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NASA NTRS1973

Part 2: Oscillations of elastic systems

Problems of oscillations of linear systems are discussed, including systems with a fractional number of degrees of freedom as well as free oscillations of a cantilever in the field of centrifugal forces. Four methods of solving the problem of the action of periodic instantaneous impulses are presented. The Tacoma catastrophe is analyzed and used as an example of aeroelastic oscillations. Problems of nonlinear system oscillations include the vibration maintenance of rotation, the Sommerfeld effect, and self-oscillations of a quasi-system with dry friction.

Source record↗

DOE OSTI2024

On plasticity-enhanced interfacial toughness in bonded joints

Here, the performance and reliability of many structures and components depend on the integrity of interfaces between dissimilar materials. Interfacial toughness Γ is the key material parameter that characterizes resistance to interfacial crack growth, and Γ is known to depend on many factors including temperature. For example, previous work showed that the toughness of an epoxy/aluminum interface decreased 40 % as the test temperature was increased from -60 °C to room temperature (RT). Interfacial integrity at elevated temperatures is of considerable practical importance. Recent measurements show that instead of continuing to decrease with increasing temperature, Γ increases when test temperature is above RT. Cohesive zone finite element calculations of an adhesively bonded, asymmetric double cantilever beam specimen of the type used to measure Γ suggest that this increase in toughness may be a result of R-curve behavior generated by plasticity-enhanced toughening during stable subcritical crack growth with interfacial toughness defined as the critical steady-state limit value. In these calculations, which used an elastic-perfectly plastic epoxy model with a temperature-dependent yield strength, the plasticity-enhanced increase in Γ above its intrinsic value Γ o depended on the ratio of interfacial strength σ* to the yield strength σyb of the bond material. There is a nonlinear relationship between Γ/Γ o and σ*/σ yb with the value Γ/Γ o increasing rapidly above a threshold value of σ*/σ yb . The predicted increase in toughness can be significant. For example, there is nearly a factor of two predicted increase in Γ/Γ o during micrometer-scale crack-growth when σ*/σ yb = 2 (a reasonable choice for σ*/σ yb ). Furthermore, contrary to other reported results, plasticity-enhanced toughening can occur prior to crack advance as the cohesive zone forms and the peak stress at the tip of the original crack tip translates to the tip of the fully formed cohesive zone. These results suggest that plasticity-enhanced toughening should be considered when modeling interfaces at elevated temperatures.

36 MATERIALS SCIENCE↗

NASA NTRS1975

Importance of helicopter dynamics to the mathematical model of the helicopter

A mathematical model of the helicopter requires appropriate representation of the constituent elements of rotor dynamics. General-purpose programs that model a variety of configurations for a broad range of operating conditions result in varying and incompatible levels of sophistication. Analysis of specific dynamic problems facilitates the identification of configuration parameters which determine system behavior. For the present analysis, the nonlinear equations of a torsionally rigid hingeless rotor are linearized about an equilibrium condition to determine flap-lag stability characteristics in hover. A collocation method was used to obtain the coupled natural frequencies and modes. These modes allow exact treatment of the effect of elastic coupling which more than compensates for the destabilizing inertial coupling. The sensitivity of damping to the number of modes was found to be small, and reasonable accuracy was obtained the first flapwise and edgewise coupled modes. The range of destabilizing precone was found to be small.

White, W. F., Jr.↗

NASA NTRS1976

A high order theory for uniform and laminated plates

A theory of plate deformation is derived which accounts for the effects of transverse shear deformation, transverse normal strain, and a nonlinear distribution of the in-plane displacements with respect to the thickness coordinate. The theory is compared with lower order plate theories through application to a particular problem involving a plate acted upon by a sinusoidal surface pressure. Comparison is also made with exact elasticity solution of this problem. It is found that when the ratio of the characteristic length of the load pattern to the plate thickness is of the order of unity, lower order theories are inadequate and the present high order theory is required to give meaningful results. Results are given for the bending of symmetric cross-ply and angle-ply laminates. Comparison with exact elasticity solutions indicates that the present plate theory is sufficiently accurate for predicting the behavior of thick laminates.

