VMCG Testing
Joined: Nov 2005
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From: USA
Nose-wheel steering explanation
From the preamble to Amdt. 25-42 to Part 25 reads in part:
Several commentators recommended that the proposal be revised to allow the use of nose wheel steering in the determination of VMCG under Sec. 25.149(e), if control is through the rudder pedals and the demonstration is made on a wet runway. The FAA does not agree. The effectiveness of nose wheel steering depends to a large degree on runway friction characteristics and the load on the nose wheel. Certification tests on a wet runway would not cover the more extreme slippery runway conditions or all possible variations in takeoff conditions and techniques likely to occur in service. The FAA therefore believes that VMCG should be determined without the use of nose wheel steering, as stated in proposed Sec. 25.149(e).
FYI -- Explanations to changes to FARs are normally found in the preambles. The preambles can be found at the same link posted above.
Several commentators recommended that the proposal be revised to allow the use of nose wheel steering in the determination of VMCG under Sec. 25.149(e), if control is through the rudder pedals and the demonstration is made on a wet runway. The FAA does not agree. The effectiveness of nose wheel steering depends to a large degree on runway friction characteristics and the load on the nose wheel. Certification tests on a wet runway would not cover the more extreme slippery runway conditions or all possible variations in takeoff conditions and techniques likely to occur in service. The FAA therefore believes that VMCG should be determined without the use of nose wheel steering, as stated in proposed Sec. 25.149(e).
FYI -- Explanations to changes to FARs are normally found in the preambles. The preambles can be found at the same link posted above.
Joined: Aug 2005
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From: Europe
NWS - take off - snow and ice
New serious incident report august 2007
During the application of take off power, there was an asymmetric build up of engine thrust causing the left engine to lag the right engine. This caused a yawing moment that resulted in a loss of directional control. The aircraft yawed approximately 40° and departed the partially snow covered runway in spite of the crew selecting engine idle, applying nose wheel steering and braking. The aircraft continued to move forward at a slow speed off the paved area and onto an area of snow-covered soft ground. The nose wheel created a large furrow as the aircraft came to a stop in snow and soil at an angle of approximately 40° to the runway centre line. The tail and the nose of the aircraft were 12 m and 35 m from the runway edge respectively. The distance from the runway centre line to the edge was 22.5 m.
Damage to the aircraft was limited to a punctured left nose wheel tyre, a separated and deformed left nose wheel hubcap and a broken nose leg taxi light.
During the application of take off power, there was an asymmetric build up of engine thrust causing the left engine to lag the right engine. This caused a yawing moment that resulted in a loss of directional control. The aircraft yawed approximately 40° and departed the partially snow covered runway in spite of the crew selecting engine idle, applying nose wheel steering and braking. The aircraft continued to move forward at a slow speed off the paved area and onto an area of snow-covered soft ground. The nose wheel created a large furrow as the aircraft came to a stop in snow and soil at an angle of approximately 40° to the runway centre line. The tail and the nose of the aircraft were 12 m and 35 m from the runway edge respectively. The distance from the runway centre line to the edge was 22.5 m.
Damage to the aircraft was limited to a punctured left nose wheel tyre, a separated and deformed left nose wheel hubcap and a broken nose leg taxi light.
Joined: Mar 2008
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From: New Delhi, India
Vmcg and V1
Hi friends,
I am new here, and read this interesting conversation on Vmcg and V1 started by IDG.
1.If you look at the factors affecting Vmcg using Vmcg tables, the values are put against two variables that is Temp and Press Altitude. These 2 variables only change the thrust and Vmcg is directly proportional to Thurst. More thrust more the Vmcg and vice a versa. Xwind has no affect on Vmcg.
2. Vmcg testing: xwind of 7kts from the worse direction is considered. Pilot should be able to control using aerodynamic controls only. Initial force applied could be 180lbs, but later the force on leg should not be more than 150lbs. The aircraft should not drift more than 30 ft from centre line + certain restriction on change of heading that I dont remember offhand. A lot more variables are used to determine like bleeds off takeoff, Minimum weight and Max aft cg. So enough built-in safety.
