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Results for “TRIGONOMETRIC FUNCTION”

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.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS2000

Closed-form integrator for the quaternion (euler angle) kinematics equations

The invention is embodied in a method of integrating kinematics equations for updating a set of vehicle attitude angles of a vehicle using 3-dimensional angular velocities of the vehicle, which includes computing an integrating factor matrix from quantities corresponding to the 3-dimensional angular velocities, computing a total integrated angular rate from the quantities corresponding to a 3-dimensional angular velocities, computing a state transition matrix as a sum of (a) a first complementary function of the total integrated angular rate and (b) the integrating factor matrix multiplied by a second complementary function of the total integrated angular rate, and updating the set of vehicle attitude angles using the state transition matrix. Preferably, the method further includes computing a quanternion vector from the quantities corresponding to the 3-dimensional angular velocities, in which case the updating of the set of vehicle attitude angles using the state transition matrix is carried out by (a) updating the quanternion vector by multiplying the quanternion vector by the state transition matrix to produce an updated quanternion vector and (b) computing an updated set of vehicle attitude angles from the updated quanternion vector. The first and second trigonometric functions are complementary, such as a sine and a cosine. The quantities corresponding to the 3-dimensional angular velocities include respective averages of the 3-dimensional angular velocities over plural time frames. The updating of the quanternion vector preserves the norm of the vector, whereby the updated set of vehicle attitude angles are virtually error-free.

Whitmore, Stephen A.↗
NASA NTRS1975

A note on an attempt at more efficient Poisson series evaluation

A substantial reduction has been achieved in the time necessary to compute lunar libration series. The method involves eliminating many of the trigonometric function calls by a suitable transformation and applying a short SNOBOL processor to the FORTRAN coding of the transformed series, which obviates many of the multiplication operations during the course of series evaluation. It is possible to accomplish similar results quite easily with other Poisson series.

Shelus, P. J.↗
NASA NTRS1982

Continuously-Variable Positive-Mesh Power Transmission

Proposed transmission with continuously-variable speed ratio couples two mechanical trigonometric-function generators. Transmission is expected to handle higher loads than conventional variable-pulley drives; and, unlike variable pulley, positive traction through entire drive train with no reliance on friction to transmit power. Able to vary speed continuously through zero and into reverse. Possible applications in instrumentation where drive-train slippage cannot be tolerated.

Johnson, J. L.↗
NASA NTRS1982

Postbuckling of orthotropic composite plates loaded in compression

The nonlinear large deflection equations of von Karman are written for 'specially' orthotropic plates. The equations are then manipulated to determine the parameters required to establish postbuckling behavior. It is found that only two new parameters are needed beyond those required for buckling. By assuming trigonometric functions in one direction, the plate equations are converted into ordinary nonlinear differential equations which are solved numerically using a two point boundary problem solver that makes use of Newton's method. The postbuckling behavior is obtained for simply supported and clamped, long, rectangular, orthotropic plates covering the complete range of dimensions and material properties.

Stein, M.↗
NASA NTRS1982

An analytic approach to two-fixed-impulse transfers between Keplerian orbits

Solutions are obtained for the two-impulse transfer of a vehicle between arbitrary inclined orbits in an inverse-square force field with the restriction that the magnitude of each of the two impulses has a fixed preassigned value. The two magnitudes need not be equal. The equations for the conservation of angular momentum and energy are augmented by the Laplace integral; these equations establish linear relationships between several of the variables. This set of linear equations and one of two quadratic equations constitute the analytically tractable part of the solution. The remaining part, consisting of finding the zeros (if they exist) of a single trigonometric function of one variable, is solved using numerical methods. Explicit lower bounds on each of the magnitudes of the two impulses are obtained by requiring the solution be real. Graphical results are presented to illustrate the solution.

Jezewski, D. J.↗
NASA NTRS1983

Postbuckling of long orthotropic plates in combined shear and compression

The nonlinear large-deflection partial differential equations of von karman for orthotropic plates loaded in combined shear and compression are converted into a set of first-order nonlinear ordinary differential equations by assuming trigonometric functions in one direction. These equations are solved numerically using a two point boundary problem solver which makes use of Newton's method. Results are obtained which determine the postbuckling behavior of rectangular plates with loading up to about three times the buckling load. Both isotropic and orthotropic composite plates are considered. Results show that orthotropic plates may behave quite differently than isotropic plates and that in-plane boundary conditions are important for plates loaded in shear.

Stein, M.↗
NASA NTRS1984

Analytical results for postbuckling behavior of plates in compression and shear

The postbuckling behavior of long rectangular isotropic and orthotropic plates is determined. By assuming trigonometric functions in one direction, the nonlinear partial differential equations of von Karman large deflection plate theory are converted into nonlinear ordinary differential equations. The ordinary differential equations are solved numerically using an available boundary value problem solver which makes use of Newton's method. Results for longitudinal compression show different postbuckling behavior between isotropic and orthotropic plates. Results for shear show that change in inplane edge constraints can cause large change in postbuckling stiffness.

Stein, M.↗
NASA NTRS1985

SPAR improved structural-fluid dynamic analysis capability

The results of a study whose objective was to improve the operation of the SPAR computer code by improving efficiency, user features, and documentation is presented. Additional capability was added to the SPAR arithmetic utility system, including trigonometric functions, numerical integration, interpolation, and matrix combinations. Improvements were made in the EIG processor. A processor was created to compute and store principal stresses in table-format data sets. An additional capability was developed and incorporated into the plot processor which permits plotting directly from table-format data sets. Documentation of all these features is provided in the form of updates to the SPAR users manual.

