Rigid Boundary Conditions Research Articles

Two modelling techniques for the vibration and transmission loss of a nonlinear vibro-acoustic system

Fahy’s and Pretlove’s modelling techniques for a vibro-acoustic system have been employed in many research works. This is the first article addressing the differences between these modelling techniques for a nonlinear vibro-acoustic system. Pretlove’s technique is used for vibro-acoustic systems with one flexible panel only (or 2D enclosures); Fahy’s is applied to vibro-acoustic systems with multiple flexible panels (or 3D enclosures). The main difference between them is that all acoustic boundary conditions can be satisfied in the 2D modelling technique, while only rigid acoustic boundary conditions can be satisfied in the 3D modelling technique. The multi-level residue harmonic balance method, which was recently developed and modified from the classical harmonic balance method, is employed here. The advantage of this method is that it requires less computation effort when solving the nonlinear equations generated in the entire solution process. The comparisons between various transmission loss and nonlinear vibration results generated from the two models are investigated.

An Exact, Steady, Purely Azimuthal Equatorial Flow with a Free Surface

AbstractThe general problem of an ocean on a rotating sphere is considered. The governing equations for an inviscid, incompressible fluid, written in spherical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions, are introduced. An exact solution of this system is presented; this describes a steady flow that is moving only in the azimuthal direction, with no variation in this direction. However, this azimuthal velocity component has an arbitrary variation with depth (i.e., radius), and so, for example, an Equatorial Undercurrent (EUC) can be accommodated. The pressure boundary condition at the free surface relates this pressure to the shape of the surface via a Bernoulli relation; this provides the constraint on the existence of a solution, although the restrictions are somewhat involved in spherical coordinates. To examine this constraint in more detail, the corresponding problems in model cylindrical coordinates (with the equator “straightened” to become a generator of the cylinder), and then in the tangent-plane version (with the β-plane approximation incorporated), are also written down. Both these possess similar exact solutions, with a Bernoulli condition that is more readily interpreted in terms of the choices available. Some simple examples of the surface pressure, and associated surface distortion, are presented. The relevance of these exact solutions to more complicated, and physically realistic, flow structures is briefly mentioned.

Numerical investigation of electrohydrodynamic instability and bifurcation in a dielectric liquid subjected to unipolar injection

The electroconvection of dielectric liquids subjected to unipolar injection is numerically studied in a two-dimensional cavity. To understand the Coulomb force driven electroconvective instabilities and bifurcations in the closed cavity, a high resolution upwind scheme is applied to carry out linear stability analysis and numerical simulation of nonlinear electrohydrodynamic equations. We focus on the strong injection case, where the non-dimensional injection parameter C is fixed at C=10. The length to height ratio of the cavity is fixed at 0.614. The numerical simulations are performed subject to free and rigid lateral wall boundary conditions. Two dimensionless mobility parameters M=5 and M=10 are considered for each kind of the lateral boundary conditions. Our linear stability analysis result is consistent with previous theoretical predictions. The nonlinear behaviors of the investigated system beyond the convection threshold are carefully examined. Abundant bifurcation phenomena, such as the pitchfork bifurcation, Hopf bifurcation, quasi-periodic Hopf bifurcation and heteroclinic bifurcation, have been observed. Some nonlinear flow features, such as the hysteresis loops, coexistence of multiple solutions and transition to chaos, have also been demonstrated. Our simulation results show that flow structure and bifurcation sequence can be greatly affected by the specified lateral boundary conditions and the chosen mobility parameters. Besides the static state, a series of steady and unsteady solutions has been calculated, like the one-cell steady state, one-cell periodic state, one-cell quasi-periodic state, and the one-cell chaotic state. All these convective states with two-cell structure are also obtained. Moreover, a periodic state oscillating between the one-cell structure and the two-cell structure has been found in flow domain bounded by rigid walls.

Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions

Abstract In this paper, the free and forced vibration analysis of circular cylindrical double-shell structures under arbitrary boundary conditions is presented. This is achieved by employing the improved Fourier series method based on Hamilton’s principle. In the formulation, each displacement component of the cylindrical shells and annular plates is invariantly expanded as the superposition of a standard Fourier series with several supplementary functions introduced to remove the potential discontinuities of the original displacement and its derives at the boundaries. With the introduction of four sets of boundary springs at the coupling interfaces and end boundaries of the shell–plate combination, both elastic and rigid coupling and end boundary conditions can be easily obtained by assigning the stiffnesses of the artificial springs to certain values. The natural frequencies and mode shapes of the structures as well as frequency responses under forced vibration are obtained with the Rayleigh–Ritz procedure. The convergence of the method is validated by comparing the present results with those obtained by the finite element method. Several numerical results including natural frequencies and mode shapes are presented to demonstrate the excellent accuracy and reliability of the current method. Finally, a number of parameter studies concerning various end and coupling boundary conditions, different dimensions of shells and annular plates are also performed.

On the shielding of sound from a source near one or two coaxial cones

The shielding of the sound from a source near a cone or two coaxial circular cones is addressed by an exact analytical method, considering free spherical waves satisfying a rigid or impedance boundary condition on the wall of the cone(s); the boundary condition is applied for impedance proportional to the distance from the vertex. The exact solution involves the eigenvalues of the problem that are generally real or complex (not integers), and coincide with the degree of the associated Legendre functions and order of the spherical Bessel functions which specify respectively the latitudinal and radial dependencies of the wave field. These eigenfunctions or ‘conical wave harmonics’ can be chosen in more than one way, e.g. (a) as standing wave modes which are finite at the vertex of the cone(s), but do not satisfy a radiation condition at infinity; (b) as propagating waves satisfying a radiation condition at infinity, and generally singular at the vertex of the cone(s). The method of calculation of eigenvalues and eigenfunctions is presented both for a single cone and two coaxial cones with the same vertex, and arbitrary aperture(s). The method specifies the eigenvalues, and the corresponding radial, azimuthal and latitudinal eigenfunctions. An asymptotic formula is obtained for the eigenvalues which gives reasonable good agreement with the exact results. The eigenvalues and eigenfunctions appear with suitable amplitudes in the Green function representing a monopole source near the vertex of the cone(s). The acoustic field is plotted also for a longitudinal and a transverse dipole and mixed quadrupole source near the vertex.

Pseudo-Casimir stresses and elasticity of a confined elastomer film.

Investigations of the elastic behavior of bulk elastomers have traditionally proceeded on the basis of classical rubber elasticity, which regards chains as thermally fluctuating but disregards the thermal fluctuations of the cross-links. Herein, we consider an incompressible and flat elastomer film of an axisymmetric shape confined between two large hard co-planar substrates, with the axis of the film perpendicular to the substrates. We address the impact that thermal fluctuations of the cross-links have on the free energy of elastic deformation of the system, subject to the requirement that the fluctuating elastomer cannot detach from the substrates. We examine the behavior of the deformation free energy for one case where a rigid pinning boundary condition is applied to a class of elastic fluctuations at the confining surfaces, and another case where the same elastic fluctuations are subjected to soft "gluing" potentials. We find that there can be significant departures (both quantitative and qualitative) from the prediction of classical rubber elasticity theory when elastic fluctuations are included. Finally, we compare the character of the attractive part of the elastic fluctuation-induced, or pseudo-Casimir, stress with the standard thermal Casimir stress in confined but non-elastomeric systems, finding the same power law decay behavior when a rigid pinning boundary condition is applied; for the case of the gluing potential, we find that the leading order correction to the attractive part of the fluctuation stress decays inversely with the inter-substrate separation.

Special Mortars for Repair Masonry: The Problem of the Optimization

Destruction masonry is a common problem. The nature of the material of masonry and the degree of preservation offers different combinations of technological methods of repair. In this regard it has formulated one of the main tasks of restoration materials science - finding the optimal composition of a material with a large number of variables in the presence of rigid boundary conditions in a short time. The solution to this problem lies in the area of mathematical modeling using the methods of planning of experiments. In this regard the aim of this study is the development of methods for optimization of mortars that allows getting the desired result in a relatively short period of time.

