Surface Normal Velocity Research Articles

Modeling high-frequency backscattering from an impedance surface using Kirchhoff approximation

The Kirchhoff approximation is based on planar wave impinging on a planar surface, which is approximated for high-frequency wave scattering from a curved object. The Kirchhoff approximation is used to assess backscattering from a curved, rigid surface when exposed to high-frequency incident waves. In this presentation, we expand the rigid surface Kirchhoff approximation to accommodate impedance surfaces. The derivation is performed on a planar wave’s incident onto an impedance flat surface, resulting in a reflection coefficient, which in turn determines both surface pressure and surface normal velocity. When these two surface quantities are integrated into a Helmholtz integral formula, it allows for the evaluation of backscattering acoustic field pressures. The BeTTSi model, featuring an impedance surface, serves as an illustration for comparing the current Kirchhoff approximation with the solution obtained by a Fast Multipole Expansion Boundary Element Method (FMBEM). The latter is a numerical solution of the corresponding Helmholtz surface integral equation. The numerical discretization involves nearly a million points, representing the degrees of freedom necessary for modeling the BeTTSi under high-frequency incident waves. The comparison involves these two numerical evaluations across various incident wave directions, encompassing scenarios with single reflection and multiple reflections due to surface geometry.

Laser-assisted see-through technology for locating sound sources inside a structure

A laser-assisted see-through technology is developed to locate sound sources inside a structure and to analyze the interior sound field. Six lasers were employed to measure simultaneously the normal velocities on the exterior surface. These input data were used to locate sound sources inside a solid structure using a passive sonic detection and ranging algorithm, and then to reconstruct the interior sound field using the Helmholtz equation least squares method, and finally to observe the changes of the interior sound field over time through computer tomography. If signals are time invariant, all these can be accomplished with two lasers, one being fixed and another moving around to measure the normal surface velocity sequentially to establish transfer function with respect to the stationary laser. Once the transfer functions are established, they can be multiplied by any segment of time-domain signals measured by the fixed laser to acquire multiple normal surface velocities, as if they were measured simultaneously. This laser-assisted see-through technology has been validated experimentally and employed to observe the aerodynamically-induced sound field generated by a blower inside a projector. This development is important as it signifies a significant advancement in sound source localization, and opens the door to a class of applications presently unattainable.

Experimental Prediction Method of Free-Field Sound Emissions Using the Boundary Element Method and Laser Scanning Vibrometry

Acoustic emissions play a major role in the usability of many product categories. Therefore, mitigating the emitted sound directly at the source is paramount to improve usability and customer satisfaction. To reliably predict acoustic emissions, numerical methods such as the boundary element method (BEM) are employed, which allow for predicting, e.g., the acoustic emission into the free field. BEM algorithms need appropriate boundary conditions to couple the sound field with the structural motion of the vibrating body. In this contribution, firstly, an interpolation scheme is presented, which allows for appropriate interpolation of arbitrary velocity data to the computational grid of the BEM. Secondly, the free-field Helmholtz problem is solved with the open-source BEM software framework NiHu. The forward coupling between the device of interest and BEM is based on the surface normal velocities (i.e., a Neumann boundary condition). The BEM simulation results are validated using a previously established aeroacoustic benchmark problem. Furthermore, an application to a medical device (knee prosthesis frame) is presented. Furthermore, the radiated sound power is evaluated and contextualized with other low-cost approximations. Regarding the validation example, very good agreements are achieved between the measurements and BEM results, with a mean effective pressure level error of 0.63 dB averaged across three microphone positions. Applying the workflow to a knee prosthesis frame, the simulation is capable of predicting the acoustic radiation to four microphone positions with a mean effective pressure level error of 1.52 dB.

Atomistic Mechanism of Friction-Force Independence on the Normal Load and Other Friction Laws for Dynamic Structural Superlubricity.

