On the contribution of circumferential resonance modes in acoustic radiation force experienced by cylindrical shells

Journal of Sound and Vibration 333 (2014) 5746–5761

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On the contribution of circumferential resonance modes in acoustic radiation force experienced by cylindrical shells Majid Rajabi, Mehdi Behzad n Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran, Iran

a r t i c l e in f o

abstract

Article history: Received 24 November 2013 Received in revised form 30 April 2014 Accepted 9 May 2014 Handling Editor: D. Juve Available online 23 June 2014

A body insonified by a constant (time-varying) intensity sound field is known to experience a steady (oscillatory) force that is called the steady-state (dynamic) acoustic radiation force. Using the classical resonance scattering theorem (RST) which suggests the scattered field as a superposition of a resonance field and a background (non-resonance) component, we show that the radiation force acting on a cylindrical shell may be synthesized as a composition of three components: background part, resonance part and their interaction. The background component reveals the pure geometrical reflection effects and illustrates a regular behavior with respect to frequency, while the others demonstrate a singular behavior near the resonance frequencies. The results illustrate that the resonance effects associated to partial waves can be isolated by the subtraction of the background component from the total (steady-state or dynamic) radiation force function (i.e., residue component). In the case of steady-state radiation force, the components are exerted on the body as static forces. For the case of oscillatory amplitude excitation, the components are exerted at the modulation frequency with frequency-dependant phase shifts. The results demonstrate the dominant contribution of the non-resonance component of dynamic radiation force at high frequencies with respect to the residue component, which offers the potential application of ultrasound stimulated vibro-acoustic spectroscopy technique in low frequency resonance spectroscopy purposes. Furthermore, the proposed formulation may be useful essentially due to its intrinsic value in physical acoustics. In addition, it may unveil the contribution of resonance modes in the dynamic radiation force experienced by the cylindrical objects and its underlying physics. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction An object standing in the path of an acoustic field is known to experience a radiation force (RF) due to the transfer of the momentum from the surrounded wave field to the object itself. This force is known to be steady when the intensity of the incident field is considered to be constant or may be dynamic (oscillatory), if the intensity of the incident field varies slowly with time (or to be modulated). The mechanical response of objects to external loads is of great interest in medical diagnosis, nondestructive evaluation of materials, and materials characterization purposes. In resonance acoustic/ultrasonic spectroscopy, the main aim is to stimulate the eigenvibrations of the object with acoustic loading in order to measure the resonance frequencies. The

n

Corresponding author. Tel.: þ98 21 66165509. E-mail addresses: [email protected] (M. Rajabi), [email protected] (M. Behzad).

http://dx.doi.org/10.1016/j.jsv.2014.05.014 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

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resonances of an object are interpreted as its signature. They are inherent characteristics of the object with known geometrical parameters and are completely independent of the source of excitation and depend only on its mechanical properties (e.g., bulk density and elastic constants). Proper identification of the resonance frequencies of an object can serve as a powerful tool in many applications such as material science, nondestructive inspection of materials, performance analysis of acoustic handling devices, remote classification of submerged targets and on-line monitoring of elastic properties of components. Resonance Scattering Theory (RST) was first introduced by Flax et al. [1] who applied the resonance formalism of the nuclear-reaction theory to the problem of sound scattering from submerged elastic bodies. In this theory, it was shown that the scattered field consists of a smooth background signal (i.e., geometrical reflection) which strongly fluctuated by the resonance behavior of the scatterer due to the excitation of its individual normal modes [2–4]. In general, when a cylindrical body is insonified by acoustic waves, a specular reflection is returned. In addition, surface waves are launched from strike points on object boundary that travel the body in spiral or helical trajectories. These surface waves emit energy into the surrounding medium. It has been established that a close relationship exists between the resonances of an elastic scatterer and the circumferential waves which propagate along its periphery [5]. If the frequency of the incident wave coincides with one of the vibrational eigen-frequencies of the cylinder, surface waves match phases upon circumnavigation so as to build up to the resonance by constructive interference. Resonance scattering theory provides a physical elucidation of the wave phenomena which are manifested in the vibro-acoustic problems. Resonance acoustic spectroscopy consists of the investigation of resonance characteristics (resonance frequency and bandwidth) present in the scattered acoustic signals from the target with the use of resonance scattering theory. The resonance features associated to the target are extracted from the amplitude [6–13] and the phase [14–17] spectra of the resonance scattered field isolated by the subtraction of the background (non-resonance) part from the total scattered field [3,4]. It is more than one decade that the resonance acoustic spectroscopy (RAS) technique has been suggested for non-destructive evaluation/testing [18] and material characterization purposes [19]; but, engineering applications are too rare due to the limitations of this technique. One of these limitations is in the excitation frequency range of the ultrasonic transducers (as the source of excitation) are commonly in the order of few to several MHz while the primary resonance frequencies of the industrial cylindrical components (e.g., rods, tubes, pipes, pressure vessels, wires, shells) are smaller than 100 kHz (e.g., for aluminum cylindrical shells with outer radius greater than 5 cm and thickness to outer radius below 0.05, all the principal resonance frequencies associated to first six vibration modes are bellow 100 kHz). Even if the transducers are set up at lower excitation frequencies, other concerns such as the insufficient spatial resolution will arise. More recently, the so-called non-contact vibro-acoustography technique (ultrasound-stimulated vibro-acoustic spectroscopy) has been presented by Fatemi and Greenleaf [20,21] for imaging the elasticity of objects and detection of resonance frequencies, with specific application in medical diagnosis such as the measurement of tissue hardness and elastic constants [22,23], assessment of bone osteoporosis [24], detection of brachytherapy metal seeds [25], in vivo quantification of liver stiffness [26], artery calcifications imaging [27], recognition of breast micro-calcifications [28], assessment of calcium deposit on heart valves [29], human calcaneus and hip [30], determining objects resonance frequencies [31,32] and porosity [33]. The concept of the ultrasound-stimulated vibro-acoustic spectroscopy technique is based on this fact that when a body is stimulated by an amplitude modulated acoustic field (i.e., generated by interfering ultrasound beams with frequencies ω1, ω2), the scattered acoustic field composes of high frequency fields with the frequencies of monochromatic incident beams, ω1,ω2 and low frequency acoustic field with the frequency of modulation, Δω¼ω1 ω2, which is the result of two main components [34]. The first is the nonlinear scattering of sound-by-sound (i.e., nonlinear interaction of primary high frequency scattered fields, especially in far-field zones) which strongly depends on the nonlinear parameter associated to the wave propagation properties in the host (surrounding) medium [34–36] and carries the high frequency information of the body. The second is the low frequency response of the body in response to a dynamic (oscillatory) radiation force experienced by the body which carries the low frequency information of the body. The complex amplitude of the low frequency acoustic field can be represented as Φ(Δω)¼YdynH(Δω)Q(Δω)þΨ(Δω), where Q (Δω) is a complex function on behalf of the low frequency mechanical frequency response of the body (i.e., considering the possible excitation of the low frequency resonances of the body, Q(Δω) may be assumed as ∑r1/(Δω ωr þiΥr/2) where ωr and Υr are resonance frequencies and their corresponding bandwidths of the body, respectively), H(Δω) means overall transfer function corresponding to the propagation medium and the receiver, Ydyn represents the amplitude of dynamic radiation force exerted on the body and Ψ(Δω) indicates the contribution of nonlinear interaction of sound-by-sound. Recording the radiated acoustic field, Φ (Δω), leads to obtain the mechanical response function of the body, Q(Δω), or at least, the low frequency resonance frequencies of the body may be detected, if the multiplier function, Ydyn(ω1,ω2)H(Δω) and added function Ψ(Δω), could be assumed as either constants functions, known functions or non-resonance (smooth) functions in the frequency range of interest (i.e., the frequency range which is swept by the modulation frequency of the transducer). The above mentioned assumptions are reasonable for Q(Δω) [31], however, they are debatable for Ydyn and especially for Ψ(Δω) where its contribution in total low frequency scattered field is probably much more pronounced in comparison with the low frequency acoustic emission [36]. In literature, there are little rigorous investigations about the theoretical or experimental analysis of the nonlinear sound-by-sound scattering and the research works are limited to the monopole scattering of rigid spheres [34–37]. In the present study, we confine our analysis to the dynamic radiation force which due to the unknown elasticity of the body, there is no estimation of generated dynamic radiation force function. On the other hand, considering the dependency of Ydyn to both of the excitation frequencies corresponding to individual beams, ω1 and ω2, the

