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Elastic Model Transitions: A Hybrid Approach Utilizing Quadratic Inequality Constrained Least Squares (LSQI) and Direct Shape Mapping (DSM)

A method for transitioning linear time invariant (LTI) models in time varying simulation is proposed that utilizes a hybrid approach for determining physical displacements by augmenting the original quadratically constrained least squares (LSQI) algorithm with Direct Shape Mapping (DSM) and modifying the energy constraints. The approach presented is applicable to simulation of the elastic behavior of launch vehicles and other structures that utilize discrete LTI finite element model (FEM) derived mode sets (eigenvalues and eigenvectors) that are propagated throughout time. The time invariant nature of the elastic data presents a problem of how to properly transition elastic states from the prior to the new model while preserving motion across the transition and ensuring there is no truncation or excitation of the system. A previous approach utilizes a LSQI algorithm with an energy constraint to effect smooth transitions between eigenvector sets with no requirement that the models be of similar dimension or have any correlation. This approach assumes energy is conserved across the transition, which results in significant non-physical transients due to changing quasi-steady state energy between mode sets, a phenomenon seen when utilizing a truncated mode set. The computational burden of simulating a full mode set is significant so a subset of modes is often selected to reduce run time. As a result of this truncation, energy between mode sets may not be constant and solutions across transitions could produce non-physical transients. In an effort to abate these transients an improved methodology was developed based on the aforementioned approach, but this new approach can handle significant changes in energy across mode set transitions. It is proposed that physical velocities due to elastic behavior be solved for using the LSQI algorithm, but solve for displacements using a two-step process that independently addresses the quasi-steady-state and non-steady-state contributions to the elastic displacement. For structures subject to large external forces, such as thrust or atmospheric drag, it is imperative to capture these forces when solving for elastic displacement. To simplify the mathematical formulation, assumptions are made regarding mass matrix normalization, constant external forcing, and constant viscous damping. These simplifications allow for direct solutions to the quasi-steady-state displacements through a process titled Direct Shape Mapping. DSM solves for the displacements using the eigenvalues of the elastic modes and the external forcing and returns a set of elastic displacements dictated by the eigenvectors of the post-transition mode set. For the non-steady-state contributions to displacement we formulate a LSQI problem that is constrained by energy of the non-steady state terms. The contributions from the quasi-steady-state and non-steady state solutions are then combined to obtain the physical displacements associated with the new set of eigenvectors. Results for the LSQI-DSM approach show significant reduction/complete removal of transients across mode set transitions while maintaining elastic motion from the prior state. For time propagation applications employing discrete elastic models that need to be transitioned in time and where running with full a full mode set is not feasible, the method developed offers a practical solution to simulating vehicle elasticity.

Hannan, Mike R.

Elastic Modulus Testing for TriTruss Struts

The TriTruss is a novel structural module developed by researchers at NASA Langley Research Center (LaRC) that can be used in space to assemble large backing structures for a variety of applications. One such application is the metering truss and primary mirror backbone support structure of an in-space assembled telescope(iSAT). For the iSAT application, the TriTruss will be supporting mirror segments, payloads, and instruments, all of which require the TriTruss to have robust structural integrity. An accurate elastic modulus value is crucial to the finite element analysis(FEA) of the structure. Initial testing indicated that the equivalent elastic modulus provided by the manufacturer and used in the analyses did not accurately predict the behavior of the structure. In this paper, the in-house testing method developed at LaRC to accurately measure the equivalent elastic modulus of the composite struts for use in the analyses is described. A tensile and compressive equivalent elastic modulus was computed from the data obtained in this test. The testing resulted in a tensile equivalent elastic modulus of 24.6 Msi (standard deviation of 0.888 Msi) or 169 GPa (standard deviation of 6.20 GPa) and a compressive equivalent elastic modulus of 22.8 Msi (standard deviation of 0.780 Msi) or 157 GPa(standard deviation of 5.38 GPa) for the single wall thickness struts. For double wall thickness struts, the testing resulted in a tensile equivalent elastic modulus of32.1 Msi (standard deviation of 1.003 Msi) or 221 GPa (standard deviation of 6.91GPa) and a compressive equivalent elastic modulus of 31.9 Msi (standard deviation of 1.123) or 220 GPa (standard deviation of 7.74 GPa).1

