SIMULATION OF NONLINEAR RANDOM VIBRATIONS USING ARTIFICIAL NEURAL NETWORKS

Thomas L. Paez* Susan Tucker

Chris O'Gorman

Sandia National Laboratories Albuquerque, New Mexico, USA

Abstract

The simulation of mechanical system random vibrations is important in structural dynamics, but it is particularly difficult when the system under consideration is nonlinear. Artificial neural networks provide a useful tool for the modeling of nonlinear systems, however, such modeling may be inefficient or insufficiently accurate when the system under consideration is complex. This paper shows that there are several transformations that can be used to uncouple and simplify the components of motion of a complex nonlinear system, thereby making its modeling and random vibration simulation, via component modeling with artificial neural networks, a much simpler problem. A numerical example is presented.

Introduction

Structural system random vibration simulations are required in a wide variety of applications. Development of techniques that can generate such simulations accurately and efficiently is important, particularly in frameworks where numerous simulations are required, frameworks like Monte Carlo analysis. In practically all situations where the excitation is Gaussian and the system under consideration is nonlinear, the responses will be nonlinear and non-Gaussian, and it is important that simulations preserve the characteristics of the response as accurately as possible.

Artificial neural networks (ANNs) have been applied to the autoregressive modeling of nonlinear system random vibrations. Investigations have shown that nonlinear structures can be modeled with ANNs, at least in the case of simple systems. (See, for example, Yamamoto, [15].) In principle, complicated systems can also be modeled using ANNs. This can be done directly (i.e., without any substantial transformation of the input or output data) using many types of ANNs. As the complexity of the system increases, an ANN that can naturally and efficiently accommodate a large number of inputs must be used for system simulation. When a mechanical system is modeled using an autoregressive ANN to directly simulate motions at a large

* This work was supported by the United States Department of Energy under contract No. DE-AC04-94AL85000. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy.

CnstRr8UllON OF THIS DOCUMENT IS UNLIM &

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liabili- ty or responsibility for the accuracy, completeness, or usefulness of any information, appa- ratus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessar- ily state or reflect those of the United States Government or any agency thereof.

number of degrees of freedom, a very large number of exemplars of motion will be required to train the ANN to accurately represent the system. The reason is that it takes a large number of exemplars to adequately populate a high dimensional input space.

This paper shows how the ANN modeling of nonlinear structures can be made more efficient and accurate when using data measured during experimental, stationary random vibration. There are a number of operations that can be performed on the data to accomplish these goals. Among these are: (1) principal component analysis, (2) localized modal filtering, (3) elimination of statistically dependent components of motion, and (4) transformation of the components of motion to statistically independent, standard normal random signals. These operations are briefly described in the following sections, along with the modeling of the components with two types of ANN - the feed forward back propagation network (BPN) and the connectionist normalized linear spline (CNLS) network. An example is included to assess the random vibration simulation capabilities of the ANNs. The accuracy of the simulations is evaluated in terms of spectral and probabilistic measures.

Data Reduction

It is important to reduce the dimensions of motion of a complex system for the reason listed in the introduction, Le., the amount of data required to train a very complex system directly is great. Further, because the CNLS net is a local approximation network, it is important to minimize the number of network inputs. The reason is that network size grows rapidly with the number of inputs. To limit the complexity of the input/output mappings required to model a complex system, the system motions can be decomposed into simple components. In general, the ANN modeling of physical systems can often be made more efficient and accurate by preprocessing the training data using any of a number of simplifying transformations. Among these are: (1) principal component analysis, (2) localized modal filtering, (3) elimination of statistically dependent components of motion, and (4) transformation of the components of motion to statistically independent, standard normal random signals. The hopes in using these transformations are that the ANN required to model a component of behavior will be simpler than a model for the entire system, and that a simpler model will be easier to train. Exactly how these transformations fit into the response simulation framework will be discussed more later, the overall framework is described in Figure 4. These operations are briefly described in the following subsections.

