Optimal location of centre of gravity for swashplateless helicopter UAV and MAV

Free vibration analysis of a rotating Timoshenko beam by differential transform method Metin O. Kaya Faculty of Aeronautics and Astronautics, Istanbul Technical University, Istanbul, Turkey Abstract Purpose – To perform the flapwise bending vibration analysis of a rotating cantilever Timoshenko beam. Design/methodology/approach – Kinetic and potential energy expressions are derived step by step. Hamiltonian approach is used to obtain the governing equations of motion. Differential transform method (DTM) is applied to solve these equations. Findings – It is observed that the rIV2 u term which is ignored by many researchers and which becomes more important as the rotational speed parameter increases must be included in the formulation. Originality/value – Kinetic and potential energy expressions for rotating Timoshenko beams are derived clearly step by step. It is the first time, for the best of author’s knowledge, that DTM has been applied to the blade type rotating Timoshenko beams. Keywords Rotation measurement, Transforms, Cantilevers Paper type Research paper by Zhou (1986) in his study about electrical circuits, is used. Recently, Banerjee (2001) has developed the Dynamic Stiffness Method for a rotating cantilever Timoshenko beam that is based on Frobenius series expansion and claims its superiority of finding more correct results. However, application of this method, as he pointed out, is not so easy. On the other hand, the advantage of the DTM is its simplicity and high accuracy. 1. Introduction There has been a growing interest in the analysis of the free vibration characteristics of elastic structures that rotate with a constant angular velocity. Numerous structural configurations such as turbine, compressor and helicopter blades, spinning spacecraft and satellite booms fall into this category. Investigation of the natural frequency variation of rotating beams originated from the work of Southwell and Gough (1921). They suggested a simple equation (known as the Southwell equation), which is based on the Rayleigh energy theorem to estimate the natural frequencies of rotating cantilever beams. Liebers (1930) and Theodorsen (1950) extended Southwell’s work. Earlier studies mainly focused on Euler Bernoulli beams. However, due to the inclusion of shear deformation and rotary inertia effects, Timoshenko beam theory is more accurate than Euler Bernoulli beam theory. Therefore, considerable research has been carried out on the free vibrations of rotating Timoshenko beams, recently (Stafford and Giurgiutiu, 1975; Yokoyama, 1988; Lee and Kuo, 1993; Du et al., 1994; Nagaraj, 1996; Bazoune et al., 1999; Banerjee, 2001). Different types of solution procedures, i.e. the finite element method, the Frobenius method of series, the Galerkin method, the Myklestad procedure, the finite differences approach, the perturbation technique, Bessel functions, etc. may be found in the literature. In this study, the differential transform method (DTM), which is a semi analytical-numerical technique that depends on Taylor series expansion and which was introduced 2. Description of the problem In Figure 1, a cantilever beam of length L, which is rigidly mounted on the periphery of a rigid hub of radius R, is shown. The hub rotates about its axis at a constant angular speed V. The origin is taken to be at the left-hand end of the beam. The x-axis coincides with the neutral axis of the beam in the undeflected position, the z-axis is parallel to the axis of rotation (but not coincidental) and the y-axis lies in the plane of rotation. The