TriTruss

Simulation and control problems in elastic robots

Computational issues associated with modeling and control of robots with revolute joints and elastic arms are considered. A manipulator with one arm and pinned at one end is considered to investigate various aspects of the modeling procedure and the model, and the effect of coupling between the rigid-body and the elastic motions. The rigid-body motion of a manipulator arm is described by means of a reference frame attached to the shadow beam, and the linear elastic operator denoting flexibility is defined with respect to this reference frame. The small elastic motion assumption coupled with the method of assumed modes is used to model the elasticity in the arm. It is shown that only terms up to quadratic in these model amplitudes need to be retained. An important aspect of the coupling between the rigid-body and the elastic motion is the centrifugal stiffening effect. This effect stiffens the elastic structure, as to be expected on physical grounds, gives rise to a time-varying inertia term for the rigid-body motion, and, in general, results in an effective inertia term smaller than the rigid-body inertia term. Simulation results are presented for an elastic beam pinned at one end and free at the other, and rotating in a horizontal plane, and control issues such as the order of the model, number of sensors, and modal extraction are examined within this context.

Tadikonda, S. S. K.

Finite element elastic-plastic-creep and cyclic life analysis of a cowl lip

Results are presented of elastic, elastic-plastic, and elastic-plastic-creep analyses of a test-rig component of an actively cooled cowl lip. A cowl lip is part of the leading edge of an engine inlet of proposed hypersonic aircraft and is subject to severe thermal loadings and gradients during flight. Values of stresses calculated by elastic analysis are well above the yield strength of the cowl lip material. Such values are highly unrealistic, and thus elastic stress analyses are inappropriate. The inelastic (elastic-plastic and elastic-plastic-creep) analyses produce more reasonable and acceptable stress and strain distributions in the component. Finally, using the results from these analyses, predictions are made for the cyclic crack initiation life of a cowl lip. A comparison of predicted cyclic lives shows the cyclic life prediction from the elastic-plastic-creep analysis to be the lowest and, hence, most realistic.

Arya, Vinod K.

Finite element elastic-plastic-creep and cyclic life analysis of a cowl lip

Results are presented of elastic, elastic-plastic and elastic-plastic-creep analyses of a test-rig component of an actively cooled cowl lip. A cowl lip is part of the leading edge of an engine inlet of proposed hypersonic aircraft and is subject to severe thermal loadings and gradients during flight. Values of stresses calculated by elastic analysis are well above the yield strength of the cowl lip material. Such values are highly unrealistic, and thus elastic stress analyses are inappropriate. The inelastic (elastic-plastic and elastic-plastic-creep) analyses produce more reasonable and acceptable stress and strain distributions in the component. Finally, using the results from these analyses, predictions are made for the cyclic crack initiation life of a cowl lip. A comparison of predicted cyclic lives shows the cyclic life prediction from the elastic-plastic-creep analysis to be the lowest and, hence, most realistic.

Arya, V. K.

Aeroelastic response and stability of tiltrotors with elastically-coupled composite rotor blades

There is a potential for improving the performance and aeroelastic stability of tiltrotors through the use of elastically-coupled composite rotor blades. To study the characteristics of tiltrotors with these types of rotor blades it is necessary to formulate a new analysis which has the capabilities of modeling both a tiltrotor configuration and an anisotropic rotor blade. Background for these formulations is established in two preliminary investigations. In the first, the influence of several system design parameters on tiltrotor aeroelastic stability is examined for the high-speed axial flight mode using a newly-developed rigid-blade analysis with an elastic wing finite element model. The second preliminary investigation addresses the accuracy of using a one-dimensional beam analysis to predict frequencies of elastically-coupled highly-twisted rotor blades. Important aspects of the new aeroelastic formulations are the inclusion of a large steady pylon angle which controls tilt of the rotor system with respect to the airflow, the inclusion of elastic pitch-lag coupling terms related to rotor precone, the inclusion of hub-related degrees of freedom which enable modeling of a gimballed rotor system and engine drive-train dynamics, and additional elastic coupling terms which enable modeling of the anisotropic features for both the rotor blades and the tiltrotor wing. Accuracy of the new tiltrotor analysis is demonstrated by a comparison of the results produced for a baseline case with analytical and experimental results reported in the open literature. Two investigations of elastically tailored blades on a baseline tiltrotor are then conducted. One investigation shows that elastic bending-twist coupling of the rotor blade is a very effective means for increasing the flutter velocity of a tiltrotor, and the magnitude of coupling required does not have an adverse effect on performance or blade loads. The second investigation shows that passive blade twist control via elastic extension-twist coupling of the rotor blade has the capability of significantly improving tiltrotor aerodynamic performance. This concept, however, is shown to have, in general, a negative impact on stability characteristics.