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Source Monochromatic Beam Generator

Source Dual Beam Generator (AM)

Fig. 1. Problem configuration, (a) plane progressive wave field (steady-state radiation force), and (b) amplitude modulated plane wave field (dynamic radiation force).

stimulation of high frequency resonances of the cylindrical body is possible which may lead to singular (non-smooth) behavior of dynamic radiation force amplitude. These issues motivate us to make an investigation on the frequency dependency of dynamic radiation force function with an emphasis on the singular effects associated to the circumferential resonance modes of the body, following the proposed method by the same authors in Ref. [38]. Here, we intend to analytically isolate the resonance effects on the dynamic radiation force experienced by a cylindrical shell from the geometrical (non-resonance) ones. This is performed with the use of the resonance scattering theory, in order to discover and formulate the contributions associated to the excited surface waves and the non-resonance reflection acoustic field.

2. Formulation In this section, we shall consider the case of dynamic acoustic radiation force acting on an elastic cylindrical shell of ρs mass density, cL and cs being the velocities of longitudinal and shear waves, with outer and inner radii of a and b, respectively, immersed in and filled with ideal compressible fluids so that the thermo-viscosity effects can be neglected. The

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properties of outer and inner fluid mediums are characterized by c1 and c2 as their dilatational wave speed, respectively, and by ρ1 and ρ2 as their mass density, respectively. The proposed formulation for the case of dynamic radiation force deals with a dual beam mode amplitude modulated wave field normally incident upon the cylindrical shell, in contrast to the steady-state radiation force phenomenon which generated by a monochromatic acoustic field. The configurations of the problems are depicted in Fig. 1. 2.1. Acoustic medium Following the standard methods of linear acoustics, the field equations for an inviscid and ideal compressible medium that cannot support shear stresses may conveniently be expressed in terms of a scalar velocity potential as [39] v ¼ ∇φ;

p ¼  iω ρ φ;

2

∇2 φ þ k φ ¼ 0;

(1)

where k¼ ω/c is the wave number for the dilatational wave, c is the speed of sound, ρ is the ambient density, v is the fluid particle velocity vector, and p is the acoustic pressure. 2.2. Acoustic radiation force definition For a body insonified by a time-harmonic wave field, the general calculation of the time averaged radiation force is performed by integrating the Brillouin radiation stress tensor over the fixed outer surface of the body at equilibrium expressed as [40,41] ! Z Z " 1ρ 1 2 2 〈F〉 ¼  〈ð∂ψ=∂tÞ 〉  ρ〈ðj∇ψj Þ〉 er þ ρ〈ðvr er þvθ eθ Þvr 〉dS; (2) 2 c2 2 S0 in whicho. 4 means time averaged over a cycle of oscillations, ρ is the density of ambient medium, ψ ¼ Reðφinc þφscatt Þ is the total velocity potential field in surrounding medium where φinc and φscatt are the velocity potential functions associated to incident and scattered acoustic fields, respectively, er and eθ are the unit vectors in radial and tangential directions, and vr ¼  ∂ψ/∂r and vθ ¼  ∂ψ/r∂θ are the radial and tangential components of the particle velocity of the surrounding fluid medium. 2.3. Dynamic acoustic radiation force: resonance and background components To generate a dynamic radiation force on a cylindrical shell, consider two incident plane progressive wave fields which propagate in the same direction toward the body, i.e. assume the body is completely located within the focal zone of the wave fields, have the same amplitude and phase and are driven at angular frequencies ω1 and ω2, which leads to the generation of an amplitude modulated beam impinged on the cylindrical body [42], as seen in Fig. 1(b). The velocity potential function for total incident field can be represented in a system of cylindrical coordinates (r,θ) by 2

φinc ¼ φ0 ∑

1

∑ εn in J n ðkm rÞ cos ðnθÞ e  iωm t ;

(3)

m¼1n¼0

where k1 ¼ω1/c1 and k2 ¼ω2/c1 are the wave numbers of the incident fields, Jn is the cylindrical Bessel function of the first kind of order n [43], symbol εn is the Neumann factor (εn ¼1 for n¼ 0, and εn ¼2 for n4 0), i¼(  1)1/2, φ0 is the amplitude of the incident wave. The scattered potential in the surrounding fluid medium 1 can be expressed as a linear combination of cylindrical waves as 2

φscatt ðr; θÞ ¼ ∑

1

 iωm t ∑ εn in Am;n H ð1Þ ; n ðkm rÞ cos ðnθÞ e

(4)

m¼1n¼0

where H ð1Þ n ðxÞ ¼ J n ðxÞ þi Y n ðxÞ is the cylindrical Hankel function of the first kind of order n [43], and Am,n are the unknown modal scattering coefficients which are determined by employing the appropriate boundary conditions, later. In order to take into account the dynamic radiation force effects, a short term time average which is defined as R t þ T=2 〈F〉 ¼ ð1=TÞ t  T=2 FðtÞdt is considered where 2π/(ω1 þω2){ T{ 2π/|ω1  ω2|. Therefore, substitution of total potential function, φinc þ φscatt, into Eq. (2) and making the integration over the outer surface of the shell and then, applying the above short term time average operator, leads to the total radiation force expressed as 〈F〉 ¼ 〈F〉1;p þ 〈F〉2;p þ 〈F〉dyn ;

(5)

where 〈F〉1,p ¼E1ScY1,p and 〈F〉2,p ¼E2ScY2,p are the steady-state (static) radiation forces originated from each individual 2 2 incident wave, Sc ¼ 2a is the cross-sectional area of the cylindrical body, E1 ¼ ð1=2Þρ1 k1 φ20 and E2 ¼ ð1=2Þρ1 k2 φ20 are the average energy indices corresponding to the incident wave fields, Y1,p and Y2,p are the steady-state dimensionless radiation force functions associated to incident wave fields given as [44]   1 2 Y m;p ¼ (6) ∑ ½αm;n þ αm;n þ 1 þ 2ðαm;n αm;n þ 1 þ βm;n βm;n þ 1 Þ; m ¼ 1; 2 km a n ¼ 0

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where αm,n and βm,n are the real and imaginary parts of Am,n. In addition to the steady-state components, 〈F〉dyn is the dynamic (oscillatory) component resulting from the interference between the wave fields expressed as 〈F〉dyn ¼ Ed Sc Y tdyn ; where [42] Y tdyn ¼ Y dyn cos ðΔωt  ϕÞ;

(7)

means the slow-time varying dimensionless dynamic radiation force, Ed ¼ ρ1 k1 k2 φ20 is a measure of incident field energy, Δω¼|ω1  ω2| is the modulation angular frequency, Ydyn ¼(Γ2 þΛ2) is the dimensionless dynamic radiation force function and ϕ ¼tan  1(Γ/Λ) is the phase-shift, where its modules are found [42] Γ¼

