Real Time Failure Detection in Smart Structures using Wavelet Analysis

Transcription

1 Real Time Failure Detection in Smart Structures using Wavelet Analysis M-F. Dandine a, D. Necsulescu b LPMI Laboratoire M.I., ENSAM, 2 bd du Ronceray, Angers, France a Department of Mechanical Engineering, University of Ottawa, Ottawa, Canada b Abstract Flexible structures equipped with piezoelectric actuators and sensors have the capability of detecting in real time the occurrence of defects in structural connections by generating flexural excitations and analysing the response. In previous work, this method used transient response and frequency analysis for computing damage index for the flexible structure. In the present paper a new approach, based on wavelet analysis is proposed. Analytical and experimental results illustrate the capability of the proposed method to detect the occurrence of failures in smart structures. Introduction High performance aeronautical structures require to include an increasing proportion of composite materials. Such structures are particularly susceptible to failures due to shocks and fatigue. Health monitoring can not be limited to ground based tests and the development of failure detection during actual operation is highly desirable. The proposed method is based on applying finite duration sinusoidal excitations with piezoelectric actuators embedded in the structure, followed by the acquisition of the structural response using embedded piezoelectric sensors [1]. The evolution in time of the response is used to detect the evolution of the mechanical properties of the structure. In this paper are presented the method, signal processing of the response to identify vibrations signatures of the structure and illustrative experimental results. Proposed Method This method is based on the fact that structural damage is associated in time by changes in the vibration frequencies from healthy to damaged state of the structure. Changes of these frequencies are not significant unless the damage is very important. For this reason, the method uses signatures for transverse vibration response to harmonic excitations of finite duration.

2 Theoretical Model The equation for transverse vibrations of a slender beam of length L, with no external force or torque applied, is [5, 6] 2 4 v(x, t) E ¼ I v(x, t) - = (1) 2 4 t ¼ a dx where: v(x,t) is the deflection, function of beam longitudinal axis x and time E is Young s elastic modulus, I is the cross-sectional area moment of inertia about the beam longitudinal axis r is mass density a is rectangular cross section area. The separation of variables approach with a solution of the form v(x,t) = p(t)¼r(x) (2) transform the above partial differential equation in two ordinary differential equations, one for p(t) and the other for r(x) d 2 p(t) / dt 2 = w 2 ¼p(t) = 0 (3) d 4 r(x) / dx 4 = b 4 ¼r(x) = 0 where: b 4 = r¼ a¼ w 2 /( E¼I) (4) These two ordinary differential equations have the solutions: p(t) = cos (w¼t + a) (5) r(x) = A¼sin (b¼x) + B¼cos (b¼x) + C¼sinh (b¼x) + D¼cosh (b¼x) (6) Four boundary conditions are required to determine the four coefficients A, B, C and D. These boundary conditions are specified by the beam configuration. The beam under study is considered clamped at x=0 and with variable state from clamped (undamaged connection) to free (totally damaged connection) at x = L, depending on the level of damage of the connection. Boundary conditions for clamped end A at x=0 are: r(x) = 0 (7) d r(x) / dx = 0 (8) while the boundary conditions for end B at x=l are dependent of the degree of tightness coefficients, x and x T, assumed to take an infinite value for clamped and zero for free state 3 r(x) ¼ I = dx E 3 ¼ k ¼ r(x) (9) 2 r(x) E ¼ I = - T ¼ k T ¼ r(x) 3 dx The last two boundary conditions correspond to the following states of end B: - x = 0 and x T = 0, free end, (10)

3 - x = 1 and x T = 1, linear spring with constant k and torsional spring with constant k T, - x and x T tending toward infinity, or 1/x and 1/x T tending toward 0, for clamped end. In the case of the clamped end A, at x=0, the solution for r(x) gives: B + D = 0 and A + C = 0 Using the above results D = - B and C = - A, for end B, at x=l, the solution for r(x), gives :k A¼[- E¼I¼b 3 ¼cos (b¼l) - E¼I¼b 3 ¼cosh (b¼l) + x¼k¼sin (b¼l) + x¼k¼sinh (b¼l)] + B¼[E¼I¼b 3 ¼sin (b¼l)-e¼i¼b 3 ¼sinh (b¼l)+x¼k¼cos (b¼l)-x¼k¼cosh(b¼l)]= 0 (11) A¼[-E¼I¼b 2 ¼sin (b¼l)+e¼i¼b 2 ¼sinh (b¼l) - x T¼k T¼cos (b¼l) - x T¼k T¼cosh (b¼l)] + B¼[-E¼I¼b 2 ¼cos (b¼l)-e¼i¼b 2 ¼cosh (b¼l)+x T¼ k T ¼sin (b¼l)+x T ¼k T ¼sinh(b¼L)]=0 (12) The last two homogenous equations give a nonzero solution to the unknowns A and B only for the case that the determinant is zero, a condition which permits the calculation of the natural frequencies of the beam function of the degree of tightness coefficient x and x T. This equation was used for calculating the natural frequencies between the extreme cases for end B, free end (x = 0 and x T = 0) and clamped end (1/x and 1/x T tending toward 0) needed for the design of the experiments. Experimental Set-up For testing the feasibility of the proposed method, a simple slender beam subject to piezoelectric actuator excitations was used. Transverse vibrations response was measured with a piezoelectric sensor. The diagram of the experimental set-up used in this study is shown in Fig.1. Clamped End A Actuators Sensor Beam End B l a l s L Figure 1 - Experimental set-up diagram End A of the beam was maintained clamped, with two tight bolts, while end B of the beam was modified gradually from clamped (representing a healthy state), to free, (representing a totally damaged state). End B in healthy state was achieved by tightening the two bolts against Belleville washers. The degradation of the end B connection was achieved by reducing gradually the clamping force applied [1]. The degree of tightness can be represented by coefficient x and x T assumed to take an