Nixon, Mark W.

Implementation of J-A Methodology Elastic-Plastic Crack Instability Analysis Capability into the WARP-3D Code

Characterization of the near crack-tip stress/strain fields is the foundation of fracture mechanics. The description of the near tip stress field and the prediction of when fracture occurs is well established for brittle materials that exhibit linear elastic behavior. However, in ductile materials or conditions that violate linear elastic assumptions (Aluminum alloys, Al 2024-T3, Al 2024- T351 etc.), the elastic-plastic crack-tip stress fields are characterized by the Hutchison-Rice-Rosengren (HRR) field. The J-integral is commonly used to characterize amplitude of the HRR field under elastic-plastic conditions. The J-integral has been demonstrated for crack-tip fields that are under high constraint conditions (i.e., small-scale plasticity where the J-dominance is maintained). However, as the external load increases, yielding changes from small- to largescale plasticity and usually a loss of constraint (i.e., reduction in the triaxial stress field along the crack front). The loss of constraint leads to the deviation of the crack-tip stress fields from that given by the HRR field. Hence, the J-dominance will be gradually lost and additional parameter(s) are required to quantify the crack-tip stress fields and predict fracture behavior. The assessment objectives were to: 1) implement a two-parameter (i.e., J-A) fracture criterion into an elastic-plastic three-dimensional (3D) finite element analysis (FEA), 2) validate the implementation by comparison with the A parameter from literature data, 3) conduct material characterization tests to quantify the material behavior and provide fracture data for validation of the J-A fracture criteria, and (4) perform evaluations to establish if the J-A criteria can be used to predict fracture in a ductile metallic material (e.g., aluminum alloys). The A parameter in these criteria is the second parameter in a three-term elastic-plastic asymptotic expansion of the neartip stress behavior. A series of extensive FEAs were performed using WARP3D software package to obtain solutions for the A parameter for different specimen configurations. The methodology needed for the estimation of the A parameter in the asymptotic expansion was developed and implemented using Matlab®. A user material (UMAT) routine was used to model the material stress-strain response using a Ramberg-Osgood power law with a hardening exponent (n) and a material coefficient (alpha). This UMAT routine was successfully implemented in WARP3D software and validated through comparison with the experimental data. Three configurations were extracted from published results: 1) center cracked plate (CCP), 2) single edge-cracked plate (SECP), and 3) double edge-cracked plate (DECP). These configurations and four other configurations (three-hole tension (THT)), three-point bend (3PTB), three-hole compact tension (3PCT), and compact tension (CT)) were analyzed to verify the methodology that was developed and implemented into WARP3D. Solutions of the A parameter were obtained for remote tension loading conditions that started with small-scale yielding and continued into the large-scale plasticity regime. The results indicate that the methodology developed can be used to calculate the elastic-plastic J-A parameters for test specimens with a range of crack geometries, material strain hardening behaviors, and loading conditions. The J-A parameters were implemented as fracture criteria and used to predict the test results. For comparison, other fracture criteria were used to predict the same test results. Major findings include: The A constraint parameter A varies with specimen type and applied load thus accurate determination is crucial in predicting the failure load, and the A parameter is asymptotic as the failure load is approached, making an accurate determination difficult (i.e., small differences in the A parameter can cause large variations in failure load) for materials exhibiting elastic-plastic behavior. The failure predictions from J-A methodology were more accurate than the traditionally used KC and J methods, and have comparable scatter to that observed when using the crack-tip opening angle (CTOA) method. However, the J-A methodology requires considerable effort (expertise level and labor) to implement and to evaluate the A parameter for different specimen types and materials, or to apply this methodology to part-through crack (e.g., 3D problems) structural applications.

Hamm, Kenneth R., Jr.

