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Journal of Sound and Vibration 458 (2019) 22e43

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Journal of Sound and Vibration

journal homepage: www.elsevier .com/locate/ jsvi

Co-existing complexity-induced traveling wave transmissionand vibration localization in Euler-Bernoulli beams

Xiangle Cheng a, Lawrence A. Bergman b, D. Michael McFarland a, b,Chin An Tan c, Alexander F. Vakakis d, Huancai Lu a, *

a Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, 18 Chaowang Road, Hangzhou,310014, Chinab Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 South Wright Street, Urbana, IL, 61801, USAc Department of Mechanical Engineering, Wayne State University, 5050 Anthony Wayne Drive, Detroit, MI, 48202, USAd Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1208 West Green Street, Urbana, IL,61801, USA

a r t i c l e i n f o

Article history:Received 23 October 2018Received in revised form 12 May 2019Accepted 1 June 2019Available online 7 June 2019

Keywords:Euler-Bernoulli beamTraveling wavesStanding wavesEvanescent wavesWave separationVibration localization

* Corresponding author.E-mail addresses: [email protected] (X. C

(C.A. Tan), [email protected] (A.F. Vakakis), huan

https://doi.org/10.1016/j.jsv.2019.06.0010022-460X/© 2019 Elsevier Ltd. All rights reserved.

a b s t r a c t

We analytically and numerically examine the dynamic behavior of an undamped, linear,uniform and homogeneous Euler-Bernoulli beam of finite length, partially supported in itsinterior by local, grounded, linear spring-dashpot pairs and subjected to a harmonicdisplacement at its pinned left boundary or at both its ends. The Euler-Bernoulli beam isknown to be dispersive and, thus, to exhibit a non-constant relationship between fre-quency and wave number. Local dissipation due to the interior support results in a non-classically damped system and, consequently, mode complexity. An analytical frame-work is developed to examine the coexistence of propagating and standing harmonicwaves in complementary regions of the beam for four distinct boundary conditions at theright end: pinned, fixed, free and linear elastic. We show that the system can be designedso that, for a particular input frequency and interior support location, nearly perfect spatialseparation of traveling and standing waves can be achieved; the imperfection is shown tobe caused by the non-oscillatory evanescent components in the solution. We furtherdemonstrate that vibration localization is achieved by satisfying necessary and sufficientwave separation conditions, which correspond to frequency- and position-dependentsupport stiffness and damping values, and that linear viscous damping in the interiorsupport, but not necessarily linear stiffness, is required to achieve the separation phe-nomenon and vibration localization.

© 2019 Elsevier Ltd. All rights reserved.

1. Introduction

Understandingwavemotion in structures is essential to the design of effective energy absorption strategies [1e5]. It is wellknown that complex modes of asynchronous motions between material points in a structure can arise due to the presence ofnon-classical damping in the distributed-parameter system [6,7]. Moreover, the non-classical damping enables the separa-tion of standing waves and traveling waves that is fundamental for the realization of one-way energy propagation.

heng), [email protected] (L.A. Bergman), [email protected] (D.M. McFarland), [email protected]@zjut.edu.cn (H. Lu).

mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.jsv.2019.06.001&domain=pdfwww.sciencedirect.com/science/journal/0022460Xwww.elsevier.com/locate/jsvihttps://doi.org/10.1016/j.jsv.2019.06.001https://doi.org/10.1016/j.jsv.2019.06.001https://doi.org/10.1016/j.jsv.2019.06.001

Nomenclature

A cross-sectional area of beam~A;A dimensional excitation amplitude, dimensionless excitation amplitude, A ¼ ~A=Lc temporal scaling factor, c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEIzÞ=ðrAL4Þ

pCt1;ct1 translational damping at ~x ¼ ~xa or ~x ¼ ~x1, ct1 ¼ ðCt1cL3Þ=ðEIzÞCt2;ct2 translational damping at ~x ¼ ~x2, ct2 ¼ ðCt2cL3Þ=ðEIzÞDinði ¼ 1;2;3; n ¼ 1;2;3;4Þ complex coefficients of solutions in the subdomainsE Young's modulus of beamEin averaged energy transfer into the systemEdiss averaged energy dissipation by the local dampersf ;g; f1;g1; f2;g2 constituent functions in the denominators of Dinfin;gin; f1;in;g1;in; f2;in;g2;in ði¼ 1;2;3; n¼ 1;2;3;4Þ constituent functions in the numerator of DinKt1;kt1 translational spring at ~x ¼ ~xa or ~x ¼ ~x1, kt1 ¼ ðKt1L3Þ=ðEIzÞKt2;kt2 translational spring at ~x ¼ ~x2 or ~x ¼ L, kt2 ¼ ðKt2L3Þ=ðEIzÞKr2;kr2 rotational spring at ~x ¼ L, kr2 ¼ ðKr2L2Þ=ðEIzÞbkt1; bkt2 complex stiffness, bkt1 ¼ kt1 þ juct1; bkt2 ¼ kt2 þ juct2L length of beamIz area moment of inertiat; t dimensional time, dimensionless time, t ¼ ctx dimensionless axial coordinate along the centroidal axis of beam at equilibriumxa; x1; x2 dimensionless position of the translational viscoelastic support, xa ¼ ~xa=L; x1 ¼ ~x1=L; x2 ¼ ~x2= Lvð$Þ; Vð$Þ deflection of beam, amplitude of the deflectionU; u dimensional excitation frequency, dimensionless excitation frequency, u ¼ U=cg wavenumberf spatial phase difference

k function in the denominators of k*t1 and c*t1

4;j functions in the denominators of D*in ði¼ 1;2; n¼ 1;2;3;4Þ for the pinned-clamped and pinned-free beamrEk ;rEp kinetic energy density, potential energy density

Reð$Þ real part of a quantityð$Þ* symbol denoting the necessary conditions for wave separation and vibration localizationfð$Þ symbol denoting certain dimensional functions and variablesc:c: complex conjugatek$k2 Euclidean norm

X. Cheng et al. / Journal of Sound and Vibration 458 (2019) 22e43 23

Hull [8] investigated the longitudinal vibration of an undamped linear elastic bar, fixed at one end and attached to aviscous damper at the other end. He showed that when the boundary damping is tuned in accordance with a specificrelationship to the wave propagation speed, complete passive energy absorption at that end is achieved; i.e., waves are notreflected at that boundary through the mechanism of wave cancellation, and the boundary damping eliminates all modes ofvibration. Blanchard et al. [7,9] and Cheng et al. [10] investigatedmode complexity andwave separation in non-dispersive anddispersive strings attached to ground in their interiors through local, linear spring-damper supports. It was shown that thereexists a unique combination of system parameters for which local damping balances the system, allowing just sufficientenergy input via harmonic motion of the left support to pass the interior support to the opposite (right) boundary, where itreflects, establishing a standing wave to the right of the interior support but not allowing energy to pass back through to thesource. Xiao et al. [11,12] demonstrated similar concepts in acoustics by investigating the sound propagation in a rigid-walledcircular duct with a harmonic pressure source at one end, a rigid, zero-velocity boundary condition at the other end, and thedamped interior support represented by a non-resonant side branch containing a layer of porous damping material and arigid cap.

The works cited above are concerned primarily with non-dispersive structures. For dispersive systems, such as the Euler-Bernoulli beam, both propagating and evanescent waves are present. Extension of previous studies to dispersive beamsystemswill not only cause the analysis to be more complicated, but also clarify whether conditions for ideal wave separationcan be realized as in the previous non-dispersive examples. Moreover, to the authors’ knowledge, the interactions of prop-agating and evanescent waves under these conditions have not been thoroughly studied.

