Academic literature on the topic 'Subharmonic spectra'
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Journal articles on the topic "Subharmonic spectra"
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Hajj, M. R., R. W. Miksad, and E. J. Powers. "Fundamental–subharmonic interaction: effect of phase relation." Journal of Fluid Mechanics 256 (November 1993): 403–26. http://dx.doi.org/10.1017/s0022112093002824.
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The effect of the phase relation (i.e. phase difference and coupling) between the fundamental and subharmonic modes on the transition to turbulence of a mixing layer is investigated. Experiments are conducted to study the development of the subharmonic and fundamental modes under different phase-controlled excitations. Higher-order spectral moments are used to measure phase differences, levels of phase coupling, and energy transfer rates between the two modes at different downstream locations. Local measurements of the wavenumber–frequency spectra are used to examine the phase-speed matching conditions required for efficient energy transfer. The results show that when the phase coupling between the fundamental and the subharmonic is high, maximum subharmonic growth is found to occur at a critical phase difference close to zero. The subharmonic growth is found to result from a resonant parametric interaction between the fundamental and the subharmonic in which phase-speed matching conditions are satisfied. In contrast, when the phase coupling level is low, the phase difference is irregular and varying, the efficiency of parametric interactions is low, phase-speed matching conditions are not met and subharmonic growth is suppressed.
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Qiu, Zhan, Wenhao Xu, Jiaxin Yuan, and Fuxin Wang. "Secondary resonances in aeroelastic response of oscillating airfoil under dynamic stall." Journal of Vibration and Control 24, no. 23 (March 26, 2018): 5665–80. http://dx.doi.org/10.1177/1077546318764191.
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To clarify the mechanism of the complex aeroelastic responses of flexible blades of helicopter rotors under dynamic stall, experiments on a 2D aeroelastic system are performed. In the spectra of the response from experiment results, special frequency components are found. Then, a numerical method based on the same aeroelastic model is introduced. Here, the flow field is solved using a zonal solver based on vorticity dynamics. When changing a system’s natural frequency, the same extra frequency components in the response spectra are found when particular ratios of natural and forcing frequencies are achieved. Secondary resonances are believed to then happen, which feature a larger response amplitude, multiple periodic motion and a subharmonic peak of driving frequency in the load spectra. With an analysis of the flow field, the 1/2 subharmonic in the airload spectra (i.e. the period doubling of the loads) is believed to be associated with the nonlinear variation of vortex structures. With a dynamic mode decomposition analysis, a counter-rotating vortex interaction instability is detected as the physical mechanism of period doubling. The coincidence of natural frequency with the odd times of the subharmonic leads to the secondary resonances.
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Meiburg, E. "On the role of subharmonic perturbations in the far wake." Journal of Fluid Mechanics 177 (April 1987): 83–107. http://dx.doi.org/10.1017/s0022112087000879.
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The possibility of an excitation of individual subharmonic perturbations in each of the shear layers forming the far wake is investigated numerically. Principal considerations allow for the existence of two equivalent subharmonic modes which by opposite routes can lead to a doubling of the wavelength in the wake. Since vortical disturbances in the far wake are amplified only convectively, the simultaneous existence of both modes in the flow field is possible, which could provide an explanation for the group structure observed experimentally in the far wake. These considerations also provide a logical explanation of the finding of a very regular vortex pairing process in forced wakes.Two-dimensional numerical simulations assuming incompressible flow and almost inviscid dynamics illustrate the opposite developments of regions dominated by the two different modes and also confirm the possibility of a resulting group structure. As an important result it is demonstrated that, if vortex pairing plays an important role in the growth of the far-wake structure, this does not have to be related to the excitation of the subharmonic peak in the frequency spectrum. Quite the contrary, it is to be expected that the subharmonic itself is of minor importance and that instead a small frequency and its multiples related to the group structure of the flow dominate the spectrum. In the light of these considerations measurements by Cimbala (1984) are discussed and frequency spectra recorded by him are analysed more closely. Various properties of these spectra seem to indicate that vortex pairing might be significant with respect to the evolution of the far-wake structure.
