Abstract

Vibration and Buckling of Rotating, Pretwisted, Preconed Beams Including Coriolis Effects The effects of pretwist, precone, setting angle and Coriolis forces on the vibration and buckling behavior of rotating, torsionally rigid, cantilevered beams are studied in this investigation. The beam is considered to be clamped on the axis of rotation in one case, and off the axis of rotation in the other. Two methods are employed for the solution of the vibration problem: one based upon a finite-difference approach using second-order central differences for solution of the equations of motion, and the other based upon the minimum of the total potential energy functional with a Ritz type of solution procedure making use of complex forms of shape functions for the dependent variables. Numerical results obtained by using these methods are compared to those existing in the literature for specialized simple cases. Results indicating the individual and collective effects of pretwist, precone, setting angle, thickness ratio, and Coriolis forces on the natural frequencies and the buckling boundaries are presented and discussed. Furthermore, it is shown that the inclusion of Coriolis effects is necessary for blades of moderate-to-large thickness ratios while these effects are not so important for small thickness ratio blades. Finally, the results show the possibility of buckling due to centrifugal softening terms for large values of precone and rotation. Introduction An important phase in the development of advanced turboprop blades currently in progress at the Lewis Research Center is the development of analytical blade models that are capable of predicting the vibration and flutter characteristics to aruacceptable degree of accuracy. The turboprop blades are of thin cross sections with large, variable sweep and pretwist, and are mounted on a rotating hub at a setting angle. Moreover, the blades are subjected to considerable centrifugal loading and develop steady-state deflections that are large compared to the cross-sectional dimensions. These blades, under certain conditions are subjected to centrifugal compressive forces on certain cross sections, and consequently the possibility of buckling cannot be ruled out. The recent research in the vibration and flutter of these blades indicate that the effects of nonlinearities, variable sweep, and pretwist must be included in the analysis for accurate prediction of the blade dynamic characteristics. The finite-element modeling of these blades is appropriate for the determination of their dynamic characteristics. However, such studies with the existing codes showed that the predicted results are satisfactory only for the first few modes and that the complicating effects included in these theories make the understanding of the individual and collective effects of the governing parameters impossible. These considerations have led to the modeling of the turboprop blade configurations using the simpler beam theories with the complicating effects successively taken into account to establish a physical understanding and to reveal the relative importance of the individual effects. A preliminary part of such study using a simplified beam model with a set of linear equations of motion is reported in this paper. In the beam model the effects of sweep on the dynamic behavior of the blade are introduced by setting the axis of the blade at an angle with respect to the plane of rotation. This angle is usually called the precone angle. The more complicated nonlinear aspects form the subject matter of a subsequent paper. An examination of the existing literature using beam theories indicates that several aspects of the vibration and buckling of rotating beams mounted off the axis of rotation were considered by several investigators Frequency Equation by Second-Order Finite-Difference Method The linear set of coupled bending-bending equations of motion for a torsionally rigid slender beam The specific objectives of the present paper are: (1) to solve the pretwisted, preconed rotating cantilever blade vibration problem allowing for the Coriolis effects for two cases of root clamping, that is, on the axis, and off the axis of rotation; and (2) to study the effects of pretwist, precone, setting angle, Coriolis forces, and the ratio of the principal moments of interia on the vibration frequencies and buckling boundaries. Two methods of solution are employed: (a) a finite difference procedure using second-order central differences for the solution of the governing equations of motion and, (b) the total potential