Effect of functionally graded materials on resonances of rotating beams

Proceedings of the IMAC-XXVIII February 1 4, 21, Jacksonville, Florida USA 21 Society for Experimental Mechanics Inc. Effect of functionally graded materials on resonances of rotating beams Arnaldo J. Mazzei, Jr. Department of Mechanical Engineering C. S. Mott Engineering and Science Center Kettering University 17 West Third Avenue Flint MI, 4854, USA Richard A. Scott Department of Mechanical Engineering University of Michigan G44 W. E. Lay Automotive Laboratory 1231 Beal Avenue Ann Arbor MI, 4819, USA NOMENCLATURE A, area of the beam cross section acmn,,,, constant numerical parameters b hub radius E, Young s modulus ( E, E, b E - distinct values of Young s modulus for different FGM cases) t Ft (), system generalized external forcing I, area moment of inertia of the beam cross section J, mass moment of inertia of the beam and hub about nn 3 K, system generalized stiffness L, beam length L, Lagrangian M, system generalized mass r, distance from hub of a generic point P on the beam (undeformed configuration) R, O beam cross section radius T, kinetic energy t, time V, potential energy V, a potential energy due to axial elongation V, s potential energy due to bending x, longitudinal coordinate (along aa 1 ) w, beam deflection in the aa 2 direction a 1, a 2, a 3 A P acceleration of a generic point P on the beam n 1, n 2, n 3 R P position of a generic point P on the beam V P velocity of a generic point P on the beam β1, λσ,, constant numerical parameters 1 η, function of time θ, hub angular position ρ, mass density ( ρ, ρ, b ρ - distinct values of mass density for different FGM cases) t

τ () t, external moment υ, Poisson s ratio ϕ, spatial function ω n, fundamental frequency of the beam Ω, angular velocity of the hub ( Ω = dddd dddd ), derivative with respect to time ( dd dddd ), derivative with respect to longitudinal coordinate ( dd dddd ABSTRACT Radially rotating beams attached to a rigid stem occur in several important engineering applications, such as helicopter and turbine blades and certain aerospace applications. In most studies the beams have been treated as homogeneous. Here, with a goal of system improvement, non-homogeneous beams made of functionally graded materials are explored. Effects on natural frequency and coupling between rigid and elastic motions are investigated. Euler-Bernoulli theory, with Young s modulus and density varying in a power law fashion, together with an axial stiffening effect, are employed. The equations of motion are derived using a variational method and an assumed mode approach. Results for the homogenous and non-homogeneous cases are treated and compared. Preliminary results show that allowing the Young s modulus and the density to vary by approximately 2.15 and 1.15 times, respectively, gives an increase of 28% in the lowest bending natural frequency of the beam, an encouraging trend. INTRODUCTION Rotating machinery form an important part of engineering and radially rotating beams constitute a major category of such systems. For instance, rotor blades, propellers and turbines fall into this category. For vibration control, it is important to identify possible system resonances and, if required and possible, change these values. Extensive work on these types of problems has been done in the aerospace literature. Comprehensive reviews can be found in the papers of Kane and Ryan [1] and Haering et al. [2]. They, and others, showed that at high speeds the rotating structure can be prone to instabilities. It is assumed here that the rotational speeds are small enough that no instabilities are encountered. There are numerous works on vibrations of radially rotating beams (uniform beams, beams including pre-twisted and tapered beams). Two classes of problems arise, namely, prescribed motions and prescribed torques. Earlier studies on the former type of problem can be found in the texts by Putter and Manor [3], Hoa [4], Hodges and Rutkwoski [5] and Hodges [6]. Putter and Manor used a finite element approach to obtain the natural frequencies and mode shapes of the beam, including shearing forces, rotary inertia and varying centrifugal forces. Hoa also utilized a finite element approach for the same objective, but effects of root radius, setting angle and tip mass were included. Hodges used asymptotic expansions to obtain an approximate value for the fundamental frequency of a uniform beam and Hodges and Rutkwoski used a finite element approach to calculate the eigenvalues and eigenvectors of the beam including different hub radii, tapered beams and beams with discontinuities. Kojima [7] investigated the transient flexural vibrations of a beam / mass system attached to a rotating rigid body. The prescribed torque problem has been studied by, for example, Yigit et al. [8], a work which the current closely follows. In that work the flexural motion of a rotating beam was investigated by using a specified torque profile to drive the rotating body (so that the rigid body motion was not known a priori). Lee et al. [9] presented experimental results confirming that centrifugal effects cannot be neglected, even at first order, when modeling these systems. Models utilizing a Timoshenko beam type approach (other than Euler-Bernoulli) are also numerous. See, for example, the work of Lin and Hsiao [1] which investigates the effect of Coriolis force on the natural frequencies of the rotating beam. More involved models including base excitation can be found in references [11] and [12].

