By radiating bifrequency primary waves from two ultrasonic emitters with changing the phases of the primary waves, we can obtain the sound fields that are different from the usual in‐phase excitation. Especially, for the excitation of out‐phase by 180 degrees the difference frequency wave has the directivity of almost uniformity near the acoustic axis. Additionally, the sound pressure levels of the harmonic components of the difference frequency and the primary waves as well are suppressed by 10 dB and more
Fiber Reinforced Plastics (FRP) has been used by Kockums' shipyard in the manufacturing of ships over 35 years, during which time is has been proven to be durable and practical. The light weight makes it a more and more attractive material as energy and material expenditure decreases are required. A special application is the Composite Superstructure Concept [1] where composite materials are added on top of a steel hull, which decreases the weight and running costs considerably, and makes it possible to even add extra levels while keeping the same center of gravity. If efficient condition monitoring systems can keep track of emerging damages of the structure, the weight may be even more reduced and the interval between maintenance inspections may be prolonged. As important steps in this process, a ship mock-up section was subjected to increased levels of explosive underwater impacts, and the damage progression in the hull was monitored by a nonlinear acoustic technique.
It is well known that transversal elastic waves in homogeneous solids satisfy a wave equation with a cubic nonlinearity. This equation with resonator boundary conditions can be transformed into a functional equation, which can be reduced to a second order partial differential equation with a cubic nonlinearity. From this equation, by specializing to steady state and integrating one step, we obtain a first order ordinary differential equation with three terms in addition to the derivative: a cubic and a linear term in the dependent variable and a known term (sinus). The coefficient of the derivative is proportional to the dissipation and assumed to be small. Among several cases the most complicated case, the coefficient of the linear term lying between zero and (0.5)(2/3)=0.63, is treated in this paper. In each period the solution has two shocks. At one side of each shock it is necessary to introduce an intermediate boundary layer between the outer region and the inner region next to the shock. The intermediate solution is matched both outwards and inwards. The actual first order ordinary differential equation is also solved numerically both in the outer region and in the neighborhood of the shocks. © 2008 American Institute of Physics.
Burgers' equation describes plane sound wave propagation through a thermoviscous fluid. If the boundary condition at the sound source is given as a pure sine wave, the exact solution is given by the Cole-Hopf transformation as a quotient between two Fourier series. Two approximate Fourier series representations of this solution are known: Fubini's (1935) solution, neglecting dissipation and valid at short distance from the sound source, and Fay's (1931) solution, valid far from the source. In the present investigation a linear system of equations is found, from which the coefficients in a series expansion of each Fourier coefficient can be derived one by one. Curves which join smoothly to Fubini's solution (valid up to slightly before shock formation) and to Fay's solution (valid for approximately three shock formation distances). Maxima for the Fourier coefficients of the higher harmonics are given. These maxima are situated in a region where neither Fubini's nor Fay's solution is valid.
Burgers' equation describes plane sound wave propagation through a thermoviscous fluid. If the boundary condition at the sound source is given as a pure sine wave, the exact solution given by the Cole-Hopftransformation is a quotient between two Fourier series. Two approximate Fourier series representations of this solution are known: Fubini's (1935) solution, neglecting dissipation and valid at short distance from the sound source, and Fay's solution, valid far from the source. In the present investigation a linear system of equations is found, from which the coefficients in a series expansion of each Fourier coefficient can be derived one by one. Curves which join smoothly to Fubini's solution (valid up to slightly before shock formation) and to Fay's solution (valid for approximately three shock formation distances). Maxima for the Fourier coefficients of the higher harmonics are given. These maxima are situated in a region where neither Fubini's nor Fay's solution is valid.
