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About this book

Topics in Modal Analysis & Testing, Volume 9: Proceedings of the 36th IMAC, A Conference and Exposition on Structural Dynamics, 2018, the ninth volume of nine from the Conference, brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Modal Analysis, including papers on:

Operational Modal & Modal Analysis Applications

Experimental Techniques

Modal Analysis, Measurements & Parameter Estimation

Modal Vectors & Modeling

Basics of Modal Analysis

Additive Manufacturing & Modal Testing of Printed Parts

Table of Contents


Chapter 1. Stochastic Modal Appropriation (SMA)

All available modal identification algorithms (for example SSI, FDD, ARMA) are based on the principle of subspaces in the sense that the pole is identified which characterizes the signal subspace and the noise subspace is taken automatically as its orthogonal complement. This is a response type characterization. For structural dynamics applications however, one is interested in different subspaces, namely the non-conservative subspace and the conservative subspace. The only currently available method that does this is the force appropriation method for in-laboratory testing. In this work, we propose an In-Operation structural identification algorithm based on a different principle, namely the anti-symmetry principle. The idea is as follows: Consider the response of a structure. Construct the anti-symmetric response depending on a certain unknown parameter and consider the resultant of both. This introduces a rotation to the original signal. By varying the parameter, one spans a lot of different subspaces and by proper (mathematically derived) operations on these subspaces and the original one it is possible to cancel out one subspace leaving the data in a single subspace from which the proper parameters can be estimated. This idea is formulated mathematically for a SDOF system subject to unmeasured white noise excitation and it is shown that it provides highly accurate modal parameter estimates.

M. Abdelghani, M. I. Friswell

Chapter 2. Derivation of Six Degree of Freedom Shaker Inputs Using Sub-structuring Techniques

Multi-degree of freedom testing is growing in popularity and in practice. This is largely due to its inherent benefits in producing realistic stresses that the test article observes in its working environment and the efficiency of testing all axes at one time instead of individually. However, deriving and applying the “correct” inputs to a test has been a challenge. This paper explores a recently developed theory into deriving rigid body accelerations as an input to a test article through sub-structuring techniques. The theory develops a transformation matrix that separates the complete system dynamics into two sub-structures, the test article and next level assembly. The transformation does this by segregating the test article’s fixed base modal coordinates and the next level assembly’s free modal coordinates. This transformation provides insight into the damage that the test article acquires from its excited fixed base shapes and how to properly excite the test article by observing the next level assembly’s rigid body motion. This paper examines using next level assembly’s rigid body motion as a direct input in a multi-degree of freedom test to excite the test article’s fixed base shapes in the same way as the working environment.

Tyler F. Schoenherr

Chapter 3. Forced Response of 2-Dof Gyroscopic Systems with Stable Eigenvalues

Gyroscopic conservative dynamical systems may exhibit flutter instability that leads to a pair of complex conjugate eigenvalues, one of which has a positive real part and thus leads to a divergent free response of the system. When dealing with non-conservative systems, the pitch fork bifurcation shifts toward the negative real part of the root locus, presenting a pair of eigenvalues with equal imaginary parts, while the real parts may or may not be negative. Several works study the stability of these systems for relevant engineering applications such as the flutter in airplane wings or suspended bridges, brake squeal, etc., and a common approach to detect the stability is the complex eigenvalue analysis that considers systems with all negative real part eigenvalues as stable systems. This paper studies the cases where the free response of these systems exhibits a transient divergent time history even if all the eigenvalues have negative real part thus usually considered as stable, and relates such a behavior to the non-orthogonality of the eigenvectors. Moreover, the forced response of these system is addressed, highlighting how and in which cases, an unexpected amplification of the forced response may occur.

O. Giannini

Chapter 4. Modal Testing of a Composite Bladed Disc Using Travelling Wave Excitation Method

This research article presents a novel application of travelling wave excitation method applied to a composite bladed disc. The objective of this work is to develop a non-contact excitation method for research applications where (i) blades are non-ferromagnetic and (ii) damping is nominally high. This goal was achieved by spinning a disc, on which 14 powerful DC magnets were installed, in front the composite bladed disc. Small DC magnets were attached near each blade root to provide repellent forces. Twenty blades were manufactured with pre-pregs IM7–8552, using unidirectional stacking sequence and were installed on a rigid metallic mounting hub. The paper will present the design and make of the bladed disc, the theoretical study of normal and tangential forces in a magnet-to-magnets configuration and, finally, the experimental validation of a 14-DC magnetic exciter. The forced responses were measured in one test case by a 3D single point LDV system and in another test case by a Scanning LDV system. This work will also present an attempt to develop a DC electromagnetic exciter with its limitation and potential.

D. Di Maio, M. Vater, R. Seidel, S. Foglia

Chapter 5. Simulation of the Dynamic Behavior of a Bi-Cable Ropeway with Modal Bases

This article presents the building of a dynamic model of a bi-cable ropeway. Thanks to the assumption of quasi-static advance of the vehicles, the calculation is performed step by step. At each step, a transient linearized dynamic solution is calculated around the quasi-static equilibrium using modal bases. The ropeway system is substructured considering the different elements: track rope, hauling rope and vehicles represented by pendulums. The results of time integration give the accelerations felt by the passengers according to the load case.

Gabriel Hurel, Jerome Laborde, Louis Jézéquel

Chapter 6. Influence of Noise in Correlation Function Estimates for Operational Modal Analysis

The modal parameters in Operational Modal Analysis (OMA) are often estimated based on non-parametric signatures of the structure’s dynamic response. For time domain OMA methods the non-parametric signatures are often correlation functions (CFs) and the pre-processing step for these methods is thus the estimation of CFs. The present paper demonstrates how measurement noise from sensors and measurement equipment affects the estimated CFs. Furthermore, the influence of the measurement noise on the modal parameter estimates is discussed. It is shown how effects of this noise can easily be avoided by ignoring the first part of the CFs when estimating the modal parameters. This is demonstrated by a theoretical review and on simulated and experimental data. The paper also addresses how to add noise to simulated data, so that it resembles a real-life scenario.

