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Recent advances in radar remote sensing of forest

On a global scale, forests represent most of the terrestrial standing biomass (80 to 90 percent). Thus, natural and anthropogenic change in forest covers can have major impacts not only on local ecosystems but also on global hydrologic, climatic, and biogeochemical cycles that involve exchange of energy, water, carbon, and other elements between the earth and atmosphere. Quantitative information on the state and dynamics of forest ecosystems and their interactions with the global cycles appear necessary to understand how the earth works as a natural system. The information required includes the lateral and vertical distribution of forest cover, the estimates of standing biomass (woody and foliar volume), the phenological and environmental variations and disturbances (clearcutting, fires, flood), and the longer term variations following deforestation (regeneration, successional stages). To this end, seasonal, annual, and decadal information is necessary in order to separate the long term effects in the global ecosystem from short term seasonal and interannual variations. Optical remote sensing has been used until now to study the forest cover at local, regional, and global scales. Radar remote sensing, which provides recent SAR data from space on a regular basis, represents an unique means of consistently monitoring different time scales, at all latitudes and in any atmospheric conditions. Also, SAR data have shown the potential to detect several forest parameters that cannot be inferred from optical data. The differences--and complementarity--lie in the penetration capabilities of SAR data and their sensitivity to dielectric and geometric properties of the canopy volume, whereas optical data are sensitive to the chemical composition of the external foliar layer of the vegetation canopy.

Letoan, Thuy↗

Design and Technology Maturation of the Stratospheric Projectile Experiment of Entry Dynamics

The supersonic and transonic dynamic stability of blunt-body reentry vehicles currently poses large risks in all of NASA’s ongoing entry missions (MSR SRL, MSR EES, and Dragonfly). These projects have allocated millions of dollars to testing and modeling efforts to buy down risk by using the current state-of-the-art (SoA) facilities at NASA’s disposal. While these facilities have heritage in supplying dynamics data to reentry missions, their availability is severely limited – particularly with the high number of concur-rent projects requesting simultaneous testing– and are costly when considering the science density per dollar. None of the current SoA facility methodologies allow the test model to have the dynamics fully develop through a flight relevant free-stream profile and as such require extrapolations with resultant high uncertainties in order to relate the test dynamics to flight expectations. SPEED is a NASA Ames Center Innovation Fund (CIF) project that is developing a highly tailorable and cost-effective test methodology to better assess the dynamic stability of blunt-body reentry vehicles via a stratospheric balloon flight. This is accomplished by dropping a suite of instrumented capsules from a stratospheric balloon to gain a statistically relevant dataset of scaled reentry vehicles in mission relevant free-flight conditions. This presentation will walk through how the test methodology is being implemented specifically for the Mars Sample Return (MSR) Earth Entry System (EES) geometry in an awarded Flight Opportunities Program (FOP) test flight in early CY24. SPEED Application to MSR: SPEED consists of three main mechanical systems: the Drop Platform, the Projectile, and the test Capsule. SPEED is being developed as a set of guidelines and recommendations for how to test with the proposed Concept of Operations (Conops) since the specific design parameters will vary depending on the specific project’s reference trajectory and entry vehicle design. As such, this presentation will walk through the development time-line as shown in Fig. 2. This is meant to serve as a blueprint for further missions as desired. Mechanical and Avionics Design. The SPEED test platform designed for the MSR-EES capsule geometry with nominal entry parameters has the ability to carry 10 Capsules to altitude instrumented with: 1. 3-Axis Accelerometer 2. IMU 3. Gyroscope 4. Magnetometer 5. Pressure Transducer cruciform 6. Uplook and Horizon Cameras To package the avionics/instrumentation suite, the capsule is approximately 1’ in diameter with the Outer Mold Line (OML) centroid-scaled from the full EES design. The internal volume is gutted and custom-shaped to fit the desired instrumentation suite, as well as to allow for the positioning of ballast mass such that the Center of Gravity is analogous to the flight vehicle. All structural components in the Capsule and Projectile are 3D printed, which significantly reduces the cost of each flight unit to around $1500 including all instrumentation, avionics, and structural components. Flight Conops. The test Capsule is accelerated to the desired altitude and Mach number while stowed in the Projectile, a missile-like vehicle consisting of steel ballast in the nose, a low-drag OML, and an Ejection Mechanism to reliably release the Capsule into the free-flow supersonic conditions. For the MSR-EES design, the capsule employs ~3kg of ballast mass at the nose to accelerate the 1.25kg test Capsule to ~Mach 1.7 at 23km altitude. This requires an initial release altitude of 40km, the quoted limit of a 80kg payload by the FOP-contracted balloon provider. Once the Ejection Mechanism avionics detect the proper conditions, the spring-loaded Ejection Mechanism will release and – guided by the sabot – expose the test Capsule to the desired test conditions for ~5 seconds of free-flight in the supersonic/transonic regimes. Dynamics in the subsonic regime will also be captured with the instrumentation suite with post-flight recovery operations aimed at recovering the high-G-load capable SD cards after the planned hard impact landings. Testing and Development: In the few months the SPEED project has worked the development of MSR-EES flight test, the team has performed lab and drone based testing which this presentation will overview. After the first design phase, the team fabricated Engineering Demonstration Units (EDUs) of all subsystems to perform validation testing shown in Fig. 5. After validation was completed on the subsystem level, a drone-drop test was performed at the recreational flight ceiling of 400ft altitude to assess the SPEED systems in a flight environment. Parameters such as in-flight stability, hard impact landing performance, and avionics performance were quantified and qualified. The FY23 CIF will culminate in a helicopter drop test aboard an Air National Guard Blackhawk. This will prepare the team for the CY24 FOP stratospheric balloon flight that should provide the final verification to begin offering the test platform for mission support. Focus of Presentation: This presentation will outline the technology maturation path of the SPEED implementation to the MSR-EES capsule baseline as well as the details regarding the mechanical system, avionics and instrumentation, and flight operations. Note that a complementary presentation is being submitted for a methodology overview of the SPEED test platform, introducing the testing technique and benefits as well as the full application space of the technology.

