Shock Amplitude Research Articles

Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws

We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable if the shock amplitude is sufficiently small. This means that an associated Evans function E:Λ→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {E}:\\Lambda \\rightarrow \\mathbb {C}$$\\end{document} with Λ⊂C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda \\subset \\mathbb {C}$$\\end{document} an open superset of the closed right half plane H+≡{κ∈C:Reκ≧0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^+\\equiv \\{\\kappa \\in \\mathbb {C}:\ ext {Re}\\,\\kappa \\geqq 0\\}$$\\end{document} has only one zero, namely, a simple zero at 0. The result is analogous to the one obtained in Freistühler and Szmolyan (Arch Ration Mech Anal 164:287–309, 2002) and Plaza and Zumbrun (Discrete Contin Dyn Syst 10(4):885–924, 2004) for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in Mascia and Zumbrun (Partial Differ Equ 34(1–3):119–136, 2009), Plaza and Zumbrun (2004) and Ueda (Math Methods Appl Sci 32(4):419–434, 2009).

A Lane Change Model Considering the Stability of Cooperative Adaptive Cruise Control Platoon Fleet

In this article, lane change models for mixed traffic flow under cooperative adaptive cruise control (CACC) platoon formation are established. The analysis begins by examining the impact of lane changes on traffic flow stability. The influences of various factors such as lane change locations, timing, and the current traffic state on stability are discussed. In this analysis, it is assumed that the lane change location and the entry position in the adjacent lane have already been selected, without considering the specific intention behind the lane change. The speeds of the involved vehicles are adjusted based on an existing lane change model, and various conditions are analyzed for traffic flow disturbances, including duration, shock amplitude, and driving delays. Numerical calculations are provided to illustrate these effects. Additionally, traffic flow stability is factored into the lane change decision-making process. By incorporating disturbances to the fleet into the lane change income model, both a lane change intention model and a lane change execution model are constructed. These models are then compared with a model that does not account for stability, leading to the corresponding conclusions.

Spatial And Temporal Modes On A Generic Tandem Configuration In Transonic Shock Buffet

Certain combinations of upstream flow conditions in transonic flight may incite a highly unsteady shock-wave/boundary-layer interaction, often referred to as transonic buffet. The coupling of periodic shock motion and intermittent separation is seen to convey a characteristic unsteadiness in the evolution of the turbulent wake. This article uses high-speed focusing Schlieren and PIV measurements at high spatial resolution to characterize the temporal and spectral modes governing the flow past a generic 2D tandem configuration. This interaction, consisting of an OAT15A supercritical airfoil as the main wing and a NACA 64A-110 profile representing the tailplane, is investigated at chord-based Reynolds numbers of 2 million and 1 million, respectively and compared against fully-established shock buffet conditions in an isolated airfoil scenario operated at identical inflow conditions. The strongest buffet is observed in both configurations for a Mach number of 0.72 and an angle of attack of 5 deg, occurring at frequencies of 112 Hz and 103 Hz. Despite an equally periodic nature of the buffet unsteadiness, both shock amplitude and frequency are seen to decrease slightly in the tandem interaction. Global spectral analysis performed on the entire field of view identifies the buffet instability as the governing mode in both flow scenarios, imposing its unsteadiness even far downstream of its origin, allowing the extraction. of relevant sensor locations for a subsequent localized assessment of the involved time scales. Except for the buffet mode, complementing DMD analyses reveal a strong influence of a vortex-shedding mode throughout the entire buffet cycle. Mode shapes and associated frequencies furthermore suggest an intermittent low-frequency/large wavelength and high-frequency/low wavelength behavior, involving leading contributions between 2000 Hz and 4000 Hz.

Chaos control in cardiac dynamics: terminating chaotic states with local minima pacing.

Current treatments of cardiac arrhythmias like ventricular fibrillation involve the application of a high-energy electric shock, that induces significant electrical currents in the myocardium and therefore involves severe side effects like possible tissue damage and post-traumatic stress. Using numerical simulations on four different models of 2D excitable media, this study demonstrates that low energy pulses applied shortly after local minima in the mean value of the transmembrane potential provide high success rates. We evaluate the performance of this approach for ten initial conditions of each model, ten spatially different stimuli, and different shock amplitudes. The investigated models of 2D excitable media cover a broad range of dominant frequencies and number of phase singularities, which demonstrates, that our findings are not limited to a specific kind of model or parameterization of it. Thus, we propose a method that incorporates the dynamics of the underlying system, even during pacing, and solely relies on a scalar observable, which is easily measurable in numerical simulations.

