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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; gkelly@uakron.edu
Received 5 September 2014; Revised 22 January 2015; Accepted 22 January 2015
Academic Editor: Rama B. Bhat
Copyright Β© 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)
wkβ1(x, t)
wkβ2(x, t)
w2(x, t)
w1(x, t)
Figure 1: Schematic representation of problem considered, π 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 π 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 πΏ. LetπΈπbe the elastic modulus, let π
πbe the mass density, let π΄
π
be the cross-sectional area, and let πΌπbe the cross-sectional
moment of inertia of the πth beamabout the neutral axis of theπth beam. Let π€
π(π₯, π‘) represent the transverse displacement
of the πth beam, where π₯ is the distance along the neutralaxis of the beam measured from its left end and π‘ representstime. Damping in each beam due to structural damping orcomplex stiffness is neglected. The viscoelastic layer betweenthe πth and π plus first layer is of the Kelvin-Voigt type and hastwo parameters, π
πrepresenting the damping property of the
layer and ππrepresenting the stiffness of the layer, such that
the force acting on the πth beam from the layer is
πΊπ= ππ(ππ€π+1
ππ‘βππ€π
ππ‘) + ππ(π€π+1β π€π) . (1)
Hamiltonβs principle is used to derive the equationsgoverning the free response of the πth beams as
πΈππΌπ
π4π€π
ππ₯4+ ππβ1(ππ€π
ππ‘βππ€πβ1
ππ‘) + ππ(ππ€π
ππ‘βππ€π+1
ππ‘)
+ ππβ1(π€πβ π€πβ1) + ππ(π€πβ π€π+1)
+ πππ΄π
π2π€π
ππ‘2= 0.
(2)
In developing (2), viscoelastic layers represented by coeffi-cients π
0, π0, ππ, and π
πare assumed to exist between the first
beam and the surroundingmedium and the πth beam and thesurrounding medium and π€
0= 0 and π€
π+1= 0.
The equations represented by (2) are nondimensionalizedby introducing
π₯β=π₯
πΏ, (3a)
π€β
π=π€π
πΏ, (3b)
π‘β= π‘β
πΈ1πΌ1
π1π΄1πΏ4. (3c)
The nondimensional variables are substituted into (2) result-ing in
ππ
π4π€π
ππ₯4+ ππβ1(π€πβ π€πβ1) + ππ(π€πβ π€π+1)
+ ]πβ1(ππ€π
ππ‘βππ€πβ1
ππ‘) + ]π(ππ€π
ππ‘βππ€π+1
ππ‘)
+ π½π
π2π€π
ππ‘2= 0,
(4)
where the ββs have been dropped from the nondimensionalvariables and
ππ=πΈππΌπ
πΈ1πΌ1
, (5a)
Advances in Acoustics and Vibration 3
ππ=πππΏ4
πΈ1πΌ1
, (5b)
]π=
πππΏ2
βπΈ1πΌ1π1π΄1
, (5c)
π½π=πππ΄π
π1π΄1
. (5d)
The differential equations have a matrix-operator formula-tion as
(K + Kπ)W + C
πW +MW = 0, (6)
where W = [π€1(π₯, π‘) π€2(π₯, π‘) β β β π€π(π₯, π‘)]π, K is a π Γ π
diagonal operator matrix with ππ,π= ππ(π4π€π/ππ₯4),M is a πΓ
π diagonal mass matrix withππ,π= π½π, and K
πis a tridiagonal
π Γ π stiffness coupling matrix with
(ππ)π,πβ1
= βππβ1, π = 2, 3, . . . , π,
(ππ)π,π= ππβ1+ ππ, π = 1, 2, . . . , π,
(ππ)π,π+1
= βππ, π = 1, 2, . . . , π β 1
(7)
and Cπis a tridiagonal π Γ π damping coupling matrix with
(ππ)π,πβ1
= β]πβ1, π = 2, 3, . . . , π,
(ππ)π,π= ]πβ1+ ]π, π = 1, 2, . . . , π,
(ππ)π,π+1
= β]π, π = 1, 2, . . . , π β 1.
