Purchase FinSimTM
AeroFinSim 4.5 NEW
Fin Flutter and Loads Analysis Software
Includes Spin Stabilization & Unsteady Torsion-Flexure Flutter

Copyright 1999-2017 John Cipolla/AeroRocket & WarpMetrics

The United States Air Force uses FinSim 4.0
FinSim used to model the CYA-100 True Angle of Attack Display System
Flutter velocity for the 2nd stage fins of the Quantum Leap
Flutter velocity comparison with MSC/NASTRAN FEA solution
Flutter velocity of a wing for a large modern airplane
Defining composite material properties
Designing spin stabilized Rockets
Drag coefficient of spin stabilized Rockets
Fin flutter and divergence velocity for supersonic flight
(or Fin flutter analysis for the Don't Debate This rocket, the correct way)
FinSim flutter velocity compared to NACA TN 4197, FEA and wind tunnel data
Defining the thickness of an equivalent rectangular fin
Latest Modifications

| MAIN PAGE | SOFTWARE LIST | AEROTESTING | MISSION | RESUME |
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By AeroRocket/John Cipolla



2 Degree of Freedom Flutter Model
featuring bending (h) and torsion (
a) springs
vibrating around the fin elastic axis (ea)

Are Your Fins Strong Enough To Survive A Wild Flight?
If you don't know how strong your fins are, you'll either be launching an unsafe rocket, or one which is overbuilt or that won't travel as high as you'd like. Can you tell simply by looking? FinSim will predict how strong your fins are. More importantly, it tells you if they will stay attached to the rocket under the extreme loads of a wild flight. FinSim can also tell you if the fins are too sturdy, meaning they are heavier and thicker than they actually need to be. If so, rocket performance will suffer.

PLEASE NOTE: FinSim s a sophisticated aeroelastic computer program when using the optional Theodorsen and U-g methods to predict flutter velocity and divergence velocity. Because FinSim uses a spring-mass-damper model in its implementation of the Theodorsen and U-g methods, fins and wings of almost any shape including winglets can be analyzed. However, FinSim is easy to use for the untrained rocketeer because the simple default Pines approximation equation displayed on this web page provides a first order approximation for flutter velocity while the complex Theodorsen and U-g methods provide more accurate flutter and divergence velocity results for high subsonic and supersonic flights.

What is FinSim?

First, FinSim is a structural analysis program. This portion of the program determines how strong the fins are by determining the aerodynamic fin loads. You can't know if your fins are strong enough unless you know the aerodynamic forces acting on them during launch. In its structural analysis mode, FinSim looks at the material of the fins, how long the fins are, their span, how thick they are, the size of the fillets, how they are attached to the rocket (through-the-wall or butt-joint), and what type of glue is used. Using this information FinSim computes the maximum allowable bending force the fin can handle without causing fin separation. Then, using the maximum angle-of-attack the rocket will attain, FinSim quickly computes aerodynamic loading based on the geometry of the fins in terms of lift and drag. Then, the program displays the highest speed that can be tolerated before the fins will shred or separate from the rocket.

Second, FinSim is an aeroelasticity program that predicts flutter and divergence velocity for up to six sets of fins. Fin flutter and divergence are vibrations of the fin caused by the coupling of free flight aerodynamic forces with lightly damped structural modes of vibration, that can range from a slight buzzing sound to instances where the oscillations are so severe the fins are stripped off the rocket. In any case, fin flutter and divergence will create excess drag, causing the rocket to lose altitude and flight speed. FinSim will predict when flutter occurs, so you can either beef up the fins or choose a different rocket motor that limits the speed of the model. Please note that for flutter/divergence and fin stress analyses the user needs to manually enter only six fin-related variables to determine flutter velocity, divergence velocity and maximum allowable rocket velocity based on allowable material strength.

To determine flutter and divergence velocity, FinSim assumes each fin is mounted on bending and torsion springs located at the fin's elastic axis (Xea) and the aerodynamic center is located at the 1/4 chord point for subsonic flight and 1/2 chord point for supersonic flight. A critical velocity will cause either a static instability (torsional divergence) or an oscillatory instability (flutter). FinSim computes divergence velocity and flutter velocity for up to 6 fin sets. FinSim models flutter and divergence using either the simple Pine's flutter approximation method or the more complex Theodorsen method and U-g method to model high speed torsion-flexure wing and fin oscillations. When using FinSim to estimate flutter velocity using the Pine's method, one should bound the flutter prediction by using the theoretical 2D lift coefficient (2p) to establish a lower flutter boundary and the 3D lift coefficient to establish an upper flutter boundary. The true flutter velocity will fall somewhere between the two flutter boundaries.