beam considered here is doubly symmetric such that the mass axis, the centroidal axis and the elastic axis are coincident. The following assumptions are made in this study: . The out-of-plane displacement of the beam is small. . The cross sections that are initially perpendicular to the neutral axis of the beam remain plane, but no longer perpendicular to the neutral axis during bending. . The beam material is homogeneous and isotropic. . Coriolis effects are not included. The current issue and full text archive of this journal is available at www.emeraldinsight.com/1748-8842.htm 3. Formulation Most of the investigators begin their studies by introducing the equations of motion directly. However, in this paper the potential and the kinetic energy expressions are derived step by step and then, equations of motion are obtained using the Hamiltonian approach. Aircraft Engineering and Aerospace Technology: An International Journal 78/3 (2006) 194– 203 q Emerald Group Publishing Limited [ISSN 1748-8842] [DOI 10.1108/17488840610663657] 194 Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 Figure 1 Configuration of a rotating cantilever Timoshenko beam 3.1 Strain displacement relations Several different classical definitions of strain may be found in literature, depending on the mathematical formulation, reference states (based on deformed or undeformed positions), and coordinate systems used. The flapwise motion of a Timoshenko beam is given in Figure 2(a) and (b). In Figure 2(a), longitudinal view of displacement of a point is introduced and in Figure 2(b), cross-sectional view of displacement of the same point is introduced. In these graphics, the point is represented by P0 before deformation and by P after deformation. The vector position of the point is defined as: ~r ¼ x~i þ y~j þ zk~ d~r0 ¼ dx~i þ dh ~j þ djk~ d~r1 ¼ ½ð1 þ u00 2 ju0 Þdx 2 u dj
~i þ dh ~j þ ðw0 dx þ djÞk~ ð5Þ where dx, dh and dj are the increments along the deformed elastic axis and two cross-sectional axes, respectively. The classical strain tensor 1ij in terms of ~r1 and ~r0 may be expressed as follows: 8 9 dx > > > > < = d~r1 · d~r1 2 d~r0 · d~r0 ¼ 2b dx dh djc ½1ij
dh > > > : dj > ; ð1Þ Keeping in mind that the subscripts ( )0 and ( )1 represent the positions of the point before and after deformation, respectively, these positions can be given as follows. . Before deformation: . ð4Þ ð6Þ Substituting equations (4) and (5) into equation (6), components of the strain tensor 1ij are obtained as: x0 ¼ R þ x ð2aÞ 21xx ¼ ð1 þ u00 2 ju 0 Þ2 þ ðw0 Þ2 2 1 ð7aÞ y0 ¼ h ð2bÞ gxh ¼ 0 ð7bÞ z0 ¼ j ð2cÞ gxj ¼ w0 2 ð1 þ u00 2 ju 0 Þu ð7cÞ x1 ¼ R þ x þ u0 2 ju ð3aÞ y1 ¼ h ð3bÞ z1 ¼ w þ j ð3cÞ After deformation: In order to obtain simple expressions for the strain components, higher order terms should be neglected, so an order of magnitude analysis is necessary. The ordering scheme is taken from Hodges and Dowell (1974) and introduced in Table I. Simplified strain components are obtained as follows by ignoring the terms which are higher than 12. ðw0 Þ2 ð8aÞ 1xx ¼ u00 2 ju 0 þ 2 Here, x is the spanwise distance of the point from the hub edge, uo is the axial displacement due to the centrifugal force, h is the transverse distance of the point from the axis of rotation, j is the vertical distance of the point from the middle plane, w is the bending displacement and u is the rotation due to bending. Knowing that ~r0 and ~r1 are the vector positions of the point on the undeformed and deformed blade, respectively, the position vector differentials can be given by: gxh ¼ 0 ð8bÞ gxj ¼ w0 2 u ð8cÞ where gxj is the loss of slope that is equal to the shear strain. 