GROUND VIBRATION TEST ON ULTRALIGHT PLANES

Transcription

1 GOUND VBATON TEST ON ULTALGHT PLANES Dr. Eric Groß, Technische Universität Hamburg-Harburg, Germany Burkhard Martyn, Technische Universität Hamburg-Harburg, Germany Abstract Modal parameters from complex structures, like airplanes or satellites, are usually extracted by applying the phase resonance method. For this method, appropriate force vectors are needed to excite the structure in a specific mode under investigation using multiple shakers at many different excitation locations. Since the effort of this technique is quite high, operational modal analysis could be a solution since the input forces need not to be known. This paper describes an operational modal analysis applied to an ultralight airplane in comparison to the standard phase resonance method. 1 ntroduction Ultralight planes are small lightweight planes for maximum two passengers and a maximum take off weight (MTOW) of 450 kg in Europe, 472,5 kg in Germany and Switzerland resp. 560 kg in nternational Standards. They can easily be recognized in Germany by means of their registration number always starting with the national letter D followed by a M (like D-MUWP in case of the present investigated plane). The manufacturer has to ensure a save operation in the capable range of speed for the airplane, what necessitates the verification of flutter stability due to aeroelastic excitation. This is done by calculation based on a ground vibration test to detect the modal parameters. 2 Phase esonance Method The phase resonance method, or normal mode testing, is been widely used for modal analysis of large structures like aircrafts, although the use of phase separation techniques increased during the past two decades. Both methods rely on knowledge of the input force by either direct measurement or by means of measured transfer functions. n both cases the structure must be excited at distinct points and, in case of an aircraft, at many different locations at the same time. Normal mode testing consists of three steps. n the first step a sinusoidal frequency sweep is performed for different excitation configuration to localise resonance frequencies and to calculate optimised force pattern and positions by means of measured transfer functions between the responses and excitation forces [1]. n a second step, each eigenmode is tuned according to the predicted force pattern by varying force level and frequency to achieve a 90 phase shift for all response sensors with respect to the input forces. Thus, the structure response in a single isolated vibration mode with the influence of closely spaced modes suppressed. The third step consists of determining the modal parameters like shape, damping and modal mass from narrow-band sweeps around the eigenfrequency under investigation.

2 The fundamental equation of the phase resonance method is based on the equation of motion of a discretized viscously-damped structure [ M ] x(t) & + [ C] x(t) & + [ K] x(t) = f (t) &, (1) where [ M ], [ C ] and [ K ] are mass, damping and stiffness matrices, x is the structural response vector and f the external force vector. f a real harmonic force vector be separated into its real and imaginary part 2 ( ω [ M] + [ K] ) Xˆ ω[ C] Xˆ = Fˆ jωt F = Fˆ e is applied, Eq.(1) can, (2) 2 ( ω [ M] + [ K] ) Xˆ ω[ C] Xˆ = 0, (3) where the subscript and indicate the real and imaginary part, respectively. Since the real part of the X components must be zero if the phase resonance criterion is fulfilled, the eigenvalue problem of the fist summand of Eq.(3) must be solved. The solution provides the natural eigenfrequency and the real normal mode information is given by the imaginary part of the response vector. From Eq.(2) follows, that in resonance the external forces just have to compensate for the internal forces. The appropriate force vector can be calculated by using the frequency response functions (FF) measured by the first step of the phase resonance method described above. Minimizing the kinetic energy of the response in phase with the excitation to that of the total response leads to an eigenvalue problem formulated in the frequency domain of the following form [2] T ( ) F( jω) T T [ H ( jω) ] F( jω) = λ [H ( jω)] [ H ( jω) ] + H ( jω)] [ H ( jω) ] [ H ( jω)] where H is the FF matrix between the responses and excitation forces F and λ is called the multivariate mode indicator function (MMF). λ will be zero or very small at the eigenfrequencies and F will be the corresponding force vector for those frequencies, which minimizes the real part of the responses. F will not be in absolute values, but it will relate the different force levels to each other. Figure 1 shows the first and second MMF and the corresponding optimised force vectors measured at the ultralight aircraft Lambada13 with two shakers attached to the wing tips. Since the eigenvectors are just valid in the vicinity of the resonance frequencies, they are marked with circles and diamonds, respectively. t can be seen, that ideally both forces should be either in phase or 1 st sym. wing bending, (4) Figure 1: MMF and optimised force vectors measures at the ultralight aircraft Lambada13 out of phase with nearly same amplitude, since both forces act at the same but opposite wing location. Exemplary, the force vectors for the first symmetrical wing bending mode is been highlighted. The appropriated force levels will be adjusted till the phase resonance criterion is fulfilled or maximized. A narrow-band sweep will be performed around each resonance frequency, keeping the