The fundamental frequency of rotating beams with pre-twist was investigated by, for example, Hu et al. ([13]). The role of Functionally Graded Materials (FGM - see for instance reference [14]) in radially rotating beams has not been fully investigated. Librescu et al. [15], in a prescribed motion problem, studied the effects of material variation through the beam thickness on the eigenfrequencies of the system. In the present work a prescribed torque problem is investigated with material properties varying along the length of the beam. The possibility of changing the fundamental frequency of vibration of a cantilever radially rotating beam is investigated. The approach includes changing the material of the beam from a homogeneous type to a FGM while keeping the physical dimensions of the beam constant. A one-term Galerkin approach is utilized and the investigation is conducted using numerical simulation. Preliminary results, using linearized versions of the equations of motion, indicate that, in the case of a beam made of Aluminum / Silicon Carbide, the fundamental frequency can be increased by approximately 28% when compared to a pure aluminum beam. This change was obtained by the use of a material that shows a Young s modulus increase of about 2.15 times over the length of the beam. The density increase is about 1.15 times. MODELING Figure 1 shows a beam with length attached to a rigid hub of radius bb. The hub rotates radially with angular velocity Ω and is subjected to an external moment ττ(tt). A set of mutually perpendicular unit vectors aa 1, aa 2 and aa 3 is attached to the undeformed configuration of the beam. A second set of mutually perpendicular unit vectors, nn 1, nn 2 and nn 3 is assumed to be the inertial reference frame. The vector nn 3 is the axis of rotation for the hub and remains parallel to aa 3 during motion. ΘΘ is the angle between the vector nn 1 and aa 1 and defines the angular position of the hub with respect to the inertial frame. Note that Ω = dddd dddd. For rotation on the plane, the position, velocity and acceleration of point P on the deformed configuration of the beam are given, respectively, by: RR pp = (bb + rr)aa 1 + wwaa 2 VV pp = wwθθ aa 1 + [(bb + rr)θθ + ww] aa 2 AA pp = [2ww θθ +(bb + rr)θθ 2 + wwθθ ] aa 1 + [ww + (bb + rr)θθ wwθθ 2 ] aa 2 (1) (2) (3) The kinetic energy of the beam can be computed from (note rr = xx ): TT = 1 2 ρρ(xx)aa 2 + 2θθ (bb + xx) + θθ 2 [(bb + xx) 2 + ww 2 ] dddd + 1 2 JJ 2 θθ (4) The potential energy of the system comes from two parts. The first part is caused by the bending elastic strain. Using Euler-Bernoulli beam theory, this can be calculated by: VV ss = 1 2 EE(xx)II ww ( 2 xx 2 )2 dddd (5)

Rotating cantilever beam v Rigid hub P ττ(tt) Figure 1 Rotating cantilever beam The second part is due to the centrifugal force acting on the beam, which causes axial elongation (see, for example, reference [12]). It is given by: The total potential energy is obtained from: VV aa = 1 ( 2 )2 ρρ(xx)aa[θθ 2 (bb + xx) + 2θθ + wwθθ ] dddd dddd xx (6) VV = VV ss + VV aa. (7) To derive the equations of motion an assumed mode approach is adopted. The following form for the one-mode approximation is assumed for the elastic form. ww(xx, tt) = φφ(xx)ηη(tt) (8) where φφ(xx) is a spatial function that satisfies the geometric boundary conditions at the clamped end of the beam.

