### 108 results for qubit oscillator frequency

Contributors: João Casaleiro, Luís B. Oliveira, Igor M. Filanovsky

Date: 2016-01-01

Relation between the **oscillation** **frequency** and the coupling strength.
...Capacitive coupled RC-**oscillators**.
...RC-**oscillators**...Coupled **oscillators**...(a) Single RC **oscillator** and (b) small-signal equivalent circuit.
...In this paper the capacitive coupling in quadrature RC-**oscillators** is investigated. The capacitive coupling has the advantages of being noiseless with a small area penalty and without increasing the power dissipation. The results show that a phase error below 1° and an amplitude mismatch lower than 1% are obtained with a coupling capacitance about 20% of the **oscillator**׳s capacitance value. Due to this kind of coupling, the phase-noise improves by 3dB (to −115.1 dBc/Hz @ 10MHz) and the increase of power requirement is only marginal leading to a figure-of-merit of −154.8dBc/Hz. This is comparable to the best state-of-the-art RC-**oscillators**, yet the dissipated power is about four times less. We present calculations of **frequency**, phase error and amplitude mismatch that are validated by simulations. The theory shows that phase error is proportional to the amplitude mismatch, indicating that an automatic phase error minimization based on the amplitude mismatches is possible. The measurements on a 2.4GHz voltage-controlled quadrature RC-**oscillator** with capacitive coupling fabricated in 130nm CMOS circuit prototypes validate the theory....Quadrature **oscillator**...Simulated **frequency**.
...Van der Pol **oscillators**...**Frequency** of **oscillation** with the **oscillators** uncoupled and coupled (CX=20fF).
... In this paper the capacitive coupling in quadrature RC-**oscillators** is investigated. The capacitive coupling has the advantages of being noiseless with a small area penalty and without increasing the power dissipation. The results show that a phase error below 1° and an amplitude mismatch lower than 1% are obtained with a coupling capacitance about 20% of the **oscillator**׳s capacitance value. Due to this kind of coupling, the phase-noise improves by 3dB (to −115.1 dBc/Hz @ 10MHz) and the increase of power requirement is only marginal leading to a figure-of-merit of −154.8dBc/Hz. This is comparable to the best state-of-the-art RC-**oscillators**, yet the dissipated power is about four times less. We present calculations of **frequency**, phase error and amplitude mismatch that are validated by simulations. The theory shows that phase error is proportional to the amplitude mismatch, indicating that an automatic phase error minimization based on the amplitude mismatches is possible. The measurements on a 2.4GHz voltage-controlled quadrature RC-**oscillator** with capacitive coupling fabricated in 130nm CMOS circuit prototypes validate the theory.

Data types:

Contributors: Shuai Zeng, Bing Li, Shaoqun Zeng, Shangbin Chen

Date: 2009-11-04

The purpose of this computational study was to investigate the possible role of voltage-gated Ca2+ channels in spontaneous Ca2+ **oscillations** of astrocytes. By incorporating different types of voltage-gated Ca2+ channels and a previous model, this study reproduced typical Ca2+ **oscillations** in silico. Our model could mimic the oscillatory phenomenon under a wide range of experimental conditions, including resting membrane potential (−75 to −60 mV), extracellular Ca2+ concentration (0.1 to 1500 μM), temperature (20 to 37°C), and blocking specific Ca2+ channels. By varying the experimental conditions, the amplitude and duration of Ca2+ **oscillations** changed slightly (both **frequency** changed significantly (∼400%). This indicates that spontaneous Ca2+ **oscillations** in astrocytes might be an all-or-none process, which might be **frequency**-encoded in signaling. Moreover, the properties of Ca2+ **oscillations** were found to be related to the dynamics of Ca2+ influx, and not only to a constant influx. Therefore, calcium channels dynamics should be used in studying Ca2+ **oscillations**. This work provides a platform to explore the still unclear mechanism of spontaneous Ca2+ **oscillations** in astrocytes....Typical spontaneous Ca2+ **oscillations** from the computational study. From top to bottom, the three plots correspond to **oscillations** in cytoplasmic Ca2+, ER Ca2+, and cytoplasmic IP3. All three variables have the same **frequency** but different peak times (details are shown in Fig. 4).
...(A) Bifurcation diagram of Ca2+ **oscillations** as a function of membrane potential. Sustained Ca2+ **oscillations** occurred in the potential range of −70.0 to −64.9 mV, where the maximum and minimum of Ca2+ **oscillations** were plotted. The dashed line refers to the unstable steady state. Out of the oscillatory domain, the system evolved into a stable steady state. (B) **Frequency** of Ca2+ **oscillations** versus membrane potential.
...Dependence of Ca2+ **oscillations** on extracellular Ca2+ concentration. Ca2+ **oscillations** stopped when the extracellular Ca2+ concentration was too low or too high. From 0.1 to 1500 μM, the **frequency** of Ca2+ **oscillations** increased with a rise in extracellular Ca2+ concentration.
...Amplitude and **frequency** of Ca2+ **oscillations** versus temperature. In the temperature range of 20–37°C, both the amplitude (indicated as an asterisk) and **frequency** (dotted line) decreased with temperature.
...The occurrence of Ca2+ **oscillation** depends on the membrane potential. When the membrane potential is −64.9 mV, there is no Ca2+ **oscillation**. Within −70.0 to −64.9 mV, the **frequency** and amplitude of Ca2+ **oscillations** change with the membrane potential.
... The purpose of this computational study was to investigate the possible role of voltage-gated Ca2+ channels in spontaneous Ca2+ **oscillations** of astrocytes. By incorporating different types of voltage-gated Ca2+ channels and a previous model, this study reproduced typical Ca2+ **oscillations** in silico. Our model could mimic the oscillatory phenomenon under a wide range of experimental conditions, including resting membrane potential (−75 to −60 mV), extracellular Ca2+ concentration (0.1 to 