IDENTIFICATION OF THE MODAL MASSES OF AN UAV STRUCTURE IN OPERATIONAL ENVIRONMENT

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1 IDENTIFICATION OF THE MODAL MASSES OF AN UAV STRUCTURE IN OPERATIONAL ENVIRONMENT M.S. Cardinale 1, M. Arras 2, G. Coppotelli 3 1 Graduated Student, University of Rome La Sapienza, mariosalvatore.cardinale@gmail.com. 2 Ph.D. Student, University of Rome La Sapienza, melissa.arras@uniroma1.it. 3 Assistant Professor, University of Rome La Sapienza, giuliano.coppotelli@uniroma1.it. ABSTRACT In the framework of Operational Modal Analysis, several methods have been developed to estimate the modal parameters, that is natural frequencies, damping ratios, and mode shapes of a structure in its operative conditions. However, it is not possible to directly estimate the modal masses associated to each mode shape due to the unknown excitation. The modal masses are usually evaluated from the analysis of the change of the modal parameters by testing the structure in correspondence of two mass configurations. In this paper the efficiency and the accuracy of two procedures for the estimate of the modal masses are assessed by performing laboratory vibration tests. Vibration response data recorded during flight tests of an unmanned aerial vehicle (UAV) are used for this purpose as well as a mass changing device was developed to produce the mass variation of the structure. Results from the traditional input/output experimental modal analysis and the operational modal analysis are compared in terms of modal parameters, modal masses, and synthesized frequency response functions. Keywords: Operational modal analysis, UAV, flight test 1. INTRODUCTION Generally, the dynamic behavior of a structure is identified by the traditional experimental modal analysis that requires to measure both the input loadings and the responses of the structure, [1]. However, this approach is not capable to identify the structure in its operative conditions. For this reason, in the recent years many authors have focused their attention on the so-called operational, or output-only modal analysis (OMA), [2, 3]. The output-only approaches allow to identify the modal parameters, i.e. natural frequencies, mode shapes, and damping ratios, of a structure in its operative conditions using the ambient excitation and measuring the responses of the system only. The lack of information on the input excitation causes the operational mode shapes to be unscaled, restricting the applicability of the outputonly approaches and requiring further investigations. The scaling factors, or modal masses, are usually

2 estimated by imposing a known mass and/or stiffness variation on the structure that modifies its modal properties, [4, 5, 6]. Commonly, only the mass is varied to produce the eigenfrequency shifts because it is easier from the practical point of view. Therefore, the estimate of the modal masses requires an initial estimate of the modal parameters by using either a frequency, [7] or a time-domain [8] output-only estimating techniques, then a second experimental test is carried out to estimate the scaling factors from the natural frequency shifts induced by the known mass variation of the structure. In this paper, two different methods will be investigated to evaluate the modal masses of the YAK-112 Airworld unmanned aerial vehicle (UAV) owned and operated by the Department of Mechanical and Aerospace Engineering of University of Rome "La Sapienza". Such approaches are based on the work presented in Refs. [4, 5] and in Ref. [6]. In order to asses the accuracy of such estimates a laboratory test activity that mimics the actual flight tests was set up. Specifically, the reference modal parameters of the UAV are estimated from operational responses obtained by exciting the structure with a random signal characterized by the same energy content as the one recorded during a typical leveled flight of the UAV. A close-loop controlled excitation is used to reproduce in laboratory the actual vibration environment of the flying UAV for this purpose. A mass changing device, to be used aboard of the UAV, was developed to investigate the capabilities of the procedures to accurately identify the modal masses for different UAV mass configurations. This device consists of a set of two elastic balloons and pipes, located at the UAV wing tips, that allows the mass variation of the wings during the flight, as a consequence of the water-flow from the balloons. A comparison between the modal parameters and frequency response functions (FRFs) gained from the operational responses and those from the traditional experimental modal analysis, in which both the input excitation and the corresponding output measurements are known, is reported for the overall accuracy assessment. By tracking the modal parameters of the resulting variable-mass UAV system useful suggestions for the actual flying test procedures are finally drawn. 2. THEORETICAL BACKGROUND 2.1. Hilbert Transform Method In order to estimate the modal parameters of the structure by using output-only data, the Hilbert Transform Method is used, [9]. The advantage in using the Hilbert transform is mainly the possibility to estimate the imaginary part of a causal function starting from its real part, [1], thus allowing the estimate of the biased FRFs. In this section, the key points of the method will be summarized. Considering the time responses of the structure at N measurement points, x i (t), i = 1, 2,..., N, the auto and cross-correlation functions are defined respectively as: R xi x i (τ) = E[x i (t)x i (t + τ)] = lim T 1 T ˆ +T/2 T/2 x i (t) x i (t + τ) dτ (1) R xi x j (τ) = E[x i (t)x j (t + τ)] = lim T 1 T ˆ +T/2 T/2 x i (t) x j (t + τ) dτ (2) where E[ ] is the expected value operator. Under the hypothesis of Gaussian stationary, ergodic with zero mean value signals, x i (t), the auto and cross spectral density functions are evaluated through the Wiener-Khintchine relations, [11]: G xi x i (ω) = ˆ + R xi x i (τ) e jωτ dτ (3) G xi x j (ω) = ˆ + R xi x j (τ) e jωτ dτ (4)

