Stunt Control System Geometry

[Howard Rush]

OK, I think I've figured out how to calculate this with Excel.  First, here are what's to be calculated, some definitions, some assumptions, and the inputs.  I included the stuff that's important to me.  Tell me what you think.  I'm sorry I don't have drawings to include.  I'll do them with my 2D CAD program, so I couldn't post them here anyhow. 


Calculate:
1. Flap and elevator deflection as functions of differential control line movement
2. Elevator deflection vs. flap deflection


Definitions:               
X = 0 at flap control horn axis               
Y = 0 at fuselage centerline               
Z = 0 at wing chord               
X is positive forward.               
Y is positive out the right wing.               
Z is positive down.               
Rotations are positive about positive axes:                
   Bellcrank rotation is positive clockwise looking from the top.            
   Flap deflection is positive clockwise looking from the left wing tip.            
   Elevator deflection is positive clockwise looking from the left wing tip.   


Assumptions:               
Airplane flies counterclockwise (looking from the top of the circle) in upright level flight.               
Wing is symmetrical about X-Y plane and has no dihedral.               
Bellcrank axis is in the X-Z plane.               
Bellcrank center (intersection of bellcrank axis and bellcrank plane) is on wing chord, Z = 0               
Leadouts guides are on wing chord, Z = 0               
Flap hinge line is straight and is on the Y axis: X = 0, Z = 0.               
Elevator hinge line is straight and is parallel to the Y axis.               
Elevator and flap deflections are relative to the airplane X-Y plane.               
Leadout attach points on bellcrank are colinear with bellcrank center               
All pushrod joints are ball joints; joint location is center of ball               
Forward pushrod goes from bellcrank to flap control horn.               
Aft pushrod goes from flap control horn to elevator control horn.               
Everything is rigid except leadouts, which are perfectly flexible.


Inputs:
X of leadout exit at wingtip
Y of aft leadout exit
Y of fwd leadout exit

Bellcrank-plane pitch angle, positive aft-end-high
Bellcrank leadout-arm radius
Bellcrank pushrod-arm radius
Bellcrank pushrod attach elevation above and perpendicular to bellcrank plane
Bellcrank leadout-arm rotation at zero flap deflection
Bellcrank pushrod-arm rotation relative to a line perpendicular to the bellcrank axis and perpendicular to the leadout arms

Flap control horn forward-pushrod-arm radius
Flap control horn aft-pushrod-arm radius
Flap control horn forward-pushrod attach point rotation at zero flap angle
Flap control horn aft-pushrod attach point rotation at zero flap angle
Flap control horn forward-pushrod attach point distance right of fuselage centerline
Flap control horn aft-pushrod attach point distance right of fuselage centerline

X of elevator axis
Z of elevator axis
Elevator control horn pushrod-arm radius
Elevator control horn pushrod attach point rotation at zero elevator angle
Elevator control horn pushrod attach point distance right of fuselage centerline
Elevator downrig angle from fuselage X axis at zero flap angle


               
         

[Tim Wescott]

• I think some of your assumptions are immaterial, or negligible (e.g. a reasonable amount of dihedral is negligible, a symmetrical wing is immaterial to calculating deflections)
• To compliment your apparent excess of assumptions, you appear to have left one out.  You may want to work in a statement about "simple hinges" assumed -- a cloth hinge rolls the LE of the surface along the TE of the wing or stab; this action is not a simple rotation about an axis.  Thus the effective radius (and offset angle, I think) from the instantaneous center of rotation will change in an amount proportional to the surface thickness, and with a complicated relationship to the surface LE shape and wing TE shape.  Tell folks to just use a lucky box and hinge the control horn!!
• The bellcrank plane pitch angle, as expressed as a rotation around the Y axis, would be positive clockwise looking from the left wing; i.e. it would be positive with the back low.  Let's stick to the right-hand rule here: http://www.xkcd.com/199/
• I got wrapped around the axle for a while on "Flap control horn forward-pushrod-arm radius", then I realized that you meant "distance from flap hinge axis to effective control horn pivot".  I don't know if you need a foot note, or to reword that, or to just smack me
• It'd be neat if you posted your calculations.  I may even try to go over them. 

