Search NASA⌕ Search

Engineering topics

Mark, W. D.

Publications and source records attributed to Mark, W. D..

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Probability density and exceedance rate functions of locally Gaussian turbulence

A locally Gaussian model of turbulence velocities is postulated which consists of the superposition of a slowly varying strictly Gaussian component representing slow temporal changes in the mean wind speed and a more rapidly varying locally Gaussian turbulence component possessing a temporally fluctuating local variance. Series expansions of the probability density and exceedance rate functions of the turbulence velocity model, based on Taylor's series, are derived. Comparisons of the resulting two-term approximations with measured probability density and exceedance rate functions of atmospheric turbulence velocity records show encouraging agreement, thereby confirming the consistency of the measured records with the locally Gaussian model. Explicit formulas are derived for computing all required expansion coefficients from measured turbulence records.

Mark, W. D.↗
NASA NTRS1988

Effects of bearing offset and flexibility on the mesh force distribution of spiral bevel gears

For straight or spiral bevel gears of nominal spherical involute design, the resultant total force vector transmitted by the gear mesh, in the absence of friction, lies in the plane of tooth contact. This force vector can be characterized by three scalar components, two orthogonal force components lying in the plane of contact and the resultant moment taken about the nominal center of the zone of contact. Equations for these three generalized force components are derived. The equations are expressed in terms of tooth pair/gear body stiffnesses, bearing/bearing support flexibility influence coefficients, the shaft input torque, deviations of the tooth running surfaces from perfect spherical involute surfaces, and bearing centerline offsets from the positions occupied by the base cone axes of the perfect involute bevel gear counterparts to the actual gears under consideration. Inertial forces arising from transverse and axial vibrations of the gear bodies are assumed to be negligible in comparison with the bearing support reaction forces.

Mark, W. D.↗
NASA NTRS1988

An extremum principle for computation of the zone of tooth contact and generalized transmission error of spiral bevel gears

For a given set of forces transmitted by the gears, each of the three components of the generalized transmission error of spiral bevel gears is shown to be stationary with respect to small independent variations in the positions of the endpoints of the lines of tooth contact about their true values. The tangential generalized transmission error component is shown to take on a minimum value at the true endpoint positions. A computational procedure based on the method of steepest descent is described for computing the true line of contact endpoint positions and the three components of the generalized transmission error. A method for computing the Fourier series coefficients of the tooth meshing harmonics of the three generalized transmission error components also is provided.

Mark, W. D.↗
NASA NTRS1987

Effects of intensity modulations on the power spectra of random processes

Intensity-modulated random processes (IMRPs), defined as the products of (1) deterministic modulating functions or processes and (2) stationary modulated processes statistically independent of (1), are investigated analytically. Instantaneous power spectra are derived for IMRPs with different classes of (1), and a spectrum series expansion with a known locally stationary approximation as its first term is obtained. Numerical results for sample IMRPs with known (deterministic), stationary, nonstationary, ergodic, onset, and bell-shaped types of (1) are presented graphically and briefly characterized.

Mark, W. D.↗
NASA NTRS1987

The generalized transmission error of spiral bevel gears

The traditional definition of the transmission error of parallel-axis gear pairs is reviewed and shown to be unsuitable for characterizing the deviation from conjugate action of bevel gear pairs for vibration excitation characterization purposes. This situation is rectified by generalizing the concept of the transmission error of parallel-axis gears to a three-component transmission error for spiral bevel gears of nominal spherical involute design. A general relationship is derived which expresses the contributions to the three-component transmission error from each gear of a meshing spiral bevel pair as a linear transformation of the six coordinates that describe the deviation of the shaft centerline position of each gear of the pair from the position of its rigid perfect involute counterpart.