energy functional and the Ritz type solution using complex shape functions for the dependent variables. The present solution methods provide a positive check on the accuracy of the results for untwisted blade cases in that the finite difference procedure gives close lower-bound solutions while the potential energy method gives upper-bound results. Further validation of the results is accomplished by comparing the present results to closed-form exact solutions wherever possible. The solution methods and parametric results provide a sufficient insight for further extensions of the present effort to account for blade sweep and second-degree nonlinearities in the analysis. Nomenclature •*{«(-£-')^)*""N"»»(T--')} It may be noted that the last two terms in equations (1) and (2) are shown with ±/=F signs. For a cantilever blade case with root clamping on the periphery of a disk of radius R and the blade protruding radially outward, the upper sign is to be taken. For the case with root clamping on the rim of a ring of radius R and the blade protruding radially inward, the lower sign is to be assumed. It is of interest to examine the coupled bending-bending equations and point out the various important terms arising due to the inclusion of precone. It may be seen that in equation (1), the term (-w sin 2 /3 Pc ) appears due to the inclusion of preconing which is a softening term. Even for the case of a rotating beam protruding radially outward and under centrifugal tension, the effect of this term is to reduce the effective stiffness, thereby providing the mechanism for instability at reasonably large values of precone and rotational speed. It may also be noted that the Coriolis terms, which are under lined in equations In seeking a solution by using a finite-difference procedure, one substitutes the finite-difference expressions for the derivatives into the differential equations, for any arbitrary station j inside the beam domain. When the beam is divided into n segments, j varies from 0, 1, 2, . . . to n, the clamped end being denoted by 0 and the free end by n. A set of n equations are written by successively assigning values for j from 1 to n. As can be seen from the expressions for the derivatives of an arbitrary variable ^ at the station j in terms of the secondorder central differences given in the appendix, one has to eliminate the evaluation of the functions at the fictitious stations outside the beam namely w_ 2 , w_,, w n+l , w" +2 , w" +3 , v^2, v~i> v" +l , v" +2 , and v" +3 . This can be accomplished by using the boundary conditions together with the recursive relations developed and discussed in detail in where ml, vol, CI, kl, k2, k3 and k4 are square matrices of order n x n each, [ w) and (t>) are column matrices containing the vector w lt w 2 . . . w"; v x , v 2 . . . v" and 0 is null matrix. For brevity elements of these matrices are not presented here. Equation Equation Assuming x = x"e^', equation where, A* = /A, i = V^T and p = ~ ifh (10) The eigenvalues of equation • • p", -p"',n being the order of M, C and K. However, eigenvectors for equations Frequency Equation by Using Energy Functionals The Lagrangian functional for a rotating, pretwisted, preconed blade including Coriolis effects but disregarding nonlinear effects can be derived from In equation (11), the last term is shown with =F/± signs as before with the convention that the upper sign is to be taken for the cantilever blade case with root clamping on the periphery of a disk and the blade protruding outward, while the lower sign is to be assumed for the case with root clamping on the rim of a ring and the blade protruding radially inward. In order to formulate the frequency equation through the Ritz process, shape functions for the variables w and v are assumed in the following complex form since the out-of-phase Coriolis effects are also included in the analysis where Aj, B jt Cj, Dj are the undetermined arbitrary real parameters in the shape functions, fj is the polynomial function defined as fj- The shape functions assumed here satisfy the boundary conditions v = w = w' = y' =0 at ij=0 and The real and imaginary parts of the shape functions are substituted into the Lagrangian functional separately, the indicated integrations are performed, the resulting functional is time averaged in the traditional manner and the Ritz process is 142/Vol. 108, APRIL 1986 Transactions of the ASME The resulting equations can be written in the following form of the eigenvalue problem after transformations similar to those described in the previous section are made: C+pD=0 where C and D are complex matrices. (17) Results and Discussion The eigenvalue problems defined by equations (9) and (17) were solved by using computer programs developed in FOR-TRAN language. These programs were run on IBM and CRAY computers at the NASA Lewis Research Center. Use was made of standard eigenvalue extraction routines Convergence. The relative convergence rates produced by the finite-difference method using second-order central dif- 144/Vol. 108, APRIL 1986 Transactions of the ASME ferences, and the potential energy method were obtained for several configurations of rotating blade cases. A typical set of such results are presented in Finally, for all the vibration and stability problems reported in this work, solutions were obtained by using the upperbound potential energy method with k = 8, and the usually lower-bound finite-difference method with n = 30. In view of the close agreement between these two sets of results observed Journal of Vibration, Acoustics, Stress, and Reliability in The effect of pretwist on nonrotating blade frequencies predicted by beam theories is well understood in the published literature. The frequencies presented in The effect of rotation on the coupling trends of pretwisted blades is illustrated in Next, the effects of pretwist and precone on rotating blade frequencies are considered. Some typical results are presented in Coriolis effects disregarded. Coriolis effects included. of pretwist, and those in 2 The effect of precone in the absence of Coriolis effects 4 The effect of varying pretwist for a given thickness ratio and precone angle are presented in 5 The effect of increasing the precone angle for low thickness ratio pretwisted blade case is shown in Buckling of Rotating Blades Mounted off the Axis of Rotation. A geometric arrangement giving rise to rotationally induced radial forces which are compressive rather than tensile is shown in An examination of the published work reveals that most of the problems in the different analytical approaches were due to some convergence problems of the solution procedure adopted. These were overcome by alternative solution methods 2 An increase in the pretwist angle for a given value of setting angle <p moves the stability boundary toward left. That is, for a given rotational speed, buckling occurs at a lower value of R for increasing 7 at a given <p. 3 An increase in the setting angle <p for a given pretwist has similar effect as discussed in (2) above. 4 For nonzero pretwist, all curves tend to meet at R = 0 as the rotational parameter, fi/Xj, tends to infinity. 5 For large values of d/b, there is a marked change in the stability boundary. Comparing <p = 0 deg, 7=15 deg curve with <p = 0 deg, 7 = 0 deg curve presented in The effects of precone, pretwist and Coriolis forces on the critical speed for causing instability are presented in Conclusions Parametric studies were conducted to ascertain the individual and collective effects of pretwist, precone, setting angle, Coriolis forces, and blade thickness ratio on the vibration frequencies and buckling boundaries of rotating beams. Two methods of solution for studying blade vibration and stability were used, namely, a finite difference procedure based upon second-order central differences that usually produces close lower-bound solutions, and the potential energy method that produces close upper-bound solutions. Results obtained by using these two methods were found to be in excellent agreement. Further validation of the present results was accomplished by comparisons to results in the literature for specialized simple cases. The parametric results show that the inclusion of Coriolis effects is necessary for blades of moderate-to-large thickness ratios while these effects are not so important for small thickness ratio blades. Thus, the linear Coriolis terms associated with precone may be neglected in the dynamic analysis of advanced turboprop blades. However, the effect of Coriolis terms in the presence of second-degree nonlinear terms is yet to be assessed. For a given thickness ratio and pretwist, an increase in rotational speed has a destabilizing effect for large precone angles. For a given thickness ratio and aspect ratio, an increase in pretwist angle has a stabilizing effect for precone angle less than 45 deg, and a destabilizing effect for precone angles greater than 45 deg. For a beam mounted off the axis of rotation, the pretwist and setting angle 148/Vol. 108, APRIL 1986 Transactions of the ASME have a significant influence on rotation-induced buckling instability. However, this influence depends markedly on the blade thickness ratio. The parametric results presented in this work are believed to be useful for future comparisons of theoretical solutions including sweep and nonlinear effects, and thereby ascertaining the importance of these complicating effects. References 1 Mostaghel, N., and Tadjbakhsh, I., "Buckling of Rotating Rods and Plates," International Journal of Mechanical Sciences, Vol. 15, No. 6, 1973, pp. 429-434. 