Vibration and Stabilit of a Composite Thin-Walled Spinning Tapered Shaft Hungwon Yoon *, Sungsoo Na, Seok-Ju Cha * Korea Universit, Seoul, Korea, 3-7 Liviu Librescu Virginia Tech, Dept. of ESM, Blacksburg, VA, USA This paper deals with the vibration and stabilit of a circular clindrical shaft, modeled as a tapered thin-walled composite beam and spinning with constant angular speed about its longitudinal ais, and subected to an aial compressive force. Hamilton s principle and the etended Galerkin Method are emploed to derive the governing equations of motion. The resulting eigenvalue problem is analzed, and the stabilit boundaries are presented for selected taper ratios and aial compressive force combinations. Taking into account the directionalit propert of fiber reinforced composite materials, it is shown that for a shaft featuring flapwise-chordwise-bending coupling, a dramatic enhancement of both the vibration and stabilit behavior can be reached. It is also found that b the structural tailoring and tapering, bending natural frequencies, stiffness and stabilit region can be significantl increased over those of uniform shafts made of the same material. In addition, the particular case of a classical beam with internal damping effect is also included. a i b i = global stiffness coefficients = global mass coefficients Nomenclature F () w s = primar warping function na() s = secondar warping function h = thickness of the thin-walled beam P = aial compressive force P = non-dimensional aial compressive force uvw,, = displacement components in, and z directions, respectivel u, v, w = translational displacement variables of the beam in, and z directions, respectivel θ, θ, φ = rigid bod rotations about the, ais and the twist about the z-ais, respectivel δ = logarithmic decrement µ = damping coefficient θ = pl angle measured from the positive s-ais toward the positive z-ais ω = natural frequenc (rad/sec) ω = non-dimensional natural frequenc Ω = spin speed (rad/sec) Ω = non-dimensional spin speed * Graduate Student, Department of Mechanical Engineering Associate Professor, Department of Mechanical Eng., AIAA Member Professor, Department of Engineering Science and Mechanics, Corresponding Author
T I. Introduction he stud of the eigenvibration and stabilit of the shaft structural sstems rotating about their longitudinal ais is an important prerequisite in the design of power transmission of aeropropulsion sstems, in helicopter drive applications, industrial machines such as steam and gas turbines, machine-tool spindles, turbogenerators, and production lines etc. All these spinning sstems have some important critical properties when the are operated at high spinning speeds. Due to the Coriolis and centrifugal forces associated with rotational motion, the natural vibrations of a rotating beam element are different from those of a non-spinning sstem. There have been a number of studies relating to the dnamic analsis of the transverse vibration of a spinning beam. Meirovitch solved groscopic sstems with a new method of solution of eigenvalue problems. Patel and Seltzer presented the solving program of comple eigenvalue analsis of rotating structures. Song and Jeong 3 studied the vibration and stabilit of pretwisted spinning thin-walled composite beams featuring bending-bending elastic coupling. Rosales and Filipich considered internal damping b means of the viscoelastic behavior of the beam material described b the Kelvin-Voigt model. Nelson proposed modified finite element models to include some effects, in the above mentioned finite element models, in addition to contributions of the translational inertia and bending stiffness, the effects of the rotator inertia, groscopic moments, shear deformations, aial loads and internal damping can be successfull taken into account. In addition, Ku and Chen effectivel utilized Nelson s model to investigate the dnamic stabilit of a shaft-disk sstem subected to an aial periodic force. Leung and Fung 7 solved a straight rotating beam problem b the finite element method, giving the eplicit element matrices. Cudne and Inman 8 determined damping mechanisms in a composite beam b eperimental modal analsis. In this paper, an important step toward the rational design of spinning shaft consists of the development of analtical models that are capable of accuratel predicting its vibration and stabilit incorporating