Chapter 2 Introductory Concepts of Wave Propagation Analysis in Structures

Transcription

1 Chater 2 Introductory Concets of Wave Proagation Analysis in Structures Wave roagation is a transient dynamic henomenon resulting from short duration loading. Such transient loadings have high frequency content. The main difference between the structural dynamics and wave roagation in structures arises due to high frequency excitations in the later case. Structures very often exerience such loadings in the forms of imact and blast loadings like gust, bird hit, tool dros etc. Aart from these, the wave roagation studies are also imortant to understand the dynamic characteristics of a structure at higher frequencies due to their various realworld alications. Structural health monitoring or detection of damage is one such imortant alication. As wave roagation deals with higher frequencies, diagnostic waves can be used to redict the resence of even minute defects, which occur at initiation of damage and roagate them till the failure of the structure. In many aircraft structures, the undesired vibration and noise transmit from the source to the other arts in form of wave roagation and this requires control or reduction, which is again an imortant alication of wave roagation studies. A structure, when subjected to dynamic loads, will exerience stresses of varying degree of severity deending uon the load magnitude and its duration. If the temoral variation of load is of large duration (of the order of seconds), the intensity of the load felt by the structure will usually be of lower severity and such roblems fall under the category of structural dynamics. For such roblems, there are two arameters which are of aramount imortance in the determination of its resonse, namely the natural frequency of the system and its normal modes (mode shaes). The total resonse of structure is obtained by the suerosition of first few normal modes. Large duration of the load makes it low on the frequency content, and hence the load will make only the first few modes to get excited. Hence, the structure could be idealized with fewer unknowns (which we call as degrees of freedom). However, when the duration of load is small (of the order of microseconds), the stress waves are set u, which starts roagating in the medium with certain velocity. Hence, the resonse is necessarily transient in nature and in the rocess, many normal modes will get excited. Hence, the model sizes will be many orders bigger than what is required for the structural dynamics roblem. Such roblems come under the category of wave roagation. S. Goalakrishnan and S. Narendar, Wave Proagation in Nanostructures, 19 NanoScience and Technology, DOI: / _2, Sringer International Publishing Switzerland 2013

2 20 2 Introductory Concets of Wave Proagation Analysis in Structures In Sect. 1.3, the need to study wave roagation in structures was highlighted. It is quite well-known that nanostructures can roagate waves of the order of terahertz. Proagation of waves in such structures can be studied using various modeling techniques highlighted in Cha. 3. For examle, modeling technique such as ab intio atomistic modeling or molecular dynamics methods can be emloyed to study wave roagation in such nanostructures. However, the main limitation of these methods is the comutational time that these methods will take. Some researchers have used finite element method couled with molecular mechanics model to solve such roblems. However, if the frequency content of the roblem is of the terahertz level, then for FE modeling, one needs a FE mesh comatible with its wavelength, which is extremal small at the terahertz frequency. These limitations can be overcomed to some extent by using Sectral Finite Element Method (SFEM), which is the FE method in the frequency domain. It uses the sectral analysis to understand the hysics of wave roagation, whose results will be directly used in the SFEM formulation. This chater mainly deals with the sectral analysis of motion, which will tell about the nature of waves that is roagating in the medium and the seed with which the waves are roagating. 2.1 Introduction to Wave Proagation The key factor in the wave roagation is the roagating velocity, level of attenuation of the resonse and its wavelengths. Hence, hase information of the resonse is one of the imortant arameters, which is of no concern in the structural dynamic roblems [1]. The wave roagation is a multi-modal henomenon, and hence the analysis becomes quite comlex when the roblem is solved in the time domain. This is because, the roblem by its nature is a high frequency content roblem. Hence, the analysis methods based on the frequency domain is normally referred for wave roagation roblems. That is, all the governing equations, boundary conditions, and the variables are transformed into the frequency domain using any of the integral transforms available. The most common transformation for transforming the roblem to the frequency domain is the Fourier Transforms. This transform has the discrete reresentation and hence amenable for numerical imlementation, which makes it very attractive for its usage in the wave roagation roblems. By transforming the roblem into frequency domain, the comlexity of the governing artial differential equation is reduced by removing the time variable out of icture, making the solution simler than the original equation. In wave roagation roblems, we are concerned about two arameters, namely the wavenumber and the seeds of the roagation (normally referred to as grou seeds). There are many tyes of waves that can be generated in structure. Wavenumber reveals the tye of waves that are generated. Hence, in wave roagation roblems, two imortant relations are very imortant, namely the sectrum relations, which is lot of the wavenumber with the frequency and the disersion relations, which is a