(8)
The vector W is an element of the vector space π = π Γ π π;
an element of π is an π-dimensional vector whose elementsall belong to π, 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 = wππππ‘, (9)
where π is a parameter and w =
[π€1(π₯) π€2(π₯) π€3(π₯) β β β π€πβ1(π₯) π€π(π₯)]π is a vector
of mode shapes corresponding to that natural frequency.Substitution of (9) into (6) leads to
(K + Kπ)w + ππC
πw = π2Mw, (10)
where the partial derivatives have been replaced by ordinaryderivatives in the definition of K.
A solution of the set of π ordinary differential equationsrepresented by (10) is assumed as
wπ (π₯) = ππ(π₯) aπ, (11)
where ππ(π₯) satisfies the equation
π4ππ
ππ₯4β π2
ππ = 0 (12)
subject to the homogeneous boundary conditions of thebeams and a
πis a vector of constants. The parameter π
πis
the πth natural frequency of an undamped beam with theappropriate end conditions. The values of π
πfor π = 1, 2, . . .
are the natural frequencies of the first beam in the seriesassuming the beam vibrates freely from the other beams andthe functions π
π(π₯) are the corresponding mode shapes.
Substitution of (12) into (10) leads to
(π2
πU + K
π) aπ+ ππC
πaπ= π
2Maπ, (13)
where U is an π Γ π diagonal matrix with π’π,π= ππ. Equation
(13) is a system of π homogeneous algebraic equations to solvefor aπ.
The differential equations governing the free vibrationsof a linear π-degree-of-freedom system with displacementvector x = [π₯1 π₯2 β β β π₯π]
π are summarized by
Mx + Cx + Kx = 0. (14)
A normal-mode solution is assumed as x = Xππππ‘ for (14),resulting in
βπ2MX + ππCX + KX = 0. (15)
Equation (15) is the same as (13) with K = π2
πU + K
π. Thus,
the same solution procedure is used to solve (13) as is used tosolve (15) for each π = 1, 2, 3, . . ..
4. General Solution
Following Kelly [1] the differential equations summarized by(11) can be rewritten as a system of 2π first-order equations ofthe form
My + Ky = 0, (16)
where
M = [0 MM C] , K = [βM 0
0 K] ,
y = [xx] .(17)
A solution to (17) is assumed as
y = Ξ¦πβπΎπ‘ (18)
which results upon substitution in
Mβ1KΞ¦ = πΎΞ¦. (19)
The values of π are related to the eigenvalues of Mβ1K by π =ππΎ. 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
4 Advances in Acoustics and Vibration
Table 1: Properties of the five beams of the example.
Beamnumber, π
Elasticmodulus, πΈ
π
(TPa)
Density, ππ
(kg/m3)
Innerradius, π
π,π
(nm)
Outerradius, π
π,π
(nm)
Area,π΄π= π(π
π,π
2β ππ,π
2)
(nm2)
Moment of inertiaπΌπ= (π/4)(π
π,π
4β ππ,π
4)
(nm4)
Length πΏπ
(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 (π₯, π‘) =β
β
π=0
(
π
β
π=1
π΅π,πXπ,ππβπΎπ,ππ‘)ππ (π₯) , (20)
where π΅π,π
are arbitrary constants of integration. When thevalues of πΎ
π,πare all complex and of the form
πΎπ,π= πΌπ,π+ ππ½π,π (21)
and the complex mode shapes have the form
Xπ,π= Uπ,π+ πVπ,π (22)
then (20) is written as
w (π₯, π‘)
=
β
β
π=0
{
π
β
π=1
πβπΌπ,ππ‘ [πΆ
π,π(Uπ,πcosπ½π,ππ‘ + Vπ,πsinπ½π,ππ‘)
+π·π,π(Vπ,πcosπ½π,ππ‘ β Uπ,πsinπ½π,ππ‘)] }
β ππ (π₯) .
(23)
In (23), πΆπ,πandπ·
π,πare constants of integration determined
from appropriate initial conditions.If a value of πΎ
π,πis real, the corresponding mode is
overdamped and there are two real values of πΎπ,π; call them
πΎπ,π,1
and πΎπ,π,2
. The real part has bifurcated into two valuesand the corresponding eigenvectors are real. The term insidethe inner summation corresponding to a real eigenvalue isπΆπ,πXπ,π,1πβπΎπ,π,1π‘ + π·
π,πXπ,π,2πβπΎπ,π,2π‘.