195 Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 Figure 2 Table I Ordering scheme u 2 R 5 Oð1 Þ v R 5 Oð1Þ w R 5 Oð1Þ Ub ¼ x R ¼ Oð1Þ h R ¼ Oð1Þ j R ¼ Oð1Þ g ¼ w 0 2 w ¼ Oð1 2 Þ C 5 Oð1Þ w 5 Oð1Þ 1 2 Z 0 L EAðu00 Þ2 dx þ 1 2 Z L EIðu0 Þ2 dx þ 0 1 2 Z 0 L EAu00 ðw0 Þ2 dx ð11Þ where: I¼ Z j2 dA A 3.2 Expression for the potential energy The strain energy due to bending, Ub, is given by: Ub ¼ ððð E12xx dV 2 is the second moment of area of the beam cross section about the y-axis, EI is the bending rigidity and EA is the axial rigidity of the beam cross section. The uniform strain 1o and the associated axial displacement uo due to the centrifugal force, T(x) is given by: ð9Þ V u00 ðxÞ ¼ 10 ðxÞ ¼ Substituting equation (8a) into equation (9) leads to: E Ub ¼ 2 Z Z A 0 L
2 1 u00 2 ju 0 þ ðw0 Þ2 dx dA 2 T ðxÞ EA ð12Þ where the centrifugal force that varies along the spanwise direction of the beam is: ð10Þ T ðxÞ ¼ Z L rAV2 ðR þ xÞdx ð13Þ x where A is the cross-sectional area and E is the Young’s modulus. Taking integration over the blade cross section gives: Substituting equation (13) into equation (12) and noting that the 196 Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 1 2 Z L 0 T 2 ðxÞ dx EA I¼ Z Z A term is constant and will be denoted as C1, we get the final form of the bending strain energy as follows: Z Z 1 L 1 L Ub ¼ EIðu 0 Þ2 dx þ T ðw0 Þ2 dx þ C 1 ð14Þ 2 0 2 0 ð15Þ where G is the shear modulus. Substituting equations (8b) and (8c) into equation (15), we get the final form of the shear strain energy as follows: Z Z Z L 1 1 L kAGðw0 2 uÞ2 dx Gðw0 2 uÞ2 dA dx ¼ Us ¼ 2 2 0 0 ðV 2x þ V 2y þ V 2z Þr dA dx where dw ¼ du ¼ 0 at t1 and t2. After integration, the equations of motion are obtained as follows: ð16Þ where k is the shear correction factor that depends on the shape of the cross section (for circular and rectangular cross sections, the values of k are 2/3 and 3/4, respectively) and that is used to take into account the variation of shear stress across the thickness and kAG is the shear rigidity. Combining all the strain energy components, the total strain energy U of the beam is found to be: Z L  1 U ¼ Ub þ Us ¼ {EIðu 0 Þ2 þ kAGðw0 2 uÞ2 þ T ðw0 Þ2 }dx 2 0 2rAw € þ ðTw0 Þ0 þ ½kAGðw0 2 uÞ
0 ¼ 0 ð24Þ 2rI u€ þ rIV2 u þ ðEI u 0 Þ0 þ kAGðw0 2 uÞ ¼ 0 ð25Þ Equations (24) and (25) define completely the free vibration of a uniform rotating Timoshenko beam. Here w is the out-ofplane displacement and u is the rotation due to bending. It must be noted that although the term rIV2 u can be important when the constant rotational speed, V is high, it is not taken into account by some authors. The physical description of this term is that as a result of the bending deformation, the radii of the elements that are symmetrically placed with respect to the mid-plane of the beam cross section are different so these elements have different centrifugal forces although the net centrifugal force is independent of the section rotation. Thus, a moment that has the value of rIV2 u appears. Primes in equations (24) and (25) mean differentiation with respect to the spanwise position, x and dots mean differentiation with respect to time, t; r is the material density and rA is the mass per unit length. Here, rA and rI are the inertia terms. The boundary conditions at x ¼ 0 and x ¼ L for equations (24) and (25) are given by: ð17Þ The first term in equation (17) represents the flexural strain energy, the second term the shear strain energy and the last term the strain energy due to the centrifugal force. 3.3 Expression for the kinetic energy The velocity of the representative point P due to rotation of the beam is given by: ð18Þ ~ can be expressed in terms of deformed The total velocity V positions as follows: L u 0 duj0 ¼ 0 ð26Þ L ½Tw0 þ kAGðw0 2 uÞ