We explore dynamic structural superlubricity for the case of a relatively large contact area, where the friction force is proportional to the area (exceeding ∼100 nm^{2}) experimentally, numerically, and theoretically. We use a setup composed of two molecular smooth incommensurate surfaces: graphene-covered tip and substrate. The experiments and molecular dynamic simulations demonstrate independence of the friction force on the normal load for a wide range of normal loads and relative surface velocities. We propose an atomistic mechanism for this phenomenon, associated with synchronic out-of-plane surface fluctuations of thermal origin, and confirm it by numerical experiments. Based on this mechanism, we develop a theory for this type of superlubricity and show that friction force increases linearly with increasing temperature and relative velocity for velocities larger than a threshold velocity. The molecular dynamic results are in a fair agreement with predictions of the theory.

Source identification of a vibrating plate using phase conjugation and interior boundary element method

Noise pollution is the most serious environmental pollution, which seriously affects people’s normal life and physical and mental health, as well as normal production work. Therefore, noise control is necessary, but before noise control, the first thing to do is to identify the location of the noise source, and the noise control work is meaningless when the noise cannot be identified. Flat plate structures are a fundamental part of complex ship structures, and their vibration, noise, and their interrelationships when excited by the outside world are receiving increasing attention. The core of the research in this paper is the identification of the sound source of the vibrating plate. A method combining phase conjugation with interior boundary element method is developed for the identification of the pressure and normal velocity distribution of a vibrating plate. An interior problem is formed by enclosing the phase conjugation array plane and the plate surface. The pressures at the array elements are phase-conjugated as the specified pressure boundary condition. The impedance relationship between the surface pressure and the surface normal velocity of the plate is utilized as a specified impedance boundary condition. The interior boundary element method is applied to solve the interior problem. The identification of the surface pressure and normal velocity distribution is studied numerically. The numerical results show that with the array located in the near field the proposed method achieves subwavelength focusing to identify the surface pressure and normal velocity distribution and clearly shows the response shapes.

Three-dimensional full-field velocity measurements in shock compression experiments using stereo digital image correlation.

Shock compression plate impact experiments conventionally rely on point-wise velocimetry measurements based on laser-based interferometric techniques. This study presents an experimental methodology to measure the free surface full-field particle velocity in shock compression experiments using high-speed imaging and three-dimensional (3D) digital image correlation (DIC). The experimental setup has a temporal resolution of 100ns with a spatial resolution varying from 90 to 200μm/pixel. Experiments were conducted under three different plate impact configurations to measure spatially resolved free surface velocity and validate the experimental technique. First, a normal impact experiment was conducted on polycarbonate to measure the macroscopic full-field normal free surface velocity. Second, an isentropic compression experiment on Y-cut quartz-tungsten carbide assembly is performed to measure the particle velocity for experiments involving ramp compression waves. To explore the capability of the technique in multiaxial loading conditions, a pressure shear plate impact experiment was conducted to measure both the normal and transverse free surface velocities under combined normal and shear loading. The velocities measured in the experiments using digital image correlation are validated against previous data obtained from laser interferometry. Numerical simulations were also performed using established material models to compare and validate the experimental velocity profiles for these different impact configurations. The novel ability of the employed experimental setup to measure full-field free surface velocities with high spatial resolutions in shock compression experiments is demonstrated for the first time in this work.

Experimental Validations of Reconstructed Excitation Forces Acting Inside a Solid Enclosure. Part I: Exterior Region

This paper presents the experimental validations of reconstructing the characteristics of the excitation forces that act inside a vibrating structure, which includes the location, type, amplitude, and spectrum, based on a single set of measurements of the normal surface velocity on the exterior surface by using the modified Helmholtz Equation Least Squares (HELS) method, as if one could see through this solid structure. Phase I of this paper shows the reconstruction of the vibroacoustic responses in the exterior region of the structure, including the field acoustic pressure, the surface acoustic pressure, the normal surface velocity or Operational Deflection Shape (ODS), the normal component of the time-averaged acoustic intensity, and the time-averaged acoustic power. Phase II of this paper illustrates the reconstruction of the excitation forces with the fluid-loading effects taken into consideration, based on the vibroacoustic responses reconstructed in the exterior region. The significance of the study, namely, the interrelationships among the excitation force, structural vibration, and acoustic radiation is discussed. The knowledge thus acquired may be important for engineers to analyze various complex noise and vibration issues in practice and to come up with the most cost-effective noise and vibration mitigation strategies.