various design parameters such as pl angles, taper ratios, etc. To the best of the authors knowledge, the present stud represents the first work investigating the stabilit of spinning structures modeled as a non-uniform thin-walled composite beam and the problem of the stabilit taking into account the internal damping effect of beam material. II. Configuration of the Thin-Walled Beam Structure The case of a straight fleible beam of length L spinning along its longitudinal Z-ais at a constant rate Ω and subected to the longitudinal compressive dead force P is considered. The linear distribution along the shaft span of the radius of curvature of the mid-line cross sections R( η ) varies according to the relationship. R( η ) = [ η( σ) ] RR () Herein σ RT / RR (. σ.) denotes the taper ratio, η z/ L is the dimensionless span-wise coordinate, where L denotes the beam span, and subscripts R and T identif its characteristics at the root and tip crosssections. The points of the beam cross-sections are identified b the global coordinates X, Y and Z, where Z is the span-wise coordinate and b a local one, n, s, and z, where n and s denote the thickness-wise coordinate normal to the beam mid-surface and the tangential one along the contour line of the beam cross-section, respectivel. (see Fig. ) Fig. Geometric configuration of the spinning shaft
III. Basic Assumptions and Kinematics A. Structural Modeling and Basic Assumptions In present stud, the tapered composite shaft is consisting of a single cell thin-walled beam and spinning with constant angular velocit Ω along its longitudinal Z-ais. Fig. shows the spinning composite thin-walled beam of a circular cross section featuring CUS(Circumferentiall Uniform Stiffness) configuration. Fig. Composite thin-walled beam of a circular cross section featuring CUS configuration The inertial reference sstem (X,Y,Z) is attached to the geometric center and ( IJK,, ) and (,, i k) denote the unit vectors associated with the coordinate sstem (X,Y,Z) and (,,z), respectivel. The equations of spinning thinwalled shaft are based on the following statements: (i) The original cross-section of the beam is preserved. (ii) Transverse shear effect, Groscopic effect and centrifugal acceleration are incorporated. (iii) The constituent material of the structures features anisotropic properties. (iv) A special la-up inducing flap wise-chord wise coupling is implemented. B. Constitutive Relationships In light of the previousl mentioned assumptions, in order to reduce the 3-D elasticit problem to an equivalent - D one, the components of the displacement vector are epressed as uzt (,, ; ) = u φ ( zt ; ) vzt (,, ; ) = v + φ ( zt ; ) d (,, ; ) = ( ; ) + θ ( ; )[ ( ) ] wzt w zt zt s n ds () d + θ( zt ; )[ s ( ) + n ] φ ( zt ; )[ Fw( s) + nas ( )] ds θ (;) zt = γ (;) zt v (;) zt z θ (;) zt = γ (;) zt u (;) zt z d d as () = s () s () ds ds (3) Eqs. () and (3) reveal that the kinematic variables, u, v, w, θ, θ and φ represente three translations in the,, z directions and three rotations about the,, z directions, respectivel. Furthermore γ z and γ z denote the transverse shear in the planes z and z respectivel, and the primes denote derivatives with respect to the z- coordinate. 3
C. Kinetic Energ For further use, additional kinematic quantities are indicated here. Among these, the position vector of a generic point M(, z, ) belonging to the deformed structure is R = ( + u) i + ( + v) + ( z+ w) k () Y Ω Z,z J R M i I Ωt X Fig. 3 Cross-section of the beam, where, and z are the Cartesian coordinates of the points of the continuum in its undeformed state. Recalling that the spin rate was assumed to be constant, and using the epressions for the time derivatives of unit vectors, the velocit and acceleration of a generic point are R = [ u Ω ( + v)] i + [ v +Ω ( + u)] + w k () R = [ u Ωv ( + u) Ω ] i + [ v + Ωu ( v) Ω ] + w k () In these equations the superposed dots denote derivatives with respect to time t. The kinetic energ K epressions for a beam are K ρ R R = d t t τ τ N L ( k ) u v w = C ρ dndsdz h( k) + + (7) k = t t t N L ( k ) d d = ρ ( u ( ) φ) + ( v + φ) + w + θ + θ F wφ + n θ θ a φ C h k dndsdz k = ds ds Carring out the indicated integrations with respect to n and s, the kinetic energ can be reduced to the compact form L K = [ bu bωvu bu Ω + bv + bωuv bv Ω + ( b + b) θ + ( b + b) θ ] dz (8) Herein various coefficients are shown in Ref.[9]. D. Strain Energ The corresponding strain energ V epressions for a beam are V = σε i id τ τ L N = [ σ ] ( ) zzεzz σszγsz σnzγnz ( k ) dndsdz + + C h k k = L N ( k ) d d = ( ) C σzz w + θ h k + θ φ Fw + n θ θ φ a k = ds ds ( k ) d d + σ c ( k ) sz ( u + θ ) + ( v + θ ) + A φ + σnz ( u + θ ) d ( v + θ ) d dndsdz ds ds β ds ds (9)