[Howard Rush]

• I think some of your assumptions are immaterial, or negligible (e.g. a reasonable amount of dihedral is negligible, a symmetrical wing is immaterial to calculating deflections)
• To compliment your apparent excess of assumptions, you appear to have left one out.  You may want to work in a statement about "simple hinges" assumed -- a cloth hinge rolls the LE of the surface along the TE of the wing or stab; this action is not a simple rotation about an axis.  Thus the effective radius (and offset angle, I think) from the instantaneous center of rotation will change in an amount proportional to the surface thickness, and with a complicated relationship to the surface LE shape and wing TE shape.  Tell folks to just use a lucky box and hinge the control horn!!
• The bellcrank plane pitch angle, as expressed as a rotation around the Y axis, would be positive clockwise looking from the left wing; i.e. it would be positive with the back low.  Let's stick to the right-hand rule here: http://www.xkcd.com/199/
• I got wrapped around the axle for a while on "Flap control horn forward-pushrod-arm radius", then I realized that you meant "distance from flap hinge axis to effective control horn pivot".  I don't know if you need a foot note, or to reword that, or to just smack me
• It'd be neat if you posted your calculations.  I may even try to go over them. 

• I have some applications in mind for this program that dihedral would mess up. 
  
  You are correct about the symmetry.  I was thinking of the X-Y plane as the plane of symmetry of the wing, but it really doesn't have to be symmetric.  This program would be inadequate for Al Rabe, but covers most anything I have in mind to build. 
  
  
• Anything I build with cloth hinges won't receive much analysis.
  
  
  • OK. I should do that to be pure, even though it would make the one in my airplane negative. 
    
    
  • Maybe some linear combination thereof
    
    
    • I'll send you the file when I write it.  The calculations themselves, as I intend to do them, would be embarrassing.  However, if you want me to do them in a more rigorous manner, you can tell me how to do the following:
      
      Pick a point on the bellcrank end of the rod: Call it (X1, Y1, Z1).  I chose X = 0, Z = 0 as the flap control horn joint, and the Y axis is the flap hinge line, so the flap control horn joint with the pushrod, (X2, Y2, Z2), is on a circle: r^2 = X2^2 +  Z2^2 .  Y2 is constant and known.  For pushrod length L, L = sqrt((X2 – X1)^2 + (Y2 – Y1)^2 +(Z2 - Z1)^2).  So there are two equations in two unknowns, X2 and Z2. Gimme a closed-form solution.  There are maybe two or four solutions.  I want the one (or two) with X2>0 and Z2>0.

[Tim Wescott]

I'm still not sure that a few degrees of dihedral would mess something like this up by any great amount, at least the way I think I'd use it.  I think that the bellcrank deflection will be off by the secant of half the dihedral, and that the flap deflection relative to the flap horn will be off by the cosine, or pretty close -- so they should cancel.  Even if they built up, for a 5 degree dihedral you'd still only be off by a percent or two.

But, if it were really important you could include the effects.

Solving for the flap horn location turns out to be direct, but tedious, if you start from a bellcrank position.  Project the bellcrank to flap pivot point onto the Y-Z plane; this can be done with a smidgen of trig if you're starting with angular deflections.  Now calculate the offset in the Y direction between the bellcrank to flap pivot and the flap to bellcrank pivot.  Use this offset to calculate the projected length of the bellcrank to flap rod onto the Y-Z plane.  Now you can find the intersections of the circles described by the flap end of the con-rod and the flap horn, and therefore the two intersections.  Then pick the reasonable one (surely z < 0, since that's above the wing centerline?).

The projected length of the flap to elevator rod never changes, so finding the elevator position once you know the flap pivot position is just a matter of inscribing your two circles and finding their intersections.

I don't know if I'm going to actually do the calculation -- the word "tedious" makes me think about all the building projects that I'm not going to have time to get to this winter.  But I've outlined it for you.

[Howard Rush]

The lad has management potential.

[Bill Little]

The lad has management potential.

You go, Howard.........

Big Bear

[John Miller]

Since I've been found lacking in the trig dept., I've always laid out the control system using my CAD system. I'll outline the steps I go through with the hope that it will be of some help to you in your quest.

I've often thought about setting up a spread sheet on my web site that would do the figuring after a few key inputs were made.

My first step relies on an assumption, I figure that I will have 50 degrees either side of nuetral for my maximum bellcrank throws.

