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DOE OSTI2025

Modal Analysis and Testing of a Cantilever Beam: Results and Guide

This report discusses modal analysis and testing of a cantilever beam including an introduction to modal testing, simulation, analysis, and a comparison of results. An accompanying presentation can be found from the LLNL CASIS website. Modal testing and analysis are essential for characterizing the response of a structure during vibration environments. Determining natural frequencies and mode shapes helps determine whether resonances will be reached during operations and if so, how the structure will respond and how the response can be tuned. The goal of this work is to convey the basics of modal testing and analysis through a simple project. This can be applied later to more complex, applicable structures. This includes a variety of work including data collection, finite element analysis and data processing. For this project a cantilever beam was chosen because it is a simple, well-characterized structure.

42 ENGINEERING↗

NASA NTRS1973

A simplified modal plotting technique for the representation of complex structural models

A plotting method has been developed that has proven to be useful in rapidly defining the mode shapes of a complex spacecraft. The reduction of the complex model to a simple stick plot is accomplished by augmenting the existing NASTRAN plot package with constrained plot elements. Both the existing NASTRAN modal plot package and the stick plots of a representative mode are presented for comparison. The NASTRAN plot package provides the capability of generating modal deformations (mode shapes) resulting from real eigenvalue analysis. These modal deformations of the structural model may be displayed in the deformed shape either alone or superimposed on the undeformed shape. Another available method of displaying the modal deformations is by means of displacement vectors at the grid points

Hanlein, S. L.↗

NASA NTRS1974

Transient Response of Shells of Revolution by Direct Integration and Modal Superposition Methods

The results of an analytical effort to obtain and evaluate transient response data for a cylindrical and a conical shell by use of two different approaches: direct integration and modal superposition are described. The inclusion of nonlinear terms is more important than the inclusion of secondary linear effects (transverse shear deformation and rotary inertia) although there are thin-shell structures where these secondary effects are important. The advantages of the direct integration approach are that geometric nonlinear and secondary effects are easy to include and high-frequency response may be calculated. In comparison to the modal superposition technique the computer storage requirements are smaller. The advantages of the modal superposition approach are that the solution is independent of the previous time history and that once the modal data are obtained, the response for repeated cases may be efficiently computed. Also, any admissible set of initial conditions can be applied.

Stephens, W. B.↗

NASA NTRS1976

Analysis of structural dynamic data from Skylab. Volume 2: Skylab analytical and test modal data

A compendium is presented of orbital configuration test modal data, analytical test modal data, analytical test correlation modal data and analytical flight configuration 1.2 modal data. Section A presents tables showing the generalized mass contributions for each of the thirty test modes. Section B presents the two dimensional mode shape plots for the thirty test modes. Tables of GMC's for the test correlated analytical modes are presented in Section C. These analytical modes were generated from a model that was adjusted to match test results by use of the methodology discussed in Sections 2.3 and 5.4 of Volume I of this report. Section D presents the two dimensional mode shape plots for the analytical modes. Sections E and F contain the uncoupled and coupled modes of the orbital flight configuration 1.2 at three development phases of the model.

Demchak, L.↗

NASA NTRS1976

Modal structure inferred from static far-field noise directivity

Turbofan noise directivity calculated for two directivity models (equal modal amplitude and equal modal power or energy) was compared with experimental blade passing frequency data from two fans at 60 and 90 percent speeds. Experimental data indicated similar directivity patterns which were well represented by a single average data curve. Calculated points using the equal amplitude model showed over-prediction near the fan axis and near the 90-deg position. Calculated points using the equal power model showed a very good match with the average data lending support to theory of equipartition of modal power from a random source such as the interaction of the rotor with inlet flow distortion. The equal modal power model also gave good agreement with individual data points.

Saule, A. V.↗

NASA NTRS1976

Modal density function and number of propagating modes in ducts

Often raised questions in duct sound propagation studies involve the total number of propagating modes, the number of propagating radial modes for a particular spinning lobe number, and the number of modes possible between two given values of cutoff ratio or eigenvalue. These questions can be answered approximately by using the modal distribution function which is the integral of the modal density function for ducts in a manner similar to that previously published for architectural acoustics. The modal density functions are derived for rectangular and circular ducts with a uniform steady flow. Results from this continuous theory are compared to the actual (discrete) modal distributions.