Mark, W. D.↗
NASA NTRS1987

Use of the generalized transmission error in the equations of motion of gear systems

The vibratory excitation arising from a gear pair is widely recognized to be a consequence of the nonuniform transmission of motion by the gear pair. In the preceding paper in this issue, it is shown that a three-component transmission error is required to describe the nonuniform transmission of motion by bevel gears for vibration excitation characterization purposes. The expression for the three-component transmission error derived in that paper is combined in the present paper with an analysis of the mesh forces and mesh elasticity to yield an equation of constraint involving the six degree-of-freedom unknown vibratory displacements of the gear shaft centerliners, the three unknown components of the generalized force transmitted by the mesh, and the geometric deviations of the tooth running surfaces from perfect involute surfaces which are assumed known. This matrix equation can be combined with the equations of motion of a gear system to predict the vibratory response of the system to the generalized transmission error excitation arising from meshing gear pairs within the system.

Mark, W. D.↗
NASA NTRS1986

Some implications of the isotropic momentarily frozen assumptions for the SPAN-MAT program

Potential tests using turbulence velocity histories measured in the SPAN-MAT Program are outlined to determine validity of the homogeneous, momentarily frozen assumptions for the vertical turbulence velocity component and the homogeneous, isotropic, momentarily frozen assumptions for the horizontal turbulence velocity components. In addition, methods are reviewed for prediction of the crosscorrelation function between any two spatially separated turbulence velocity components using the homogeneous, isotropic, momentarily frozen assumptions and measurements of the transverse and longitudinal turbulence velocity components.

Mark, W. D.↗
NASA NTRS1984

Maximum likelihood estimation of turbulence spectrum parameters

Estimation of the integral scale and intensity of a generic turbulence record is treated as a statistical problem of parameter estimation. Properties of parameter estimators and the method of maximum likelihood are reviewed. Likelihood equations are derived for estimation of the integral scale and intensity applicable to a general class of turbulence spectra that includes the von Karman and Dryden transverse and longitudinal spectra as special cases. The method is extended to include the Bullen transverse and longitudinal spectra. Coefficients of variation are given for maximum likelihood estimates of the integral scale and intensity of the von Karman spectra. Application of the method is illustrated by estimating the integral scale and intensity of an atmospheric turbulence vertical velocity record assumed to be governed by the von Karman transverse spectrum.

Mark, W. D.↗
NASA NTRS1982

The transfer function method for gear system dynamics applied to conventional and minimum excitation gearing designs

A transfer function method for predicting the dynamic responses of gear systems with more than one gear mesh is developed and applied to the NASA Lewis four-square gear fatigue test apparatus. Methods for computing bearing-support force spectra and temporal histories of the total force transmitted by a gear mesh, the force transmitted by a single pair of teeth, and the maximum root stress in a single tooth are developed. Dynamic effects arising from other gear meshes in the system are included. A profile modification design method to minimize the vibration excitation arising from a pair of meshing gears is reviewed and extended. Families of tooth loading functions required for such designs are developed and examined for potential excitation of individual tooth vibrations. The profile modification design method is applied to a pair of test gears.

Mark, W. D.↗
NASA NTRS1982

Turbofan noise generation. Volume 1: Analysis

Computer programs were developed which calculate the in-duct acoustic modes excited by a fan/stator stae operating at subsonic tip speed. Three noise source mechanisms are included: (1) sound generated by the rotor blades interacting with turbulence ingested into, or generated within, the inlet duct; (2) sound generated by the stator vanes interacting with the turbulent wakes of the rotors blades; and (3) sound generated by the stator vanes interacting with the mean velocity deficit wakes of the rotor blades. The fan/stator stage is modeled as an ensemble of blades and vanes of zero camber and thickness enclosed within an infinite hard-walled annular duct. Turbulence drawn into or generated within the inlet duct is modeled as nonhomogeneous and anisotropic random fluid motion, superimposed upon a uniform axial mean flow, and convected with that flow. Equations for the duct mode amplitudes, or expected values of the amplitudes, are derived.

Ventres, C. S.↗
NASA NTRS1982

Turbofan noise generation. Volume 2: Computer programs

The use of a package of computer programs developed to calculate the in duct acoustic mods excited by a fan/stator stage operating at subsonic tip speed is described. The following three noise source mechanisms are included: (1) sound generated by the rotor blades interacting with turbulence ingested into, or generated within, the inlet duct; (2) sound generated by the stator vanes interacting with the turbulent wakes of the rotor blades; and (3) sound generated by the stator vanes interacting with the velocity deficits in the mean wakes of the rotor blades. The computations for three different noise mechanisms are coded as three separate computer program packages. The computer codes are described by means of block diagrams, tables of data and variables, and example program executions; FORTRAN listings are included.