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Zhang, Liang, Mariano Méndez, Diego Altamirano, Jinlu Qu, Li Chen, Konstantinos Karpouzas, Tomaso M. Belloni, et al. "A systematic analysis of the phase lags associated with the type-C quasi-periodic oscillation in GRS 1915+105." Monthly Notices of the Royal Astronomical Society 494, no. 1 (March 24, 2020): 1375–86. http://dx.doi.org/10.1093/mnras/staa797.
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ABSTRACT We present a systematic analysis of the phase lags associated with the type-C quasi-periodic oscillations (QPOs) in GRS 1915+105 using RXTE data. Our sample comprises 620 RXTE observations with type-C QPOs ranging from ∼0.4 to ∼6.3 Hz. Based on our analysis, we confirm that the QPO phase lags decrease with QPO frequency, and change sign from positive to negative at a QPO frequency of ∼2 Hz. In addition, we find that the slope of this relation is significantly different between QPOs below and above 2 Hz. The relation between the QPO lags and QPO rms can be well fitted with a broken line: as the QPO lags go from negative to positive, the QPO rms first increases, reaching its maximum at around zero lag, and then decreases. The phase-lag behaviour of the subharmonic of the QPO is similar to that of the QPO fundamental, where the subharmonic lags decrease with subharmonic frequency and change sign from positive to negative at a subharmonic frequency of ∼1 Hz; on the contrary, the second harmonic of the QPO shows a quite different phase-lag behaviour, where all the second harmonics show hard lags that remain more or less constant. For both the QPO and its (sub)harmonics, the slope of the lag–energy spectra shows a similar evolution with frequency as the average phase lags. This suggests that the lag–energy spectra drive the average phase lags. We discuss the possibility for the change in lag sign, and the physical origin of the QPO lags.
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Kahraman, A., and G. W. Blankenship. "Experiments on Nonlinear Dynamic Behavior of an Oscillator With Clearance and Periodically Time-Varying Parameters." Journal of Applied Mechanics 64, no. 1 (March 1, 1997): 217–26. http://dx.doi.org/10.1115/1.2787276.
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A number of experiments on a physical system with clearance having combined parametric and external forcing excitation are presented. The data demonstrate several nonlinear phenomena that exist in periodically excited oscillators subject to clearance at a contact interface. Jump discontinuities in forced response curves are shown to exist and subharmonic resonances which are formed by period-n (nT) motions with n ≥ 2 are demonstrated. Long-period subharmonic motions with periods as long as nine times the excitation period are reported. Finally, several cases of measured chaotic motions are also illustrated through time-traces, phase portraits, Fourier spectra, and Poincare plots which form strange attractors.
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Spalart, Philippe R., and Kyung-Soo Yang. "Numerical study of ribbon-induced transition in Blasius flow." Journal of Fluid Mechanics 178 (May 1987): 345–65. http://dx.doi.org/10.1017/s0022112087001253.
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The early three-dimensional stages of transition in the Blasius boundary layer are studied by numerical solution of the Navier-Stokes equations. A finite-amplitude two-dimensional wave and low-amplitude three-dimensional random disturbances are introduced. Rapid amplification of the three-dimensional components is observed and leads to transition. For intermediate amplitudes of the two-dimensional wave the breakdown is of subharmonic type, and the dominant spanwise wavenumber increases with the amplitude. For high amplitudes the energy of the fundamental mode is comparable to the energy of the subharmonic mode, but never dominates it; the breakdown is of mixed type. Visualizations, energy histories, and spectra are presented. The sensitivity of the results to various physical and numerical parameters is studied. The agreement with experimental and theoretical results is discussed.
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Ali, Mohamed G. S., Nour Z. Elsayed, and Ebtsam A. Eid. "A Simulation for Detecting Nonlinear Echoes from Microbubbles Packets." Archives of Acoustics 40, no. 2 (June 1, 2015): 151–57. http://dx.doi.org/10.1515/aoa-2015-0017.
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Abstract This work presents a simulation of the response of packets of microbubbles in an ultrasonic pulse-echo scan line. Rayleigh-Plesset equation has been used to predict the echo from numerically obtained radial dynamics of microbubbles. Varying the number of scattering microbubbles on the pulse wave form has been discussed. To improve microbubble-specific imaging at high frequencies, the subharmonic and second harmonic signals from individual microbubbles as well as microbubbles packets were simulated as a function of size and pressure. Two different modes of harmonic generation have been distinguished. The strength and bandwidth of the subharmonic component in the scattering spectrum of microbubbles is greater than that of the second harmonic. The pressure spectra provide quantitative and detailed information on the dynamic behaviour of ultrasound contrast agent microbubbles packet.