Rice, E. J.↗

NASA NTRS1978

Modal analysis of turborotors using planar modes - Theory

The generalized dynamic equations of motion have been obtained by the direct stiffness method for multimass flexible rotor bearing systems including the effects of gyroscopic moments, disk skew, and rotor acceleration. A set of undamped critical speed mode shapes calculated from the average horizontal and vertical bearing stiffness is used to transform the equations of motion into a set of coupled modal equations of motion. The modal equations are coupled by the generalized bearing coefficients and the gyroscopic moments. An analysis using only undamped critical speeds or decoupled modal analysis assuming proportional damping may lead to erroneous results. This paper presents a rapid method of calculating rotor resonance speeds with their corresponding amplification factors, stability and unbalance response of turborotors. Examples of the application of this modal approach are presented and results are compared to those of other methods such as matrix transfer analysis.

Gunter, E. J.↗

NASA NTRS1978

Modal content of noise generated by a coaxial jet in a pipe

Noise generated by air flow through a coaxial obstruction in a long, straight pipe was investigated with concentration on the modal characteristics of the noise field inside the pipe and downstream of the restriction. Two measurement techniques were developed for separation of the noise into the acoustic duct modes. The instantaneous mode separation technique uses four microphones, equally spaced in the circumferential direction, at the same axial location. The time-averaged mode separation technique uses three microphones mounted at the same axial location. A matrix operation on time-averaged data produces the modal pressure levels. This technique requires the restrictive assumption that the acoustic modes are uncorrelated with each other. The measured modal pressure spectra were converted to modal power spectra and integrated over the frequency range 200-6000 Hz. The acoustic efficiency levels (acoustic power normalized by jet kinetic energy flow), when plotted vs. jet Mach number, showed a strong dependence on the ratio of restriction diameter to pipe diameter. The acoustic energy flow analyses based on the thermodynamic energy equation and on the results of Mohring both resulted in orthogonality properties for the eigenfunctions of the radial mode shape equation. These orthogonality relationships involve the eigenvalues and derivatives of the radial mode shape functions.

Kerschen, E. J.↗

NASA NTRS1979

Approaches to adaptive digital control focusing on the second order modal descriptions of large, flexible spacecraft dynamics

The widespread modal analysis of flexible spacecraft and recognition of the poor a priori parameterization possible of the modal descriptions of individual structures have prompted the consideration of adaptive modal control strategies for distributed parameter systems. The current major approaches to computationally efficient adaptive digital control useful in these endeavors are explained in an original, lucid manner using modal second order structure dynamics for algorithm explication. Difficulties in extending these lumped-parameter techniques to distributed-parameter system expansion control are cited.

Johnson, C. R., Jr.↗

NASA NTRS1983

Acoustical modal analysis of the pressure field in the tailpipe of a turbofan engine

The results of a modal analysis of the pressure field in the tailpipe of a turbofan engine are presented. Modal amplitudes, at the tailpipe inlet and exit, are presented, as a function of frequency, for several operating conditions. The modal amplitudes were obtained using an optimization routine to obtain a best fit between measured cross spectra and an analytical expression for the cross spectra between pressures at circumferentially spaced locations. The measured pressure field was decomposed into a set of five modal amplitudes corresponding to the (0,0), (1,0), (2,0), (3,0), and (4,0) modes. The analysis was limited to frequencies below 1500 Hz where higher order modes are cutoff. The results of the analysis showed that at low frequencies, up to the cuton frequency of the (1,0) mode, the (0,0) mode (plane wave) dominated the pressure field. The frequency range from the cuton of the (1,0) mode to the cuton of the (2,0) mode was dominated by the (1,0) mode. The (2,0) mode dominated from its cuton frequency to the upper limit of the analysis, i.e., 1500 Hz. The contribution of modes other than the dominant mode was usually small.

Krejsa, E. A.↗

NASA NTRS1983

A modal analysis of flexible aircraft dynamics with handling qualities implications

A multivariable modal analysis technique is presented for evaluating flexible aircraft dynamics, focusing on meaningful vehicle responses to pilot inputs and atmospheric turbulence. Although modal analysis is the tool, vehicle time response is emphasized, and the analysis is performed on the linear, time-domain vehicle model. In evaluating previously obtained experimental pitch tracking data for a family of vehicle dynamic models, it is shown that flexible aeroelastic effects can significantly affect pitch attitude handling qualities. Consideration of the eigenvalues alone, of both rigid-body and aeroelastic modes, does not explain the simulation results. Modal analysis revealed, however, that although the lowest aeroelastic mode frequency was still three times greater than the short-period frequency, the rigid-body attitude response was dominated by this aeroelastic mode. This dominance was defined in terms of the relative magnitudes of the modal residues in selected vehicle responses.

