Identification of Time-Variant Systems Using Wavelet Analysis of Force and Acceleration Response Signals

The identification of dynamic parameters is of prime importance in vibration analysis. A number of different time and frequency domain methods have been developed for vibration analysis of time-invariant systems. However, many engineering systems exhibit timevariant behavior. This paper presents a simulation example of a time-variant approach used for parameter identification in a simple vibration system with time-varying mass characteristics, which is a steel beam traversed by a mass travelling at constant velocity. Force excitation and acceleration response signals are analysed to reveal time-varying nature of the system. Wavelet analysis is used for system identification. The method is based on a recently developed direct identification algorithm. 2 IOMAC'11 – 4 th International Operational Modal Analysis Conference 2 CONTINUOUS WAVELET TRANSFORM ALGORITHM OF A FUNCTION’S INTEGRATION 2.1 Continuous Wavelet Transform Theory The CWT of a function of time ) (t y can be defined as       dt a b t t y a b a y W ) ( ) ( 1 ) , }( {   (1) where a is a scale parameter, typically a positive real number;b is a shift parameter, indicating locality of transformation; ) (t  is a mother wavelet function, and the overbar indicates complex conjugate. The notation ) , }( { b a y W indicates that the function ) (t y is mapped to the ) , ( b a plane by the wavelet transform with the mother wavelet function ) (t  . In this work, we assume that the mother wavelet function ) (t  satisfies the two following conditions 1) the mother wavelet function has at least two vanishing moments, i.e. 1 , 0 , 0 ) (       i dt t t  (2) 2) the mother wavelet has the first and second integrals ) ( 1 t  and ) ( 2 t  decaying fast and vanishing at infinity, i.e. 0 ) ( ) ( ) ( 2 1          (3) 2.2 Continuous Wavelet Transform Algorithm for Functional Integration Assuming that the first integral ) ( 1 t Y of the function ) (t y exists, the wavelet transform of this integral can be defined as        } ) ( { 1 ) , )}( ) ( ( { 0 1 y t Y a b a dt t y W dt a b t ) (   (4) where 0 y is a constant. Partial integration of the right hand side of equation (4) leads to the following result        a a dt a b t y t Y ) ( } ) ( { 0 1  dt a b t t y a a a b t t Y ) ( ) ( ) ( ) ( 1 1 1                 