free and forced vibrations of elastically connected structures

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2010, Article ID 984361, 11 pagesdoi:10.1155/2010/984361

Research Article

Free and Forced Vibrations of Elastically Connected Structures

S. Graham Kelly

Department of Mechanical Engineering, The University of Akron, Akron, Oh 44235, USA

Correspondence should be addressed to S. Graham Kelly, [email protected]

Received 21 October 2010; Accepted 11 November 2010

Academic Editor: K. M. Liew

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.

A general theory for the free and forced responses of n elastically connected parallel structures is developed. It is shown that if thestiffness operator for an individual structure is self-adjoint with respect to an inner product defined for Ck[0, 1], then the stiffnessoperator for the set of elastically connected structures is self-adjoint with respect to an inner product defined onU = Rn×Ck[0, 1].This leads to the definition of energy inner products defined on U . When a normal mode solution is used to develop the freeresponse, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is self-adjoint withrespect to the energy inner product. The completeness of the eigenvectors in W is used to develop a forced response. Special casesare considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapesreduces 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 acoupling 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 forcedresponse of elastically connected strings [9, 10], Euler-Bernoulli beams [11–14], and Timoshenko beams [15].These analyses focused on a pair of elastically connectedstructures using a normal-mode solution for the freeresponse and a modal analysis for the forced response. Eachof these papers uses a normal-mode solution or a modalanalysis specific to the problem to obtain a solution.

Ru [16–18] proposed that model for multiwalled carbonnanotubes to be modeled by elastically connected structureswith the elastic layers representing interatomic vanDer Waalsforces. Ru [16] proposed a model of concentric beamsconnected by elastic layers to model buckling of carbonnanotubes and elastic shell models [17, 18]. Yoon et al. [19]and Li and Chou [20] modeled free vibrations of multiwallednanotubes by a series of concentric elastically connectedEuler-Bernoulli beams, while Yoon et al. [21, 22] modelednanotubes as concentric Timoshenko beams connected byan elastic layer. Xu et al. [23] modeled the nonliearity ofthe vanDer Waals forces. Elishakoff and Pentaras [24] gave

approximate formulas for the natural frequencies of double-walled nanotubes noting that if developed from the eigen-value relation, the computations can be compuitationallyintensive and difficult.

Kelly and Srinivas [25] developed a Rayleigh-Ritz meth-od for elastically connected stretched structures.

This paper develops a general theory within which afinite set of parallel structures connected by elastic layers ofa Winkler type can be analyzed. The theory shows that thedetermination of the natural frequencies for uniform parallelstructures such as shafts and Euler-Bernoulli beams can bereduced to matrix eigenvalue problems. The general theoryis also used to develop a modal analysis for forced responseof a set of parallel structures.

2. Problem Formulation

The problem considered is that of n structural elements inparallel but connected by elastic layers. Each elastic layeris modeled by a Winkler foundation, a layer of distributedstiffness across the span of the element. For generality, it isassumed that the outermost structures are connected to fixedfoundations through elastic layers, as illustrated in Figure 1.

2 Advances in Acoustics and Vibration

Each layer has a uniform stiffness per unit length ki i =0, 1, 2, . . . ,n.

Let wi(x) represent the displacement of the ith structure.If isolated from the system, the nondimensional differentialequation governing the time-dependent motion of thisstructure is written as

Liwi(x) + Miwi = Gi(x, t), (1)

where Li is the stiffness operator for the element, Mi is aninertia operator for the element, and Gi(x, t) is the forceper unit length acting on the structure which includes theforces from the elastic layer as well as any externally appliedforces. The stiffness operator is a differential operator oforder k (k = 2 for strings and shafts, k = 4 for Euler-Bernoulli beams), where the inertia operator is a function ofthe independent variable x.

Each structure has the same end supports, and thereforetheir differential equations are subject to the same boundaryconditions. Let S be the subspace of Ck[0, 1] defined by theboundary conditions; all elements in S satisfy all boundaryconditions.

The external forces acing on the ith structure are

Gi(x, t)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

−λ0w1 − λ1(w1 −w2) + F1(x, t) i = 1,

−λi−1(wi −wi−1)

−λi(wi −wi+1) + Fi(x, t) i = 2, 3, . . . ,n− 1,

−λn−1(wn −wn−1)− λnwn + Fn(x, t) i = n,(2)

where λi, i = 0, 1, . . . ,n are the nondimensional stiffnesscoefficients connecting the ith and i plus first structures.Substitution of (2) into (1) leads to a coupled set ofdifferential equations which are written in a matrix form as

(K + Kc)W + MW = F, (3)

where W =[

w1(x, t) w2(x, t) · · · wn(x, t)]T

, F =[

F1(x, t) F2(x, t) · · · Fn(x, t)]T

, K is an n × n diagonaloperator matrix with k i,i = Li, M is an n× n diagonal massmatrix with mi,i = Mi, and Kc is a tridiagonal n× n stiffnesscoupling matrix with

(kc)i,i−1 = −λi−1 i = 2, 3, . . . ,

(kc)i,i = λi−1 + λi i = 1, 2, . . . ,n,

(kc)i,i+1 = −λi i = 1, 2, . . . ,n− 1.

