Beat Phenomenon Research Articles

An Experimental Study on the Effects of Nonuniform Vane Spacing on Forced Response Reduction in a Mistuned Rotor Blisk in a Multistage Axial Compressor

Abstract Stators with nonuniform vane spacing (NUVS) are known to effectively reduce the forced response of adjacent rotors by spreading the primary engine order (EO) excitation at a certain speed to a series of weaker excitations at nearby EOs over a wider speed range. To be used in a real gas turbine engine, it is essential to know the forced response vibratory reduction that can be achieved for a specific NUVS configuration at different operating conditions. The classical estimation method to predict the blade response reduction is to quantify the reduction of the forcing function at a potential resonance crossing by conducting a circumferential Fourier analysis of the asymmetric flow field. However, besides excitation reduction, the blade forced response also depends on other factors, such as damping and blade-to-blade interactions due to mistuning. To study the blade forced response reduction in a realistic environment, a comprehensive experimental study was conducted in the Purdue three-stage axial research compressor at three different loading conditions. The vibratory response of the first torsion (1 T) mode forced response of rotor 2 was measured by strain gages (SGs) for two different upstream stator 1 configurations: the symmetric 38-vaned stator 1 configuration and the NUVS stator 1 configuration with 18–20 vane halves. As expected, the strong 38EO-1T response from the symmetric stator reduced to a series of weaker responses from 35EO to 41EO in the NUVS configuration. The overlap of adjacent EO responses caused considerable beating phenomena in the blade SG time–history data. There was substantial blade-to-blade variation in blade response for both symmetric and asymmetric stator 1 configurations due to the inherent nonintentional mistuning in the rotor 2 blisk. This, in turn, causes a large blade-to-blade variation in the reduction factors. While higher responding blades tend to have larger reduction factors, some blades show almost no forced response reduction when switching to the NUVS stator. In addition, the reduction factors for each blade and the maximum reduction factor for the whole blisk also change significantly with loading conditions. The measured reduction factors and the peak response EO are not well predicted by the classical estimation method. This indicates that both damping and mistuning effects need to be considered in the NUVS reduction factor prediction.

Density Waves of Vortex Fluids on a Sphere

Abstract We aim to find one highly nontrivial example of the solutions to the vortex fluid dynamical equation on the unit sphere (S^2) and compare it with the numerical simulation. Since the rigid rotating steady solution for vortex fluids on S^2 is
already known to us, we consider the perturbations above it. After decomposing the perturbation of the vortex number density and vortex charge density into spherical harmonics, we find that the perturbations are propagating waves. To
be precise, the velocities for different single-mode vortex number density waves are all the same, while the velocities for single-mode vortex charge density waves depend on the degree of the spherical harmonics l, which is a signal of the existence of dispersion. Meanwhile, we find that there is a beat phenomenon for the positive (or negative) vortex density wave. Numerical simulation based on the canonical equations for point vortex model agrees perfectly with our theoretical calculations.

Assessing Leg Blood Flow and Cardiac Output During Running Using Thermodilution.

Cardiac output (Q̇C) and leg blood flow (Q̇LEG) can be measured simultaneously with high accuracy using transpulmonary and femoral vein thermodilution with a single-bolus injection. The invasive measure has offered important insight into leg hemodynamics and blood flow distribution during exercise. Despite being the natural modality of exercise in humans, there has been no direct measure of Q̇LEG while running in humans. We sought to determine the feasibility of the thermodilution technique for measuring Q̇LEG and conductance during high-intensity running, in an exploratory case study. A trained runner (30 years male) completed two maximal incremental tests on a cycle ergometer and motorized treadmill. Q̇LEG and Q̇C were determined using the single-bolus thermodilution technique. Arterial and venous blood were sampled throughout exercise, with continuous monitoring of metabolism, intra-arterial and venous pressure, and temperature. The participant reached a greater peak oxygen uptake (V̇O2peak) during running relative to cycling (74 vs. 68 mL/kg/min) with comparable Q̇LEG (19.0 vs. 19.5 L/min) and Q̇C (27.4 vs. 26.2 L/min). Leg vascular conductance was greater during high-intensity running relative to cycling (82 vs. 70 mL/min/mmHg @ ~80% V̇O2peak). The "beat phenomenon" was apparent in femoral flow while running, producing large gradients in conductance (62-90 mL/min/mmHg @ 70% V̇O2peak). In summary, we present the first direct measure of Q̇LEG and conductance in a running human. Our findings corroborate several assumptions about Q̇LEG during running compared with cycling. Importantly, we demonstrate that using thermodilution in running exercise can be completed effectively and safely.

