Dynamic simulation of planetary gear set with flexible spur ring gear

Ring gear is a key element for vibration transmission and noise radiation in the planetary gear system which has been widely employed in different areas, such as wind turbine transmissions. Its flexibility has a great influence on the mesh stiffness of internal gear pair and the dynamic response of the planetary gear system, especially for the thin ring cases. In this paper, the flexibility of the internal ring gear is considered based on the uniformly curved Timoshenko beam theory. The ring deformation is coupled into the mesh stiffness model, which enables the investigation on the effects of the ring flexibility on the mesh stiffness and the dynamic responses of the planetary gear. A method about how to synthesize the total mesh stiffness of the internal gear pairs in multi-tooth region together with the ring deformation and the tooth errors is proposed. Numerical results demonstrate that the ring thickness has a great impact on the shape and magnitude of the mesh stiffness of the internal gear pair. It is noted that the dynamic responses of the planetary gear set with equally spaced supports for the ring gear are modulated due to the cyclic variation of the mesh stiffness resulted from the presence of the supports, which adds more complexity in the frequency structure.     

Effect of tooth mesh stiffness asymmetric nonlinearity for drive and coast sides on hypoid gear dynamics

A nonlinear time-varying dynamic model of a hypoid gear pair system with time-dependent nonlinear mesh stiffness, mesh damping and backlash properties is formulated to study the effect of mesh stiffness asymmetry for drive and coast sides on dynamic response. The asymmetric characteristic is the result of the inherent curvilinear tooth form and pinion offset in hypoid set. Using the proposed nonlinear time-varying dynamic model, effects of asymmetric mesh stiffness parameters that include mean mesh stiffness ratio, mesh stiffness variation and mesh stiffness phase angle on the dynamic mesh force response and tooth impact regions are examined systematically. Specifically, the dynamic models with only asymmetric mesh stiffness nonlinearity, with only backlash nonlinearity and with both asymmetric mesh stiffness and backlash nonlinearities are analyzed and compared. Using the parameters of a typical hypoid gear set, the extent of the effect of asymmetry in the mesh coupling on gear pair dynamics is quantified numerically. The results show that the increase in the mean mesh stiffness ratio tends to worsen the dynamic response amplitude, and the mesh stiffness parameters for drive side have more effect on dynamic response than those of the coast side one.     

Spectral fine-structures of low-frequency modulated distortion product otoacoustic emissions

Biasing of the cochlear partition with a low-frequency tone can produce an amplitude modulation of distortion product otoacoustic emissions (DPOAEs) in gerbils. In the time domain, odd- versus even-order DPOAEs demonstrated different modulation patterns depending on the bias tone phase. In the frequency domain, multiple sidebands are presented on either side of each DPOAE component. These sidebands were located at harmonic multiples of the biasing frequency from the DPOAE component. For odd-order DPOAEs, sidebands at the even-multiples of the biasing frequency were enhanced, while for even-order DPOAEs, the sidebands at the odd-multiples were elevated. When a modulation in DPOAE magnitude was presented, the magnitudes of the sidebands were enhanced and even greater than the DPOAEs. The amplitudes of these sidebands varied with the levels of the bias tone and two primary tones. The results indicate that the maximal amplitude modulations of DPOAEs occur at a confined bias and primary level space. This can provide a guide for optimal selections of signal conditions for better recordings of low-frequency modulated DPOAEs in future research and applications. Spectral fine-structure and its unique relation to the DPOAE modulation pattern may be useful for direct acquisition of cochlear transducer nonlinearity from a simple spectral analysis.     

