William D. Mark

Analysis of the vibratory excitation of gear systems: Basic theory

Formulation of the equations of motion of a generic gear system in the frequency domain is shown to require the Fourier‐series coefficients of the components of vibration excitation; these components are the static transmission errors of the individual pairs of meshing gears in the system. A general expression for the static transmission error is derived and decomposed into components attributable to elastic tooth deformations and to deviations of tooth faces from perfect involute surfaces with uniform lead and spacing. The component due to tooth‐face deviations is further decomposed into appropriately defined mean and random components. The harmonic components of the static transmission error that occur at integral multiples of the tooth‐meshing frequency are shown to be caused by tooth deformations and mean deviations of the tooth faces from perfect involute surfaces. Harmonic components that occur at the remaining multiples of gear‐rotation frequencies are shown to be caused by the random components of...

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

The second part of a theory is presented 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. The first part of the theory [J. Acoust. Soc. Am. 63, 1409–1430 (1978)] provides expressions for the Fourier‐series coefficients of the vibratory excitation, which was shown to be described by the static transmission errors of all pairs of meshing gears in a system. The present paper provides expressions for these Fourier‐series coefficients in terms of easily interpreted gear tooth metrics that are readily evaluated from tooth‐face measurements. Detailed 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 ...

Contributions to the vibratory excitation of gear systems from periodic undulations on tooth running surfaces

The transmission error is widely recognized to be the principal source of vibratory excitation arising from meshing gear pairs. Periodic machining errors (undulation errors) on gear teeth can provide an important contribution to the transmission error of wide‐face helical gears. A method is described for using complex sinusoids to represent such periodic errors on individual gear teeth. Expressions are derived for the Fourier series expansion coefficients of the contributions from such errors to the transmission error. A model describing the relative phase relation on successive teeth of the individual sinusoidal components of such errors is postulated which is believed to provide a valid representation of the phase of most, if not all, such errors generated by imperfections in the rotating elements of gear manufacturing apparatus. For particular parameter values of the phase model, it is shown that such periodic errors can contribute to: (1) tooth meshing harmonics only, (2) rotational harmonics only, or...

Power spectral representation of nonstationary random processes defined over semi‐infinite intervals

A new sequence of locally time‐averaged power spectra is defined that describes the time evolution of the frequency content of nonstationary random processes and linear system impulse response functions. The exact input‐response relations for these spectral sequences are shown to be finite discrete convolutions of the input and system spectral sequences. Expressions for the coefficients of a Laguerre function expansion of the time‐varying mean square response are derived from the response spectral sequence, and the time resolution and convergence of the expansion are discussed. Simple formulas are derived for generation of the spectra from recorded sample functions using Fourier transform computational algorithms. The relationship of the spectra to the Laplace transform is also developed. Physical interpretation of the spectra is discussed in detail and related to the uncertainty principle for Fourier transforms.Subject Classification: [43]45.40; [43]60.20; [43]20.50; [43]30.30; [43]40.35.

Synthesis of helicopter rotor tips for less noise

Theoretical and computational studies of rotor‐tip sound radiation have been conducted for the purpose of designing rotor tips that radiate less sound in specified frequency bands. Consideration is given to radiation due to lift and thickness effects. Effects of unsteady vortex shedding on lift radiation are examined. It is shown that lift radiation is generally negligible in comparison with thickness radiation. A computational algorithm is developed for the synthesis of tip shapes that cause minimum thickness radiation in specified frequency bands. Numerical results are obtained for tip shapes that minimize high‐frequency radiation, and a substantial reduction of radiation in comparison with existing shapes is shown. The uncertainty principle is used to establish a fundamental relationship between the tip section chord length and the minimum possible cutoff frequency for effective suppression of high‐frequency sound. Factors that affect tradeoffs between choices on airfoil section and planform are discussed.

Statistics of Multipath Transmission of Finite Bandwidth Signals

The statistical properties of a random variable E that denotes either the time‐averaged power in a received band of noise or the total energy in a received transient waveform are studied for situations where the variablity in E is due to uncertainty in the detailed structure of the multipath transmission channel. The transmission channel impulse response function is modeled by a nonstationary Gaussian random process—a model that predicts the Rayleigh probability density for received amplitudes of transmitted pure tones. General expressions are derived for the mean and variance of E. These expressions are reduced to the practically important case of impulse response functions that are δ‐correlated in time. Approximate expressions and bounds are derived for the relative variance of E. Special cases of the results are shown to be in essential agreement with results previously obtained in studies of the responses of rooms to bands of noise.

Probability density approximations of linear system responses to intensity‐modulated Gaussian processes

An intensity‐modulated Gaussian process {x(t)} is defined as the product of a non‐negative stationary random process {σx(t)} and a stationary Gaussian process {z(t)} that is statistically independent of {σx(t)}. When such an intensity‐modulated process is the excitation to a linear time‐invariant system, the response process {y(t)}, conditioned on the excitation‐modulating function σx(t), is a generally nonstationary Gaussian process. This property is used to develop an approximation to the probability density function (PDF) of the system response by assuming that the time‐varying conditional response variance σ2y(t), conditioned on the excitation‐modulating function σx(t), is governed by the gamma PDF. The resulting response process PDF is characterized by two parameters: the mean and variance of the conditional response variance σ2y(t). An alternative approximation to the response PDF, characterized by these same two parameters, is developed by using a Taylor’s series representation in the variable σ2y ...

Effects of intensity modulations on the power spectra of random processes

An intensity‐modulated random process {(x(t)} is defined as the product of a deterministic modulating function a σ(t) or modulating process {σ(t)} and a stationary‐modulated process {z(t)} that is statistically independent of {σ(t)}. General expressions for the instantaneous power spectra of intensity‐modulated processes are presented for various classes of modulating functions and processes. A series expansion of the instantaneous power spectrum of intensity‐modulated processes is presented which has for its first term a well‐known locally stationary spectrum approximation. This expansion is especially useful when the fluctuation scales Tσ of the modulating functions are large in comparison with the fluctuation scales Tz of the modulated processes. The expansion can be interpreted as an asymptotic series in the parameter Tσ/Tz. For given scales Tσ and Tz, it is shown that modulating processes {σ(t)} possessing no first derivative have a substantially larger effect on the power spectrum of modulated proce...

Statistical dependence of subpulse echoes in active sonar systems using frequency diversity

Active sonar systems using waveforms consisting of M subpulses of different frequencies are a common method for counteracting the Rayleigh fading statistics of a single pulse echo. When frequency differences between adjacent subpulses are not large in comparison with the reciprocal of target‐impulse response durations, the various subpulse echoes are not statistically independent. This lack of independence reduces the 2M statistical df that are associated with statistically independent echoes. Probability density functions of squared and summed subpulse echoes plus reverberation and noise are determined as functions of target‐impulse response durations and subpulse frequency spacings. Limiting cases are shown to coincide with the classical chi‐square distribution results. Effects of time constants of reverberation and noise‐estimation circuits on receiver operating characteristics also will be discussed.

Frequency decomposition of the vibratory excitation of gear systems

The principal source of vibratory excitation of gear systems is the static transmission error which measures the deviation from uniform relative rates of (slow) rotation of pairs of meshing gears. The static transmission error can be decomposed into tooth bending, profile modification, and machining error components. The contributions of each of these components to the frequency spectrum of the static transmission error, and the effects of the mesh geometry parameters of helical gears on the spectrum, will be discussed.