(4)

The vector W is an element of the vector space U = S × Rn;an element of U is an n-dimensional vector, whose elementsall belong to S.

3. General Theory

Let f (x) and g(x) be arbitrary elements of S. A standardinner product on S is defined as

(f , g

)

S =∫ 1

0f (x)g(x)dx. (5)

x

kn

Wnkn−1

Wn−kn−2

W3k2

W2k1

W1k0

......

......

......

......

......

......

......

......

Figure 1: Set of n elastically connected parallel structures.

If the stiffness operator is self-adjoint [(Li f , g)S =( f , Lig)S] and positive definite [(Li f , f )s ≥ 0 and (Li f , f )s =0 if and only if f = 0] with respect to the standard innerproduct, then a potential energy inner product is defined as

(f , g

)

Li= (Li f , g

)

S. (6)

Clearly each Mi is positive definite and self-adjoint withrespect to the standard inner product, and thus a kineticenergy inner product can be defined as

(f , g

)

Mi=∫ 1

0

(Mi f

)gdx, (7)

for any f and g in S.The standard inner product on U is defined as

(f , g)

U =∫ 1

0gTfdx, (8)

for any f and g in U . It is easy to show that M is self-adjointwith respect to the inner product of (8) and a kinetic energyinner product on U is defined as

(f , g)

M =(

Mf , g)

U. (9)

Define K = K + Kc. Since K is self-adjoint with respectto the standard inner product on S and Kc is a symmetricmatrix, it can be shown that K is self-adjoint with respect tothe standard inner product on U . The positive definitenessof K with respect to the standard inner product on U isdetermined by considering (Kf , f)U = (Kf , f)U + (Kcf , f)U .If Kc is a positive definite matrix with respect to the standardinner product on Rn, then (Kcf , f)U ≥ 0 and (Kcf , f)U = 0 ifand only if f = 0. If each of the operators Li, i = 1, 2, . . . ,nis positive definite with respect to the standard inner producton Ck[0, 1] then (Kf , f)U ≥ 0 and (Kf , f)U = 0 if and onlyif f = 0. Thus, K is positive definite with respect to thestandard inner product on U if either Kc is a positive definite

Advances in Acoustics and Vibration 3

matrix with respect to the standard inner product on Rn oreach of the operators Li, i = 1, 2, . . . ,n is positive definitewith respect to the standard inner product onCk[0, 1]. Undereither of these conditions, a potential energy inner product isdefined on U by

(f , g)

K =(

Kf , g)

U. (10)

The operator K is not positive definite only when thestructures are unrestrained and λ0 = 0 and λn = 0.

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

4. Free Response

First consider the free response of the structures, F = 0. Anormal-mode solution is assumed as

W = weiωt, (11)

where ω is a natural frequency and w =[

w1(x) w2(x) w3(x) · · · wn−1(x) wn(x)]T

is a vectorof mode shapes corresponding to that natural frequency.Substitution of (11) into (3) leads to

M−1(K + Kc)w = ω2w, (12)

where the partial derivatives have been replaced by ordinaryderivatives in the definition of K. From (12), it is clear thatthe natural frequencies are the square roots of the eigenvaluesof D = M−1(K + Kc), and the mode shape vectors are thecorresponding eigenvectors.

It is well known [18] that eigenvalues of a self-adjointoperator are all real and that eigenvectors corresponding todistinct eigenvalues are orthogonal with respect to the innerproduct for which the operator is self-adjoint. Thus, if ωi andωj are distinct natural frequencies with corresponding modeshape vectors wi and w j , respectively, then

(

wi, w j

)

M= 0,

(

wi, w j

)

K= 0.

(13)

The mode shape vectors can be normalized by requiring

(wi, wi)M = 1, (14)

which then leads to

(wi, wi)K = ω2i . (15)

5. Forced Response

Since D is a self-adjoint operator, its eigenvectors, the modeshape vectors, can be shown to be complete in U . An

expansion theorem then implies that for any f in U thereexists coefficients αi, i = 1, 2, . . ., such that

f =∑

i

αiwi, (16)

where wi are the normalized mode shape vectors, and thesummation is carried out over all modes. For a given f inU , the coefficients are calculated by

αi = (f , wi)M. (17)

Let W(t) represent the response due to the force vectorF(t). Since W must be in U , the expansion theorem may beapplied at any t, leading to

W(x, t) =∑

i

ci(t)wi(x). (18)

Substitution of (18) into (3) results in∑

i

ci(t)Mwi +∑

i

ci(t)Kwi = 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 usingmode-shape orthogonality properties of (13)–(15) leads toan uncoupled set of differential equations of the form

c j(t) + ω2j c j(t) =

(

F, w j

)

U. (20)

A convolution integral solution of (20) is

cj(t) = 1ωj

∫ t

0

(

F(x, τ), w j(x))

Usin[

ωj(t − τ)]

dτ

= 1ωj

∫ t

0

∫ 1

0wTj (x)F(x, τ) sin

[

ωj(t − τ)]

dx dτ.