Third, In addition to flutter and fin stress analyses, FinSim has the ability to determine the stability of spin stabilized rockets that use canted fins to achieve rocket rotation. FinSim computes the center of pressure location of a spin stabilized rocket by applying the principals of gyroscopic motion. In addition, the SpinSim routine computes precession angle, added moment coefficient due to spin stabilization, total pitch moment coefficient with spin stabilization, rotary speed, precession speed and total drag coefficient (Cd) due to spin stabilization. The SpinSim routine takes information from either mass properties CSV export files or from manually entered inputs. Both the FinSim Manual and the SpinSim Manual are included during installation and are accessible from within the program's HELP routine. Please note that for a spin-stabilized rocket the center of pressure location (XCp) should be at least one body diameter behind the center of gravity (CG). This very same stability criterion is used to define the static stability of all fin-stabilized rockets and is referred to as the static margin (XCp-Xcg). Please note the SpinSim routine is and always was a subroutine in FinSim and is not a separate computer program.

FIRST, SOME CONCEPTS ABOUT AEROELASTICITY
DIVERGENCE VELOCITY: Fin or wing divergence is an example of a steady-state aeroelastic instability. If a wing in steady flight is accidentally deformed an aerodynamic moment will generally be induced which tends to twist the fin/wing. Fin/wing twisting is resisted by the restoring elastic moment along the elastic axis (ea). However, since the elastic stiffness is independent of the flight speed, whereas the aerodynamic moment is proportional to the square of the flight speed, there may exist a critical speed, at which the elastic stiffness is barely sufficient to hold the fin in the disturbed position. Above such a critical speed, an accidental deformation of the fin/wing will lead to a large angle of twist (torsion). This critical speed is called the divergence speed, and the fin/wing is said to be torsionally divergent. Rocket fins should be designed so the divergence speed is never exceeded at any altitude during the flight.


Where,
qD = Divergence velocity, Ka  = Torsion spring stiffness
S = Fin surface area,
e = Xea - Xac. ∂CL/∂a = Fin lift slope  = CLa (2p  for 2-D fins)
Note: This equation is an approximation for subsonic divergence velocity.

FLUTTER VELOCITY: Flutter is a dynamic instability of an elastic body (wing or fin) in an airstream and like divergence the only forces necessary to produce flutter are those due to the deflection of an elastic structure from its initially un-deformed state. The flutter velocity or critical speed UF and frequency wF are defined respectively as the lowest airspeed and corresponding circular frequency at which an elastic body flying at a given atmospheric pressure and temperature will exhibit sustained harmonic oscillation. When there is no flow and the rocket's fin is disturbed, say, by a poke with a rod, oscillation or vibration occurs, which is damped (reduction of amplitude caused by structural resistance) gradually over successive vibration cycles. When the speed of flow is gradually increased, the rate of damping of the oscillation of the disturbed fin increases at first. With further increase in rocket velocity, however, a point is reached at which the damping rapidly decreases. At the critical flutter velocity, an oscillation can just maintain itself with steady amplitude. At speeds above this critical condition (UF), any small accidental disturbance of the fin from a gust of wind can serve as a trigger to initiate an oscillation of great violence that will rip the fin right off the rocket causing an unstable flight condition. Rocket fins should be designed so the flutter velocity and divergence velocity is never exceeded. Please note that no flutter velocity exists for center of gravity positions (Xcg) forward of the elastic axis (Xea) of the fin/wing. Please note the two equations presented here are an approximation based on steady flow flutter assumptions and are only valid for wa/wh > 1 and mass ratio (m) < 10. Where, wa/wh is the ratio of the natural torsion frequency to the natural bending frequency.

For a more precise analyses of the critical flutter velocity and divergence velocity use either the
Theodorsen method or U-g method located on the Torsion-Flexure (2-D) Unsteady Flutter screen.


Where, U = Flutter velocity, wa = Uncoupled torsion frequency, b = Average fin half-chord
m = Fin mass, S = Fin surface area, ra = Fin radius of gyration, e = Xea - Xac
∂CL/∂a = Fin lift slope = CLa (2p  for 2-D fins), xa = Xcg - Xea.
Note: This equation is an approximation for subsonic flutter velocity.

TORSION-FLEXURE (2-D) UNSTEADY FLUTTER ANALYSIS
The discussion in the previous section of the Pines' flutter velocity approximation (used on the main screen) is based on quasi-steady aerodynamic assumptions. Therefore, as stated in An Introduction to the Theory of Elasticity, the Pines' approximation is practical for determining flutter velocity of low speed aircraft and model rockets. However, high speed aircraft and model rockets require the linearized aerodynamic theory represented by Theodoren's function, F(k) + i G(k) and implemented on the new Torsion-Flexure (2-D) Unsteady Flutter analysis screen in FinSim 4. Simply stated, the aerodynamic forces of the linearized theory are coupled with the assumption of a two-dimensional standard airfoil, that is an airfoil having two degrees of freedom: a bending or flexure degree of freedom, h measured around the elastic axis and a pitching or torsion degree of freedom, a measured around the elastic axis of the airfoil.