The spatially distributed mode shapes satisfy an orthogo-nality condition, which for a uniform beam is
β«
1
0
ππππdπ₯ = 0, π = π. (24)
Let w(π₯, 0) be a vector of initial conditions. Then
w (π₯, 0) =β
β
π=0
[
π
β
π=1
(πΆπ,πUπ,π+ π·π,πVπ,π)] ππ (π₯) . (25)
Multiplying both sides of (25) by ππ(π₯) for an arbitrary value
of π, integrating from 0 to 1, and using the orthogonalitycondition lead to the equation:
π
β
π=1
(πΆπ,πUπ,π+ π·π,πVπ,π) = β«
1
0
w (π₯, 0) ππ (π₯) dπ₯. (26)
A similar procedure is used for the vector of initial velocitiesw(π₯, 0) yielding
π
β
π=1
[πΆπ,π(βπΌπ,πUπ,π+ π½π,1Vπ,1)
β π·π,π(πΌπ,πVπ,π+ π½π,1Uπ,1)]
= β«
1
0
w (π₯, 0) ππ (π₯) dπ₯.
(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 πthbeam is π΄
π= π(π
π,π
2β ππ,π
2), where π
π,πis the outer radius
of the ith beam and ππ,πis the inner radius of the πth beam
which is the outer radius of the π-1st beam.The cross-sectionalmoment of inertia of the πth beam is πΌ
π= (π/4)(π
π,π
4β ππ,π
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:
ππ=366.67 (2π
π,π)
0.16π2erg/cm2, (28)
whereπ = 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.
Advances in Acoustics and Vibration 5
Table 2: Properties of layers in example.
Layer, π Stiffness parameter,ππ(TPa)
Damping parameter,ππ(Nβ 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
ππ (π₯) = cosβπ
ππ₯ β coshβπ
ππ₯
+ πΌπ(sinhβπ
ππ₯ β sinβπ
ππ₯) ,
(29)
where
πΌπ=cosβπ
πβ coshβπ
π
sinβππβ sinhβπ
π
(30)
and ππis the πth positive solution of
tanβππ= tanhβπ
π. (31)
The first five solutions of (31) are given in Table 4.The free vibration response is given by (23), where the
values of πΎπ,π
for π = 1, 2, . . . are determined using (13).Choosing π = 3, (13) is written as
π2
[[[[[
[
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
]]]]]
]
[[[[[
[
π3,1
π3,2
π3,3
π3,4
π3,5
]]]]]
]
+ ππ102
[[[[[
[
5.31 β5.31 0 0 0
β5.31 12.42 β7.11 0 0
0 β7.11 16.03 β8.92 0
0 0 β8.92 19.69 β10.72
0 0 0 β10.72 10.72
]]]]]
]
β
[[[[[
[
π3,1
π3,2
π3,3
π3,4
π3,5
]]]]]
]
+ 105
[[[[[
[
2.54 β2.54 0 0 0
β2.54 5.73 β3.18 0 0
0 β3.18 7.02 β3.89 0
0 0 β3.89 8.31 β4.51
0 0 0 β4.51 4.54
]]]]]
]
β
[[[[[
[
π3,1
π3,2
π3,3
π3,4
π3,5
]]]]]
]
=
[[[[[
[
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 π = 3 as
w3 (π₯, π‘)
=
{{{{{
{{{{{
{
πβ2.29π‘
{{{{{
{{{{{
{
πΆ3,1
[[[[[
[
[[[[[
[
2.64
2.49
2.27
2.01
1.92
]]]]]
]
cos (176.9π‘)
+
[[[[[
[
1.45
1.28
1.05
0.823
0.690
]]]]]
]
sin (176.9π‘)]]]]]
]
+ π·3,1
[[[[[
[
β
[[[[[
[
2.64
2.49
2.27
2.01
1.92
]]]]]
]
sin (176.9π‘)
+
[[[[[
[
1.45
1.28
1.05
0.823
0.690
]]]]]
]
cos (176.9π‘)]]]]]
]
}}}}}
}}}}}
}
+ πβ9.77π‘
{{{{{
{{{{{
{
πΆ3,2
[[[[[
[
[[[[[
[
1.77
1.03
0.0017
β0.832
β1.26
]]]]]
]
cos (326.9π‘)
+
[[[[[
[
1.09
0.704
0.185
β0.285
β0.562
]]]]]
]
sin (326.9π‘)]]]]]
]
+ π·3,2
[[[[[
[
β
[[[[[
[
1.77
1.03
0.0017
β0.832
β1.26
]]]]]
]
sin (326.9π‘)
+
[[[[[
[
1.09
0.704
0.185
β0.285
β0.562
]]]]]
]
cos (326.9π‘)]]]]]
]
}}}}}
}}}}}
}
+ πβ347.3π‘
{{{{{
{{{{{
{
πΆ3,3
[[[[[
[
[[[[[
[
β1.05
0.312
0.934
0.298
β0.606
]]]]]
]
cos (466.2π‘)
6 Advances in Acoustics and Vibration
Table 3: Nondimensional parameters.