Demonstrating beat phenomenon and Lissajous figures with hard disk drive voice-coil motors

Voice coil motors (VCMs) in mechanical hard disk drives can be used as the adjustable-electric tuning fork driven by alternating signals due to their unique structure and movement. The VCM component is modified to pull the laser or mirror for phase-adjustable simple harmonic motion (SHM). Lissajous experiments using the traditional tuning fork for active vibration can be achieved by VCMs with the help of a laser and a scrollable phosphorescent screen. The demonstrations of beat and Lissajous figures with the double oscillating mirrors method are analysed. The VCM can not only drive the mirror but also drive the laser or another VCM component. Based on this carrying capacity, the demonstration method of a single oscillating mirror has been proposed. Furthermore, one VCM can support another with a laser, which will exhibit the composition of two rotary SHMs more intuitively. The new method is beneficial for students to understand and practice the composition of SHMs from a new perspective.

A mechanical device to demonstrate the composition of two simple harmonic motions

The slot link mechanism can project uniform rotation to a simple harmonic motion (SHM), based on which, the device including two sliding table systems has been prepared. These two sliding table systems can be stacked up and down, and each of them can produce a tunable mechanical SHM using a stepper motor. When the upper sliding table system is attached to the lower one and both are turned on, the motion of the sliding table of the upper component is the composition of the two SHMs. With the help of a self-powered LED, a violet laser, and a movable phosphorescent foil, this mechanical device can visualize the composition of the SHMs in the same and mutual perpendicular directions including the beat phenomenon and Lissajous figures. In addition to demonstrating several typical compositions of two SHMs, this device can also highlight the vector characteristics of mechanical vibration and demonstrate the tilted Lissajous figures.

Beat phenomena of oscillating readouts.

To demonstrate slowly varying, erroneous magnetic field gradients for oscillating readouts due to the mechanically resonant behavior of gradient systems. Projections of a static phantom were acquired using a one-dimensional (1D) EPI sequence with varying EPI frequencies ranging from 1121 to 1580 Hz on clinical 3T systems (30 mT/m, 200 T/m/s). Phase due to static B0 inhomogeneities was eliminated by a complex division of two separate scans with different polarities of the EPI readout. The temporal evolution of phase was evaluated and related to the mechanical resonances of the gradient systems derived from the gradient modulation transfer function. Additionally, the impact of temporally varying mechanical resonance effects on EPI was evaluated using an echo-planar spectroscopic imaging sequence. A beat phenomenon resulting in a slowly varying phase was observed. Its temporal frequency was given by the difference between the EPI frequency and the mechanical resonance frequency of the activated gradient axis. The maximum erroneous, oscillating phase during phase encoding was ±0.5 rad for an EPI frequency of 1281 Hz. Echo-planar spectroscopic imaging images showed the resulting time-dependent stretching/compression of the FOV. Oscillating readouts such as those used in EPI can result in low-frequency, erroneous phase contributions, which are explained by the beat phenomenon. Therefore, EPI phase-correction approaches may need to include beat effects for accurate image reconstruction.

Coupled oscillations of the Wilberforce pendulum unveiled by smartphones

The Wilberforce pendulum illustrates important properties of coupled oscillators including normal modes and beat phenomena. When helical spring is attached to a mass to create the Wilberforce pendulum, the longitudinal and torsional oscillations are coupled. A Wilberforce can be constructed simply from a standard laboratory spring, and a smartphone's accelerometer and gyroscope can be used to monitor the oscillations. We show that the resulting time-series data match theoretical predictions, and we share the procedures for observing both normal modes and beats.