3 The spectral density matrix of the response signals G xx (ω) C N N can be written as: G x1 x 1 (ω) G x1 x 2 (ω)... G x1 x N (ω) G x2 x 1 (ω) G x2 x 2 (ω)... G x2 x N (ω) G xx (ω) =.... G xn x 1 (ω) G xn x 2 (ω)... G xn x N (ω) (5) Such matrix can be expressed in terms of the FRF matrix of the system H(ω) C N N i, N i being the number of inputs, by: G xx (ω) = H(ω) G ff (ω) H H (ω) (6) where G ff (ω) C N i N i is the spectral density matrix of the input, and [ ] H represents the hermitian operator. In operational modal analysis the forcing actions are not known so, ideally, the term G ff (ω) is unknown but it is possible to assume it to be derived from a white noise excitation. This implies that G ff (ω) is frequency independent so G ff (ω) = G ff where G ff is a diagonal matrix when the input excitation is uncorrelated in the space domain. Considering the polar representation of a driving point FRF (amplitude and phase) it is possible to directly relate the amplitude and the phase to the spectral density matrix of the response as follow (see [9] for further details): H ii (ω) 2 =: H ii (ω) 2 G fi f i = G x i x i (ω) G fi f i (7) φ ii (ω) = 1 2 H[ln(G x i x i (ω))] (8) where the unknown PSD function G fi f i gives the bias on H ii (ω) with respect to H ii (ω). The bias FRF in the i-th driving point is given by: H ii (ω) = G xi x i (ω) e jφ ii(ω) (9) Once the driving points bias FRF is estimated it is possible to evaluate the off-diagonal terms of the bias FRF matrix, [12]: H ij (ω) = G x i x j (ω) H ii (ω) (1) It is important to underline that while on the one hand the estimated functions are unbiased since the input forces are unknown, on the other hand the bias constant on the operational FRF does not affect the estimate of the modal parameters since they are not dependent on the biasing level. Natural frequencies, damping ratios and mode shapes are further estimated by using a residue/pole-based technique commonly available in most commercial numerical softwares, [13] Modal masses estimate Output only techniques do not allow to estimate the modal masses, or scaling factors, because the excitation is not measured. As a consequence, the so-called operational mode shapes remain unscaled restricting the applicability of the operational modal models. In the last years, several methods have been developed to evaluate the modal scaling factors based on the analysis of the variation of the modal parameters of a structure due to a variation of its dynamic properties. From the evaluation of the modal

