# Free Vibrations of a Series of Beams Connected by Viscoelastic Layers

Authors: S. Graham Kelly; Clint Nicely
Journal: Advances in Acoustics and Vibration
 (2015)

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## Abstract

An exact solution for free vibrations of a series of uniform Euler-Bernoulli beams connected by Kelvin-Voigt 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.


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## Body

## 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 nonuniform but they have the same support. 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 et al. [10] and Li and Chou [11] proposed that free vibrations of multiwalled carbon nanotubes can be modeled by elastically connected Euler-Bernoulli beams. They employed normal-mode solutions, showing that multiwalled nanotubes have an infinite series of noncoaxial modes. Yoon et al. [12] modeled free vibrations of nanotubes with concentric Timoshenko beams connected by an elastic layer. Xu et al. [13] modeled nonlinear vibrations in the elastically connected structures modeling nanotubes by considering the nonlinearity of the van der 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 conditions 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 Euler-Bernoulli 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

Considern Euler-Bernoulli beams connected by viscoelastic layers as shown in Figure 1. Each beam is assumed to have its own neutral axis. All beams are uniform of length L. Let Ei be the elastic modulus, let ρi be the mass density, let Ai be the cross-sectional area, and let Ii be the cross-sectional moment of inertia of the ith beam about the neutral axis of the ith 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
(1)Gi=ci∂wi+1∂t-∂wi∂t+kiwi+1-wi.Figure 1
Schematic representation of problem considered,k beams in parallel connected by viscoelastic layers of the Kelvin-Voigt type.Hamilton’s principle is used to derive the equations governing the free response of theith beams as
(2)EiIi∂4wi∂x4+ci-1∂wi∂t-∂wi-1∂t+ci∂wi∂t-∂wi+1∂t+ki-1wi-wi-1+kiwi-wi+1+ρiAi∂2wi∂t2=0.
In developing (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 nth beam and the surrounding medium and w0=0 and wn+1=0.The equations represented by (2) are nondimensionalized by introducing
(3a)x*=xL,(3b)wi*=wiL,(3c)t*=tE1I1ρ1A1L4.
The nondimensional variables are substituted into (2) resulting in
(4)μi∂4wi∂x4+ηi-1wi-wi-1+ηiwi-wi+1+νi-1∂wi∂t-∂wi-1∂t+νi∂wi∂t-∂wi+1∂t+βi∂2wi∂t2=0,
where the *’s have been dropped from the nondimensional variables and
(5a)μi=EiIiE1I1,(5b)ηi=kiL4E1I1,(5c)νi=ciL2E1I1ρ1A1,(5d)βi=ρiAiρ1A1.
The differential equations have a matrix-operator formulation as
(6)K+KcW+CcW˙+MW¨=0,
where W=w1x,tw2x,t⋯wnx,tT, K is a n×n diagonal operator matrix with ki,i=μi(∂4wi/∂x4), M is a n×n diagonal mass matrix with mi,i=βi, and Kc is a tridiagonal n×n stiffness coupling matrix with
(7)kci,i-1=-ηi-1,i=2,3,…,n,kci,i=ηi-1+ηi,i=1,2,…,n,kci,i+1=-ηi,i=1,2,…,n-1
and Cc is a tridiagonal n×n damping coupling matrix with
(8)cci,i-1=-νi-1,i=2,3,…,n,cci,i=νi-1+νi,i=1,2,…,n,cci,i+1=-νi,i=1,2,…,n-1.
The vector W is an element of the vector space U=S×Rn; 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 (6) is assumed as
(9)W=weiωt,
where ω is a parameter and w=w1xw2xw3x⋯wn-1xwnxT is a vector of mode shapes corresponding to that natural frequency. Substitution of (9) into (6) leads to
(10)K+Kcw+iωCcw=ω2Mw,
where the partial derivatives have been replaced by ordinary derivatives in the definition of K.A solution of the set ofn ordinary differential equations represented by (10) is assumed as
(11)wkx=ϕkxak,
where ϕk(x) satisfies the equation
(12)d4ϕkdx4-ξk2ϕ=0
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 (12) into (10) leads to
(13)ξk2U+Kcak+iωCcak=ω2Mak,
where U is an n×n diagonal matrix with ui,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 linearn-degree-of-freedom system with displacement vector x=x1x2⋯xnT are summarized by
(14)Mx¨+Cx˙+Kx=0.
A normal-mode solution is assumed as x=Xeiωt for (14), resulting in
(15)-ω2MX+iωCX+KX=0.
Equation (15) is the same as (13) with K=ξk2U+Kc. Thus, the same solution procedure is used to solve (13) as is used to solve (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 of the form
(16)M~y˙+K~y=0,
where
(17)M~=0MMC,K~=-M00K,y=x˙x.
A solution to (17) is assumed as
(18)y=Φe-γt
which results upon substitution in
(19)M~-1K~Φ=γΦ.
The values of ω are related to the eigenvalues of M~-1K~ by ω=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(20)wx,t=∑k=0∞∑i=1nBk,iXk,ie-γk,itϕkx,
where Bi,k are arbitrary constants of integration. When the values of γi,k are all complex and of the form
(21)γk,i=αk,i+iβk,i
and the complex mode shapes have the form
(22)Xk,i=Uk,i+iVk,i
then (20) is written as
(23)wx,t=∑k=0∞∑i=1ne-αk,itCk,iUk,icos⁡⁡βk,it+Vk,isin⁡βk,it00000000000000+Dk,iVk,icos⁡⁡βk,it-Uk,isin⁡βk,it∑i=1n00000·ϕkx.
In (23),  Ck,i and Dk,i are constants of integration determined from appropriate initial conditions.If a value ofγk,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,iXk,i,1e-γk,i,1t+Dk,iXk,i,2e-γk,i,2t.The spatially distributed mode shapes satisfy an orthogonality condition, which for a uniform beam is(24)∫01ϕjϕkdx=0,j≠k.
Let w(x,0) be a vector of initial conditions. Then
(25)wx,0=∑k=0∞∑i=1nCk,iUk,i+Dk,iVk,iϕkx.
Multiplying both sides of (25) by ϕj(x) for an arbitrary value of j, integrating from 0 to 1, and using the orthogonality condition lead to the equation:
(26)∑i=1nCj,iUj,i+Dj,iVj,i=∫01wx,0ϕjxdx.
A similar procedure is used for the vector of initial velocities w˙x,0 yielding
(27)∑i=1nCj,i-αk,iUj,i+βj,1Vj,100000-Dj,iαk,iVj,i+βj,1Uj,1=∫01w˙x,0ϕjxdx.