Pearson, M. L.↗
NASA NTRS1986

An approximate buckling analysis for rectangular orthotropic plates with centrally located cutouts

An approximate analysis for predicting buckling of rectangular orthotropic composite plates with centrally located cutouts is presented. In this analysis, prebuckling and buckling problems are converted from a two-dimensional to a one-dimensional system of linear differential equations with variable coefficients. The conversion is accomplished by expressing the displacements as series with each element containing a trigonometric function of one coordinate and a coefficient that is an arbitrary function of the other coordinate. Ordinary differential equations are then obtained from a variational principle. Analytical results obtained from the approximate analysis are compared with finite element analyses for isotropic plates and for specially orthotropic plates with central circular cutouts of various sizes. Experimental results for the specially orthotropic plates are also presented. In nearly all cases, the approximate analysis predicts the buckling mode shapes correctly and predicts the buckling loads to within a few percent of the finite element and experimental results.

Nemeth, M. P.↗
NASA NTRS1985

Analytical results for post-buckling behaviour of plates in compression and in shear

The postbuckling behavior of long rectangular isotropic and orthotropic plates is determined. By assuming trigonometric functions in one direction, the nonlinear partial differential equations of von Karman large deflection plate theory are converted into nonlinear ordinary differential equations. The ordinary differential equations are solved numerically using an available boundary value problem solver which makes use of Newton's method. Results for longitudinal compression show different postbuckling behavior between isotropic and orthotropic plates. Results for shear show that change in inplane edge constraints can cause large change in postbuckling stiffness.

Stein, M.↗
NASA NTRS1987

Quaternions as astrometric plate constants

A new method for solving problems in relative astrometry is proposed. In it, the relationship between the measured quantities and the components of the position vector of a star is modeled using quaternions, in effect replacing the plate constants of a standard four-plate-constant solution with the four components of a quaternion. The method allows a direct solution for the position vectors of the stars, and hence for the equatorial coordinates. Distortions, magnitude, and color effects are readily incorporated into the formalism, and the method is directly applicable to overlapping-plate problems. The advantages of the method include the simplicity of the resulting equations, their freedom from singularities, and the fact that trigonometric functions and tangential point transformations are not needed to model the plate material. A global solution over the entire sky is possible.

Jefferys, William H.↗
NASA NTRS1987

Stability of viscous flow past a circular cylinder

A spectral method which employs trigonometric functions and Chebyshev polynomials is used to compute the steady, incompressible laminar flow past a circular cylinder. Linear stability methods are used to formulate a pair of decoupled generalized eigenvalue problems for the growth of symmetric and asymmetric (about the dividing streamline) perturbations. It is shown that, while the symmetric disturbances are stable, the asymmetric perturbations become unstable at a Reynolds number about 40 with a Strouhal number about 0.12. The critical conditions are found to depend on the size of the computational domain in a manner similar to that observed in the laboratory.

Zebib, A.↗
NASA NTRS1988

Free vibration of rectangular plates with a small initial curvature

The method of matched asymptotic expansions is used to solve the transverse free vibration of a slightly curved, thin rectangular plate. Analytical results for natural frequencies and mode shapes are presented in the limit when the dimensionless bending rigidity, epsilon, is small compared with in-plane forces. Results for different boundary conditions are obtained when the initial deflection is: (1) a polynomial in both directions, and (2) the product of a polynomial and a trigonometric function, and arbitrary. For the arbitrary initial deflection case, the Fourier series technique is used to define the initial deflection. The results obtained show that the natural frequencies of vibration of slightly curved plates are coincident with those of perfectly flat, prestressed rectangular plates. However, the eigenmodes are very different from those of initially flat prestressed rectangular plates. The total deflection is found to be the sum of the initial deflection, the deflection resulting from the solution of the flat plate problem, and the deflection resulting from the static problem.

Adeniji-Fashola, A. A.↗
NASA NTRS1988

Stress and vibraton analyses of anisotropic shells of revolution

An efficient computational strategy is presented for reducing the cost of the stress and free vibration analyses of laminated anisotropic shells of revolution. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight strain components, the eight stress resultants and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions (Fourier series) in the circumferential co-ordinate, and a three-field mixed finite element model is used for the discretization in the meridional direction. The shell response associated with a range of Fourier harmonics is approximated by a linear combination of a few global approximation vectors, which are generated at a particular value of the Fourier harmonic, within that range. The full equations of the finite element model are solved for only a single Fourier harmonic, and the response corresponding to the other Fourier harmonics is generated using a reduced system of equations with considerably fewer degrees of freedom.

Noor, Ahmed K.↗
NASA NTRS1991

Elastic stress analysis of three-dimensional solids with small holes by BEM

An advanced level development of the boundary element method is presented for the elastic stress analysis of a three-dimensional solid containing a large number of small diameter, tubular shaped holes. The formulation has been developed such that these holes can be modeled by a system of curvilinear line elements, resulting in substantial savings in both data preparation and computing costs. This is accomplished by assuming a variation in the displacement field along the circumference defined in terms of trigonometric functions together with a linear or quadratic variation of displacements along the longitudinal direction.

Henry, D. P., Jr.↗
NASA NTRS1991

Ordered fast Fourier transforms on a massively parallel hypercube multiprocessor

The present evaluation of alternative, massively parallel hypercube processor-applicable designs for ordered radix-2 decimation-in-frequency FFT algorithms gives attention to the reduction of computation time-dominating communication. A combination of the order and computational phases of the FFT is accordingly employed, in conjunction with sequence-to-processor maps which reduce communication. Two orderings, 'standard' and 'cyclic', in which the order of the transform is the same as that of the input sequence, can be implemented with ease on the Connection Machine (where orderings are determined by geometries and priorities. A parallel method for trigonometric coefficient computation is presented which does not employ trigonometric functions or interprocessor communication.

Tong, Charles↗