INPUT VARIABLES FOR TORSION-FLEXURE (2-D) UNSTEADY FLUTTER
g = Structural damping coefficient, usually having a value between 0.005 and 0.05 for metalic structures
m = mass ratio = m/(p r b2) = (4/p) (rm/rair) (t/c) = Ratio of the mass of the wing to the mass of a cylinder of air of a diameter equal to the chord of the wing.
ah = Axis of rotation (elastic axis) location from the wing/fin center-chord = 2 Xea - 1
xa = C.G. location aft of the axis of rotation (ah) location = (2 Xcg - 1) - ah
ra = Radius of gyration about the elastic axis = SQR[Ia / (m b2)]
wa = Natural angular frequency of torsional vibration around 'a' in vacuum (rad/sec)
wh = Natural angular frequency of wing in flexure (bending) in vacuum (rad/sec)
b = Half chord, used as a reference unit length (inches)

Where:
Xcg = Center of gravity location measured from the airfoil leading edge divided by the chord length (c).
Xea = Elastic axis location measured from the airfoil leading edge divided by the chord length (c).
and
ra (radius of gyration) is made non-dimensional by dividing by b (half chord), c = chord length and t = thickness.
k =
wa b / U = Reduced frequency or Strouhal number represents the ratio of the characteristic length of the body (b) to the wave length of the disturbance. Where U is the mean speed of the flow and wa is the fundamental frequency of the wing in torsional oscillation in still air (rad/sec).


The Standard 2-D Wing Section
 
DEFINING THE THICKNESS OF AN EQUIVALENT RECTANGULAR FIN: BACK

Arbitrary cross-section, Sdesign = design area

FinSim's Rectangular fin cross-section

Planform geometry where b = fin semi-span
 

FinSim
numerically defines its fin cross-section as rectangular having fin thickness (t), chord (c) and semi-span (b). Where, fin cross-sectional area is S = c t and fin mass is m =
rmS b. For most model and high power rockets, fin thickness may be assumed constant. However, for double-wedge fins and aerodynamically shaped fins, like the example above, fin thickness inserted into FinSim is more complicated. Because fin flutter depends on torsional frequency when torsional frequency is less than bending frequency, finding an equivalent fin thickness based on the design mass of the fin will allow FinSim to model torsional fin flutter more accurately. Finding a reasonable value for fin thickness that maintains the original fin design mass is done by simply finding an equivalent rectangular fin having the same cross-sectional area and mass as the design fin. Therefore, fin thickness is simply, t = S_design/c. Where, the fin thickness (t) input into FinSim uses the design cross-sectional area (S_design) and design chord (c) to maintain the same mass as the original design. If fin thickness varies linearly in the span direction, use an average fin thickness in the span direction that maintains the original design mass. For information purposes, torsional frequency, fa = SQRT(Ka/Ia) where Ka is the torsional stiffness and Ia is the mass moment of inertia.
 
FINSIM FEATURES
1) Determine fin flutter critical velocity (UF) and fin divergence critical velocity (UD) using the Pines' approximate method.
2) Define aerodynamic loads using either the 3-dimensional Barrowman lift-slope (CN_alpha) or the 2-dimensional lift slope (CN_alpha).
3) Define up to six fin-sets using only five variables to define fin geometry.
4) Easily define rocket angle of attack, flight altitude, fin fillet radius and butt-joint or thru-the-wall fin mounts using simple options buttons.
5) Specify from a list of 25 standard and composite materials or manually enter modulus of elasticity, material density, poissons ratio and bending yield strength.
6) Specify from a list of 12 common adhesives or manually enter the adhesive allowable strength.
7) Fin allowable and adhesive allowable is displayed for comparison purposes.
8) Plot fin stress verses rocket velocity and see the maximum allowable velocity as limited by either the fin material or adhesive material strength.
9) Plot each fin set by simply clicking one of up to 6 fin-set option buttons.
10) By clicking SHOW or HIDE in the Additional Results menu in the toolbar, display Stress Concentration Factor due to fillets, Torsional Frequency, Bending Frequency, Fin-Tip Deflection and Maximum Fin Bending Moment.
11) Determine the stability margin (XCp-XCG) of spin stabilized rockets using canted fins to achieve rotational velocity.
12) FinSim instructions and SpinSim instructions are included with purchase and are accessible from within the program's HELP routine.
13) Specify a title on the main screen to differentiate between the various input data files. NOTE: Fin flutter and stress analysis files have the .FIN specification
14) Use the Classical 2-D Lift Slope, Barrowman 3-D Lift Slope or the new Supersonic Airfoil Lift Slope to define fin loads for flutter and stress analyses.
15) Location of the aerodynamic center (A.C.) automatically changes to the 25% chord length position for subsonic airfoils (Classical 2-D Lift Slope and Barrowman 3-D Lift Slope) and automatically changes to the 50% chord length position for supersonic airfoils (Supersonic Airfoil Lift Slope). Mach number is inserted or modified in the STRESS routine.
 
