Waves in periodic structures with imperfections

PII: S 1359-8368(96)00025-X

ELSEVIER

Composites Part B 25B (1997) 143-152 © 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/97/$17.00

Waves in periodic structures with imperfections

H. Benaroya Department of Mechanical & Aerospace Engineering, Rutgers University, PO Box 909, Piscataway, NJ 08855-0909. USA (Received 3 January 1996; revised 3 March 1996) The study of wave propagation in structures and discrete or continuous media has a significant history which follows from the examination of the dynamics of atomic and molecular lattices. Relatively recently, some of these ideas have been transferred and applied to the dynamic behavior of engineered structures. In particular, structures with periodic and almost periodic topologies and material properties have been extensively studied and important conclusions drawn regarding their energy-transmission properties. The attraction to wave propagation models is due to the efficient nature of the analytical tools available to study how energies of different frequency content are propagated or filtered by the structure. Such properties of a structure are profoundly affected by any imperfections or 'near-periodicities'. It has been found that imperfections will have the effect of localizing energies about them, thus not allowing the development of normal modes of vibration as would be observed when assuming a perfect structure. It is envisioned that such understanding will permit the analyst to take advantage of localization effects to isolate locations experiencing loading. Additional applications possibly include the modeling of composite and layered structures and cracks. Also, one expects that structures with periodic boundary conditions will experience some sort of localization of energies in certain frequency ranges. Of particular interest there is the possibility that these ideas may apply to the concept of functionally graded materials and composites. © 1997 Elsevier Science Limited. All rights reserved (Keywords: periodic structures; imperfections; localization)

1 BACKGROUND AND INTRODUCTION The study of wave p r o p a g a t i o n in structures and discrete or continuous media has a significant history which emanates from the examination of the dynamics of atomic and molecular lattices 1. Relatively recently, some of these ideas have been transferred and applied to the dynamic behavior of engineered structures 2. In particular, structures with periodic and almost periodic topologies and material properties have been extensively studied and important conclusions drawn regarding their energy-transmission properties. The attraction to wave propagation models is due to the efficient nature of the analytical tools available to study how energies of different frequency content are propagated or filtered by the structure 3. Such properties of a structure are profoundly affected by any imperfections or 'near-periodicities'. O f particular interest here is to examine how significant such deviations from perfect periodicity affect engineering concerns. Also, the inverse problem of how to design imperfections leading to a desired macroscopic behavior is of great interest. Detailed reviews of the linear problem have been published, including, a m o n g others, by Li and Benaroya 4

and Mester and Benaroya 5. A special issue of the A S M E Applied Mechanics Reviews 6 on the subject has just been published.

2 P H Y S I C S OF T H E P R O B L E M F r o m Roseau 7 follows a concise introduction to a simple mechanical model of vibration in a 'perfect' lattice. Consider a system of identical masses m arrayed in a one dimensional configuration, linked by identical springs with stiffnesses k, and distance a apart. The equation of motion for any of these masses is m ~d2un = k(Un+l + un_l - 2Un) ' where u, represents the displacement of the nth mass from its equilibrium position, and the position of mass n is a the x-coordinate x~ -- na. It is a straightforward exercise to show that the m a x i m u m frequency of energy that such a system can support and propagate is O.;max 2 ( k / m ) 1/2. I f instead the lattice contains mass elements which alternate between mass ml and m2 then it will be found that there will be an allowed lower (acoustic) z