dwj0 ¼ 0 ð19Þ Substituting the derivatives of equations (3a)-(3c) with respect to time, t, into equation (19), the velocity components are obtained as follows: V x ¼ 2ju_ 2 hV ð20aÞ V y ¼ ðR þ x þ u0 2 juÞV ð20bÞ Vz ¼ w _ ð20cÞ ð21Þ t1 A ~ ¼ ð_x1 2 Vy1 Þ~i þ ð_y1 þ Vx1 Þ~j þ z_ 1 k~ V 0 3.4 Derivation of the governing differential equations of motion The governing differential equations of motion and the boundary conditions can be derived by means of the Hamiltonian approach, which can be stated in the following form: Z t2 ð23Þ ðdI 2 dU Þdt ¼ 0 A ~ ¼ d~r þ Vk~ £ ~r1 V dt L Substituting the velocity components introduced in equations (20a)-(20c) into equation (21), the final form of the kinetic energy expression is Z obtained. 1 L I¼ ð22Þ ½rAw _ 2 þ rI u_2 þ rIV2 u2
dx 2 0 The strain energy due to shear, U s ; is given by:   Z Z L  ððð G g 2 þ g 2  xh xj 1 G gx2h þ gx2j dA dx Us ¼ dV ¼ 2 2 0 V 1 2 ð27Þ 3.5 Free vibration analysis A sinusoidal variation of nðx; tÞ and uðx; tÞ with circular frequency v can be given by: wðx; tÞ ¼ W ðxÞeivt uðx; tÞ ¼ u
ðxÞe ivt ð28Þ ð29Þ Substituting equations (28) and (29) into equations (24) and (25), equations of motion are expressed as follows: The kinetic energy is given by: 197 Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 rI v2 u
þ rIV2 u
þ EI u
00 þ kAGðW 0 2 u
Þ ¼ 0 ð30Þ mv2 W þ ðTw0 Þ0 þ kAGðW 00 2 u
0 Þ ¼ 0 ð31Þ Combining equations (36) and (37), one obtains:   1 X ðx 2 x0 Þk dk f ðxÞ f ðxÞ ¼ k! dxk x¼x0 k¼0 Considering equation (38), it is noticed that the concept of differential transform is derived from Taylor series expansion. However, the method does not evaluate the derivatives symbolically. In actual applications, the function f(x) is expressed by a finite series and equation (38) can be written as follows:   n X ðx 2 x0 Þk dk f ðxÞ f ðxÞ ¼ ð39Þ dxk x¼x0 k! k¼0 3.6 Nondimensionalizing of the problem The dimensionless parameters that are used to simplify the equations and to make comparisons with the studies in literature can be given as follows: j¼ x ; L d¼ EI ; s ¼ kAGL2 2 R ; L W ðj Þ ¼ W ; L rAL4 V2 h ¼ ; EI 2 r2 ¼ I ; AL2 ð32Þ rAL4 v2 m ¼ EI which implies that: 2 f ðxÞ ¼ Here d is the hub radius parameter, r is the rotary inertia parameter, s is the shear deformation parameter, h is the rotational speed parameter and m is the frequency parameter. Using the first two dimensionless parameters, dimensionless expression for the centrifugal force can be written as follows:   ð1 2 j2 Þ T ðjÞ ¼ rAV2 L2 dð1 2 jÞ þ ð33Þ 2   1 X ðx 2 x0 Þk dk f ðxÞ dxk x¼x0 k! k¼nþ1 is negligibly small. In this study, the convergence of the natural frequencies determines the value of n. Theorems that are frequently used in the transformation of the equations of motion and the boundary conditions are introduced in Tables II and III, respectively. 5. Solution with DTM Substituting equations (32) and (33) into equations (30) and (31), the dimensionless form of the equations of motion are obtained   v2 1 u 00 þ h2 r 2 1 þ 2 u þ 2 ðw0 2 uÞ ¼ 0 ð34Þ s V
