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Uncertainty Quantification of Inducer Natural Frequency using Conditional Assessment of Modeling and Modal Testing of Simpler Systems

The low pressure fuel pump inducer of the new Space Launch System RS25 core stage engine operates in a highly complex environment that substantially affects its modal characteristics. Some of the more important effects are fluid-added mass resulting from operation within a light liquid (Hydrogen), and the magnification of this effect due to tight tip clearance. Since higher order cavitation has been identified as a significant harmonic driver, knowledge of the natural frequency of potentially excitable modes is critical for safe operation, but this frequency cannot be measured during the severe operational environment. A comprehensive testing and analysis program has therefore been performed over the last four years to identify the nominal value and uncertainty of the frequency by modeling and testing four simpler structures which share some of the characteristics of the operational inducer. This testing was used to assess and adjust modeling techniques and excellent correlation was achieved. Identification of the uncertainty in the inducer frequency itself was still problematic, however. This difficulty led to an investigation of Bayesian uncertainty quantification techniques, and to the application of the relatively simple technique of Multi-Variate Normal conditional distributions to calculate the inducer natural frequency uncertainty. Assumptions on prior distributions of uncertainty of the fluid-added mass and tip clearance effect are initially applied to models of each of the simple structures and the inducer itself, and these uncertainties are propagated to generate natural frequencies using design of experiments. Simple response surfaces are then created from this data in order to calculate a Covariance Matrix relating all of these natural frequencies. Finally, the results from modal test of the simple structures are considered to be observations and used to calculate the conditional variance of the desired inducer frequencies. As this method is less rigorous than more complicated Bayesian methods reported in the literature, a conservative factor is applied to the result, but the resulting uncertainty is still significantly less than originally estimated and will greatly assist certification of the inducer for use in the engine.

Uncertainty Quantification

Uncertainty Quantification of Inducer Eigenvalues using Conditional Assessment of Models and Modal Test of Simpler Systems

The low pressure fuel pump inducer of the new Space Launch System RS25 core stage engine operates in a highly complex environment that substantially affects its modal characteristics. Some of the more important effects are fluid-added mass resulting from operation within a light liquid (Hydrogen), and the magnification of this effect due to tight tip clearance. Since higher order cavitation has been identified as a significant harmonic driver, knowledge of the natural frequency of potentially excitable modes is critical for safe operation, but this frequency cannot be measured during the severe operational environment. A comprehensive testing and analysis program has therefore been performed over the last four years to identify the nominal value and uncertainty of the frequency by modeling and testing four simpler structures which share some of the characteristics of the operational inducer. This testing was used to assess and adjust modeling techniques and excellent correlation was achieved. Identification of the uncertainty in the inducer frequency itself was still problematic, however. This difficulty led to an investigation of Bayesian uncertainty quantification techniques, and to the application of the relatively simple technique of Multi-Variate Normal conditional distributions to calculate the inducer natural frequency uncertainty. Assumptions on prior distributions of uncertainty of the fluid-added mass and tip clearance effect are initially applied to models of each of the simple structures and the inducer itself, and these uncertainties are propagated to generate natural frequencies using design of experiments. Simple response surfaces are then created from this data in order to calculate a Covariance Matrix relating all of these natural frequencies. Finally, the results from modal test of the simple structures are considered to be observations and used to calculate the conditional variance of the desired inducer frequencies. As this method is less rigorous than more complicated Bayesian methods reported in the literature, a conservative factor is applied to the result, but the resulting uncertainty is still significantly less than originally estimated and will greatly assist certification of the inducer for use in the engine.

Uncertainty Quantification

Aeroelastic stability analyses of two counter rotating propfan designs for a cruise missile model

Aeroelastic stability analyses were performed to insure structural integrity of two counterrotating propfan blade designs for a NAVY/Air Force/NASA cruise missile model wind tunnel test. This analysis predicted if the propfan designs would be flutter free at the operating conditions of the wind tunnel test. Calculated stability results are presented for the two blade designs with rotational speed and Mach number as the parameters. A aeroelastic analysis code ASTROP2 (Aeroelastic Stability and Response of Propulsion Systems - 2 Dimensional Analysis), developed at LeRC, was used in this project. The aeroelastic analysis is a modal method and uses the combination of a finite element structural model and two dimensional steady and unsteady cascade aerodynamic models. This code was developed to analyze single rotation propfans but was modified and applied to counterrotating propfans for the present work. Modifications were made to transform the geometry and rotation of the aft rotor to the same reference frame as the forward rotor, to input a non-uniform inflow into the rotor being analyzed, and to automatically converge to the least stable aeroelastic mode.

