Free and Forced Vibrations of Elastically Connected Structures

Abstract

A general theory for the free and forced responses of 𝑛 elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for 𝐶^𝑘[0,1], then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on 𝑈=𝑅^𝑛×𝐶^𝑘[0,1]. This leads to the definition of energy inner products defined on 𝑈. 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 self-adjoint with respect to the energy inner product. The completeness of the eigenvectors in 𝑊 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 𝑛 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], Euler-Bernoulli beams [11–14], and Timoshenko beams [15]. These analyses focused on a pair of elastically connected structures using a normal-mode solution for the free response and a modal analysis for the forced response. Each of these papers uses a normal-mode 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 Euler-Bernoulli 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 double-walled nanotubes noting that if developed from the eigenvalue relation, the computations can be compuitationally intensive and difficult.

Kelly and Srinivas [25] developed a Rayleigh-Ritz 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 Euler-Bernoulli 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 𝑛 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 𝑘_𝑖𝑖=0,1,2,…,𝑛.

Figure 1 

Set of 𝑛 elastically connected parallel structures.

Let 𝑤_𝑖(𝑥) represent the displacement of the ith structure. If isolated from the system, the nondimensional differential equation governing the time-dependent motion of this structure is written as 𝐋_𝑖𝑤_𝑖(𝑥)+𝐌_𝑖𝑤̈_𝑖=𝐺_𝑖(𝑥,𝑡), (1) where 𝐋_𝑖 is the stiffness operator for the element, 𝐌_𝑖 is an inertia operator for the element, and 𝐺_𝑖(𝑥,𝑡) 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 𝑘  (𝑘=2 for strings and shafts, 𝑘=4 for Euler-Bernoulli beams), where the inertia operator is a function of the independent variable 𝑥.

Each structure has the same end supports, and therefore their differential equations are subject to the same boundary conditions. Let 𝑆 be the subspace of 𝐶^𝑘[0,1] defined by the boundary conditions; all elements in 𝑆 satisfy all boundary conditions.

The external forces acing on the 𝑖th structure are𝐺_𝑖(𝑥,𝑡)=−𝜆₀𝑤₁−𝜆₁𝑤₁−𝑤₂+𝐹₁(𝑥,𝑡)𝑖=1,−𝜆_𝑖−1𝑤_𝑖−𝑤_𝑖−1−𝜆_𝑖𝑤_𝑖−𝑤_𝑖+1+𝐹_𝑖(𝑥,𝑡)𝑖=2,3,…,𝑛−1,−𝜆_𝑛−1𝑤_𝑛−𝑤_𝑛−1−𝜆_𝑛𝑤_𝑛+𝐹_𝑛(𝑥,𝑡)𝑖=𝑛, (2) where 𝜆_𝑖,𝑖=0,1,…,𝑛 are the nondimensional stiffness coefficients connecting the 𝑖th and 𝑖 plus first structures. Substitution of (2) into (1) leads to a coupled set of differential equations which are written in a matrix form as 𝐊+𝐊_𝑐𝐖+𝐌𝐖̈=𝐅, (3) where 𝐖=[𝑤₁(𝑥,𝑡)𝑤₂(𝑥,𝑡)⋯𝑤_𝑛(𝑥,𝑡)]^𝑇, 𝐅=[𝐹₁(𝑥,𝑡)𝐹₂(𝑥,𝑡)⋯𝐹_𝑛(𝑥,𝑡)]^𝑇, 𝐊 is an 𝑛×𝑛 diagonal operator matrix with 𝑘_𝑖,𝑖=𝐋_𝑖, 𝐌 is an 𝑛×𝑛 diagonal mass matrix with 𝑚_𝑖,𝑖=𝐌_𝑖, and 𝐊_𝑐 is a tridiagonal 𝑛×𝑛 stiffness coupling matrix with 𝑘_{𝑐𝑖,𝑖−1}=−𝜆_𝑖−1𝑘𝑖=2,3,…,_{𝑐𝑖,𝑖}=𝜆_𝑖−1+𝜆_𝑖𝑘𝑖=1,2,…,𝑛,_{𝑐𝑖,𝑖+1}=−𝜆_𝑖𝑖=1,2,…,𝑛−1. (4) The vector 𝐖 is an element of the vector space 𝑈=𝑆×𝑅^𝑛; an element of 𝑈 is an 𝑛-dimensional vector, whose elements all belong to 𝑆.

