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NASA NTRS1980

A semi-analytic approach to the self induced motion of vortex sheets

The rolling up of the trailing vortex sheet produced by a wing of finite span was calculated as a series expansion in time. For a vorticity distribution corresponding to a wing with cusped tips, the shape of the sheet was found by summing the series using Pade approximants. The sheet remains analytic for some time but ultimately develops an exponential spiral at the tips. The centroid of vorticity was conserved to high accuracy.

Schwartz, L. W.↗

NASA NTRS1980

On the pressure field of nonlinear standing water waves

The pressure field produced by two dimensional nonlinear time and space periodic standing waves was calculated as a series expansion in the wave height. The high order series was summed by the use of Pade approximants. Calculations included the pressure variation at great depth, which was considered to be a likely cause of microseismic activity, and the pressure distribution on a vertical barrier or breakwater.

Schwartz, L. W.↗

NASA NTRS1981

A semi-analytic approach to the self-induced motion of vortex sheets

The rolling-up of the trailing vortex sheet produced by a wing of finite span is calculated as a series expansion in time. For a vorticity distribution corresponding to a wing with cusped tips, the shape of the sheet is found by summing the series using Pade approximants. The sheet remains analytic for some time but ultimately develops an exponential spiral at the tips. The centroid of vorticity is conserved to high accuracy.

Schwartz, L. W.↗

NASA NTRS1981

Vibration-rotation intensities of SiO

Dipole moment matrix elements have been computed for a large number of transitions of astrophysical interest for the more abundant isotopes of SiO. The wave functions utilized were obtained from a direct solution of the Schroedinger equation with an accurate RKR potential. The dipole moment function, in the form of a Pade approximant, was chosen to reproduce the experimental measurements near equilibrium, to have the proper united and separated atom limits, and to have the correct long-range asymptotic dependence on internuclear separation. Because of the large number of transitions involved, and to facilitate applications, the squares of the dipole moment matrix elements were fitted by a least-squares procedure to polynomials in v and J. In addition, Einstein A coefficients are given for observed maser transitions and for selected vibrational bands. These latter are compared with previous calculations, and it is concluded that for the higher Delta v transitions, the present results represent a significant improvement.

Tipping, R. H.↗

NASA NTRS1983

Calculation of wing response to gusts and blast waves with vortex lift effect

A numerical study of the response of aircraft wings to atmospheric gusts and to nuclear explosions when flying at subsonic speeds is presented. The method is based upon unsteady quasi-vortex lattice method, unsteady suction analogy and Pade approximant. The calculated results, showing vortex lag effect, yield reasonable agreement with experimental data for incremental lift on wings in gust penetration and due to nuclear blast waves.

Chao, D. C.↗

NASA NTRS1983

Perturbation analysis of the limit cycle of the free van der Pol equation

A power series expansion in the damping parameter, epsilon, of the limit cycle of the free van der Pol equation is constructed and analyzed. Coefficients in the expansion are computed in exact rational arithmetic using the symbolic manipulation system MACSYMA and using a FORTRAN program. The series is analyzed using Pade approximants. The convergence of the series for the maximum amplitude of the limit cycle is limited by two pair of complex conjugate singularities in the complex epsilon-plane. A new expansion parameter is introduced which maps these singularities to infinity and leads to a new expansion for the amplitude which converges for all real values of epsilon. Amplitudes computed from this transformed series agree very well with reported numerical and asymptotic results. For the limit cycle itself, convergence of the series expansion is limited by three pair of complex conjugate branch point singularities. Two pair remain fixed throughout the cycle, and correspond to the singularities found in the maximum amplitude series, while the third pair moves in the epsilon-plane as a function of t from one of the fixed pairs to the other. The limit cycle series is transformed using a new expansion parameter, which leads to a new series that converges for larger values of epsilon.

Dadfar, M. B.↗

NASA NTRS1983

Calculation of wing response to gusts and blast waves with vortex lift effect

A numerical study of the response of aircraft wings to atmospheric gusts and to nuclear explosions when flying at subsonic speeds is presented. The method is based upon unsteady quasi-vortex-lattice method, unsteady suction analogy, and Pade approximate. The calculated results, showing vortex lag effect, yield reasonable agreement with experimental data for incremental lift on wings in gust penetration and due to nuclear blast waves.

