Free Vibrations of a Series of Beams Connected by
Viscoelastic Layers
S. Graham Kelly
Clint Nicely
The University of Akron
ABSTRACT
An exact solution for free vibrations of a series of uniform Euler-Bernoulli beams connected by KelvinVoigt is developed. The beams have the same length and end conditions but can have different material
or geometric properties. An example of five concentric beams connected by viscoelastic layers is
considered.
Keywords: viscoealstic layers, elastically coupled systems
1. Introduction
This paper presents an exact solution to the problem of the free vibrations of an arbitrary
number of beams connected by viscoelastic layers of the Kelvin-Voigt type. The beams and the
layers may have different properties but the beams must have the same length and the same
end conditions.
The general theory for the free and forced response of strings, shafts, beams, and axially
loaded beams is well documented [1]. Oniszczuk [2]-[3] investigated the free and forced
responses of elastically connected strings. Using a normal-mode solution he analyzed two
coupled second order ordinary differential equations to determine the natural frequencies. He
used a modal analysis to determine the forced response. Selig and Hoppmann [4], Osborne [5]
and Oniszczuk [6] studied the free or forced response of elastically connected Euler-Bernoulli
beams. They each used a normal mode analysis resulting in coupled sets of fourth-order
differential equations whose eigenvalues were related to the natural frequencies. Rao [7] also
employed a normal mode solution to compute the natural frequencies of elastically connected
Timoshenko beams. Each study did not consider damping of the beams or damping in the
elastic connection.
Kelly [8] developed a general theory for the exact solution of free vibrations of
elastically coupled structures without damping. The structures may have different properties or
even be non-uniform but they have the same supports. He applied the theory to Euler-Bernoulli
beams and concentric torsional shafts. Kelly and Srinivas [9] developed a Rayleigh-Ritz method
for elastically connected stretched Euler-Bernoulli beams.
Yoon, Ru and Mioduchowski [10] and Li and Chou [11] proposed that free vibrations of
multi-walled carbon nanotubes can be modeled by elastically connected Euler-Bernoulli beams.
They employed normal-mode solutions, showing that multi-walled nanotubes have an infinite
series of non-coaxial modes. Yoon, Ru, and Mioduchowski [12] modeled free vibrations of
nanotubes with concentric Timoshenko beams connected by an elastic layer. Xu, Guo, and Ru
[13] modeled nonlinear vibrations the in elastically connected structures modeling nanotubes
by considering the nonlinearity of the vanDer Waals forces. They analyzed the nonlinear free
vibrations by employing a Galerkin method. Elishakoff and Pentaras [14] developed
approximate formulas for the natural frequencies of double walled nanotubes modeled as
concentric elastically coupled beams, noting that if developed from the eigenvalue relation the
computations can be computationally intensive and difficult.
Damped vibrations of elastically connected structures have been studied by few
authors. Oniszczuk [15] used a normal mode solution in considering the vibration of two strings
connected by a viscoelastic layer of the Kelvin-Voigt type. Palmeri and Adhikari [16] used a
Galerkin method to analyze the vibrations of a double-beam system connected by a viscoelastic
layer of the Maxwell type. Jun and Hongxing [17] used a dynamic stiffness matrix to analyze
free vibrations of three beams connected by viscoelastic layers. Their analysis does not require
the beams to have the same end condi tions, but does require the use of computational tools to
determine the natural frequencies.
An exact solution for the free vibration of a series of elastically connected EulerBernoulli beams is considered in this paper. The elastic layers are viscoelastic with damping of
the Kelvin-Voigt type. The results are applied to a series of five concentric beams .
2. Problem formulation
Consider n Euler-Bernoulli beams connected by viscoelastic layers as shown in Fig ure 1.