3. General Theory

Let 𝑓(𝑥) and 𝑔(𝑥) be arbitrary elements of 𝑆. A standard inner product on 𝑆 is defined as 𝑓,𝑔_𝑆=¹₀𝑓(𝑥)𝑔(𝑥)𝑑𝑥. (5)

If the stiffness operator is self-adjoint [(𝐋_𝑖𝑓,𝑔)_𝑆=(𝑓,𝐋_𝑖𝑔)_𝑆] and positive definite [(𝐋_𝑖𝑓,𝑓)_𝑠≥0 and (𝐋_𝑖𝑓,𝑓)_𝑠=0 if and only if 𝑓=0] with respect to the standard inner product, then a potential energy inner product is defined as𝑓,𝑔_𝐋𝑖=𝐋_𝑖𝑓,𝑔_S. (6)

Clearly each 𝐌_𝑖 is positive definite and self-adjoint with respect to the standard inner product, and thus a kinetic energy inner product can be defined as𝑓,𝑔_𝑀𝑖=¹₀𝐌_𝑖𝑓𝑔𝑑𝑥, (7) for any 𝑓 and 𝑔 in 𝑆.

The standard inner product on 𝑈 is defined as(𝐟,𝐠)_𝑈=¹₀𝐠^𝐓𝐟𝑑𝑥, (8) for any 𝐟 and 𝐠 in 𝑈. It is easy to show that 𝐌 is self-adjoint with respect to the inner product of (8) and a kinetic energy inner product on 𝑈 is defined as (𝐟,𝐠)_𝐌=(𝐌𝐟,𝐠)_𝑈. (9)

Define 𝐊=𝐊+𝐊_𝑐. Since 𝐊 is self-adjoint with respect to the standard inner product on 𝑆 and 𝐊_𝑐 is a symmetric matrix, it can be shown that 𝐊 is self-adjoint with respect to the standard inner product on 𝑈. The positive definiteness of 𝐊 with respect to the standard inner product on 𝑈 is determined by considering (𝐊𝐟,𝐟)_𝑈=(𝐊𝐟,𝐟)_𝑈+(𝐊_𝑐𝐟,𝐟)_𝑈. If 𝐊_𝑐 is a positive definite matrix with respect to the standard inner product on 𝑅^𝑛, then (𝐊_𝑐𝐟,𝐟)_𝑈≥0 and (𝐊_𝑐𝐟,𝐟)_𝑈=0 if and only if 𝐟=𝟎. If each of the operators 𝐋_𝑖,𝑖=1,2,…,𝑛 is positive definite with respect to the standard inner product on 𝐶^𝑘[0,1] then (𝐊𝐟,𝐟)_𝑈≥0 and (𝐊𝐟,𝐟)_𝑈=0 if and only if 𝐟=𝟎. Thus, 𝐊 is positive definite with respect to the standard inner product on 𝑈 if either 𝐊_𝑐 is a positive definite matrix with respect to the standard inner product on 𝑅^𝑛 or each of the operators 𝐋_𝑖,𝑖=1,2,…,𝑛 is positive definite with respect to the standard inner product on 𝐶^𝑘[0,1]. Under either of these conditions, a potential energy inner product is defined on 𝑈 by(𝐟,𝐠)_𝐊=(𝐊𝐟,𝐠)_𝑈. (10) The operator 𝐊 is not positive definite only when the structures are unrestrained and 𝜆₀=0 and 𝜆_𝑛=0.

Define 𝐃=𝐌^−𝟏𝐊. It is possible to show that 𝐃 is self-adjoint with respect to the kinetic energy inner product of (9) and the potential energy inner product of (10). If 𝐊 is positive definite with respect to the standard inner product, then 𝐃 is positive definite with respect to both inner products.

4. Free Response

First consider the free response of the structures, 𝐅=𝟎. A normal-mode solution is assumed as𝐖=𝐰𝑒^𝑖𝜔𝑡, (11) where 𝜔 is a natural frequency and 𝐰=[𝑤₁(𝑥)𝑤₂(𝑥)𝑤₃(𝑥)⋯𝑤_𝑛−1(𝑥)𝑤_𝑛(𝑥)]^𝑇 is a vector of mode shapes corresponding to that natural frequency. Substitution of (11) into (3) leads to𝐌^−𝟏𝐊+𝐊_𝑐𝐰=𝜔^𝟐𝐰, (12) where the partial derivatives have been replaced by ordinary derivatives in the definition of 𝐊. From (12), it is clear that the natural frequencies are the square roots of the eigenvalues of 𝐃=𝐌^−𝟏(𝐊+𝐊_𝑐), and the mode shape vectors are the corresponding eigenvectors.