Mahajan, Aparajit J.

The Role of Structural Dynamics and Testing in the Shuttle Flowliner Crack Investigation

During a normal inspection of the main propulsion system at Kennedy Space Center, small cracks were noticed near a slotted region of a gimbal joint flowliner located just upstream from one of the Space Shuttle Main Engines (SSME). These small cracks sparked an investigation of the entire Space Shuttle fleet main propulsion feedlines. The investigation was initiated to determine the cause of the small cracks and a repair method that would be needed to return the Shuttle fleet back to operation safely. The cracks were found to be initiated by structural resonance caused by flow fluctuations from the SSME low pressure fuel turbopump interacting with the flowliner. The pump induced backward traveling wakes that excited the liner and duct acoustics which also caused the liner to vibrate in complex mode shapes. The investigation involved an extensive effort by a team of engineers from the NASA civil servant and contractor workforce with the goal to characterize the root cause of the cracking behavior of the fuel side gimbal joint flowliners. In addition to working to identify the root cause, a parallel path was taken to characterize the material properties and fatigue capabilities of the liner material such that the life of the liners could be ascertained. As the characterization of the material and the most probable cause matured, the combination of the two with pump speed restrictions provided a means to return the Shuttle to flight in a safe manner. This paper traces the flowliner investigation results with respect to the structural dynamics analysis, component level testing and hot-fire flow testing on a static testbed. The paper will address the unique aspects of a very complex problem involving backflow from a high performance pump that has never been characterized nor understood to such detail. In addition, the paper will briefly address the flow phenomena that excited the liners, the unique structural dynamic modal characteristics and the variability of SSME operation which has ultimately ensured the safe and reliable operation of the shuttle main engines for each flight.

Frady, Gregory P.

Natural Frequency Testing and Model Correlation of Rocket Engine Structures in Liquid Hydrogen - Phase I, Cantilever Beam

Many structures in the launch vehicle industry operate in liquid hydrogen (LH2), from the hydrogen fuel tanks through the ducts and valves and into the pump sides of the turbopumps. Calculating the structural dynamic response of these structures is critical for successful qualification of this hardware, but accurate knowledge of the natural frequencies is based entirely on numerical or analytical predictions of frequency reduction due to the added-fluid-mass effect because testing in LH2 has always been considered too difficult and dangerous. This fluid effect is predicted to be approximately 4-5% using analytical formulations for simple cantilever beams. As part of a comprehensive test/analysis program to more accurately assess pump inducers operating in LH2, a series of frequency tests in LH2 were performed at NASA/Marshall Space Flight Center's unique cryogenic test facility. These frequency tests are coupled with modal tests in air and water to provide critical information not only on the mass effect of LH2, but also the cryogenic temperature effect on Young's Modulus for which the data is not extensive. The authors are unaware of any other reported natural frequency testing in this media. In addition to the inducer, a simple cantilever beam was also tested in the tank to provide a more easily modeled geometry as well as one that has an analytical solution for the mass effect. This data will prove critical for accurate structural dynamic analysis of these structures, which operate in a highly-dynamic environment.

Brown, Andrew M.

Space Shuttle Main Engine structural analysis and data reduction/evaluation. Volume 7: High pressure fuel turbo-pump third stage impeller analysis

This volume summarizes the analysis used to assess the structural life of the Space Shuttle Main Engine (SSME) High Pressure Fuel Turbo-Pump (HPFTP) Third Stage Impeller. This analysis was performed in three phases, all using the DIAL finite element code. The first phase was a static stress analysis to determine the mean (non-varying) stress and static margin of safety for the part. The loads involved were steady state pressure and centrifugal force due to spinning. The second phase of the analysis was a modal survey to determine the vibrational modes and natural frequencies of the impeller. The third phase was a dynamic response analysis to determine the alternating component of the stress due to time varying pressure impulses at the outlet (diffuser) side of the impeller. The results of the three phases of the analysis show that the Third Stage Impeller operates very near the upper limits of its capability at full power level (FPL) loading. The static loading alone creates stresses in some areas of the shroud which exceed the yield point of the material. Additional cyclic loading due to the dynamic force could lead to a significant reduction in the life of this part. The cyclic stresses determined in the dynamic response phase of this study are based on an assumption regarding the magnitude of the forcing function.

