Heavy Fluid Loading Conditions Research Articles

A method for atomic force microscopy cantilever stiffness calibration under heavy fluid loading

This work presents a method for force calibration of rectangular atomic force microscopy (AFM) microcantilevers under heavy fluid loading. Theoretical modeling of the thermal response of microcantilevers is discussed including a fluid-structure interaction model of the cantilever-fluid system that incorporates the results of the fluctuation-dissipation theorem. This model is curve fit to the measured thermal response of a cantilever in de-ionized water and a cost function is used to quantify the difference between the theoretical model and measured data. The curve fit is performed in a way that restricts the search space to parameters that reflect heavy fluid loading conditions. The resulting fitting parameters are used to calibrate the cantilever. For comparison, cantilevers are calibrated using Sader's method in air and the thermal noise method in both air and water. For a set of eight cantilevers ranging in stiffness from 0.050 to 5.8 N/m, the maximum difference between Sader's calibration performed in air and the new method performed in water was 9.4%. A set of three cantilevers that violate the aspect ratio assumption associated with the fluid loading model (length-to-width ratios less than 3.5) ranged in stiffness from 0.85 to 4.7 N/m and yielded differences as high as 17.8%.

Flow-induced vibrations of an elastic cylindrical shell conveying a compressible fluid

The paper addresses the effects of generation and transmission of the vibro-acoustic energy in an elastic cylindrical shell filled with water. The energy input is produced by velocity and pressure pulsations due to the presence of an obstacle in a flowing fluid. It is assumed that there is no back-reaction of the vibro-acoustic response on the driving turbulent field. The concept of modal Green's functions and modal boundary integral equations is introduced and the role of elastic deformations of a tube is highlighted in view of the energy re-distribution between the structural and the acoustical transmission paths. Another important aspect of the problem of ‘flow-induced sound generation’, which is addressed in this paper, is the treatment of a CFD pressure output for the modelling of vibro-acoustic field generation. For an elastic shell in heavy fluid loading conditions, it is indicated that only the pressure distribution at the surface of a shell provided by a CFD solver is requested in order to adequately predict its coupled vibro-acoustic response. The ‘inner’ and ‘outer’ acousto-elastic domains are introduced and the low-order system of linear algebraic equations is set up, which is constituted by modal boundary equations and continuity conditions. Several ‘generic’ excitation cases are studied and the results of analysis are explained.

A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading

In this paper, we derive weakly nonlinear equations for the dynamics of a thin elastic plate of large extent under conditions of heavy fluid loading. Two situations are then considered. First, we consider the case in which transverse motion of the plate generates a weaker in-plane motion, which is in turn coupled back to the evolution of the transverse motion. This results in the familiar nonlinear Schrödinger equation for the amplitude of a transverse plane wave, and we show that solitary-wave solutions are possible over the range of (non-dimensional) frequencies ω > ω c , which depends on the material properties. Dimensional values of ω c are physically realizable for a typical composite material underwater. Second, we consider the case in which the amplitudes of the transverse and in-plane motion are of the same order of magnitude, possible at a single resonant frequency, which leads to an evolution equation of rather novel type. We find a range of travelling-wave solutions, including cases in which incident in-plane waves can generate localized regions of transverse displacement.

FORCED AND FREE VIBRATIONS OF RECTANGULAR SANDWICH PLATES WITH PARAMETRIC STIFFNESS MODULATION

An active control of the resonant vibrations of a rectangular sandwich plate performed by the parametric stiffness modulation is analyzed. The controlled vibrations are those of the dominantly flexural type excited by the transverse force acting at the first resonant frequency of dominantly flexural vibrations. The stiffness modulation is performed at a comparatively high frequency identified by the resonance of a mode of the dominantly shear type. The method of direct partition of motions is used that predicts an existence of the modal interaction between these two modes of vibrations due to the parametric stiffness modulation. It is shown that such a parametric control can provide a significant shift of the first eigenfrequency of a controlled plate (the one subjected to the stiffness modulation) from its nominal value for an uncontrolled plate. Heavy fluid loading conditions are accounted for as well as the energy dissipation in the material of a plate. It is demonstrated that although heavy fluid loading reduces resonant frequencies of forced vibrations, the suggested mechanism of control remains valid in these cases. Dynamics of an elementary two-degree-of-freedom model mechanical system is considered to illustrate the mechanism of modal interaction, which is involved in the suggested way of an active control of vibrations of sandwich plates.

