Research Article  Open Access
S. Graham Kelly, "Free and Forced Vibrations of Elastically Connected Structures", Advances in Acoustics and Vibration, vol. 2010, Article ID 984361, 11 pages, 2010. https://doi.org/10.1155/2010/984361
Free and Forced Vibrations of Elastically Connected Structures
Abstract
A general theory for the free and forced responses of elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is selfadjoint with respect to an inner product defined for , then the stiffness operator for the set of elastically connected structures is selfadjoint with respect to an inner product defined on . This leads to the definition of energy inner products defined on . When a normal mode solution is used to develop the free response, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is selfadjoint with respect to the energy inner product. The completeness of the eigenvectors in is used to develop a forced response. Special cases are considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapes reduces to a matrix eigenvalue problem, and it is shown that for each spatial mode there is a set of intramodal mode shapes. When the structures are identical, uniform, or nonuniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.
1. Introduction
The general theory for the free and forced response of strings, shafts, beams, and axially loaded beams is well documented [1–8]. Investigators have examined the free and forced response of elastically connected strings [9, 10], EulerBernoulli beams [11–14], and Timoshenko beams [15]. These analyses focused on a pair of elastically connected structures using a normalmode solution for the free response and a modal analysis for the forced response. Each of these papers uses a normalmode solution or a modal analysis specific to the problem to obtain a solution.
Ru [16–18] proposed that model for multiwalled carbon nanotubes to be modeled by elastically connected structures with the elastic layers representing interatomic vanDer Waals forces. Ru [16] proposed a model of concentric beams connected by elastic layers to model buckling of carbon nanotubes and elastic shell models [17, 18]. Yoon et al. [19] and Li and Chou [20] modeled free vibrations of multiwalled nanotubes by a series of concentric elastically connected EulerBernoulli beams, while Yoon et al. [21, 22] modeled nanotubes as concentric Timoshenko beams connected by an elastic layer. Xu et al. [23] modeled the nonliearity of the vanDer Waals forces. Elishakoff and Pentaras [24] gave approximate formulas for the natural frequencies of doublewalled nanotubes noting that if developed from the eigenvalue relation, the computations can be compuitationally intensive and difficult.
Kelly and Srinivas [25] developed a RayleighRitz method for elastically connected stretched structures.
This paper develops a general theory within which a finite set of parallel structures connected by elastic layers of a Winkler type can be analyzed. The theory shows that the determination of the natural frequencies for uniform parallel structures such as shafts and EulerBernoulli beams can be reduced to matrix eigenvalue problems. The general theory is also used to develop a modal analysis for forced response of a set of parallel structures.
2. Problem Formulation
The problem considered is that of structural elements in parallel but connected by elastic layers. Each elastic layer is modeled by a Winkler foundation, a layer of distributed stiffness across the span of the element. For generality, it is assumed that the outermost structures are connected to fixed foundations through elastic layers, as illustrated in Figure 1. Each layer has a uniform stiffness per unit length .
Let represent the displacement of the ith structure. If isolated from the system, the nondimensional differential equation governing the timedependent motion of this structure is written as where is the stiffness operator for the element, is an inertia operator for the element, and is the force per unit length acting on the structure which includes the forces from the elastic layer as well as any externally applied forces. The stiffness operator is a differential operator of order ( for strings and shafts, for EulerBernoulli beams), where the inertia operator is a function of the independent variable .
Each structure has the same end supports, and therefore their differential equations are subject to the same boundary conditions. Let be the subspace of defined by the boundary conditions; all elements in satisfy all boundary conditions.
The external forces acing on the th structure are where are the nondimensional stiffness coefficients connecting the th and plus first structures. Substitution of (2) into (1) leads to a coupled set of differential equations which are written in a matrix form as where , , is an diagonal operator matrix with , is an diagonal mass matrix with , and is a tridiagonal stiffness coupling matrix with The vector is an element of the vector space ; an element of is an dimensional vector, whose elements all belong to .
3. General Theory
Let and be arbitrary elements of . A standard inner product on is defined as
If the stiffness operator is selfadjoint and positive definite and if and only if with respect to the standard inner product, then a potential energy inner product is defined as
Clearly each is positive definite and selfadjoint with respect to the standard inner product, and thus a kinetic energy inner product can be defined as for any and in .
