# MODAL v3 - Abstracts

NOTE: The journal articles are listed alphabetically by first author.

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Abrahamsson, T. Modal-parameter extraction for nonproportionally damped linear structures. 3(2):62-68; Apr 1988.

ABSTRACT - An efficient dynamic analysis of a built-up structure can often be performed setting out from the modal parameters of each of its substructures. Some of these sets of modal parameters of each of its substructures. Some of these sets of modal parameters can be experimentally measure; others have to be extracted from a analytically determined behavior of a substructure in the frequency domain. This paper advances a method for extracting the modal parameters of the latter type of (sub)structure. This structure may be composed of various types of continuous and discrete elements. The damping over the structure may be nonproportionally distributed. Poles and eigenvectors are found by solving the nonlinear eigenvalue problem associate with the transcendentally frequency-dependent stiffness matrix of the structure. Starting with the poles of the corresponding undamped structure, as found by use of the Wittrick-Williams algorithm, the poles of the damped structure are found by use of an algorithm which replaces the nonlinear problem with a sequence of linear problems. Residues at poles are derived from the calculated poles and eigenvectors and from the given mass and damping properties of the structure. Two numerical examples are included.

Abrahamsson, T.; Lundblad, M. Transient vibration analysis of nonproportionally damped linear structures using modal parameters. 3(3):108-114;Jul 1988.

ABSTRACT - The modal parameters (generally complex) from a known eigensolution are used to transform the coupled differential equations of motion of a structure into a set of uncoupled first-order differential equations. Solutions for forcing functions whose dependence on time is piece wise polynomial, exponential, or sinusoidal (or a product of these) are given by exact explicit expressions. Initial modal displacements are fitted to a given set of physical initial displacements and velocities (nonzero) through use of a weighted minimum two-norm method. The contribution to the structural response from truncated high-frequency modes is approximately accounted for by adding residual flexibility terms into the transient solution. Three numerical examples are included. Both discrete and continuous structures are treated.

Braccesi, C.; Carfagni, M.; Rissone, P. Using vibratory and modal analysis to reduce vibrations and noise in an automotive engine. 3(3):89-95; Jul 1988.

ABSTRACT - An investigation was undertaken to determine mechanical noise in an internal-combustion engine. Noise detection was carried out by characterizing the transmission paths of piston slap and bearing-crankshaft impacts; the characterization was then recontraucted by modal analysis. Thereafter,using the structural modification technique, it was possible to modify the engine block and thus achieve a 5-dB(A) reduction in overall engine-noise level.

Carne, T.G.; Lauffer, J.P.; Gomez, A.J.; Benjannet, H. Modal testing an immense flexible structure using natural and artificial excitation. 3(4):117-122; Oct 1988.

ABSTRACT - Modal testing of immense and very flexible structures poses a number of problems. It requires innovative excitation techniques since the modal frequencies of these structures can be quite low. Also, substantial energy must be input to the structure to obtain reasonable levels of response. In this paper, results are presented from a modal test of the 110-m tall EOLE wind turbine which had four modal frequencies below l.0 Hz. Step-relaxation and wind were used to excite the structure.

Carne, T.G.; Lobitz, D.W.; Nord, A.R.; Watson, R.A. Finite-element analysis and modal testing of a rotating wind turbine. 3(l):32-41; Jan 1988.

ABSTRACT - A finite-element procedure which includes geometric stiffening, and centrifugal and Coriolis terms resulting from the use of a rotating coordinate system, has been developed to compute the mode shapes and frequencies of rotating structures. Special application of this capability has been made to Darrieus, vertical-axis wind turbines. In a parallel development effort, a technique for the modal testing of a rotating vertical-axis wind turbine has been established to measure modal parameters directly. Results from the predictive and experimental techniques for the modal frequencies and mode shapes are compared over a wide range of rotational speeds.

Cobb, R.E.; Mitchell, L.D. A method for the unbiased estimate of system FRFs in the presence of multiple-correlated inputs. 3(4):123-128; Oct 1988.

ABSTRACT - The two currently used frequency-response-function (FRF) estimates, (super)1 H(hat) and (super)2 H(hat), product biased estimates when there is uncorrelated content present in the measurement system. Multiple-input FRF estimates have been based on (super)1 H(hat). Such estimates are biased in the presence of noise. An unbiased estimate, (super)c H(hat), have been developed for single input/output systems using a third channel of data and is computed by forming a ratio of cross spectra. This paper presents the application of this method to obtain unbiased FRF estimates in multiple input/output systems. Computer simulations illustrate the strengths of this method. Good estimates of FRFs have been obtained where the inputs are 99.8-percent correlated. In addition, good estimates have been obtained when coherence is less than 0.05!

