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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2012, Article ID 191042, 9 pages doi:10.1155/2012/191042 Research Article Analysis of Static Instability of an Asymmetric, Rotating Sand-Wich Beam 1 2 1 1 P. R. Dash, K. Ray, S. K. Sarangi, and P. K. Pradhan Mechanical Engineering Department, VSSUT Burla, Orissa 768018, India Mechanical Engineering Department, IIT Kharagpur, Kharagpur 721302, India Correspondence should be addressed to P. K. Pradhan, prasant2001uce@gmail.com Received 16 May 2011; Accepted 8 August 2011 Academic Editor: Joseph CS Lai Copyright © 2012 P. R. Dash et al. 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. The static stability of an asymmetric, rotating sandwich beam subjected to an axial pulsating load has been investigated for pinned- pinned and ﬁxed-free boundary conditions. The equations of motion and associated boundary conditions have been obtained by using the Hamilton’s energy principle. Then, these equations of motion and the associated boundary conditions have been nondimensionalised. A set of Hill’s equations are obtained from the nondimensional equations of motion by the application of the general Galerkin method. The static buckling loads have been obtained from Hill’s equations. The inﬂuences of geometric parameters and rotation parameters on the nondimensional static buckling loads have been investigated. 1. Introduction and found that the most common and dangerous parametric instabilities arose as a result of combination resonance. Kam- The dynamic behavior of rotating beams is of great practical mer and Schlack [4] obtained the boundaries between the interest in the design of steam and gas turbine blades and stable and unstable regions of an Euler beam rotating with helicopter blades. Sandwich constructions with high strength an angular velocity which has a small periodic component. facings and a light weight core have been very popular in Sri Namachchivaya [5] derived explicit stability conditions aerospace applications. Typical sandwich members used var- for a rotating shaft under parametric excitation comprising ied from structural panels in aircraft to the helicopter a combination of harmonic terms and stationary stochastic rotor blades. In general, high modulus and light weight processes using the Routh-Hurwitz criterion and found that characteristics of the sandwich construction normally have the addition of nonwhite noise excitation has a stabilizing great advantages of high movability, power saving, and eﬀect on harmonically excited shafts. The vibration behavior high strength in robotics applications. Extensive publications of a rotating beam, oriented perpendicular to the axis of spin, have been available concerning the design and analysis of was investigated by Bauer and Eidel [6]. It was observed that sandwich structures. the speed of rotation has a very pronounced inﬂuence of Bhat [1] studied the natural frequencies and mode shapes the rotating beam and an increase in the speed of rotation of a rotating uniform cantilever beam with tip mass for may increase or decrease the natural frequencies depending diﬀerent values of rotational speed, hub radius, and setting on the boundary conditions. Abbas and Thomas [7, 8] angle using beam characteristic orthogonal polynomials in studied the eﬀect of rotational speed and root ﬂexibilities the Rayleigh-Ritz method. Liu and Yeh [2] presented the on the ﬁrst-order simple resonance zones of a rotating natural frequencies of a nonuniform rotating beam with a Timoshenko’s beam by using the ﬁnite element method. restrained base for various values of restraint parameters and Bauchau and Hong [9] utilized the same method to analyze rotational speed. Unger and Brull [3]madeananalytical the eﬀect of viscous damping on the response and stability investigation to determine the simple and combination res- of