Experimental Validations of Reconstructed Excitation Forces Acting Inside a Solid Enclosure. Part II: Interior Region

Part II of this paper discusses experimental validations of the reconstructed excitation forces acting inside a vibrating structure with the fluid-loading effect taken into consideration. Specifically, the characteristics of the excitation forces such as their locations, types, amplitudes, and spectra are reconstructed by using the modified Helmholtz Equation Least Squares (HELS) method, based on a single set of measurements of the normal surface velocity on the exterior surfaces, as if one could see through such a solid structure. Since the fluid-loading effect has a direct impact on the vibration responses of a structure, it is not possible to derive analytic solutions to vibration responses of the structure. Therefore, numerical solutions are sought by using the boundary element method (BEM). The fluid-loading effect, a.k.a., the reverberation sound field inside the structure is calculated based on the absorption coefficients and surface areas of the interior objects. The reconstructed excitation forces are then compared to the benchmark values and satisfactory agreements are obtained.

On the Efficiency of Staggered C-Grid Discretization for the Inviscid Shallow Water Equations from the Perspective of Nonstandard Calculus

This paper provides a rationale for the commonly observed numerical efficiency of staggered C-grid discretizations for solving the inviscid shallow water equations. In particular, using the key concepts of nonstandard calculus, we aim to show that the grid staggering of the primitive variables (surface elevation and normal velocity components) is capable of dealing with flow discontinuities. After a brief introduction of hyperreals through the notion of infinitesimal increments, a nonstandard rendition of the governing equations is derived that essentially turns into a finite procedure and permits a convenient way of modeling the hydraulic jumps in open channel flow. A central result of this paper is that the discrete formulations thus obtained are distinguished by the topological structures of the solution fields and subsequently provide a natural framework for the staggered discretization of the governing equations. Another key of the present study is to demonstrate that the discretization naturally regularizes the solution of the inviscid flow passing through the hydraulic jump without the need of non-physical dissipation. The underlying justification is provided by analytically studying the distributions of the flow variables across an infinitesimal thin hydraulic jump along with the use of hyperreal Heaviside step functions. This main finding is shown to be useful to comprehend the importance of the application of staggered finite difference schemes to accurately predict rapidly varying free-surface flows. A numerical experiment is provided to confirm this result.

A Modified Helmholtz Equation Least Squares Method for Reconstructing Vibroacoustic Quantities on an Arbitrarily Shaped Vibrating Structure

A modified Helmholtz equation least-square (HELS) method is developed to reconstruct vibroacoustic quantities on an arbitrarily shaped vibrating structure. Unlike the traditional nearfield acoustical holography that relies on the acoustic pressures collected on a hologram surface at a short stand-off distance to a target structure, this modified HELS method takes the partial normal surface velocities and partial acoustic pressures as the input data. The advantages of this approach include but not limited to: (1) The normal surface velocities that represent the nearfield effects are collected directly, which lead to a more accurate reconstruction of the normal surface velocity distribution; (2) The field acoustic pressures are also measured, which leads to a more accurate reconstruction of the acoustic pressure on the source surface as well as in the field; and (3) There is no need to measure the normal surface velocities over the entire surface, which makes this approach quite appealing in practice because most vibrating structures do not allow for measuring the normal surface velocities over the entire source surface as there are always obstacles or constrains around a target structure. Needless to say, regularization is necessary in reconstruction process since all inverse problems are mathematically ill-posed. To validate this approach, both numerical simulations and experimental results are presented. An optimal reconstruction scheme is developed via numerical simulations to achieve the most cost-effective reconstruction results for practical applications.