It is well known [18] that eigenvalues of a self-adjoint operator are all real and that eigenvectors corresponding to distinct eigenvalues are orthogonal with respect to the inner product for which the operator is self-adjoint. Thus, if 𝜔_𝑖 and 𝜔_𝑗 are distinct natural frequencies with corresponding mode shape vectors 𝐰_𝑖 and 𝐰_𝑗, respectively, then𝐰_𝑖,𝐰_𝑗𝐌𝐰=0,_𝑖,𝐰_𝑗𝐊=0. (13)

The mode shape vectors can be normalized by requiring𝐰_𝑖,𝐰_𝑖𝐌=1, (14) which then leads to 𝐰_𝑖,𝐰_𝑖𝐊=𝜔²_𝑖. (15)

5. Forced Response

Since 𝐃 is a self-adjoint operator, its eigenvectors, the mode shape vectors, can be shown to be complete in 𝑈. An expansion theorem then implies that for any 𝐟 in 𝑈 there exists coefficients 𝛼_𝑖, 𝑖=1,2,…, such that 𝐟=_𝑖𝛼_𝑖𝐰_𝑖, (16) where 𝐰_𝑖 are the normalized mode shape vectors, and the summation is carried out over all modes. For a given 𝐟 in 𝑈, the coefficients are calculated by 𝛼_𝑖=𝐟,𝐰_𝑖𝐌. (17)

Let 𝐖(𝑡) represent the response due to the force vector 𝐅(𝑡). Since 𝐖 must be in 𝑈, the expansion theorem may be applied at any 𝑡, leading to 𝐖(𝑥,𝑡)=_𝑖𝑐_𝑖(𝑡)𝐰_𝑖(𝑥). (18) Substitution of (18) into (3) results in 𝑐_𝑖̈_𝑖(𝑡)𝐌𝐰_𝑖+_𝑖𝑐_𝑖(𝑡)𝐊𝐰_𝑖=𝐅(𝑥,𝑡). (19) Taking the standard inner product on 𝑈 of both sides of (26) with 𝐰_𝑗(𝑥) for an arbitrary 𝑗=1,2,…,𝑛 and using mode-shape orthogonality properties of (13)–(15) leads to an uncoupled set of differential equations of the form 𝑐_𝑗̈(𝑡)+𝜔²_𝑗𝑐_𝑗(𝑡)=𝐅,𝐰_𝑗𝑈. (20) A convolution integral solution of (20) is𝑐_𝑗(𝑡)=1𝜔_𝑗^𝑡₀𝐅(𝑥,𝜏),𝐰_𝑗(𝑥)_𝑈sin𝜔_𝑗(=1𝑡−𝜏)𝑑𝜏𝜔_𝑗^𝑡₀¹₀𝐰^𝐓_𝑗(𝑥)𝐅(𝑥,𝜏)sin𝜔_𝑗(𝑡−𝜏)𝑑𝑥𝑑𝜏. (21)

6. Uniform Structures with L_𝐢 Proportional to L₁

A special case occurs when the structures are uniform and operators for the individual structural elements are proportional to one another, 𝐋_𝑖=𝜇_𝑖𝐋_𝟏. (22) In this case, the component of the stiffness operator due to the elasticity of the structural elements becomes𝐊=𝐀𝐋_𝟏=𝜇₁000⋯00𝜇₂00⋯000𝜇₃0⋯0000𝜇₄⋯0⋮⋮⋮⋮⋱⋮0000⋯𝜇_𝑛𝐋_𝟏. (23)

The system may be nondimensionalized such that 𝐌_𝟏=1. Then the differential equation governing the free response of the first structural element, if isolated from the remainder of the system, isΦ̈₁+𝐋_𝟏Φ₁=0. (24) A normal-mode solution of (32) of the form Φ(𝑥,𝑡)=𝜙(𝑥)𝑒^𝑖𝛿𝑡 leads to the eigenvalue-eigenvector problem 𝐋_𝟏𝜙=𝛿²𝜙. (25) There are an infinite, but countable, number of natural frequencies for the system of (25), 𝛿₁,𝛿₂,…,𝛿_𝑘−1,𝛿_𝑘,𝛿_𝑘+1,… with corresponding mode shapes 𝜙₁,𝜙₂,…,𝜙_𝑘−1,𝜙_𝑘,𝜙_𝑘+1,… The mode shapes are normalized by requiring (𝜙_𝑖,𝜙_𝑖)_𝑆=1.