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BREUER, KENNETH S., JACOB COHEN, and JOSEPH H. HARITONIDIS. "The late stages of transition induced by a low-amplitude wavepacket in a laminar boundary layer." Journal of Fluid Mechanics 340 (June 10, 1997): 395–411. http://dx.doi.org/10.1017/s0022112097005417.
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The evolution of a wavepacket in a laminar boundary layer is studied experimentally, paying particular attention to the stage just prior to the formation of a turbulent spot. The initial stages of development are found to be in very good agreement with previous results and indicate a stage in which the disturbance grows according to linear theory followed by a weakly nonlinear stage in which the subharmonic grows, apparently through a parametric resonance mechanism. In a third stage, strong non-linear interactions are observed in which the disturbance develops a streaky structure and the corresponding wavenumber–frequency spectra exhibit an organized cascade mechanism in which spectral peaks appear with increasing spanwise wavenumber and with frequencies which alternate between zero and the subharmonic frequency. Higher harmonics are also observed, although with lower amplitude than the low-frequency peaks. The final (breakdown) stage is characterized by the appearance of high-frequency oscillations with random phase, located at low-speed ‘spike’ regions of the primary disturbance. Wavelet transforms are used to analyse the structure of both coherent and random small-scale structure of the disturbance. In particular, the breakdown oscillations are also observed to have a wavepacket character riding on the large-amplitude primary disturbance.
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Zhao, J. Y., I. W. Linnett, and L. J. McLean. "Stability and Bifurcation of Unbalanced Response of a Squeeze Film Damped Flexible Rotor." Journal of Tribology 116, no. 2 (April 1, 1994): 361–68. http://dx.doi.org/10.1115/1.2927236.
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The stability and bifurcation of the unbalance response of a squeeze film damper-mounted flexible rotor are investigated based on the assumption of an incompressible lubricant together with the short bearing approximation and the “π” film cavitation model. The unbalanced rotor response is determined by the trigonometric collocation method and the stability of these solutions is then investigated using the Floquet transition matrix method. Numerical examples are given for both concentric and eccentric damper operations. Jump phenomenon, subharmonic, and quasi-periodic vibrations are predicted for a range of bearing and unbalance parameters. The predicted jump phenomenon, subharmonic and quasi-periodic vibrations are further examined by using a numerical integration scheme to predict damper trajectories, calculate Poincare´ maps and power spectra. It is concluded that the introduction of unpressurized squeeze film dampers may promote undesirable nonsynchronous vibrations.
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Hebboul, S. E., and J. C. Garland. "rf power dependence of subharmonic voltage spectra of two-dimensional Josephson-junction arrays." Physical Review B 47, no. 9 (March 1, 1993): 5190–95. http://dx.doi.org/10.1103/physrevb.47.5190.
Full textDissertations / Theses on the topic "Subharmonic spectra"
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Bunnell, Leah M. "SUBHARMONIC FREQUENCIES IN GUITAR SPECTRA." Cleveland State University / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=csu1624389777728368.
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Moolman, Ruan. "On the role of subharmonic functions in the spectral theory of general Banach algebras." Thesis, 2010. http://hdl.handle.net/10210/3026.
Full textConference papers on the topic "Subharmonic spectra"
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Ehrich, Fredric. "Spontaneous Sidebanding in High Speed Rotordynamics." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0225.
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Abstract Several observations have been made in the Fourier spectra of high speed rotordynamic response of uniformly spaced frequency spikes on either side of key synchronous or subharmonic or superharmonic response frequencies. In instances where this so-called “sidebanding” could not readily be explained as the nonlinear interaction or combination tones of two distinct responses at slightly different frequencies, we have referred to this class of phenomena as spontaneous sidebanding. It is invariably noted that the sideband spacing frequency appears to be a whole number fraction (1/J) of the operating speed which suggests that the wave form is periodic and completes a full cycle every J rotations of the rotor. Using a numerical model of a rotor which simulates local contact with a stator in close proximity as a bilinear spring, several studies have been carried out to explore the circumstances for this spontaneous sidebanding. Two general classes of this type of response have been found. A) For highly nonlinear systems, the chaotic-like response in transition zones between successive orders of subharmonic and superharmonic operation is actually periodic and results in spontaneous sidebands clustered around the key subharmonic or superharmonic frequencies. B) In trans-critical operation of highly nonlinear and lightly damped systems, a major sideband frequency spike is noted at a frequency which is approximately the system’s natural frequency. Recognition of this fact permits a simple estimate of the repetition index (J). All these observations from operation of the numerical model have been compared with experimental data derived from incidents of spontaneous sidebanding on aircraft gas turbine rotors. Excellent qualitative agreement has been found in most instances.