4 parameters corresponding to two structural configurations (one before and one after a mass/stiffness variation) it is possible to calculate the scaling factor as will be presented in this section. The most used technique to evaluate the modal masses considers only the variation of mass because it can be varied easily, whereas the variation of the stiffness is quite hard since it requires tools able to change the stiffness properties without affecting the constraints and the boundary conditions of the structure. Two different methods capable to evaluate the modal scaling factors will be presented in the following subsections, the Mass Variation Method (MVM), [4], and the Receptance Based Mode Normalization Approach (RBN), [6]. Such approaches assume the structure behaving as a linear system whose dynamics is described by the vector x R N of the lagrangean (and observed) degrees of freedom, whereas the mass and the stiffness properties are described in a finite element representation by the matrices M R N N and K R N N respectively Mass Variation Method The Mass Variation Method (MVM), [4], allows the estimate of the modal masses from the estimate of the change of the natural frequency corresponding to a change of the mass distribution of the structure, M, under the hypothesis that the mode shapes are practically insensitive to such mass modification. The corresponding shifts of the natural frequencies must be measurable, but not too high, and the mode shapes variation is checked by calculating the Modal Assurance Criterion (MAC) 1. The following properties have to be defined in the design phase of the test: the total amount of mass to add to the structure the points in which the masses will be applied and so the mass variation matrix M Considering the spatial model described by Mẍ + Kx = f and applying a variation of both the mass and stiffness matrices, M and K, the associated eigenvalues problem is written as, [5]: ( (K + K) ψ (n) + ψ (n)) ( = (M + M) ψ (n) + ψ (n)) ( ω 2 n + ωn 2 ) (11) it is worthwhile noting that in this paper the n th mode ψ (n) is estimated in the framework of the operational modal analysis. Pre-multiplying Eq. (11) by ψ (n)t, neglecting high order terms, and assuming that the mode shapes are not modified by the considered change in the mass distribution, the following relation is obtained: ψ (n)t M ψ (n) ω 2 n = ψ (n)t K ψ (n) ω 2 n ψ (n)t M ψ (n) (12) The modal mass associated to the n th mode shape is: m n = ψ (n)t M ψ (n) = ψ(n)t K ψ (n) ω 2 n ψ (n)t M ψ (n) ω 2 n (13) 1 The Modal Assurance Criterion, MAC, identifies a real scalar between zero and unity and is defined between mode φ (n) and mode ψ (n) by: MAC(φ (n), ψ (n) ) := φ (n)h ψ (n) 2 (φ (n)h φ (n) ) (ψ (n)h ψ (n) )

5 Once the modal mass is known, the n th scaled mode shape can be obtained as: φ (n) = 1 m n ψ (n) (14) When the variation of mass produces small shifts of the natural frequencies, the scaling factors m n predicted by Eq. (13) (with K = ), may be inaccurate. In order to improve such accuracy, the following first order approximation is introduced: m n = 1 2 ω n ψ (n)t M ψ (n) (15) Finally, the FRF matrix could be evaluated from the estimated modal model. If the viscous damping is considered, then the ij th element of the FRF matrix is synthesized by the following well-known relationship: H ij (ω) = N r=1 φ (r) i φ (r) j ωn 2 r ω 2 + 2jζ r ω nr ω LR + UR (16) ω2 where LR and U R are the lower and upper residuals, respectively, that take into account the contribution of the modes that are outside of the band of interest Receptance Based Mode Normalization Approach The Receptance Based Mode Normalization Approach method (RBN), [6], estimates the modal masses from the change of the natural frequencies of the structure as a result of a change in its mass distribution as the previous method. Let λ j and ψ (j) be the j th eigenvalue and the unscaled eigenvector of the original structure, whereas λ j and ψ (j) be the j th eigenvalue and the unscaled eigenvector of the modified structure. Again, the eigenvalues problem associated to the modified structure is: K ψ (j) = (M + M) ψ (j) λj j = 1, 2,..., M (17) that could be rewritten as: [ K λj M ] ψ (j) = M ψ (j) λj (18) or [ K λj M ] 1 M ψ(j) λj = ψ (j) (19) where the matrix [ K λ j M ] 1 is the receptance matrix, H C N N, of the original system. By introducing the expression of the receptance in terms of the modal parameters of the system, that is H ij (ω) = N n=1 φ (n) i φ (n) j m n (ω 2 n ω 2 + 2jζ n ω n ω) for jω = λ j, (2)

