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Free and Forced Vibrations of Elastically Connected Structures

Free and Forced Vibrations of Elastically Connected Structures Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 984361, 11 pages doi:10.1155/2010/984361 Research Article S. Graham Kelly Department of Mechanical Engineering, The University of Akron, Akron, Oh 44235, USA Correspondence should be addressed to S. Graham Kelly, gkelly@uakron.edu 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 the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for C [0, 1], then the stiffness n k operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on U = R ×C [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 approximate formulas for the natural frequencies of double- walled nanotubes noting that if developed from the eigen- The general theory for the free and forced response of strings, value relation, the computations can be compuitationally shafts, beams, and axially loaded beams is well documented intensive and difficult. [1–8]. Investigators have examined the free and forced Kelly and Srinivas [25] developed a Rayleigh-Ritz meth- response of elastically connected strings [9, 10], Euler- od for elastically connected stretched structures. Bernoulli beams [11–14], and Timoshenko beams [15]. This paper develops a general theory within which a These analyses focused on a pair of elastically connected finite set of parallel structures connected by elastic layers of structures using a normal-mode solution for the free a Winkler type can be analyzed. The theory shows that the response and a modal analysis for the forced response. Each determination of the natural frequencies for uniform parallel of these papers uses a normal-mode solution or a modal structures such as shafts and Euler-Bernoulli beams can be analysis specific to the problem to obtain a solution. reduced to matrix eigenvalue problems. The general theory Ru [16–18] proposed that model for multiwalled carbon is also used to develop a modal analysis for forced response nanotubes to be modeled by elastically connected structures of a set of parallel 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 2. Problem Formulation nanotubes and elastic shell models [17, 18]. Yoon et al. [19] andLiand Chou [20] modeled free vibrations of multiwalled The problem considered is that of n structural elements in nanotubes by a series of concentric elastically connected parallel but connected by elastic layers. Each elastic layer Euler-Bernoulli beams, while Yoon et al. [21, 22] modeled is modeled by a Winkler foundation, a layer of distributed nanotubes as concentric Timoshenko beams connected by stiffness across the span of the element. For generality, it is an elastic layer. Xu et al. [23] modeled the nonliearity of assumed that the outermost structures are connected to fixed the vanDer Waals forces. Elishakoff and Pentaras [24]gave foundations through elastic layers, as illustrated in Figure 1. 2 Advances in Acoustics and Vibration Each layer has a uniform stiffness per unit length k i = 0, 1, 2,... , n. Let w (x) represent the displacement of the ith structure. k If isolated from the system, the nondimensional differential equation governing the time-dependent motion of this n−1 structure is written as n− L w (x) + M w ¨ = G (x, t),(1) i i i i i n−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . where L is the stiffness operator for the element, M is an i i . . . . . . . . . . . . . . . . inertia operator for the element, and G (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 = 4for Euler- 1 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 k Figure 1: Set of n elastically connected parallel structures. conditions. Let S be the subspace of C [0, 1] defined by the boundary conditions; all elements in S satisfy all boundary conditions. If the stiffness operator is self-adjoint [(L f , g ) = The external forces acing on the ith structure are i S ⎧ ( f , L g ) ] and positive definite [(L f , f ) ≥ 0and (L f , f ) = i i i S s ⎪ −λ w − λ (w − w ) + F (x, t) i = 1, 0 1 1 1 2 1 0 if and only if f = 0] with respect to the standard inner ⎪ product, then a potential energy inner product is defined as −λ (w − w ) i−1 i i−1 G (x, t)= f , g = L f , g . (6) ⎪ i ⎪ −λ (w − w ) + F (x, t) i = 2, 3,... , n − 1, L S i i i+1 i i Clearly each M is positive definite and self-adjoint with ( ) ( ) −λ w − w − λ w + F x, t i = n, i n−1 n n−1 n n n respect to the standard inner product, and thus a kinetic (2) energy inner product can be defined as where λ , i = 0, 1,... , n are the nondimensional stiffness coefficients connecting the ith and i plus first structures. f , g = M f gdx,(7) Substitution of (2) into (1) leads to a coupled set of differential equations which are written in a matrix form as for any f and g in S. (K + K )W + MW = F,(3) c The standard inner product on U is defined as where W = w (x, t) w (x, t) ··· w (x, t) , F = 1 2 n T f , g = g fdx,(8) F (x, t) F (x, t) ··· F (x, t) , K is an n × n diagonal 1 2 n for any f and g in U . It is easy to show that M is self-adjoint operator matrix with k = L , M is an n × n diagonal mass i,i with respect to the inner product of (8) and a kinetic energy matrix with m = M ,and K is a tridiagonal n × n stiffness i,i i c coupling matrix with inner product on U is defined as (k ) =−λ i = 2, 3,... , c i,i−1 i−1 f , g = Mf , g . (9) M U (k ) = λ + λ i = 1, 2,... , n, (4) c i−1 i i,i Define K = K + K . Since K is self-adjoint with respect to the standard inner product on S and K is a symmetric (k ) =−λ i = 1, 2,... , n − 1. c i i,i+1 matrix, it can be shown that K is self-adjoint with respect to The vector W is an element of the vector space U = S × R ; the standard inner product on U . The positive definiteness an element of U is an n-dimensional vector, whose elements of K with respect to the standard inner product on U is all belong to S. determined by considering (Kf , f ) = (Kf , f ) +(K f , f ) . U U c U If K is a positive definite matrix with respect to the standard 3. General Theory inner product on R , then (K f , f ) ≥ 0and (K f , f ) = 0if c c U U and only if f = 0. If each of the operators L , i = 1, 2,... , n Let f (x)and g (x) be arbitrary elements of S. A standard is positive definite with respect to the standard inner product inner product on S is defined as on C [0, 1] then (Kf , f ) ≥ 0and (Kf , f ) = 0ifand only U U if f = 0.Thus, K is positive definite with respect to the ( ) ( ) f , g = f x g x dx. (5) standard inner product on U if either K is a positive definite 0 Advances in Acoustics and Vibration 3 matrix with respect to the standard inner product on R or expansion theorem then implies that for any f in U there each of the operators L , i = 1, 2,... , n is positive definite exists coefficients α , i = 1, 2,..., such that i i with respect to the standard inner product on C [0, 1]. Under f = α w , (16) i i either of these conditions, a potential energy inner product is defined on U by where w are the normalized mode shape vectors, and the f , g = Kf , g . (10) K U summation is carried out over all modes. For a given f in U , the coefficients are calculated by The operator K is not positive definite only when the structures are unrestrained and λ = 0and λ = 0. 0 n α = (f , w ) . (17) i i −1 Define D = M K. It is possible to show that D is self-adjoint with respect to the kinetic energy inner product Let W(t) represent the response due to the force vector of (9) and the potential energy inner product of (10). If F(t). Since W must be in U , the expansion theorem may be K is positive definite with respect to the standard inner applied at any t, leading to product, then D is positive definite with respect to both inner products. W(x, t) = c (t)w (x). (18) i i 4. Free Response Substitution of (18) into (3) results in First consider the free response of the structures, F = 0.A c ¨ (t)Mw + c (t)Kw = F(x, t). (19) i i i i normal-mode solution is assumed as i i iωt Taking the standard inner product on U of both sides of W = we , (11) (26)with w (x) for an arbitrary j = 1, 2,... , n and using where ω is a natural frequency and w = mode-shape orthogonality properties of (13)–(15) leads to an uncoupled set of differential equations of the form w (x) w (x) w (x) ··· w (x) w (x) is a vector 1 2 3 n−1 n of mode shapes corresponding to that natural frequency. ¨ ( ) ( ) c t + ω c t = F, w . (20) j j j Substitution of (11) into (3)leads to −1 2 A convolution integral solution of (20)is M (K + K )w = ω w, (12) where the partial derivatives have been replaced by ordinary c (t) = F(x, τ ), w (x) sin ω (t − τ ) dτ j j j j 0 derivatives in the definition of K.From(12), it is clear that (21) the natural frequencies are the square roots of the eigenvalues t 1 −1 T of D = M (K + K ), and the mode shape vectors are the c = w (x)F(x, τ ) sin ω (t − τ ) dx dτ. j 0 0 corresponding eigenvectors. It is well known [18] that eigenvalues of a self-adjoint operator are all real and that eigenvectors corresponding to 6. Uniform Structures with L