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NASA NTRS2019

Properties of Rubble-Pile Asteroid (101955) Bennu from OSIRIS-REx Imaging and Thermal Analysis

Establishing the abundance and physical properties of regolith and boulders on asteroids is crucial for understanding the formation and degradation mechanisms at work on their surfaces. Using images and thermal data from NASA's Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer (OSIRIS-REx) spacecraft, we show that asteroid (101955) Bennu's surface is globally rough, dense with boulders, and low in albedo. The number of boulders is surprising given Bennu's moderate thermal inertia, suggesting that simple models linking thermal inertia to particle size do not adequately capture the complexity relating these properties. At the same time, we find evidence for a wide range of particle sizes with distinct albedo characteristics. Our findings imply that ages of Bennu's surface particles span from the disruption of the asteroid's parent body (boulders) to recent in situ production (micrometre-scale particles).

DellaGiustina, D. N.↗
NASA NTRSDate not supplied

Corrugated Rollable Tubular Booms

A promising candidate for a new class of deployable composite boom is the COrrugated ROllable TUbular Boom (COROTUB), which is to be employed on future large space structures by the National Aeronautics and Space Administration (NASA). This is due to its two corrugated thin-shells forming a closed-section,which yields high bending and torsion stiffness, allowing for high dimensional stability. The corrugation geometry that dictates the boom cross-section shape is completely defined by closed-form analytical equations given a set of key input parameters available to the designer. Parametric studies changing two geometric variables are used to evaluate which parameters most influence the cross-section’s area moment of inertia and torsional constant. The number of corrugations, flattened height of the boom, and concave and convex corrugation radii are varied to study the complete boom design space. Deeper corrugations increase the local buckling loads (crimpling) of the thin-shell boom but reduce the area moment of inertia and therefore the global Euler buckling load of the compressive member,and thus a fine balance is sought. Finite element analyses of pure bending and compression load cases for 1.22 m boom specimens were carried out to study this design trade-off in detail. Based on previous work, several thin-ply composite laminates have been selected for the different shells that form the cross-section. Feasible boom designs with a maximized stiffness and load-carrying capacity will be manufactured to study their rollability and for experimental characterization. Structural characterization tests and additional numerical analyses will provide guidelines towards future corrugated boom designs and adequate boundary conditions to maximize the boom bending strength.

Deployable Booms↗
NASA NTRSDate not supplied

Packed Bed Reactor Experiment (PBRE)-2: Pressure Drop Measurements in Microgravity

Single and two-phase flow through porous media are encountered on ground and in space in numerous applications related to life support systems, fuel cells, chemical/materials processing and transporting nutrients to plants. These systems operate differently in the microgravity environment encountered in space travel because the density differences no longer cause the phases to separate or “drain” under the body force of gravity. In the absence of gravity, the interfacial or capillary forces play a more significant role in determining the operational variables such as phase distribution, liquid holdup, and pressure drop especially when the liquid inertia and viscous forces are at minimum. Lower liquid flow rates in air-water two-phase flows represent the case for most of the space processes that depend on two-phase flows in porous media. The Packed Bed Reactor Experiment (PBRE) was developed and flown on the International Space Station (ISS) by NASA to extend a series of fundamental studies of gas-liquid flows through porous media. The goal of the this series of flight experiments (PBRE-1 and PBRE-2) is to better understand the hydrodynamics of two phase flows in porous media in flow regimes dominated by inertia and viscous effects and understanding the role of the interfacial forces and their role in determining the flow dynamics. The flight experiments were preceded by a series of low gravity experiments performed on the low gravity aircraft. These experiments led to the development of a semi-empirical two-phase pressure drop and flow pattern transition in support of a number of reactor beds planned for water reclamation processes onboard of the international space station. The PBRE flight experiment was designed to deliver a wide range of tightly controlled gas (nitrogen) and liquid (water) flows to one of two interchangeable test sections differing only in the type of internal packing material. In this paper, we describe the PBRE-2 experiment and present preliminary low gravity data on pressure drop and compare with semi-empirical model pressure drop predictions. This work is supported by the National Aeronautics and Space Administration (NASA) under grant no. 80NSSC20K0830

Suratman Number↗
NASA NTRSDate not supplied

Characterization of Collection Efficiency of the Common Research Model Midspan Wing Section in the IRT