(21)

6. Uniform Structures with Li Proportional to L1

A special case occurs when the structures are uniformand operators for the individual structural elements areproportional to one another,

Li = μiL1. (22)

In this case, the component of the stiffness operator due tothe elasticity of the structural elements becomes

K = AL1 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

μ1 0 0 0 · · · 0

0 μ2 0 0 · · · 0

0 0 μ3 0 · · · 0

0 0 0 μ4 · · · 0

......

......

. . ....

0 0 0 0 · · · μn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

L1. (23)

The system may be nondimensionalized such that M1 =1. Then the differential equation governing the free response


of the first structural element, if isolated from the remainderof the system, is

Φ1 + L1Φ1 = 0. (24)

A normal-mode solution of (32) of the form Φ(x, t) =φ(x)eiδt leads to the eigenvalue-eigenvector problem

L1φ = δ2φ. (25)

There are an infinite, but countable, number of naturalfrequencies for the system of (25), δ1, δ2, . . . , δk−1, δk, δk+1, . . .with corresponding mode shapes φ1,φ2, . . . ,φk−1,φk,φk+1, . . .The mode shapes are normalized by requiring (φi,φi)S = 1.

Assume a solution to (12) of the form

w = aφk(x), (26)

where a is an n × 1 vector of constants. Substitution of (26)into (12) using (24) and (25) leads to

M−1(

δ2kA + Kc

)

a = ω2a. (27)

Equation (27) implies that, for this special case, the deter-mination of the natural frequencies of the set of elasticallyconnected structures is reduced to the determination of theeigenvalues of the matrix M−1(δ2

kA + Kc). For each k thereare n natural frequencies. The natural frequencies can thusbe indexed as ωk, j for k = 1, 2, . . . and j = 1, 2, . . . ,n. Thecorresponding mode shapes are written as

wk, j(x) = ak, jφk(x), (28)

where ak, j is the eigenvector of M−1(δ2kA+Kc) corresponding

to the natural frequency ωk, j .The function φk(x) represents the kth mode shape of a

single structure. For each k, there are n natural frequenciesand n corresponding mode shapes. Such a set of modeshapes, which are referred to as intramodal modes, havethe same spatial behavior, but their dependence across thestructures varies. Two mode shapes wk, j(x) and wp,q(x) forwhich p /= k are referred to as intermodal modes.

Note that since M and δ2kA + K are both symmetric

matrices, the matrix M−1(δ2kA + Kc) is self-adjoint with

respect to a kinetic energy inner product. The intramodalmode shapes satisfy an orthogonality condition on Rn givenby

(

ak,i, ak, j

)

Mn=(

ak, j

)TMak,i = 0 for i /= j. (29)

All mode shapes satisfy the orthogonality condition on U ,

(

wk,i, w�, j

)

M=∫ 1

0

(

a�, j

)TMak,iφk(x)φ�(x)dx

= 0 when k /= � or i /= j.

(30)

Consider the case when Kc is singular (λ0 = 0 and λn =0). Then zero is the smallest eigenvalue of Kc and b is its

corresponding eigenvector, b = c[

1 1 · · · 1]

, where c is

Table 1: First five sets of intramodal natural frequencies of fourelastically connected fixed free shafts, ω(k, j).

jk

1 2 3 4 5

1 1.5708 4.7124 7.8540 10.9956 14.1372

2 1.5763 4.7142 7.8551 10.9964 14.1378

3 1.5965 4.7210 7.8592 10.9993 14.1400

4 1.8811 4.8247 7.9219 11.0442 14.1750

a normalization constant. Suppose that M−1A = I, then (27)becomes

(

δ2kI + Kc

)

a = ω2a. (31)

Note that the solution of (31) corresponds to ω = δk anda = b. Thus, for this special case, the lowest natural frequencyfor each set of intramodal modes is equal to the naturalfrequency for the spatial mode of one structural element andits corresponding mode shape is the null space of Kc.

To examine the most general case when Kc is singular,take the standard inner product for Rn of both sides of (27)with b. Using properties of inner products and noting that A,Kc, and M are symmetric leads to

(

a,(

δ2kA− ω2M

)

b)

Rn= 0. (32)

Application of the orthogonality condition of (32) leads to

n∑

i=1

ai(

δ2kμi − ω2βi

)

= 0. (33)

The expansion theorem is used to assume a forcedresponse of the form

W(x, t) =∞∑

k=1

n∑

j=1

ck, j(t)ak, jφk(x). (34)

Use of (34) in (3) leads to differential equations of the form

ck, j + ω2k, j ck, j =

(

F, ak, jφk(x))

U. (35)

A special case occurs when uniform structures areidentical such that A = I and M = I. Then, (27) becomes

(

δ2kI + Kc

)

a = ω2a. (36)

Equation (36) can be rewritten as

Kca =(

ω2 − δ2k

)

a. (37)

Let κj j = 1, 2, . . . ,n be the eigenvalues of Kc. Then,

ω2k, j =

(

δ2k + κj

)

. (38)

In addition, since the same set of eigenvalues is used tocalculate the natural frequencies for each intramodal sets, theintramodal mode shape vectors are the same for each set andare the eigenvectors of Kc, that is

ak, j = a�, j = v jk, � = 1, 2, . . . , j = 1, 2, . . . ,n, (39)

where v j j = 1, 2, . . . ,n are the eigenvectors of Kc.


