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.
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
Consider 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 . Let be the elastic modulus, let be the mass density, let be the cross-sectional area, and let be the cross-sectional moment of inertia of the th beam about the neutral axis of the th beam. Let represent the transverse displacement of the th beam, where is the distance along the neutral axis of the beam measured from its left end and represents time. Damping in each beam due to structural damping or complex stiffness is neglected. The viscoelastic layer between the th and plus first layer is of the Kelvin-Voigt type and has two parameters, representing the damping property of the layer and representing the stiffness of the layer, such that the force acting on the th beam from the layer is
Hamilton’s principle is used to derive the equations governing the free response of the th beams as In developing (2), viscoelastic layers represented by coefficients , , , and are assumed to exist between the first beam and the surrounding medium and the th beam and the surrounding medium and and .
The equations represented by (2) are nondimensionalized by introducing The nondimensional variables are substituted into (2) resulting in where the ’s have been dropped from the nondimensional variables and The differential equations have a matrix-operator formulation as where , is a diagonal operator matrix with , is a diagonal mass matrix with , and is a tridiagonal stiffness coupling matrix with and is a tridiagonal damping coupling matrix with The vector is an element of the vector space ; an element of is an -dimensional vector whose elements all belong to , 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 where is a parameter and is a vector of mode shapes corresponding to that natural frequency. Substitution of (9) into (6) leads to where the partial derivatives have been replaced by ordinary derivatives in the definition of .
A solution of the set of ordinary differential equations represented by (10) is assumed as where satisfies the equation subject to the homogeneous boundary conditions of the beams and is a vector of constants. The parameter is the th natural frequency of an undamped beam with the appropriate end conditions. The values of for are the natural frequencies of the first beam in the series assuming the beam vibrates freely from the other beams and the functions are the corresponding mode shapes.
Substitution of (12) into (10) leads to where is an diagonal matrix with . Equation (13) is a system of homogeneous algebraic equations to solve for .
The differential equations governing the free vibrations of a linear -degree-of-freedom system with displacement vector are summarized by A normal-mode solution is assumed as for (14), resulting in Equation (15) is the same as (13) with . Thus, the same solution procedure is used to solve (13) as is used to solve (15) for each .
4. General Solution
Following Kelly [1] the differential equations summarized by (11) can be rewritten as a system of 2 first-order equations of the form where A solution to (17) is assumed as which results upon substitution in The values of are related to the eigenvalues of by . 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 where are arbitrary constants of integration. When the values of are all complex and of the form and the complex mode shapes have the form then (20) is written as In (23),   and are constants of integration determined from appropriate initial conditions.
If a value of is real, the corresponding mode is overdamped and there are two real values of ; call them and . 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 .
The spatially distributed mode shapes satisfy an orthogonality condition, which for a uniform beam is Let be a vector of initial conditions. Then Multiplying both sides of (25) by for an arbitrary value of , integrating from 0 to 1, and using the orthogonality condition lead to the equation: A similar procedure is used for the vector of initial velocities yielding
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 the th beam is , where is the outer radius of the ith beam and is the inner radius of the th beam which is the outer radius of the -1st beam. The cross-sectional moment of inertia of the th beam is . 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 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: where  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.
The mode shapes of a fixed-pinned beam are where and is the th positive solution of The first five solutions of (31) are given in Table 4.
The free vibration response is given by (23), where the values of for are determined using (13). Choosing ,  (13) is written as Table 5 presents the five intermodal frequencies corresponding to the five lowest intra-modal frequencies.
A solution of the form of (23) is applied resulting in the portion of the solution of (10) corresponding to as The parameters for and for all five beams are presented in Table 6. For these damping properties, all parameters are complex except for . 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 is overdamped.
Let 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 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 for each mode versus . 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 for which the bifurcation occurs is larger for lower modes. The value of does not bifurcate but reaches a maximum value and then decreases.
The imaginary part of for each mode is plotted against in Figure 3. The higher modes vibrate at higher frequencies for small . 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 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding publication of this paper.