THERMALLY INDUCED VIBRATION CONTROL OF COMPOSITE THIN-WALLED ROTATING BLADE

Transcription

1 4 H INERNAIONAL CONGRESS OF HE AERONAUICAL SCIENCES HERMALLY INDUCED VIBRAION CONROL OF COMPOSIE HIN-WALLED ROAING BLADE Sungsoo Na*, Liviu Librescu**, Hoedo Jung*, Injoo Jeong* *Korea Universit, **VA ech Kewor: thin-walled rotating beam, composite material, thermall induced vibration control Abstract his paper deals with thermall induced dnamic response control analsis of a rotating composite blade, modeled as a tapered thinwalled beam induced b heat flu. he displaed results reveal that the thermal environment iel a detrimental repercussions upon their dnamic responses. he blade consists of host graphite epo laminate and spanwise distributed transversel isotropic (PZ-4) sensors and actuators. he controller is implemented via the combined negative displacement and velocit feedback control methodolog, which prove to overcome the deleterious effect associated with the thermall induced dnamic response. he structure is modeled as a composite thin-walled beam incorporating a number of nonclassical features such as transverse shear, secondar warping, anisotrop of constituent materials, and rotar inertias. 1 Introduction he increasing use of fiber-reinforced, composite, thin-walled beam construction for rotor blades used in helicopter, tilt rotor aircraft, turbo engine, spacecraft boom and other applications, has generated a great deal of research activit aimed at enhancing their dnamic response performances. For reasons of efficienc involving gas dnamics and weight, the must be thin, et to operate in severe thermal environments and at higher rotational spee. Bole[1] was the first to include inertia effects in calculating the thermal-structural response of a beam subject to rapid heating and presented the governing equations for the problem of thermall induced vibrations. Seibert and Rice [] investigated coupled thermoelastic effect for Euler-Bernoulli and imoshenko beam model. Johnston and hornton [3] analed the effects of thermall induced structural disturbances of an appendage on the dnamics of a simple spacecraft. hornton and Kim [4] developed an analtical approach to determine the thermalstructural response of a fleible rolled-up solar arra due to a sudden increase in the eternal heating. he coupled thermal-structural responses were compared with the uncoupled analsis results. I. Yoon [5] investigated thermall induced vibration of composite thinwalled beam. he structure is modeled as a circular thin-walled beam of closed cross section and has constant cross area ratio. Although of an evident importance, to the best of authors knowledge, no such studies including thermall-induced dnamic response control of nonuniform composite thin-walled rotating blade, have been found. he rotating beam ma be constitutes part of a rotating spacecraft boom. Both the dnamic equations involving the temperature effects and the related boundar conditions are obtained via the application of Hamilton s variational principle. In its modeling the effects of anisotrop of constituent materials, transverse shear, warping, rotar inertia, etc are incorporated. In addition, in order to induce elastic couplings between flapwise bending and chordwise bending, a special pl-angle distribution achieved via the usual helicall wounding fiber-reinforced technolog is 1