1500 μM), temperature (20 to 37°C), and blocking specific Ca2+ channels. By varying the experimental conditions, the amplitude and duration of Ca2+ **oscillations** changed slightly (both <25%), while the **frequency** changed significantly (∼400%). This indicates that spontaneous Ca2+ **oscillations** in astrocytes might be an all-or-none process, which might be **frequency**-encoded in signaling. Moreover, the properties of Ca2+ **oscillations** were found to be related to the dynamics of Ca2+ influx, and not only to a constant influx. Therefore, calcium channels dynamics should be used in studying Ca2+ **oscillations**. This work provides a platform to explore the still unclear mechanism of spontaneous Ca2+ **oscillations** in astrocytes.

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Contributors: R. Kalter, M.J. Tummers, S. Kenjereš, B.W. Righolt, C.R. Kleijn

Date: 2014-01-01

(a) Time series for the first chrono-mode of the POD, a1(t), for the three different forcings with vin=0.4m/s (Re=3.1in×103, N=0.02). (b) Power spectra of the chrono-modes a1(t). **Frequency** peaks are found at fPOD=0.027Hz (FL0). The values of the **frequency** peaks are in reasonable agreement with the **frequencies** found for the free surface fluctuations, fTS.
...(a–c) Profiles of the turbulence kinetic energy kturb,2D. (d–f) Profiles of the kinetic energy associated with the large-scale **oscillations** kosc,2D. The inlet velocity is vin=0.4m/s (Rein=3.1×103, N=0.02).
...Amplitude A and **frequency** fTS of the free surface **oscillation** at a monitoring point at x=0.175m for the three different forcings (Rein=3.1×103, N=0.02). Dominant **frequency** fPOD from the power spectrum of the first chrono mode of the POD.
...The effect of Lorentz forcing on self-sustained **oscillations** of turbulent jets (Re=3.1×103) issuing from a submerged bifurcated nozzle into a thin rectangular liquid filled cavity was investigated using free surface visualization and time-resolved particle image velocimetry (PIV). A Lorentz force is produced by applying an electrical current across the width of the cavity in conjunction with a magnetic field. As a working fluid a saline solution is used. The Lorentz force can be directed downward (FL0), to weaken or strengthen the self-sustained jet **oscillations**. The low **frequency** self-sustained jet **oscillations** induce a free surface **oscillation**. When FLoscillation is reduced by a factor of 6 and when FL>0 the free surface **oscillation** amplitude is enhanced by a factor of 1.5....Self-sustained **oscillations** ... The effect of Lorentz forcing on self-sustained **oscillations** of turbulent jets (Re=3.1×103) issuing from a submerged bifurcated nozzle into a thin rectangular liquid filled cavity was investigated using free surface visualization and time-resolved particle image velocimetry (PIV). A Lorentz force is produced by applying an electrical current across the width of the cavity in conjunction with a magnetic field. As a working fluid a saline solution is used. The Lorentz force can be directed downward (FL0), to weaken or strengthen the self-sustained jet **oscillations**. The low **frequency** self-sustained jet **oscillations** induce a free surface **oscillation**. When FL0 the free surface **oscillation** amplitude is enhanced by a factor of 1.5.