Experimental Testing of Bend-Twist Coupled Composite Shafts

Bend-twist coupling is a property of certain specially designed composite beams that when subjected to loading conditions that would normally result in pure bending, undergo both bending and twisting. This phenomenon is called bend-twist coupling. To fully characterize the degree of bend-twist coupling in the beam, shear center per unit length (e y /L) must be determined. This paper proposes an experimental method for quantifying e y /L. The specific method for manufacturing these composite shafts is detailed. A special apparatus was constructed to load the tip of a cantilever sample with an adjustable torque. Digital image correlation (DIC) was used to measure tip rotation and deflection. From these measurements shear center values were determined. Repeating this process for different shaft lengths and orientations allows for a final e y /L measurement. Techniques are provided to reduce the various uncertainties in these measurements. The experiment utilizes three dimensional DIC, a rigid boundary condition such as a vice to simulate a cantilever condition, and an adjustable moment arm on which to load the specimen. The special case of a cylindrical carbon fiber beam is presented here, but the experimental methods can be generalized to tapered beams, non-circular cross-sections, and other orthotropic materials. The results are compared against finite element and analytical predictions. The final average experimental e y /L for all four shafts is 8 % higher than what the analytical method and FEA predict.

Microwave-heating laboratory experiments for planetary mantle convection

Thermal evolution of telluric planets is mainly controlled by secular cooling and internal heating due to the decay of radioactive isotopes, two processes that are equivalent from the standpoint of convection dynamics. In a fluid cooled from above and volumetrically heated, convection is dominated by instabilities of the top boundary layer and the interior thermal structure is non-isentropic. Here we present innovative laboratory experiments where microwave radiation is used to generate uniform internal heat in fluids at high Prandtl number (${>}300$) and high Rayleigh–Roberts number (ranging from $10^{4}$ to $10^{7}$), appropriate for planetary mantle convection. Non-invasive techniques are employed to determine both temperature and velocity fields. We successfully validate the experimental results by conducting numerical simulations in three-dimensional Cartesian geometry that reproduce the experimental conditions. Scaling laws relating key characteristics of the thermal boundary layer, namely its thickness and temperature drop, to the Rayleigh–Roberts number have been established for both rigid and free-slip boundary conditions. A robust conclusion is that for rigid boundary conditions the internal temperature is significantly higher than for free-slip boundary conditions. Our scaling laws, coupled with plausible physical parameters entering the Rayleigh–Roberts number, enable us to calculate the mantle potential temperature for the Earth and Venus, two telluric planets with different mechanical boundary conditions at their surface.

Analytical zigzag formulation with 3D elasticity corrections for bending and stress analysis of circular/annular composite sandwich plates with auxetic cores

In the present paper, distributions of lateral deflections and in-plane normal and transverse shear stresses of circular/annular sandwich plates with orthotropic composite face sheets and auxetic (with negative Poisson’s ratio) cores are determined, for the first time, employing a zigzag theory whose results are corrected based on the three-dimensional theory of elasticity. The governing equations are developed based on the principle of minimum total potential energy and solved using a differential transform whose center is located at the outer radius of the plate. A comprehensive parametric study including evaluation of effects of various geometric and material parameters, lamination schemes, and boundary conditions is accomplished. Accuracy of the results is verified by results of the 3D theory of elasticity extracted from the ABAQUS code. Results reveal that the core auxeticity increases the rigidity of the soft core drastically and consequently, decreases the lateral deflection of the plate and moderates the interfacial jumps in the through-thickness distribution of the transverse shear stress, so that a semi-continuous parabolic distribution may be established. The core auxeticity may considerably reduce the maximum radial stresses of the face sheets and influence of the auxeticity is more remarkable for plates with more rigid constructions or boundary conditions.