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Wang, Ruimin, Fengde Zong, and Yang Yang. "Influence of Parametric Resonance on a Bubble Driven by Intensive Sound During Stable Cavitation." In ASME 2008 International Mechanical Engineering Congress and Exposition. ASMEDC, 2008. http://dx.doi.org/10.1115/imece2008-68610.
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Ultrasonic cavitation is a well-known phenomenon that plays an important role in several physical systems and its applications are commonly utilized in different fields of physics and technology. The cavitation phenomena can be described by means of a field theory that should be able to predict the values of the macroscopic quantities, introducing physical parameters specifically for the bubbly liquid to be considered as a continuum; while on the other hand, the goal is to solve the problem of single bubble dynamics in an ultrasonic field as a starting point towards a multibubble theory. Usually the theory of single bubble dynamics in ultrasonic cavitation is constructed by primarily imposing the conditions of spherical symmetry on the bubble interface and a viscoelastic liquid, thus obtaining a significant simplification of the equations of motion and a single nonlinear equation for the interface. This approach can be satisfactory in several cases, but the situations in which the bubble deviates from its spherical shape (i.e. the collapse on a rigid boundary) and the problem of the stability of the interface motion, which turns out to be very important in sonoluminescence, cannot be treated by this theory. In the field of ultrasonic cavitation numerical analysis is a further means of investigation besides the analytical approach and experimental measurements, and it is necessary at least for two reasons. Specifically, an exact analytical treatment of the equations that model this phenomenon is substantially impossible due to their high nonlinearity; and furthermore the typical order of magnitude of the measurable quantities (object sizes in the range of microns, time intervals in the range of microseconds with nanosecond resolution) makes experiments difficult to perform. Hence we numerically analyze the relationships between amplitude and frequency by the use of SPECTRA PLUS software. The method is tested analyzing forced oscillations of cavitation bubbles excited by ultrasonic standing waves at different pressure amplitudes, showing characteristic behaviour of nonlinear dynamical systems; frequency spectra are obtained, stability analysis is performed. It is important to note that we observe subharmonic behaviour of the volume mode of the bubble prior to the instabilities due to shape modes. If one further increases the value of pressure amplitudes, one can clearly observe surface instabilities and deformations that lead to the destruction of the bubble. This evidence may suggest that the subharmonic behaviour leads to chaos in ultrasonic cavitation.
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Shen, Yanfeng, Nipon Roy, Junzhen Wang, Zixuan Liu, Danyu Rao, and Wu Xu. "Amplitude and Sweeping Direction Dependent Nonlinear Ultrasonic Resonance Spectroscopy for Fatigue Crack Detection." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-86221.
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This paper investigates the amplitude and sweeping direction dependent behavior of nonlinear ultrasonic resonance spectroscopy for fatigue crack detection. The Contact Acoustic Nonlinearity (CAN) and the nonlinear resonance phenomena are illuminated via a reduced-order bilinear oscillator model. Unlike conventional linear ultrasonic spectroscopy, which would not change its pattern under different amplitudes of excitation or the frequency sweeping direction, the nonlinear resonance spectroscopy, on the other hand, may be noticeably influenced by both the wave amplitude and the loading history. Both up-tuning and down-tuning sweeping active sensing tests with various levels of excitation amplitudes are performed on a fatigued specimen. Short time Fourier transform is adopted to obtain the time-frequency features of the sensing signal. Corresponding to each excitation frequency, a nonlinear resonance index can be established based on the amplitude ratio between the superhamronic, the subharmonic, the mixed-frequency response components and the fundament frequency. The measured nonlinear resonance spectroscopy for a certain amplitude and frequency sweeping direction can be readily used to establish an instantaneous baseline. The spectroscopy of a different amplitude or frequency sweeping direction can be compared with such an instantaneous baseline and a Damage Index (DI) is obtained by measuring the deviation between the two spectra. Experimental investigations using an aluminum plate with rivet hole nucleated fatigue cracks are performed. A series of nonlinear spectroscopies are analyzed for both the pristine case and the damaged case. The spectral features for both cases are obtained to demonstrate the proposed fatigue crack detection methodology which may find its application for structural health monitoring (SHM). The paper finishes with summary, concluding remarks, and suggestions for future work.