Assume a solution to (12) of the form 𝐰=𝐚𝜙_𝑘(𝑥), (26) where 𝐚 is an 𝑛×1 vector of constants. Substitution of (26) into (12) using (24) and (25) leads to 𝐌^−𝟏𝛿²_𝑘𝐀+𝐊_𝑐𝐚=𝜔²𝐚. (27)

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 𝐌^−𝟏(𝛿²_𝑘𝐀+𝐊_𝑐). For each 𝑘 there are 𝑛 natural frequencies. The natural frequencies can thus be indexed as 𝜔_𝑘,𝑗 for 𝑘=1,2,… and 𝑗=1,2,…,𝑛. The corresponding mode shapes are written as 𝐰_𝑘,𝑗(𝑥)=𝐚_𝑘,𝑗𝜙_𝑘(𝑥), (28) where 𝐚_𝑘,𝑗 is the eigenvector of 𝐌^−𝟏(𝛿²_𝑘𝐀+𝐊_𝑐) corresponding to the natural frequency 𝜔_𝑘,𝑗.

The function 𝜙_𝑘(𝑥) represents the kth mode shape of a single structure. For each 𝑘, there are 𝑛 natural frequencies and 𝑛 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 𝐰_𝑘,𝑗(𝑥) and 𝐰_𝑝,𝑞(𝑥) for which 𝑝≠𝑘 are referred to as intermodal modes.

Note that since 𝐌 and 𝛿²_𝑘𝐀+𝐊 are both symmetric matrices, the matrix 𝐌^−𝟏(𝛿²_𝑘𝐀+𝐊_𝑐) is self-adjoint with respect to a kinetic energy inner product. The intramodal mode shapes satisfy an orthogonality condition on 𝑅^𝑛 given by 𝐚_𝑘,𝑖,𝐚_{𝑘,𝑗𝑀𝑛}=𝐚_𝑘,𝑗^𝑇𝐌𝐚_𝑘,𝑖=0for𝑖≠𝑗. (29) All mode shapes satisfy the orthogonality condition on 𝑈,𝐰_𝑘,𝑖,𝐰_ℓ,𝑗𝑀=¹₀𝐚_ℓ,𝑗^𝑇𝐌𝐚_𝑘,𝑖𝜙_𝑘(𝑥)𝜙_ℓ(𝑥)𝑑𝑥=0when𝑘≠ℓor𝑖≠𝑗. (30)

Consider the case when 𝐊_𝑐 is singular (𝜆₀=0 and 𝜆_𝑛=0). Then zero is the smallest eigenvalue of 𝐊_𝑐 and 𝐛 is its corresponding eigenvector, 𝐛=𝑐[11⋯1], where 𝑐 is a normalization constant. Suppose that 𝐌^−𝟏𝐀=𝐈, then (27) becomes 𝛿²_𝑘𝐈+𝐊_𝑐𝐚=𝜔^𝟐𝐚. (31) Note that the solution of (31) corresponds to 𝜔=𝛿_𝑘 and 𝐚=𝐛. 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 𝐊_𝑐.

To examine the most general case when 𝐊_𝑐 is singular, take the standard inner product for 𝑅^𝑛 of both sides of (27) with 𝐛. Using properties of inner products and noting that 𝐀, 𝐊_𝑐, and 𝐌 are symmetric leads to𝐚,𝛿^𝟐_𝑘𝐀−𝜔^𝟐𝐌𝐛_𝑅𝑛=0. (32) Application of the orthogonality condition of (32) leads to^𝑛_𝑖=1𝑎_𝑖𝛿²_𝑘𝜇_𝑖−𝜔²𝛽_𝑖=0. (33)

The expansion theorem is used to assume a forced response of the form𝐖(𝑥,𝑡)=^∞_𝑘=1^𝑛_𝑗=1𝑐_𝑘,𝑗(𝑡)𝐚_𝑘,𝑗𝜙_𝑘(𝑥). (34) Use of (34) in (3) leads to differential equations of the form𝑐̈_𝑘,𝑗+𝜔²_𝑘,𝑗𝑐_𝑘,𝑗=𝐅,𝐚_𝑘,𝑗𝜙_𝑘(𝑥)_𝑈. (35)