The Lagrangian [16] can then be written as: L = TT VV = 1 2 ρρ(xx)aa φφ2 ηη 2 + 2θθ φφηη (bb + xx) + θθ 2 [(bb + xx) 2 + φφ 2 ηη 2 ] dddd + 1 2 JJθθ 2 1 2 EE(xx)IIφφ"2 ηη 2 + φφ 2 ηη 2 ρρ(xx)aa[θθ 2 (bb + xx) + 2θθ φφηη + φφφφθθ ] dddd dddd xx (9) In a prescribed motion problem, θθ is known a priori and then one equation of motion is obtained from (9). When a torque is prescribed θθ is a generalized coordinate and two coupled non-linear equations of motion are obtained. Solutions to these equations are computationally intensive and, to obtain preliminary results, simplified versions of the equations are utilized in the following. From the Lagrangian, linearized 1 equations of motion can be obtained as follows. ρρ (xx)aaφφ 2 dddd ηη + ρρ ρρ (xx)aaaa(bb + xx)dddd θθ + EE(xx)IIφφ" 2 dddd ηη = (xx)aaaa(bb + xx)dddd ηη + JJ + ρρ (xx)aa(bb + xx) 2 dddd θθ = ττ(tt) (1) (11) Note that here, as done in reference [8], the rotation of the hub is not assumed to be prescribed. Next equation (11) is solved for θθ and the result is substituted into equation (1). This leads to: MMηη + KKKK = FF(tt) (12) where MM = ρρ(xx)aaφφ 2 dddd ρρ(xx)aaaa(bb + xx)dddd 2 JJ + ρρ(xx)aa(bb + xx) 2 dddd, KK = EE(xx)IIφφ" 2 dddd, FF(tt) = ρρ(xx)aaaa(bb + xx)dddd ττ(tt) JJ + ρρ(xx)aa(bb + xx) 2 dddd It follows from equation (12) that the fundamental frequency is: ωω nn = KK MM (13) 1 2 Following Yigit et al [8], terms involving θθ are neglected. With this simplification equation (11) can be solved for θθ and substituted into equation (1).

In the preceding, no assumption was made concerning the material type in the derivation of the equations of motion. In the following both homogeneous and non-homogeneous material types are considered. The nonhomogeneous materials utilized here are FGMs. Two different models are employed, one involving power law variations (aluminum / silicon carbide) and another based on a volume fraction approach (aluminum / steel). Results for the fundamental frequency of the FGM beams are compared to those for a pure aluminum beam. FGM I The first FGM model is based on the one described by Chiu and Erdogan [17]. The material is assumed to be isotropic and non-homogeneous with properties given by: x x ( ) = ( + 1) m, ρ( x) = ρ( a + 1) n L Ex E a L (14) where a, m and n are arbitrary real constants with a > 1. E and ρ are the Young s modulus and mass density at x =. In the sequel the FGM utilized is a composite made from aluminum and silicon carbide. The properties of the material are given in Table 1 and are taken from reference [17]. Aluminum / Silicon Carbide E ( GPa ) 15.197 3 ρ ( kg / m ) 271. a 1.14568 m 1. n.17611 υ.33 Table 1 Material properties for Al / SiC FGM Note that in this model Poisson s ratio is taken to be a constant. FGM II The second FGM model is developed by assuming that its composition is derived from of a mixture of two materials, with the material variation given by a power-law gradient (see, for instance, reference [18]). The effective material properties of the beam are given by: x λ x Ex ( ) = Eb + ( Et Eb)( ) L, ρ( x) = ρb + ( ρt ρb)( ) L λ (15) where λλ is a non-negative constant describing the volume fraction, which can be determined experimentally ([18]). The subscripts bb and tt refer to the value of the parameter at xx = and xx =, respectively. These values are the ones for the pure materials involved in the composition of the FGM and are obtained from tables or manufacturer s specifications. Also, Poisson s ratio varies slightly but it is usually taken to be constant.

