Randomising the most effective parameters ( and )

As discussed earlier, the major problem in the uncertainty analysis of industrial-scale finite element models is the issue of computational workload. Although the perturbation method can sufficiently overcome this issue, reducing the number of uncertain variables can help further. If a sensitivity analysis reveals that some of the selected input variables are of low activity in a particular model, excluding them from the uncertainty analysis will be a right decision. Even if there is a high level of uncertainty about one variable, it can be excluded when its influence on a particular unstable mode is negligible. Both the levels of sensitivity and uncertainty should be considered when right uncertain input parameters are selected. To illustrate this point, it is worthwhile to re-consider the results of the sensitivity analysis. Table 7.1 shows that the real part of the first eigenvalue which is in the vicinity of instability is highly sensitive to the friction coefficient ( ). Its sensitivity to the contact stiffness ( ) is also considerable. However, for example, the contact damping has a minor effect on this particular mode. Although contact damping is sometimes considered one of the major sources of uncertainty, here it is not necessary to include this variable in the uncertainty analysis.

In order to get a better understanding of what is being discussed, the uncertainty analysis is re-performed in this section. However, only the variations of and are considered this time. Since the first eigenvalue is at high risk to become unstable, only the probability distribution of its real part is shown here. Figure 7.5 displays the distribution of the real part of the first eigenvalue.

Two interesting points may be concluded based on these results. First, when the parameters of low activity are set aside from the analysis, the results are almost the same with less computational workload. The mean value and variance of the output distribution for the real part of the first eigenvalue is and in Figure 7.5, while they were and in Figure 7.3. Secondly, the number of samples in the conventional Monte Carlo simulation is 10,000 this time. Consequently, the output histogram is not as smooth as the one presented in the first example. In fact, this is another drawback of the conventional Monte Carlo simulation. Although the convergence in the values of the mean and variance may occur when enough number of samples is collected, a smoother distribution will be achieved when the number of samples is significantly

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increased. Incidentally, even collecting 10,000 samples is not feasible in many practical applications.

Figure ‎7.5. Distribution of the real part of the 1st eigenvalue

7.7.

Reliability analysis

In general, a reliability analysis is performed for evaluating the robustness of a design. In fact, the reliability analysis quantifies the failure probability of the design. The term “failure” is widely used in engineering for various undesired states of a system. In this example, failure means that one of the system eigenvalues becomes unstable, i.e. the sign of the associated real part turns positive. For brake analysts, it is very important to know what percentage of a brake design will „fail‟ either for production variability or due to usage and aging effects. Then, the design modifications which are usually done to reduce brake noises can be carried out for reducing the likelihood of unstable vibration. The failure probability is obtained as:

[{ }] _(7.39) Among the four modes of the system shown in Figure 7.1, only the first one exceeds the stable region under the influence of the uncertainties. Figure 7.6 shows the distribution of the real part of the first eigenvalue under the influence of uncertainty of all selected variables.

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Figure ‎7.6. Reliability analysis

In Figure 7.6, the green triangles highlight the border of the unstable region. By the use of the results of Monte Carlo simulation, the percentage of failure for the first mode is 35.7. If the output distribution is approximated by Gaussian probability density function, the predicted probability of failure is calculated by

[ ] (7.40)

where ( )

√ ∫ ( ₎

and . Accordingly, the approximated failure

probability of the first mode is 35.2 percent. This percentage is calculated by the estimated mean value and standard deviation via the perturbation method.

Moreover, one may say that the failure probability is quite large for the system under this study. In fact, the design point is deliberately set slightly below the critical friction coefficient in order to demonstrate how bad a design can be if the variability and uncertainty of the inputs are not considered carefully.

7.8.

Conclusions

The statistics of the complex eigenvalues are studied in this chapter with the application to friction-induced vibration problems. In order to deal with non-proportional damping and the asymmetry of the stiffness matrix, the state-space equations are used. The second-order perturbation method is then extended for incorporating the variability and uncertainty of input variables. A few expressions for calculating the mean value, variance and pseudo-

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variance of the complex eigenvalues are derived. The mean value of the eigenvalues is, in fact, a complex-valued number which provides the mean values of the real parts and also the mean value of the imaginary parts. The variance and pseudo-variance of the complex eigenvalues include information of the variance of the real parts, the variance of imaginary parts and the covariance of the real and imaginary parts. The correlation between the real and imaginary parts of the eigenvalues can also be obtained by these statistical measures.

The most important outcome of this study is that the distributions of the real and imaginary parts of the complex eigenvalues can efficiently be approximated without decomposing them into two real-valued numbers (real and imaginary parts). In this way, the results are produced in just one-run, while Monte Carlo simulation requires a large number of samples to find the statistical measures of the outputs.

This study also allows the probability of unstable vibration to be predicted and then provides a very useful tool for design.

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8.

Surrogate Modelling

The idea behind surrogate modelling is fully explained in this chapter. In order to illustrate how a surrogate model is constructed, a few simple examples are given in this chapter. There are two major steps for constructing a surrogate model: making a sampling plan and training the replacement model. Several careful considerations must be taken for these two steps. These considerations will be fully discussed here.

Note that making a sampling plan is also termed as “Design of Experiment”. This term is well-known in different disciplines of science and engineering and is abbreviated to “DOE”. However, since in this study “Design and Analysis of Computer Experiment” is abbreviated to “DACE”, it is preferred to use the term “sampling plan” in place of “DOE” only to avoid confusion. However, the concept of these two terms is the same.

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