The objectives of this paper are to examine complex motions induced by non-classical damping in Euler-Bernoulli beams,and to understand complexity-induced transmission of traveling waves and vibration localization obtained by parameter

X. Cheng et al. / Journal of Sound and Vibration 458 (2019) 22e4324

tuning. The beam is excited harmonically at one end and is attached to ground through a local linear spring-dashpot at anintermediate point. Four boundary conditions at the opposite end (pinned, clamped, free, and linear elastic) are considered inorder to assess the effects of boundary conditions on spatial wave separation and vibration mode localization. An additionalexample of the pinned-pinned beam, which is subject to synchronous support motions at both ends andwith two viscoelasticsupports in its interior, is shown to verify that the vibration confinement strategy remains valid for the undamped Euler-Bernoulli beam in the presence of multiple spring-damper supports.

It is noted that complex modes in damped systems can also arise from gyroscopic, aerodynamic, and nonlinear effects[13,14]. Numerical discretizationmethods are generally applied to solve those problems. For example, Krenk [15,16] employeda state-space formulation, which required solving a complex, symmetric eigenvalue problem, to examine optimal dampingand tuning of complex damped modes of taut cables and beams. For damped discrete structures often used for typicalbuilding models [17], an interpolation scheme was developed to find the free vibration solution of the damped system,approximated from the undamped and rigidly constrained eigensolutions. For the problem of a cantilever beam supported bylocal dampers [18], root locus and transfer function approaches were applied to predict the attainable damping ratio asso-ciated with the relative frequency shift of the structure with and without local dampers.

Damping-induced complex modes in structures have many important applications, especially in the design of passivesupplemental energy dissipation devices [19,20]. These are well documented in the literature, with various analytical andnumerical methods applied to solve the basic problem of beams with locally attached dampers [21e26]. Recently, non-reciprocal wave scattering in string-coupled oscillator systems and acoustic wave guides was investigated [27,28], whichmay have some application to studies of the human auditory system. It appears that the human basilar membrane of thecochlea works as a viscous damping element to passively receive the incident sound [29,30]. Wave absorption via the basilarmembrane in the ear canal [31,32] is thus conceptually similar to generation of traveling waves in structures.

The remainder of this paper is organized as follows: Section 2 formulates the general boundary value problem, focusing onthe pinned-pinned beamwith harmonic base motion applied at the left support, and connected to ground at an interior pointthrough a linear spring-dashpot. The other cases studied merely require a change in the boundary conditions at the rightsupport or a modification of the equilibrium conditions at the attachment location, since the solution procedure remains thesame. In Section 3, necessary and sufficient conditions for vibration localization through separation of the traveling wavesfrom the standing waves are established for the pinned-pinned case. A discussion of complexity-induced vibration locali-zation is also presented. Section 4 illustrates the major simulation results for the remaining three boundary casesdclamped,free, and linear elasticdwith pertinent mathematical results summarized in the Appendix. The necessary formulations andsimulation results for the beamwith two viscoelastic supports are presented as well. Conclusions from this study are given inSection 5.

2. Problem formulation

Consider an undamped Euler-Bernoulli beam with ~x denoting axial position along the centroidal axis at equilibrium. Thebeam is of length L, Young's modulus E, cross-sectional area A, area moment of inertia Iz, and mass density r. Four cases,representing the four different boundary conditions at ~x ¼ L, are studied in this paper. In each case, the beam is pinned andgiven a harmonic displacement at its left end, ~x ¼ 0. A Voigt element, consisting of a translational linear spring Kt1 and aviscous damper Ct1 in parallel, connects the beam to ground at an interior point ~xa. At its right end, ~x ¼ L, the beam can bepinned, fixed, free, or connected to ground through linear translational and rotational springs, Kt2 and Kr2, respectively.

The transverse response of the beam is represented by the displacement, ~vð~x;tÞ, and the harmonic excitation applied to theboundary ~x ¼ 0 is ~vð0; tÞ ¼ ~AejUt , where ~A is the amplitude and U is the forcing frequency.

The non-dimensional parameters are defined as

x ¼ ~xL; v ¼ ~v

L; A ¼

~AL; c ¼

ffiffiffiffiffiffiffiffiffiffiEIzrAL4

s; t ¼ ct; u ¼ U

c;

kt1 ¼Kt1L

3

EIz; ct1 ¼

Ct1cL3

EIz; kt2 ¼

Kt2L3

EIz; kr2 ¼

Kr2L2

EIz;

(1)

where c is a temporal scaling factor with units sec�1. With this scaling, the coefficients of the governing differential equationsare unity.

Employing the comma convention to represent partial differentiation, the normalized equations of motion of the systemare

v1;xxxxðx; tÞ þ v1;ttðx; tÞ ¼ 0; 0 � x � x�av2;xxxxðx; tÞ þ v2;ttðx; tÞ ¼ 0; xþa � x � 1; t�0; (2)

and the initial conditions are assumed to be homogeneous such that vðx;0Þ ¼ v;tðx;0Þ ¼ 0. In what follows, the steady-statesolutions corresponding to each of the three homogeneous boundary conditions (pinned, clamped, free), as well as the linear


elastic boundary condition at x ¼ 1, are determined. The results for the pinned problemwill first be derived with expositionof both the boundary value problem and steady-state solution. The formulations for each of the remaining three cases (fixed,free and linear elastic), which differ from the pinned example only by the boundary conditions at x ¼ 1, are summarized inthe Appendix.

The first case of interest, the pinned-pinned beam in normalized coordinates, is shown in Fig. 1. The boundary conditionsare given by

v1ð0; tÞ¼Aejut; v1;xxð0; tÞ ¼ 0; v2ð1; tÞ ¼ 0; v2;xxð1; tÞ ¼ 0; (3)

and the continuity and equilibrium conditions are

v1�x�a ; t

� ¼ v2�xþa ; t�; v1;x�x�a ; t� ¼ v2;x�xþa ; t�;v1;xx

�x�a ; t

� ¼ v2;xx�xþa ; t�; v2;xxx�xþa ; t�� v1;xxx�x�a ; t� ¼ ��kt1v1�x�a ; t�þ ct1v1;t�x�a ; t��: (4)

Assuming a steady-state solution of the form

viðx; tÞ¼ViðxÞejut; i ¼ 1;2; (5)

and substituting Eq. (5) into Eqs. (2)e(4) results in the boundary value problem

V0000i ðxÞ � g4ViðxÞ ¼ 0; i ¼ 1;2;

V1ð0Þ ¼ A; V001ð0Þ ¼ 0; V2ð1Þ ¼ 0; V

002ð1Þ ¼ 0;

V1�x�a

� ¼ V2�xþa �; V 01�x�a � ¼ V 02�xþa �; V 001�x�a � ¼ V 002�xþa �; V 0002�xþa �� V 0001�x�a � ¼ �ðkt1 þ juct1ÞV1�x�a �; (6)

where g4 ¼ u2, and prime denotes ordinary differentiationwith respect to x. The steady-state amplitude of the displacementin each subdomain of the beam can then be written as

V1ðxÞ ¼ D11ejgx þ D12e�jgx þ D13egx þ D14e�gx; 0 � x � x�a ;V2ðxÞ ¼ D21ejgð1�xÞ þ D22e�jgð1�xÞ þ D23egð1�xÞ þ D24e�gð1�xÞ; xþa � x � 1:

(7)

The leading two terms in each expression in Eq. (7) represent the harmonic or far-field components while the remainingtwo terms are the evanescent or near-field components in the two subdomain solutions.

Substitution of Eq. (7) into the boundary, continuity and equilibrium conditions of Eq. (6) results in the solutions for theeight complex coefficients, written in the compact form

Fig. 1. Schematic for the pinned case in normalized coordinates. Euler-Bernoulli beamwith a pinned support at x ¼ 0 subject to an applied displacement vð0;tÞ ¼Aejut , an interior viscoelastic support at x ¼ xaconnecting the beam to ground, and a pinned support at x ¼ 1.


