Free Vibrations of a Series of Beams Connected by Viscoelastic ...

Research ArticleFree Vibrations of a Series of Beams Connected byViscoelastic Layers

S. Graham Kelly and Clint Nicely

The University of Akron, Akron, OH 44235, USA

Correspondence should be addressed to S. Graham Kelly; [email protected]

Received 5 September 2014; Revised 22 January 2015; Accepted 22 January 2015

Academic Editor: Rama B. Bhat

Copyright © 2015 S. G. Kelly and C. Nicely. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

An exact solution for free vibrations of a series of uniformEuler-Bernoulli beams connected byKelvin-Voigt is developed.Thebeamshave the same length and end conditions but can have different material or geometric properties. An example of five concentricbeams connected by viscoelastic layers is considered.

1. Introduction

This paper presents an exact solution to the problem of thefree vibrations of an arbitrary number of beams connectedby viscoelastic layers of the Kelvin-Voigt type.The beams andthe layers may have different properties but the beams musthave the same length and the same end conditions.

The general theory for the free and forced responseof strings, shafts, beams, and axially loaded beams is welldocumented [1]. Oniszczuk [2, 3] investigated the free andforced responses of elastically connected strings. Using anormal-mode solution, he analyzed two coupled second-order ordinary differential equations to determine the naturalfrequencies. He used a modal analysis to determine theforced response. Selig and Hoppmann [4], Osborne [5], andOniszczuk [6] studied the free or forced response of elasticallyconnected Euler-Bernoulli beams. They each used a normal-mode analysis resulting in coupled sets of fourth-orderdifferential equations whose eigenvalues were related to thenatural frequencies. Rao [7] also employed a normal-modesolution to compute the natural frequencies of elasticallyconnected Timoshenko beams. Each study did not considerdamping of the beams or damping in the elastic connection.

Kelly [8] developed a general theory for the exact solutionof free vibrations of elastically coupled structures withoutdamping. The structures may have different properties oreven be nonuniform but they have the same support. He

applied the theory to Euler-Bernoulli beams and concen-tric torsional shafts. Kelly and Srinivas [9] developed aRayleigh-Ritz method for elastically connected stretchedEuler-Bernoulli beams.

Yoon et al. [10] and Li and Chou [11] proposed that freevibrations of multiwalled carbon nanotubes can be mod-eled by elastically connected Euler-Bernoulli beams. Theyemployed normal-mode solutions, showing that multiwallednanotubes have an infinite series of noncoaxial modes.Yoon et al. [12] modeled free vibrations of nanotubes withconcentric Timoshenko beams connected by an elastic layer.Xu et al. [13] modeled nonlinear vibrations in the elasticallyconnected structures modeling nanotubes by considering thenonlinearity of the van der Waals forces. They analyzed thenonlinear free vibrations by employing a Galerkin method.Elishakoff and Pentaras [14] developed approximate formulasfor the natural frequencies of double walled nanotubesmodeled as concentric elastically coupled beams, noting thatif developed from the eigenvalue relation the computationscan be computationally intensive and difficult.

Damped vibrations of elastically connected structureshave been studied by few authors. Oniszczuk [15] used anormal-mode solution in considering the vibration of twostrings connected by a viscoelastic layer of the Kelvin-Voigttype. Palmeri and Adhikari [16] used a Galerkin methodto analyze the vibrations of a double-beam system con-nected by a viscoelastic layer of the Maxwell type. Jun and

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2015, Article ID 976841, 8 pageshttp://dx.doi.org/10.1155/2015/976841

2 Advances in Acoustics and Vibration

wk(x, t)

wk−1(x, t)

wk−2(x, t)

w2(x, t)

w1(x, t)

Figure 1: Schematic representation of problem considered, 𝑘 beams in parallel connected by viscoelastic layers of the Kelvin-Voigt type.

Hongxing [17] used a dynamic stiffness matrix to analyze freevibrations of three beams connected by viscoelastic layers.Their analysis does not require the beams to have the sameend conditions but does require the use of computationaltools to determine the natural frequencies.

An exact solution for the free vibration of a series ofelastically connected Euler-Bernoulli beams is considered inthis paper. The elastic layers are viscoelastic with damping ofthe Kelvin-Voigt type.The results are applied to a series of fiveconcentric beams.