7. Nonuniform Structures

The case when one or more of the structures is nonuniform ismore difficult in that the differential equations of (12) havevariable coefficients. Consider the case when the structuresare nonuniform, but identical. In this case, the coupled set ofdifferential equations can be written as

ILw + Kcw − ω2IMw = 0, (40)

where L and M are the stiffness and inertia operators,respectively, for any of the structures.

Let κ1, κ2, . . . , κn be the eigenvalues of Kc and letz1, z2, . . . , zn be their corresponding eigenvectors normalizedwith respect to the standard inner product on Rn (zT

i zi =1). Let P be the matrix whose columns are the normalizedeigenvectors. Since Kc is symmetric, it can be shown thatPTP = I and PTKcP = Δ, where Δ is a diagonal matrix withΔi, j = κi. Defining

w = Pq (41)

and substituting into (40) and then premultiplying by PT

leads to

ILq + Δq− ω2IMq = 0. (42)

Equation (42) represents a set of uncoupled differentialequations, each of the form

Lqj + κjq j − ω2Mqj = 0 j = 1, 2, . . . ,n. (43)

Equation (43) represents, along with appropriate homo-geneous boundary conditions, an eigenvalue problem todetermine the natural frequencies. For each j, there arean infinite, but countable, number of natural frequencies.Thus, the natural frequencies can be indexed by ωj,k j =1, 2, . . . ,n k = 1, 2, . . ..

It still may not be possible to solve (43) in closedform; however, it is now known how to index the naturalfrequencies. For a set of identical structures, there are aninfinite number of natural frequencies corresponding to eacheigenvalue of Kc. The term intramodal is not appropriate forthis set of natural frequencies as they do not correspond tothe same mode. Indeed, there are not necessarily intramodalfrequencies for the nonuniform case.

If Kc is singular and thus has zero as its lowest eigenvalue,then (43) shows that one set of natural frequencies is identi-cal to the natural frequencies of the individual structures.

8. Examples

8.1. Shafts. Consider n concentric shafts of equal lengthconnected by elastic layers. The stiffness and inertia operatorsfor uniform shafts are, respectively,

Liθi = −μi ∂2θi∂x2

,

Miθi = βiθi.

(44)

The stiffness operators are of the form of those consid-ered in Section 6. Hence, the natural frequencies and modeshapes can be determined from solving a matrix eigenvalueproblem. If all shafts are made of the same material thenβi = μi leading to M−1K = I, and thus if Kc is singularthen the lowest natural frequency for each set of intramodalmodes is the same as the modal natural frequency for theinnermost shaft, and the mode shapes corresponding to eachfrequency are the eigenvector of Kc corresponding to its zeroeigenvalue 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 forδk k = 1, 2, . . . , 5 and the corresponding normalized modeshapes φk(x) are listed in Table 1 for various end conditions.

The matrix eigenvalue problem for a set of intramodalfrequencies is of the form

Ka = ω2Ma, (46)

where K is a tridiagonal matrix whose elements are

ki,i = μiδ2 + λi−1 + λi i = 1, 2, . . . ,n,

ki,i−1 = −λi−1 i = 2, 3, . . . ,n,

ki,i+1 = −λi i = 1, 2, . . . ,n− 1,

(47)

and M is a diagonal matrix with mi,i = βi.

8.2. Natural Frequencies of Four Concentric Fixed-Free Shafts.As a numerical example, consider four concentric fixed-freeshafts connected by layers of torsional stiffness. Solving (45)subject to φk(0) = 0 and φ′k(1) = 0,

δk = (2k − 1)π2

φk(x) =√

2 sin[

(2k − 1)π2

x]

. (48)

Each shaft is made of the same material. The inner shaft issolid of radius r. The outer shafts are each of thickness r.The thickness of each elastic layer is negligible. This leads toμ1 = 1 μ2 = 15 μ3 = 65, μ4 = 175, β1 = 1 β2 = 15 β3 =65, β4 = 175. The torsional stiffness of each elastic layeris the same and is taken such that λi = 1. The inner shaftis solid, thus, λ0 = 0. The outer radius of the outer shaftis unrestrained from rotation, hence, λ4 = 0. The matrixeigenvalue problems become


⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 +(

(2k − 1)π2

)2

−1 0 0

−1 2 + 15(

(2k − 1)π2

)2

−1 0

0 −1 2 + 65(

(2k − 1)π2

)2

−1

0 0 −1 1 + 175(

(2k − 1)π2

)2

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

ak,1

ak,2

ak,3

ak,4

⎤

⎥⎥⎥⎥⎥⎥⎦

= ω2

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 16 0 0

0 0 81 0

0 0 0 256

⎤

⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

ak,1

ak,2

ak,3

ak,4

⎤

⎥⎥⎥⎥⎥⎥⎦

.

