Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia

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Mechanism and Machine Theory 44 (2009) 272–288

Mechanism and Machine Theory www.elsevier.com/locate/mechmt

Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia S.A.A. Hosseini, S.E. Khadem * Department of Mechanical Engineering, Tarbiat Modarres University, P.O. Box 14115-177, Tehran, Iran Received 23 April 2007; received in revised form 4 December 2007; accepted 21 January 2008 Available online 14 March 2008

Abstract In this paper the free vibrations of an in-extensional simply supported rotating shaft with nonlinear curvature and inertia are considered. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. To analyze the free vibrations of the shaft, the method of multiple scales is used. This method is applied to the discretized equations, and directly to the partial differential equations of motion, which demonstrates the same results. An expression is derived which describes the nonlinear free vibrations of the rotating shaft in two transverse planes. It is found that in this case, both forward and backward nonlinear natural frequencies are being excited. The results of perturbation method are validated with numerical simulations. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Rotating shaft; Free vibrations analysis; Large amplitude vibrations; Nonlinear curvature and inertia; Multiple scales method

1. Introduction Rotating shafts are used for power transmission in many modern machines. Accurate prediction of dynamics of rotating shafts is necessary for a successful design. Free vibrations analysis is one of the important steps in rotor-dynamics. Grybos [1] considered the effect of shear deformation and rotary inertia of a rotor on its critical speeds. Choi et al. [2] presented the consistent derivation of a set of governing differential equations describing the flexural and the torsional vibrations of a rotating shaft where a constant compressive axial load was acted on it. Jei and Leh [3] investigated the whirl speeds and mode shapes of a uniform asymmetrical Rayleigh shaft with asymmetrical rigid disks and isotropic bearings. Free damped flexural vibrations analysis of composite cylindrical tubes was carried out by Singh and Gupta [4], where they used beam and shell theories. Sturla and Argento [5] studied the free and forced response of a viscoelastic spinning Rayleigh shaft. Melanson and Zu [6] studied the free vibrations and stability of internally damped rotating shafts with general boundary conditions. Kim et al. [7] studied the free vibrations of a rotating tapered composite Timoshenko shaft.

*

Corresponding author. Tel./fax: +98 21 88011001 3388. E-mail address: [email protected] (S.E. Khadem).

