Damage detection and quantification in an 8 DOF system exhibiting non-linear behavior

Introduction. In this paper we consider an 8 DOF system which can be altered to simulate damage. A simple sketch of the system is shown below.

The system consists of 8 masses connected with springs. The first mass is acted on by a forcing input. Between masses 5 and 6, a "bumper" is installed so as to restrict compression of spring 5. The clearance between the bumper and mass 5 can be adjusted to simulate various levels of non-linearity. For the purposes of this paper, the there will be three levels of non-linearity. The first is actually linear; that is, the bumper never contacts mass 5. The second is allowing 0.015 inch clearance between the bumper and mass 5. This is meant to represent a low level of damage. The third and final level allows only 0.008 inch clearance between the bumper and mass 5. This is meant to represent a higher level of damage. Five, 8 second (4096 point) tests were performed for each damage level. A shaker provided random input. Each mass was instrumented with an accelerometer. Given a certain amount of training data, the primary objective is to predict the probability of damage and possibly the level of damage of the system, given novel data.

The first thing that must be done in a classification problem such as this is to choose the features of the data. In a related experiment done previously, linear predictive coefficients (LPC's) were used as features. As will be discussed in later sections, LPC's and ARMA coefficients are both investigated as appropriate features.

Once the features are chosen, the classification can begin. Previously, a linear discriminant operator has been used to project the data into a single dimension. A Bayesian classification is then used to classify novel data. Advantages and disadvantages to using a linear discriminant operator will also be presented.

Feature selection. As stated above, a related experiment was done in which LPC's were chosen as features (Nix). Essentially, LPC's are used to predict the next value of a signal given previous values of the signal. That is,

where G and a_i are the LPC's. The previous experiment consisted of the same 8 DOF system, but without a bumper. Instead, certain springs were changed to simulate damage. In addition, the system was excited with an impulse input, rather than a random input. Note that the LPC's do not explicity reflect the input signal. While this is fine for an impulse input, where the input is zero almost everywhere, intuition tells us that we could obtain better results with ARMA coefficients in the presence of a random input. An ARMA model attempts to express the relationship between the input and output signal in manner similar to the LPC case. That is,

where a_i and b_j are the ARMA coefficients.

The advantage that the LPC's offer is faster computation. However, they do not directly reflect the input signal, whereas ARMA coefficients do.

Classification. As stated previously, a linear discriminant operator (Fisher's discriminant) was used in the classification procedure in (Nix). Essentially, Fisher's discriminant attempts to find the vector w such that for two classes of data, A and B, the function

is maximized, where y_k = w^Tx_k,   and . Once w is determined, and assume that the prior for A is discrete, the probability that new data, y_new = w^Tx_new came from class A is given by

  (1)

The use of Fisher's discriminant does have some disadvantages. First, it assumes Gaussian distribution for A and B. Second, as presented it can only be used in the classification of two data sets. In addition, it is possible for it to produce correct, yet somewhat misleading results, as evidenced by the following example.

B

A

The topmost heavy dot on the line orthogonal to the line the distributions are projected onto is new data point 1. The bottommost dot is new data point 4.

Clearly, the all the new data points get projected to the same point when Fisher's discriminant is used. Thus, if one were asked to classify data point 4, Fisher's discriminant does not offer much help. However, if (1) is simply applied to the two dimensional data, one could say with a relatively high degree of certainty that data point 4 belongs to class B. In addition, simply applying (1) to the full dimensioned data, there is no assumption of normality, nor is the number of classes restricted to two.

Further, if normality is assumed, (1) is just as easy to implement in higher dimensions as it is in one dimension.

Application to non-linear 8 DOF data.