3. V1mcg. We all know that V1 cannot be lower than Vmcg. So the lowest V1 is always higher to cater for the max x-wind component allowed for the aircraft.
IDG has noted that using LPC the speeds have gone down quite a lot. The operation/performance engineering department of the airlines can choose between three V1. Minimum V1, Maximum V1 and Optimum V1. May be IDGs airline has chosen a V1 that is lower than Optimum V1. Minimum V1 gives better safety margin and Maximum gives better takeoff performance. In my airliner this kind of revision has taken place many a times.
Thanks for bearing such a long one.
Moch
A330-200
I am new here, and read this interesting conversation on Vmcg and V1 started by IDG.
1.If you look at the factors affecting Vmcg using Vmcg tables, the values are put against two variables that is Temp and Press Altitude. These 2 variables only change the thrust and Vmcg is directly proportional to Thurst. More thrust more the Vmcg and vice a versa. Xwind has no affect on Vmcg.
2. Vmcg testing: xwind of 7kts from the worse direction is considered. Pilot should be able to control using aerodynamic controls only. Initial force applied could be 180lbs, but later the force on leg should not be more than 150lbs. The aircraft should not drift more than 30 ft from centre line + certain restriction on change of heading that I dont remember offhand. A lot more variables are used to determine like bleeds off takeoff, Minimum weight and Max aft cg. So enough built-in safety.
3. V1mcg. We all know that V1 cannot be lower than Vmcg. So the lowest V1 is always higher to cater for the max x-wind component allowed for the aircraft.
IDG has noted that using LPC the speeds have gone down quite a lot. The operation/performance engineering department of the airlines can choose between three V1. Minimum V1, Maximum V1 and Optimum V1. May be IDGs airline has chosen a V1 that is lower than Optimum V1. Minimum V1 gives better safety margin and Maximum gives better takeoff performance. In my airliner this kind of revision has taken place many a times.
Thanks for bearing such a long one.
Moch
A330-200
Joined: Dec 2003
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From: Oslo, Norway
Vmcg
And why does an aft CG have any influence on the calculation of Vmcg? Remember according to rules, no nosewheel steering!
In real life NWS would make a difference of course. But as soon as you try to counteract, you will start skidding, and forget about traction and directional control. Even more valid on contaminated runways of course!
In real life NWS would make a difference of course. But as soon as you try to counteract, you will start skidding, and forget about traction and directional control. Even more valid on contaminated runways of course!
Joined: Apr 2005
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From: UK
Aft C of G = closer to the rudder, so with a constant aerodynamic force created, the turning moment to keep the aircraft straight will be less.
You therefore need a higher airspeed to increase the turning moment of the rudder to counteract the yaw from the live engine, so Vmcg increases.
You therefore need a higher airspeed to increase the turning moment of the rudder to counteract the yaw from the live engine, so Vmcg increases.
Joined: Dec 2003
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From: Oslo, Norway
Vmcg turning moment
Wouldn't that apply to the turning moment of the remaining engine(s) as well?
Less abrupt yaw, - better rudder control, if you (read me!) are a little slow applying rudder?
Basically, I would think, the yaw moment (force X arm), with assymetric thrust is not dependent upon CG, and the yaw moment of the rudder would only be dependent upon deflection and speed? Aft CG would only result in a slower yaw (tail sideways) acceleration?
Wow, this was complicated, and it's getting worse!
Less abrupt yaw, - better rudder control, if you (read me!) are a little slow applying rudder?
Basically, I would think, the yaw moment (force X arm), with assymetric thrust is not dependent upon CG, and the yaw moment of the rudder would only be dependent upon deflection and speed? Aft CG would only result in a slower yaw (tail sideways) acceleration?
Wow, this was complicated, and it's getting worse!
Joined: Sep 1999
Posts: 5,552
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From: ME
Its always nice to see interesting threads start again... 
Yes it does.
Please quote the "present day" regulation showing this is the case.