0094-114X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2008.01.007

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Karunendiran and Zu [8] analyzed free vibrations analysis of a shaft on resilient bearings. Free and forced vibrations analysis of a rotating disk-shaft system with linear elastic bearings was investigated by Shabaneh and Zu [9]. Bearings were mounted on viscoelastic suspensions. El-Mahdy and Gadelrab [10] studied the free vibrations of unidirectional fiber reinforcement composite rotor. Raffa and Vatta [11] derived the equations of motion for an asymmetric Timoshenko shaft with unequal principal moments of inertia. The critical speeds and mode shapes of a spinning Rayleigh beam with six general boundary conditions are investigated analytically by Sheu and Yang [12]. Gubran and Gupta [13] studied the effect of stacking sequence and coupling mechanisms on the natural frequencies of composite shafts. To simplify the analysis, researchers often try to use the linear analysis. But, application of nonlinear analysis is sometimes inevitable. Many phenomena should be described with nonlinear equations which are not explainable with linear analysis. Using the Hopf bifurcation theory, Kurnik [14] analyzed self-excited vibrations of a rotating geometrically nonlinear shaft caused by internal friction. Shaw and Shaw [15] analyzed stability and bifurcations of a rotating shaft made of a viscoelastic material. Using the general theory of bar bending, Leinonen [16] presented a nonlinear model to describe the bending behavior of a rotating shaft. Kurnik [17] analyzed the stability and self-excited postcritical whirling of a rotating shaft with the aid of bifurcation theory. The shaft was made of a material with elastic and viscous nonlinearities. Vibrations of a spinning rotor with nonlinear elastic and geometric properties were considered by Cveticanin [18]. The method of multiple scales was applied to analyze the free and forced vibration of nonlinear rotor-bearing systems by Ji and Zu [19]. They used a nonlinear spring and linear damping to model the nonlinear bearing pedestal. A geometrically nonlinear model of a rotating shaft was introduced by Luczko [20]. The model included Von-Karman nonlinearity, nonlinear curvature effects, large displacements and rotations as well as gyroscopic and shear effects. Viana Serra Villa et al. [21] used the invariant manifold approach to explore the dynamics of a nonlinear rotor. They constructed a reduced order model with the aid of nonlinear normal modes and evaluated its performance. Cveticanin [22] considered the free vibrations of a Jeffcott rotor with cubic nonlinear elastic property. He applied the Krylov–Bogolubov method to solve the nonlinear equations of motion. Lately, present authors studied free vibrations of a rotating beam with random properties [23], and vibrations and reliability of a rotating beam with random properties under random excitations [24]. To study uncertainty, stochastic finite element method based on the second order perturbation method was applied. In this paper, the equations of motion of a continuous simply supported rotating shaft with nonlinear curvature and inertia are derived. Rotary inertia and gyroscopic effects are included but, shear deformation is neglected. Using the in-extensionality assumption, the equations of motion are derived with the aid of Hamilton principal. To solve theses gyroscopic nonlinear equations of motion approximately, the multiple scales method is used. This method is applied directly to the partial differential equation of motion and to the discretized equations. Some researches have shown that applying the multiple scales method to the discretized equations might produce quantitative and qualitative errors (for example, [25]). The systems that they considered were nongyroscopic. Here, it is shown that in our gyroscopic system, resulting reduced equations from two approaches are the same. An expression is derived which describes the nonlinear free vibration of the rotating shaft in two transverse planes. Some authors have used only forward whirling frequency to study the nonlinear free vibrations of a rotating shaft with gyroscopic effects (for example, [19]). Here, it is shown that in the nonlinear free vibrations of a rotating shaft with gyroscopic effects, both forward and backward nonlinear natural frequencies are excited. So, if one takes into account only the forward natural frequencies, the results become incorrect. Effects of rotary inertia, external damping coefficient and rotating speed on the nonlinear amplitude and natural frequencies of first two modes of a shaft are examined. The results of perturbation method are validated with numerical simulations. 2. Equations of motion The schematic of a continuous rotating shaft has been shown in Fig. 1. The length of the undeformed shaft center line is l. Displacements of any particle of the shaft are described in inertial frame X–Y–Z. The x–y–z constitute a local coordinate which are principal axes of the beam cross section. The axes are attached to the center line of the deformed shaft (Fig. 1) at position x. Displacements of a particle in arbitrary location x along X-, Y- and Z-axes are u(x, t), v(x, t) and w(x, t), respectively, and torsional angle is /(x, t). Following

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Fig. 1. Schematic of the rotating shaft and coordinates X–Y–Z and x–y–z.

assumptions are employed: (1) the shaft has uniform circular cross section, and spins about longitudinal axis X with a constant speed X; (2) the effect of gravity is neglected; (3) the shaft is slender and consequently, shear deformation is neglected; (4) the shaft is simply supported; (5) support O is fixed but support O0 is free to move along the X-axis (Fig. 1). This assumption implies that the stretching effect is negligible. This situation is more realistic than some previous works which nonlinearity was due to the stretching of the shaft centerline [15]; (6) external viscose damping is the only dissipating mechanism in the system; (7) vibrations of the rotating shaft are large amplitude and shortening effect due to in-extensibility assumption is considered [26,27]. Therefore, only nonlinear effects of curvature and inertia are studied, here. 2.1. Kinetic and potential energy The relation between the original frame X–Y–Z and the deformed frame x–y–z can be described by three successive Euler-angle rotations [26]. Here, a 3–2–1 body rotation with angles of rotation w(x, t), h(x, t) and

Fig. 2. Three-axis Euler-angle rotation 3–2–1.

