Matlab

MAE 384 Advanced Mathematical Methods for Engineers Homework #4

Due: February 18, Thursday in class Prob. 1 By-hand + Matlab

The resistance to flow through pipes and tubes is parameterized by a dimensionless number called the friction factor (denoted as f, ranging from about 0.008 to 0.08), which can estimated using

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§ ��

fDf Re 51.2

7.3 log0.2

1 10

H

where İ = the roughness (m), D = diameter (m), and Re = the Reynolds number defined as

P UVD

Re

with ȡ = the fluid's density (kg/m3), V = the fluid's velocity, and ȝ = dynamic viscosity (N·s/m2).

Assume ȡ = 1.23 kg/m3, ȝ = 1.79×10-5 N·s/m2, D = 0.005 m, V = 40 m/s, and İ = 1.5×10-6 m. Part I: By-hand (a) Determine the friction factor f of the system using the Secant method. Start with initial guesses f = 0.01 and

0.02, and perform two iterations.

Part II: Matlab: Find the friction factor f by performing iterations until the approximate relative error

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� i

ii a f

ff H falls below 2%

(b) Use the Newton's method with initial guess f = 0.01; (c) Use the Secant method with initial guesses f = 0.01 and 0.02

Prob. 2 By-hand

The magnitude of the resultant force from the three forces acting on the bracket is 450 N directed in the positive u. To determine the direction angle ׋ of F1, and the magnitude of F3, the following system of equations is obtained:

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°° ®

­

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�

225 13 12

cos500

190 13 5

sin500

3

3

F

F

I

I

Using Newton’s method for a system of nonlinear equations, determine ׋ and F3. Perform TWO iterations starting with initial values 0 = ׋ (in radians) and F3 = 100 N.