信号在时变等离子体中的传播特性

高速飞行器等离子鞘套由于飞行姿态调整、湍流、非均匀烧蚀等因素的影响,使其等离子体参数存在时变特性,针对这种传输介质的快速时变特性引起的电波幅度、相位上的寄生调制效应,本文利用大面积辉光放电等离子体产生装置,搭建了等离子体中信号传输实验系统,进行了S频段的单频信号与调制信号传输实验,观测验证了调制效应的存在,且其调制频率与等离子体变化频率具有一致性,进一步分析了等离子参数与寄生调制效应的关系,理论和实验结果表明:即使当载波频率大于等离子频率时,时变等离子引起的寄生调制效应也会使传输信号的星座图发生旋转,造成其判决容差裕度变小,通信可靠性下降,并且载波频率越接近等离子频率时,其寄生调制效应越强烈。     

Localized proton spectroscopy without water suppression: removal of gradient induced frequency modulations by modulus signal selection

Most Magnetic Resonance Spectroscopy (MRS) localization methods can generate gradient vibrations at acoustic frequencies and/or magnetic field oscillation, which can cause a time-varying magnetic field superimposed onto the static one. This effect can produce frequency modulations of the spectral resonances. When localized MRS data are acquired without water suppression, the associated frequency modulations are manifested as a manifold of spurious peaks, called sidebands, which occur symmetrically around the water resonance. These sidebands can be larger than the small metabolite resonances and can present a problem for the quantitation of the spectra, especially at short echo times. Furthermore, the resonance lineshapes may be distorted if any low frequency modulations are present. A simple solution is presented which consists of selecting the modulus of the acquired Free Induction Decay (FID) signal. Since the frequency modulations affect only the phase of the FID signal, the obtained real spectrum of the modulus is free from the spurious peaks where quantitative results may be directly obtained. Using this method, the distortions caused by the sidebands are removed. This is demonstrated by processing proton MRS spectra acquired without water suppression collected from a phantom containing metabolites at concentrations comparable to those in human brain and from a human subject using two different localization methods (PRESS and Chemical Shift Imaging PRESS-(CSI)). The results obtained illustrate the ability of this approach to remove the spurious peaks. The corrected spectra can then be fit accurately. This is confirmed by the results obtained from both the relative and the absolute metabolites concentrations in phantoms and in vivo.     

Nonlinear dynamics of planetary gears using analytical and finite element models

Vibration-induced gear noise and dynamic loads remain key concerns in many transmission applications that use planetary gears. Tooth separations at large vibrations introduce nonlinearity in geared systems. The present work examines the complex, nonlinear dynamic behavior of spur planetary gears using two models: (i) a lumped-parameter model, and (ii) a finite element model. The two-dimensional (2D) lumped-parameter model represents the gears as lumped inertias, the gear meshes as nonlinear springs with tooth contact loss and periodically varying stiffness due to changing tooth contact conditions, and the supports as linear springs. The 2D finite element model is developed from a unique finite element-contact analysis solver specialized for gear dynamics. Mesh stiffness variation excitation, corner contact, and gear tooth contact loss are all intrinsically considered in the finite element analysis. The dynamics of planetary gears show a rich spectrum of nonlinear phenomena. Nonlinear jumps, chaotic motions, and period-doubling bifurcations occur when the mesh frequency or any of its higher harmonics are near a natural frequency of the system. Responses from the dynamic analysis using analytical and finite element models are successfully compared qualitatively and quantitatively. These comparisons validate the effectiveness of the lumped-parameter model to simulate the dynamics of planetary gears. Mesh phasing rules to suppress rotational and translational vibrations in planetary gears are valid even when nonlinearity from tooth contact loss occurs. These mesh phasing rules, however, are not valid in the chaotic and period-doubling regions.     