NUMERICAL RESULTS For the numerical simulations, the beam is taken to have a circular cross section with a radius given by RR = 1127 mm. The length of the beam is =.896 mm and the radius of the hub is bb =.5 mm. Also, initially, the mass moments of inertia of the rigid and flexible parts are taken to be related in the following manner: JJ = (.5)( ρρ(xx)aa(bb + xx) 2 dddd) (this corresponds to an inertia ratio of.5). The spatial function chosen to be used in equation (12) is the first mode shape of a non-rotating cantilever beam (see, for example, [19]): CC(cosh(ββ 1 xx) cos(ββ 1 xx) σσ 1 [sinh(ββ 1 xx) sin(ββ 1 xx)]) where CC is a constant, ββ 1 = 1.875147 and σσ 1 = (sinh(ββ 1 ) sin(ββ 1 )) (cosh(ββ 1 ) + cos(ββ 1 ) ). (16) Homogeneous material beam For the homogeneous material, aluminum was chosen. The properties are: 3 ρ = 271 Kg/m. 2 E =.71e11 N/m and In this case the fundamental frequency of the rotating beam, given by equation (13), is ωω nn = 36.98 HHHH. Note that, if one assumes prescribed motion (θθ = constant) the frequency would be ωω nn = 22.5 HHHH. This is the fundamental frequency of a cantilever beam with the dimensions given in this example. The higher frequency obtained via equation (13) shows the stiffening effect that the rotation produces. Figure 2 Fundamental frequency variation of homogeneous beam as a function of the inertia ratio Figure 2 shows the variation of the fundamental frequency, for this case, as a function of the inertia ratio. It is seen that as the inertia ratio increases the fundamental frequency decreases substantially.

FGM I beam For a beam made of Aluminum / Silicon Carbide, the fundamental frequency is ωω nn = 47.4HHHH. When compared to the pure aluminum beam, this result shows a frequency increase of about 28%. The variation of this frequency as a function of the inertia ratio is shown in Figure 3. As in the previous case, inertia ratio increase causes a decrease in the frequency. Figure 3 Fundamental frequency variation of FGM I beam as a function of inertia ratio FGM II beam Next the Aluminum / Steel case is considered. The beam is assumed to be aluminum rich at xx = and steel rich at xx =. Poisson s ratio is taken to be constant: υυ =.33 and λλ = 1 is used. The fundamental frequency for this case is ωω nn = 27.77 HHHH, which shows a frequency decrease of about 25% when compared to the pure aluminum beam. The frequency variation as a function of the inertia ratio is shown in Figure 4. Equation (12) also allows for a comparison of the generalized masses for each case. By computing the masses, the following observations can be made. The aluminum / steel beam is 2.5 times heavier than the aluminum one. On the other hand the aluminum / silicon carbide beam is only 11% heavier than its aluminum counterpart. Clearly these results show the advantages of using FGM I, when the objective sought is to increase the fundamental frequency of the system while minimizing overall weight gain.

