FSF model

The full 3D fluid-structure-fracture (FSF) model has been first developed to simulate rapid crack propagation in plastic pipes [6], and is adopted in the present work. Apart from fluid-solid coupling issues described elsewhere [3,4,7], there are two main issues that require special care in order to develop predictive model of failures of plastic containers ... 

The EWF results were obtained for two HDPE grades at various test speeds. In addition to conventional EWF results, a special scaling analysis was performed to obtain CZM parameters. Some uncertainties due to the large scatter in measured separation distances were experienced, in particular at intermediate test speeds around 10 mm/s. However, the CZM data will only be used as the initial guess values in the numerical simulations of EWF tests. These simulations are designed for accurate calibration of the CZM parameters. Once the parameters are calibrated, they will be embedded in the predictive FSF model of the drop impact tests. 

This chapter consists of six sections. Section 4.2 introduces the FSF model structure. Section 4.3 examines the properties of the FSF model with a fast data sampling rate. Section 4.4 introduces the concept of a reduced order FSF model. Section 4.5 discusses the use of least squares for estimating the FSF model parameters from input-output data. Section 4.6 excunines the nature of the correlation matrix that arises when using a least squares estimator with an FSF model and the relationship between the elements of this matrix and the energy content of the input signal. 

To derive the frequency sampling filter (FSF) model, we first make use of the inverse Discrete Fourier IVansform (DFT) relationship between the process frequency response and its impulse response, under the assumption that N is an odd number... 

One other point worth making is the relationship between the FSF model parameters and the entire process frequency response. As stated earlier, the parameters of the FSF model are the values of the discrete-time process... 

The properties of the FSF model with respect to choice of the sampling interval are summarized in the theorem below. 

The parameters of the FSF model correspond to the continuous-time process frequency response evaluated at w = 0,radians/time for a fixed value of Tg. Therefore, as At decreases, the number of parameters associated with the model increases, but only in the high frequency region. 

Example 4.1. This example compares a discrete-time, rational transfer function model with the FSF model in terms of the effect of sampling rate on the model parameters. Consider a third order system with time delay given by... 

The FSF model parameters correspond to the frequency rraponse of the discrete-time transfer function models presented in Equations (4.12)-(4.14) at three sets of frequencies ... 

Figure 4.4 Continuous-time frequency response (solid) and FSF model parameters for Example 4-1 ( + At = 0.5/ o At = 5 At = 10 ...

Based on these properties, we will refer to n as the reduced order of the FSF model, which represents the number of significant parameters in the FSF model. We must qualify the use of the term reduced order to distinguish the number of significant parameters in the FSF model (n) from the order of the individual FSF filters (JV). This reduced nth order FSF model can be written in the following form... 

Table 4.1 Reduced FSF model orders corresponding to different truncation levels for the second order system...

The reduced FSF model order n corresponding to maximum truncation levels of 10%, 5% and 1% are listed in Table 4.1 for the above examples, where the neglected frequency response coefficients all have magnitudes less them the indicated percentage of the steady state gain. Note that the reduced orders given in Table 4.1 are independent of the time constant T,... 

Effect of FSF Model Reduction on Process Frequency Response... 

We can attempt to construct the process frequency response using the reduced order FSF model given in Equation (4.15) by letting = and evaluating... 

Figure 4.5 Construction of frequency responses using reduced order FSF models for Example 4-3 (solid true response dash-dotted n = 3 dotted n = 5)...

This section deals with the problem of estimating the parameters of a reduced order FSF model from process input-output data using the least squares algorithm. 

We assume that the process being identified is stable, linear and time invariant and can be accurately represented by a reduced nth order FSF model. For an arbitrary process input u k) and the measured process output y k), the frequency sampling filter model can be written as... 

The least squares estimates of the FSF model parameters are given by... 

For each process output, the p inputs are denoted as ui k), U2 k),..., Up k), the times to steady state for the individual subsystems are given by Ni, N2,- Np, and the reduced orders for each subsystem represented by its own FSF model are chosen to be ni, ri2,..., np. In the matrix representation for this MISO system, the first input ui k) is passed through a set of n frequency sampling filters based on N to form the first n columns in the data matrix, followed by passing the second input U2 k) through a set of U2 frequency sampling filters based on N2 to form the next U2 columns in the data matrix, etc. The parameter vector 9 contains the n firequency response parameters associated with the first subsystem, followed by the n2... 

Figure 4.8 Elements of the correlation matrix using an FSF model structure (dimension 199 X 199 for Example 4-4- Upper diagram the diagonal elements of the correlation matrix lower diagram row sums of the absolute values of the off-diagonal elements...

OBTAINING A STEP RESPONSE MODEL FROM THE FSF MODEL... 

Equation (5.4) gives an explicit relationship between the parameters of the FSF model and the step response coefficients. The step response coefficients, evaluated using a reduced order FSF model, can be obtained from... 

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