This paper presents a preliminary study for the characterization of collection efficiency from icing tests conducted in the Icing Research Tunnel at NASA Glenn Research Center. A test method previously developed for measuring the attachment line maximum collection efficiency of a swept NACA 0012 airfoil model at zero angle of attack was applied to the leading-edge region of a 65%-scale version of the Common Research Model midspan wing section. A correlation for the stagnation line maximum collection efficiency as a function of the modified inertia parameter was obtained with LEWICE3D simulations utilizing a discrete number of drop diameters. It was then compared with the collection efficiency measurement data obtained in the IRT. For the experimental collection efficiency, two ice shape digitization procedures were utilized to extract 2-D chord-wise ice shape profiles, i.e., the Maximum Combined Cross Section or MCCS, and the Minimum Combined Cross Section or Min CCS, at selected span-wise locations from the 3-D scanned ice shapes. The preliminary result showed that the rime ice thickness method can be used to characterize the collection efficiency distribution, for conditions free of ice erosion, in the main ice shape region. However, a high-order statistical processing of the ice scan is needed for estimating the mean ice thickness in areas where feathers are prevalent. From the limited comparison of the experimental and LEWICE3D collection efficiency data in the CRM65 MS model leading edge area, it was shown that the attachment line maximum collection efficiency is reasonably estimated by the best curve-fit correlation in the range of modified inertia parameter tested. A tighter alignment control of the iced and cold clean model scans is needed to improve the comparison. Further evaluation of this correlation is recommended to assess its applicability for Common Research Model type swept wing icing scaling analysis.

Ice Scailing↗
NASA NTRSDate not supplied

Characterization of Collection Efficiency of the Common Research Model Midspan Wing Section in the IRT

This paper presents a preliminary study for the characterization of collection efficiency from icing tests conducted in the Icing Research Tunnel at NASA Glenn Research Center. A test method previously developed for measuring the attachment line maximum collection efficiency of a swept NACA 0012 airfoil model at zero angle of attack was applied to the leading-edge region of a 65%-scale version of the Common Research Model midspan wing section. A correlation for the stagnation line maximum collection efficiency as a function of the modified inertia parameter was obtained with LEWICE3D simulations utilizing a discrete number of drop diameters. It was then compared with the collection efficiency measurement data obtained in the IRT. For the experimental collection efficiency, two ice shape digitization procedures were utilized to extract 2-D chord-wise ice shape profiles, i.e., the Maximum Combined Cross Section or MCCS, and the Minimum Combined Cross Section or MinCCS, at selected span-wise locations from the 3-D scanned ice shapes. The preliminary result showed that the rime ice thickness method can be used to characterize the collection efficiency distribution, for conditions free of ice erosion, in the main ice shape region. However, a high-order statistical processing of the ice scan is needed for estimating the mean ice thickness in areas where feathers are prevalent. From the limited comparison of the experimental and LEWICE3D collection efficiency data in the CRM65 MS model leading edge area, it was shown that the attachment line maximum collection efficiency is reasonably estimated by the best curve-fit correlation in the range of modified inertia parameter tested. A tighter alignment control of the iced and cold clean model scans as well as the model angle of attack are needed to improve the comparison. Further evaluation of this correlation is recommended to assess its applicability for Common Research Model type swept wing icing scaling analysis.

Icing Scaling↗
NASA NTRSDate not supplied

Pressure Gradient and Gas Hold-Up in Two-Phase Flows through Porous Media in Microgravity

Single and two-phase flow through porous media are encountered on ground and in space in numerous applications related to life support systems, fuel cells, chemical/materials processing and transporting nutrients to plants. These systems operate differently in the microgravity environment encountered in space travel because the density differences no longer cause the phases to separate or “drain” under the body force of gravity. In the absence of gravity, the interfacial or capillary forces play a more significant role in determining the operational variables such as phase distribution, liquid holdup, and pressure drop especially when the liquid inertia and viscous forces are at minimum. Lower liquid flow rates in air-water two-phase flows represent the case for most of the space processes that depend on two-phase flows in porous media. The Packed Bed Reactor Experiment (PBRE) was developed and flown on the International Space Station (ISS) by NASA to extend a series of fundamental studies of gas-liquid flows through porous media. The goal of the this series of flight experiments (PBRE-1 and PBRE-2) is to better understand the hydrodynamics of two phase flows in porous media in flow regimes dominated by inertia and viscous effects and understanding the role of the interfacial forces and their role in determining the flow dynamics. The flight experiments were preceded by a series of low gravity experiments performed on the low gravity aircraft. These experiments led to the development of a semi-empirical two-phase pressure drop and flow pattern transition in support of a number of reactor beds planned for water reclamation processes onboard of the international space station. The PBRE flight experiment was designed to deliver a wide range of tightly controlled gas (nitrogen) and liquid (water) flows to one of two interchangeable test sections differing only in the type of internal packing material. In this presentation, we describe the PBRE-2 experiment and present preliminary low gravity data on pressure drop and gas hold-up and compare with semi-empirical model pressure drop predictions.