6 Eq. (19) becomes: N m k=1 ψ (k) (ψ (k)t M ψ (p) ) λp λ k λ p γ k = ψ (p) p = 1, 2,..., N m (21) where γ k = α 2 k = 1 m k are the unknown scaling factors. If N N responses are available and N m M modes are investigated, the overall equations are N m N and the unknown quantities are N m so the solution of the over-determined linear system of equation Eq. (21) in term of the normalization constants is found using least squares numerical approaches, [6]. Such a formulation holds for both damped and undamped systems without keeping the same level of accuracy of the solution. 3. EXPERIMENTAL INVESTIGATION In this section the previously described approaches are applied to evaluate the modal scaling factors of a real structure. First, the modal parameters of the investigated structure are estimated by performing both a traditional input-output experimental modal analysis and an operational modal analysis, then the effects of the mass loading in predicting the generalized masses is analyzed. Finally, the comparison between the FRFs evaluated from the output only approaches and the ones estimated from the traditional input/output analysis will be presented. The analyzed structure is the YAK-112 Airworld UAV, Fig. 1. The physical properties of the wing, the horizontal tail, and the vertical tail are reported in Table 1, Table 2 and Table 3 respectively. Figure 1: YAK-112 Airworld UAV Mass Modification System In order to evaluate the modal masses of the UAV, a system capable to modify the mass of the structure has been designed. As previously mentioned, some masses will be added to the structure in order to produce a shift of the natural frequencies. A preliminary check aimed to find the best locations to add the masses is numerically performed with the aid of the Finite Element analysis. The lumped masses should

7 Property Symbol Value Wing Span b m Chord c.4 m Planform S m 2 Aspect Ratio λ Dihedral Angle 1.5 deg Sweep Angle at c/4 Λ c/4 deg Fuselage length l f m Table 1: Wing properties of the YAK-112 Airworld UAV. Property Symbol Value Wing Span b t.9 m Chord c t.256 m Planform S t.2319 m 2 Aspect Ratio λ t Dihedral Angle t deg Sweep Angle at c/4 Λ tc/4 deg Angle of Incidence i t deg Table 2: Horizontal tail properties of the YAK-112 Airworld UAV. Property Symbol Value Wing Span b v.339 m Mean Chord c v.256 m Planform S v.817 m 2 Aspect Ratio λ v Sweep Angle at leading edge Λ LE 25 deg Rudder Mean Chord c r.18 m Table 3: Vertical tail properties of the YAK-112 Airworld UAV. be placed at different points of the structure so that natural frequency shifts could be observed keeping the mode shapes practically unchanged. Such a numerical analysis suggested to place the masses at the tip of the wings, Fig. 2. The device that allows to modify the mass properties of the UAV consists of elastic balloons filled with water. They were inserted into the wings at the mentioned position, and connected to tubes that allowed both to refill and drain the water by two valves positioned in the fuselage, Fig. 3. The maximum capacity of each elastic balloon is 3 ml (.3 kg) Impact test The UAV model has been initially tested with a traditional input/output test, an impact test, in order to identify its modal parameters. This step was fundamental in order to have a reference modal base to be used to verify the goodness of the output only estimates. The UAV has been tested in the same constraint conditions chosen to perform the environmental test, Fig. 4. The structure has been excited at the right wing tip in Z direction, in the -128 Hz frequency range, and 19 measurement points were considered, Fig. 5, whose corresponding responses were measured through five roving accelerometers. The modal parameters have been evaluated by using PolyMax, [14], an LMS Test.Lab tool, in the -64 Hz frequency range because the most visible and well defined peaks of resonance were placed in this band, Fig. 6. From the stabilization diagram, Fig. 7, it is possible to choose the stable poles, corresponding to the natural frequencies of the system, and calculate the mode shapes.

8 Figure 2: Positioning of the mass: (left) schematization; (right) balloon inserted in the wing. Figure 3: Positioning of the valves. Then two different tests aimed to check the effects on the modal parameters of the mass variation are carried out. The first test was performed by adding.15 kg of water in each wing and the second one by adding.3 kg of water in each wing. The resulting change in the dynamic behavior of the structure is synthesized in Fig. 8 where the sum of the estimated FRFs, for each of the mass configurations, are compared. Recalling that the variation of mass should not change the mode shapes of the structure, a further check on the mode shapes is carried out. The Modal Assurance Criterion evaluated between the mode shapes of the reference structure, considered with no added masses, and the corresponding ones of the massmodified structure, for each mass condition, is carried out. A mass variation of.3 kg in each wing was considered the best mass modification from the experimental point of view. Fig. 9 shows the MAC between the reference structure and the structure with the chosen loading condition Environmental test In order to accurately simulate the flight conditions, the UAV structure has been excited with a random signal whose power spectral density value was almost equal to the PSD value of the dynamic excitation acting on the UAV in its operative conditions. The UAV was subjected to closed-loop base excitation car-