On Forced-Vibroacoustic Component Analyses of an Arbitrarily Shaped Vibrating Structure

This paper presents a methodology that enables one to trace back the forced-vibroacoustic components (F-VAC) that are responsible for acoustic radiation from an arbitrarily shaped structure. This methodology relies on a modified Helmholtz equation least-square (HELS) method that takes partial normal surface velocities obtained by using a laser vibrometer and partial acoustic pressures measured using a small array of microphones as input to reconstruct the distributions of the normal surface velocity and acoustic pressure over the entire source surface. Next, the normal component of the time-averaged acoustic intensity distribution is reconstructed and linked to the field acoustic pressure. Finally, the singular value decomposition (SVD) technique is used to reveal the most critical components of the structural vibrations that are responsible for acoustic radiation. The advantages of using partial normal surface velocities and acoustic pressures as input data are that: (1) it enables one to acquire the rich near-field information embedded in the normal surface velocities; (2) it is a noncontact and noninvasive measurement approach using a laser vibrometer or scanning laser and a small microphone array at a remote location; (3) it enables one to obtain a comprehensive picture of the vibroacoustic fields that traditional near-field acoustic holography (NAH) cannot offer; (4) it allows for correlating the near-field effects to the far-field acoustic pressures and revealing the root causes of structure-borne sound radiation; (5) it provides an insight into developing the most cost-effective noise reduction measures; and (6) it is suited for engineering applications since in practice there are always areas that are inaccessible for measurements. Finally, it is emphasized that although sound may be produced by structural vibrations, not all structural vibrations may produce sound. Therefore, all one needs to do is to suppress the most critical components of structural vibrations that can emit sound. Experimental validations of using F-VAC analyses to reduce sound radiation from an arbitrarily shaped structure are demonstrated.

Obtaining flow curve for viscoplastic fluids through inclined open-channel apparatus

Non-Newtonian fluids are commonly seen in industrial processes, such as those of the oil and mining industry, and in natural flows, like dam ruptures, landslides or mud flows. The hydrodynamic modeling of such processes/phenomena is directly linked to the rheological properties of the flowing fluid, usually characterized through rheometers. The high cost of rheometers and possible inaccessibility for certain applications demand for research of alternative rheometric methods. In order to assess the problem, the present work discusses a detailed experimental methodology to evaluate if the steady and uniform flow in an inclined channel is able to produce the flow curve for the test fluid carbopol 996 gel and work as an alternative rheometer. In order to estimate the shear rates and shear stresses, we measured the normal depth (ultrasonic technique), specific discharge (manual gravimetric method) and free surface velocity (manually and with laser barrier sensors). Based on the theoretical solutions, a simplified fitting procedure was adopted to make possible the assessment of shear rate and shear stress through the experimental data. The obtained flow curves were then compared with the reference flow curve, determined by a commercial R/S rheometer. Results showed that the experimental methods were able to provide the flow curves within acceptable uncertainty and the defined methodology detailed in the work can estimate satisfactorily the flow curve of non-Newtonian fluids. Finally, we highlighted that the wide channel hypothesis is the strongest condition to be guaranteed in order to obtain precise flow curves through the methodology present in this work.