Chao, D. C.↗

NASA NTRS1983

Flutter and forced response of mistuned rotors using standing wave analysis

A standing wave approach is applied to the analysis of the flutter and forced response of tuned and mistuned rotors. The traditional traveling wave cascade airforces are recast into standing wave arbitrary motion form using Pade approximants, and the resulting equations of motion are written in the matrix form. Applications for vibration modes, flutter, and forced response are discussed. It is noted that the standing wave methods may prove to be more versatile for dealing with certain applications, such as coupling flutter with forced response and dynamic shaft problems, transient impulses on the rotor, low-order engine excitation, bearing motions, and mistuning effects in rotors.

Dugundji, J.↗

NASA NTRS1983

Flutter and forced response of mistuned rotors using standing wave analysis

Bundas, D. J.↗

NASA NTRS1985

Viscous effects on transonic airfoil stability and response

Viscous effects on transonic airfoil stability and response are investigated using an integral boundary layer model coupled to the inviscid XTRAN2L transonic small disturbance code. Unsteady transonic airloads required for stability analyses are computed using a pulse transfer function analysis including viscous effects. The pulse analysis provides unsteady aerodynamic forces for a wide range of reduced frequency in a single flow field computation. Nonlinear time marching aeroelastic solutions are presented which show the effects of viscosity on airfoil response behavior and flutter. Effects of amplitude on time marching responses are demonstrated. A state space aeroelastic model employing Pade approximants to describe the unsteady airloads is used to study the effects of viscosity on transonic airfoil stability. State space dynamic pressure root loci are in good overall agreement with time marching damping and frequency estimates. Parallel sets of results with and without viscous effects reveal the effects of viscosity on transonic unsteady airloads and aeroelastic characteristics of airfoils.

Berry, H. M.↗

NASA NTRS1985

Viscous effects on transonic airfoil stability and response

An integral boundary layer model coupled to the inviscid XTRAN2L transonic small disturbance code is used to investigate viscous effects on transonic airfoil stability and response. A pulse transfer function analysis is employed for the computation of unsteady transonic airloads which yields forces for a wide range of reduced frequencies in a single flowfield computation. Nonlinear, time-marching aeroelastic solutions are obtained which exhibit the effects of viscosity on airfoil response behavior and flutter. A state-space aeroelastic model using Pade approximants to describe unsteady airloads is used to study viscosity effects on transonic airfoil stability. The state-space dynamic pressure root loci are in good general agreement with time marching damping and frequency estimates.

Berry, H. M.↗

NASA NTRS1985

Finite-time arbitrary-motion unsteady cascade airfoil theory for helicopter rotors in hover

A complete and detailed derivation of finite-time arbitrary-motion cascade theory is presented for both Laplace and frequency domains. This theory includes the effect of returning wakes for both single- and multibladed rotors. The generalized cascade lift-deficiency function (CLDF) is shown to be consistent with the generalized Theodorsen lift-deficiency function when the wake spacing approaches infinity or when the reduced frequency tends to infinity. This function predicts correct zero-reduced-frequency limit. Accurate and efficient numerical procedures are presented for the evaluation of the CLDF. Numerical examples comparing the CLDF with Loewy's lift deficiency function in frequency domain are presented. Accurate Pade approximants of the CLDF are constructed using a Bode-plot approach which allows for complex poles.

Dinyavari, M. A. H.↗

NASA NTRS1988

The PAWS and STEM reliability analysis programs

The PAWS and STEM programs are new design/validation tools. These programs provide a flexible, user-friendly, language-based interface for the input of Markov models describing the behavior of fault-tolerant computer systems. These programs produce exact solutions of the probability of system failure and provide a conservative estimate of the number of significant digits in the solution. PAWS uses a Pade approximation as a solution technique; STEM uses a Taylor series as a solution technique. Both programs have the capability to solve numerically stiff models. PAWS and STEM possess complementary properties with regard to their input space; and, an additional strength of these programs is that they accept input compatible with the SURE program. If used in conjunction with SURE, PAWS and STEM provide a powerful suite of programs to analyze the reliability of fault-tolerant computer systems.

