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Author Topic: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas  (Read 3958 times)

Online Brent Williams

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M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« on: February 26, 2017, 09:22:39 PM »
I have been using these formulas to calculate the M.A.C.(Mean Aerodynamic Chord) and T.V.C.(Tail Volume Coefficient) and would just like to confirm if they are correct.  Please correct any errors if there are any and I'll update this post.  Thanks, Brent.



1.   Mean Aerodynamic Chord:    ~All measurements are including flaps~
          rc = Root Chord,    
           t = Taper Ratio = (Tip Chord ÷ Root Chord)

          M.A.C. = (rc x 2/3) x [( 1 + t + t˛ ) ÷ ( 1 + t )]
------------------------------------------------------------------------

2.   Aerodynamic Center:  25% of M.A.C.   ~Measured back from the leading edge.~

------------------------------------------------------------------------

3.   Tail Moment Arm:   1/4 Chord of wing M.A.C. to 1/4 Chord of Stab/Elevator M.A.C.
                                  
                                                              
   ~aerodynamic center to aerodynamic center~

------------------------------------------------------------------------

4.   Tail Volume Coefficient:    (Tail Area X Tail Moment Arm) ÷ (Wing Area x Wing M.A.C)

------------------------------------------------------------------------


Updated to show proper formula nomenclature
« Last Edit: February 27, 2017, 12:50:02 AM by Brent Williams »
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Offline Howard Rush

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment" Formulas
« Reply #1 on: February 26, 2017, 10:48:05 PM »
Yep, except that what you call "tail moment" should be "tail moment arm".

For control line airplanes the mean aerodynamic chord calculation should be tweaked to account for the outside wing going faster, particularly if you are calculating its spanwise position.  We also have the habit of making different sized right and left wings.  See http://stunthanger.com/smf/engineering-board/design-parameters-ii/ reply #8 for a pickier definition.  Mean aerodynamic chord is kinda approximate, so you don't need to be really picky.
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Online Brent Williams

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #2 on: February 26, 2017, 11:02:30 PM »
Thank you Howard!  I updated the original post with your suggestions.
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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #3 on: February 27, 2017, 07:48:12 AM »
Ok you know how to measure that stuff but what are the numbers for stunt supposed to be at. Must be a component for size, people tell me smaller planes need longer tail moments.


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Offline Air Ministry .

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #4 on: February 27, 2017, 09:47:39 PM »
Al uses Stab Hingeline , will dig out his data / document .

Offline Jim Svitko

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #5 on: February 28, 2017, 02:35:51 PM »
For longitudinal stability, should the aircraft center of gravity be located forward of the wing aerodynamic center?

Offline Howard Rush

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #6 on: February 28, 2017, 03:02:38 PM »
For longitudinal stability, should the aircraft center of gravity be located forward of the wing aerodynamic center?

Not necessarily. That's why they call the aft wing a stabilizer.
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Offline Chuck_Smith

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #7 on: March 01, 2017, 05:24:05 AM »
For longitudinal stability, should the aircraft center of gravity be located forward of the wing aerodynamic center?

You need the change in pitching moment WRT change in angle of attack to be negative. That's for static stability. For dynamic stability it gets a little more convoluted.
Also, no airplane will truly 100% stable. It will have a damped oscillation. Most airplanes have imaginary roots for the discriminate of the static stability formula which tells you they will have an oscillating response to a disturbance.

There's both a "short period" and "long period" response. The long period is so long you might not notice it. The short period is the "bobble" almost all airplanes do to varying extents.

Luckily for CLPA, because our airframes are so light the mass of the air displaced damps most of the wiggly tendencies to the point we can ignore them.
« Last Edit: March 01, 2017, 06:15:47 AM by Chuck_Smith »
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Offline phil c

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #8 on: March 01, 2017, 08:54:50 AM »
Brent, as long as you are doing stability equations you should include the Neutral Point-  the spot along the tail moment arm where the plane has neutral static stability.  It's normally a significant distance back of the MAC.  I don't have the equations handy, but it's essentially a teeter totter between the wing and the tail. Balanced at the neutral point the plane won't react to being disturbed at all.  The CG is always ahead of the NP, except in some full scale fighter like the F-22 and other fly-by-wire designs where a computer actually runs the controls and controls the stability.  It's also why most of the bigger stunters can balance around the MAC(25% of the mean chord).  The CG is still far enough forward to provide adequate stability.  Of course, in bigger planes, and in all planes the effects of other sufaces(rudder, fuselage area, control inputs, gusts, propwash, etc.) the NP can easily move around when the plane is not "static".

