Appendix 1
e. General Procedures for Control System Modes. The experimental determination of control surface and tab rotation modes about their hinge lines may be difficult due to inherent friction within the system or the masking of these modes by structural interaction. On this basis, extra care is required for proper identification of the system's characteristics.
For conventional aileron or elevator Systems, the rotation modes may be successfully measured by applying a single excitation force to either the righthand or lefthand surface. However, multiple shakers are preferred, particularly if the right and left surfaces are operated from separate control systems. Likely shaker positions are on the trailing edge at midspan or on the horn leading edge. Tab rotation may be determined from the control surface excitation but usually a direct excitation on the tab surface is required with the control surface (aileron, elevator, or rudder) blocked to its main surface.
A transducer placed on the control being excited is used to monitor the response and determine peak frequencies by the same technique described for airframe modes. To define the modes excited, it is generally necessary to follow any or all of the following procedures:
(1) Monitor the phase between the right and left surface, the control column, or the attaching structure.
(2) Conduct a detailed survey of the surface, spanwise and chordwise, to define any structural modes. If the surface has a very long span or wide chord, these modes) bending and torsion, are likely to be dominant.
(3) Visually monitor the surface under excitation.
(4) Simple rationalization to distinguish the excited modes from previously defined airframe modes.
In the performance of these tests, the shakers and/or transducers may contribute sufficient weight to the surface being tested to significantly affect the frequency of the surface. This is particularly true for tabs with very small mass and rotational inertias. Dunkerly's equation, presented in reference 8, provides an acceptable method for correcting the measured frequency to the true surface frequency.
A check on experimentally determined modes may be facilitated by calculating rotational frequencies from measured inertias and system stiffness properties obtained from static tests.
For extremely light weight structures, another method that may be used to eliminate the shaker influence is to use an air shaker or other device which does not directly attach to the control surface or tab.
Appendix 1
f. Control Surface Rotation. Symmetric aileron rotation, the normal opposed operational mode, with control stick fixed or free, is defined as the peak frequency at which both ailerons are rotating in phase.
Antisymmetric rotation, the normal operation mode, generally has zero or very low stiffness.
Rudder rotation in the normal operation mode with pedals free occurs when the rudder and pedals are out of phase.
Elevator and all movable tailplane rotation modes should be determined with the pilot's controls fixed and free. Elevator rotation with the stick fixed is defined as the peak frequency at which both elevators are in phase for symmetric rotation, and out of phase for antisymmetric rotation. For all moving tailplanes or elevators with stability augmentation systems (control column bob weights and down springs), normal opposed operation with stick free will occur when the control stick and elevator are responding out of phase.
The effect of variations in control cable tension should be investigated.
g. Tab Rotation. Rotational modes for irreversible trim and servo tabs are determined experimentally to supplement the calculated frequency obtained from measured stiffness in the free play tests. Tab rotation frequency will usually vary with angular deflection and is determined at maximum trailing edge up1 neutral, and maximum trailing edge down positions to determine the range of tab frequencies. For geared tabs, the rotation frequency is usually determined with the control surface at maximum deflections and at neutral.
Large tabs, either wide chord or very long with a single actuator, often tend to be difficult to measure in a resonance test. Wide chord tabs often become significantly involved with "plate modes" of their carrying surfaces, while long narrow tabs may have their lowest frequency in a torsional mode rather than rotation. On this basis, it may be necessary to survey each response frequency rather extensively to properly define each mode.
Test requirements for spring tabs are dependent upon the tab control system design. In general, the following tests should be conducted to provide the required data for a mathematical representation of a spring tab system.
(These tests are similar to those discussed in the previous paragraph for all moving tailplane systems.)
Appendix 1
(1)For a preloading spring, tests should be performed for several amplitudes including complete removal of the preload, if practical.
(2) Frequency of the control surface, with tab locked to surface and pilot's control column blocked, against the elastic restraint of the control system. A stick fixed mode.
(3) Frequency of the control column) with the control surface locked to its main surface, against the elastic restraint of the control system. A stick free mode.
(4) Frequency of the tab, with the control system cables disconnected and the control surface blocked to its supporting structure, against the elastic restraint of the springs in the tab system.
Spring loaded tabs are non-linear systems which are usually quite sensitive to small parameter changes making the design of these Systems to preclude flutter most difficult. It is advisable to avoid their use unless extensive flutter analyses, including detail parameter evaluations, are conducted.
h. Structural Damping Measurements. Structural damping of each significant mode surveyed should be measured. The most commonly used procedure is based on the measurement of the rate of decay of oscillation.