Proportional to L i 1 distinct eigenvalues are orthogonal with respect to the inner A special case occurs when the structures are uniform product for which the operator is self-adjoint. Thus, if ω and and operators for the individual structural elements are ω are distinct natural frequencies with corresponding mode proportional to one another, shape vectors w and w respectively, then i j , L = μ L . (22) i i 1 w , w = 0, i j (13) In this case, the component of the stiffness operator due to w , w = 0. i j the elasticity of the structural elements becomes ⎡ ⎤ The mode shape vectors can be normalized by requiring μ 000 ··· 0 ⎢ ⎥ ⎢ ⎥ 0 μ 00 ··· 0 ⎢ ⎥ (w , w ) = 1, (14) i i ⎢ ⎥ ⎢ ⎥ 00 μ 0 ··· 0 ⎢ 3 ⎥ ⎢ ⎥ which then leads to K = AL = ⎢ ⎥ L . (23) 1 1 000 μ ··· 0 ⎢ 4 ⎥ ⎢ ⎥ ⎢ ⎥ (w , w ) = ω . (15) i i K i . . . . . ⎢ . ⎥ . . . . . . ⎢ ⎥ . . . . . ⎣ ⎦ 0000 ··· μ 5. Forced Response n Since D is a self-adjoint operator, its eigenvectors, the mode The system may be nondimensionalized such that M = shape vectors, can be shown to be complete in U.An 1. Then the differential equation governing the free response 4 Advances in Acoustics and Vibration of the first structural element, if isolated from the remainder Table 1: First five sets of intramodal natural frequencies of four elastically connected fixed free shafts, ω . (k,j ) of the system, is Φ + L Φ = 0. (24) 1 1 1 123 4 5 A normal-mode solution of (32) of the form Φ(x, t) = 1 1.5708 4.7124 7.8540 10.9956 14.1372 iδt φ(x)e leads to the eigenvalue-eigenvector problem 2 1.5763 4.7142 7.8551 10.9964 14.1378 3 1.5965 4.7210 7.8592 10.9993 14.1400 L φ = δ φ. (25) 4 1.8811 4.8247 7.9219 11.0442 14.1750 There are an infinite, but countable, number of natural frequencies for the system of (25), δ , δ ,... , δ , δ , δ ,... 1 2 k−1 k k+1 −1 a normalization constant. Suppose that M A = I, then (27) with corresponding mode shapes φ , φ ,... , φ , φ , φ ,... 1 2 k−1 k k+1 becomes The mode shapes are normalized by requiring (φ , φ ) = 1. i i 2 2 Assume a solution to (12) of the form δ I + K a = ω a. (31) Note that the solution of (31) corresponds to ω = δ and w = aφ (x), (26) k a = b. Thus, for this special case, the lowest natural frequency where a is an n × 1 vector of constants. Substitution of (26) for each set of intramodal modes is equal to the natural frequency for the spatial mode of one structural element and into (12) using (24)and (25)leads to its corresponding mode shape is the null space of K . −1 2 2 To examine the most general case when K is singular, M δ A + K a = ω a. (27) c take the standard inner product for R of both sides of (27) with b. Using properties of inner products and noting that A, Equation (27) implies that, for this special case, the deter- K ,and M aresymmetric leadsto mination of the natural frequencies of the set of elastically c connected structures is reduced to the determination of the 2 2 −1 2 a, δ A − ω M b = 0. (32) eigenvalues of the matrix M (δ A + K ). For each k there n c R are n natural frequencies. The natural frequencies can thus Application of the orthogonality condition of (32)leads to be indexed as ω for k = 1, 2,... and j = 1, 2,... , n.The k,j corresponding mode shapes are written as 2 2 a δ μ − ω β = 0. (33) i i i i=1 w (x) = a φ (x), (28) k,j k,j k The expansion theorem is used to assume a forced −1 where a is the eigenvector of M (δ A + K ) corresponding k,j c response of the form to the natural frequency ω . k,j ∞ n The function φ (x) represents the kth mode shape of a W(x, t) = c (t)a φ (x). (34) k,j k,j k single structure. For each k, there are n natural frequencies k=1 j=1 and n corresponding mode shapes. Such a set of mode Use of (34)in(3) leads to differential equations of the form shapes, which are referred to as intramodal modes, have the same spatial behavior, but their dependence across the c ¨ + ω c = F, a φ (x) . (35) k,j k,j k,j k k,j structures varies. Two mode shapes w (x)and w (x)for k,j p,q U which p= k are referred to as intermodal modes. A special case occurs when uniform structures are Note that since M and δ A + K are both symmetric identical such that A = I and M = I. Then, (27)becomes −1 matrices, the matrix M (δ A + K ) is self-adjoint with 2 2 respect to a kinetic energy inner product. The intramodal δ I + K a = ω a. (36) mode shapes satisfy an orthogonality condition on R given Equation (36)can be rewrittenas by 2 2 T K a = ω − δ a. (37) a , a = a Ma =0for i= j. (29) k,i k,j k,j k,i / Let κ j = 1, 2,... , n be the eigenvalues of K .Then, j c All mode shapes satisfy the orthogonality condition on U , 2 2 ω = δ + κ . (38) k,j k w , w = a Ma φ (x)φ (x)dx k,i ,j ,j k,i k  In addition, since the same set of eigenvalues is used to (30) calculate the natural frequencies for each intramodal sets, the intramodal mode shape vectors are the same for each set and = 0 when k =  or i= j. / / are the eigenvectors of K , that is Consider the case when K is singular (λ = 0and λ = c 0 n a = a = v k,  = 1, 2,... , j = 1, 2,... , n, (39) k,j ,j j 0). Then zero is the smallest eigenvalue of K and b is its corresponding eigenvector, b = c 11 ··· 1 ,where c is where v j = 1, 2,... , n are the eigenvectors of K . j c Advances in Acoustics and Vibration 5 7. Nonuniform Structures The stiffness operators are of the form of those consid- ered in Section 6. Hence, the natural frequencies and mode The case when one or more of the structures is nonuniform is shapes can be determined from solving a matrix eigenvalue more difficult in that the differential equations of (12)have problem. If all shafts are made of the same material then variable coefficients. Consider the case when the structures −1 β = μ leading to M K = I, and thus if K is singular i i c are nonuniform, but identical. In this case, the coupled set of then the lowest natural frequency for each set of intramodal differential equations can be written as modes is the same as the modal natural frequency for the innermost shaft, and the mode shapes corresponding to each ILw + K w − ω IMw = 0, (40) frequency are the eigenvector of K corresponding to its zero eigenvalue times, the spatial mode shape for the shaft. where L and M are the stiffness and inertia operators, The eigenvalue problem for the innermost shaft is respectively, for any of the structures. Let κ , κ ,... , κ be the eigenvalues of K and let 1 2 n c z , z ,... , z be their corresponding eigenvectors normalized 1 2 n φ (x) + δ φ(x) = 0, (45) n T with respect to the standard inner product on R (z z = 1). Let P be the matrix whose columns are the normalized eigenvectors. Since K is symmetric, it can be shown that T T subject to appropriate boundary conditions. The values for P P = I and P K P = Δ,where Δ is a diagonal matrix with δ k = 1, 2,... , 5 and the corresponding normalized mode Δ = κ . Defining i,j i shapes φ (x) are listed in Table 1 for various end conditions. The matrix eigenvalue problem for a set of intramodal w = Pq (41) frequencies is of the form and substituting into (40) and then premultiplying by P leads to Ka = ω Ma, (46) ILq + Δq − ω IMq = 0. (42) Equation (42) represents a set of uncoupled differential where K is a tridiagonal matrix whose elements are equations, each of the form Lq + κ q − ω Mq = 0 j = 1, 2,... , n. (43) j j j j k = μ δ + λ + λ i = 1, 2,... , n, i,i i i−1 i Equation (43) represents, along with appropriate homo- k =−λ i = 2, 3,... , n, (47) i,i−1 i−1 geneous boundary conditions, an eigenvalue problem to determine the natural frequencies. For each j, there are k =−λ i = 1, 2,... , n − 1, i,i+1 i an infinite, but countable, number of natural frequencies. Thus, the natural frequencies can be indexed by ω j = j ,k 1, 2,... ,nk = 1, 2,.... and M is a diagonal matrix with m = β . i,i i 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 8.2. Natural Frequencies of Four Concentric Fixed-Free Shafts. infinite number of natural frequencies corresponding to each As a numerical example, consider four concentric fixed-free eigenvalue of K . The term intramodal is not appropriate for shafts connected by layers of torsional stiffness. Solving (45) this set of natural frequencies as they do not correspond to subject to φ (0) = 0and φ (1) = 0, the same mode. Indeed, there are not necessarily intramodal frequencies for the nonuniform case. If K is singular and thus has zero as its lowest eigenvalue, (2k − 1)π (2k − 1)π then (43) shows that one set of natural frequencies is identi- δ = φ (x) = 2sin x . (48) k k 2 2 cal to the natural frequencies of the individual structures. 8. Examples Each shaft is made of the same material. The inner shaft is solid of radius r . The outer shafts are each of thickness r . 