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Zhang, Weihao, Zhengping Zou, and Jian Ye. "Large Eddy Simulations of Separated Shear Layer Behaviors in a High-Lift LP Turbine." In ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/gt2015-43054.
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Separated shear layer behaviors under the conditions without and with elevated inlet freestream turbulence intensity (FSTI=2.5%) are investigated by using large eddy simulations (LES) in this paper. The study is performed on a low-Re high-lift LP turbine blade (T106D-EIZ) at Reynolds number (Re) of 60,154 (based on the chord and outflow velocity) and isentropic exit Mach number of 0.402. A number of detection points are set along the dividing streamlines and long time series of data are gathered. The time-mean flow and the transient vortex structures are discussed, which indicated the FSTI can suppress the open separation and thus improve the aeroperformance. Power spectra and wavelet analysis are applied to axial velocity and pressure signals of the detection points. The results show that although the vortices pairing are missed, the subharmonic of the roll-up frequency is still observable under the elevated FSTI condition. It is also found that both of the locations and power of the vortices roll-up, as well as vortices pairing, vary with time in large scales and these phenomena are weakened under the elevated FSTI condition.
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Guyot, Daniel, Christian Oliver Paschereit, and Surya Raghu. "A Fluidic Actuator for Active Combustion Control." In ASME Turbo Expo 2008: Power for Land, Sea, and Air. ASMEDC, 2008. http://dx.doi.org/10.1115/gt2008-50797.
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In this work the performance of a fluidic actuator in an active combustion control scheme is demonstrated. The actuator was tested in two different burner configurations, a bluff body burner and a generic swirl-stabilized burner, where it modulated parts of the fuel flow. The oscillation frequency was controlled by varying the inlet mass flow of the actuator. Fluidic actuators are of special interest for fuel-based active control schemes featuring high frequency fuel flow modulation, as they are much more durable then conventional valves due to the absence of fast moving parts. Hot wire measurements were performed to investigate the fluidic actuator’s oscillation characteristics without combustion. The actuator was then incorporated into a bluff body burner and a swirl-stabilized burner, respectively, where it modulated parts of the fuel flow blended with nitrogen. Pressure and heat release fluctuations in the combustor were recorded and images of the flame were taken. For both burners the heat release response of the flame to fuel flow modulation was first studied during stable combustion. The spectra of the heat release signals showed a clear peak corresponding to the fluidics’ oscillation frequency, thus validating the ability of the actuator to influence the combustion process. As the next step, each of the two combustors was operated at conditions that featured a strong low-frequency combustion instability when no fuel was modulated. In case of the bluff body burner applying fuel modulation resulted in attenuation of the combustion instability for some oscillation frequencies. The attenuation was highest when modulating the fuel flow in between the fundamental instability frequency and its subharmonic. Modulating the fuel flow at the subharmonic, however, resulted in an amplification of the instable mode. Also when applied to the swirl burner, the fludics’ fuel flow modulation caused a significant reduction of the pressure oscillations, although the actuator could only be operated at oscillation frequencies much lower than the instability frequency due to the attached tubes. The results obtained in this work show that the fluidic actuator in use allows for fuel modulation and hence combustion control without the need for complex and fast moving parts, thus ensuring a long actuator lifetime. This makes the fluidic actuator highly appropriate for application in industrial gas turbines.
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Esipov, Igor B., Andrey N. Vilman, Andrey V. Fokin, Bengt Enflo, Claes M. Hedberg, and Leif Kari. "Subharmonics Spectra of Acoustics Excitation in Granular Medium." In NONLINEAR ACOUSTICS - FUNDAMENTALS AND APPLICATIONS: 18th International Symposium on Nonlinear Acoustics - ISNA 18. AIP, 2008. http://dx.doi.org/10.1063/1.2956204.
