Sometimes IFR aircraft (aircraft on flight plans that allow them to fly through clouds) have to stay in one place and wait, while they are flying. Typically they'd be waiting for their turn to land at the airport, and the way air traffic controllers say "take a number" is by issuing a hold clearance.
A "hold" is an oblong path: two parallel straightaways connected at the ends by semicircles. It is defined by a point called a "fix," a direction to that fix, an altitude, and whether the turns are to be made to the left or the right. The pilot identifies the fix using radionavigation equipment. The oblong is flown so as to put the fix at the end of one of the straightaways, just before a turn. Sometimes the length of the straightaway is defined in miles, but usually the length is defined in terms of time: the inbound leg is supposed to take one minute, or, if the hold is flown at an altitude above fourteen thousand feet, a minute and a half. With no wind, the opposite straightaway would require the same time. In wind, the timing of the outbound leg must be adjusted so as to satisfy the timing of the inbound leg.Pilots' web has a great picture.
Imagine the wind is such that the aircraft experiences a tailwind during the outbound leg. That will make it cover distance faster, so that when one minute has elapsed and it turns inbound, it is further from the fix than it would be in still air. But if there is a tailwind outbound, there is a headwind inbound, so the airplane not only has further to go, but covers the distance at a lower speed. Reaching the fix, the pilot will discover that the inbound leg took more than one minute. On the next leg, she flies outbound for a shorter time, to compensate. But how much shorter?
There are many rules of thumb. During my initial training, I learned that if the inbound leg took more than a minute, to take two thirds of the excess inbound time off of the outbound time on the next leg. If the inbound leg takes less than a minute, add one and a third the excess to the next outbound leg. A pilot I respect, who can talk mathematical rings around me, subtracts half the excess or adds all the shortfall. As long as you're correcting in the right direction at each turn in the hold, you'll eventually converge on an outbound time that gives you a one minute inbound. It's a cockpit not a calculus class.
This being the internet, not the cockpit, I have filled a few sheets of paper with calculations, and determined that the distance travelled in the direction of the inbound track during a ninety degree rate one turn away from the inbound track is equal to the definite integral from t=0 to t=30 of (v cos (3t) + wt) dt, where v is the speed of the airplane and w is the wind speed. That ground to a stunning halt when I realized that whatever part of my brain had once known how to integrate trigonometric functions has found a new life, possibly storing and cross referencing the details of Friends episodes.
If someone has a more cooperative calculus-related brain segment, and wants to help out here, here are a couple more things to know about the path in the hold.
According to AIP-RAC 10.2, we "are expected to make all turns to achieve an average bank angle of at least 25 degrees or a rate of turn of three degrees per second, whichever requires the lesser bank." That "three degrees per second" turn is a common requirement, known as a rate one turn. We even have a rule of thumb to determine the bank angle in degrees that will produce it: ten percent of airspeed (in knots) plus seven. So for an aircraft flying at an indicated airspeed of 175 knots (the maximum permitted in a hold for any propeller-driven aircraft), the bank angle for a rate one turn is 175/10 + 7, or 24.5 degrees. Therefore, for all propeller-driven aircraft the 180 degree turns at the hold ends will each take one minute.
Timing of the outbound leg starts abeam the fix or when the wings are level after the turn, whichever comes later. Timing of the inbound leg starts with wings level. I have deliberately neglected the effects of wind not parallel to the inbound track.
Can't do you on your calculus. Thanks for your comment on last night's Airbus question. Oooh you ned more responses esle I'll seem like a sad persom, whicj I am not
Can't do you on your calculus. Thanks for your comment on last night's Airbus question. Oooh you need more responses else I'll seem like a sad person, which I am not
Hope your Badger or whatever hunting went better after that.
I've just stumbled across your blog and am slowly working my way from the beginning to the end. I am up to month three now (sometime in early 2005) so I still have a way to go. It provides good respite from study for my ATPLs.
Funny you talk about hold timings and the like because I was just taught that the other day. I ended up doing some maths myself to check the degrees to go for a monitored turn (as I couldn't get them to work in the sim) and it turned out that what we had been taught was wrong. I didn't need to do any calculus though, just trig.
If my calculations are correct I think you will find that the distance travelled in the direction of the inbound leg during the 90° rate one turn away from the inbound can be found using the following.
r = the radius of the turn
d = the diameter of the turn (which is the same as 2r)
c = the circumference of the turning circle
From a diagram we can deduce that the distance travelled inbound is the same as r.
From circle geometry we know that:
c = 2d = 2πr
So r = c / 2π
c in this case is the distance travelled in a full 360° rate one turn.
distance = d = st = 2s/60 (since to do 360° in a rate one turn takes 2 minutes or 2/60 hours)
Now we know d = c = 2s/60.
So r = 2s/60 / 2π = s/60π.
And now we can calculate the distance travelled inbound during half of the turn onto the outbound leg, s/60π, which makes sense as the distance inbound will depend on how fast you are travelling.
Not sure if that's what you were after but I sure had fun doing it.
Okay, sorry, just kept reading and saw your post from the next day where you mention the solution. Hope my post wasn't patronising then.
Anyway, off to do a VOR holds and approachs flight now.
Not at all. It's just one of those weird things I did way back when.
When you get to the gaps in the blog, e-mail me.
Ah, the gaps. (I'm supposed to be studying for the last six of my ATPL subjects (and I am) but occasionally I get distracted and jump online to read a month's worth of back issues of Cockpit Conversation. Recently I've noticed that it's gotten a lot quicker to get through a month's worth. ;-)
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