2 Sungsoo Na, Liviu Librescu, Hoedo Jung, Injoo Jeong implemented. he numerical simulations displa eigenfrequencies and deflection timehistor as a function of the fiber orientation of the composite materials, rotating speed, taper ratio. hermal Analsis.1 Basic Assumptions A thin-walled beam of radius R and wall thickness h is considered (Fig. 1). he blade is subjected to a known incident heat flu S applied at time t=. he associated problem is to determine the transient temperature response of the tip of the blade. o this end, the following assumptions are adopted. 1. Heat is conducted onl in the circumferential direction, impling that the heat conduction along the blade length is negligible,. hermal energ losses at the cantilevered support at = are neglected, and thermal energ is emitted from eternal surface of the blade assuming diffuse radiation, but internal radiation within the blade is neglected, 3. he temperature field is assumed to be uniform across the beam thickness, impling that the temperature gradient across it is neglected, 4. Convection heat transfer inside and outside the beam is negligible, (a) Heat Flu (b) Beam Cross Section Fig.1 Heat flu for coupled thermal-structural analsis. hermal Analsis In such a contet, the thermodnamic equation of heat conduction and radiation is k σε 4 αs + = δ cosφcos( β + θ ) (1) t ρcr φ ρch ρch In Eq. (1), ª(,φ,t) is the absolute temperature at an arbitrar point of beam, k is the thermal conductivit, ρ and c are the weight densit and the specific heat of the material, respectivel, t is time coordinate and δ is unit for the values of circumferential coordinate φ corresponding to the portion of beam surface eposed to radiation and ero otherwise. he heat flu intensit of radiation source at an angle β with respect to the direction normal to the undeflected beam ais, S, is related to the counterpart one at an arbitrar point of the deflected beam surface, S, b S = S cos( ) β + θ () he thermodnamic equation of heatconduction-radiation can be linearied. As a result, one can represent as (, φ,) t = + 1 (, φ,) t (3) where 1 (,φ,t) is the disturbance temperature, and is the stead-state absolute temperature fulfilling the condition 1 (,φ,t). Further, we will consider ˆ 1(, φ,) t = (,)cos t φ (4) where ˆ is the maimum disturbance temperature. As a result of Eq. (3), 4 (,φ,t) intervening in Eq. (1) is epressed as a truncated binomial series epansion about in the form 4 4 (, φ,) t + 4 3ˆ (,)cos t φ (5) Moreover, the heat flu distribution on the right of Eq. (1) is represented as: δ cosφ = + + cosφ (6) π 4π B virtue of Eqs. (5) and (6), from Eq. (1) one obtains 1/4 3 αso cos β ss = (7) π σε and ˆ * 1 ˆ + = cos( β + θ ) (8) t τ τ where 3 1 k 4σε = +, * σ S = + τ (9) τ ρcr ρch 4π ρch

3 HERMALLY INDUCED VIBRAION CONROL OF COMPOSIE HIN-WALLED ROAING BLADE denote a characteristic time and the timeindependent maimum disturbance temperature, respectivel, when the blade is deflected staticall. Assuming ero initial conditions, from Eq. (8) one obtains t / τ * e t / ˆ(,) p t t = e cos( β + θ ) dp τ (1) where p is a dumm time variable. It is readil seen that the disturbance temperature as epressed b Eq. (1) depen nonlinearl on θ. Assuming θ to be small, one can linearie ˆ(, ) tas to become t / τ * e t / ˆ(, ) p t t = e (cos β sin θ ) dp τ (11) 3 Formulation of he Composite hin-walled Beam Model 3.1 Basic Assumptions and Kinematics of the Modeling Formulation he tapered composite blade consisting of a single cell thin-walled beam is mounted on a rigid hub (radius R ) that rotates with constant angular velocit Ω about origin O (Fig. ). X O Y O Z R Z Ω R c R b R s n L A A c b section A-A h c(η) b(η) Fig. Geometric Configuration of the Rotating Blade he inertial reference sstem (X, Y, Z) is attached to the center of the hub O. B (i, j, k) and (I, J, K), we define the unit vectors associated with the coordinate sstems (,, ) and (X, Y, Z), respectivel. he equations of rotating thin-walled beam are based on the following statements [6, 7, 8]: (i) the original cross-section of the beam is preserved; (ii) the secondar warping effects are included; (iii) transverse shear, Coriolis effect, and centrifugal acceleration are incorporated; and finall, (iv) the constituent material of the structure features thermomechanical anisotropic properties. he linear distribution of the chord c( η) and height b( η) of the mid-line cross-section profiles along the beam span is considered as c( η) cr = [ 1 η(1 σ) ] (1) b( η) br Herein σ c / cr denotes the taper ratio, η / L is the dimensionless spanwise coordinate, where L denotes the beam semi-span, and subscripts R and identif its characteristics at the root and tip cross-sections, respectivel. In the same contet, the radius of curvature of the circular arc associated with the midline contour at section η along the beam span varies according to the relationship: R( η) = [ 1 η(1 σ) ] RR (13) he points of the beam cross-sections are identified b the global coordinates, and, where is the spanwise coordinate and b a local one, n, s, and, where n and s denote the thicknesswise coordinate normal to the beam mid-surface and the tangential one along the contour line of the beam cross-section, respectivel. (see Fig. ) In accordance with the above assumptions and in order to reduce the 3-D problem to an equivalent 1-D, the components of the displacement vector are epressed as [6] ut (,,, ) = u φ ( t, ) vt (,,, ) = v + φ ( t, ) d (,,, ) = (, ) + (, )[ ( ) ] + wt w t θ t s n d θ(, t)[ ( s) + n ] φ (, t)[ Fw( s) + na( s)] θ(,) t = γ (,) t v (,) t θ (,) t = γ (,) t u (,) t (14) (15) 3

