Ani sıcaklık değişimi etkisindeki türbin kanadı nonlineer titreşimlerinin sonlu elemanlar yöntemi ile incelenmesi

Bu çalışmada değişik sıcaklık etkileri göz önüne alınarak dönen türbin kanadının lineer ve nonlineer titreşim karakteristikleri Rayleigh çubuğu hali, ve boyuna-eğilme girişimi ve büyük yer değişimleri göz önüne alınarak sonlu elemanlar yöntemi ile incelenmiştir. Bunun yanısıra iki yüzey arasındaki basınç farkı göz önüne alınmış ve sıcaklık etkilerinin zorlanmış titreşim sonuçlarına olan tesiri araştırılmıştır. Sıcaklık etkileri; 1. iki yüzey arasındaki sıcaklık farkı, 2. Kanat boyunca sıcaklık değişimi, 3. Ani sıcaklık değişimi başlıkları altında incelenirken kanat mekanik özelliklerinin sıcaklıkla değişiminde göz önüne alınmıştır. Bu sıcaklık etkilerinin nonlineer titreşim karakteristikleri hesaplanırken; a. Maksimum kesit dönmesi kullanılarak, b. Kesit dönmesi sürekli değişim formunda ele alınarak farklı çözümler geliştirilmiştir. Nonlineer titreşimlere; 1. Boyuna-eğilme girişimi, 2. Büyük yer değişimleri ilave edilmiştir. Bütün bunların yanı sıra Rayleigh çubuğunda olduğu gibi kesit dönmesi etkileri göz önüne alınarak çözümler yapılmıştır. Türbin kanadı titreşim karakteristiklerine etki eden temel büyüklüklerden, disk yarıçapı, dönme hızı, kanat konum açısı, iki yüzey arasındaki sıcaklık farkı ve kanat boyunca sıcaklık farkı gibi parametrelerin beraberce göz önüne alınmasıyla çözümler elde edilmiştir. Kanadın iki yüzeyi arasındaki basınç farkı ise yayılı yük olarak düşünülmüş ve bu yük virtüel işler prensibi kullanılarak her elemana etkiyen eşdeğer kuvvet vektörüne indirgenmiştir. Bu eşdeğer kuvvetlerin etkisiyle meydana gelen titreşim genliklerine kanat boyunca sıcaklık değişiminin etkileri de dahil edilmiştir. Hesaplar ankestre-serbest, ankestre-ankestre, basit-basit sınır şartları için yapılmıştır. Bulunan sonuçlar daha önceki yazarların kullandığı yöntemlerle karşılaştırılmış, ani sıcaklık değişimlerini ve kanat boyunca ve iki yüzeyi arasındaki sıcaklık değişiminin, dönen türbin kanatlarında göz önüne alınması gerektiği sayısal değerlerle gösterilmiştir. In this work, the vibrational characteristics and harmonic response of rotating beam attached to a disc, under linear thermal gradient along the beam, are studied by FEM. Effects of nonlinear bending and longitudinal displacement on the frequency is also considered. The numerical results are presented to illustrate the effects of linear thermal gradient and nonlinear bending and longitudinal displacement upon the dynamic behaviour of the system in practice. For this purpose, turbine blade is considered as a rectangular beam which is attached to rotating disc. For a beam undergoing lateral vibration, if w is the deflection in the z direction of an arbitrary point on the middle plane (xy plane) of the beam (Fig. 2.1. and Fig. 2. 2), then instantaneous location in the x,y, z system of co-ordinates can be taken as 3 2 W=C X +C X +c x+c ( 1 ) 4 3 2 1 The radial component of the centrifugal force per unit volume acting on an element of the beam at x is given by F = pQ2(r+x) (2) r where p is the mass per unit volume of the beam. A displacement of w along the z direction would result in a component (-wSin8) in the Y direction and a component (wCosS) in the Z direction (Fig. 2. 3). The force F can be resolved into components along the y and z directions as follows: F = -pn2wSin9Cos9 (3) y F = pQ2wSin28 (4) z Under the effects of these forces, the equation of motion of the system, using Hamilton principle can be written as follows: 5j (T-U)dt+ J SW dt=0 (5) t t v 1 1After the required mathematical steps the equation of motion of the system becomes [[K]-w2[M]]{5>={0> (5. a) The mass and stiffness matrices are found trough the expressions of potential and kinetic energy and by writing the virtual work expression in Hamilton Principle. For a non-rotating beam undergoing transverse vibration, the potential energy is given by the expression V =(EI/2)J ( a2w/Sx2)2dx (6) o where x represents the co-ordinate for the beam. In the finite element formulation, the beam is divided into elements and the above expression becomes,. ri V =(EI/2)J ( Ö2w/a«:2)2da: (6. a) o where w = [D] {5} (6.b) The equation (6. a) yields the following: V = (l/2){5}T[Ki]ew{5> (7) Under the effect of the rotational speed, the beam is- stiffened due to additional stresses created by centrifugal forces. These forces create stresses in the neutral surface and the additional strain energy stored in the element can be written as: V =(1/2) J {tr (aw/d«02+2T (aw/3«H aw/flu) }d* (8) The stresses due to axial forces F can be obtained as follows; m the axial force acting on any section at a distance T[K2lew{5> (10) XIIwhere [K ] is called the centrifugal stiffness matrix 2 ew ö If the gradient of