Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Galerkin method”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
NASA NTRS1976

Large amplitude free vibrations of tapered beams

The Galerkin method is used to investigate the large-amplitude free vibrations of simply supported and clamped tapered beams of rectangular cross-section which are frequently encountered in practical structures. Two types of linear tapers are considered: breadth and depth tapers. Two solutions are obtained for each type of taper, using trigonometric and polynomial displacement distributions. Frequency-amplitude relationships are obtained for all these cases for the fundamental flexural mode. The results indicate good agreement between trigonometric and polynomial solutions for all cases. The nonlinearity is noted to be always of the hardening type and, as expected, it is severe for beams with depth taper as compared with beams with breadth taper.

Raju, L. S.↗
NASA NTRS1976

Stability of a viscous fluid in a rectangular cavity in the presence of a magnetic field

The stability of an electrically conducting fluid subjected to two dimensional disturbance was investigated. A physical system consisting of two parallel infinite vertical plates which are thermally insulated was studied. An external magnetic field of constant strength was applied to normal plates. The fluid was heated from below so that a steady temperature gradient was maintained in the fluid. The governing equations were derived by perturbation technique, and solutions were obtained by a modified Galerkin method. It was found that the presence of the magnetic field increases the stability of the physical system and instability can occur in the form of neutral or oscillatory instability.

Liang, C. Y.↗
NASA NTRS1977

Modal characteristics of crossed rectangular waveguides

An integral-eigenvalue problem is formulated for a crossed rectangular waveguide and solved numerically by applying the Ritz-Galerkin method. Theoretical formulas for determining cutoff frequencies and modal-field expressions are obtained for the specific case of a symmetrical rectangular waveguide, cutoff frequencies are calculated numerically, and the results are verified by comparison with available experimental data. The modal fields are expressed in terms of Fourier series for both TE and TM modes. It is found that the bandwidth can be increased to a maximum of 38% when the waveguide dimensions are properly selected and that the numerical results are in agreement with those computed by the method of partial regions. Some practical applications of the modal-field equations are briefly noted.

Lin, F.-L. C.↗
NASA NTRS1978

An integral solution to a nonlinear diffusion problem

The generalized Galerkin method (or the method of integral relations) is applied to the type of problem described by quasilinear parabolic equations. As an example the problem of nonlinear transient slab diffusion with a general reservoir boundary condition is worked out. The integral relations are given for an arbitrary number of strips, and solutions using up to seven strips have been obtained in order to investigate the convergence of the method.

Hussaini, M. Y.↗
NASA NTRS1977

A finite element analysis for acoustic transmission in nonuniform ducts

A multimodal finite-element analysis for two-dimensional and axisymmetric ducts with flow has been adopted to predict acoustic transmission in nonuniform ducts carrying a nonuniform flow (i.e., turbofan aircraft engine conditions). The analysis is based on a Galerkin method with a finite-element discretization, and multimodal matching at the duct ends. The element mesh and the number of modes required at each end are found to have a strong influence on the analysis. The numerical results compare well with results obtained from other techniques, including the method of weighted residuals.

Astley, R. J.↗
NASA NTRS1979

Stability analysis of a liquid fuel annular combustion chamber

The problems of combustion instability in an annular combustion chamber are investigated. A modified Galerkin method was used to produce a set of modal amplitude equations from the general nonlinear partial differential acoustic wave equation. From these modal amplitude equations, the two variable perturbation method was used to develop a set of approximate equations of a given order of magnitude. These equations were modeled to show the effects of velocity sensitive combustion instabilities by evaluating the effects of certain parameters in the given set of equations. By evaluating these effects, parameters which cause instabilities to occur in the combustion chamber can be ascertained. It is assumed that in the annular combustion chamber, the liquid propellants are injected uniformly across the injector face, the combustion processes are distributed throughout the combustion chamber, and that no time delay occurs in the combustion processes.

Mcdonald, G. H.↗
NASA NTRS1978

Stability analysis of a liquid fuel annular combustion chamber

High frequency combustion instability problems in a liquid fuel annular combustion chamber are examined. A modified Galerkin method was used to produce a set of modal amplitude equations from the general nonlinear partial differential acoustic wave equation in order to analyze the problem of instability. From these modal amplitude equations, the two variable perturbation method was used to develop a set of approximate equations of a given order of magnitude. These equations were modeled to show the effects of velocity sensitive combustion instabilities by evaluating the effects of certain parameters in the given set of equations.