Lo, K. H.↗

DOE OSTI2021

Potential vorticity transport in weakly and strongly magnetized plasmas

Tangled magnetic fields, often coexisting with an ordered mean field, have a major impact on turbulence and momentum transport in many plasmas, including those found in the solar tachocline and magnetic confinement devices. We present a novel mean field theory of potential vorticity mixing in β-plane magnetohydrodynamic (MHD) and drift wave turbulence. Our results show that mean square stochastic fields strongly reduce Reynolds stress coherence. This decoherence of potential vorticity flux due to stochastic field scattering leads to suppression of momentum transport and zonal flow formation. A simple calculation suggests that the breaking of the shear-eddy tilting feedback loop by stochastic fields is the key underlying physics mechanism. Furthermore, a dimensionless parameter that quantifies the increment in power threshold is identified and used to assess the impact of stochastic field on the L-H transition. We discuss a model of stochastic fields as a resisto-elastic network.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1978

The dynamics and control of large flexible space structures. Part B: Development of continuum model and computer simulation

The equations of motion of an arbitrary flexible body in orbit were derived. The model includes the effects of gravity with all its higher harmonics. As a specific example, the motion of a long, slender, uniform beam in circular orbit was modelled. The example considers both the inplane and three dimensional motion of the beam in orbit. In the case of planar motion with only flexible vibrations, the pitch motion is not influenced by the elastic motion of the beam. For large values of the square of the ratio of the structural modal frequency to the orbital angular rate the elastic motion was decoupled from the pitch motion. However, for small values of the ratio and small amplitude pitch motion, the elastic motion was governed by a Hill's 3 term equation. Numerical simulation of the equation indicates the possibilities of instability for very low values of the square of the ratio of the modal frequency to the orbit angular rate. Also numerical simulations of the first order nonlinear equations of motion for a long flexible beam in orbit were performed. The effect of varying the initial conditions and the number of modes was demonstrated.

Bainum, P. M.↗

NASA NTRS1982

Calculation of exact vibration modes for plane grillages by the dynamic stiffness method

A dynamic stiffness method is developed for the calculation of the exact modal parameters for plane grillages which consist of straight and uniform beams with coincident elastic and inertial axes. Elementary bending-torsion beam theory is utilized, and bending translation is restricted to one direction. The exact bending-torsion dynamic stiffness matrix is obtained for a straight and uniform beam element with coincident elastic and inertial axes. The element stiffness matrices are assembled using the standard procedure of the static stiffness method to form the dynamic stiffness matrix of the complete grillage. The exact natural frequencies, mode shapes, and generalized masses of the grillage are then calculated by solving a nonlinear eigenvalue problem based on the dynamic stiffness matrix. The exact modal solutions for an example grillage are calculated and compared with the approximate solutions obtained by using the finite element method.

Hallauer, W. L., Jr.↗

NASA NTRS1996

Computation of Static Shapes and Voltages for Micromachined Deformable Mirrors with Nonlinear Electrostatic Actuators

In modeling micromachined deformable mirrors with electrostatic actuators whose gap spacings are of the same order of magnitude as those of the surface deformations, it is necessary to use nonlinear models for the actuators. In this paper, we consider micromachined deformable mirrors modeled by a membrane or plate equation with nonlinear electrostatic actuator characteristics. Numerical methods for computing the mirror deformation due to given actuator voltages and the actuator voltages required for producing the desired deformations at the actuator locations are presented. The application of the proposed methods to circular deformable mirrors whose surfaces are modeled by elastic membranes is discussed in detail. Numerical results are obtained for a typical circular micromachined mirror with electrostatic actuators.

Wang, P. K. C.↗

DOE OSTI2020

Modeling and Detection of Future Cyber-Enabled DSM Data Attacks

Demand-Side Management (DSM) is an essential tool to ensure power system reliability and stability. In future smart grids, certain portions of a customer’s load usage could be under the automatic control of a cyber-enabled DSM program, which selectively schedules loads as a function of electricity prices to improve power balance and grid stability. In this scenario, the security of DSM cyberinfrastructure will be critical as advanced metering infrastructure and communication systems are susceptible to cyber-attacks. Such attacks, in the form of false data injections, can manipulate customer load profiles and cause metering chaos and energy losses in the grid. The feedback mechanism between load management on the consumer side and dynamic price schemes employed by independent system operators can further exacerbate attacks. To study how this feedback mechanism may worsen attacks in future cyber-enabled DSM programs, we propose a novel mathematical framework for (i) modeling the nonlinear relationship between load management and real-time pricing, (ii) simulating residential load data and prices, (iii) creating cyber-attacks, and (iv) detecting said attacks. In this framework, we first develop time-series forecasts to model load demand and use them as inputs to an elasticity model for the price-demand relationship in the DSM loop. This work then investigates the behavior of such a feedback loop under intentional cyber-attacks. We simulate and examine load-price data under different DSM-participation levels with three types of random additive attacks: ramp, sudden, and point attacks. We conduct two investigations for the detection of DSM attacks. The first studies a supervised learning approach, with various classification models, and the second studies the performance of parametric and nonparametric change point detectors. Results conclude that higher amounts of DSM participation can exacerbate ramp and sudden attacks leading to better detection of such attacks, especially with supervised learning classifiers. We also find that nonparametric detection outperforms parametric for smaller user pools, and random point attacks are the hardest to detect with any method.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1988