Not correct.
You forgot about balanced V1.
Good Night.....will expand on these points when sober...
Mutt
Xwind has no affect on Vmcg.
Vmcg testing: xwind of 7kts from the worse direction is considered
So the lowest V1 is always higher to cater for the max x-wind component allowed for the aircraft.
between three V1
Good Night.....will expand on these points when sober...
Mutt
Joined: Dec 2006
Posts: 2,486
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From: The No Transgression Zone
Here it is right from the Administrator [page 145]
http://rgl.faa.gov/Regulatory_and_Gu...9?OpenDocument
http://rgl.faa.gov/Regulatory_and_Gu...9?OpenDocument
Joined: Sep 2002
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From: La Belle Province
For those as easily confused as I was, I believe the post above means you to look at p145 (actually 146 I think 
) as numbered on the pages. Which is p118 per the pdf file numbering.
If you look at the 145th page of the pdf file, you'll get a couple of graphs relating stall speed and CL to various factors, which certainly threw me until I realised the pdf pages don't line up with the document pages....
) as numbered on the pages. Which is p118 per the pdf file numbering.If you look at the 145th page of the pdf file, you'll get a couple of graphs relating stall speed and CL to various factors, which certainly threw me until I realised the pdf pages don't line up with the document pages....
Joined: Sep 2002
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From: La Belle Province
@Airborne Viking
Nope.
The moment due to the engine is basically net thrust of the asymmetric live engine * the distance from centreline - which is essentially independent of forward or aft cg.
The rudder moment is taken about the cg, so must be affected by fore/aft cg movement. So the rudder is less effective as the cg goes aft.
There are secondary effects (the aircraft will be less directionally stiff at aft cg, and likely have more weight on mainwheels, both of which will have a small effect on the motion) but the big effect is aft cg = less effective rudder control, so harder to fight the engine yawing moment.
Wouldn't that apply to the turning moment of the remaining engine(s) as well?
Less abrupt yaw, - better rudder control, if you (read me!) are a little slow applying rudder?
Basically, I would think, the yaw moment (force X arm), with assymetric thrust is not dependent upon CG, and the yaw moment of the rudder would only be dependent upon deflection and speed? Aft CG would only result in a slower yaw (tail sideways) acceleration?
Wow, this was complicated, and it's getting worse!
Less abrupt yaw, - better rudder control, if you (read me!) are a little slow applying rudder?
Basically, I would think, the yaw moment (force X arm), with assymetric thrust is not dependent upon CG, and the yaw moment of the rudder would only be dependent upon deflection and speed? Aft CG would only result in a slower yaw (tail sideways) acceleration?
Wow, this was complicated, and it's getting worse!
The moment due to the engine is basically net thrust of the asymmetric live engine * the distance from centreline - which is essentially independent of forward or aft cg.
The rudder moment is taken about the cg, so must be affected by fore/aft cg movement. So the rudder is less effective as the cg goes aft.
There are secondary effects (the aircraft will be less directionally stiff at aft cg, and likely have more weight on mainwheels, both of which will have a small effect on the motion) but the big effect is aft cg = less effective rudder control, so harder to fight the engine yawing moment.
Joined: Sep 2002
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From: La Belle Province
@ Mutt
If I were being incredibly pedantic I'd agree with the statement. Vmcg is defined, for certification, for a specific (zero) xwind. So, technically, the declared Vmcg doesn't change with Xwind.
But, the real speed at which the equivalent conditions of Vmcg are met certainly does vary with Xwind, which is what you are saying, and I'll heartily agree with that; only a bloody idiot would go do Vmcg testing in any kind of significant Xwind.
Xwind has no affect on Vmcg.
But, the real speed at which the equivalent conditions of Vmcg are met certainly does vary with Xwind, which is what you are saying, and I'll heartily agree with that; only a bloody idiot would go do Vmcg testing in any kind of significant Xwind.
Fleet Manager
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From: various places .....
.. which brings us to the main concerns ..