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Contributors: Tara A. Whitten, Adam M. Hughes, Clayton T. Dickson, Jeremy B. Caplan

Date: 2011-01-15

**Oscillation** detection in a single electrode with weak alpha. The electrode was selected from the same subject as in Figs. 2 and 4. (A) The 256-electrode array with the selected electrode highlighted in yellow. (B) Background wavelet power spectrum mean and standard deviation (blue), and the linear regression fit to the background (green). (C) **Oscillations** detected across all **frequencies** by the oscillatory episode detection method. Red vertical lines indicate when participants were instructed to close their eyes and black vertical lines indicate when participants were instructed to open their eyes. (D) The proportion of time (Pepisode) during the eyes -closed condition (red) and eyes-open condition (black) that **oscillations** were detected at each **frequency**. (E) The raw signal from the chosen electrode, with detected **oscillations** at the peak alpha **frequency** (9.5Hz) highlighted in red. Vertical lines are the same as above. (F) An expansion of the highlighted section in E, to show the spindle-like appearance of the alpha **oscillation**.
...Oscillatory activity is a principal mode of operation in the brain. Despite an intense resurgence of interest in the mechanisms and functions of brain rhythms, methods for the detection and analysis of oscillatory activity in neurophysiological recordings are still highly variable across studies. We recently proposed a method for detecting oscillatory activity from time series data, which we call the BOSC (Better **OSCillation** detection) method. This method produces systematic, objective, and consistent results across **frequencies**, brain regions and tasks. It does so by modeling the functional form of the background spectrum by fitting the empirically observed spectrum at the recording site. This minimizes bias in **oscillation** detection across **frequency**, region and task. Here we show that the method is also robust to dramatic changes in state that are known to influence the shape of the power spectrum, namely, the presence versus absence of the alpha rhythm, and can be applied to independent components, which are thought to reflect underlying sources, in addition to individual raw signals. This suggests that the BOSC method is an effective tool for measuring changes in rhythmic activity in the more common research scenario wherein state is unknown....Temporal independence of two alpha components. (A) An 8-s epoch from the alpha component shown in Fig. 2, with detected alpha-**frequency** **oscillations** highlighted in red. (B) The same time segment as in A, from the alpha component in Fig. 6. Note the alpha **oscillation** is maximal in B when the **oscillation** is at a minimum in A, demonstrating why these were extracted as temporally independent components.
...Lateralized alpha component. From the same subject as Figs. 2 and 4–5. (A) The spline-interpolated scalp distribution of an alpha component extracted by ICA. Color scale denotes electrode weight (unitless). (B) Background wavelet power spectrum mean and standard deviation (blue), and the linear regression fit to the background (green). (C) **Oscillations** detected across all **frequencies** by the oscillatory episode detection method. Red vertical lines indicate when participants were instructed to close their eyes and black vertical lines indicate when participants were instructed to open their eyes. (D) The proportion of time (Pepisode) during the eyes-closed condition (red) and eyes-open condition (black) that **oscillations** were detected at each **frequency**. (E) The time-domain representation of the chosen component, with detected **oscillations** at the peak alpha **frequency** (9.5Hz) highlighted in red. Vertical lines are the same as above. (F) An expansion of the highlighted section in E.
...**Oscillation** detection in an ICA alpha component. (A) The spline-interpolated scalp distribution of an alpha component extracted by ICA. Color scale denotes electrode weight (unitless). (B) Background wavelet power spectrum mean and standard deviation (blue) and the linear regression fit to the background (green). (C) **Oscillations** detected across all **frequencies** by the oscillatory episode detection method. Red vertical lines indicate when participants were instructed to close their eyes and black vertical lines indicate when participants were instructed to open their eyes. (D) The proportion of time (Pepisode) during the eyes-closed condition (red) and eyes-open condition (black) that **oscillations** were detected at each **frequency**. (E) The time-domain representation of the chosen component, with detected **oscillations** at the peak alpha **frequency** (9.5Hz) highlighted in red. Vertical lines are the same as above. (F) An expansion of the highlighted section in E, to show the spindle-like appearance of the alpha **oscillation**.
...**Oscillation**...**Oscillation** detection in a single electrode with strong alpha. The electrode was selected from the same subject as in Fig. 2. (A) The 256-electrode array with the selected electrode highlighted in yellow. (B) Background wavelet power spectrum mean and standard deviation (blue), and the linear regression fit to the background (green). (C) **Oscillations** detected across all **frequencies** by the oscillatory episode detection method. Red vertical lines indicate when participants were instructed to close their eyes and black vertical lines indicate when participants were instructed to open their eyes. (D) The proportion of time (Pepisode) during the eyes-closed condition (red) and eyes-open condition (black) that **oscillations** were detected at each **frequency**. (E) The raw signal from the chosen electrode, with detected **oscillations** at the peak alpha **frequency** (9.5Hz) highlighted in red. Vertical lines are the same as above. (F) An expansion of the highlighted section in E to show the spindle-like appearance of the alpha **oscillation**.
... Oscillatory activity is a principal mode of operation in the brain. Despite an intense resurgence of interest in the mechanisms and functions of brain rhythms, methods for the detection and analysis of oscillatory activity in neurophysiological recordings are still highly variable across studies. We recently proposed a method for detecting oscillatory activity from time series data, which we call the BOSC (Better **OSCillation** detection) method. This method produces systematic, objective, and consistent results across **frequencies**, brain regions and tasks. It does so by modeling the functional form of the background spectrum by fitting the empirically observed spectrum at the recording site. This minimizes bias in **oscillation** detection across **frequency**, region and task. Here we show that the method is also robust to dramatic changes in state that are known to influence the shape of the power spectrum, namely, the presence versus absence of the alpha rhythm, and can be applied to independent components, which are thought to reflect underlying sources, in addition to individual raw signals. This suggests that the BOSC method is an effective tool for measuring changes in rhythmic activity in the more common research scenario wherein state is unknown.