## 5. Example

Consider five concentric fixed-pinned beams connected by viscoelastic layers of the Kelvin-Voigt type of negligible thickness. The cross-sectional moment of inertia of theith beam is Ai=π(ri,o2-ri,i2), 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 of the ith beam is Ii=(π/4)(ri,o4-ri,i4). The properties of each of the five beams are given in Table 1.Table 1
Properties of the five beams of the example.
Beam number,i
Elastic modulus,Ei (TPa)
Density,ρi (kg/m3)
Inner radius,ri,i (nm)
Outer radius,ri,o (nm)
Area,Ai=π(ri,o2-ri,i2) (nm2)
Moment of inertiaIi=(π/4)(ri,o4-ri,i4) (nm4)
LengthLi (nm)
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
20Each layer has two parameters. The stiffness parameters, given in Table2, are consistent with those generated by the van der Waals forces between atoms in a carbon nanotube and are given by a formula derived using the data of Girifalco and Lad [18] and the Lennerd-Jones potential function:
(28)ki=366.672ri,i0.16d2erg/cm2,
where d=0.147 nm is the interatomic distance between bond lengths. 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
0
1
0.277
0.1
2
0.347
0.134
3
0.418
0.168
4
0.493
0.202
5
0
0Table 3
Nondimensional parameters.
i
μ
i
=
E
i
I
i
/
E
1
I
1
β
i
=
ρ
i
A
i
/
ρ
1
A
1
η
i
=
k
i
L
4
/
E
1
I
1
ν
i
=
c
i
L
2
/
E
1
I
1
ρ
1
A
1
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
0The mode shapes of a fixed-pinned beam are(29)ϕkx=cos⁡⁡ξkx-cosh⁡⁡ξkx+αksinh⁡⁡ξkx-sin⁡ξkx,
where
(30)αk=cos⁡⁡ξk-cosh⁡⁡ξksin⁡ξk-sinh⁡⁡ξk
and ξk is the kth positive solution of
(31)tan⁡ξk=tanh⁡ξk.
The first five solutions of (31) are given in Table 4.Table 4
Five lowest solutions oftan⁡δ=tanh⁡δ.
i
δ
i
1
15.42
2
49.96
3
104.2
4
178.3
5
272.0The 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
(32)ω21000001.291000001.581000001.991000002.317a3,1a3,2a3,3a3,4a3,5+iω1025.31-5.31000-5.3112.42-7.11000-7.1116.03-8.92000-8.9219.69-10.72000-10.7210.72·a3,1a3,2a3,3a3,4a3,5+1052.54-2.54000-2.545.73-3.18000-3.187.02-3.89000-3.898.31-4.51000-4.514.54·a3,1a3,2a3,3a3,4a3,5=00000.
Table 5 presents the five intermodal frequencies corresponding to the five lowest intra-modal frequencies.Table 5
Undamped natural frequenciesωk,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
ω
1
,
i
ω
2
,
i
ω
3
,
i
ω
4
,
i
ω
5
,
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 × 102A solution of the form of (23) is applied resulting in the portion of the solution of (10) corresponding to k=3 as