(a) Vmcg is one of those certification beasties which have their purpose in life but shouldn't be nipping at the heels of line pilots ... unfortunately, Vmcg is one which does do just that in some circumstances... which is why Mutt, MFS, I and others like to see it come up periodically in the sandpit .. although more frequently in Tech Log than FT.
(b) most of the time Vmcg (and crosswind effects on the "real world" Vmcg) is irrelevant to line operations as it is way, way below the V1 for the day. This brings into play that dreadful animal called line pilot complacency. Added to training bias, this can set up such folk for embarrassment .. easier for the TP who is more prepared to think on the run and abandon a takeoff post-V1 when things are obviously going pear shaped.
(c) if the V1 of the day is not too far above Vmcg, then crosswind needs to be in the line pilot's mind.
(d) for those occasions where one has little option (short runway etc) then it may be reasonable to accept the crosswind risk and go (although perhaps one could consider delaying until the wind abates ?) .. on the real world basis that the risk of a failure at/near a critical stage of the takeoff is pretty low .. and the chances of all the holes lining up (CG etc) are pretty low as well
(e) however, the concern we have relates to the case where options ARE available but the system sets the line pilot up for a fall. Consider the case of a ferry flight at very low TOW. If one is fat, dumb and happy, it is too easy to use a very low speed schedule appropriate to the weight. If the crosswind is a problem .. then it may be quite easy to ramp up the speed schedule to make the directional control problem go away. In the absence of specific Type data, a starting point is to aim for a pad in excess of 0.5 kt/kt for (rear-mounted) twins increasing to 1 kt/kt for quads (and, perhaps, wing-mounted twins ?)...
One would feel an awful goose .. sitting in the mud off to one side ... with the bird all bent and mangled ... knowing that one could have elected to use a V1 many tens of knots higher than the min speed schedule V1 one did, in fact, use ... with spectacularly unsuccessful results ...
Is this a problem we should lose a lot of sleep over ? .. probably not .. but I did some twin training for one largish operator which routinely used very high overspeed takeoff speed schedules (for appropriate reasons) but had a revenue sequence which included a short hop ferry to position the aircraft. So here we had all these fine folk doing all their training at humungous V1s .. but quite frequently doing this short hop ferry with a V1 down in the weeds near Vmcg .. is that not an inadvertant set up ? (As far as I knew, there was no specific training directed to this end of the operational spectrum)
For those command upgrades I put through I wasn't too worried .. they could handle sim exercises at min weight, min speed, aft CG etc .. and track on the centreline using the other end localiser .. but, judging from a few routine renewal checks I got roped into doing .. some of the other folk had the odd problem or two with low speed cuts ... not too hard to get an experienced pilot up to speed with the problem in several practice cuts .. but one really doesn't want a chap (or lady, for that matter) to find out .. on the line .. at V1 .. that his (or her) training missed that small point.
(a) Vmcg is one of those certification beasties which have their purpose in life but shouldn't be nipping at the heels of line pilots ... unfortunately, Vmcg is one which does do just that in some circumstances... which is why Mutt, MFS, I and others like to see it come up periodically in the sandpit .. although more frequently in Tech Log than FT.
(b) most of the time Vmcg (and crosswind effects on the "real world" Vmcg) is irrelevant to line operations as it is way, way below the V1 for the day. This brings into play that dreadful animal called line pilot complacency. Added to training bias, this can set up such folk for embarrassment .. easier for the TP who is more prepared to think on the run and abandon a takeoff post-V1 when things are obviously going pear shaped.
(c) if the V1 of the day is not too far above Vmcg, then crosswind needs to be in the line pilot's mind.