A Lomcevak(forward tumble) is an example of where the plane loses most of it's aerodynamic forces and the pilot forces it to tumble mostly from prop precession.
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Online Brett Buck

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #9 on: March 01, 2017, 09:27:10 AM »
You need the change in pitching moment WRT change in angle of attack to be negative. That's for static stability. For dynamic stability it gets a little more convoluted.
Also, no airplane will truly 100% stable. It will have a damped oscillation. Most airplanes have imaginary roots for the discriminate of the static stability formula which tells you they will have an oscillating response to a disturbance.

There's both a "short period" and "long period" response. The long period is so long you might not notice it. The short period is the "bobble" almost all airplanes do to varying extents.

Luckily for CLPA, because our airframes are so light the mass of the air displaced damps most of the wiggly tendencies to the point we can ignore them.

    If you want to be super-picky, even a very lightly-damped oscillation is still "stable". As long as it goes away eventually, it's stable. Maybe it doesn't go away fast enough, but that's a performance issue, not stability. Only of the poles get into the right half-plane/oscillations grow with time is it actually unstable.

     Brett

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #10 on: March 01, 2017, 09:30:29 AM »
Not necessarily. That's why they call the aft wing a stabilizer.

  Going to incur someone's wrath for that one!  Last time I pointed that out, I got a 5000-word response about how stupid I was and that I didn't even know what was in the "Handbook for Naval Aviators".l That might have been the same one where I said something less than charitable like "flying them only proves you know where the switch for the yaw damper is, not that you know how it works".

    Brett

Offline Fredvon4

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #11 on: March 01, 2017, 10:36:26 AM »
Very loose engine mount bolts made me wish I has a "Yaw Damper" and a "Pitch Damper" switch* one day....and tis true...if there was one (or both), I would not have known, nor cared, HOW it worked

Brett...I am bored, please post a link to any 5000 word dissertations about you being stupid.... I enjoy the mental effort needed to read gibberish and false science and try to apply what I know and what I learn from the exercise

*Twas my job years ago to understand and fix SCAS** systems on Army Helicopters

**STABILITY AND CONTROL AUGMENTATION SYSTEM

The SCAS is a three-axis***, limited authority rate reference augmentation system. The SCAS cancels undesired motion of the helicopter during flight. This is accomplished by inducing an electrical input into the flight control system to augment the pilot mechanical input.

***Roll, Pitch and Yaw

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Online Brett Buck

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #12 on: March 01, 2017, 12:17:53 PM »
Ok you know how to measure that stuff but what are the numbers for stunt supposed to be at. Must be a component for size, people tell me smaller planes need longer tail moments.


   They tell you something, and there is nothing more rehashed than "stunt numbers" but you still have to have a concept of what you are trying to achieve or correct in an existing design. A few very excellent airplanes have been built using the "stunt numbers" and a huge number of bad to mediocre airplanes have been built using the same numbers.

     Brett

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #13 on: March 02, 2017, 05:19:37 AM »
    If you want to be super-picky, even a very lightly-damped oscillation is still "stable". As long as it goes away eventually, it's stable. Maybe it doesn't go away fast enough, but that's a performance issue, not stability. Only of the poles get into the right half-plane/oscillations grow with time is it actually unstable.

     Brett

Well Brett,

If we want to be super-super picky we can argue this is not a stick-free response but stick-fixed.
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Offline Scott Richlen

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #14 on: March 07, 2017, 06:31:04 AM »
So, may we conclude that the best design process for the rest of us (who have to remove our shoes to get to "20") is to get a copy of the Stargazer plans and then change the shape of the rudder (being careful to not change its area) and move the canopy forward or back by an inch or two?   ;D

Of course, I haven't figured out how I am going to power it with my LA-46....

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #15 on: April 04, 2017, 10:02:38 PM »
Quote
As far as I know, the 1973 Mustang V was the first stunt ship design taking tail volume calculations into consideration for determining moments and areas.  First, I measured all of my airplane’s moments and areas to determine their tail volume and related this to their handling characteristics.  This gave me an approximate design goal for the Mustang V.  To calculate a tail volume, I use the entire width of the wing including flaps, and emphanage (stab plus elevators) as shown in the plan view to determine the mean aerodynamic chord (MAC) of these surfaces.  The MAC is the wing chord location where the areas of the wing, both inboard and outboard of that chord, are equal.  The calculation takes into consideration the width of the wing and emphanage MACs, the areas of the wing and emphanage and the distance from the center of gravity to the emphanage, quarter chord   This formula tells us that for a given value of tail volume we can vary the area of the tail, or length of the tail moment in our calculations.  Specific TVCs can be calculated by varying the chords, moments or areas within the formula.  The engineering need for using MAC lies in stability calculations for airplanes with swept wings where the MAC may be a long way from the root chord where the wing enters the fuselage.
 