NEW FinSim 4.0 Features
16) Added the ability to model
unsteady torsion-flexure wing oscillations using the Theodorsen method or U-g method to determine critical flutter velocity and divergence velocity. Also, included six test cases with reference pages.
17) Manually enter experimentally derived aeroelastic data on the Theodorsen and U-g method screen.
18) Increased the altitude corresponding to atmospheric density and pressure from 10K feet to 50K feet greatly increasing the atmospheric affect on flutter and divergence velocity.
19) In the Additional Results section on the main screen added output of material properties including Modulus of elasticity (E), Shear modulus (G), Poissons ratio, and Material density (r) in addition to the uncoupled bending frequency (wh) and torsion frequency (wa) of fin/wing vibration.
20) Improved accuracy of the Pines' approximate method for determining critical flutter and divergence velocity.
21) Fixed a few errors in the material properties data base, specifically the Polystyrene material.
22) Save 1/k, F(k), G(k), X1r(k), X2r(k), X1i(k), X2i(k) for the SQR(X) verses 1/k analysis to a CSV file. Also, Save k, F(k), G(k), UF(k), g(k) for the U verses g analysis to a CSV file for later use in Excel or other spreadsheet programs. The Theodorsen aerodynamic function is F(k) + i G(k).
 
NEW FinSim 4.5 Features
23) For the User Defined Materials option in the Fin Materials pull down menu, added the option to save user defined modulus of elasticity (E), density (r), poissons ratio (n) and bending yield strength in material files having an .MAT extension. New material data can be opened or saved in the Fin Materials section by selecting File and then Open New Material in MAT Format or Save New Material in MAT Format and then clicking Insert New Data.
24) Increased altitude corresponding to atmospheric density and pressure from 50K feet to 200K feet for stress analyses, flutter velocity and divergence velocity.
25) Increased numerical accuracy and speed of the flutter velocity and divergence velocity calculations.
26) Under the CN-Alpha pull-down menu, the option Supersonic Airfoil Lift Slope has been replaced by a method from NACA TN 4197 (see description) for determining main screen flutter velocity and divergence velocity. The new flutter velocity and divergence velocity method is called the NACA TN 4197 METHOD and is located in the CN-alpha pull-down menu.

NOW, SOME FINSIM TEST CASES

Please Note:
Click the icon located on the main FinSim Flutter screen to access the new Torsion-Flexure (2-D) Unsteady Flutter analysis screen for modeling unsteady torsion-flexure wing oscillations using the Theodorsen method or U-g method to determine flutter and divergence velocity.


ToolBar located on the main screen

(1) FLUTTER VELOCITY FOR THE 2ND STAGE FINS OF THE QUANTUM LEAP: BACK
The following FinSim analysis predicts flutter and divergence velocity for the second stage fins of the PML Quantum Leap. This FinSim unsteady Torsion-Flexure flutter analysis indicates the Quantum Leaps' second stage fins will flutter at approximately 0.76 Mach (see Figure-4) and become fully divergent at 0.96 Mach. In-flight video seems to indicate the second stage fins of the Quantum Leap will flutter at 0.90 Mach when fiber glassed. The FinSim critical flutter velocity result of 0.76 Mach defines the earliest possible onset of flutter when the oscillations can just maintain themselves at small steady amplitude and the divergence velocity of 0.96 Mach completely bounds the observed result. Above the critical flutter velocity any accidental disturbance can initiate oscillations of great amplitude. Therefore, the large oscillations observed at 0.90 Mach were probably triggered by an accidental disturbance of the airflow (gust of wind?) after exceeding the critical flutter velocity, explaining why the oscillations were observed at 0.90 Mach although flutter may have been occurring earlier in the flight as predicted by FinSim's result.