Esben Orlowitz, Anders Brandt

Chapter 7. Comparison of Two (Geometric) Algorithms for Auto OMA

In this paper we compare two geometric algorithms for automatic Operational Modal Analysis(OMA). The compared algorithms are the Shortest Path Algorithm (SPA) that considers shortest paths in the set of poles and the Smallest Sphere Algorithm (SSA) that operates on the set of identified poles to find the set of smallest spheres, containing physical poles. Both algorithm are based on sliding filter stability diagrams recently introduced by Olsen et al. We show how the two algorithms identify system parameters of a simulated system, and illustrate the difference between the identified parameters. The two algorithms are compared and illustrated on simulated data. Different choices of distance measures are discussed and evaluated. It is illustrated how a simple distance measure outperforms traditional distance measures from other Auto OMA algorithms. Traditional measures are unable to discriminate between modes and noise.

Martin Juul, Peter Olsen, Ole Balling, Sandro Amador, Rune Brincker

Chapter 8. Operational Modal Analysis on Wind Turbine Hub

In this paper we present a dataset for Operational Modal Analysis (OMA) on a Vestas V112 wind turbine hub and illustrate how reliable Automatic OMA (AOMA) can be done on this structure. The chosen AOMA algorithm is called the Smallest Sphere Algorithm (SSA), and is based on the sliding filter stability diagram recently introduced by Olsen et al. We show how the SSA is able to identify all modes of the hub dataset reliable, including many closely spaced modes. The algorithm also successfully identifies two triple modes automatically. We show how a set of manually identified mode shapes compares to the Finite Element (FE) model. In this context it is discussed how the phenomenon of closely spaced modes is handled by rotating the identified modes into a basis defined by the FE model. A method for mode shape merging for multi setup testing is discussed.

Martin Juul, Ole Balling, Rune Brincker

Chapter 9. The Influence of Edge Boundary Conditions and Cracks on Vibrational Modes of Multilayer Ceramic Capacitors

Electrical failure of layered capacitors is often a limiting factor in the design of many important electronic devices. Manufacturing processes, soldering, and service conditions have been shown to induce cracks in the dielectric material of the capacitor, providing conductive pathways that result in electrical leakage. In addition to the crack itself, edge boundary conditions can cause modal stress concentrations in a particular region of the capacitor that can initiate new cracks or propagate existing ones. Small but potentially damaging cracks can be very difficult to detect, and their presence may only become evident when they grow large enough to impact the capacitor’s performance. Recent experimental studies have demonstrated that cracks in layered capacitors can be detected nondestructively by measuring a shift in the resonant frequency of the structure via ferroelectric transduction. This study seeks to extend these recent findings by developing finite element models of layered capacitors in order to determine the level of influence that cracks and edge boundary conditions have on their frequency spectrum and their localized stress fields. Of particular interest is determining if ferroelectrically excited modes will be sensitive to cracks that can commonly appear near either restrained or traction-free corners. Computational investigation is intended to supplement future experiments on these structures, with the eventual goal of merging and analyzing the results from theoretical predictions and physical measurements. Specifically, three different crack types were simulated (endcap crack, inner crack, corner crack) along with three boundary conditions (free-free, fixed surface, and soldered). Results indicated that a fractured capacitor yielded lower natural frequency values in comparable modes to an uncracked capacitor, and that this shift in natural frequency values could be magnified depending on the applied boundary conditions. This finding is an important contribution toward the effort of non-destructively detecting cracks in the MLCCs and for future research to confidently utilize MLCCs in future applications.

Jonel Ortiz, Giuliana Davis, Kevin Troyer, Paul Heyliger

Chapter 10. The Cross Spectrum in Multiple Input Multiple Response Vibration Testing

Random vibration tests have been conducted for over 5 decades using vibration machines which excite a test item in uniaxial motion. With the advent of multi shaker test systems, excitation in multiple axes and/or at multiple locations is feasible. For random vibration testing, both the auto spectrum of the individual controls and the cross spectrum, which defines the relationship between the controls, define the test environment. This is a striking contrast to uniaxial testing where only the control auto spectrum is defined.In a vibration test the energy flow proceeds from drive excitation voltages to control acceleration auto and cross spectral densities and finally, to response auto and cross spectral densities. This paper examines these relationships, which are encoded in the frequency response function. Following the presentation of a complete system diagram, examination of the relationships between the excitation and control spectral density matrices is clarified. It is generally assumed that the control auto spectra are known from field measurements, but the control cross spectra may be unknown or uncertain. Given these constraints, control algorithms often prioritize replication of the field auto spectrum. The system dynamics determine the cross spectrum. The Nearly Independent Drive Algorithm, described herein, is one approach.A further issue in Multi Input Multi Response testing is the link between cross spectrum at one set of locations and auto spectra at a second set of locations. The effect of excitation cross spectra on control auto spectra is one important case, encountered in every test. The effect of control cross spectra on response auto spectra is important since we may desire to adjust control cross spectra to achieve some desired response auto spectra. The relationships between cross spectra at one set of locations and auto spectra at another set of locations is examined with the goal of elucidating the advantages and limitations of using control cross spectra to define response auto spectra.