This is best expressed in terms of logarithmic decrement, the natural logarithm of the ratio of two successive amplitudes. Records of the response of a reference transducer, while driving the structure at a specific frequency and obtained immediately before and after power to the shaker is cut off, provide the amplitude relationships required.
Appendix 1 The log decrement, , is then equal to In (X1/X2); or as 0.693 where n=no. of cycles to 1/2 amplitude; i.e.: X1/X2 = 2.
small values of damping, the damping factor, y or C/Co, can be estimated as d/2m and the structural damping g= d/m (References 2 and 8).
Balance Weight Attachment. For control surfaces with balance weights mounted at one end of a cantilevered moment arm, the resonant frequency of the balance weight attachment arm should be at least 50% greater than the highest frequency of the fixed surface with which the control surface may couple. The control surface should be mounted in a jig and the vibrator attached to the balance weight. The input frequency is varied upward and the response of a reference transducer mounted on the balance weight is monitored to define the peak response.
All balance weight supporting structure should be designed for a limit static load of 24g normal to a plane containing the hinge and the weight, and 12g within that plane parallel with the hinge. The balance weight loads should be able to be carried by the control surface and by the fittings and their attachments on both sides of the hinge. Proof of these criteria can be accomplished by relatively simple static tests of the control surface mounted in a jig.
11 (and 12)
Appendix 1 Page intentionally left blank
Appendix 2 APPENDIX 2. FLUTTER ANALYSTS
1. INTRODUCTION. The objective of this appendix is to provide those persons responsible for predicting the flutter characteristics of Part 23 airplanes with some general guidelines for conducting a rational analysis. Two-Degreeof-Freedom, Three-Degree-of-Freedom, and Multi-Degree-of-Freedom Systems and Whirl Mode Analysis are considered briefly. The scheme of analyses outlined here makes use of un-coupled bending and torsion modes. This information herein should assist the analyst in determining the type of analysis suited to a given situation but is not sufficient to permit an analysis without a thorough study of the references.
Compressibility effects on the flutter speed should be considered at and above Mach 0.6.
2. TWO-DEGREES-OF-FREEDON. The flutter characteristics of straight wings (or tails) of large aspect ratio can be predicted fairly well by considering a "Representative Airfoil" that has two-degrees-of-freedom, translation and pitch. This representative airfoil is usually given the geometric and inertial properties of the station three-quarters of the way from the centerline to the tip.
Information regarding this approach is contained in reference 3, and is outlined below:
Appendix 2 Where: m= Mass of section Sa = mbxa = Static Unbalance about Elastic Axis Ia = Inertia about Elastic Axis The equations of motion are:
mun?
Qa Assuming harmonic motion, one obtains:
m Sa 2 h 4 2 S 10 • a Using the aerodynamic expressions of Air Force Technical Report 4798, the expressions for L and My become :
I =-rp b302 {In5+lLa-In (1/2+a))a} +(Ma- (La+My) (1/2 +а) +In (1/2 +a))0} 2
Appendix 2 Where Ih, La, Mh, and Mo are functions of V/bw and obtainable from reference 4.
In this approach, the translation motions, h, are usually assumed to originate from the fundamental bending mode and the pitch motions, a, from the fundamental torsion mode. Higher modes or control surface rotations are not considered.
The value of velocity and damping obtained by solving the equations at each value of V/bw are usually plotted to obtain the velocity at which the damping goes to zero.
3. THREE-DEGREE-OF-FREEDOM. Flutter mechanisms involving control surface rotation are many times more critical than those involving just bending-torsion. At least three degrees of freedom are required to analyze this phenomenon.
The procedure most commonly used is presented in references 2 and 4. The problem is first considered two-dimensional and then extended to the three-dimensional case.
a. Two-Dimensional Flutter Theory.