Determining structural damping and vibroacoustic characteristics of a non-symmetrical vibrating plate in free boundary conditions using the modified Helmholtz equation least squares method

This paper presents a comprehensive study to determine the structural damping and vibroacoustic responses of an arbitrarily shaped planar structure subject to non-contact acoustic excitations under free boundary conditions using a modified HELS (Helmholtz equation least squares) method. The input data consist of the normal surface velocities measured by a laser vibrometer at a discrete number of points on the source surface, and the acoustic pressures measured by a small array of microphones in the field. The normal surface velocity distribution over the entire surface of the plate is then reconstructed by the HELS method and compared to benchmark data. Similarly, the reconstructed acoustic power level spectra are compared to those measured by the array of microphones. The reconstructed vibroacoustic quantities using the modified HELS method are interrogated, and the method's accuracy evaluated. Specifically, a limited number of normal velocity data points on the surface of the target structure are taken as input to the HELS formulations. The reconstructed velocity distributions on the entire surface of the structure with much higher density even for areas of the target structure with little or no input data were compared to the benchmark results. It is worth noticing that no other vibroacoustic technologies are available that allow for complete reconstruction of the normal surface velocity distributions based on limited input data. The dimensionless damping ratio of the structure is also determined. Results indicate that the dimensionless damping ratio for metals, for instance steel, is frequency dependent rather than a constant. Moreover, the empirical formulation developed in this study enables one to get the dimensionless damping ratios continuously over the frequency range from 0 to 10,000 Hz.

Modeling Vibration-Induced Tire-Pavement Interaction Noise in the Mid-frequency Range

ABSTRACT Tire-pavement interaction noise (TPIN) is one of the main sources of exterior noise produced by vehicles traveling at greater than 50 kph. The dominant frequency content is typically within 500–1500 Hz. Structural tire vibrations are among the principal TPIN mechanisms. In this work, the structure of the tire is modeled and a new wave propagation solution to find its response is proposed. Multiple physical effects are accounted for in the formulation. In an effort to analyze the effects of curvature, a flat plate and a cylindrical shell model are presented. Orthotropic and nonuniform structural properties along the tire's transversal direction are included to account for differences between its sidewalls and belt. Finally, the effects of rotation and inflation pressure are also included in the formulation. Modeled frequency response functions are analyzed and validated. In addition, a new frequency-domain formulation is presented for the computation of input tread pattern contact forces. Finally, the rolling tire's normal surface velocity response is coupled with a boundary element model to demonstrate the radiated noise at the leading and trailing edge locations. These results are then compared with experimental data measured with an on-board sound intensity system.

Axisymmetric granular flow on a bounded conical heap: Kinematics and size segregation

Free surface flows of granular materials in bounded axisymmetric geometries such as a cylindrical silo are poorly understood. In particular, a detailed description of the local three-dimensional velocity field and predictive models for segregation are lacking. Here, the details of the kinematics of flow in a rising conical heap formed by a centrally fed mixture of spherical particles are investigated using Discrete Element Method (DEM) simulations in a wedge-shaped geometry with periodic azimuthal boundaries. The dependence of the streamwise and surface normal velocity components on the inlet flow rate and silo radius for size-bidisperse and monodisperse mixtures of glass spheres is characterized. Compared to a wedge with frictional sidewalls, the flowing layer is much thicker on average in the axisymmetric case. Using the scalings for kinematics obtained from the DEM simulations, a modified continuum advection-diffusion-segregation model accurately predicts steady-state segregation for axisymmetric conical heap flows of size-bidisperse mixtures.

Gear shift simulation by using virtual prototype

The presented paper is focused on the simulation of gear shift and prediction of whole gearbox dynamic behavior during gear shift. For this complex task the methodology for simplified gearbox was developed first. The whole methodology is divided into three different level of numerical model, where each of them was developed and validated by technical experiment. Afterwards the methodology was applied on more complex gearboxes, in presented case the heavy-duty gearbox concept. The virtual prototype was extended by gear shift option. The numerical sensitivity study of different gear shift time, which are simulated by different clutch activation and deactivation, was performed and the surface normal velocity and torque were evaluated and compared. The simulation was performed to find the optimal gear shift progress with minimal affects vehicle comfort. For this purpose, the surface normal velocity is evaluated, which is directly connected to emitted noise of whole gearbox.