FRF-estimate notation becomes complicated in a multiple input/output system. This paper proposes a system of notation for FRF specification where the type of estimator, number of averages, input location, and output location are clearly indicated. The new notation is used in the beginning of this abstract.

Dimas, D.J.; Pardown, G.C. Extrapolation of modal-analysis techniques to nonlinear damped systems. 3(3):81-88; Jul 1988.

ABSTRACT - The resistive forces of objects that move through fluids in mechanical systems can be proportional to their velocity raised to powers other then one. This type of resistive force, that is generically termed Nth power velocity damping, yields a governing differential equation which is nonlinear. This investigation considered the effects of the damping nonlinearities by numerically integrating the equation of motion for a nonlinear one-DOF system subjected to a series of harmonic forcing functions. The results of these analyses were displayed in the frequency domain by a series of three-dimensional response surfaces using interactive graphics software(PATRAN). The numerical-integration results were (a) curve fit in the frequency domain using a rational fraction polynomial technique to estimate the one-DOF modal parameters and (b) compared to the actual values used in the numerical integrations. The conclusions provide bounds on the salient parameters for this type of nonlinear system in which the results of traditional modal-analysis procedures can be extrapolated.

Doyle, J.F. A spectrally formulated finite element for longitudinal wave propagation. 3(1):1-5; Jan 1988.

ABSTRACT - By recasting the dynamics of rods in spectral terms and using the FFT for inversion, a description emerges that requires information only at the end points. This description is eminently suited for a finite-element formulation because the chosen length of the element can be large. This affords a substantial reduction in the size of the system of equations to be solved.

Henried, A.G.; Lau, K.S. Dynamic response of secondary systems in structures subjected to earthquake excitation. 3(1):2025; Jan 1988.

ABSTRACT - The dynamic response of lightweight equipment (secondary system) attached to a heavier structure (primary system) which is subjected to earthquake ground motion is investigated. A recently developed technique for the determination of secondary subsystem response, which avoids the shortcomings of either a numerical integration of the combined system equations or various ad hoc approaches is summarized. The results of an intensive series of numerical experiments on several primary-secondary systems subjected to earthquake excitation are presented. These numerical experiments shed new light on the dynamic response of secondary systems in structures subjected to earthquake excitation as well as on the proposed technique for the determination of this response.

Huang, T.C.; Huang, X.L. Fixed-interface substructural modal perturbation method. 3(1):6-11; Jan 1988.

ABSTRACT - By combining the modal-perturbation technique and substructural modal analysis, a new substructural-modal-perturbation method is developed.

Huo, Q.Z.; Zhang, D.J. Application of impact testing in the modal analysis of a large structure. 3(2):57-61; Apr 1988.

ABSTRACT - In order to use less equipment in the modal testing of a large structure, an excitation method based on a random series of impacts was studied analytically and experimentally. The divisional excitation and measurement technique was also considered. The methods were successfully applied in the modal testing of a diesel-locomotive body. The first seven normal modes and other modal parameters were obtained.

Lee, G.M.; Trethewey, M.W. Modal-parameter quality assessment from time-domain data. 3(4):129-136; Oct 1988.

ABSTRACT - This paper will explore the application of digital filtering methods for the evaluation of modal-parameter quality for the finite-difference least-squares time-domain modalidentification method. The approach uses a comparison of the modal-model response to the actual response in a family of band-limited frequency regions. The band-limited responses are generated through the application of finite impulse response filters to both the measured and modal model time series. The calculation of root mean square error between the series provides a quantitative measure of the modal parameter quality in narrow band frequency regions. The fundamentals of the approach are presented along with practical details of its implementation. The approach is applied to both simulated and experimental structural-vibration data to examine the utility of the approach. The results illustrate that the approach can quantitatively evaluate the modal-parameter quality for modes either with a low-level response or which are masked by extraneous background noise.

Leonard, F. ZMODAL: a new modal-identification technique. 3(2):69-76; Apr 1988.