parametrically excited beams undergoing large deﬂections onance regions of shafts mounted in a long and short and rotations. Dynamic stability of an ordinary rotating bearings subjected to pulsating torque applied at the ends beam with various boundary conditions was studied by Kar 2 Advances in Acoustics and Vibration and Sujata [10]. The same authors [11] also studied the angular velocity Ω and is capable of oscillating in the x-z stability of a rotating, pretwisted and preconed cantilever plane. It is asymmetric with respect to the x-y plane as layer beam. Parametric instability of a rotating pretwisted beam 1 is not same as layer 3 both geometrically and materially. subjected to sinusoidal compressive axial loads was addressed Note that some authors might call this a “rotating column” by Tan et al. [12]. conﬁguration. However, it is also customary in the literature As it has been pointed out by many investigators the shear to call it a “rotating beam”. This is followed throughout the deformation of the core plays an important role in the ﬂex- paper. ural and dynamic behavior of a sandwich beam; therefore, The top layer of the beam is made of an elastic material of the ﬂexural rigidity in the core and shear deformation of thickness 2h and Young’s modulus E , and the bottom layer 1 1 the facings were neglected in many analysis. The parametric is made of an elastic material of thickness 2h and Young’s instability of a cantilever beam with magnetic ﬁeld and modulus E . The core is made of a linearly viscoelastic periodic axial load has been studied by Pratiher and Dwivedy material with a shear modulus G = G (1 + jη), where G 2 2 [13]. The results obtained from perturbation analysis have is the in-phase shear modulus, η is the core loss factor and been veriﬁed by solving the temporal equation of motion j = −1. using fourth-order Runge-Kutta method. Dwivedy et al. A pulsating axial load P(t) = P + P cos(ωt)isapplied at 0 1 [14] have investigated the parametric instability regions for the end x = b +l of the beam. P and P are, respectively, the 0 1 simple supported, clamp-guided, clamp-free riveted, and static and dynamic load amplitudes, ω is the frequency of the clamped-free end condition by modiﬁed Hsu’s method and dynamic load, and t is time. found that zones of instability are aﬀected by the shear parameter, core-loss factor and the ratio of core thickness to 2.2. Derivation. Thefollowing assumptionsare made for skin thickness. Zheng et al. [15] investigated the instability deriving the equations of motion. of a moving load system with the load approximated as a single-axle mass-spring-damper system. The author showed (1) The beam deﬂection w(x, t), parallel to z-axisissmall that instability occurs for a lower mass as compression and is the same at all points of a given cross-section. axial force increases. Metrikine and Verichev [16] introduced two degrees of freedom load and approximated the rail (2) The layers are perfectly bonded so that displacements as a Timoshenko beam. They have observed that under are continuous across the interfaces. The elastic face these approximations, larger mass of the load lowers the layers obey the Euler-Bernoulli beam theory. velocity at which instability is observed. The same authors (3) The allowance for rotary inertia is neglected while [17] have considered the eﬀect of periodic variations of the calculating the kinetic energy of the system. foundation stiﬀness on instability. The parametric instability of viscoelastically supported asymmetric sandwich beam (4) Shear deformation of the facings are neglected. have been studied by Ghosh et al. [18] and found that the (5) Damping in the viscoelastic core is predominantly viscoelastic constraints improve the stability of the system. The dynamic stability of a rotating sandwich beam using the due to shear. Bending and extensional eﬀects in the ﬁnite element method has been