Critical modulation frequency based on detection of AM versus FM tones

The ratios between the modulation index (eta) for just noticeable FM of a sinusoidally modulated pure tone and the degree of modulation (m) for just noticeable AM at the same carrier and the same modulation frequency were measured at carrier frequencies of 0.125, 0.25, 0.5, 1, 2, 4, and 8 kHz. Signal levels were 20 dB SL and 50 dB SPL or 80 dB SPL. At low modulation frequencies, for example, 8 Hz, AM and FM elicit very different auditory sensations (i.e., a fluctuation in loudness or pitch, respectively). In this case, eta and m show different values for just noticeable modulation. Since both stimuli have almost equal amplitude spectra if eta equals m (m less than 0.3), the difference in detection thresholds reflects differences in the phase relation between carrier and sidebands in AM and FM. With increasing modulation frequency, the eta-m ratio decreases and reaches unity at a modulation frequency called the "critical modulation frequency" (CMF). At modulation frequencies above the CMF, the same modulation thresholds are obtained for AM and FM. Therefore, it can be concluded that the difference in phase between the two types of stimuli is not perceived in this range. At center frequencies below 1 kHz, where phase errors caused by headphones and ear canal presumably are small, the CMF is useful in estimating critical bandwidth.     

Dynamic analysis for a planetary gear with time-varying pressure angles and contact ratios

In this study, the dynamic responses of a planetary gear are analyzed when component gears have time-varying pressure angles and contact ratios caused by bearing deformations. For this purpose, this study proposes a new dynamic model of the planetary gear, in which the pressure angles and contact ratios change with time. The main difference from previous studies is that the present study regards the pressure angles and contact ratios as time-varying variables, while previous studies regarded them as constants. After nonlinear equations of motion for the planetary gear are derived, the dynamic responses are computed by applying the Newmark time integration method. The time responses for the present and previous studies are compared to show the effects of the time-varying pressure angles and contact ratios on the dynamic behaviors of a planetary gear. In addition, the effects of bearing stiffness on the pressure angles and contact ratios are also analyzed.     

Analytical investigation on unstable vibrations of single-mesh gear systems with velocity-modulated time-varying stiffness

Time-varying mesh stiffness parametrically excites gear systems and causes severe vibrations and instabilities. Taking speed fluctuations into account, the time-varying mesh stiffness is frequency modulated, and more complex instabilities might arise. Considering two different speed fluctuation models, parametric instability associated with velocity-modulated time-varying stiffness is analytically investigated using a typical single-mesh gear system model. Closed-form approximations are obtained by perturbation analysis, and verified by numerical analysis. The effects of the amplitude of the mesh stiffness variation, the characteristics of speed fluctuations and damping on parametric instability are systematically examined.     

Vibration signal models for fault diagnosis of planetary gearboxes

A thorough understanding of the spectral structure of planetary gear system vibration signals is helpful to fault diagnosis of planetary gearboxes. Considering both the amplitude modulation and the frequency modulation effects due to gear damage and periodically time variant working condition, as well as the effect of vibration transfer path, signal models of gear damage for fault diagnosis of planetary gearboxes are given and the spectral characteristics are summarized in closed form. Meanwhile, explicit equations for calculating the characteristic frequency of local and distributed gear fault are deduced. The theoretical derivations are validated using both experimental and industrial signals. According to the theoretical basis derived, manually created local gear damage of different levels and naturally developed gear damage in a planetary gearbox can be detected and located.     

噪声免疫腔增强光外差分子饱和光谱线型的理论分析

噪声免疫腔增强光外差分子光谱(NICE-OHMS)作为世界上最灵敏的光谱技术可以被应用到痕量气体检测、频率标准、原子分子光谱以及超灵敏引力波测量等领域中,高精细度谐振腔吸收池的使用在增长激光与腔内物质相互作用路径的同时,极大的提高了腔内激光功率,这就会饱和低气压下的气态样品吸收线从而获得亚多普勒光谱结构,因此NICE-OHMS技术不仅具有高灵敏、还具有超高分辨的优点。该研究基于光与二能级分子相互作用的密度矩阵理论对NICE-OHMS技术中包含亚多普勒的多普勒展宽光谱线型进行了理论推导,获得了光谱线型的表达式,同时以该表达式对光谱线型进行了数值模拟,其中调制频率、饱和参量、频率调制系数分别设置为384 MHz,10和0.2。由模拟结果可见吸收光谱由两个边带的吸收信号构成,在包络上存在四个亚多普勒饱和结构;色散光谱由载频以及边带的色散三者决定,并在包络上存在五个亚多普勒饱和结构,获得了与已有实验一致的结果。最后重点分析了不同探测相位、不同饱和参量下的NICE-OHMS光谱线型尤其是亚多普勒结构的变化,由于饱和参量按照调制系数分配给载频和边带,因此虽然载频饱和参量很大,但NICE-OHMS吸收光谱幅度变化不大,主要是由于该光谱信号只与边带饱和参量有关,可以看出NICE-OHMS多普勒展宽信号具有饱和效应免疫的特性,与已有实验结果也符合较好。为更进一步的实验研究提供了必要的理论基础。     