pitch damping coefficient↗

Results of tests in the NASA/LaRC 31 inch CFHT on an 0.010-scale model (32-OT) of the space shuttle configuration 3 to obtain hypersonic aerodynamic characteristics for second stage operation during nominal boost and the abort RTLS mode (IA58)

Tests were conducted to obtain hypersonic aerodynamic forces and moments on an 0.010-scale model of the space shuttle vehicle configuration 3. Hypersonic stability data were obtained from tests at Mach 10.3 and dynamic pressure of 150 psf for the integrated orbiter and external tank, orbiter alone, and external tank alone. The effects of solid plume simulation from the main propulsion system as well as elevon, aileron, and rudder deflections were also investigated.

Thornton, D. E.↗

Galaxy collisions - A preliminary study

Collisions of spherical galaxies were studied in a series of numerical experiments to see what happens when galaxies collide. Each experiment starts with two model galaxies, each consisting of 50,000 stars, moving toward each other along a specified orbit. The series of experiments provides a systematic sampling of the parameter space spanned by the initial orbital energy and the initial angular momentum. Deeply penetrating collisions are emphasized. The collisions reported here scale to relative velocities as great as 500 km/s, well into the range for collisions within clusters of galaxies. It is found that: (1) the galaxies contract momentarily to about half their original sizes shortly after close passage; and (2) the initial galaxies blend into a single dynamical system while they are near each other.

Miller, R. H.↗

Overview of NASA Lewis Research Center free-piston Stirling engine activities

A generic free-piston Stirling technology project is being conducted to develop technologies generic to both space power and terrestrial heat pump applications in a cooperative, cost-shared effort. The generic technology effort includes extensive parametric testing of a 1 kW free-piston Stirling engine (RE-1000), development of a free-piston Stirling performance computer code, design and fabrication under contract of a hydraulic output modification for RE-1000 engine tests, and a 1000-hour endurance test, under contract, of a 3 kWe free-piston Stirling/alternator engine. A newly initiated space power technology feasibility demonstration effort addresses the capability of scaling a free-piston Stirling/alternator system to about 25 kWe; developing thermodynamic cycle efficiency or equal to 70 percent of Carnot at temperature ratios in the order of 1.5 to 2.0; achieving a power conversion unit specific weight of 6 kg/kWe; operating with noncontacting gas bearings; and dynamically balancing the system. Planned engine and component design and test efforts are described.