Mcdonald, G. H.↗
NASA NTRS1980

Response of a panel to a supersonic turbulent boundary layer - Studies on a theoretical model

A study of the response of a clamped panel to a supersonic turbulent boundary layer, based on a recently developed theoretical model, is presented. It is demonstrated that the model incorporates the effect of coupling between the panel motion and the flow of the surrounding fluid. Further, a Ritz-Galerkin method is used to obtain approximate solutions for the statistics of the panel response to the turbulence. The numerical results are compared with previous experimental data and the theoretical model is assessed.

Yen, D. H. Y.↗
NASA NTRS1980

Rotor blade aeroelastic stability and response in forward flight

The aeroelastic stability and response problem of the coupled flap-lag-torsional dynamics of a hingeless rotor blade in forward flight is treated in a comprehensive manner. The spatial dependence of the partial differential, nonlinear, equations of motion is discretized using a multimodal Galerkin method. The aeroelastic problem is coupled with the trim state of the helicopter obtained from improved, representative, trim procedures. The nonlinear time dependent equilibrium position, or response, about which the equations are linearized is obtained by solving a sequence of linear periodic response problems, using quasi-linearization. Numerous results illustrating blade behavior in forward flight are presented.

Friedmann, P. P.↗
NASA NTRS1982

Exact triple integrals of beam functions

Definite triple integrals encountered in applying the Galerkin method to the problem of heat and mass transfer across rectangular enclosures are discussed. Rather than evaluating them numerically, the technique described by Reid and Harris (1958) was extended to obtain the exact solution of the integrals. In the process, four linear simultaneous equations with triple integrals as unknowns were obtained. These equations were then solved exactly to obtain the closed form solution. Since closed form representations of this type have been shown to be useful in solving nonlinear hydrodynamic problems by series expansion, the integrals are presented here in general form.

Jhaveri, B. S.↗
NASA NTRS1982

Incompressible Navier-Stokes and parabolized Navier-Stokes solution procedures and computational techniques

Recent developments with finite-difference techniques are emphasized. The quotation marks reflect the fact that any finite discretization procedure can be included in this category. Many so-called finite element collocation and galerkin methods can be reproduced by appropriate forms of the differential equations and discretization formulas. Many of the difficulties encountered in early Navier-Stokes calculations were inherent not only in the choice of the different equations (accuracy), but also in the method of solution or choice of algorithm (convergence and stability, in the manner in which the dependent variables or discretized equations are related (coupling), in the manner that boundary conditions are applied, in the manner that the coordinate mesh is specified (grid generation), and finally, in recognizing that for many high Reynolds number flows not all contributions to the Navier-Stokes equations are necessarily of equal importance (parabolization, preferred direction, pressure interaction, asymptotic and mathematical character). It is these elements that are reviewed. Several Navier-Stokes and parabolized Navier-Stokes formulations are also presented.

Rubin, S. G.↗
NASA NTRS1983

Character and stability of axisymmetric thermal convection in spheres and spherical shells

The influence of shell size and mode of heating on the behavior and stability of axisymmetric, infinite Prandtl number convection in a spherical geometry is studied. Heating from within and below features convection onset governed by a self-adjoint system of equations and boundary conditions. For heating only from within or from below, linearized equations and boundary conditions are non-self-adjoint. Identification of the parameter which initiates the departure from self-adjointness, together with the properties of the self-adjoint solution, provide a basis for calculating the heat transfer characteristics of the non-self-adjoint situations. The investigations are an effort to develop a model for heat transfer in planetary interiors. Further development of the technique by modifying the Galerkin method by the introduction of diagonal mode truncation is suggested to permit the consideration of higher values of the Rayleigh numbers, i.e., those more commensurate with terrestrial planet mantles.

Zebib, A.↗
NASA NTRS1982

Coupled flap-lag-torsional dynamics of hingeless rotor blades in forward flight

The aeroelastic stability and response problem of the coupled flap-lag-torsional dynamics of an isolated hingeless rotor blade in forward flight is considered. Linear, quasi-steady aerodynamics below stall is included. The spatial dependence of the partial differential nonliner equations of motion is discretized using a multimodal Galerkin method. The nonlinear time dependent equilibrium position (steady state response) about which the equations are linearized is obtained by solving a sequence of linear periodic response problems (quasilinearization). Results illustrating blade behavior in forward flight for both soft-in-plane and stiff-in-plane designs are presented. Quasilinearization provides a clear indication of the cases when nonlinear terms due to moderate deflections are important. The results indicate that nonlinearities affect system stability much more than system response. This implies that for obtaining blade vibrations and loads, approximate analytical models based on linearized formulations could be employed, whereas for determining blade stability a more accurate nonlinear analysis would be required.