Endothelial, cardiac muscle and skeletal muscle exhibit different viscous and elastic properties as determined by atomic force microscopy

This study evaluated the hypothesis that, due to functional and structural differences, the apparent elastic modulus and viscous behavior of cardiac and skeletal muscle and vascular endothelium would differ. To accurately determine the elastic modulus, the contribution of probe velocity, indentation depth, and the assumed shape of the probe were examined. Hysteresis was observed at high indentation velocities arising from viscous effects. Irreversible deformation was not observed for endothelial cells and hysteresis was negligible below 1 microm/s. For skeletal muscle and cardiac muscle cells, hysteresis was negligible below 0.25 microm/s. Viscous dissipation for endothelial and cardiac muscle cells was higher than for skeletal muscle cells. The calculated elastic modulus was most sensitive to the assumed probe geometry for the first 60 nm of indentation for the three cell types. Modeling the probe as a blunt cone-spherical cap resulted in variation in elastic modulus with indentation depth that was less than that calculated by treating the probe as a conical tip. Substrate contributions were negligible since the elastic modulus reached a steady value for indentations above 60 nm and the probe never indented more than 10% of the cell thickness. Cardiac cells were the stiffest (100.3+/-10.7 kPa), the skeletal muscle cells were intermediate (24.7+/-3.5 kPa), and the endothelial cells were the softest with a range of elastic moduli (1.4+/-0.1 to 6.8+/-0.4 kPa) depending on the location of the cell surface tested. Cardiac and skeletal muscle exhibited nonlinear elastic behavior. These passive mechanical properties are generally consistent with the function of these different cell types.

Non-NASA Center

Intramuscular pressures beneath elastic and inelastic leggings

Leg compression devices have been used extensively by patients to combat chronic venous insufficiency and by astronauts to counteract orthostatic intolerance following spaceflight. However, the effects of elastic and inelastic leggings on the calf muscle pump have not been compared. The purpose of this study was to compare in normal subjects the effects of elastic and inelastic compression on leg intramuscular pressure (IMP), an objective index of calf muscle pump function. IMP in soleus and tibialis anterior muscles was measured with transducer-tipped catheters. Surface compression between each legging and the skin was recorded with an air bladder. Subjects were studied under three conditions: (1) control (no legging), (2) elastic legging, and (3) inelastic legging. Pressure data were recorded for each condition during recumbency, sitting, standing, walking, and running. Elastic leggings applied significantly greater surface compression during recumbency (20 +/- 1 mm Hg, mean +/- SE) than inelastic leggings (13 +/- 2 mm Hg). During recumbency, elastic leggings produced significantly higher soleus IMP of 25 +/- 1 mm Hg and tibialis anterior IMP of 28 +/- 1 mm Hg compared to 17 +/- 1 mm Hg and 20 +/- 2 mm Hg, respectively, generated by inelastic leggings and 8 +/- 1 mm Hg and 11 +/- 1 mm Hg, respectively, without leggings. During sitting, walking, and running, however, peak IMPs generated in the muscular compartments by elastic and inelastic leggings were similar. Our results suggest that elastic leg compression applied over a long period in the recumbent posture may impede microcirculation and jeopardize tissue viability.(ABSTRACT TRUNCATED AT 250 WORDS).

Clinical Trial

Air and ground resonance of helicopters with elastically tailored composite rotor blades

The aeromechanical stability, including air resonance in hover, air resonance in forward flight, and ground resonance, of a helicopter with elastically tailored composite rotor blades is investigated. Five soft-inplane hingeless rotor configurations, featuring elastic pitch-lag, pitch-flap and extension-torsion couplings, are analyzed. Elastic couplings introduced through tailored composite blade spars can have a powerful effect on both air and ground resonance behavior. Elastic pitch-flap couplings (positive and negative) strongly affect body, rotor and dynamic inflow modes. Air resonance stability is diminished by elastic pitch-flap couplings in hover and forwrad flight. Negative pitch-lag elastic coupling has a stabilizing effect on the regressive lag mode in hover and forward flight. The negative pitch-lag coupling has a detrimental effect on ground resonance stability. Extension-torsion elastic coupling (blade pitch decreases due to tension) decreases regressive lag mode stability in both airborne and ground contact conditions. Increasing thrust levels has a beneficial influence on ground resonance stability for rotors with pitch-flap and extension-torsion coupling and is only marginally effective in improving stability of rotors with pitch-lag coupling.