(49)

The natural frequencies for k = 1, 2, . . . , 5 are given inTable 1. Since Kc is singular and M−1Kb = I, the lowestnatural frequency in each intramodal set is δk. Each modeshape in a set of intramodal mode shapes corresponds tothe same spatial mode φk(x). The difference in intramodalmode shapes is in the relative magnitude and signs ofthe displacements of the individual shafts. The normalizedmode shapes of Figure 2 correspond to the first modeshape in the intramodal set for the first spatial modeand illustrate the mode in which the shafts rotate as ifthey are rigidly connected. The mode shapes of Figure 3correspond to the third intramodal mode for the firstspatial mode and illustrate that when the rotations of thefirst, second, and fourth shafts are counterclockwise, therotation of the third shaft is clockwise. Figures 4 and 5illustrate mode shapes corresponding to the third spatialmode. All mode shapes in the intramodal set for this modehave two nodes across the length of the shaft. Note thatfor the second intramodal frequency there is a change inthe direction of rotation of the shafts between the thirdand fourth shafts. Thus, there is a cylindrical surface ofnodes between these shafts. There are two changes in thedirection of rotation for the third intramodal frequency,between the second and third shafts and between the thirdand fourth shafts, leading to two cylindrical surfaces ofnodes.

8.3. Forced Response of Four Concentric Shafts. Suppose thatthe midspan of the outer shaft is subject to a constanttorque, T0, such that the nondimensional applied torques areT1(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 aconvolution integral solution of the form of (21) leadingto

ck, j(t) = 1ωk, j

∫ t

0

∫ 1

0aTk, jF(x, τ)φk(x) sin

[

ωk, j(t − τ)]

dxdτ

=(

ak, j

)

4φk(1/2)

ω2k, j

[

1− cos(

ωk, j t)]

.

(50)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

w

0 0.2 0.4 0.6 0.8 1

x

w1

w2

w3

w4

Mode shapes for ω1.1 = 1.5708

Figure 2: Set of intramodal mode shapes of elastically connectedfixed-free torsional shafts with k = 1 and j = 1. The mode shapescorrespond to rigid-body motion across the set of shafts.

Substitution of (50) into (18) leads to

W(x, t)=∞∑

k=1

⎛

⎝4∑

j=1

(

ak, j

)

4

ω2k, j

ak, j

[

1− cos(

ωk, j t)]⎞

⎠φk

(12

)

φk(x).

(51)

Equation (51) is evaluated leading to the time dependence ofthe response at x = 1/2 and x = 1 illustrated in Figures 6 and7.

8.4. Nonuniform Shafts. Now consider the same set of shafts,except that each has a taper, such that the differentialequation for the innermost shaft when isolated from thesystem is

d

dx

[

(1− 0.1x)2 dθ

dx

]

+ ω2(1− 0.1x)2θ = 0. (52)


−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

w

0 0.2 0.4 0.6 0.8 1

x

w1

w2

w3

w4


Figure 3: Set of intramodal mode shapes of elastically connectedfixed-free torsional shafts with k = 1 and j = 3.

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

w

0 0.2 0.4 0.6 0.8 1

x

w1

w2

w3

w4



Along with the boundary for a fixed-free shaft, (52) hasa Bessel function solution leading to the characteristicequation for the shaft’s natural frequencies as

j′0(9ω)yo(10ω)− y′0(9ω) j0(10ω) = 0, (53)

where jn(x) and yn(x) are spherical Bessel functions ofthe first and second kinds of order n and argument x.

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

w

0 0.2 0.4 0.6 0.8 1

x

w1

w2

w3

w4



The mode shape (nonnormalized) corresponding to a nat-ural frequency ωk is

θk(x) = jo[10ωk(1− 0.1x)]− j′0[9ωk]y0[10ωk(1− 0.1x)].(54)

The differential equations for the elastically coupledshafts become

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 15 0 0

0 0 65 0

0 0 0 175

⎤

⎥⎥⎥⎥⎥⎥⎦

d

dx

[

(1− 0.1x)2 d

dx

]

⎡

⎢⎢⎢⎢⎢⎢⎣

θ1

θ2

θ3

θ4

⎤

⎥⎥⎥⎥⎥⎥⎦

−

⎡

⎢⎢⎢⎢⎢⎢⎣

1 −1 0 0

−1 2 −1 0

0 −1 2 −1

0 0 −1 1

⎤

⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

θ1

θ2

θ3

θ4

⎤

⎥⎥⎥⎥⎥⎥⎦

+ ω2

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 15 0 0

0 0 65 0

0 0 0 175

⎤

⎥⎥⎥⎥⎥⎥⎦

× (1− 0.1x)2

⎡

⎢⎢⎢⎢⎢⎢⎣

θ1

θ2

θ3

θ4

⎤

⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎣

0

0

0

0

⎤

⎥⎥⎥⎥⎥⎥⎦

.

