Formulas for nth order derivatives of hyperbolic and trigonometric functions
Formulas for nth order derivatives of hyperbolic and trigonometric functions used in evaluating Fourier sine and cosine integrals
Wintucky, E. G.↗
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Indefinite integrals of products of some exponential and trigonometric functions
Two integrals have been developed which are needed to solve certain problems of mathematical physics related to the Bessel and Euler equations. Integrals of this type have known application in problems using cylindrical coordinates.
Rostafinski, W. A.↗
Automated Testcase Generation for Numerical Support Functions in Embedded Systems
We present a tool for the automatic generation of test stimuli for small numerical support functions, e.g., code for trigonometric functions, quaternions, filters, or table lookup. Our tool is based on KLEE to produce a set of test stimuli for full path coverage. We use a method of iterative deepening over abstractions to deal with floating-point values. During actual testing the stimuli exercise the code against a reference implementation. We illustrate our approach with results of experiments with low-level trigonometric functions, interpolation routines, and mathematical support functions from an open source UAS autopilot.
Metrics↗
A table of integrals of the error function. II - Additions and corrections.
Integrals of products of error functions with other functions are presented, taking into account a combination of the error function with powers, a combination of the error function with exponentials and powers, a combination of the error function with exponentials of more complicated arguments, definite integrals from Laplace transforms, and a combination of the error function with trigonometric functions. Other integrals considered include a combination of the error function with logarithms and powers, a combination of two error functions, and a combination of the error function with other special functions.
Geller, M.↗
A method of weighted residuals with trigonometric basis functions for sound transmission in circular ducts
The method of weighted residuals (MWR) in the form of a modified Galerkin method with trigonometric basis functions is used to compute the transmission of sound in an axisymmetric duct. The method is used to generate the axial wave number for uniform ducts. These are compared with exact solutions generated by a formal eigenvalue routine in the hard-wall case and a Runge-Kutta integration eigenvalue scheme in the soft-wall case. The method is applicable to both flow and no-flow cases.
Vo, P. T.↗
Newtonian analogues of the trigonometric and exponential functions.
Newtonian analogues of trigonometric and exponential functions studied by series
NEWTON THEORY↗
An analytic solution for the orbital perturbations of the Venus Radar Mapper due to gravitational harmonics
Hill's variational equations are solved analytically for the orbital perturbations of a spacecraft nominally in an elliptic orbit around a non-spherical body. The rotation of the central planet about its spin-axis is not considered in the analysis. The perturbations are restricted to the planetary gravitational harmonics only. An extremely simple algorithm is derived to transform the spherical harmonic potentials to the orbital coordinate system, and the resulting accelerations are shown to be simply trigonometric functions of the true anomaly. With the principal matrix solution for the differential equations of the adjoint system given in closed form, the orthogonality of the trigonometric functions makes it possible to obtain an analytic solution for the non-homogeneous problem, at intervals of 2 pi in true anomaly. The solution for orbital perturbations can be extended over several revolutions by applying well-known results from Floquet's theory. The technique is demonstrated with results presented on the spacecraft periapsis altitude for the forthcoming Venus Radar Mapper Mission.
Vijayaraghavan, A.↗
Sunspot analysis and prediction
An attempt is made to develop an accurate functional representation, using common trigonometric functions, of all existing sunspot data, both quantitative and qualitative, ancient and modern. It is concluded that the three periods of high sunspot activity (1935 to 1970, 1835 to 1870, and 1755 to 1790) are independent populations. It is also concluded that these populations have long periods of approximately 400, 500, and 610 years, respectively. The difficulties in assuming a periodicity of seven 11-year cycles of approximately 80 years are discussed.
Steyer, C. C.↗
Adaptation of a Control Center Development Environment for Industrial Process Control
In the control center, raw telemetry data is received for storage, display, and analysis. This raw data must be combined and manipulated in various ways by mathematical computations to facilitate analysis, provide diversified fault detection mechanisms, and enhance display readability. A development tool called the Graphical Computation Builder (GCB) has been implemented which provides flight controllers with the capability to implement computations for use in the control center. The GCB provides a language that contains both general programming constructs and language elements specifically tailored for the control center environment. The GCB concept allows staff who are not skilled in computer programming to author and maintain computer programs. The GCB user is isolated from the details of external subsystem interfaces and has access to high-level functions such as matrix operators, trigonometric functions, and unit conversion macros. The GCB provides a high level of feedback during computation development that improves upon the often cryptic errors produced by computer language compilers. An equivalent need can be identified in the industrial data acquisition and process control domain: that of an integrated graphical development tool tailored to the application to hide the operating system, computer language, and data acquisition interface details. The GCB features a modular design which makes it suitable for technology transfer without significant rework. Control center-specific language elements can be replaced by elements specific to industrial process control.