Smith, Edward C.

Effect of a hypergravity environment on cortical bone elasticity in rats

There is considerable interest in determining whether hypergravity can be used as a countermeasure for microgravity-induced bone loss. This study was conducted on 20 immature male rats in order to investigate possible elastic adaptations of cortical bone in rapidly growing rats exposed to chronic hypergravity. Ten rats were continuously centrifuged for 14 days at twice gravitational acceleration (2G) on a 12.75 foot radius centrifuge and 10 rats concurrently acted as stationary controls. The effect of hypergravity on the elastic characteristics of cortical bone was quantified via ultrasonic wave propagation. Propagation velocities of longitudinal and shear waves were measured through cubic cortical specimens from the posterior femoral diaphyses. Density was measured with an Archimedes' technique. The orthotropic elastic properties were calculated and used to compare the difference between groups. Results showed an average increase in both the Young's moduli (Eii, + 2.2%) and shear moduli (Gij, + 4.3%) with a statistically significant increase only in G12 (+15.7%, P = 0.046). The ratio of transverse to axial strain (Poisson's ratio, nuij) demonstrated statistically significant changes in nu12, nu21, nu13, and nu31 (P < 0.05). These findings suggest that although slight elastic changes were incurred via a hypergravity environment, the treatment level or duration in this study do not dramatically perturb the normal elastic behavior of cortical bone and that dramatic biomechanical differences noted in previous studies were due more to structural changes than material elasticity changes. Hypergravity applied post facto to a microgravity environment would offer further illucidation of this method as treatment for a degenerative spaceflight experience.

Non-NASA Center

An elastic-perfectly plastic analysis of the bending of the lithosphere at a trench

A number of authors have modeled the flexure of the lithosphere at an oceanic trench using a thin elastic plate with a hydrostatic restoring force. In some cases good agreement with observed topography is obtained but in other cases the slope of the lithosphere within the trench is greater than that predicted by the elastic theory. In this paper the bending of a thin plate is considered using an elastic-perfectly plastic rheology. It is found that the lithosphere behaves elastically seaward of the trench, but that plasticity decreases the radius of curvature within the trench. The results are compared with a number of observed trench profiles. The elastic-perfectly plastic profiles are in excellent agreement with those profiles that deviate from elastic behavior.

Turcotte, D. L.

Elastic-plastic finite-element analyses of thermally cycled single-edge wedge specimens

Elastic-plastic stress-strain analyses were performed for single-edge wedge alloys subjected to thermal cycling in fluidized beds. Three cases (NASA TAZ-8A alloy under one cycling condition and 316 stainless steel alloy under two cycling conditions) were analyzed by using the MARC nonlinear, finite-element computer program. Elastic solutions from MARC showed good agreement with previously reported solutions that used the NASTRAN and ISO3DQ computer programs. The NASA TAZ-8A case exhibited no plastic strains, and the elastic and elastic-plastic analyses gave identical results. Elastic-plastic analyses of the 316 stainless steel alloy showed plastic strain reversal with a shift of the mean stresses in the compressive direction. The maximum equivalent total strain ranges for these cases were 13 to 22 percent greater than that calculated from elastic analyses.

Kaufman, A.

Elastic-plastic finite-element analyses of thermally cycled double-edge wedge specimens

Elastic-plastic stress-strain analyses were performed for double-edge wedge specimens subjected to thermal cycling in fluidized beds at 316 and 1088 C. Four cases involving different nickel-base alloys (IN 100, Mar M-200, NASA TAZ-8A, and Rene 80) were analyzed by using the MARC nonlinear, finite element computer program. Elastic solutions from MARC showed good agreement with previously reported solutions obtained by using the NASTRAN and ISO3DQ computer programs. Equivalent total strain ranges at the critical locations calculated by elastic analyses agreed within 3 percent with those calculated from elastic-plastic analyses. The elastic analyses always resulted in compressive mean stresses at the critical locations. However, elastic-plastic analyses showed tensile mean stresses for two of the four alloys and an increase in the compressive mean stress for the highest plastic strain case.

Kaufman, A.