A special case occurs when uniform structures are identical such that 𝐀=𝐈 and 𝐌=𝐈. Then, (27) becomes𝛿²_𝑘𝐈+𝐊_𝑐𝐚=𝜔^𝟐𝐚. (36) Equation (36) can be rewritten as𝐊_𝑐𝐚=𝜔^𝟐−𝛿²_𝑘𝐚. (37) Let 𝜅_𝑗  𝑗=1,2,…,𝑛 be the eigenvalues of 𝐊_𝑐. Then, 𝜔²_𝑘,𝑗=𝛿²_𝑘+𝜅_𝑗. (38) 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 𝐊_𝑐, that is 𝐚_𝑘,𝑗=𝐚_ℓ,𝑗=𝐯_𝑗𝑘,ℓ=1,2,…,𝑗=1,2,…,𝑛, (39) where 𝐯_𝑗𝑗=1,2,…,𝑛 are the eigenvectors of 𝐊_𝑐.

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 as𝐈𝐋𝐰+𝐊_𝑐𝐰−𝜔^𝟐𝐈𝐌𝐰=𝟎, (40) where 𝐋 and 𝐌 are the stiffness and inertia operators, respectively, for any of the structures.

Let 𝜅₁,𝜅₂,…,𝜅_𝑛 be the eigenvalues of 𝐊_𝑐 and let 𝐳_𝟏,𝐳_𝟐,…,𝐳_𝐧 be their corresponding eigenvectors normalized with respect to the standard inner product on 𝑅^𝑛  (𝐳^𝐓_𝑖𝐳_𝑖=1). Let 𝐏 be the matrix whose columns are the normalized eigenvectors. Since 𝐊_𝑐 is symmetric, it can be shown that 𝐏^𝐓𝐏=𝐈 and 𝐏^𝐓𝐊_𝑐𝐏=Δ, where Δ is a diagonal matrix with Δ_𝑖,𝑗=𝜅_𝑖. Defining 𝐰=𝐏𝐪 (41) and substituting into (40) and then premultiplying by 𝐏^𝐓 leads to 𝐈𝐋𝐪+Δ𝐪−𝜔^𝟐𝐈𝐌𝐪=𝟎. (42) Equation (42) represents a set of uncoupled differential equations, each of the form 𝐋𝑞_𝑗+𝜅_𝑗𝑞_𝑗−𝜔²𝐌𝑞_𝑗=0𝑗=1,2,…,𝑛. (43) Equation (43) represents, along with appropriate homogeneous boundary conditions, an eigenvalue problem to determine the natural frequencies. For each 𝑗, there are an infinite, but countable, number of natural frequencies. Thus, the natural frequencies can be indexed by 𝜔_𝑗,𝑘𝑗=1,2,…,𝑛  𝑘=1,2,….

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 𝐊_𝑐. 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 𝐊_𝑐 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 𝑛 concentric shafts of equal length connected by elastic layers. The stiffness and inertia operators for uniform shafts are, respectively,𝐋_𝑖𝜃_𝑖=−𝜇_𝑖𝜕²𝜃_𝑖𝜕𝑥²,𝐌_𝑖𝜃_𝑖=𝛽_𝑖𝜃_𝑖. (44)

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 𝛽_𝑖=𝜇_𝑖 leading to 𝐌⁻¹𝐊=𝐈, and thus if 𝐊_𝑐 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 𝐊_𝑐 corresponding to its zero eigenvalue times, the spatial mode shape for the shaft.

The eigenvalue problem for the innermost shaft is 𝜙^′′(𝑥)+𝛿²𝜙(𝑥)=0, (45) subject to appropriate boundary conditions. The values for 𝛿_𝑘𝑘=1,2,…,5 and the corresponding normalized mode shapes 𝜙_𝑘(𝑥) are listed in Table 1 for various end conditions.

Table 1 

First five sets of intramodal natural frequencies of four elastically connected fixed free shafts, 𝜔_(𝑘,𝑗).

The matrix eigenvalue problem for a set of intramodal frequencies is of the form 𝐊𝐚=𝜔^𝟐𝐌𝐚, (46) where 𝐊 is a tridiagonal matrix whose elements are𝑘_𝑖,𝑖=𝜇_𝑖𝛿²+𝜆_𝑖−1+𝜆_𝑖𝑘𝑖=1,2,…,𝑛,_{𝑖,𝑖−1}=−𝜆_𝑖−1𝑘𝑖=2,3,…,𝑛,_𝑖,𝑖+1=−𝜆_𝑖𝑖=1,2,…,𝑛−1, (47) and 𝐌 is a diagonal matrix with 𝑚_𝑖,𝑖=𝛽_𝑖.