2. Problem Formulation

Consider 𝑛 Euler-Bernoulli beams connected by viscoelasticlayers as shown in Figure 1. Each beam is assumed to haveits own neutral axis. All beams are uniform of length 𝐿. Let𝐸𝑖be the elastic modulus, let 𝜌

𝑖be the mass density, let 𝐴

𝑖

be the cross-sectional area, and let 𝐼𝑖be the cross-sectional

moment of inertia of the 𝑖th beamabout the neutral axis of the𝑖th beam. Let 𝑤

𝑖(𝑥, 𝑡) represent the transverse displacement

of the 𝑖th beam, where 𝑥 is the distance along the neutralaxis of the beam measured from its left end and 𝑡 representstime. Damping in each beam due to structural damping orcomplex stiffness is neglected. The viscoelastic layer betweenthe 𝑖th and 𝑖 plus first layer is of the Kelvin-Voigt type and hastwo parameters, 𝑐

𝑖representing the damping property of the

layer and 𝑘𝑖representing the stiffness of the layer, such that

the force acting on the 𝑖th beam from the layer is

𝐺𝑖= 𝑐𝑖(𝜕𝑤𝑖+1

𝜕𝑡−𝜕𝑤𝑖

𝜕𝑡) + 𝑘𝑖(𝑤𝑖+1− 𝑤𝑖) . (1)

Hamilton’s principle is used to derive the equationsgoverning the free response of the 𝑖th beams as

𝐸𝑖𝐼𝑖

𝜕4𝑤𝑖

𝜕𝑥4+ 𝑐𝑖−1(𝜕𝑤𝑖

𝜕𝑡−𝜕𝑤𝑖−1

𝜕𝑡) + 𝑐𝑖(𝜕𝑤𝑖

𝜕𝑡−𝜕𝑤𝑖+1

𝜕𝑡)

+ 𝑘𝑖−1(𝑤𝑖− 𝑤𝑖−1) + 𝑘𝑖(𝑤𝑖− 𝑤𝑖+1)

+ 𝜌𝑖𝐴𝑖

𝜕2𝑤𝑖

𝜕𝑡2= 0.

(2)

In developing (2), viscoelastic layers represented by coeffi-cients 𝑐

0, 𝑘0, 𝑐𝑛, and 𝑘

𝑛are assumed to exist between the first

beam and the surroundingmedium and the 𝑛th beam and thesurrounding medium and 𝑤

0= 0 and 𝑤

𝑛+1= 0.

The equations represented by (2) are nondimensionalizedby introducing

𝑥∗=𝑥

𝐿, (3a)

𝑤∗

𝑖=𝑤𝑖

𝐿, (3b)

𝑡∗= 𝑡√

𝐸1𝐼1

𝜌1𝐴1𝐿4. (3c)

The nondimensional variables are substituted into (2) result-ing in

𝜇𝑖

𝜕4𝑤𝑖

𝜕𝑥4+ 𝜂𝑖−1(𝑤𝑖− 𝑤𝑖−1) + 𝜂𝑖(𝑤𝑖− 𝑤𝑖+1)

+ ]𝑖−1(𝜕𝑤𝑖

𝜕𝑡−𝜕𝑤𝑖−1

𝜕𝑡) + ]𝑖(𝜕𝑤𝑖

𝜕𝑡−𝜕𝑤𝑖+1

𝜕𝑡)

+ 𝛽𝑖

𝜕2𝑤𝑖

𝜕𝑡2= 0,

(4)

where the ∗’s have been dropped from the nondimensionalvariables and

𝜇𝑖=𝐸𝑖𝐼𝑖

𝐸1𝐼1

, (5a)

Advances in Acoustics and Vibration 3

𝜂𝑖=𝑘𝑖𝐿4

𝐸1𝐼1

, (5b)

]𝑖=

𝑐𝑖𝐿2

√𝐸1𝐼1𝜌1𝐴1

, (5c)

𝛽𝑖=𝜌𝑖𝐴𝑖

𝜌1𝐴1

. (5d)

The differential equations have a matrix-operator formula-tion as

(K + K𝑐)W + C

𝑐W +MW = 0, (6)

where W = [𝑤1(𝑥, 𝑡) 𝑤2(𝑥, 𝑡) ⋅ ⋅ ⋅ 𝑤𝑛(𝑥, 𝑡)]𝑇, K is a 𝑛 × 𝑛

diagonal operator matrix with 𝑘𝑖,𝑖= 𝜇𝑖(𝜕4𝑤𝑖/𝜕𝑥4),M is a 𝑛×

𝑛 diagonal mass matrix with𝑚𝑖,𝑖= 𝛽𝑖, and K

𝑐is a tridiagonal

𝑛 × 𝑛 stiffness coupling matrix with

(𝑘𝑐)𝑖,𝑖−1

= −𝜂𝑖−1, 𝑖 = 2, 3, . . . , 𝑛,

(𝑘𝑐)𝑖,𝑖= 𝜂𝑖−1+ 𝜂𝑖, 𝑖 = 1, 2, . . . , 𝑛,

(𝑘𝑐)𝑖,𝑖+1

= −𝜂𝑖, 𝑖 = 1, 2, . . . , 𝑛 − 1

(7)