8.1. Shafts. Consider n concentric shafts of equal length The thickness of each elastic layer is negligible. This leads to connected by elastic layers. The stiffness and inertia operators μ = 1 μ = 15 μ = 65, μ = 175, β = 1 β = 15 β = 1 2 3 4 1 2 3 for uniform shafts are, respectively, 65, β = 175. The torsional stiffness of each elastic layer is the same and is taken such that λ = 1. The inner shaft ∂ θ L θ =−μ , is solid, thus, λ = 0. The outer radius of the outer shaft i i i ∂x (44) is unrestrained from rotation, hence, λ = 0. The matrix eigenvalue problems become M θ = β θ . i i i i 6 Advances in Acoustics and Vibration ⎡ ⎤ (2k − 1)π 1+ −10 0 ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ k,1 ⎢ ⎥ ⎢ ⎥ (2k − 1)π ⎢ ⎥ ⎢ ⎥ −12+15 −10 ⎢ ⎥ ⎢ k,2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ (2k − 1)π ⎢ ⎥ ⎢ a ⎥ k,3 ⎢ ⎥ ⎣ ⎦ 0 −12+65 −1 ⎢ ⎥ ⎢ ⎥ 2 k,4 ⎣ ⎦ (2k − 1)π (49) 00 −1 1 + 175 ⎡ ⎤ ⎡ ⎤ 10 0 0 a k,1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 016 0 0 ⎥ ⎢ a ⎥ k,2 ⎢ ⎥ ⎢ ⎥ = ω . ⎢ ⎥ ⎢ ⎥ ⎢ 0081 0 ⎥ ⎢ a ⎥ k,3 ⎣ ⎦ ⎣ ⎦ 00 0256 a k,4 Mode shapes for ω = 1.5708 The natural frequencies for k = 1, 2,... ,5 are given in 1.1 −1 0.07 Table 1. Since K is singular and M K = I, the lowest c b natural frequency in each intramodal set is δ .Eachmode 0.06 shape in a set of intramodal mode shapes corresponds to the same spatial mode φ (x). The difference in intramodal 0.05 mode shapes is in the relative magnitude and signs of the displacements of the individual shafts. The normalized 0.04 mode shapes of Figure 2 correspond to the first mode shape in the intramodal set for the first spatial mode 0.03 and illustrate the mode in which the shafts rotate as if they are rigidly connected. The mode shapes of Figure 3 0.02 correspond to the third intramodal mode for the first spatial mode and illustrate that when the rotations of the 0.01 first, second, and fourth shafts are counterclockwise, the rotation of the third shaft is clockwise. Figures 4 and 5 00.20.40.60.81 illustrate mode shapes corresponding to the third spatial mode. All mode shapes in the intramodal set for this mode x have two nodes across the length of the shaft. Note that w w 1 3 for the second intramodal frequency there is a change in w w 2 4 the direction of rotation of the shafts between the third and fourth shafts. Thus, there is a cylindrical surface of Figure 2: Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with k = 1and j = 1. Themodeshapes nodes between these shafts. There are two changes in the correspond to rigid-body motion across the set of shafts. 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. Substitution of (50) into (18)leads to ⎛ ⎞ ∞ 4 k,j 8.3. Forced Response of Four Concentric Shafts. Suppose that ⎝ ⎠ ( ) ( ) W x, t = a 1 − cos ω t φ φ x . k,j k,j k k ω 2 the midspan of the outer shaft is subject to a constant k,j k=1 j=1 torque, T , such that the nondimensional applied torques are 0 (51) T (x, t) = T (x, t) = T (x, t) = 0, T (x, t) = δ(x − 1/2). 1 2 3 4 Equation (51) is evaluated leading to the time dependence of The forced response of the system is calculated by using a the response at x = 1/2and x = 1 illustrated in Figures 6 and convolution integral solution of the form of (21) leading to 7. t 1 1 8.4. Nonuniform Shafts. Now consider the same set of shafts, c (t) = a F(x, τ )φ (x) sin ω (t − τ ) dxdτ k,j k k,j k,j except that each has a taper, such that the differential 0 0 k,j equation for the innermost shaft when isolated from the ( )  system is a φ 1/2 k,j k = 1 − cos ω t . k,j k,j d dθ 2 2 (1 − 0.1x) + ω (1 − 0.1x) θ = 0. (52) (50) dx dx Advances in Acoustics and Vibration 7 Mode shapes for ω = 1.5965 Mode shapes for ω = 7.8592 1.3 3.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.05 0.1 w w 0.05 −0.05 −0.1 −0.15 −0.05 −0.2 −0.1 −0.25 00.20.40.60.81 00.20.40.60.81 x x w w w w 1 3 1 3 w w w w 2 4 2 4 Figure 5: Set of intramodal mode shapes of elastically connected Figure 3: Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with k = 3and j = 3. fixed-free torsional shafts with k = 1and j = 3. The mode shape (nonnormalized) corresponding to a nat- Mode shapes for ω = 7.8551 3.2 0.15 ural frequency ω is 0.1 θ (x) = j [10ω (1 − 0.1x)] − j [9ω ]y [10ω (1 − 0.1x)]. k o k k 0 k 0.05 (54) The differential equations for the elastically coupled −0.05 shafts become −0.1 ⎡ ⎤ ⎡ ⎤ 10 0 0 θ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥  ⎢ ⎥ −0.15 015 0 0 θ ⎢ ⎥ ⎢ 2⎥ d d ⎢ ⎥ ⎢ ⎥ (1 − 0.1x) ⎢ ⎥ ⎢ ⎥ dx dx ⎢ 0065 0 ⎥ ⎢ θ ⎥ −0.2 ⎣ ⎦ ⎣ ⎦ 00.20.40.60.81 00 0175 θ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 −10 0 θ 10 0 0 w w 1 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ w w ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 4 −12 −10 θ 015 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − + ω ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 −12 −1⎥ ⎢ θ ⎥ ⎢ 0065 0 ⎥ Figure 4: Set of intramodal mode shapes of elastically connected ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ fixed-free torsional shafts with k = 3and j = 2. 00 −11 θ 00 0175 ⎡ ⎤ ⎡ ⎤ θ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ θ 0 ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ × (1 − 0.1x) = . ⎢ ⎥ ⎢ ⎥ Along with the boundary for a fixed-free shaft, (52)has ⎢ θ ⎥ ⎢ 0⎥ ⎣ ⎦ ⎣ ⎦ a Bessel function solution leading to the characteristic θ 0 equation for the shaft’s natural frequencies as (55) j (9ω)y (10ω) − y (9ω)j (10ω) = 0, (53) o 0 0 0 Even though the shafts are not identical, the differential where j (x)and y (x) are spherical Bessel functions of equations in (55) may still be decoupled because the stiffness n n −1 the first and second kinds of order n and argument x. and inertia matrices are the same. The eigenvalues of M K c 8 Advances in Acoustics and Vibration −3 are κ = 0, κ = 0.0172, κ = 0.0815, and κ = 1.0712. The ×10 1 2 3 4 Forced response at x = 0.5 corresponding matrix of eigenvectors is ⎡ ⎤ 0.0625 0.1191 0.2292 0.9640 ⎢ ⎥ ⎢ ⎥ 0.0625 0.1171 0.2106 −0.0686 ⎢ ⎥ ⎢ ⎥ P = . (56) ⎢ ⎥ 10 ⎢ 0.0625 0.0848 −0.0654 0.0010 ⎥ ⎣ ⎦ −6 0.0625 −0.0422 0.0049 −5.44 × 10 The columns of P have been normalized such that P MP = I and P K P = Δ. Following the same procedure as in the derivation of (43), the uncoupled differential equations become d dq 2 2 2 −2 (1 − 0.1x) + ω (1 − 0.1x) q = 0, dx dx −4 0 1 2 345 dq 2 2 2 (1 − 0.1x) + ω (1 − 0.1x) − 0.0172 q = 0, 2 t dx dx w (t) w (t) 1 3 d dq 2 2 w (t) w (t) (1 − 0.1x) + ω (1 − 0.1x) − 0.0815 q = 0, 2 4 dx dx Figure 6: Forced response of elastically connected torsional shafts d dq 2 2 (1 − 0.1x) + ω (1 − 0.1x) − 1.0712 q = 0. at x = 0.5 due to constant concentrated torque applied to outer dx dx shaft at x = 0.5. (57) The solutions of (57)are −3 ×10 Forced response at x = 1 q (x) = C j [ω(1 − 0.1x)] + C y [ω(1 − 0.1x)], 1 1 0 2 0 16 ( ) [ ( )] [ ( )] q x = C j ω 1 − 0.1x + C y ω 1 − 0.1x , 2 1 0.904 2 0.904 q (x) = C j [ω(1 − 0.1x)] + C y [ω(1 − 0.1x)], 3 1 2.398 2 2.398 q (x) = C j [ω(1 − 0.1x)] + C y [ω(1 − 0.1x)]. 4 1 9.862 2 9.862 (58) The characteristic equations to determine the natural fre- quencies are j 10ω y 9ω − y (10ω )j 9ω = 0, 0 k,1 1 k,1 0 1 1 k,1 y 9ω 0.904 k,2 −2 j 10ω 0.904 − y 9ω 0,904 k,2 1.904 k,2 9ω k,2 −4 0 1 2 345 j 9ω 0.904 k,2 − y 10ω 0.904 − j 9ω = 0, 0,904 k,2 1.904 k,2 9ω k,2 w (t) w (t) 1 3 w (t) w (t) 2 4 y 9ω 2.398 k,3 j 10ω 2.398 − y 9ω 2.398 k,3 3.398 k,3 9ω k,3 Figure 7: Forced response of elastically connected torsional shafts at x = 1 due to constant concentrated torque applied to outer shaft j 9ω 2.398 k,3 − y 10ω 2.398 − j 9ω = 0, at x = 0.5. 2.398 k,3 3.398 k,3 9ω k,3 y 9ω 9.862 k,4 j 10ω 9.862 − y 9ω 9.862 k,4 10.862 k,4 9ω k,4 8.5. Euler-Bernoulli Beams. Consider a set of n parallel Euler- j 9ω 9.862 k,4 Bernoulli beams connected by elastic layers. For uniform − y 10ω 9.862 − j 9ω = 0. 9.862 k,4 10.962 k,4 4 4 9ω beams, the mass and stiffness operators are L = μ (∂ /∂x ) k,4 i i (59) and M = β .The differential eigenvalue problem for the first i i beam in the set is The characteristic equation for the first set of frequencies is identical to (53). The first five frequencies for each j are given d φ − δ φ = 0. (60) in Table 2. dx W (1, t) W (0.5, t) Advances in Acoustics and Vibration 9 Table 2: First five sets of frequencies for set of linearly tapered Table 3: First four sets of intramodal natural frequencies for a set shafts. of five elastically connected fixed-fixed Euler-Bernoulli beams. 123 4 5 3.5100 22.0300 61.7000 120.90 1 1.639 4.736 7.868 11.006 14.145 1 3.4630 16.7195 44.0798 85.7222 2 1.645 4.738 7.869 11.007 14.146 2 6.6167 20.6044 51.6490 99.3829 3 1.667 4.745 7.874 11.010 14.148 3 8.5529 23.4906 62.2901 121.1925 4 1.981 4.861 7.944 11.010 14.187 4 12.7319 24.1708 62.4633 121.3072 5 14.8317 28.0564 71.9859 139.9667 The transverse displacements of Euler-Bernoulli beams is Mode shapes for ω = 24.1708 2.4 another example of the special case discussed in Section 6. 