3 force level constant to minimize non-linearity effects in the measurements. This is been done by a feed-forward shaker control, taking interactions between the attached shakers into account. Modal damping and modal mass can be determined using the complex input power. Separated into real and imaginary part, the input power is defined in terms of response accelerations as P T = F ( jω) X & ( jω) and T P = F ( jω) X& ( jω ). (5) Plotting Eq.5 over a narrow-band frequency range in the vicinity of a resonance frequency leads to two characteristic curves. The real part has a zero crossing with a positive slope, since very little energy is needed to excite the structure, while the imaginary part has its maximum at resonance. With the slope of P and the maximum of P the modal damping and modal mass at the eigenfrequency ω 0 can be expressed as [3] P ( jω0 ) D =, dp ( jω) ω0 dω ω=ω0 (6), ω0 dp ( jω) M = 2 2 X& ( jω ) dω (7) max 0 ω=ω0 with & ( jω ) as the maximum acceleration level at resonance for modal mass normalization. X max 0 Together with the eigenfrequency ω 0 and the normalized imaginary part of the response sensors as the mode shape, all modal parameters are determined. t is important to note that these equations relay on the assumption of real forces, i.e. all force must be in phase or ±180 out of phase to fulfil the phase resonance criterion. 3 Operational Modal Analysis As pointed out above it is obvious to see that the well proved phase resonance method is a time consuming task and especially to find the optimal shaker positions to excite all mode shapes necessitates some experience. So for a very light airplane structure an acoustic excitation and an Operational Modal Analysis (OMA) should be reasonable [4], [5]. Due to considerable problems to separate closely spaced mode shapes, as they occur on airplane structures, the frequency domain methods implemented in Operational Modal Analysis fail to identify all modes in the frequency range of interest, so that the investigation is focused on the Stochastic Subspace dentification (SS) techniques, where a parametric model is fitted directly to the time data. The Principal Component (PC) and the Canonical Variate Analysis (CVA) have been used. For the investigated ultralight airplane this means to capture time data over an appropriate time period based on the lowest frequency of the first structural mode. This first fundamental wing bending mode is known in the present case to be close to 4 Hz resulting in a measurement time of approx. 2 min. following the rule of thumb to chose 500 times the time period of the lowest frequency of interest. This recording time proved to be sufficient. To get an excitation random in time and random in space for the acoustic excitation two independent loudspeaker systems were used placed in two setups. The use of two setups proved to be extremely important, because using only one loudspeaker setup deteriorated the results significantly. Another very important hint is the use of projection channels, otherwise closely spaced modes could not be separated in the particular case.