Each beam is assumed to have its own neutral axis. All beams are uniform of length 𝐿. Let 𝐸𝑖 be
the elastic modulus,𝜌𝑖 be the mass density, 𝐴𝑖 be the cross-sectional area and 𝐼𝑖 be the cross
sectional moment of inertia of the i th beam about the neutral axis of the i th beam. Let 𝑤𝑖 (𝑥, 𝑡)
represent the transverse displacement of the ith beam, where x is the distance along the neutral
axis of the beam measured from its left end and 𝑡 represents time. Damping in each beam due
to structural damping or complex stiffness is neglected. The viscoelastic layer between the ith
and i plus first layer is of the Kelvin-Voigt type and has two parameters, 𝑐𝑖 representing the
damping property of the layer and 𝑘𝑖 representing the stiffness of the layer, such that the force
acting on the ith beam from the layer is
𝐺𝑖 = 𝑐𝑖
𝜕𝑤𝑖+1 𝜕𝑤𝑖
−
+ 𝑘𝑖 𝑤𝑖+1 − 𝑤𝑖
𝜕𝑡
𝜕𝑡
(1)
Hamilton’s principle is used to derive the equations governing the free response of the
ith beams as
𝐸𝑖 𝐼𝑖
𝜕4 𝑤𝑖
𝜕𝑤𝑖 𝜕𝑤𝑖−1
𝜕𝑤𝑖 𝜕𝑤𝑖+1
+ 𝑐𝑖−1
−
+ 𝑐𝑖
−
+ 𝑘𝑖−1 𝑤𝑖 − 𝑤𝑖−1
4
𝜕𝑥
𝜕𝑡
𝜕𝑡
𝜕𝑡
𝜕𝑡
+ 𝑘𝑖 𝑤𝑖 − 𝑤𝑖+1 + 𝜌𝑖 𝐴𝑖
(2)
𝜕2 𝑤 𝑖
=0
𝜕𝑡 2
In developing equation (2), viscoelastic layers represented by coefficients 𝑐0 , 𝑘0 , 𝑐𝑛 and 𝑘𝑛 are
assumed to exist between the first beam and the surrounding medium and the n th beam a nd
the surrounding medium and 𝑤0 = 0 and𝑤𝑛 +1 = 0.
The equations represented by equation (2) are non-dimensionalized by introducing
𝑥∗ = 𝑥 𝐿
𝑤𝑖∗ = 𝑤𝑖 𝐿
𝑡∗ = 𝑡
𝐸1 𝐼1
𝜌1 𝐴1 𝐿4
The non-dimensional variables are substituted into Eq. (2) resulting in
3𝑎
3𝑏
3𝑐
𝜇𝑖
𝜕4 𝑤 𝑖
𝜕𝑤𝑖 𝜕𝑤𝑖−1
+
𝜂
𝑤
−
𝑤
+
𝜂
𝑤
−
𝑤
+
𝜈
−
𝑖
−1
𝑖
𝑖−1
𝑖
𝑖
𝑖+1
𝑖
−1
𝜕𝑥 4
𝜕𝑡
𝜕𝑡
+ 𝜈𝑖
(4)
𝜕𝑤𝑖 𝜕𝑤𝑖+1
𝜕2 𝑤 𝑖
−
+ 𝛽𝑖
=0
𝜕𝑡
𝜕𝑡
𝜕𝑡 2
where the *s have been dropped from the non-dimensional variables and
𝜇𝑖 =
𝜂𝑖 =
𝜈𝑖 =
𝛽𝑖 =
𝐸𝑖 𝐼𝑖
𝐸1 𝐼1
𝑘𝑖 𝐿4
𝐸1 𝐼1
𝑐𝑖 𝐿2
𝐸1 𝐼1 𝜌1 𝐴1
𝜌𝑖 𝐴𝑖
𝜌1 𝐴1
5𝑎
5𝑏
(5𝑐)
(5𝑑)
The differential equations have a matrix-operator formulation as
  MW
  0
K  K c W  Cc W
(6)
where W  w1 ( x, t ) w2 ( x, t )  wn ( x, t ) , K is a nxn diagonal operator matrix with
T
 4 wi
k i ,i   i
, M is a nxn diagonal mass matrix with mi ,i   i , K c is a tri-diagonal nxn stiffness
x 4
coupling matrix with
kc i ,i1  i1
kc i ,i  i1  i
kc i ,i1  i
i  2,3,  n
i  1,2, , n
i  1,2,  n  1
(7)
and C c is a tri-diagonal nxn damping coupling matrix with
cc i ,i1   i1
cc i ,i   i1   i
cc i ,i1   i
i  2,3,  n
i  1,2, , n
(8)
i  1,2,  n  1
The vector W is an element of the vector space U=SxRn; an element of U is an n-dimensional
vector whose elements all belong to S, the space of functions which satisfy the homogeneous
boundary conditions of each beam.