4 Sungsoo Na, Liviu Librescu, Hoedo Jung, Injoo Jeong Eqs. (14) and (15) reveal that the kinematic variables, u (,t), v (,t), w (,t), θ (,t), θ (,t) and φ(,t) representing three translations in the,, directions and three rotations about the,, directions, respectivel are used to define the displacement components, u, v and w, while following coordinates description, θ (,t) and θ (,t) denote the rotations about aes and respectivel, while γ and γ denote the transverse shear in the planes and respectivel and the primes denote derivatives with respect to the -coordinate, respectivel. Notice that the - ais is located as to coincide with the locus of smmetrical points of the cross-section along the wing span. he kinetic energ K, and potential energ V, epressions for a beam are 1 K = ρ(r i R i) dτ τ 1 V = bij bijd σ ε τ τ N 1 L = [ σ ε + σ γ + σ γ ] C h( k) k = 1 b b bs bs bn bn ( k) dnd (16) he epressions for the virtual work done b eternall applied forces are L Wf = f(,) t δ v(,) t d (17) In these equations dτ( dnd) denotes the differential volume element and the position vector R R(,,,t) relative to a fied origin is defined as: R = R + r+ (18) In Eq. (18), r( i+j+k) defines the undeformed position of a point measured in the beam coordinate sstem and ( ui+vj+wk) denotes the displacement vectors of the points of the blades, while R = R k. 3. he Equations of Motion and Boundar conditions Emploment of constitutive equations and strain-displacement relationships in the Eq. (16), and carring out the indicated integrations with respect to n and s, one can obtain simplified kinetic and potential energ, which is applied to the etended Hamilton s principle in order to obtain the coupled bending equations of adaptive rotating beams and the associated boundar conditions. t1 ( δk δv + δw) dt = t (19) δ u = δ v = δθ = δθ = at t = t1, t Herein K and V denote the kinetic and strain energ, respectivel, δw is the virtual work of eternal forces, t 1 and t are two arbitrar instants of time, while δ is the variational operator. he Equations governing the (flap-lag) Bending-ransverse Shear Motion: [ a θ + a ( u + θ )] +Ω { P( ) u } bω u bu h =, θ θ +Ω 1 5 = [ a a ( v )] { P( ) v } bv h, [ a θ + a ( v + θ )] a ( u + θ ) a θ b5 b 15 θ θ h h4 ( + )( Ω ) + =, [ a33θ + a34( u + θ )] a55( v + θ) a5θ ( b + b )( θ Ω θ ) h + h = (a-d) he associated BCs for the rotating beams clamped at = and free at =L are: At =, u = v = θ = θ = (1a-d) At =L, a θ + a ( u + θ ) = h, a a v h a a ( v ) h M, a a u h M θ + 55( + θ) = 5, a θ θ = a 33θ + 34( + θ) = 3 ( a-d) In Eqs. () through (), a ij, b i denote global stiffness and mass quantities, respectivel. P() is obtained as L P ( ) = b( ) ( R+ ) d (3) 1 and h i ( h i (,t)) denote the thermal stressresultants and thermal stress-couples defined as: d d h = ( N1 + N4 ), h = N C 4 C d d h = ( N 3 1 N4 ), h = N C 5 C (4a-d) 3.3 Pieoelectric Distribution and the Control Law 4