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b(x, t) is used as shown in Fig. 2. It should be noted that b(x, t) is total rotation angle due to torsional deformation /(x, t) and spin angle Xt bðx; tÞ ¼ /ðx; tÞ þ Xt:

ð1Þ

The kinetic energy for a rotating shaft can be written [28] Z 1 l T ¼ mðu_ 2 þ v_ 2 þ w_ 2 Þ þ I 1 x21 þ I 2 ðx22 þ x23 Þdx: 2 0

ð2Þ

Mass per unit length m, polar and diametrical mass moment of inertia I1 and I2 are Z Z Z Z m ¼ q dA; I 1 ¼ qðy 2 þ z2 ÞdA; I 2 ¼ qy 2 dA ¼ qz2 dA;

ð3Þ

where q is mass density. The angular velocities of the frame x–y–z with respect to the frame X–Y–Z are (Fig. 2) x ¼ x1 e1 þ x2 e2 þ x3 e3 ¼ ðb_  w_ sin hÞe1 þ ðw_ sin b cos h þ h_ cos bÞe2 þ ðw_ cos b cos h  h_ sin bÞe3 :

ð4Þ

If shear deformation is negligible, the strain energy for a rotating shaft with isotropic and linear material properties becomes [29] Z l dP ¼ ðA11 e de þ D11 q1 dq1 þ D22 q2 dq2 þ D22 q3 dq3 Þdx; ð5Þ 0

where the strain along the center line of the shaft is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e ¼ ð1 þ u0 Þ þ v02 þ w02  1 and A11 ¼

Z E dA;

D11 ¼

Z

Gðy 2 þ z2 ÞdA;

D22 ¼

ð6Þ Z

Ey 2 dA ¼

Z

Ez2 dA:

ð7Þ

In the above equations, E and G are elasticity and shear modulus, respectively, and qi (i = 1, . . . , 3) are shaft curvatures. Using Love’s kinetic analogy [29], shaft curvatures qi (i = 1, . . . , 3) can be computed as q ¼ q 1 e1 þ q 2 e2 þ q 3 e3 ¼ ð/0  w0 sin hÞe1 þ ðw0 sin / cos h þ h0 cos /Þe2 þ ðw0 cos / cos h  h0 sin /Þe3 :

ð8Þ

Because the shear deformation is negligible, angles w and h can be related to the displacements (Fig. 2): v0 w ¼ sin1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ð1 þ u0 Þ þ v02

w0 h ¼ sin1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 ð1 þ u0 Þ þ v02 þ w02

ð9Þ

2.2. In-extensionality assumption Eqs. (2) and (5) are expressions for kinetic and strain energy of an isotropic rotating shaft. It was noted earlier that support O0 in Fig. 1 is movable in X-direction. So, the in-extensionality assumption can be employed, which implies that the strain along the shaft center line is zero [26,29]. Eq. (6) gives e ¼ 0 ! ð1 þ u0 Þ2 þ v02 þ w02 ¼ 1: Expanding Eq. (10) into a Taylor series pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u0 ¼ 1  v02  w02  1 ’  ðv02 þ w02 Þ þ    : 2

ð10Þ

ð11Þ

Therefore, if v = O(e) and w = O(e), then u = O(e2), where e  1 is a bookkeeping parameter. Substituting Eq. (9) into Eqs. (4) and (8), expanding the outcomes in Taylor series and retaining terms up to O(e3), one can

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compute curvatures and angular velocities up to O(e3). Substituting these curvatures and angular velocities into Eqs. (2) and (5), and using Eq. (11), the final form of kinetic and strain energy is obtained. Applying Hamilton principal to these kinetic and strain energies, one may obtain differential equations of motion governing the nonlinear bending–bending–torsional vibration of a rotating shaft. These equations and associated boundary conditions are presented in Appendix A. These differential equations can be simplified using following assumptions: (1) the shaft is circular; so, its fundamental torsional frequency is much higher than the frequency of flexural modes. Consequently, the torsional inertia terms can be neglected in comparison with the flexural inertia and stiffness terms [27]. (2) The shaft is slender; so, the rotary inertia is small and the nonlinear terms that involve the rotary inertia can be neglected [29]. Now, following nondimensional quantities are defined (I1 = 2I2): rffiffiffiffiffiffiffiffi D22 t; x ¼ x=l; v ¼ v=l; w ¼ w=l; t ¼ ml4 sffiffiffiffiffiffiffiffi ð12Þ pffiffiffiffiffiffiffiffiffiffiffi ml4 2 2    c ¼ cl = mD22 ; I 2 ¼ I 2 =ðml Þ; X ¼ X: D22 Applying the above assumptions and using the nondimensional quantities (12), as shown in Appendix A, one may obtain the following equations of motion governing the vibrations of an in-extensional rotating shaft with nonlinearities in curvature and inertia: Z x Z xZ x €v þ v0 € 02 w0 Þdx þ v00 € 0 w0 Þdx dx þ c_v  I 2 ð2Xw_ 00 þ €v00 Þ ð_v02 þ €v0 v0 þ w_ 02 þ w ð_v02 þ €v0 v0 þ w_ 02 þ w 0