Performing various integrations step b step from previous work, the potential energ can be shown to have the form L V = [ a3θ u a ( u θ u ) aθ v a ( v θ v ) aθ a( v θ θ ) + + + + + + + + θ () a ( u θ + θ ) a θ + a θ + a ( u θ + θ θ ) a ( v θ + θ θ ) a θ θ ] dz 3 33 3 Herein various coefficients are shown in Ref.[9]. IV. The Dnamic Equations of the Composite Shaft A. CUS(Circumferentiall Uniform Stiffness) Configuration In the present paper a special case of pl-angle distribution inducing special elastic couplings will be considered with the following lamination scheme (see Fig. ). θ ( ) = θ ( ) () The present pl-angle configuration was referred to as the circumferentiall uniform stiffness configuration, resulting in an eact decoupling between flapwise-chordwise-transverse shear, on one hand, and etention-torsion motion, on the other hand. θ θ Fig. CUS(Circumferentiall Uniform Stiffness) configuration B. Governing Equations and their Associated Boundar Conditions To derive the coupled bending equations of spinning beams and the associated boundar conditions, the etended Hamilton s principle is used. t ( δk δv + δw) dt = t () δ u = δ v = δθ = δθ = at t = t, t Herein K and V denote the kinetic and strain energ, respectivel, δ W is the virtual work of eternal forces, t and t are two arbitrar instants of time, while δ is the variational operator. The equations governing the (flapwise-chordwise) bending-transverse shear motion : δu :[ a θ + a ( u ' + θ )] Pu bu + bω v + bω u = (3a) 3 δv :[ a θ ' + a ( v ' + θ )]' Pv bv bω u + bω u = (3b) δθ :[ a θ ' + a ( v ' + θ )]' a ( u ' + θ ) a θ ' ( b + b ) θ = (3c) 3 δθ :[ a θ ' + a ( u ' + θ )]' a ( v ' + θ ) a θ ' ( b + b ) θ = (3d) 33 3 The associated boundar conditions for the spinning beams clamped at z = and free at z = L are : at z = ; u = v = θ = θ = (a) at z = L ;
δu : a θ ' + a ( u ' + θ ) Pu = (b) 3 δ v : a θ ' + a ( v ' + θ ) Pu = (c) δθ : a θ ' + a ( v ' + θ ) = (d) δθ : a θ ' + a ( u ' + θ ) = (e) 33 3 The coefficients ai = ai and b i appearing in these equations denote stiffness and reduced mass terms, respectivel. Their epressions are shown in Ref. 9. P is the constant aial force, positive in compression. C. Internal Damping Effect In special case, this paper deals with the problem of the stabilit taking into account the internal damping effect of beam material. Internal damping can be modeled b eamining the various forces and moments involved in deriving the equations of motion. One possible choice for material damping is to assign a viscous damping proportional to the rate of strain in the beam. Considering Euler-Bernoulli theor, equation of motion are epressed as EOM : δ u :[ a u + µ a u ] + Pu + bu ( b + b ) u bωv bω u = (a) δ v :[ a v + µ a v ] + Pv + bv ( b + b ) v + bωu bω v = (b) 33 33 BC s : au + µ au + Pu ( b + b) u = (a) a v + µ a v + Pv ( b + b ) v = (b) 33 33 au = or u = (c) a33v = or v = (d) The coefficient µ in these equations denotes damping coefficient. Cudne and Inman performed the eperimental verification of these equations of motion 8. V. Solution Methodolog of the Eigenvalue Problem For the purpose of solving the eigenbvalue problem of the groscopic sstem as given b Eq (3). assuming snchronous motions, the generalized displacements are represented in the form t ( u(,), zt v(,), zt θ (,), zt θ (,)) zt = ( U(), z V(), z Y(), z X()) z e λ (7a) N ( U( z), V( z), Y( z), X( z)) = ( a u ( z), b v ( z), c ( z), d ( z)) (7b) where λ is the comple eigenvalue and u, v, =, are trial functions which have to fulfil all of the kinematic boundar conditions, whereas a, b, c, d are constant vectors. Replacement of the representations in Eqn. (7) in the variational integral (), carring out the indicated variations t with respect to the spanwise z-coordinate and time t, and dividing through b e λ, the sstem of governing equations is cast in matri form as [[ M] λ + [ G] λ+ [ K]]{ X} = {} (8) { X} T = { u, v,, } Herein, M and K are the smmetric mass matri and the stiffness matri, respectivel, G is the skew-smmetric groscopic matri.