Figure-1: Main FinSim analysis screen displaying the Pines' approximate flutter results (Windows 10 image displayed)



Figure-2: FinSim Input geometry screen



Figure-3: FinSim fin stress analysis screen

Click the icon located on the main FinSim toolbar to access the new Torsion-Flexure (2-D) Unsteady Flutter analysis

Figure-4: Flutter and divergence velocity using the U-g method


(2) FLUTTER
VELOCITY COMPARISON WITH MSC/NASTRAN SOLUTION: BACK
The following is an unsteady Torsion-Flexure flutter validation of an airfoil mounted on bending and torsion springs located aft of the aerodynamic center of a fin or wing. A critical velocity will be found that will cause either a static instability (torsion divergence) or an oscillatory instability (flutter). Both divergence and flutter speeds of the airfoil are determined and compared to exact theory and a separate MSC/NASTRAN finite element analysis (FEA) technique using the K-method based on the exact Theodorsen function. The following table illustrates the usefulness of FinSim for accurately determining critical flutter and divergence velocity of typical cruciform model rocket fins. This example uses the Theodorsen and U-g methods to predict flutter and divergence velocity as described in NACA Report 685, Mechanism of Flutter by Theodorsen and Garrick on page 542 of the report. Comparison between FinSim results and the paper's results are excellent.

Fin Aeroelastic Data
g
(structural damping) = 0.0
m (mass ratio) = 20.0
a
(elastic axis location) = -0.2
xa (c.g. location) = 0.1
r
a (radius of gyration) = 0.5
wa (torsion frequency, rad/sec) = 25
wh (bending frequency, rad/sec) = 10
b
(half chord, inches) = 36.0

TORSION-FLEXURE AIRFOIL FLUTTER AND DIVERGENCE VELOCITY VALIDATION

Results

Flutter Velocity (UF)

 Difference

Divergence Velocity (VD)

 Difference

Exact Theory

 169 ft/sec

 -

 216 ft/sec

 -

MSC/NASTRAN

 166 ft/sec

 -1.8%

 216 ft/sec

 + 0.0%

FinSim 4.0

 166 ft/sec

 -1.8%

 217 ft/sec

 + 0.5%

Click the icon located on the main FinSim toolbar to access the new Torsion-Flexure (2-D) Unsteady Flutter analysis

Figure-5: Flutter and divergence velocity using the U-g method

Click the icon located on the main FinSim toolbar to access the new Torsion-Flexure (2-D) Unsteady Flutter analysis

Figure-6: Flutter velocity using the Theodorsen method (SQR(X) verses 1/k)


(3) FLUTTER VELOCITY OF A WING FOR A LARGE MODERN AIRPLANE: BACK

This example uses the U-g method to predict critical flutter velocity for a wing described in NACA Report 685 as being for a large modern airplane. Please reference, NACA Report 685 Mechanism of Flutter by Theodorsen and Garrick on page 108 of the report. The parameters supplied by the report are as follows. Comparison between FinSim results and the report's results are excellent.

Fin Aeroelastic Data
g (structural damping) = 0.0
m (mass ratio) = 4.0
a
(elastic axis location) = -0.4
xa (c.g. location) = 0.2
r
a (radius of gyration) = 0.5.
wa (torsion frequency, rad/sec) = 90
wh (bending frequency, rad/sec) = 22.5
b
(half chord, inches) = 72.0

UNSTEADY TORSION-FLEXURE FLUTTER VALIDATION

Results

Flutter Velocity (UF)

 Difference

SQR(X) Difference 1/k Difference

NACA Report 685, page 108

 567.0 mph

-

1.594 - 2.460 -

FinSim 4.0

 568.65 mph

+0.29%

1.592 -.0125% 2.460 0.0%

Click the icon located on the main FinSim toolbar to access the new Torsion-Flexure (2-D) Unsteady Flutter analysis

Figure-7: Flutter and divergence velocity using the U-g method

 

FINSIM USES THE FLUTTER VELOCITY METHOD FROM NACA TN 4197 (FINSIM 4.5.3 OR LATER): BACK
Under the CN-Alpha pull-down menu, the option Supersonic Airfoil Lift Slope has been replaced by Equation-18 from NACA TN 4197 for determining main screen flutter velocity and divergence velocity. The Classical 2-D Lift Slope and Barrowman 3-D Lift Slope methods are not affected and are very accurate over a wide range of Mach numbers. This modification was necessary to more accurately estimate high subsonic and supersonic flutter and divergence velocity. The Theodorsen and U-g flutter velocity results are not affected and are extremely reliable. An extensive investigation represented by the following table (Table-1) and plot was conducted to validate the method from NACA TN 4197. Because the results from NACA TN 4197 reasonably matched the Theodorsen and U-g flutter velocity methods, the decision was made to replace the Supersonic Airfoil Lift Slope option. The new flutter velocity and divergence velocity method is called the NACA TN 4197 METHOD and is located in the CN-alpha pull-down menu. In the illustration below please note that FinSim_Ug has excellent correlation between results using NASTRAN FEA and actual wind tunnel measurement (ASROC Test-38).