Planar wave propagation in nonlinear acoustics is modeled by the Burgers equation, which is exactly soluble. Spherical wave propagation is modeled by a generalized Burgers equation, in which the dissipative parameter of the plane wave Burgers equation is replaced by an exponentially growing function of the variable symbolizing the travelled length of the wave. A procedure previously used in 1998 by B.O. Enflo [1] on cylindrical short pulses is now used on spherical short pulses, which are originally N-waves. The procedure consists of the four steps: 1) A shock solution of the generalized Burgers equation is found by asymptotic matching. The shock fades in the region where the nonlinear term in the equation can be neglected. 2) The linear equation in step 1) is rescaled. It is identically solved by an integral representation containing an unknown function. 3) The integral representation found in step 2) is evaluated by the steepest descent method in the fading shock region introduced in step 1). The unknown function introduced in step 2) is determined by comparing the result of this evaluation with the fading shock solution found in step 1). 4) The integral representation with the unknown function determined is evaluated approximately asymptotically for large values of the original length (or time) variables in the original generalized Burgers equation (old-age regime). The result of this procedure is an old-age solution, controlled by numerical calculations. Curves of analytical and numerical solutions are given
This book presents theoretical nonlinear acoustics in fluids with equal stress on physical foundations and mathematical methods. From first principles in fluid mechanics and thermodynamics a universal mathematical model (Kuznetsov's equation) of nonlinear acoustics is developed. This model is applied to problems such as nonlinear generation of higher harmonics and combination frequencies, the shockwave from a supersonic projectile, propagation of shocks in acoustic beams and nonlinear standing waves in resonators. Special for the book is the coherent account of nonlinear acoustic theory from a unified point of view and the detailed presentations of the mathematical techniques for solving the nonlinear acoustic model equations. The book differs from mathematical books on nonlinear wave equations by its stress on their origin in physical principles and their use for physical applications. It differs from books on applications of nonlinear acoustics by its ambition to explain all steps in mathematical derivations of physical results. It is useful for practicians and researchers in acoustics feeling the need for more theoretical understanding. It can be used as a textbook for graduate or advanced undergraduate students with an adequate background in physics and mathematical analysis, specializing in acoustics, mechanics or applied mathematics. See also http://www.wkap.nl/prod/b/1-4020-0572-5.
Simplified nonlinear evolution equations describing nonsteady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach is used based on a nonlinear functional equation. This approach is shown to be equivalent to the version of the successive approximation method developed in 1964 by Chester. It is explained how the acoustic field in the cavity is described as a sum of counterpropagating waves with no cross-interaction. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. Some results from a perturbation calculation of the wave profile are given.
Simplified nonlinear evolution equations describing non-steady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach based on a nonlinear functional equation is used. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically on the base of exact nonlinear solutions for different laws of periodic motion of the wall. For harmonic excitation the wave profiles are described by Mathieu functions, and their mean energy characteristics by the corresponding eigenvalues. The sawtooth-shaped motion of the boundary leads to a similar process of evolution of the profile, but the solution has a very simple form. Some possibilities to enhance the Q-factor of a nonlinear system by suppression of nonlinear energy losses are discussed. (C) 2005 Acoustical Society of America.
The paper has three parts. In the first part a cubically nonlinear equation is derived for a transverse finite-amplitude wave in an isotropic solid. The cubic nonlinearity is expressed in terms of elastic constants. In the second part a simplified approach for a resonator filled by a cubically nonlinear medium results in functional equations. The frequency response shows the dependence of the amplitude of the resonance on the difference between one of the resonator's eigenfrequencies and the driving frequency. The frequency response curves are plotted for different values of the dissipation and differ very much for quadratic and cubic nonlinearities. In the third part a propagating N-wave is studied, which fulfils a modified Burgers' equation with a cubic nonlinearity. Approximate solutions to this equation are found for new parts of the wave profile.
Simplified nonlinear evolution equations describing nonsteady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach is used based on a nonlinear functional equation. This approach is shown to be equivalent to the version of the successive approximation method developed in 1964 by Chester. It is explained how the acoustic field in the cavity is described as a sum of counterpropagating waves with no cross-interaction. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically for three different types of periodic motion of the wall: harmonic excitation, sawtooth-shaped motion and "inverse saw motion".
This paper describes test results of a nonlinear wave modulation spectroscopy (NWMS) technique. These techniques are based on the strong connection between nonlinearity and presence of micro or macro-cracks. NWMS includes existence of higher harmonics, the presence of sidebands and ratio between excitation frequency and sideband components. Tests of different nonlinear techniques have been performed and compared to linear techniques. Tensile beams have been used as specimens. Piezoceramics were applied and a high frequency signal transmitted into the specimen simultaneously as a low frequency. Sideband around the high frequency and higher harmonics were then studied to detect nonlinearities. If there are cracks present, the NWMS show nonlinearities by combining the different frequencies shown as sidebands from the high frequency component, and the higher harmonics' amplitudes will increase. NWMS techniques are more sensitive than linear as they can detect smaller cracks. NWMS is faster to evaluate than a linear scanning method, as it can interrogate a complete object in just one measurement, and NWMS is not as limited by complicated geometry of the specimen.
