Catenary - Computational Methods - Lecture Slides, Slides of Calculus for Engineers

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Engineering 25

Catenary

Tutorial Part-

UNloaded Cable → Catenary

 Consider a cable uniformly loaded by the cable itself, e.g., a cable hanging under its own weight.

T = T 02 + w^2 s^2 = w^2 ( T 02 w^2 + s^2 ) = w c^2 + s^2

• With loading on the cable from lowest point C to a point D given by W = ws , the Force Triangle on segment CD reveals the internal tension force magnitude, T
• Where c^ = T 0^ w

UNloaded Cable → Catenary (3)

 Factoring Out c

 Integrate Both Sides using Dummy Variables of Integration:

• σ: 0→x η: 0→s

ds c c c s c

c ds c s

c dx (^) 2 2 2 2 2 2

 Finally the Integral Eqn

ds s c

dx (^) 2 2 1

UNloaded Cable → Catenary (4)

 Using σ: 0→x η: 0→s

 Now the R.H.S. AntiDerivative is the argSINH

 Noting that

∫ ∫

=

= = =^ +

x s d c

d

η η

σ

σ 0 σ^0 1 η 2 2 η

[ ]

s s x x c

d c c

d

=

=

=

=

= = 

∫ =^ = ∫

η

η

η η

σ σ

σ σ

(^0 ) 0 0 argsinh 1

arg sinh ( ) 0 = sinh−^1 ( ) 0 = 0

UNloaded Cable → Catenary (6)

 Finally, Eliminate s in favor of x & y. From the Diagram

 So the Differential Eqn

 From the Force Triangle

dy = dx tan θ

0

tan T

W

 And From Before

W = ws and T 0 = wc

dx c

s dx wc

ws dx T

W dy = dx = = = 0

tan θ

UNloaded Cable → Catenary (7)

 Recall the Previous Integration That Relates x and s

 Integrating with Dummy Variables:

• Ω: c→y σ: 0→x

[ ]

x x y c

y c d c d c c

=

=

=

Ω= Ω=

Ω= Ω = 

  

  

  

 =  

  

Ω = Ω =  ∫ ∫

σ

σ

σ σ

σ σ σ (^0 ) sinh cosh

 

  



c

x s c sinh

 Using s(x) above in the last ODE

dx c

dx x c

c x c

sdx c

dy dx  

  

 =  

  

= tan θ =^1 =^1 sinh sinh

Catenary Tension, T ( y )

 With Hyperbolic-Trig ID: cosh 2 – sinh 2 = 1

 Recall From the Differential Geometry

 Thus:

y = c cosh ( x c )

( ) ( ) (^222) [ 2 ( ) 2 ( )] 2

2 2 2 2 2 2

cosh sinh

cosh sinh y s c x c x c c

y s c x c c x c ∴ − = − =

− = −

y^2 − s^2 = c^2 or c^2 + s^2 = y^2

T ( c , s ) = w c^2 + s^2 = w y^2 = wy = T ( y )

c = T 0 w

T^ (^ y )^ = wy

• Catenary Cabling Contraption  Shape is defined by the Catenary Equation

y = c cosh ( x c )

• Note that the ORIGIN for y is the Distance “c” below the HORIZONTAL Tangent Point

y = c