Generalized nonlinear hybrid-norm parallel sparse filtering for bearing fault diagnosis under complex interference

Abstract Considerable attention has been garnered by the bearing fault diagnostic method, founded on sparse feature learning, for its exceptional robustness and adaptability to noise. Being an effective sparse learning method, parallel sparse filtering (PSF) has found successful applications in intelligent fault diagnosis. However, influenced by mechanical structural collisions and complex transmission paths, the obtained bearing signals come with abnormal impacts and noise, resulting in a reduced reliability of the PSF. Additionally, PSF constrains features and samples with l 1 / 2 norm, which is significantly disturbed by noise. To address these issues, generalized nonlinear hybrid-norm PSF (GNHNPSF) is proposed. Firstly, the Sigmoid nonlinear function is examined to suppress the amplitude of the anomalous shock. Subsequently, the hybrid-norm constraints, which involve the application of a l 1 / 2 norm to the column features of the input signal and a l 1 / 3 norm to the row features, can be employed to effectively extract distinctive information from the original data. The GNHNPSF is demonstrated to improve the accuracy of bearing fault diagnosis under complex interference, as evidenced by the results of simulation and experiments.

Bar-driven Gas Dynamics of M31

The large-scale gaseous shocks in the bulge of M31 can be naturally explained by a rotating stellar bar. We use gas dynamical models to provide an independent measurement of the bar pattern speed in M31. The gravitational potentials of our simulations are from a set of made-to-measure models constrained by stellar photometry and kinematics. If the inclination of the gas disk is fixed at i = 77°, we find that a low pattern speed of 16–20 km s−1 kpc−1 is needed to match the observed position and amplitude of the shock features, as shock positions are too close to the bar major axis in high Ω b models. The pattern speed can increase to 20–30 km s−1 kpc−1 if the inner gas disk has a slightly smaller inclination angle compared with the outer one. Including subgrid physics such as star formation and stellar feedback has minor effects on the shock amplitude, and does not change the shock position significantly. If the inner gas disk is allowed to follow a varying inclination similar to the H i and ionized gas observations, the gas models with a pattern speed of 38 km s−1 kpc−1, which is consistent with stellar-dynamical models, can match both the shock features and the central gas features.

Effects of Nonextensive Electrons on Dust-Ion Acoustic Waves in a Collisional Dusty Plasma with Negative Ions.

The effects of nonextensive electrons on nonlinear ion acoustic waves in dusty negative ion plasmas with ion-dust collisions are investigated. Analytical results show that both solitary and shock waves are supported in this system. The wave propagation is governed by a Korteweg-de Vries Burgers-type equation. The coefficients of this equation are modified by the nonextensive parameter q. Numerical calculations indicate that the amplitude of solitary wave and oscillatory shock can be obviously modified by the nonextensive electrons, but the monotonic shock is little affected.

Dust-ion-acoustic shock waves in the presence of (r, q) distributed electrons

A three-component dusty plasma system, with non-Maxwellian electrons, is investigated. The ions are considered to be mobile, the (spherical) dust is assumed to be stationary with charge- fluctuations, and the electrons follow the generalized velocity distribution function (VDF). The respective Burger’s equation, which manifests a shock structure, is derived by employing a reductive perturbation technique (RPT). It is deduced that dust charge fluctuations induce shock potential in a collisionless plasma system. The resultant perturbed potential is evaluated as affected by different plasma parameters, e.g. the ratios of electron to ion temperature (σ i ) and equilibrium densities (μ), and the non-thermal indices r and q. The shock amplitude is seen to decrease when the value of σ i is increased, and vice versa, whereas an opposite trend is observed for μ. It is found that for r = 0 and increasing values of spectral index q, amplitude of the potential distribution enhances while its width decreases. It is further noted that non-Maxwellian VDF significantly modifies the potential distribution. For the limiting case, namely, r = 0 and q → ∞ , the corresponding Maxwellian results are retrieved which benchmarks our findings. Present work will be useful in understanding the nonlinear propagation of dust-ion-acoustic shock waves in system where non-thermal population—as caused by various physical processes—of electrons are observed.