Analysis and control of vibrations of honeycomb plates by parametricstiffness modulations

This paper addresses an active control of the resonant vibrations ofa sandwich plate with the honeycomb composition of a core ply performed by aparametric stiffness modulation. The controlled vibrations are those of thedominantly flexural type excited by a transverse force acting at a lowresonant frequency. The stiffness modulation is introduced by some fairlysmall changes in an orientation of plates composing cell elements of a coreply. It is performed at a comparatively high frequency identified by theresonance of a mode of the dominantly shear type. The method of directpartition of motions is used that predicts an existence of the modalinteraction between the low-frequency and the high-frequency motions. It isshown that such a parametric control can provide a significant favourableshift of the first eigenfrequency of a controlled beam (the one subjected tothe stiffness modulation) from its nominal value for an uncontrolled beam.Asymptotic results are checked by direct time-marching integration. The energyof the `micro-motions' is calculated and it is compared with the energies offlexural vibrations of a plate with and without control. Heavy fluid loadingconditions are accounted for as well as material losses in a structure. It isdemonstrated that although heavy fluid loading reduces the resonantfrequencies of forced vibrations, the suggested mechanism of controlremains valid in these cases.

The active control of vibrations of composite beams by parametric stiffness modulation

The paper addresses an active control of the resonant vibrations of composite beams performed by a parametric stiffness modulation. A sandwich beam composition with the continuous core is considered. The stiffness modulation is introduced by some fairly small changes in an orientation of elements of the microstructure of a core ply. The controlled vibrations are those of the dominantly flexural type excited by a transverse force acting at a low resonant frequency, whereas the stiffness modulation is performed at a comparatively high frequency identified by the resonance of a mode of the dominantly shear type. This difference in time scales of the controlled vibrations and the input signal facilitates a use of the method of direct partition of motion that predicts an existence of the modal interaction between the low-frequency and the high-frequency motions due to so-called vibrational forces. It is shown that such a parametric control can provide a significant favourable shift of the first eigenfrequency of a controlled beam (the one subjected to the stiffness modulation) from its nominal value for an uncontrolled beam. Heavy fluid loading conditions are accounted for as well as material losses in a structure. Then instead of analysis of eigenfrequencies, a problem of forced vibrations is posed and the forced frequency–amplitude response is analysed. It is demonstrated that although heavy fluid loading reduces resonant frequencies of forced vibrations, the suggested mechanism of control remains valid in these cases.

NON-LINEAR OSCILLATIONS OF A BAFFLED ELASTIC PLATE IN HEAVY FLUID LOADING CONDITIONS

Non-linear vibrations of an elastic plate in heavy fluid loading conditions are considered. The structural non-linearity is taken into account along with the fluid non-linearity in a full Bernoulli integral formulation for the contact acoustic pressure and with the non-linearity in the formulation of the continuity condition. The modal analysis in spatial co-ordinates is used along with the method of multiple scales to search for a stationary response in the time domain. Resonant frequencies of a fluid-loaded plate are detected in a coupled formulation of structural acoustics and typical excitation conditions (a weak resonant excitation, hard monochromatic sub- and super-harmonic excitations, an excitation by two driving forces) are explored. The roles of the structural non-linearity and the non-linearity in the formulation of the fluid response are compared.

