Appendix 2 Kw. Ko ky = Radii of gyration of primary surface about its CG, control surface about its CG, and tab about its CG respectively.
The K matrix is a matrix of stiffness influence coefficients. When referred to the elastic axis, the K matrix for one mass panel becomes:
khh Kha khB кп8 KOB коб KBh kốn Many times it is easier to actually measure flexibility influence coefficients, [G] , and then obtain | k by:
[x] = [0]=] The • matrix is a mode shape matrix, the eigenvectors from solving the equation of motion without aerodynamics. For each mass panel 32 3 ....
° 2 "3 " ..an 83 ° 63 ° -§ Where n is the number of modes.
Appendix 2 The FA matrix represents the generalized aerodynamic forces, obtained from:
Where [AIG ] represents the matrix of aerodynamic influence coefficients. There are several in use, but in general they will contain the following (as presented in Reference 5):
Ina Ing LAZE] - M "ви "ва Іва 988 These are referenced to the wing quarter chord, the control surface hinge line, and the tab hinge line. A transformation is required to relate these to the elastic axis.
Using the above explanations and assuming harmonic motion, the generalized equation of motion may be written for each coupled frequency as:
+ (1+18) TR Where: a = gelwt @ * Harmonic frequency F(p) = A scalar function of density [A] - [8] [AIC] [O] Rearranging the flutter equation, we may write Where the eigenvalue A is defined by 1 = + • Re (Real Part) (Imaginary Part) 9
Appendix 2 Hence the flutter frequency becomes WE = Re 1 And the damping g 8 = w? Im = Re And the flutter speed is defined by:
where the equations have been non-dimensionalized and br equals a reference length and kr equals a reference value of reduced frequency, Wb.
V
5. WHIRL MODE.
It is the intent of this section to outline the general procedure for setting up the mathematics involved in a whirl mode analysis. Familiarity with whirl mode analysis is assumed. This analysis includes only engine cantilevered modes. Coupling of these modes with a flexible airplane's modes may be accomplished by the superposition of modes using the method outlined in paragraph 4.
Appendix 2 The whirl mode stability equation for cantilevered power package is:
B= [EM] + D[FD + FG + FQ2] + FK + FQ1] = 0 where FM [GM(M) , generalized masses (1b. sec.?/in.)
FK (2N)= [FREQ(M))?] [GMIM)], generalized stiffnesses (lb./in.)
FD (.02) (2 T) [FREQ (M)] [GM(M)] , generalized damping (lb. sec./in.)
(.02 is the assumed structural damping) га = [Ри(1, м) ]} [а] [рнІ(1,м)] F22 = RADOV: [PHI (I, M)) I [02] [PHI (I,M)] M = Number of modes FREQ(M) = Frequency (Hz) PHI (I,M) = Prop displacement matrix RAD = Relative Density Ratio, Q/Co NOTE: A special purpose computer program may be required for each power package to determine its vibration mode shapes and frequencies and the generalized masses for each mode.
Appendix 2 The equation for gyroscopic forces is shown below.
GYROSCOPIC MATRIX - G P Ip-2 = Y P
- Ips₴ = Prop. vertical force e = Prop. pitching moment P Prop. side force P Prop. yawing moment Z Prop. vertical displacement P P = Prop. pitch angle P = Prop. lateral displacement Ip = Prop. yaw angle 12 = Rotational velocity (radians/sec.)
38 = Prop. polar mass moment of inertia (1b. sec.2 in.)
Appendix 2 The prop aerodynamic force equation for each set of prop coefficients are shown PROP AERODYNANIC MATRIX - Q2 FIR
- Poco0 21 ·p PROP AERODYNAMIC MATRIX - Q1 FOp 0 < FU R - Po CHu 2 Sc where propeller disk area (sq. in.)
propeller diameter (in.)
Po air density at sea level (1b. sec./in.4) preo relative air density V = true air speed (in./sec.)
below.
Appendix 2 In terms of the basic propeller aerodynamic derivative defined in reference 13:
Vertical force derivative due to pitch angle C26 = -Gy = CZIH Pitching moment derivative due to pitch rate Gon = Cos = CUủ = CMQ Pitching moment derivative due to yaw angle Cmy = Coy = - CHO = CMPSI Pitching moment derivative due to pitch angle Сте = Све = Суу = 'CMTH Vertical force derivative due to yaw angle CZY = GrE = CAPSI The propeller coefficients must be furnished in a comparable form to those found in reference 13. For each true air speed, the coefficients are found as a function of prop blade angle determined from the advance ratio (forward velocity divided by prop tip velocity).