(55)

Even though the shafts are not identical, the differentialequations in (55) may still be decoupled because the stiffnessand inertia matrices are the same. The eigenvalues of M−1Kc


are κ1 = 0, κ2 = 0.0172, κ3 = 0.0815, and κ4 = 1.0712. Thecorresponding matrix of eigenvectors is

P =

⎡

⎢⎢⎢⎢⎢⎢⎣

0.0625 0.1191 0.2292 0.9640

0.0625 0.1171 0.2106 −0.0686

0.0625 0.0848 −0.0654 0.0010

0.0625 −0.0422 0.0049 −5.44× 10−6

⎤

⎥⎥⎥⎥⎥⎥⎦

. (56)

The columns of P have been normalized such that PTMP =I and PTKcP = Δ. Following the same procedure as inthe derivation of (43), the uncoupled differential equationsbecome

d

dx

[

(1− 0.1x)2 dq1

dx

]

+ ω2(1− 0.1x)2q1 = 0,

d

dx

[

(1− 0.1x)2 dq2

dx

]

+[

ω2(1− 0.1x)2 − 0.0172]

q2 = 0,

d

dx

[

(1− 0.1x)2 dq3

dx

]

+[

ω2(1− 0.1x)2 − 0.0815]

q3 = 0,

d

dx

[

(1− 0.1x)2 dq4

dx

]

+[

ω2(1− 0.1x)2 − 1.0712]

q4 = 0.

(57)

The solutions of (57) are

q1(x) = C1 j0[ω(1− 0.1x)] + C2y0[ω(1− 0.1x)],

q2(x) = C1 j0.904[ω(1− 0.1x)] + C2y0.904[ω(1− 0.1x)],

q3(x) = C1 j2.398[ω(1− 0.1x)] + C2y2.398[ω(1− 0.1x)],

q4(x) = C1 j9.862[ω(1− 0.1x)] + C2y9.862[ω(1− 0.1x)].(58)

The characteristic equations to determine the natural fre-quencies are

j0(10ωk,1

)y1(9ωk,1

)− y0(10ω1) j1(9ωk,1

) = 0,

j0,904(10ωk,2

)[

0.904y0.904

(9ωk,2

)

9ωk,2− y1.904

(9ωk,2

)]

− y0,904(10ωk,2

)[

0.904j0.904

(9ωk,2

)

9ωk,2− j1.904

(9ωk,2

)]

= 0,

j2.398(10ωk,3

)[

2.398y2.398

(9ωk,3

)

9ωk,3− y3.398

(9ωk,3

)]

− y2.398(10ωk,3

)[

2.398j2.398

(9ωk,3

)

9ωk,3− j3.398

(9ωk,3

)]

= 0,

j9.862(10ωk,4

)[

9.862y9.862

(9ωk,4

)

9ωk,4− y10.862

(9ωk,4

)]

− y9.862(10ωk,4

)[

9.862j9.862

(9ωk,4

)

9ωk,4− j10.962

(9ωk,4

)]

=0.

(59)

The characteristic equation for the first set of frequencies isidentical to (53). The first five frequencies for each j are givenin Table 2.

−4

−2

0

2

4

6

8

10

12

14

16

W(0.5

,t)

×10−3

0 1 2 3 4 5

t

w1(t)w2(t)

w3(t)w4(t)

Forced response at x = 0.5

Figure 6: Forced response of elastically connected torsional shaftsat x = 0.5 due to constant concentrated torque applied to outershaft at x = 0.5.

−4

−2

0

2

4

6

8

10

12

14

16

W(1

,t)

×10−3

0 1 2 3 4 5

t

w1(t)w2(t)

w3(t)w4(t)

Forced response at x = 1

Figure 7: Forced response of elastically connected torsional shaftsat x = 1 due to constant concentrated torque applied to outer shaftat x = 0.5.

8.5. Euler-Bernoulli Beams. Consider a set of n parallel Euler-Bernoulli beams connected by elastic layers. For uniformbeams, the mass and stiffness operators are Li = μi(∂4/∂x4)and Mi = βi. The differential eigenvalue problem for the firstbeam in the set is

d4φ

dx4− δ2φ = 0. (60)


Table 2: First five sets of frequencies for set of linearly taperedshafts.

jk

1 2 3 4 5

1 1.639 4.736 7.868 11.006 14.145

2 1.645 4.738 7.869 11.007 14.146

3 1.667 4.745 7.874 11.010 14.148

4 1.981 4.861 7.944 11.010 14.187

The transverse displacements of Euler-Bernoulli beams isanother example of the special case discussed in Section 6.

If the beams are identical (μi = 1 and βi = 1) andif Kc is singular, then the lowest natural frequency for thekth set of intramodal modes is δk with the mode shape,such that each beam has the same displacement and thesprings are unstrectched. Otherwise, the mode shape for thelowest natural frequency of each intramodal set satisfies theorthogonality condition of (31).