8.2. Natural Frequencies of Four Concentric Fixed-Free Shafts

As a numerical example, consider four concentric fixed-free shafts connected by layers of torsional stiffness. Solving (45) subject to 𝜙_𝑘(0)=0 and 𝜙^′_𝑘(1)=0, 𝛿_𝑘=(2𝑘−1)𝜋2𝜙_𝑘(𝑥)=√()𝜋2sin2𝑘−12𝑥. (48) Each shaft is made of the same material. The inner shaft is solid of radius 𝑟. The outer shafts are each of thickness 𝑟. The thickness of each elastic layer is negligible. This leads to 𝜇₁=1𝜇₂=15𝜇₃=65,  𝜇₄=175,  𝛽₁=1𝛽₂=15𝛽₃=65,  𝛽₄=175. The torsional stiffness of each elastic layer is the same and is taken such that 𝜆_𝑖=1. The inner shaft is solid, thus, 𝜆₀=0. The outer radius of the outer shaft is unrestrained from rotation, hence, 𝜆₄=0. The matrix eigenvalue problems become1+(2𝑘−1)𝜋2²−100−12+15(2𝑘−1)𝜋2²−100−12+65(2𝑘−1)𝜋2²−100−11+175(2𝑘−1)𝜋2²𝑎_𝑘,1𝑎_𝑘,2𝑎_𝑘,3𝑎_𝑘,4=𝜔²1000𝑎0160000810000256^𝑘,1𝑎_𝑘,2𝑎_𝑘,3𝑎_𝑘,4. (49)The natural frequencies for 𝑘=1,2,…,5 are given in Table 1. Since 𝐊_𝑐 is singular and 𝐌^−𝟏𝐊_𝐛=𝐈, the lowest natural frequency in each intramodal set is 𝛿_𝑘. Each mode shape in a set of intramodal mode shapes corresponds to the same spatial mode 𝜙_𝑘(𝑥). 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.

Figure 2 

Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with 𝑘=1 and 𝑗=1. The mode shapes correspond to rigid-body motion across the set of shafts.

Figure 3 

Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with 𝑘=1 and 𝑗=3.

Figure 4 

Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with 𝑘=3 and 𝑗=2.

Figure 5 

Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with 𝑘=3 and 𝑗=3.

8.3. Forced Response of Four Concentric Shafts

Suppose that the midspan of the outer shaft is subject to a constant torque, 𝑇₀, such that the nondimensional applied torques are 𝑇₁(𝑥,𝑡)=𝑇₂(𝑥,𝑡)=𝑇₃(𝑥,𝑡)=0,𝑇₄(𝑥,𝑡)=𝛿(𝑥−1/2). The forced response of the system is calculated by using a convolution integral solution of the form of (21) leading to 𝑐_𝑘,𝑗(𝑡)=1𝜔_𝑘,𝑗^𝑡₀¹₀𝐚^𝑇_𝑘,𝑗𝐅(𝑥,𝜏)𝜙_𝑘(𝑥)sin𝜔_𝑘,𝑗(=𝐚𝑡−𝜏)𝑑𝑥𝑑𝜏_𝑘,𝑗4𝜙_𝑘(1/2)𝜔²_𝑘,𝑗1−cos𝜔_𝑘,𝑗𝑡. (50) Substitution of (50) into (18) leads to 𝐖(𝑥,𝑡)=^∞_𝑘=1⁴_𝑗=1𝐚_𝑘,𝑗4𝜔²_𝑘,𝑗𝐚_𝑘,𝑗1−cos𝜔_𝑘,𝑗𝑡𝜙_𝑘12𝜙_𝑘(𝑥). (51) Equation (51) is evaluated leading to the time dependence of the response at 𝑥=1/2 and 𝑥=1 illustrated in Figures 6 and 7.

Figure 6 

Forced response of elastically connected torsional shafts at 𝑥=0.5 due to constant concentrated torque applied to outer shaft at 𝑥=0.5.

Figure 7 

Forced response of elastically connected torsional shafts at 𝑥=1 due to constant concentrated torque applied to outer shaft at 𝑥=0.5.