Is there a builder who hasn’t considered the possibility that if enough is good, then might not more be better?  The problem with adding both tail area and tail moment, or a lot of either, is that it creates very powerful pitching moments, which is another way of saying very sensitive and responsive controls.  Strong pitching moments and sensitive controls make airplanes turn easily with small control deflections.  Large tail volumes though, don’t work well for semi-scale airplanes because powerful controls limit control deflections.  Semi-scale airplanes with reduced wing area depend on airfoils and flaps designed to produce the maximum possible lift, and this only occurs with relatively large control deflections.  Practically speaking, I can get the same corner radius and control sensitivity with a different configuration, but need a carefully considered tail volume and larger line spacing at the handle.  Tail volume coefficients must be big enough for longitudinal stability and small enough to allow large control deflections for maximum lift.  My design TVC is .4.
 
What about moving the CG back to reduce longitudinal stability a bit for sharp corners with only a twitch of the wrist?  This is an example of how builders can make a seemingly minor and logical change which has dramatic effect.  Moving the CG back 1/2” has a negligible effect on the tail volume.  This would typically be less than .01 or, for example, from .42 to .41.  That same 1/2” happens to be a very large reduction in the distance between the center of gravity and the center of lift for a substantial loss of stability.   We could reduce elevator sensitivity by making the elevator area or deflection smaller.  I can only guess what, if any, desirable effect this reduction in stability and less effective elevators might have. 
 
Area limited airplanes must have bellcrank/flap horn ratios large enough to deflect the flaps.  Flap/elevator ratios, elevator area, CG location and tail volume can all be varied, but with results difficult to predict.  I stick with relatively forward locations of the CG, and target TVCs, for adequate stability and use approximately 50-50 elevator/stab areas.  This puts me in the ball park with enough pitch for a competitive corner, and enough flap deflection for the necessary lift.  Performance is fine tuned with trim adjustments.
 
Personally, I consider tail volume calculations necessary only for the design of stunt ships which depart from conventional values.  My goal is to make a grooving airplane with a competitive corner without over sizing moments or areas.  The calculated value is useful, but at best, it is only a rough estimate in determining the distance between the wing and tail or an appropriate area for the tail.  In any case, when designing the Mustang, I drew the fuselage using the desired outlines and tail moment, and used the formula to determine the area of the tail.
 
If you are interested, the formula for tail volume coefficient is: area of the tail divided by the area of the wing, times, length of the tail divided by the wing MAC.  A value between .4 and .45 is usually enough.

NOW You Know !

Mucho Gracious to Al Rabe , for digging this out and forwarding , Earlier .
Just found it inadvertantly searching for otherthings , as it wouldnt
materialise when I WAS looking for IT ! .

 H^^

Offline Double Deuce

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #16 on: April 09, 2017, 08:27:53 PM »
As a product of the Louisiana School System, and a Control line Speed Flier, what do I know? So be gentle with me when I ask this question. In comments nr 15, Mr Rabe states, " The Mean Aerodynamic Chord location is where the areas of the wing, inboard and outboard of that chord line are equal in area." So, my question is this. "Is this a true statement?"  I tend to think not but reread line one.

Double Deuce

Offline Serge_Krauss

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #17 on: April 11, 2017, 08:23:56 PM »
You're correct; Al missed this one by a smidgen. The MAC doesn't divide the half-span wing area into two equal areas. However, the "good" news is that for mildly tapered wings, like most stunters there isn't a lot of difference between the areas inboard and outboard of the MAC. With any taper at all, the MAC moves inboard, and the difference between inner and outer areas increases. The most extreme taper occurs with a pointed tip. Such triangular wings have an MAC of 2/3 the root chord, located 1/3 of the way out. On this extreme wing, the ratio of the inboard area to the outboard area is 5/4 for a 25% greater inboard area. For less severe tapers, the ratio is lower. For a rectangular wing, the MAC equals the root chord and is located in the center of the half-span, making the two areas equal. All other tapers to a smaller tip chord give values between these two.

Here's a more typical tapered-wing example. If the tip chord is 70% of the root chord (Ct = .7 Cr), then again theoretically, the MAC will be 47.1% of the half-span out from the root, and the ratio of "inner" area to "outer" area will be 1.063. So the inner part will have 6.3% greater area than the outer part.

The defining equations used for these computations are based on the simplified idea that lift is equal at all points of the wing, which isn't true. In reality, more lift will take place inboard, even for a "Hershey-Bar" wing, because of tip losses. This moves the actual aerodynamic center further inboard. However, other complexities interfere to make these approximations pretty accurate, it seems. SO,...we just use these for "ball-park" starting points.