The second step is to determine how much flap throw I want. I often go for 30 degrees up and down, as maximum throws. It's a simple ratio, 50/30 = 1.66666.

If I use a radious of, for simplicities sake, 1 inch for the bellcrank, to attach the flap pushrod to. then the flap horn pushrod attachment will have to be on a radious of 1.666 inches from the center of rotation.

Now that the ratio is established, I have 2 of the needed dimensions, 90 degrees at the atachment point, and the legnth of the short arm or radius legnth, to plug into the Trig so I can determine the angle of tilt. 

Since I'm using the CAD system to lay this out. I measure the distancefrom the pushrod attachment point, to the center of rotation for the horn. Trig can be used to come up with the tilt angle.

Now, realistically, the flap pushrod is some distance above the airfoil chord line, so there is a minor discrepancy between what I lay out, and what I come up with using the math. i know I just need to input a couple more points of data to correct it, but the graphic solves so much faster for me.

In any case, I draw a line tangent to the flap horn arc, to the attachment point for the flap pushrod. Another line from the tangent point to the center of rotation of the flap horn. I use the CAD dimensioning routine to measue the angles needed.

I figure you already know this information, but offer it for whatever it's might be useful to your project.

[Howard Rush]

Thanks, folks. I sorta did John's procedure with my last airplane, but I think I can come up with a quicker way to get the plots (and to calculate some weirder things), particularly considering my glacial CAD pace (it took me all day to do the attached drawing).  Here is an update incorporating some of Tim's corrections and some other changes.  I included the variable names below, because I use them on the drawing, but the spacing came out messed up.  Now I am chugging along on the program.  I did the hard part: figuring out where the bellcrank parts are in x, y, and z.  Now I'm calculating leadout and pushrod lengths at zero flap deflection. Then will come the fun part: figuring positions of the moving parts as functions of leadout travel.


Definitions:   
X = 0 at flap control horn axis   
Y = 0 at fuselage centerline   
Z = 0 at wing chord   
X is positive forward.   
Y is positive out the right wing.   
Z is positive down.   
Rotations are positive about positive axes:    
   Bellcrank rotation is positive clockwise looking from the top.
   Flap deflection is positive clockwise looking from the left wing tip.
   Elevator deflection is positive clockwise looking from the left wing tip.
   
   
Assumptions:   
Airplane flies counterclockwise (looking from the top of the circle) in upright level flight.   
Wing has no dihedral.   
Bellcrank axis is parallel to the X-Z plane.   
Bellcrank center (intersection of bellcrank axis and bellcrank plane) is on wing chord, Z = 0   
Leadouts guides are on wing chord, Z = 0   
Flap hinge line is straight and is on the Y axis: X = 0, Z = 0.   
Elevator hinge line is straight and is parallel to the Y axis.   
Elevator and flap deflections are relative to the airplane X-Y plane.   
Leadout attach points on bellcrank are colinear with bellcrank center   
All pushrod joints are ball joints; joint location is center of ball   
Forward pushrod goes from bellcrank to flap control horn.   
Aft pushrod goes from flap control horn to elevator control horn.   
Everything is rigid except leadouts, which are perfectly flexible.   


Inputs:
X of fwd leadout exit   XTipFwdLeadout
X of aft leadout exit   XTipAftLeadout
Y of leadout exit at wingtip   YTipLeadout
   
X of bellcrank center   XBellcrank
Y of bellcrank center   YBellcrank
Bellcrank-plane pitch angle, positive fwd-end high   BellcrankPitch
Bellcrank leadout-arm radius   RLeadoutArm
Bellcrank pushrod-arm radius   RBellcrankPushrodArm
Bellcrank pushrod attach elevation above and perpendicular to bellcrank plane   HBellcrankPushrodArm
Bellcrank leadout-arm rotation at zero flap deflection   DeltaBellcrank0
Bellcrank pushrod-arm rotation relative to a plane including the bellcrank axis and perpendicular to the leadout arms   DeltaBellcrankPushrodArm0
   
Flap control horn forward-pushrod-arm radius   RFlapPushrodArmFwd
Flap control horn aft-pushrod-arm radius   RFlapPushrodArmAft
Flap control horn forward-pushrod attach point rotation at zero flap angle   DeltaFlapPushrodArmFwd0
Flap control horn aft-pushrod attach point rotation at zero flap angle   DeltaFlapPushrodArmAft0
Flap control horn forward-pushrod attach point distance right of fuselage centerline   YFlapPushrodArmFwd
Flap control horn aft-pushrod attach point distance right of fuselage centerline   YFlapPushrodArmAft
   