 ð38Þ In the solution stage, the DTM is applied to the equations (34) and (35) by using the rules given in Table II and the following expressions are obtained.   1 1 d þ þ 2 2 ðk þ 2Þðk þ 1ÞW ðk þ 2Þ 2 dðk þ 1Þ2 W ðk þ 1Þ 2 s h ð40Þ  2  v kðk þ 1Þ W ðkÞ 2 ðk þ 1Þuðk þ 1Þ ¼ 0 2 þ 2 V2     v2 1 ðk þ 2Þðk þ 1Þuðk þ 2Þ þ h2 r 2 1 þ 2 2 2 uðkÞ s V ð41Þ 1 þ 2 ðk þ 1ÞW ðk þ 1Þ ¼ 0 s  0  2  ð1 2 j2 Þ 0 v 1 0 00 w þ dð1 2 jÞ þ 2 w þ s2 h2 ðw 2 u Þ ¼ 0 ð35Þ 2 V 4. Differential transform method The DTM is one of the useful techniques to solve the ordinary differential equations with fast convergence rate and small calculation error. One advantage of this method over the integral transformation methods is its ability to handle nonlinear differential equations, either initial value problems or boundary value problems. It was introduced by Zhou (1986) in his study about electrical circuit. Recently, it has gained much attention by researchers (Arikoglu and Ozkol, 2004, 2005; Bert and Zeng, 2004; Chen and Ju, 2004; Ozdemir and Kaya, 2005). Let f(x) be analytic in a domain D and let x ¼ x0 represent any point in D. The function f(x) is then represented by one power series whose center is located at x0. The differential transform of the function f(x) is defined as follows:   1 dk f ðxÞ F½k
¼ ð36Þ k! dxk x¼x0 Applying the DTM to equations (26) and (27), the boundary conditions are given by: at j ¼ 0 ) uð0Þ ¼ W ð0Þ ¼ 0 1 X at j ¼ 1 ) kuðkÞ ¼ 0 ð42Þ ð43Þ k¼0 1 X ½kW ðkÞ 2 uðkÞ
¼ 0 ð44Þ k¼0 Table II Basic theorems of DTM for equations of motion where f(x) is the original function and F[k ] is the transformed function. The inverse transformation is defined as: 1 X f ðxÞ ¼ ðx 2 x0 Þk F½k
ð37Þ Original function Transformed functions f ðxÞ ¼ gðxÞ ^ hðxÞ f ðxÞ ¼ lgðxÞ f ðxÞ ¼ gðxÞhðxÞ f ðxÞ ¼ d dgxðnxÞ F ½k
¼ G½k
^ H½k
F ½k
¼ l G½k
P F ½k
¼ kl¼0 G½k 2 l
H½l
f ðxÞ ¼ x n F ½k
¼ d ðk 2 nÞ ¼ n k¼0 198 F ½k
¼ ðkþk!nÞ! G½k þ n
( 0 if k – n 1 if k ¼ n Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 Table III DTM theorems for boundary conditions x50 x51 Original BC Transformed BC Original BC f ð0Þ ¼ 0 df dx ð0Þ ¼ 0 F ½0
¼ 0 F ½1
¼ 0 f ð1Þ ¼ 0 df dx ð1Þ ¼ 0 d2 f dx 2 d3 f dx 3 ð0Þ ¼ 0 F ½2
¼ 0 ð0Þ ¼ 0 F ½3
¼ 0 d2 f dx 2 d3 f dx 3 In equations (40)-(44), W(k) and u(k) are the differential transforms of w(j) and u(j), respectively. Using equations (40) and (41), W(k) and u(k) values for k ¼ 2; 3; 4. . . can now be evaluated in terms of v2 ; c1 and c2. These values were achieved by using the Mathematica computer package. The results are introduced below for the values d ¼ 0; r ¼ 0:04; s ¼ 2r and h ¼ 8: uð2Þ ¼ 278:125c2 uð4Þ ¼ 2c2 ð174:47 2 0:045v2 Þ W ð2Þ ¼ 0:415007c1 W ð3Þ ¼ 2c2 ð21:558 þ 0:000885v2 Þ W ð4Þ ¼ c1 ð0:95 2 0:00024v2 Þ where c1 and c2 are constants. Substituting all W(i) and u(i) terms into boundary condition expressions, i.e. equations (42) and (43), the following equation is obtained. ðnÞ ðnÞ ð45Þ ðnÞ ðnÞ Aj1 ðvÞ; Aj2 ðvÞ where are polynomials of v corresponding to nth term. When equation (45) is written in matrix form, we get: " n #( ) ( ) A11 ðvÞ An12 ðvÞ c1 0 ¼ ð46Þ n n A21 ðvÞ A22 ðvÞ c2 0 The eigenvalue equation is obtained from equation (46) as follows: An11 ðvÞ An12 ðvÞ An21 ðvÞ An22 ðvÞ ¼0 ð47Þ ðnÞ Solving equation (47), we get v ¼ vj where j ¼ 1; 2; 3; . . .n: ðnÞ Here, vj is the jth estimated eigenvalue corresponding to n. The value of n is obtained by the following equation: ðnÞ ðn21Þ vj 2 vj #1 ð1Þ ¼ 0 natural frequencies and to plot the related graphics. In order to validate the computed results, an illustrative example taken from Banerjee (2001) is solved and the results are compared with the ones in the same reference paper. The results of Table IV illustrate the effect of the rotational speed parameter, h, on the fundamental natural frequency of the Timoshenko beam. Present study and Banerjee (2001) show good agreement up to the fourth digit. However, the results of Lee and Kuo (1993) differ from the results of this study and the difference increases with the increasing rotational speed parameter, h, due to the lack of rI V2 u