Numerical study on reducing the vibration of spur gear pairs with phasing

A new method of reducing gear vibration was analyzed using a simple spur gear pair with phasing. This new method is based on reducing the variation in mesh stiffness by adding another pair of gears with half-pitch phasing. This reduces the variation in the mesh stiffness of the final (phasing) gear, because each gear compensates for the variation in the other's mesh stiffness. A single gear pair model with a time-varying rectangular-type mesh stiffness function and backlash was used, and the dynamic response over a wide range of speeds was obtained by numerical integration. Because of the reduced variation in mesh stiffness and the double frequency, the phasing gear greatly reduced the dynamic response and nonlinear behavior of the normal gears. The results of the analysis indicate the possibility of reducing vibration of spur gear pairs using the proposed method.     

1H spectroscopy without solvent suppression: characterization of signal modulations at short echo times.

While most proton ((1)H) spectra acquired in vivo utilize selective suppression of the solvent signal for more sensitive detection of signals from the dilute metabolites, recent reports have demonstrated the feasibility and advantages of collecting in vivo data without solvent attenuation. When these acquisitions are performed at short echo times, the presence of frequency modulations of the water resonance may become an obstacle to the identification and quantitation of metabolite resonances. The present report addresses the characteristics, origin, and elimination of these sidebands. Sideband amplitudes were measured as a function of delay time between gradient pulse and data collection, as a function of gradient pulse amplitude, and as a function of spatial location of the sample for each of the three orthogonal gradient sets. Acoustic acquisitions were performed to demonstrate the correlation between mechanical vibration resonances and the frequencies of MR sidebands. A mathematical framework is developed and compared with the experimental results. This derivation is based on the theory that these frequency modulations are induced by magnetic field fluctuations generated by the transient oscillations of gradient coils.     

PLANETARY GEAR PARAMETRIC INSTABILITY CAUSED BY MESH STIFFNESS VARIATION

Parametric instability is investigated for planetary gears where fluctuating stiffness results from the changing contact conditions at the multiple tooth meshes. The time-varying mesh stiffnesses of the sun-planet and ring-planet meshes are modelled as rectangular waveforms with different contact ratios and mesh phasing. The operating conditions leading to parametric instability are analytically identified. Using the well-defined properties of planetary gear vibration modes, the boundaries separating stable and unstable conditions are obtained as simple expressions in terms of mesh parameters. These expressions allow one to suppress particular instabilities by adjusting the contact ratios and mesh phasing. Tooth separation from parametric instability is numerically simulated to show the strong impact of this non-linearity on the response.     

利用法布里-珀罗腔抑制电光相位调制中的剩余幅度调制

激光经电光相位调制后,由于剩余幅度调制的存在造成调制边带幅度不相等。利用法布里-珀罗腔的透射特性和Pound-Drever-Hall技术对通过法布里-珀罗腔的调制光正、负一级边带的幅度产生不同的衰减,使得调制边带的幅度相等,从而实现对电光相位调制中剩余幅度调制的抑制。采用该方法,理论上计算了调制光经法布里-珀罗腔后的光外差光谱信号,获得锁定法布里-珀罗腔后调制边带幅度的不对称度较腔锁定前减小四个数量级。实验研究了调制光经法布里-珀罗腔透射的光外差光谱,结果表明将法布里-珀罗腔锁定于该透射光外差光谱中心零位时,对剩余幅度调制的抑制程度可达45 dB。     

