Amplitude Response Curves Research Articles

Stability and bifurcation analysis for a model of a nonlinear coupled pitch–roll ship

Using a combination of analytical and numerical methods, the paper studies the stability and bifurcations for a model of a nonlinear coupled pitch–roll ship. The model represents a two-degree-of-freedom system with quadratic coupling subjected to a modulated sinusoidal excitation. Three types of critical points for the bifurcation response equations near the combination resonance in the presence of internal resonance are considered. These points are characterized by a double zero and two negative eigenvalues, a double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues, respectively. For each case, the stability regions for the initial equilibrium solution and the critical bifurcation curves are obtained. The amplitude response curves with respect to detuning/damping parameters for critical bifurcation parameters are obtained for the first two types, and detailed analyses of the eigenvalues of the linearized system for each parametric region are given. For the third type, with the aid of normal form theory, the explicit expressions of the critical bifurcation curves leading to incipient and secondary bifurcations are obtained. Bifurcations leading to 2D, 3D tori and their stability conditions are also investigated. Some new dynamical behaviors are presented for this system.

Amplitude Response Curves of Frequency-Locked Rotations

This letter studies frequency-locked rotations in Nonlinear Phenomena and Analysis a phase-locked loop (PLL) circuit as FM demodulator. A rotation represents a desynchronized steady state in the PLL circuit and is regarded as another type of self-excited oscillations with natural rotation frequencies. The rotation frequency can be locked at driving frequencies of modulation signals. This letter shows response curves for harmonic amplitude of frequency-locked rotations. They have several different features from response curves of van der Pol oscillator.

Principal parametric resonances of a slender cantilever beam subject to axial narrow-band random excitation of its base

The non-linear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated. The method of multiple scales is used to determine a uniform first-order expansion of the solution of equations. According to solvability conditions, the non-linear modulation equations for the principal parametric resonance are obtained. Firstly, The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is quantificationally resolved, in which, the modified Bessel function of the first kind is introduced. Results show that the increase of the bandwidth facilitates the almost sure stability of the trivial response and stabilizes the system for a lower acceleration oscillating amplitude but intensifies the instability of the trivial response for a higher one. Secondly, the first and second order non-trivial steady state response of the system is obtained by perturbation method and the corresponding amplitude–frequency curves are calculated when the bandwidth is very small. Results show that the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the hardening type for the first mode, whereas for the second mode the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the softening type. Finally, the stochastic jump and bifurcation is investigated for the first and second modal parametric principal resonance. The basic jump phenomena indicate that, under the conditions of system parameters with a smaller bandwidth, the most probable motion is around the non-trivial branch of the amplitude response curve, whereas with a higher bandwidth, the most probable motion is around the trivial one of the amplitude response curve. However, the stochastic jump is sometimes more sensitive to the change of the bandwidth, in other words, a small change of bandwidth may induce a series of stochastic jump and bifurcation.

Pallidal stimulation modifies after-effects of paired associative stimulation on motor cortex excitability in primary generalised dystonia

Objective To determine the effect of globus pallidus internus (GPi) deep brain stimulation (DBS) on motor cortex plasticity in patients with primary generalised dystonia. Methods We studied 10 patients with primary generalised dystonia (5 DYT1+, 5 idiopathic, 5 female, mean age 42) following GPi DBS and 10 healthy subjects. Motor cortex plasticity was assessed using transcranial magnetic stimulation (TMS) paired associative stimulation (PAS) of motor cortex and median nerve, a method which has been shown in healthy subjects to produce LTP-like effects. Thresholds and TMS intensity to produce a resting motor evoked potential (MEP) of 1 mV were determined. Resting MEP amplitude and stimulus response curves were recorded before and after PAS. Patients were recorded ON and OFF DBS in separate sessions. Results The mean TMS intensity to produce a resting MEP of 1 mV was 54% of maximum stimulator output when OFF and 52% ON DBS. Fifteen minutes after PAS the resting MEP amplitude increased in patients OFF DBS and in control subjects whereas it decreased in patients ON DBS. Similarly, after PAS, the mean amplitude of the stimulus response curve increased OFF DBS, but this effect was abolished with DBS ON. Furthermore, patients who had the largest clinical response to chronic DBS also had the largest difference in the effect of PAS with DBS ON vs. OFF. Conclusions After PAS, patients with primary generalised dystonia showed a similar pattern of increased motor cortex excitability as healthy subjects when GPi DBS was OFF but not with GPi DBS ON. These results suggest that GPi DBS may reduce LTP-like motor cortex plasticity, which could contribute to its mechanism of action in dystonia.