Like many materials, granite exhibits both nonlinear acoustic distortion and slow nonequilibrium dynamics. Measurements to date have shown a response from both phenomena simultaneously, thus crosscontaminating the results. In this Letter, constant strain frequency sweep measurements eliminate the slow dynamics and, for the first time, permit evaluation of nonlinearity by itself characterized by lower resonance frequencies and a steeper slope. Measurements such as these are necessary for the fundamental understanding of material dynamics, and for the creation and validation of descriptive models.
When making acoustic measurements on materials with cracks, there exist two types of behavior that influence the sound velocity monitored through the resonance frequency of the object. One is the material's nonlinearity, and the other is a slow recovery process of the material parameters towards equilibrium called Slow Dynamics. The former is a wave distortion taking place in the presence of the wave while the latter is a slow recovery process that makes the time history of the material state count. For the understanding of the dynamics of these solids it is necessary to be able to separate the effects of nonlinearity and slow dynamics. In this work, this has been accomplished by making measurements on steel at steady-state through keeping the strain constant. Normal frequency sweeps at different strains are compared to constant strain sweeps. As every material state parameter can induce a slow dynamic response it is important to keep control of humidity and temperature. Measurements performed at different temperatures give different results. An example of this is the presented resonance frequency plots for the temperatures 20, 25 and 30 degrees Celcius. © 2008 American Institute of Physics.
Non-Destructive Evaluation has been carried out on three different test objects, with three different methods based on exhibits of slow dynamics and nonlinear effects. The three diverse objects were cast iron, ceramic semi-conductors on circuit boards, and rubber. The three approaches were Higher Harmonics detection (HH), Nonlinear Wave Modulation Spectroscopy (NWMS), and Slow Dynamics (SD). For all of the objects the three approaches were tried. The results showed that for each of the objects, a different method worked the best. The cast iron worked best with nonlinear wave modulation, the ceramic semi-conductors worked well with the higher harmonics detection, while the rubber showed best results with slow dynamics.
The interconnection between variations of elasticity and dielectric permittivity of mesoscopic solid systems under exposure to ultrasound is experimentally observed. A phenomenological theory generalizing Debye’s approach for polar fluids is developed to explain the measured data. The substitution of acoustic measurements by dielectric ones not only simplifies the procedure, but offers new possibilities to remotely evaluate the mechanical properties of materials and natural media.
A generalization of the single frequency Cole-Mendousse solution for the Burgers equation is shown. The solution is in the same form - a ratio between two Fourier series containing Bessel functions. The input is given as an arbitrary number of frequency components which can have any amplitude, frequency and phase. The solution is valid for any distance.
An investigation of time reversal of nonlinear and non-dissipative or dissipative plane sound waves was made. Propagation and back propagation is time reversal invariant only when the dissipation is zero and the wave has not been shocked. A wave that has shocked has irretrievably lost part of its content. Still, if the wave and its derivatives are considered continuous there remain in theory forever information about the original signal. Factors like numerical accuracy and noise naturally set limits in practical situations.
A nonlinear layer can be a model for a cloud of gas bubbles in a liquid, a crack or split plane in a solid, or contact between two tighted surfaces. Solutions were derived for media under strong load. Numerical calculations based on Preisach-Mayergoyz space description (nonlinear stress-strain relationships typical for solids containing mesoscopic inhomogeneities or defects) have given results like those obtained from LISA (see results by Delsanto). IMVITED. Non-uniform massdistribution of grains immersed into a vibrating fluid create internal forces which is responsible for generation of higher harmonics. Tests on slow dynamics were performed.
A solution for multifrequency plane waves propagating through a dissipative and nonlinear medium is presented. It originates from the well-known Bessel function series ratio for a pure sinusiodal wave, introduced by Cole and Mendousse. The solution is exact. The only limitation, inherited from the single-frequency solution, is the slow convergence of the series when the nonlinearity is very large compared to the dissipation. Otherwise any frequencies, amplitudes and phases can be introduced in the original wave and the solution is valid for any propagated distance.