One by product of this effort has been the development of a divergence velocity estimate based on methods presented in NACA TN 4197. The AeroRocket derived expression for divergence velocity (UD) and the expression presented in NACA TN 4197 for flutter velocity (UF) appears in FinSim as a new option to compute divergence velocity. The discussion below explains in more detail how the new expression for divergence velocity was derived. This new equation for divergence velocity will appear in future versions of AeroRockets Excel flutter velocity and divergence velocity spreadsheet, NACA TN 4197 (33.8 KB).

FinSim 4.5
Project Name
Aspect Ratio
(AR)
MF
NACA TN 4197
MF
POF-291
MF
FinSim Ug
MF
FinSim Main
CN-alpha
Main Method
Rocket
Altitude
Velocity
UNITS
E.A. C.G. Fin Materials
 NASTRAN EXAMPLE 0.167 0.604 0.855 0.15 (0.15*) 0.24 Classical 2-D Sea Level MACH 0.50 0.50 Steel
ASROC Flutter (Test 38) 0.446 1.319 1.866 1.20 (1.3**) 1.33 NACA TN 4197 29K FT MACH 0.50 0.694 Aluminum
N5800 Project (Don't debate This) 0.561 3.891 5.504 3.28 3.89 NACA TN 4197 11K FT MACH 0.50 0.658 Aluminum
Apogee Test 0.704 0.504 0.713 0.48 0.52 Classical 2-D 3K FT MACH 0.50 0.50 User Defined Balsa Wood
Tomahawk Rocket (2.933 span) 0.875 1.493 2.112 0.93 1.49 NACA TN 4197 1K FT MACH 0.50 0.50 User Defined Balsa Wood
Quantum Leap Flutter 0.972 0.437 0.618 0.38 0.44 Classical 2-D 15K FT MACH 0.50 0.64 G-10 Fiberglass
Tomahawk Rocket (5" span) 1.492 0.739 1.045 0.62 0.74 NACA TN 4197 1K FT MACH 0.50 0.50 User Defined Balsa Wood
Table-1, FinSim input data and output flutter velocity results compared to supplementary information. * NASTRAN FEA flutter result, ** Wind tunnel test result.


FinSim flutter velocity compared to NACA TN 4197, POF-291 and NASTRAN FEA/experimental data.

Discussion of flutter velocity from NACA TN 4197 and derivation of the new AeroRocket equation for divergence velocity.


DEFINING COMPOSITE MATERIAL PROPERTIES IN FINSIM: BACK
FinSim has a built-in library of standard materials, adhesive materials and composite materials. The list of materials are located in FinSim_Theory.pdf, which tabulates E, r, n and S for the provided materials. Also, user defined materials may be specified in the Fin Materials pull-down menu where modulus of elasticity, material density, Poissons ratio and yield strength in bending can be inserted into the analysis. However, occasionally the user may need to determine composite material properties based on specific properties of the fiber and matrix materials used for fin construction. In this case the Rule of Mixtures or the volume fraction rule is used to estimate composite material properties as a function of the fiber and matrix constituents and their volume fractions. The volume fraction rule can be stated as follows. The modulus of elasticity of a composite material equals the modulus of elasticity of the first phase times the volume fraction of that phase plus the modulus of elasticity of the second phase times the volume fraction of that phase. A similar rule applies to the other properties of a composite material. Using the Rule of Mixtures the relationship for longitudinal modulus of elasticity (E), material density (r), Poissons ratio (n) and allowable stress (S) are specified as follows:

Composite material modulus of elasticity
E = f Ef + (1-f) Em
Which says that the longitudinal modulus of elasticity (E) is proportional to the volume fraction of the fiber material (f) and the volume fraction of the matrix material (1-f).

Composite material density
r = f rf + (1-f) rm

Composite material Poissons ratio
n = f nf + (1-f) nm

Composite material allowable stress
S = f Sf + (1-f) Sm

NEW DOWNLOAD: To illustrate how to define E, r, n and S for FinSim, a new Microsoft Excel spreadsheet based on the Rule of Mixtures (49 KB) is provided as a free download. The user simply inserts results from the spreadsheet into the User Defined Material section of the Fin Materials pull-down menu on the FinSim main screen. The example illustrated in the spreadsheet is for the construction of Glass Fiber Reinforced Plastic (GFRP) fins that use epoxy for the matrix material and fiberglass for the fiber material. This spreadsheet analysis uses GFRP an example so it is recommended the user consult other sources for more exotic constituent material properties.