Brian Motil↗
NASA NTRS2021

The Determination of the Rotational State and Interior Structure of Venus with VERITAS

Understanding the processes that led Venus to its current state and will drive its future evolution is a major objective of the next generation of orbiters. In this work we analyze the retrieval of the spin vector, the tidal response and the moment of inertia of Venus with VERITAS, a NASA Discovery-class mission. By simulating a systematic joint analysis of Doppler tracking data and tie points provided by the onboard synthetic aperture radar we show that VERITAS will provide accuracies (3σ) in the estimates of the tidal Love number k_2 to 4.6×10^(-4), its tidal phase lag to 0.05°, and the moment of inertia factor to 9.8×10^(-4) (0.3% of the expected value). Applying these results to recent models of the Venus interior, we show that VERITAS will provide much improved constraints on the interior structure of the planet.

G Cascioli↗
NASA NTRS2022

Thermal Hydraulic Disaggregation of SMAP Soil Moisture Over the Continental United States

Thermal Hydraulic disaggregation of Soil Moisture (THySM) algorithm was implemented to downscale NASA’s Soil Moisture Active Passive (SMAP) Enhanced soil moisture (SM) product to 1 km over the continental United States (CONUS). This algorithm was developed by combining thermal inertia theory with a soil hydraulic-based approach that considers fine-scale SM spatial distribution driven by both heat fluxes and hydraulic conductivity in soils. Relative soil wetness values were estimated using land surface temperature and normalized difference vegetation index for the thermal inertia model and using soil properties for the hydraulic model. The relative soil wetness values at 1 km from both models were then combined by using weighting functions whereby the spatial distribution of SM was governed more by thermal fluxes during times of strong heat transport and infiltration during moisture abundant soil conditions. THySM values were evaluated using in situ SM measurements from SMAP Core Validation Sites (CVS), the USDA Soil Climate Analysis Network, and the NOAA Climate Reference Network over CONUS. THySM shows higher accuracy than the SMAP / Sentinel-1 (SPL2SMAP_S) 1 km SM product when compared to in situ measurements. The accuracy of THySM is 0.048 m3/m3 based on unbiased root mean square error (ubRMSE), outperforming SPL2SMAP_S by 0.01-0.02 m3/m3. The ubRMSE of THySM 1km SM over the SMAP grassland/rangeland-dominated CVS sites is better than 0.04 m3/m3, which meets the SMAP mission SM accuracy requirement applied at 9 and 36 km.

SMAP↗
NASA NTRS2012

Gravity Field and Internal Structure of Mercury From MESSENGER

Radio tracking of the MESSENGER spacecraft has provided a model of Mercury’s gravity field. In the northern hemisphere, several large gravity anomalies, including candidate mass concentrations (mascons), exceed 100 milli-Galileos (mgal). Mercury’s northern hemisphere crust is thicker at low latitudes and thinner in the polar region and shows evidence for thinning beneath some impact basins. The low-degree gravity field, combined with planetary spin parameters, yields the moment of inertia C/MR(exp 2) = 0.353 ± 0.017, where M and R are Mercury’s mass and radius, and a ratio of the moment of inertia of Mercury’s solid outer shell to that of the planet of C(sub m)/C = 0.452 ± 0.035. A model for Mercury’s radial density distribution consistent with these results includes a solid silicate crust and mantle overlying a solid iron-sulfide layer and an iron-rich liquid outer core and perhaps a solid inner core.