2 If the beams are identical (μ = 1and β = 1) and i i if K is singular, then the lowest natural frequency for the 1.5 kth set of intramodal modes is δ with the mode shape, such that each beam has the same displacement and the springs are unstrectched. Otherwise, the mode shape for the 0.5 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 −0.5 Equation (60) subject to the boundary conditions φ(0) = 0, φ (0) = 0, φ (1) = 0, and φ (1) = 0leads to −1 cos δ +cosh δ k k −1.5 φ (x)= cosh δ x − cos δ x − k k k 00.20.40.60.81 sin δ + sinh δ k k w w × sinh δ x − sin δ x , 1 4 k k (61) Figure 8: Intramodal mode shapes for a set of five elastically where δ is the kth solution of cos( δ )cosh( δ ) =−1. k k k connected Euler-Bernoulli beams. Numerical values used in the computations are μ = 1, μ = 2, μ = 0.5, μ = 1, μ = 0.25, β = 1, β = 1.5, 2 3 4 5 1 2 β = 0.75, β = 1, β = 0.5, λ = 0, λ = 100, λ = 50, 3 4 5 0 1 2 natural frequency of an Euler-Bernoulli beam increases with λ = 50, λ = 20, and λ = 0 3 4 5 stiffness. Since the first beam is stiffer than several other Using these numerical values, the matrix eigenvalue beams in the set, some intramodal frequencies are lower than problem for a set of intramodal frequencies and mode shapes δ . The mode shapes for k = 2and j = 4illustrated in is Figure 8 show one spatial node and three cylindrical surfaces ⎡ ⎤ δ + 100 −100 0 0 0 of nodes. ⎢ ⎥ ⎢ 2 ⎥ −100 2δ + 150 −50 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 9. Conclusion 0 −50 0.5δ + 100 −50 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A general theory is developed for the free and forced response ⎢ ⎥ 00 −50 δ +70 −20 ⎣ ⎦ of elastically connected structures. The following has been 00 0 −20 0.25δ +20 shown. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ a 10 0 0 0 a 1 1 (i) The general problem can be formulated using the ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ vector space U = R × S,where S is a subspace of ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a 01.500 0 a ⎢ 2⎥ ⎢ ⎥ ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ C [0, 1] defined by the system’s end conditions. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ × ⎢ a ⎥ = ω ⎢ 000.75 0 0 ⎥ ⎢ a ⎥ . 3 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (ii) If the differential stiffness operator for a single struc- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a 00 0 1 0 a 4 4 ture is self-adjoint with respect to a standard inner ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ product on S, then the general stiffness operator is a 00 0 0 0.5 a 5 5 self-adjoint with respect to a standard inner product (62) on U . The sets of intramodal frequencies are listed in Table 3. (iii) Kinetic and potential energy inner products are Recall that δ is the natural frequency of the first beam. The defined on both S and U . k 10 Advances in Acoustics and Vibration (iv) A normal-mode solution for the free response leads References to the formulationofaneigenvalueproblem defined [1] S. G. Kelly, Fundamentals of Mechanical Vibrations, McGraw- for a matrix of operators. Hill, Boston, Mass, USA, 2nd edition, 2000. (v) The operator is self-adjoint with respect to the energy [2] S. S. Rao, Mechanical Vibrations, Pearson/Prentice Hall, Upper inner products leading to the development of an Saddle River, NJ, USA, 4th edition, 2004. orthogonality condition. [3] L. Mierovitch, Fundamentals of Vibrations, McGraw-Hill, Bos- ton, Mass, USA, 2001. (vi) The expansion theorem is used to develop a modal [4] B. Balachandran and E. McGrab, Vibrations, Thomson, Tor- analysis for the forced response. onto, Canada, 2003. (vii) The case where the structures are uniform and the [5] J. Ginsburg, Mechanical and Structural Vibrations: Theory and individual stiffness operators are proportional is a Applications, Wiley, New York, NY, USA, 2001. special case in which the determination of natural [6] D. Inman, Engineering Vibrations, Prentice Hall, Upper Saddle frequencies and mode shapes can be reduced to River, NJ, USA, 3rd edition, 2007. eigenvalue problems for matrices on R . [7] S.G.Kelly, Advanced Vibration Analysis, CRC Press/Taylor and Francis Group, Boca Raton, Fla, USA, 2007. (viii) When the stiffness operators are proportional, the [8] L. Meirovitch, Principles and Techniques of Vibration, Prentice natural frequencies and mode shapes are indexed Hall, Upper Saddle River, NJ, USA, 1997. with two indices, the first representing the spatial [9] Z. Oniszczuk, “Transverse vibrations of elastically connected mode shape, the second representing the intramodal double-string complex system. Part I: free vibrations,” Journal mode shapes. of Sound and Vibration, vol. 232, no. 2, pp. 355–366, 2000. (ix) If the uniform structures are identical, then a simple [10] Z. Oniszczuk, “Transverse vibrations of elastically connected formula can be derived for the sets of intramodal nat- double-string complex system. Part II: forced vibrations,” ural frequencies using the eigenvalues of the coupling Journal of Sound and Vibration, vol. 232, no. 2, pp. 367–386, stiffness matrix. The intramodal mode shapes for each spatial mode are the eigenvectors of the coupling [11] J. M. Selig and W. H. Hoppmann, “Normal mode vibrations stiffness martrix. of systems of elastically connected parallel bars,” Journalofthe Acoustical Society of America, vol. 36, pp. 93–99, 1964. (x) An iterative solution must be applied to determine [12] E. Osborne, “Computations of bending modes and mode the natural frequencies for the most general case of shapes of single and double beams,” Journal of the Society the uniform structure. for Industrial and Applied Mathematics, vol. 10, pp. 329–338, (xi) The differential equations for the coupling of iden- tical structures, uniform, or nonuniform can be [13] Z. Oniszczuk, “Free transverse vibrations of elastically con- uncoupled through diagonalization of the coupling nected simply supported double-beam complex system,” stiffness matrix. Journal of Sound and Vibration, vol. 232, no. 2, pp. 387–403, (xii) Elastically connected uniform strings and elastically [14] Z. Oniszczuk, “Forced transverse vibrations of an elastically connected uniform concentric shafts are applications connected complex simply supported double-beam system,” in which the stiffness operators are proportional. Journal of Sound and Vibration, vol. 264, no. 2, pp. 273–286, (xiii) The differential equations for the concentric shafts, even though they are not identical, can be decoupled [15] S. S. Rao, “Natural frequencies of systems of elastically con- when each individual stiffness operator is the same as nected Timoshenko beams,” Journal of the Acoustical Society of America, vol. 55, no. 6, pp. 1232–1237, 1974. the individual mass operator. [16] C. Q. Ru, “Column buckling of multiwalled carbon nanotubes (xiv) The individual stiffness operators for uniform Euler- with interlayer radial displacements,” Physical Review B, vol. Bernoulli beams are proportional implying that their 62, no. 24, pp. 16962–16967, 2000. natural frequencies can be indexed as an infinite [17] C. Q. Ru, “Axially compressed buckling of a doublewalled number of sets of intramodal frequencies. carbon nanotube embedded in an elastic medium,” Journal of the Mechanics and Physics of Solids, vol. 49, no. 6, pp. 1265– The general method is applied here only for undamped 1279, 2001. systems. However, it can be applied to certain damped sys- [18] C. Q. Ru, “Effect of van der Waals forces on axial buckling of tems as well. If the structures are undamped but the Winkler a double-walled carbon nanotube,” Journal of Applied Physics, layers have viscous damping, the same δ and φ for the k k vol. 87, no. 10, pp. 7227–7231, 2000. undamped system may be used, but an eigenvalue problem [19] J. Yoon, C. Q. Ru, and A. Mioduchowski, “Noncoaxial is obtained involving complex numbers. If the structures are resonance of an isolated multiwall carbon nanotube,” Physical damped but the Winkler layers are undamped, the choice Review B, vol. 66, no. 23, Article ID 233402, 4 pages, 2002. of δ and φ is modified to include viscous damping, but k k [20] C. Li and T. W. Chou, “Vibrational behaviors of multiwalled- again a complex eigenvalue problem is obtained. If the carbon-nanotube-based nanomechanical resonators,” Applied entire system (both the individual structures and the Winkler Physics Letters, vol. 84, no. 1, pp. 121–123, 2004. layers) is subject to proportional damping, the eigenvalues [21] J. Yoon, C. Q. Ru, and A. Mioduchowski, “Terahertz vibration and the eigenvectors of the undamped system can be used to of short carbon nanotubes modeled as Timoshenko beams,” uncouple the forced vibrations equations. Journal of Applied Mechanics, vol. 72, no. 1, pp. 10–17, 2005. Advances in Acoustics and Vibration 11 [22] J. Yoon, C. Q. Ru, and A. Mioduchowski, “Sound wave prop- agation in multiwall carbon nanotubes,” Journal of Applied Physics, vol. 93, no. 8, pp. 4801–4806, 2003. [23] K. Y. Xu, X. N. Gao, and C. Q. Ru, “Vibration of a double walled carbon nanotube aroused by nonlinear Van der Waals forces,” 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 and Vibration, vol. 322, no. 2, pp. 652–664, 2009. [25] S. G. Kelly and S. Srinivas, “Free vibrations of elastically connected stretched beams,” Journal of Sound and Vibration, vol. 326, no. 3–5, pp. 883–893, 2009. 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Free and Forced Vibrations of Elastically Connected Structures

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Hindawi Publishing Corporation
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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.