Norman F. Hunter, Kevin R. Cross, Garrett Nelson

Chapter 11. A Systematic Evaluation of Test Specification Derivation Methods for Multi-axis Vibration Testing

In the past decade, multi-axis vibration testing has progressed from its early research stages towards becoming a viable technology which can be used to simulate more realistic environmental conditions. The benefits of multi-axis vibration simulation over traditional uniaxial testing methods have been demonstrated by numerous authors. However, many challenges still exist to best utilize this new technology. Specifically, methods to obtain accurate and reliable multi-axis vibration specifications based on data acquired from field tests is of great interest. Traditional single axis derivation approaches may be inadequate for multi-axis vibration as they may not constrain profiles to adhere to proper cross-axis relationships—they may introduce behavior that is neither controllable nor representative of the field environment. A variety of numerical procedures have been developed and studied by previous authors. The intent of this research is to benchmark the performance of these different methods in a well-controlled lab setting to provide guidance for their usage in a general context. Through a combination of experimental and analytical work, the primary questions investigated are as follows: (1) In the absence of part-to-part variability and changes to the boundary condition, which specification derivation method performs the best? (2) Is it possible to optimize the sensor selection from field data to maximize the quality/accuracy of derived multi-axis vibration specifications? (3) Does the presence of response energy in field data which did not originate due to rigid body motion degrade the accuracy of multi-axis vibration specifications obtained via these derivation methods?

Garrett Nelson

Chapter 12. Designing Hardware for the Boundary Condition Round Robin Challenge

Qualification of products to their vibration and shock requirements in a laboratory setting consists of two basic steps. The first is the quantification of the product’s mechanical environment in the field. The second is the process of testing the product in the laboratory to ensure it is robust enough to survive the field environment. The latter part is the subject of the “Boundary Condition for Component Qualification” challenge problem. This paper describes the challenges in determining the appropriate boundary conditions and input stimulus required to qualify the product. This paper also describes the steps and analyses that were taken to design a set of hardware that demonstrates the issue and can be used by round robin challenge participants to investigate the problem.

David E. Soine, Richard J. Jones, Julie M. Harvie, Troy J. Skousen, Tyler F. Schoenherr

Chapter 13. Modal Comparison of Stock and Performance Brake Rotors

Although generally similar in appearance, automobile brake rotors vary widely in terms of cost, material, and quality. Modal analysis was conducted on a variety of stock and performance used and new brake rotors to determine whether vibration analysis could be an effective method for evaluating quality, performance, and/or remaining life of a given brake rotor. Regular and slotted performance brake rotors were tested using an impact hammer and tri-axial accelerometer, and natural frequencies were evaluated up to 12,000 Hz. Testing methodology was varied to determine what parameters were most effective to highlight differences between rotors. The overall finding was that the most significant difference between rotors was due to the reduced mass of a used rotor; i.e., different types of rotors for the same vehicle showed greater similarity in dynamic response than a new rotor and a used rotor of the same type. Additionally, comparison of new and used rotors can show a correlation between remaining rotor life and dynamic response. While simple mass measurement of the rotor could provide similar information, vibration data can be taken with the brake rotor in place on the vehicle, and could provide information about the amount of brake rotor used in high performance driving events during the event, as well as the condition of the rotor. Testing is ongoing, and may also include identification of dynamic response for cracked rotors. Further testing is required to verify the differences in rotor wear for rotors installed on the vehicle.

Kaitlin Spak

Chapter 14. On the Veering Phenomenon Potential in High Speed Gears Design

In modern industry, the design process of most mechanical components is aimed at reducing their mass and increasing the performance, especially in weight-critical applications like aero-space engines. This approach often results in components that can have resonances in the operative range that could cause excessive vibrations and a consequential reduction of the life of the component itself. For this reason, a modal analysis check is always performed and the design process is iterated until also the dynamic behavior is acceptable. However, this approach is non-optimal, as not all resonances are excited during operation. Hence a design process that aims also to reduce the real dynamic response is proposed in this paper. By changing the geometry of thin-webbed high speed helical gears the dynamic response is altered to modify the mode shapes to obtain ones that are non-excitable by the external forces in the operative range, thus greatly improving its dynamic response. The original contribution of the paper relies on the improvement of the dynamic design process by means of the optimal balance between geometrical parameters and dynamic behavior governing phenomena like veering and crossing. Due to the high flexibility of these components, the design process is developed considering the stress-stiffening effect of the centrifugal force on the resonance frequencies and also the high influence of gyroscopic effects. To prove the effectiveness of this approach, a test case is presented with the main rotation-induced phenomena considered and the results obtained highlight the importance of the veering phenomenon in the switching between different mode shapes, and the great reduction in the response of the component.

Carlo Rosso, Elvio Bonisoli, Fabio Bruzzone

Chapter 15. Modal Truncation in Experimental Modal Analysis

Some methods in experimental modal analysis rely on a finite set of modes and they neglect the higher modes. However, this approach causes a truncation of the modal decomposition and the modal truncation introduces errors of unknown magnitude. In this paper the effect of modal truncation is investigated on a test specimen in the laboratory. It is found that the system response is dependent of the frequency and the distribution of the load. Modal truncation can introduce significant errors if the set of mode shapes does not efficiently span the spatial distribution of the load.

Marius Tarpø, Michael Vigsø, Rune Brincker

Chapter 16. Combined Mechanical Environments for Design and Qualification

Aerospace systems and components are designed and qualified against several operational environments. Some of these environments are climatic, mechanical, and electrical in nature. Traditionally, mechanical test specifications are derived with the goal of qualifying a system or component to a suite of independent mechanical environments in series. True operational environments, however, are composed of complex, combined events. This work examines the effect of combined mechanical shock and vibration environments on response of a dynamic system. Responses under combined environments are compared to those under single environments, and the adequacy/limitations of conventional, single environment test approaches (shock only or vibration only) will be assessed. Test integration strategies for combined shock and vibration environments are also discussed.