Butler, Ricky W.↗

NASA NTRS1989

Time domain unsteady incompressible cascade airfoil theory for helicopter rotors in hover

A detailed derivation of a finite-time arbitrary-motion incompressible cascade theory is presented for both Laplace and frequency domains. The generalized cascade lift deficiency function is shown to be consistent with the generalized Theodorsen's lift deficiency function when the wake spacing approaches infinity or when the reduced frequency tends to infinity; it also yields a correct value for the zero reduced frequency limit. Efficient numerical procedures are presented for the evaluation of the cascade lift deficiency function. Numerical examples comparing the cascade lift deficiency function with Loewy's (1957) rotary-wing lift deficiency function in frequency domain are presented. Pade approximants of the cascade lift deficiency function are constructed using a Bode plot approach that allows for complex poles. A general-purpose optimization program is used to determine the coefficients of the approximant.

Dinyavari, M. A. H.↗

NASA NTRS1991

Programs For Modeling Fault-Tolerant Computing Systems

Pade Approximation with Scaling, (PAWS) and Scaling Taylor Exponential Matrix (STEM) computer programs are software tools for design and validation. Provide flexible, user-friendly, language-based interface for input of Markov mathematical methods describing behaviors of fault-tolerant computer systems. Markov models include both recovery from faults via reconfiguration and behaviors of such systems when faults occur. PAWS and STEM produce exact solutions of probability of system failure and provide conservative estimate of number of significant digits in solution. Written in PASCAL and FORTRAN.

Butler, Ricky W.↗

NASA NTRS1992

Renormalization group analysis of the Reynolds stress transport equation

The pressure velocity correlation and return to isotropy term in the Reynolds stress transport equation are analyzed using the Yakhot-Orszag renormalization group. The perturbation series for the relevant correlations, evaluated to lowest order in the epsilon-expansion of the Yakhot-Orszag theory, are infinite series in tensor product powers of the mean velocity gradient and its transpose. Formal lowest order Pade approximations to the sums of these series produce a fast pressure strain model of the form proposed by Launder, Reece, and Rodi, and a return to isotropy model of the form proposed by Rotta. In both cases, the model constant are computed theoretically. The predicted Reynolds stress ratios in simple shear flows are evaluated and compared with experimental data. The possibility is discussed of driving higher order nonlinear models by approximating the sums more accurately.

Rubinstein, R.↗

NASA NTRS1992

Order reduction of z-transfer functions via multipoint Jordan continued-fraction expansion

The order reduction problem of z-transfer functions is solved by using the multipoint Jordan continued-fraction expansion (MJCFE) technique. An efficient algorithm that does not require the use of complex algebra is presented for obtaining an MJCFE from a stable z-transfer function with expansion points selected from the unit circle and/or the positive real axis of the z-plane. The reduced-order models are exactly the multipoint Pade approximants of the original system and, therefore, they match the (weighted) time-moments of the impulse response and preserve the frequency responses of the system at some characteristic frequencies, such as gain crossover frequency, phase crossover frequency, bandwidth, etc.

Lee, Ying-Chin↗

NASA NTRS1992

Vision-based stereo ranging as an optimal control problem

The recent interest in the use of machine vision for flight vehicle guidance is motivated by the need to automate the nap-of-the-earth flight regime of helicopters. Vision-based stereo ranging problem is cast as an optimal control problem in this paper. A quadratic performance index consisting of the integral of the error between observed image irradiances and those predicted by a Pade approximation of the correspondence hypothesis is then used to define an optimization problem. The necessary conditions for optimality yield a set of linear two-point boundary-value problems. These two-point boundary-value problems are solved in feedback form using a version of the backward sweep method. Application of the ranging algorithm is illustrated using a laboratory image pair.

Menon, P. K. A.↗