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1687-6261
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1687-627X
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10.1155/2010/984361
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 984361, 11 pages doi:10.1155/2010/984361 Research Article S. Graham Kelly Department of Mechanical Engineering, The University of Akron, Akron, Oh 44235, USA Correspondence should be addressed to S. Graham Kelly, gkelly@uakron.edu 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 the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for C [0, 1], then the stiffness n k operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on U = R ×C [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 approximate formulas for the natural frequencies of double- walled nanotubes noting that if developed from the eigen- The general theory for the free and forced response of strings, value relation, the computations can be compuitationally shafts, beams, and axially loaded beams is well documented intensive and difficult. [1–8]. Investigators have examined the free and forced Kelly and Srinivas [25] developed a Rayleigh-Ritz meth- response of elastically connected strings [9, 10], Euler- od for elastically connected stretched structures. Bernoulli beams [11–14], and Timoshenko beams [15]. This paper develops a general theory within which a These analyses focused on a pair of elastically connected finite set of parallel structures connected by elastic layers of structures using a normal-mode solution for the free a Winkler type can be analyzed. The theory shows that the response and a modal analysis for the forced response. Each determination of the natural frequencies for uniform parallel of these papers uses a normal-mode solution or a modal structures such as shafts and Euler-Bernoulli beams can be analysis specific to the problem to obtain a solution. reduced to matrix eigenvalue problems. The general theory Ru [16–18] proposed that model for multiwalled carbon is also used to develop a modal analysis for forced response nanotubes to be modeled by elastically connected structures of a set of parallel 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 2. Problem Formulation nanotubes and elastic shell models [17, 18]. Yoon et al. [19] andLiand Chou [20] modeled free vibrations of multiwalled The problem considered is that of n structural elements in nanotubes by a series of concentric elastically connected parallel but connected by elastic layers. Each elastic layer Euler-Bernoulli beams, while Yoon et al. [21, 22] modeled is modeled by a Winkler foundation, a layer of distributed nanotubes as concentric Timoshenko beams connected by stiffness across the span of the element. For generality, it is an elastic layer. Xu et al. [23] modeled the nonliearity of assumed that the outermost structures are connected to fixed the vanDer Waals forces. Elishakoff and Pentaras [24]gave foundations through elastic layers, as illustrated in Figure 1. 2 Advances in Acoustics and Vibration Each layer has a uniform stiffness per unit length k i = 0, 1, 2,... , n. Let w (x) represent the displacement of the ith structure. k If isolated from the system, the nondimensional differential equation governing the time-dependent motion of this n−1 structure is written as n− L w (x) + M w ¨ = G (x, t),(1) i i i i i n−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . where L is the stiffness operator for the element, M is an i i . . . . . . . . . . . . . . . . inertia operator for the element, and G (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 = 4for Euler- 1 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 k Figure 1: Set of n elastically connected parallel structures. conditions. Let S be the subspace of C [0, 1] defined by the boundary conditions; all elements in S satisfy all boundary conditions. If the stiffness operator is self-adjoint [(L f , g ) = The external forces acing on the ith structure are i S ⎧ ( f , L g ) ] and positive definite [(L f , f ) ≥ 0and (L f , f ) = i i i S s ⎪ −λ w − λ (w − w ) + F (x, t) i = 1, 0 1 1 1 2 1 0 if and only if f = 0] with respect to the standard inner ⎪ product, then a potential energy inner product is defined as −λ (w − w ) i−1 i i−1 G (x, t)= f , g = L f , g . (6) ⎪ i ⎪ −λ (w − w ) + F (x, t) i = 2, 3,... , n − 1, L S i i i+1 i i Clearly each M is positive definite and self-adjoint with ( ) ( ) −λ w − w − λ w + F x, t i = n, i n−1 n n−1 n n n respect to the standard inner product, and thus a kinetic (2) energy inner product can be defined as where λ , i = 0, 1,... , n are the nondimensional stiffness coefficients connecting the ith and i plus first structures. f , g = M f gdx,(7) Substitution of (2) into (1) leads to a coupled set of differential equations which are written in a matrix form as for any f and g in S. (K + K )W + MW = F,(3) c The standard inner product on U is defined as where W = w (x, t) w (x, t) ··· w (x, t) , F = 1 2 n T f , g = g fdx,(8) F (x, t) F (x, t) ··· F (x, t) , K is an n × n diagonal 1 2 n for any f and g in U . It is easy to show that M is self-adjoint operator matrix with k = L , M is an n × n diagonal mass i,i with respect to the inner product of (8) and a kinetic energy matrix with m = M ,and K is a tridiagonal n × n stiffness i,i i c coupling matrix with inner product on U is defined as (k ) =−λ i = 2, 3,... , c i,i−1 i−1 f , g = Mf , g . (9) M U (k ) = λ + λ i = 1, 2,... , n, (4) c i−1 i i,i Define K = K + K . Since K is self-adjoint with respect to the standard inner product on S and K is a symmetric (k ) =−λ i = 1, 2,... , n − 1. c i i,i+1 matrix, it can be shown that K is self-adjoint with respect to The vector W is an element of the vector space U = S × R ; the standard inner product on U . The positive definiteness an element of U is an n-dimensional vector, whose elements of K with respect to the standard inner product on U is all belong to S. determined by considering (Kf , f ) = (Kf , f ) +(K f , f ) . U U c U If K is a positive definite matrix with respect to the standard 3. General Theory inner product on R , then (K f , f ) ≥ 0and (K f , f ) = 0if c c U U and only if f = 0. If each of the operators L , i = 1, 2,... , n Let f (x)and g (x) be arbitrary elements of S. A standard is positive definite with respect to the standard inner product inner product on S is defined as on C [0, 1] then (Kf , f ) ≥ 0and (Kf , f ) = 0ifand only U U if f = 0.Thus, K is positive definite with respect to the ( ) ( ) f , g = f x g x dx. (5) standard inner product on U if either K is a positive definite 0 Advances in Acoustics and Vibration 3 matrix with respect to the standard inner product on R or expansion theorem then implies that for any f in U there each of the operators L , i = 1, 2,... , n is positive definite exists coefficients α , i = 1, 2,..., such that i i with respect to the standard inner product on C [0, 1]. Under f = α w , (16) i i either of these conditions, a potential energy inner product is defined on U by where w are the normalized mode shape vectors, and the f , g = Kf , g . (10) K U summation is carried out over all modes. For a given f in U , the coefficients are calculated by The operator K is not positive definite only when the structures are unrestrained and λ = 0and λ = 0. 