Engineering 25
Problem 10-25
Catenary Tutorial
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
ENGR/MTH/PHYS25: Computational Methods
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt
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
ENGR/MTH/PHYS25: Computational Methods
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt
Catenary Length (2)

Next, relate
horizontal distance,
x, to cable-length s
dx  ds cos 

Then
1
ds 
dx  secdx
cos 

Recall Trig ID:
ENGR/MTH/PHYS25: Computational Methods
3
sec   1  tan 
2
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt
Catenary Length (3)
 Using Trig ID in ds
Equation
ds  secdx
 1  tan  dx
b
2
 Now find Length, L,
between pts a & b
by integrating ds
L
  sb
  sa
d  
ENGR/MTH/PHYS25: Computational Methods
4
x b
x a
a
dL  ds   dL   ds
1  tan 2  dx
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt
Catenary Length (4)
 Now Eliminate θ
 From Differential
Diagram note:
dy
tan  
dx
 Sub Out tanθ in the definite Integral for L:
L
x b
x a
1  tan  dx  
ENGR/MTH/PHYS25: Computational Methods
5
2
x b
x a
2
 dy 
1    dx
 dx 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt
Catenary Length (5)
yO
 Finally
L
x b
x a
2
 dy 
1    dx
 dx 
 Now in the Case of
Prob10-25
 x  20 
y  10 cosh

 10 
for 0  x  50
 An Analytical Soln
for L is possible as
ENGR/MTH/PHYS25: Computational Methods
6
d
cosh z  sinh z
dz
 But it’s a bit Tedious
so Let’s have
MATLAB do it
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt
Catenary Length (6)
 MATLAB
SOLUTION PLAN
yO
• 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)
• Find numerical value
for L between 0 & 50
using double
command
• Set a = 0, b =50
ENGR/MTH/PHYS25: Computational Methods
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt
ENGR/MTH/PHYS25: Computational Methods
8
MATLAB Code
% Bruce Mayer, PE
% ENGR25 * 03Jan08
% file = Prob10_25_Symbolic_Soln_0801.m
%
% Solve P10.25
%
% 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)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Prob_10-25_Catenary_Solution.ppt.ppt