3 2.1 Introduction to Wave Proagation 21 lot of wave velocity with the frequency. These relations reveal the characteristics of different waves that are generated in a given structure. 2.2 Sectral Analysis Sectral analysis determines the local wave behavior for different waveguides and hence the wave characteristics, namely the sectrum and the disersion relation. These local characteristics are synthesized over large number of frequencies to get the global wave behavior. Sectral analysis uses discrete Fourier transform to reresent a field variable (say dislacement) as a finite series involving a set of coefficients, which requires to be determined based on the boundary conditions of the roblem [1 3]. Sectral analysis enables the determination of two imortant wave arameters, namely the wavenumbers and the grou seeds (discussed in next section). These arameters are not only required for sectral element formulation [1], but also to understand the wave mechanics in a given waveguide. These arameters enable us to know whether the wave mode is a roagating mode or a daming mode or a combination of these two (roagation as well as wave amlitude attenuation). If the wave is roagating, the wavenumber exression will let us know whether it is nondisersive (that is, the wave retains its shae as it roagates) or disersive (when the wave changes its shae as it roagates). More details on sectral analysis and alications to several roblems can be found in Refs. [1, 4, 5]. In the next subsection, we will define some of the commonly used wave roagation terminologies. 2.3 Wave Proagation Terminologies 1. Waveguide Any structural element is called a waveguide as it guides the wave in a articular manner. For examle, a bulk or a nano rod essentially suorts only the axial motion and hence it is called axial or longitudinal waveguide. In the case of a beam, only bending motion is ossible and hence the beam is called the flexural waveguide. In the case of shafts, the only ossible motion is the twist and hence they are called torsional waveguide. In case of laminated comosite beam, due to stiffness couling both axial and flexural motions are ossible. In general, if there are n highly couled governing artial differential equations, then such a waveguide can suort n different motions. 2. Wavenumber This is a frequency-deendent arameter that determines the following: Whether the wave is roagating or nonroagating or will roagate after certain frequency.

4 22 2 Introductory Concets of Wave Proagation Analysis in Structures It also determines the tye of wave, namely disersive or nondisersive wave. Nondisersive waves are those that retain its shae as it roagates, while the disersive waves are those that comletely change its shae as it roagates. That is if the wavenumber (k) is exressed as a linear function of frequency (ω) as k = c 1 ω,(c 1 is a constant) then the waves will be nondisersive in nature. Wavenumber in rods and in general for most second-order system, will be of this form and hence the waves will be nondisersive in nature. However, if the wave number is of the form k = c 2 ω n,(c 2 is a constant) the waves will essentially be disersive. Such a behavior can be seen in higher order systems such as beams and lates. In such cases, the wave seeds will change with the frequencies. The lot of wavenumber with the frequency is usually referred to as sectrum relations. 3. Phase seed These are the seeds of the individual articles that roagate in the structure. They are related to the wavenumber through the relation C = ω Real(k) (2.1) If the waves are nondisersive in nature (that is k = aω ), then the hase seeds are constant and indeendent of frequency. Conversely, if the hase seeds are constant, then such a system is nondisersive system. Phase seed is not associated with transfer of any hysical quantity (e.g., mass, momentum or energy) in a waveguide. 4. Grou seed Grou velocity is associated with the roagation of a grou of waves of similar frequency. During the roagation of waves, grous of articles, travel in bundles. The seeds of each of these bundle are called the grou seed of the wave. They are mathematically exressed as C g = ω k (2.2) Again here, for nondisersive system, the grou seeds are constant and indeendent of frequency. Hence, the time of arrival of all waves will be based on this frequency. The lot of hase/grou seeds with the frequency is called the disersion relations. This is a velocity of the energy transortation and it must be bounded. Monograh [1] derives the exression for the wavenumbers and grou seeds for some commonly used metallic and comosite waveguides in engineering. 5. Relation between grou and hase seeds Since from Eq. (2.1) wehave k = ω (2.3) C

