White-noise excitation

Impulse-response and transfer functions can be measured not only by pulse excitation, but also by excitation with monochromatic, continuous waves (CW), and with continuous noise or stochastic excitation. In general, the transformation executed by the system can be described by an expansion of the acquiired response signal in a series of convolutions of the impulse-response functions with different powers of the excitation [Marl, Schl]. Given the excitation and response functions, the impulse-response functions can be retrieved by deconvolution of the signals. For white noise excitation, deconvolution is equivalent to cross-correlation [Leel]. 

Nonlinear cross-correlation of the system response y(t) (4.2.4) with different powers of a white-noise excitation x(t) yields multi-dimensional impulse-response functions hn (tTl,. . . , CTn),... 

They differ from the kernels it (ti, ..., r ) of the Volterra series only by a faster signal decay with increasing time arguments [Bliil]. For coinciding time arguments the crosscorrelation function is the sum of the n-dimensional impulse-response function h with the impulse-response functions hm of lower orders m < n. The stochastic impulse-response functions h are the kernels of an expansion of the system response y(t) similar to the Volterra series (4.2.4) but with functionals orthogonalized for white-noise excitation x t) [Bliil, Marl, Leel, Schl], This expansion is known by the name Wiener series, and the h are referred to as Wiener kernels. 

A single-degree-of-freedom (SDOF) system is subjected to zero-mean stationary white noise excitation with spectral intensity Spo = 0.0048 s. The mass, damping coefficient and stiffness are taken to be Af = 1.0 kg, C = 0.4ar N s/m, and K = (4 r) N/m so that the natural frequency and damping ratio of the system are 2.0 Hz and 5.0%, respectively. Its velocity is sampled at a rate of 100 Hz for 10 s. To generate the time history of the velocity measurement, 10% of the rms noise is superimposed onto the velocity time history, i.e., the root-mean-square (rms) of the measurement noise is equal to 10% of the rms value of the noise-free velocity of... 

The second example uses a ten-story shear-building model. It has equal floor mass and interstory stiffness distributed over all stories. The building is subjected to base acceleration adequately modeled by stationary Gaussian white noise excitation with spectral intensity... 

Although the above formulation was presented for displacement time history, it can be easily modified for velocity or acceleration measurements. In this case the right hand side of Equations (3.4) or (3.17) can be modified with the corresponding expressions for velocity or acceleration. Of course, the case of relative acceleration with white noise excitation is not realistic since the response variance is infinity. However, the absolute acceleration measurements can be considered for ground excitation. Another choice is to utilize a band-limited excitation model. 

Proppe, C. Exact stationary probability density functions fo" nonlinear systems under Poisson white noise excitation, bitemational Journal of NrmlinearMechanics 38(4) (2003), 557-564. 

The testing procedure comprises two types of tests. The first raie is a low level white noise test to obtain the main dynamic characteristics of the wall (natural modes and frequencies, damping ratio). This test lasts about 3 min, to reach a stationary vibration state. A random white noise excitation with frequency content between 1 and 100 Hz and at Root-Mean-Square (RMS) level of about 0.1 g is generated using an Advantest R9211C Spectmm Analyser. 

Detailed evaluation of the load transfer mechanism and of the contact length was also presented. The identification of the natural vibration modes under low level white noise excitation raises the question of the accuracy of the values of the shear modulus usually suggested in the main reference documents, this modulus seeming to be significantly overestimated for the studied type of masonry. 

Based on the method illustrated in Figure 5, resonance frequencies calculated for seven White Noise excitations are shown in Figure 6. Firstly, it can been seen that, the resonance frequency of soft rock model (1 model) is smaller than that of hard rock model (2 model). Then another observation is that, as test goes on (along with the increasing excitation intensity), the resonance frequency of two model slopes decreases. The sharp drop begins in White 5 excitation, which indicates a sudden change of the internal structure of the model slope, when the excitation intensity increases up to 0.5 g. 

Figure 5. Model slope transfer functions of horizontal (X) shakings (a), based on the White Noise excitation (b) at the first stage of test.

These factors have been considered in more detail by Gabrielli et al [6], who compare sine wave correlation methods with those employing random (white noise) excitation signals. 