Figure 4 Fundamental frequency variation of FGM II beam as a function of inertia ratio CONCLUSIONS The use of FGM for radially rotating cantilever beams can change the fundamental frequency of vibration significantly. For a beam made of aluminum / steel, the frequency decreases by approximately 25% when compared to an aluminum one. This is followed by an increase of 2.5 times in overall weight. The Young s modulus increase is about 3 times over the length of the beam and the density increase is approximately 2.9 times. For this case the use of such FGM may not be practical, or desirable, for the application at hand. For a beam made of aluminum / silicon carbide, the frequency can be increased by approximately 28% when compared to an aluminum beam. This is achieved by the use of a material where the Young s modulus increases about 2.15 times over the length of the beam, whereas density increases about 1.15 times. These changes are considered reasonable. Moreover, these results are achieved with only an 11% increase in weight when compared to an aluminum beam. Here these are considered encouraging results warranting further investigation. REFERENCES [1] Kane, T. R.; Ryan, R. R. and Banerjee, A. K., Dynamics of a cantilever beam attached to a moving base, Journal of Guidance, Control and Dynamics, vol. 1, pp. 139-151, 1987. [2] Haering, W. J.; Ryan, R. R. and Scott, R. A., New formulation for flexible beams undergoing large overall plane motion, Journal of Guidance, Control and Dynamics, vol. 17, pp. 76-83, 1994. [3] Putter, S. and Manor, H., Natural frequencies of radial rotating beams, Journal of Sound and Vibration, vol. 56, pp. 175-185, 1978. [4] Hoa, S. V., Vibration of a rotating beam with tip mass, Journal of Sound and Vibration, vol. 67, pp. 369-381, 1979. [5] Hodges, D. H. and Rutkowski, M. J., Free-vibration analysis of rotating beams by a variable-order finiteelement method, AIAA Journal, vol. 19, pp. 1459-1466, 1981. [6] Hodges, D. H., An approximate formula for the fundamental frequency of a uniform rotating beam clamped off the axis of rotation, Journal of Sound and Vibration, vol. 77, pp. 11-18, 1981.

[7] Kojima, H., Transient vibrations of a beam / mass system fixed to a rotating body, Journal of sound and vibration, vol. 17, pp. 149-154, 1986. [8] Yigit, A.; Scott, R. A. and Ulsoy, A. G., Flexural Motion of a Radially Rotating Beam Attached to a Rigid Body, Journal of Sound and Vibration, vol. 121, pp. 21-21, 1988. [9] Lee, C. L.; Al-Salem, M. F. and Woehrle, T. G., Natural frequency measurements for rotating spanwise uniform cantilever beams, Journal of Sound and Vibration, vol. 24, pp. 957-961, 21. [1] Lin, S. C. and Hsiao, K. M., Vibration analysis of a rotating Timoshenko beam, Journal of Sound and Vibration, vol. 24, pp. 33-322, 21. [11] Hyun, S. H. and Yoo, H. H., Dynamic modelling and stability analysis of axially oscillating cantilever beams, Journal of Sound and Vibration, vol. 228, pp. 543-558, 1999. [12] Tan, T. H.; Lee, H. P. and Leng, G. S. B., Dynamic stability of radially rotating beam subjected to base excitation, Computer methods in applied mechanics and engineering, vol. 146, pp. 265-279, 1997. [13] Hu, X. X.; Sakiyama, T.; Matsuda, H. and Morita, C., Fundamental vibration of rotating cantilever blades with pre-twist, Journal of Sound and Vibration, vol. 271, pp. 47-66, 24. [14] Miyamoto, Y.; Kaysser, W. A.; Rabin, B. H.; Kawasaki, A. and Ford, R. G., Functionally graded materials: design, processing and applications, 1st ed: Springer, 1999. [15] Librescu, L.; Oh, S.-Y.; Song, O. and Kang, H.-S., Dynamics of advanced rotating blades made of functionally graded materials and operating in high-temperature field, Journal of Engineering Mathematics, vol. 61, pp. 1-16, 28. [16] Greenwood, D. T., Principles of Dynamics, Second ed. Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1988. [17] Chiu, T.-C. and Erdogan, F., One-dimensional wave propagation in a functionally graded elastic medium, Journal of Sound and Vibration, vol. 222, pp. 453-487, 1999. [18] Li, X.-F., A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams, Journal of Sound and Vibration, vol. 318, pp. 121-1229, 28. [19] Meirovitch, L., Fundamentals of Vibrations: McGraw-Hill, 21.