As a numerical example, consider a set of five fixed-freeelastically connected Euler-Bernoulli beams. The solution ofEquation (60) subject to the boundary conditions φ(0) = 0,φ′(0) = 0, φ′′(1) = 0, and φ′′′(1) = 0 leads to

φk(x)=cosh(√

δkx)

− cos(√

δkx)

−cos(√δk)

+ cosh(√δk)

sin(√δk)

+ sinh(√δk)

×[

sinh(√

δkx)

− sin(√

δkx)]

,

(61)

where δk is the kth solution of cos(√δk) cosh(

√δk) = −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 eigenvalueproblem for a set of intramodal frequencies and mode shapesis⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

δ2k + 100 −100 0 0 0

−100 2δ2k + 150 −50 0 0

0 −50 0.5δ2k + 100 −50 0

0 0 −50 δ2k + 70 −20

0 0 0 −20 0.25δ2k + 20

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a1

a2

a3

a4

a5

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

= ω2

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0

0 1.5 0 0 0

0 0 0.75 0 0

0 0 0 1 0

0 0 0 0 0.5

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a1

a2

a3

a4

a5

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(62)

The sets of intramodal frequencies are listed in Table 3.Recall that δk is the natural frequency of the first beam. The

Table 3: First four sets of intramodal natural frequencies for a setof five elastically connected fixed-fixed Euler-Bernoulli beams.

jδk

3.5100 22.0300 61.7000 120.90

1 3.4630 16.7195 44.0798 85.7222

2 6.6167 20.6044 51.6490 99.3829

3 8.5529 23.4906 62.2901 121.1925

4 12.7319 24.1708 62.4633 121.3072

5 14.8317 28.0564 71.9859 139.9667

−1.5

−1

−0.5

0

0.5

1

1.5

2

w

0 0.2 0.4 0.6 0.8 1x

w1

w2

w3

w4w5


Figure 8: Intramodal mode shapes for a set of five elasticallyconnected Euler-Bernoulli beams.

natural frequency of an Euler-Bernoulli beam increases withstiffness. Since the first beam is stiffer than several otherbeams in the set, some intramodal frequencies are lower thanδk. The mode shapes for k = 2 and j = 4 illustrated inFigure 8 show one spatial node and three cylindrical surfacesof nodes.

9. Conclusion

A general theory is developed for the free and forced responseof elastically connected structures. The following has beenshown.

(i) The general problem can be formulated using thevector space U = Rn × S, where S is a subspace ofCk[0, 1] defined by the system’s end conditions.

(ii) If the differential stiffness operator for a single struc-ture is self-adjoint with respect to a standard innerproduct on S, then the general stiffness operator isself-adjoint with respect to a standard inner producton U .

(iii) Kinetic and potential energy inner products aredefined on both S and U .


(iv) A normal-mode solution for the free response leadsto the formulation of an eigenvalue problem definedfor a matrix of operators.

(v) The operator is self-adjoint with respect to the energyinner products leading to the development of anorthogonality condition.

(vi) The expansion theorem is used to develop a modalanalysis for the forced response.

(vii) The case where the structures are uniform and theindividual stiffness operators are proportional is aspecial case in which the determination of naturalfrequencies and mode shapes can be reduced toeigenvalue problems for matrices on Rn.

(viii) When the stiffness operators are proportional, thenatural frequencies and mode shapes are indexedwith two indices, the first representing the spatialmode shape, the second representing the intramodalmode shapes.

(ix) If the uniform structures are identical, then a simpleformula can be derived for the sets of intramodal nat-ural frequencies using the eigenvalues of the couplingstiffness matrix. The intramodal mode shapes foreach spatial mode are the eigenvectors of the couplingstiffness martrix.

(x) An iterative solution must be applied to determinethe natural frequencies for the most general case ofthe uniform structure.

(xi) The differential equations for the coupling of iden-tical structures, uniform, or nonuniform can beuncoupled through diagonalization of the couplingstiffness matrix.

(xii) Elastically connected uniform strings and elasticallyconnected uniform concentric shafts are applicationsin which the stiffness operators are proportional.

(xiii) The differential equations for the concentric shafts,even though they are not identical, can be decoupledwhen each individual stiffness operator is the same asthe individual mass operator.

(xiv) The individual stiffness operators for uniform Euler-Bernoulli beams are proportional implying that theirnatural frequencies can be indexed as an infinitenumber of sets of intramodal frequencies.

The general method is applied here only for undampedsystems. However, it can be applied to certain damped sys-tems as well. If the structures are undamped but the Winklerlayers have viscous damping, the same δk and φk for theundamped system may be used, but an eigenvalue problemis obtained involving complex numbers. If the structures aredamped but the Winkler layers are undamped, the choiceof δk and φk is modified to include viscous damping, butagain a complex eigenvalue problem is obtained. If theentire system (both the individual structures and the Winklerlayers) is subject to proportional damping, the eigenvaluesand the eigenvectors of the undamped system can be used touncouple the forced vibrations equations.