143

Waves in structures with imperfections: H. Benaroya frequency spectrum and an allowed upper (optical) frequency spectrum, and these spectra will be separated by a forbidden range of propagation frequencies. A variety of mass distributions can be considered and qualitatively similar characteristics of system behavior will be found. Details regarding perfect lattice dynamics are available 8. It becomes very difficult, however, to quantify systems of dimension greater than one. This is partially due to the limited understanding of the physics of these systems. Generalizing to the nonlinear domain, again for perfect lattices, it is discovered that the nonlinear lattice can transmit 'solitary' waves or solitons 9. As an example, for the above equation with a nonlinear potential, e.g., ¢(r) = krT/2 + kar3/3 l°, where ~ provides a measure of the nonlinearity, one finds a so-called recurrence phenomenon where, if the energy is not large and if the initial wave form is smooth enough, then almost all the initial energy will go back to the initial mode of vibration. While the study of exact simplified problems generally provides the analyst with insight of more complex problems, it turns out that for physically realistic problems, that is, problems where natural or engineered imperfections exist, modal behavior predicted for perfect lattices cannot exist. Rather, an imperfect lattice will 'localize' energy around imperfections. This is true for linear lattices n, and also true to some extent for nonlinear lattices. For the nonlinear imperfect case, however, it has been found that very special waveforms, solitary waves, may still propagate through the complete structure. All of this has been verified and discussed in the literature 1°, in particular the solid state physics literature, from where most of our understanding about linear lattice dynamics and 'Anderson localization' has been derived. Such interesting and complex behavior has significant and crucial implications on engineering analysis and design. For example, many structures are idealized as periodic: turbomachinery, space frames, ribbed structures such as aircraft, and layered and composite materials. It is then expected that the localization of energy will have a profound impact on the dynamic characteristics of these structures. In particular, localization of energy may be misinterpreted for material damping. Furthermore, many new materials exist for design in the nonlinear domain. The nonlinearities may appear for geometric as well as material reasons. It now appears that for such systems, one cannot casually extrapolate linear imperfect lattice behavior to the nonlinear range. Completely new mechanisms govern, and it will be important for engineers to understand this as they analyze and design complex systems. Therefore, perturbation methods cannot be arbitrarily used 12 and it is shown that, even for a linear lattice with imperfections, the shift or spread in the frequency content can be significant and not small. This important physical point has been emphasized by others as well. See, for example, Pierre 13. We would like to explore whether the existence of localization can be utilized in applications such as the analysis and design of functionally graded materials and

144

composites. As materials-related technologies become more sophisticated, one can envision a manufacturing process where each product is unique to its particular application. This implies the ability to tailor the material in quite subtle detail. First, some results are summarized and then the possibility of capitalizing on these phenomena is discussed.

3 DISCRETE L I N E A R I M P E R F E C T STRUCTURES One-dimensional systems are initially examined to reduce the complexity of the analysis so that the fundamental effects of the disorder can be more readily seen. To determine the effects of disorder, we first need to analyze the ideal case of the perfectly periodic structure, so that effects of disorder can be compared to the ideal. Wave propagation in the perfectly periodic one-dimensional chain of harmonic oscillators has been analyzed by Ghatak and Kothari 8, among others. The equation of motion for any mass, M, is Mi)p -- k [%+1 - 2Vp + Vp-1], where a is the spacing between masses and k is the stiffness between masses. The dispersion relation between w and ~; is derived to be /~a

w = w0 sin-~--, where w0 is the natural frequency of an oscillator and ~ is the wavenumber. The dispersion curve is well known. This curve indicates that waves with a frequency greater than w0 cannot propagate through the chain. It also shows that many waves with different wavelengths can propagate at the same frequency. For a given wave vector ~0, the waves that travel at the same frequency have wave vectors that are 2mr/a + x~o where n is an integer. Repetitiveness in structures can be used to reduce the amount of computation required to analyze them. However, it has been observed that slight deviations from a perfectly periodic pattern in weakly coupled structures causes localization which is characterized by an increased response amplitude in particular locations while other parts of the structure remain undisturbed. This paper provides an overview of key parameters and characteristics, in particular how imperfections affect mode shapes and overall response of a simple 'near'periodic structure. It is found that imperfections which increase the stiffness have a greater impact on the higher mode shapes than imperfections which decrease the stiffness. It is also noted that for a structure with randomly distributed parameters, the lower modes tend to localize at the bay with the lowest natural frequencies, while the higher modes tend to localize at bays with the highest natural frequencies. However, the overall response of a structure with a single imperfection to initial conditions is that the imperfect bay reflects the energy generated by the initial condition. In the case of a

Waves in structures with imperfections: H. Benaroya 'near'-periodic structure the response is highly localized in the bay at which the initial condition is not zero.

4 RESULTS The following results are only aflavor of many results by many workers. Here some recent results are summarized and highlighted.