studied by Lien and Chen core are neglected. [19, 20]. The parametric instability of a rotating asymmetric (6) The Kerwin’s assumption [23]isused. sandwich beam with viscoelastic core and subjected to a pulsating axial load has been studied by Dash et al. [21]. According to the above assumption, E A U + 1 1 1,x The present work deals with the static stability of E A U = 0 3 3 3,x a rotating asymmetric sandwich beam subjected to axial For the system pulsating load, since it has not been studied till now. The equations of motion for transverse vibrations of the beam are obtained using Hamilton’s principle. The general Galerkin T = m w dx, method is used to reduce the nondimensional equations of 0 motion to a set of coupled Hill’s equations with complex l l 1 1 2 2 coeﬃcients. The static buckling loads are obtained from V = E A U dx + E A U dx 1 1 3 3 1,x 3,x 2 2 0 0 Hill’s equations [22]. The eﬀect of rotation parameters and 2 2 geometric parameters on the nondimensional static buckling l 2 l 2 1 ∂ w 1 ∂ w + E I dx + E I dx loads is investigated for the pinned-pinned and ﬁxed-free 1 1 3 3 2 2 2 ∂x 2 ∂x 0 0 boundary conditions. l (1) ∗ 2 + G A ν dx 2 2 2. Formulation of the Problem l x 1 ∂w + mΩ (b + x) dx dx, 2.1. System Conﬁguration. A viscoelastic sandwich beam of 0 2 ∂x 0 0 length l and width B,set oﬀ at a distance b from the axis of rotation (z -axis) and oriented along the x-axis, perpendicular to the axis of rotation is shown in Figure 1. ( ) W = P t w dx, The beam rotates about the vertical z -axis at a constant 0 Advances in Acoustics and Vibration 3 b l 2h P(t) x 2h 2h Figure 1: System conﬁguration. where ν = ((U −U −cw )/2h ), which is derived according or, 2 1 3 x 2 to Kerwin’s assumption, where T is the kinetic energy of the system, V is the strain energy of the system and W is the work w = 0 done on the system by the external force. (5) Using Hamilton’s principle, u =0or u = 0. δ (T − V + W)dt = 0. (2) Introducing the nondimensional variables x = x/l, w = The equations of motion and the associated boundary w/l, u = u /l,and t = t/t ,where t = ml /(E I + E I ) 1 1 0 0 1 1 3 3 conditions are obtained as follows: and simplifying, the following nondimensional equations of motion are obtained: w ¨ + (E I + E I ) 1 1 3 3 ⎡ ⎤ 1 3Ω 0 2 2 3 + f − (x + b) w λ 1+ E h 31 2 0 31 f m 2(E A + E A ) 1 1 3 3 ¨ ⎣ ⎦ w + 1+ − x + b w l (1+ E h ) l 31 31 2 E I + E I 1 1 3 3 −3 Ω (x + b)w E A + E A 1 1 3 3 2 3 2λ 1+ E h 0 31 31 E I + E I 1 − x + b w 1 1 3 3 2 2 2 + − 3 Ω − Ω f − (x + b) ( ) l 1+ E l 0 0 31 31 h1 (3) 2(E A + E A ) 2 1 1 3 3 ∗ 2 G A c P(t) 1 ⎡ 2 2 × + w + Ω (x + b)w λ 1+ E h 0 2 2 0 31 31 f m 2 2 m(2h ) + − λ − x + b l (1+ E l ) 31 31 h1 ( ) G A c 1+ α + u = 0 m(2h ) ⎤ h + h 12 32 ∗ ∗ G A (1+ α) −3g 1+ + P t w 2 2 ( ) 2 u − 1+ α u − cw = 0. 1 2 (2h ) (E A + α E A ) 2 1 1 3 3 The boundary conditions at x = 0and x = are 3 h + h 12 32 2 ∗ + λ x + b w + g l h 1+ h1 12 2 2 w =0or w = 0, × (1+ α)u = 0, ∂ 3mΩ 2 1 (E I + E I ) 1+ f − (x + b) w 1 1 3 3 ∂x 2(E A + E A ) 1 1 3 3 ∗ 3 1+ E h 2 31 u − h (1+ α) 1 12 2 ∗ 2 2 4 1+ α E h mΩ G A c 2 31 31 0 2 2 − f − (x + b) + + P(t) w (2h ) (1+ ((h + h )/2)) 12 32 ∗ × (1+ α)u − 2 w = 0. G A c(1+ α) 2 2 ( ) l h h1 12 + u = 0, (2h ) (6) (4) 4 Advances in Acoustics and Vibration 700 180 Mode 3 Mode 3 Mode 2 80 Mode 2 Mode 1 Mode 1 0 0 12 3 6 12 3 6 4 5 4 5 h h 31 31 (a) (b) Figure 2: (a) Variation static buckling loads with h (pinned-pinned). (b) Variation static buckling loads with h (clamped-free). 31 31 The associated nondimensional boundary conditions at where g being the shear parameter x = 0and x = 1are mΩ l λ = , w =0or w = 0, 0 ( ) 2 E I + E I 1 1 3 3 (10) ∂ f 1+ λ − x + b w 1 2 3mΩ l ∂x l 0 λ = . 2(E A + E A ) 1 1 3 3 − λ − x + b w (7) These are rotation parameters P t = P + P cos ωt 0 1 h + h 12 32 +3g 1+ + P t w (11) P(t)l = . 