David E Smith↗
NASA NTRSDate not supplied

Attitude Control Approach for Solar Cruiser, a Large, Deep Space Solar Sail

Solar Cruiser is a small satellite Technology Demonstration Mission (TDM) to mature solar sail propulsion technology using a solar sail larger than 1600 square meters, demonstrating performance as a propulsion system and a stable pointing platform for science observations in an artificial halo orbit sunward of the Sun-Earth Lagrange Point 1 (sub-L1). For the “sailcraft” to meet mission ob- jectives, there are several unique attitude control challenges that the Attitude De- termination and Control System (ADCS) must overcome. Large disturbance tor- ques, primarily due to sail deformations coupled with off-sun pointing angles, make it more difficult to maintain adequate controls performance and manage accumulated momentum on the control actuators. The large-amplitude, low- frequency flexible body modes of the sail, in concert with noisy sensors and ac- tuators, make it challenging to mitigate control-structure interactions and main- tain fine pointing capabilities. Stability and control performance during sail de- ployment is complicated by rapidly and widely varying inertias. The Solar Cruiser ADCS effectively addresses these challenges using a simple, traditional control system – including momentum management actuators that use bang- bang control to constrain internal accumulated momentum, a low-pass controls filter and an attitude determination Kalman Filter (KF) that blends multiple star tracker solutions, and a reaction wheel assembly (RWA) with controller gains tuned to specific configurations or inertias. Solar Cruiser’s attitude control ap- proach, and lessons learned from its development, establishes a state of the art of high value to future solar sail missions and other small spacecraft with large de- ployable structures operating in deep space.

solar sail↗
NASA NTRSDate not supplied

Attitude Control Approach for Solar Cruiser, a Large, Deep Space Solar Sail

Solar Cruiser is a small satellite Technology Demonstration Mission (TDM) to mature solar sail propulsion technology using a solar sail larger than 1600 square meters, demonstrating performance as a propulsion system and a stable pointing platform for science observations in an artificial halo orbit sunward of the Sun-Earth Lagrange Point 1 (sub-L1). For the “sailcraft” to meet mission objectives, there are several unique attitude control challenges that the Attitude Determination and Control System (ADCS) must overcome. Large disturbance torques, primarily due to sail deformations coupled with off-sun pointing angles, make it more difficult to maintain adequate controls performance and manage accumulated momentum on the control actuators. The large-amplitude, low-frequency flexible body modes of the sail, in concert with noisy sensors and actuators, make it challenging to mitigate control-structure interactions and maintain fine pointing capabilities. Stability and control performance during sail deployment is complicated by rapidly and widely varying inertias. The Solar Cruiser ADCS effectively addresses these challenges using a simple, traditional control system – including momentum management actuators that use bang-bang control to constrain internal accumulated momentum, a low-pass controls filter and an attitude determination Kalman Filter (KF) that blends multiple star tracker solutions, and a reaction wheel assembly (RWA) with controller gains tuned to specific configurations or inertias. Solar Cruiser’s attitude control approach, and lessons learned from its development, establishes a state of the art of high value to future solar sail missions and other small spacecraft with large deployable structures operating in deep space.