Scaling laws for bubble collapse driven by an impulsive shock wave

Upon interaction with underwater shock waves, bubbles can collapse and produce high-speed liquid jets in the direction of the wave propagation. This work experimentally investigates the impact of laser-induced underwater impulsive shock waves, i.e. shock waves with a short, finite width, of variable peak pressure on bubbles of radii in the range 10–500 $\mathrm {\mu }$ m. The high-speed visualisations provide new benchmarking of remarkable quality for the validation of numerical simulations and the derivation of scaling laws. The experimental results support scaling laws describing the collapse time and the jet speed of bubbles driven by impulsive shock waves as a function of the impulse provided by the wave. In particular, the collapse time and the jet speed are found to be, respectively, inversely and directly proportional to the time integral of the pressure waveform for bubbles with a collapse time longer than the duration of shock interaction and for shock amplitudes sufficient to trigger a nonlinear bubble collapse. These results provide a criterion for the shock parameters that delimits the jetting and non-jetting behaviour for bubbles having a shock width-to-bubble size ratio smaller than one. Jetting is, however, never observed below a peak pressure value of 14 MPa. This limit, where the pressure becomes insufficient to yield a nonlinear bubble collapse, is likely the result of the time scale of the shock wave passage over the bubble becoming very short with respect to the bubble collapse time scale, resulting in the bubble effectively feeling the shock wave as a spatially uniform change in pressure, and in an (almost) spherical bubble collapse.

Evaluation of shock migration performance for a multi-stable mechanical metamaterial

Studies on multi-stable metamaterials mainly focus on the quasi-static performance and the viscoelastic properties of the substrate are ignored, which could cause great deviations in shock migration performance evaluation. In the study, a multi-stable mechanical metamaterial prototype with 2 × 4 cells are investigated and fabricated by thermoplastic polyurethanes (TPU). An analytical, experimental and numerical model was developed to evaluate the dynamic characteristics of the TPU substrate from 0.001 s−1 to 33 s−1, which illustrates significant higher stress levels at higher strain rates. A generalized Maxwell viscoelastic constitutive model was constructed for numerical analysis. Experiments involving quasi-static compression and drop-impact were done to evaluate the prototype's energy absorption and shock reduction capabilities. Quasi-static compression tests and simulations revealed that the peak force of the prototype is about 2.20kN, which presented great repeatability at different loading velocities. The prototype showed significant shock mitigation ability, which could reduce shock acceleration amplitude from 147.02 g to 22.54 g. But the peak reaction force obtained through the acceleration response curve was 3.21 kN, which was 50 % larger than those obtained by quasi-static experiments and simulations. Numerical simulation considering viscoelasticity of the substrate could accurately predict the response of this type of prototype with different shock amplitude, which demonstrates an effective method for the design of protective facilities with specified requirements.

Comparative Characterization of Nonlinear Ultrasound Fields Generated by Sonalleve V1 and V2 MR-HIFU Systems.

A Sonalleve magnetic resonance-guided high-intensity focused ultrasound (MR-HIFU) clinical system (Profound Medical, Mississauga, ON, Canada) has been shown to generate nonlinear ultrasound fields with shocks up to 100 MPa at the focus as required for HIFU applications such as boiling histotripsy of hepatic and renal tumors. The Sonalleve system has two versions V1 and V2 of the therapeutic array, with differences in focusing angle, focus depth, arrangement of elements, and the size of a central opening that is twice larger in the V2 system compared to the V1. The goal of this study was to compare the performance of the V1 and V2 transducers for generating high-amplitude shock-wave fields and to reveal the impact of different array geometries on shock amplitudes at the focus. Nonlinear modeling of the field in water using boundary conditions reconstructed from holography measurements shows that at the same power output, the V2 array generates 10-15-MPa lower shock amplitudes at the focus. Consequently, substantially higher power levels are required for the V2 system to reach the same shock-wave exposure conditions in histotripsy-type treatments. Although this difference is mainly caused by the smaller focusing angle of the V2 array, the larger central opening of the V2 array has a nontrivial impact. By excluding coherently interacting weakly focused waves coming from the central part of the source, the presence of the central opening results in a somewhat higher effective focusing angle and thus higher shock amplitudes at the focus. Axisymmetric equivalent source models were constructed for both arrays, and the importance of including the central opening was demonstrated. These models can be used in the "HIFU beam" software for simulating nonlinear fields of the Sonalleve V1 and V2 systems in water and flat-layered biological tissues.

Mechanical Response of MEMS Suspended Inductors under Shock Using the Transfer Matrix Method.

MEMS suspended inductors are susceptible to deformation under external forces, which can lead to the degradation of their electrical properties. The mechanical response of the inductor to a shock load is usually solved by a numerical method, such as the finite element method (FEM). In this paper, the transfer matrix method of linear multibody system (MSTMM) is used to solve the problem. The natural frequencies and mode shapes of the system are obtained first, then the dynamic response by modal superposition. The time and position of the maximum displacement response and the maximum Von Mises stress are determined theoretically and independently of the shock. Furthermore, the effects of shock amplitude and frequency on the response are discussed. These MSTMM results agree well with those determined using the FEM. We achieved an accurate analysis of the mechanical behaviors of the MEMS inductor under shock load.