Brian C. Owens, Julie M. Harvie

Chapter 17. Comparing Free-Free and Shaker Table Model Correlation Methods Using Jim Beam

Finite element model correlation as part of a spacecraft program has always been a challenge. For any NASA spacecraft, a Coupled Loads Analysis (CLA) is used to predict the coupled system response of the spacecraft and launch vehicle. The accuracy of the CLA is highly dependent on the precision of the frequencies and mode shapes extracted from the spacecraft model. NASA standards require the spacecraft model used in the final Verification Loads Cycle to be correlated by a modal test. Due to budgetary and time constraints, most programs opt to correlate the spacecraft dynamic model during the environmental qualification test, conducted on a large shaker table.For any model correlation effort, the key has always been finding a proper definition of the boundary conditions. This paper is a correlation case study to investigate the difference in responses of a simple structure using a free-free boundary, a “fixed” boundary on the shaker table, and a base-drive vibration test, all using identical instrumentation. The NAVCON Jim Beam test structure, featured in the IMAC XXVII round robin modal test of 2009, was selected as a simple, well recognized and well characterized structure to conduct this investigation.First, a free-free impact modal test of the Jim Beam was done as an experimental control. Second, the Jim Beam was mounted to a large 22,000 lbf shaker, and an impact modal test in this pseudo-fixed configuration was conducted. Lastly, a vibration test of the Jim Beam was conducted on the shaker table. The free-free impact test, the “fixed” impact test, and the base-drive test were used to assess the effect of the shaker modes, evaluate the validity of fixed-base modeling assumptions, and compare final model correlation results between these boundary conditions.

James Ristow, Kenneth Wayne Smith, Nathaniel Johnson, Jackson Kinney

Chapter 18. Real-Time Hybrid Testing: Challenges and Experiences from a Teaching Point of View

Hybrid simulation techniques, which combine real physical models with virtual simulation models, have developed significantly in the last decades. Continuing scientific work and steady progress in simulation and modeling techniques together with powerful automatic code generation tools have pushed this development. For an engineer, this technology offers the possibility of significant savings, faster product development, reduced design uncertainties and reliable component testing without any risk. Due to these advantages there is an increased industrial acceptance and consequently the integration in engineering education is required. From a scientific point of view, hybrid testing demands advanced knowledge in the fields of modeling and simulation, real-time integration, model-order reduction, measurement and signal processing. Furthermore, since coupled systems generally result in a closed loop structure, profound understanding of control theory as well as sensors and actuators is essential. In order to manage this diversity of requirements, several experiments of varying complexity have been developed for illustration of the theoretical background. The laboratory experiments presented comprise dynamic absorber testing and hybrid testing of mechatronic systems like a quadcopter during complex flight operations. The aim is a basic understanding of hybrid testing, its challenges and potentials, and the ability recognize and implement possible application in science and industry.

Markus J. Hochrainer

Chapter 19. Comparison of Computational Generalized and Standard Eigenvalue Solutions of Rotating Systems

Modal analysis is regularly used to compute natural frequencies and mode shapes of structures via eigenvalue solutions in vibration engineering. In this paper, the eigenvalue problem of a 6 degrees of freedom rotating system with gyroscopic effects, including axial, torsional and lateral motion, is investigated using Timoshenko beam theory. The main focus thereby is the investigation of the computational time and the numerical errors in generalized and standard eigenvalue solutions of rotating systems. The finite element method is employed to compute the global stiffness, mass and gyroscopic matrices of the rotating system. The equations of motion is expressed in the state space form to convert the quadratic eigenvalue problem into the generalized and standard forms. The number of elements in the finite element model was varied to investigate the convergence of the natural frequencies and the computational performance of the two eigenvalue solutions. The numerical analyses show that the standard eigenvalue solution is significantly faster than the generalized one with increasing number of elements and the generalized eigenvalue solution can yield wrong solutions when using higher numbers of elements due to the ill-conditioning phenomenon. In this regard, the standard eigenvalue solution gives more reliable results and uses less computational time than the generalized one.

Ali Tatar, Loic Salles, Alexander H. Haslam, Christoph W. Schwingshackl

Chapter 20. Residual States for Modal Models Identified from Accelerance Data

Residual stiffness and mass terms are often employed in frequency response synthesis to compensate for outside band eigenmodes in the identification of modal models from test data. For structures that have strongly participating modes above the test frequency band, it has been observed that in particular direct accelerances with strong outside-band modal contribution tend to render modal models that give poor fit to test data. For such problems it may be insufficient to just add residual mass and stiffness terms to the accelerance modal series to get a sufficiently improved fit. For accelerance, such residual terms are constant and quadratic in frequency. Another, residual term that is quasi-linear over the frequency range of interest has been found to augment the identified model. In this paper that complementary term is added to the constant and quadratic terms in a state-space model identification with a subspace state-space identification method. A comparison is performed to an alternative residualisation method. The methods’ results are compared on simulated finite element test data from of an automotive component.

Mladen Gibanica, Thomas J. S. Abrahamsson, Randall J. Allemang

Chapter 21. Comparison of Time-Domain Objective Functions in Dynamic Fixture Optimization

Differences in impedance are usually observed when components are tested in fixtures at lower levels of assembly from those in which they are fielded. In this work, the Kansas City National Security Campus (KCNSC) test bed hardware geometry is used to explore the sensitivity of the form of the objective function on the adequate reproduction of relevant response characteristics at the next level of assembly. Inverse methods within Sandia National Laboratories’ Sierra/SD code suite along with the Rapid Optimization Library (ROL) are used for identifying an unknown material (variable shear and bulk modulus) distributed across a predefined fixture volume. Comparisons of the results between time-domain based objective functions are presented. The development of the objective functions, solution sensitivity, and solution convergence will be discussed in the context of the practical considerations required for creating a realizable set of test hardware based on the variable-modulus optimized solutions.