2 Lakin, W. D., and Nachman, A., "Unstable Vibrations and Buckling of Rotating Flexible Rods," Quarterly of Applied Mathematics, Vol. 35, No. 4, Jan. 1978, pp. 479-493. 3 Lakin, W. D., and Nachman, A., "Vibration and Buckling of Rotating Flexible Rods at Transitional Parameter Values," Journal of Engineering Mathematics, Vol. 13, 1979, pp. 339-346. 4 White, W. F. Jr., Kvaternik, R. O., and Kaza, K. R. V., "Buckling of Rotating Beams," International Journal of Mechanical Sciences, Vol. 21, No. 12, 1979, pp. 739-745. 5 Wang, J. T. S., "On the Buckling of Rotating Rods," International Journal of Mechanical Sciences, Vol. 18, No. 7-8, 1976, pp. 407-411. 6 Kvaternik, R. G" White W. F., and Kaza, K. R. V., "Nonlinear FlapLag-Axial Equations of a Rotating Beam with Arbitrary Precone Angle," AIAA Paper, 78-491, 1978. 7 Peters, D. A., and Hodges, O. H., "In-Plane Vibration and Buckling of a Rotating Beam Clamped Off the Axis of Rotation," ASME Journal of Applied Mechanics, Vol. 47, No. 2, June 1980, pp. 398-402. 8 Sreenivasamurthy, S., and Ramamurthi, V., "Coriolis Effect on the Vibration of Flat Rotating Low Aspect Ratio Cantilever Plates," Journal of Strain Analysis, Vol. 16, No. 2, Apr. 1981, pp. 97-106. 9 Leissa, A., and Co, C, "Coriolis Effects on the Vibrations of Rotating Beams and Plates," Proceedings XII SECTAM Conference, Vol. II, Auburn Univ., AL, 1984, pp. 508-513. 10 Kaza, K. R. V., and Kvaternik, R. G., "Nonlinear Aeroelastic Equations for Combined Flapwise Bending, Chordwise Bending, Torsion, and Extension of Twisted Nonuniform Rotor Blades in Forward Flight," NASA TM-74059, 1977. 11 Subrahmanyam, K. B., and Kaza, K. R. V., "An Improved FiniteDifference Analysis of Uncoupled Vibrations of Tapered Cantilever Beams," NASA TM-83495, 1983. 12 Subrahmanyam, K. B., and Kaza, K. R. V., "Improved Methods of Vibration Analysis of Pretwisted Airfoil Blades," NASA TM-83735, 1984. 13 Gupta, K. K., "Free Vibration Analysis of Spinning Structural Systems," International Journal for Numerical Methods in Engineering, Vol. 5, No. 3, Jan.-Feb. 1973, pp. 395-418. 14 The International Mathematical and Statistical Library (IMSL), Houston, Texas, 1975. 15 Subrahmanyam, K. B., and Kaza, K. R. V., "An Improved FiniteDifference Vibration Analysis of Pretwisted, Tapered Beams," Proceedings of the XII Southeastern Conference, Vol. I, Auburn Univ., AL, 1984, pp. 118-126. 16 Richardson, L. F., "The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, With an Application to the Stresses in a Masonry Dam," Philosophical Transactions, Royal Society of London, Series A, Vol. 210, 1911, pp. 307-357. 17 Salvadori, M. G., "Numerical Computation of Buckling Loads by Finite Differences," Proceedings of the American Society of Civil Engineers, Vol. 75, No. 10, Dec. 1949, pp. 1441-1475 18 Subrahmanyam, K. B., Kulkarni, S. V., and Rao, J. S., "Application of the Reissner Method to Derive the Coupled Bending-Torsion Equations of Dynamic Motion of Rotating Pretwisted Cantilever Blading with Allowance for Shear Deflection, Rotary Inertia, Warping and Thermal Effects," Journal of Sound and Vibration, Vol. 84, No. 2, Sept. 22, 1982, pp. 223-240. 19 Dokumaci, E., Thomas, J., and Carnegie, W., "Matrix Displacement Analysis of Coupled Bending-Bending Vibrations of Pretwisted Blading, '' Journal of Mechanical Engineering Science, Vol. 9, No. 4, Oct. 1967, pp. 247-254. 20 Rosard, D. D., "Natural Frequencies of Twisted Cantilever Beams," ASME Journal of Applied Mechanics, Vol. 20, No. 2, June 1953, pp. 241-244. 21 Leissa, A. W., Macbain, J. C, and Kielb, R. E., "Vibration of Twisted Cantilever Plates-Summary of Previous and Current Studies," Journal of Sound and Vibration, Vol. 96, No. 2, Sept. 22, 1984, pp. 159-173. 22 Using the expressions for the derivatives given by equations (Al) to (A3) and invoking the conditions of symmetry as discussed in w"+i-w"-i= 2 ( w '«-w"-i); y«+i-««-i=2(^"-u"_ 1 ) w" +2 -M'"_ 2 =4(M'"-w"_,); w" + 3-w"-3=6K,-M'"_ 1 ); Vn + 3-Vn-3=6(Vn-Vn-l) vc_! = W[, w_2 = ve 2 , y"! = v it v_ 2 = v 2 (A10) Equations (A7) to (AlO) can be used to eliminate the fictitious stations outside the beam domain.