π 1 ∑ f½J ðx2 ÞJ n þ 1 ðx1 Þ  J n ðx1 ÞJ n þ 1 ðx2 Þð2 þ α1;n þ α1;n þ 1 þ α2;n þ α2;n þ 1 Þ 2n¼0 n þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þðβ1;n þ β1;n þ 1 Þ þ ½J n þ 1 ðx1 ÞY n ðx2 Þ J n ðx1 ÞY n þ 1 ðx2 Þðβ2;n þ β2;n þ 1 Þ þ ½J n ðx2 ÞJ n þ 1 ðx1 Þ  J n ðx1 ÞJ n þ 1 ðx2 Þ þ Y n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞY n þ 1 ðx2 Þ ðα1;n α2;n þ 1 þ α2;n α1;n þ 1 þ β1;n β2;n þ 1 þβ2;n β1;n þ 1 Þ þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þ þ J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þ ðα2;n β1;n þ 1 þ α2;n þ 1 β1;n  α1;n β2;n þ 1  α1;n þ 1 β2;n Þg;

(8a)

π 1 ∑ f½J ðx1 ÞJ n þ 1 ðx2 Þ J n ðx2 ÞJ n þ 1 ðx1 Þðβ1;n þβ1;n þ 1  β2;n β2;n þ 1 Þ 2n¼0 n þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þðα1;n þ α1;n þ 1 Þ þ ½J n ðx1 ÞY n þ 1 ðx2 Þ Y n ðx2 ÞJ n þ 1 ðx1 Þðα2;n þ α2;n þ 1 Þ þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þ þ J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þ ðα1;n α2;n þ 1 þ α2;n α1;n þ 1 þ β1;n β2;n þ 1 þβ2;n β1;n þ 1 Þ þ ½J n ðx2 ÞJ n þ 1 ðx1 Þ  J n ðx1 ÞJ n þ 1 ðx2 Þ þ Y n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞY n þ 1 ðx2 Þ ðα1;n β2;n þ 1 þ α1;n þ 1 β2;n  α2;n þ 1 β1;n  α2;n β1;n þ 1 Þg;

(8b)

Λ¼

where xm ¼ kma. The detailed derivation process of the above expressions is given in [42]. The resonance scattering theory ðbÞ suggests the scattered field as a superposition of a background field and a resonance field [1–4], i.e. Am;n ¼ AðresÞ m;n þAm;n where ðresÞ ðbÞ Am;n and Am;n denote the modal resonance and background scattering coefficients, respectively (which are determined by employing an appropriate background theory, later). Utilizing the resonance scattering theory and after some manipulations, the dynamic radiation force modules are represented as Λ ¼ Λb þ Λr þΛi ; Γ ¼ Γb þ Γr þ Γi ;

(9a,b)

where the background, (Λb,Γb), the resonance, (Λr,Γr), and the interaction, (Λi,Γi), modules are related to the resonance and background scattering coefficients as Γb ¼

π 1 ∑ f½J ðx2 ÞJ n þ 1 ðx1 Þ  J n ðx1 ÞJ n þ 1 ðx2 Þð2 þ αb1;n þ αb1;n þ 1 þ αb2;n þ αb2;n þ 1 Þ 2n¼0 n þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þðβb1;n þ βb1;n þ 1 Þ þ ½J n þ 1 ðx1 ÞY n ðx2 Þ J n ðx1 ÞY n þ 1 ðx2 Þðβb2;n þ βb2;n þ 1 Þ

þ ½J n ðx2 ÞJ n þ 1 ðx1 Þ  J n ðx1 ÞJ n þ 1 ðx2 Þ þ Y n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞY n þ 1 ðx2 Þ ðαb1;n αb2;n þ 1 þ αb2;n αb1;n þ 1 þ βb1;n βb2;n þ 1 þβb2;n βb1;n þ 1 Þ þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þ þ J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þ ðαb2;n βb1;n þ 1 þ αb2;n þ 1 βb1;n  αb1;n βb2;n þ 1  αb1;n þ 1 βb2;n Þg; Λb ¼

(10a)

π 1 ∑ f½J ðx1 ÞJ n þ 1 ðx2 Þ J n ðx2 ÞJ n þ 1 ðx1 Þðβb1;n þβb1;n þ 1  βb2;n βb2;n þ 1 Þ 2n¼0 n þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þðαb1;n þ αb1;n þ 1 Þ þ ½J n ðx1 ÞY n þ 1 ðx2 Þ Y n ðx2 ÞJ n þ 1 ðx1 Þðαb2;n þ αb2;n þ 1 Þ

þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þ þ J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þ ðαb1;n αb2;n þ 1 þ αb2;n αb1;n þ 1 þ βb1;n βb2;n þ 1 þβb2;n βb1;n þ 1 Þ þ ½J n ðx2 ÞJ n þ 1 ðx1 Þ  J n ðx1 ÞJ n þ 1 ðx2 Þ þ Y n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞY n þ 1 ðx2 Þ ðαb1;n βb2;n þ 1 þ αb1;n þ 1 βb2;n  αb2;n þ 1 βb1;n  αb2;n βb1;n þ 1 Þg; Γr ¼

π 1 ∑ f½J ðx2 ÞJ n þ 1 ðx1 Þ J n ðx1 ÞJ n þ 1 ðx2 Þðαr1;n þαr1;n þ 1 þ αr2;n þ αr2;n þ 1 Þ 2n¼0 n þ ½J n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þðβr1;n þ βr1;n þ 1 Þ

(10b)

M. Rajabi, M. Behzad / Journal of Sound and Vibration 333 (2014) 5746–5761

5751

þ ½J n þ 1 ðx1 ÞY n ðx2 Þ J n ðx1 ÞY n þ 1 ðx2 Þðβr2;n þ βr2;n þ 1 Þ þ ½J n ðx2 ÞJ n þ 1 ðx1 Þ J n ðx1 ÞJ n þ 1 ðx2 Þ þ Y n ðx2 ÞY n þ 1 ðx1 Þ  Y n ðx1 ÞY n þ 1 ðx2 Þ ðαr1;n αr2;n þ 1 þαr2;n αr1;n þ 1 þβr1;n βr2;n þ 1 þ βr2;n βr1;n þ 1 Þ þ ½J n ðx2 ÞY n þ 1 ðx1 Þ  Y n ðx1 ÞJ n þ 1 ðx2 Þ þ J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þ ðαr2;n βr1;n þ 1 þ αr2;n þ 1 βr1;n  αr1;n βr2;n þ 1 αr1;n þ 1 βr2;n Þg; Λr ¼

(10c)

π 1 ∑ f½J ðx1 ÞJ n þ 1 ðx2 Þ J n ðx2 ÞJ n þ 1 ðx1 Þðβr1;n þ βr1;n þ 1 βr2;n βr2;n þ 1 Þ 2n¼0 n þ ½J n ðx2 ÞY n þ 1 ðx1 Þ  Y n ðx1 ÞJ n þ 1 ðx2 Þðαr1;n þ αr1;n þ 1 Þ þ ½J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þðαr2;n þ αr2;n þ 1 Þ

þ ½J n ðx2 ÞY n þ 1 ðx1 Þ  Y n ðx1 ÞJ n þ 1 ðx2 Þ þ J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þ ðαr1;n αr2;n þ 1 þαr2;n αr1;n þ 1 þβr1;n βr2;n þ 1 þ βr2;n βr1;n þ 1 Þ þ ½J n ðx2 ÞJ n þ 1 ðx1 Þ J n ðx1 ÞJ n þ 1 ðx2 Þ þ Y n ðx2 ÞY n þ 1 ðx1 Þ  Y n ðx1 ÞY n þ 1 ðx2 Þ ðαr1;n βr2;n þ 1 þ αr1;n þ 1 βr2;n  αr2;n þ 1 βr1;n αr2;n βr1;n þ 1 Þg; Γi ¼

(10d)