Known models with huge nonlinear response of grainy media are based on nonlinear stress-strain relationships typical for solids containing mesoscopic inhomogeneities or defects in their structure. Such nonlinear behaviour is well-defined at significant local deformation caused by high applied load, even if the load does not vary in time. However, there exist a different type of nonlinearity which manifests itself due to inertial forces between grains. Such forces appear in a moving noninertial frame of reference, in particular, if a small system of interacting particles is placed into a vibrating fluid. High spatial gradients of internal forces are determined by the non-uniform distribution of mass. These gradients varying in time can excite the internal degrees of freedom. So, even at harmonic vibration of the fluid caused by sound, the nonlinear internal dynamics can be responsible for generation of higher harmonics. As an example of nonlinearity of inertial type, a model of grainy medium is developed where the grains are immersed into a vibrating fluid. The inertial attractive forces have hydrodynamic origin, and the repulsive forces are caused by deformation of colliding grains. Dynamic and stochastic motion is studied, resonances are discovered and nonlinear response is evaluated.
Time-reversed nonlinear acoustics has shown that a propagated signal retains information about the original signal after shock formation and even after merging of shocks. This information is dependent on the accuracy of the measured or calculated data . Presented here is a more detailed analysis of the estimation of dimensionless and real parameters in the problem - the original amplitude, the distance traveled and the dissipation over non- linearity ratio. It is shown that the methods are fairly insensitive to noise.
The solution for multi-frequency plane waves propagating through a dissipative and nonlinear medium is shown for some examples of periodic conditions. The expression may for any given condition be expressed analytically as a ratio of Fourier series with Bessel function coefficients. In the examples are shown how the final appearance of any initial wave always is a pure periodic wave in the lowest frequency existing in the problem - the period of the condition.
A new method for monitoring the time dependent dynamics of materials is proposed and implemented. By completely separating the conditioning (here ultrasound), the probing (here gravity), and the material state indicator (here deflection), the details of this dynamic process becomes apparent. The method allows both continuous monitoring of the material state without cross-interaction by the measuring process on the results, as well as complete freedom of conditioning and probing. It was successfully tested for sensitivity and repeatability when applied on a horizontally suspended beam of gabbro rock, which was observed to sag when subjected to ultrasound. These introductory tests have given new insights. The beam rises back, against the force of gravity, after the ultrasound is turned off. The deflection motions are fast both at the onset and at the termination of ultrasound, with the subsequent continuations being much slower. This new method is able to provide the higher accuracy needed for the advancement of the theoretical framework for material property time dependent dynamics.
A Carbon Fibre Reinforced Plastic plate was manufactured to have internal damages of different types. A nonlinear ultrasound technique was used to scan the plate. Non-contact transmitters of own design were used as transducer, and a contact sensor was used to measure the wave in the composite. Scanning was made perpendicularly with the sensor being on the same side as the transducer. The technique can be adapted in accuracy and speed.
Two ways in which the concept of open resonator can be used in connection with nondestructive evaluation will be described. In the first, a non-contact transducer's efficiency of transferring energy into an object can be increased by utilizing the resonance of the gap between the transducer and object. By choosing the gap distance, frequency and transducer width a resonant wave appears which will have a considerably larger amplitude at the object surface. Secondly, by fitting the critical parameters of open resonators for thin objects, like extended plates, the wave field inside the object can be localized. Thus some Nonlinear Elastic Wave Spectroscopy methods may be used to determine the location of damage.
High power ultrasonic transducers are widely used for detecting objects and measuring distances, etc. In actual applications, for the air coupling ones the impedance mismatch between the transducer surface and air is large, so there is only a small amount of energy transmitted. Most of the energy loss occurs during the transformation of energy from the transducer radial cone to the air. The purpose of this study is to investigate different designs of the radial cone of the transducer and analyze their vibration in order to suggest design variations for improved vibration amplitudes. Concerning a piezoelectric ceramic transducer, we focus on two parameters of the transducer radial cone. This report presents schemes for increasing pressure levels.
For certain resonant high-power air transducers exists a radiating cone which is in some way connected to a forcing piezoelectric ceramic. This study focuses purely on the cone structure itself, and the form of excitation that gives desired results on the vibration modes of the cone. In the new design three different parameters are changed from existing devices. They are: 1) the change of the round piezoelectric disc center position to a piezoelectric ring shape, while making the cone center fixed, 2) the cutting of the cone into leaves, and 3) the radial thickness decreasing with radius. In simulations, the new design yields considerably higher vibration amplitudes.