Principal Component Analysis - SVD

Principal component analysis of complex structural system motions is aimed at decomposing the motions into their essential constituent parts. A special example of this is the modal decomposition of linear systems, and analogous decompositions can be defined for nonlinear systems using, for example, singular value decomposition (SVD), or a principal component analysis ANN.

SVD is described in detail, for example, in Golub and Van Loan [4]. It can be used to clecompose linear or nonlinear structural motions in the following way. Let X be an n x N matrix representing the motion of a structural system at N transducer locations and at n consecutive times. The form of the SVD is

(1) T x = UWVT iz uwv

V is an N x N matrix; its columns describe the characteristic shapes present in the rows of X . W is an N x N diagonal matrix whose nonnegative elements characterize the amplitudes of the corresponding shapes in V. The elements in Ware normally arranged in descending order. Its largest elements correspond to the most important components in the representation. U is an n x N ma&, its columns are filtered versions of the motions represented by the columns in X. The reason that U is said to be a filtered form of the motions in X is that the columns of both V and U are orthonormal with respect to themselves. Therefore,

u = xvw-' (2)

and VW-' serves as a filtering coefficient. Because some of the elements of W may be zero or nearly zero, indicating components that do not contribute substantially to the characterization of X, the elements of W-' are taken as the inverses of the diagonal elements in W that are greater than a cutoff level; Zen, or near zero elements in W are replaced with zeros in W-l . The approximate equality on the right side of Eq. (1) indicates that some components of the representation can be zeroed, and still maintain a good approximation to X. In the experimental framework, the components of W whose ratio to the maximum value is lower than the experiment noise-to-signal ratio are set to zero. The matrices u, w , and Y are the matrices U, W , and V with components removed.

The columns of the matrix u are the principal components of the representation. It is the evolution of the system represented in the columns of u which we seek to simulate. Once models are established to simulate system response through simulation of the columns of u , the models can be used along with system initial conditions to predict structural response. The predicted response can be used, along with Eq. (l), to synthesize response predictions in the original measurement space.

Principal Component Analysis - ANN

ANNs can be used in a number of application frameworks, and one of the focuses of this paper is to show how components of a complex system motion can be modeled with ANNs. However, a particular ANN can also be used as a means for decomposing complex system motions into simpler components. This ANN is the principal component ANN (PCANN). (Baldi and Hornik, [2]) The PCANN is simply a multi-layer network of perceptrons like the standard BPN (Freeman and Skapura, [3]), but it has a particular geometry,

shown in Figure 1. Let (x,) be a row vector of Nelements, the jth row of data in the matrix X defied above. Then the collection of all the rows of X provide n exemplars - both input and output - for training the PCANN in Figure 1. Some important features of the ANN in Figure 1 are that (1) it is a BPN with one hidden layer, and (2) the number of neurons, R, in the hidden layer is smaller than N, the number of columns in the measurement matrix. The idea behind the PCANN is that it compresses the information in the input layer into the information present in the hidden layer, then uses this information to reconstruct, as accurately as possible, the original signal on the output layer. To obtain optimal effect from the PCANN, sigmoidal activation functions would normally be used in the hidden and output layer neurons. However, when linear activation functions are used in the hidden and output layer neurons, the ANN weights are related to the components of the SVD, Eq. (1).

A

A

x2

A

XN

Figure 1. Geometry of the principal component artificial neural network.

Let the jth row of u be the hidden layer outputs, (u,) in Figure 1. The columns of u are the principal components of the PCANN representation. It is the evolution of the system represented in the columns of u which we seek to simulate. Once models are established to simulate system response through simulation of the columns of u, the models can be used along with system initial conditions to predict structural response. The predicted response can be used, along with the portion of the PCANN to the right of the hidden layer, to synthesize response predictions in the original measurement space.

Normally, only one of the principal component analyses described in this and the previous sections would be applied to the data.

Modal Filtering

Modal decomposition of complex system motions is often used for the simplification of mechanical system response when the model for the system is assumed linear or approximately linear. In fact though, data-based modal

.

decomposition can be used on any collection of data; its purpose is to break the data into simple narrowband components, thereby simplifying system characterization, and perhaps simulation. When such a decomposition is used on nonlinear system or general system data, it is often referred to as modal filtering.