4 4 Experiment Setup The test object for comparison of both methods was the ultralight aircraft Lambada13 with a span of 13m. Fuselage and wings are made out of glass fiber reinforced plastic. The aircraft was suspended on two soft springs to simulate a free-free boundary condition (see Figure 2). The suspension mode was three times less then the first fundamental wing bending mode to avoid mode interactions. A measurement grid of 69 accelerometers was placed on both of the wings, tailplane, fuselage and on all control surfaces. The control was locked during the test to prevent the control surfaces from twisting around the control fixtures. For the phase resonance method the aircraft was excited sinusoidal by up to four electro-dynamic shakers simultaneously on ten different positions spread over the structure. Typical positions are wing tip for bending modes, leading edge of the wings for torsion, in horizontal direction at the wings for sway modes, horizontal and vertical tailplane for coupled wingtailplane modes and at the fuselage itself, to support fuselage bending modes. Figure 2: Measurement setup with electro dynamic shakers The excitation for the operational modal analysis was made by use of two 15 inch subwoofer loudspeaker systems placed underneath the wings and tailplane (see Figure 3). The loudspeakers were placed in two different setups to change the incident angle to excite vertical tailplane and wing sway modes as well, the time data from both setups was concatenated for analysis. The excitation signal was band limited pink noise (upper frequency 150 Hz due to the frequency dividing network for the woofer system), both speakers were supplied by independent noise generators and power amplifiers (375 W continuous rated power each). The sensor grid for the operational modal analysis was slightly refined compared to the phase resonance measurements using 79 accelerometers on a 80-channel front end. Speaker 1 Speaker 2 Figure 3: Test setup with Subwoofers and sensor grid for the Operational Modal Analysis with successively loudspeaker positions and

5 5 esults For the operational modal analysis stable modes are found around 4 Hz as expected from the first fundamental wing bending modes (mode 1 and mode 2 according to Table 1) and furthermore above 15 Hz as can be seen in the following stabilization diagram (Figure 4). Table 1 compares the mode shapes, frequencies and damping coefficients. t is remarkable, that even the closely spaced torsional modes of the wings at 25,18 Hz and 25,21 Hz have been well separated (see Figure 5), the overall small differences in the resulting frequencies between phase resonance method and operational modal analysis are negligible because they are in the range of temperature effects of the airplane structure. Above 15 Hz all modes could be identified, but in contrast modes between 5 Hz and 10 Hz in operational modal analysis using acoustic excitation were simply not present. Figure 4: Stabilization diagram using Principal Components (PC) technique with 5 projection channels Table 1: Modal parameters in comparison for both phase resonance and OMA method Phase esonance OMA f [Hz] M [kgm 2 ] ζ [%] f [Hz] ζ [%] from ζ [%] to Mode 4,15 15,27 1,42 4,08 0,77 1,42 Mode 1: 1 st sym. wing bending 4,79 33,60 1,17 4,61 1,38 1,76 Mode 2: 1 st asym. wing bending 7,99 48,65 1,59 n. a. - - Mode 3: Sym.fuselag bending & 1.sym. wing sway 8,21 67,03 1,52 n. a. - - Mode 4: 1 st asym.wing bending, tailplane out of phase 8,98 15,21 1,83 n. a. - - Mode 5: 1 st asym. horizontal tailplane 9,43 17,03 1,30 n. a. - - Mode 6: 1 st asym.wing bending, tailplane in phase 10,25 20,36 2,12 n. a. - - Mode 7: 1 st sym. wing sway 15,25 12,38 2,19 15,41 3,08 4,13 Mode 8: 2 nd sym. wing bending 23,64 5,76 1,01 23,43 3,04 5,69 Mode 9: 1 st asym. horizontal tailplane 24,80 17,71 1,14 25,18 1,91 3,33 Mode 10: Sym. torsion 24,82 20,12 0,75 25,21 1,99 3,64 Mode 11: Asym. torsion 26,52 30,40 1,64 26,24 4,27 5,15 Mode 12: 2 nd asym. wing bending 32,15 1,61 1,74 33,18 2,72 4,97 Mode 13: Sym. horizontal tailplane 34,00 17,62 1,39 33,94 3,66 5,16 Mode 14: 3 rd sym. wing bending 34,80 1,77 3,53 35,50 2,13 4,59 Mode 15; Sym. horizontal tailplane & elevator 38,60 29,27 2,74 38,12 3,21 3,73 Mode 16: 2 nd asym. wing sway 48,00 3,98 1,28 48,51 0,92 3,41 Mode 17: 3 rd asym. wing bending