9 Figure 4: Detail of the "UAV-shaker" connection. Figure 5: Impact point and position of the 19 measurement points. ried out with the environmental testing facility available at the Department of Mechanical and Aerospace Engineering Department of the University of Rome La Sapienza. A custom-made support properly designed and manufactured provided the connection between the head expander of the shaker and the UAV structure, Fig. 4. The time responses of the structure corresponding to 19 measurement points were recorded for 9 s, Fig. 1, and the auto and cross-spectral densities were evaluated in the -128 Hz frequency range using 32 data blocks records of 234 sampling points. Note that 5 roving accelerometers were used to collect the data from the 19 measurement points, as did for the previously reported impact test, Fig. 5. The measured time signals have been processed with a MATLAB-based tool called Natural Input Modal Analysis - N.I.M.A. developed by the Department of Mechanical and Aerospace Engineering of University of Rome "La Sapienza". This tool allows to process the time response of a structure with different OMA methods such as the Frequency Domain Decomposition (FDD), [7], the Balance Realization (BR), [15], the Stochastic Subspace Identification (SSI), [8], and the Hilbert Transform Method (HTM), [12]. In this work the time signals have been processed by using the HTM method.

10 Figure 6: Sum Frequency Response Function of the UAV, impact test. Figure 7: Stabilization diagram. Once the spectral density matrix was evaluated, the biased FRF matrix was estimated using the HTM approach, see Fig. 11 for the amplitude/phase plot of one of the driving point FRF evaluated in the -128 Hz frequency range. Then the modal parameters are estimated by applying the residue/pole curve fitting approach with a maximum order of 3, Fig. 12. A comparison between the modal parameters from both the I/O and the output only modal analyses, as well as the correspondent error (in percentage) are reported in Table 4. An average good correlation between the two estimates is reported although the effects of the operational conditions induce a difference higher than 5% in estimating the 2nd and the 3rd mode. It is important to highlight that these frequencies refer to the reference structure, i.e. the structure without the added masses into the wings. In order to estimate the modal masses of the structure, a second test was performed by modifying the mass

11 Sum FRF (6g added mass) Sum FRF (3g added mass) Sum FRF (no added mass) g/n db 1 Hz 64 Figure 8: Comparison between the FRFs obtained in the three different loading conditions. Figure 9: MAC between the reference structure and the structure with the chosen loading condition of.3 kg in each wing, impact test. of the UAV. As for the impact test, a mass of.3 kg was added to each wing through the above-described device. Table 5 shows the natural frequencies estimated from the OMA analysis for the reference and the modified structure. As one can see, the considered variation of mass allowed to evaluate clearly the shift of the natural frequencies. The effect of the mass loading on the mode shape was taken into consideration by evaluating the MAC between the corresponding modes from the two mass configuration. From Fig. 13 it turned out that modes 1, 2, 3, 4, 1, 12 were less affected by the mass variation, and so such modes were considered for the evaluation of the corresponding masses. Such last estimates were compared with the modal masses obtained by the traditional I/O experimental test, Table 6. The modal masses estimated by using the operational modal parameters are really close to the values

12 1 Response [m/s 2 ] Time [s] Figure 1: Time responses of the structure in 4 of the 19 measurement points. -2 H of Selected Chanel (17) db Frequency [Hz] 5 phase(h) of Selected Chanel (17) deg Frequency [Hz] Figure 11: Estimate of the biased driving point FRF (Channel 17) of the UAV using HTM technique (channel 17 as example). given by the input/output test for mode 1 and mode 2, whereas modes 3, 4, and 12 show significant deviations. The modal mass of mode 12 estimated with I/O test is much higher with respect to the modal