Friedmann, P. P.↗
NASA NTRS1984

Analysis of rectangular microstrip antennas

The problem of microstrip antennas covered by a dielectric substrate is formulated in terms of coupled integro-differential equations with the current distribution on the conducting patch as an unknown quantity. The Galerkin method is used to solve for the unknown patch current. Using the present formulation, the radiation pattern, the resonant frequency, and the bandwidth of a rectangular microstrip antenna are computed. Design data for a rectangular microstrip antenna are also presented.

Bailey, M. C.↗
NASA NTRS1985

Nonlinear flap-lag-extensional vibrations of rotating, pretwisted, preconed beams including Coriolis effects

The effects of pretwist, precone, setting angle, Coriolis forces and second degree geometric nonlinearities on the natural frequencies, steady state deflections and mode shapes of rotating, torsionally rigid, cantilevered beams were studied. The governing coupled equations of flap lag extensional motion are derived including the effects of large precone and retaining geometric nonlinearities up to second degree. The Galerkin method, with nonrotating normal modes, is used for the solution of both steady state nonlinear equations and linear perturbation equations. Parametric indicating the individual and collective effects of pretwist, precone, Coriolis forces and second degree geometric nonlinearities on the steady state deflection, natural frequencies and mode shapes of rotating blades are presented. It is indicated that the second degree geometric nonlinear terms, which vanish for zero precone, can produce frequency changes of engineering significance. Further confirmation of the validity of including those generated by MSC NASTRAN. It is indicated that the linear and nonlinear Coriolis effects must be included in analyzing thick blades. The Coriolis effects are significant on the first flatwise and the first edgewise modes.

Subrahmanyam, K. B.↗
NASA NTRS1984

Analysis of an infinite array of rectangular microstrip patches with idealized probe feeds

A solution is presented to the problem of an infinite array of microstrip patches fed by idealized current probes. The input reflection coefficient is calculated versus scan angle in an arbitrary scan plane, and the effects of substrate parameters and grid spacing are considered. It is pointed out that even when a Galerkin method is used the impedance matrix is not symmetric due to phasing through a unit cell, as required for scanning. The mechanism by which scan blindness can occur is discussed. Measurement results are presented for the reflection coefficient magnitude variation with angle for E-plane, H-plane, and D-plane scans, for various substrate parameters. Measured results from waveguide simulators are also presented, and the scan blindness phenomenon is observed and discussed in terms of forced surface waves and a modified grating lobe diagram.

Pozar, D. M.↗
NASA NTRS1985

Dispersion characteristics for wide slotlines on low-permittivity substrates

A spectral-domain Galerkin method is used to calculate the dispersion characteristics of wide slotlines on low-permittivity substrates that are electrically thin. The range of parameters covered are d/lambda sub 0 = 0.006-0.06, W/lambda sub 0 = 0.005-2.0, and epsilon sub r = 2.22, 3.0, 3.8, and 9.8. The choice of expansion and testing functions makes possible the extraction of the asymptotic form of the infinite integrals, resulting in a computationally efficient solution. The agreement between calculated and measured slot wavelengths on narrow and wide slots is generally within 2.0 percent over 2-6 GHz.

Janaswamy, R.↗
NASA NTRS1986

Nonlinear bending-torsional vibration and stability of rotating, pretwisted, preconed blades including Coriolis effects

The coupled bending-bending-torsional equations of dynamic motion of rotating, linearly pretwisted blades are derived including large precone, second degree geometric nonlinearities and Coriolis effects. The equations are solved by the Galerkin method and a linear perturbation technique. Accuracy of the present method is verified by comparisons of predicted frequencies and steady state deflections with those from MSC/NASTRAN and from experiments. Parametric results are generated to establish where inclusion of only the second degree geometric nonlinearities is adequate. The nonlinear terms causing torsional divergence in thin blades are identified. The effects of Coriolis terms and several other structurally nonlinear terms are studied, and their relative importance is examined.

Subrahmanyam, K. B.↗