Experimental tests of protective covers to obtain optimal transmission loss in porous materials

Porous resistive materials are widely used to increase the wide-band transmission loss and reduce shell vibrations and noise propagation. For industrial applications, porous materials need to be covered to protect them from environmental contaminants. The goal of our experimental studies is to determine suitable covers which are acoustically transparent and offer this needed protection. Expansion volumes were designed for transmission loss tests with two accurate modeled limits related to two rigid boundary conditions. Seven different covers were tested in this study tests. Best results were achieved using thin porous steel woven cloth. Tyvek, Mylar, thick steel woven cloth, and the Nomex were not as effective. The porous materials and covering materials were tested and then retested after being soaked in oil to explore transmission loss that might occur in actual industrial conditions. These independent measurements of the cover materials were used to determine the material property in the FEA models.

A Numerical and experimental study on the crash behavior of the extruded aluminum crash box with elastic support

Thin-walled structures like crash boxes may be used as energy absorption members in automotive chassis. There have been many studies addressing the behaviors of energy absorption members on frontal crash. These researches have attempted to predict the energy absorption and maximum impact load in shell structures. The energy absorption and maximum impact load depend on many parameters including boundary condition, history of plastic deformation during metalworking, geometry; and material and impact energy (i.e. mass and velocity of the striker). This study examined the crash behavior of square tube made of the extruded aluminum alloy 6063T4 using an elastic boundary condition -instead of rigid boundary condition- on the bottom of a crash box, In addition, the effect of elastic boundary condition on energy absorption of square tubes under the impact load was investigated both numerically and experimentally. Results showed that using elastic boundary could change the deformation mode and decrease the maximum impact load. Further, a high correlation between the numerical and experimental results validated the findings of the study.

Viscous and potential forces on an advancing surface-piercing flat plate with a fixed drift angle

Transverse force and yaw moment acting on a surface-piercing flat plate with yaw angle and forward speed are studied. Following the Froude hypothesis, the problem is decomposed into a tail-separated forward flow and a bottom-tip-separated cross-flow parts. The objective is to quantify the role of different components in the resulted force and moment acting on the plate. A 3D potential flow code using distribution of Rankine sources and dipoles is used to solve the linear potential flow problem. The free-surface boundary condition is linearized based on the undisturbed flow velocity (Neumann–Kelvin linearization). The plate’s trailing-edge flow separation is modeled using a dipole distribution behind the plate on a linearized vortex sheet. The force and moment due to the cross-flow separation from the plate’s bottom tip are calculated by means of a slender body cross-flow method with a rigid free-surface boundary condition. Hence the free-surface effects are confined in the forward flow problem and neglected in the cross-flow problem. Simulations are carried out for different aspect ratios and drift angles. The plate’s thickness is taken into account. Convergence and sensitivity studies are performed carefully. Results are compared with previous experimental and numerical studies. The importance of the second-order forces and validity of the linear assumptions are touched upon. The overall agreement of the results is acceptable. The relative importance of the two viscous and potential components are studied. It is shown that the decomposed problem could follow the experimental results up to a relatively high Froude number.

Numerical nonlinear analysis of subcritical Rayleigh-Bénard convection in a horizontal confined enclosure filled with non-Newtonian fluids

Non-Newtonian subcritical convection in a Rayleigh-Bénard configuration was investigated numerically, using rigid boundary conditions. The flow configuration consisted of a finite aspect ratio enclosure subject to a vertical temperature gradient, which was established by heating and cooling the lower and the upper walls by either a constant heat flux or by applying constant temperatures. The non-Newtonian fluid viscosity was modeled using the Carreau-Yasuda model. The convective flow was governed by the conservation equations, which were solved numerically using a finite difference method with a time-accurate scheme. For a shallow enclosure, when the walls were subject to constant heat fluxes, an asymptotic solution was derived assuming a parallel flow behavior. A comparison between the numerical and asymptotic solutions was performed. The effects of the controlling parameters, namely, the Rayleigh and the Prandtl numbers, and the fluid rheological parameters on the onset of subcritical convective flow were investigated. The threshold for the onset of subcritical convection was found to be well below the threshold of stationary convection and decreased considerably (with the fluid rheological parameters variation) as the fluid became more and more shear thinning. Depending on the governing parameters, steady and unsteady-periodic flow solutions were possible. Within a square enclosure with slip or no-slip vertical walls, maintaining the active walls at a constant heat flux or at constant temperature, the rheological subcritical flow behavior remained qualitatively the same, but led to different subcritical convection thresholds.