3. Free vibrations
A normal-mode solution of Eq. (6) is assumed as
𝐖 = 𝐰ei𝜔𝑡
(9)
where  is a parameter and w  w1 ( x) w2 ( x) w3 ( x)  wn1 ( x) wn ( x) is a vector of
T
mode shapes corresponding to that natural frequency. Substitution of Eq. (9) into Eq. (6) leads
to
K  K c w  iC c w   2 Mw
(10)
where the partial derivatives have been replaced by ordinary derivatives in the definition of K.
A solution of the set of n ordinary differential equations represented by Eq. (10) is
assumed as
𝐰𝐤 𝑥 = 𝜙𝑘 𝑥 𝐚𝐤
(11)
where 𝜙𝑘 (𝑥) satisfies the equation
𝑑 4 𝜙𝑘
− 𝜉𝑘2 𝜙 = 0
𝑑𝑥 4
(12)
subject to the homogeneous boundary conditions of the beams and 𝐚𝒌 is a vector of constants.
The parameter 𝜉𝑘 is the kth natural frequency of an undamped beam with the appropriate end
conditions. The values of 𝜉𝑘 for k=1,2,… are the natural frequencies of the first beam in the
series assuming the beam vibrates freely from the other beams and the functions 𝜙𝑘 (𝑥) are the
corresponding mode shapes.
Substitution of Eq. (12) into Eq. (10) leads to

2
k

U  K c a k  iC c a k   2 Ma k
(13)
where U is an nxn diagonal matrix with 𝑢 𝑖,𝑖 = 𝜇𝑖 . Equation (13) is a system of n homogeneous
algebraic equations to solve for 𝐚𝑘 .
The differential equations governing the free vibrations of a linear n-degree-of-freedom
system with displacement vector 𝐱 = 𝑥 1 𝑥 2 …
𝑥 𝑛 𝑇are summarized by
𝐌𝐱 + 𝐂𝐱 + 𝐊𝐱 = 𝟎
(14)
A normal mode solution is assumed as 𝐱 = 𝐗ei𝜔𝑡 for Eq. (14) as, resulting in
−𝜔2 𝐌𝐗 + i𝜔𝐂𝐗 + 𝐊𝐗 = 𝟎
(15)
Equation (15) is the same as equation (13) with K   k2 U  K c . Thus the same solution
procedure is used to solve Equation (13) as is used to solve Equation (15) for each k=1,2,3,….