Intraoperative evoked potentials monitoring for cochlear implantation in common cavities

Objectives: To determine the efficacy of intraoperative evoked compound action potential (ECAP) monitoring for optimal electrode placement and postoperative programming in cochlear implantation for patients with common cavity malformations. Methods: Retrospective case review of 1 child with a common cavity malformation undergoing revision cochlear implantation with a Clarion HiRes 90K™ implant. Outcome measures included neural response amplitude curves from intraoperative ECAP for each of 16 independent output circuits. Postoperative ECAP aided in mapping configuration to adjust threshold (T) and most comfortable loudness (M) levels across hearing frequencies. Results: Intraoperative ECAP confirmed proximity of the electrode to the neural response elements in this child with a common cavity defect by demonstrating expected neural response amplitudes across 10 of the 16 independent output circuits. No intraoperative repositioning of the electrode was required. Postoperative computed tomography imaging confirmed the location of electrodes with present ECAP responses to be along the medial most wall of the cavity. Postoperative ECAP aided the adjustment of mapping configuration to optimize threshold levels across frequencies for this child. Conclusion: Intraoperative ECAP can aid placement of a cochlear electrode in patients with common cavity malformations, particularly when the location of the neural response elements within the malformed otic capsule are unknown. ECAP can also assist in optimizing the postoperative mapping configurations in these patients.

Multiple components in distortion product otoacoustic emissions from the amphibian ear

Distortion product otoacoustic emissions (DPOAEs) are weak intermodulation distortions generated in the inner ear in response to two stimulus tones. Both the amphibian papilla and the basilar papilla in the inner of the frog may generate DPOAEs [Van Dijk and Manley, Hear. Res. 153, 14–22 (2001)]. Here, we measured the level and phase of DPOAEs in the Leopard frog, Rana pipiens, in response to stimulus tones between 40- and 90-dB SPL. Results show that for stimulus tones in the amphibian papilla frequency range, two components contribute to DPOAEs. One component dominates for stimulus levels below about 70-dB SPL, while the other is most prominent at higher levels. The transition between both components is accompanied by a conspicuous phase change, and sometimes by a notch in the amplitude response curve. Similar results were obtained in the basilar papilla frequency range, but in addition a third component was present for stimuli below about 55-dB SPL. With the exception of this third component, our findings are remarkably similar to those in mammals, despite the structural differences between the mammalian and amphibian inner ear. [Work supported by NWO.]

Analog simulation of the dynamics of a van der Pol oscillator coupled to a Duffing oscillator

This paper studies, with analog simulators, the dynamics of a system consisting of a van der Pol oscillator coupled to a Duffing oscillator. Amplitude-response curves are obtained in the case of internal resonance. The jump or hysteresis phenomenon is found. Various bifurcations are observed, and it is found that chaos can appear suddenly, through period-adding or through torus breakdown. The experimental results are compared with the numerical solutions.

Evidence for active, nonlinear, negative feedback in the vibration response of the apical region of the in-vivo guinea-pig cochlea

The transverse vibration response of the organ of Corti near the apical end of the guinea-pig cochlea was measured in vivo. For cochleae in good physiological condition, as ascertained with threshold compound action potentials and the endocochlear potential, increasing amounts of attenuation and phase lag were found as the intensity was decreased below 80 dB SPL. These nonlinear phenomena disappeared post mortem. The data suggest that an active, nonlinear damping mechanism exists at low intensities at the apex of the cochlea. The phase nonlinearity, evident at all frequencies except at the best frequency (BF), was limited to a total phase change of 0.25 cycles, implying negative feedback of electromechanical force from the outer hair cells into a compliant organ of Corti. The amplitude nonlinearity was largest above BF, possibly due to interaction with a second vibration mode. The high-frequency flank of the amplitude response curve was shifted to lower frequencies by as much as 0.6 octave (oct) for a 50-dB reduction of sound intensity; the reduction of BF was 0.3 oct, but there was no change of relative bandwidth (Q 10 dB). Detailed frequency responses measured at 60 dB SPL were consistent with non-dispersive, travelling-wave motion: travel time to the place of BF (400 Hz at 60 dB SPL) was 2.9 ms, Q 10 dB was 1.0; standing-wave motion occurred above 600 Hz. Based on comparison with neural and mechanical data from the base of the cochlea, amplitudes at the apex appear to be sufficient to yield behavioural thresholds. It is concluded that active negative feedback may be a hallmark of the entire cochlea at low stimulus frequencies and that, in contrast to the base, the apex does not require active amplification.