The standard inner product on is defined as for any and in . It is easy to show that is selfadjoint with respect to the inner product of (8) and a kinetic energy inner product on is defined as
Define . Since is selfadjoint with respect to the standard inner product on and is a symmetric matrix, it can be shown that is selfadjoint with respect to the standard inner product on . The positive definiteness of with respect to the standard inner product on is determined by considering . If is a positive definite matrix with respect to the standard inner product on , then and if and only if . If each of the operators is positive definite with respect to the standard inner product on then and if and only if . Thus, is positive definite with respect to the standard inner product on if either is a positive definite matrix with respect to the standard inner product on or each of the operators is positive definite with respect to the standard inner product on . Under either of these conditions, a potential energy inner product is defined on by The operator is not positive definite only when the structures are unrestrained and and .
Define . It is possible to show that is selfadjoint with respect to the kinetic energy inner product of (9) and the potential energy inner product of (10). If is positive definite with respect to the standard inner product, then is positive definite with respect to both inner products.
4. Free Response
First consider the free response of the structures, . A normalmode solution is assumed as where is a natural frequency and is a vector of mode shapes corresponding to that natural frequency. Substitution of (11) into (3) leads to where the partial derivatives have been replaced by ordinary derivatives in the definition of . From (12), it is clear that the natural frequencies are the square roots of the eigenvalues of , and the mode shape vectors are the corresponding eigenvectors.
It is well known [18] that eigenvalues of a selfadjoint operator are all real and that eigenvectors corresponding to distinct eigenvalues are orthogonal with respect to the inner product for which the operator is selfadjoint. Thus, if and are distinct natural frequencies with corresponding mode shape vectors and _{,} respectively, then
The mode shape vectors can be normalized by requiring which then leads to
5. Forced Response
Since is a selfadjoint operator, its eigenvectors, the mode shape vectors, can be shown to be complete in . An expansion theorem then implies that for any in there exists coefficients , , such that where are the normalized mode shape vectors, and the summation is carried out over all modes. For a given in , the coefficients are calculated by
Let represent the response due to the force vector . Since must be in , the expansion theorem may be applied at any , leading to Substitution of (18) into (3) results in Taking the standard inner product on of both sides of (26) with for an arbitrary and using modeshape orthogonality properties of (13)–(15) leads to an uncoupled set of differential equations of the form A convolution integral solution of (20) is
6. Uniform Structures with Proportional to
A special case occurs when the structures are uniform and operators for the individual structural elements are proportional to one another, In this case, the component of the stiffness operator due to the elasticity of the structural elements becomes
The system may be nondimensionalized such that . Then the differential equation governing the free response of the first structural element, if isolated from the remainder of the system, is A normalmode solution of (32) of the form leads to the eigenvalueeigenvector problem There are an infinite, but countable, number of natural frequencies for the system of (25), with corresponding mode shapes The mode shapes are normalized by requiring .
Assume a solution to (12) of the form where is an vector of constants. Substitution of (26) into (12) using (24) and (25) leads to
Equation (27) implies that, for this special case, the determination of the natural frequencies of the set of elastically connected structures is reduced to the determination of the eigenvalues of the matrix . For each there are natural frequencies. The natural frequencies can thus be indexed as for and . The corresponding mode shapes are written as where is the eigenvector of corresponding to the natural frequency .
The function represents the kth mode shape of a single structure. For each , there are natural frequencies and corresponding mode shapes. Such a set of mode shapes, which are referred to as intramodal modes, have the same spatial behavior, but their dependence across the structures varies. Two mode shapes and for which are referred to as intermodal modes.
Note that since and are both symmetric matrices, the matrix is selfadjoint with respect to a kinetic energy inner product. The intramodal mode shapes satisfy an orthogonality condition on given by All mode shapes satisfy the orthogonality condition on ,
Consider the case when is singular ( and ). Then zero is the smallest eigenvalue of and is its corresponding eigenvector, , where is a normalization constant. Suppose that , then (27) becomes Note that the solution of (31) corresponds to and . Thus, for this special case, the lowest natural frequency for each set of intramodal modes is equal to the natural frequency for the spatial mode of one structural element and its corresponding mode shape is the null space of .
To examine the most general case when is singular, take the standard inner product for of both sides of (27) with . Using properties of inner products and noting that , , and are symmetric leads to Application of the orthogonality condition of (32) leads to
The expansion theorem is used to assume a forced response of the form Use of (34) in (3) leads to differential equations of the form
A special case occurs when uniform structures are identical such that and . Then, (27) becomes Equation (36) can be rewritten as Let be the eigenvalues of . Then, In addition, since the same set of eigenvalues is used to calculate the natural frequencies for each intramodal sets, the intramodal mode shape vectors are the same for each set and are the eigenvectors of , that is where are the eigenvectors of .