4. General solution
Following Kelly [1] the differential equations summarized by Eq. (11) can be rewritten as
a system of 2n first-order equations of the form
𝐌𝐲 + 𝐊𝐲 = 𝟎
(16)
−𝐌 𝟎
𝟎 𝐊
(17)
where
𝐌=
𝟎
𝐌
𝐌
𝐂
𝐊=
𝐲= 𝐱
𝐱
A solution to Eq. (17) is assumed as
𝐲 = 𝚽e−𝛾𝑡
(18)
which results upon substitution in
𝐌 −𝟏 𝐊𝚽 = 𝛾𝚽
(19)
The values of 𝜔 are related to the eigenvalues of 𝐌 −𝟏 𝐊 by𝜔 = i𝛾. The resulting problem has, in
general, complex eigenvalues. The corresponding mode shape vectors are also complex. The
real part of an eigenvalue is negative and is an indication of the damping properties of that
mode. When complex eigenvalues occur they occur in complex conjugate pairs. The imaginary
part is the frequency of the mode. The mode shape vectors corresponding to complex
conjugate eigenvalues are also complex conjugates of one another. When the general solution
is written as a linear combination over all mode shapes the complex eigenvalues and the
complex eigenvectors combine leading to terms involving the sine and cosine of the imaginary
part of the eigenvalues
The general solution of the partial differential equations is
∞
𝑛
𝐵𝑘,𝑖 𝐗 𝑘,𝑖 𝑒 −𝛾𝑘 ,𝑖 𝑡 𝜙𝑘 (𝑥)
𝐰 𝑥, 𝑡 =
𝑘=0
(20)
𝑖=1
where 𝐵𝑖,𝑘 are arbitrary constants of integration. When the values of 𝛾𝑖 ,𝑘 are all complex and of
the form
𝛾𝑘 ,𝑖 = 𝛼𝑘,𝑖 + i𝛽𝑘,𝑖
(21)
and the complex mode shapes have the form
𝐗 𝑘,𝑖 = 𝐔𝑘 ,𝑖 + i𝐕𝑘,𝑖
(22)
then Eq. (20) is written as
∞
𝑛
𝑒 −𝛼𝑘 ,𝑖 𝑡 𝐶𝑘,𝑖 𝐔𝑘,𝑖 cos 𝛽𝑘,𝑖 𝑡 + 𝐕𝑘,𝑖 sin 𝛽𝑘,𝑖 𝑡
𝐰 𝑥, 𝑡 =
𝑘=0 𝑖 =1
+ 𝐷𝑘,𝑖 𝐕𝑘,𝑖 cos 𝛽𝑘,𝑖 𝑡 − 𝐔𝑘,𝑖 sin 𝛽𝑘,𝑖 𝑡
𝜙𝑘 (𝑥)
(23)
In Eq. (23) 𝐶𝑘,𝑖 and 𝐷𝑘,𝑖 are constants of integration determined from appropriate initial
conditions.
If a value of 𝛾𝑘 ,𝑖 is 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 values and the
corresponding eigenvectors are real. The term inside the inner summation corresponding to a
real eigenvalue is 𝐶𝑘,𝑖 𝐗 𝑘,𝑖 ,1 𝑒 −𝛾𝑘 ,𝑖,1 𝑡 + 𝐷𝑘,𝑖 𝐗 𝑘,𝑖,2 𝑒 −𝛾𝑘 ,𝑖,2 𝑡 .
The spatially distributed mode shapes satisfy an orthogonality condition, which for a
uniform beam is
1
0
𝜙𝑗 𝜙𝑘 d𝑥 = 0
𝑗≠𝑘
(24)
Let 𝐰(𝑥, 0) be a vector of initial conditions. Then
∞
𝑛
𝐰 𝑥, 0 =
𝐶𝑘,𝑖 𝐔𝑘 ,𝑖 + 𝐷𝑘,𝑖 𝐕𝑘,𝑖
𝜙𝑘 (𝑥)
(25)
𝑘=0 𝑖=1
Multiplying both sides of Eq. (25) by 𝜙𝑗 (𝑥) for an arbitrary value of j, integrating from 0 to 1
and using the orthogonality condition leads to the equations
𝑛
1
𝐶𝑗,𝑖 𝐔𝑗 ,𝑖 + 𝐷𝑗,𝑖 𝐕𝑗,𝑖 =
𝑖 =1
0
𝐰 𝑥, 0 𝜙𝑗 𝑥 d𝑥
(26)
A similar procedure is used for the vector of initial velocities 𝐰 𝑥, 0 yielding
𝑛
1
𝐶𝑗,𝑖 −𝛼𝑘,𝑖 𝑈𝑗,𝑖 + 𝛽𝑗,1 𝑉𝑗,1 − 𝐷𝑗,𝑖 𝛼𝑘,𝑖 𝑉𝑗,𝑖 + 𝛽𝑗 ,1 𝑈𝑗,1
𝑖 =1
=
0
𝐰 𝑥, 0 𝜙𝑗 𝑥 d𝑥
(27)
5. Example
Consider five-concentric fixed-pinned beams connected by viscoelastic layers of the KelvinVoigt type of negligible thickness. The cross-sectional moment of inertia of the ith beam is