ABSTRACT - This paper describes a new tool for modal identification and proposes a novel approach for evaluating the real mode-shape accuracy. The Z-plane modal-analysis (ZMODAL) algorithm is based on topological characteristics of amplitude and phase in the Z plane of the Z transform of damped sinusoids. Since damping is a dimension of the signal representation space, it is easy to disassociate frequency modes that are close but have different damping coefficients. In the Z-plane representation space, poles corresponding to modes have less energy coupling between them than the frequency lobes in Fourier representation space. This, together with the use of spectral windows, improves mode-energy separation. In this context of good cross-modal rejection (pole energy versus coupling energy with another pole). MDOF and SDOF algorithms give similar results although for this study the latter was selected for its shorter CPU time, robustness and simplicity of implementation. Like the lbrahim time-domain (ITD) algorithm, it is based on free-decay responses and does not call for force inputs. The algorithm has been tested for different experimental and analytical results and shows good agreement with ITD results. The paper also proposes a method for quantitative evaluation of the real mode-shape accuracy based on the phase output; the phase standard deviation. The mathematical demonstration of this significant new tool is given and comparisons based on it are presented.

Mitchell, L.D. A perspective view of modal analysis. 3(2):45-48; Apr 1988.

ABSTRACT - Historically, experimental modal analysis (EMA) grew to prominence because the engineering community was incapable of properly analyzing the dynamics of commercially significant structures. As experimental modal analysis developed the capacity to generate mathematical models from experimental data, the finite-element-analysis (FEA) community established significant dynamic-analysis capability. Accelerated development in both fields followed. Experimental modal analysis had the new task of verifying analytical-modal-analysis (AMA) results. The trend now is toward using EMA to help pinpoint the modeling inadequacies in FEA. Activity in sales of, equipment for, and new horizons in experimental modal analysis seem to have hit a plateau. This paper traces these developments, relates its connection to the CAD/CAM/CAE revolution, overviews the roadblocks, assesses the current state, and probes the future. Computer-aided engineering (CAE), in general, and experimental/analytical modal analysis, in particular, face a series of challenges. When met, these challenges will form the base for a major expansion of the technology and its application. Moreover, merging experimental and analytical modal analysis will solidify the contribution of modal analysis to the betterment of the overall engineering-design process. An improved and more efficient design process will serve society for years to come.

Mitchell, L.D.; Cobb, R.E.; Deel, J.C.; Luk, Y.W. An unbiased frequency-response-function estimator. 3(1):12-19; Jan 1988.

ABSTRACT - In the past several years a number of alternative methods have been introduced for the estimation of frequency-response functions (FRF) of structures using averaged auto and cross spectra of the force and response signals. All of these methods coincide in the noise-free environment, but each produce a biased estimate unless certain assumptions about the force and response signal-to-noise ratios are satisfied. Since these noise characteristics are frequency dependent and are generally unknown, it has been difficult to properly apply these methods in practice. This paper develops a little-known FRF estimate, denoted by H(hat) (super)c , which avoid these problems. Because this estimate is compute entirely from averaged cross spectra, it avoids the errors which uncorrelated sources produce in the autospectra used in all the other currently available FRF estimates. Examples are presented that compare H(hat) (super)c with other estimates computed from exactly the same date. The effect of averaging on the convergence of |H(hat) (super)c| is investigated.

Nariboli, G.A.; McConnell, K.G. Curvature coupling of catenary cable equations. 3(2):49-56; Apr 1988.

ABSTRACT - Not currently available

Olausson, H. L.; Torby, B.J. Turbine design using complex modes and substructuring. 3(4):148-157; Oct 1988

ABSTRACT - This paper presents a complex modal-analysis method for studying the behavior of a turbine near its design speed. In this method gyroscopic moments as well as non-symmetric bearing effects are accounted for in the modal calculations. The complex mode shapes that are found here more accurately describe steady-state orbital motion than planar modes. Transient motion, induced by some disturbance from a dynamic equilibrium state, is formulated through a series expansion of the complex normal-coordinate terms associated with these mode shapes. The ensuing equations of motion can then be integrated and the solution brought back to the problem’s original generalized coordinates. Since in normal-coordinate description no distinction can be made between modes obtained mathematically and those found experimentally, the paper also describes how the latter mode shapes can easily be coupled to the system equation when they are considered as substructure component modes. This use of component-mode synthesis allows the turbine model to incorporate several rotors turning at different speeds, and even torsional motion. Results are given from typical calculations performed for a 650 MW ASEA STAL turbine installation.

Park, Y.S.; Park,, H.S.; Lee, S.S. Weighted-error-matrix application to detect stiffness damage by dynamic-characteristic measurement. 3(3):101-107; Jul 1988.