3 h + h (E I + E I ) 1 1 3 3 12 32 + g l h (1+ α) 1+ u = 0, h1 12 1 2 2 This is the nondimensional load. or 2.3. Approximate Solution. Approximate solution to the w = 0, nondimensional equations of motion are assumed as (8) u =0or u = 0 1 i=N w x, t = f t w (x), i i The various parameters. are deﬁned as i=1 (12) k=2N E = , u x, t = f t u (x), 1 k 1k k=N+1 where f (r = 1, 2,... ,2N) are the generalized coordinates h = , r and w and u are the coordinate functions satisfying as i 1k many boundary conditions as possible [24]. For the pinned- l = , h1 pinned case, the shape functions chosen are (9) ( ) ( ) w x = sin iπx , h i h = , u (x) = cos kπx , 1k (13) g = g 1+ jη ∗ 2 k = k − N. G h l h1 = , E 1+ E h For i = 1, 2,... , N,and k = N +1, N +2,... ,2N. 1 31 31 (P ) 0 crit (P ) 0 crit Advances in Acoustics and Vibration 5 For the clamped-free case, the approximating functions where are −1 T i+1 i+2 i+2 [k] = k − [k ][k ] [k ] , w (x) = (i +2)(i +3)x − 2i(i +3)x + i(i +1)x , 12 22 12 (14) 1 k k+1 u (x) = x − x . k = 1+ λ − x + b w w dx 1k ij 1 i j k +1 0 Substitution of the series solutions in the nondimen- + λ − x + b × w w dx 0 i j (21) sional equations of motion and subsequent application of the general Galerkin method [24] leads to the following matrix h + h equations of motion: 12 32 +3g 1+ w w dx, i j 2 0 [ ] [ ] [ ] m f + k f + k f ={0}, (15) j 11 j 12 l H = w w dx. ij i j [k ] f + [k ] f ={0}, (16) 0 22 l 21 j where j = 1, 2,... , N and l = N +1,... ,2N. The various 2.4. Static Buckling Loads. Substitution of P =0and { f }= 1 j matrix elements are given by −1 {0} in (20) leads to the eigen value problem [k] [H]{ f }= 1/P { f }. The static buckling loads (P ) for the ﬁrst few m = w w dx, 0 j 0 crit ij i j modes are obtained as the real parts of the reciprocals of the −1 eigen values of [k] [H]. k = 1+ λ − x + b w w dx 11ij 1 i j 3. Numerical Results and Discussion + λ − x + b w w dx 0 2 i j Numerical results were obtained for various values of the non-dimensional geometric parameters h , h , l , b, the 31 12 h1 h + h 12 32 modulus ratio G /E and the rotation parameter λ .The ∗ 2 1 1 + 3g 1+ − P t w w dx , i j following values of parameters are used unless otherwise stated: η = 0.05, h = h = 1.0, l = 40, b = 0.05, 31 12 h1 3 h + h 12 32 G /E = 0.001, g = 0.1, λ = 0.05, and λ = 0.1. 2 1 0 1 k =− g l h (1+ α) 1+ 12jl h1 12 2 2 It should be noted that Figures 2(a), 3(a), 4(a), 5(a), 6(a),and 7(a) are for the pinned-pinned case and Figures × u w dx , 2(b), 3(b), 4(b), 5(b), 6(b),and 7(b) are for the clamped-free 1l j conﬁguration. Figures from 2(a), 2(b)–7(a),and 7(b) show the variation of the ﬁrst three nondimensional static buckling 3 h + h 12 32 k =− g l h (1+ α) 1+ 21li h1 12 loads, (P ) with h , h , l , b, G /E ,and λ ,respectively. 0 31 12 h1 2 1 1 crit 2 2 The variation of these loads with g and λ are similar to those for G /E and l , respectively, and are not shown. 2 1 h1 × u w dx , 1l With an increase in h (Figure 2(a)), the buckling loads are seen to decrease monotonically for the pinned-pinned 2 1 1+ α E h case. For the clamped-free case (Figure 2(b)), these ﬁrst 31 31 k = 3l u u dx 22kl h1 1k 1l increase and then decrease. The rate of decrease is maximum 1+ E h 31 31 for the highest mode for moderate h values. As h (Figures 3(a) and 3(b)) is increased, (P ) for 12 0 crit ∗ 2 2 + g l h (1+ α) u u dx . 1k 1l 12 the three modes shows a decreasing trend for low values of h1 h . It remains fairly constant for moderate values of h and 12 12 (17) increases for higher values. From (16) It is observed that (P ) increases nonlinearly with l 0 crit h1 for three modes (Figures 4(a) and 4(b)). −1 f =−[k ] [k ] f , (18) l 22 21 j The critical loads are almost constant over the entire range of b (Figures 5(a) and 5(b)). where For increasing the G /E , Figures 6(a) and 6(b), the 2 1 [k ] = [k ] . (19) 21 12 (P ) values are initially constant and rise appreciably for 0 crit higher values of G /E , the rate of increase being higher for 2 1 Substitution of above in (15) and subsequent simpliﬁcation the higher modes. leads to It is seen that (P ) is almost constant over a wide range 0 crit [m] f + [k] − P [H] f − P cos t [H] f ={0}, j 0 j 1 j of λ (Figures 7(a) and 7(b)) and decrease slightly for the values of λ greater than 0.1. (20) 1 6 Advances in Acoustics and Vibration Mode 3 Mode 3 Mode 2 Mode 2 Mode 1 Mode 1 0 0 0.01 0.1 1 10 100 0.01 0.1 1 10 100 h h 12 12 (a) (b) Figure 3: (a) Variation static buckling loads with h (pinned-pinned). (b) Variation static buckling loads with h (clamped-free). 