What is aeroelastic fin flutter? Simply put, fin flutter is an effect where the physical properties of the fin material will cause an amplification feedback loop that will eventually cause large oscillations and physical failure of the fin medium. Or by textbook:

“A dynamic instability associated with the interaction of aerodynamic, elastic and inertial forces.” [1]

or

"Flutter of an elastic body is a self-excited vibration of that body while immersed in fluid flow. In the case of a rocket, flutter is associated with the aero-elastic characteristics of the fin/body combination while flying through the atmosphere. Flutter is typically the coupled motion of fin twist (torsion), fin plunge (bending) and possibly the rigid and/or flexible motions of the rocket body." [2]

There are a few methods of calculation differing in complexity and relying more and more on specific physical properties that are exactly defined. Much of what we work with we can give estimations on, but variances of up to 10-20% can be expected. Therefor it is recommended that a high safety margin by utilized to minimize the risk of failure.

We will be using the Flutter Boundary Equation based on a technical report on NASA's research server [3].

The equation is as follows [1] [3]:

We will step by step go through the calculations necessary, please see further down the post or look at [1] and [3] for more details on this formula, but know that :

• G = materials shear modulus (strength of material in PSI)
• P = Atmospheric pressure, see [4] (in PSI)

We will calculate the fin flutter for my last rocket which utilized a fin that was:

• Root chord: 9" (length of fin on body)
• Sweep: 5" (distance of tip of fin from start of fin)
• Tip chord: 0" (came to a point at end of fin)
• Thickness: 0.25"
• Semi-span: 3" (distance furthest from the rocket body)
• Shear Modulus: 2,020,000 PSI (derived from reference from Finsim software [5])
• Altitude: 2,000ft (Altitude of max velocity, not apogee of rocket)

We can now calculate the fin surface area:

The fin aspect ratio:

The ratio of tip to root chord:

Temperature at altitude (see [4]):

Speed of sound at altitude:

To make calculations a bit easier we will now brake apart the original equation into three parts. (NACA TN 4197, EQ 18):

The actual flutter velocity will now be:

We can now convert this to mach by:

These stubby little fins are actually quite strong due to the limiting sub span length. If we increase these fins from 3" to 4" we get the following:

If we add a 25% safety margin we reduce this down to mach 2.496.

It needs to be taken into consideration that materials properties are much more exact in metals, less so in composites, and even less so in wood. This variance can easily cause huge discrepancies.

One can also automatically calculate these values utilizing more exact computational fluid dynamic simulations on software known as "Finsim" [5] I urge those interested to sign up for the software and enjoy it.

Any questions are welcome, hope this illuminated some of the math in rocketry for you.

[1] https://www.apogeerockets.com/education/downloads/Newsletter291.pdf

[2] https://drive.google.com/file/d/0ByUEkb0cIAHncUN4ejhEZWtsRWc/view

[3] - http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930085030.pdf

[4] http://www.grc.nasa.gov/WWW/K-12/airplane/atmos.html

[5] http://www.aerorocket.com/finsim.html