LOGO IOMAC'11 4th International Operational Modal Analysis Conference Identification of Time-Variant Systems Using Wavelet Analysis of Force and Acceleration Response Signals X. Xu 1,, W. J. Staszewski 1,3 Z. Y. Shi, S. Fassois 4 1. University of Sheffield, UK. Nanjing University of Aeronautics & Astronautics, China 3. AGH University of Science and Technology, Poland 4. University of Patras, Greece 011-5
contents 1 introduction formulation 3 simulation 4 conclusion
introduction Many engineering structures and systems have time-varying dynamic parameters and exhibit timevarying dynamic properties. Aircraft Structures
introduction Many engineering structures and systems have time-varying dynamic parameters and exhibit timevarying dynamic properties. Cranes
introduction Many engineering structures and systems have time-varying dynamic parameters and exhibit timevarying dynamic properties. Aerospace structures
introduction Many engineering structures and systems have time-varying dynamic parameters and exhibit timevarying dynamic properties. Vehicles (trains) moving on continuous bridges.
introduction Many engineering structures and systems have time-varying dynamic parameters and exhibit timevarying dynamic properties. Skyscrapers
introduction Many engineering structures and systems have time-varying dynamic parameters and exhibit timevarying dynamic properties. Robot arms
introduction Many engineering structures and systems have time-varying dynamic parameters and exhibit timevarying dynamic properties. Tanks
introduction LTV identification methods empirical mode decomposition Functional Series Time-dependent Auto-Regressive Moving Average Wavelet transform
introduction A time-invariant systems identification method has been proposed in reference[1]. Functional integration to obtain velocities and displacements from accelerations responses.
formulation 1. Continuous wavelet transform a b ( 1 t b { W y}( a, b) y( ( ) dt a a is a scale parameter, typically a positive real number; (1) is a shift parameter, indicating locality of transformation; The overbar indicates complex conjugate. is a mother wavelet function and satisfies two conditions: 1) ) Assuming t i ( dt 0, i 0,1 ( ) 1 ( ) ( ) 0 ( has the first and second integrals ( ) ( t ) 1 t () (3)
formulation. CWT algorithm for functional integration Assuming that the first integral Y 1( t ) of the function y( exists 1 t b { W ( y( d}( a, b) { Y1 ( y0} ( ) dt (4) y a 0is a constant a Partial integration of the right hand side t b a t b a t b Y 1( y } ( ) dt Y 1( 1 ( ) y( 1 ( ) dt a a a a a { 0 1 a t y ( 0 b ) dt a (5) Based on equations () and (3), two terms are equal to zero.
formulation { W ( y( d}( a, b) { W( Y1 ( y0)}( a, b) a{ W y}( a, b) 1 a t b y t dt a ( ) 1 ( ) a (6) The CWT algorithm for functional integration can be used by applying the wavelet transform to the functiony ( with 1( t ) used as the mother wavelet. Double functional integration can be performed using the same approach: { W ( y( dtdt )}( a, b) { W( Y ( y1t a { W y}( a, b) y)}( a, b) a t b y t dt a ( ) ( ) a (7) Y ( ) is the doubled integral of ( ; and are arbitrary constants. t. CWT algorithm for functional integration y y1 y
formulation 3. LTV identification procedure p degree-of-freedom (DOF) linear time-variant system M( x( E( x ( K( x( f( (8) p M (, E ( and K( are ( ) time-variant mass, damping p and stiffness matrices. x( t x ( dt 0 x(0) (9) x( t x( dt x (0) x(0) x(0 ) 0 x(0) t t 0 0 x( dt x (0) t x(0) (10), and are constant vectors determined by initial conditions.
formulation 3. LTV identification procedure Assuming M (, E ( and K( matrices are approximately constant in a very short time. M( { W x }( a, b) ae( { W x }( a, b) a 1 { W x} 1 a { W x} M( E( K( a{ W x} { W f} K( { W x }( a, b) { W Equation (1) is a set of linear algebraic equations. Considering a sliding time window of proper length, a LS problem may be set up and solved to extract the system physical parameters. f}( a, b) (11) (1)
simulation 1. A bridge with a heavy traversing vehicle steel beam: 670 50 0 mm steel sliding mass element: 170 50 50 mm constant speed: v 0.3 m/s
simulation. The beam is modelled using finite element analysis Mass: M ( e M e ( Stiffness: K ( K e ( Damping: C ( e C e ( e 156 L 54 13L e e LA L 4L 13L 3L M ( AM c 40 54 13L 156 L M e ( K e ( and C e ( are element 13L 3L L 4L, mass, stiffness and damping 1 6L 1 6L matrices. e e E( I 6L 4L 6L L K ( E( IK c 3 L is the length of element. L 1L 6L 1 6L 6L L 6L 4L 6 C e e e ( M ( K ( is Raleigh proportional damping. M K M K 10 10 6
simulation 3. Prepare for identification 1. excitation---------- zero-mean, Gaussian, random force. responses---------- Newmark 3. all initial conditions---------- zero 4. sampling frequency---------- f 1000Hz 4 5. Mexican hat mother wavelet --------- ( ( t 6t 3) e 6. wavelet scale and shift parameters: scale a in which s 0.01 j 0. 01 j shift b (16k 8) k 0: 0.03:8 extract frequency components below 50 Hz j 60, 70 t above 50 Hz
simulation 4. Identification results Solid red lines are True value ; Dotted ones are estimated value. force and acceleration response signals of all four masses are needed; 1000 samples points are used for setting up the linear LS problem.
simulation 5. About the results Can not be correctly identified for the beginning and end of the analysed time records due to the edge effects related to the CWT calculations. Define the signal-to-noise ratio Srs SNR Srs Sn ------ pseudo (aggregate over the time duration) standard deviation of the original response signal. Sn ------ pseudo standard deviation of the added noise. Define mean absolute percentage error MAPE N P i Pˆ 1 i 100% N 1 P i i P i Pˆi ----the true frequency ----the estimated frequency
simulation 5. About the results Table 1: MAPE errors of natural frequency identification with different SNR SNR values f1 (%) f (%) f3 (%) 100 1.4845 1.5064 1.0686 90 1.7963 1.956 1.3649 80.8430 4.0550.067 50 3.830 4.18 3.46 no noise 0.5151 1.084 0.3964 MAPE error is in all cases smaller than 5%
conclusion 1 A LTV identification method based on Continuous Wavelet Transform is implemented and applied. This method has been used in LTI identification. 3 The simulation illustrates the performance of the method with respect to noise contamination. Further research work - Confirm the good performance and the antinoise ability.
acknowledgements The work described in this paper was supported by the Jang Su Province in China (innovative project for postgraduate students, grant No. CX10B 088Z) and by the Foundation for Polish Science (project FNP-WELCOME.010-3/).
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