Pool, Kirby V.

Preliminary measurements of aerodynamic damping of a transonic compressor rotor

The aeroelastic behavior of a transonic compressor rotor operated in the MIT Blowdown Compressor Facility has been examined by means of piezoelectric motion sensors at the base of each of the 23 blades. Excitation has been observed due to rotating stall, due to an incipient flutter, and due to the facility startup transient. A method has been found for determining the aerodynamic damping force by modal analysis of the blade motion. Application of this technique to the example of excitation by rotating stall has led to the conclusions that the blade loading decreases in the stall cell, and that the damping force on the blades in the clean flow is in phase with blade velocity but opposite it in sign, leading to a logarithmic decrement of 0.2. This method of force derivation has quite general applicability as it requires only blade motion data such as are routinely acquired with strain gages. It is argued that models are needed for aerodynamic damping which focus on the effects of near neighbors of a given blade, since flutter often results in large response of isolated blades or small groups of blades.

Crawley, E. F.

The Urban Deployment Model: A Toolset for the Simulation and Performance Characterization of Radiation Detector Deployments in Urban Environments

Static and mobile radiation detectors can be deployed in urban environments for a range of nuclear security applications, including radiological source search-and-tracking scenarios. Modeling detector performance for such applications is challenging, as it does not depend solely on the detector capabilities themselves. Many factors must be taken into consideration, including specific source and background signatures, the topology and constraints of the deployment environment, the presence of nuisance sources, and whether detectors are mobile or static. When considering the simultaneous deployment of multiple, heterogeneous detectors, assessment of the system-wide performance requires the simulation of the individual detectors, and a system-level analysis of the detection performance. In radiological source search-and-tracking scenarios, performance is mostly dominated by the probability of encounter, which depends on the specifics of a given deployment, e.g., static vs. mobile detectors or a combination of both modalities, the number of detectors deployed, the dynamic vs. static setting of false alarm rates, and individual vs. networked operation. The Urban Deployment Model (UDM) toolset was specifically developed to cover the gap in the available generic frameworks for the simulation of radiation detector deployments at city scales. UDM provides a unified and modular framework to support the simulation and performance characterization of heterogeneous detector deployments in urban environments. This paper presents the key components along the UDM workflow.

98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL P

Large Space Shuttle Flight Experiment

It appears practical to challenge the Orbiter DAP with a large, attached structure. The definition of this capability is a fundamental step in the development of nearly all large space systems currently under consideration. Experiment features may be incorporated that apply to control systems for large space systems such as modal damping devices. In a relatively simple deployable structure, the correlation of flight test results with ground test and analysis should provide a basis for extrapolation to more complex structures. Initial experiment concepts will provide a starting point for the examination of antenna feed mast requirements with the objective changing the design to produce a representative test article. Correlation of construction operations with ground simulations will provide for better task and time-line definition. EVA needs to be a direct benefit to the conduct of the experiment. Early consideration of safety issues is a precaution against defining an unacceptable experiment concept. Integration of many objectives seems feasible and is generally perceived as the only way to justify a relatively expensive experiment.

Jenkins, L. M.

Uncertainty Modeling for Structural Control Analysis and Synthesis

The development of an accurate model of uncertainties for the control of structures that undergo a change in operational environment, based solely on modeling and experimentation in the original environment is studied. The application used throughout this work is the development of an on-orbit uncertainty model based on ground modeling and experimentation. A ground based uncertainty model consisting of mean errors and bounds on critical structural parameters is developed. The uncertainty model is created using multiple data sets to observe all relevant uncertainties in the system. The Discrete Extended Kalman Filter is used as an identification/parameter estimation method for each data set, in addition to providing a covariance matrix which aids in the development of the uncertainty model. Once ground based modal uncertainties have been developed, they are localized to specific degrees of freedom in the form of mass and stiffness uncertainties. Two techniques are presented: a matrix method which develops the mass and stiffness uncertainties in a mathematical manner; and a sensitivity method which assumes a form for the mass and stiffness uncertainties in macroelements and scaling factors. This form allows the derivation of mass and stiffness uncertainties in a more physical manner. The mass and stiffness uncertainties of the ground based system are then mapped onto the on-orbit system, and projected to create an analogous on-orbit uncertainty model in the form of mean errors and bounds on critical parameters. The Middeck Active Control Experiment is introduced as experimental verification for the localization and projection methods developed. In addition, closed loop results from on-orbit operations of the experiment verify the use of the uncertainty model for control analysis and synthesis in space.