term in the equations of Lee and Kuo (1993). Additionally, variation of the fundamental natural frequency of a rotating Timoshenko beam with cantilever end condition with respect to various values of Sð¼ 1=rÞ and h is introduced in Table V. As it is seen in this table, the agreement between the results of the present study and Banerjee (2001) is excellent. In the case of a nonrotating beam ðh ¼ 0Þ; a good agreement with Lee and Kuo (1993) is observed, but in the case of rotation ðh ¼ 5Þ; the results do not match due to the reason mention before. Moreover, in Table VI, variation of the fundamental natural frequency of rotating Timoshenko beam is given for various values of the Timoshenko effect parameter, r, and the rotational speed parameter, h. As expected, the values of the natural frequencies increase with the increasing rotational speed parameter due to the stiffening effect of the centrifugal force and the natural frequencies decrease as the Timoshenko effect is increased because the shear deformation has a decreasing effect on the natural frequencies. The results of the present study and Banerjee (2001) agree completely. However, due to missing term mentioned before, the difference between Du et al. (1994) increases with the increasing rotation speed parameter h. In Figure 3, variation of the first five natural frequencies of a rotating beam with respect to the Timoshenko effect parameter, r, is given. For all modes, the natural frequencies decrease with increasing r because shear deformation has a decreasing effect on the natural frequencies, but the Timoshenko effect is more dominant on higher modes as expected so the higher mode frequencies of the rotating beam decrease remarkably on account of the rotary inertia parameter while the lower modes are nearly unaffected. Furthermore, nondimensional frequency variation with respect to the Timoshenko effect, r, is given in Figure 4. Nondimensionalization is made with respect to natural frequency parameter of Bernoulli-Euler beam, m0. As discussed above, decreasing of natural frequency for higher modes is evident. uð3Þ ¼ 2c1 ð24:40967 þ 0:000267v2 Þ Aj1 ðvÞc1 þ Aj2 ðvÞc2 ¼ 0; j ¼ 1; 2; 3; . . .n ð1Þ ¼ 0 Transformed BC P1 k¼0 F ½k
¼ 0 P1 k¼0 kF ½k
¼ 0 P1 k¼0 k ðk 2 1ÞF ½k
¼ 0 P1 k¼0 k ðk 2 1Þðk 2 2ÞF ½k
¼ 0 ð48Þ where 1 is the tolerance parameter. If equation (48) is satisfied, then we have jth eigenvalue ðnÞ ðnÞ vj : In general, vj are conjugated complex values, and can ðnÞ be written as vj ¼ aj þ ibj . Neglecting the small imaginary part bj, we have the jth natural frequency aj. 6. Results and discussion The computer package Mathematica is used to code the expressions obtained by using DTM, to calculate the 199 Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 Table IV Variation of the fundamental natural frequencies of a rotating Timoshenko beam with cantilever end condition for various values of the rotational speed parameter h (d ¼ 0; r ¼ 1=30; E=kG ¼ 3:059) h Present study 0 1 2 3 4 5 10 3.4798 3.6445 4.0971 4.7516 5.5314 6.3858 11.0643 Fundamental natural frequency m1 Banerjee (2001) Lee and Kuo (1993) 3.4798 3.6445 4.0971 4.7516 5.5314 6.3858 – 3.4798 3.6452 4.0994 4.7558 5.5375 6.3934 – Table V Variation of the fundamental natural frequency of a rotating Timoshenko beam with cantilever end condition with respect to the inverse of the Timoshenko effect parameter, Sð¼ 1=rÞ; and the rotational speed parameter, h (d ¼ 0; E=kG ¼ 3:059) Fundamental natural frequency parameter (m) h50 S 20 30 40 50 80 100 150 200 300 500 1,000 h55 Present study Banerjee Lee Kuo Present study Banerjee Lee Kuo 3.4364 3.4798 3.4955 3.5028 3.5108 3.5127 