13 H of Selected Chanel (17) db Frequency [Hz] 5 phase(h) of Selected Chanel (17) deg Frequency [Hz] Figure 12: Estimate of the natural frequencies of the UAV using HTM technique. Table 4: Natural frequencies of the reference UAV estimated with both impact test and output only modal analysis. Frequency Mode Impact Test OMA Error [Hz] [Hz] % mass obtained with both the operational methods. These differences are due to some difficulties in the estimation process of the mode shapes, whose contribution in the calculation of the modal masses is essential, Eq. 13 and Eq FRF Synthesis The capabilities of the considered method to properly identify the dynamic system is further investigated by comparing the FRFs synthesized using output-only data with the corresponding ones from the Input/Output analysis. The synthesis of the FRFs is performed by using Eq. 16 with all the modal parameters estimated by applying the different OMA methods. In Fig. 15, and Fig. 16 three different FRFs are compared. An overall acceptable agreement is obtained especially in the neighborhood of the first considered natural frequencies. At higher frequencies the effects of the uncertainties in the mode shape identification seems to be more relevant. Furthermore, it is expected a general increase in the

14 Table 5: Natural frequencies shift due to the mass variation of the UAV. Frequency Mode Reference structure Modified structure [Hz] [Hz] n.a n.a Figure 13: MAC between the reference structure and the structure with the chosen loading condition of.3 kg in each wing, environmental test Table 6: Comparison between the modal masses estimated from OMA and I/O test. Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 12 n [kg] n [kg] n [kg] m I/O m MV M m RBN correlation by reducing the effects of the uncertainties in the damping ratio estimates, responsible for the difference in the amplitude of the FRF at the natural frequencies, and by including the lower and upper residual terms in the estimates. Finally, both the considered methods for the estimate of the modal masses

15 showed practically the same behavior with respect to the synthesis of the FRFs. The only remarkable difference is reported in correspondence of the third mode where an order of magnitude of difference between the corresponding modal masses is obtained I/O FRF MVM Synthetized FRF (OMA Empty-Full) RBN Synthetized FRF (OMA Empty-Full) Amplitude [db] Phase [rad] Frequency [Hz] Figure 14: Comparison between the I/O and OMA FRFs for measurement point 17 (right wing tip) i=8 and j=14 (impact point) I/O FRF MVM Synthetized FRF (OMA Empty-Full) RBN Synthetized FRF (OMA Empty-Full) Amplitude [db] Phase [rad] Frequency [Hz] Figure 15: Comparison between the I/O and OMA FRFs for measurement point 8 (left wing tip).

16 I/O FRF MVM Synthetized FRF (OMA Empty-Full) RBN Synthetized FRF (OMA Empty-Full) Amplitude [db] Phase [rad] Frequency [Hz] Figure 16: Comparison between the I/O and OMA FRFs for measurement point 11 (left wing tip), 4. CONCLUDING REMARKS In this paper, two procedures for the identification of the modal masses of a UAV have been proposed and their accuracy investigated. Different tests have been carried out in order to assess the efficiency of the methodology. In particular, both a traditional input/output impact test and an environmental test have been performed. The environmental test, that allows to identify the structure in its operative conditions, has been completed by exciting the structure with a random excitation with the same energy content as the one recorded during a typical UAV levelled flight and recording the response at certain measurement points. In order to focus on the identification process as much as possible, the vibration environment has been obtained using a close-loop controlled excitation. The evaluation of the modal masses requires to execute two different tests in which the mass properties of the structure are modified in order to move the natural frequencies keeping the mode shapes unchanged. This operation has been performed by designing a mass changing device to be used aboard of the UAV. Such device consists of a set of two elastic balloons and pipes, located at the UAV wing tips, that allows the variation of mass of the wing, as a consequence of the water-flow from the balloons. The experimental work showed that the Operational Modal Analysis has an acceptable reliability with respect to the traditional modal analysis. Indeed, the modal parameters estimated by applying a frequency domain OMA method, the Hilbert Transform Method, are well correlated with those coming from the input/output test. Moreover, the modal masses estimated with the two proposed procedures are in acceptable agreement with those calculated by the traditional test. Finally, the comparison between the input/output FRFs and the synthesized ones from the operational modal parameters shows differences due to the missing information about the lower and upper residuals and suffered from the uncertainties in the damping ratio estimates. REFERENCES [1] Ewins, D.J. (1984) Modal Testing: Theory and Practice, Research Studies Press. [2] Brincker, R., Ventura, C.E. and Andersen, P. (23) Why output-only modal testing is a desirable tool for a wide range of practical application, In: Proc. XXI IMAC (pp ). Orlando, FL,

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