and C𝑐is a tridiagonal 𝑛 × 𝑛 damping coupling matrix with

(𝑐𝑐)𝑖,𝑖−1

= −]𝑖−1, 𝑖 = 2, 3, . . . , 𝑛,

(𝑐𝑐)𝑖,𝑖= ]𝑖−1+ ]𝑖, 𝑖 = 1, 2, . . . , 𝑛,

(𝑐𝑐)𝑖,𝑖+1

= −]𝑖, 𝑖 = 1, 2, . . . , 𝑛 − 1.

(8)

The vector W is an element of the vector space 𝑈 = 𝑆 × 𝑅𝑛;

an element of 𝑈 is an 𝑛-dimensional vector whose elementsall belong to 𝑆, the space of functions which satisfy thehomogeneous boundary conditions of each beam.

3. Free Vibrations

A normal-mode solution of (6) is assumed as

W = w𝑒𝑖𝜔𝑡, (9)

where 𝜔 is a parameter and w =

[𝑤1(𝑥) 𝑤2(𝑥) 𝑤3(𝑥) ⋅ ⋅ ⋅ 𝑤𝑛−1(𝑥) 𝑤𝑛(𝑥)]𝑇 is a vector

of mode shapes corresponding to that natural frequency.Substitution of (9) into (6) leads to

(K + K𝑐)w + 𝑖𝜔C

𝑐w = 𝜔2Mw, (10)

where the partial derivatives have been replaced by ordinaryderivatives in the definition of K.

A solution of the set of 𝑛 ordinary differential equationsrepresented by (10) is assumed as

w𝑘 (𝑥) = 𝜙𝑘(𝑥) a𝑘, (11)

where 𝜙𝑘(𝑥) satisfies the equation

𝑑4𝜙𝑘

𝑑𝑥4− 𝜉2

𝑘𝜙 = 0 (12)

subject to the homogeneous boundary conditions of thebeams and a

𝑘is a vector of constants. The parameter 𝜉

𝑘is

the 𝑘th natural frequency of an undamped beam with theappropriate end conditions. The values of 𝜉

𝑘for 𝑘 = 1, 2, . . .

are the natural frequencies of the first beam in the seriesassuming the beam vibrates freely from the other beams andthe functions 𝜙

𝑘(𝑥) are the corresponding mode shapes.

Substitution of (12) into (10) leads to

(𝜉2

𝑘U + K

𝑐) a𝑘+ 𝑖𝜔C

𝑐a𝑘= 𝜔

2Ma𝑘, (13)

where U is an 𝑛 × 𝑛 diagonal matrix with 𝑢𝑖,𝑖= 𝜇𝑖. Equation

(13) is a system of 𝑛 homogeneous algebraic equations to solvefor a𝑘.

The differential equations governing the free vibrationsof a linear 𝑛-degree-of-freedom system with displacementvector x = [𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛]

𝑇 are summarized by

Mx + Cx + Kx = 0. (14)

A normal-mode solution is assumed as x = X𝑒𝑖𝜔𝑡 for (14),resulting in

−𝜔2MX + 𝑖𝜔CX + KX = 0. (15)

Equation (15) is the same as (13) with K = 𝜉2

𝑘U + K

𝑐. Thus,

the same solution procedure is used to solve (13) as is used tosolve (15) for each 𝑘 = 1, 2, 3, . . ..

4. General Solution

Following Kelly [1] the differential equations summarized by(11) can be rewritten as a system of 2𝑛 first-order equations ofthe form

My + Ky = 0, (16)

where

M = [0 MM C] , K = [−M 0

0 K] ,

y = [xx] .(17)

A solution to (17) is assumed as

y = Φ𝑒−𝛾𝑡 (18)

which results upon substitution in

M−1KΦ = 𝛾Φ. (19)

The values of 𝜔 are related to the eigenvalues of M−1K by 𝜔 =𝑖𝛾. The resulting problem has, in general, complex eigenval-ues.The correspondingmode shape vectors are also complex.The real part of an eigenvalue is negative and is an indica-tion of the damping properties of that mode. When complexeigenvalues occur, they occur in complex conjugate pairs.Theimaginary part is the frequency of themode.Themode shapevectors corresponding to complex conjugate eigenvalues are


Table 1: Properties of the five beams of the example.