modulus of elasticity along the blade is linear, then the modulus of elasticity E can be incorporated into the integral. If the gradient of modulus is e then the modulus of elasticity E at a section x can be written as follows: E=E (1-ex) = E (1-a x/L) (11) o o In this case, the stifness matrix [K ] is obtained instead of the matrice [K ] i ew [Vewt={1-e(n-im[Vew -EoIe^ * tDw^Dw]d* (12) where e=o/L (13) T[K ] AS} (18) nli wet where *2 r [KKL1]wet=AE0^ J4[(l-e(n-l)«-e«r]^Tp3^d« (18.a) XIIIAn alternative approach to the above formulation is as follows: 4^-=[D«H5> (13) dec w w If this equation is calculated for each element using the values of the first mode eigenvector, equation (20) is obtained, 9W =f(w, w', w, w') (20) 9 [D'] [D']{8} dT[K 1 {«} (24), 4 ew where [K4]ew=^A and by F during an arbitrary virtual displacament must be equal. If {5} is the vector of virtual nodal displacements corresponding to w the virtual displacement w within the element, then from equation (B.b) wV=[D 1V (25) w w Equating the work done by {F > and F gives =J wVF F dv (26) w e v z Since this relation is true for any arbitrary virtual displacement, by substituting F in eq. (4) and w from eq. (6.b) into equation (26), {F > can be calculated as, e {F >= Pn2ASin2eJ. [D ]T[D ] da; {5}V (27) WW w Using eq. (27), the additional stiffness matrix including the effect of setting angle to the elements of stiffness matrix will be [K ] = pn2ASin29j. [D ]T[D ]2[M]]{5}={0> (35) and solved numerically. In the solutions, the effect of each assumption on the eigenvalue of the system is also investigated. The effects of rotating speed on vibrational characteristics is calculated and then compared with the exact solutions of [24] and [58]. The effect of setting angle on vibration is the same with the results of ref. [58] and [24] for uniform blade cases. The frequency parameters for several rotating speeds are also given. The temperature variation along the blade is assummed to be linear. If the temperature is 32 C and 870 C at the root and at the free end of the blade, respectively; then, the thermal gradient of modulus of elasticity a becomes a =(E(0) -E(L))/E(0) (36) where a =[E(320C)-E(870°C)]/E(32°C) (37) E(x) =E (1-a x/L) (38) o The values of E(T) is given ref. [57]. Present results are compared with the results in ref. [59]. A comparison has also been given for different thermal gradient of modulus of elasticity. The finite element results are in good agreement with [59]. The results, for a turbine blade subjected to transient heat flux example, for a stepwise temperature change, are given in several tables, for nonrotating and rotating cases. The instantaneous frequencies are calculated for certain blade temperature points assuming a constant temperature at that particular point. For increased rotational speed, the tempareture effects on the vibrational frequencies are reduced by an amount of 6.6%-10% for higher frequencies, compared to nonrotating blades. XVIThe effect of average stresses caused by the temperature difference between the two surfaces of the blade, is modelled by an average cr along the x direction. The results are obtained using eq. (24), for a setting angle of 49,3. A 3% reduction in the natural frequencies is observed. The effects of linear thermal gradient including the other factors are also studied. There are small, but important variations in the vibrational frequencies due to thermal gradient effects with increasing speed. The first three natural frequencies of a nonrotating blade decrease by 3.1%, 6.8%, 8% respectively. The first three natural frequencies of a rotating blade at 1000 rad/sec decrease by 0.4%, 4.3%, 6.74% respectively. Nonlinear harmonic response of the system is found by iteration. In each iteration step, the system is considered to be conservative. The results for the nonlinear harmonic response of the blade subjected to thermal gradient are also tabulated. In conclusion, the nonlinear vibration characteristics and harmonic response of rotating turbine blade subjected to transient heat flux and thermal gradient are investigated by FEM. The importance of the subject has been shown by several examples chosen from the practice. The combined effect of the factors studied in this thesis on the first three vibrational frequencies of a model varies between 6.6%-16.3%. The effect of the thermal gradient is significant on the natural vibration frequencies (5.88-7.37%). The increase of the rotating speed has, also, a significant effect on these frequencies (about 4%). The effects of the thermal gradient on the nonlinear vibrational frequencies are, also, considerable (3.07, 6.34, 7.33% respectively). Inclusion of the transient heat flux into the nonlinear theory has a slight effect on the vibration characteristics. 131