References

[1] S. G. Kelly, Fundamentals of Mechanical Vibrations, McGraw-Hill, Boston, Mass, USA, 2nd edition, 2000.

[2] S. S. Rao, Mechanical Vibrations, Pearson/Prentice Hall, UpperSaddle River, NJ, USA, 4th edition, 2004.

[3] L. Mierovitch, Fundamentals of Vibrations, McGraw-Hill, Bos-ton, Mass, USA, 2001.

[4] B. Balachandran and E. McGrab, Vibrations, Thomson, Tor-onto, Canada, 2003.

[5] J. Ginsburg, Mechanical and Structural Vibrations: Theory andApplications, Wiley, New York, NY, USA, 2001.

[6] D. Inman, Engineering Vibrations, Prentice Hall, Upper SaddleRiver, NJ, USA, 3rd edition, 2007.

[7] S. G. Kelly, Advanced Vibration Analysis, CRC Press/Taylor andFrancis Group, Boca Raton, Fla, USA, 2007.

[8] L. Meirovitch, Principles and Techniques of Vibration, PrenticeHall, Upper Saddle River, NJ, USA, 1997.

[9] Z. Oniszczuk, “Transverse vibrations of elastically connecteddouble-string complex system. Part I: free vibrations,” Journalof Sound and Vibration, vol. 232, no. 2, pp. 355–366, 2000.

[10] Z. Oniszczuk, “Transverse vibrations of elastically connecteddouble-string complex system. Part II: forced vibrations,”Journal of Sound and Vibration, vol. 232, no. 2, pp. 367–386,2000.

[11] J. M. Selig and W. H. Hoppmann, “Normal mode vibrationsof systems of elastically connected parallel bars,” Journal of theAcoustical Society of America, vol. 36, pp. 93–99, 1964.

[12] E. Osborne, “Computations of bending modes and modeshapes of single and double beams,” Journal of the Societyfor Industrial and Applied Mathematics, vol. 10, pp. 329–338,1962.

[13] Z. Oniszczuk, “Free transverse vibrations of elastically con-nected simply supported double-beam complex system,”Journal of Sound and Vibration, vol. 232, no. 2, pp. 387–403,2000.

[14] Z. Oniszczuk, “Forced transverse vibrations of an elasticallyconnected complex simply supported double-beam system,”Journal of Sound and Vibration, vol. 264, no. 2, pp. 273–286,2003.

[15] S. S. Rao, “Natural frequencies of systems of elastically con-nected Timoshenko beams,” Journal of the Acoustical Society ofAmerica, vol. 55, no. 6, pp. 1232–1237, 1974.

[16] C. Q. Ru, “Column buckling of multiwalled carbon nanotubeswith interlayer radial displacements,” Physical Review B, vol.62, no. 24, pp. 16962–16967, 2000.

[17] C. Q. Ru, “Axially compressed buckling of a doublewalledcarbon nanotube embedded in an elastic medium,” Journal ofthe Mechanics and Physics of Solids, vol. 49, no. 6, pp. 1265–1279, 2001.

[18] C. Q. Ru, “Effect of van der Waals forces on axial buckling ofa double-walled carbon nanotube,” Journal of Applied Physics,vol. 87, no. 10, pp. 7227–7231, 2000.

[19] J. Yoon, C. Q. Ru, and A. Mioduchowski, “Noncoaxialresonance of an isolated multiwall carbon nanotube,” PhysicalReview B, vol. 66, no. 23, Article ID 233402, 4 pages, 2002.

[20] C. Li and T. W. Chou, “Vibrational behaviors of multiwalled-carbon-nanotube-based nanomechanical resonators,” AppliedPhysics Letters, vol. 84, no. 1, pp. 121–123, 2004.

[21] J. Yoon, C. Q. Ru, and A. Mioduchowski, “Terahertz vibrationof short carbon nanotubes modeled as Timoshenko beams,”Journal of Applied Mechanics, vol. 72, no. 1, pp. 10–17, 2005.


[22] J. Yoon, C. Q. Ru, and A. Mioduchowski, “Sound wave prop-agation in multiwall carbon nanotubes,” Journal of AppliedPhysics, vol. 93, no. 8, pp. 4801–4806, 2003.

[23] K. Y. Xu, X. N. Gao, and C. Q. Ru, “Vibration of a doublewalled carbon nanotube aroused by nonlinear Van der Waalsforces,” Journal of Applied Physics, vol. 99, p. 604033, 2006.

[24] I. Elishakoff and D. Pentaras, “Fundamental natural frequen-cies of double walled carbon nanotubes,” Journal of Sound andVibration, vol. 322, no. 2, pp. 652–664, 2009.

[25] S. G. Kelly and S. Srinivas, “Free vibrations of elasticallyconnected stretched beams,” Journal of Sound and Vibration,vol. 326, no. 3–5, pp. 883–893, 2009.

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