This work is devoted to the investigation of evolution of intense quasi-harmonic signals in the case of infinite acoustic Reynolds numbers. The consideration is based on the zero viscocity limit solution of the Burgers equation, which reduces the Cole-Hopf solution to a "maximum" principle. This limit solution permits an easy way to get the profile of the waves, postition of shocks and their velocities at arbitrary times. The process of transformation of an initial quasi-monochromatic wave into s sawtooth wave is considered. It is shown that the nonlinearity leads to suppression of the initial amplitude modulation and to the transformation of the initial frequency modulation inot a shock amplitude modulation. The amplitude of the low frequency component generated by a quasi-mono-chromatic wave is found. It is shown that the interaction of this component with high frequency waves leads to phase modulation, which increases with distance. The amplitudes of the new components of the spectrum are found. Is is show n that when the value of phase modulation is small, the amplitudes of the satellites do not depend on the distance or the number of harmonics of the primary wave.
There exist two concepts of open resonators that can be combined into a device for acoustic non-destructive testing. The device is non-contact and based on the fact that an airgap between a transducer surface and the test object can be set into resonance. This occurs for the conditions that makes this an open resonator (open because the air volume is not closed), and the wave field is amplified so that a larger amplitude is reached at the object surface. Let us assume that the object is a plate. When the acoustic wave enters the plate, one can let the frequency be the resonant frequency for one of the modes for the plate thickness. A limited part of the plate may be seen as a resonator, open to the sides. When the conditions for an open resonator is fulfilled for the plate thickness, the wave field will be greatly amplified within a region close to the insonified plate surface, and thus the linear or nonlinear response from this excitation is local. The device may then be used to scan the plate for material parameter changes.
Slow Dynamics is a specific material property, which for example is connected to the degree of damage. It is therefore of importance to be able to attain proper measurements of it. Usually it has been monitored by acoustic resonance methods which have very high sensitivity as such. However, because the acoustic wave is acting both as conditioner and as probe, the measurement is affecting the result which leads to a mixing of the fast nonlinear response to the excitation and the slow dynamics material recovery. In this article a method is introduced which, for the first time, removes the fast dynamics from the process and allows the behavior of the slow dynamics to be monitored by itself. The new method has the ability to measure at the shortest possible recovery times, and at very small conditioning strains. For the lowest strains the sound speed increases with strain, while at higher strains a linear decreasing dependence is observed. This is the first method and test that has been able to monitor the true material state recovery process.
This work is the second part of three that presents new tools to be used for damage localization in plates by nonlinear acoustical methods. It introduces an important in-plane localization technique, which is based on the existence of resonant spatially localized wave fields. The wave from the transducer is acting as a dynamic influence on the plate surface, making the waves reflect in a non-ideal way. The non-ideal reflections make the modes underneath the transducer have different resonant frequencies than the modes beside the insonified area. They appear both for contact and non-contact sources. In the nonlinear damage localization application, the trapped mode wave field interacts with another signal at lower frequency. This results in sidebands around the high frequency whose amplitudes are related to the amount of damage underneath the transducer.
This is the first of three articles that deals with the goal of using non-contact nonlinear acoustic methods for the damage localization in plates. In this first part, the resonant air gap wave field between one active and one passive reflecting plate was investigated experimentally. Of particular interest is the wave field amplitude strength, and its distribution on the passive surface. The wave amplitude may be increased by choosing one plate to be concave, overcoming the nonlinear damping taking place for two flat plates. Applications are connected with an increased wave field having advantages for processes taking place under open conditions and for a non-contact transducers' efficiency of transferring energy into an object. A double resonance gives highest air pressure, while a triple provides most energy in the passive object. By choosing the gap distance, frequency and transducer width, the appearing resonant wave will have a considerably larger amplitude at the object surface. The work through the interface from air to object is investigated and the wave field in a resonant air-plate system is shown.
Parametric loudspeakers are transmitting two high power ultrasound frequencies. During propagation through the air, nonlinear interaction creates a narrow sound beam at the difference frequency, similar to a light beam from a torch. In this work is added the physical phenomenon of propagation cancellation, leaving a limited region within which the sound can be heard—a 1 meter long cylinder with diameter 8 cm. It is equivalent to a torch which would only illuminate objects within 1 meter. The concept is demonstrated both in simulation and in experiment.