DESIGNING SPIN STABILIZED ROCKETS (SPINSIM): BACK
In addition to flutter and fin stress analyses, FinSim has the ability to determine the stability of spin stabilized rockets that use canted fins to achieve rotation. FinSim computes the center of pressure location of a spin stabilized rocket by applying the principals of gyroscopic motion. In addition, the SpinSim routine computes precession angle, added moment coefficient due to spin stabilization, total pitch moment coefficient with spin stabilization, rotary speed, precession speed and total drag coefficient (Cd) due to spin stabilization. The SpinSim routine takes information from either mass properties CSV export files or from manually entered inputs. Both the FinSim Manual and the SpinSim Manual are included during installation and are accessible from within the program's HELP routine. Please note that for a spin-stabilized rocket the center of pressure location (XCp) should be at least one body diameter behind the center of gravity (CG). This very same stability criterion is used to define the static stability of  all fin-stabilized rockets and is referred to as the static margin (XCp-Xcg).

SpinSim uses fins, fixed at a constant angle of inclination, to induce rotation during flight. Spin stabilization is achieved when external aerodynamic forces change the rocket's angular momentum, L in time dt by an amount, dL. During this time interval the aerodynamic forces applied at the center of pressure (Cp), exert a restoring torque given as M = dL/dt around the center of gravity. The incremental moment caused by the restoring torque moves the effective Cp aft by an amount determined by the separation of the Cg and Cp and the value of the incremental moment. For more information about the technical aspects of spin stabilization and a step-by-step procedure please read the Spin Stabilization pdf instructions.
 

GENERAL PROCEDURE TO RUN SPINSIM
To generate the SpinSim results illustrated in Figure-8 use the following general procedure.

1) To use SpinSim the rocket's fin geometry must first be defined. On the main FinSim screen click the left icon with ToolBar description, Open FinSim Project in FIN format. For this example click Saturn Rocket.FIN which is a file that describes the fin geometry for Apogee's spin stabilized Saturn model rocket.


ToolBar located on the main screen

2) Click the forth icon from the left with ToolBar description, SpinSim - Spin stabilization analysis to enter the SpinSim analysis screen. Then, under File click Open SpinSim data in TXT format. For this example click Saturn Spin Data-1.TXT to open a file that further describes the spin stabilized characteristics of Apogee's spin stabilized Saturn model rocket. Required data for a SpinSim analysis is summarized in the Basic SpinSim Data and Rocket Properties sections. Most of the information required for a SpinSim analysis is pretty simple to determine except for Radial moment of inertia (Ixx) and Longitudinal moment of inertia (Izz) which are rather complicated. The rocketeer has two options to determine Ixx and Izz. The most accurate way to determine Ixx and Izz is to use a torsional pendulum as detailed in the Sprint Assembly Instructions report (no longer available) describing how John Cipolla used a torsional pendulum to measure Ixx and Izz for his exact scale Sprint ABM model rocket. The other method involves using one of the commercially available model rocket simulation programs that produce values for Ixx and Izz that may or may not be accurate enough for a typical SpinSim analysis. In this author's opinion these programs were not accurate enough to determine Ixx and Izz for the Saturn rocket and Sprint ABM spin stabilization analyses.


Figure-8: SpinSim Spin Stabilization Screen


DRAG COEFFICIENT OF SPIN STABILIZED ROCKETS: BACK

SpinSim
determines the zero lift drag coefficient (Cd) for finless projectiles based on experimental results from Fluid Dynamic Drag, page 3-13. As defined in SpinSim drag coefficient is a function of circumferential velocity ratio (u/V). Where, projectile circumferential velocity (u) around an axis parallel to the direction of flight and projectile speed (V) define the velocity ratio.

Projectile rotation has the following influences on finless projectile aerodynamics.
a) Projectile rotation causes additional drag because the added speed component, u thickens the boundary layer. Where, u =
p D nx and nx is the speed of projectile rotation (rev/sec).
b) The thickened boundary layer causes flow to separate from the aft end of the projectile causing additional form drag.
c) Centrifugal forces in the rotating boundary layer around the projectile increase base flow separation and base drag. Boat tail designs can help mitigate this situation.
d) Depending on projectile shape the critical Reynolds number may be reduced decreasing the velocity of transition from laminar to turbulent flow.

The equation displayed in Figure-9 is a curve fit approximation for Cd verses u/V based on experimental data for a finless projectile. Where, Cd0 is finless projectile drag coefficient for zero rotation and u/V is circumferential velocity ratio. SpinSim uses this equation to determine the total drag coefficient of a finned rotating projectile that includes canted fins and base drag. Figure-10 displays experimental data from Fluid Dynamic Drag (red dots) plotted verses the curve fit equation displayed in Figure-9 over a very wide range of velocity ratio (u/V).

