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Characterizing electrodynamic shakers

Description: An electrodynamic shaker is modeled as a mixed electrical/mechanical system with an experimentally derived two port network characterization. The model characterizes the shaker in a manner that the performance of the shaker with a mounted load (test item and fixture) can be predicted. The characterization depends on the measurements of shaker input voltage and current, and on the acceleration of the shaker armature with several mounted loads. The force into the load is also required, and can be measured directly or inferred from the load apparent mass.
Date: December 31, 1996
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Multiple Shaker Random Vibration Control--An Update

Description: The theory of the control of multiple shakers driving a single test item is reviewed. Several improvements that have been introduced since the original papers on the subject will be discussed. The improvements include: (1) specification of the control spectra; (2) the control of non-square systems (the number of shakers does not have to be equal to the number of control points); (3) the connection between sine testing, waveform control, and random control; (4) improvements in feedback control; (5) overlap-add versus time domain randomization; and (6) reproduction of non-Gaussian waveforms.
Date: February 18, 1999
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Generation of time histories with a specified auto spectral density and probability density function

Description: It is recognized that some dynamic and noise environments are characterized by time histories which are not Gaussian. An example is high intensity acoustic noise. Another example is some transportation vibration. A better simulation of these environments can be generated if a zero mean non-Gaussian time history can be reproduced with a specified auto (or power) spectral density (ASD or PSD) and a specified probability density function (pdf). After the required time history is synthesized, the waveform can be used for simulation purposes. For example, modem waveform reproduction techniques can be used to reproduce the waveform on electrodynamic or electrohydraulic shakers. Or the waveforms can be used in digital simulations. A method is presented for the generation of realizations of zero mean non-Gaussian random time histories with a specified ASD, and pdf. First a Gaussian time history with the specified auto (or power) spectral density (ASD) is generated. A monotonic nonlinear function relating the Gaussian waveform to the desired realization is then established based on the Cumulative Distribution Function (CDF) of the desired waveform and the known CDF of a Gaussian waveform. The established function is used to transform the Gaussian waveform to a realization of the desired waveform. Since the transformation preserves the zero-crossings and peaks of the original Gaussian waveform, and does not introduce any substantial discontinuities, the ASD is not substantially changed. Several methods are available to generate a realization of a Gaussian distributed waveform with a known ASD. The method of Smallwood and Paez (1993) is an example. However, the generation of random noise with a specified ASD but with a non-Gaussian distribution is less well known.
Date: August 1, 1996
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Generation of partially coherent stationary time histories with non-Gaussian distributions

Description: In a previous paper Smallwood and Paez (1991) showed how to generate realizations of partially coherent stationary normal time histories with a specified cross-spectral density matrix. This procedure is generalized for the case of multiple inputs with a specified cross-spectral density function and a specified marginal probability density function (pdf) for each of the inputs. The specified pdfs are not required to be Gaussian. A zero memory nonlinear (ZMNL) function is developed for each input to transform a Gaussian or normal time history into a time history with a specified non-Gaussian distribution. The transformation functions have the property that a transformed time history will have nearly the same auto spectral density as the original time history. A vector of Gaussian time histories are then generated with the specified cross-spectral density matrix. These waveforms are then transformed into the required time history realizations using the ZMNL function.
Date: August 27, 1996
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Shaker force measurements using voltage and current

Description: In a previous paper (Smallwood and Coleman, 1993), equations were developed which would allow the force into a test item during a vibration test to be measured using voltage and current measurements from the input to an electrodynamic shaker. To accomplish this, the voltage and current required to drive the shaker with no load and with a known mass were required. In this paper, the equations are generalized to cover the case where the measurements are made with several (at least 2) load conditions. It is not required that one of the conditions be the no load condition. The equations are written in a form where the known loads are not required to be a simple mass, but can be a complex impedance. For the case of more than two load conditions, a least squares solution is found.
Date: October 1, 1996
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Correcting numerical integration errors caused by small aliasing errors

Description: Small sampling errors can have a large effect on numerically integrated waveforms. An example is the integration of acceleration to compute velocity and displacement waveforms. These large integration errors complicate checking the suitability of the acceleration waveform for reproduction on shakers. For waveforms typically used for shaker reproduction, the errors become significant when the frequency content of the waveform spans a large frequency range. It is shown that these errors are essentially independent of the numerical integration method used, and are caused by small aliasing errors from the frequency components near the Nyquist frequency. A method to repair the integrated waveforms is presented. The method involves using a model of the acceleration error, and fitting this model to the acceleration, velocity, and displacement waveforms to force the waveforms to fit the assumed initial and final values. The correction is then subtracted from the acceleration before integration. The method is effective where the errors are isolated to a small section of the time history. It is shown that the common method to repair these errors using a high pass filter is sometimes ineffective for this class of problem.
Date: November 1, 1997
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Using singular value decomposition to compute the conditioned cross-spectral density matrix and coherence functions

Description: It is shown that the usual method for computing the coherence functions (ordinary, partial, and multiple) for a general multiple-input/multiple-output problem can be expressed as a modified form of Cholesky decomposition of the cross spectral density matrix of the inputs and outputs. The modified form of Cholesky decomposition used is G{sub zz} = LCL{prime}, where G is the cross spectral density matrix of inputs and outputs, L is a lower; triangular matrix with ones on the diagonal, and C is a diagonal matrix, and the symbol {prime} denotes the conjugate transpose. If a diagonal element of C is zero, the off diagonal elements in the corresponding column of L are set to zero. It is shown that the results can be equivalently obtained using singular value decomposition (SVD) of G{sub zz}. The formulation as a SVD problem suggests a way to order the inputs when a natural physical order of the inputs is absent.
Date: August 7, 1995
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Generation of time histories with a specified auto spectral density, skewness, and kurtosis