Effects of Nozzle Geometry on Turbulent Characteristics and Structure of Surface Attaching Jets

The effects of nozzle exit geometry on the characteristics of a surface attaching jet were investigated experimentally. Three different types of nozzle geometries were studied, including a circular, square and rectangular nozzle with the same exit area. The jets were discharged with an offset height of two nozzle width at a Reynolds number of 5500. A particle image velocimetry system was used to measure the flow. Instantaneous visualizations revealed that the largest enhancements in the near field mixing and entrainment occur in the minor plane of the rectangular nozzle compared to the square and circular nozzles resulting in more rapid expansion of the shear layer. This also caused a faster deflection of the jet towards the free surface, maximum velocity decay and spread rates of the rectangular jet with minor axis orientation. The jet-surface interaction was examined using surface velocity, vorticity thickness and surface turbulence intensities. It was found that the damping of the surface-normal velocity fluctuations at the free surface is more severe in the minor plane of the rectangular jet than the square and circular nozzles. The influence of the free surface was also felt in the profiles of mean velocity and Reynolds stresses. The attenuation of Reynolds shear stress due to confinement was more dramatic in the minor plane of the rectangular nozzle than the other geometries. To quantify the influence of nozzle geometry on the coherent structures in the interaction region, two-point correlations of the velocity fluctuations and swirling strength; and proper orthogonal decomposition were performed.

Unsteady free surface flow above a moving circular cylinder

A problem on non-stationary free surface flow of infinitely deep ideal fluid generated by the motion of a submerged body is considered. The water wave problem is reduced to the integral–differential system of equations for the functions defining free surface shape, normal, and tangential velocity components on the free boundary. Small-time asymptotic solution is constructed for the case of circular cylinder that moves with constant acceleration from rest. The role of non-linearity is clarified by analysis of this approximate solution which describes the formation of added mass layers, splash jets, and finite amplitude surface waves.

Parameter effecting the experimental determination of modal properties

The investigation of structure dynamic behavior can be done by finding modal properties, which are based on the relation of force and vibration response. The information about exact force value is in most application unknown, but the investigation can be done for unit magnitude of force. In case of using modal exciter, the force applied to structure is transferred by connection between modal exciter and structure. The stiffness of connection part can affect the dynamic behavior and change the modal properties, or response of investigated part, because of additional modal properties of connection part. This article is focused on the influence of two different rod stiffness during modal properties identification. The modal exciter is used to find potential critical frequency during decreasing frequency. The affect is evaluated on the surface normal velocity as well as acoustic pressure. The stiffness of rod significantly affects the dynamic response of structure and based on that can add frequency, when the emitted noise is high.

Reconstructing non-stationary surface normal velocity of a planar structure using pressure-velocity probes

An acoustic method based on the time domain plane wave superposition method is proposed to reconstruct the non-stationary surface normal velocity of an impacted planar structure by measuring the normal particle velocity. In this method, the time-evolving normal particle velocity on the hologram plane is first measured by pressure-velocity probes; then, the normal particle velocity spectrum on a virtual source plane is used to establish the relationship between the time-evolving normal particle velocity on the hologram plane and the time-evolving surface normal velocity on the structural plane; finally, the normal particle velocity spectrums can be solved by an iterative solving process and are used to calculate the non-stationary surface normal velocity of the planar structure. An experiment of a planar steel plate impacted by a steel ball is presented to examine the ability of the proposed method, where the time-evolving normal particle velocity and pressure on the hologram plane measured by pressure-velocity probes are used as the inputs of the proposed method and the pressure-based reconstruction method, respectively, and a laser Doppler vibrometry is used to measure the surface normal velocity of the plate as the reference for comparisons. The comparison results demonstrate that the proposed method is effective in reconstructing the non-stationary surface normal velocity in both time and space domains and can provide more accurate results than that of the pressure-based reconstruction method.