Slaby, J. G.↗

Overview of NASA Lewis Research Center free-piston Stirling engine activities

A generic free-piston Stirling technology project is being conducted to develop technologies generic to both space power and terrestrial heat pump applications in a cooperative, cost-shared effort. The generic technology effort includes extensive parametric testing of a 1 kW free-piston Stirling engine (RE-1000), development of a free-piston Stirling performance computer code, design and fabrication under contract of a hydraulic output modification for RE-1000 engine tests, and a 1000-hour endurance test, under contract, of a 3 kWe free-piston Stirling/alternator engine. A newly initiated space power technology feasibility demonstration effort addresses the capability of scaling a free-piston Stirling/alternator system to about 25 kWe; developing thermodynamic cycle efficiency or equal to 70 percent of Carnot at temperature ratios in the order of 1.5 to 2.0; achieving a power conversion unit specific weight of 6 kg/kWe; operating with noncontacting gas bearings; and dynamically balancing the system. Planned engine and component design and test efforts are described.

Slaby, J. G.↗

The Distribution of Carbon Monoxide in the GOCART Model

Carbon monoxide (CO) is an important trace gas because it is a significant source of tropospheric Ozone (O3) as well as a major sink for atmospheric hydroxyl radical (OH). The distribution of CO is set by a balance between the emissions, transport, and chemical processes in the atmosphere. The Georgia Tech/Goddard Global Ozone Chemistry Aerosol Radiation and Transport (GOCART) model is used to simulate the atmospheric distribution of CO. The GOCART model is driven by the assimilated meteorological data from the Goddard Earth Observing System Data Assimilation System (GEOS DAS) in an off-line mode. We study the distribution of CO on three time scales: (1) day to day fluctuation produced by the synoptic waves; (2) seasonal changes due to the annual cycle of CO sources and sinks; and (3) interannual variability induced by dynamics. Comparison of model results with ground based and remote sensing measurements will also be presented.

Fan, Xiaobiao↗

Structural Analysis and Test Comparison of a 20-Meter Inflation-Deployed Solar Sail

Under the direction of the NASA In-Space Propulsion Technology Office, the team of L Garde, NASA Jet Propulsion Laboratory, Ball Aerospace, and NASA Langley Research Center has been developing a scalable solar sail configuration to address NASA s future space propulsion needs. Prior to a flight experiment of a full-scale solar sail, a comprehensive test program was implemented to advance the technology readiness level of the solar sail design. These tests consisted of solar sail component, subsystem, and sub-scale system ground tests that simulated the aspects of the space environment such as vacuum and thermal conditions. In July 2005, a 20-m four-quadrant solar sail system test article was tested in the NASA Glenn Research Center s Space Power Facility to measure its static and dynamic structural responses. Key to the maturation of solar sail technology is the development of validated finite element analysis (FEA) models that can be used for design and analysis of solar sails. A major objective of the program was to utilize the test data to validate the FEA models simulating the solar sail ground tests. The FEA software, ABAQUS, was used to perform the structural analyses to simulate the ground tests performed on the 20-m solar sail test article. This paper presents the details of the FEA modeling, the structural analyses simulating the ground tests, and a comparison of the pretest and post-test analysis predictions with the ground test results for the 20-m solar sail system test article. The structural responses that are compared in the paper include load-deflection curves and natural frequencies for the beam structural assembly and static shape, natural frequencies, and mode shapes for the solar sail membrane. The analysis predictions were in reasonable agreement with the test data. Factors that precluded better correlation of the analyses and the tests were unmeasured initial conditions in the test set-up.

Sleight, David W.↗

Structural Test and Analysis of a Hybrid Inflatable Antenna

NASA is developing ultra-lightweight structures technology for communication antennas for space missions. One of the research goals is to evaluate the structural characteristics of inflatable and rigidizable antennas through test and analysis. Being able to test and analyze the structural characteristics of a full scale antenna is important to enable the simulation of various mission scenarios to determine system performance in space. Recent work completed to evaluate a Hybrid Inflatable Antenna concept will be discussed. Tests were completed on a 2-m prototype to optimize its static shape and identify its modal dynamics that are important for analytical model validation. These test results were used to evaluate a preliminary finite element model of the antenna, and this model development and correlation activity is also described in the paper.