For non-trivial solution for { X }, the determinant formed b the coefficients of Eq. (8) must be equal to zero det([ M] λ [ G] λ [ K]) + + = (9) which can be solved for the eigenvalue λ r, where λ r = σ r ± iω r is the comple eigenvalue and ω r is the natural frequenc of the sstem. The logarithmic decrement δ r is defined as πσ r δr = () ω δ represents the instabilit threshold of the sstem when δ <. r r r VI. Numerical Simulations and Discussions The numerical simulations are performed b considering the beam geometrical characteristics as L=.3m, h=. - m and R R =.7m. The mechanical characteristics of the beam correspond to the graphite/epo material whose on-ais elastic characteristics are as follows: Table Mechanical characteristics of the beam Parameter Value E.8 N/m E =E 3.7 9 N/m G 3. 9 N/m G 3 =G 3. 9 N/m µ =µ 3. µ 3. ρ 8. kg/m 3 A. Critical Speed Analsis Before the numerical stud of the anisotropic composite structural model, the spinning beam without transverse shear is compared with the Ref. 7 in Fig.. This figure shows the first and second two dimensionless natural frequencies of the bending mode with respect to the spinning rate. The numerical result is almost coincided. ω ρal / EI Natural frequenc 3 Present Ref. [7] Fw Bw Fw nd bending mode st bending mode Bw 3 Spinning rate Ω ρ AL / EI Fig. Non-dimensionl natural frequenc of the spinning shaft vs. the spinning speed rate ω, ω 8 σ =. σ =. σ =. θ=9 θ= σ=. θ= σ=. 3 7 Spinning speed rate σ=. Fig. Variations of the natural frequenc ratios vs. the spinning speed rate Ω for several pl angles and different taper ratios. Ω 7
For the fied compressive force, the minimum spin rate at which the natural frequenc becomes zero valued corresponds to the critical spinning speed, denoted as Ω cr. In Fig. pictorial representations of the variation of the fundamental natural frequenc ω ( ω / ˆ ω) versus the spin speed Ω Ω ( / ω) for the case of the unloaded beam ( P = ) as well as for various pl-angles ( θ =,,9 ) and for taper ratios ( σ =.,.,.) are displaed. The normalized factor ˆω is the fundamental frequenc of the non-spinning, characterized b θ =, σ =. and P =. For Ω =, in the absence of groscopic effects, it is seen that the sstem is characterized, for each pl angle and taper ratio, b a single fundamental frequenc, while with the increase of the pl angle θ, an increase of the non-rotating natural frequencies is obtained. This trend is attributed to the increase of the bending stiffness a 33( = a ) that is associated with the increase of the pl angle θ 9. As soon as the rotation starts, a fact which is accompanied b the generation of groscopic forces, a bifurcation of natural frequencies is eperienced. In other words, due the effect of the Coriolis force, two distinct frequenc branches of free bending vibration, namel the forward(fw) and backward(bw) frequenc branches are produced. The minimum spinning speed rate at which the lowest natural frequenc becomes zero valued, is called the critical spinning speed, denoted as Ω cr, that corresponds to the divergence instabilit. Throughout these results it becomes apparent that at each pl angle and taper ratio there is a specific critical spinning speed and that, the minimum and maimum ones occur at θ = and θ = 9 respectivel. Furthermore, from these figures it is readil seen that while the frequencies of the forward whirling increase with the increase of the spinning speed, the frequencies of the backward whirling decrease with the increase of spinning speed. The minimum spin rate at which the lowest rotating natural frequenc becomes zero valued is referred to as the critical spinning speed, and corresponds to the divergence instabilit. B. Critical Force Analsis Figs. 7 and 8 depict the variation of the forward(fw) and backward(bw) whirling natural frequenc branches for selected values of the pl angle and spinning speed rate versus the increase of the dimensionless aial compress force P( PL / aˆ 33), where â 33 is the normalized bending stiffness corresponding to the pl angle θ = and taper ratio σ =.. According to increasing the aial compressive force, forward and backward whirling natural frequencies decrease. The compressive force at which the lowest natural frequenc becomes zero valued, is called the critical force, denoted as P cr. Throughout these results it becomes apparent that at each pl angle and spinning speed there is a specific critical force and that, the minimum and maimum ones occur at θ = and θ = 9 respectivel.