Michael J. Starr, Timothy Walsh

Chapter 22. Advanced Hammer Excitation Technique for Impact Modal Testing on Lightweight Materials Using Scalable Automatic Modal Hammer

Experimental Modal Analysis (EMA) on a lightweight material has proven to be very challenging in the recent past. The applications of these materials have increased invariably in various fields and so have a high demand for Research & Development (R&D). A lightweight material is very sensitive in terms of vibration. EMA on these materials in free - free boundary condition is very complicated as the hammer excitation becomes very difficult. In order to acquire valid results, the conditions are modified, and in consequence, obtain inaccurate dynamic characteristics.Some of the major challenges faced are: (a) material getting displaced from its original position after every hit, (b) difficulties in obtaining a single hit, (c) reproducing the same excitation force level for averaging output response. Overcoming these crucial challenges can result in reducing the inaccuracies in the results. Scalable Automatic Modal hammer (SAM) is developed to overcome these challenges and enables the ability to reproduce the same force level of excitation. This advanced hammer excitation technique has the capability to avoid the double hit, adjust the repeatability of force level and automatizes the entire excitation process.In this research paper, a light weight material is experimented under free-free boundary condition and the obtained results are analyzed. The input hammer excitation is provided by SAM and the output contactless response is measured by Scanning Laser Doppler Vibrometer (SLDV).The conclusions provided will reflect the importance of repeatability and reproducibility of hammer excitation force level in order to acquire accurate results. The controlling of SAM, by changing various parameters, in order to precisely excite lightweight structures will be demonstrated.

Tarun Teja Mallareddy, Sarah Schneider, Peter G. Blaschke

Chapter 23. Evaluation of MIMO Input Derivations and Their Physical Context

Multi-Input-Multi-Output (MIMO) vibration testing is considered to be more representative of the actual loads on many articles of interest. Since the derivation of MIMO inputs involve a matrix inversion process using N × M transfer functions corresponding to N input and M output locations, it is affected by both mathematical and physical parameters. The derived input loads can differ depending on the inversion and convergence algorithm used as well as on the accuracy of the underlying data.A study was conducted using data from a benchtop MIMO test. The objective of the study was to understand how the convergence criteria and the underlying data used in MIMO algorithms affect the derived inputs. Two different input derivation methods were used. Of particular interest is the influence of the Tikhonov regularization used in the matrix inversion process and the consequence thereof in achieving the desired outputs as well as the input loads. The paper will present the results of these studies with the intent of providing a better understanding of their implications in tests and analyses associated with MIMO.

Arup Maji

Chapter 24. Using Modal Substructuring to Improve Shock & Vibration Qualification

Qualification of complex systems often involves shock and vibration testing at the component level to ensure each component is robust enough to survive the specified environments. In order for the component testing to adequately satisfy the system requirements, the component must exhibit a similar dynamic response between the laboratory component test and system test. There are several aspects of conventional testing techniques that may impair this objective. Modal substructuring provides a framework to accurately assess the level of impairment introduced in the laboratory setup. If the component response is described in terms of fixed-base modes in both the laboratory and system configurations, we can gain insight into whether the laboratory test is exercising the appropriate damage potential. Further, the fixed-base component response in the system can be used to determine the correct rigid body laboratory fixture input to overcome the errors seen in the standard component test. In this paper, we investigate the effectiveness of reproducing a system shock environment on a simple beam model with an essentially rigid fixture.

Julie M. Harvie

Chapter 25. Off-Axis Input Characterization of Random Vibration Laboratory Data for Model Credibility

The goal of this work is to build model credibility of a structural dynamics model by comparing simulated responses to measured responses in random vibration environments, with limited knowledge of the true test input. Oftentimes off-axis excitations can be introduced during single axis vibration testing in the laboratory due to shaker or test fixture dynamics and interface variation. Model credibility cannot be improved by comparing predicted responses to measured responses with unknown excitation profiles. In the absence of sufficient time domain response measurements, the true multi-degree-of-freedom input cannot be exactly characterized for a fair comparison between the model and experiment. Methods exist, however, to estimate multi-degree-of-freedom (MDOF) inputs required to replicate field test data in the laboratory Ross et al.: 6-DOF Shaker Test Input Derivation from Field Test. In: Proceedings of the 35th IMAC, A Conference and Exposition on Structural Dynamics, Bethel (2017). This work focuses on utilizing one of these methods to approximately characterize the off-axis excitation present during laboratory random vibration testing. The method selects a sub-set of the experimental output spectral density matrix, in combination with the system transmissibility matrix, to estimate the input spectral density matrix required to drive the selected measurement responses. Using the estimated multi-degree-of-freedom input generated from this method, the error between simulated predictions and measured responses was significantly reduced across the frequency range of interest, compared to the error computed between experimental data to simulated responses generated assuming single axis excitation.

Jill Blecke, James Freymiller, Michael Ross

Chapter 26. Modal Analysis of a Brake-Ruess Beam and Computational Modeling at the Undergraduate Level

This paper outlines ongoing efforts to develop labs for undergraduate students at New Mexico Tech (NMT) from research projects sponsored by Sandia National Laboratories (SNL) [1, 2] to quantify the low frequency, high displacement damping characteristics of a Brake-Reuss beam.

Aaron J. Misla, Curtis J. O’Malley, Siavash Norouzi

Chapter 27. A Primer on Multiple Degree of Freedom Vibration Test for Aerospace and Military Applications

The aerospace and military communities have performed vibration tests for decades using a single axis or single degree of freedom (SDOF) approach. In recent years, military standards have recognized a multiple exciter or multiple degree of freedom (MDOF) approach for conducting vibration testing. This primer on MDOF vibration testing serves to introduce the topic to the IMAC community and fits with IMAC-XXXVI’s theme: Engineering Extremes.This presentation will review how SDOF and MDOF vibration environmental definitions are obtained. The concept of cloud plots will be reviewed and approaches for determining acceptable test levels will be discussed. The common approach in the MIL-STD-810 community for accelerating testing using the Palmgren-Miner Hypothesis will be reviewed.Next, the presentation will shift from the actual real-world vibration environment to the laboratory environment with special consideration reviewed such as: non-linearity effects, boundary conditions, controllability and observability.Analogies between the power spectral density for an SDOF test and the spectral density matrix (SDM) for an MDOF test will be reviewed. The author’s previous work will be adapted for illustrating how the SDM can take on a truly random nature for MDOF environments.