5 2.3 Wave Proagation Terminologies 23 Substituting Eq. (2.1) into Eq. (2.2)gives [ ( )] [ ω 1 dω C g = dω d = dω ω dc ] 1 [ C C C 2 = C 2 C ω dc ] 1 dω (2.4) Using ω = 2π f, [ C g = C 2 C ( ft) dc ] 1 (2.5) d( ft) where ft denotes the frequency times thickness. When the derivative of C with resect to ft becomes zero, C g = C. As the derivative of C with resect to ftaroaches infinity (that is, cut-off frequency), C g aroaches zero. 6. Cut-off Frequency In some waveguides, some waves will start roagating only after certain frequency called the cut-off frequency. The wavenumber and grou seeds before this frequency will be imaginary and zero, resectively. In the next section, we will outline a rocedure to comute the cut-off frequency for the second- and fourth-order systems. 2.4 Sectrum and Disersion Relations Here, two imortant frequency-deendent wave characteristics, namely, sectrum and disersion relations, are obtained for a generalized system defined by the secondand fourth-order artial differential equations (PDE). These relations are the frequency variation of the wave arameters termed as wavenumbers and wave seeds, resectively. These arameters are essential to understand the wave mechanics in a given waveguide and are also required for SFEM formulation. These arameters rovide information like whether the wave mode is a roagating mode or a daming mode or a combination of these two (roagation as well as wave amlitude attenuation). Next, for a roagating mode, the nature of frequency variation of wavenumbers gives information whether the mode is nondisersive, i.e., the wave retains its shae as it roagates or disersive where the shae changes with roagation. In this section, these arameters are exlained using the examle of a generalized one-dimensional second- and fourth-order systems.

6 24 2 Introductory Concets of Wave Proagation Analysis in Structures Second-Order PDE The sectral analysis starts with the artial differential equation governing the waveguide. Considering a generalized second-order artial differential equation given by 2 u x 2 + q u x = r 2 u t 2 (2.6) where, q, and r are known constants deending on the material roerties and geometry of the waveguide. u(x, t) is the field variable to be solved for with x being the satial dimension and t the temoral dimension. First, u(x, t) is transformed to frequency domain using discrete Fourier transform (DFT) as u(x, t) = N 1 n=1 û n (x,ω n )e jω nt (2.7) where ω n is the discrete circular frequency in rad/sec and N is the total number of frequency oints used in the transformation. The ω n is related to the time window by ω n = nδω = nω f N = n NΔt = n (2.8) T where Δt is the time samling rate and ω f is the highest frequency catured by Δt. The frequency content of the load decides N and consideration of the wra around and aliasing roblem decides Δω. More details and associated roblems are given in Ref. [1]. Here, û n is the nth DFT coefficient and can also be referred to as the coefficient at frequency ω n. û n varies only with x. Substituting Eq. (2.7) into Eq. (2.6), we get d2 û n dx 2 + q dû n dx + rω2 nûn = 0, n = 0, 1,...N 1 (2.9) Thus, through DFT, the governing PDE given by Eq. (2.6) is reduced to N ODE. Equation (2.9) being constant coefficient ODEs, have a solution of the form û n (x) = A n e jk n x, where A n are the unknown constants, which will be comuted from the boundary values and k n is called the wavenumbers corresonding to the frequency ω n. Substituting the above solution into Eq. (2.9), we get the following characteristic equation to determine k n, ( ) kn 2 + jqk n + rωn 2 A n = 0 (2.10) The subscrit n is droed hereafter for simlified notations. The above equation is quadratic in k and has two roots corresonding to the incident and reflected waves. If the wavenumbers are real, then the wave is called roagating mode. On the

