1. Introduction This paper presents an exact solution to the problem
Transcription
1. Introduction This paper presents an exact solution to the problem
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 ๐ฟ. Let ๐ธ๐ be the elastic modulus,๐๐ be the mass density, ๐ด๐ be the cross-sectional area and ๐ผ๐ be the cross sectional moment of inertia of the i th beam about the neutral axis of the i th beam. Let ๐ค๐ (๐ฅ, ๐ก) 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 ๐ก 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, ๐๐ representing the damping property of the layer and ๐๐ representing the stiffness of the layer, such that the force acting on the ith beam from the layer is ๐บ๐ = ๐๐ ๐๐ค๐+1 ๐๐ค๐ โ + ๐๐ ๐ค๐+1 โ ๐ค๐ ๐๐ก ๐๐ก (1) Hamiltonโs principle is used to derive the equations governing the free response of the ith beams as ๐ธ๐ ๐ผ๐ ๐4 ๐ค๐ ๐๐ค๐ ๐๐ค๐โ1 ๐๐ค๐ ๐๐ค๐+1 + ๐๐โ1 โ + ๐๐ โ + ๐๐โ1 ๐ค๐ โ ๐ค๐โ1 4 ๐๐ฅ ๐๐ก ๐๐ก ๐๐ก ๐๐ก + ๐๐ ๐ค๐ โ ๐ค๐+1 + ๐๐ ๐ด๐ (2) ๐2 ๐ค ๐ =0 ๐๐ก 2 In developing equation (2), viscoelastic layers represented by coefficients ๐0 , ๐0 , ๐๐ and ๐๐ are assumed to exist between the first beam and the surrounding medium and the n th beam a nd the surrounding medium and ๐ค0 = 0 and๐ค๐ +1 = 0. The equations represented by equation (2) are non-dimensionalized by introducing ๐ฅโ = ๐ฅ ๐ฟ ๐ค๐โ = ๐ค๐ ๐ฟ ๐กโ = ๐ก ๐ธ1 ๐ผ1 ๐1 ๐ด1 ๐ฟ4 The non-dimensional variables are substituted into Eq. (2) resulting in 3๐ 3๐ 3๐ ๐๐ ๐4 ๐ค ๐ ๐๐ค๐ ๐๐ค๐โ1 + ๐ ๐ค โ ๐ค + ๐ ๐ค โ ๐ค + ๐ โ ๐ โ1 ๐ ๐โ1 ๐ ๐ ๐+1 ๐ โ1 ๐๐ฅ 4 ๐๐ก ๐๐ก + ๐๐ (4) ๐๐ค๐ ๐๐ค๐+1 ๐2 ๐ค ๐ โ + ๐ฝ๐ =0 ๐๐ก ๐๐ก ๐๐ก 2 where the *s have been dropped from the non-dimensional variables and ๐๐ = ๐๐ = ๐๐ = ๐ฝ๐ = ๐ธ๐ ๐ผ๐ ๐ธ1 ๐ผ1 ๐๐ ๐ฟ4 ๐ธ1 ๐ผ1 ๐๐ ๐ฟ2 ๐ธ1 ๐ผ1 ๐1 ๐ด1 ๐๐ ๐ด๐ ๐1 ๐ด1 5๐ 5๐ (5๐) (5๐) The differential equations have a matrix-operator formulation as ๏ฆ ๏ซ MW ๏ฆ๏ฆ ๏ฝ 0 ๏จK ๏ซ K c ๏ฉW ๏ซ Cc W (6) where W ๏ฝ ๏w1 ( x, t ) w2 ( x, t ) ๏ wn ( x, t )๏ , K is a nxn diagonal operator matrix with T ๏ถ 4 wi k i ,i ๏ฝ ๏ญ i , M is a nxn diagonal mass matrix with mi ,i ๏ฝ ๏ข i , K c is a tri-diagonal nxn stiffness ๏ถx 4 coupling matrix with ๏จkc ๏ฉi ,i๏ญ1 ๏ฝ ๏ญ๏จi๏ญ1 ๏จkc ๏ฉi ,i ๏ฝ ๏จi๏ญ1 ๏ซ ๏จi ๏จkc ๏ฉi ,i๏ซ1 ๏ฝ ๏ญ๏จi i ๏ฝ 2,3, ๏ n i ๏ฝ 1,2, ๏, n i ๏ฝ 1,2, ๏ n ๏ญ 1 (7) and C c is a tri-diagonal nxn damping coupling matrix with ๏จcc ๏ฉi ,i๏ญ1 ๏ฝ ๏ญ๏ฎ i๏ญ1 ๏จcc ๏ฉi ,i ๏ฝ ๏ฎ i๏ญ1 ๏ซ ๏ฎ i ๏จcc ๏ฉi ,i๏ซ1 ๏ฝ ๏ญ๏ฎ i i ๏ฝ 2,3, ๏ n i ๏ฝ 1,2, ๏, n (8) i ๏ฝ 1,2, ๏ n ๏ญ 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 ๐ = ๐ฐei๐๐ก (9) where ๏ท is a parameter and w ๏ฝ ๏w1 ( x) w2 ( x) w3 ( x) ๏ wn๏ญ1 ( x) wn ( x)๏ is a vector of T mode shapes corresponding to that natural frequency. Substitution of Eq. (9) into Eq. (6) leads to ๏จK ๏ซ K c ๏ฉw ๏ซ i๏ทC c w ๏ฝ ๏ท 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 ๐ฐ๐ค ๐ฅ = ๐๐ ๐ฅ ๐๐ค (11) where ๐๐ (๐ฅ) satisfies the equation ๐ 4 ๐๐ โ ๐๐2 ๐ = 0 ๐๐ฅ 4 (12) subject to the homogeneous boundary conditions of the beams and ๐๐ is a vector of constants. The parameter ๐๐ is the kth natural frequency of an undamped beam with the appropriate end conditions. The values of ๐๐ for k=1,2,โฆ are the natural frequencies of the first beam in the series assuming the beam vibrates freely from the other beams and the functions ๐๐ (๐ฅ) are the corresponding mode shapes. Substitution of Eq. (12) into Eq. (10) leads to ๏จ๏ธ 2 k ๏ฉ U ๏ซ K c a k ๏ซ i๏ทC c a k ๏ฝ ๏ท 2 Ma k (13) where U is an nxn diagonal matrix with ๐ข ๐,๐ = ๐๐ . Equation (13) is a system of n homogeneous algebraic equations to solve for ๐๐ . The differential equations governing the free vibrations of a linear n-degree-of-freedom system with displacement vector ๐ฑ = ๐ฅ 1 ๐ฅ 2 โฆ ๐ฅ ๐ ๐are summarized by ๐๐ฑ + ๐๐ฑ + ๐๐ฑ = ๐ (14) A normal mode solution is assumed as ๐ฑ = ๐ei๐๐ก for Eq. (14) as, resulting in โ๐2 ๐๐ + i๐๐๐ + ๐๐ = ๐ (15) Equation (15) is the same as equation (13) with K ๏ฝ ๏ธ k2 U ๏ซ 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,โฆ. 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 ๐๐ฒ + ๐๐ฒ = ๐ (16) โ๐ ๐ ๐ ๐ (17) where ๐= ๐ ๐ ๐ ๐ ๐= ๐ฒ= ๐ฑ ๐ฑ A solution to Eq. (17) is assumed as ๐ฒ = ๐ฝeโ๐พ๐ก (18) which results upon substitution in ๐ โ๐ ๐๐ฝ = ๐พ๐ฝ (19) The values of ๐ are related to the eigenvalues of ๐ โ๐ ๐ by๐ = i๐พ. 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 โ ๐ ๐ต๐,๐ ๐ ๐,๐ ๐ โ๐พ๐ ,๐ ๐ก ๐๐ (๐ฅ) ๐ฐ ๐ฅ, ๐ก = ๐=0 (20) ๐=1 where ๐ต๐,๐ are arbitrary constants of integration. When the values of ๐พ๐ ,๐ are all complex and of the form ๐พ๐ ,๐ = ๐ผ๐,๐ + i๐ฝ๐,๐ (21) and the complex mode shapes have the form ๐ ๐,๐ = ๐๐ ,๐ + i๐๐,๐ (22) then Eq. (20) is written as โ ๐ ๐ โ๐ผ๐ ,๐ ๐ก ๐ถ๐,๐ ๐๐,๐ cos ๐ฝ๐,๐ ๐ก + ๐๐,๐ sin ๐ฝ๐,๐ ๐ก ๐ฐ ๐ฅ, ๐ก = ๐=0 ๐ =1 + ๐ท๐,๐ ๐๐,๐ cos ๐ฝ๐,๐ ๐ก โ ๐๐,๐ sin ๐ฝ๐,๐ ๐ก ๐๐ (๐ฅ) (23) In Eq. (23) ๐ถ๐,๐ and ๐ท๐,๐ are constants of integration determined from appropriate initial conditions. If a value of ๐พ๐ ,๐ is real the corresponding mode is overdamped and there are two real values of ๐พ๐ ,๐ , call them ๐พ๐ ,๐,1 and ๐พ๐ ,๐,2 . 