Ventres, C. S.↗
NASA NTRS1981

Characterization, parameter estimation, and aircraft response statistics of atmospheric turbulence

A nonGaussian three component model of atmospheric turbulence is postulated that accounts for readily observable features of turbulence velocity records, their autocorrelation functions, and their spectra. Methods for computing probability density functions and mean exceedance rates of a generic aircraft response variable are developed using nonGaussian turbulence characterizations readily extracted from velocity recordings. A maximum likelihood method is developed for optimal estimation of the integral scale and intensity of records possessing von Karman transverse of longitudinal spectra. Formulas for the variances of such parameter estimates are developed. The maximum likelihood and least-square approaches are combined to yield a method for estimating the autocorrelation function parameters of a two component model for turbulence.

Mark, W. D.↗
NASA NTRS1981

Statistics of some atmospheric turbulence records relevant to aircraft response calculations

Methods for characterizing atmospheric turbulence are described. The methods illustrated include maximum likelihood estimation of the integral scale and intensity of records obeying the von Karman transverse power spectral form, constrained least-squares estimation of the parameters of a parametric representation of autocorrelation functions, estimation of the power spectra density of the instantaneous variance of a record with temporally fluctuating variance, and estimation of the probability density functions of various turbulence components. Descriptions of the computer programs used in the computations are given, and a full listing of these programs is included.

Mark, W. D.↗
NASA NTRS1979

Analysis of the vibratory excitation of gear systems. II - Tooth error representations, approximations, and application

The second part of a theory for predicting the vibratory excitation of gear systems from fundamental descriptions of gear tooth elastic properties and deviations of tooth faces from perfect involute surfaces is presented. The first part of the theory provides expressions for the Fourier-series coefficients of the vibratory excitation, and this paper gives expressions for these Fourier-series coefficients in terms of easily interpreted gear tooth metrics that are readily evaluated from tooth-face measurements. Results are given for rectangular tooth-face contact regions using two-dimensional Legendre polynomial expansions of local tooth-pair stiffnesses and stiffness-weighted deviations of tooth faces from perfect involute surfaces. A rigorous transfer function approach is developed that permits separation of the effects of gear tooth errors and gear design parameters; the theory is applicable to helical and spur gears and is illustrated with measurements of tooth-spacing errors and tooth profiles obtained from a pair of spur gears.

Mark, W. D.↗
NASA NTRS1979

Analysis of Vibratory Excitation of Gear Systems as a Contributor to Aircraft Interior Noise

Application of the transfer function approach to predict the resulting interior noise contribution requires gearbox vibration sources and paths to be characterized in the frequency domain. Tooth-face deviations from perfect involute surfaces were represented in terms of Legendre polynomials which may be directly interpreted in terms of tooth-spacing errors, mean and random deviations associated with involute slope and fullness, lead mismatch and crowning, and analogous higher-order components. The contributions of these components to the spectrum of the static transmission error is discussed and illustrated using a set of measurements made on a pair of helicopter spur gears. The general methodology presented is applicable to both spur and helical gears.

Mark, W. D.↗
NASA NTRS1977

Characterization of nonGaussian atmospheric turbulence for prediction of aircraft response statistics

Mathematical expressions were derived for the exceedance rates and probability density functions of aircraft response variables using a turbulence model that consists of a low frequency component plus a variance modulated Gaussian turbulence component. The functional form of experimentally observed concave exceedance curves was predicted theoretically, the strength of the concave contribution being governed by the coefficient of variation of the time fluctuating variance of the turbulence. Differences in the functional forms of response exceedance curves and probability densities also were shown to depend primarily on this same coefficient of variation. Criteria were established for the validity of the local stationary assumption that is required in the derivations of the exceedance curves and probability density functions. These criteria are shown to depend on the relative time scale of the fluctuations in the variance, the fluctuations in the turbulence itself, and on the nominal duration of the relevant aircraft impulse response function. Metrics that can be generated from turbulence recordings for testing the validity of the local stationary assumption were developed.

Mark, W. D.↗