[email protected] ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt 1 Bruce Mayer, PE...
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[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt 1 Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods Bruce Mayer, PE Registered Electrical & Mechanical Engineer [email protected] Engineering 25 Problem 10-25 Problem 10-25 Catenary Tutorial Catenary Tutorial
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Transcript of [email protected] ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt 1 Bruce Mayer, PE...
[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt1
Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods
Bruce Mayer, PERegistered Electrical & Mechanical Engineer
Engineering 25
Problem 10-Problem 10-2525
Catenary Catenary TutorialTutorial
[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt2
Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods
Catenary LengthCatenary Length Consider a cable uniformly loaded
by the cable itself, e.g., a cable hanging under its own weight.
We would like to find the Curve-Length of the cable, s, as function of x alone• Use Differential Analysis
[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt3
Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods
Catenary Length (2)Catenary Length (2) Next, relate
horizontal distance, x, to cable-length s
cosdsdx Then
Recall Trig ID:
dxdxds
seccos
1
22 1 tansec
[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt4
Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods
Catenary Length (3)Catenary Length (3) Using Trig ID in ds
Equation
Now find Length, L, between pts a & b by integrating ds
dx
dxds
21 tan
sec
a
b
dxdLbx
ax
s
s
b
a
21 tan
dsdLdsdL
[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt5
Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods
Catenary Length (4)Catenary Length (4) Now Eliminate θ
From Differential Diagram note:
dx
dytan
Sub Out tanθ in the definite Integral for L:
dxdx
dydxL
bx
ax
bx
ax
22 11 tan
[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt6
Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods
Catenary Length (5)Catenary Length (5) Finally
Oy
Now in the Case of Prob10-25
500for
10
2010
x
xy cosh
dxdx
dyL
bx
ax
2
1
zzdz
dsinhcosh
An Analytical Soln for L is possible as
But it’s a bit Tedious so Let’s have MATLAB do it
[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt7
Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods
Catenary Length (6)Catenary Length (6) MATLAB
SOLUTION PLAN• syms for x, a, b
• Set y = 10*cosh[(x-20)/10]
• Take dydx = diff(y)
• Find L = int(sqrt(1+dydx^2),a,b)
• Set a = 0, b =50
Oy
• Find numerical value for L between 0 & 50 using double command
[email protected] • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt8
Bruce Mayer, PE ENGR/MTH/PHYS25: Computational Methods
MA
TL
AB
Co
de
MA
TL
AB
Co
de
% Bruce Mayer, PE% ENGR25 * 03Jan08% file = Prob10_25_Symbolic_Soln_0801.m%% Solve P10.25 %% Declare x, a, b as symbolicsyms x a b%% Define Catenary y(x)y = 10*cosh((x-20)/10)%% Take dy/dx symbolicallydydx = diff(y)%% Find L SymbolicallyL = int(sqrt(1+dydx^2),a,b)pretty(L)%% display Ldisp(' ')disp('DISPLAYING L(a,b) - HIT ANY KEY TO CONTINUE')disp(' ')pause% % calc L(0,50)anum = 0; bnum = 50;Lnum = double(int(sqrt(1+dydx^2),anum,bnum));disp('L from 0 to 50 = ')disp(Lnum)