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 is𝑑)𝑑𝑥(1−0.1𝑥²𝑑𝜃𝑑𝑥+𝜔²(1−0.1𝑥)²𝜃=0. (52) Along with the boundary for a fixed-free shaft, (52) has a Bessel function solution leading to the characteristic equation for the shaft’s natural frequencies as𝑗^′₀(9𝜔)𝑦_𝑜(10𝜔)−𝑦^′₀(9𝜔)𝑗₀(10𝜔)=0, (53) where 𝑗_𝑛(𝑥) and 𝑦_𝑛(𝑥) are spherical Bessel functions of the first and second kinds of order 𝑛 and argument 𝑥. The mode shape (nonnormalized) corresponding to a natural frequency 𝜔_𝑘 is𝜃_𝑘(𝑥)=𝑗_𝑜10𝜔_𝑘(1−0.1𝑥)−𝑗^′₀9𝜔_𝑘𝑦₀10𝜔_𝑘(1−0.1𝑥). (54)

The differential equations for the elastically coupled shafts become𝑑10000150000650000175)𝑑𝑥(1−0.1𝑥²𝑑𝜃𝑑𝑥¹𝜃₂𝜃₃𝜃₄−1−100𝜃−12−100−12−100−11¹𝜃₂𝜃₃𝜃₄+𝜔²×()100001500006500001751−0.1𝑥²𝜃₁𝜃₂𝜃₃𝜃₄=0000. (55) 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 𝐌^−𝟏𝐊_𝑐 are 𝜅₁=0,𝜅₂=0.0172,𝜅₃=0.0815, and 𝜅₄=1.0712. The corresponding matrix of eigenvectors is𝐏=0.06250.11910.22920.96400.06250.11710.2106−0.06860.06250.0848−0.06540.00100.0625−0.04220.0049−5.44×10⁻⁶. (56) The columns of 𝐏 have been normalized such that 𝐏^𝐓𝐌𝐏=𝐈 and 𝐏^𝐓𝐊_𝑐𝐏=Δ. Following the same procedure as in the derivation of (43), the uncoupled differential equations become𝑑)𝑑𝑥(1−0.1𝑥²𝑑𝑞₁𝑑𝑥+𝜔²(1−0.1𝑥)²𝑞₁𝑑=0,)𝑑𝑥(1−0.1𝑥²𝑑𝑞₂+𝜔𝑑𝑥²(1−0.1𝑥)²−0.0172𝑞₂𝑑=0,)𝑑𝑥(1−0.1𝑥²𝑑𝑞₃+𝜔𝑑𝑥²(1−0.1𝑥)²−0.0815𝑞₃𝑑=0,)𝑑𝑥(1−0.1𝑥²𝑑𝑞₄+𝜔𝑑𝑥²(1−0.1𝑥)²−1.0712𝑞₄=0. (57) The solutions of (57) are𝑞₁(𝑥)=𝐶₁𝑗₀[𝜔(1−0.1𝑥)]+𝐶₂𝑦₀[𝜔(,𝑞1−0.1𝑥)]₂(𝑥)=𝐶₁𝑗_0.904[𝜔(1−0.1𝑥)]+𝐶₂𝑦_0.904[𝜔(,𝑞1−0.1𝑥)]₃(𝑥)=𝐶₁𝑗_2.398[𝜔(1−0.1𝑥)]+𝐶₂𝑦_2.398[𝜔(,𝑞1−0.1𝑥)]₄(𝑥)=𝐶₁𝑗_9.862[𝜔(1−0.1𝑥)]+𝐶₂𝑦_9.862[𝜔(1−0.1𝑥)]. (58) The characteristic equations to determine the natural frequencies are𝑗₀10𝜔_𝑘,1𝑦₁9𝜔_𝑘,1−𝑦₀10𝜔₁𝑗₁9𝜔_𝑘,1𝑗=0,_0,90410𝜔_𝑘,2𝑦0.904^0.9049𝜔_𝑘,29𝜔_𝑘,2−𝑦_1.9049𝜔_𝑘,2−𝑦_0,90410𝜔_𝑘,2𝑗0.904^0.9049𝜔_𝑘,29𝜔_𝑘,2−𝑗_1.9049𝜔_𝑘,2𝑗=0,_2.39810𝜔_𝑘,3𝑦2.398^2.3989𝜔_𝑘,39𝜔_𝑘,3−𝑦_3.3989𝜔_𝑘,3−𝑦_2.39810𝜔_𝑘,3𝑗2.398^2.3989𝜔_𝑘,39𝜔_𝑘,3−𝑗_3.3989𝜔_𝑘,3𝑗=0,_9.86210𝜔_𝑘,4𝑦9.862^9.8629𝜔_𝑘,49𝜔_𝑘,4−𝑦_10.8629𝜔_𝑘,4−𝑦_9.86210𝜔_𝑘,4𝑗9.862^9.8629𝜔_𝑘,49𝜔_𝑘,4−𝑗_10.9629𝜔_𝑘,4=0. (59) The characteristic equation for the first set of frequencies is identical to (53). The first five frequencies for each 𝑗 are given in Table 2.