5 HERMALLY INDUCED VIBRAION CONROL OF COMPOSIE HIN-WALLED ROAING BLADE For the general case, the epression of the pieoelectricall induced flap and lag bending moment is given b, respectivel [ 1 ] [ ] a M (,) t = CVt () H( ) H( ) a M (,) t = CVt () H( ) H( ) 1 (5a,b) where C, C are constants dependent on the mechanical and geometrical properties of the pieoactuator and host structure and V(t) is the applied input voltage that is equal and opposite in sign in the upper and lower pieoactuators(out-of-phase actuation). H( ) denotes the Heaviside function representing the actuator distribution. (Fig. 3) X O Z s B Fig. 3 Distribution of Pieoactutors B S a section B-B In the previousl displaed equations, due to the special distribution of pieoactuators, it was shown that the pieoelectricall induced moment intervenes solel in the boundar conditions associated with the bending motion, prescribed at the beam tip, and hence it plas the role of the boundar moment control. Within the adopted feedback control law the pieoelectricall induced bending moment at the blade tip is epressed as a M ( L) = k vθ( L) + kpθ( L) a M ( L) = k θ ( L) + k θ ( L) v p t a (6a,b) Herein k, k denote the feedback gains, and in the numerical simulations nondimensional counterpart of k, k is K, K defined b K = k L / a, K = k L / a v v 33 p p 33 v = v /, p = p / K k L a K k L a 4. Results and Discussions (7a,b) composite thin-walled blade eposed to an incident heat flu applied instantaneousl at t=. he data on which basis the numerical simulations have been generated are supplied in able 1. Figs. 4 and 5 show the average, perturbation temperature response of the composite blade, while Fig. 6 displas temperature distribution along the blade crosssection from the uncoupled analsis. Fig. 4 shows average temperature is approimatel 33.8K, when is 9K. emperature profiles for the upper one-half of the blade cross-section at various times are shown in Fig. 6, which proves the development of the blade thermal gradients Fig.4 Average temperature response of a composite blade = ss Fig. 5 Perturbation temperature response of a composite blade A numerical stud was performed to investigate the quasi-static and the dnamic response of the sstem consisting of a rotating 5

6 Sungsoo Na, Liviu Librescu, Hoedo Jung, Injoo Jeong t=s t=5s t=1s t=s t=35s(t ss ) σ=1. σ=1.5 σ=. θ=9 o 35 4 θ=75 o θ=6 o 3 θ=45 o θ= o φ (Degree) Fig.6 emperature distribution for a composite blade From Figs. 7 to 9 the plots highlight the effects of angular velocit, taper ratio and pl angle orientation on the natural frequencies of the coupled flap-lag bending motion. From the results it becomes evident that as the beam taper increases (i.e. when σ goes above 1.) the first and second coupled natural frequencies decrease whereas the third one increases. he plots also displa the sensitivit of natural frequencies to pl angle orientation. A general remark emerging from Figs. 7 to 9 is that the stiffening effect due to beam rotation contributes to the increase of natural frequencies for all taper ratios σ=1. σ=1.5 σ=. θ=9 o θ=75 o θ=6 o Fig. 8 Second coupled flap-lag bending frequenc vs. Ω for different pl angles σ=1. σ=1.5 σ=. θ=75 o θ=9 o θ=6 o 5 o θ= θ=45 o Fig. 9 hird coupled flap-lag bending frequenc vs. Ω for different pl angles Figs. 1 and 11 highlight the effect of the incident angle of heat flu and of taper ratio on the dnamic response behavior. he results reveal that taper ratio plas a significant role in confining the deflection response and the increase of incidence angle can decrease both the flapwise and chordwise response. 3 θ=45 o 1 θ= o Fig. 7 First coupled flap-lag bending frequenc vs. Ω for different pl angles U β= o β=15 o β=3 o β= o σ=1. σ=. β= o Fig. 1 Nondimensional flapping response for various heat incident angle,θ=3 o, Ω= rad/s 6

7 HERMALLY INDUCED VIBRAION CONROL OF COMPOSIE HIN-WALLED ROAING BLADE V β= o β= o σ=1. β= o β=15 o β=3 o σ=. Fig. 11 Nondimensional flapping response for various heat incident angle, θ=3 o, Ω= rad/s Fig. 1 displas the time-histor of transversal deflection response. his graph highlights the strong effect plaed b the angular velocit to flapping dnamic response. In Figs. 13 through 15 there are depictions of the effects of angular velocit and pl angle orientation on the uncontrolled and controlled natural frequencies of the coupled flap-lag bending motions. he results reveal the potential role plaed b the pieoelectric actuation upon the enhancement of eigenfrequencies. Figs 16 and 17 displaed uncontrolled and controlled time histor of lagging and flapping response of a blade subjected to heat flu, respectivel. he results reveal negative displacement feedback control provides a powerful tool for improved structural responses, relativel, in low range of angular velocit 1 8 Uncontrolled Controlled(K v =.1,K p =5) θ=9 o V Ω =8 Ω = Ω =1 Ω =5 6 4 θ=75 o θ=6 o θ=45 o Ω = θ= o Fig. 1 Nondimensional flapping response for various angular velocit, σ=, θ=3 o, β=15 o Fig. 14 Uncontrolled and controlled second coupled flap-lag bending frequenc vs. Ω for different pl angles, σ= 3 5 Uncontrolled Controlled(K v =.1,K p =5) θ=9 o θ=75 o 15 1 θ=6 o 5 θ=45 o θ= o Fig. 13 Uncontrolled and controlled first coupled flap-lag bending frequenc vs. Ω for different pl angles, σ= Fig. 15 Uncontrolled and controlled third coupled flap-lag bending frequenc vs. Ω for different pl angles, σ= 7