Some of the top guys, like Ted and Al do seem to use the half-and-half approximation or a mid-point position of the MAC (not the same). They do OK.

'best I can do. - SK

Offline Ted Fancher

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #18 on: April 11, 2017, 10:49:04 PM »
 

Some of the top guys, like Ted and Al do seem to use the half-and-half approximation or a mid-point position of the MAC (not the same). They do OK.

'best I can do. - SK

[/quote]

That's a true statement, Serge, as far as it goes.  I have...however...always made it clear that the "half-span" simplification of the MAC isn't a technically accurate location but, with roughly 95+% of the world's modestly tapered wing stunt designs throughout recorded history, it's close enough to allow effective trim decisions to be "wagged" and doesn't clutter up Stunt Hangar and Stuka stunt with lengthy attempts to pin down precise numbers with which 90+% of the world's stunt flyers will never come to grips.  Such a WAG allows a stunt flyer...if not an aerodynamic engineer...to determine and work with a common reference point with which to make adjustments that are reasonably replicable and responsive to pitch changes that result from fore and/or aft movement of the CG.  All of my designs utilize a commonly accepted graphic method of determining the MAC but that's only because I enjoy doing so.  Graphics I provided for the old Imitation included a drawing illustrating that method.

As generally happens with these more esoteric discussions on a site populated primarily by aerodynamic lay persons the, result is a plethora of responses frequently too helter skelter to provide information which results in adequately informed readers interested solely in making their version of the 500th rebirth of the Nobler perform better. :-D

Ted

Offline Ted Fancher

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #19 on: April 12, 2017, 12:29:55 PM »
To amplify the above just a bit.

"Numbers" per se are of value primarily in the design or modification of a stunt ship (most especially if it is "significantly" different from its more or less cut and paste forebears).  they are important mostly for insuring that a knowledge-based WAG of a "very likely" ultimate CG location can be part of the planning process and drawing the plan from which the new ship will be constructed.  Most important, by far, are the likely location of the CG and, therefore, the best location for the adjustable leadout guides.  Also important is planning a location at which (hopefully, modest amounts of) weights can be added if necessary to refine the real world example from the drawing board paper!  One thing which might not spring immediately to mind, for instance is the location of leadout holes in the inboard wing ribs and/or the location of the foam spars in a foamy so as not to require unsightly "adjustments to their location" after the 20 point finish is applied.

In the final analysis these "numbers" are needed so as to allow at the flying site minor adjustments to achieve maximum inflight performance most of which will be determined by assessing inflight performance performance and determining the need to alter either CG and/or leadout exit location; done by balancing the ship on your left and right index fingers and determining where the CG is versus where you think it might be better and simultaneously where the leadouts exit in relation to the current CG.  We then add or subtract weight as suggested by that test and adjust the leadouts if needed and recheck it: all while balanced on our finger tips.  I don't think even Howard or Al probably took their drawing board to the field (although I wouldn't put it past Howie to have taken his lap top!) for trimming flights but rather flew; evaluated; adjusted via finger tip WAG and reflew.  All of this is relevant to the discussion of tail volume/CG location/flap size and travel, etc. that has been beaten to death in this and myriad other posts on-line.  I think this subject needs a bit of a primer that will hopefully provide basic understanding of what is being discussed...to death

I wish I could remember exactly where (maybe on-line or in one or another magazine) but I once wrote a fairly lengthy piece addressing CG location and tail feathers  (think Stab and Elevators). Here's a shortened rehash.


The experiment starts out with, say, a 36"X1/4" steel rod (best choice as its CG will automatically be at 50% of its length) or a straight stick with a CG located at mid length.  Throw it or drop it out an upstairs window and note that it will "not" fly straight but flail randomly (about its CG, by the way) and its "landing attitude" will be unpredictable.  

Add weight to one end, however, and throw/drop it again and the weighted end will always land first.  The more weight the more pronounced the "stabilizing" effect, just like a spear! This illustrates the effects of CG location on the stability "equation".  It "wants" to go in one direction.  Aha!  So that's why we add nose weight!

Next, we can remove the weight (making the rod/stick once again unstable) and, instead add "significant" tail feathers" to it so it looks like an arrow.  Once again throw it or drop it out of the window.  Voila, the non-feathered end will again repeatedly arrive first, thus illustrating the "neutral point" of the--now, due to its complex structure--vehicle!  The drag on the "vehicle" is now behind the CG!  The more tail feathers you add the greater will be the degree of stability.  In addition, the farther aft you locate the tail-feathers the greater will be the stability (think tail moment "arm").