Schmidt, D. K.↗

NASA NTRS1984

Evaluation of modal testing methods

Modal tests are playing an increasingly important role in structural dynamics efforts which are in need of analytical model verification or trouble shootings. In the meantime, the existing modal testing methods are undergoing great changes as well as new methods are being created. Although devoted advocates of each method can be found to argue the relative advantages and disadvantages, the general superiority, if any, of one or the other is not yet evident. The Galileo spacecraft, a realistic, complex structural system, will be used as a test article for performing modal tests by various methods. The results will be used to evaluate the relative merits of the various modal testing methods.

Chen, J.-C.↗

NASA NTRS1986

Quasi-modal vibration control by means of active control bearings

This paper investigates a design method of an active control bearing system with only velocity feedback. The study provides a new quasi-modal control method for a control system design of an active control bearing system in which feedback coefficients are determined on the basis of a modal analysis. Although the number of sensors and actuators is small, this quasi-modal control method produces a control effect close to an ideal modal control.

Nonami, K.↗

NASA NTRS1984

A comparison of measured spacecraft modal damping values

The prediction and the measurement of modal damping values of spacecrafts for use in the determination of design loads were summarized. Realistic transient forcing functions were used. The solution of complex eigenvalue response solutions resulting from modal synthesis of subsystems in which each subsystem is assigned modal damping values is discussed. Damping data on both the Voyager and the Galileo spacecrafts using up to eight different types of techniques to analyze modal data are presented. It is concluded that it is difficult to estimate the true damping of a structure and the measured damping values which are dependent on the analysis method of the data.

Wada, B. K.↗

NASA NTRS1985

Structural analysis model validation using modal test data

The design of a structure which can survive a prescribed dynamic environment is generally based on the use of an analytical model of the structure. The modal test is employed to verify the analytical model. This paper is concerned with the involved validation process, taking into account modal test results from realistic complex spacecraft structural systems as examples to demonstrate the proposed validation procedure. Attention is given to a load analysis model definition, a pretest analysis and test/analysis model, and the correlation parameters. Coordinate transformation is considered along with mode identification, frequency orthogonality, effective mass, mode shape, local kinetic energy, modal force, strain energy, and modal forces by acceleration measurements. Two examples are used to illustrate the procedure.

Chen, J.-C.↗

NASA NTRS1986

Illustration of the use of modal assurance criterion to detect structural changes in an Orbiter test specimen

The analysis of empirical modal models of structural components using techniques based on the modal assurance criterion (MAC) defined by Ewins (1984) is discussed and demonstrated for the case of the Space Shuttle Orbiter aft bulkhead. Modal models of the bulkhead are constructed before and after environmental acoustic tests equivalent to 30 or 100 missions and compared (by computing their MACs) to detect significant structural changes; subdivision of the modal vector followed by computation of the MAC for each part is used to localize the changes. Numerical data demonstrating the usefulness of the MAC approach are presented in tables and diagrams.

West, Walter M.↗

NASA NTRS1986

Studies of modal identification performance using hybrid data

Modal identification accuracy can be affected significantly in practice by high modal density and nonlinearities. It may be difficult to determine the true number of modes, and many of the identified mode shapes may be questionable. Results can vary considerably using different analysis techniques. This paper introduces a new method for studying modal identification performance under such circumstances. A difficult set of test data from an erectable truss structure was modified numerically by adding two artificial modes with known parameters. These 'hybrid data' were then analyzed with the Polyreference technique and the Eigensystem Realization Algorithm (ERA). Identification performance is studied by comparing the identified modal parameters for the artificial modes with their known values.

Pappa, Richard S.↗

NASA NTRS1987

On sine dwell or broadband methods for modal testing

For large, complex spacecraft structural systems, the objectives of the modal test are outlined. Based on these objectives, the comparison criteria for the modal test methods, namely, the broadband excitation and the sine dwell methods are established. Using the Galileo spacecraft modal test and the Centaur G Prime upper stage vehicle modal test as examples, the relative advantages or disadvantages of each method are examined. The usefulness or shortcoming of the methods are given from a practicing engineer's view point.

Chen, Jay-Chung↗