7. Nonuniform Structures
The case when one or more of the structures is nonuniform is more difficult in that the differential equations of (12) have variable coefficients. Consider the case when the structures are nonuniform, but identical. In this case, the coupled set of differential equations can be written as where and are the stiffness and inertia operators, respectively, for any of the structures.
Let be the eigenvalues of and let be their corresponding eigenvectors normalized with respect to the standard inner product on . Let be the matrix whose columns are the normalized eigenvectors. Since is symmetric, it can be shown that and , where is a diagonal matrix with . Defining and substituting into (40) and then premultiplying by leads to Equation (42) represents a set of uncoupled differential equations, each of the form Equation (43) represents, along with appropriate homogeneous boundary conditions, an eigenvalue problem to determine the natural frequencies. For each , there are an infinite, but countable, number of natural frequencies. Thus, the natural frequencies can be indexed by .
It still may not be possible to solve (43) in closed form; however, it is now known how to index the natural frequencies. For a set of identical structures, there are an infinite number of natural frequencies corresponding to each eigenvalue of . The term intramodal is not appropriate for this set of natural frequencies as they do not correspond to the same mode. Indeed, there are not necessarily intramodal frequencies for the nonuniform case.
If is singular and thus has zero as its lowest eigenvalue, then (43) shows that one set of natural frequencies is identical to the natural frequencies of the individual structures.
8. Examples
8.1. Shafts
Consider concentric shafts of equal length connected by elastic layers. The stiffness and inertia operators for uniform shafts are, respectively,
The stiffness operators are of the form of those considered in Section 6. Hence, the natural frequencies and mode shapes can be determined from solving a matrix eigenvalue problem. If all shafts are made of the same material then leading to , and thus if is singular then the lowest natural frequency for each set of intramodal modes is the same as the modal natural frequency for the innermost shaft, and the mode shapes corresponding to each frequency are the eigenvector of corresponding to its zero eigenvalue times, the spatial mode shape for the shaft.
The eigenvalue problem for the innermost shaft is subject to appropriate boundary conditions. The values for and the corresponding normalized mode shapes are listed in Table 1 for various end conditions.

The matrix eigenvalue problem for a set of intramodal frequencies is of the form where is a tridiagonal matrix whose elements are and is a diagonal matrix with .
8.2. Natural Frequencies of Four Concentric FixedFree Shafts
As a numerical example, consider four concentric fixedfree shafts connected by layers of torsional stiffness. Solving (45) subject to and , Each shaft is made of the same material. The inner shaft is solid of radius . The outer shafts are each of thickness . The thickness of each elastic layer is negligible. This leads to , , , . The torsional stiffness of each elastic layer is the same and is taken such that . The inner shaft is solid, thus, . The outer radius of the outer shaft is unrestrained from rotation, hence, . The matrix eigenvalue problems becomeThe natural frequencies for are given in Table 1. Since is singular and , the lowest natural frequency in each intramodal set is . Each mode shape in a set of intramodal mode shapes corresponds to the same spatial mode . The difference in intramodal mode shapes is in the relative magnitude and signs of the displacements of the individual shafts. The normalized mode shapes of Figure 2 correspond to the first mode shape in the intramodal set for the first spatial mode and illustrate the mode in which the shafts rotate as if they are rigidly connected. The mode shapes of Figure 3 correspond to the third intramodal mode for the first spatial mode and illustrate that when the rotations of the first, second, and fourth shafts are counterclockwise, the rotation of the third shaft is clockwise. Figures 4 and 5 illustrate mode shapes corresponding to the third spatial mode. All mode shapes in the intramodal set for this mode have two nodes across the length of the shaft. Note that for the second intramodal frequency there is a change in the direction of rotation of the shafts between the third and fourth shafts. Thus, there is a cylindrical surface of nodes between these shafts. There are two changes in the direction of rotation for the third intramodal frequency, between the second and third shafts and between the third and fourth shafts, leading to two cylindrical surfaces of nodes.
8.3. Forced Response of Four Concentric Shafts
Suppose that the midspan of the outer shaft is subject to a constant torque, , such that the nondimensional applied torques are . The forced response of the system is calculated by using a convolution integral solution of the form of (21) leading to Substitution of (50) into (18) leads to Equation (51) is evaluated leading to the time dependence of the response at and illustrated in Figures 6 and 7.