Campbell, Mark E.

Characterization of the Modal Characteristics of Structures Operating in Dense Liquid Oxygen Turbopumps

A number of valuable conclusions can be drawn from this study. First, knockdown factors for a specific fluid are not constant but instead are dependent on the mode shape, although the largest this variability gets is about 10% for LOX, the densest fluid. The factors decrease the most for lower frequency shapes and less for higher ones. It follows, therefore, that mode number mismatch between air and fluid operation becomes not only possible, but common, as a knockdown factor for a particular mode shape may be higher than for another mode shape. Since this is a function of added mass, the mismatch is more prevalent for higher density fluids, but it initiates even for very low density ones. Another important conclusion reached is that it appears that the basic mode shapes of a structure do not change if it is fully symmetric, which includes its geometry and boundary conditions. There is some indication of small changes in the relative magnitudes within the mode shape. This conclusion is evident in the results from the cantilever rectangular plate and the inducer, which are not symmetric, and the fixed-fixed plate and the annular disk, which are. For non-symmetric structures, though, the mode shapes almost universally change for dense fluids, as shown by the very low MAC calculations. For the inducer in particular, the changes follow a trend of reduced parabolic and sine wavelengths with increasing density. It is critical to recognize the change in mode shape for several reasons. First, model updating with modal test becomes problematic if the shapes change. Second, design to avoid resonance is highly critical on the mode shape for modes other than the primary ones, as resonance is only a factor when the excitation shape matches the mode shape. Finally, application of the modal superposition method of forced response analysis is dependent on the use of accurate mode shapes. A more-refined assessment of the "knockdown" factor values and ranges than any previously reported in the literature for a realistic engineering structure is also presented in this paper. This data is of tremendous benefit for preliminary analysis and design, where a quick estimate is necessary. These results are important not just for rocket engine turbomachinery, but for water pumps and turbines, propellers, and any other structure operating in a heavy fluid with dynamic excitation. The clear avenue for future work for this endeavor is to expand the analytical techniques discussed in the literature to develop analytical expressions and justification for the mode shape changes and associated frequency knockdowns. These expressions must be able to accurately predict the functional relationship to the shapes, which will enable accurate tracing of the mode number from vacuum analysis (or testing in air) to analysis and operation in the intended fluid environment.

Chiu, Joseph

Aeroacoustic Analysis of Turbofan Noise Generation

This report provides an updated version of analytical documentation for the V072 Rotor Wake/Stator Interaction Code. It presents the theoretical derivation of the equations used in the code and, where necessary, it documents the enhancements and changes made to the original code since its first release. V072 is a package of FORTRAN computer programs which calculate the in-duct acoustic modes excited by a fan/stator stage operating in a subsonic mean flow. Sound is generated by the stator vanes interacting with the mean wakes of the rotor blades. In this updated version, only the tonal noise produced at the blade passing frequency and its harmonics, is described. The broadband noise component analysis, which was part of the original report, is not included here. The code provides outputs of modal pressure and power amplitudes generated by the rotor-wake/stator interaction. The rotor/stator stage is modeled as an ensemble of blades and vanes of zero camber and thickness enclosed within an infinite hard-walled annular duct. The amplitude of each propagating mode is computed and summed to obtain the harmonics of sound power flux within the duct for both upstream and downstream propagating modes.

Meyer, Harold D.

