Catenary equation [Solved!]

X

Hello, in your article titled "<a href="http://www.intmath.com/applications-integration/11-arc-length-curve.php">Arc Length of a Curve using Integration</a>", in example 3 regarding the Golden Gate Bridge cables. May you please elaborate how you "guessed and checked" the catenary equation of the cables. I am doing a math exploration internal assessment (for ib) about this topic and its mandatory to explain every equation I use, hence "guess and check" is not sufficient.

Relevant page

<a href="/applications-integration/11-arc-length-curve.php">11. Arc Length of a Curve using Integration</a>

What I've done so far

I plugged in the point (152,640) and tried solving for a, but there is no answer

X

Hello Shahad

Well, it's best if you figure out how to do it (rather than me just telling you :-)

Let's start with the equation for the simplest catenary: `y=(e^[x]+e^[-x])/(2)`.

I said in the solution on that page "For convenience, we'll place the origin at the lowest point of the cable."

In the equation `y=(e^[x]+e^[-x])/(2)`, when `x=0`, what is the value of `y` (this is the lowest point)?

X

I am aware of that, so we also have point (0,0). so do I need to graph this simplest equation and then try transforming it? Or what I do exactly. I want to model the main cable of the Golden Gate Bridge, and it's actual vertex is 0,0. So I only need to do horizontal and vertical stretches and compressions right?

X

I'm taking you step by step through the process I used to come up with the equation for the ropes.

When `x=0`, the equation `y=(e^[x]+e^[-x])/(2)` doesn't give us `(0,0)`!

X

Seems like Shahad has disappeared. Anyone else like to have a go at finishing this?

X

I'm brave and I'll give it a go.

At `x=0`, 

`y=(e^[x]+e^[-x])/(2)=(e^[0]+e^[-0])/(2)=(1+1)/2=1`

So it passes through (0,1) at the lowest point.

X

Yes, Stephen, you are correct.

This will also be the case if our function is, say,

`y=(e^[3x]+e^[-3x])/(2)`,

or in fact, any number `b` in there, like

`y=(e^[bx]+e^[-bx])/(2)`.

They all will pass through (0,1).

That's where the "`-1`" comes from in my equation (since I'm assuming the curve passes through (0,0), as stated in the solution).

So I knew it had to be of the form:

`y=a((e^[bx]+e^[-bx])/(2)-1)`.

OK so far?

X

Yes, it makes sense.

Now we have 2 equations in 2 unknowns, right?

The curve passes through (-640,152) and (640,152), so

When `x=-640`,

`152=a((e^[-640b]+e^[640b])/(2)-1)`

and

When `x=640`,

`152=a((e^[640b]+e^[-640b])/(2)-1)`

Putting them equal and cancelling off the `a`'s gives:

`(e^[-640b]+e^[640b])/(2)-1 = (e^[640b]+e^[-640b])/(2)-1`

Adding 1 to both sides gives the obvious statement whcich can't be solved:

`(e^[-640b]+e^[640b])/(2) = (e^[640b]+e^[-640b])/(2)`

So I'm stuck.

X

Good reply, Stephen - and so fast!

You're right - that approach does lead to a dead end.

We need to go back to this:

`y=a((e^[bx]+e^[-bx])/(2)-1)`

We need to choose an `a` and `b` that passes through the given points, AND has close to the correct shape for the Golden Gate Bridge.

In fact, there are an infinite number of catenaries that pass through the required points - we just need one that is close.

I can't remember now, but I probably used <a href="https://www.desmos.com/calculator">Desmos</a> or <a href="http://www.wolframalpha.com/">Wolfram|Alpha</a> to do my "Guessing and checking".

In this first image, you can see 3 curves.

The red one is `y=100(\cosh (x/406.2)-1)`

The blue one is `y=1280(\cosh (x/1326)-1)` (this is the one I ended up using)

As you can see, there is very little difference between those two.

The green one has a very small `a` value, for comparison:

`y=0.1(\cosh (x/79.6)-1)`

<img src="/forum/uploads/imf-3249-golden-gate.png" width="328" height="83" alt="Applications of Integration" />

Having chosen some (random, at first) value for `a`, I just tweaked the value for `b` until it passed through the required points (-640,152) and (640,152). Quick and easy to do using Desmos. (I could have <a href="http://bit.ly/2h1xZZZ">solved it for b using Wolfram|Alpha</a>, directly.)

When overlaying the 3 curves on an actual photo of the Golden Gate Bridge, we can see the first 2 are very close (basically indistinguishable from the actual shape), but the third is way off.

<img src="/forum/uploads/imf-3315-golden-gate.jpg" width="400" height="97" alt="Applications of Integration" />

[Bridge <a href="https://en.wikipedia.org/wiki/GoldenGateBridge">image source</a>]

So Stephen and Shahad, sometimes when we have 2 unknowns, a bit of guess and check is necessary!

X

Hello, sorry for disappearing!
Thank you very much Murray and Stephen for solving this problem.

X

Hello! I wanted to know why inside your general formula of the catenary equation you take `bx` as in:

`y = a ((e^(bx) + e^(-bx))/2-1 )` 

and not `x/a` as in:

`y = a ((e^(x/a) + e^(-x/a))/2-1 )` which is the accurate substitution of `cosh (x/a)`

Thank you

X

@Sofia: Fair enough question. I have added some more detail in the <a href="/applications-integration/11-arc-length-curve.php">original answer</a> to explain we are assuming it's a flattened catenary.

BTW, the formula for a (simple) catenary is `y=a cosh (x/a)` (with the `a` out front), which is equal to 

`y =a ((e^(x/a) + e^(-x/a))/2 )`

I have `-1` in my model since I'm placing the cable on either side of the `y`-axis..

X

Hi Murray. 

Can you help me with my math exploration for IB about catenary equations?

X

@belaldaq<span></span>qah: Feel free to post a new question here on the IntMath Forum. Don't forget to show your working!

X

Hi, 

Wanted to just ask about what Sofia said, why are the values for a different (1280 vs 1326)?
Is it because its a flattened catenary? In what ways does a flattened catenary differ from a simple one?

Thanks

X

The cables of the Golden Gate Bridge are suspended between two towers and undergo tension from their own weight and the weight of the roadway and the vehicles on it. The cable shape must be such that the tension at every point along the cable is perpendicular to the curve. The only curve that satisfies this condition is a catenary curve.