12 12 Mode 3 Mode 3 140 90 Mode 2 Mode 2 Mode 1 Mode 1 30 20 40 60 80 100 20 40 60 80 100 l l h1 h1 (a) (b) Figure 4: (a) Variation static buckling loads with l (pinned-pinned). (b) Variation static buckling loads with l (clamped-free). h1 h1 Mode 3 Mode 3 Mode 2 Mode 2 30 8 4 Mode 1 Mode 1 0 0.5 1 1.5 2 012 b b (a) (b) Figure 5: (a) Variation static buckling loads with b (pinned-pinned). (b) Variation static buckling loads with b (clamped-free). (P ) (P ) 0 crit 0 crit (P ) 0 crit (P ) 0 crit (P ) (P ) 0 crit 0 crit Advances in Acoustics and Vibration 7 700 200 Mode 3 Mode 3 Mode 2 Mode 2 Mode 1 Mode 1 0 0 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 G /E G /E 2 1 2 1 (a) (b) Figure 6: (a) Variation static buckling loads with G /E (pinned-pinned). (b) Variation static buckling loads with G /E (clamped-free). 2 1 2 1 Mode 3 Mode 3 Mode 2 Mode 2 Mode 1 Mode 1 0.0001 0.1 1 0.0001 0.1 1 0.01 0.01 0.001 0.001 λ λ 1 1 (a) (b) Figure 7: (a) Variation static buckling loads with λ (pinned-pinned). (b) Variation static buckling loads with λ (clamped-free). 1 1 4. Conclusions Nomenclature A (i = 1, 2, 3): Areas of cross section of a three In this paper, the static stability of an asymmetric three- layered beam, i = 1, 2, and 3 for top, layered rotating sandwich beam with viscoelastic core has middle and bottom layer, been considered. The beam is subjected to an axial pul- respectively, sating load. Both pinned-pinned and clamped-free con- B: Width of beam ﬁgurations have been considered. An increase in h is b: Distance of nearer end of the beam seen to have a detrimental eﬀect on the nondimensional from the axis of rotation static buckling loads. Hence, a symmetric beam is seen b: b/l to have better resistance against static buckling. Over a c: h +2h + h wide range of values, h is seen to have little eﬀect upon 1 2 3 E (i = 1, 2, 3): Young’s modulli, i = 1, 2, 3 for top, the buckling loads. However, for small values, it has a i middle and bottom layer, detrimental eﬀect, and for large values, it improves the respectively, buckling loads. l , G /E , λ ,and g improve static stability, h1 2 1 0 especially for large values of the parameters. The buck- f : ∂ f /∂t j j ling loads are seen to be almost independent of b and G : In-phase shear modulus of the λ . viscoelastic core (P ) 0 crit (P ) 0 crit (P ) (P ) 0 crit 0 crit 8 Advances in Acoustics and Vibration G : G (1 + jη), complex shear modulus of [3] A. Unger and M. A. Brull, “Parametric instability of a rotating core shaft due to pulsating torque,” Journal of Applied Mechanics, Transactions ASME, vol. 48, no. 4, pp. 948–958, 1981. g : g(1 + jη), complex shear parameter [4] D. C. Kammer and A. L. Schlack, “Dynamic response of a g: Shear parameter radial beam with nonconstant angular velocity,” Journal of 2h (i = 1, 2, 3): Thickness of the ith layer, i = 1, 2 and 3 Vibration, Acoustics, Stress, and Reliability in Design, vol. 109, for top, middle and bottom layer, no. 2, pp. 138–143, 1987. respectively [5] N. Sri Namachchivaya, “Mean square stability of a rotating h : h /h 12 1 2 shaft under combined harmonic and stochastic excitations,” h : h /h 31 3 1 Journal of Sound and Vibration, vol. 133, no. 2, pp. 323–336, I (i = 1, 2, 3): Second moments of area of cross section about a relevant axis, i = 1, 2 [6] H. F. Bauer and W. Eidel, “Vibration of a rotating uniform