4 infinite value for healthy state and zero for totally damaged state. These coefficients x and x T were also used in the theoretical model. Other measurements concerning twice bending and stretching of the beam were pursued; the experimental device was slightly different with the employment of only two transducers. A transducer for applying locally bending and stretching of the beam, and just on the other side of the beam an other transducer for the measurement of the corresponding deformation of the material. Data Acquisition and Actuators Control Scheme PC DSP Beam, Actuators, Sensor DAQ Board and Driver Figure 2 - Block diagram Data Acquisition hardware and software for PC were developed at ENSAM [1, 3]. Data acquisition board has analog and digital inputs compatible with the bandwidth of the dynamics of the experimental set-up. Data analysis was carried out using both wavelet method and Fourier transform. Given the transitory nature of the harmonic responses obtained, wavelet method proved particularly interesting [1]. Harmonic Excitation of the Structure Harmonic excitation was generated by a piezoelectric actuator glued to the beam. Excitation frequency was chosen to be a resonant frequency of the clamped clamped beam. A mode of vibration, which was not too close to other adjacent modes of vibration, was chosen. The reason is the requirement that the slide of natural frequency from clamped-clamped to clamped-free beam will not pass another clamped-clamped natural frequency. The excitation was applied for a finite time. Fig. 3 shows a typical measurement result of the excitation and the response relaxation for the experimental set-up used in this study. The following domains would be observed : - Excitation domain, with the duration chosen such that the initial transient is followed by a long enough steady state part,

5 Excitation Relaxation Fig.3 Response to finite time sinusoidal excitation - Relaxation domain, which starts after the excitation source is turned off and is characterised by duration and amplitude variation strongly dependent of the boundary conditions, excitation type and position of actuators. Experimental Results For the experimental set-up used in this study, the deflections due to transversal vibrations were measured and analysed. Preliminary experiments show important time variation of the amplitude of the response and even amplitude discontinuities. For this reason, the study focused on using wavelet method rather than Fourier transform for response analysis [1, 3]. Fig. 4 and Fig. 5 show, on the left hand side, the responses recorded versus time and, on the right hand side, the corresponding results from wavelet analysis, for a decreasing tightness of the connection at end B of the beam, i.e. for a increasing values of the corresponding tightness coefficients, x and x T. For Fig. 4 (concerning only bending) these values are only qualitatively ordered. The quantification will be investigated later. The shape of the time variation of the amplitude changes significantly as the tightness decreases. Moreover, wavelet analysis results are distinctly different for various degrees of tightness, in terms of frequency versus time dependence, and this feature can be used for discriminating between a healthy and a damaged structure. After these first experiments, and in sort to increase efficiency, it was preferred to applied simultaneously bending and stretching actions, and to measure the deflection at the same beam section. Fig.5 concerns some experimental results. The discontinuity of the amplitude variation versus time is also particularly visible in the wavelet analysis results. Such a processing way permits to increase the sensivity of the response to frequency variations. Two significant indices has been proposed: - The first one concerning amplitudes discontinuities [1], - The other concerning the frequency modulations of the receiving signal.

6 Fig.4 Experimental results with only transverse vibrations; Connections: A clamped, 1) B clamped ; 2) B damaged ; 3) B free, Fig.5 Experimental results with transverse and longitudinal vibrations. Connections: A clamped, 1) B clamped; 2) B lightly damaged; 3) B strongly damaged,

7 Now, we prefer strongly to use these shapes indices than amplitudes indices (which are more difficult to take into account). Conclusions Real time failure detection in smart structures can be achieved using wavelet analysis of the harmonic response. This approach permits to discriminate between cases with various degrees of damage of the structural connections using the slide of the natural frequency with the level of the connection tightness. Now the continuation of this work concerns different aspects: - Damage localization, - Detection and localization of intern material defects, - Investigation concerning composite materials, - Application of this method to plates, - Optimization of transducer positions l a and l s, - Measurement and data processing automation, References [1] Dandine M-F.,. Necsulescu D., Benillan S. and Cadou S., Détection des défauts de liaison et structuraux dans les structures aéronautiques, Proceedings of the Third International Conference on Composites (CANCOM 2001), Technomic Publishing, pp [2] Necsulescu D., De Abreu R.F., Bakhtiari-Nejad F., Smart Structure Integrity Monitoring using Transient Response, Computational Methods for Smart Structures and Materials, (Santini P., Marchetti M., Brebbia C. Ed), WTT Press, 1998, pp [3] Dandine M-F., Yelle H., Koffi D.,Etude du comportement vibratoire des engrenages en plastique par analyse acoustique, Proceedings of CANCAM Conference, Université Laval, Quebec, Canada, Juin 1997 [4] Plantier G., Guigou C., Nicolas J., Piaud J.B. and Charette F., Variational Analysis of a Thin Finite Beam Excitation with a Single Piezoelectric Actuator Including Boundary Layer and Dynamical Effects. Acta Acustica April 1995, p [5] Gorman D. J., Free Vibration Analysis of Beams and Shafts, J. Wiley & Sons, 1975 [6] Inman D. J., Engineering Vibration, Prentice Hall, 1994 [7] Gandhi M.V. and Thomson B. S., Smart Materials and Structures, Chapman & Hall, 1992