Dynamics of a simple third-order autonomous MLC circuit

Abstract This paper reports a new simple third-order memristive circuit only containing three elements of inductor, capacitor, and active generalized memristor, from which rich dynamical behaviors are generated. With a dimensionless system model, the performed analyses show that the proposed memristive circuit only has an unstable equilibrium point of saddle-focus-type. The antimonotonicity makes the system exhibits coexisting chaotic and periodic bubbling single-parameter bifurcation routes. Moreover, the quasiperiodic torus, various bursting and beat phenomena with chaotic and periodic oscillations are demonstrated by numerical simulations. The analog circuit implementations are further presented to show the phase portraits and time sequences of the generated attractors.

Influences of spatial structure of plasma shock wave on electromagnetic wave propagation characteristics

Plasma shock waves represent high-speed, nonlinear motion states of plasma in which the physical parameters of fluid, such as density, temperature, and velocity, vary dramatically within a limited space. These variations make studying electromagnetic wave propagation in plasma shock waves difficult. In this study, we calculate the spatial distribution of the plasma frequency and collision frequency in the plasma shock layer based on the one-dimensional structure of the plasma shock wave. In the process from upstream to downstream of the plasma shock wave, the plasma frequency increases as a function of electron density; downstream, the plasma frequency increases as a function of the free stream Mach number, while the collision frequency decreases first and then increases. We also use the recursive convolution finite-difference time-domain method to calculate the propagation of the electromagnetic wave in the plasma shock layer. The absorption of the plasma shock layer to the electromagnetic wave decreases gradually as a function of the electromagnetic wave frequency. The absorption of the plasma shock layer downstream of the electromagnetic wave is greater than that upstream owing to the larger plasma frequency. In the case of low-Mach numbers, the wavelength of the electromagnetic wave downstream is larger than that upstream. In the case of high-Mach numbers, the thickness of the entire shock layer is much greater than that at low-Mach numbers. Due to the nonlinear effects of electromagnetic processes in plasmas, a modulation phenomenon occurs when electromagnetic waves propagate in the shock layer. In the case of low frequencies, the electromagnetic wave gave rise to a modulation phenomenon that resembled the beat phenomenon. When the electromagnetic wave frequency increased, a modulation phenomenon gradually appeared that resembled the oscillation superimposed on a sine wave. Further increases in the electromagnetic wave frequency led to the gradual disappearance of the modulation phenomenon.

Spin-valley switch and conductance oscillations in antiferromagnetic transition metal dichalcogenides

Antiferromagnetic materials are regarded as the outstanding candidates for the next generation of spintronics applications thanks to the numerous interesting features. We theoretically study the spin and valley transport in transition metal dichalcogenides in the present of antiferromagnetic exchange field. It is found that the spin and valley dependent band gap can be controlled by the exchange field. The system could become a spin-valley half metal, where a certain spin-valley electron is metallic state and other electrons are insulating states. The normal/antiferromagnetic/normal junction could work as an effective spin-valley switch controlled by the gate voltage. In the normal/antiferromagnetic/normal/ferromagnetic/normal junction, quantum beats occur in the oscillations of total conductance. The beat phenomenon results from the interference of two different spin-valley dependent conductances with similar frequencies. In addition, the junction can also work as a magnetoresistance device.