Luke A. Martin

Chapter 28. A Study on the Generation and Propagation of Traveling Waves in Strings

Traveling and standing waves occurring in mechanical systems are the results of the interplay of excitation sources, locations, and boundary conditions. While traveling waves carry energy throughout a system, standing waves keep such energy within a located area associated with the modes of excitation. Depending on the desired characteristics of a system, it is crucial to understand the wave propagation and what parameters affect wave propagations. In the present work, one-dimensional string equations are studied with fixed-fixed boundary condition with the purpose of generating steady-state traveling waves. Two excitation forces at various frequencies of excitation are applied to a string near opposing boundaries to understand the generation and propagation of traveling and standing waves. The work focuses on how parameters affect the wave propagation on a string under the fixed-fixed boundary condition and their quality examined. Understanding the effects of these parameters on the wave propagation of a string can lead to better understanding of microorganism behaviors, such as the propulsion of flagella or biological counter parts such as the basilar membrane (2D string) in the ear’s cochlea that exhibit non-reflective waves.

Isil Anakok, V. V. N. Sriram Malladi, Pablo A. Tarazaga

Chapter 29. A Color-Coded Complex Mode Indicator Function for Selecting a Final Mode Set

Many test articles exhibit slight nonlinearities which result in natural frequencies shifting between data from different references. This shifting can confound mode fitting algorithms because a single mode can appear as multiple modes when the data from multiple references are combined in a single data set. For this reason, modal test engineers at Sandia National Laboratories often fit data from each reference separately. However, this creates complexity when selecting a final set of modes, because a given mode may be fit from a number of reference data sets. The color-coded complex mode indicator function was developed as a tool that could be used to reduce a complex data set into a manageable figure that displays the number of modes in a given frequency range and also the reference that best excites the mode. The tool is wrapped in a graphical user interface that allows the test engineer to easily iterate on the selected set of modes, visualize the MAC matrix, quickly resynthesize data to check fits, and export the modes to a report-ready table. This tool has proven valuable, and has been used on very complex modal tests with hundreds of response channels and a handful of reference locations.

Randy L. Mayes, Daniel P. Rohe

Chapter 30. Using Manual Excitation for Large Displacement on a Highly Damped System

Small-scale, quickly executed modal surveys can yield an incredible amount of useful information. Electrodynamic shakers and especially impact hammers have been the excitation methods of choice for these tests because of their ease of use. However, large low-frequency structures with highly damped modes are difficult to excite due to exciter stroke limitations or inability to apply long duration pulse inputs. Hydraulic actuation allows large displacements at low frequencies, but it is expensive and challenging to install in a short amount of time. Similarly, while step-relaxation techniques allow large force and displacement inputs, setup, using a reaction structure, and execution is complicated by the need to reset the system multiple times. In order to overcome some of these drawbacks, ATA developed a modal handle to apply a manual excitation while measuring the input levels of the excitation force. This paper presents results of using this modal handle to perform a modal survey on a test article to evaluate the effect of different dampers and damper configurations on the test article primary torsion mode. This inexpensive and efficient excitation method proved to be successful in quickly completing over eighty test configurations needed to down select to a final damper configuration.

Douglas J. Osterholt, David E. Cloutier

Chapter 31. Traveling Wave Generation on a Clamped, Thin Plate with Flush-Mounted Piezoelectric Actuators

Structural traveling waves have significant potential in applications such as propulsion and drag reduction. Using the two-mode excitation method, traveling waves which are both steady-state and open-loop controlled can be generated on various structures such as beams, plates, and cylinders. In order to develop this method further for applications such as drag reduction, more representative cases must be investigated. This work models and experimentally validates traveling waves generated on a thin, fully clamped plate with flush-mounted piezoelectric actuators. A two-step experimental modal analysis is conducted on a free and then clamped plated to validate and update a finite-element model of the plate. The finite element model, which has been developed in-house, accounts for piezoelectric actuators, pre-stresses in the plate, and non-ideal (elastic) clamped boundary conditions. The validated model is then used to compare experimental- and model-generated traveling waves.

Patrick F. Musgrave, Mohammad I. Albakri, Pablo A. Tarazaga

Chapter 32. Experimental Modal Analysis of an Aircraft Fuselage Panel: Part II

Hypersonic aircraft structures must operate in complex loading conditions and very high temperatures, making the design of a robust and reusable platform very challenging. An analytical and experimental test program was developed by the Air Force Research Laboratory (AFRL) and industry. The objective of the program is to review the design process of a thin skinned aircraft panel subjected to combined thermal-acoustic-mechanical loading, through a series of laboratory experiments at the AFRL’s Structural Dynamics Laboratory.This paper is a continuation of previously presented work on a series of modal tests, performed to characterize the dynamics of a hat-stiffened fuselage panel designed by industry. In the previous work the modal test was performed under free-free boundary conditions. In order to assess the effect of the operating boundary conditions on the modal parameters, a roving impact test was performed after its installation in the Combined Environment Acoustic Chamber (CEAC) facility at AFRL. The article and fixture supports were instrumented in order to assess the effect of the support system. A description of the modal test will be included in the paper along with a discussion of the data obtained.