Data types:

Contributors: Andrada Ianuş, Bernard Siow, Ivana Drobnjak, Hui Zhang, Daniel C. Alexander

Date: 2013-02-01

**Oscillating** gradients provide an optimal probe of small pore sizes in diffusion MRI. While sinusoidal **oscillations** have been popular for some time, recent work suggests additional benefits of square or trapezoidal **oscillating** waveforms. This paper presents analytical expressions of the free and restricted diffusion signal for trapezoidal and square **oscillating** gradient spin echo (OGSE) sequences using the Gaussian phase distribution (GPD) approximation and generalises existing similar expressions for sinusoidal OGSE. Accurate analytical models are necessary for exploitation of these pulse sequences in imaging studies, as they allow model fitting and parameter estimation in reasonable computation times. We evaluate the accuracy of the approximation against synthesised data from the Monte Carlo (MC) diffusion simulator in Camino and Callaghan’s matrix method and we show that the accuracy of the approximation is within a few percent of the signal, while providing several orders of magnitude faster computation. Moreover, since the expressions for trapezoidal wave are complex, we test sine and square wave approximations to the trapezoidal OGSE signal. The best approximations depend on the gradient amplitude and the **oscillation** **frequency** and are accurate to within a few percent. Finally, we explore broader applications of trapezoidal OGSE, in particular for non-model based applications, such as apparent diffusion coefficient estimation, where only sinusoidal waveforms have been considered previously. We show that with the right apodisation, trapezoidal waves also have benefits by virtue of the higher diffusion weighting they provide compared to sinusoidal gradients....(a) Diffusion signal for different waveforms: square with 90° phase, apodised cosine and apodised trapezoid as a function of **oscillation** **frequency** for four different sizes of the restricted compartment; (b) corresponding extracted ADC values. The diffusion signal and ADC for apodised trapezoid and square wave are very similar and are plotted on top of each other.
...**Oscillating** gradient...(a) Average signal difference between square and sine approximations and the full trapezoidal expressions as a function of α for R=2μm and 10μm. (b) Diffusion signal for R=5μm for the three waveforms with gradient strength G=60mT/m and 200mT/m as a function of **oscillation** **frequency**.
...(a) Average signal difference between square and sine approximations and the full trapezoidal expression considering: I – same amplitude, II – same area under the curves, III – same area under the squared curves and IV – same b value per **oscillation**. (b) Difference between square and sine approximations and the full trapezoidal expressions with SR=200T/m/s as a function of n for all data points with R=5μm.
...Restricted diffusion signal as a function of **oscillation** **frequency** for (a) several values of Δ, R=5μm and G=0.1T/m; (b) several gradient strengths, R=5μm and Δ=25ms. In (a) and (b) the filled markers indicate waveforms with integer number of **oscillations**. Restricted diffusion as a function of (c) gradient strength for several **frequencies**, R=5μm and Δ=45ms; (d) cylinder radius for several **frequencies**, G=0.1T/m and Δ=45ms. The markers show the MC simulation and the solid lines are the GPD approximations. The vertical bar separates different scales on the x-axis.
...Square wave **oscillations** ... **Oscillating** gradients provide an optimal probe of small pore sizes in diffusion MRI. While sinusoidal **oscillations** have been popular for some time, recent work suggests additional benefits of square or trapezoidal **oscillating** waveforms. This paper presents analytical expressions of the free and restricted diffusion signal for trapezoidal and square **oscillating** gradient spin echo (OGSE) sequences using the Gaussian phase distribution (GPD) approximation and generalises existing similar expressions for sinusoidal OGSE. Accurate analytical models are necessary for exploitation of these pulse sequences in imaging studies, as they allow model fitting and parameter estimation in reasonable computation times. We evaluate the accuracy of the approximation against synthesised data from the Monte Carlo (MC) diffusion simulator in Camino and Callaghan’s matrix method and we show that the accuracy of the approximation is within a few percent of the signal, while providing several orders of magnitude faster computation. Moreover, since the expressions for trapezoidal wave are complex, we test sine and square wave approximations to the trapezoidal OGSE signal. The best approximations depend on the gradient amplitude and the **oscillation** **frequency** and are accurate to within a few percent. Finally, we explore broader applications of trapezoidal OGSE, in particular for non-model based applications, such as apparent diffusion coefficient estimation, where only sinusoidal waveforms have been considered previously. We show that with the right apodisation, trapezoidal waves also have benefits by virtue of the higher diffusion weighting they provide compared to sinusoidal gradients.

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Contributors: Tolga Esat Özkurt, Alfons Schnitzler

Cross-**frequency** coupling...Simulation results. Coupling portraits for simulated data for (A) clean (SNR=20dB) and (B) noisy cases (SNR=−10dB). PAC was generated to be at 60–80Hz (amplitude **frequency**) and 15Hz (phase **frequency**). The simulated signal contained also **oscillations** (at 20, 25, 30, 40 and 100Hz) having no coupling relation. These portraits show the mean PAC estimates over 100 repetitions for each method. Only methods robust enough were presented: direct PAC estimate, GLM with spurious term removed, MI with statistics and raw MI without statistics (ordered from left to right). Notice that the first two methods yield very similar outputs identifying PAC correctly and they are robust to both non-coupled **oscillations** and noise.
...**Oscillations** ... Recent studies have observed the ubiquity of phase–amplitude coupling (PAC) phenomenon in human and animal brain recordings. While various methods were performed to quantify it, a rigorous analytical definition of PAC is lacking. This paper yields an analytical definition and accordingly offers theoretical insights into some of the current methods. A direct PAC estimator based on the given definition is presented and shown theoretically to be superior to some of the previous methods such as general linear model (GLM) estimator. It is also shown that the proposed PAC estimator is equivalent to GLM estimator when a constant term is removed from its formulation. The validity of the derivations is demonstrated with simulated data of varying noise levels and local field potentials recorded from the subthalamic nucleus of a Parkinson's disease patient.