D1n ¼bkt1f1nðu;g; xaÞ � g1nðu;g; xaÞbkt1f ðu;g; xaÞ � gðu;g; xaÞ ;

D2n ¼bkt1f2nðu;g; xaÞ � g2nðu;g; xaÞbkt1f ðu;g; xaÞ � gðu;g; xaÞ

n ¼ 1; :::;4; (8)

where bkt1 ¼ kt1 þ juct1. The complex functions f1n; f2n; f and g1n; g2n; g ðn¼ 1; :::; 4Þ are given below. Focusing on the left

subdomain, 0� x � x�a , the component functions in Eq. (8) are found to be

f11 ¼ Ae�ð1þjÞgnð1� jÞ � ð1þ jÞe2g þ eð1þjÞgð2�xaÞ þ eð1þjÞgxa þ e2gð1þj�jxaÞ

�eg½2�ð1�jÞxa� þ je2gð1�xaÞ � e2jgð1�xaÞ þ je2gxa � eg½2jþð1�jÞxa�o;

g11 ¼ �8jAe�jgg3 sinhðgÞ;f12 ¼ Aeð1þjÞg

n� ð1� jÞ þ ð1þ jÞe�2g � e�ð1þjÞgð2�xaÞ � e�ð1þjÞgxa � e�2jgð1�j�xaÞ

þe�g½2�ð1�jÞxa� � je�2gð1�xaÞ þ e�2jgð1�xaÞ � je�2gxa þ e�g½2jþð1�jÞxa �o;

g12 ¼ 8jAejgg3 sinhðgÞ;f13 ¼ Ae�ð1þjÞg

nð1� jÞ þ ð1þ jÞe2jg � jeð1þjÞgð2�xaÞ � jeð1þjÞgxa � je2gð1þj�xaÞ

þjeg½2�ð1�jÞxa� þ je2gð1�xaÞ � e2jgð1�xaÞ � e2jgxa þ jeg½2jþð1�jÞxa�o;

g13 ¼ 8Ae�gg3 sinðgÞ;f14 ¼ Aeð1þjÞg

n� ð1� jÞ � ð1þ jÞe�2jg þ je�ð1þjÞgð2�xaÞ þ je�ð1þjÞgxa þ je�2gð1þj�xaÞ

�je�g½2�ð1�jÞxa� � je�2gð1�xaÞ þ e�2jgð1�xaÞ � e�2jgxa � je�g½2jþð1�jÞxa�o;

g14 ¼ �8Aegg3 sinðgÞ;

(9)

and the functions in the right subdomain, xþa � x � 1,

f21 ¼ 4jA½sinðgxaÞ þ sinhðgxaÞ�sinh½gð1� xaÞ�;g21 ¼ 8jAg3 sinhðgÞ;f22 ¼ �4jA½sinðgxaÞ þ sinhðgxaÞ�sinh½gð1� xaÞ�;g22 ¼ �8jAg3 sinhðgÞ;f23 ¼ 4A½sinðgxaÞ þ sinhðgxaÞ�sin½gð1� xaÞ�;g23 ¼ �8Ag3 sinðgÞ;f24 ¼ �4A½sinðgxaÞ þ sinhðgxaÞ�sin½gð1� xaÞ�;g24 ¼ 8Ag3 sinðgÞ:

(10)

The functions f and g in the denominator are

f ¼ 16�sinðgÞcoshðgÞsinh2ðgxaÞ þ sinhðgÞfsinðgxaÞsin½gð1� xaÞ�

�sinðgÞsinhðgxaÞcoshðgxaÞgÞ;g ¼ �32g3 sinðgÞsinhðgÞ:

(11)

3. Vibration localization in the pinned-pinned beam resulting from complexity induced by local damping at theinterior support

A solution to the boundary value problem defined previously is sought such that the vibration is localized to the segmentof the beam to the right of the viscoelastic interior support; i.e., xþa � x � 1. This ideally requires that energy input via aharmonic displacement at x ¼ 0 be transmitted from left to right by a traveling wave, pass through the interior support at x ¼xa without reflection, and continue on to the pinned support at x ¼ 1, where it is perfectly reflected back toward the interiorsupport with no energy passing through to the source. To achieve ideal vibration localization at steady state, the right-movingand left-movingwaves at the right of the interior support must be at the samewavenumber g, such that interference betweenthem creates a standing wave, with excess energy dissipated at the interior support. This ideal behavior can be observed in,for example, non-dispersive systems such as strings and rigid circular acoustic ducts [7,9,11,12]. However, the Euler-Bernoullibeam is a dispersive system, exhibiting non-oscillatory, rapidly decaying evanescent waves confined to regions near thesupports. Thus, perfect localization is generally not possible in such systems, and the regions of the beam where theevanescent modes contribute exhibit boundary layers of complexity which locally distort the trivial phase relations in thestanding wave.


Imposing the boundary conditions of the prescribed displacement and zero bending moment at x ¼ 0 leads to the har-monic and evanescent conditions, respectively

D11 þD12 ¼A2; D13 þ D14 ¼

A2: (12)

�
Then, a purely right-moving traveling wave in the subdomain 0� x � xa can be achieved by setting the complex coef-ficient corresponding to the left-moving harmonic wave in that subdomain to zero; i.e., D11 ¼ 0. Combining this requirementwith the two conditions of Eq. (12) gives
D12 þD13 þ D14 ¼ A; (13)

which is a necessary condition for localization for all the boundary cases to be examined.

Employing the formulae in Eq. (8) and solving the resulting equation yields a set of specific values of bkt1, denoted by bk*t1,that represent the solutions to the complex equation

bk*t1 b k*t1 þ juc*t1 ¼ g11ðu;g; xaÞf11ðu;g; xaÞ : (14)

This can be reduced to two real equations for the interior linear support stiffness and damping values that localize vi-

bration at specific values of frequency u and interior support location xa. The explicit expressions for k*t1 and c

*t1 are given by

k*t1 ¼�2g3k

ðsinhð2gxaÞ � sin½2gð1� xaÞ� � sin½gð2� xaÞ�sinhð2gÞsinhðgxaÞ

þsinðgxaÞsinhð2gÞsinhðgxaÞ þ 2 sinhð2gÞsinh2ðgxaÞþcoshð2gÞfsin½2gð1� xaÞ� � sinhð2gxaÞgþ2 coshðgxaÞsinh2ðgÞfsin½gð2� xaÞ� � sinðgxaÞg

�;

(15)

* 8g sinhðgÞsin½gð1� xaÞ�
ct1 ¼ k fsinhðgÞsin½gð1� xaÞ� þ sinh½gð1� xaÞ�sinðgÞg; (16)
where

k ¼ d21 þ d22¼ ðsinhðgÞsin½2gð1� xaÞ� � fsinðgxaÞ þ 2 sinhðgxaÞ � sin½gð2� xaÞ�gsinh½gð1� xaÞ�Þ2

þ�2 sinhðgÞsin2½gð1� xaÞ� þ fcosðgxaÞ � cos½gð2� xaÞ�gsinh½gð1� xaÞ�

�2:

(17)

Critical values of the stiffness and damping of the interior viscoelastic support leading to spatial separation of traveling andstanding waves to achieve vibration localization to the right of the interior support are plotted in Fig. 2. In Fig. 2(a), thenecessary values of support stiffness and damping are plotted versus frequency with the location of the interior support fixedat xa ¼ 0:4, while in Fig. 2(b) the corresponding values are shown versus support location xa for a fixed frequency u ¼ 100p.It is noted that, in both figures, interior support stiffness and damping values can go to zero at irregularly spaced points.However, this is not acceptable for damping, as complexity due to local damping in the interior support is required to achievethe spatial wave separation phenomenon leading to vibration localization. These points must, in fact, be excluded to guar-antee that the given conditions are both necessary and sufficient. Since c*t1 must be strictly non-negative, and the function k in