Modal filtering of measured data can be performed in one of two frameworks. First, a single modal analysis can be used to filter measured data into its component parts. The problem with this is that when a system is nonlinear, its characteristics can change with response magnitude, and the number of principal components (modes) also changes. It is difficult for a single model to accurately capture such changes. The second alternative is to specify multiple modal filters, and use each one in a particular range of response amplitudes. A very simple realization of this type of analysis would involve the use of two modal models. One would be applied to data below a particular threshold, and the other would be applied to data above the threshold. A more complicated application creates a linear modal model at each step in a system analysis. Such a model is described in Hunter [7].

The form of a modal filter is similar to the SVD, but the means for obtaining the filter factors is much different. Meirovitch [9] describes the theoretical operations included in the definition and use of a modal filter. Allemang and Brown [l] outline practical means for performing data-based modal analysis. The form of the modal filter can be expressed as

T T X = U S @ G U S $ (3)

where X is the same mechanical system motion representation as above. The columns of I p represent the characteristic shapes of the system, and the diagonal matrix S contains normalizing factors. When X comes fkom a linear system the columns of U are the linear modal Components of the system motion. As with the S W , the approximate equality on the right indicates that some components of the representation can be eliminated, and still maintain a good approximation to X. It is the evolution of the columns in u (a reduced form of U) which we seek to simulate. Once models are established to simulate system response through simulation of the columns of u, the models can be used along with system initial conditions to predict structural response. The predicted response can be used, along with Eq. (3), to synthesize response predictions in the original measurement space.

When multiple decompositions are used to filter the motions of a complex system, then multiple expressions like Eq. (3) are used to obtain modal components .

Elimination of Statistically Dependent Components

Under certain circumstances, some of the components produced during principal component analysis are completely or highly statistically dependent upon others. For the sake of efficiency, we seek to eliminate statistically dependent components from the set to be modeled, then reintroduce these

components during physical system simulation. In this way, ANN modeling of structural behavior is simplified.

When a dependency exists, it can be characterized using the conditional expected value of the variables in one column of u given values in another column of u. This requires approximation of a joint probability density function (pdf) of the data, and this can be obtained using the kernel density estimator. (See Silverman, [13].) The pdf approximation is known as the kernel density estimator (kde). Let (ui), j = 1, ..., n, denote the row vectors of the matrix u. The kde of the random source u is given by

where a is an N x l variate vector, K(.) is a kernel function, and h is a window width parameter of the kernel function. The kernel function can be any standard probability density function, and often the pdf of a multivariate standard normal random vector is used. That is

where x is the N x 1 variate vector. Using the kde in Eq. (4), the estimator for the conditional pdf of elements in one column of u given the values in another column of u can be obtained using the standard formulas. (See, for example, Papoulis [ 1 13 .)

A statistical dependency between two columns of u can be detected by forming the bivariate pdf estimator of the random source of the two columns using the kde with data from the columns in question, then evaluating and plotting the conditional expected value of one variable, given a range of values of the other variable. At each point where the conditional expected value is evaluated, the conditional variance can also be evaluated. The conditional expected value and variance can be evaluated for situations in which the data in the two columns of u are lagged with respect to one another. If a lag is found where the conditional variance is uniformly small, Le., small at all locations defined by the conditioning variable, then a statistical dependency has been detected, and the functional form of the dependency is defined by the conditional expected value. The dependent variable can be eliminated from modeling consideration, When modeling has been completed and it is necessary to restore the eliminated component, this can be accomplished using the conditional expected value developed here.

The effect of eliminating components of motion that are completely dependent on other components is to eliminate some columns in the matrix u. Denote the reduced matrix ur; our objective is to model the evolution of the columns of u, with an ANN.

Rosenblatt Transform

The previous step produces a description of the motion of a complex structure in terms of a set of components, none of which is completely statistically dependent on others. We can further transform the components, the columns of ur, into signals that are statistically independent with Gaussian distributions. The transformation that accomplishes this is the Rosenblatt transform. (See Rosenblatt, [ 121.) The Rosenblatt transform has the following form.