0

l

þ v02 vðIV Þ þ v0 wðIV Þ w0 þ 3w000 w00 v0 þ v00 w002 þ v003 þ vðIV Þ þ w000 v00 w0 þ 4v0 v00 v000 ¼ 0; Z x Z xZ x 0 02 0 0 02 0 0 00 € þw € w Þdx þ w € 0 w0 Þdx dx þ cw_ þ I 2 ð2X_v00  w € 00 Þ w ð_v þ €v v þ w_ þ w ð_v02 þ €v0 v0 þ w_ 02 þ w 0

l

0

þ w00 v002 þ wðIV Þ þ v0 v000 w00 þ 3w0 v00 v000 þ w0 v0 vðIV Þ þ w02 wðIV Þ þ w003 þ 4w0 w00 w000 ¼ 0: ð13Þ The boundary conditions are v ¼ 0;

v00 ¼ 0;

w ¼ 0;

w00 ¼ 0

at x ¼ 0 and x ¼ 1:

ð14Þ

In Eq. (13), c is (external) damping coefficient. For ease of notation, the asterisks in the above equations have been dropped. 3. Method of multiple scales In this section, the multiple scales method is used to study the free vibrations of the rotating shaft [30]. In general, there exist two approaches for application the method of multiple scales to the equations of motion. In the first approach, the partial differential equations are directly attacked by the multiple scales method. In the second approach, the partial differential equations are discretized by a suitable method, e.g. Galerkin method. Then, the resulted ordinary differential equations are attacked by the multiple scales method. Some researches have shown that applying the multiple scales method to the discretized equations may produce quantitative and qualitative errors (for example, [25]). Here, two approaches are applied to the equations of motion and associated boundary conditions derived in Section 2.2, i.e. Eqs. (13) and (14). 3.1. Application of multiple scales method to the partial differential equations of motion To apply the multiple scales method, v and w are expanded in the form vðx; tÞ ¼ ev1 ðx; T 0 ; T 2 Þ þ e3 v3 ðx; T 0 ; T 2 Þ þ    ; wðx; tÞ ¼ ew1 ðx; T 0 ; T 2 Þ þ e3 w3 ðx; T 0 ; T 2 Þ þ    ;

ð15Þ

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where e is a small dimensionless parameter, T0 = t and T2 = e2t are fast and slow time scales, respectively. Damping should be scaled, so that its effects are balanced with nonlinearities. So, c is replaced with ce2. Using the chain rule, time derivatives in terms of T0 and T2 become o=ot ¼ D0 þ e2 D2 þ    ;

o=ot2 ¼ D20 þ 2e2 D2 D0 þ    ;

ð16Þ

where Dn = o/oTn (n = 0, 2). Substituting Eqs. (15) and (16) into Eq. (13), and equating the coefficients of the same power of e; following equations are obtained: O(e) ðIV Þ

D20 v1 þ v1

 I 2 D20 v001  2I 2 XD0 w001 ¼ 0;

ðIV Þ

D20 w1 þ w1

 I 2 D20 w001 þ 2I 2 XD0 v001 ¼ 0:

ð17Þ

3

O(e ) ðIV Þ

D20 v3 þ v3

 2I 2 XD0 w003  I 2 D20 v003 ¼ W1 ;

ðIV Þ

D20 w3 þ w3

þ 2I 2 XD0 v003  I 2 D20 w003 ¼ W2 ;