0 n α = (f , w ) . (17) i i −1 Define D = M K. It is possible to show that D is self-adjoint with respect to the kinetic energy inner product Let W(t) represent the response due to the force vector of (9) and the potential energy inner product of (10). If F(t). Since W must be in U , the expansion theorem may be K is positive definite with respect to the standard inner applied at any t, leading to product, then D is positive definite with respect to both inner products. W(x, t) = c (t)w (x). (18) i i 4. Free Response Substitution of (18) into (3) results in First consider the free response of the structures, F = 0.A c ¨ (t)Mw + c (t)Kw = F(x, t). (19) i i i i normal-mode solution is assumed as i i iωt Taking the standard inner product on U of both sides of W = we , (11) (26)with w (x) for an arbitrary j = 1, 2,... , n and using where ω is a natural frequency and w = mode-shape orthogonality properties of (13)–(15) leads to an uncoupled set of differential equations of the form w (x) w (x) w (x) ··· w (x) w (x) is a vector 1 2 3 n−1 n of mode shapes corresponding to that natural frequency. ¨ ( ) ( ) c t + ω c t = F, w . (20) j j j Substitution of (11) into (3)leads to −1 2 A convolution integral solution of (20)is M (K + K )w = ω w, (12) where the partial derivatives have been replaced by ordinary c (t) = F(x, τ ), w (x) sin ω (t − τ ) dτ j j j j 0 derivatives in the definition of K.From(12), it is clear that (21) the natural frequencies are the square roots of the eigenvalues t 1 −1 T of D = M (K + K ), and the mode shape vectors are the c = w (x)F(x, τ ) sin ω (t − τ ) dx dτ. j 0 0 corresponding eigenvectors. It is well known [18] that eigenvalues of a self-adjoint operator are all real and that eigenvectors corresponding to 6. Uniform Structures with L Proportional to L i 1 distinct eigenvalues are orthogonal with respect to the inner A special case occurs when the structures are uniform product for which the operator is self-adjoint. Thus, if ω and and operators for the individual structural elements are ω are distinct natural frequencies with corresponding mode proportional to one another, shape vectors w and w respectively, then i j , L = μ L . (22) i i 1 w , w = 0, i j (13) In this case, the component of the stiffness operator due to w , w = 0. i j the elasticity of the structural elements becomes ⎡ ⎤ The mode shape vectors can be normalized by requiring μ 000 ··· 0 ⎢ ⎥ ⎢ ⎥ 0 μ 00 ··· 0 ⎢ ⎥ (w , w ) = 1, (14) i i ⎢ ⎥ ⎢ ⎥ 00 μ 0 ··· 0 ⎢ 3 ⎥ ⎢ ⎥ which then leads to K = AL = ⎢ ⎥ L . (23) 1 1 000 μ ··· 0 ⎢ 4 ⎥ ⎢ ⎥ ⎢ ⎥ (w , w ) = ω . (15) i i K i . . . . . ⎢ . ⎥ . . . . . . ⎢ ⎥ . . . . . ⎣ ⎦ 0000 ··· μ 5. Forced Response n Since D is a self-adjoint operator, its eigenvectors, the mode The system may be nondimensionalized such that M = shape vectors, can be shown to be complete in U.An 1. Then the differential equation governing the free response 4 Advances in Acoustics and Vibration of the first structural element, if isolated from the remainder Table 1: First five sets of intramodal natural frequencies of four elastically connected fixed free shafts, ω . (k,j ) of the system, is Φ + L Φ = 0. (24) 1 1 1 123 4 5 A normal-mode solution of (32) of the form Φ(x, t) = 1 1.5708 4.7124 7.8540 10.9956 14.1372 iδt φ(x)e leads to the eigenvalue-eigenvector problem 2 1.5763 4.7142 7.8551 10.9964 14.1378 3 1.5965 4.7210 7.8592 10.9993 14.1400 L φ = δ φ. (25) 4 1.8811 4.8247 7.9219 11.0442 14.1750 There are an infinite, but countable, number of natural frequencies for the system of (25), δ , δ ,... , δ , δ , δ ,... 1 2 k−1 k k+1 −1 a normalization constant. Suppose that M A = I, then (27) with corresponding mode shapes φ , φ ,... , φ , φ , φ ,... 1 2 k−1 k k+1 becomes The mode shapes are normalized by requiring (φ , φ ) = 1. i i 2 2 Assume a solution to (12) of the form δ I + K a = ω a. (31) Note that the solution of (31) corresponds to ω = δ and w = aφ (x), (26) k a = b. Thus, for this special case, the lowest natural frequency where a is an n × 1 vector of constants. Substitution of (26) for each set of intramodal modes is equal to the natural frequency for the spatial mode of one structural element and into (12) using (24)and (25)leads to its corresponding mode shape is the null space of K . −1 2 2 To examine the most general case when K is singular, M δ A + K a = ω a. (27) c take the standard inner product for R of both sides of (27) with b. Using properties of inner products and noting that A, Equation (27) implies that, for this special case, the deter- K ,and M aresymmetric leadsto mination of the natural frequencies of the set of elastically c connected structures is reduced to the determination of the 2 2 −1 2 a, δ A − ω M b = 0. (32) eigenvalues of the matrix M (δ A + K ). For each k there n c R are n natural frequencies. The natural frequencies can thus Application of the orthogonality condition of (32)leads to be indexed as ω for k = 1, 2,... and j = 1, 2,... , n.The k,j corresponding mode shapes are written as 2 2 a δ μ − ω β = 0. (33) i i i i=1 w (x) = a φ (x), (28) k,j k,j k The expansion theorem is used to assume a forced −1 where a is the eigenvector of M (δ A + K ) corresponding k,j c response of the form to the natural frequency ω . k,j ∞ n The function φ (x) represents the kth mode shape of a W(x, t) = c (t)a φ (x). (34) k,j k,j k single structure. For each k, there are n natural frequencies k=1 j=1 and n corresponding mode shapes. Such a set of mode Use of (34)in(3) leads to differential equations of the form shapes, which are referred to as intramodal modes, have the same spatial behavior, but their dependence across the c ¨ + ω c = F, a φ (x) . (35) k,j k,j k,j k k,j structures varies. Two mode shapes w (x)and w (x)for k,j p,q U which p= k are referred to as intermodal modes. A special case occurs when uniform structures are Note that since M and δ A + K are both symmetric identical such that A = I and M = I. Then, (27)becomes −1 matrices, the matrix M (δ A + K ) is self-adjoint with 2 2 respect to a kinetic energy inner product. The intramodal δ I + K a = ω a. (36) mode shapes satisfy an orthogonality condition on R given Equation (36)can be rewrittenas by 2 2 T K a = ω − δ a. (37) a , a = a Ma =0for i= j. (29) k,i k,j k,j k,i / Let κ j = 1, 2,... , n be the eigenvalues of K .Then, j c All mode shapes satisfy the orthogonality condition on U , 2 2 ω = δ + κ . (38) k,j k w , w = a Ma φ (x)φ (x)dx k,i ,j ,j k,i k  In addition, since the same set of eigenvalues is used to (30) calculate the natural frequencies for each intramodal sets, the intramodal mode shape vectors are the same for each set and = 0 when k =  or i= j. / / are the eigenvectors of K , that is Consider the case when K is singular (λ = 0and λ = c 0 n a = a = v k,  = 1, 2,... , j = 1, 2,... , n, (39) k,j ,j j 0). Then zero is the smallest eigenvalue of K and b is its corresponding eigenvector, b = c 11 ··· 1 ,where c is where v j = 1, 2,... , n are the eigenvectors of K . j c Advances in Acoustics and Vibration 5 7. Nonuniform Structures The stiffness operators are of the form of those consid- ered in Section 6. Hence, the natural frequencies and mode The case when one or more of the structures is nonuniform is shapes can be determined from solving a matrix eigenvalue more difficult in that the differential equations of (12)have problem. If all shafts are made of the same material then variable coefficients. Consider the case when the structures −1 β = μ leading to M K = I, and thus if K is singular i i c are nonuniform, but identical. In this case, the coupled set of then the lowest natural frequency for each set of intramodal differential equations can be written as modes is the same as the modal natural frequency for the innermost shaft, and the mode shapes corresponding to each ILw + K w − ω IMw = 0, (40) frequency are the eigenvector of K corresponding to its zero eigenvalue times, the spatial mode shape for the shaft. where L and M are the stiffness and inertia operators, The eigenvalue problem for the innermost shaft is respectively, for any of the structures. Let κ , κ ,... , κ be the eigenvalues of K and let 1 2 n c z , z ,... , z be their corresponding eigenvectors normalized 1 2 n φ (x) + δ φ(x) = 0, (45) n T with respect to the standard inner product on R (z z = 1). Let P be the matrix whose columns are the normalized eigenvectors. Since