Modeling circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators: phase shifts and phase response curves.

Circadian rhythm generation in the suprachiasmatic nucleus was modeled by locally coupled self-sustained oscillators. The model is composed of 10,000 oscillators, arranged in a square array. Coupling between oscillators and standard deviation of (randomly determined) intrinsic oscillator periods were varied. A stable overall rhythm emerged. The model behavior was investigated for phase shifts of a 24-h zeitgeber cycle. Prolongation of either the dark or the light phase resulted in a lengthening of the period, whereas shortening of the dark or the light phase shortened the period. The model's response to shifts in the light-dark cycle was dependent only on the extent of the shift and was insensitive to changes in parameters. Phase response curves (PRC) and amplitude response curves were determined for single and triple 5-h light pulses (1000 lux). Single pulses lead to type 1 PRCs with larger phase shifts for weak coupling. Triple pulses generally evoked type 1 PRCs with the exception of weak coupling, where a type 0 PRC was observed.

AN EFFECT OF TURBULENCE ON FLUIDELASTIC INSTABILITY IN TUBE BUNDLES: A NONLINEAR ANALYSIS

This paper is concerned with the behaviour of a tube bundle subjected to combined fluidelastic and turbulence excitation. Here, we formulate the fluidelastic forces based on a simplified, nonlinear model for a single flexible tube surrounded by rigid neighbours and constrained to move transverse to the mean flow. We use a flat power spectral density function to express the turbulence excitation. The resulting system we first examine heuristically, based on a superposition of both excitation mechanisms. We then assess the merits of this approach via direct numerical integration of the equation of motion. Lastly, we perform a nonlinear investigation into the sensitivity of the fluidelastic stability boundary on variations in the random field of turbulence and generate a stability map. The analysis shows that the fluidelastic stability boundary defined by an unstable bifurcation may be reduced by turbulence; for long-term operation, the threshold reduction may approach the size of a hysteresis region. This effect increases with turbulence intensity and decreases with unstable-limit-cycle amplitude. For a stable bifurcation, the fluidelastic stability boundary is virtually unaffected by turbulence. In the latter case, the effect of turbulence is through practical stability definitions made using amplitude–response curves.

An Experimental Investigation of Multimode Responses in a Cantilever Beam

An experimental investigation into the planar, multimode response of a cantilever metallic beam to a transverse harmonic excitation is presented. The cantilever beam was tested in both the horizontal and vertical directions. In the vertical configuration, the beam was excited near its fourth natural frequency and energy was transferred from the directly excited fourth mode to a low-frequency mode through both resonant and nonresonant modal interactions. In the latter case, the response contained contributions from the first and fourth modes, whereas in the former case, the response contained contributions from the fourth, second, and fifth modes. The mechanism responsible for the resonant interaction involving the three modes is a subcombination internal resonance of the additive type; that is, Ω ≈ ω4 ≈ 1/2(ω2 + ω5), where the ωi are the natural frequencies of the beam. This type of resonance occurs in systems with a dominant cubic nonlinearity. In the horizontal configuration, the beam was excited near its fourth and sixth natural frequencies. Again, energy was transferred to a low-frequency mode, but in this case only through a resonant interaction due to a subcombination internal resonance of the additive type. Power spectra, time series, and frequency- and amplitude-response curves were obtained for characterization of the dynamic multimode responses.

Investigation of relative effects of mass and damping in vortex-induced vibration of a circular cylinder

Abstract An experimental study of vortex-induced vibration of a circular cylinder was conducted in a unique experimental facility especially designed for low mass and low damping. In this low mass-damping regime, two separate regions of the high amplitude response curve were identified, which we label “upper” and “lower” branches of response. A systematic study including twelve responses was performed to determine the effect of separately varying the nondimensional mass and varying the nondimensional damping. It seems apparent over the range of our present investigation that the mass - damping (the product of mass and damping) collapses well the amplitude of response, indicating two distinct amplitude curves, rather than the single, unique relationship previously supposed.