The use of composite superstructures on current or newly built steel hulls is a recently emerged technology. The economic estimations predict that the extra costs for putting composite superstructures, with the present safety margins, on steel ships will be paid back in only 2-3 years. This also makes the ships having smaller ecological footprints with less fuel consumption and CO2 emissions. In this stage of development it is needed to ensure the durability of the joints between the steel and glass fiber reinforced plastic. The first step is that the joints must first be proven to withstand fatigue. In this test a 4-meter beam, which represents the joint, were investigated for fatigue progression by a four-point-bending fatigue test. In order to show that ultrasonic material monitoring techniques can be used to monitor the damage progression, the beam was measured during the tests until failure. The test was successful both in showing that the joint could withstand high levels of mechanical exposure, and in that the ultrasonic techniques accompanied the damage progression which means that they may be used on vessels during operation
This book comes as a result of the research work developed in the framework of two large international projects: the European Science Foundation (ESF) supported program NATEMIS (Nonlinear Acoustic Techniques for Micro-Scale Damage Diagnostics) (of which Professor Delsanto was the European coordinator, 2000-2004) and a Los Alamos-based network headed by Dr. P.A. Johnson. The main topic of both programs and of this book is the description of the phenomenology, theory and applications of nonclassical Nonlinearity (NCNL). In fact NCNL techniques have been found in recent years to be extremely powerful (up to more than 1000 times with respect to the corresponding linear techniques) in a wide range of applications, including Elasticity, Material Characterization, Ultrasonics, Geophysics to Maintenance and Restoration of artifacts (paintings, stone buildings, etc.). The book is divided into three parts: Part I - defines and describes the concept of NCNL and its universality and reviews several fields to which it may apply; Part II - describes the phenomenology, theory, modelling and virtual experiments (simulations); Part III -discusses some of the most relevant experimental techniques and applications.
One of the most important sections of nonlinear wave theory is related to the collisions of single pulses. These often exhibit corpuscular properties. For example, it is well known that solitons described by the Korteweg–de Vries equation and a few other conservative model equations exhibit properties of elastic particles, while shock waves described by dissipative models like Burgers’ equation stick together as absolutely inelastic particles when colliding. The interactions of single pulses in media with modular nonlinearity considered here reveal new physical features that are still poorly understood. There is an analogy between the single pulses collision and the interaction of clots of chemical reactants, such as fuel and oxidant, where the smaller component disappears and the larger one decreases after a reaction. At equal “masses” both clots can be annihilated. In this work various interactions of two and three pulses are considered. The conditions for which a complete annihilation of the pulses occurs are indicated. © 2017 The Author(s)
Irreversible processes taking place during nonlinear acoustic wave propagation are considered using a representation by loops in a thermodynamic parameter space. For viscous and heat conducting media, the loops are constructed for quasi-harmonic and sawtooth waves and the descriptive equations are formulated. The linear and nonlinear absorptions are compared. For relaxing media, the processes are frequency-dependent. The loops broadens, narrows, and bends. The linear and nonlinear relaxation losses of wave energy are shown. Residual stresses and irreversible strains appear for hysteretic media, and here, a generalization of Rayleigh loops is pictured which takes into account the nonlinearly frequency-dependent hereditary properties. These describe the dynamic behavior, for which new equations are derived.
A simple mechanical system containing a low-frequency vibration mode and set of high-frequency acoustic modes is considered. The frequency response is calculated. Nonlinear behaviour and interaction between modes is described by system of functional equations. Two types of nonlinearities are taken into account. The first one is caused by the finite displacement of a movable boundary, and the second one is the volume nonlinearity of gas. New mathematical models based on nonlinear equations are suggested. Some examples of nonlinear phenomena are discussed on the base of derived solutions.
A simple analytical theory is developed for the description of non-steady state response of a thin nonlinear layer, which differs markedly in its linear properties from the surrounding medium. Such a layer can model the behavior of real inhomogeneities like a cloud of gas bubbles in a liquid, a crack or split plane in a solid, or the contact between two slightly tighted rough surfaces. Both weakly nonlinear pulse and harmonic responses are calculated and the general properties of the spectral and temporal structure of the scattered field are discussed. The exact strongly nonlinear solutions are derived for a special type of stress-strain relationship corresponding to the behavior of real condensed media under strong load. Profiles and spectra shown are in conformity with experimental results. The pulse response on the short delta-pulse shaped incident wave is calculated for arbitrary nonlinear properties of the layer. The possibilities to apply the sets of data on measured characteristics of pulse response in the solution of inverse problems are briefly discussed.