Combine the weight added to the nose and the tail feathers and you've now replicated an "arrow" which is great if you have a bow or a strong arm with which to launch it.  It will not, however, ever win the Walker Cup!  For that you now need the last part of a stunt ship; a source of lift.  We like to call that a wing and some wings will have a "integral" component we call flaps.  Note these are not "separate" elements but part of a single lift structure whose configuration allows its lifting capability to be altered in response to lift needs.

We now have a complete vehicle which can fly and/or  (in our case given inflight control capability) maneuver if CG, Lift and Tail feathers are appropriately located.

A final significant factor re lift and stability.  

Where the CG is located with respect to the "central location" that LIFT is generated (routinely referred to as the center of lift although the better informed might quibble) is the next most important factor in our ability to do tricks with our aerodynamically stable stunter.  If the CG is too far forward we'll have inadequate stabilizer/elevator authority no matter the tail's size or elevator deflection, with which to do our tricks.  If the CG is too far aft tricks will be impossible because the vehicle has aerodynamically reverted to its "initial" status (the unstable rod) and is no longer capable of controlled flight let alone maneuver on command.  

The point at which the aft motion of CG reaches this state is at or in close vicinity to the NEUTRAL POINT (NP), a less familiar term addressed in a few of the later posts.  The NP is  the point at which the CG is in the same longitudinal (lengthwise)  location as ALL of the aerodynamic forces acting on the entire vehicle.  You can throw it now (or attempt to fly it) as hard as you can and the outcome will end up in the trash can.  Lacking modern fly-by-wire computerization it isn't going to fly straight and level let alone win the Walker Cup.

This is too long already and I could be writing to an empty auditorium so I'll quit.  If anyone is interested in more pontification from a near contemporary of the Wright brothers write a post to that effect.


Offline Double Deuce

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #20 on: April 12, 2017, 03:15:36 PM »
Serge and Ted

Thank you for taking time out of your laughing to do a come back to my alleged missive. What I say here probably should not be attached to this thread but, honestly, I only read thru most of the comments herein and do not understand totally the procedures to be used.

I am slightly short of 82 years on this Earth and still am trying to catch up on the education I am so sorely lacking. I am truly amazed at what I see when Ted, and Phil G. and Brett and David, and obviously others, do their thing on the hallowed Napa Circle. Impressed somewhat.

Speed vehicles are quite simple subjects. I also ran tether cars for more than a few years. Aerodynamics play little use in either of these vehicles. This is not to say one should slaughter the physics of it all. Neither maneuvers, at least intentionally, and both depend so much on the tank. I do believe one will never truly understand how a tank works until they are successful with a tether car. Remember the tank goes from 1 G to near 90 G. Essentially the same is true with a speed model. What I understand about tanks occurs in a sorta consistent attitude. You guys with a pattern to perform do amaze me.

My interest in aerodynamics, other than just wanting to know, primarily consists of wanting to be able to set things up, your version of trimming?, so the tank will do its thing. Line drag, leadout position, balance point, and tank position vs needle position have to be right. So a lot of thinking on my part has gone into asking myself why did we wander off years ago in plane design and leave the vertical fin at home? I remember an article  by the man who built the first replica of the GeeBee, complete with wheel pants a yard or so long. He said the pants acted like a rudder on the wrong end of the fuselage . Now I am starting to tell myself what was causing me some difficulty. Next move was to go back to the vertical fin. Surely effected the pull. The cowing was the rudder on the wrong end and acted so.

I need to know more on the side view. Free flighters refer to Center of Lateral area. Some say it is real, some say it is not. Won't go there though I do spend time tweaking fin area and arm to trim for the cowl.

Aerodynamics, to me, really is a search for making things right to allow the tank to do it. I will honestly admit I do not understand how you all can make the tank work from start to finish. I will keep watching but apparently I am not sitting close enough.

Ted, from my view you will never lecture to an empty room. This Cajun Boy will always pay attention to your comments. And may even understand some of them.Do hope to see you in Napa

Luke

 

Offline Ted Fancher

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #21 on: April 13, 2017, 10:26:40 AM »
Thanks Double Deuce,

I appreciate your nice comments.  However, the way the thread is dying makes it pretty clear that further pontification from an old geezer isn't at the top of most lists.  Since the lengthy posts take a measurable amount of time I'll let this one drop.

Ted

Offline Double Deuce

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Re: M.A.C., T.V.C, Aerodynamic Center and "Tail Moment Arm" Formulas
« Reply #22 on: April 13, 2017, 11:05:54 AM »
Ted

Agreed

Luke


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