Vibrations of and sound radiation from sandwich plates in heavy fluid loading conditions

A problem of forced vibrations of a sandwich plate submerged in a fluid (an acoustic medium) is formulated as a coupled problem of structural acoustics. Two-dimensional (plane) formulation relevant to cylindrical bending of a plate is explored. Dynamics of a sandwich beam (a sandwich plate in cylindrical bending) is described in the framework of the sixth-order theory of multi-layered plates. Asymptotic analysis of the dispersion polynomial is performed in the low frequency and the low module limits for an isolated infinitely long beam. Green's functions for flexural vibrations are obtained analytically and thus explicitly contain the principal parameters of beam's composition. Forced vibrations of a fluid-loaded beam in a rigid baffle are considered. Both the interaction between an acoustic medium and a plate and the interaction between a plate and its boundaries are described by boundary integral equations assembled in a two-level system. Eigenfrequencies of a fluid-loaded beam are detected by maximum of the radiated acoustic power. A semi-analytical sensitivity analysis of the objective function selected as a radiated acoustic power to the parameters of beam's composition is performed.

MODELLING OF NON-LINEAR OSCILLATIONS OF ELASTIC STRUCTURES IN HEAVY FLUID LOADING CONDITIONS

In a problem of structural acoustics with non-liner formulation of structural dynamics, a linearized compatibility condition at the fluid–structure interface is used along with the linear wave equation for the acoustic medium [1–7]. This approach is referred to as a light acoustic loading limit. Another approach is to formulate a compatibility condition at the moving boundary and solve a non-linear wave equation in a volume. As it is shown in references [3–5], a solution for this problem predicts shock wave formation at a certain distance from a vibrating surface. In the present paper, one more model of interaction between an acoustic medium and a non-linear structure is suggested for heavy fluid loading conditions. In this model, propagation of acoustic waves is described by a linear wave equation, but the continuity condition is formulated at the moving boundary and the contact acoustic pressure acting at the vibrating non-linear structure is calculated by the Bernoulli integral with a quadratic velocity term retained. Two model problems of coupled structural acoustics are considered—oscillations of a fluid-loaded piston and oscillations of an infinitely long periodically supported elastic plate. A method of multiple scales is used for analysis of the local non-linear dynamics of the model systems, whilst matched asymptotic expansions are used to model the fluid's motion. Several specific effects of structural vibrations generated by the non-linearity of fluid-structure interaction, rather than by structural non-linearity are demonstrated.

The generation of sound by turbulent and separated flows

An overview is given of the mechanisms of sound generation by free field turbulence and by vorticity interacting with a body, with particular reference to flows at low Mach number. The theory is illustrated by discussions of the periodic sound produced by a cylinder in a uniform stream (Æolian tones) and of the noise generated by turbulence flowing over the trailing edge of an airfoil. In the latter case consideration is also given to the influence of surface compliance: structural modes of vibration are excited by the edge-flow; in conditions of heavy fluid loading, which typically occur in underwater applications, the energy imparted to the structure leads to the generation of substantial levels of additional sound when “surface waves” are scattered at mechanical discontinuities.

Structural and acoustic noise produced by turbulent flow over an elastic trailing edge

An analysis is made of the sound and vibration produced by turbulent flow at low Mach number over the trailing edge of an elastic plate. The trailing edges of airfoils and other flow control surfaces are known to be important sources of high frequency sound. When the surface is compliant the turbulent edge-flow also excites structural modes of vibration. In conditions of heavy fluid loading, which typically occurs in underwater applications, the energy imparted to the structural motions can be large, and the subsequent scattering of ‘surface waves’ at mechanical discontinuities is frequently an important secondary source of sound. In this paper general formulae are developed for the structural and acoustic edge-noise when the control surface is modelled by a semi-infinite, thin elastic plate which can support bending waves. Numerical results are given for steel plates in air and in water. In the latter case it is shown that, when the frequency is smaller than the coincidence frequency the bending wave power exceeds the total sound power generated at the edge by 20–40 dB, independently of the mean flow velocity, so that sound generated by secondary scattering may then be the dominant source of acoustic radiation.