..I

where the zi, i = 1, ..., IVY are unconelated, standard normal random variables, @(.) is the cumulative distribution function (cdf) of a standard normal random variable, @-'(.) is its inverse, and the k(.) are the estimated marginal and conditiond cdf's of the random variables that are the sources of the columns of u. These approximate cdf s can be obtained by integrating Eq. (4), and this can be accomplished directly when the kernel used in Eq. (4) is Eq. (5).

The Rosenblatt transformation is uniquely invertible because the exact and approximate cdf s used in Eq. (6) are monotone increasing. The data in the matrix u can be transformed to the standard normal space by using it in Eq. (6) in place of the a's. The matrix z is composed of the elements zi, i = 1, ..., A', and is the same size as the matrix u, with the same number of nonzero columns.

It is the evolution of the values in these columns that is to be simulated with ANNs. Because the columns in z are statistically independent, we need only to create ANN models for signals in individual columns. Once models are established to simulate system response through simulation of the columns of z , the models can be used along with system initial conditions to predict structural response. The predicted response can be used, along with the inverse form of Eqs. (6), to synthesize response predictions in the original measurement space.

Modeling of Component Motion with ANNs

Our ultimate objective is to simulate complex system motion, and we aim to do this by simulating the components of system motion obtained using the decompositions and transformations described above. Many ANNs are suitable for this task. The two that we consider in this paper are the feed forward back propagation network (BPN) and the connectionist normalized linear spline

(CNLS) network. The BPN is the most widely used ANN and it is described in detail in many texts and papers, for example Freeman and Skapura [3], and Haykin [5]. The BPN is very general in the sense that it can approximate mappings of relatively low or very high input dimension. It has been shown that, given sufficient'training data, a BPN with at least one hidden layer and sufficient neurons can approximate a mapping to arbitrary accuracy (Hornik, Stinchcombe, and White, [6]).

The CNLS network is an extension of the radial basis function neural network (Moody and Darken, [lo]). It is described in detail in Jones, et.al., [8]. It is designed to approximate a functional mapping by superimposing the effects of basis functions that approximate the mapping in local regions of the input space. Because it is a local approximation neural network, we cannot use the CNLS network to accurately approximate mappings involving a large number of inputs. The CNLS network has not been widely used for the simulation of oscillatory system behavior.

To simulate a column in z using either of the ANNs described above, we configure the net in an autoregressive framework. This configuration uses as inputs previous response values and the independent excitation, and yields on output, the current response. Figure 2 shows such an application schematically. The quantity zii denotes an element in the ith column of z at the jth time index. The quantity q, denotes the excitation at the jth time index. Lj denotes a lag index. There are rn system response input terms; there are M+2 excitation terms The configuration shown in Figure 2 implies our belief that there is a mapping

and that the ANNs can identify that mapping. The subscript i on the function g(. ) indicates that the functional mapping varies from one column of z to the next, and a different ANN models each mapping. It is normally anticipated that the time increment, At, separating system motion measurements that are the rows of the mamx X is small relative to the period of motion of the highest frequency component we intend to simulate.

We seek to train both types of ANN to model the behavior of the oscillations represented in the columns of z. One ANN of each type (BPN and CNLS net) is used to model each column of z . The inputs to the ANN are current and lagged (past) values of the transformed response and the one-step- into-the-future value, the current value, and lagged values of the excitation. The ANN output is the transformed response one step in the future. Both the ANNs are trained using the scheme described in Figure 3. The ANN inputs are transformed using a feed forward operation. The ANN output is compared to the desired output, and the error is used to modify the ANN parameters. The BPN uses a back propagation and gradient descent scheme in each training step. The CNLS network uses least mean square (LMS) plus random sampling scheme to identify its parameters. The desired effect of training in both types of

ANN is to'modify the parameters of the network to diminish the error of representation of the input/output mapping.

Zj,i +

+ A zj-Ll ,i

...

-j Zj+l,i j-L,,, ,i + N

q j + l + N 4 j + ...

qj-LM + - Figure 2. Schematic of ANN in autoregressive configuration.