Gaspar, James L.↗

Shallow and Deep Latent Heating Modes Over Tropical Oceans Observed with TRMM PR Spectral Latent Heating Data

The global hydrological cycle is central to the Earth's climate system, with rainfall and the physics of its formation acting as the key links in the cycle. Two-thirds of global rainfall occurs in the Tropics. Associated with this rainfall is a vast amount of heat, which is known as latent heat. It arises mainly due to the phase change of water vapor condensing into liquid droplets; three-fourths of the total heat energy available to the Earth's atmosphere comes from tropical rainfall. In addition, fresh water provided by tropical rainfall and its variability exerts a large impact upon the structure and motions of the upper ocean layer. Three-dimensional distributions of latent heating estimated from Tropical Rainfall Measuring Mission Precipitation Radar (TRMM PR)utilizing the Spectral Latent Heating (SLH) algorithm are analyzed. Mass-weighted and vertically integrated latent heating averaged over the tropical oceans is estimated as approx.72.6 J/s (approx.2.51 mm/day), and that over tropical land is approx.73.7 J/s (approx.2.55 mm/day), for 30degN-30degS. It is shown that non-drizzle precipitation over tropical and subtropical oceans consists of two dominant modes of rainfall systems, deep systems and congestus. A rough estimate of shallow mode contribution against the total heating is about 46.7 % for the average tropical oceans, which is substantially larger than 23.7 % over tropical land. While cumulus congestus heating linearly correlates with the SST, deep mode is dynamically bounded by large-scale subsidence. It is notable that substantial amount of rain, as large as 2.38 mm day-1 in average, is brought from congestus clouds under the large-scale subsiding circulation. It is also notable that even in the region with SST warmer than 28 oC, large-scale subsidence effectively suppresses the deep convection, remaining the heating by congestus clouds. Our results support that the entrainment of mid-to-lower-tropospheric dry air, which accompanies the large-scale subsidence is the major factor suppressing the deep convection. Therefore, representation of the realistic entrainment is very important for proper reproduction of precipitation distribution and resultant large-scale circulation.

Takayabu, Yukari N.↗

Satellite Ocean Colour: Current Status and Future Perspective

Spectrally resolved water-leaving radiances (ocean colour) and inferred chlorophyll concentration are key to studying phytoplankton dynamics at seasonal and inter-annual scales, for a better understanding of the role of phytoplankton in marine biogeochemistry; the global carbon cycle; and the response of marine ecosystems to climate variability, change and feedback processes. Ocean colour data also have a critical role in operational observation systems monitoring coastal eutrophication, harmful algal blooms, and sediment plumes. The contiguous ocean-colour record reached 21 years in 2018; however, it is comprised of a number of one-off missions such that creating a consistent time-series of ocean-colour data requires merging of the individual sensors (including MERIS, Aqua-MODIS, SeaWiFS, VIIRS, and OLCI) with differing sensor characteristics, without introducing artefacts. By contrast, the next decade will see consistent observations from operational ocean colour series with sensors of similar design and with a replacement strategy. Also, by 2029 the record will start to be of sufficient duration to discriminate climate change impacts from natural variability, at least in some regions. This paper describes the current status and future prospects in the field of ocean colour focusing on large to medium resolution observations of oceans and coastal seas. It reviews the user requirements in terms of products and uncertainty characteristics and then describes features of current and future satellite ocean-colour sensors, both operational and innovative. The key role of in situ validation and calibration is highlighted as are ground segments that process the data received from the ocean-colour sensors and deliver analysis-ready products to end-users. Example applications of the ocean-colour data are presented, focusing on the climate data record and operational applications including water quality and assimilation into numerical models. Current capacity building and training activities pertinent to ocean colour are described and finally a summary of future perspectives is provided.

ocean colour↗

Adaptive Control for Linear Uncertain Systems with Unmodeled Dynamics Revisited via Optimal Control Modification

This paper presents the optimal control modification for linear uncertain plants. The Lyapunov analysis shows that the modification parameter has a limiting value depending on the nature of the uncertainty. The optimal control modification exhibits a linear asymptotic property that enables it to be analyzed in a linear time invariant framework for linear uncertain plants. The linear asymptotic property shows that the closed-loop plants in the limit possess a scaled input-output mapping. Using this property, we can derive an analytical closed-loop transfer function in the limit as the adaptive gain tends to infinity. The paper revisits the Rohrs counterexample problem that illustrates the nature of non-robustness of model-reference adaptive control in the presence of unmodeled dynamics. An analytical approach is developed to compute exactly the modification parameter for the optimal control modification that stabilizes the plant in the Rohrs counterexample. The linear asymptotic property is also used to address output feedback adaptive control for non-minimum phase plants with a relative degree 1.