(d) for those occasions where one has little option (short runway etc) then it may be reasonable to accept the crosswind risk and go (although perhaps one could consider delaying until the wind abates ?) .. on the real world basis that the risk of a failure at/near a critical stage of the takeoff is pretty low .. and the chances of all the holes lining up (CG etc) are pretty low as well
(e) however, the concern we have relates to the case where options ARE available but the system sets the line pilot up for a fall. Consider the case of a ferry flight at very low TOW. If one is fat, dumb and happy, it is too easy to use a very low speed schedule appropriate to the weight. If the crosswind is a problem .. then it may be quite easy to ramp up the speed schedule to make the directional control problem go away. In the absence of specific Type data, a starting point is to aim for a pad in excess of 0.5 kt/kt for (rear-mounted) twins increasing to 1 kt/kt for quads (and, perhaps, wing-mounted twins ?)...
One would feel an awful goose .. sitting in the mud off to one side ... with the bird all bent and mangled ... knowing that one could have elected to use a V1 many tens of knots higher than the min speed schedule V1 one did, in fact, use ... with spectacularly unsuccessful results ...
Is this a problem we should lose a lot of sleep over ? .. probably not .. but I did some twin training for one largish operator which routinely used very high overspeed takeoff speed schedules (for appropriate reasons) but had a revenue sequence which included a short hop ferry to position the aircraft. So here we had all these fine folk doing all their training at humungous V1s .. but quite frequently doing this short hop ferry with a V1 down in the weeds near Vmcg .. is that not an inadvertant set up ? (As far as I knew, there was no specific training directed to this end of the operational spectrum)
For those command upgrades I put through I wasn't too worried .. they could handle sim exercises at min weight, min speed, aft CG etc .. and track on the centreline using the other end localiser .. but, judging from a few routine renewal checks I got roped into doing .. some of the other folk had the odd problem or two with low speed cuts ... not too hard to get an experienced pilot up to speed with the problem in several practice cuts .. but one really doesn't want a chap (or lady, for that matter) to find out .. on the line .. at V1 .. that his (or her) training missed that small point.
Joined: Sep 1999
Posts: 5,552
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From: ME
I'm disappointed that Moch330 didnt reply 
MFS, thanks for not putting on your pedantic hat
The problem with rules of thumb, is trying to justify them later. I remember when we had this discussion before, I got a PM from someone in the FAA that stated they didn’t consider V1=VMCG with a crosswind to be a problem as you had the nose steering to counter the effect. It’s probably a valid argument on a dry runway, but i would hazard a guess that when the runway is wet/slippery and you are near Vmu, then the nose steering isn’t effective. Unfortunately this message doesn’t appear to be getting distributed to the general population.
The comment from BelArgUSA was extremely fitting.....
Nowadays, in 747 simulators, I often take advantage of available time to practice engine "failures" myself, near V1 speed. Generally it is no problem for control at heavy weights and high speeds, but it is not the case if practiced at light weights, and low V1 speeds, selecting 20 kts crosswind, and a full aft CG and V1 is equal to VmcG, using max EPR (or N1) - Instructors, if you wish to "bust" a snotty captain in a simulator, just try that recipe...
Mutt
MFS, thanks for not putting on your pedantic hat
In the absence of specific Type data, a starting point is to aim for a pad in excess of 0.5 kt/kt for (rear-mounted) twins increasing to 1 kt/kt for quads (and, perhaps, wing-mounted twins ?)
The comment from BelArgUSA was extremely fitting.....
Nowadays, in 747 simulators, I often take advantage of available time to practice engine "failures" myself, near V1 speed. Generally it is no problem for control at heavy weights and high speeds, but it is not the case if practiced at light weights, and low V1 speeds, selecting 20 kts crosswind, and a full aft CG and V1 is equal to VmcG, using max EPR (or N1) - Instructors, if you wish to "bust" a snotty captain in a simulator, just try that recipe...
Mutt
Fleet Manager
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From: various places .....
The problem with rules of thumb, is trying to justify them later
True .. but, for the record .. 0.5 kt/kt comes from Douglas for the DC9-33F (I probably still have the fax on file somewhere) and I use that as the bottom figure (the Diesel 9 being held to track on the runway like it's on rails .. which it certainly DOESN'T near Vmcg .. and, probably, I still have the videos to back up that statement .. DC9 exiting the field of view left or right as the case may have been .. with my good mate CJF busily treading the light fantastic on the pedals ..) and the B747 is a bit above 1.0 kt/kt as I recall from some OEM paperwork I read years ago ... hence my rule of thumb range. Is it a guarantee ? .. of course not .. there aren't any in this life.