Neuroscience Investigations: An Overview of Studies Conducted

The neural processes that mediate human spatial orientation and adaptive changes occurring in response to the sensory rearrangement encountered during orbital flight are primarily studied through second and third order responses. In the Extended Duration Orbiter Medical Project (EDOMP) neuroscience investigations, the following were measured: (1) eye movements during acquisition of either static or moving visual targets, (2) postural and locomotor responses provoked by unexpected movement of the support surface, changes in the interaction of visual, proprioceptive, and vestibular information, changes in the major postural muscles via descending pathways, or changes in locomotor pathways, and (3) verbal reports of perceived self-orientation and self-motion which enhance and complement conclusions drawn from the analysis of oculomotor, postural, and locomotor responses. In spaceflight operations, spatial orientation can be defined as situational awareness, where crew member perception of attitude, position, or motion of the spacecraft or other objects in three-dimensional space, including orientation of one's own body, is congruent with actual physical events. Perception of spatial orientation is determined by integrating information from several sensory modalities. This involves higher levels of processing within the central nervous system that control eye movements, locomotion, and stable posture. Spaceflight operational problems occur when responses to the incorrectly perceived spatial orientation are compensatory in nature. Neuroscience investigations were conducted in conjunction with U. S. Space Shuttle flights to evaluate possible changes in the ability of an astronaut to land the Shuttle or effectively perform an emergency post-landing egress following microgravity adaptation during space flights of variable length. While the results of various sensory motor and spatial orientation tests could have an impact on future space flights, our knowledge of sensorimotor adaptation to spaceflight is limited, and the future application of effective countermeasures depends, in large part, on the results from appropriate neuroscience investigations. Therefore, the objective of the neuroscience investigations could have a negative effect on mission success. The Neuroscience Laboratory, Johnson Space Center (JSC), implemented three integrated Detailed Supplementary Objectives (DSO) designed to investigate spatial orientation and the associated compensatory responses as a part of the EDOMP. The four primary goals were (1) to establish a normative database of vestibular and associated sensory changes in response to spaceflight, (2) to determine the underlying etiology of neurovestibular and sensory motor changes associated with exposure to microgravity and the subsequent return to Earth, (3) to provide immediate feedback to spaceflight crews regarding potential countermeasures that could improve performance and safety during and after flight, and (4) to take under consideration appropriate designs for preflight, in-flight, and postflight countermeasures that could be implemented for future flights.

Reschke, Millard F.

Advanced Multimodal Solutions for Information Presentation

High-workload, fast-paced, and degraded sensory environments are the likeliest candidates to benefit from multimodal information presentation. For example, during EVA (Extra-Vehicular Activity) and telerobotic operations, the sensory restrictions associated with a space environment provide a major challenge to maintaining the situation awareness (SA) required for safe operations. Multimodal displays hold promise to enhance situation awareness and task performance by utilizing different sensory modalities and maximizing their effectiveness based on appropriate interaction between modalities. During EVA, the visual and auditory channels are likely to be the most utilized with tasks such as monitoring the visual environment, attending visual and auditory displays, and maintaining multichannel auditory communications. Previous studies have shown that compared to unimodal displays (spatial auditory or 2D visual), bimodal presentation of information can improve operator performance during simulated extravehicular activity on planetary surfaces for tasks as diverse as orientation, localization or docking, particularly when the visual environment is degraded or workload is increased. Tactile displays offer a third sensory channel that may both offload information processing effort and provide a means to capture attention when urgently required. For example, recent studies suggest that including tactile cues may result in increased orientation and alerting accuracy, improved task response time and decreased workload, as well as provide self-orientation cues in microgravity on the ISS (International Space Station). An important overall issue is that context-dependent factors like task complexity, sensory degradation, peripersonal vs. extrapersonal space operations, workload, experience level, and operator fatigue tend to vary greatly in complex real-world environments and it will be difficult to design a multimodal interface that performs well under all conditions. As a possible solution, adaptive systems have been proposed in which the information presented to the user changes as a function of taskcontext-dependent factors. However, this presupposes that adequate methods for detecting andor predicting such factors are developed. Further, research in adaptive systems for aviation suggests that they can sometimes serve to increase workload and reduce situational awareness. It will be critical to develop multimodal display guidelines that include consideration of smart systems that can select the best display method for a particular contextsituation.The scope of the current work is an analysis of potential multimodal display technologies for long duration missions and, in particular, will focus on their potential role in EVA activities. The review will address multimodal (combined visual, auditory andor tactile) displays investigated by NASA, industry, and DoD (Dept. of Defense). It also considers the need for adaptive information systems to accommodate a variety of operational contexts such as crew status (e.g., fatigue, workload level) and task environment (e.g., EVA, habitat, rover, spacecraft). Current approaches to guidelines and best practices for combining modalities for the most effective information displays are also reviewed. Potential issues in developing interface guidelines for the Exploration Information System (EIS) are briefly considered.

