THE EFFECT OF THE FIBER ORIENTATION ON THE DYNAMIC BEHAVIOUR OF ROTORS IN WOUNDING-SHAFT

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1 THE EFFECT OF THE FIBER ORIENTATION ON THE DYNAMIC BEHAVIOUR OF ROTORS IN WOUNDING-SHAFT José C. Pereira Universidade Federal de Santa Catarina Campus Universitário Trindade Caixa Postal Florianópolis SC, Brasil - jcarlos@grante.ufsc.br Abstract. The purpose of this work is to analse the dnamic behaviour of simple-supported rotors in which its shaft are made of fibre/resin in a wounding process. The orientation of the wounding angle is an important parameter on determing the properties of the section such as the equivalent bending stiffness <EI> and the equivalent torsional stiffness <GJ>, which can modif the strain energ of the shaft. The Campbell Diagram, in which the bending and torsion modes of the rotor is included, can be appreciabl changed with the evolution of the orientation of the wounding angle. This analsis can be used in an optimisation process maximiing the critical velocities distance from the operating rotation of the rotor. Kewords: Rotor, Composite, Homogeneit, Campbell Diagram.. INTRODUCTION In rotordnamics prediction, we currentil emplo the finite element method to analse this tpe of structure in searching the undamped natural frequencies (Campbell Diagram), the steadstate response to unbalance, the transient response to unbalance and external driving forces as shown in Rossi et al. (989). Nelson et al. (976) and Ögüven et al. (984) using the finite element method introduced different effects as the rotator inertia, groscopic moments, axial load, etc. Steffen et al. (987) linked a finite element package with an optimiation program in order to perform the optimal structural design with the purpose of to maximiing the distance of the critical speeds from it-selves. In these studies seen earlier the material of the shaft is considered isotropic. In this work it is introduced a new parameter in rotordnamics analsis given b the fiber orientation in the case of rotors in which the shaft is made of fiber/resin in a wounding process. In this case, the homogeneit properties of the cross section of the shaft <EI> and <GJ> in determining the strain energ in bending and torsion are used. As a first approach, the Raleigh- Rit method is used and onl the first mode in bending and in torsion are observed The introdution of the parameter wounding angle of the shaft can modif the behaviour in

2 bending and torsion of the rotor and this effect can be included in optimiation techniques in searching the optimal design of the rotor machine.. THE EQUATIONS OF ENERGY OF THE ROTOR. The kinetic energ of the disc From the Fig., we can deduce the instantaneous vector of rotation in the reference coordinate sstem, alanne et al. (986): r r r r ω = ψ& + θ& x + φ& () where r, x r, r are unitar vecteurs. Z φ θ ψ& θ ψ φ & = Ω X x ψ φ θ & x x w u Y Figure - Reference coordinate sstem (x,, ) The angular velocit of the disc is φ & and the compounds of ω r in the reference coordinate sstem is: ω ω ω x = & cos θ sen φ + θ& cos φ φ & + ψ sen θ ψ& cos θ cos φ + θ& sen φ ψ () The kinetic energ of the disc can be expressed b:

3 T D ( u + w ) + ( I ω + I ω + I ω ) = M D & & D D D (3) where u and w are the coordinates of the center of inertia of the disc and I Dx, I D and I D are the inertia moments of the disc in the reference coordinate sstem. Taking in account that the angles θ and ψ are small, the velocit of rotation is φ & = Ω and the smetr of the disc implies I Dx = I D, we ma obtain from Eq. (3) that: T D ( u + w& ) + I ( θ& ψ& ) + I Ω ψ θ + I Ω = M D & Dx D & D (4) From the application of the Eq. (4) in the agrange s equation, in which the generalised coordinates are u, w, θ, ψ, we can identif the inertia effect and the giroscopic effects of the disc.. The strain energ of the shaft in bending The general expression for the strain energ is: U = ε τ t σ dτ (5) where σ = E ε : et be u* and w* be components of the displacement of a point P in the cross section in the reference coordinate sstem, Fig.. If we consider onl the linear effect of the longitudinal strain: ε = x u* w* Z (6) Ωt P w * w x X x u * u Figure The reference coordinate sstem of the shaft

4 The final expression of the strain energ is: w * U = E x τ u * dτ (7) The displacements u* and w* in the global coordinate sstem are: u* w* = w sen Ω t + u cos Ω t = w cos Ω t + u sen Ω t (8) The Eq. (7) in terms of u and w is: w u U = E I d + (9) From the application of the Eq. (9) in to the agrange s equation, in which the generalised coordinates are u, w, we can identif the stiffness of the shaft in bending..3 The movement in torsion of the shaft The general expression for the kinetic energ of the shaft in torsion is, alanne et al. (986): T = ρ J ϕ& d () where ϕ is the torsion angle. The general expression for the strain energ of the shaft in torsion is: ϕ U = G J d () B using Eq. () and Eq. () in the agrange s equations, in which the generalised coordinate is ϕ, we can deduce the movement of torsion of the shaft. 3. RAYEIGH-RITZ METHOD With a reasonable aproximation of the displacement field, we can deduce the equations of the movement in bending and in torsion using the Raileigh-Rit Method. In this work, we search the first frequencies in bending and the first frequenc in torsion for a simple-supported rotor, as shown in the Fig. 3.