Eq. (17) can be shown to be finite since it is continuous and defined in a compact metric space [33], the zeros of c*t1 must resultfrom one or both of the bracketed terms in the numerator of Eq. (16). The first term gives sin½gð1 � xaÞ� ¼ 0, the nontrivial

solutions of which are gr ¼ r p1�xa; r ¼ 1; 2;::::, and the corresponding frequencies are ur ¼ r2

p1�xa

2; r ¼ 1; 2;::::. The second

term is given by

sinhðgÞsin½gð1� xaÞ�þ sinh½gð1� xaÞ�sinðgÞ ¼ 0; (18)

which must be solved numerically. For the parameters employed to obtain Fig. 2(a), the first equation yields the set of fre-quencies ur¼ 27.4156, 109.6623, 246.7401, 438.6491, /, while the second gives uc¼ 25.2319, 110.1228, 246.7400, 438.6352,/, verifying the values in the figure. Note that the trivial frequencies (ur¼uc¼ 0), excluded from the aforementioned so-lutions, can be used to interpret the static deflection of the beam under a constant point load. For the parameters used toobtain Fig. 2(b), the first equation yields the set of positions xar¼ 0.1138, 0.2910, 0.4683, 0.6455, 0.8228, /, while the second

Fig. 2. Depiction of parameter values k*t1 and c*t1 for wave separation in the pinned-pinned beam of Fig. 1: (a) separation conditions versus excitation frequency u

for attachment location xa ¼ 0:4; (b) separation conditions versus attachment location xa for excitation frequency u ¼ 100p.


gives xac¼ 0.1199, 0.2907, 0.4683, 0.6455, 0.8228,/, verifying the values in the figure. The zeros (roots of c*t1 ¼ 0) indicate theboundaries of the regions in which spatial wave separation cannot be achieved. These regions are closely spaced but do notcoincide even with very fine numerical resolution in the root computations. Note that the two sets of frequencies ur and ucincrease alternately at a fixed attachment location xa ¼ 0:4, and the two sets of positions xarand xac increase alternately at afixed excitation frequency u ¼ 100p. When the viscoelastic support moves close to the pinned right end, satisfaction of Eqs.(15) e (17) requires the spring stiffness k*t1 to be negative; thus, spatial wave separation in that region is not physicallyrealizable, as indicated in Fig. 2(b). Since strong constraint forces arising from the boundaries are imposed on the passiveattachment, “dead zones” occur at the beam ends where a support cannot be placed and induce separation.

Imposing the pinned right boundary condition at x ¼ 1 leads to the relations.

D*21 ¼ � D*22; D*23 ¼ �D*24: (19)

The steady-state solution can then be written as

v1ðx; tÞ ¼�A2e�jgx þ A

2e�gx þ 2D*13 sinhðgxÞ

�ejut; 0 � x � x�a ;

v2ðx; tÞ ¼ 2

jD*21 sin½gð1� xÞ� þ D*23 sinh½gð1� xÞ�

�ejut; xþa � x � 1;

(20)

where

D*13 ¼ �A4sinh½gð1� xaÞ�e�jg þ e�g sin½gð1� xaÞ�

sinhðgÞsin½gð1� xaÞ� ;

D*21 ¼A4j

e�jgxa

sin½gð1� xaÞ�; D*23 ¼

A4sin½gð1� xaÞ� � sinhðgxaÞe�jg

sinhðgÞsin½gð1� xaÞ� :(21)

The function sin½gð1�xaÞ� appearing in the numerator of c*t1 also appears in the denominator of the coefficients D*13 andD*23. Thus, the magnitudes of D

*13 and D

*23 go to infinity at the aforementioned set of frequencies ur , as shown in Fig. 3. This

explains the mechanism by which the beam becomes resonant when the interior support damping becomes zero. Since k*t1and c*t1 cannot approach zero simultaneously, the frequencies u ¼ ur are associated with the natural frequencies of a pinned-pinned beam with an interior elastic support of stiffness k*t1ður; xaÞ. Interestingly, the frequency equation sin½gð1�xaÞ� ¼ 0corresponds to that of a pinned-pinned beam of length ð1 � xaÞ, which is conceptualized as a beam of unit length with a verystiff elastic support at x ¼ xa, which is consistent with Fig. 2. Note that D*13 and D*23 tend to zero with increasing g, ensuringthat v1ðx; tÞ is dominated by the outgoing traveling wave and v2ðx; tÞ by the standingwave. This is demonstrated by the results

Fig. 3. Plots of the magnitudes of the coefficients D*13 and D*23 versus the excitation frequency u for wave separation in the pinned-pinned beamwith attachment

location xa ¼ 0:4 andA ¼ 1.

Fig. 4. Steady-state response of the pinned-pinned beam with an interior linear viscoelastic support: (a) displacements; (b) spatial phase of the displacementsnormalized by p; (c) spatio-temporal evolution of the kinetic energy density; (d) spatio-temporal evolution of the potential energy density. Parameters areu¼ 100p; xa ¼ 0:4; A ¼ 1; k*t1 ¼ 2377:43 and c*t1 ¼ 39:00. The vertical dashed lines identify the location of the interior support.


of a simulated case, shown in Fig. 4, with parameters u¼ 100p; xa ¼ 0:4; A ¼ 1; k*t1 ¼ 2377:43 and c*t1 ¼ 39:00. To under-stand energy propagation associated with the wave separation dynamics, and adopting the previously defined normaliza-tions, the kinetic and potential energy densities are defined, respectively, as


rEk ¼12Re½vtðx; tÞ�2 ; rEp ¼

12Re½vxxðx; tÞ�2; (22)

where Re½$� represents the real part of the quantity in the square brackets.Examination of the displacement, phase and energy density evolution plots, in Fig. 4(a) e 4(d), reveals spatial separation,

with the traveling wave to the left and standing wave to the right of the interior support. The traveling wave is characterizedby nearly linear variation of phase, with mild distortions near the source and interior support, while the standing wave ischaracterized by trivial phase differences everywhere except the neighborhood of the interior support. These variations aredue to boundary layers of complexity induced by the evanescent components of the solution arising from dispersion. This canbe further demonstrated by decomposing the harmonic and evanescent components of the response, as shown in Fig. 5.

Figures 5(a), (b) and (e) do, in fact, confirm that perfect separation of traveling and standing waves and complete local-ization of vibration can be achieved when considering only the harmonic component, while Figs. 5(c), (d) and (f) define

Fig. 5. Decomposition of the harmonic and evanescent parts of the responses in Fig. 4(a) e 4(c): (a) harmonic component of displacements; (b) spatial phase ofthe harmonic component normalized by p; (c) evanescent component of displacements; (d) spatial phase of the evanescent component normalized by p; (e)harmonic component of kinetic energy density; (f) evanescent component of the kinetic energy density. The vertical dashed lines identify the location of theinterior linear viscoelastic support.


precisely the deviations from the ideal that appear as boundary layers of complexity at the source and interior support in thetotal solution of Fig. 4.

From Fig. 2, it is apparent that either the interior stiffness or damping can be zero with the variation of external excitationfrequency u and attachment location xa. The specific regions, corresponding to zero damping, should be interpreted asinsufficient conditions for realizing wave separation. However, for regions of zero stiffness, the dashpot alone is sufficient togenerate the separation phenomenon, which is well known since pure damping-induced traveling waves have been observedin the nondispersive string and bar [7,8] wherein the local attachment is placed at an otherwise free boundary. At u ¼ 100pand xa ¼ 0:60, the simulation results with pure damping c*t1 ¼ 70:51 are given in Fig. 6(a) and Fig. 6(b), fromwhich we findthat wave separation is still attainable. When the support is moved to xa ¼ 0:65, Eq. (14) requires the system to be undampedwith k*t1 ¼ 22273:40, which results in a global resonant mode with trivial spatial phase distributution, as seen from Fig. 6(c)and (d).

4. Simulation results for additional examples

The necessary and sufficient conditions for spatial separation of traveling waves and standing waves, and the specificvibration localization dynamics, in a pinned-pinned Euler-Bernoulli beam connected to ground through a local linear spring-damper have been studied in detail in Sections 2 and 3. In this section, we extend the discussion to Euler-Bernoulli beams forthree additional boundary conditions at the right end: clamped, free, and linear elastic, to understand the effects of non-dissipative boundary conditions on the vibration localization. However, with no need to duplicate the mathematical deri-vations step by step, the main formulations and equations are summarized in the Appendix, illustrating the basic differencesfrom the pinned-pinned problem. In fact, the dynamics in terms of wave separation as well as vibration localization areindependent of the boundary conditions to be considered. The key simulation results are shown and discussed in thefollowing subsections.