ABSTRACT - A nondestructive method to detect stiffness damages in a structure is studied. The method is based on the fact that dynamic characteristics vary depending on the locations of stiffness defect. Close examinations of measured dynamic characteristics enable us to study the defects. First the error-matrix method was tested. Since the error matrix originates from the differences of measured natural frequencies, and mode shapes, before and after damages, it is generally influenced by measurement errors and the number of measurable modes. The error-matrix method was applied to two sample structures to search for damaged areas. The method was found useful when the stiffness changes were large. But cracks and small defects in structures most often cause only small changes in local stiffness. The error matrix method is thus not effective for searching out small local defects. Next, the weighted-error-matrix (WEM) method was tested to magnify the effect of small stiffness loss in the error matrix. The idea of WEM is to magnify the damaged area in the error matrix by adding the information from the eigenproperties’changing patterns through sensitivity analysis. Applying the WEM to the two same sample structures, it was found that WEM is a far better method for identifying defects than the error-matrix method, especially when the defect is small. A step by step method is suggested in order to locate the defects in a large structure economically. First the structure is divided into a few numbers of large elements and the damage is studied by WEM techniques. Next only the probable damaged element, which is found in the first step, is divided into finer elements, and the damage is studies within the narrow-downed element. Applying the step by step method to a plate structure, it was found that the method is an economical way to study the damaged areas.

Patton, M.E.; Trethewey, M.W. External-intensity-modulated fiber-optic sensors for structural-vibration measurements. 3(4):137-147; Oct 1988.

ABSTRACT - External-intensity-modulated fiber-optic systems have gained considerable use recently for the measurement of dynamic mechanical phenomena. The systems posses many outstanding characteristics. However, their principle of operation makes them susceptible to a variety of measurement errors if proper care is not exercised in the experimental setup and data interpretation. The objective of this paper is to examine the transducer’s characteristics and present procedures for their proper implementation for modal-analysis applications. The paper examines several potential errors associated with these sensors and discusses the effect on experimental data. Specifically, the difficulties examined are positioning accuracy, target-reflectivity differences, target curvature, nonlinear sensor response, and inherent dynamic-range effects of displacement-based modal data. Each of these topics is investigated to examine the effects which these errors have on the sensitivity, linearity, and data consistency. Procedures to either eliminate or minimize the effects of the errors are described.

Starkey, J.M.; Durchholz, A.M. A dynamic substructure algorithm to investigate the feasibility of adaptable engine mounts. 3(1):26-31; Jan 1988.

ABSTRACT - This paper presents a technique for predicting the effects of variable engine-mount properties on the natural frequencies and mode shapes of a light-truck engine/chassis structure. Modal testing was used to develop chassis models and to validate the substructuring procedure. Engine-mount sensitivity studies indicate that engine mounts with adjustable damping and stiffness can significantly reduce chassis vibrations at idle.

Steffen. V., Jr.; Marcelin, J.L. On the optimization of vibration frequencies of rotors. 3(3):77-80; Jul 1988.

ABSTRACT - At the design stage of construction of rotating machinery, the natural frequencies and critical speeds much be optimized in such a way that the operating speeds must be optimized in such a way that the operating speed of rotation is safely far from the criticals. This can be performed by manipulating design variables taking into account technological constraints. This paper presents an optimizing program coupled to a rotor-dynamics finite-elements code to perform the optimization of rotors. Three different applications for which natural frequencies optimization were performed are presented.

Unlusoy, Y.S. Mobility analysis of helical coil springs. 3(3):96-100;Jul 1988.

ABSTRACT - Dynamic behavior of helical compression springs as represented by their point-mobility functions is investigated. Mass and stiffness matrices for the springs are obtained using straight-beam finite elements with six degrees of freedom at each end. The effects of static axial load (i.e., preload) on the dynamic behavior are also taken into account. The eigenvalue problem is solved by the subspace iteration method. The undamped natural frequencies together with the corresponding modal vectors are determined. Theoretical point-mobility functions are then calculated by the mode superposition method for excitation in the direction of longitudinal spring axis at one end of the spring while the other end is fixed. The calculated and experimentally obtained point-mobility functions for three automobile suspension springs, drawn in the frequency range from 3 to 200 Hz, display fairly good agreement. The shapes of the first ten modes are also plotted and the major categories of vibration modes are identified. The effects of preload on the natural frequencies corresponding to each category of modes are identified. The results allow an assessment of the applicability of the widely used distributed-parameter spring model which considers longitudinal spring vibrations only.