Beamnumber, 𝑖

Elasticmodulus, 𝐸

𝑖

(TPa)

Density, 𝜌𝑖

(kg/m3)

Innerradius, 𝑟

𝑖,𝑖

(nm)

Outerradius, 𝑟

𝑖,𝑜

(nm)

Area,𝐴𝑖= 𝜋(𝑟

𝑖,𝑜

2− 𝑟𝑖,𝑖

2)

(nm2)

Moment of inertia𝐼𝑖= (𝜋/4)(𝑟

𝑖,𝑜

4− 𝑟𝑖,𝑖

4)

(nm4)

Length 𝐿𝑖

(nm)

1 1 1300 1.0 1.34 2.50 1.75 202 1 1300 1.34 1.68 3.23 3.73 203 1 1300 1.68 2.02 3.95 6.82 204 1 1300 2.02 2.36 4.98 12.12 205 1 1300 2.36 2.70 5.79 19.07 20

also complex conjugates of one another. When the generalsolution is written as a linear combination over all modeshapes the complex eigenvalues and the complex eigenvectorscombine leading to terms involving the sine and cosine of theimaginary part of the eigenvalues.

The general solution of the partial differential equationsis

w (𝑥, 𝑡) =∞

∑

𝑘=0

(

𝑛

∑

𝑖=1

𝐵𝑘,𝑖X𝑘,𝑖𝑒−𝛾𝑘,𝑖𝑡)𝜙𝑘 (𝑥) , (20)

where 𝐵𝑖,𝑘

are arbitrary constants of integration. When thevalues of 𝛾

𝑖,𝑘are all complex and of the form

𝛾𝑘,𝑖= 𝛼𝑘,𝑖+ 𝑖𝛽𝑘,𝑖 (21)

and the complex mode shapes have the form

X𝑘,𝑖= U𝑘,𝑖+ 𝑖V𝑘,𝑖 (22)

then (20) is written as

w (𝑥, 𝑡)

=

∞

∑

𝑘=0

{

𝑛

∑

𝑖=1

𝑒−𝛼𝑘,𝑖𝑡 [𝐶

𝑘,𝑖(U𝑘,𝑖cos𝛽𝑘,𝑖𝑡 + V𝑘,𝑖sin𝛽𝑘,𝑖𝑡)

+𝐷𝑘,𝑖(V𝑘,𝑖cos𝛽𝑘,𝑖𝑡 − U𝑘,𝑖sin𝛽𝑘,𝑖𝑡)] }

⋅ 𝜙𝑘 (𝑥) .

(23)

In (23), 𝐶𝑘,𝑖and𝐷

𝑘,𝑖are constants of integration determined

from appropriate initial conditions.If a value of 𝛾

𝑘,𝑖is real, the corresponding mode is

overdamped and there are two real values of 𝛾𝑘,𝑖; call them

𝛾𝑘,𝑖,1

and 𝛾𝑘,𝑖,2

. The real part has bifurcated into two valuesand the corresponding eigenvectors are real. The term insidethe inner summation corresponding to a real eigenvalue is𝐶𝑘,𝑖X𝑘,𝑖,1𝑒−𝛾𝑘,𝑖,1𝑡 + 𝐷

𝑘,𝑖X𝑘,𝑖,2𝑒−𝛾𝑘,𝑖,2𝑡.

The spatially distributed mode shapes satisfy an orthogo-nality condition, which for a uniform beam is

∫

1

0

𝜙𝑗𝜙𝑘d𝑥 = 0, 𝑗 = 𝑘. (24)

Let w(𝑥, 0) be a vector of initial conditions. Then

w (𝑥, 0) =∞

∑

𝑘=0

[

𝑛

∑

𝑖=1

(𝐶𝑘,𝑖U𝑘,𝑖+ 𝐷𝑘,𝑖V𝑘,𝑖)] 𝜙𝑘 (𝑥) . (25)

Multiplying both sides of (25) by 𝜙𝑗(𝑥) for an arbitrary value

of 𝑗, integrating from 0 to 1, and using the orthogonalitycondition lead to the equation:

𝑛

∑

𝑖=1

(𝐶𝑗,𝑖U𝑗,𝑖+ 𝐷𝑗,𝑖V𝑗,𝑖) = ∫

1

0

w (𝑥, 0) 𝜙𝑗 (𝑥) d𝑥. (26)

A similar procedure is used for the vector of initial velocitiesw(𝑥, 0) yielding

𝑛

∑

𝑖=1

[𝐶𝑗,𝑖(−𝛼𝑘,𝑖U𝑗,𝑖+ 𝛽𝑗,1V𝑗,1)

− 𝐷𝑗,𝑖(𝛼𝑘,𝑖V𝑗,𝑖+ 𝛽𝑗,1U𝑗,1)]

= ∫

1

0

w (𝑥, 0) 𝜙𝑗 (𝑥) d𝑥.