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Liu, Baojiang, Litang Yan, Qihan Li, and Zigen Zhu. "Vibration of a Rotor With Extremely Nonlinear Stiffness Passing Through Critical Speeds." In ASME 1993 International Gas Turbine and Aeroengine Congress and Exposition. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/93-gt-027.
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Soft spring nonlinearity refers to supporting stiffness becomes smaller with increase of vibratory displacements. In this paper, transient response and subharmonic vibration of a rotor with extremely soft nonlinear spring characteristics are investigated. A simple but exact computer model of the phenomena has been evolved based on the numerical integration of a finite difference formulation. Response curves and wave forms of the rotor vibration passing through critical speeds are computed. The subharmonic vibration of the rotor with soft nonlinear spring characteristics has been observed by spectral analysis of vibratory response. It is shown that the rotor will tend to bounce when its vibratory displacements exceed the offset distance of the system with stiffness K1 when it is accelerated for passing through critical speeds and the vibratory motion after bouncing is predominantly at the forcing frequency. Maximum amplitude of transient response is affected mainly by damping ratio ξ, the degree of nonlinearity β and the offset distance h. A series of subharmonic pseudo critical peaks will appear in the acceleration of a rotor with soft nonlinear spring characteristics. But the general nature of the subharmonic vibration is much different from that with hard ones. Finally some computational results are demonstrated by an experiment.
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Leindecker, Nicholas, Alireza Marandi, Konstantin L. Vodopyanov, and Robert L. Byer. "Mid-IR spectral comb with broad instantaneous bandwidth using subharmonic OPO." In SPIE LASE, edited by Konstantin L. Vodopyanov. SPIE, 2011. http://dx.doi.org/10.1117/12.876203.
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Butikov, Eugene I. "Complicated Regular and Chaotic Motions of the Parametrically Excited Pendulum." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84388.
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Several new types of regular and chaotic behavior of the parametrically driven pendulum are discovered with the help of computer simulations. A simple physical explanation is suggested to the phenomenon of subharmonic resonances. The boundaries of these resonances in the parameter space and the spectral composition of corresponding stationary oscillations are determined theoretically and verified experimentally. A close relationship between the upper limit of stability of the dynamically stabilized inverted pendulum and parametric resonance of the non-inverted pendulum is established. Most of the newly discovered modes are still waiting a plausible physical explanation.
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Ehrich, Fredric. "Observations of Subcritical Superharmonic and Chaotic Response in Rotordynamics." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0222.
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Abstract When a rotor, excited by unbalance, is operating eccentrically within a clearance and in local contact with the stator it behaves as a bilinear oscillator with a natural periodic motion that resembles bouncing. When excited by unbalance at a subcritical rotative speed which is exactly or nearly 1/Nsuper times its natural frequency, the nonlinear system will respond by bouncing at or nearly at its natural frequency, or super harmonically at a frequency exactly Nsuper times the operating speed, or forcing frequency. As in supercritical subharmonic response, there is a zone of with characteristics of chaotic behavior in the transition zone between any order of superharmonic response and the next highest order superharmonic response. There is also an intricate pattern of progressive bifurcations of the orbit on entry into this characteristically chaotic region and a reverse progression on exit from this region. The response is a mirror image or reciprocal set of the more thoughly studied supercritical subharmonic response of the same bilinear oscillator system which, when excited by unbalance at a supercritical rotative speed which is exactly or nearly a whole number Nsub times its natural frequency, the nonlinear system will respond by bouncing at exactly or nearly its natural frequency at a frequency exactly 1/Nsub times the operating speed or forcing frequency. Such supercritical subharmonic response is also characterized by the appearance of characteristically chaotic behavior in the transition zone between successive orders of subharmonic response and by patterns of progressive bifurcations of the orbit on entry into and exit from each region of characteristically chaotic response. Various aspects of subcritical superharmonic response are studied in a numerical model of the nonlinear system, and are compared to data taken on the core spool of an aircraft engine gas turbine. The engine data show many of the unique characteristics of response, wave form and spectral content predicted by the numerical model of the bilinear oscillator when operating at subcritical rotative speed.