Killough, Ronnie L.↗
Expressions Module for the Satellite Orbit Analysis Program
The Expressions Module is a software module that has been incorporated into the Satellite Orbit Analysis Program (SOAP). The module includes an expressions- parser submodule built on top of an analytical system, enabling the user to define logical and numerical variables and constants. The variables can capture output from SOAP orbital-prediction and geometric-engine computations. The module can combine variables and constants with built-in logical operators (such as Boolean AND, OR, and NOT), relational operators (such as >, <, or =), and mathematical operators (such as addition, subtraction, multiplication, division, modulus, exponentiation, differentiation, and integration). Parentheses can be used to specify precedence of operations. The module contains a library of mathematical functions and operations, including logarithms, trigonometric functions, Bessel functions, minimum/ maximum operations, and floating- point-to-integer conversions. The module supports combinations of time, distance, and angular units and has a dimensional- analysis component that checks for correct usage of units. A parser based on the Flex language and the Bison program looks for and indicates errors in syntax. SOAP expressions can be built using other expressions as arguments, thus enabling the user to build analytical trees. A graphical user interface facilitates use.
Edmonds, Karina↗
Research study to develop methods for solving problems associated with space vehicles Monthly progress report /Final/
Trigonometric function algorithms for algebraic simplification in computer programs
COMPUTER PROGRAM↗
A universal formulation for conic trajectories. Basic variables and relationships
Truncated trigonometric functions for conic trajectory formulation in space flight applications
TRUNCATION↗
Sunspot analysis and prediction
Evaluation of several procedures that apply rarely used and exotic mathematical functions yields a unique method of application of common trigonometric functions. These functions appear to produce results in the development of a mathematical model capable of describing all available sunspot data.
Source record↗
Application of a trigonometric finite difference procedure to numerical analysis of compressive and shear buckling of orthotropic panels
A numerical analysis developed for the buckling of rectangular orthotropic layered panels under combined shear and compression is described. This analysis uses a central finite difference procedure based on trigonometric functions instead of using the conventional finite differences which are based on polynomial functions. Inasmuch as the buckle mode shape is usually trigonometric in nature, the analysis using trigonometric finite differences can be made to exhibit a much faster convergence rate than that using conventional differences. Also, the trigonometric finite difference procedure leads to difference equations having the same form as conventional finite differences; thereby allowing available conventional finite difference formulations to be converted readily to trigonometric form. For two-dimensional problems, the procedure introduces two numerical parameters into the analysis. Engineering approaches for the selection of these parameters are presented and the analysis procedure is demonstrated by application to several isotropic and orthotropic panel buckling problems. Among these problems is the shear buckling of stiffened isotropic and filamentary composite panels in which the stiffener is broken. Results indicate that a break may degrade the effect of the stiffener to the extent that the panel will not carry much more load than if the stiffener were absent.
Stein, M.↗
Kepler Equation solver
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.
Markley, F. Landis↗
The Incorporation of Truncated Fourier Series into Finite Difference Approximations of Structural Stability Equations
A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half-wavelength of the buckled mode shape. It was found from a modal analysis that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions.
Hannah, S. R.↗
Spacecraft Attitude Representations
The direction cosine matrix or attitude matrix is the most fundamental representation of the attitude, but it is very inefficient: It has six redundant parameters, it is difficult to enforce the six (orthogonality) constraints. the four-component quaternion representation is very convenient: it has only one redundant parameter, it is easy to enforce the normalization constraint, the attitude matrix is a homogeneous quadratic function of q, quaternion kinematics are bilinear in q and m. Euler angles are extensively used: they often have a physical interpretation, they provide a natural description of some spacecraft motions (COBE, MAP), but kinematics and attitude matrix involve trigonometric functions, "gimbal lock" for certain values of the angles. Other minimum (three-parameter) representations: Gibbs vector is infinite for 180 deg rotations, but useful for analysis, Modified Rodrigues Parameters are nonsingular, no trig functions, Rotation vector phi is nonsingular, but requires trig functions.
Markley, F. Landis↗
Reactant conversion in homogeneous turbulence: Mathematical modeling, computational validations and practical applications
Closed form analytical expressions are obtained for predicting the limited rate of reactant conversion in a binary reaction of the type F + rO yields (1 + r) Product in unpremixed homogeneous turbulence. These relations are obtained by means of a single point Probability Density Function (PDF) method based on the Amplitude Mapping Closure. It is demonstrated that with this model, the maximum rate of the reactants' decay can be conveniently expressed in terms of definite integrals of the Parabolic Cylinder Functions. For the cases with complete initial segregation, it is shown that the results agree very closely with those predicted by employing a Beta density of the first kind for an appropriately defined Shvab-Zeldovich scalar variable. With this assumption, the final results can also be expressed in terms of closed form analytical expressions which are based on the Incomplete Beta Functions. With both models, the dependence of the results on the stoichiometric coefficient and the equivalence ratio can be expressed in an explicit manner. For a stoichiometric mixture, the analytical results simplify significantly. In the mapping closure, these results are expressed in terms of simple trigonometric functions. For the Beta density model, they are in the form of Gamma Functions. In all the cases considered, the results are shown to agree well with data generated by Direct Numerical Simulations (DNS). Due to the simplicity of these expressions and because of nice mathematical features of the Parabolic Cylinder and the Incomplete Beta Functions, these models are recommended for estimating the limiting rate of reactant conversion in homogeneous reacting flows. These results also provide useful insights in assessing the extent of validity of turbulence closures in the modeling of unpremixed reacting flows. Some discussions are provided on the extension of the model for treating more complicated reacting systems including realistic kinetics schemes and multi-scalar mixing with finite rate chemical reactions in more complex configurations.
Madnia, C. K.↗