Strain intensity factor approach for predicting the strength of continuously reinforced metal matrix composites

A method was previously developed to predict the fracture toughness (stress intensity factor at failure) of composites in terms of the elastic constants and the tensile failing strain of the fibers. The method was applied to boron/aluminum composites made with various proportions of 0 to + or - 45 deg plies. Predicted values of fracture toughness were in gross error because widespread yielding of the aluminum matrix made the compliance very nonlinear. An alternate method was developed to predict the strain intensity factor at failure rather than the stress intensity factor because the singular strain field was not affected by yielding as much as the stress field. Strengths of specimens containing crack-like slits were calculated from predicted failing strains using uniaxial stress-strain curves. Predicted strengths were in good agreement with experimental values, even for the very nonlinear laminates that contained only + or - 45 deg plies. This approach should be valid for other metal matrix composites that have continuous fibers.

Poe, C. C., Jr.↗

NASA NTRS1989

Strain intensity factor approach for predicting the strength of continuously reinforced metal matrix composites

A method was previously developed to predict the fracture toughness (stress intensity factor at failure) of composites in terms of the elastic constants and the tensile failing strain of the fibers. The method was applied to boron/aluminum composites made with various proportions of 0 deg and +/- 45 deg plies. Predicted values of fracture toughness were in gross error because widespread yielding of the aluminum matrix made the compliance very nonlinear. An alternate method was develolped to predict the strain intensity factor at failure rather than the stress intensity factor because the singular strain field was not affected by yielding as much as the stress field. Far-field strains at failure were calculated from the strain intensity factor, and then strengths were calculated from the far-field strains using uniaxial stress-strain curves. The predicted strengths were in good agreement with experimental values, even for the very nonlinear laminates that contained only +/- 45 deg plies. This approach should be valid for other metal matrix composites that have continuous fibers.

Poe, Clarence C., Jr.↗

NASA NTRS2001

The Nonlinear Response of Cracked Aluminum Shells Subjected to Combined Loads

The results of a numerical study of the nonlinear response of thin unstiffened aluminum cylindrical shells with a longitudinal crack are presented. The shells are analyzed with a nonlinear shelf analysis code that accurately accounts for global and structural response phenomena. The effects of initial crack length on the prebuckling, buckling and postbuckling responses of a typical shell subjected to axial compression loads, and subjected to combined internal pressure and axial compression loads are described. Both elastic and elastic-plastic analyses are conducted. Numerical results for a fixed initial crack length indicate that the buckling load decreases as the crack length increases for a given pressure load, and that the buckling load increases as the internal pressure load increases for a given crack length. Furthermore, results indicate that predictions from an elastic analysis for the initial buckling load of a cracked shell subjected to combined axial compression and internal pressure loads can be unconservative. In addition, the effect of crack extension on the initial buckling load is presented.

Rose, Cheryl A.↗

NASA NTRS2012

Quasi 1D Modeling of Mixed Compression Supersonic Inlets

The AeroServoElasticity task under the NASA Supersonics Project is developing dynamic models of the propulsion system and the vehicle in order to conduct research for integrated vehicle dynamic performance. As part of this effort, a nonlinear quasi 1-dimensional model of the 2-dimensional bifurcated mixed compression supersonic inlet is being developed. The model utilizes computational fluid dynamics for both the supersonic and subsonic diffusers. The oblique shocks are modeled utilizing compressible flow equations. This model also implements variable geometry required to control the normal shock position. The model is flexible and can also be utilized to simulate other mixed compression supersonic inlet designs. The model was validated both in time and in the frequency domain against the legacy LArge Perturbation INlet code, which has been previously verified using test data. This legacy code written in FORTRAN is quite extensive and complex in terms of the amount of software and number of subroutines. Further, the legacy code is not suitable for closed loop feedback controls design, and the simulation environment is not amenable to systems integration. Therefore, a solution is to develop an innovative, more simplified, mixed compression inlet model with the same steady state and dynamic performance as the legacy code that also can be used for controls design. The new nonlinear dynamic model is implemented in MATLAB Simulink. This environment allows easier development of linear models for controls design for shock positioning. The new model is also well suited for integration with a propulsion system model to study inlet/propulsion system performance, and integration with an aero-servo-elastic system model to study integrated vehicle ride quality, vehicle stability, and efficiency.