(27)

5. Example

Consider five concentric fixed-pinned beams connected byviscoelastic layers of the Kelvin-Voigt type of negligiblethickness. The cross-sectional moment of inertia of the 𝑖thbeam is 𝐴

𝑖= 𝜋(𝑟

𝑖,𝑜

2− 𝑟𝑖,𝑖

2), where 𝑟

𝑖,𝑜is the outer radius

of the ith beam and 𝑟𝑖,𝑖is the inner radius of the 𝑖th beam

which is the outer radius of the 𝑖-1st beam.The cross-sectionalmoment of inertia of the 𝑖th beam is 𝐼

𝑖= (𝜋/4)(𝑟

𝑖,𝑜

4− 𝑟𝑖,𝑖

4).

The properties of each of the five beams are given in Table 1.Each layer has two parameters. The stiffness parameters,

given in Table 2, are consistent with those generated by thevan der Waals forces between atoms in a carbon nanotubeand are given by a formula derived using the data of Girifalcoand Lad [18] and the Lennerd-Jones potential function:

𝑘𝑖=366.67 (2𝑟

𝑖,𝑖)

0.16𝑑2erg/cm2, (28)

where𝑑 = 0.147 nm is the interatomic distance between bondlengths. The damping parameters are assumed. The non-dimensional parameters for each beam are given in Table 3.


Table 2: Properties of layers in example.

Layer, 𝑖 Stiffness parameter,𝑘𝑖(TPa)

Damping parameter,𝑐𝑖(N⋅s/m2)

0 0 01 0.277 0.12 0.347 0.1343 0.418 0.1684 0.493 0.2025 0 0

The mode shapes of a fixed-pinned beam are

𝜙𝑘 (𝑥) = cos√𝜉

𝑘𝑥 − cosh√𝜉

𝑘𝑥

+ 𝛼𝑘(sinh√𝜉

𝑘𝑥 − sin√𝜉

𝑘𝑥) ,

(29)

where

𝛼𝑘=cos√𝜉

𝑘− cosh√𝜉

𝑘

sin√𝜉𝑘− sinh√𝜉

𝑘

(30)

and 𝜉𝑘is the 𝑘th positive solution of

tan√𝜉𝑘= tanh√𝜉

𝑘. (31)

The first five solutions of (31) are given in Table 4.The free vibration response is given by (23), where the

values of 𝛾𝑘,𝑖

for 𝑘 = 1, 2, . . . are determined using (13).Choosing 𝑘 = 3, (13) is written as

𝜔2

[[[[[

[

1 0 0 0 0

0 1.291 0 0 0

0 0 1.581 0 0

0 0 0 1.991 0

0 0 0 0 2.317

]]]]]

]

[[[[[

[

𝑎3,1

𝑎3,2

𝑎3,3

𝑎3,4

𝑎3,5

]]]]]

]

+ 𝑖𝜔102

[[[[[

[

5.31 −5.31 0 0 0

−5.31 12.42 −7.11 0 0

0 −7.11 16.03 −8.92 0

0 0 −8.92 19.69 −10.72

0 0 0 −10.72 10.72

]]]]]

]

⋅

[[[[[

[

𝑎3,1

𝑎3,2

𝑎3,3

𝑎3,4

𝑎3,5

]]]]]

]

+ 105

[[[[[

[

2.54 −2.54 0 0 0

−2.54 5.73 −3.18 0 0

0 −3.18 7.02 −3.89 0

0 0 −3.89 8.31 −4.51

0 0 0 −4.51 4.54

]]]]]

]

⋅

[[[[[

[

𝑎3,1

𝑎3,2

𝑎3,3

𝑎3,4

𝑎3,5

]]]]]

]

=

[[[[[

[

0

0

0

0

0

]]]]]

]

.

(32)

Table 5 presents the five intermodal frequencies correspond-ing to the five lowest intra-modal frequencies.