solar sail↗
NASA NTRS2023

Phyllosilicate Decomposition on Bennu Due to Prolonged Surface Exposure

The Origins, Spectral Interpretation, Resource Identification, and Security–Regolith Explorer (OSIRIS-REx) mission to carbonaceous asteroid (101955) Bennu performed detailed mapping with a suite of instruments to characterize the composition and geology of its surface. Here we use data from the OSIRIS-REx Thermal Emission Spectrometer (OTES) instrument to investigate the relationship of OTES-derived spectral indices to other derived data products from OTES, the OSIRIS-REx Camera Suite (OCAMS), and the OSIRIS-REx Visible and InfraRed Spectrometer (OVIRS) at global and local scales. We quantitatively confirm that high values of the OTES silicate stretching slope (from ∼10 to ∼12 μm) in midday spectra that are indicative of thin and/or patchy dust cover are strongly associated with low thermal inertia (high porosity), low albedo boulders on Bennu. These high porosity boulders have brecciated textures with embedded clasts that likely originated on Bennu's parent body or during its disruption. The high porosity of these boulders is a key factor in the local production of the dust or its entrapment, as some large, brecciated boulders with a lower porosity have little evidence of dust. A second OTES spectral parameter, the silicate bending band depth near 22.7 μm applied to early evening spectra, is not correlated to thermal inertia, but is weakly to strongly correlated to albedo, OVIRS-derived 1.05 μm and 2.74 μm band depths, OVIRS-derived hydrogen abundance, and modeled nanophase troilite abundance. In several regions on Bennu there is a strong spatial relationship between these parameters, whereby areas with shallower silicate bending bands also have shallower 1.05 μm and 2.74 μm bands and lower albedo with higher nanophase troilite abundances. These correlations, combined with analysis of the silicate bending band in laboratory experiments of space weathered and mildly heated carbonaceous chondrites, suggests that decreased silicate bending band depths signify decomposition of phyllosilicates, likely Fe-bearing, due to space weathering or mild heating (<600 °C) via solar radiation during Bennu's time in near-Earth space. There is a strong association of larger silicate bending band depths in areas dominated by small rocks and unresolved material and in areas with small (≤ 25 m) craters identified as the spectrally reddest on Bennu, suggesting that this material has been more recently exposed due to impact and/or mass wasting processes. The shallowest silicate bending depths are associated with larger rocks and boulders that appear to have the longest surface exposure history, although there is band depth variation among them suggesting either initial composition variation that resulted in different responses to space weathering or heating, or varied exposure history of individual boulders themselves. We predict that any grains returned from Bennu with a significant surface exposure history will be characterized by shallower 22.7 μm, 1.05 μm and 2.7 μm band depths and increased sulfide (troilite) abundance, as well as textural and chemical evidence for phyllosilicate dehydration.

Asteroids, surfaces↗
NASA NTRS2024

Subsurface Thermophysical Properties of Europa's Leading and Trailing Hemispheres as Revealed by ALMA

We present best-fit values of porosity—and the corresponding effective thermal inertiae—determined from three different depths in Europa's near-subsurface (∼1–20 cm). The porosity of the upper ∼20 cm of Europa's subsurface varies between 75% and 50% (Γeff ≈ 50–140 J m −2 K −1 s −1/2 ) on the leading hemisphere and 50%–40% (Γeff ≈ 140–180 J m −2 K −1 s −1/2 ) on the trailing hemisphere. Residual maps produced by comparison with these models reveal thermally anomalous features that cannot be reproduced by globally homogeneous porosity models. These regions are compared to Europa's surface terrain and known compositional variations. We find that some instances of warm thermal anomalies are co-located with known geographical or compositional features on both the leading and trailing hemisphere; cool temperature anomalies are well correlated with surfaces previously observed to contain pure, crystalline water ice and the expansive rays of Pwyll crater. Anomalous regions correspond to locations with subsurface properties different from those of our best-fit models, such as potentially elevated thermal inertia, decreased emissivity, or more porous regolith. We also find that ALMA observations at ∼3 mm sound below the thermal skin depth of Europa (∼10–15 cm) for a range of porosity values, and thus do not exhibit features indicative of diurnal variability or residuals similar to other frequency bands. Future observations of Europa at higher angular resolution may reveal additional locations of variable subsurface thermophysical properties, while those at other wavelengths will inform our understanding of the regolith compaction length and the effects of external processes on the shallow subsurface.