6 Figure 5: Symmetric (left) and asymmetric (right) torsional wing mode extracted with OMA A look at the acceleration power spectral density (APSD) analysis of the sensor signals gives the answer to this problem. Off course, even using powerful loudspeakers, a drop in sound pressure level to low frequencies is unavoidable. Due to the stochastic random vibration excitation there is still sufficient energy brought into the structure by the acoustic excitation for the large wing area. Normal to these large surfaces, even for frequencies below the lowest radiated sound wave, sufficient excitation is found, especially for the very easy to excite fundamental wing bending mode. For the wing sway modes (the sharp edges of the wing) there is simply not enough area to be excited (mode No. 3 and 7 at 7,99 Hz and 10,25 Hz in Table 1), the same problem occurs to the tailplane with relative small areas (see dashed curve in Figure 6), where in the acceleration level there is an obvious gap compared to the wing sensors at low frequencies. For this reason, the partial modes of the tailplane between 8,21 Hz and 9,43 Hz could also not be identified (too few acoustic energy for a too small area). Figure 6: Acceleration power spectral density plot of one accelerometer on the wing structure (full line) and one accelerometer on the tailplane (dashed)

7 t is always good practice in modal analysis to check the orthogonality of the analyzed mode shapes by the Modal Assurance Criteria (MAC). Figure 7 as an example states the good separation even of the closely spaced modes of the investigated airplane. For the different Stochastic Subspace dentification techniques the conformance in the MAC is something diminishing at higher frequencies (see Figure 8) resulting in a range of damping coefficients for each mode shape as listed in Table 1. n general, with exception of the first wing bending modes, the damping is overestimated using operational modal analysis and acoustic excitation. For this appearance there is no explanation right now, so future work shall imply the determination of damping coefficients in operational modal analysis. Despite this small problems the major advantage of the investigation of the modal properties in the present case for the ultralight airplane is a much faster way to detect the mode shapes and eigenfrequencies. For the proceeding flutter calculation besides the eigenfrequencies the modal mass has to be extracted. This only will work using a known excitation, so shaker tests are still unavoidable. But especially the first step necessary in any case in the normal mode testing, the sinusoidal frequency sweep over the full frequency range of interest for different excitation configuration to localise resonance frequencies and optimised positions for shaker connections, could be substituted by the much faster and, even in the case of closely spaced eigenmodes, reliable operational modal analysis. Figure 7: Orthogonality check using the MAC criterion Figure 8: MAC comparison PC versus CVA 6 Conclusion An operational modal analysis of an ultralight airplane using acoustic excitation is a powerful time saving tool to analyze the eigenfrequencies and mode shapes at a glance. All mode shapes of the wings are identified. Nevertheless, using acoustic excitation this method will not necessarily identify all mode shapes especially for wing sway and for the tailplane and the damping coefficients are

8 generally overestimated, so this is not a substitute for the phase resonance method or normal mode testing using known excitation forces driven by shakers. Especially for proceeding flutter calculation the modal masses have to be calculated, what is only possible with known excitation forces, but for the first step to search for the eigenfrequencies and mode shapes it is possible to get these results much faster. 7 eferences [1] Williams,., Vold, H., Multiphase Step-Sine Method for Experimental Modal Analysis, int. Journal of Analytical and Experimental Modal Analysis, Vol. 1., No.2, pp 25-34, 1986 [2] Homes, P.S., Cooper, J.E., Wright, J.., Normal Mode Estimation from Non- Proportional Damped Systems, SMA 21, pp , 1996 [3] Knan, G, Standschwingversuch und Flatterrechnungen mit dem Segelflugzeug B13, nternal eport, DL B J 03, 1995 [4] Møller, N., Gade, S. and Herlufsen, H. Stochastic Subspace dentification in Operational Modal Analysis, Proc. 1 th nternational Operational Modal Analysis Conference OMAC, 2005 [5] Møller, N. and Herlufsen, H. Estimating Modal Parameters using Acoustic Excitation, Proc. 19 th nternational Modal Analysis Conference MAC XX, 2001