Nonlinear vibrations in beams and frames: The effect of the deformed equilibrium state

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.

Transit-time limited high-frequency response characteristics of MSM photodetectors

A simplified, but useful, one-dimensional transit-time limited frequency domain current response model of MSM photodetectors is presented. The model includes the effects of unequal velocities and lifetimes governing the lateral transport of photogenerated holes and electrons in the absorption layer. Normalized frequency response curves depicting the amplitude and phase versus the frequency are presented using the transit-time ratio as a parameter. The usefulness of the analytical response amplitude curves for the extraction of hole and electron transit times has been demonstrated by regressively fitting them with the measured results obtained from practical MSM photodetectors.

Correction circuit and method for improving the transient behavior of a two-way loudspeaker system

A circuit for improving the transient behavior of a two-way loudspeaker system includes a crossover circuit with high selectivity, amplitude and phase correction circuitry for separately correcting the amplitude and phase responses of the high and low frequency drivers in their mounting environment, and correction circuitry for correcting the composite amplitude and phase response of the overall loudspeaker system after insertion of the crossover. A further phase offset technique and circuit provides for introducing frequency dependent phase shift in the loudspeaker system's high or low frequency channels for offsetting the phase responses of the high and low frequency drivers within the crossover frequency range. According to the phase offset technique of the invention, phase shift is added, preferably in the high frequency channel, until composite amplitude response curves observed on-axis and at different vertical angles off-axis are forced to be consistent. After consistency is achieved the deterioration of the amplitude response resulting from the phase offset is corrected to a flat response by means of a forced series amplitude correction circuit inserted before the crossover. The result is improved transient response off-axis as well as on-axis.

Simple models for the geochemical response of the ocean to climatic and tectonic forcing

Simple models for the geochemical response of the ocean to periodic forcing are used as a framework to discuss the shortest periods of change resolvable by present analytical precision of elemental or isotopic ratio measurements of the chemical evolution of seawater. For chemical properties whose budget is dominated by fluxes into and out of the ocean, we show that the geochemical response to changing fluxes is strongly attenuated at periods shorter than the residence time, and that the response at these periods is also phase shifted so as to lag the forcing by one quarter cycle. The amplitude response as a function of period is used to estimate the periods of forcing resolvable by present analytical precision in measuring the 87Sr/ 86Sr and 234U/ 238U evolution of seawater, and we find that the shortest periods resolvable are of the order of 1 myr. If variations at periods shorter than this are found, we would argue that these are very likely of local, as opposed to global, origin. We illustrate the importance of phase shifts as a function of forcing period by discussing some recently published data that show a high degree of correlation (over the last 300 kyr) between changes in 87Sr/ 86Sr and δ 18O measured in the same sediment core. At first sight this correlation might be taken as good evidence that both 87Sr/ 86Sr and δ 18O are responding to some common climatic forcing, but given the long residence time of Sr in the ocean, Sr and O should in fact have different phase shifts relative to any proposed common forcing. Thus the apparent correlation between 87Sr/ 86Sr and δ 18O over the past several 100 Ma is not necessarily an indication that they are responding to a common cause. Amplitude response curves are also used to place limits on permissible global seawater 234U/ 238U variations, and these are then compared with published data to identify local (diagenetic) effects. Application to 187Os/ 186Os and U records in seawater detectors is proposed.

Two-degree-of-freedom systems with quadratic non-linearities subjected to parametric and self excitation

The method of multiple scales is used to study the response of two-degree-of-freedom systems with quadratic non-linearities under the simultaneous effects of a harmonic parametric excitation and self excitation. The principal parametric resonance of the first mode and a three-to-one internal resonance is considered, followed by the case of internal and parametric resonance of the second mode. In both cases the stability of the system is also studied. Amplitude and frequency response curves are presented for both cases. The character of stability and the mode in which the system loses its stability is also discussed.