不同宽带脉冲在钕玻璃放大器中的幅频效应

宽带脉冲在钕玻璃放大器中传输时,由于受到增益窄化和增益饱和效应的影响,脉冲频谱将受到调制,从而产生边带丢失或者频谱两侧失衡,进而导致脉冲时间波形产生调制。针对不同的宽带光源模型,数值模拟了放大器输出波形的幅度调制,分析了增益带宽、输入脉冲带宽及增益曲线中心波长相对输入脉冲中心波长的偏差等因素对输出脉冲调制度的影响。结果表明:不同的宽带光源脉冲经过钕玻璃放大器后,均会产生不同程度的幅度调制;啁啾堆积脉冲比正弦相位调制脉冲产生的幅度调制小。     

The influence of carrier level and frequency on modulation and beat-detection thresholds for sinusoidal carriers

This paper is concerned with modulation and beat detection for sinusoidal carriers. In the first experiment, temporal modulation transfer functions (TMTFs) were measured for carrier frequencies between 1 and 10 kHz. Modulation rates covered the range from 10 Hz to about the rate equaling the critical bandwidth at the carrier frequency. In experiment 2, TMTFs for three carrier frequencies were obtained as a function of the carrier level. In the final experiment, thresholds for the detection of either the lower or the upper modulation sideband (beat detection) were measured for "carrier" frequencies of 5 and 10 kHz, using the same range of modulation rates as in experiment 1. The TMTFs for carrier frequencies of 2 kHz and higher remained flat up to a modulation rate of about 100-130 Hz and had similar values across carrier frequencies. For higher rates, modulation thresholds initially increased and then decreased rapidly, reflecting the subjects' ability to resolve the sidebands spectrally. Detection thresholds generally improved with increasing carrier level, but large variations in the exact level dependence were observed, across subjects as well as across carrier frequencies. For beat rates up to about 70 Hz (at 5 kHz) and 100 Hz (at 10 kHz), beat detection thresholds were the same for the upper and the lower sidebands and were about 6 dB higher than the level per sideband at the modulation-detection threshold. At higher rates the threshold for both sidebands increased, but the increase was larger for the lower sideband. This reflects an asymmetry in masking with more masking towards lower frequencies. Only at rates well beyond the maximum of the TMTF did detection for the lower sideband start to be better than that for the upper sideband. The asymmetry at intermediate frequency separations can be explained by assuming that detection always takes place in filters centered above the stimulus spectrum. The shape of the TMTF and the beat-detection data reflects a limitation in resolving fast amplitude variations, which must occur central to the inner-ear filtering. Its characteristic resembles that of a first-order low-pass filter with a cutoff frequency of about 150 Hz.     

The continuous wave electron paramagnetic resonance experiment revisited

When the modulation frequency used in continuous wave electron paramagnetic resonance (cw EPR) spectroscopy exceeds the linewidth, modulation sidebands appear in the spectrum. It is shown theoretically and experimentally that these sidebands are actually multiple photon transitions, sigma(+)+kxpi, where one microwave (mw) sigma(+) photon is absorbed from the mw radiation field and an arbitrary number k of radio frequency (rf) pi photons are absorbed from or emitted to the modulation rf field. Furthermore, it is demonstrated that both the derivative shape of the lines in standard cw EPR spectra and the distortions due to overmodulation are caused by the unresolved sideband pattern of these lines. The single-photon transition does not even give a contribution to the first-harmonic cw EPR signal. Multiple photon transitions are described semiclassically in a toggling frame and their existence is proven using second quantization. With the toggling frame approach and perturbation theory an effective Hamiltonian for an arbitrary sideband transition is derived. Based on the effective Hamiltonians an expression for the steady-state density operator in the singly rotating frame is derived, completely describing all sidebands in all modulation frequency harmonics of the cw EPR signal. The relative intensities of the sidebands are found to depend in a very sensitive way on the actual rf amplitude and the saturation of single sidebands is shown to depend strongly on the effective field amplitude of the multiple photon transitions. By comparison with the analogous solutions for frequency-modulation EPR it is shown that the field-modulation and the frequency-modulation technique are not equivalent. The experimental data fully verify the theoretical predictions with respect to intensities and lineshapes.     