K is symmetric, it can be shown that T T subject to appropriate boundary conditions. The values for P P = I and P K P = Δ,where Δ is a diagonal matrix with δ k = 1, 2,... , 5 and the corresponding normalized mode Δ = κ . Defining i,j i shapes φ (x) are listed in Table 1 for various end conditions. The matrix eigenvalue problem for a set of intramodal w = Pq (41) frequencies is of the form and substituting into (40) and then premultiplying by P leads to Ka = ω Ma, (46) ILq + Δq − ω IMq = 0. (42) Equation (42) represents a set of uncoupled differential where K is a tridiagonal matrix whose elements are equations, each of the form Lq + κ q − ω Mq = 0 j = 1, 2,... , n. (43) j j j j k = μ δ + λ + λ i = 1, 2,... , n, i,i i i−1 i Equation (43) represents, along with appropriate homo- k =−λ i = 2, 3,... , n, (47) i,i−1 i−1 geneous boundary conditions, an eigenvalue problem to determine the natural frequencies. For each j, there are k =−λ i = 1, 2,... , n − 1, i,i+1 i an infinite, but countable, number of natural frequencies. Thus, the natural frequencies can be indexed by ω j = j ,k 1, 2,... ,nk = 1, 2,.... and M is a diagonal matrix with m = β . i,i i 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 8.2. Natural Frequencies of Four Concentric Fixed-Free Shafts. infinite number of natural frequencies corresponding to each As a numerical example, consider four concentric fixed-free eigenvalue of K . The term intramodal is not appropriate for shafts connected by layers of torsional stiffness. Solving (45) this set of natural frequencies as they do not correspond to subject to φ (0) = 0and φ (1) = 0, the same mode. Indeed, there are not necessarily intramodal frequencies for the nonuniform case. If K is singular and thus has zero as its lowest eigenvalue, (2k − 1)π (2k − 1)π then (43) shows that one set of natural frequencies is identi- δ = φ (x) = 2sin x . (48) k k 2 2 cal to the natural frequencies of the individual structures. 8. Examples Each shaft is made of the same material. The inner shaft is solid of radius r . The outer shafts are each of thickness r . 8.1. Shafts. Consider n concentric shafts of equal length The thickness of each elastic layer is negligible. This leads to connected by elastic layers. The stiffness and inertia operators μ = 1 μ = 15 μ = 65, μ = 175, β = 1 β = 15 β = 1 2 3 4 1 2 3 for uniform shafts are, respectively, 65, β = 175. The torsional stiffness of each elastic layer is the same and is taken such that λ = 1. The inner shaft ∂ θ L θ =−μ , is solid, thus, λ = 0. The outer radius of the outer shaft i i i ∂x (44) is unrestrained from rotation, hence, λ = 0. The matrix eigenvalue problems become M θ = β θ . i i i i 6 Advances in Acoustics and Vibration ⎡ ⎤ (2k − 1)π 1+ −10 0 ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ k,1 ⎢ ⎥ ⎢ ⎥ (2k − 1)π ⎢ ⎥ ⎢ ⎥ −12+15 −10 ⎢ ⎥ ⎢ k,2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ (2k − 1)π ⎢ ⎥ ⎢ a ⎥ k,3 ⎢ ⎥ ⎣ ⎦ 0 −12+65 −1 ⎢ ⎥ ⎢ ⎥ 2 k,4 ⎣ ⎦ (2k − 1)π (49) 00 −1 1 + 175 ⎡ ⎤ ⎡ ⎤ 10 0 0 a k,1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 016 0 0 ⎥ ⎢ a ⎥ k,2 ⎢ ⎥ ⎢ ⎥ = ω . ⎢ ⎥ ⎢ ⎥ ⎢ 0081 0 ⎥ ⎢ a ⎥ k,3 ⎣ ⎦ ⎣ ⎦ 00 0256 a k,4 Mode shapes for ω = 1.5708 The natural frequencies for k = 1, 2,... ,5 are given in 1.1 −1 0.07 Table 1. Since K is singular and M K = I, the lowest c b natural frequency in each intramodal set is δ .Eachmode 0.06 shape in a set of intramodal mode shapes corresponds to the same spatial mode φ (x). The difference in intramodal 0.05 mode shapes is in the relative magnitude and signs of the displacements of the individual shafts. The normalized 0.04 mode shapes of Figure 2 correspond to the first mode shape in the intramodal set for the first spatial mode 0.03 and illustrate the mode in which the shafts rotate as if they are rigidly connected. The mode shapes of Figure 3 0.02 correspond to the third intramodal mode for the first spatial mode and illustrate that when the rotations of the 0.01 first, second, and fourth shafts are counterclockwise, the rotation of the third shaft is clockwise. Figures 4 and 5 00.20.40.60.81 illustrate mode shapes corresponding to the third spatial mode. All mode shapes in the intramodal set for this mode x have two nodes across the length of the shaft. Note that w w 1 3 for the second intramodal frequency there is a change in w w 2 4 the direction of rotation of the shafts between the third and fourth shafts. Thus, there is a cylindrical surface of Figure 2: Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with k = 1and j = 1. Themodeshapes nodes between these shafts. There are two changes in the correspond to rigid-body motion across the set of shafts. 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. Substitution of (50) into (18)leads to ⎛ ⎞ ∞ 4 k,j 8.3. Forced Response of Four Concentric Shafts. Suppose that ⎝ ⎠ ( ) ( ) W x, t = a 1 − cos ω t φ φ x . k,j k,j k k ω 2 the midspan of the outer shaft is subject to a constant k,j k=1 j=1 torque, T , such that the nondimensional applied torques are 0 (51) T (x, t) = T (x, t) = T (x, t) = 0, T (x, t) = δ(x − 1/2). 1 2 3 4 Equation (51) is evaluated leading to the time dependence of The forced response of the system is calculated by using a the response at x = 1/2and x = 1 illustrated in Figures 6 and convolution integral solution of the form of (21) leading to 7. t 1 1 8.4. Nonuniform Shafts. Now consider the same set of shafts, c (t) = a F(x, τ )φ (x) sin ω (t − τ ) dxdτ k,j k k,j k,j except that each has a taper, such that the differential 0 0 k,j equation for the innermost shaft when isolated from the ( )  system is a φ 1/2 k,j k = 1 − cos ω t . k,j k,j d dθ 2 2 (1 − 0.1x) + ω (1 − 0.1x) θ = 0. (52) (50) dx dx Advances in Acoustics and Vibration 7 Mode shapes for ω = 1.5965 Mode shapes for ω = 7.8592 1.3 3.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.05 0.1 w w 0.05 −0.05 −0.1 −0.15 −0.05 −0.2 −0.1 −0.25 00.20.40.60.81 00.20.40.60.81 x x w w w w 1 3 1 3 w w w w 2 4 2 4 Figure 5: Set of intramodal mode shapes of elastically connected Figure 3: Set of intramodal mode shapes of elastically connected fixed-free torsional shafts with k = 3and j = 3. fixed-free torsional shafts with k = 1and j = 3. The mode shape (nonnormalized) corresponding to a nat- Mode shapes for ω = 7.8551 3.2 0.15 ural frequency ω is 0.1 θ (x) = j [10ω (1 − 0.1x)] − j [9ω ]y [10ω (1 − 0.1x)]. k o k k 0 k 0.05 (54) The differential equations for the elastically coupled −0.05 shafts become −0.1 ⎡ ⎤ ⎡ ⎤ 10 0 0 θ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥  ⎢ ⎥ −0.15 015 0 0 θ ⎢ ⎥ ⎢ 2⎥ d d ⎢ ⎥ ⎢ ⎥ (1 − 0.1x) ⎢ ⎥ ⎢ ⎥ dx dx ⎢ 0065 0 ⎥ ⎢ θ ⎥ −0.2 ⎣ ⎦ ⎣ ⎦ 00.20.40.60.81 00 0175 θ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 −10 0 θ 10 0 0 w w 1 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ w w ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 4 −12 −10 θ 015 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − + ω ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 −12 −1⎥ ⎢ θ ⎥ ⎢ 0065 0 ⎥ Figure 4: Set of intramodal mode shapes of elastically connected ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ fixed-free torsional shafts with k = 3and j = 2. 