multimodal display

Operating Deflection Shapes for the Space Shuttle Partial Stack Rollout

In November of 2003 a rollout test was performed to gain a better understanding of the dynamic environment for the Space Shuttle during transportation from the Vehicle Assembly Building to the launch pad. This was part of a study evaluating the methodology for including the rollout dynamic loads in the Space Shuttle fatigue life predictions. The rollout test was conducted with a partial stack consisting of the Crawler Transporter, Mobile Launch Platform, and the Solid Rocket Boosters with an interconnecting crossbeam. Instrumentation included over 100 accelerometers. Data was recorded for steady state speeds, start-ups and stops, and ambient wind excitations with the vehicle at idle. This paper will describe the operating deflection shape analysis performed using the measured acceleration response data. The response data for the steady state speed runs were dominated by harmonics of the forcing frequencies, which were proportional to the vehicle speed. Assuming a broadband excitation for the wind, analyses of the data sets with the vehicle at idle were used to estimate the natural frequencies and corresponding mode shapes. Comparisons of the measured modal properties with numerical predictions are presented.

Buehrle, Ralph D.

Approach to large space structure control system design using traditional tools

This paper illustrates how traditional analysis and design tools can be successfully applied, with insights derived from recent research on control-structure interaction, to preliminary control system design for a large space structure such as the Space Station Freedom. The control system regulates attitude during construction, normal on-orbit operation, reboost, and docking using control-moment gyros and thrusters. The tools include finite-element modeling, model reduction based on modal truncation, single-input/single-output Bode plot design, position and rate feedback, roll-off and phase filter, and simulation. The resulting system is characterized by simple architecture and moderate bandwidth. The large configuration changes during the construction period are handled by gain scheduling.

Chu, P. Y.

Design Spectrum Analysis in NASTRAN

The utility of Design Spectrum Analysis is to give a mode by mode characterization of the behavior of a design under a given loading. The theory of design spectrum is discussed after operations are explained. User instructions are taken up here in three parts: Transient Preface, Maximum Envelope Spectrum, and RMS Average Spectrum followed by a Summary Table. A single DMAP ALTER packet will provide for all parts of the design spectrum operations. The starting point for getting a modal break-down of the response to acceleration loading is the Modal Transient rigid format. After eigenvalue extraction, modal vectors need to be isolated in the full set of physical coordinates (P-sized as opposed to the D-sized vectors in RF 12). After integration for transient response the results are scanned over the solution time interval for the peak values and for the times that they occur. A module called SCAN was written to do this job, that organizes these maxima into a diagonal output matrix. The maximum amplifier in each mode is applied to the eigenvector of each mode which then reveals the maximum displacements, stresses, forces and boundary reactions that the structure will experience for a load history, mode by mode. The standard NASTRAN output processors have been modified for this task. It is required that modes be normalized to mass.

Butler, T. G.

Identification of Rotorcraft Structural Dynamics from Flight and Wind Tunnel Data

Excessive vibration remains one one of the most difficult problems that faces the helicopter industry today, affecting all production helicopters at some phase of their development. Vibrations in rotating structures may arise from external periodic dynamic airloads whose frequencies are are close to the natural frequencies of the rotating system itself. The goal for the structures engineer would thus be to design a structure as free from resonance effects as possible. In the case of a helicopter rotor blade these dynamic loads are a consequence of asymmetric airload distribution on the rotor blade in forward flight, leading to a rich collection of higher harmonic airloads that force rotor and airframe response. Accurate prediction of the dynamic characteristics of a helicopter rotor blade will provide the opportunity to affect in a positive manner noise intensity, vibration level, durability, reliability and operating costs by reducing objectionable frequencies or moving them to a different frequency range and thus providing us with a lower vibration rotor. In fact, the dynamic characteristics tend to define the operating limits of a rotorcraft. As computing power has increased greatly over the last decade, researchers and engineers have turned to analyzing the vibrational characteristics of aerospace structures at the design and development stage of the production of an aircraft. Modern rotor blade construction methods lead to products with low mass and low inherent damping so careful design and analysis is required to avoid resonance and an undesirable dynamic performance. In addition, accurate modal analysis is necessary for several current approaches in elastic system identification and active control.

McKillip, Robert M., Jr.