Fig. 6. Steady-state response of the pinned-pinned beam with an interior viscoelastic support at u ¼ 100p and A ¼ 1: (a) displacements and (b) spatial phase ofthe displacements normalized by p with xa ¼ 0:60; k*t1 ¼ 0:00 and c*t1 ¼ 70:51; (c) displacements and (d) spatial phase of the displacements normalized by pwith xa ¼ 0:65; k*t1 ¼ 22273:39 and c*t1 ¼ 0:00. The vertical dashed lines identify the location of the interior support.


4.1. Clamped boundary at x ¼ 1

The formulation in normalized coordinates with a clamped boundary condition at x ¼ 1 differs from the previous pinned-pinned example only in the latter of the two boundary conditions,

v2ð1; tÞ¼0; v2;xð1; tÞ ¼ 0: (23)

Following the same solution procedure as for the previous example, the boundary value problem, similar to Eq. (6), can beobtained and solved. For this case, the complex coefficients Di;n ði¼ 1;2; n¼ 1; :::; 4Þ from the solution of Eq. (7) can beemployed to find the necessary and sufficient conditions for vibration localization, as given by the specific values of k*t1 and c

*t1

in Eqs. (A.1) e (A.3). Provided that the conditions for vibration localization are established, the system responses can besimplified to Eqs. (A.4) e (A.6). Figure 7 displays the variation of the critical values of stiffness and damping with excitationfrequency u and support location xa necessary for wave separation and vibration localization, based on Eqs. (A.1) e (A.3). Thedips in the plots of c*t1 versus u and xa in Figs. 7(a) and (b) can be found by numerically solving the expression c

*t1 ¼ 0, as was

done in Section 3.Note that, when the support location xa is fixed and c*t1 is equal to zero, the roots (u ¼ur) denote the natural frequencies of

a pinned-clamped beam on an interior elastic support with stiffness k*t1ður ;xaÞ. Comparing the dips in the curves of c*t1 versusu in Fig. 2(a) and Fig. 7(a), the natural frequencies are higher in the clamped case as it is stiffer than the pinned case. In termsof wave separation, given a fixed excitation frequency u, Fig. 7(b) shows that the effective region of support location shrinkscompared to that of the previous case given in Fig. 2(b). The clamped boundary imposes a stronger constraint on the system sothat energies carried by reflected waves from the boundaries leak through the interior support as the support location movestoward the ends of the beam.

According to Eqs. (A.4) e (A.6), with the excitation frequency at u ¼ 100p and the excitation amplitude at A ¼ 1, properselection of the spring constant and damping coefficient in the interior support results in separation of traveling and standingwaves for xa ¼ 0:40, as shown in Figs. 8(a) and (b), and globally resonant vibration for xa ¼ 0:60, as shown in Figs. 8(c) and (d).Examination of Figs. 8(a) and (b) reveals that, due to evanescent effects, imperfect traveling waves are generated in the vi-cinity of the source and the interior support. Parameter values in the caption of Figs. 8(a) and (b) differ from those in Figs. 4(a)and (b) only because the fixed right boundary makes the beam stiffer. As a result, Eqs. (A.4) e (A.6) require the parameters ofthe interior support to be larger in order to realize wave separation.

Fig. 7. Depiction of parameter values k*t1 and c*t1 for wave separation in the pinned-clamped beam: (a) separation conditions versus excitation frequency u for

attachment location xa ¼ 0:4; (b) separation conditions versus attachment location xa for excitation frequency u ¼ 100p.

Fig. 8. Steady-state response of the pinned-clamped beamwith an interior viscoelastic support at u ¼ 100p and A ¼ 1: (a) displacements and (b) spatial phase ofthe displacements normalized by pwith xa ¼ 0:4; k*t1 ¼ 31238:57and c*t1 ¼ 135:68; (c) displacements and (d) spatial phase of the displacements normalized by pwith xa ¼ 0:60; k*t1 ¼ 22300:18 and c*t1 ¼ 0:00. The vertical dashed lines identify the location of the interior support.


4.2. Free boundary at x ¼ 1

We repeat the problem formulation and solution scheme outlined in Section 4.1 to consider a free boundary at the rightend. The boundary conditions are

v2;xxð1; tÞ¼0; v2;xxxð1; tÞ ¼ 0: (24)

The major formulations that consist of the necessary conditions and the resulting localized responses are highlighted inthe Appendix, as given by Eqs. (A.7) e (A.9) and Eqs. (A.10) e (A.12), respectively. In particular, Fig. 9 shows the plots of theseparation conditions.

Comparing Eqs. (A.7)e (A.9) and Eqs. (A.1)e (A.3), the separation conditions for the free-boundary case differ only slightlyfrom the those of the clamped case. Therefore, Fig. 9(a) nearly coincides with Fig. 7(a) when the right boundary is altered fromclamped to free. As mentioned in the pinned-pinned example, the dips in the plots of c*t1 versus u indicate where globalresonances occur. It is known that the natural frequencies of a pinned-clamped beam are the same as those of the pinned-freebeam with the exception of the rigid-body rotation (trivial natural frequency) in the pinned-free case [3,5]. Therefore, ex-amination of prefect coincidence of natural frequencies in the two standard cases can be used to interpret the complexity-induced dynamics, in particular for the similarity of the conditions for wave separation and vibration localization.

Note that Fig. 9(b) indicates that a physical combination of k*t1 and c*t1 is realizable when the viscoelastic interior support

moves to the free right boundary; i.e., xa ¼ 1 . In this situation, the necessary conditions for eliminating the left-movingharmonic wave (D11 ¼ 0) are found to be

Fig. 9. Depiction of parameter values k*t1 and c*t1 for wave separation in the pinned-free beam: (a) separation conditions versus excitation frequency u for

attachment location xa ¼ 0:4; (b) separation conditions versus attachment location xa for excitation frequency u ¼ 100p.


k*t1 ¼12g3 csch gðcosh g� cos gÞ;

c*t1 ¼12g csch gðsinh g� sin gÞ:

(25)

When the excitation frequency is chosen as u ¼ 100p, Eq. (25) gives k*t1 ¼ 2784:16 and c*t1 ¼ 8:86, resulting in nearly puretraveling waves distributed along the entire beam. This special case demonstrates that boundary impedance matching nearlyenables the input energy to be transmitted in one direction [7,8], which allows the Euler-Bernoulli beam to act as a one-dimensional waveguide. If the beam is sufficiently long to enable the evanescent waves to decay rapidly except at theboundaries, Fig. 10(a) indicates that nearly pure traveling waves can be observed along the beam.

Based on Eqs. (A.10) e (A.12), Fig. 11 depicts simulation results for two cases. In Fig. 11(a) the wave separation that takesplace for strictly positive damping is demonstrated, and in Fig. 11(b), the global resonant mode that results in the case of zerodamping is found. The parameter settings used in these simulations are shown in the caption of Fig. 11.

Fig. 10. Steady-state response of the Euler-Bernoulli beam pinned and subject to a harmonic excitation Aejut at x ¼ 0 and supported by a linear translationalspring-dashpot at x ¼ 1 : (a) displacements and (b) spatial phase of the displacements normalized by p with u ¼ 100p; A ¼ 1, k*t1 ¼ 2784:16 and c*t1 ¼ 8:86.

Fig. 11. Steady-state response of the pinned-free beamwith an interior viscoelastic support at u ¼ 100p and A ¼ 1: (a) displacements and (b) spatial phase of thedisplacements normalized by p with xa ¼ 0:4; k*t1 ¼ 31227:18and c*t1 ¼ 135:72; (c) displacements and (d) spatial phase of the displacements normalized by pwith xa ¼ 0:60; k*t1 ¼ 22246:54and c*t1 ¼ 0:00. The vertical dashed lines identify the location of the interior support.