π ππ= πΈππΌπ/πΈ1πΌ1
π½π= πππ΄π/π1π΄1
ππ= πππΏ4/πΈ1πΌ1
]π= πππΏ2/βπΈ1πΌ1π1π΄1
1 1 1 2.54 Γ 105 5.31 Γ 102
2 2.13 1.29 3.19 Γ 105 7.11 Γ 102
3 3.90 1.58 3.83 Γ 105 8.14 Γ 102
4 6.84 1.99 4.51 Γ 105 1.07 Γ 103
5 10.92 2.31 0 0
+
[[[[[
[
β4.06
1.01
4.10
1.27
β2.92
]]]]]
]
sin (466.2π‘)]]]]]
]
+ π·3,3
[[[[[
[
β
[[[[[
[
β1.05
0.312
0.934
0.298
β0.606
]]]]]
]
sin (466.2π‘)
+
[[[[[
[
β4.06
1.01
4.10
1.27
β2.92
]]]]]
]
cos (466.2π‘)]]]]]
]
}}}}}
}}}}}
}
+ πβ638.1π‘
{{{{{
{{{{{
{
πΆ3,4
[[[[[
[
[[[[[
[
5.38
β7.95
0.798
5.78
β3.37
]]]]]
]
cos (437.7π‘)
+
[[[[[
[
2.64
β3.32
β0.357
2.93
β1.61
]]]]]
]
sin (437.7π‘)]]]]]
]
+ π·3,4
[[[[[
[
β
[[[[[
[
5.38
β7.95
0.798
5.78
β3.37
]]]]]
]
sin (437.7π‘)
+
[[[[[
[
2.64
β3.32
β0.357
2.93
β1.61
]]]]]
]
cos (437.72π‘)]]]]]
]
}}}}}
}}}}}
}
+ πβ896.9π‘
{{{{{
{{{{{
{
πΆ3,5
[[[[[
[
[[[[[
[
0.985
β1.366
0.331
0.382
β0.196
]]]]]
]
cos (103.6π‘)
Table 4: Five lowest solutions of tanβπΏ = tanhβπΏ.
π πΏπ
1 15.422 49.963 104.24 178.35 272.0
+
[[[[[
[
β2.13
5.54
β7.28
5.53
β1.95
]]]]]
]
sin (103.6π‘)]]]]]
]
+ π·3,5
[[[[[
[
β
[[[[[
[
0.985
β1.366
0.331
0.382
β0.196
]]]]]
]
sin (103.6π‘)
+
[[[[[
[
β2.13
5.54
β7.28
5.53
β1.95
]]]]]
]
cos (103.6π‘)]]]]]
]
}}}}}
}}}}}
}
}}}}}
}}}}}
}
β βcos 10.2π₯ β cosh 10.2π₯
+πΌ3 (sinh 10.2π₯ β sin 10.2π₯)β .
(33)
The parameters πΎπ,π
for π = 1, 2, . . . , 5 and for all fivebeams are presented inTable 6. For these damping properties,all parameters are complex except for πΎ
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 πΎ
1,5is overdamped.
Let πΏ 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 π 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.
Advances in Acoustics and Vibration 7
Table 5: Undamped natural frequencies ππ,π, for example. π
π,πfor a fixed π and π = 1, 2, . . . , 5 is a set of intramodal frequencies, whereas π
π,π
and ππ,πrepresent intermodal frequencies.