3.5145 3.5152 3.5156 3.5159 3.5160 3.4364 3.4798 3.4955 3.5028 3.5108 3.5127 3.5145 3.5152 3.5156 – – 3.4364 3.5798 3.4954 3.5028 3.5108 3.5126 3.5144 3.5152 3.5155 – – 6.3126 6.3858 6.4131 6.4260 6.4403 6.4436 6.4469 6.4481 6.4489 6.4493 6.4495 6.3126 6.3858 6.4131 6.4260 6.4403 6.4436 6.4469 6.4481 6.4489 – – 6.3241 6.3934 6.4179 6.4294 6.4418 6.4446 6.4476 6.4485 6.4493 – – Table VI Variaton of the fundamental natural frequency of a rotating Timoshenko beam with cantilever end condition with respect to the Timoshenko effect r and the rotational speed parameter h (d ¼ 0; k ¼ 2=3; E=G ¼ 8=3) Fundamental natural frequency parameter (m) h54 h58 h 5 12 Present study Banerjee Du et al. Present study Banerjee Du et al. Present study Banerjee Du et al. Present study Banerjee Du et al. h50 r 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 3.5160 3.5119 3.4998 3.4799 3.4527 3.4187 3.3787 3.3335 3.2837 3.2302 3.1738 2.8692 3.5160 3.5119 3.4998 3.4799 3.4527 3.4187 3.3787 3.3335 3.2837 3.2302 3.1738 – 3.516 3.512 3.500 3.480 3.453 3.419 3.379 3.333 3.248 3.230 3.174 – 5.5850 5.5791 5.5616 5.5332 5.4951 5.4487 5.3954 5.3370 5.2749 5.2104 5.1448 4.8262 5.5850 5.5791 5.5616 5.5332 5.4951 5.4487 5.3954 5.3370 5.2749 5.2104 5.1448 – 5.585 5.580 5.564 5.539 5.505 5.463 5.415 5.363 5.307 5.249 5.191 – 200 9.2568 9.2447 9.2096 9.1549 9.0854 9.0060 8.9208 8.8333 8.7456 8.6588 8.5735 8.1406 9.2568 9.2447 9.2096 9.1549 9.0854 9.0060 8.9208 8.8333 8.7456 8.6588 8.5735 – 9.257 9.246 9.215 9.167 9.106 9.036 8.963 8.889 8.815 8.744 8.677 – 13.170 13.148 13.087 12.998 12.893 12.783 12.672 12.564 12.458 12.353 12.247 11.398 13.170 13.148 13.087 12.998 12.893 12.783 12.672 12.564 12.458 12.353 12.247 – 13.170 13.150 13.087 13.015 12.923 12.827 12.734 12.646 12.564 12.487 12.415 – Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 Figure 3 The first five natural frequencies of a rotating Timoshenko beam Figure 4 The first five natural frequencies of a rotating Timoshenko beam ðh ¼ 4Þ (m0 corresponds to the natural frequencies of Bernoulli-Euler beam) 7. Conclusions The variation of first five natural frequencies with respect to rotational speed, h, is given in Figure 5. As expected, due to the centrifugal stiffening effect, natural frequencies increase with increasing rotational speed, h. The natural frequency increase due to the centrifugal stiffening is much evident for lower modes than the higher ones. Moreover, variation of the natural frequencies with respect to the hub radius parameter, d, is introduced in Figure 6. The hub radius has an increasing effect on the value of the centrifugal force so the hub radius parameter has an increasing effect on the natural frequencies. A new and semi-analytical technique called the DTM is applied to the problem of a rotating Timoshenko beam in a simple and accurate way, the natural frequencies are calculated and related graphics are plotted. The effects of the hub radius, rotary inertia, shear deformation and rotational speed are investigated. The numerical results indicate that the natural frequencies increase with the rotational speed and hub radius while they decrease with the rotary inertia (and shear deformation). The effect 201 Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 Figure 5 The effect of rotational speed on the first five natural frequencies of a rotating Timoshenko beam ðr ¼ 0:04Þ Figure 6 Effect of hub radius on the first five natural frequencies of a rotating Timoshenko beam of the rotary inertia (and shear deformation) is more dominant on the higher modes. The calculated results are compared with the ones in literature and great agreement