π 1 ∑ f½J ðx2 ÞJ n þ 1 ðx1 Þ  J n ðx1 ÞJ n þ 1 ðx2 Þ þ Y n ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞY n þ 1 ðx2 Þ 2n¼0 n

ðαr1;n αb2;n þ 1 þ αb1;n αr2;n þ 1 þ αr2;n αb1;n þ 1 þ αb2;n αr1;n þ 1 þβr1;n βb2;n þ 1 þβb1;n βr2;n þ 1 þ βr2;n βb1;n þ 1 þ βb2;n βr1;n þ 1 Þ þ ½J n ðx2 ÞY n þ 1 ðx1 Þ  Y n ðx1 ÞJ n þ 1 ðx2 Þ þ J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þ ðαr2;n βb1;n þ 1 þ αb2;n βr1;n þ 1 þ αr2;n þ 1 βb1;n þ αb2;n þ 1 βr1;n  αr1;n βb2;n þ 1 αb1;n βr2;n þ 1 αr1;n þ 1 βb2;n αb1;n þ 1 βr2;n Þg; Λi ¼

(10e)

π 1 ∑ f½J ðx2 ÞY n þ 1 ðx1 Þ Y n ðx1 ÞJ n þ 1 ðx2 Þ þ J n ðx1 ÞY n þ 1 ðx2 Þ  Y n ðx2 ÞJ n þ 1 ðx1 Þ 2n¼0 n

ðαr1;n αb2;n þ 1 þ αb1;n αr2;n þ 1 þ αr2;n αb1;n þ 1 þ αb2;n αr1;n þ 1 þβr1;n βb2;n þ 1 þβb1;n βr2;n þ 1 þ βr2;n βb1;n þ 1 þ βb2;n βr1;n þ 1 Þ þ ½J n ðx2 ÞJ n þ 1 ðx1 Þ J n ðx1 ÞJ n þ 1 ðx2 Þ þ Y n ðx2 ÞY n þ 1 ðx1 Þ  Y n ðx1 ÞY n þ 1 ðx2 Þ ðαr1;n βb2;n þ 1 þ αb1;n βr2;n þ 1 þ αr1;n þ 1 βb2;n þ αb1;n þ 1 βr2;n  αr2;n þ 1 βb1;n αb2;n þ 1 βr1;n αr2;n βb1;n þ 1 αb2;n βr1;n þ 1 Þg:

(10f)

Combination of Eq. (7) with Eq. (9) leads to a novel definition of the slow time-varying dimensionless dynamic radiation force as Y tdyn ¼ Y ðbÞ cos ðΔωt  ϕb Þ þ Y ðresÞ cos ðΔωt ϕr Þ þ Y ðintÞ cos ðΔωt  ϕi Þ; (11) dyn dyn dyn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðbÞ ðresÞ ðintÞ b 1 r 1 2 2 2 2 2 2 where Y dyn ¼ Γ b þ Λb , Y dyn ¼ Γ r þ Λr and Y dyn ¼ Γ i þΛi mean the amplitudes and ϕ ¼ tan (Γb/Λb),ϕ ¼tan (Γr/Λr) and ϕi ¼tan  1(Γi/Λi) mean the phase shift of background, resonance and interaction components of dimensionless dynamic radiation force function, respectively. Combination of Eqs. (7) and (11) yields n Y dyn ¼ ðY ðbÞ Þ2 þ ðY ðresÞ Þ2 þðY ðintÞ Þ2 dyn dyn dyn o1=2 þ2 Y ðbÞ Y ðresÞ cos ðϕb  ϕr Þ þ 2 Y ðresÞ Y ðintÞ cos ðϕr  ϕi Þ þ 2 Y ðintÞ Y ðbÞ cos ðϕi  ϕb Þ ; (12) dyn dyn dyn dyn dyn dyn The above description of the amplitude of the dynamic radiation force as a function of the amplitudes and the phases of its background, resonance and interaction components may be used to illustrate how the resonance effects appear as maxima or minima in the spectra of dynamic radiation force frequency function. Clearly, this appearance is a function of both excitation frequencies, ω1 and ω2, (or the modulation frequency, Δω¼ω1  ω2), since ϕb ¼ϕb(ω1,ω2), ϕr ¼ϕr(ω1,ω2), ðbÞ ðrÞ ϕi ¼ϕi(ω1,ω2) and Y dyn ¼ Y ðbÞ ðω1 ; ω2 Þ; Y dyn ¼ Y ðrÞ ðω1 ; ω2 Þ; dyn dyn ¼ Y ðiÞ ðω1 ; ω2 Þ: Y ðiÞ dyn dyn In analogy with the resonance scattering formalism, the residue part of dynamic radiation force may be defined as the ðresidueÞ subtraction of the background part from the total dynamic radiation force, Y dyn ¼ Y tdyn  Y ðbÞ cos ðΔωt  ϕb Þ, which leads dyn to ðresidueÞ Y dyn ¼ Y ðresiÞ cos ðΔωt  ϕresi Þ; (13) dyn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðresiÞ where Y dyn ¼ Γ 2resi þ Λ2resi and ϕresi ¼tan  1(Γresi/Λresi) in which Γresi ¼ Γ Γb and Λresi ¼Λ  Λb. In addition, another interesting case may occur in the above formulation when the frequency of incident fields is identical (i.e., ω1 ¼ω2). In this case, the incident wave field is equivalent to a plane progressive wave field with double amplitude; thus, the expression of dynamic radiation force, Ydyn, reduces to the steady-state one, Yp, and the expressions associated to the modules of dynamic radiation force become Γ(ω1 ¼ω2) ¼0, Λ(ω1 ¼ω2) ¼Yp. Furthermore, the amplitudes ðresÞ ðresÞ and the modules of the components are Λb ðω1 ¼ ω2 Þ ¼ Y ðbÞ ¼ Y ðbÞ and Λi ðω1 ¼ ω2 Þ ¼ Y ðintÞ ¼ Y ðintÞ p ; Λr ðω1 ¼ ω2 Þ ¼ Y dyn ¼ Y p p dyn dyn ðresÞ ðintÞ where Y ðbÞ ; Y and Y are background, resonance and interaction components of the steady-state radiation force given p p p

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as [38]  2 1 b ∑ ½α þ αbn þ 1 þ 2ðαbn αbn þ 1 þ βbn βbn þ 1 Þ; ka n ¼ 0 n   2 1 r ∑ ½α þ αrn þ 1 þ 2ðαrn αrn þ 1 þ βrn βrn þ 1 Þ; Y ðresÞ ¼ p ka n ¼ 0 n   4 1 r b ¼ þ αbn αrn þ 1 þβrn βbn þ 1 þ βbn βrn þ 1 ; Y ðintÞ ∑ ½α α p ka n ¼ 0 n n þ 1 Y ðbÞ p ¼



(14a-c)

þ Y ðintÞ [38]. where Y p ¼ Y pðbÞ þ Y ðresÞ p p The expression of the dynamic radiation force function, Eq. (11), reveals that, similar to the steady-state radiation force function [38], the dynamic radiation force is composed of three physical components: background, resonance and a third term which originates from their interaction due to the nonlinearity effects. All the components are driven at the modulation frequency, but with different frequency-dependent phase shifts. In addition to the above mathematical formalism of the presented decomposition, a clear physical interpretation is involved in the extracted components. The background component reveals the pure effect of the geometrical reflection from the body which leads to a non-resonance (smooth) function of the bulk properties (i.e., density and geometry) of the body and acoustical properties (i.e., density and speed of sound) of the inner and outer fluid mediums. Similarly, the resonance component reflects the pure contribution of the circumferential resonance modes of the body in the absence of the geometrical (background) reflection field, in the dynamic radiation force exerted on the body. The last term, represents the radiation force function experienced by the body from the interaction of the resonance and background fields in the surrounding fluid medium. In other words, the above formulation makes isolation between the contributions of the geometrical and the resonance effects which are caused by the stimulation of the eigen-vibrations of the body. In general application, this matter that how the excitation of resonances of a target may affect the radiation force experienced by the target is a fundamental issue, and a model based answer to this question may serve in many applications such as vibro-acoustography based nondestructive evaluation/testing and resonance detection purposes, object manipulation etc. The presented formulation gives an analytical estimation of the contribution of the resonance and geometrical (non-resonance) effects.