(see Figs. 7, 9). θ = o θ = o θ = o.. Ω = Ω =. Ω =.3 Ω =.. Ω =. ω, ω Fw.8 ω, ω Ω =. Bw.878 3.9.8 8 Aial compressive force P. Ω =.... Aial compressive force Fig. 7 Variations of the natural frequenc ratios vs. Fig. 8 Variations of the natural frequenc ratios vs. aial compressive force for several pl angles. aial compressive force for selected spinning ( Ω=., σ =.) speeds. ( θ =, σ =.) P 8
3 σ =. σ =. σ =.. Fw ω, ω 3 σ=. ω, ω. Flutter occurs σ=. Bw A A σ=. shift shift 8 Aial compressive force P Fig. 9 Variations of the natural frequenc ratios vs. aial compressive force for several spinning speeds with selected taper ratios. ( Ω=., θ = 9 ). 99.3.3 8 8 9 9 Aial compressive force P Fig. Detail at A-A Section. In Figs. 9 and the variation of the forward(fw) and backward(bw) whirling natural frequenc branches for selected values of the taper ratio σ =.,.,. versus the increase of the dimensionless aial compress force P are depicted. Throughout these results it becomes apparent that at each taper ratio there is a specific critical force and that, according to increasing the taper ratio, the points of the critical force shift lager ones. C. Stabilit Analsis Combination of compressive force and spinning speed rate ielding two natural frequencies to coalesce constitute a flutter condition. Increasing either of these two parameters beond the value of the force or spinning speed rate corresponding to the flutter boundar results in comple conugate eigenvalues and correspondingl, to bending oscillations with eponentiall increasing amplitudes. Before the numerical stud of the anisotropic composite structural model, the spinning beam without transverse shear is compared with the Ref. in Fig.. This figure shows the divergence boundar, stabilit and flutter region and the numerical results are almost coincided. ρ AL / EI 8 Present Ref. [] Flutter Ω Spinning rate Divergence 9.87 8 Aial compressive force P Fig. Non-dimensional natural frequenc vs. aial compressive force ; without transverse shear. 9
. Flutter 8 Ω =. Ω =. Ω = 3. Ω =. Divergence Ω P flutter Ω =.. θ = o θ = o θ = o 8 Ω =. Aial compressive force P Pl Angle (θ) Fig. Stabilit plot in the Ω P plane displaing Fig. 3 Variations of the flutter boundaries vs. the domains of stabilit, divergence instabilit boundar, the pl angle for several spinning speeds. and flutter for selected pl angles. ( σ =.) ( σ =.) Flutter θ = 7 o o θ = 8 θ = 8 o θ = 9 o θ = 9 o 8 θ = 8 o Ω 3 Divergence P flutter θ = 8 o θ = 7 o σ =. shift σ =. 8.. Aial compressive force P Taper Ratio (σ) Fig. Stabilit plot in the Ω P plane displaing Fig. Variations of the flutter boundaries vs. the domains of stabilit, divergence instabilit boundar, taper ratios for several pl angles. and flutter for different taper ratios. ( θ = 9 ) In Figs. - there is depicted a stabilit plot of the spinning sstem in the Ω P plane for the selected pl angle and taper ratio. For Ω and P equal to zero, the sstem is stable. With the increase of Ω and P, instabilities b divergence or flutter ma occur. Due to the fact that the sstem is conservative, initial instabilit will alwas be of a divergence tpe, characterized b ω r =. The plots reveal that in the plane Ω P the instabilit boundar separates two stable regions. In all of these plots, the results reveal that the increase of the pl angle and taper ratio ields a considerable increase of the stabilit domains. D. Internal Damping Analsis In special case, this paper deals with the problem of the stabilit taking into account the internal damping effect of beam material. For a validation of mathematical modeling, the spinning beam without transverse shear is compared with the Ref. in Fig.. This figure shows the first dimensionless natural frequencies of the bending mode with respect to the spinning rate and the numerical result is almost coincided.