Temporal modulation transfer functions obtained using sinusoidal carriers with normally hearing and hearing-impaired listeners

Temporal modulation transfer functions were obtained using sinusoidal carriers for four normally hearing subjects and three subjects with mild to moderate cochlear hearing loss. Carrier frequencies were 1000, 2000 and 5000 Hz, and modulation frequencies ranged from 10 to 640 Hz in one-octave steps. The normally hearing subjects were tested using levels of 30 and 80 dB SPL. For the higher level, modulation detection thresholds varied only slightly with modulation frequency for frequencies up to 80 Hz, but decreased for high modulation frequencies. The decrease can be attributed to the detection of spectral sidebands. For the lower level, thresholds varied little with modulation frequency for all three carrier frequencies. The absence of a decrease in the threshold for large modulation frequencies can be explained by the low sensation level of the spectral sidebands. The hearing-impaired subjects were tested at 80 dB SPL, except for two cases where the absolute threshold at the carrier frequency was greater than 70 dB SPL; in these cases a level of 90 dB was used. The results were consistent with the idea that spectral sidebands were less detectable for the hearing-impaired than for the normally hearing subjects. For the two lower carrier frequencies, there were no large decreases in threshold with increasing modulation frequency, and where decreases did occur, this happened only between 320 and 640 Hz. For the 5000-Hz carrier, thresholds were roughly constant for modulation frequencies from 10 to 80 or 160 Hz, and then increased monotonically, becoming unmeasurable at 640 Hz. The results for this carrier may reflect "pure" effects of temporal resolution, without any influence from the detection of spectral sidebands. The results suggest that temporal resolution for deterministic stimuli is similar for normally hearing and hearing-impaired listeners.     

Second-order temporal modulation transfer functions

Detection thresholds were measured for a sinusoidal modulation applied to the modulation depth of a sinusoidally amplitude-modulated (SAM) white noise carrier as a function of the frequency of the modulation applied to the modulation depth (referred to as f'm). The SAM noise acted therefore as a "carrier" stimulus of frequency fm, and sinusoidal modulation of the SAM-noise modulation depth generated two additional components in the modulation spectrum: fm-f'm and fm+f'm. The tracking variable was the modulation depth of the sinusoidal variation applied to the "carrier" modulation depth. The resulting "second-order" temporal modulation transfer functions (TMTFs) measured on four listeners for "carrier" modulation frequencies fm of 16, 64, and 256 Hz display a low-pass segment followed by a plateau. This indicates that sensitivity to fluctuations in the strength of amplitude modulation is best for fluctuation rates f'm below about 2-4 Hz when using broadband noise carriers. Measurements of masked modulation detection thresholds for the lower and upper modulation sideband suggest that this capacity is possibly related to the detection of a beat in the sound's temporal envelope. The results appear qualitatively consistent with the predictions of an envelope detector model consisting of a low-pass filtering stage followed by a decision stage. Unlike listeners' performance, a modulation filterbank model using Q values > or = 2 should predict that second-order modulation detection thresholds should decrease at high values of f'm due to the spectral resolution of the modulation sidebands (in the modulation domain). This suggests that, if such modulation filters do exist, their selectivity is poor. In the latter case, the Q value of modulation filters would have to be less than 2. This estimate of modulation filter selectivity is consistent with the results of a previous study using a modulation-masking paradigm [S. D. Ewert and T. Dau, J. Acoust. Soc. Am. 108, 1181-1196 (2000)].