Midchord- Aileron Hinge b Wing Bending Wing Torsion Spring Spring at Rotation Point eb eb Aileron Torsion Spring at Aileron Hinge The motion of the system can be represented as above, where:
b = semichord cb = distance between midchord and aileron hinge, positive if aft of midchord eb = distance between midchord and aileron leading edge, positive aft of midchord ab = distance between rotation point (elastic axis) and midchord h = bending deflection of rotation point (elastic axis), positive downward
• = angular deflection about rotation point (elastic axis), positive for leading edge up ß = angular deflection of aileron about aileron hinge relative to wing chord, positive for aileron leading edge up 3
Appendix 2 Assuming linearized harmonic motion, the equations of motion become:
А Д + Ва + СВ = 0 DI +ÈO + FB = O Where the coefficients are given in references 2 and 4. For each value of V/b? , the determinant of the coefficients matrix is set equal to zero and the resulting value of artificial damping plotted versus airspeed.
The limitations of a two-dimensional flutter theory are delineated in reference 4.
Cl) All spanwise elements are considered identical with respect to all their flutter parameters.
(2) The vibration amplitudes in each mode do not vary with the spanwise location of the element under consideration.
(3) The effective aspect ratio approaches infinity.
(4) Aerodynamic flow over the oscillating airfoil is not disrupted by interferences.
In general, these limitations are prohibitive and some form of threedimensional analysis is required.
b. Three-Dimensional Flutter Theory. The typical three-dimensional analysis accounts for spanwise variations in mass, geometry and mode shape, but does not account for aspect ratio and aerodynamic interference effects.
The mathematics is presented in Scanlan and Rosenbaum (ref.2). To use this approach one needs the spanwise distribution of the following parameters:
m(×) = Mass per unit span of wing and aileron In (x) = Mass moment of inertia per unit span of wing and aileron, referred to elastic axis So (x) = Static unbalance per unit span of wing and aileron, referred to elastic axis Ig (x) = Aileron mass moment of inertia per unit span, referred to aileron hinge line Sg (x) = Aileron static unbalance per unit span, referred to hinge line b = Semichord length (c-a)b = Distance between the elastic axis and aileron hinge line 4
Appendix 2 The motion at flutter is assumed to be a superposition of the wing first uncoupled bending mode, the wing first uncoupled torsion mode and the aileron rotation mode. In Scanlan and Rosenbaum the mode shapes are treated as continuous functions of span and integration performed over the span of the wing. The vibratory motion at flutter is considered unchanged by the aerodynamic forces.
In practice, the integration is performed numerically, rather than continuously. The wing is divided into a number of spanwise panels and approached from a lumped parameter concept.
4. MULTI-DEGREE-OF-FREEDOM, THREE DIMENSIONAL FLUTTER THEORY. It is not the intent of this section to present a detailed explanation, but rather to outline the general procedure for setting up the mathematics involved in a multi-degree-of-freedom flutter analysis. Familiarity with threedegree-of-freedom flutter analysis and matrix algebra is assumed. The system shown in the sketch is for each mass panel:
Appendix 2 First, the general flutter equation with uncoupled modes is solved with the aerodynamic forces set to zero.
[M]fä}+ (1+18) [R] {9} = [EA! = zero [М] = [ @]І[М] [Ф Generalized mass (non diagonal) PŘ] = [ш2ц] [MaJ Generalized stiffness (diagonal) Where: [@- = Uncoupled assumed modal matrix [Ma] = diagonal elements of [M] [w'ul = uncoupled frequency matrix [M] = Mass Matrix {g} = Vector of generalized coordinates The modal solution to this equation results in the coupled modes and the coupled frequencies which can now be used to solve the equation with airforces.
Md{a}+ (1+18) [w2]MM] {a} = (FA!
[M] = [@]][M][@] = Generalized mass matrix (diagonal) Where: [FA] = Generalized aerodynamic forces [@ ] = Coupled modal matrix Гw2] = Coupled frequency matrix (diagonal)
Appendix 2 The equation of motion including the aerodynamic forces (4) is solved for selected values of bu to predict the damping in the same manner as previously discussed.
The [M] matrix is a mass matrix, usually referred to the local elastic axis, and includes the necessary transformation terms for a control surface or tab. For each mass panel, i, the [M | matrix is:
Sar Są Sa Ia PaB Pas SB Paß IB S6 PB& Where:
W- We + We + We So = We Jet We (le + »e) So = We # : +53315637 2+430 2 + 5c And: Is = We ( 2 + Ke Ww, We. W, = Wts of primary surface, control surface, and tab, respectively.
X = Distance from surface CG to EA - Positive aft Xe = Distance from control hinge to control CG X, = Distance from tab hinge to tab CG 1c = Distance from EA to control hinge 11 = Distance from control hinge to tab hinge 7