Travis A. Wyen, Ricardo A. Perez, Jonathan Knox, Joshua J. Schoettelkotte, Thomas G. Eason

Chapter 33. Combining Virtual Simulation with Hands-on Experiments for Teaching Mechanical Vibrations

Mechanical vibration is one of the important courses offered in the systems and dynamics area. Traditionally this course is taught theoretically with little emphasis on experimental verification. Since non-intuitive concepts are explained involving complex mathematical manner, students usually find this to be a one of the difficult course in the curriculum. To address these issues, we have developed an integrated approach which combines virtual simulation of each concept with the laboratory experimentation. The course material is built and structured around a standard vibration course material to make it easy to implement in a traditional curriculum while enhancing the learning experience of the students. This combines the theory with hands-on, easy-to-set experiments and interesting simulations. The material is structured similar to a typical vibration text book. Students can perform a number of experiments, which include; simple single degree of freedom systems, multi-degree of freedom systems and continuous beams with different boundary conditions. Both free and forced vibration experiments can be performed in torsional and translational configurations, with and without damping. We also provide an appealing virtual environment for each experiment so that students can graphically see and sense the effects of changing the system parameters. An example is seeing the effect of damping on vibration amplitude in different frequency regions such as stiffness controlled, mass controlled, and around resonances. The complete package includes a full set of cost-effective data acquisition system, wireless sensors and comprehensive user-friendly software.

Changrui Bai, Surendra (Suri) Ganeriwala

Chapter 34. Finite Element Model Updating Using the Local Correspondence Principle

In this, paper an overview of a Finite Element (FE) model updating technique based on the Local Correspondence (LC) principle is presented. The main idea behind the LC technique is to update the FE model by replacing the mode shape vectors and natural frequencies with their corresponding experimental counterparts obtained from an output-only modal testing. This is accomplished by taking advantage of the fact that the inverse mass and stiffness matrices can be expressed as a linear combination of outer products of the mode shape vectors. Aiming at discussing the LC technique from a practical perspective, a simulation study is presented to illustrate its ability to improve the Maximum Assurance Criterion (MAC) between the FE and experimental mode shape vectors so that it gets close to unity.

Sandro Amador, Martin Juul, Tobias Friis, Rune Brincker

Chapter 35. Approximate General Responses of Tuned and Mistuned 4-Degree-of-Freedom Systems with Parametric Stiffness

The purpose of this study is to find approximate solutions to tuned and mistuned 4-DOF systems with parametric stiffness. In this work, the solution and stability of four-degree-of-freedom Mathieu-type system will be investigated. To find the broken-symmetry system response, Floquet theory with harmonic balance will be used. A Floquet-type solution is composed of a periodic and an exponential part. The harmonic balance is applied to the original differential equation of motion. The analysis brings about an eigenvalue problem. By solving this, the Floquet characteristic exponents and the corresponding eigenvectors that give the Fourier coefficients are found in terms of the system parameters. The stability transition curve can be found by analyzing the real parts of the characteristic exponents. The frequency content can be determined by analyzing imaginary parts at the exponents. A response that involves single Floquet exponent (and its complex conjugate) can be generated with a specific set of initial conditions, and can be regarded as a modal response. The method is applied to both tuned and detuned four-degree-of-freedom examples.

Ayse Sapmaz, Gizem D. Acar, Brian F. Feeny

Chapter 36. Modal Analysis of a Vertical-Axis Darrieus Wind Turbine Blade with a Troposkein Shape

Darrieus wind turbines with troposkein shaped blades are an important type of vertical axis wind turbines. When designing these blades it is important to consider vibration properties and how vibrations may affect failure and safe load margins. In order to avoid the occurrence of resonance, the natural frequencies of the blade structure needs to be determined and designed relative to the turbine operating ranges.In this work we examine the modal characteristics of Darrieus style vertical-axis wind-turbine blades. We build a finite element model of a spinning Sandia straight-circular-straight blade, and compare the results of our model with those of Sandia. We then use the model to study the stationary blade, and the blade with a true troposkein blade shape. The modeling of the blades using slender beam theory is sketched for comparison with the finite element model and for its eventual use for low-order modeling.

Amr Saleh, Brian F. Feeny

Chapter 37. Floquet-Type Analysis of Transient Vibrations of a Horizontal Axis Wind Turbine

General responses of coupled blade-hub equations of a horizontal axis wind turbine (HAWT) are studied. HAWT blades have parametric stiffness terms due to gravity. The blade equations are coupled through the rotor equation, and blade stiffness varies cyclically with the hub angle. In this study, the equations of motion are transformed from the time domain to the hub angle domain, and a scaling scheme is used which results in interdependent blade equations and eliminates the rotor equation. With hub angle as the independent coordinate, the blade equations have parametric stiffness. Then, assuming a Floquet-type solution, unforced dynamics of a turbine is investigated. The assumed solution is a product between an exponential and a periodic part. Plugging into the equations of motion, and applying harmonic balance method, an eigenvalue problem is obtained in terms of system parameters, where the eigenvalues are the characteristic exponents in the exponential part of the assumed solution. The solution to the eigenvalue problem provides parametric modal solutions. The response to an arbitrary initial condition is approximated by combining the modal solutions. Additionally, stability of the solutions is investigated by examining the characteristic exponents. The results are then compared to numerical solutions for verification.

Gizem D. Acar, Brian F. Feeny

Chapter 38. Evaluation of Traveling Wave Models for Carangiform Swimming Based on Complex Modes

The research problem we considered is to evaluate the accuracy of traveling wave model proposed in the literature as the kinematic model for fish midline motions during straight forward carangiform swimming. Almost all the literature uses a sinusoidal traveling wave model with constant wavelength and frequency for the model of lateral movements of body. We acquired raw data of midline lateral movements for three Carangiform fish from the resources available in the literature. On the other hand, we built the traveling wave models based on the format used in literature. We used COD (complex orthogonal decomposition) to decompose the total motion associated with the raw data and with the traveling wave model into complex modes and derive the wave properties. Through this analysis we evaluated the traveling wave model accuracy. The criteria we chose for comparison was the dominant modes’ shape and their number, frequencies and wavelength associated to each mode. As a result of this analysis, we found that both the lab data and the traveling wave model, have a single dominant mode. The main difference between these two was that the phase change rate with respect to location and with respect to time is not constant in raw data, however in the traveling wave model we used constant frequency and wavelength.