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Contributors: G.D. Gkikas, G.A. Athanassoulis

Date: 2014-04-01

The same as Fig. 17, but for a monochromatic wave excitation of **frequency** f2=0.2Hz and amplitude A2=0.50m.
...**Oscillating** water column...The same as Fig. 18, but for a monochromatic wave excitation of **frequency** f2=0.2Hz and amplitude A2=0.50m.
...The same as Fig. 22, but for a monochromatic wave excitation of **frequency** f2=0.2Hz and amplitude A2=0.50m.
...In this work, we present a nonlinear system identification method, modeling the pressure fluctuation inside the chamber of an **oscillating** water column wave energy converter (OWC–WEC) under monochromatic excitation. The systemic scheme, upon which the identification is based, is a Wiener–Hammerstein cascade and thus the functional analogue of the model is a truncated Volterra series....The same as Fig. 3, but for different constant **frequency**, i.e., f2=0.05Hz.
...**Frequency**-amplitude domain...Maximum dynamic pressure, max(pD(t;Aexc,Texc)), against (mean wave elevation) **oscillation's** amplitude Aexc.
... In this work, we present a nonlinear system identification method, modeling the pressure fluctuation inside the chamber of an **oscillating** water column wave energy converter (OWC–WEC) under monochromatic excitation. The systemic scheme, upon which the identification is based, is a Wiener–Hammerstein cascade and thus the functional analogue of the model is a truncated Volterra series.

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Contributors: Banafsheh Seyed-Aghazadeh, Collin Budz, Yahya Modarres-Sadeghi

Date: 2015-09-29

Flow forces acting on an **oscillating** cylinder.
...Vortex-induced vibration (VIV) of a curved circular cylinder (a quarter of a ring, with no extension added to either end) free to **oscillate** in the crossflow direction was studied experimentally. Both the concave and the convex orientations (with respect to the oncoming flow direction) were considered. As expected, the amplitude of **oscillations** in both configurations was decreased compared to a vertical cylinder with the same mass ratio. Flow visualizations showed that the vortices were shed in parallel to the curved cylinder, when the cylinder was free to **oscillate**. The sudden jump in the phase difference between the flow forces and the cylinder displacement observed in the VIV of vertical cylinders was not observed in the curved cylinders. Higher harmonic force components at **frequencies** twice and three times the **frequency** of **oscillations** were observed in flow forces acting on the vertical cylinder, as well as the curved cylinder. Asymmetry in the wake was responsible for the 2nd harmonic force component and the relative velocity of the structure with respect to the oncoming flow was responsible for the 3rd harmonic force component. The lock-in occurred over the same range of reduced velocities for the curved cylinder in the convex orientation as for a vertical cylinder, but it was extended to higher reduced velocities for a curved cylinder in the concave orientation. Higher harmonic force components were found to be responsible for the extended lock-in range in the concave orientation. Within this range, the higher harmonic forces were even larger than the first harmonic force and the structure was being excited mainly by these higher harmonic forces....Dimensionless (a) amplitude (A*=A/D) and (b) **frequency** (f*=fos/fna) of the crossflow **oscillations** versus the reduced velocity for a curved cylinder in the convex configuration (■) and a vertical cylinder (○).
...Flow visualizations in the wake of a curved cylinder for the fixed (a) convex and (b) concave configurations, and free-to-**oscillate** (c) convex and (d) concave configurations. Flow is from left to right.
...Dimensionless (a) amplitude (A*=A/D) and (b) **frequency** (f*=fos/fna) of the crossflow **oscillations** versus the reduced velocity for a curved cylinder in the concave configuration (▲) and a vertical cylinder (○).
... Vortex-induced vibration (VIV) of a curved circular cylinder (a quarter of a ring, with no extension added to either end) free to **oscillate** in the crossflow direction was studied experimentally. Both the concave and the convex orientations (with respect to the oncoming flow direction) were considered. As expected, the amplitude of **oscillations** in both configurations was decreased compared to a vertical cylinder with the same mass ratio. Flow visualizations showed that the vortices were shed in parallel to the curved cylinder, when the cylinder was free to **oscillate**. The sudden jump in the phase difference between the flow forces and the cylinder displacement observed in the VIV of vertical cylinders was not observed in the curved cylinders. Higher harmonic force components at **frequencies** twice and three times the **frequency** of **oscillations** were observed in flow forces acting on the vertical cylinder, as well as the curved cylinder. Asymmetry in the wake was responsible for the 2nd harmonic force component and the relative velocity of the structure with respect to the oncoming flow was responsible for the 3rd harmonic force component. The lock-in occurred over the same range of reduced velocities for the curved cylinder in the convex orientation as for a vertical cylinder, but it was extended to higher reduced velocities for a curved cylinder in the concave orientation. Higher harmonic force components were found to be responsible for the extended lock-in range in the concave orientation. Within this range, the higher harmonic forces were even larger than the first harmonic force and the structure was being excited mainly by these higher harmonic forces.