Characteristic Elastic Systems of Time-Limited Optimal Maneuvers

Optimizing an elastic system and its active control is discussed. Maneuvers from an initial state to a final state in a finite time interval are considered. An active generalized control force that accomplishes the desired maneuver of a prespecified system is optimal if it minimizes a given quadratic cost function. By also varying a set of design parameters, the elastic system can be determined so as to further minimize the cost function. Here, the elastic system that minimizes the actual control cost is compared with the system that minimizes the ratio of actual cost to the cost of optimally maneuvering a rigid system of the same inertial properties. It is shown that an elastic system corresponding to an extremum of the ratio is actually a characteristic of the time-limited maneuver. Because both the spatial domain and the time interval are fixed, a characteristic elastic system is tuned to the specified temporal boundary conditions. The implication for rest-to-rest, spinup, and spin reversal maneuvers of spacecraft is that the optimal control for a characteristic elastic spacecraft is identical to the optimal control for the same spacecraft as if it were rigid.

Hale, A. L.

Elastic electron scattering cross sections for molecular hydrogen

Using an electron-beam - molecular-beam apparatus and employing the relative flow technique, ratios of the differential elastic scattering cross sections (DCSs of H2 to He were measured at incident electron energies of 15-100 eV and over the angular range of 10-125 degrees. From these ratios, the absolute elastic DCSs for H2 were determined by normalization to accurate, available elastic DCSs of He. Since pure rotational structure was not resolved in this work, the DCSs reported are the sum of elastic and rotational excitations of H2 at room temperature. The reliability of the relative flow normalization to He was checked at each energy and angle by performing similar elastic DCS measurements on Ne (for which the cross sections are known). The resulting absolute Ne DCSs were found to be in good agreement (within 10 percent with the Ne elastic DCSs measured previously (Register and Trajmar, 1984). From the DCSs, integral and momentumtransfer cross sections were calculated. The present results are compared with other recent measurements.

Khakoo, M. A.

Flexural models of trench/outer rise topography of coronae on Venus with axisymmetric spherical shell elastic plates

Magellan altimetry has revealed that many coronae on Venus have trenches or moats around their peripheries and rises outboard of the trenches. This trench/outer rise topographic signature is generally associated with the tectonic annulus of the corona. Sandwell and Schubert have interpreted the trench/outer rise topography and the associated tectonic annulus around coronae to be the result of elastic bending of the Venus lithosphere (though the tectonic structures are consequences of inelastic deformation of the lithosphere). They used two-dimensional elastic plate flexure theory to fit topographic profiles across a number of large coronae and inferred elastic lithosphere thicknesses between about 15 and 40 km, similar to inferred values of elastic thickness for the Earth's lithosphere at subduction zones around the Pacific Ocean. Here, we report the results of using axisymmetric elastic flexure theory for the deformation of thin spherical shell plates to interpret the trench/outer rise topography of the large coronae modeled by Sandwell and Schubert and of coronae as small as 250 km in diameter. In the case of a corona only a few hundred kilometers in diameter, the model accounts for the small planform radius of the moat and the nonradial orientation of altimetric traces across the corona. By fitting the flexural topography of coronae we determine the elastic thickness and loading necessary to account for the observed flexure. We calculate the associated bending moment and determine whether the corona interior topographic load can provide the required moment. We also calculate surface stresses and compare the stress distribution with the location of annular tectonic features.

Moore, W.

Evaluation of a Hybrid Elastic EVA Glove

The hybrid elastic design is based upon an American Society for Engineering Education (ASEE) glove designed by at the Space Systems Laboratory (SSL) in 1985. This design uses an elastic restraint layer instead of convolute joints to achieve greater dexterity and mobility during EVA (extravehicular activity). Two pilot studies and a main study were conducted using the hybrid elastic glove and 4000-series EMU (extravehicular activity unit) glove. Data on dexterity performance, joint range of motion, grip strength and perceived exertion was assessed for the EMU and hybrid elastic gloves with correlations to a barehanded condition. During this study, 30 test subjects performed multiple test sessions using a hybrid elastic glove and a 4000- series shuttle glove in a 4.3psid pressure environment. Test results to date indicate that the hybrid elastic glove performance is approximately similar to the performance of the 4000-series glove.

Korona, F. Adam