𝐴𝑖 = 𝜋 𝑟𝑖,𝑜 2 − 𝑟𝑖,𝑖 2 where 𝑟𝑖,𝑜 is the outer radius of the ith beam and 𝑟𝑖,𝑖 is the inner radius of
the ith beam which is the outer radius of the i-1st beam. The cross-sectional moment of inertia
𝜋
of the ith beam is 𝐼𝑖 = 4 𝑟𝑖,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
those generated by the van der Waals forces between atoms in a carbon nanotube and is given
by a formula derived using the data of Girifalco and Lad [18] and the Lennerd-Jones potential
function
𝑘𝑖 =
366.67 2𝑟𝑖,𝑖 erg
0.16𝑑 2 cm2
(28)
where 𝑑 = 0.147 nm is the inter-atomic distance between bond lengths. The damping
parameters are assumed.
The mode shapes of a fixed-pinned beam are
𝜙𝑘 𝑥 = cos 𝜉𝑘 𝑥 − cosh 𝜉𝑘 𝑥 + 𝛼𝑘 sinh 𝜉𝑘 𝑥 − sin 𝜉𝑘 𝑥
(29)
where
𝛼𝑘 =
cos 𝜉𝑘 −cosh 𝜉𝑘
sin 𝜉𝑘 − sinh 𝜉𝑘
(30)
and 𝜉𝑘 is the kth positive solution of
tan 𝜉𝑘 = tanh 𝜉𝑘
The first five solutions of Eq. (31) are given in Table 4.
(31)
The free vibration response is given by Eq. (23) where the values of 𝛾𝑘 ,𝑖 for k=1,2… are
determined using Eq.(13). Choosing k=3 Eq. (13) is written as
1
0
𝜔2 0
0
0
5.31
−5.31
𝑖𝜔102
0
0
0
2.54
−2.54
105
0
0
0
−2.54
5.73
−3.18
0
0
0 0
0 0
0
0
1.991
0
0
2.317
𝑎3,1
𝑎3,2
𝑎3,3 +
𝑎3,4
𝑎3,5
0 0 0
−7.11 0 0
16.03 −8.92
0
−8.92 19.69 −10.72
0
−10.72 10.72
𝑎3,1
𝑎3,2
𝑎3,3 +
𝑎3,4
𝑎3,5
0
1.291
0
0
0
−5.31
12.42
−7.11
0
0
0 0 0
−3.18 0 0
7.02 −3.89
0
−3.89 8.31 −4.51
0
−4.51 4.54
1.581
0
0
0
0
𝑎3,1
0
𝑎3,2
0
𝑎3,3 = 0
𝑎3,4
0
𝑎3,5
0
(32)
A solution of the form of Eq. (23) is applied resulting in the portion of the solution of Eq. (10)
corresponding to k=3 as
𝐰3 𝑥, 𝑡 = 𝑒 −2.29𝑡 𝐶3,1
+ 𝐷3,1
2.64
1.45
2.49
1.28
2.27
−
sin 176.9𝑡 + 1.05 cos 176.9𝑡
2.01
0.823
1.92
0.690
+ 𝑒 −9.77𝑡
+ 𝐷3,2
2.64
1.45
2.49
1.28
2.27 cos 176.9𝑡 + 1.05 sin 176.9𝑡
2.01
0.823
1.92
0.690
𝐶3,2
1.09
1.77
1.03
0.704
− 0.0017 sin 326.9𝑡 + 0.185 cos 326.9𝑡
−0.832
−0.285
−1.26
−0.562
+ 𝑒 −347 .3𝑡 𝐶3,3
+ 𝐷3,3
1.09
1.77
1.03
0.704
0.0017 cos 326.9𝑡 + 0.185 sin 326.9𝑡
−0.832
−0.285
−1.26
−0.562
−1.05
−4.06
1.01
0.312
0.934 cos 466.2𝑡 + 4.10 sin 466.2𝑡
0.298
1.27
−0.606
−2.92
−1.05
−4.06
1.01
0.312
− 0.934 sin 466.2𝑡 + 4.10 cos 466.2𝑡
0.298
1.27
−0.606
−2.92
+𝑒 −638 .1𝑡 𝐶3,4
+ 𝐷3,4
2.64
5.38
−3.32
−7.95
− 0.798 sin 437.7𝑡 + −0.357 cos 437.72𝑡
5.78
2.93
−3.37
−1.61
+ 𝑒 −896.9𝑡 𝐶3,5
+ 𝐷3,5
2.64
5.38
−3.32
−7.95
0.798 cos 437.7𝑡 + −0.357 sin 437.7𝑡
5.78
2.93
−3.37
−1.61
0.985
−2.13
5.54
−1.366
0.331 cos 103.6𝑡 + −7.28 sin 103.6𝑡
0.382
5.53
−0.196
−1.95
0.985
−2.13
5.54
−1.366
− 0.331 sin 103.6𝑡 + −7.28 cos 103.6𝑡
0.382
5.53
−0.196
−1.95
cos 10.2𝑥
− cosh10.2𝑥 + 𝛼3 sinh 10.2𝑥 − − sin 10.2𝑥
(33)
The parameters 𝛾𝑘 ,𝑖 for k=1,2,…5 and for all five beams are presented in Table 6. For these