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Contributors: Gianluca Gatti, Michael J. Brennan

Date: 2011-08-29

Numerical validation of the FRCs shown in the corresponding plots of Fig. 4. Stable FRC solution (blue solid line), unstable FRC solution (red dashed line), numerical solution (black circles). In (e) and (f) the approximate analytical expression for the FRC fails to predict the response, which is not harmonic (NH) in a small **frequency** range around 1. Approximate expressions for the jump **frequencies** given in Table 3 are shown as vertical dashed lines. (a) Region I, (b) Region II′, (c) Region IIIb, (d) Region IIIa′, (e) Region IV and (f) Region V. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
...This paper discusses the dynamic behaviour of a nonlinear two degree-of-freedom system consisting of a harmonically excited linear **oscillator** weakly connected to a nonlinear attachment having linear and cubic restoring forces. The effects of the system parameters on the shape of the **frequency**-response curve are investigated, in particular those yielding the appearance and disappearance of outer and inner detached resonance curves. In contrast to the case when the linear stiffness of the attachment is zero, it is found that multivaluedness occurs at low **frequencies** as the resonant peak bends to the right. It is also found that as the coefficient of the linear term increases, the range of parameters yielding detached curves reduces. Compared to the case when the attached system has no linear stiffness term, this range of parameters corresponds to smaller values of the damping and nonlinear coefficients. Approximate analytical expressions for the jump-up and jump-down **frequencies** of the system under investigation are also derived....Approximate expressions for the jump-up and jump-down **frequencies** together with their regions of applicability.
...Three-dimensional plots illustrating the relationship between the bifurcation curves and the FRCs, for ω0=0.3 and different combinations of ζ and γ: ζ=0.002 and (a) γ=2×10−9, (b) γ=2×10−6, (c) γ=2×10−5; ζ=0.02 and (d) γ=5×10−4, (e) γ=4×10−3 and (f) γ=2×10−2. On the Ω−γ plane, γ1 is indicated by the upper thin solid line, γ2 by the lower one. On the Ω−W plane, the FRC is plotted with the stable solution (blue solid line) and the unstable solution (red dashed line). The intersections between the Ω−W plane containing the FRC and the bifurcation curves on the Ω−γ plane indicate the expected values for the jump **frequencies**. (a) Region I, (b) Region II′, (c) Region IIIb, (d) Region IIIa′, (e) Region IV and (f) Region V. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
... This paper discusses the dynamic behaviour of a nonlinear two degree-of-freedom system consisting of a harmonically excited linear **oscillator** weakly connected to a nonlinear attachment having linear and cubic restoring forces. The effects of the system parameters on the shape of the **frequency**-response curve are investigated, in particular those yielding the appearance and disappearance of outer and inner detached resonance curves. In contrast to the case when the linear stiffness of the attachment is zero, it is found that multivaluedness occurs at low **frequencies** as the resonant peak bends to the right. It is also found that as the coefficient of the linear term increases, the range of parameters yielding detached curves reduces. Compared to the case when the attached system has no linear stiffness term, this range of parameters corresponds to smaller values of the damping and nonlinear coefficients. Approximate analytical expressions for the jump-up and jump-down **frequencies** of the system under investigation are also derived.

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Contributors: G. Catanzaro, F. Leone, I. Busá, P. Romano

Date: 2008-02-01

In this figure we report as a function of optical depth the computed **frequencies**, center of mass (γ0), amplitudes and phases derived from the fit of velocities for each of the selected lines. Meaning of the symbols is: circles (red) carbon lines, stars (magenta) silicon lines, triangles (blues) oxygen lines and boxes (green) nitrogen lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
...Pulsations, **oscillations**, and stellar seismology...**Frequencies** in **oscillating** β Cephei stars are usually inferred by means of radial velocities measured from the Siiii triplet λλ 4552–4574Å. These lines, relatively insensitive to the variation of Teff through a pulsation cycle, show small equivalent width variations. ... **Frequencies** in **oscillating** β Cephei stars are usually inferred by means of radial velocities measured from the Siiii triplet λλ 4552–4574Å. These lines, relatively insensitive to the variation of Teff through a pulsation cycle, show small equivalent width variations.

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