Unmodeled Dynamics↗

Imaging Science Requirements for a Uranus Flagship Mission

Introduction: Our presentation will summarize the requirements for UV/Visible/Near-IR remote-sensing imaging science instruments (including mapping spectrometers) envisioned for the Uranus Orbiter and Probe (UOP) mission recommended by the recent planetary decadal survey. The Uranus system offers many targets for imaging investigations. Imaging science investigations are crucial to answering fundamental questions about various physical processes that shape the Uranian system and drive its evolution. Each science objective addressed through imaging has requirements for observation and illumination geometries as well as coverage and resolution in the spatial, spectral and temporal dimensions. By combining the requirements for the diverse science targets in the Uranian system, we present a set of overarching imaging science requirements for UOP. Atmospheric Science: The study of Uranus’ atmosphere is crucial in understanding the planet's energy balance and interior heat flux. Imaging observations map how the atmosphere scatters incident solar radiation. Imaging is key to characterizing planetary-scale energy/material transport though global atmospheric dynamics and local meteorology. Hyperspectral mapping combined with radiative transfer analysis reveals the vertical distribution of clouds and hazes. Chronicling temporal variabilities of atmospheric features is key to understanding the various processes that operate within the dynamic Uranian atmosphere. A comprehensive understanding of Uranus' atmosphere is essential in deciphering the planet's origins and evolution as well as providing insight into the broader field of planetary and exoplanetary science. Magnetospheric Interactions: Imaging science is critical to understand the elusive Uranian aurorae. Mapping the aurorae and their temporal dynamics will reveal the interaction of Uranus’ complex magnetosphere with the solar wind. The apparent decades-long cooling of Uranus’ upper atmosphere is thought to be due to changes in energy input from the overlying magnetosphere, and so mapping and quantifying the auroral evidence for this energy flux is particularly relevant for upper atmospheric science In addition, Uranus’ large moons may generate auroral footprints, which can be used to constrain the planet’s unusual magnetic field structure. Satellite Interiors: Imaging science can address the potential habitability of the larger moons of Uranus by measuring their shapes and rotational states (including librations), which combined with gravity data are crucial to understanding their internal density structure. Satellite Origins: Origin markers such as organics and volatiles may be preserved on the satellite surfaces, which may be sensed by UV/IR mapping. Imaging combined with the moons’ densities may reveal whether the moons formed in Uranus’ circumplanetary disk, rings, or from ejecta of a giant impact on Uranus. Satellite Geology: Imaging investigations, including stereo imaging to obtain topography, will enable constraining each satellite’s geologic history by investigating the distributions and morphologies of impact craters, tectonic and potential cryovolcanic features, and regolith and mass wasting deposits. These records will help unravel the past complex history of orbital resonances that occurred between Uranian satellites, which may have driven geologic activity. Analyses of these features will also point to resurfacing styles and modification processes important for understanding each moon’s thermal evolution, and any present-day geological activity, including potential subsurface deep oceans or possible current activity. Additionally, hyperspectral imaging will enable mapping the surface composition and identifying the potential presence of material that may point to a subsurface ocean, such as ammonia-bearing species. Rings and Small Moons: Imaging observations elucidate the physical processes that contribute to the complex ring-moon system’s inferred unstable state. Imaging surveys will seek smaller moons and other debris that could provide evidence of past collisions, while precise astrometric data of moons will determine the current dynamical state of the system. Many of Uranus' rings are exceptionally narrow, and images of the rings covering a broad range of longitudes, and high-resolution images of features within the rings will reveal how these rings are confined and maintained, as well as illustrating the rings' internal structure and dynamics. Images of the rings and moons obtained at a variety of wavelengths and lighting conditions are also needed to quantify the composition and the size distribution of the small satellites and ring particles.