(33)w3x,t=e-2.29tC3,12.642.492.272.011.92cos⁡⁡176.9t000000000000000+1.451.281.050.8230.690sin⁡176.9t00000000000+D3,1-2.642.492.272.011.92sin⁡176.9t00000000000000000000000+1.451.281.050.8230.690cos⁡⁡176.9t00000+e-9.77tC3,21.771.030.0017-0.832-1.26cos⁡⁡326.9t0000000000000000000+1.090.7040.185-0.285-0.562sin⁡326.9t000000000000000+D3,2-1.771.030.0017-0.832-1.26sin⁡326.9t00000000000000000000000+1.090.7040.185-0.285-0.562cos⁡⁡326.9t00000+e-347.3tC3,3-1.050.3120.9340.298-0.606cos⁡⁡466.2t00000000000000000000+-4.061.014.101.27-2.92sin⁡466.2t000000000000000+D3,3--1.050.3120.9340.298-0.606sin⁡466.2t00000000000000000000000+-4.061.014.101.27-2.92cos⁡⁡466.2t00000+e-638.1tC3,45.38-7.950.7985.78-3.37cos⁡⁡437.7t00000000000000000000+2.64-3.32-0.3572.93-1.61sin⁡437.7t000000000000000+D3,4-5.38-7.950.7985.78-3.37sin⁡437.7t0000000000000000000000+2.64-3.32-0.3572.93-1.61cos⁡⁡437.72t00000+e-896.9tC3,50.985-1.3660.3310.382-0.196cos⁡⁡103.6t00000000000000000000+-2.135.54-7.285.53-1.95sin⁡103.6t00000000000000+D3,5-0.985-1.3660.3310.382-0.196sin⁡103.6t000000000000000000000+-2.135.54-7.285.53-1.95cos⁡⁡103.6t·cos⁡⁡10.2x-cosh⁡⁡10.2x00000+α3sinh⁡⁡10.2x-sin⁡10.2x.
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.Table 6
Values ofγk,i for k=1,2,…,5, for example.
i
γ
1
,
i
γ
2
,
i
γ
3
,
i
γ
4
,
i
γ
5
,
i
1
1.22
×
10
-
3
±
268.8
i
0.1319
±
8.65
i
2.291
±
176.3
i
15.17
±
286.7
i
48.16
±
397.9
i
2
110.0
±
280.6
i
99.79
±
290.8
i
97.68
±
326.9
i
87.15
±
409.8
i
70.17
±
562.8
i
3
344.6
±
435.6
i
344.4
±
442.3
i
343.7
±
466.2
i
341.7
±
521.3
i
337.4
±
617.3
i
4
638.7
±
405.3
i
638.6
±
412.4
i
638.1
±
437.7
i
636.8
±
495.5
i
634.5
±
595.5
i
5
759.71.031 × 103
785.71.006 × 103
896.9
±
103.6
i
897.8
±
256.6
i
894.8
±
414.8
iLetδ 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.Figure2 shows the real parts of γ3,i for each mode versus δ. The real part 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.Figure 2
Real part ofγ3,i for each mode versus δ, for example. All values except the lowest have a bifurcation for some value of δ. When a bifurcation occurs, the mode is critically damped.The imaginary part ofγ3,i 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.Figure 3
Imaginary part ofγ3,i for each mode versus δ, for example.


## 6. Conclusions

The free vibrations of a set ofn 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 system connected by Kelvin-Voigt layers.A Kelvin-Voigt model was assumed for layers between multiwalled 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 multiwalled nanotube with linear damping in the nanotubes can be analyzed using the method presented.



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