Superharmonic Frequency Research Articles

Spontaneous Sidebanding in High Speed Rotordynamics

Several observations have been made in the Fourier spectra of high speed rotor-dynamic 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 stimuli 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 in systems that are effectively single-degree-of-freedom: (A) For highly nonlinear systems, the chaotic-like response in transition zones between successive orders of subharmonic and superharmonic operation is actually periodic, with a repetition index (J), and results in spontaneous sidebands clustered around the key subharmonic or superharmonic frequencies. No systematic relationship has been determined for the value of (J). (B) In transcritical operation of highly nonlinear and very 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.

Local methods in predicting occurrence of chaos in two-well potential systems: Superharmonic frequency region

Approximate criteria for chaos in two-well potential systems excited by harmonic force are considered. The excitation frequency is assumed to be close to 0·5 the natural frequency of the Small Orbit: i.e., close to the second order superharmonic resonance. Computer simulations reveal a scenario of transition to chaos significantly different from that at the principal resonance. Here both periodic attractors, of the Small and Large Orbits, lose stability and jointly turn into a strange attractor. Classical analytical approximate techniques are applied to analyze approximate periodic solutions and their higher order instabilities. The symmetry breaking instabilities of the Large Orbit and the first order instability of the approximate non-resonant Small Orbit solution are used to estimate the parameter region in which chaos may occur.

Microwave phase locking of Josephson-junction fluxon oscillators.

Application of the classic McLaughlin-Scott soliton perturbation theory to a Josephson-junction fluxon subjected to a microwave field that interacts with the fluxon only at the junction boundaries reduces the problem of phase locking of the fluxon oscillation to the study of a two-dimensional functional map. Phase-locked states correspond to fixed points of the map. For junctions of in-line geometry, the existence and stability of such fixed points can be studied analytically. Study of overlap-geometry junctions requires the numerical inversion of a functional equation, but the results are qualitatively very similar. The map predicts significantly different behaviors for locking at odd and even subharmonic frequencies and at superharmonic frequencies. It also gives indications regarding hysteresis in the current-voltage characteristic, the existence of zero-crossing steps, and a description of the locking process in the frequency domain. The principal merit of the map is that it captures much of the experimental phenomenology at a very low computational cost.

Nonlinear resonant interactions in internal cavity flows

F the suggestion of Flandro and Jacobs that periodic vortex shedding from internal obstructions could produce significant oscillating pressure levels in turbulent duct flow, Culick and Magiawala, Dunlop and Brown, and Brown et at. presented a variety of results verifying the phenomenon of acoustic energy production by vortex shedding and interaction. In further investigation of some of the flow mechanisms involved in acoustic energy production by vortex shedding, Isaacson and Marshall presented measurements of oscillating velocity frequency spectra which indicated that in certain flow regimes, velocity'oscillations occur which satisfy the frequency resonant interaction. This Note presents results of bispectral cross-power measurements of trios of frequencies occurring in an internal cavity in a turbulent flow which provide additional evidence that these trios of frequencies are of quadratic interaction origin. These measurements were made in a subsonic wind tunnel which is driven by a centrifugal fan, driving air into a settling chamber which is 0.61 m on each side and 1.22 m long. The settling chamber is segmented and contains four damping screens, which act to reduce the turbulence of the flow into the test section. A converging section connects the settling chamber to the test section which is 0.178 m on each side and 2.44m long. The internal flow cavity is made up of four plexiglas restrictors, with the forward set of restrictors located 1.89 m downstream of the entrance to the test section. A schematic diagram of the flow cavity is shown in Fig. 1. For the measurements reported here, the gap between the forward pair of restrictors was 53.3 mm and the distance between the two pairs of restrictors was 101.6 mm. A TSI Model 1050 constant-temperature hot-film anemometer, together with a TSI Model 1055 linearizer, and a TSI Model 1057 signal conditioner were used for these measurements. The single sensor, located a distance of 16 mm behind the forward restrictor pair along .the flow axis of zero vertical velocity, was used to measure the velocity frequency spectrum presented in Fig. 2. The output of this sensor was also used to generate the bispectral measurements indicated in Fig. 3. The frequency spectra were obtained with a HewlettPackard Model 3590A wave analyzer and an HP Model 3594A sweeping local oscillator. The analog output of the analyzer was digitally processed with, a Hewlett-Packard Model 2240A measurement and control processor, with an HP-85 computer as the controller. The data were plotted on a Hewlett-Packard Model 7225A plotter. Note that the cavity configuration used in this experiment produces two converging, almost parallel, shear layers at the forward baffles. Kelly has shown that shear layers of this type produce instabilities of both subharmonic and superharmonic frequencies relative to the most unstable frequency determined by the Orr-Sommerfeld equations. Phillips has shown that waves of this type can be turned away from the critical region of the shear layer and

Nonlinear Response of a Straight Beam to Narrow-Band Excitation

The response of a nonlinear oscillator to narrow-band random excitation is investigated experimentally using a straight beam with clamped ends excited by motion of its supporting base. Multivalued or “jump”-type response behavior is observed for a range of excitation levels centered near the first-mode resonant frequency. The time spent in response levels and the magnitudes of the levels are studied and compared with harmonic response. Large peaks in displacement response are seen for excitation centered at the third-order superharmonic frequency.