Spin current and internal Zeeman field in spin-orbit coupled rings

We investigate the one-dimensional quantum ring constructed by the spin-orbit coupled material, in which the quantum spin-Hall Bernevig-Zhang (BZ) Hamiltonian and Rashba-Dirac (RD) type spin-orbit coupling are taken into account (called RD-BZ Hamiltonian in this paper). It is known that the curvature of the ring generates an out-of-plane effective magnetic field, acting as an internal Zeeman field. We find that only the BZ coupling can change the strength of the internal Zeeman field, which enables us to detect the effect of the internal Zeeman field. Furthermore, we find that the total angular momentum is conserved in the RD-BZ Hamiltonian, and the energy eigenvalue and wave function must be modified to fit the conserved quantity, which are ignored in the previous studies. The conductance without leads is discussed. Different from the previous results, we find that the conductance behaves like a beat phenomenon resulting from the interplay between the magnetic flux and Aharonov-Casher (AC) phase, and thus, it can oscillate without passing through the insulating state in some regimes of magnetic flux or AC phase. Importantly, we find that the conductance with integer and half-integer magnetic flux provides us a method to measure the AC phase. The cancellation of the internal Zeeman field due to the BZ coupling can be detected by using specific fractional magnetic flux. In the ring with nonvanishing RD and BZ couplings, the conductance can exhibit a quasiplateau near the small RD coupling. The increase in the strength of BZ coupling would result in wider quasiplateau in conductance, which implies that the ring could remain insulating state (or conducting state) regardless of the small change in RD coupling. The thermal average of spin and charge currents are calculated at low temperatures without impurities. Importantly, we find that the persistent spin and charge currents as a function of the magnetic flux exhibit nodelike lines, which amazingly are perpendicular to each other. As a consequence, the persistent spin current could be nonzero even when the charge current vanishes at nonzero magnetic flux. The result suggests a pure spin current in quantum rings, and its direction can be reversed by changing the magnetic flux. The increase in the BZ coupling would exhibit the plateaulike pure spin current.

The material flow and texture-weakening mechanism in double-sided friction stir welded Mg alloy

In conventional friction stir welding of magnesium alloys, a strong (0001) texture at the welded zone is inevitable, which deteriorates the joint mechanical properties. A novel method called ‘double-sided friction stir welding’ solved this problem by weakening the texture. However, the texture-weakening mechanism remains unclear. In this study, the mechanism of the texture weakening in double-sided friction stir welded Mg alloy was systematically analysed. The welding temperature and the material flow mode were revealed to control the (0001) texture randomisation simultaneously. The unique material flow at the mid-thickness layer caused by the double-sided friction stir welding was revealed to match the ‘beat phenomenon’ for the first time, which was revealed as the dominant mechanism for the (0001) texture randomisation.

A novel distance protection scheme using single‐ended transient voltage based on generalized S transform

Concerning the distance protection steady-state over-reach or under-reach caused by the transition resistance, a novel distance protection scheme based on transient information is proposed. First, multiple fault features are extracted using the Generalized Stockwell Transform (GST), and a criterion for identifying low-resistance and high-resistance ground faults is constructed. Then, a GST coefficient matrix is established for the single-ended transient voltage of the line, the attenuation coefficient and delay coefficient are derived by the least-squares method and the beat phenomenon, and the transition resistance is calculated. On this basis, adjust the operation boundary in the +R axis direction of the operation zone according to the transition resistance value. Finally, test results on real-time digital simulation system (RTDS) demonstrate that the protection scheme operates reliably in different fault cases and is highly immune to fault resistance. Compared with the traditional distance protection scheme, this scheme performs better when the transition resistance is large.