7 2.4 Sectrum and Disersion Relations 25 other hand, if the wavenumbers are urely imaginary, then the wave dams out as it roagates and hence is called evanescent mode. If wavenumbers are comlex having both the real and imaginary arts, then the wave with such a wavenumber will attenuate as they roagate. These waves are normally referred to as inhomogeneous waves. The set of the wavenumbers obtained by solving the characteristic Eq. (2.10) is given as k 1 = j q 2 + q rω2 k 2 = j q 2 q rω2 (2.11) Equation (2.11) is the generalized exression for the determination of the wavenumbers. Different wave behaviors are ossible deending uon the values of the radical rω 2 q As an examle, for a case with q = 0, the wavenumbers are given as r k 1 = ω r k 2 = ω (2.12) For such a case, the wavenumbers are real and hence the corresonding waves are roagating. When rω 2 / < q 2 /4 2, then the wavenumber is urely imaginary and the system will not allow any way to roagate. However, when rω 2 / > q 2 /4 2, the wavenumber will be comlex and in this case, the waves will attenuate as they roagate. Next, two other imortant wave arameters, namely, hase seed (C ) and grou seed (C g ), are briefly exlained. Let us consider the revious examle where wavenumbers vary linearly with frequency as given by Eq. (2.12). Corresondingly, the wave seeds are obtained as follows: C = ω k = r C g = dω dk = r (2.13) We find that both grou and hase seeds are constant and equal. Hence, when wavenumbers vary linearly with frequency ω, the wave retains its shae as it roagates. Such waves are called nondisersive waves. When wavenumbers have a nonlinear variation with frequency, the hase and grou seeds will not be constant but will be function of frequency. As a result, each frequency comonent will travel

8 26 2 Introductory Concets of Wave Proagation Analysis in Structures with different seeds and the wave shae will not be reserved with wave roagation. Such waves are called disersive waves. For nonzero values of, q and r, the exression for hase and grou seeds become C = Re C g = Re rω 2 ω rω 2 q2 4 2 rω q2 4 2 (2.14) Here Re reresents real art of the exression. Thus, it can be seen that the wave seeds C and C g are not the same and hence the waves are disersive in nature. The value of the radical, however, deends on frequency and there can be a frequency after which the wavenumbers transit from being urely imaginary to comlex or real wavenumbers resulting in roagation of the wave mode. This transition frequency is called the cut-off frequency (ω c ) and can be derived by equating the radical to zero. The exression for the cut-off frequency for this second-order system is given as ω c = q 2 r (2.15) Once the wavenumbers are determined the solution of the transformed ODEs given by Eq. (2.9) can be written as û(x,ω)= A 1 e jk 1x + A 2 e jk 2x (2.16) The unknown constants A 1 and A 2 can be evaluated in terms of the hysical boundary conditions of the one-dimensional waveguide. This can be done in a formal manner using the sectral finite element technique which will be exlained in details later. For q = 0 the above equation is of the form, û(x,ω)= A 1 e jkx + A 2 e jkx, r k = ω (2.17) where A 1 reresent the incident wave coefficient while A 2 stands for the reflected wave coefficient Fourth Order PDE Next, let us consider a fourth-order system and study its wave behavior. Consider the following governing PDE

9 2.4 Sectrum and Disersion Relations 27 4 w x 4 + qw + r 2 w t 2 = 0 (2.18) where w(x, t) is the field variable and, q, and r are arbitrary known constants deending on the material and geometric roerties of the waveguide as in the case of the second-order system. The above equation is similar to the equation of motion of a Euler Bernoulli beam on elastic foundation. The DFT of w(x, t) can be written in a similar form as Eq. (2.7), w(x, t) = N 1 n=1 ŵ n (x,ω n )e jω nt (2.19) Substituting Eq. (2.19) into the governing PDE given by Eq. (2.18) we get the reduced ODEs as d4 ŵ ( ) n dx 4 + q rωn 2 ŵ n = 0 (2.20) The above ODEs have constant coefficients and hence the solution will be of the form ŵ n (x) = A n e jk n x. Substituting this solution into Eq. (2.20) we get the characteristic equation for solution of the wavenumbers. Again the subscrit n is droed hereafter for simlified notations and all the following equations have to be derived for n varying from 0 to N-1. The characteristic equation is of the form ( r k 4 + q rω 2 = 0 or k 4 ω2 q ) = 0 (2.21) This is a fourth-order equation and will give two sets of wavenumbers. The tye of wave is deendent uon the numerical value of r ω2 q.for r ω2 > q, the solution of Eq. (2.21) will give the following wavenumbers, k 1 =+α, k 2 = α k 3 =+jα, k 4 = jα (2.22) ( ) where α = r 1/4. ω 2 q In the above equation, k1 and k 2 reresent the roagating wave modes while k 3 and k 4 are the daming or evanescent modes. From the above equations, we find that the wavenumbers are nonlinear functions of the frequency, and hence the corresonding waves are exected to be highly disersive in nature. Also, using the above exression we can find the hase and grou seeds for the roagating mode from Eq. (2.13). Next, consider the case when r ω2 < q. For such conditions, the wavenumbers are given by