Model of a stationary thermal detonation wave in the “liquid lead – water” system for safety analysis of NPP with the reactor BREST-OD-300 during heat exchanger tube break accident

This paper presents a mathematical model of a stationary wave of thermal detonation in the “liquid lead - water” system, which can occur after the rupture of a steam generator tube of the reactor BREST-OD-300. The model is based on the mechanics of multiphase fluid flows. The system under study includes a continuous phase of liquid lead, in which drops of water surrounded by a steam film are dispersed. Heat exchange between high-temperature molten lead and water drops is carried out in the film boiling mode. A shock wave propagating in this multiphase system causes all phases in motion. Due to the significant difference in density between liquid lead and water, a difference in velocities arises between dispersed water drops and liquid lead, which cause the fragmentation of water droplets. The shock wave velocity and amplitude are determined by constructing the shock adiabat and the Hugoniot adiabat for the parameters of the studied multiphase mixture. To describe the fragmentation of water droplets, heat transfer between phases, and interfacial friction, the empirical correlations of the thermal detonation theory are used. The obtained numerical solutions describe the distributions of parameters in the zone of melt – water interaction.

Spectral stability of weak dispersive shock profiles for quantum hydrodynamics with nonlinear viscosity

This paper studies the stability of weak dispersive shock profiles for a quantum hydrodynamics system in one space dimension with nonlinear viscosity and dispersive (quantum) effects due to a Bohm potential. It is shown that, if the shock amplitude is sufficiently small, then the profiles are spectrally stable. This analytical result is consistent with numerical estimations of the location of the spectrum [43]. The proof is based on energy estimates at the spectral level, on the choice of an appropriate weighted energy function for the perturbations involving both the dispersive potential and the nonlinear viscosity, and on the montonicity of the dispersive profiles in the small-amplitude regime.

Investigation on double shock in vibration response of the gear pair with spalling defects

As an important component, the gear pair is frequently used in the mechanical transmission. If there is a localized spalling defect on the gear pair, the double shock phenomenon in the vibration response can be observed. In order to reveal the mechanism, an 8DOFs dynamic model for the gear pair is established, the symmetric and asymmetric rectangular spalling defects are considered, the torsional deformation is introduced. Meanwhile, the change law of the double shock during the defect expansion is explored. From the results, the double shock is caused by multiple changes in meshing stiffness, the amplitude and distribution range are closely related to the defect sizes. When the localized defect is extended along the length and depth directions, the amplitude of the double shock is significantly increased. If the defect area is asymmetric along the center plane, the torsional deformation of the gear pair will be generated. Then, the meshing stiffness is reduced, and the double shock becomes more obvious. Finally, through the finite element method and the experiment, the established model is confirmed.

Dynamic crack propagation under thermal impact

This paper addresses the dynamic crack growth in an isotropic media subjected to a non-classical thermal shock by implementing the Green-Naghdi (GN) generalized thermoelasticity theory in the eXtended Finite Element Method (XFEM). To satisfy the crack tip singularity for modeling dynamic crack growth via XFEM, proper enrichment functions for field variables are needed, which are extracted from the asymptotic fields for a dynamically growing crack in a thermoelastic media. These near tip fields of a dynamically moving crack are also used in the formulations of the maximum circumferential stress criterion and the equivalent stress intensity factor. Finally, numerical examples of the dynamic crack propagation under thermal shock are given out, and the effect of the dissipation coefficient in the GN model and the thermal shock amplitude on the crack growth path is studied in detail. In addition, the crack propagation under GN thermal shock is compared with that under classic thermal shock. The main difference between them is discussed in a numerical example.

Performance of a shock isolator inspired by skeletal muscles

Nonlinear vibration isolators with low dynamic stiffness have advantages compared to linear isolators, but they also have limitations in terms of the maximum allowable displacement around a chosen equilibrium position. To overcome this issue, a novel nonlinear isolator is proposed that outperforms the classical quasi-zero stiffness nonlinear isolator when large inputs are applied. In this paper, the response of the isolator when subject to shock excitation is investigated. The isolator suspension is designed to exhibit a quasi-static force-deflection curve with sigmoidal shape, which has been observed in many studies reporting the characteristics of skeletal muscles. The shape of the force-deflection curve is such that the suspension system can store greater elastic potential energy. Simulations show that the proposed four-spring system outperforms the classical nonlinear three-spring configuration, in terms of maximum displacement of a suspended mass, when the shock amplitude is relatively large. This is achieved by exploiting the softening effect for large deformations.

