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Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollegePowerPoint Presentation

Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

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Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

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Engineering 25

Catenary Tutorial Part-1

Bruce Mayer, PE

Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

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

- 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

- Next, relate horizontal distance, x, to cable-length s

- But by Force Balance Triangle

- Also From last slide recall

- Thus

- Factoring Out c

- Finally the Integral Eqn

- Integrate Both Sides using Dummy Variables of Integration:
- σ: 0→x η: 0→s

- Using σ: 0→x η: 0→s

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

- Noting that

- Thus the Solution to the Integral Eqn

- Then

- Solving for s in terms of x

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

- From the Force Triangle

- And From Before

- So the Differential Eqn

- Recall the Previous Integration That Relates x and s

- Using s(x) above in the last ODE

- Integrating with Dummy Variables:
- Ω: c→yσ: 0→x

- Noting that cosh(0) = 1

- Solving for y yields theCatenary Equation in x&y:

- Where
- c = T0/w
- T0 = the 100% laterally directed force at the ymin point
- w = the lineal unit weight of the cable (lb/ft or N/m)

- With Hyperbolic-Trig ID: cosh2 – sinh2 = 1

- Thus:

- Recall From the Differential Geometry

Catenary Cabling Contraption

- Shape is defined by the Catenary Equation

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

y = c

An 8m length of chain has a lineal unit mass of 3.72 kg/m. The chain is attached to the Beam at pt-A, and passes over a small, low frictionpulley at pt-B.

Determine the value(s) of distance a for which the chain is in equilibrium (does not move)