X of elevator axis   XElevator
Z of elevator axis   ZElevator
Elevator control horn pushrod-arm radius   RElevatorPushrodArm
Elevator control horn pushrod attach point rotation at zero elevator angle   DeltaElevatorPushrodArm0
Elevator control horn pushrod attach point distance right of fuselage centerline   YElevatorPushrodArm
Elevator deflection from fuselage X axis at zero flap deflection   DeltaElevator0



   
Calculated at Zero Flap Deflection:   
Fwd leadout wire length inside wing   LFwdLeadout
Aft leadout wire length inside wing   LAftLeadout
Forward pushrod length   LPushrodFwd
Aft pushrod length   LPushrodAft

[Allan Perret]

Looks like you have 4~5° stab incidence.
 
Exaggerated ??

[Howard Rush]

Looks like you have 4~5° stab incidence.
  
Exaggerated ??

Yes.  I was hoping somebody would notice.  That's to show that I'm measuring elevator position relative to the body axis, rather than to the stabilizer chord line.  Either would have been OK, as long as which I picked was clear.

[Tim Wescott]

 Does that also mean that the bellcrank is intentionally bent, and not just an artifact of your CAD skills?

[Howard Rush]

Does that also mean that the bellcrank is intentionally bent, and not just an artifact of your CAD skills?

Yes.  I do this sometimes, particularly with airplanes with a big ratio of line diameter to airplane weight (F2D planes).  The leadouts have a lot of sweep, so I offset the pushrod attach point.  That's one of those extra inputs you were complaining about.

[Howard Rush]

I am chugging along, with the occasional hall-decking break.  The program now takes max and min flap deflections and calculates max and min leadout lengths, bellcrank deflections, and elevator deflections for a set of inputs above.  That was a little more tedious than I expected.  One interesting detour was caused by my confusion over which way +X is.  Structures guys measure from the front of the airplane back, while stability and control guys think of +X as being forward.  Although I put a big label on the axis, my long-ago structures memory caused me to put in a positive number for X for the elevator.  Had I realized sooner that I was dealing with a canard, it wouldn't have taken so long.  I think I'm almost finished.  This is fun.

[Howard Rush]

I got it working.  Now I can plot control geometry as a function of all those variables on the input sheet.  Here is an example.  I made up some input numbers, shown in attachment 3.  Attachment 1 is the plot that interests me.  For one thing, the flatter you can make the green line, the better the hinge moment per unit of differential line tension.  Attachment 2 is what folks seem to worry about.  Yes, the offset is intentional.

Next I might throw some loops on it to plot flaps vs. elevator for six different bellcrank tilts, for example. 

[Serge_Krauss]

Next I might throw some loops on it to plot flaps vs. elevator for six different bellcrank tilts, for example. 

I  would like to see this especially, perhaps for a couple or more other parameter variations. I found with the other spreadsheet, which did not allow for tilt, that I could get zero bias at a certain flap deflection (i.e. equal +/- flap deflections with equal +/- bellcrank rotations), but that there would be biases for both less and greater deflections. The bias would reverse at that value, and diverge rapidly for greater deflections. You've seen the graphs. So I've wondered what effect the tilted bellcrank would have on that. One thing I think happens is that the nonlinearities of the system sometimes cancel to some extent. At least one asymmetry in placement of the bellcrank just moved the graphed curve, and I'm wondering whether tilting might do the same; is it the same effect as raising the bellcrank?  Pretty cool, Howard!

SK

[Igor Burger]

Great, and where is the input variable for pivot position of my logarithmic unit on flaps??? 

[Howard Rush]

Great, and where is the input variable for pivot position of my logarithmic unit on flaps??? 

It's coming.  It's coming.  I have to put lights on the Christmas tree first.

[Howard Rush]

Great, and where is the input variable for pivot position of my logarithmic unit on flaps??? 

Right here on my calculating engine.  Here is a sample calculation.

[Igor Burger]

Great ... now we know what you got under the Christmas tree ... one big box of .... free time for programming :- ))))))))))))

... now I must wait till next Christmas to get such box as well ... for filling all those parameters and playing