2.4. Scattering coefficient and its resonance and background components Consider a plane progressive wave field driven at the angular frequency of ωm. Solving the classical Navier equation which governs the elastic motion of cylindrical shells, along with the acoustic field equations and the application of boundary conditions (i.e., continuity of normal fluid and solid velocities, normal stress and fluid pressure, and vanishing of tangential stress at inner and outer surface of the cylindrical shell), leads to the scattering coefficients, Am,n, as [3,4] Am;n ¼ 

xm J 0n ðxm Þ  J n ðxm ÞF n ðωm Þ

ð1Þ xm H ð1Þ0 n ðxm Þ H n ðxm ÞF n ðωm Þ

;

(15)

where the prime (.)0 indicates the differentiation with respect to the argument, xm ¼kma, in which km ¼ωm/c1 and Fn(ωm)¼  ρ1(Kma)(Zm,n/Wm,n) where Km ¼ωm/c2, Zn and Wn are the determinants of two 5n5 matrices. The elements of these matrices are given in the Appendix. The function Fn(ωm) is called the modal accelerance function of the shell in the literature [45,46], which implicitly depends on the characteristics of the bulk waves in the shell. In order to obtain the background scattering coefficients, AðbÞ m;n , the “inherent background approach” [45,46] is used. In this approach, the non-resonance component is extracted from the zero frequency limit of the modal accelerance function of the analogous liquid shell, due to the negligible interaction between the resonance and background signals near zero frequency. The inherent background scattering coefficients are expressed as AðbÞ m;n ¼ 

xm J 0n ðxm Þ J n ðxm ÞF n ð0 þ Þ

; þ xm H nð1Þ0 ðxm Þ  H ð1Þ n ðxm ÞF n ð0 Þ

where the constant modal accelerance of a fluid-filled analogous liquid shell is given [45,47] n 2 ðnρs =ρ2 Þ 4ρ1 : F n ð0 þ Þ ¼ ρρ1 nQþþQðnρ =ρ Þ n a 0ρ  4ρ lnðb=aÞn ¼ 0 s

s

2

2

s

(16)

(17)

in which Q¼n((1þ(b/a)2n)/(1  (b/a)2n)). Finally, the resonance scattering coefficients are obtained as the subtraction of the background scattering coefficient from the scattering coefficients as ðbÞ AðresÞ m;n ¼ Am;n Am;n :

(18)

This completes the necessary background required for the frequency analysis of the steady-state and dynamic radiation force experienced by a submerged and fluid filled cylindrical shell via resonance scattering theorem. We consider some numerical examples in the following section.

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3. Numerical results In this section, we consider some numerical examples in order to demonstrate the nature and general behavior of the proposed approach. Focusing on the main concept of the present work, we confine the calculations to a particular model. An aluminum shell is assumed to be surrounded by water (ρ1 ¼1000 kg/m3, c1 ¼1480 m/s) and filled with air (ρ2 ¼1.2 kg/m3, c2 ¼344 m/s) at atmospheric pressure and ambient temperature. The mass density of the aluminum shell is specified as ρs ¼2706 kg/m3, and the velocities of longitudinal and shear waves are set as cL ¼6401 m/s and cs ¼3031 m/s. The thickness parameter of the aluminum shell is b/a¼0.96, and the excitation frequency covers the range 0 ox1 ¼k1ao25 for which a collection of resonances associated to zero-order symmetric Lamb waves and fluid born A-type waves are anticipated to be excited. A MATLABs code was constructed for solving Eqs. (15)–(18) in order to calculate the unknown total, background and resonance scattering coefficients as a function of the nondimensional frequency, x1 ¼k1a, and subsequently, evaluating the dimensionless steady-state and dynamic radiation force function and their components and modules. The computations were performed on a Pentium IV personal computer with a frequency resolution of Δx1 ¼ Δk1a¼ 0.005 and maximum truncation constant of nmax ¼x1,max þ25; especially selected to assure convergence of the calculations. The convergence was systematically checked in a simple trial and error manner, by increasing the truncation constant, n, (i.e., by including more modes in all summations) while looking for stability in the numerical value of the solutions. Before presenting our numerical results, we shall establish the overall validity of the calculations. We first computed the dimensionless steady-state radiation force, Yp, versus dimensionless frequency, x1, for a water-submerged and water-filled homogeneous non-absorbent lucite shell, by setting b/a ¼0.9, ρs ¼1191 kg/m3, cL ¼2690 m/s, cs ¼1340 m/s in our general MATLABs code. The result is shown in Fig. 2(a) exhibits an excellent agreement with that displayed in Fig. 1 of Ref. [48]. Furthermore, the dimensionless dynamic radiation force, Ydyn, versus dimensionless frequency, x1, is computed for an aluminum cylinder, b/a¼0, submerged in water. The result of Fig. 2(b) is in complete agreement of Fig. 12 of Ref. [49]. For the analysis of the acoustic radiation force phenomenon we need to study the resonance scattering characteristics associated to the object. For this purpose, Fig. 3(a) illustrates the backscattering resonance amplitude spectrum of a b/a¼ ðresÞ ðbÞ 0.96 aluminum shell, defined as jf 1 ðθ ¼ π; k1 aÞj  jf 1 ðθ ¼ π; k1 aÞ f 1 ðθ ¼ π; k1 aÞj;where pffiffiffiffiffiffiffiffiffiffi |f1(θ¼π,k1a)| represents the backscattering far-filed form function amplitude defined as jf 1 ðθ ¼ π; k1 aÞj  lim 2r=a e  ik1 r jφscatt ðr;θ ¼ π;k1 aÞ=φinc j [2–4], r-1 which reduces to jf 1 ðθ ¼ π; x1 Þj ¼ j∑1 n ¼ 0 f n ðθ ¼ π; x1 Þj; upon using the high-frequency asymptotic form of cylindrical Hankel

1.4 1.2

Yp

1 0.8 0.6 0.4

Lucite Shell, b/a=0.9 Mitri's result [48]

0.2 0

0

5

10

15

20

25

30

k 1a

1.3

Ydyn

1.2 1.1 1 Aluminium Cylinder, b/a=0 Mitri & Fatemi 's result [49]

0.9 0.8

5

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

k 1a Fig. 2. The frequency variation of (a) the total dimensionless radiation force, Yp, versus dimensionless frequency, k1a, for a b/a ¼0.9 thickness watersubmerged and water-filled homogeneous non-absorbent lucite shell, and (b) the total dynamic radiation force, Ydyn, versus dimensionless frequency, x1, for a b/a¼ 0 thickness aluminum cylinder submerged in water.