The scattering of sound from fluid-loaded plates

The scattering of sound waves from a finite strip plate under conditions of heavy fluid loading is numerically simulated. Both baffled and unbaffled geometries are considered, and results for the nearfields and farfields are obtained. Under conditions of heavy fluid loading, scattering effects due to the plate elasticity are generally small unless the excitation frequency is close to a resonance frequency of the fluid–plate system. The general behavior of the fluid–plate system and the frequencies, radiation efficiencies, widths, and strengths of the resonances are determined. With these results, one can determine the relevance of such phenomena in a variety of applications. The reliability of predictions for the resonance parameters based on simple modeling techniques is evaluated. The results indicate that the frequencies and effective masses of the resonances can be reasonably approximated. However, the radiation efficiencies are difficult to estimate because of the effects of the strong fluid loading. In the unbaffled case, modeling techniques based on vacuum mode shapes tend to substantially underestimate the radiation efficiency; in the baffled case, such techniques tend to substantially overestimate the radiation efficiency.

The modes, resonances and forced response of elastic structures under heavy fluid loading

This paper reports analytical studies of problems that involve the motion of plane elastic structures under conditions of heavy fluid loading. The main aspect concerns the description of the vibration response of a thin elastic plate (or membrane), of finite extent in at least one dimension, when the structure is excited by concentrated mechanical drive along a line or at a point; and as part of this the possibility of resonant response is discussed, and the resonance conditions and free modes of oscillation are obtained. There is also some discussion of the acoustic fields radiated by the structures under localized mechanical excitation. The analysis makes extensive use of results for the reflection of a structural wave (subject to heavy fluid loading) at an edge, and the paper gives results for that reflection process covering waves incident normally on eight different edge configurations and waves incident obliquely on two edge configurations. These results include the reflection coefficient (whose magnitude is unity in the leading-order approximation of low-frequency heavy fluid loading), and the amplitude and directivity of the edge-scattered sound. By using the argument that edge reflection is a local process, the response is then calculated for a strip plate, under both line and point forcing, and the response is, for the first time, obtained for structures finite in both dimensions and subject to heavy fluid loading. Specifically, solutions are given here for a circular plate with eccentric drive, and for a membrane model of a rectangular panel, with central point drive. For some conditions and geometries expressions in simple form are found for the natural frequencies and mode shapes, and for the off-resonance forced response. Expressions for the drive admittances are found which display a variety of interesting features.

Influence of Fluid Loading on the Radiation from Infinite Plates below the Critical Frequency

Fluid loading influences the total acoustic power and the directivity of the radiation from a large, flat plate driven by localized forces. The frequency range considered here is that below the “critical” frequency. In this range, the in vacuo speed of plate flexural waves is less than the speed of sound in the adjacent fluid medium. For a point force, the directivity of the radiation field is that of a monopole for light fluid loading, but changes to that of a dipole for heavy fluid loading. Analogous behavior occurs in the case of a line force. An equivalent acoustic-dipole source can be specified, and one can apply this equivalent-source formalism to calculate the power radiated by point- and line-moment excitations under conditions of heavy fluid loading. The radiated power for a point-moment excitation computed by this approach agrees with the earlier results of Thompson and Rattayya. [J. Acoust. Soc. Am. 36, 1488–1490 (1964)].

Acoustic Radiation from Locally Driven Infinite Plates with Fluid Loading

Fluid loading influences the total acoustic power and the directivity of the radiation from a large flat plate driven by localized forces. The frequency range considered here is that below the “critical” frequency. In this range, the in vacuo speed of plate flexural waves is less than the speed of sound in the adjacent fluid medium. For a point force, the directivity of the radiation field is that of a monopole for light fluid loading, but changes to that of a dipole for heavy fluid loading. Analogous behavior occcurs in the case of a line force. An equivalent acoustic dipole source can be specified, and one can apply this equivalent source formalism to calculate the power radiated by point and line moment excitations under conditions of heavy fluid loading. The radiated power for a point moment excitation computed by this approach agrees with the earlier results of Thompson and Rattayya [J. Acoust. Soc. Am. 36, 1488–1490 (1964)]. [Work supported by the Ship Silencing Branch of the U.S.N. Bureau of Ships.]