The higher-order nonlinearity and parametric effects on dust-ion-acoustic shock waves

In the case of obliquely propagated small-amplitude shock waves, in this study, we investigate the impacts of higher-order nonlinearity as well as various parameters (such as dust concentrations, viscosity, trapping parameters, etc) on the shock wave structures. The considered magnetized plasma system consists of three components, such as inertial positive ions (mobile), trapped electrons, and immobile negatively charged dust particles. The modified Burgers equation with a dominating dissipative term (in which the viscous effect is significant) is derived initially to examine the lower-order nonlinear and dissipative effects, and then, to the best of our knowledge, the modified Burgers-type linear inhomogeneous equation is derived for the first time to observe the higher-order nonlinear effects on shock waves while the plasma contains trapped electrons. The reductive perturbation method is used for the derivation of the equations, whereas the Abel’s theorem and the method of variation of parameters are used for adding the higher-order effect. From the theoretical investigation, we observe that the higher-order nonlinearity has an increasing effect on the shock amplitude. Furthermore, the viscosity and dust concentration increase the shock width and the phase speed, respectively.

The impact of the central opening on nonlinear effects in ultrasound fields generated by Sonalleve V1 and V2 MR-HIFU systems

Previous studies have shown that the Sonalleve MR-HIFU clinical system can generate shocks of 80–105 MPa at the focus, which is sufficient for boiling histotripsy. Two versions of the system (V1 and V2) have therapeutic transducers that differ in focusing angle, arrangement of elements, and the size of a central opening. The goal here was to reveal the impact of different array geometries on shock amplitudes at the focus. Nonlinear modeling of the array’s field in water using boundary conditions reconstructed from holography shows that at the same power output, the V2 array generates 10–15 MPa lower shock amplitudes at the focus. Although this difference is mainly caused by the smaller focusing angle of the V2 array, the larger central opening of the V2 array has a nontrivial impact. By suppressing a direct wave, the central opening produces a somewhat higher shock amplitude. Axisymmetric equivalent source models were constructed for both arrays and the importance of including the central opening was demonstrated. These models can be used in the “HIFU beam” software for simulating nonlinear fields of the Sonalleve V1 and V2 systems in water and flat-layered biological tissues. [Work supported by RSF-22-72-00047 and NIH R01EB025187.]

Characterization Of Shock Buffet On A Supercritical 2D Airfoil In Transonic Flow

The transonic flow over the supercritical OAT15A airfoil is investigated experimentally. Using time-resolved focusing schlieren sequences, we extracted the temporal evolution of the chordwise shock location for a wide range of combinations of Mach number and angle of attack. Based on this data, we identified the conditions of pre-buffet, buffet onset, fully-established buffet, and buffet-offset behavior. Key parameters characterizing most pronounced buffet behavior are presented and quantified in terms of the shock amplitude, its variation over multiple buffet cycles, and the associated frequency of the periodic shock oscillation. Fully-established buffet is identified at a Mach number of 0.72 and angle of attack of 5 deg. It is demonstrated that these operating conditions are characterized by a quasi-sinusoidal shock oscillation at a distinct frequency of 113 Hz, in good agreement with previous studies. This analysis reveals that the harmonic shock oscillation spanning a range of about 16.5% of the airfoil chord is accompanied by a coupled large-scale variation of the global flow topology. Both instantaneous and mean velocity components in the chordwise and vertical directions are discussed for representative phases of the buffet cycle. Instantaneous vector fields of both components provide insight into the flow organization throughout the buffet cycle and allow to identify the streamwise distance traveled by the shock wave within a full cycle. We can thus precisely capture the state of the shock wave and quantify the evolution of the shock-induced separation and the size of the turbulent wake. Severe variations of the flow topology are induced by the large-scale upstream and downstream shock displacement. The results show that incidents of massive flow separation are observed for shock positions being close to the upstream limit, and more general, that the wake influence is more pronounced when induced by an upstream traveling shock wave. Phase-averaged velocity profiles at significant locations along the airfoil suction side complement this analysis. Characteristic sets of velocity profiles are discussed in detail that allow to uniquely describe the development of the flow in each phase of the buffet cycle. Detailed contours and profiles of both velocity and Reynolds shear stress contributions across the wake region highlight the strong downstream influence the shock-induced turbulent wake. Phases of severe flow separation result in reverse flow persisting until the trailing edge which further substantiates the induced momentum deficit.