5 /3 /3 Figure. 3 Rotor simple-supported on the ends An approach for the displacement in bending for this configuration is: u(, t) w(, t) π = sin π = sin p p () and, w π π θ(, t) = = cos p u π π ψ(, t) = = cos p (3) Using Eq. () and Eq. (3) in Eq. (4), the kinetic energ of the disc is given b: (4) T D = M D π sin + I Dx π π cos ( p& p& π π + ) I D Ω cos p& p Using Eq. () in Eq. (9), the strain energ of the shaft ma be written as: 4 ( p p ) π π U = < EI > sin d + (5) The application of the agrange s equation in Eq. (4) and (5) leads to: a && p + b Ω p& + c p = (6) a & p b Ω p& + c p =

6 3 π π π π π π π with: a = M D sin + I Dx cos, b = I D cos, c =< EI > The solution for the Eq. (6) are: n n rt p = P e (7) where r = ± j ω( Ω) are the natural frequencies in bending on each velocit of rotation of the rotor. Using Eq. (7) in Eq. (6) we obtain the characteristic equation for the rotor in which the roots are the natural frequencies: 4 r c b c + + Ω r + = (8) a a a An approach for the displacement in torsion for this configuration is: ϕ (, t) = (a). q (9) B replacing Eq. (9) in Eq. (), the kinetic energ of the shaft is given as: T = Jρ q& () 3 B Using Eq. () in Eq. (3), the strain energ of the shaft is given as: < GJ > U = q () The application of the agrange s equation in Eq. () and () leads to: ω = π 3 < GJ > J ρ () where ω is the natural frequenc in torsion of the rotor. 4. HOMOGENEITY PROPERTIES OF THE SHAFT In Eq. (5) and Eq. () we can identif the properties of the cross section of the shaft as the equivalent bending stiffness <EI> and the equivalent torsional stiffness <GJ>. In the case of

7 rotors in which the shaft are made of fibre/resin in a wounding process, the orientation of the wounding angle can modif the strain energ of the shaft. So, the Campbell Diagram, can be appreciabl changed with the evolution of the orientation of the wounding angle. In this work we use the equivalent stiffness <EI> and <GJ> as a function of the wounding angle for a hollow shaft in kevlar/epox and glass/epox as shown in Pereira (999). In this case the external diameter and the internal diameter of the wounding-shaft are 4 mm and 3 mm, and the lenght is,8 m. The configuration of the laers in the thickness direction of the shaft is [θ, θ] 4. As demonstred b Pereira (999), the equivalent bending stiffness <EI> and equivalent torsional stiffness <GJ> can be seen in Fig. 4 for a kevlar/epox wounding-shaft and in Fig. 5 for a glass/epox wounding-shaft. Ν/m,7,6,5,4,3,, α <EI>e4 <GJ>e3 Figure 4 - Evolution of <EI> and <GJ> as function of the wounding angle α kevlar/epox N/m,,8,6,4, α <EI>e4 <GJ>e3 Figure 5 - Evolution of <EI> and <GJ> as function of the wounding angle α glass/epox 5. ΑPPICATION AND CONCUSIONS In this section we plotted the Campbell Diagram for the simple-supported rotor shown in Fig. 3 for different wounding angles. The equivalent stiffness <EI> and <GJ> are from Fig. 4 and from Fig. 5. The properties of the disc are: I Dx =.5 kg.m, I D =.45 kg.m and M D = 7,85 kg. The bending modes are in full lines and the torsional modes are in doted lines.

8 5 ω(h) 5 ω=ω Ω(rps) (,- ) 4 (3,-3 ) 4 (4,-4 ) 4 x (5,-5 ) 4 * (6,-6 ) 4 (7,-7 ) 4 Figure 6 Campbell Diagram - kevlar/epox wounding shaft ω=ω 5 ω(h) 5 (,- ) 4 (3,-3 ) 4 (4,-4 ) 4 x (5,-5 ) 4 * (6,-6 ) 4 (7,-7 ) Ω(rps) Figure 7 Campbell Diagram - glass/epox wounding shaft This work shows the application of composite materials in rotor dnamics. As we can see in the Fig. 6 and Fig. 7, the Campbell Diagram changes with the orientation of the wounding angle. This effect are more appreciabl on low velocit of rotation Ω in bending modes. We can also identif a wounding angle in which we have the highest distance between the bending modes and the torsional modes on the same velocit of rotation Ω, let be 45. Using composite materials in rotor dnamic analsis can introduce additional design variables which ma be searched for the optimal performance, as the stiffness and the damping of the material, the wounding angle, the number of laers and the weight.

9 REFERENCES alanne, M., Berthier, P., Der Hagopian, J., 986, Mécanique des vibrations ineaires, Masson. Nelson, H. D., McVaugh, J. M. 976, The dnamics of rotor-bearing sstems using finite elements, Journal of Engineering for Insdustr, Ma. Pereira, J. C., 999, A numerical approach on determinig the equivalent torsional stiffness of wounding-tubes, th Iberian atin-american Congress of Computational Methods in Engineering, São Paulo. (a aparecer). Rossi, M. A., Squaroni, A. D., 986, Finite element modal approach to large rotor-bearing sstem analsis. Ögüven, H. N., Ökan Z.., 984, Whril speeds and unbalance response of multibearing rotors usinf finie elements, Journal of Vibration, Acoustics, Stress, and Reliabilit in Design, Vol. 6, Januar. Steffen Jr., V., Marcelin, J.., 987, Dnamic Optimiation of Rotors, 9 th Brailian Congress of Mechanical Engineering, Florianópolis.