4.3. Linear elastic boundary at x ¼ 1

The beam vibration problem with a linear elastic boundary is important for developing useful guidelines for passivecontrol strategies. The compliant boundary condition can be employed to simulate any of the three cases already discussed bysetting the translational spring stiffness kt2 and rotational spring stiffness kr2 to their appropriate limiting values. Employingthe normalized parameters in Eq. (1) for kt2 and kr2, the compliant boundary conditions are shown to be

v2;xxð1; tÞ¼ � kr2 v2;xð1; tÞ; v2;xxxð1; tÞ ¼ kt2 v2ð1; tÞ: (26)

However, the wave separation conditions with respect to k*t1 and c*t1, along with the steady-state responses v1ðx; tÞ and

v2ðx; tÞ, are quite lengthy; therefore, we present only the key results. By means of numerical extraction of the real andimaginary parts of bk*t1, the curves describing the necessary and sufficient conditions for the wave separation and vibrationlocalization are shown in Fig. 12.

Continuing, the displacement and phase plots are given in Figs. 13(a) e (d). The linear elastic support at the right end isnon-dissipative and, thus, provides total energy reflection. Exclusive of the neighborhood of the interior support, the rightportion of the beam exhibits a purely standing-wavemotion, which is characterized by trivial phase difference, as can be seenfrom Figs. 13(a) and (b).

However, when separating the harmonic and evanescent components from the total displacements and spatial phase inFigs.13(a) and (b), the individual displacement components fail to satisfy the boundary condition at the right end. Focusing onthe spatial phase for the far-field component and near-field component at x ¼ 1, and noting that ffar ¼ 4near ¼ 0:7432pwhilef ¼ � 1:2568p, we observe that the in-phase motion at the right end indicates that the displacement components, instead ofbeing canceled, are superposed to yield the total displacement, as can be seen from Fig. 13(a), Fig. 14(a) and Fig. 14(c).

Fig. 12. Plots of the parameter values k*t1 and c*t1 for wave separation in the compliant case with kt2 ¼ kr2 ¼ 2000: (a) separation conditions versus excitation

frequency u for attachment location xa ¼ 0:4; (b) separation conditions versus attachment location xa for excitation frequency u ¼ 100p.

Fig. 13. Steady-state response of the beam with an interior viscoelastic support and a compliant support at the right end, with u ¼ 100p and A ¼ 1: (a) dis-placements and (b) spatial phase of the displacements normalized by p with xa ¼ 0:4; k*t1 ¼ 10636:44 and c*t1 ¼ 10:43; (c) displacements and (d) spatial phase ofthe displacements normalized by p with xa ¼ 0:72; k*t1 ¼ 22342:51 and c*t1 ¼ 0:00. The vertical dashed lines identify the location of the interior support.


Fig. 14. Decomposition of the harmonic and evanescent parts of the responses in Fig. 13(a) and Fig. 13(b): (a) harmonic component of displacements; (b) spatialphase of the harmonic component normalized by p; (c) evanescent component of the displacements; (d) spatial phase of the evanescent component normalizedby p. The vertical dashed lines identify the location of the interior support.


4.4. Beam with two interior viscoelastic supports

While the previous examples were composed of a beam with a single viscoelastic support, it is worthwhile to furtherconsider the use of two viscoelastic supports to limit the flexural waves to a region of the beam.We examine a pinned-pinnedEuler-Bernoulli beam attached to two translational spring-dashpot pairs to verify that the flexural waves become standing

Fig. 15. Schematic of an Euler-Bernoulli beam in normalized coordinates, with pinned supports at x ¼ 0 and x ¼ 1 subject to identical forced displacements Aejut .Two interior viscoelastic supports at x ¼ x1 and x ¼ x2 connect the beam to ground.


waves locally confined between the dampers and traveling waves are directed from the energy sources towards the dampers.This is similar to the findings for the non-dispersive string system in Ref. [9]. In non-dimensional coordinates, the attach-ments are placed asymmetrically at the locations x ¼ x1 and x ¼ x2. Both ends of the beam are subject to identical harmonicmotions Aejut, as depicted in Fig. 15.

The system is partitioned into three segments by the local springs and dampers. The normalized equations of motion ineach subdomain are formulated similarly as in the pinned-pinned case with a single spring-dashpot, resulting in

v1;xxxxðx; tÞ þ v1;ttðx; tÞ ¼ 0; 0 � x � x�1v2;xxxxðx; tÞ þ v2;ttðx; tÞ ¼ 0; xþ1 � x � x�2 ;v3;xxxxðx; tÞ þ v3;ttðx; tÞ ¼ 0; xþ2 � x � 1

t � 0; (27)

where the resulting subsystems are indexed from left to right. The boundary conditions are given by

v1ð0; tÞ¼Aejut; v1;xxð0; tÞ ¼ 0; v3ð1; tÞ ¼ Aejut; v3;xxð1; tÞ ¼ 0; (28)and the continuity and equilibrium conditions at x ¼ x1 and x ¼ x2 are

v1�x�1 ; t

� ¼ v2�xþ1 ; t�; v1;x�x�1 ; t� ¼ v2;x�xþ1 ; t�; v1;xx�x�1 ; t� ¼ v2;xx�xþ1 ; t�;v2;xxx

�xþ1 ; t

�� v1;xxx�x�1 ; t� ¼ ��kt1v1�x�1 ; t�þ ct1v1;t�x�1 ; t��;v2�x�2 ; t

� ¼ v3�xþ2 ; t�; v2;x�x�2 ; t� ¼ v3;x�xþ2 ; t�; v2;xx�x�2 ; t� ¼ v3;xx�xþ2 ; t�;v3;xxx

�xþ2 ; t

�� v2;xxx�x�2 ; t� ¼ ��kt2v2�x�2 ; t�þ ct2v2;t�x�2 ; t��:(29)

jut
Substituting the steady-state assumptions viðx; tÞ ¼ ViðxÞe ði ¼ 1;2;3Þ into Eqs. (27) e (29) gives the boundary valueproblems
V0000i ðxÞ � g4ViðxÞ ¼ 0; i ¼ 1;2;3

V1ð0Þ ¼ A; V001ð0Þ ¼ 0; V3ð1Þ ¼ A; V

003ð1Þ ¼ 0;

V1�x�1

� ¼ V2�xþ1 �; V 01�x�1 � ¼ V 02�xþ1 �; V 001�x�1 � ¼ V 002�xþ1 �;V

0002�xþ1

�� V 0001�x�1 � ¼ �ðkt1 þ juct1ÞV1�x�1 �;V2

�x�2

� ¼ V3�xþ2 �; V 02�x�2 � ¼ V 03�xþ2 �; V 002�x�2 � ¼ V 003�xþ2 �;V

0003�xþ2

�� V 0002�x�2 � ¼ �ðkt2 þ juct2ÞV2�x�2 �;(30)

which lead to the steady-state amplitudes of the displacements in the form

V1ðxÞ ¼ D11ejgx þ D12e�jgx þ D13egx þ D14e�gx; 0 � x � x�1 ;V2ðxÞ ¼ D21ejgx þ D22e�jgx þ D23egx þ D24e�gx; xþ1 � x � x�2 ;

V3ðxÞ ¼ D31ejgð1�xÞ þ D32e�jgð1�xÞ þ D33egð1�xÞ þ D34e�gð1�xÞ; xþ2 � x � 1:(31)

The amplitude of the displacement in each subdomain of the beam can then be written in the compact form.

Din ¼bkt1bkt2f1;in þ bkt1g1;in þ bkt2f2;in þ g2;inbkt1bkt2f1 þ bkt1g1 þ bkt2f2 þ g2 ; i ¼ 1;/; 3; n ¼ 1; :::; 4; (32)

where bkt1 ¼ kt1 þ juct1 and bkt2 ¼ kt2 þ juct2. Note that upon simplification f1;in; f2;in; f1; f2 and g1;in; g2;in; g1; g2 ði ¼1;/;3; n ¼ 1; :::; 4Þ are defined as functions of u; g; x1 and x2, similar to the linear elastic case in Section 4.3.