8.4. Nonuniform Shafts
Now consider the same set of shafts, except that each has a taper, such that the differential equation for the innermost shaft when isolated from the system is Along with the boundary for a fixedfree shaft, (52) has a Bessel function solution leading to the characteristic equation for the shaft’s natural frequencies as where and are spherical Bessel functions of the first and second kinds of order and argument . The mode shape (nonnormalized) corresponding to a natural frequency is
The differential equations for the elastically coupled shafts become Even though the shafts are not identical, the differential equations in (55) may still be decoupled because the stiffness and inertia matrices are the same. The eigenvalues of are , and . The corresponding matrix of eigenvectors is The columns of have been normalized such that and . Following the same procedure as in the derivation of (43), the uncoupled differential equations become The solutions of (57) are The characteristic equations to determine the natural frequencies are The characteristic equation for the first set of frequencies is identical to (53). The first five frequencies for each are given in Table 2.

8.5. EulerBernoulli Beams
Consider a set of parallel EulerBernoulli beams connected by elastic layers. For uniform beams, the mass and stiffness operators are and . The differential eigenvalue problem for the first beam in the set is The transverse displacements of EulerBernoulli beams is another example of the special case discussed in Section 6.
If the beams are identical ( and ) and if is singular, then the lowest natural frequency for the th set of intramodal modes is with the mode shape, such that each beam has the same displacement and the springs are unstrectched. Otherwise, the mode shape for the lowest natural frequency of each intramodal set satisfies the orthogonality condition of (31).
As a numerical example, consider a set of five fixedfree elastically connected EulerBernoulli beams. The solution of Equation (60) subject to the boundary conditions , , , and leads to where is the th solution of . Numerical values used in the computations are , , , , , , , , , , , , , , , and
Using these numerical values, the matrix eigenvalue problem for a set of intramodal frequencies and mode shapes is
The sets of intramodal frequencies are listed in Table 3. Recall that is the natural frequency of the first beam. The natural frequency of an EulerBernoulli beam increases with stiffness. Since the first beam is stiffer than several other beams in the set, some intramodal frequencies are lower than . The mode shapes for and illustrated in Figure 8 show one spatial node and three cylindrical surfaces of nodes.

9. Conclusion
A general theory is developed for the free and forced response of elastically connected structures. The following has been shown.(i)The general problem can be formulated using the vector space , where is a subspace of defined by the system’s end conditions.(ii)If the differential stiffness operator for a single structure is selfadjoint with respect to a standard inner product on , then the general stiffness operator is selfadjoint with respect to a standard inner product on .(iii)Kinetic and potential energy inner products are defined on both and . (iv)A normalmode solution for the free response leads to the formulation of an eigenvalue problem defined for a matrix of operators.(v)The operator is selfadjoint with respect to the energy inner products leading to the development of an orthogonality condition.(vi)The expansion theorem is used to develop a modal analysis for the forced response.(vii)The case where the structures are uniform and the individual stiffness operators are proportional is a special case in which the determination of natural frequencies and mode shapes can be reduced to eigenvalue problems for matrices on . (viii)When the stiffness operators are proportional, the natural frequencies and mode shapes are indexed with two indices, the first representing the spatial mode shape, the second representing the intramodal mode shapes.(ix)If the uniform structures are identical, then a simple formula can be derived for the sets of intramodal natural frequencies using the eigenvalues of the coupling stiffness matrix. The intramodal mode shapes for each spatial mode are the eigenvectors of the coupling stiffness martrix.(x)An iterative solution must be applied to determine the natural frequencies for the most general case of the uniform structure.(xi)The differential equations for the coupling of identical structures, uniform, or nonuniform can be uncoupled through diagonalization of the coupling stiffness matrix.(xii) Elastically connected uniform strings and elastically connected uniform concentric shafts are applications in which the stiffness operators are proportional.(xiii)The differential equations for the concentric shafts, even though they are not identical, can be decoupled when each individual stiffness operator is the same as the individual mass operator.(xiv)The individual stiffness operators for uniform EulerBernoulli beams are proportional implying that their natural frequencies can be indexed as an infinite number of sets of intramodal frequencies.
The general method is applied here only for undamped systems. However, it can be applied to certain damped systems as well. If the structures are undamped but the Winkler layers have viscous damping, the same and for the undamped system may be used, but an eigenvalue problem is obtained involving complex numbers. If the structures are damped but the Winkler layers are undamped, the choice of and is modified to include viscous damping, but again a complex eigenvalue problem is obtained. If the entire system (both the individual structures and the Winkler layers) is subject to proportional damping, the eigenvalues and the eigenvectors of the undamped system can be used to uncouple the forced vibrations equations.
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Copyright
Copyright © 2010 S. Graham Kelly. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.