damping properties all parameters are complex except for 𝛾1,5 . The real part represents the
amount of damping a mode has while the imaginary part is the damped natural frequency of
the mode. The mode represented by 𝛾1,5 is overdamped.
Let 𝛿 represent the damping coefficient of the first layer and assume the damping
parameter of each layer is proportional to the stiffness of the layer. The damping does not
constitute proportional damping (Rayleigh damping) for a specific value of 𝑘 as the stiffness
matrix is a combination of the coupling stiffness matrix due to the viscoelastic layers and the
diagonal bending stiffness matrix whereas the damping matrix is just from the viscoelastic
layers.
Figure 2 shows the real parts of 𝛾3,𝑖 for each mode versus 𝛿. The real parts starts at zero
(the undamped solution) and increases until (except for the lowest mode) it bifurcates when
the mode becomes overdamped. The value of 𝛿 for which the bifurcation occurs is larger for
lower modes. The value of 𝛾3 ,1 does not bifurcate, but reaches a maximum value and then
decreases.
The imaginary part of 𝛾3 ,𝑖 is for each mode is plotted against 𝛿 in Figure 3. The higher
modes vibrate at higher frequencies for small 𝛿. For higher delta the imaginary part goes to
zero except for the lowest mode which approaches a constant value.
6. Conclusions
The free vibrations of a set of n beams connected by viscoelastic layers of the Kelvin-Voigt
type are considered. The beams have the same length and are subject to the same end
conditions, but may have different properties. The equations of motion are derived and nondimensionalized. A normal mode solution is assumed. When substituted into the partial
differential equations it leads to a set of ordinary differential equations 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 each beam is proportional to the bending
stiffness of the first beam, however it is not necessary that all properties of the beams are
proportional. The result is, for each mode, a matrix equation which is similar to the matrix
equation governing a discrete linear system with damping. The method used to find the free
response of a discrete linear system is used to solve for the parameters governing the
vibrations of a continuous systems connected by Kelvin-Voigt layers.
A Kelvin-Voigt model was assumed for layers between multi walled nanotubes with the
elasticity representing the van der Waals forces between atoms. The damping was assumed to
present an example. However the method can be used for any form of linear damping in the
beams or in the layers. Thus a model of a multi-walled nanotube with linear damping in the
nanotubes can be analyzed using the method presented.