Table 2 

First five sets of frequencies for set of linearly tapered shafts.

8.5. Euler-Bernoulli Beams

Consider a set of 𝑛 parallel Euler-Bernoulli beams connected by elastic layers. For uniform beams, the mass and stiffness operators are 𝐋_𝑖=𝜇_𝑖(𝜕⁴/𝜕𝑥⁴) and 𝐌_𝑖=𝛽_𝑖. The differential eigenvalue problem for the first beam in the set is 𝑑⁴𝜙𝑑𝑥⁴−𝛿²𝜙=0. (60) The transverse displacements of Euler-Bernoulli beams is another example of the special case discussed in Section 6.

If the beams are identical (𝜇_𝑖=1 and 𝛽_𝑖=1) and if 𝐊_𝑐 is singular, then the lowest natural frequency for the 𝑘th set of intramodal modes is 𝛿_𝑘 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 fixed-free elastically connected Euler-Bernoulli beams. The solution of Equation (60) subject to the boundary conditions 𝜙(0)=0, 𝜙′(0)=0, 𝜙^′′(1)=0, and 𝜙^′′′(1)=0 leads to 𝜙_𝑘(𝑥)=cosh𝛿_𝑘𝑥−cos𝛿_𝑘𝑥−cos𝛿_𝑘+cosh𝛿_𝑘sin𝛿_𝑘+sinh𝛿_𝑘×sinh𝛿_𝑘𝑥−sin𝛿_𝑘𝑥, (61) where 𝛿_𝑘 is the 𝑘th solution of cos(𝛿_𝑘)cosh(𝛿_𝑘)=−1. Numerical values used in the computations are 𝜇₁=1, 𝜇₂=2, 𝜇₃=0.5, 𝜇₄=1, 𝜇₅=0.25, 𝛽₁=1, 𝛽₂=1.5, 𝛽₃=0.75, 𝛽₄=1, 𝛽₅=0.5, 𝜆₀=0, 𝜆₁=100, 𝜆₂=50, 𝜆₃=50, 𝜆₄=20, and 𝜆₅=0

Using these numerical values, the matrix eigenvalue problem for a set of intramodal frequencies and mode shapes is𝛿²_𝑘+100−100000−1002𝛿²_𝑘+150−50000−500.5𝛿²_𝑘+100−50000−50𝛿²_𝑘+70−20000−200.25𝛿²_𝑘×𝑎+20₁𝑎₂𝑎₃𝑎₄𝑎₅=𝜔²10000𝑎01.5000000.75000001000000.5¹𝑎₂𝑎₃𝑎₄𝑎₅. (62)

The sets of intramodal frequencies are listed in Table 3. Recall that 𝛿_𝑘 is the natural frequency of the first beam. The natural frequency of an Euler-Bernoulli beam increases with stiffness. Since the first beam is stiffer than several other beams in the set, some intramodal frequencies are lower than 𝛿_𝑘. The mode shapes for 𝑘=2 and 𝑗=4 illustrated in Figure 8 show one spatial node and three cylindrical surfaces of nodes.

Table 3 

First four sets of intramodal natural frequencies for a set of five elastically connected fixed-fixed Euler-Bernoulli beams.

Figure 8 

Intramodal mode shapes for a set of five elastically connected Euler-Bernoulli beams.

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 𝑈=𝑅^𝑛×𝑆, where 𝑆 is a subspace of 𝐶^𝑘[0,1] defined by the system’s end conditions.(ii)If the differential stiffness operator for a single structure is self-adjoint with respect to a standard inner product on 𝑆, then the general stiffness operator is self-adjoint with respect to a standard inner product on 𝑈.(iii)Kinetic and potential energy inner products are defined on both 𝑆 and 𝑈. (iv)A normal-mode solution for the free response leads to the formulation of an eigenvalue problem defined for a matrix of operators.(v)The operator is self-adjoint 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 𝑅^𝑛. (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 Euler-Bernoulli 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 𝛿_𝑘 and 𝜙_𝑘 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 𝛿_𝑘 and 𝜙_𝑘 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.