The solution to the whirl mode stability equation is found as shown below.
(1) Summing the matrix coefficients lets the equation read:
P A + PB + C = O
(2) Inverting matrix A and premultiplying the equation by A-l yields:
в - р [н1] - [н2] = 0 where [н1] -А-B [нг] -A-C 14
Appendix 2
(3) If we let an arbitrary vector, X1, postmultiply each term
(4) Now, let [H2] x1 = p ×2 '
(5) Then, by substitution, xI - Px2 = 0.
(6) Dividing by p, the above equation becomes
(7) Rearranging terms in (6) and combining with equation (4) in a single equation of twice the order, we obtain
(8) The above equation is solved, using the Laguerre method described in Reference 14, for its eigenvalues, P, which come in complex conjugate pairs. The frequency of the whirl mode is determined by ≤ = 277 PIMAG i hence, the roots with negative imaginary values are discarded. The damping required to produce instability is determined by 8 = ZPREAL/(PREAL + PIMAG'" ;
where positive values are unstable.
Appendix 2
(9) The eigenvector is found by substituting the eigenvalue solution in (1) and combining terms and postmultiplying by y to obtain DY = O
(10) Then, by rearranging the first row y (1) = where N = the number of modes.
(11) And by rearranging the remaining rows where I and J go from 2 to N.
(12) If we let y(1) = (1.0,0), recalling that it is complex, then where I and J go from 2 to N.
(13) The mode shape is then determined by PHIWM(L) = > (PHI (1,I)) • y (I).
I = 1,N
(14) This complex vector is normalized so that the largest translation (L=l or 3) has an amplitude of unity with the phase relationship preserved.
Appendix 3 APPENDIX 3. FLIGHT FLUTTER TESTING
1. INTRODUCTION. This appendix presents a general discussion of acceptable procedures for conducting flight flutter tests intended as final proof of flutter free operation for new or modified airplanes. The methods described herein do not represent a comprehensive survey of existing techniques, but rather represent methods which have been proven to be particularly adaptable to general aviation aircraft.
Paragraph 23.629(c) permits the use of flight tests as the only means of showing freedom from flutter. However, it is recommended that these tests be conducted only after appropriate analyses, defining the critical conditions and severity of flutter onset, have been performed. Both the risk and scope of testing required to substantiate the total airplane is significantly increased without the benefit of reference analyses.
In-flight excitation of only the critical mode(s) is, generally, all that is necessary for final demonstration of flutter free operation if preceded by rational flutter analyses. However, without these studies, all modes of the airplane, or those modes affected by the modification of an altered airplane, must be excited in flight. All test airplanes, whether the objective is to support analyses or to serve as the singular means of flutter substantiation, should include proper instrumentation for recording airplane response.
2. DETERMINATION OF VIBRATION CHARACTERISTICS. Paragraph 23.629(a) requires a determination of the natural frequencies of main structural components by vibration tests or other approved methods. This must be done regardless of the flutter substantiation method selected; i.e., (a) rational analysis, (b) flight flutter tests, (c) simplified criteria, or (d) combinations of these. This determination must be made for all new airplanes and for existing airplanes, before and after any major modification to assess the effects of these structural changes. Engineering judgement should be exercised in determining whether the effects of the modification on aerodynamics, stiffness, or mass are sufficient to warrant flutter reinvestigation. The effects of variations in fuel loading, airplane weight, and center of gravity should also be assessed. It is recommended that mode shapes, as well as frequency, be determined by either ground vibration tests or analytical methods, if adequately supported by test.
For modified airplanes with no available analyses, the degree of frequency change requiring flight investigation is dependent upon, in part, the nature of the modification, the relative change of bending-torsion frequency ratios, and the relationship of the structural modes to control surface modes. Shifts in node lines may also dictate flight checks.
3. AIRCRAFT EXCITATION METHODS. The airframe modes and frequencies can be excited in flight by any number of techniques. The important criteria for technique selection is that the modes and frequencies of interest must be 1
Appendix 3 adequately excited to allow for proper modal response. To illustrate, most general aviation aircraft use cable or push rod control systems which have high levels of coulomb damping. The coulomb damping gill cause a non-linear control response. At low amplitudes the damping will be high and the system stable; whereas, at higher amplitudes, the coulomb damping will be reduced and the control system could be unstable. A proper level of modal excitation will, therefore, produce lower system damping and an earlier indication of a developing flutter mechanism. Consequently, without proper excitation) the test engineer may have very little warning of developing flutter.