The beating effect in uniaxial oriented polymer materials

Elastic-relaxation properties of uniaxial oriented polymer materials under dynamic deformation mode are investigated. A theoretical explanation of the phenomenon of the occurrence of beats is proposed. Experimental confirmation of the obtained theoretical results is given. Using the barrier theory and applying the balance equation of the number of cluster transitions through the energy barrier, taking into account the transition time, a refined constitutive equation of the viscoelasticity of the polymer materials is obtained. Experimental studies in the dynamic mode were carried out by the method of the longitudinal low-amplitude oscillations. For a polymer material, taking into account the time of transition of the cluster through the energy barrier, a refined constitutive equation is obtained. The resulting equation is a second-order differential equation and admits a periodic solution. The application of the obtained equation to the study of free longitudinal low-amplitude oscillations in uniaxial oriented polymer materials is considered. It is shown that the solution of the equation admits two close complex roots, the existence of which leads to the effect of the observed beats. The relations between the parameters of the oscillatory process and viscoelastic characteristics are obtained. The experiment showed that the polymer materials under study have a complex form of the oscillatory process, similar to the phenomenon of beats, in a certain range of load. The dependence of the tangent of the mechanical losses angle calculated from the main frequency on the stress has an acute maximum in the stress range where beats are observed. According to the obtained theoretical ratios and experimentally determined attenuation coefficient and main frequency, the time of transition of the cluster through the energy barrier or the lifetime of the cluster in this energy state is determined. The use of additional information obtained during experiments in static modes made it possible to determine the relaxation time for this material. The calculation on the example of polyethyleneterephthalate (PET) filament showed that the theory is consistent with the experiment. Based on the experimental data obtained during the study of free longitudinal oscillations and the solution of the refined constitutive equation, it is possible to determine the necessary viscoelastic characteristics which makes it possible to predict dynamic deformation processes in polymer materials.

Interaction of two-photon NOON state with ultrasonic wave

The paper presents a classical and quantum description of the diffraction of two light beams incident on an ultrasonic wave at a positive and negative Bragg angle. The quantum description of the interaction with the ultrasonic wave was carried out for the two-photon NOON state. Study of diffraction of entangled pair of photon was performed in Mach–Zehnder interferometer wherein an output beam splitter is replaced by ultrasonic wave. It has been shown theoretically and experimentally that when a pair of photons generated from a parametric down-conversion was incident on two input ports of Mach–Zehnder interferometer with an acousto-optical beam splitter, then two-photon beats are observed at the output of the interferometer. The phenomenon of two-photon beats with the double frequency of the ultrasonic wave is the result of the Doppler effect on the ultrasonic wave. Time-correlated single-photon counting method was used to register the phenomenon of two-photon beats. The method of calibrating the interferometer with the use of an additional light source was presented in detail, which guarantees the observation of interaction of two-photon NOON state with ultrasonic wave.Graphical abstract

A method of generating car sounds based on granular synthesis algorithm

The active sound generation technique for automobiles, where car sounds can be synthesized by means of electronic sound production, is one of the effective methods to achieve the target sound. A method for active sound generation of automobile based on the granular synthesis algorithm is put forward here to avoid the broadband beating phenomena that occur due to the mismatch of parameters of sound granular signals. The comparison of the function expression of sound signal and Hilbert transformation is performed based on the principle of overlap and add; moreover, those parameters (phase, frequency and amplitude) of sound signals are interpolated by means of the Hermite interpolation algorithm which can ensure the continuity of the phase, frequency and amplitude curves. Thus, the transition audio is constructed by means of the sound signal function here to splice the adjacent sound granules. Our simulations show that our method can be applied to solve the current broadband beating issue for splicing sound granules and achieve natural continuity of synthesized car sounds. The subjective test results also indicate that our transition audio can produce high quality audio restitution.

Stability study of centrifugal pendulum vibration absorbers in gravity field

In this study, Lyapunov stability theory was used to systematically investigate the stability of centrifugal pendulum vibration absorber (CPVA) in terms of the shape of the absorber path, mistuning levels, damping coefficient, number of absorbers, and force of gravity. Our approach improved upon previous studies by introducing a gravity term, optimising the calculation method, and using a function diagram to represent the asymptotically stable region to produce more accurate results. Our finding that the influence of gravity on the stability of the CPVA system is reflected in the change of the tuning order prompted us to define new terminology—’gravity mistuning factor ([Formula: see text])’ and ‘zone yielding to gravity ([Formula: see text])’—to intuitively reflect the influence of gravity. Comparison of the results obtained from numerical simulation based on a fully nonlinear equation of motion with analytical results revealed nonlinear dynamic characteristics such as bifurcation to asynchrony response, nonlinear jump, and beat phenomena. The approach and results of this study are more practical in settings in which gravity plays an important role, including vehicle engine idling, low-speed locomotive or marine engines, and drum washing machines.