Description: Some dynamic environments are characterized by time histories that are not Gaussian. A more accurate simulation of these environments can be generated if a realization of a non Gaussian time history can be reproduced which has a specified auto spectral density (also called power spectral density) and a specified skewness and kurtosis (not necessarily the skewness and kurtosis of a Gaussian time history). The mean square of the waveform is reproduced if the spectrum is reproduced. Modern waveform reproduction techniques can be used to reproduce the realized waveform on an electrodynamic or electrohydraulic shaker. A method is presented for the generation of realizations of zero mean non Gaussian random time histories with a specified auto spectral density, skewness, and kurtosis. Kurtosis, defined in this paper as E[{chi}{sup 4}]/E{sup 2}[{chi}{sup 2}], greater than 3 can be realized. Realizations of the random process are generated with a generalization of shot noise.
Date: February 1, 1996
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Qualification of frequency response functions using the rigid-body response

Description: The response of a structure at low frequencies with free boundary conditions is dominated by the rigid-body modes. The displacement shapes obtained from the low frequency values of the frequency response functions can be compared with ideal rigid-body motion to point out errors in the measurements. Insight is enhanced when the comparisons are made in the coordinate system of the measurements. Without this procedure intuition can rarely determine the proper rigid-body response at each measurement location. Typical errors identified are scaling errors, errors in location or direction, measurements with poor dynamic range and other instrumentation problems. The procedure is particularly useful when the test object is multidimensional, has a complicated geometry, has measurements in other than rectangular coordinates, and where more than one rigid-body mode is excited. It is suggested that data qualification using this method would be a useful addition to most modal tests. A least squares approach, to determine the proper rigid-body response, is reviewed and several experimental examples are given. 4 refs., 12 figs.
Date: January 1, 1987
Creator: Smallwood, D.O. & Lauffer, J.P.
Partner: UNT Libraries Government Documents Department

Salvaging pyrotechnic data with minor overloads and offsets

Description: The authors are sometimes presented with data with serious flaws, like saturation, over-range, zero shifts, and impulsive noise, including much of the available pyrotechnic data. Obviously, these data should not be used if at all possible. However, they are sometimes forced to use these data as the only data available. A method to salvage these data using wavelets is discussed. The results must be accepted with the understanding that the answers are credible, not necessarily correct. None of the methods will recover information lost due to saturation and over-range with the subsequent nonlinear behavior of the data acquisition system. The results are illustrated using analytical examples and flawed pyrotechnic data.
Date: January 27, 1998
Creator: Smallwood, D.O. & Cap, J.S.
Partner: UNT Libraries Government Documents Department

Salvaging transient data with overloads and zero offsets

Description: The authors are sometimes presented with data with serious flaws, like overloads, zero shifts, and impulse noise, including much of the available pyrotechnic data. Obviously, these data should not be used if at all possible. However, they are sometimes forced to use these data as the only data available. Methods to salvage these data are discussed. Using the methods requires judgment, and the results must be accepted with the understanding that the answers are credible, not necessarily correct. None of the methods will recover information lost due to overloads or non-linearities of the data system. The best that can be accomplished is the recovery of data, after the data system has recovered from the overload. Several correction methods are discussed: high pass filtering of the data, correction with two forms of an exponential function, and a correction with the form t exp({minus}{alpha}t). Examples showing the results of the methods will be given using flawed pyrotechnic data.
Date: October 21, 1997
Creator: Smallwood, D.O. & Cap, J.S.
Partner: UNT Libraries Government Documents Department

A methodology for defining shock tests based on shock response spectra and temporal moments

Description: Defining acceptable tolerances for a shock test has always been a problem due in large part to the use of Shock Response Spectra (SRS) as the sole description of the shock. While SRS do contain a wealth of information if one knows what to look for, it is commonly accepted that different agencies can generate vastly different time domain test inputs whose SRS all satisfy the test requirements within a stated tolerance band. At an even more basic level, the laboratory test specifications often fail to resemble the field environment even though the SRS appear to be similar. A concise means of bounding the time domain inputs would be of great benefit in reducing the variation in the resulting shock tests. This paper describes a methodology that uses temporal moments to improve the repeatability of shock test specifications.
Date: November 1, 1997
Creator: Cap, J.S. & Smallwood, D.O.
Partner: UNT Libraries Government Documents Department

Characterization and simulation of gunfire with wavelets

Description: Gunfire is used as an example to show how the wavelet transform can be used to characterize and simulate nonstationary random events when an ensemble of events is available. The response of a structure to nearby firing of a high-firing rate gun has been characterized in several ways as a nonstationary random process. The methods all used some form of the discrete fourier transform. The current paper will explore a simpler method to describe the nonstationary random process in terms of a wavelet transform. As was done previously, the gunfire record is broken up into a sequence of transient waveforms each representing the response to the firing of a single round. The wavelet transform is performed on each of these records. The mean and standard deviation of the resulting wavelet coefficients describe the composite characteristics of the entire waveform. It is shown that the distribution of the wavelet coefficients is approximately Gaussian with a nonzero mean and that the standard deviation of the coefficients at different times and levels are approximately independent. The gunfire is simulated by generating realizations of records of a single-round firing by computing the inverse wavelet transform from Gaussian random coefficients with the same mean and standard deviation as those estimated from the previously discussed gunfire record. The individual realizations are then assembled into a realization of a time history of many rounds firing. A second-order correction of the probability density function (pdf) is accomplished with a zero memory nonlinear (ZMNL) function. The method is straightforward, easy to implement, and produces a simulated record very much like the original measured gunfire record.
Date: September 1, 1998
Creator: Smallwood, D.O.
Partner: UNT Libraries Government Documents Department