00 −11 θ 00 0175 ⎡ ⎤ ⎡ ⎤ θ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ θ 0 ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ × (1 − 0.1x) = . ⎢ ⎥ ⎢ ⎥ Along with the boundary for a fixed-free shaft, (52)has ⎢ θ ⎥ ⎢ 0⎥ ⎣ ⎦ ⎣ ⎦ a Bessel function solution leading to the characteristic θ 0 equation for the shaft’s natural frequencies as (55) j (9ω)y (10ω) − y (9ω)j (10ω) = 0, (53) o 0 0 0 Even though the shafts are not identical, the differential where j (x)and y (x) are spherical Bessel functions of equations in (55) may still be decoupled because the stiffness n n −1 the first and second kinds of order n and argument x. and inertia matrices are the same. The eigenvalues of M K c 8 Advances in Acoustics and Vibration −3 are κ = 0, κ = 0.0172, κ = 0.0815, and κ = 1.0712. The ×10 1 2 3 4 Forced response at x = 0.5 corresponding matrix of eigenvectors is ⎡ ⎤ 0.0625 0.1191 0.2292 0.9640 ⎢ ⎥ ⎢ ⎥ 0.0625 0.1171 0.2106 −0.0686 ⎢ ⎥ ⎢ ⎥ P = . (56) ⎢ ⎥ 10 ⎢ 0.0625 0.0848 −0.0654 0.0010 ⎥ ⎣ ⎦ −6 0.0625 −0.0422 0.0049 −5.44 × 10 The columns of P have been normalized such that P MP = I and P K P = Δ. Following the same procedure as in the derivation of (43), the uncoupled differential equations become d dq 2 2 2 −2 (1 − 0.1x) + ω (1 − 0.1x) q = 0, dx dx −4 0 1 2 345 dq 2 2 2 (1 − 0.1x) + ω (1 − 0.1x) − 0.0172 q = 0, 2 t dx dx w (t) w (t) 1 3 d dq 2 2 w (t) w (t) (1 − 0.1x) + ω (1 − 0.1x) − 0.0815 q = 0, 2 4 dx dx Figure 6: Forced response of elastically connected torsional shafts d dq 2 2 (1 − 0.1x) + ω (1 − 0.1x) − 1.0712 q = 0. at x = 0.5 due to constant concentrated torque applied to outer dx dx shaft at x = 0.5. (57) The solutions of (57)are −3 ×10 Forced response at x = 1 q (x) = C j [ω(1 − 0.1x)] + C y [ω(1 − 0.1x)], 1 1 0 2 0 16 ( ) [ ( )] [ ( )] q x = C j ω 1 − 0.1x + C y ω 1 − 0.1x , 2 1 0.904 2 0.904 q (x) = C j [ω(1 − 0.1x)] + C y [ω(1 − 0.1x)], 3 1 2.398 2 2.398 q (x) = C j [ω(1 − 0.1x)] + C y [ω(1 − 0.1x)]. 4 1 9.862 2 9.862 (58) The characteristic equations to determine the natural fre- quencies are j 10ω y 9ω − y (10ω )j 9ω = 0, 0 k,1 1 k,1 0 1 1 k,1 y 9ω 0.904 k,2 −2 j 10ω 0.904 − y 9ω 0,904 k,2 1.904 k,2 9ω k,2 −4 0 1 2 345 j 9ω 0.904 k,2 − y 10ω 0.904 − j 9ω = 0, 0,904 k,2 1.904 k,2 9ω k,2 w (t) w (t) 1 3 w (t) w (t) 2 4 y 9ω 2.398 k,3 j 10ω 2.398 − y 9ω 2.398 k,3 3.398 k,3 9ω k,3 Figure 7: Forced response of elastically connected torsional shafts at x = 1 due to constant concentrated torque applied to outer shaft j 9ω 2.398 k,3 − y 10ω 2.398 − j 9ω = 0, at x = 0.5. 2.398 k,3 3.398 k,3 9ω k,3 y 9ω 9.862 k,4 j 10ω 9.862 − y 9ω 9.862 k,4 10.862 k,4 9ω k,4 8.5. Euler-Bernoulli Beams. Consider a set of n parallel Euler- j 9ω 9.862 k,4 Bernoulli beams connected by elastic layers. For uniform − y 10ω 9.862 − j 9ω = 0. 9.862 k,4 10.962 k,4 4 4 9ω beams, the mass and stiffness operators are L = μ (∂ /∂x ) k,4 i i (59) and M = β .The differential eigenvalue problem for the first i i beam in the set is The characteristic equation for the first set of frequencies is identical to (53). The first five frequencies for each j are given d φ − δ φ = 0. (60) in Table 2. dx W (1, t) W (0.5, t) Advances in Acoustics and Vibration 9 Table 2: First five sets of frequencies for set of linearly tapered Table 3: First four sets of intramodal natural frequencies for a set shafts. of five elastically connected fixed-fixed Euler-Bernoulli beams. 123 4 5 3.5100 22.0300 61.7000 120.90 1 1.639 4.736 7.868 11.006 14.145 1 3.4630 16.7195 44.0798 85.7222 2 1.645 4.738 7.869 11.007 14.146 2 6.6167 20.6044 51.6490 99.3829 3 1.667 4.745 7.874 11.010 14.148 3 8.5529 23.4906 62.2901 121.1925 4 1.981 4.861 7.944 11.010 14.187 4 12.7319 24.1708 62.4633 121.3072 5 14.8317 28.0564 71.9859 139.9667 The transverse displacements of Euler-Bernoulli beams is Mode shapes for ω = 24.1708 2.4 another example of the special case discussed in Section 6. 2 If the beams are identical (μ = 1and β = 1) and i i if K is singular, then the lowest natural frequency for the 1.5 kth set of intramodal modes is δ with the mode shape, such that each beam has the same displacement and the springs are unstrectched. Otherwise, the mode shape for the 0.5 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 −0.5 Equation (60) subject to the boundary conditions φ(0) = 0, φ (0) = 0, φ (1) = 0, and φ (1) = 0leads to −1 cos δ +cosh δ k k −1.5 φ (x)= cosh δ x − cos δ x − k k k 00.20.40.60.81 sin δ + sinh δ k k w w × sinh δ x − sin δ x , 1 4 k k (61) Figure 8: Intramodal mode shapes for a set of five elastically where δ is the kth solution of cos( δ )cosh( δ ) =−1. k k k connected Euler-Bernoulli beams. Numerical values used in the computations are μ = 1, μ = 2, μ = 0.5, μ = 1, μ = 0.25, β = 1, β = 1.5, 2 3 4 5 1 2 β = 0.75, β = 1, β = 0.5, λ = 0, λ = 100, λ = 50, 3 4 5 0 1 2 natural frequency of an Euler-Bernoulli beam increases with λ = 50, λ = 20, and λ = 0 3 4 5 stiffness. Since the first beam is stiffer than several other Using these numerical values, the matrix eigenvalue beams in the set, some intramodal frequencies are lower than problem for a set of intramodal frequencies and mode shapes δ . The mode shapes for k = 2and j = 4illustrated in is Figure 8 show one spatial node and three cylindrical surfaces ⎡ ⎤ δ + 100 −100 0 0 0 of nodes. ⎢ ⎥ ⎢ 2 ⎥ −100 2δ + 150 −50 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 9. Conclusion 0 −50 0.5δ + 100 −50 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A general theory is developed for the free and forced response ⎢ ⎥ 00 −50 δ +70 −20 ⎣ ⎦ of elastically connected structures. The following has been 00 0 −20 0.25δ +20 shown. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ a 10 0 0 0 a 1 1 (i) The general problem can be formulated using the ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ vector space U = R × S,where S is a subspace of ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a 01.500 0 a ⎢ 2⎥ ⎢ ⎥ ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ C [0, 1] defined by the system’s end conditions. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ × ⎢ a ⎥ = ω ⎢ 000.75 0 0 ⎥ ⎢ a ⎥ . 3 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (ii) If the differential stiffness operator for a single struc- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a 00 0 1 0 a 4 4 ture is self-adjoint with respect to a standard inner ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ product on S, then the general stiffness operator is a 00 0 0 0.5 a 5 5 self-adjoint with respect to a standard inner product (62) on U . The sets of intramodal frequencies are listed in Table 3. (iii) Kinetic and potential energy inner products are Recall that δ is the natural frequency of the first beam. The defined on both S and U . k 10 Advances in Acoustics and Vibration (iv) A normal-mode solution for the free response leads References to the formulationofaneigenvalueproblem defined [1] S. G. Kelly, Fundamentals of Mechanical Vibrations, McGraw- for a matrix of operators. Hill, Boston, Mass, USA, 2nd edition, 2000. (v) The operator is self-adjoint with respect to the energy [2] S. S. Rao, Mechanical Vibrations, Pearson/Prentice Hall, Upper inner products leading to the development of an Saddle River, NJ, USA, 4th edition, 2004. orthogonality condition. [3] L. Mierovitch, Fundamentals of Vibrations, McGraw-Hill, Bos- ton, Mass, USA, 2001. (vi) The expansion theorem is used to develop a modal [4] B. Balachandran and E. McGrab, Vibrations, Thomson, Tor- analysis for the forced response. onto, Canada, 2003. (vii) The case where the structures are uniform and the [5] J. 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Part I: free vibrations,” Journal mode shapes. of Sound and Vibration, vol. 232, no. 2, pp. 355–366, 2000. (ix) If the uniform structures are identical, then a simple [10] Z. Oniszczuk, “Transverse vibrations of elastically connected formula can be derived for the sets of intramodal nat- double-string complex system. Part II: forced vibrations,” ural frequencies using the eigenvalues of the coupling Journal of Sound and Vibration, vol. 232, no. 2, pp. 367–386, stiffness matrix. The intramodal mode shapes for each spatial mode are the eigenvectors of the coupling [11] J. M. Selig and W. H. Hoppmann, “Normal mode vibrations stiffness martrix. of systems of elastically connected parallel bars,” Journalofthe Acoustical Society of America, vol. 36, pp. 93–99, 1964. (x) An iterative solution must be applied to determine [12] E. 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