References
[1] S.G. Kelly, Advanced Vibration Analysis. CRC Press, Taylor and Francis Group; Boca Raton,
FL, 2007.
[2] Z. Oniszczuk, Transverse vibrations of elastically connected double string complex system.
Part 1 free vibrations. Journal of Sound and Vibration, 232 (2000) pp. 355-366.
[3] Z. Oniszczuk, Transverse vibrations of elastically connected double string complex system.
Part II forced vibrations. Journal of Sound and Vibration, 232 (2000) pp. 367-386.
[4] J.M. Selig, W.H. Hoppmann, Normal mode vibrations of systems of elastically connected
parallel bars. Journal of the Acoustical Society of America 36 (1964) pp. 93-99.
[5] E. Osborne, Computations of bending modes and mode shapes of single and double beams .
Journal of the Society of Industrial and Applied Mathematics 10 (1962) pp. 329-338.
[6] Z. Oniszczuk, Free transverse vibrations of elastically connected simply supported doublebeam complex systems. Journal of Sound and Vibration, 232, (2000), pp. 387-403.
[7] S.S. Rao, Natural frequencies of systems of elastically connected Timoshenko beams ,
Journal of the Acoustical Society of America, 55 (1974) 1232-1237
[8] Kelly, S.G., Free and forced vibrations of elastically connected structures, Advances in
Acoustics and Vibrations, 2010, 984361, 11 pages.
[9] Kelly, S.G., Srinivas, S., Free vibration of elastically-connected stretched beams, Journal of
Sound and Vibration, 326 (2009), 883-893.
[10] J. Yoon, C.Q. Ru, A. Mioduchowski, Non coaxial resonance of an isolated multi-walled
nanotube. Physics Review B, 66 (2002) 233402-1-233403-4.
[11] C. Li, T. Chou, Vibrational behaviors of multiwalled-carbon naotube-based nanomechanical
resonators, Applied Physics Letters, 84 (2004), 121-123.
[12]J. Yoon, C.Q. Ru, A. Mioduchowski, Terahertz vibration of short carbon nanotubes
modeled as Timoshenko beams, Journal of Applied Mechanics., 72 (2005), 10-17.
[13] K.Y. Xu, X.N. Guo, C.Q. Ru, Vibration of a double-walled carbon nanotube aroused by
nonlinear intertube van der Waals forces, Journal of Applied Physics, 99 (2006), 063403-063410.
[14] I. Elishakoff, D. Pentaras, Fundamental natural frequencies of double-walled carbon
nanotubes, Journal of Sound and Vibration, 322 (2009), 652-664.
[15] Z. Oniszczuk, Damped vibration of an elastically connected complex double-string system,
Journal of Sound and Vibration, 264 (2003), 253-271.
[16] A. Palmeri, S.A. Adhikari, Galerkin state-space approach for transverse vibrations of
slender double-beam systems with viscoelastic inner layer, Journal of Sound and Vibration, 330
(2011) 6372-6386.
[17] L. Jun, H. Hongxing, Dynamic stiffness vibration analysis of an elastically connected threebeam system, Applied Acoustics, 69 (2008), 591-600.
[18] L. Girifalco, R. Lad, Energy of cohesion, compressibility, and the potential energy of the
graphite system, Journal of Chemical Physics,55, (1956) 693-697.
Figure 1: Schematic representation of problem considered, k beams in parallel connected by
viscoelastic layers of the Kelvin-Voigt type.
.
8
10
6
10
4
 3,i
10
2
10
0
10
-2
10
-3
10
-2
10
-1
0
10
10
1
10
2
10

.
Figure 2: Real part of 𝛾3 ,𝑖 for each mode versus 𝛿 for example. All values except the lowest has
a bifurcation for some value of 𝛿. When a bifurcation occurs the mode is critically damped.