Mahdieh Tanha, Brian F. Feeny

Chapter 39. Application of Frequency Domain Decomposition Identification Technique to Half Spectral Densities

Because of its simplicity and robustness, the Frequency Domain Decomposition (FDD) identification technique have become very popular in the operational modal analysis community. The basic idea behind this technique consists of computing the singular value decomposition of the power spectral densities estimated with the periodogram (also known as “Welch’s” periodogram) approach to identify the natural frequencies and mode shape vectors. In this paper, the benefits of the application of the FDD technique to half spectral densities – the power spectral densities estimated from the positive part of the correlation functions – are investigated. In order to illustrate such benefits from a practical perspective, the FDD identification results obtained from the half spectral densities, of both simulated and real structures, are compared to those from the classical periodogram-driven FDD.

Sandro Amador, Martin Ørum, Tobias Friis, Rune Brincker

Chapter 40. Modal Survey of the MPCV Orion European Service Module Structural Test Article Using a Multi-axis Shake Table

In 2016, the Orion European Service Module Structural Test Article (E-STA) underwent sine vibration testing using the multi-axis shaker system at NASA Glenn Research Center’s (GRC) Plum Brook Station (PBS) Space Power Facility (SPF) Mechanical Vibration Facility (MVF). An innovative approach using measured constraint shapes at the interface of E-STA to the MVF Table allowed high-quality fixed base modal parameters of the E-STA to be extracted, which have been used to update the E-STA finite element model (FEM), without the need for a traditional fixed base modal survey. This innovative approach provided considerable program cost and test schedule savings. This paper documents this modal survey, which includes the modal pretest analysis sensor selection, the fixed base methodology using measured constraint shapes as virtual references and measured frequency response functions, and post-survey comparison between measured and analysis fixed base modal parameters.

James P. Winkel, James C. Akers, Vicente J. Suarez, Lucas D. Staab, Kevin L. Napolitano

Chapter 41. Modal Analysis of Healthy and Cracked Isotropic Plates in Peridynamics

Modal analysis is widely used to validate numerical models, for quality control in manufacturing, and to detect structural damage. However, it is hard to compute modes of a structure with cracks, because the partial differential equations used in continuum mechanics are undefined along discontinuities in the deformation field. Peridynamic theory is a nonlocal extension of continuum mechanics that uses integral equations, which are defined in presence of cracks. Therefore, it is can be used to analyze changes in natural frequencies and mode shapes due to cracking. In this study, we compute the first five modes of 3D isotropic plates made from poly(methyl methacrylate) with free-free boundary conditions in peridynamics and compare them to experimental and finite-element analysis results. Afterwards, we introduce a crack in the cross-section and again compare the peridynamic and experimental results. Peridynamic natural frequencies of both healthy and cracked plates are within 3% of the experimental results and show similar frequency shifts due to damage. PD mode shapes match the experimental ones in both healthy and cracked cases.

Andris Freimanis, Ainārs Paeglītis

Chapter 42. Initial Modal Results and Operating Data Acquisition of Shock/Vibration Fixture

This paper presents the initial experimental and FEA based modal analysis results obtained on a test assembly developed specifically to study the effects of component boundary conditions and excitation techniques on test damage potential during component qualification testing. This assembly was developed as a platform with a simple “component” and “next assembly” that allows the component to be removed and attached via a fixture to shock or vibration test equipment. All data and results will be made publicly available for other groups wishing to study the test assembly in pursuit of insight into how to define appropriate boundary conditions for component testing.

William Larsen, Jason R. Blough, James P. DeClerck, Charles D. VanKarsen, David E. Soine, Richard Jones

Chapter 43. Effects of Variable Thickness Circular Plates on Frequency Response Functions and Shock Response Spectrum

Resonant plates used for shock testing are typically a constant thickness. Prior research demonstrated that circular plates utilize symmetry to limit the number of contributing modes, although more design control is necessary to achieve target shock response spectra (SRS). Analytical modeling results show that variable thickness plates provide more flexibility to meet a target SRS. The first membrane mode of a circular plate correlates with the knee frequency in the shock response spectrum. Higher order membrane modes can cause the SRS to occur outside of the target band. Concave plates decrease the frequency band between first membrane mode and higher order membrane modes, while convex plates show the opposite effect. Using this theory, resonant plate cross section can be altered to tune resonant plate natural frequencies in order to achieve target SRS.

William Larsen, Jason R. Blough, James DeClerck, Charles VanKarsen, David Soine, Richard Jones

Chapter 44. Inverse Force Estimation for Resonant Shock Plate Application

Resonant shock plate testing uses a projectile and programmer material to deliver and tune an impulsive force. Typically, the force level is too high to directly measure with conventional force sensors, so the spectral and temporal characteristics of these forces are not well understood. Non-linear simulations of the projectile, programmer, shock plate, and fixture are currently used to predict the results and design a resonant shock plate. A linear model of the resonant shock plate and fixture could be used if a reasonable representation of the applied force was known.This paper explores the use of inverse force estimation to estimate the spectral content of the force applied from the projectile through the programmer material. The process involves de-convolving the resonant plate response and the impulse response of the resonant plate/fixture system. A spectral representation of the force can be obtained by dividing the linear spectrum of the resonant plate/fixture response and its frequency response function.

William Larsen, Jason R. Blough, James DeClerck, Charles VanKarsen, David Soine, Richard Jones
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