First, linear elastic SDOF and MDOF structures with at rest initial conditions are subjected to white noise excitation. It is found that the mean up-crossing rates obtained using FORM are in very good agreement with available closed-form solutions (Lutes Sarkani 1997) as shown in Figures 9(a) and (b) for SDOF and MDOF systems, respectively, when a sufficiently small time-interval, dt, is used in discretizing the white noise excitation process. 

Finally f(t) is the external force vector, whose only non-zero entry is the white noise exciting the Kanai-Tajimi filter ... 

It can be shown, for example, with white noise excitation if the "true" linearization coefficients are used in the linear model, then the first two moments of the linearized system will be identical to the first two moments of the true non-linear system. Thus, it is of interest for us to study more closely the question of the coefficients in the linearized model. 

In this present paper, we shall establish a somewhat surprising result from parameter estimation theory. Indeed in the white noise excitation case, it has been established for the stationary case that the maximum likelihood estimates of the coefficients will converge almost surely to the true parameter values for linear or non-linear Ito equations [ 9], [10]. Let us now pose the following question ... 

The unexpected answer that we establish in this paper is that these parameter estimates will converge to the "true" statistical. linearization coefficients for the white noise excitation case. Even of more conceptual interest is that the specific form of the non-linearity does not have to be known. Thus, in order to construct the linear model, we only require the model order, as well as the observed response data from the true system. 

Let us now turn to the question of estimation of parameters where observed data are basic to carrying out the model fitting procedures. For gaussian white noise excited systems the parameter estimation problem is quite well understood at this time [see [10], and references therein contained]. 

There is, moreover, an extremely important point to stress here. We need only observe the response vector (y(t), t [0,T] and evaluate (3.3) with x(s) replaced by y(s). Thus, in the white noise case, the actual form of the non-linearity does not have to be known. Indeed, we can obtain our statistically equivalent linear system without any assumptions on the nature of the true non -linearity. All the information that is required will reside in the response, y(t), for the white noise excited nonlinear system. 

Exact Solutions for Nonlinear Systems under Parametric and External White-Noise Excitations... 

When parametric (or multiplicative) white noise excitations are also present, namely, the excitations also appear in the coefficients of the unknowns in the equations of motion, solution to a reduced Fokker-Planck equation becomes... 

Yong, Y. and Lin, Y. K., Exact stationary response solutions for second order nonlinear systems under parametric and e cternal white-noise excitations, J. of Appl. Mechanics, 54 (1987) 414-418. 

Lin, Y. K. and Cai, G. Q., Exact solution for nonlinear systems under parametric and external white-noise excitations. Part II, to appear. 

Fig. 8 (a) The measured system responses and (b) the modal responses recovered by CP of the experimental model subject to white noise excitation... 

The probability density evolution method is outlined and illustrated in this entry. It is concluded that (1) the thought of physical stochastic systems provides a new perspective to stochastic dynamics and (2) the probability density evolution method shows its versatility in stochastic dynamics, particularly for MDOF nonlinear systems subjected to non-white noise excitations. However, improvements and extension of the physical stochastic models of dynamic excitations and more robust and efficient numerical algorithms are still needed. 

Here w(t) is modeled as a zero-mean white noise excitation with autocovariance w t)w(t+ t)) = e t) is a deterministic envelope function which... 

It can be noted, as demonstrated by Lin and Cai (1995), that for an SDOF system having nonlinear stiffness, linear damping, and exposed to a Gaussian white-noise excitation, the stationary displacement (x) and the velocity (i) are independent random variables. 

The FPK equation that describes the evolution of the response s probability density (PD) of a nonlinear system excited by an external white noise can be solved numerically by path integral (PI) solution procedures. In essence the PI method is a stepwise calculation of the joint probability density function of a set of state space variables describing a white-noise-excited nonlinear dynamic system. Among the first efforts to develop the PI method into numerical tools are those of Wehner and Wolfer (1983), Sun and Hsu (1990), and Naess and Johnsen (1993). 

The response of a linear system to a broadband excitation is different to one due to a harmonic excitation, since the former happens with the natural frequency(ies) of the system. In cases when the broadband excitation has an almost constant spectrum over the range of the natural frequencies, it is convenient to replace such an excitation with its liming cases - a white noise. Without loss of generality consider a linear TDOF system with the primary mass subjected to a white noise excitation ... 

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