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 ๐ถ๐,๐ ๐ ๐,๐ ,1 ๐ โ๐พ๐ ,๐,1 ๐ก + ๐ท๐,๐ ๐ ๐,๐,2 ๐ โ๐พ๐ ,๐,2 ๐ก . The spatially distributed mode shapes satisfy an orthogonality condition, which for a uniform beam is 1 0 ๐๐ ๐๐ d๐ฅ = 0 ๐โ ๐ (24) Let ๐ฐ(๐ฅ, 0) be a vector of initial conditions. Then โ ๐ ๐ฐ ๐ฅ, 0 = ๐ถ๐,๐ ๐๐ ,๐ + ๐ท๐,๐ ๐๐,๐ ๐๐ (๐ฅ) (25) ๐=0 ๐=1 Multiplying both sides of Eq. (25) by ๐๐ (๐ฅ) for an arbitrary value of j, integrating from 0 to 1 and using the orthogonality condition leads to the equations ๐ 1 ๐ถ๐,๐ ๐๐ ,๐ + ๐ท๐,๐ ๐๐,๐ = ๐ =1 0 ๐ฐ ๐ฅ, 0 ๐๐ ๐ฅ d๐ฅ (26) A similar procedure is used for the vector of initial velocities ๐ฐ ๐ฅ, 0 yielding ๐ 1 ๐ถ๐,๐ โ๐ผ๐,๐ ๐๐,๐ + ๐ฝ๐,1 ๐๐,1 โ ๐ท๐,๐ ๐ผ๐,๐ ๐๐,๐ + ๐ฝ๐ ,1 ๐๐,1 ๐ =1 = 0 ๐ฐ ๐ฅ, 0 ๐๐ ๐ฅ d๐ฅ (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 ๐ด๐ = ๐ ๐๐,๐ 2 โ ๐๐,๐ 2 where ๐๐,๐ is the outer radius of the ith beam and ๐๐,๐ is the inner radius of the ith beam which is the outer radius of the i-1st beam. The cross-sectional moment of inertia ๐ of the ith beam is ๐ผ๐ = 4 ๐๐,๐ 4 โ ๐๐,๐ 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 ๐๐ = 366.67 2๐๐,๐ erg 0.16๐ 2 cm2 (28) where ๐ = 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 ๐๐ ๐ฅ = cos ๐๐ ๐ฅ โ cosh ๐๐ ๐ฅ + ๐ผ๐ sinh ๐๐ ๐ฅ โ sin ๐๐ ๐ฅ (29) where ๐ผ๐ = cos ๐๐ โcosh ๐๐ sin ๐๐ โ sinh ๐๐ (30) and ๐๐ is the kth positive solution of tan ๐๐ = tanh ๐๐ 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 ๐พ๐ ,๐ for k=1,2โฆ are determined using Eq.(13). Choosing k=3 Eq. (13) is written as 1 0 ๐2 0 0 0 5.31 โ5.31 ๐๐102 0 0 0 2.54 โ2.54 105 0 0 0 โ2.54 5.73 โ3.18 0 0 0 0 0 0 0 0 1.991 0 0 2.317 ๐3,1 ๐3,2 ๐3,3 + ๐3,4 ๐3,5 0 0 0 โ7.11 0 0 16.03 โ8.92 0 โ8.92 19.69 โ10.72 0 โ10.72 10.72 ๐3,1 ๐3,2 ๐3,3 + ๐3,4 ๐3,5 0 1.291 0 0 0 โ5.31 12.42 โ7.11 0 0 0 0 0 โ3.18 0 0 7.02 โ3.89 0 โ3.89 8.31 โ4.51 0 โ4.51 4.54 1.581 0 0 0 0 ๐3,1 0 ๐3,2 0 ๐3,3 = 0 ๐3,4 0 ๐3,5 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 ๐ฐ3 ๐ฅ, ๐ก = ๐ โ2.29๐ก ๐ถ3,1 + ๐ท3,1 2.64 1.45 2.49 1.28 2.27 โ sin 176.9๐ก + 1.05 cos 176.9๐ก 2.01 0.823 1.92 0.690 + ๐ โ9.77๐ก + ๐ท3,2 2.64 1.45 2.49 1.28 2.27 cos 176.9๐ก + 1.05 sin 176.9๐ก 2.01 0.823 1.92 0.690 ๐ถ3,2 1.09 1.77 1.03 0.704 โ 0.0017 sin 326.9๐ก + 0.185 cos 326.9๐ก โ0.832 โ0.285 โ1.26 โ0.562 + ๐ โ347 .3๐ก ๐ถ3,3 + ๐ท3,3 1.09 1.77 1.03 0.704 0.0017 cos 326.9๐ก + 0.185 sin 326.9๐ก โ0.832 โ0.285 โ1.26 โ0.562 โ1.05 โ4.06 1.01 0.312 0.934 cos 466.2๐ก + 4.10 sin 466.2๐ก 0.298 1.27 โ0.606 โ2.92 โ1.05 โ4.06 1.01 0.312 โ 0.934 sin 466.2๐ก + 4.10 cos 466.2๐ก 0.298 1.27 โ0.606 โ2.92 +๐ โ638 .1๐ก ๐ถ3,4 + ๐ท3,4 2.64 5.38 โ3.32 โ7.95 โ 0.798 sin 437.7๐ก + โ0.357 cos 437.72๐ก 5.78 2.93 โ3.37 โ1.61 + ๐ โ896.9๐ก ๐ถ3,5 + ๐ท3,5 2.64 5.38 โ3.32 โ7.95 0.798 cos 437.7๐ก + โ0.357 sin 437.7๐ก 5.78 2.93 โ3.37 โ1.61 0.985 โ2.13 5.54 โ1.366 0.331 cos 103.6๐ก + โ7.28 sin 103.6๐ก 0.382 5.53 โ0.196 โ1.95 0.985 โ2.13 5.54 โ1.366 โ 0.331 sin 103.6๐ก + โ7.28 cos 103.6๐ก 0.382 5.53 โ0.196 โ1.95 cos 10.2๐ฅ โ cosh10.2๐ฅ + ๐ผ3 sinh 10.2๐ฅ โ โ sin 10.2๐ฅ (33) The parameters ๐พ๐ ,๐ for k=1,2,โฆ5 and for all five beams are presented in Table 6. For these damping properties all parameters are complex except for ๐พ1,5 . 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 ๐พ1,5 is overdamped. Let ๐ฟ 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 ๐ 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 ๐พ3,๐ for each mode versus ๐ฟ. 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 ๐ฟ for which the bifurcation occurs is larger for lower modes. The value of ๐พ3 ,1 does not bifurcate, but reaches a maximum value and then decreases. The imaginary part of ๐พ3 ,๐ is for each mode is plotted against ๐ฟ in Figure 3. The higher modes vibrate at higher frequencies for small ๐ฟ. 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 ๏ก 3,i 10 2 10 0 10 -2 10 -3 10 -2 10 -1 0 10 10 1 10 2 10 ๏ค . Figure 2: Real part of ๐พ3 ,๐ for each mode versus ๐ฟ for example. All values except the lowest has a bifurcation for some value of ๐ฟ. When a bifurcation occurs the mode is critically damped. 1000 900 800 700 ๏ข3,i 600 500 400 300 200 100 0 0 0.05 0.1 0.15 0.2 0.25 0.3 ๏ค Figure 3: Imaginary part of ๐พ3,๐ for each mode versus ๐ฟ for example. 0.35 0.4 Table 1 Properties of the five beams of the example Beam Elastic Density, number, modulus, ๐๐ ๐ ๐ธ๐ (TPa) (kg/m3) Inner Outer Area, Moment Length radius, radius, ๐ด๐ = of inertia ๐ฟ ๐ (nm) ๐๐,๐ (nm) ๐๐,๐ (nm) ๐ ๐๐,๐ 2 โ ๐๐,๐ 2 ๐ผ๐ = (nm2) ๐ 4 ๐๐,๐ 4 โ ๐๐,๐ 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, ๐ Stiffness parameter, ๐๐ (TPa) Damping parameter, ๐๐ (Nโs/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 ๐ ๐๐ = ๐ธ๐ ๐ผ๐ ๐ธ1 ๐ผ1 ๐ฝ๐ = ๐๐ ๐ด๐ ๐1 ๐ด1 ๐๐ ๐ฟ4 ๐๐ = ๐ธ1 ๐ผ1 ๐๐ = ๐๐ ๐ฟ2 ๐ธ1 ๐ผ1 ๐1 ๐ด1 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 ๐ฟ = tanh ๐ฟ ๐ ๐ฟ๐ 1 15.42 2 49.96 3 104.2 4 178.3 5 272.0 Table 5 Undamped natural frequencies ๐๐,๐ for example. ๐๐,๐ for a fixed k and i=1,2,โฆ,5 is a set of intramodal frequencies whereas ๐๐,๐ and ๐๐,๐ represent intermodal frequencies. ๐ ๐1,๐ ๐2,๐ ๐3,๐ ๐4,๐ ๐5,๐ 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 ๐พ๐ ,๐ for ๐ = 1,2 โฆ ,5 for example ๐ 1 ๐พ1,๐ 1.22x10-3 ๐พ2 ,๐ ๐พ3 ,๐ ๐พ4,๐ ๐พ5 ,๐ 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.