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2 1.5 S10

1

S20

S30

0.5

A19 A20

A18

0

0

5

21

10

A

S40

A23 A22

S50

A24

15

S60

25

A

27 A26 A

A28

A29

20

25

k1a S10

1

S30

S20

A20

4

5 22 A21 S0 A A23 A24 S0

A25

6

A26 A27 S0

A28

A29

0.8 0.6 0.4 0.2 0

0

5

10

15

20

25

k1a ðresÞ

ðresÞ

Fig. 3. (a) The frequency variation of the backscattering resonance scattering amplitude, jf 1 ðθ ¼ π; k1 aÞj ¼ j∑1 ðθ ¼ π; k1 aÞj; and (b) the isolated n ¼ 0f n modal resonance scattering amplitude, jAðresÞ j, for selected mode numbers, of a b/a¼ 0.96 thickness aluminum shell versus dimensionless frequency, k1a. n

pffiffiffiffiffiffiffiffiffiffiffi function; i.e., in H nð1Þ ðk1 rÞ  ð2= πik1 r Þeik1 r ðk1 r-1Þ, where the modal form function are given by 2εn f n ðθ; k1 aÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiAn;1 cos ðnθÞ e  iω t : πik1 a ðbÞ

(19) ðbÞ

and the far-fieldpffiffiffiffiffiffiffiffiffiffiffi backscattering background amplitude is defined as jf 1 ðθ ¼ π; k1 aÞj ¼ j∑1 n ¼ 0 f n ðθ ¼ π; x1 Þj; where ffi ðbÞ ðbÞ f n ðθ; k1 aÞ ¼ ð2εn = πik1 aÞAn cos ðnθÞ e  iω t : Considering Eq. (18), the resonance back-scattering amplitude is obtained as ðresÞ ðresÞ jf 1 ðθ ¼ π; k1 aÞj ¼ j∑1 ðθ ¼ π; x1 Þj;where n ¼ 0f n ðresÞ

fn

2εn ðθ; k1 aÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiAðresÞ cos ðnθÞ e  iω t : n πik1 a

(20)

For resonance identification purposes, the amplitude of modal resonance scattering coefficients, AðresÞ , are also calculated n and illustrated in Fig. 3(b). The spectra are plotted with a frequency resolution of Δx1 ¼Δk1a¼0.00005 in order to capture the resonance peaks which is anticipated to follow a Breit–Wigner form of AðresÞ ¼ ∑l ððiϒ n;l =2Þ=ðx  xn;l þ iϒ n;l =2ÞÞ, where n x ¼k1a is the dimensionless frequency, l ¼1,2,3,… is to identify the overtones associated with the fundamental mode of the vibration, n ¼0,1,2,…, whose members appear in all the partial waves, shifting to higher frequencies from one partial wave to the next one. The quantity xn,l indicates the real value of vibration frequency and Υn,l represents the bandwidth of the corresponding resonance and is a measure of lifetime of the resonance. At the resonance frequencies, the amplitudes of the corresponding modal resonance scattering coefficients are expected to be equal to one. Making comparison with the information of resonance scattering spectrum of a b/a¼0.96 thickness aluminum shell provided by Veksler [4], the regularly spaced resonances are attributable to the zero-order symmetric Lamb waves (i.e., symbolized by Sn0 corresponding to the nth vibration mode) originated from constructive interference of repeated circumnavigations. These resonances are approximately spaced at intervals of Δxres 1 ¼ cg =c1 where cg is the group velocity which for zero-order symmetric Lamb wave is roughly equal to cph (phase velocity) and the plate wave speed cPL [50]. For our mentioned aluminum case, Δxres 1  3:7. In addition to the symmetric type Lamb wave, the resonances corresponding to the fluid-borne Lamb waves, symbolized by An, are also observed in Fig. 3(a). A simple calculation of the peripheral phase speed of the excited fluid born A-type wave at their resonance state, cph/c¼(ka)res/n, categorizes them as the subsonic type (i.e., their phase velocities are less than the speed of sound in water) and their propagation is accompanied with the coupling of the incident plane field and the

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Fig. 4. The frequency variation of (a) total stationary radiation force function, Yst, and background component, Y(b), (b) residue term, Y(residue) ¼Y  Y(b), (c) resonance components, Y(res), (d) interaction component, Y(int), for a b/a ¼0.96 thickness aluminum shell versus dimensionless frequency, x1.

scattered waves into these surface resonance modes by tunneling through the evanescent field adjacent to the shell [50]. Considering their ultra-narrow nature (too small bandwidth), these resonances are observable only for n Z18 for the considered frequency resolution Δk1a ¼0.005. Smaller frequency resolutions may lead to complicated and busy figures which are unessential for our analysis. Fig. 4(a) shows the frequency dependency of the steady-state radiation force function, Yp, and its background component, Y ðbÞ p ; versus dimensionless frequency, x1. Fig. 4(b) illustrates the frequency dependency of the residue component (i.e., the subtraction of background component from the total stationary radiation force or in other word, the superposition of resonance and interaction components), Y pðresiÞ ¼ Y p  Y ðbÞ p ; which reflects the total contribution of resonance circumferential modes. In addition, Figs. 4(c) and (d) depict the variation of the resonance components, Y ðresÞ ; and the interaction p component, Y ðintÞ ; with respect to the dimensionless frequency, respectively. p As Fig. 4(a) and (b) clearly shows, the radiation force function consists of a resonance spectrum superimposed on the proposed smooth (non-resonance) background component so that the subtraction of background (geometrical) effects leads to the resonance manifestation in the residue component in Fig. 4(b). Comparing the trends of Fig. 4(a) and (b) with the information illustrated in the resonance spectrum in Fig. 3(a), leads to the identification of the appeared dips and peaks as the resonances of S10 through S60 , and A20 through A29. Here, this conclusion might be established that the resonances of the body can be isolated by the subtraction of the proper background component from the steady-state radiation force spectrum experienced by the body, as well as what is common in the resonance acoustic spectroscopy technique [1–4]. Further exploration in Fig. 4 indicates that the resonance and interaction components of radiation force function take the negative values in some frequency ranges, which indicates the pulling effects in spite of the usual pushing effects of the total steady-state radiation force due to the plane progressive incidence [51]. In the mentioned example, the residue part which is the resultant of both resonance and interaction components (all the resonance effects), show an attraction (pulling) effect at resonance frequencies associated to the zero-order symmetric Lamb waves and an repulsive (pushing) effect at resonance frequencies associated to the fluid-born A-type waves. The negative or positive values corresponding to the resonance and interaction components might be related to the directivity patterns of the resonance and the background scattered acoustic pressure around the body as well as the resultant transported momentum to the body and their interaction. Moreover, the decreasing or increasing effects of the resonances might be interpreted due to the out-of-phase or in-phase interactions of the circumnavigating surface waves on the body and the radiated bulk wave into the surrounding acoustic medium. In general, considering the regular positive values of the background component in the plane progressive incident case, the appearance of a specific resonance as maxima or minima in the radiation force spectrum is the result of its increasing effects (pushing) or decreasing (pulling) effects.

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Fig. 5. The frequency variation of (a) total dynamic radiation force function, Ydyn, and its phase, ϕ, (b) dynamic background component, Y ðbÞ ; and its phase, dyn ϕb, (c) dynamic resonance components, Y ðresÞ ; and its phase, ϕr, (d) dynamic interaction component, Y ðintÞ ; and its phase, ϕi, with respect to dimensionless dyn dyn frequency, x1, at modulation frequency δx ¼0.2, for a b/a ¼0.96 thickness aluminum shell. In all figures, the solid line denotes the variation of the amplitude and the dotted line indicates the variation corresponding to phase.

Fig. 6. The frequency variation of dynamic radiation force modules and their components (a) Λ and Λb, (b) Λresi ¼Λ  Λb, (c) Λr, (d) Λi, (e) Γ and Γb, (f) Γresi ¼ Γ  Γb, (g) Γr, (h) Γi, with respect to dimensionless frequency, x1, at modulation frequency δx ¼0.2, for a b/a¼ 0.96 thickness aluminum shell.