1000
900
800
700
3,i
600
500
400
300
200
100
0
0
0.05
0.1
0.15
0.2
0.25
0.3

Figure 3: Imaginary part of 𝛾3,𝑖 for each mode versus 𝛿 for example.
0.35
0.4
Table 1 Properties of the five beams of the example
Beam
Elastic
Density,
number, modulus, 𝜌𝑖
𝑖
𝐸𝑖 (TPa)
(kg/m3)
Inner
Outer
Area,
Moment
Length
radius,
radius,
𝐴𝑖 =
of inertia
𝐿 𝑖 (nm)
𝑟𝑖,𝑖 (nm)
𝑟𝑖,𝑜 (nm)
𝜋 𝑟𝑖,𝑜 2 − 𝑟𝑖,𝑖 2
𝐼𝑖 =
(nm2)
𝜋
4
𝑟𝑖,𝑜 4 −
𝑟𝑖,𝑖 4
(nm4)
1
1
1300
1.0
1.34
2.50
1.75
20
2
1
1300
1.34
1.68
3.23
3.73
20
3
1
1300
1.68
2.02
3.95
6.82
20
4
1
1300
2.02
2.36
4.98
12.12
20
5
1
1300
2.36
2.70
5.79
19.07
20
Table 2 Properties of layers in example
Layer, 𝑖
Stiffness parameter, 𝑘𝑖 (TPa)
Damping parameter, 𝑐𝑖
(N∙s/m2)
0
0
0
1
0.277
0.1
2
0.347
0.134
3
0.418
0.168
4
0.493
0.202
5
0
0
Table 3 Non-dimensional parameters
𝑖
𝜇𝑖 =
𝐸𝑖 𝐼𝑖
𝐸1 𝐼1
𝛽𝑖 =
𝜌𝑖 𝐴𝑖
𝜌1 𝐴1
𝑘𝑖 𝐿4
𝜂𝑖 =
𝐸1 𝐼1
𝜈𝑖
=
𝑐𝑖 𝐿2
𝐸1 𝐼1 𝜌1 𝐴1
1
1
1
2.54x105
5.31x102
2
2.13
1.29
3.19x105
7.11x102
3
3.90
1.58
3.83x105
8.14x102
4
6.84
1.99
4.51x105
1.07x103
5
10.92
2.31
0
0
Table 4 Five lowest solutions of tan 𝛿 = tanh 𝛿
𝑖
𝛿𝑖
1
15.42
2
49.96
3
104.2
4
178.3
5
272.0
Table 5 Undamped natural frequencies 𝜔𝑘,𝑖 for example. 𝜔𝑘,𝑖 for a fixed k and i=1,2,…,5 is a set
of intramodal frequencies whereas 𝜔𝑘,𝑖 and 𝜔𝑗,𝑖 represent intermodal frequencies.
𝑖
𝜔1,𝑖
𝜔2,𝑖
𝜔3,𝑖
𝜔4,𝑖
𝜔5,𝑖
1
2.688x101
8.647x101
1.754x102
2.797x102
3.866x102
2
2.978x102
3.075x102
3.425x102
4.275x102
5.961x102
3
5.540x102
5.594x102
5.790x102
6.261x102
7.153x102
4
7.538x102
7.578x102
7.724x102
8.075x102
8.755x102
5
8.906x102
8.937x102
9.017x102
9.34x102
9.942x102
Table 6 Values of 𝛾𝑘 ,𝑖 for 𝑘 = 1,2 … ,5 for example
𝑖
1
𝛾1,𝑖
1.22x10-3
𝛾2 ,𝑖
𝛾3 ,𝑖
𝛾4,𝑖
𝛾5 ,𝑖
0.1319±8.65i
2.291±176.3i
15.17±286.7i
48.16±397.9i
±268.8i
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
759.7
785.7
896.9±103.6i
897.8±256.6i
894.8±414.8i
1.031x103
1.006x103
The authors declare that there is no conflict of interest regarding publication of this article.