My concern is that of risk .. why put the aircraft in that sort of difficult speed range .. if you don't need to ? .. and then, given the flexibility to do so .. I'd include a few extras knots for mum and the kids.
they didn’t consider V1=VMCG with a crosswind to be a problem
.. and neither it is .. so long as you don't have an engine failure with the wind from the wrong side. AFAIK, Oz is the only country to have looked at the problem in detail ... I wouldn't give a fig for NWS based on some of the videos I have taken from the upwind threshhold during testing ... did I mention how it's very illuminating not to see an aircraft .. after it's departed the video viewfinder during critical cuts ? .. centreline tracking during a failure is one of those folkloric myths the pilot fraternity likes to perpetuate ...
I often take advantage of available time to practice engine "failures" myself
.. only the one caveat .. depends on the fidelity of the particular simulator in that area of the operational envelope. I recall a 732 box which was dreadfully and unrealistically boring before, but illuminating and electrifying following .. a software upgrade tweak in compliance with the FAA's stance on rudder modelling ...
True .. but, for the record .. 0.5 kt/kt comes from Douglas for the DC9-33F (I probably still have the fax on file somewhere) and I use that as the bottom figure (the Diesel 9 being held to track on the runway like it's on rails .. which it certainly DOESN'T near Vmcg .. and, probably, I still have the videos to back up that statement .. DC9 exiting the field of view left or right as the case may have been .. with my good mate CJF busily treading the light fantastic on the pedals ..) and the B747 is a bit above 1.0 kt/kt as I recall from some OEM paperwork I read years ago ... hence my rule of thumb range. Is it a guarantee ? .. of course not .. there aren't any in this life.
My concern is that of risk .. why put the aircraft in that sort of difficult speed range .. if you don't need to ? .. and then, given the flexibility to do so .. I'd include a few extras knots for mum and the kids.
they didn’t consider V1=VMCG with a crosswind to be a problem
.. and neither it is .. so long as you don't have an engine failure with the wind from the wrong side. AFAIK, Oz is the only country to have looked at the problem in detail ... I wouldn't give a fig for NWS based on some of the videos I have taken from the upwind threshhold during testing ... did I mention how it's very illuminating not to see an aircraft .. after it's departed the video viewfinder during critical cuts ? .. centreline tracking during a failure is one of those folkloric myths the pilot fraternity likes to perpetuate ...
I often take advantage of available time to practice engine "failures" myself
.. only the one caveat .. depends on the fidelity of the particular simulator in that area of the operational envelope. I recall a 732 box which was dreadfully and unrealistically boring before, but illuminating and electrifying following .. a software upgrade tweak in compliance with the FAA's stance on rudder modelling ...
Joined: Feb 2005
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From: KPHL
Mad Flight Scientist wrote:
MFS, I have no doubt you're correct about the standard rules of calculating moments within the industry, however moments that are calculated about different datums cannot simply be added together. To use them, you have to convert them to the same data, which would require knowing c of g for the assymetric thrust moment.
The other thing to consider is that the moment itself is not that interesting without knowing the moment of inertia. Of course, moment of inertia is dependant on mass distribution.
I guess the only caveat here is actual experience with this condition. I.e. Does the c of g actually affect Vmcg? Certification people seem to think so. What is the experience of the ppruners who have flown these tests?
Matthew.
The moment due to the engine is basically net thrust of the asymmetric live engine * the distance from centreline - which is essentially independent of forward or aft cg.
The rudder moment is taken about the cg, so must be affected by fore/aft cg movement. So the rudder is less effective as the cg goes aft.
The rudder moment is taken about the cg, so must be affected by fore/aft cg movement. So the rudder is less effective as the cg goes aft.