Alexander E Thelen↗
NASA NTRSDate not supplied

Comparison of Orion Flywheel to Commercially Available Flywheel Device

INTRODUCTION: The chosen exercise device for the Orion Multi-purpose Crew Vehicle (MPCV) for Artemis missions is the Orion Flywheel (OFW), which was selected in 2017 from a pool of exercise devices. The Orion Flywheel is to be installed in the MPCV for its first use during the Artemis II mission and will be utilized for subsequent Artemis missions. The OFW provides the capability to perform both resistive and aerobic exercises while taking up a small volume and not requiring power. Aerobic exercise consists of rowing while resistive exercise includes the deadlift, harness squat, and deadlift high pull among others. The efficacy of the OFW as an exercise countermeasure to spaceflight deconditioning has not yet been fully evaluated. A future HRP-funded study using head-down tilt bed rest is planned to evaluate the efficacy of the OFW to mitigate aerobic and muscular deconditioning. The OFW, made by KBR, is a specialized device that poses high costs for purchase and maintenance. The aim of the study is to determine if a commercially available alternative could be used as a backup for the OFW during the bed rest study or as an alternative in other studies. This study seeks to compare the loading profile and mechanics of a commercial flywheel device (kBox Pro) with the three resistance settings of the OFW using a controlled, instrumented setup to validate its acceptability as a backup. METHODS: An experimental setup was designed to simulate human use of the OFW. Flywheel devices can have varied actuated inertial mass and strap length. The inertial mass refers to the flywheel itself or the object that spins as the result of the movement of the strap. The OFW has three gears actuated with a lever, with the low gear used for aerobic rowing and the medium and high gears used for resistance exercises. These gears correspond to increasing flywheel inertia, which corresponds to a certain number and sizes of inertial plates on a commercial flywheel. The kBox Pro has varied sizes of inertial discs that are installed to the external shaft of the device and can be configured to roughly match the inertia seen in the OFW per gear setting. The strap length can be adjusted to allow for different ranges of motion. The experimental setup allows for the simulation of rowing, harness squats, and deadlift high pulls with corresponding exercise strap lengths. All three OFW gears are to be tested, as well as multiple combinations of kBox discs. The setup was created using a system of pulleys and cables to minimize friction loss. The system is actuated through the dropping of a connected mass which, through the pulley system, pulls the strap of the exercise device upwards as if a human was exercising. A string potentiometer and tensile force transducer are used to measure strap displacement and strap force, respectively. The masses dropped are exercise chains that simulate the decrease in force observed throughout a typical exercise. As the chains are lowered to the ground, they begin to spool and therefore are no longer subjecting the exercise device’s strap to their load. This setup was designed to provide realistic loads and protect the OFW from damage, an important precaution because it’s the only available unit for research and ground operations. Data is to be collected across simulated exercises and with different strap lengths, with the same conditions collected on both the OFW and the kBox Pro. RESULTS: Data collection is ongoing for the kBox and the Orion Flywheel. CONCLUSIONS: Study results will inform if the OFW and a kBox Pro device can be deemed mechanically similar. Depending on the results, additional data collection may be required to assess the user experience between the devices. A key result of this study is a mechanical characterization of the OFW device, demonstrating the load and force profile using a repeatable test method. This data will guide next steps on potential kBox Pro modifications needed to produce a comparable exercise experience to the OFW.

Exercise↗
NASA NTRS2023

Hydrodynamics of bubble flow through a porous medium with applications to packed bed reactors

Gas-liquid flows through packed bed reactors (PBRs) are challenging to predict due to the tortuous flow paths that fluid interfaces must traverse. Experiments at the International Space Station showed that bubble and pulse flows are predominately observed under microgravity conditions, while the trickle and spray flows observed under terrestrial conditions are not present in microgravity. To understand the physics behind the former experiments, we simulate bubble flow through a PBR for different packing-particle-diameter-based Weber numbers and under different gravity conditions. We demonstrate different pore-scale mechanisms, such as capillary entrapment, buoyancy entrapment, and inertia-induced bubble displacement. Then, we perform a quantitative analysis by introducing new dynamic scales, dependent upon the evolving gas-liquid interfacial area, to understand the dynamic trade-offs between the inertia, capillary, and buoyancy forces on a bubble passing through a PBR. This analysis leads us to define new dimensionless Weber-like numbers that delineate bubble entrapment from bubble displacement.

Pranay P Nagrani↗
NASA NTRS1961

Method for Approximating the Vacuum Motions of Spinning Symmetrical Bodies with Nonconstant Spin Rates

A method for approximating the vacuum motions of spinning rigid symmetrical bodies with varying spin rates and inertias has been completed. The analysis includes the effects of time varying thrust misalignments, mass unbalance, and jet damping. Results are given in the form of equations for space­ referenced Euler angles, flight-path angles, body­ referenced attitude rates, and earth-referenced vehicle-trajectory coordinates. The method consists of dividing the problem into intervals during which the time-dependent variables are assumed constant at their mean interval value. In order to check this procedure, solutions for various interval sizes are compared with solutions obtained with numerical methods. Although the method is somewhat lengthy for accurate hand computation in most cases, it is readily programed for machine solutions. Probably more important, the general solutions give insight into the separate effects of the variables and, in many cases, can be quickly used to determine the approxi­mate ranges of the variables required for the desired solution to a given problem. In this respect, equations for determining maximum wobble have been derived for certain input conditions. The method has been shown to compare closely with the numerical solutions of two sample problems. The sample problems also illustrated the relatively large effect of pitch and yaw jet damping on body motions.

VACUUM↗