The conditions D11 ¼ 0 and D31 ¼ 0 are enforced to produce the steady-state solution that admits nearly pure travelingwave propagation from the energy sources to the dampers. These two conditions lead to expressions for the stiffness anddamping of the individual attachments,

bk*t1 ¼ � bk*t2f2;11 þ g2;11bk*t2f1;11 þ g1;11 and bk*

t2 ¼ �bk*t1g1;31 þ g2;31bk*t1f1;31 þ f2;31 : (33)

Tuning the parameters of the attachments for traveling wave realization corresponds to solving Eq. (33) simultaneously,which reduces to solving a quadratic equation for each complex stiffness. Physically, the parameter values of each attachment

should be positive so energy can be dissipated. We choose the solutions where both k*t1; c*t1; k

*t2 and c

*t2 are non-negative

and plot the these parameters versus external frequency u for one asymmetric case, x1 ¼ 0:30 and x2 ¼ 0:80, as shown inFig. 16. Different from the previous cases with only one interior support, given the fixed attachment locations, Fig. 16 showsthat the effective frequency range for wave separation is narrower, which is understandable since the enforced conditions forwave separation are more restrictive. In addition, the two damping coefficients do not go to zero simultaneously over the

Fig. 16. Depiction of the non-negative parameter values of k*t1; c*t1; k

*t2 and c

*t2 versus frequency u for wave separation in the pinned-pinned beam for the

support placement x1 ¼ 0:30 and x2 ¼ 0:80.


frequency range of interest, which indicates that the global vibration localization phenomenon cannot be realized during thisparameter tuning process.

When the excitation frequency is specified as u ¼ 700, the parameter values of the attachments are found to bek*t1 ¼ 1:03� 105; c*t1 ¼ 102:74; k*t2 ¼ 2:66� 105 and c*t2 ¼ 84:41, leading to the development of traveling waves in the beamsub-spans 0� x � x1 and x2 � x � 1. Note that the spatial phase of displacement v2 is non-trivial in such a case, involvingsmall complexity and giving modulated standing waves between the spring-damper supports; while v1 and v3 are dominatedby nearly linear phases, with variations only at the boundaries, a result of the near-field evanescent waves. Figures 17(a) and

Fig. 17. Steady-state response of the pinned-pinned beamwith two interior linear viscoelastic supports: (a) displacements; (b) spatial phase of the displacementsnormalized by p. Parameters are u¼ 700; x1 ¼ 0:30; x2 ¼ 0:80; k*t1 ¼ 1:03� 105; c*t1 ¼ 102:74; k*t2 ¼ 2:66� 105 and c*t2 ¼ 84:41: The vertical dashed linesidentify the locations of the interior supports.


(b) imply that the input energies are channeled in a one-way fashion and then confined to the region between the dampers inthe undamped Euler-Bernoulli beam.

The applied displacements vð0; tÞ ¼ Aejut and vð1; tÞ ¼ Aejut on the boundaries of the beam are the only energy inputs tothe system. The energies are re-distributed to satisfy the condition for spatial separation of traveling and standing waves inthe presence of properly tuned coefficients of the damping. In complex notation, we employ the averaged power flow percycle [5] to calculate the energy transfer from x ¼ 0 and x ¼ 1, giving

Ein ¼12Refc:c:½vxxxðx¼0; tÞ�vtðx¼0; tÞg þ 12Refc:c:½vxxxðx¼1; tÞ�vtðx¼1; tÞg; (34)

while the averaged energy dissipation per cycle by the dampers at x ¼ x1 and x ¼ x2 is expressed by

Ediss ¼12Re

hc*t1kvtðx ¼ x1; tÞk22

iþ 12Re

hc*t2kvtðx ¼ x2; tÞk22

i; (35)

where c.c. stands for the complex conjugate operator and k$k2 for the Euclidean norm. These calculations provide a measureof the energy input and dissipation as the ratio ε¼ Ein=Ediss ¼ 1 for all the examples we examined, which implies that the localdampers balance the input energy exactly through the appropriate design.

5. Conclusions

In this paper, we have presented an exact analysis of an Euler-Bernoulli beam connected to ground in its interior by local,linear viscous spring-dampers, pinned and harmonically excited at one end or at both ends and having one of four differentboundary conditions (pinned, clamped, free, and linear elastic) at the other end. The local damping introduces non-classicaldamping to the system so that no classical normal modes exist. This study extends earlier works on spatial wave separationinduced by such local damping in non-dispersive media, with the aim to further understand complexity-induced waveseparation and vibration localization in dispersive media. Our analysis has shown that by tuning the parameter values of thelocal damping and stiffness elements, spatial wave separation, as well as vibration response localization in the undampedsystem, can be obtained. Nearly pure traveling waves, instead of pure traveling waves, are observed due to the evanescenteffects in the dispersion relation.

It is shown that the mechanism of complexity-induced vibration localization is characterized by the elimination of one ofthe left- or right-propagating wave components in the beam response, and a linearly-varying spatial phase distribution in theregion where traveling waves predominate. For the first four systems considered with one single spring-dashpot pair,

combinations of values of the stiffness and damping of the local attachment, denoted by k*t1 and c*t1, respectively, that give

necessary and sufficient conditions for separating the traveling and standing waves are numerically computed and plotted asfunctions of the excitation frequency u and the attachment location xa. For some specified values of u and xa, when the localdamping vanishes, c*t1 ¼ 0, the entire system is undamped and global resonance is observed. The exact values of u and xa thatcorrespond to the vanishing of the local damping, c*t1 ¼ 0, for wave separationwere analytically and numerically computed. Itis shown that, for either a fixed interior support location or excitation frequency, two sets of values are obtained. These valuesoccur in pairs that are close to each other but never coincide, and they increase in an alternating order. Eliminating points inthe damping versus support location and frequency curves that correspond to the so-called dips makes these conditionsnecessary and sufficient for spatial wave separation. For some other specified values of u and xa, where the local stiffness

k*t1 ¼ 0 but the local damping c*t1s0, these conditions require non-classical damping to guarantee the realization of waveseparation. As an extension to the previous cases, the last example considered demonstrates that vibration confinementthrough wave separation is still attainable with multiple translational spring-dampers connected to the beam.

This work presents possibilities for new passive control strategies in dispersive structures via the mechanisms of sepa-ration of standing and traveling waves (one-way energy propagation) and vibration localization. It lays the foundation foreffective designs of passive energy dissipation and vibration energy harvesting strategies.

Acknowledgments

This research was supported by the Ministry of Science and Technology of China through Grant No. 2017YFC0306202 (H.L.) and the Department of Science and Technology of Zhejiang Province, China, through Grant No. 2018C04018 (H. L.).

Appendix. Constituent functions for the Euler-Bernoulli beamwith clamped or free boundary conditions at the rightend

The boundary value problem for the Euler-Bernoulli beam with a clamped right boundary can be obtained by modifyingthe pinned-pinned case in Section 2. Solving the resulting boundary value problem yields the complex coefficients Di;n ði ¼1;2; n ¼ 1; :::;4Þ. Following the solution procedure outlined in Section 2 to establish the necessary and sufficient conditions


for vibration localization, the explicit expressions for k*t1 and c*t1, analogous to those defined in Eq. (14)e (16), are shown in Eq.

