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Unformatted text preview: Nonlinear System Analysis and Identification 1 1. SISO cubic nonlinear system subject to a Gaussian stationary random input 1.1 Output moments through a squarer with sign The general problem to be analyzed for wave forces on small-diameter fixed structures such as vertical piles is shown in Fig. 1. The structure is assumed to be subject to Gaussian stationary random ocean waves represented by the wave velocity input x ( t ) at a specified depth. This input record produces a wave force output at the same depth denoted by the output record y ( t ). Morison’s equation, which is widely used to study such wave forces on structures, is given by ) ( ) ( ) ( ) ( ) ( ) ( 2 1 t x t x C t x C t d t m t y + = + = & (1) where m ( t ) is the linear inertial force, d ( t ) is the nonlinear drag force proportional to ) ( ) ( t x t x , a squarer with sign, and C i can vary with frequency. A generalization of Morison’s equation, that is a nonlinear wave force model, is drawn in Fig. 2, where the frequency response functions ) ( ) 2 ( ) ( 1 f C f j f H π = and ) ( ) ( 2 f C f A = . The problem of concern is to determine the spectral properties of the linear inertial component m ( t ) and the nonlinear drag component d ( t ) from measurements of the input wave velocity x ( t ) and the total wave force output y ( t ). 1 Julius S. Bendat, Nonlinear System Analysis and Identification from Random Data , John Wiley & Sons, 1990 ) ( t x ) ( f H ) ( f A ) ( t m ) ( t d ) ( t n ) ( t y ) ( t v Squarer with sign Fig. 2 Nonlinear wave force model with parallel linear and nonlinear systems ) ( t x ) ( t y Wave velocity input Wave force output Fig . 1 Nonlinear wave force problem MAE 591 RANDOM DATA C. W. Lee Square-law systems with sign are of great importance in fluid dynamics models involving “dynamic pressure,” defined as x x v ρ ) 2 / 1 ( ~ = where ρ is the fluid density and x is the fluid velocity. This quantity ) ( ~ t v is a key parameter in the description of boundary-layer pressures and drag forces due to the fluid flow over structures. The output probability density function through a squarer with sign, i.e. ) ( ) ( ) ( t x t x t v = , becomes, for zero mean value Gaussian input x , − = = = 2 2 exp 2 2 1 2 ) ( 2 ) ( ) ( x x v v v v p x x p v p σ π σ (2) which is sketched in Fig. 3. The output moments through a square-law system with sign are given by 2 ] [ ] [ = = = x x E v E v µ 4 4 2 3 ] [ ] [ ) ( x vv x E v E R σ = = = ( 3 ) 6 6 3 15 ] [ ] [ x x E v E σ = = v p ( v ) Fig. 3 Output PDF for Gaussian data through a square-law system with sign 1.2 Third-order polynomial least-squares approximation The least-squares approximation to general zero-memory nonlinear system by the third-order polynomial of the form 3 3 2 2 1 x a x a x a v + + ≅ ( 4 ) can be determined from minimization of the mean square error, with respect to i a , defined by 2 Note that all odd-order moments of the zero mean value Gaussian input data are zero. The fourth-order Note that all odd-order moments of the zero mean value Gaussian input data are zero....
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This note was uploaded on 11/21/2009 for the course ME . taught by Professor . during the Spring '09 term at Korea University.
- Spring '09