A study of tyre cavity resonance and noise reduction using inner trim

A study of tyre inner trim as a method for reducing tyre cavity resonance noise is presented. The tyre is modelled as a rectangular toroid where only the outside shell is flexible. A modal series solution of the sound pressure frequency response inside the tyre cavity is derived from the wave equation using modal superposition. In the solution with the rigid and flexible wall boundary condition, the effect of placing a trim layer onto the inner surface of the tyre tread plate wall is reflected by adding a damping loss term in the sound pressure frequency response function. The numerical simulation result was then compared with the result obtained from a roving impact test performed on a tyre. The results show that selective trim material may be effective for reducing the structure-borne noise magnitude resulting from the tyre cavity resonance.

Importance of initial buoyancy field on evolution of mantle thermal structure: Implications of surface boundary conditions

Although there has been significant progress in the seismic imaging of mantle heterogeneity, the outstanding issue that remains to be resolved is the unknown distribution of mantle temperature anomalies in the distant geological past that give rise to the present-day anomalies inferred by global tomography models. To address this question, we present 3-D convection models in compressible and self-gravitating mantle initialised by different hypothetical temperature patterns. A notable feature of our forward convection modelling is the use of self-consistent coupling of the motion of surface tectonic plates to the underlying mantle flow, without imposing prescribed surface velocities (i.e., plate-like boundary condition). As an approximation for the surface mechanical conditions before plate tectonics began to operate we employ the no-slip (rigid) boundary condition. A rigid boundary condition demonstrates that the initial thermally-dominated structure is preserved, and its geographical location is fixed during the evolution of mantle flow. Considering the impact of different assumed surface boundary conditions (rigid and plate-like) on the evolution of thermal heterogeneity in the mantle we suggest that the intrinsic buoyancy of seven superplumes is most-likely resolved in the tomographic images of present-day mantle thermal structure. Our convection simulations with a plate-like boundary condition reveal that the evolution of an initial cold anomaly beneath the Java-Indonesian trench system yields a long-term, stable pattern of thermal heterogeneity in the lowermost mantle that resembles the present-day Large Low Shear Velocity Provinces (LLSVPs), especially below the Pacific. The evolution of subduction zones may be, however, influenced by the mantle-wide flow driven by deeply-rooted and long-lived superplumes since Archean times. These convection models also detect the intrinsic buoyancy of the Perm Anomaly that has been identified as a unique slow feature distinct from the two principal LLSVPs. We find there is no need for dense chemical ‘piles’ in the lower mantle to generate a stable distribution of temperature anomalies that are correlated to the LLSVPs and the Perm Anomaly. Our tomography-based convection simulations also demonstrate that intraplate volcanism in the south-east Pacific may be interpreted in terms of shallow small-scale convection triggered by a superplume beneath the East Pacific Rise.

A Method of Function Space for Vertical Impedance Function of a Circular Rigid Foundation on a Transversely Isotropic Ground

This paper is concerned with investigation of vertical impedance function of a surface rigid circular foundation resting on a semi-infinite transversely isotropic alluvium. To this end, the equations of motion in cylindrical coordinate system, which because of axissymmetry are two coupled equations, are converted into one partial differential equation using a method of potential function. The governing partial differential equation for the potential function is solved via implementing Hankel integral transforms in radial direction. The vertical and radial components of displacement vector are determined with the use of transformed displacement-potential function relationships. The mixed boundary conditions at the surface are satisfied by specifying the traction between the rigid foundation and the underneath alluvium in a special function space introduced in this paper, where the vertical displacements are forced to satisfy the rigid boundary condition. Through exercising these restraints, the normal traction and then the vertical impedance function are obtained. The results are then compared with the existing results in the literature for the simpler case of isotropic half-space, which shows an excellent agreement. Eventually, the impedance functions are presented in terms of dimensionless frequency for different materials. The method presented here may be used to obtain the impedance function in any other direction as well as in buried footing in layered media.