π π1,π
π2,π
π3,π
π4,π
π5,π
1 2.688 Γ 101 8.647 Γ 101 1.754 Γ 102 2.797 Γ 102 3.866 Γ 102
2 2.978 Γ 102 3.075 Γ 102 3.425 Γ 102 4.275 Γ 102 5.961 Γ 102
3 5.540 Γ 102 5.594 Γ 102 5.790 Γ 102 6.261 Γ 102 7.153 Γ 102
4 7.538 Γ 102 7.578 Γ 102 7.724 Γ 102 8.075 Γ 102 8.755 Γ 102
5 8.906 Γ 102 8.937 Γ 102 9.017 Γ 102 9.34 Γ 102 9.942 Γ 102
Table 6: Values of πΎπ,πfor π = 1, 2, . . . , 5, for example.
π πΎ1,π
πΎ2,π
πΎ3,π
πΎ4,π
πΎ5,π
1 1.22 Γ 10β3Β± 268.8π 0.1319 Β± 8.65π 2.291 Β± 176.3π 15.17 Β± 286.7π 48.16 Β± 397.9π
2 110.0 Β± 280.6π 99.79 Β± 290.8π 97.68 Β± 326.9π 87.15 Β± 409.8π 70.17 Β± 562.8π
3 344.6 Β± 435.6π 344.4 Β± 442.3π 343.7 Β± 466.2π 341.7 Β± 521.3π 337.4 Β± 617.3π
4 638.7 Β± 405.3π 638.6 Β± 412.4π 638.1 Β± 437.7π 636.8 Β± 495.5π 634.5 Β± 595.5π
5 759.71.031 Γ 103
785.71.006 Γ 103
896.9 Β± 103.6π 897.8 Β± 256.6π 894.8 Β± 414.8π
πΌ3,i
108
106
104
102
100
10β2
10β3 10β2 10β1 100 101 102
πΏ
Figure 2: Real part of πΎ3,πfor each mode versus πΏ, for example. All
values except the lowest have a bifurcation for some value of πΏ.Whena bifurcation occurs, the mode is critically damped.
Figure 2 shows the real parts of πΎ3,πfor each mode versus
πΏ. The real part starts at zero (the undamped solution) andincreases until (except for the lowestmode) it bifurcates whenthe mode becomes overdamped.The value of πΏ for which thebifurcation occurs is larger for lower modes. The value of πΎ
3,1
does not bifurcate but reaches a maximum value and thendecreases.
The imaginary part of πΎ3,πfor eachmode is plotted against
πΏ in Figure 3.The higher modes vibrate at higher frequenciesfor small πΏ. 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
πΏ
π½3,i
Figure 3: Imaginary part of πΎ3,πfor eachmode versus πΏ, for example.
zero except for the lowest mode which approaches a constantvalue.
6. Conclusions
The free vibrations of a set of π 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
8 Advances in Acoustics and Vibration
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
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[3] Z. Oniszczuk, βTransverse vibrations of elastically connecteddouble-string complex system. Part II: forced vibrations,β Jour-nal of Sound and Vibration, vol. 232, no. 2, pp. 367β386, 2000.
[4] J. M. Selig and W. H. Hoppmann, βNormal mode vibrationsof systems of elastically connected parallel bars,β Journal of theAcoustical Society of America, vol. 36, pp. 93β99, 1964.
[5] E. E. Osborne, βComputations of bending modes and modeshapes of single and double beams,β Journal of the Society forIndustrial and Applied Mathematics, vol. 10, no. 2, pp. 329β338,1962.
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[13] K. Y. Xu, X. N. Guo, and C. Q. Ru, βVibration of a double-walled carbon nanotube aroused by nonlinear intertube van derWaals forces,β Journal of Applied Physics, vol. 99, no. 6, ArticleID 064303, 2006.
[14] I. Elishakoff andD. Pentaras, βFundamental natural frequenciesof double-walled carbon nanotubes,β Journal of Sound andVibration, vol. 322, no. 4-5, pp. 652β664, 2009.
[15] Z. Oniszczuk, βDamped vibration analysis of an elastically con-nected complex double-string system,β Journal of Sound andVibration, vol. 264, no. 2, pp. 253β271, 2003.
[16] A. Palmeri and S. Adhikari, βA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beam sys-tems with viscoelastic inner layer,β Journal of Sound and Vibra-tion, vol. 330, no. 26, pp. 6372β6386, 2011.
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