Suppression of Beat Phenomenon for Electrolytic Capacitorless Motor Drives Accounting for Sampling Delay of DC-Link Voltage

Beat phenomenon is an important issue for the practical application of electrolytic capacitorless motor drives. In this study, the characteristics of stator current harmonics caused by the voltage sampling delay are analyzed. Further, the beat phenomenon generated from the interaction between the harmonics and the fundamental currents is investigated, and the envelope feature of the stator current is derived mathematically. For the purpose of suppressing the beat phenomenon, a voltage reconstruction strategy is proposed to reduce the influence of dc-link voltage sampling delay at the six times of grid frequency. By utilizing the reconstructed dc-link voltage for calculation of PWM duty ratio, the beat phenomenon can be attenuated significantly. The proposed method can be integrated into motor control system easily. Experimental results show the effectiveness of the proposed strategy in a prototype of electrolytic capacitorless permanent magnet synchronous motor drive.

Vibrational Characteristics of Au‐Doped Si Nanowires: A Molecular Dynamics Study with a Modified Embedded Atom Method Potential Developed for Si‐H‐Au Systems

Silicon nanowires (SiNWs) are promising structures as resonators for applications in ultrasensitive force and mass transducers. In the present work, large‐scale molecular dynamics simulations were conducted to explore the effect of gold (Au), a commonly used catalyst, on the vibrational properties of [1 0 0] and [1 1 0] SiNWs. The force field was established using the modified embedded atom method (MEAM) formalism. The parameterization focused on the properties of the hydrogen‐covered SiNWs doped with Au, involving the formation energy as well as the structural and vibrational characteristics. The vibrational characteristics of a clamped‐clamped SiNW were examined through excitation using a sinusoidal velocity field after energy minimization and thermal equilibration. Different doping concentrations and distributions can significantly change the frequency response, quality factor Q, and beat phenomenon. The natural frequency f0 decreased with increasing Au concentration, whereas the effect of the impurity distribution on f0 was negligible. Furthermore, Q was sensitive to the Au concentration and distribution, and the increased concentration led to an overall increment in Q, accompanied by considerably increased scattering. Besides, the beat period of [1 1 0] SiNWs showed a strong positive correlation with the concentration. The vibration along the elementary axes cannot produce the beat phenomenon. Deflection of the elementary axes in a very small range would lead to a higher energy dissipation rate. In addition, owing to the considerable difference between the atomic masses of Au and Si, randomly distributed Au atoms significantly disturb the symmetry of the SiNWs. However, the elementary axes of the [1 1 0] SiNWs remained stable, even in the models with extremely radial or longitudinal impurity segregation. In contrast, we failed to capture the distinguishable elementary axes for most of the doped [1 0 0] samples. The features described above indicate that the impurity Au can effectively tune the resonant frequency of the [1 1 0] SiNWs, meanwhile, causes no large frequency shift that could be potentially caused by the deflection of the elementary coordinate system.

Hydraulic coupling vibration characteristics and control of hydropower station with upstream and downstream surge tanks

This paper investigates the hydraulic coupling vibration characteristics and control of hydropower station with upstream and downstream surge tanks (UDST). The basic equations of hydropower station with UDST are established. The state equation and transfer function of hydropower station are derived. The hydraulic coupling vibration domain is proposed and determined. The hydraulic coupling vibration characteristics are analyzed based on vibration domain. The interference and resonance characteristics of UDST are revealed. The control method for avoiding hydraulic resonance is proposed. The results indicate that the hydraulic coupling vibration domain is related to the stable domain. In Domain Ⅰ and Domain Ⅱ, the dynamic response processes (DRPs) are superposed by three vibrations and two vibrations, respectively. In Domain Ⅱ, the coupling effect of the vibrations of upstream surge tank (UST) and downstream surge tank (DST) is strong and can change the frequencies of DRPs. The vibrations of UST and DST have the same frequency after coupling effect. The interference of UDST shows a beat phenomenon. The amplitudes of DRPs present a kind of persistent and periodic variation. The selections of sectional areas of UDST should avoid the hydraulic resonance domain.