and 3 for top, middle, and bottom layer beam, Part-II: orientation perpendicular to the axis of rota- j: −1 tion,” Journal of Sound and Vibration, vol. 122, no. 2, pp. 357– l:Beamlength 375, 1988. l : l/h h1 1 [7] B. A. H. Abbas, “Vibrations of Timoshenko beams with m: Mass/unit length of beam elastically restrained ends,” Journal of Sound and Vibration, P : Nondimensional amplitude for the 1 vol. 97, no. 4, pp. 541–548, 1984. dynamic loading [8] B. A. H. Abbas and J. Thomas, “Dynamic stability of (P ) : Nondimensional static critical buckling Timoshenko beams resting on an elastic foundation,” Journal crit of Sound and Vibration, vol. 108, pp. 25–32, 1986. load t:Time [9] O. A. Bauchau and C. H. Hong, “Nonlinear response and stability analysis of beams using ﬁnite elements in time,” AIAA t: Nondimensional time Journal, vol. 26, no. 9, pp. 1135–1142, 1988. u (x, t), U (x, t): Axial displacement at the middle of the 1 1 [10] R. C. Kar and T. Sujata, “Dynamic stability of a rotating beam top layer of beam with various boundary conditions,” Computers and Structures, u (x, t), U (x, t): Axial displacement at the middle of the 3 3 vol. 40, no. 3, pp. 753–773, 1991. bottom layer of beam [11] R. C. Kar and T. Sujata, “Dynamic stability of a rotating, w(x, t): Transverse deﬂection of beam pretwisted and preconed cantilever beam including coriolis w : ∂w/∂x eﬀects,” Computers and Structures, vol. 42, no. 5, pp. 741–750, 2 2 w : ∂ w/∂x w: w/l [12] T. H. Tan, H. P. Lee, and G. S. B. Leng, “Parametric instability w: ∂ w/∂t of spinning pretwisted beams subjected to sinusoidal com- 2 2 w : ∂ w/∂x pressive axial loads,” Computers and Structures, vol. 66, no. 6, t : ml /(E I + E I ) pp. 745–764, 1998. 0 1 1 3 3 u : ∂u /∂x 1 [13] B. Pratiher and S. K. Dwivedy, “Parametric instability of a cantilever beam with magnetic ﬁeld and periodic axial load,” u : ∂ u /∂x 1 1 Journal of Sound and Vibration, vol. 305, no. 4-5, pp. 904–917, f:(l + b) , for beam free at 3 2 x = l, = (l /3) + b + bl for all other [14] S. K. Dwivedy, K. C. Sahu, and S. Babu, “Parametric instability cases regions of three-layered soft-cored sandwich beam using η: Core-loss factor higher-order theory,” Journal of Sound and Vibration, vol. 304, λ : Rotation parameter no. 1-2, pp. 326–344, 2007. λ : Rotation parameter [15] D. Y. Zheng, F. T. K. Au, and Y. K. Cheung, “Vibration of (Φ): A null matrix vehicle on compressed rail on viscoelastic foundation,” Journal ω: ωt of Engineering Mechanics, vol. 126, no. 11, pp. 1141–1147, ω: Frequency of forcing function ω: Nondimensional forcing frequency [16] A. V. Metrikine and S. N. Verichev, “Instability of vibrations Ω : Speeds of rotation of a moving two-mass oscillator on a ﬂexibly supported Ω : Nondimensional rotation parameters 0 Timoshenko beam,” Archive of Applied Mechanics, vol. 71, no. α: E A /E A . 9, pp. 613–624, 2001. 1 1 3 3 [17] A. V. Metrikine and S. N. 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Advances in Acoustics and Vibration 9 [20] C. Y. Lin and L. W. Chen, “Dynamic stability of a rotating beam with a constrained damping layer,” Journal of Sound and Vibration, vol. 267, no. 2, pp. 209–225, 2003. [21] P. R. Dash, B. B. Maharathy, R. Mallick, B. B. Pani, and K. Ray, “Parametric instability of an asymmetric rotating sandwich beam,” Journal of Aerospace Sciences and Technologies, vol. 60, no. 4, pp. 292–309, 2008. [22] H. Saito and K. Otomi, “Parametric response of viscoelasti- cally supported beams,” JournalofSound andVibration, vol. 63, no. 2, pp. 169–178, 1979. [23] E. M. Kerwin Jr., “Damping of ﬂexural waves by a constrained viscoelastic layer,” TheJournal of theAcousticalsociety of America, vol. 31, pp. 952–962, 1959. [24] H. Leipholz, Stability Theory, John Wiley & Sons, Chichester, UK, 2nd edition, 1987. 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Published: Oct 16, 2011
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