Kopasakis, George↗

NASA NTRS1974

Insulation of nonlinear and random vibrations in the mining industry

The insulation of nonlinear and random vibrations is considered for some ore preparing and sorting implements: rotary crushers, resonance screens, hammer mills, etc. The appearance of subharmonic vibrations is analyzed, and the conditions for their appearance are determined. A method is given for calculating the insulation of these vibrations by means of elastic elements made of rubber. The insulation of the random vibrations produced by Symons crushers is calculated by determining the transmissability and deformation of the insulation system for a narrow band random response.

Zeveleanu, C.↗

NASA NTRS1976

Some inconsistencies of the finite element method as applied to inelastic response

The inadequacy of a two noded beam-column element with a linear axial and a cubic transverse displacement field for inelastic analysis is demonstrated. For complete equilibrium satisfaction in the linear elastic range a three noded beam-column element is shown to be consistent. Next, the sensitivity of the inelastic response to numerical solutions of the inelastic response of a cantilever beam resulting from approximate integration of strain energy are brought out and finally, consequences of this on the nonlinear transient response of structures are considered.

Kamat, M. P.↗

NASA NTRS1982

Anharmonic properties of solids from measurements of the stress acoustic constant

The equations of elastic motion and their solutions are generalized in order to include nonzero homogeneous initial stresses and redefine the stress acoustic constants to include the effect of initial stress. In deriving the relationship between the stress acoustic constants and the strain-generalized Gruneisen parameters, implications in material anharmonicity and nonlinear thermoelasticity are discovered. It is found that the linear change in velocity obtained from these measurements is a nonlinear process resulting from anharmonicity in the interatomic potential. In addition, the thermal expansion coefficient can also be expressed in terms of linear combinations.

Cantrell, J. H., Jr.↗

NASA NTRS1989

A review of some problems in global-local stress analysis

The various types of local-global finite-element problems point out the need to develop a new generation of software. First, this new software needs to have a complete analysis capability, encompassing linear and nonlinear analysis of 1-, 2-, and 3-dimensional finite-element models, as well as mixed dimensional models. The software must be capable of treating static and dynamic (vibration and transient response) problems, including the stability effects of initial stress, and the software should be able to treat both elastic and elasto-plastic materials. The software should carry a set of optional diagnostics to assist the program user during model generation in order to help avoid obvious structural modeling errors. In addition, the program software should be well documented so the user has a complete technical reference for each type of element contained in the program library, including information on such topics as the type of numerical integration, use of underintegration, and inclusion of incompatible modes, etc. Some packaged information should also be available to assist the user in building mixed-dimensional models. An important advancement in finite-element software should be in the development of program modularity, so that the user can select from a menu various basic operations in matrix structural analysis.

Nelson, Richard B.↗

NASA NTRS1962

BUCKLING AND POST-BUCKLING OF ELASTIC SHELLS

Some theoretical investigations of buckling of elastic shells are surveyed in this report. Only geometrically perfect shells are considered; initial dents and out-of-roundness are not taken into account. Several questions raised by the studies are: (a) Under what conditions is the infinitesimal theory of buckling of shells adequate? (b) How does the energy theory of buckling of shells correlate with the method based on equilibrium equations for bending moments, tensions, and shears in a buckled configuration? (c) How important are nonlinear terms in the tangential displacements u, v in the strain-displacement relations for buckling and post-buckling studies? (d) How important are the boundary conditions for u, v in affecting stability? (e) If a condition of snap-through is approached, how much external work is required to push the shell "over the hump" into the buckled configuration? (f) How reliable are mathematical approximations used previously in the infinitesimal theory of buckling of shells? Tentative and incomplete answers to some of these questions are suggested.

ELASTIC BUCKLING↗