Catenary 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
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Catenary Length - Computational Methods - Lecture Slides, Slides of Calculus for Engineers
These are the Lecture Slides of Computational Methods which includes Thévenin’s Equivalent Circuit, Circuit Simplification, Analysis of Power Transfer, Voltage Division, Analytical Game Plan, Array Operation, Element Operations, Number of Allowable Values etc.Key important points are: Catenary Length, Curve-Length of Cable, Differential Analysis, Matlab Solution Plan, Numerical Value, Double Command, Matlab Code, Ummy Variables of Integration, Design of Program
Typology: Slides
2012/2013
1 / 7
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Catenary 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
Next, relate
horizontal distance,
x, to cable-length s
dx ds cos
Then
Recall Trig ID:
ds dx dx
sec
cos
1
sec 1 tan
• Now Eliminate θ
From Differential
Diagram note:
dx
dy
tan
Sub Out tan θ in the definite Integral for L :
dx
dx
dy
L dx
x b
x a
x b
x a
1 tan 1
• Finally
yO
Now in the Case of
Prob10-
for 0 50
x
x
y cosh
dx
dx
dy
L
x b
x a
z z
dz
d
cosh sinh
An Analytical Soln
for L is possible as
But it’s a bit Tedious
so Let’s have
MATLAB do it
MATLAB Code
% Bruce Mayer, PE % ENGR25 * 03Jan % file = Prob10_25_Symbolic_Soln_0801.m % % Solve P10. % % Declare x, a, b as symbolic syms x a b % % Define Catenary y(x) y = 10*cosh((x-20)/10) % % Take dy/dx symbolically dydx = diff(y) % % Find L Symbolically L = int(sqrt(1+dydx^2),a,b) pretty(L) % % display L disp(' ') 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)