4 4 Sy s tern ANN 2

Modify parameters

Train 4

Figure 3. Schematic describing training sequence for ANNs.

Summary

Figure 4 summarizes the decomposition, simulation, and modeling of structural motion described in the previous sections. The principal component analysis in the second box in the top row refers to one of the following: SVD, PCANN. or modal filtering. The svnthesis in the fourth box in the second row refers to the correspondiniinverse' operation - Eq. ( l), the right half of Figure 1, or Eq. (3).

In the example that follows, one form of principal component analysis will be combined with ANN simulations to model a nonlinear structure's random vibrations.

Decomposition and Modeling Operations

Struct. .) Principal + Elimof + Transform Model Motions Comp Dependent to indep, std Comps w/

+

Analysis Comps norm comps A N N S

Simulation of Response

Simulate + Inverse .)- Restore .)- Synthesize .)- Struct Comps w/ Rosenblau Dependent Using Motion ANNs Transfonn Comps APProP Eq Simulation

Figure 4. Summary of operations in system simulation.

Numerical Example

We simulate in this example the motion of a simple, nonlinear 10 degree- of-freedom system excited with a Gaussian white noise. Figure 5 shows a schematic of the system. The damping connecting the masses is linear viscous. The springs have a restoring force that is a tangent function. The system physical parameters are summarized in Table 1. Training data for the neural networks were generated by computing response over 8192 time steps (box number one on the top line in Figure 4); excitation and responses at ten locations were recorded. The time increment between response realizations is 0.04 second. Figure 6 shows the displacement response at the 10th mass. This is the location where the simulation-to-experiment comparisons are made in the present example, and where the simulation yielded the poorest match to the experimental results.

The responses were placed in a matrix X as referred to in the previous sections, and its SVD was computed (box number two on the top line of Figure 4). The singular values of the response indicate that accuracy of about 89% should be achieved by simulating the system response with its first four components. The four components were not strongly statistically dependent, so none was removed. The kde of the four components indicate that none is highly non-Gaussian, therefore, the Rosenblatt transform was not used. The first four components of the response were modeled with both BPN and CNLS nets (box number five on the top line in Figure 4).

The entire system was tested in autoregressive operation, as described in Figure 2, using data generated over 1000 steps of response computation. The initial conditions and excitation were used to start then execute a random vibration response simulation with ANNs in the space of u (box number one on the second line of Figure 4). The test was iterated, i.e., the estimated responses at step j were used as initial values for response predictions at time indices greater than j. The first and most significant column of u from the test data and the first column from the ANN simulated data are compared in Figures 7a and 7b. This is the dominant component of the response. The match is good, particularly in view of the fact that the simulation is iterated. Note that although these and later response predictions remain fairly well in phase with

.. .

the test responses, this is not usually the case. Typically, we hope that the simulated response amplitudes match the test responses well, and accept the fact that phase will usually be lost. Figures 8a and 8b compare the spectral densities of the signals shown in Figures 7a and 7b. (These were computed using the technique described, for example, in Wirsching, Paez, and Ortiz, [14].) A block size of 256 data points was used, along with a Hanning window, and an overlap factor of 0.55. The first harmonic of motion, at 0.25 Hz, is very well matched. A third harmonic of motion appears to be present in both the test and simulated signals; the BPN provides a better match of the third harmonic than the CNLS net. Figures 9a and 9b compare the kde’s of the signals shown in Figures 7a and 7b. The responses are clearly non-Gaussian, as anticipated because of nonlinearity, and to some extent the simulated component responses match the character of the test response. This match needs to be improved to maximize the quality of the simulation. (Some of the mismatch is caused by the limited data - lo00 points - upon which the comparison is based.) However, as will be seen, the simulated synthesized responses have kde’s that match test response kde’s quite well.

Table 1. Parameters of the test system.