Two planar ultrasound pro jectors having identical rectangular apertures were placed side by side. Both pro jectors radiated bifrequency primary waves in air. The frequencies were 26 and 28 kHz, and the initial phases were diﬀerent. Two driving modes were considered, namely, conventional in-phase driving and phase-inversion driving. The spatial proﬁles of sound pressure ﬁelds were measured along and across the sound beam axis for the primary waves and a diﬀerence in frequency waves of 2 kHz. The second and third harmonic components of the diﬀerence frequency waves were also measured. The pressure levels of the primary waves were considerably suppressed near the beam axis owing to phase cancellation when the driving signals were phase-inversed, i.e., 180 degrees out of phase. The beam pattern of the diﬀerence frequency was, however, almost the same as that for the case in which the signals were in phase. Interestingly, the harmonic pressure amplitudes of the diﬀerence frequency were reduced by more than 10 dB. The validity of the experimental results were conﬁrmed based on their good agreement with the theoretical predictions based on the Khokhlov-Zabolotskaya-Kuznetsov equation.
The temperature influence on the acoustically amplitude dependent sound speed of a marble rod was investigated. The sound speed was monitored through the resonance frequency by a series of ultrasonic frequency sweeps with successively increasing amplitudes. For the temperatures from 15 C to 60 C, the resonance frequency was measured in 5-degree increments as a function of the resonant acoustic amplitude inside the marble rod. In most of the curves the marble exhibits a softening (i.e. the sound speed decreases) with higher amplitude, but for each test run there exist one notable exception - for one temperature - where the marble gets stiffer (i.e the sound speed increases). The test also shows that the average sound velocity level first, as expected, decreases with temperature, but for the higher temperatures it increases - to well past the starting value.
A new experimental technique for evaluating Young’s (or elastic) modulus of a vibrating thin film from a dynamic measurement is presented. The technique utilizes bending resonance from a remote acoustic excitation to determine Young’s modulus. Equations relating the natural frequencies to the mechanical properties are obtained, and Young’s modulus is subsequently determined. Young’s modulus values from dynamic test are compared with those (static) obtained by a standard tensile test, and consistent results are obtained. The proposed technique is relatively simple and could be used to determine Young’s modulus of a wide variety of sheet materials initially having no bending stiffness. It can also be used for determining other mechanical properties, such as compliance methods in connections with fracture mechanical testing, fatigue and damage measurements. This work emphasizes the feasibility of a damage assessment of components in-service by evaluating changes in the material characteristics.
A simple method of damage severity assessment on sheet materials is suggested and proved by theory and experiment. The investigated defect types are in forms of added mass and crack. The method is based on the frequency shift measurement of a material vibrating as a membrane subjected to static tension and irradiated by an acoustic wave. It is shown both theoretically and experimentally that the natural frequency of the damaged membrane is shifted relative to its position in the ideal material. A local increase in thickness (or addition of mass) shifts the natural frequency down, while a crack shifts the frequency up. The method can be considered as acoustic weighting through the frequency shift. The sensitivity of this method can be high because frequency measurement is one of the most accurate measurements in physics and metrology.
The feasibility of a remote monitoring of structures for a progressive damage assessment as well as material characterization using a simple and inexpensive experimental setup is discussed. The method is based on a remote acoustic excitation of transverse vibrations on a membrane using an ordinary broadband low frequency loudspeaker, and the measurement of the response using a Laser Doppler Vibrometer (LDV). Theoretical modeling is also developed to correlate the experimental results obtained, and this yields a new method for Non Destructive Testing (NDT) of sheet-like materials. The function generator provides an input voltage of a sine signal to the loudspeaker, and laser detection of the surface vibrational response of the sample is accomplished with the laser vibrometer.