A solution of the form of (23) is applied resulting in theportion of the solution of (10) corresponding to 𝑘 = 3 as

w3 (𝑥, 𝑡)

=

{{{{{

{{{{{

{

𝑒−2.29𝑡

{{{{{

{{{{{

{

𝐶3,1

[[[[[

[

[[[[[

[

2.64

2.49

2.27

2.01

1.92

]]]]]

]

cos (176.9𝑡)

+

[[[[[

[

1.45

1.28

1.05

0.823

0.690

]]]]]

]

sin (176.9𝑡)]]]]]

]

+ 𝐷3,1

[[[[[

[

−

[[[[[

[

2.64

2.49

2.27

2.01

1.92

]]]]]

]

sin (176.9𝑡)

+

[[[[[

[

1.45

1.28

1.05

0.823

0.690

]]]]]

]

cos (176.9𝑡)]]]]]

]

}}}}}

}}}}}

}

+ 𝑒−9.77𝑡

{{{{{

{{{{{

{

𝐶3,2

[[[[[

[

[[[[[

[

1.77

1.03

0.0017

−0.832

−1.26

]]]]]

]

cos (326.9𝑡)

+

[[[[[

[

1.09

0.704

0.185

−0.285

−0.562

]]]]]

]

sin (326.9𝑡)]]]]]

]

+ 𝐷3,2

[[[[[

[

−

[[[[[

[

1.77

1.03

0.0017

−0.832

−1.26

]]]]]

]

sin (326.9𝑡)

+

[[[[[

[

1.09

0.704

0.185

−0.285

−0.562

]]]]]

]

cos (326.9𝑡)]]]]]

]

}}}}}

}}}}}

}

+ 𝑒−347.3𝑡

{{{{{

{{{{{

{

𝐶3,3

[[[[[

[

[[[[[

[

−1.05

0.312

0.934

0.298

−0.606

]]]]]

]

cos (466.2𝑡)


Table 3: Nondimensional parameters.

𝑖 𝜇𝑖= 𝐸𝑖𝐼𝑖/𝐸1𝐼1

𝛽𝑖= 𝜌𝑖𝐴𝑖/𝜌1𝐴1

𝜂𝑖= 𝑘𝑖𝐿4/𝐸1𝐼1

]𝑖= 𝑐𝑖𝐿2/√𝐸1𝐼1𝜌1𝐴1

1 1 1 2.54 × 105 5.31 × 102

2 2.13 1.29 3.19 × 105 7.11 × 102

3 3.90 1.58 3.83 × 105 8.14 × 102

4 6.84 1.99 4.51 × 105 1.07 × 103

5 10.92 2.31 0 0

+

[[[[[

[

−4.06

1.01

4.10

1.27

−2.92

]]]]]

]

sin (466.2𝑡)]]]]]

]

+ 𝐷3,3

[[[[[

[

−

[[[[[

[

−1.05

0.312

0.934

0.298

−0.606

]]]]]

]

sin (466.2𝑡)

+

[[[[[

[

−4.06

1.01

4.10

1.27

−2.92

]]]]]

]

cos (466.2𝑡)]]]]]

]

}}}}}

}}}}}

}

+ 𝑒−638.1𝑡

{{{{{

{{{{{

{

𝐶3,4

[[[[[

[

[[[[[

[

5.38

−7.95

0.798

5.78

−3.37

]]]]]

]

cos (437.7𝑡)

+

[[[[[

[

2.64

−3.32

−0.357

2.93

−1.61

]]]]]

]

sin (437.7𝑡)]]]]]

]

+ 𝐷3,4

[[[[[

[

−

[[[[[

[

5.38

−7.95

0.798

5.78

−3.37

]]]]]

]

sin (437.7𝑡)

+

[[[[[

[

2.64

−3.32

−0.357

2.93

−1.61

]]]]]

]

cos (437.72𝑡)]]]]]

]

}}}}}

}}}}}

}

+ 𝑒−896.9𝑡

{{{{{

{{{{{

{

𝐶3,5

[[[[[

[

[[[[[

[

0.985

−1.366

0.331

0.382

−0.196

]]]]]

]

cos (103.6𝑡)

Table 4: Five lowest solutions of tan√𝛿 = tanh√𝛿.

𝑖 𝛿𝑖

1 15.422 49.963 104.24 178.35 272.0

+

[[[[[

[

−2.13

5.54

−7.28

5.53

−1.95

]]]]]

]

sin (103.6𝑡)]]]]]

]

+ 𝐷3,5

[[[[[

[

−

[[[[[

[

0.985

−1.366

0.331

0.382

−0.196

]]]]]

]

sin (103.6𝑡)

+

[[[[[

[

−2.13

5.54

−7.28

5.53

−1.95

]]]]]

]

cos (103.6𝑡)]]]]]

]

}}}}}

}}}}}

}

}}}}}

}}}}}

}

⋅ ⌈cos 10.2𝑥 − cosh 10.2𝑥

+𝛼3 (sinh 10.2𝑥 − sin 10.2𝑥)⌉ .