The other thing to consider is that the moment itself is not that interesting without knowing the moment of inertia. Of course, moment of inertia is dependant on mass distribution.
I guess the only caveat here is actual experience with this condition. I.e. Does the c of g actually affect Vmcg? Certification people seem to think so. What is the experience of the ppruners who have flown these tests?
Matthew.
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From: La Belle Province
Actually, those are moments about the same position, to a first approximation.
If you assume:
That the thrust vector from the live engine is along the centreline; and
The aircraft cg lies on the centreline
Then the moment due to the engine relative to any point on the centreline is the same - it's the net thrust of the engine times the lateral offset of the thrustline from centreline.
Therefore whether the c.g. moves forward or aft does not affect the thrust moment - only lateral motion of the cg would make a difference.
Similarly, since the sideforce generated by the rudder is considered to act purely sideways, the yawing moment due to the rudder does not vary with lateral motion of the cg - all that matters is the distance from the point of action of the force to the cg, meausred at right angles to the force.
Now, I'll grant that in practice the thrust line is often slightly canted in or out, and that the cg is not perfectly on the centreline, and the real direction of the rudder force may even be slightly canted (though by definition that is accounted for in another force component). But these effects are largely secondary to the main effects.
With regard to moments of inertia: if looking at a simple force/moment balance, then inertia terms don't actually matter - rudder moment = engine moment is a calculation that can be done independent of inertia terms. Where inertias come into consideration are dynamic events (and Vmcg can certainly be dynamic in real life). Although Vmcg is often tested at light weights, the actual reg states "most unfavourable weight"*, and AC25-7A suggests "VMCG testing should be conducted at the heaviest weight where VMCG may impact the AFM V1 schedule." My impression has always been that weight/inertia is at best a second order effect on VMCG - while the heavier aircraft is slower to diverge, it's also slower to return to centreline and the two effects are closely cancelling.
* Incidentally, the current FAR 25.149(c) states that VMCL uses " (4) The most favorable weight, or, at the option of the applicant, as a function of weight;" which is certainly an interesting typo!
If you assume:
That the thrust vector from the live engine is along the centreline; and
The aircraft cg lies on the centreline
Then the moment due to the engine relative to any point on the centreline is the same - it's the net thrust of the engine times the lateral offset of the thrustline from centreline.
Therefore whether the c.g. moves forward or aft does not affect the thrust moment - only lateral motion of the cg would make a difference.
Similarly, since the sideforce generated by the rudder is considered to act purely sideways, the yawing moment due to the rudder does not vary with lateral motion of the cg - all that matters is the distance from the point of action of the force to the cg, meausred at right angles to the force.
Now, I'll grant that in practice the thrust line is often slightly canted in or out, and that the cg is not perfectly on the centreline, and the real direction of the rudder force may even be slightly canted (though by definition that is accounted for in another force component). But these effects are largely secondary to the main effects.
With regard to moments of inertia: if looking at a simple force/moment balance, then inertia terms don't actually matter - rudder moment = engine moment is a calculation that can be done independent of inertia terms. Where inertias come into consideration are dynamic events (and Vmcg can certainly be dynamic in real life). Although Vmcg is often tested at light weights, the actual reg states "most unfavourable weight"*, and AC25-7A suggests "VMCG testing should be conducted at the heaviest weight where VMCG may impact the AFM V1 schedule." My impression has always been that weight/inertia is at best a second order effect on VMCG - while the heavier aircraft is slower to diverge, it's also slower to return to centreline and the two effects are closely cancelling.
* Incidentally, the current FAR 25.149(c) states that VMCL uses " (4) The most favorable weight, or, at the option of the applicant, as a function of weight;" which is certainly an interesting typo!
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From: Estonia
The rudder moment is taken about the cg, so must be affected by fore/aft cg movement. So the rudder is less effective as the cg goes aft.
If main wheels, then the rudder moment arm should be unaffected by CoG.
Obviously, the "loss" of an outboard engine causes asymmetric weight as well - those high-bypass engines are heavy, so CoG would move sidewards away from the lost engine.