8 Sungsoo Na, Liviu Librescu, Hoedo Jung, Injoo Jeong U V Uncontrolled Controlled(K p =5) Ω=5 rad/s Ω= rad/s Fig. 16 Uncontrolled and controlled lagging response for various angular velocit, σ=, θ=3, β= Uncontrolled Controlled(K p =5) Ω= rad/s Ω=5 rad/s Fig. 17 Uncontrolled and controlled flapping response for various angular velocit, σ=, θ=3, β=15 5. Conclusions A comprehensive structural model of composite thin-walled rotating blade was developed and the problem of the thermall induced vibration was addressed. he effects of the heat incident angle, rotating speed, blade taper ratios and pl angles of composite materials to the dnamic response of the blade structure are studied b using the coupled thermal-structural analsis. he implications of a number of factors such as blade taper ratios, pl angles of composite materials, rotational speed, heat incident angle and pieoelectric actuation ma improve the quasi-static and dnamic deflection of the blade. Acknowledgment Sungsoo Na would like to acknowledge the financial support b the Basic Research Program of the Korea Science & Engineering Foundation, Grant No. R References [1] B. A. Bole, J. H. Weiner, heor of hermal Stress, John Wile & Sons, Inc., pp 3-355, 196. [] A. G. Seibert, J. S. Rice, Coupled hermall Induced Vibrations and Beams, AIAA Journal, Vol. 7, No. 7, pp , [3] J. D. Johnson, E. A. hornton, hermall Induced Attitude Dnamics of a Spacecraft with Fleible Appendage, Journal of Guidance, Control and Dnamics, Vol. 1, No. 4, pp , [4] E. A. hornton, Y. A. Kim, hermall Induced Bending Vibration of a Fleible Rolled-Up Solar Arra, Journal of Spacecraft and Rockets, Vol. 3, No. 4, pp , [5] I. Yoon, hermall Induced Bending Vibration of Composite Spacecraft Booms Subjected to Solar Heating, Ph. D hesis, Chung-Nam National Universit, Korea. [6] Song, O. and Librescu, L., Structural Modeling and Free Vibration Analsis of Rotating Composite hin- Walled Beams, J. of the American Helicopter Societ, Vol. 4, No. 4, pp , [7] Song, O., Librescu, L. Oh, S.Y., Dnamics of Pretwisted Rotating hin-walled Beams Operating in a emperature Environment, Journal of hermal Stresses, Vol. 4, pp 55-79, 1. [8] Na, S., and Librescu, L., Modeling and Vibration Feedback Control of Rotating apered Beams Incorporating Adaptive Capabilities, ASME, PVP- Vol. 415, Recent Advances in Soli and Structures, pp 35-53,. [9] O. Song, Modeling and Response Analsis of hin- Walled Beam Structures Constructed of Advanced Composite Materials, Ph. D hesis, VPI&SU, USA. [1] M. Muroono, S. Sumi, hermall-induced Bending Vibration of hin-walled Beam with Closed Section Caused b Radient Heating, Memoirs of the facult of Engineering, Kushu Universit, Vol. 49, No. 4, pp 73-9, [11] S. S. Na, Control of Dnamic Response of hin- Walled Composite Beams Using Structural ailing and Pieoelectric Actuation, Ph. D hesis, VPI&SU, USA,

9 HERMALLY INDUCED VIBRAION CONROL OF COMPOSIE HIN-WALLED ROAING BLADE Appendi able. 1 Material and geometric properties of composite material (Graphite/Epo) Parameter Value L.3 m h.35e-4 m R.54 m E 1.68E11 N/m E =E E9 N/m G E9 N/m G 3 =G E9 N/m µ 1 =µ 3 =µ 13.5 ρ kg/m 3 α.9 α 1 1.1E-6 K -1 α 5.E-6 K -1 ε.84 σ 5.67E-8 W/m K 4 k W/mK c 144 J/kgK S 1.35E3 W/m 9