall allowing for the determination of the fluid level in the tank. The procedure was to fill the tank with water while the orifice was plugged. The plug was then removed and the water was allowed to flow through the orifice. The water level in the tank, hexp in meters, was recorded as a function of time, t. The experimental data are shown in Table P8.1.
Table P8.1 hexp vs. time
hexp (m) t(s) hexp (m) t(s)
0.288 0 0.080 110
0.258 10 0.065 120
0.234 20 0.053 130
0.215 30 0.041 140
0.196 40 0.031 150
0.178 50 0.022 160
0.160 60 0.013 170
0.142 70 0.006 180
0.125 80 0.002 190
0.110 90 0.000 200
0.095 100
The diameters of the orifice and the tank are Do = 0.0055 m and Dt = 0.146 m, respectively. The free surface elevation, heq,o, at t = 0 is 0.288 m. The gravitational constant, g = 9.81 m/s2.
Use the mse as defined by Equation (8.12) to determine the value for Cd that best fits the data. Vary Cd from 0.3 to 0.9 in steps of 0.01 and evaluate the mse for each Cd selected, where
mse N h teq i h ti
i N
=
-∑
=1
1
[ ( ) exp( )]2 (P8.1b)
where
N = the number of data points.
h teq i( ) = the water level in the tank at as deteermined by Equation (P8.1a).ti the water
h texp( ) =i level in the tank at as determined by experti iiment.
For the Cd with the lowest mse, create a plot of heq vs. t (solid line) and superimpose hexp vs. t as little x’s onto the plot of heq vs. t. Also print out the value of Cd that gives the lowest mse.
Also create a 30-line table of mse vs. Cd.
Project 8.2
This project involves determining the best fit polynomial approximating curve to the (H vs. Q) data obtained from a pump manufacturer’s catalog (units changed to SI units). The data points of the (H vs. Q) curve are shown in Table P8.2.
Table P8.2 H vs. Q Data from the Pump Manufacturer
Q H Q H
(m3/h) (m) (m3/h) (m)
3.3 43.3 61.6 40.8
6.9 43.4 68.5 39.6
13.7 43.6 75.3 38.7
20.5 43.6 82.2 37.2
27.4 43.3 89.0 36.3
34.2 43.0 95.8 34.4
41.1 42.7 102.7 32.6
Try degree polynomials of 2 through 4 to determine which degree polynomial will give the smallest mse. Use MATLAB’s function polyfit, which returns the coef-ficients for each of the three polynomials. Then use MATLAB’s function polyval to create for each polynomial:
(a) A table containing Q, Hc, and H, where Hc is the approximating curve for H vs. Q.
(b) A plot of Hc vs. Q (solid line) and H vs. Q (small circles), all plots on the same page.
171
Optimization
9.1 Introduction
The objective of optimization is to maximize or minimize some function f. The function f is called the object function. For example, suppose there is an electronics company that manufactures several different types of circuit boards. Each circuit board must pass through several different departments (such as drilling, compo-nent assembly, testing, etc.) before shipping. The time required for each circuit board to pass through the various departments is also known. There is a minimum production quantity per month that the company must produce. However, the company is capable of producing more than the minimum production require-ment for each type of circuit board each month. The profit the company will make on each circuit board it produces is known. The problem is to determine the pro-duction amount of each type of circuit board per month that will result in the maximum profit. A similar type of problem may be one in which the object is to minimize the cost of producing a particular product. These types of optimiza-tion problems are discussed in greater detail later in this chapter. In most optimi-zation problems, the object function, f, will depend on several variables—x1, x2, x3, …, xn. These are called the control variables because their values can be selected.
Optimization theory develops methods for selecting optimal values for the control variables, x1, x2, x3, …, xn, that either maximizes (or minimizes) the objective func-tion f. In many cases, the choice of values for x1, x2, x3, …, xn is not entirely free, but is subject to some constraints.