is considered. Arikoglu, A. and Özkol, I. (2005), “Analysis for slip over a single free disk with heat transfer”, Journal of Fluids Engineering, Vol. 127. Banerjee, J.R. (2001), “Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timoshenko beams”, Journal of Sound and Vibration, Vol. 247 No. 1, pp. 97-115. Bazoune, A., Khulief, Y.A. and Stephen, N.C. (1999), “Further results for modal characteristics of rotating tapered Timoshenko beams”, Journal of Sound and Vibration, Vol. 219, pp. 157-74. References Arikoglu, A. and Ozkol, I. (2004), “Solution of boundary value problems for integro-differential equations by using differential transform method”, Applied Mathematics and Computation. 202 Free vibration analysis of a rotating Timoshenko beam Aircraft Engineering and Aerospace Technology: An International Journal Metin O. Kaya Volume 78 · Number 3 · 2006 · 194 –203 Bert, C.W. and Zeng, H. (2004), “Analysis of axial vibration of compound bars by differential transformation method”, Journal of Sound and Vibration, Vol. 275, pp. 641-7. Chen, C.K. and Ju, S.P. (2004), “Application of differential transformation to transient advective-dispersive transport equation”, Applied Mathematics and Computation, Vol. 155, pp. 25-38. Du, H., Lim, M.K. and Liew, K.M. (1994), “A power series solution for vibrations of a rotating Timoshenko beam”, Journal of Sound and Vibration, Vol. 175 No. 4, pp. 505-23. Hodges, D.H. and Dowell, E.H. (1974), “Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades”, NASA TN D-7818. Lee, S.Y. and Kuo, Y.H. (1993), “Bending frequency of a rotating Timoshenko beam with general elastically restrained root”, Journal of Sound and Vibration, Vol. 162, pp. 243-50. Liebers, F. (1930), “Contribution to the theory of propeller vibrations”, NACA TM No. 568. Nagaraj, V.T. (1996), “Approximate formula for the frequencies of a rotating Timoshenko beam”, Journal of Aircraft, Vol. 33, pp. 637-9. Ozdemir, O. and Kaya, M.O. (2005), “Flapwise bending vibration analysis of a rotating tapered cantilevered Bernoulli-Euler beam by differential transform method”, Journal of Sound and Vibration, Vol. 289 Nos 1/2, pp. 413-20. Southwell and Gough (1921), “The free transverse vibration of airscrew blades”, British A.R.C. Report and Memoranda No. 766. Stafford, R.O. and Giurgiutiu, V. (1975), “Semi-analytic method for rotating Timoshenko beams”, International Journal of Mechanical Sciences, Vol. 17, pp. 719-27. Theodorsen, T. (1950), “Propeller vibrations and the effect of centrifugal force”, NACA TN No.516. Yokoyama, T. (1988), “Free vibration characteristics of rotating Timoshenko beams”, International Journal of Mechanical Sciences, Vol. 30, pp. 743-55. Zhou, J.K. (1986), Differential Transformation and its Application for Electrical Circuits, Huazhong University Press, Wuhan. Further reading Lin, S.C. and Hsiao, K.M. (2001), “Vibration analysis of a rotating Timoshenko beam”, Journal of Sound and Vibration, Vol. 240 No. 2, pp. 303-22. Novozhilov, V.V. (1999), Foundations of the Nonlinear Theory of Elasticity, Dover Publications, Mineola, NY. About the author Metin O. Kaya was born in Darmstadt, Germany in 1962. He is working in Istanbul Technical University at the Faculty of Aeronautics and Astronautics as an Associate Professor. His main research areas are aeroelasticity (helicopter and aircraft), rotating beams and applied mathematics. Minor areas are fluid mechanics and nanotechnology. Metin O. Kaya can be contacted at: kayam@itu.edu.tr To purchase reprints of this article please e-mail: reprints@emeraldinsight.com Or visit our web site for further details: www.emeraldinsight.com/reprints 203