Consideration should also be given to the possible influence of the excitation system on the flutter characteristics of the airplane.
Excitation methods presently being used for general aviation airplanes are discussed briefly in the following paragraphs.
a. PILOT INDUCED CONTROL SURFACE IMPULSES. Pulsing the controls may be used to excite modes, generally, below 10 Hz, but is not recommended above 10 Hz. The effectiveness of the method is usually limited by either the ability of the pilot to impart a pulse of proper duration or the ability of the control to transmit the pulse to the primary surface. Three degrees of control rotation are normally sufficient if the duration is short enough to encompass all harmonics of interest.
b. SINUSOIDAL EXCITATION USING ROTATION MASSES. This technique has been used successfully for exciting airplane modes between 10 and 50 Hz.
Rotating eccentric mass shakers mounted in the wing tips and/or the tail section of the fuselage will usually produce wing and empennage modes of concern. Although shakers in each wing tip and both vertical and lateral mounted fuselage shakers may be required to excite both symmetric and antisymmetric modes) single shakers have been used successfully for both symmetries where adequate frequency separation exists. The eccentric mass should be large enough to excite the 10 Hz modes and the shaker supporting structure strong enough to withstand the shaker force at 50 Hz. Shaker forces of up to 300 pounds at 50 Hz have been used and produced very good results. In general, the larger the exciting forces the safer the test since the non-linearities will be minimized. The shaker force should be adequate to assure that modal response is easily distinguishable from random or buffet excitation.
A simple inertial shaker system can be constructed using off-the-shelf shop components. The system consists of a container of compressed nitrogen, pressure regulator, line gate valve, air hose and shop drill motor. The RPM of the drill motor is controlled by varying the line pressure and can be monitored by a tach generator attached to the drill motor shaft. The only parts requiring design and fabrication are the eccentric mass and appropriate mountiag brackets.
Appendix 3
c. EXCITATION USING AUTOPILOT AND OTHER METHODS. Sinusoidal excitation using the autopilot will produce modal responses similar to the inertial system but has the advantage of supplying a stronger input to the fundamental modes. A restriction is its ability to transmit energy into the control surfaces at the higher frequencies due to control system flexibility.
Other methods such as flutter vanes and rocket impulse units may be used.
Regardless of the method used, the same principles of frequency response and mode identification apply; i.e., adequate response of the modes and frequencies will produce the desired indication of any developing flutter mechanism.
4. AIRCRAFT INSTRUMENTATION. The aircraft instrumentation required to adequately monitor vibration characteristics will vary greatly depending on the extent of the test (number of modes being investigated) and the special design characteristics. As a minimum, transducers that measure acceleration or velocity should be installed on the tips of the aerodynamic surfaces of concern; i.e., for wing and horizontal stabilizers, front and rear spars on one side for bending and torsion response and on one spar of the opposite side as an indicator of symmetry. The frequency response characteristics of the installed transducer should be checked to assure adequate sensitivity throughout the test frequency range. Strain gages or accelerometers should also be installed on each control surface. Care should be taken to assure control balance is not disturbed by the instrumentation.
Unless workload makes it prohibitive, it is preferred that the in-flight data recorder and exciters be operable by the pilot to eliminate any need for additional personnel on board the airplane during the test. This is particularly desirable if testing is performed without supporting analysis.
Although various telemetry systems exist for recording and transmitting data to the ground, an on-board oscillograph or magnetic tape recorder are more commonly used. A magnetic tape recorder is advantageous over an oscillograph if filtering, sensitivity change, and speed variations are available on playback to aid in determining frequency and damping. An oscilloscope, or other means to monitor input frequency in real time, should also be available to the crew.
The complete excitation and data recording system should be thoroughly checked and calibrated, on the ground and as mounted in the airplane, to assure excitation of the desired modes and in turn to establish baseline amplitude and damping data. The response to engine noise and aerodynamic buffeting can seriously distort the data and, as a general rule, the signalto-noise ratio should be at least 4 to 1. The unwanted signals can be filtered or minimized by increasing the exciter force. However, the maximum level of excitation should be limited to prevent structural damage from dynamic overload.