Curve fit approximation
Figure-9: Curve fit approximation for Cd verses u/V based on Fluid Dynamic Drag experimental data

Curve fit plot verses data

Figure-10: Drag coefficient (Cd) of spin stabilized projectiles as a function of circumferential velocity ratio (u/V)

 

MODIFICATIONS AND REVISIONS: BACK
FinSim 4.5.3 Features (10/26/2014)

1) Under the CN-Alpha pull-down menu, replaced Supersonic Airfoil Lift Slope with the NACA TN 7149 Method for determining main screen flutter velocity and divergence velocity. The existing Classical 2-D Lift Slope and Barrowman 3-D Lift Slope methods were not affected. This modification was necessary to more accurately estimate supersonic flutter and divergence velocity. The Theodorsen and U-g flutter velocity results were not affected.
2) When performing a Fin Bending Stress Analysis, the flutter velocity computed on the main screen is now inserted into the Maximum rocket velocity input box in selected units of FT/SEC, MPH, M/SEC or MACH. Previously, a fixed velocity was inserted which was confusing when MACH units were also selected. In addition, fixed an error which caused confusion when the displayed Lift Slope (CN-a) on the main screen would change by simply changing the Maximum rocket velocity when the Supersonic Airfoil lift Slope option was selected. The Theodorsen and U-g flutter velocity results were not affected.
3) By clicking Additional Results and then SHOW, FinSim now displays local speed of sound for the selected flight altitude. Speed of sound information is necessary for being able to double check results when changing units.
4) Double checked that fin bending stress, fin deflection, bending frequency and torsion frequency were correct by using MathCAD for several test cases.

FinSim 4.5.2 Features (10/23/2014)

1) For the User Defined Materials option in the Fin Materials pull down menu, added the option to save user defined modulus of elasticity (E), density (r), poissons ratio (n) and bending yield strength in material files having an .MAT extension. New material data can be opened or saved in the Fin Materials section by selecting File and then Open New Material in MAT Format or Save New Material in MAT Format and then clicking Insert New Data.
2) Fixed an error that occurred in the Fin Bending Stress Analysis section when velocity in Mach units and Supersonic Airfoil Lift Slope (CLa) caused a feedback situation with the Pines method flutter analysis on the main screen causing flutter velocity results and stress analysis results to be totally inaccurate. The Theodorsen and U-g flutter velocity results were not affected.


FinSim 4.5.1 Features (03/27/2013)

1)
Increased altitude corresponding to atmospheric density and pressure from 50K feet to 200K feet for stress analyses, flutter velocity and divergence velocity.
2) Increased numerical accuracy and speed of the flutter velocity and divergence velocity calculations.


FinSim 4.0.3 Features (10/05/2009)
1) At a purchasers request added display of fin force due to flight angle of attack and canted fins for each fin-set selected in the Fin Stress plot. Also, added display of fin moment due to angle of attack and canted fins for each fin-set selected in the Fin Stress plot. Fin forces and fin moments were required for designing a rocket-motor spin stabilized rocket. The new display of fin forces and fin moments are located on the Fin bending stress analysis screen and are accessed by clicking Additional Results then selecting SHOW.

FinSim 4.0.2 Features (09/14/2009)

1) For FinSim, fixed all input data text boxes for 32 bit and 64 bit Windows Vista. When operating earlier versions of FinSim in Windows Vista the input data text boxes failed to show their borders making it difficult to separate each input data field from adjacent input data fields.

FinSim 4.0.1 Features
1) Added the ability to model unsteady torsion-flexure wing oscillations for determining flutter and divergence velocity.
2)
Increased the altitude corresponding to atmospheric density and pressure from 10K feet to 50K feet.
3)
In the Additional Results section added output of material properties in addition to the uncoupled bending frequency and torsion frequency.
4) Improved accuracy of the Pines' approximate method for determining critical flutter and divergence velocity.
5) Fixed a few errors in the material properties data base specifically the Polystyrene material.
6) Save SQR(X) verses 1/k, U verses g and the Theodorsen aerodynamic coefficients to a CSV file.

FinSim 3.1 Features
1) Included a Supersonic Airfoil Lift Slope for the analysis of fins in supersonic flow (above Mach 1).
2) Corrected an error in the computation of fin bending frequency.
3) Added a colorful and more descriptive illustration describing the aerodynamic center (A.C.), elastic axis (E.A.) and center of gravity (C.G.).


FinSim 3.0 Features
1) Made FinSim a stand alone computer program that no longer relies on output files (either XML or CSV) from other flight simulation programs.
2) Displayed several variables that are useful for aeroelastic evaluations.

 

AeroFinSim Minimum System Requirements
(1) Screen resolution: 1024 X 768
(2) System: Windows 98, XP, Vista, Windows 7/8 (32 bit and 64 bit)
(3) Processor Speed: Pentium 3 or 4
(4) Memory: 64 MB RAM
(5) English (United States) Language

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