Bob DalyApril 11, 2020, 10:20pm
When researching a question regarding flutter and Vne I came across this:
https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/ADA955270.xhtml
The report contains a method for determining flutter resistance based on a wing torsional flexibility factor compared with the design dive speed.
The wing torsional flexibility factor, F, should be less than 200/V2D.
Where F = ∫ Θi C2 ds and
𝜃𝑖 = Wing twist at station 𝑖, per unit torsional moment applied at a wing station outboard of the end of the aileron, (radians/ft-lb)
𝐶𝑖 = Wing chord length at station 𝑖, (ft)
𝑑𝑠 = Increment of span (ft)
𝑉𝑑 = Design dive speed (IAS)
The report describes "applying a torsional couple" near the wing tip and measuring the wing twist to find Θi at several locations along the span. Then preparing a table for calculating F. This is an attempt to do this for the Minimax mathematically.
Because the wing is constrained in torsion by the struts, I chose to calculate F for the 75" (6.25') cantilever section.
For the torsional couple I chose a 200 lbs force pushing up on the end of the main spar and a simultaneous 200 lbs down force on the rear spar end. Then the torque on the wing is:
Tw = 200 lbs x 28" between the spars / 12 inches per foot = 467 ft-lbs
To get the wing twist, Θ, we need to apply the formula for beam deflection:
y = P L3 / 3EI - (P L2 / 2EI) x + (P/6EI) x3
Where P is the 200 lb end force
L is the length of the beam, 6.25'
E is the modulus of elasticity, 1460 kpsi for western white pine.
I is the moment of inertia for the "C" channel spar cross section
Here are some helpful web calculators: https://amesweb.info/Beam/cantilever-beam-with-point-load-at-free-end-calculator.aspx
https://calcresource.com/moment-of-inertia-channel.html
Since the report method specifies the chord and span increment in feet, we'll be careful to calculate I in ft4 and convert E to lbs/ft2. Alternatively, we could work in inches and factor the result by 0.007 (1/144) ft2 per in2.
With the known values and calculated moments of inertia the beam deflection formulas become:
yfront = 0.1306 - 0.0313 x + 0.000267 x3
yrear = 0.2210 - 0.0530 x + 0.000453 x3
The wing twist in radians is the sum of the spar deflections divided by the distance between the spars, then combining like terms of the two deflection formulas into a single formula and dividing each term by 2.33 feet, the twist at any point x inboard from the tip is:
Θ(x) = 0.15 - 0.0362 x + 0.00031 x3
and dividing by the wing torsion, Tw, to get the twist per unit torsion:
Θ(x) = 0.00032 - 0.0000775 x + 0.000000664 x3
Then F = 4.52 0∫6.25 (0.00032 - 0.0000775 x + 0.000000664 x3) dx
Integrating: F = 20.25( 0.00032 x - 0.00003875 x2 + 0.000000166 x4 ) ]06.25
F = 0.015
Then VDmax = √(200/F) = 115 mph and Vne = 104 mph (90% VD)
Note that this result only takes into account the beam stiffness of the two spars. It doesn't account for the additional beam stiffness nor the torsional stiffness supplied by the main spar 'D' tube.
https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/ADA955270.xhtml
The report contains a method for determining flutter resistance based on a wing torsional flexibility factor compared with the design dive speed.
The wing torsional flexibility factor, F, should be less than 200/V2D.
Where F = ∫ Θi C2 ds and
𝜃𝑖 = Wing twist at station 𝑖, per unit torsional moment applied at a wing station outboard of the end of the aileron, (radians/ft-lb)
𝐶𝑖 = Wing chord length at station 𝑖, (ft)
𝑑𝑠 = Increment of span (ft)
𝑉𝑑 = Design dive speed (IAS)
The report describes "applying a torsional couple" near the wing tip and measuring the wing twist to find Θi at several locations along the span. Then preparing a table for calculating F. This is an attempt to do this for the Minimax mathematically.
Because the wing is constrained in torsion by the struts, I chose to calculate F for the 75" (6.25') cantilever section.
For the torsional couple I chose a 200 lbs force pushing up on the end of the main spar and a simultaneous 200 lbs down force on the rear spar end. Then the torque on the wing is:
Tw = 200 lbs x 28" between the spars / 12 inches per foot = 467 ft-lbs
To get the wing twist, Θ, we need to apply the formula for beam deflection:
y = P L3 / 3EI - (P L2 / 2EI) x + (P/6EI) x3
Where P is the 200 lb end force
L is the length of the beam, 6.25'
E is the modulus of elasticity, 1460 kpsi for western white pine.
I is the moment of inertia for the "C" channel spar cross section
Here are some helpful web calculators: https://amesweb.info/Beam/cantilever-beam-with-point-load-at-free-end-calculator.aspx
https://calcresource.com/moment-of-inertia-channel.html
Since the report method specifies the chord and span increment in feet, we'll be careful to calculate I in ft4 and convert E to lbs/ft2. Alternatively, we could work in inches and factor the result by 0.007 (1/144) ft2 per in2.
With the known values and calculated moments of inertia the beam deflection formulas become:
yfront = 0.1306 - 0.0313 x + 0.000267 x3
yrear = 0.2210 - 0.0530 x + 0.000453 x3
The wing twist in radians is the sum of the spar deflections divided by the distance between the spars, then combining like terms of the two deflection formulas into a single formula and dividing each term by 2.33 feet, the twist at any point x inboard from the tip is:
Θ(x) = 0.15 - 0.0362 x + 0.00031 x3
and dividing by the wing torsion, Tw, to get the twist per unit torsion:
Θ(x) = 0.00032 - 0.0000775 x + 0.000000664 x3
Then F = 4.52 0∫6.25 (0.00032 - 0.0000775 x + 0.000000664 x3) dx
Integrating: F = 20.25( 0.00032 x - 0.00003875 x2 + 0.000000166 x4 ) ]06.25
F = 0.015
Then VDmax = √(200/F) = 115 mph and Vne = 104 mph (90% VD)
Note that this result only takes into account the beam stiffness of the two spars. It doesn't account for the additional beam stiffness nor the torsional stiffness supplied by the main spar 'D' tube.