(33)

The parameters 𝛾𝑘,𝑖

for 𝑘 = 1, 2, . . . , 5 and for all fivebeams are presented inTable 6. For these damping properties,all parameters are complex except for 𝛾

1,5. The real part

represents the amount of damping a mode has while theimaginary part is the damped natural frequency of the mode.The mode represented by 𝛾

1,5is overdamped.

Let 𝛿 represent the damping coefficient of the firstlayer and assume the damping parameter of each layer isproportional to the stiffness of the layer. The damping doesnot constitute proportional damping (Rayleigh damping) fora specific value of 𝑘 as the stiffness matrix is a combination ofthe coupling stiffnessmatrix due to the viscoelastic layers andthe diagonal bending stiffness matrix, whereas the dampingmatrix is just from the viscoelastic layers.


Table 5: Undamped natural frequencies 𝜔𝑘,𝑖, for example. 𝜔

𝑘,𝑖for a fixed 𝑘 and 𝑖 = 1, 2, . . . , 5 is a set of intramodal frequencies, whereas 𝜔

𝑘,𝑖

and 𝜔𝑗,𝑖represent intermodal frequencies.

𝑖 𝜔1,𝑖

𝜔2,𝑖

𝜔3,𝑖

𝜔4,𝑖

𝜔5,𝑖

1 2.688 × 101 8.647 × 101 1.754 × 102 2.797 × 102 3.866 × 102

2 2.978 × 102 3.075 × 102 3.425 × 102 4.275 × 102 5.961 × 102

3 5.540 × 102 5.594 × 102 5.790 × 102 6.261 × 102 7.153 × 102

4 7.538 × 102 7.578 × 102 7.724 × 102 8.075 × 102 8.755 × 102

5 8.906 × 102 8.937 × 102 9.017 × 102 9.34 × 102 9.942 × 102

Table 6: Values of 𝛾𝑘,𝑖for 𝑘 = 1, 2, . . . , 5, for example.

𝑖 𝛾1,𝑖

𝛾2,𝑖

𝛾3,𝑖

𝛾4,𝑖

𝛾5,𝑖

1 1.22 × 10−3± 268.8𝑖 0.1319 ± 8.65𝑖 2.291 ± 176.3𝑖 15.17 ± 286.7𝑖 48.16 ± 397.9𝑖

2 110.0 ± 280.6𝑖 99.79 ± 290.8𝑖 97.68 ± 326.9𝑖 87.15 ± 409.8𝑖 70.17 ± 562.8𝑖

3 344.6 ± 435.6𝑖 344.4 ± 442.3𝑖 343.7 ± 466.2𝑖 341.7 ± 521.3𝑖 337.4 ± 617.3𝑖

4 638.7 ± 405.3𝑖 638.6 ± 412.4𝑖 638.1 ± 437.7𝑖 636.8 ± 495.5𝑖 634.5 ± 595.5𝑖

5 759.71.031 × 103

785.71.006 × 103

896.9 ± 103.6𝑖 897.8 ± 256.6𝑖 894.8 ± 414.8𝑖

𝛼3,i

108

106

104

102

100

10−2

10−3 10−2 10−1 100 101 102

𝛿

Figure 2: Real part of 𝛾3,𝑖for each mode versus 𝛿, for example. All

values except the lowest have a bifurcation for some value of 𝛿.Whena bifurcation occurs, the mode is critically damped.

Figure 2 shows the real parts of 𝛾3,𝑖for each mode versus

𝛿. The real part starts at zero (the undamped solution) andincreases until (except for the lowestmode) it bifurcates whenthe mode becomes overdamped.The value of 𝛿 for which thebifurcation occurs is larger for lower modes. The value of 𝛾

3,1

does not bifurcate but reaches a maximum value and thendecreases.

The imaginary part of 𝛾3,𝑖for eachmode is plotted against

𝛿 in Figure 3.The higher modes vibrate at higher frequenciesfor small 𝛿. For higher delta, the imaginary part goes to

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

100

200

300

400

500

600

700

800

900

1000

𝛿

𝛽3,i

Figure 3: Imaginary part of 𝛾3,𝑖for eachmode versus 𝛿, for example.

zero except for the lowest mode which approaches a constantvalue.

6. Conclusions

The free vibrations of a set of 𝑛 beams connected by vis-coelastic layers of the Kelvin-Voigt type are considered. Thebeams have the same length and are subject to the same endconditions but may have different properties. The equationsof motion are derived and nondimensionalized. A normal-mode solution is assumed. When substituted into the partialdifferential equations, it leads to a set of ordinary differentialequations which is solved by assuming the solution is a vector


times the undamped spatial mode shape of the first beam.This solution is valid because the bending stiffness of eachbeam is proportional to the bending stiffness of the first beam;however, it is not necessary that all properties of the beams areproportional. The result is, for each mode, a matrix equationwhich is similar to the matrix equation governing a discretelinear systemwith damping.Themethod used to find the freeresponse of a discrete linear system is used to solve for theparameters governing the vibrations of a continuous systemconnected by Kelvin-Voigt layers.