The fracture behavior of paper board (100 μm) used in food packaging material is studied. The plane stress fracture toughness is measured based on a centered crack panel. Different crack sizes have been tested. A compromise (crack length) was found, at which Strip Yield Model as well as Linear Elastic Fracture Mechanics allow the validation of experimental results. Meanwhile, accurate results are obtained using the Strip Yield Model with a geometric correction. Besides, detection of damage in food packaging material is an interesting feature in quality control of the product. Therefore the material is investigated using an acoustic method. The method consists of a vibration-based damage assessment and leads to a first level differentiation between damaged and non-damaged specimens. The fracture behavior of paper board (100 μm) used in food packaging material is studied. The plane stress fracture toughness is measured based on a centered crack panel. Different crack sizes have been tested. A compromise (crack length) was found, at which Strip Yield Model as well as Linear Elastic Fracture Mechanics allow the validation of experimental results. Meanwhile, accurate results are obtained using the Strip Yield Model with a geometric correction. Besides, detection of damage in food packaging material is an interesting feature in quality control of the product. Therefore the material is investigated using an acoustic method. The method consists of a vibration-based damage assessment and leads to a first level differentiation between damaged and non-damaged specimens.
Understanding the change in material properties in time is necessary for in-service diagnosis of structures and prevention of accidents. Therefore, a new experimental technique for evaluating Young’s (or elastic) modulus of a vibrating thin sheet from a dynamic measurement is presented. The technique utilizes bending resonance from a remote acoustic excitation. Equations relating the natural frequencies to the mechanical properties are obtained, and Young’s modulus is subsequently determined experimentally using the implemented dynamic measurement method. Young’s modulus values from the dynamic test are then compared with those (static) obtained by a standard tensile test and those obtained by the theory of laminated materials. The proposed technique appears relatively simple and is applied in this paper to laminates initially having no (or negligible) bending stiffness, and used in packaging industries. This work emphasizes the feasibility of a remote condition monitoring of components in-service by evaluating changes in the material properties.
A series of uniaxial tensile tests was performed for sheet materials like paperboard, polyethylene and packing layered composites. These sheets can be considered as membranes. In parallel with a tensile test, the natural frequency was measured through an acoustical excitation. Firstly, it was shown both theoretically and experimentally that, at a given load, the frequency is sensitive to the local deviation in the standard thickness or to the presence of cracks inside the material. It means that this acoustic measurement can be used as one of the methods of damage assessment, or nondestructive testing in general. Secondly, the resonance frequency shift was continuously monitored for increasing strain on polyethylene and paperboard, and the curves obtained were compared to the stress-strain curves for material characterization. They were not the same and showed a non-monotonic stiffness variation for the polyethylene. It was shown that the resonance frequency shift measurement can successfully replace the stress-strain curve for material characterization under tensile test. During a long time under load an irreversible plastic deformation of the sample takes place, and the frequency shift can also serve as a new method for evaluating the residual strain of the material.
The parametric array is formed by two finite-amplitude ultrasound beams with neighboring frequencies. The nonlinear interaction of the beams generates a highly directive audible sound at the difference frequency. Up to now, we have analyzed theoretically parametric sound fields in a free space. This study proposes a numerical simulation method of nonlinear ultrasound propagation to estimate the parametric sound field in the time domain. Using the finite-difference time-domain method based on the Yee algorithm with operator splitting, axisymmetric nonlinear propagation was simulated on the basis of equations for a compressible viscous fluid. As model application, a lengthlimited parametric sound beam, which is formed by four finite-amplitude ultrasound beams with different frequencies and controlled by phases and amplitudes of sound sources and has a truncated array length [C.M. Hedberg et al., Acoust. Phys. 56, 637–639 (2010)], was numerically simulated. The simulation showed a spatially restricted and very narrow difference frequency sound beam similar to their experiment. In addition, to investigate the dependence of amplitude and phase on the lengthlimited beam profiles, parametric sound fields were numerically estimated by varying the amplitude and phase slightly from the condition of the length-limited beam. Numerical results indicated that the changes of the amplitude and phase affected the beam length and width, in particular, amplitude changes drastically deformed the beam shape from that of the length-limited beam. This result suggests that precisely controlled sound sources are necessary to obtain the best length-limited sound beam.
This study proposes a numerical simulation method that predicts nonlinear propagation of ultrasound beams in order to estimate the sound field of parametric array in the time domain. Basically, the method resorts to solving numerically and compactly the governing equations in a compressible viscous fluid using the Yee algorithm finite-difference time domain method. The simulation indicates a narrow audible sound beam which is a feature of the parametric array. Additionally, a length-limited parametric sound beam, which is proposed by Hedberg et al. [C. M. Hedberg et al., Acoust. Phys. 56, 637-639 (2010)], is numerically simulated as a model application. A pair of parametric sound sources generates the length-limited sound beam by control of the amplitudes and initial phases of the sources. The simulation shows a narrow truncated array length-limited sound beam.