10 28 2 Introductory Concets of Wave Proagation Analysis in Structures k 1 =+ 1 + j 2 α, k 2 = 1 + j 2 α k 3 =+ 1 + j α, k 4 = 1 + j α (2.23) 2 2 From the above equation, we see that the change of sign of r ω2 q has comletely changed the wave behavior. Now, all the wavenumbers have both real and imaginary arts. Hence, all the wave modes are roagating as well as attenuating. The initial evanescent mode also becomes a roagating mode after the cut-off frequency ω c. The exression for the cut-off frequency obtained by equating r ω2 q to zero is ω c = q (2.24) r Again, if q = 0, the cut-off frequency vanishes and the wave behavior is similar to the first case, i.e., it will have roagating and daming modes. In all cases, however, the waves will be highly disersive in nature. The solution of the fourth-order governing Eq. (2.20) can be written as ŵ(x,ω)= A 1 e jαx + B 1 e αx + A 2 e jαx + B 2 e αx (2.25) As in the revious case, A 1, B 1 are the incident wave coefficients and A 2, B 2 are the reflected wave coefficients. These unknown constants can be determined in terms of the hysical boundary conditions of the beam. From the above discussion, we see that the sectral analysis gives an insight into the wave mechanics of a system defined by its governing differential equation. Though sectral analysis can be done similarly using wavelet transform, it is not as straightforward as Fourier transform-based analysis. This is because the wavelet basis functions are bounded both in time and frequency unlike the basis for Fourier transform which is unbounded in time. This has been exlained in greater detail with alication to nanostructures in the other chaters. In the last section, the arameters, wavenumber, and wave seed were exlained with the examles of generalized second- and fourth-order artial differential wave equations. The wavenumbers k were obtained as a function of frequency by solving second- and fourth-order olynomial equations, resectively. The comutation of wavenumbers is, however, not so straightforward for structures with higher comlexities (esecially for the cases when the characteristic equation for solution of wavenumbers is of the order greater than 3). For one-dimensional structures such cases arise when the governing equation is a set of couled PDEs and a coule of such examles are Timoshenko beam and other higher order beams. In a Timoshenko beam, the governing equations consist of two couled PDEs with transverse and shear dislacements as the variables. Another common examle of structure having a set of couled PDEs is the governing equations for a comosite beam with asymmetric ly lay-u resulting in elastic couling. In addition to the different one-dimensional structures, comutation of wavenumbers for two-dimensional structures is also dif-

11 2.4 Sectrum and Disersion Relations 29 ficult rimarily, because the wavenumbers here are a function of both frequency and wavenumber in the other direction. In order to handle such roblems, generalized and comutationally imlementable methods have been roosed to calculate the wavenumbers and associated wave amlitude. The two different aroaches to solve the roblem are based on singular value decomosition (SVD) and olynomial eigenvalue roblem (PEP) methods. The methods are described briefly in refs. [1, 4, 5]. 2.5 Summary In this chater, a brief introduction to wave roagation is given. First, the different terminologies used in wave roagation are defined. This is followed by a brief descrition of wave characteristics for systems defined by second- and fourth-order PDEs. The wave behavior is described based on wavenumbers and grou seeds. Existence of cut-off frequencies in these two systems is also highlighted. The PDEs that govern the nano waveguides described by nonlocal elasticity (which is the focus of this book) are normally defined by either second- or fourthorder PDEs. However, their form is entirely different than what is described in this chater. The concets outlined in this chater will be very imortant and necessary for the reader to understand the wave roagation in different nano waveguides that are outlined in the later chaters of the book. References 1. S. Goalakrishnan, A. Chakraborty, D. Roy Mahaatra, Sectral Finite Element Method (Sringer, London, 2008) 2. J.F. Doyle, Wave roagation in structures (Sringer, New York, 1997) 3. A. Graff, Wave Motion in Elastic Solids (Dover Publications, New york, 1995) 4. S. Goalakrishnan, M. Mitra, Wavelet Methods for Dynamical Problems (Taylor and Francis Grou, LLC, Boca Raton, FL, 2010) 5. U. Lee, Sectral Finite Element Method, (Cambridge University Press, Cambridge, 2009)

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