A Kelvin-Voigt model was assumed for layers betweenmultiwalled nanotubes with the elasticity representing thevan der Waals forces between atoms. The damping wasassumed to present an example. However, the method can beused for any form of linear damping in the beams or in thelayers. Thus, a model of a multiwalled nanotube with lineardamping in the nanotubes can be analyzed using the methodpresented.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding publication of this paper.

References

[1] S. G. Kelly, Advanced Vibration Analysis, CRC Press; Taylor &Francis, Boca Raton, Fla, USA, 2007.

[2] Z. Oniszczuk, “Transverse vibrations of elastically connecteddouble-string complex system, Part I: free vibrations,” Journalof Sound and Vibration, vol. 232, no. 2, pp. 355–366, 2000.

[3] Z. Oniszczuk, “Transverse vibrations of elastically connecteddouble-string complex system. Part II: forced vibrations,” Jour-nal of Sound and Vibration, vol. 232, no. 2, pp. 367–386, 2000.

[4] J. M. Selig and W. H. Hoppmann, “Normal mode vibrationsof systems of elastically connected parallel bars,” Journal of theAcoustical Society of America, vol. 36, pp. 93–99, 1964.

[5] E. E. Osborne, “Computations of bending modes and modeshapes of single and double beams,” Journal of the Society forIndustrial and Applied Mathematics, vol. 10, no. 2, pp. 329–338,1962.

[6] Z. Oniszczuk, “Free transverse vibrations of elastically con-nected simply supported double-beam complex system,” Jour-nal of Sound and Vibration, vol. 232, no. 2, pp. 387–403, 2000.

[7] S. S. Rao, “Natural vibrations of systems of elastically connectedTimoshenko beams ,” Journal of the Acoustical Society ofAmerica, vol. 55, no. 6, pp. 1232–1237, 1974.

[8] S. G. Kelly, “Free and forced vibrations of elastically connectedstructures,” Advances in Acoustics and Vibration, vol. 2010,Article ID 984361, 11 pages, 2010.

[9] S. G. Kelly and S. Srinivas, “Free vibrations of elastically con-nected stretched beams,” Journal of Sound and Vibration, vol.326, no. 3–5, pp. 883–893, 2009.

[10] J. Yoon, C.Q. Ru, andA.Mioduchowski, “Noncoaxial resonanceof an isolatedmultiwall carbon nanotube,”Physics ReviewB, vol.66, Article ID 233402, 4 pages, 2002.

[11] C. Li and T.-W. Chou, “Vibrational behaviors of multiwalled-carbon-nanotube-based nanomechanical resonators,” AppliedPhysics Letters, vol. 84, no. 1, pp. 121–123, 2004.

[12] J. Yoon, C. Q. Ru, and A. Mioduchowski, “Terahertz vibrationof short carbon nanotubes modeled as Timoshenko beams,”Transactions ASME—Journal of Applied Mechanics, vol. 72, no.1, pp. 10–17, 2005.

[13] K. Y. Xu, X. N. Guo, and C. Q. Ru, “Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van derWaals forces,” Journal of Applied Physics, vol. 99, no. 6, ArticleID 064303, 2006.

[14] I. Elishakoff andD. Pentaras, “Fundamental natural frequenciesof double-walled carbon nanotubes,” Journal of Sound andVibration, vol. 322, no. 4-5, pp. 652–664, 2009.

[15] Z. Oniszczuk, “Damped vibration analysis of an elastically con-nected complex double-string system,” Journal of Sound andVibration, vol. 264, no. 2, pp. 253–271, 2003.

[16] A. Palmeri and S. Adhikari, “A Galerkin-type state-spaceapproach for transverse vibrations of slender double-beam sys-tems with viscoelastic inner layer,” Journal of Sound and Vibra-tion, vol. 330, no. 26, pp. 6372–6386, 2011.

[17] L. Jun and H. Hongxing, “Dynamic stiffness vibration analysisof an elastically connected three-beam system,” Applied Acous-tics, vol. 69, no. 7, pp. 591–600, 2008.

[18] L.A.Girifalco andR.A. Lad, “Energy of cohesion, compressibil-ity, and the potential energy functions of the graphite system,”The Journal of Chemical Physics, vol. 25, no. 4, pp. 693–697, 1956.

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