Index Mass 1 1 .o 2 0.95 3 0.90 4 0.85 5 0.80 6 0.75 7 0.70 8 0.65 9 0.60

10 0.55 11

Tangent Stiffness

40 38 36 34 32 30 28 26 24 22 2

Maximum Deformation

1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o

Viscous Damping

0.40 0.38 0.36 0.34 0.32 0.30 0.28 0.26 0.24 0.22 0.02

The simulated responses are now reconstructed by substituting into Eq. (1) using the simulated u (box number four on the second line of Figure 4), and the results are compared to the test response at mass 10 (box number five on the second line of Figure 4). The result from the BPN simulation is shown in Figure loa; the result from the CNLS net simulation is shown in Figure lob. Of course, the matches are quite good since the dominant component is well simulated by both ANNs. Figures 1 la and 11 b compare the spectral densities of the signals shown in Figures 10a and lob. The spectral densities estimated from the simulated signals match those of the test signals well up to the frequency where components are no longer simulated; the CNLS net does a slightly better job of matching the test spectral density than the BPN. At mass 10 the rms response of the test signal is 0.45 in, and the rms values of the simulated responses are 0.47 in and 0.40 in for the BPN and CNLS net, respectively. The corresponding kde’s of test and simulated responses were computed and are shown in Figures 12a and 12b. The probabilistic character of the responses appears to be matched well by the ANN simulations.

Figure 5. A nonlinear spring-mass system.

1

0.5

- c v

z o x

-0.5

(I .I, -1 0 10 20 30 40

Time. sec

Figures 6. Response at mass 10 to Gaussian white noise input.

0.02

0.01 - c v + v 7 - 0 F 3 3

-0.01

-0.02

-0.03 ' I 0 10 20 30 40

Time, sec

0.02 I

0.01

0

-0.01

I -0.02 '

0 10 20 30 40 Time, sec

Figures 7a and 7b. Comparison of test and ANN simulated responses - Component 1. BPN simulation on left; CNLS simulation on right. ANN simulation - solid line; test data - dashed line. ,

lo-' 1 oo Frequency, Hz

10 1 1 0-1 1 oo 1 o1

Frequency, Hz Figures 8a and 8b. Comparison of the spectral density estimates of test and ANN simulated responses - Component 1. BPN simulation on left; CNLS net simulation on right, ANN simulation - solid line; test data - dashed line.

50

40

30

- 20

10

7

v 3

0 -0.05 0 0.05 -0v.05 0 0.05

u l Ul

Figures 9a and 9b. Comparison of the kernel density estimators of test and ANN simulated responses - Component 1. BPN simulation on left; CNLS net simulation on right. ANN simulation - solid line; test data - dashed line.

1 1

0.5 0.5 - - v L Y

s ! o X

2 0 X

-0.5 -0.5

-1 -1

-1.5 I 0 10 20 30 40

Time, sec

-1.5' I 0 10 20 30 40

Time, sec

Figures 10a and lob. Test (dashed) and simulated (solid) responses at a mass 10 in the simple system - BPN (left), CNLS net (right).

c a, U

a, 13

n ‘lb \

10’ 1 oo 10’ Frequency, Hz Frequency, Hz

Figures 1 la and 1 1 b. Test (dashed) and simulated (solid) estimated response spectral densities at mass 10 in the simple system - BPN (left), CNLS net (right).

0.2 1 a I \

/ \ I 0 1 2

0 -2 -1

x10

0.2 I I li I

0 -2 -1 0 1 2

XlO

/

Figures 12a and 12b. Test (dashed) and simulated (solid) response kde’s at mass 10 in the simple system - BPN (left), CNLS net (right).

Conclusions

A sequence of operations leading to the simulation of nonlinear structural random vibrations with ANNs is described in this paper. Such simulations are desirable because of their efficiency and relative accuracy. It is argued that if the motions can be decomposed and transformed into simple components, then the simulation will be simpler and more accurate. A numerical example confirms that relatively simple motions can, indeed, be modeled with ANNs - the BPN and CNLS net (a local approximation network) in particular. Given sufficient training data accurate simulations of simple components should always be possible, though obtaining satisfactory accuracy may require substantial effort. Accurate simulations should correctly reflect probabilistic, spectral, and all other characteristics of the simulated component responses. When component responses are correctly modeled then system level responses will be simulated accurately.

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