Autosomal dominant rod-cone dysplasia in the Rdy cat : 2. Electrophysiological findings

Electroretinography was performed on cats affected with autosomal dominant rod-cone dysplasia (gene symbol Rdy). In normal kittens it was found that retinal sensitivity increased and rod thresholds decreased as the animals matured. Electroretinogram (ERG) amplitudes were mature by 4·5 weeks and adult timing was attained by 6 weeks of age, consistent with the findings of other workers. In Rdy-affected heterozygous kittens the ERG was absent or barely recordable using conventional corneal contact lens electrodes. However, the enhanced sensitivity of an intravitreal needle electrode permitted the recording of ERGs from affected kittens aged 4·5 weeks and older. The intravitreally recorded scotopic ERG in Rdy-affected kittens was a very low amplitude, largely negative response with prolonged a- and b-wave times-to-peak (two to threefold longer than in comparable recordings from an age-matched normal kitten). The b-wave lacked oscillatory potentials and was relatively small so that the ERG was a-wave dominated. This was attributed to delayed and defective synaptogenesis in the outer plexiform layer of dystrophic retinas. In contrast to normal kittens, the b-wave threshold was higher than that of the a-wave in affected kittens. Photopic responses were unrecordable. The intravitreal ERG was barely recordable in a 5-month-old Rdy-affected cat and was apparently extinguished by 7 months of age. In vitro electroretinography permitted a comparison of the photoreceptor responses (fast PIII) from the isolated retinas of 6-week-old control and Rdy-affected heterozygous kittens. Maximum fast PIII amplitudes were reduced by about 75% in affected retinas compared with age-matched normal retinas ( P < 0·005). The mean fast PIII time-to-peak, at maximum light intensity, in Rdy-affected retinas was prolonged by about 15 msec and was approximately twofold longer than the time-to-peak of normal retinas ( P < 0·005). The fast PIII response threshold (light intensity at 5 μV response) was increased by 1·0 log unit in affected retinas, and there was also a 0·94 log unit reduction in photoreceptor sensitivity (σ), indicated by the shift of the amplitude-response curve ( V-log I function) towards higher light intensities and a corresponding increase in semi-saturation luminance. These findings correlate well with the paucity of photoreceptor outer segment material, and therefore reduced rhodopsin content, seen histologically in affected retinas. The altered slope of the fast PIII V-log I function of Rdy-affected retinas indicated a smaller increase in relative fast PIII amplitude per log unit light intensity in dystrophic photoreceptors. The fast PIII time-to-peak vs. light intensity curve ( T-log I function) of Rdy-affected retinas had a steeper slope with relatively greater prolongations in time-to-peak at lower luminances compared with normal retinas ( P < 0·025). These changes in temporal characteristics may be explained either by severe disorganization of photoreceptor outer segments or by altered phototransduction kinetics.

Force Oscillator Model for Rectangular Cylinder

In 1970, Hartlen and Currie published a theory which described the transverse oscillations of a circular cylinder in a normal steady wind. This theory is used here to successfully describe the longitudinal oscillations of a rectangular cylinder with a side length ratio of 10 and standing at an angle of attack of 30% to the oncoming flow of water. This configuration is similar to that sometimes found in trashracks. The model is based on the notion that the fluctuating longitudinal force associated with vortex shedding and exciting the oscillations may be considered to arise from the action of something analogous to an oscillator in the separated flow. The model fndicates a dynamic interaction between the vortex shedding and the cylinder motion. Good agreement between measured amplitude and frequency response curves was obtained. This agreement suggests that a fundamental similarity exists in the physics of the transverse oscillations of Hartlen's and Currie's circular cylinder and the longitudinal oscillations of the rectangular cylinder investigated in the present study.

Reflection perceptibility threshold on NTSC signals and its measurement technique in a CATV system

The subjective effects of reflection impairment on NTSC pictures are established. Reflections in a CATV system can be analyzed using a broadband technique. The proposed RF model formulates a simple representation to describe the impairment caused by reflections. In the RF domain, reflections manifest themselves as ripple in the amplitude vs. frequency response curve. As the reflection gets closer in time to the transmitted signal, the number of voltage ripples in a given bandwidth decreases. By incorporating adequate computing and processing power in a CATV sweep system receiver, reflections can be characterized on site. Though such an instrument does not exist in the market, the author believes that this instrument, if available, would assist the CATV operator in carrying out preventive maintenance, such as locating mediocre cable connections/splices or devices with poor return loss more efficiently. >