4.1 Discrete model Figures 1-5 and Table 1 pertain to a discrete model. Figure I is a schematic of a 10-bay structure. Each bay is coupled to the adjacent one via stiffnesses k i and coupled to 'ground' via bay stiffnesses Ki. We define the following parameters:

Coupling stiffness ratio (CSR): which is equal to the ratio of the stiffness between bays to the bay stiffness

ki/Ki,

K 1

K S

K S

K 7

K

9

Stiffness imperfection ratio (SIR): which is equal to the ratio of the differences in bay stiffnesses to the nominal or average stiffness I(Kj - Ki)/Ki[, where Ki is taken to be the nominal. Figure 2 depicts modes 1, 3, 5, 7, and 10, for the case where the stiffness properties of each bay were selected randomly about a mean using a uniform probability distribution. The uniform assumption implies that we can reasonably estimate the upper and lower bounds of the stiffness, but cannot say which values in the range are more likely. Results are presented for the various CSR values, for SIR = 10%. Table 1 shows how the natural frequency of each bay changes significantly as the coupling stiffness ratio (CSR) changes by 1000 orders of magnitude, and due to the randomness in Ki between bays. Figure 3 demonstrates the time histories of each of the ten masses, for the case of a perfectly periodic structure subjected to the initial condition xl = 1. Figure 4 demonstrates the differences, and localization, for the case where bay 5 is 'imperfect', and Figure 5 for the case where properties are randomly distributed. In summary, this discrete model study shows that slight deviations from a perfectly ordered structure cause substantial changes in the structural modes shapes. Even the first mode can localize from the presence of most ('engineering') imperfections. As a result, the actual response of a structure may be quite different than that predicted by a periodic analysis method. Thus, in designing and/or controlling a structure, it is important to account for imperfections. Also it has been shown that

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145

Waves in structures with imperfections: H. Benaroya Table 1 Natural bay frequencies of the stochastic structure Natural frequency of bay no. CSR%

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9.918 3.057 0.990 0.294

10.187 3.300 1.047 0.318

9.446 3.345 1.058 0.331

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9.983 3.328 0.976 0.321

9.724 3.130 1.002 0.309

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146

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and last modes are more sensitive to imperfections, as they are in the deterministic case. The nth mode localizes in the bay which has the nth lowest frequency of all the bays. Additional results may be found in Refs 5 and 11. 4.2 Continuous model

Continuous rod models have also been studied and

may be viewed as more realistic for certain structures and applications. It is shown that for a strongly-coupled elastic system, a disorder will likely produce one or two additional localized modes in each stop-band, while multiple disorders lead to more localized modes in each stop-band in a more complex way. Higher mode frequencies are more sensitive to disorders and susceptible to being pushed into stop-bands and localized. Both the

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Monte Carlo and the perturbation method are applied to fully understand the effects of disorders on normal frequencies and normal modes of randomly-disordered periodic rods. By understanding the basic behavior of such a system, it is hoped ultimately that some insights can be achieved where closed-form results are not possible, especially for large-scale systems to be found in practice. Figures 6-12 refer to these studies. Figure 6 is a schematic of a typical regular and a disordered bay of a periodic semi-infinite rod. Note that the bay is asymmetric in material properties. Figure 7 shows the distributions of the natural frequencies of the semi-infinite periodic rod for the cases of the perfect rod and the rod with imperfections within 5 bays. One may note the increase in the

148

number of frequencies which are in the stop-bands of the imperfect structure. Figure 8 demonstrates this increase as a function of disorder size (standard deviation). Figures 9 and 10 plot the standard deviations of the frequencies of the disordered rod. The greatest standard deviations are in the stop-bands. Figure 11 provides the probability distributions of the number of normal modes in two stop-bands for a 20-bay clamped-clamped rod with randomly disordered properties. Finally, Figure 12 demonstrates how extended modes become localized when small disorder is introduced. Thus, our numerical studies show that a single boundary disorder will rarely produce additional localized modes. The location of the single disorder generally does

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not affect the number of additional localized modes as long as the disorder is not located too near the boundary region. We have found that increasing the disorder strength beyond certain values will generally not increase the number of localized modes in the stop-bands. Doing this most likely shifts one frequency lying in each passband into a nearby stop-band and simultaneously shifts one frequency lying in each stop-band into a nearby pass-band. However, only when the disorder strength

reaches a certain level can this kind of frequency-shift be great enough to push the frequency nearest a lower stopband into that lower stop-band. This implies that there is a threshold of the disorder strength beyond which one or two more localized modes are likely generated in some stop-bands, but the number of modes lying in stop-bands does not increase with the disorder strength beyond that point. Additional results are available in Li and Benaroya 15.