Comment on Falcke et al., “Polymerization, bending, tension: What happens at the leading edge of motile cells?”

Commentary on the contribution by M. Falcke and J. Zimmermann (1) in this special issue. In their contribution, the authors review the implications of their model of cytoske- leton-membrane interaction for the force velocity relation of moving cells, as well as for cell oscillations and protrusion formation. Their model, which neglects spatial degrees of freedom, couples a small, anterior region of the lamellipodium, where a loose brush of actin filaments makes contacts with the membrane, to a gel-like region behind in the lamella. The model accounts in detail for several distinct processes and it is impressive in its power to reproduce experimental data. Below we give a series of comments, questions and suggestions, related mostly to cell motility. We do not expect the authors to have clear-cut answers, of course, the aim is rather to stimulate further discussion and future research. The gel and large deformations. The gel is only accounted for via crosslinking rates and in the retro- grade flow. The force transfer is not very transparent: is it sticking to the substrate or gliding? Most importantly, should it not enter the force balance as an elastic degree of freedom? For free protrusions this may be safely neglected (rigid boundary condition due to infinite stiffness compared to the semiflexible region). However, it may have important effects on the semiflexible region in geometries with large deformations, as occurring when a keratocyte is crawling against a AFM cantilever: one could measure, to some extent, the gel and not the semiflexible region. Possibly, this issue could be addressed in an extended model including spatial degrees of freedom? Dynamic adhesion. An actively debated question in cell motility is the effect of adhesion on the motility modes and phenotype switches. Barnhart et al. (2) have shown that keratocytes show their "archetypical" persistent movement with constant crescent-like shape only for an intermediate adhesiveness of the substrate.

Effects of alloy disorder and confinement on phonon modes and Raman scattering in SixGe1−x nanocrystals: A microscopic modeling

Confinement and alloy disorder effects on the lattice dynamics and Raman scattering in Si1−xGex nanocrystals (NCs) are investigated numerically employing two different empirical inter-atomic potentials. Relaxed NCs of different compositions (x) were built using the Molecular Dynamics method and applying rigid boundary conditions mimicking the effect of surrounding matrix. The resulting variation of bond lengths with x was checked against Vegard's law and the NC phonon modes were calculated using the same inter-atomic potential. The localization of the principal Raman-active (Si-Si, Si-Ge, and Ge-Ge) modes is investigated by analysing representative eigenvectors and their inverse participation ratio. The dependence of the position and intensity of these modes upon x and NC size is presented and compared to previous calculated results and available experimental data. In particular, it is argued that the composition dependence of the intensity of the Si-Ge and Ge-Ge modes does not follow the fraction of the corresponding nearest-neighbour bonds as it was suggested by some authors. Possible effects of alloy segregation are considered by comparing the calculated properties of random and clustered SixGe1−x NCs. It is found that the Si-Si mode and Ge-Ge mode are enhanced and blue-shifted (by several cm−1for the Si-Si mode), while the intensity of the Si-Ge Raman mode is strongly suppressed by clustering.

Seismic Response Analysis of Twin-Tower Building with Enlarged Base

This paper adopts large finite element software ANSYS to establish finite element model of twin-tower building with enlarged base, uses dynamic time history analysis method for seismic response calculation, compare and analyze the calculation results of twin-tower building with enlarged base under elastic boundary conditions and rigid boundary conditions. The results showe that dynamic response for model under elastic boundary conditions is larger than dynamic response for model under rigid boundary conditions, and elastic boundary conditions is more close to the actual situation.