ω ρal / EI Present, P = Present, P = - Ref. [], P = Ref. [], P = - µ = µ =.7-3 µ =. -3 µ = Natural frequenc ω, ω 8 µ =. -3 µ =. -3 µ = Spinning rate Ω ρ AL / EI 8 Spinning speed rate Ω Fig. Non-dimensional natural frequenc of Fig. 7 Variations of the natural frequenc ratios vs. the spinning shaft vs. spinning speed rate the spinning speed rate Ω with internal damping ; with internal damping ; without transverse shear. without transverse shear. ( µ E I / ρ AL =.) ( θ = 9, σ =., P = ) 7 7 Divergence Divergence Ω 3 Flutter Ω 3 Flutter µ =, Divergence µ =, Flutter 8 µ =. -3, Divergence µ =. -3, Flutter 8 Aial compressive force P Aial compressive force P Fig. 8 Stabilit plot in the Ω P plane displaing Fig. 9 Stabilit plot in the Ω P plane displaing the domains of stabilit, divergence instabilit the domains of stabilit, divergence instabilit boundar, and flutter for undamped case ; boundar, and flutter for damped case ; without transverse shear. ( θ = 9, σ =.) without transverse shear. ( θ = 9, σ =.) In Fig. 7, the effect of internal damping is involved. For the case of damping coefficients 3 3 ( µ =,.7,. ), pl angle θ = 9, and taper ratio σ =., pictorial representations of the variation of the fundamental natural frequenc ω ( ω / ˆ ω) versus the spin speed Ω Ω ( / ω), and for the case of the unloaded beam ( P = ) are displaed. As soon as the rotation starts, a fact which is accompanied b the generation of groscopic forces, a bifurcation of natural frequencies is eperienced. The minimum spinning speed rate at which the lowest natural frequenc becomes zero valued, is called the critical spinning speed, denoted as Ω cr, that corresponds to the divergence instabilit. Critical spinning speeds are same values for various damping coefficients. Fig. 8-9 displa the effect of internal damping comparing undamped case. From those results it is readil seen that internal damping is stabilizing at speeds below the critical speed but destabilizing at speeds above the critical speed.
VII. Conclusions An analtical stud devoted to the mathematical modeling of spinning circular shafts, modeled as a non-uniform thin-walled beam, subected to an aial compressive force was presented. As shown, in the conditions described in the paper, the spinning shaft can eperience instabilities b flutter and divergence. Among others, the results reveal that structural tailoring and tapering can be successfull emploed to enhance their behavior b increasing the critical spinning speed rate, and b shifting the domains of divergence and flutter instabilit toward larger spinning speed rates. In particular, taking into account the internal damping effect, the sstem is more stable in the stable region, but on the other hand, the stable region is reduced as compared with the undamped case. Acknowledgments Sungsoo Na acknowledges the support of the work b the Basic Research Program of the Korea Science and Engineering Foundation, Grant No. R---9-. References Periodicals L. Meirovitch, A new method of solution of eigenvalue problems for groscopic sstems, AIAA J., 97, pp.337-3. Patel, J.S. and Seltzer, S.M., Comple Eigenvalue Analsis of Rotating Structures, in NASTRAN ; User s Eperiences NASA TMS-37, 97, pp.97-3. 3 O. Song and N. H. Jeong, Vibration and Stabilit of pretwisted spinning thin-walled composite beams featuring bending-bending elastic coupling,, pp.3-33. M. B. ROSALES and C.P. FILIPICH, Dnamic Stabilit of a Spinning Beam Carring an Aial Dead Load, Journal of Sound and Vibration, 3(), 993, pp.83-9 Zorzi, E. S., Nelson, H.D., Finite Element simulation of rotor-bearing sstems with internal damping, Trans. ASME, Journal of Engineering for Power, 977, pp.7-7. Ku, D.M. and Chen, L.W., Stabilit and whirl speeds of rotating shaft under aial loads, International Journal of Analtical and Eperimental Modal Analsis, 9(), 99, pp. -3 7 Leung, A.Y.T. and Fung, T.C., Spinning finite elements, Journal of Sound and Vibration (3), 988, pp.3-37 8 Cudne H. H. and Inman D. J., Determining Damping Mechanisms in a Composite Beam b Eperimental Modal Analsis, International Journal of Analtical and Eperimental Modal Analsis, Vol. (), 988, pp.38-3. 9 Na, S.S. and Librescu, L., Dnamic Response Control of Elasticall Tailored Adaptive Cantilevered Nonuniform Cross Section Eposed to Blast Pressure Pulses, International Journal of Impact Eng., No.,, pp.87-87 Wettergren, H., Material damping in composite rotors, Journal of Composite Materials, Vol. 3, No.7, 998, pp.-3.