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Figure l l Probability distribution o f the number of normal modes in the first and tenth stop-bands for a clamped-clamped 20-bay randomly disordered periodic rod with 10% material parameter standard deviation

5 L O C A L I Z A T I O N AND F U N C T I O N A L L Y G R A D E D M A T E R I A L S AND COMPOSITES 200

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150

Recently, interest has developed in the materials community on the concept o f f u n c t i o n a l l y graded composites. For example, see Aboudi et al. 16 and Pindera et al. 17. This new class of materials is developed by tailoring its microstructure to meet specific applications. Such tailoring is accomplished by distributing reinforcement in a nonuniform manner so that at the macroscopic level the composite structure has customized mechanical properties. The effectiveness of such materials design depends on the ability to physically create such complex topologies as well as on the ability to predict their behavior. Here we only address the latter in a narrow sense. We look at the vibration-based discipline discussed above known as localization and note that the periodicity-breaking properties that are at its cause may be taken advantage of in a design procedure leading to a functionally graded composite. That the origin of the study of localization was with the examination of crystal lattice dynamics gives us some confidence that the application suggested here has merit. A glance at a fiber matrix cross-section of a composite structural element such as a plate will visually confirm the periodic nature of the unit cell comprised of a fiber plus surrounding region. It may be initially concluded that imperfections in this periodicity can lead to vibration localization, even in 'two-dimensional' structures such as plates. Given this, the question arises whether the designer can intentionally adjust spacings, symmetries, and material properties so that a specific localization can be designed into the material for a specific application. It was this sort of application that drew our interest to the study of localization in the first place. Initially, we needed to examine and understand the localization of energies. Then we saw this as a vehicle to the passive control of structures, in particular space structures where little structaral and almost no aerodynamic damping occurs.

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This paper was prepared with the intention of drawing the attention of the composites community to the behavior known as vibration localization. The actual implementation of localization properties for the purposes proposed here will require much effort, both analytical and experimental. Cross-fertilization between the two communities appears to be a fruitful activity in the further development of functionally-graded materials and composites. ACKNOWLEDGMENTS

The author gratefully acknowledges the support provided by the Federal Aviation Administration Technical

Center and the CMAS Center at Rutgers, and the invitation by Professor Pindera to prepare this manuscript. The thoughtful and thorough recommendations of one of the reviewers are appreciated and have been adopted. The results summarized herein are the product of the author's work with two graduate students who are now in the near-periodic world. REFERENCES 1 2

Hori, J. Spectral Properties of Disordered Chains and Lattices, Pergamon Press, 1968 von Flotow, A. Disturbance Propagation in Structural Networks; Control of Large Space Structures, Ph.D. Thesis, Stanford, 1984

1 51

Waves in structures with imperfections." H. Benaroya 3 4 5 6 7 8 9 10 11

152

Doyle, J.F. 'Wave Propagation in Structures', Springer-Verlag, 1989 Li, D. and Benaroya, H. Dynamics of periodic and near-periodic structures. Applied Mechanics Reviews 1992, 45, 447-459 Mester, S. and Benaroya, H. Periodic and near-periodic structures. Shock and Vibration Journal 1995, 2, 69-95 Benaroya, H. (Ed.) Localization and the Effects of Irregularities in Structures, Applied Mechanics Reviews 1996, 49, 55-135 Roseau, M. 'Vibrations in Mechanical Systems', Springer-Verlag, 1987 Ghatak, A.K. and Kothari, L.S. 'An Introduction to Lattice Dynamics', Addison-Wesley, 1972 Ablowitz, M.J. and Segur, H. 'Solitons and the Inverse Scattering Transform', SIAM, 1981 Toda, M. 'Theory of Nonlinear Lattices', Second Edition, Springer-Verlag, 1988 Ishii, K. Localization of eigenstates and transport phenomena in the one-dimensional disordered system. Progress of Theoretical Physics 1973, 53, 77-138

12 13 14 15 16 17

Montroll, E.W. and Potts, R.B. Effect of defects on lattice vibrations. Physical Review 1955, 1~, 525-543 Pierre, C. Mode localization and eigenvalue loci veering phenomena in disordered structures. Journal of Sound and Vibration 1988, 126, 485-502 Mester, S. and Benaroya, H. A parameter study of localization. Shock and Vibration Journal 1996, 3, 1-9 Li, D. and Benaroya, H. Waves, Normal Modes and Frequencies in Periodic and Near-Periodic Rods, Part I and Part II. Wave Motion 1994, 20, 315-338, 339-358 Aboudi, J., Arnold, S.M. and Pindera, M.-J. Response of functionally graded composites to thermal gradients. Composites Engineering 1994, 4, 1-18 Pindera, M.-J., Aboudi, J. and Arnold, S.M. Limitations of the uncoupled, RVE-based micromechanical approach in the analysis of functionally graded composites. Mechanics of Materials 1995, 20, 77-94