K M Sayanagi↗

Early Spring Post-Fire Snow Albedo Dynamics in High Latitude Boreal Forests Using Landsat-8 OLI Data

Taking advantage of the improved radiometric resolution of Landsat-8 OLI which, unlike previous Landsat sensors, does not saturate over snow, the progress of fire recovery progress at the landscape scale (less than 100 m) is examined. High quality Landsat-8 albedo retrievals can now capture the true reflective and layered character of snow cover over a full range of land surface conditions and vegetation densities. This new capability particularly improves the assessment of post-fire vegetation dynamics across low- to high-burn severity gradients in Arctic and boreal regions in the early spring, when the albedos during recovery show the greatest variation. We use 30 m resolution Landsat-8 surface reflectances with concurrent coarser resolution (500 m) MODIS high quality full inversion surface Bidirectional Reflectance Distribution Functions (BRDF) products to produce higher resolution values of surface albedo. The high resolution full expression shortwave blue sky albedo product performs well with an overall RMSE of 0.0267 between tower and satellite measures under both snow-free and snow-covered conditions. While the importance of post-fire albedo recovery can be discerned from the MODIS albedo product at regional and global scales, our study addresses the particular importance of early spring post-fire albedo recovery at the landscape scale by considering the significant spatial heterogeneity of burn severity, and the impact of snow on the early spring albedo of various vegetation recovery types. We found that variations in early spring albedo within a single MODIS gridded pixel can be larger than 0.6. Since the frequency and severity of wildfires in Arctic and boreal systems is expected to increase in the coming decades, the dynamics of albedo in response to these rapid surface changes will increasingly impact the energy balance and contribute to other climate processes and physical feedback mechanisms. Surface radiation products derived from Landsat-8 data will thus play an important role in characterizing the carbon cycle and ecosystem processes of high latitude systems.

Landsat-8 snow albedo↗

Evaluation of an aeroelastic model technique for predicting airplane buffet loads

A wind-tunnel technique which makes use of a dynamically scaled aeroelastic model to predict full-scale airplane buffet loads during buffet boundary penetration is evaluated. A 1/8-scale flutter model of a fighter airplane with remotely controllable variable-sweep wings and trimming surfaces was used for the evaluation. The model was flown on a cable-mount system which permitted high lift forces comparable to those in maneuvering flight. Bending moments and accelerations due to buffet were measured on the flutter model and compared with those measured on the full-scale airplane in an independent flight buffet research study. It is concluded that the technique can provide valuable information on airplane buffet load characteristics not available from any other source except flight test.

Hanson, P. W.↗

Linear regulator design for stochastic systems by a multiple time scales method

This paper develops a hierarchically-structured, suboptimal controller for a linear stochastic system composed of fast and slow subsystems. It is proved that the controller is optimal in the limit as the separation of time scales of the subsystems becomes infinite. The methodology is illustrated by design of a controller to suppress the phugoid and short period modes of the longitudinal dynamics of the F-8 aircraft.

Teneketzis, D.↗

Finite-difference fluid dynamics computer mathematical models for the design and interpretation of experiments for space flight

Numerical methods are used to design a spherical baroclinic flow model experiment of the large scale atmosphere flow for Spacelab. The dielectric simulation of radial gravity is only dominant in a low gravity environment. Computer codes are developed to study the processes at work in crystal growing systems which are also candidates for space flight. Crystalline materials rarely achieve their potential properties because of imperfections and component concentration variations. Thermosolutal convection in the liquid melt can be the cause of these imperfections. Such convection is suppressed in a low gravity environment. Two and three dimensional finite difference codes are being used for this work. Nonuniform meshes and implicit iterative methods are used. The iterative method for steady solutions is based on time stepping but has the options of different time steps for velocity and temperature and of a time step varying smoothly with position according to specified powers of the mesh spacings. This allows for more rapid convergence. The code being developed for the crystal growth studies allows for growth of the crystal as the solid-liquid interface. The moving interface is followed using finite differences; shape variations are permitted. For convenience in applying finite differences in the solid and liquid, a time dependent coordinate transformation is used to make this interface a coordinate surface.

Roberts, G. O.↗

How gravity and joint scaling affect dynamics response

If a scale model of a space structure is ground tested to establish dynamic characteristics of full-scale space structure, two factors need to be taken into consideration: the joint tolerances may not be scaled, and joint preloads (from gravity) may be present. In some situations, these differences may significantly affect dynamic response so that the model is not representative of the structure in space. In this study, joint effects are investigated by considering the response of a single-degree-of-freedom system that is restrained by a joint linkage. The joint is characterized by its initial stiffness, the load at which it slips, its hardening rate, and the distance it can slip (joint 'slop' or gap). The steady-state response of the system is calculated using a harmonic balance approach. It is found that the system response is multivalued for certain ranges of gap dimensions. When this is the case, small changes in system properties can lead to large differences in dynamic response and make it difficult to obtain meaningful results directly from model testing.

Hsu, S. T.↗