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At a very low frequency, a notably high peak is observed in the background and resonance components. This peak is due to the radial (i.e., monopole n¼0) resonance effects of the cylindrical bubble of air in the water [52,53]. Similar situation has been reported in Ref. [54] in a spherical bubble case where a “giant monopole” resonance is manifested. This resonance does not manifest in the radiation force function graph, but clearly observed in the resonant component. Fig. 5(a)–(d) shows the variation of the dynamic radiation force function, Ydyn, and its phase, ϕ, the dynamic background component, Y ðbÞ ; and its phase, ϕb, the dynamic resonance components, Y ðresÞ ; and its phase, ϕr, and the dynamic interaction dyn dyn ðintÞ i component, Y dyn ; and its phase, ϕ , with respect to the dimensionless frequency, x1, respectively, for the previously mentioned b/a¼0.96 thickness aluminum shell. It is shown in [49] that as the modulation frequency increases, the dynamic radiation force deviates from the steady-state one so that a resonance frequency splitting phenomenon occurs. Therefore, in the present work, the frequency of modulation is chosen to be constant as δx¼Δω/c1 ¼0.2. This value is selected for the dimensionless modulation frequency, so that the required condition of averaging process (i.e., Δω{ ω1,ω2 or δx{ x1,x2) is met in the entire frequency range of interest, 0 ox1 ¼ k1a o25. In all figures, the solid line denotes the variation of the amplitude and the dotted line indicates the variation of the phase. Comparing Fig. 5(a)–(d) with the general trend of steady-state radiation force and its components depicted in Fig. 4(a)– (d), illustrates the same general behavior. It is seen in Fig. 5(c) that the resonance frequencies are much more detectable in the resonance components, Y ðresÞ , while as it was expected, a smooth (regular) behavior associated to the background dyn component, Y ðbÞ ; and its phase shift, ϕb, is apparent. As seen in Fig. 5(a), the phase shift of dynamic radiation force, ϕ, is zero dyn in all frequency range except the resonance region where non-zero values are observed. Considering the measurable nature of the dynamic radiation force and its phase, the gradient of the phase might be proposed for precise resonance frequency detection similar to the phase of scattered resonance acoustic field [17]. In addition, the phase shifts associated to the resonance component, ϕr, and interaction component, ϕi, show highly non-regular pattern so that sudden π rad jumps occurred. In a simple interpretation, these jumps mean a sudden change in the direction of the corresponding force component; i.e., pushing effect turn to pulling effect and vice versa. They may be interrogated from the directivity pattern of the resonance and background scattered fields as well as their interaction with the resultant transported momentum of surrounding fluid to the body; but in fact, the physical origin of this phenomenon is not clear and must be studied in future. Fig. 6(a)–(d) and (e)–(h) make another examination of the dynamic radiation force function by studying of frequency dependency of its modules, Λ and Γ, their background components, Λb and Γb, their residue terms, Λresi ¼Λ Λb and Γresi ¼Γ Γb, their resonance components, Λr and Γr, and their interaction components, Λi and Γi, with respect to dimensionless frequency, x1, respectively. For non-zero modulation frequency, Λ deviates from the steady-state radiation force, Yp, and Γ takes non-zero values, as it is expected. Due to the small value of modulation frequency (i.e., δx¼0.2), the general trend of the first dynamic radiation force module, Λ, is close to the steady-state radiation force depicted in Fig. 4. As it is clear, a singular behavior is seen near the resonance frequencies of the object in both of the modules and their resonance components, interaction component and obviously their residue terms. This behavior is more clearly observed in the dynamic module, Γ. As it is seen, the resonance frequencies are clearly distinguishable. The higher sensitivity of this module may candidate it as an observant asset for resonance detection purposes. Furthermore, the splitting of resonance frequencies is observed in Γ. 1.4

Ydyn Ydyn (b) Ydyn (resi)

1.2 1 0.8 0.6 0.4 0.2 0 10

20

30

40

50

60

70

80

90

100

x1 Fig. 7. The frequency variation of the dynamic radiation force function, Ydyn, the dynamic background component, Y ðbÞ ; and the dynamic residue part, dyn Y ðresidueÞ ¼ Y dyn  Y ðbÞ with respect to dimensionless frequency, 0r x1 r100, at modulation frequency δx¼ 5, for a b/a¼ 0.96 thickness aluminum shell. dyn dyn ,

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Another common observation, which is seen in the interaction components of both steady-state and dynamic radiation force amplitude and modules, is the suppression of some of the resonances, e.g. A28 and A29 in Fig. 6(d) and (h). This occurrence may be due to the possible destructive interconnection effects between the resonance and the background terms. As it is stated before, the idea of application of the ultrasonic stimulated vibro-acoustic spectroscopy technique for nondestructive evaluation/testing purposes is weighted down due to the singular behavior and drastic frequency dependency of dynamic radiate force amplitude. In the following, we intend to examine and compare the contributions of the nonresonance and the resonance effects on the dynamic radiation force amplitude for the mentioned example, with a chosen modulation frequency in the range of 0 rδxr5 in order to capture the first fundamental resonance frequency of symmetric Lamb wave, S10 . The excitation frequency of the source is chosen in the range of 10 ox1 ¼k1a o100 (remember that the minimum value of the frequency range is selected so that the condition of δxrx1,x2 is met). Fig. 7 shows the frequency variations of the dynamic radiation force, Ydyn, its background component amplitude, Y ðbÞ ; and the residue part amplitude, dyn Y ðresiÞ . The most important observation is that the amplitude of the residue part, which holds all the resonance effects, dyn demonstrates a decreasing manner, for the frequencies greater than x1 480, where the ratio of the residue amplitude to the dynamic radiation force amplitude becomes smaller than 0.02. This diminishing behavior may lead to the approximation of ðbÞ Y dyn  Y dyn in the high frequency ranges. This is a promising result, considering the known features of the background component (i.e., amplitude and phase) which is only a function of the material density and the geometrical parameters and is independent of the body elasticity and resonance characteristics. It could be shown that this manner is also expected for the case of steady-state radiation force. The diminishing contribution of the resonance effects (i.e., dominant contribution of the background component) with respect to frequency may be interpreted mathematically, as follows. For simplicity, we just present the formulation for the case of steady-state radiation force. The definition of the resonance and interaction components of the radiation force function, Eqs. 14(a) and 14(c), are ðresÞ ðintÞ ðintÞ re-written as Y ðresÞ ¼ ∑1 ¼ ∑1 p n ¼ 0 Y p;n and Y p n ¼ 0 Y p;n where the modal contributor terms are obtained as Y ðresÞ n

 (   2 r  fαn 2 þ ðαrn  1 þ αrn þ 1 Þ þβrn ðβrn  1 þ βrn þ 1 Þg ka ¼   2 r ðα0 þ αr0 αr1 þ βr0 βr1 Þ ka

na0 n¼0

;

(21)

1

(b) bn

α n βb n

0.5 0 -0.5 -1

0

2

4

6

8

10

12

14

16

18

20

x1

1

(res) rn

2 0

0.5

α n βr n

0 -0.5 -1

0

2

4

6

8

10

12

14

16

18

20

x1 b b Fig. 8. The frequency variation of (a) the modal background scattering amplitude, jAðbÞ n j, its real part, αn , and its imaginary part, β n , and (b) the modal r r resonance scattering amplitude, jAðresÞ j, its real part, α , and its imaginary part, β , for selected mode number, n¼ 2, with respect to dimensionless n n n frequency, 0 rk1a r20, for a b/a¼ 0.96 thickness aluminum shell.