(A.1) e (A.3),

k*t1 ¼ �8e2gg3

kfcosh½gð1� 2xaÞ�sinðgÞ þ coshðgxaÞsin½gð2� xaÞ�

�cosh½gð2� xaÞ�sinðgxaÞ þ coshðgÞsin½gð1� 2xaÞ� þ sin½2gð1� xaÞ�þ2 cosh½gð1þ xaÞ�sin½gð1� xaÞ� � 2 cosðgÞsinhðgÞ � cos½gð1� 2xaÞ�sinhðgÞ�sinhð2gÞ � cos½2gð1� xaÞ�sinhð2gÞ � cos½gð2� xaÞ�sinh½gð2� xaÞ��4 cos½gð1� xaÞ�sinh½gð1� xaÞ� þ 5 cosðgxaÞsinhðgxaÞþ2 cos½gð1� xaÞ�sinh½gð1þ xaÞ� þ cosðgÞsinh½gð1� 2xaÞ� þ sinh½2gð1� xaÞ�g;

(A.1)

16e2g g

c*t1 ¼ k fcoshðgÞsin½gð1� xaÞ� � cos½gð1� xaÞ�sinhðgÞ þ sinhðgxaÞg

�fcosh½gð1� xaÞ�sinðgÞ � sinðgxaÞ þ coshðgÞsin½gð1� xaÞ��cos½gð1� xaÞ�sinhðgÞ þ sinhðgxaÞ � cosðgÞsinh½gð1� xaÞ�g;

(A.2)

where

k ¼ d21 þ d22 ¼�1� e2g � e2gxa � egð2�xaÞ cosðgxaÞ

þegxa cos½gð2� xaÞ� þ cos½2gð1� xaÞ��egxa sinðgxaÞ þ egð2�xaÞ sin½gð2� xaÞ� þ e2g sin½2gð1� xaÞ�þ4eg sinhðgxaÞfcos½gð1� xaÞ� � sin½gð1� xaÞ�gÞ2þ�1� e2g þ e2ð1�xaÞg þ egxa cosðgxaÞ

�egð2�xaÞ cos½gð2� xaÞ� � e2g cos½2gð1� xaÞ��egð2�xaÞ sinðgxaÞ þ egxa sin½gð2� xaÞ� þ sin½2gð1� xaÞ�þ4eg sinhðgxaÞfcos½gð1� xaÞ� þ sin½gð1� xaÞ�gÞ2:

(A.3)

Given the wave separation conditions, the steady-state solution for this case, comparable to Eq. (20), is given by Eq. (A.4)



�ejut; 0 � x � x�a ;

v2ðx; tÞ ¼hD*21e

�jgð1�xÞ þ D*22 ejgð1�xÞ þ D*23e�gð1�xÞ þ D*24 egð1�xÞiejut; xþa � x � 1;

(A.4)

where

D*13 ¼A4

nð1� jÞe�jgxa � ð1� jÞe�gxa �

hegð1�xaÞ � je�gð1�xaÞ�e�jg þ e�g½e�jgð1�xaÞ � jejgð1�xaÞ

i o;

D*21 ¼ �A4

n�eg þ je�gÞe�jgxa � ð1� jÞ

�egxa � e�gxa Þe�jg þ ð1þ jÞe�jgð1�xaÞ

o;

D*22 ¼A4

hjege�jgxa þ e�ge�jgxa þ ð1þ jÞejgð1�xaÞ

i;

D*23 ¼ �A4

nejgð1�xaÞ � je�jgð1�xaÞ þ

�egxa � e�gxa Þe�jg þ ð1� jÞe�ge�jgxa

o;

D*24 ¼ �A4

njejgð1�xaÞ � e�jgð1�xaÞ � jðegxa � e�gxa Þe�jg � ð1� jÞege�jgxa

o;

(A.5)

and

4 ¼ 4ð1� jÞfcoshðgÞsin½gð1� xaÞ � � cos½gð1� xaÞ �sinhðgÞ þ sinhðgxaÞ g: (A.6)

For the case with a free boundary at the right end, upon solving the boundary value problem the necessary and sufficientconditions for vibration localization are obtained as


k*t1 ¼ �8e2gg3

kf � cosh½gð1� 2xaÞ�sinðgÞ þ coshðgxaÞsin½gð2� xaÞ�

�cosh½gð2� xaÞ�sinðgxaÞ � coshðgÞsin½gð1� 2xaÞ� þ sin½2gð1� xaÞ��2 cosh½gð1þ xaÞ�sin½gð1� xaÞ� þ 2 cosðgÞsinhðgÞ þ cos½gð1� 2xaÞ�sinhðgÞ�sinhð2gÞ � cos½2gð1� xaÞ�sinhð2gÞ � cos½gð2� xaÞ�sinh½gð2� xaÞ�þ4 cos½gð1� xaÞ�sinh½gð1� xaÞ� þ 5 cosðgxaÞsinhðgxaÞ�2 cos½gð1� xaÞ�sinh½gð1þ xaÞ� � cosðgÞsinh½gð1� 2xaÞ� þ sinh½2gð1� xaÞ�g;

(A.7)

* 16e2gg

ct1 ¼ k fcoshðgÞsin½gð1� xaÞ� � cos½gð1� xaÞ�sinhðgÞ � sinhðgxaÞg

�fcosh½gð1� xaÞ�sinðgÞ þ sinðgxaÞ þ coshðgÞsin½gð1� xaÞ��cos½gð1� xaÞ�sinhðgÞ � sinhðgxaÞ � cosðgÞsinh½gð1� xaÞ�g;

(A.8)

where

k ¼ d21 þ d22 ¼�1� e2g � e2gxa � egð2�xaÞ cosðgxaÞ

þegxa cos½gð2� xaÞ� þ cos½2gð1� xaÞ��egxa sinðgxaÞ þ egð2�xaÞ sin½gð2� xaÞ� þ e2g sin½2gð1� xaÞ��4eg sinhðgxaÞfcos½gð1� xaÞ� � sin½gð1� xaÞ�gÞ2þ�1� e2g þ e2gð1�xaÞ þ egxa cosðgxaÞ

�egð2�xaÞ cos½gð2� xaÞ� � e2g cos½2gð1� xaÞ��egð2�xaÞ sinðgxaÞ þ egxa sin½gð2� xaÞ� þ sin½2gð1� xaÞ��4eg sinhðgxaÞfcos½gð1� xaÞ� þ sin½gð1� xaÞ�gÞ2:

(A.9)

Substituting Eqs. (A.7) e (A.9) into the steady-state solution form Eq. (4) and simplifying the resulting equations yield thesteady-state solution for this case,



�ejut; 0 � x � x�a ;

v2ðx; tÞ ¼hD*21e

�jgð1�xÞ þ D*22 ejgð1�xÞ þ D*23e�gð1�xÞ þ D*24 egð1�xÞiejut; xþa � x � 1;

(A.10)

where

D*13 ¼Aj

n� ð1� jÞe�jgxa þ ð1� jÞe�gxa �

hegð1�xaÞ � je�gð1�xaÞ�e�jg þ e�g½e�jgð1�xaÞ � jejgð1�xaÞ

io;

D*21 ¼ �Aj

n�eg þ je�gÞe�jgxa þ ð1� jÞ

�egxa � e�gxa Þe�jg � ð1þ jÞe�jgð1�xaÞ

o;

D*22 ¼Aj

hjege�jgxa þ e�ge�jgxa � ð1þ jÞejgð1�xaÞ

i;

D*23 ¼ �Aj

nejgð1�xaÞ � je�jgð1�xaÞ þ

�egxa � e�gxa Þe�jg � ð1� jÞe�ge�jgxa

o;

D*24 ¼ �Aj

njejgð1�xaÞ � e�jgð1�xaÞ � jðegxa � e�gxa Þe�jg þ ð1� jÞege�jgxa

o;

(A.11)

and

j ¼ 4ð1� jÞfcoshðgÞsin½gð1� xaÞ� � cos½gð1� xaÞ�sinhðgÞ � sinhðgxaÞg: (A.12)

In the last two examples we considered an Euler-Bernoulli beam with a compliant boundary at its right end and twoviscoelastic supports in its interior, respectively. However, the necessary and sufficient conditions on vibration localizationand the final steady-state solution are too lengthy to append. Therefore, we resort to numerical approaches to produce theresults given in Sections 4.3 and 4.4.


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Co-existing complexity-induced traveling wave transmission and vibration localization in Euler-Bernoulli beams1. Introduction2. Problem formulation3. Vibration localization in the pinned-pinned beam resulting from complexity induced by local damping at the interior support4. Simulation results for additional examples4.1. Clamped boundary at x=14.2. Free boundary at x=14.3. Linear elastic boundary at x=14.4. Beam with two interior viscoelastic supports

5. ConclusionsAcknowledgmentsAppendix. Constituent functions for the Euler-Bernoulli beam with clamped or free boundary conditions at the right endReferences

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