M. Rajabi, M. Behzad / Journal of Sound and Vibration 333 (2014) 5746–5761

Y ðintÞ n

8  <  2 fαrn ðαbn  1 þαbn þ 1 Þ þαbn ðαrn  1 þ αrn þ 1 Þ þ βbn ðβrn  1 þ βrn þ 1 Þ þ βrn ðβbn  1 þβbn þ 1 Þg ka ¼   2 r b : ka ðα0 α1 þ αb0 αr1 þ βb0 βr1 þ βr0 βb1 Þ

5759

na0 n¼0

;

(22)

In order to focus on the contribution of an individual resonance mode, we just take into account the case of isolated (dominant) mth partial mode of vibration where the interference of other (especially neighborhoods) eigenvibrations of the body in its frequency band is negligible (i.e., αrm þ 1 ; αrm  1 ; βrm þ 1 ; βrm  1 -0). The modal resonance and interaction contributors are rewritten as (   4 r αm m a0 ka ðresÞ Y m ¼   2 r ; (23) α0 m ¼ 0 ka Y ðintÞ m

8  <  2 fαrm ðαbm  1 þ αbm þ 1 Þ þ βrm ðβbm  1 þβbm þ 1 Þg ka ¼   2 r b : ka ðα0 α1 þ βr0 βb1 Þ

ma0 m¼0

;

(24) ðbÞ

ðbÞ ðbÞ 2iδn Considering the description of the background scattering coefficients, Eq. (16), as AðbÞ n ¼ ð1=2ÞðSn 1Þ where Sn ¼ e denotes the partial wave scattering function and δnðbÞ means the background phase shift, it may be easily shown that 2 ðbÞ ðbÞ ðbÞ b b 0 r jAðbÞ n j ¼ j sin ðδn Þj r 1, 1 r αn ¼  sin ðδn Þ r 0 and  1=2 r βn ¼ ð1=2Þ sin ð2δn Þ r 1=2. Furthermore, taking into account a Breit–Wigner approximation for mth resonance scattering coefficients as AðresÞ ¼ ∑l ðiϒ m;l =2Þ=ðx1  xm;l þiϒ m;l =2Þ, m the relations of 1 rαrm r 1 and 1 rβrm r1 are obtained. Fig. 8(a) and (b) depicts the variations of the modal background and the modal resonance scattering amplitudes and their real and imaginary parts versus dimensionless frequency, respectively. Now, turning back to Eqs. (23) and (24) with the above governing bounds, it is concluded that the weight of the resonance scattering components in the resonance and interaction contributors are decreased as the stimulated resonance frequency increases (i.e., lim Y ðresÞ ¼ 0; lim Y ðintÞ m m ¼ 0). ka-1

ka-1

4. Conclusions In the present work, exploiting the classical resonance scattering theorem (RST) which suggests the scattered field as a superposition of a resonant field and a background (non-resonant) component, it has been shown that the radiation force function acting on a cylindrical shell can be synthesized as the composition of three components: background component, resonance component and a third term which originates from their interaction due to the nonlinearity effects. In contrary to the static nature associated to the components of radiation force produced by a monochromatic harmonic wave field, the components of dynamic radiation force (which may be generated by an amplitude modulated beam or two coplanar wave fields driven at close frequencies), are driven at the modulation frequency with frequency-dependent phase shifts. The phase shifts of the components of the dynamic radiation force interpret the type of emergence of a specific resonance as a maxima or minima in its frequency spectrum. In both cases, the background component revealed the pure geometrical reflection effects and illustrates a smooth behavior with respect to frequency, while the others demonstrate a singular behavior near the resonance frequencies. The resonant and interaction components influence the purely non-resonance component so that the magnitude of the radiation force function changes quickly in the vicinity of the resonance frequencies, and slowly out of the resonance frequencies. The resonance effects associated to partial waves can be isolated by the subtraction of the background component from the total radiation force function (i.e., residue part). Investigating the frequency functions of the dynamic radiation force components, it is shown that the contribution of the background component becomes the dominant part of the dynamic radiation force and the contribution of the residue part tends to zero, as the frequency increases and the approximation of Y dyn  Y ðbÞ is met. Considering the limitations of the dyn ultrasound stimulated vibro-acoustic spectroscopy technique for non-destructive evaluation/testing purposes, it is a promising outcome, due to the known features of the background component (i.e., amplitude and phase) which is only a function of the material density and the geometrical parameters and is independent of the body elasticity and resonance characteristics. The proposed formulation may be helpful essentially due to its inherent value as a canonical subject in physical acoustics and may help to isolate the resonance effects on the steady-state or dynamic radiation force function. Appendix

Zn ð1; 1Þ ¼  ξ1 J 0n ðξ1 Þ; Zn ð1; 2Þ ¼  ξ1 Y 0n ðξ1 Þ; Zn ð1; 3Þ ¼ nJ n ðξ2 Þ; Zn ð1; 4Þ ¼ nY n ðξ2 Þ; Zn ð1; 5Þ ¼ 0;

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 1 2 ξ2 ξ21 J n ðξ1 Þ  ξ21 J ″n ðξ1 Þ ; 2 

 1 2 ξ2 ξ21 Y n ðξ1 Þ ξ21 Y ″n ðξ1 Þ ; Wn ð1; 2Þ ¼ 2ρs 2 Wn ð1; 3Þ ¼ 2ρs n½  J n ðξ2 Þ þξ2 J 0n ðξ2 Þ;

Wn ð1; 1Þ ¼ 2ρs

Wn ð1; 4Þ ¼ 2ρs n½ Y n ðξ2 Þ þξ2 Y 0n ðξ2 Þ; Wn ð1; 5Þ ¼ 0;

Zn ð2; 1Þ ¼ Wn ð2; 1Þ ¼ 2n½J n ðξ1 Þ  ξ1 J 0n ðξ1 Þ; Zn ð2; 2Þ ¼ Wn ð2; 2Þ ¼ 2n½Y n ðξ1 Þ  ξ1 Y 0n ðξ1 Þ; Zn ð2; 3Þ ¼ Wn ð2; 3Þ ¼ n2 J n ðξ2 Þ  ξ2 J 0n ðξ2 Þ þ ξ22 J ″n ðξ2 Þ; Zn ð2; 4Þ ¼ Wn ð2; 4Þ ¼ n2 Y n ðξ2 Þ  ξ2 Y 0n ðξ2 Þ þ ξ22 Y ″n ðξ2 Þ; Zn ð2; 5Þ ¼ Wn ð2; 5Þ ¼ 0; 

 1 2 ζ 2  ζ 21 J n ðζ 1 Þ ζ 21 J ″n ðζ 1 Þ ; 2 

 1 2 ζ 2 ζ 21 Y n ðζ 1 Þ  ζ 21 Y ″n ðζ 1 Þ ; Zn ð3; 2Þ ¼ Wn ð3; 2Þ ¼ 2ρs 2 Zn ð3; 3Þ ¼ Wn ð3; 3Þ ¼ 2ρs n½  J n ðζ 2 Þ þ ζ 2 J 0n ðζ 2 Þ;

Zn ð3; 1Þ ¼ Wn ð3; 1Þ ¼ 2ρs

Zn ð3; 4Þ ¼ Wn ð3; 4Þ ¼ 2ρs n½ Y n ðζ 2 Þ þ ζ 2 Y 0n ðζ 2 Þ; Zn ð3; 5Þ ¼ Wn ð3; 5Þ ¼  ζ22 ρ2 J n ðζ 3 Þ; Zn ð4; 1Þ ¼ Wn ð4; 1Þ ¼ ζ 1 J 0n ðζ 1 Þ; Zn ð4; 2Þ ¼ Wn ð4; 2Þ ¼  ζ 1 Y 0n ðζ 1 Þ; Zn ð4; 3Þ ¼ Wn ð4; 3Þ ¼ nJ n ðζ 2 Þ; Zn ð4; 4Þ ¼ Wn ð4; 4Þ ¼ nY n ðζ 2 Þ; Zn ð4; 5Þ ¼ Wn ð4; 5Þ ¼ y3 J 0n ðζ 3 Þ; Zn ð5; 1Þ ¼ Wn ð5; 1Þ ¼ 2n½J n ðζ 1 Þ ζ 1 J 0n ðζ 1 Þ; Zn ð5; 2Þ ¼ Wn ð5; 2Þ ¼ 2n½Y n ðζ 1 Þ  ζ 1 Y 0n ðζ 1 Þ;

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