ð18Þ

where W1 and W2 are defined in Appendix B. The boundary conditions in any order are the same as Eq. (14), except in O(ei) (i = 1, 3), variables v and w are replaced with vi and wi (i = 1, 3), respectively. Solution of Eq. (17) can be written as pffiffiffi   v1 ðs; T 0 ; T 2 Þ ¼ 2 sin npx F 1 ðT 2 Þebf T 0 I þ F 1 ðT 2 Þebf T 0 I þ F 2 ðT 2 Þebb T 0 I þ F 2 ðT 2 Þebb T 0 I ; ð19Þ pffiffiffi   w1 ðs; T 0 ; T 2 Þ ¼ 2 sin npx IF 1 ðT 2 Þebf T 0 I þ IF 1 ðT 2 Þebf T 0 I þ IF 2 ðT 2 Þebb T 0 I  IF 2 ðT 2 Þebb T 0 I ; pffiffiffiffiffiffiffi where I ¼ 1 and n is mode number. F1(T2) and F2(T2) are complex-valued functions which will be determined at higher order levels of approximation; bf and bb are forward and backward linear natural frequencies, respectively, defined as

bf ¼ n2 p2

I 2X þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 I 22 þ n2 I 2 p2 þ 1 n2 I 2 p2 þ 1

;

bb ¼ n2 p2

I 2 X þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 I 22 þ n2 I 2 p2 þ 1 n2 I 2 p2 þ 1

:

ð20Þ

It should be noted that in Eq. (19) both forward and backward frequencies have been considered. In some papers, only forward whirling frequency (bf) have been accounted in the nonlinear free vibration of a rotating shaft with gyroscopic effects (for example, [19]). Indeed, they have assumed that F2(T2) = 0. It is realized later that both F1(T2) and F2(T2) have nonzero values. Substituting of Eq. (19) into Eq. (18) gives ðIV Þ

D20 v3  I 2 D20 v003  2I 2 XD0 w003 þ v3

¼ G1 ðx; T 2 Þebf T 0 I þ H 1 ðx; T 2 Þebb T 0 I þ CC þ NST;

ðIV Þ

D20 w3  I 2 D20 w003 þ 2I 2 XD0 v003 þ w3

¼ G2 ðx; T 2 Þebf T 0 I þ H 2 ðx; T 2 Þebb T 0 I þ CC þ NST:

ð21Þ

where G1(x, T2), G2(x, T2), H1(x, T2) and H2(x, T2) are defined in [28]; ‘‘NST” and ‘‘CC” stands for ‘‘Non-Secular Term” and ‘‘Complex Conjugate”, respectively. If the homogeneous parts of Eq. (21) have nontrivial solutions, the inhomogeneous Eq. (21) have solution only if a solvability condition is satisfied [30]. The solvability conditions demand that the right side of Eq. (21) be orthogonal to every solution of the adjoint problem. It can be proven that the homogeneous parts of Eq. (21) are a set of self-adjoint equations. Therefore, solvability conditions can be written as Z Z

1

pffiffiffi 2 sinðnpxÞ½G1 ðx; T 2 Þ þ G2 ðx; T 2 Þdx ¼ 0;

1

pffiffiffi 2 sinðnpxÞ½H 1 ðx; T 2 Þ þ H 2 ðx; T 2 Þdx ¼ 0:

0

0

ð22Þ

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After substitution, final form of solvability conditions is obtained 2

IK1 D2 F 1 ðT 2 Þ  2IcF 1 ðT 2 Þbf  8n6 p6 F 1 ðT 2 ÞF 1 ðT 2 Þ þ CF 1 ðT 2 ÞF 2 ðT 2 ÞF 2 ðT 2 Þ ¼ 0; 2

IK2 D2 F 2 ðT 2 Þ  2IcF 2 ðT 2 Þbb  8n6 p6 F 2 ðT 2 ÞF 2 ðT 2 Þ þ CF 2 ðT 2 ÞF 1 ðT 2 ÞF 1 ðT 2 Þ ¼ 0; where

  3 2 2 4 4 4 2 C ¼  n p  n p ðbf þ bb Þ  16n6 p6 ; 2 3 K1 ¼ ð4n2 p2 XI 2  4n2 p2 bf I 2  4bf Þ;

ð23Þ

ð24Þ

K2 ¼ ð4n2 p2 bb I 2 þ 4bb þ 4n2 p2 XI 2 Þ:

3.2. Application of multiple scales method to the discretized differential equations of motion In this approach, before applying the multiple scales method, partial differential equations of motion are discretized. Here, a single mode Galerkin method is used vðx; tÞ ¼ /n ðxÞV ðtÞ;

wðx; tÞ ¼ /n ðxÞW ðtÞ;

where n is the mode number and /n(x) is the linear mode shape of the shaft pffiffiffi /n ðxÞ ¼ 2 sin npx:

ð25Þ

ð26Þ

Substituting Eq. (25) into Eq. (13), taking the inner product of each equation with its corresponding mode shape, and using the orthogonality properties of the mode shapes, the following discretized equations of motion are obtained:   3 2 2 1 4 4 2 2 4 4 2 2 6 6 3 € _ _ pn  pn ð1 þ n p I 2 ÞW þ cW þ p n W  2p n I 2 XV þ n p W  8 3 2 2 2 6 6 2 € þ VW V€ Þ þ n p V W ¼ 0;  ðW V_ þ W W_ þ W W   ð27Þ 3 2 2 1 4 4 2 2 4 4 2 2 6 6 3 € _ _ pn  pn ð1 þ n p I 2 ÞV þ cV þ p n V þ 2p n I 2 XW þ n p V  8 3 2 2 2 6 6 2 € þ V V€ Þ þ n p VW ¼ 0:  ðV V_ þ V W_ þ WV W Because the effects of damping should be balanced with nonlinearities, c is replaced with ce2. Expanding V and W in the form V ðtÞ ¼ eV 1 ðT 0 ; T 2 Þ þ e3 V 3 ðT 0 ; T 2 Þ þ    ; W ðtÞ ¼ eW 1 ðT 0 ; T 2 Þ þ e3 W 3 ðT 0 ; T 2 Þ þ    :

ð28Þ

Substituting Eq. (28) into Eq. (27), using Eq. (16), and equating the coefficients of the same power of e, following equations are obtained: O(e) ð1 þ n2 p2 I 2 ÞD20 V 1 þ n4 p4 V 1 þ 2n2 p2 I 2 XD0 W 1 ¼ 0; ð1 þ n2 p2 I 2 ÞD20 W 1 þ n4 p4 W 1  2n2 p2 I 2 XD0 V 1 ¼ 0:

ð29Þ

O(e3) ð1 þ n2 p2 I 2 ÞD20 V 3 þ n4 p4 V 3 þ 2n2 p2 I 2 XD0 W 3 ¼ W3 ; ð1 þ n2 p2 I 2 ÞD20 W 3 þ n4 p4 W 3  2n2 p2 I 2 XD0 V 3 ¼ W4 ;

ð30Þ

where W3 and W4 are defined in Appendix B. Solution of equations in O(e) is W 1 ðT 0 ; T 2 Þ ¼ IF 1 ðT 2 Þebf T 0 I þ IF 2 ðT 2 Þebb T 0 I þ IF 1 ðT 2 Þebf T 0 I  IF 2 ðT 2 Þebb T 0 I ; V 1 ðT 0 ; T 2 Þ ¼ F 1 ðT 2 Þebf T 0 I þ F 2 ðT 2 Þebb T 0 I þ F 1 ðT 2 Þebf T 0 I þ F 2 ðT 2 Þebb T 0 I :

ð31Þ

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Substituting Eq. (31) into Eq. (30), so ð1 þ n2 p2 I 2 ÞD20 V 3 þ n4 p4 V 3 þ 2n2 p2 I 2 XD0 W 3 ¼ G3 ðT 2 Þebf T 0 I þ H 3 ðT 2 Þebb T 0 I þ CC þ NST; ð1 þ n2 p2 I 2 ÞD20 W 3 þ n4 p4 W 3  2n2 p2 I 2 XD0 V 3 ¼ G4 ðT 2 Þebf T 0 I þ H 4 ðT 2 Þebb T 0 I þ CC þ NST;

ð32Þ

where G3(T2), G4(T2), H3(T2) and H4(T2) are defined in Appendix B. Eq. (32) are a set of gyroscopic ordinary differential equations. To find the solvability conditions V3and W3 are expressed in the form [30] V 3 ðT 0 ; T 2 Þ ¼ F 11 ðT 2 Þebf T 0 I þ F 12 ðT 2 Þebb T 0 I ;

ð33Þ

W 3 ðT 0 ; T 2 Þ ¼ F 21 ðT 2 Þebf T 0 I þ F 22 ðT 2 Þebb T 0 I :

Substituting Eq. (33) into Eq. (32), and equating the coefficient of ebf T 0 I in both sides of Eq. (32), one can obtain ðb2f  n2 p2 I 2 b2f þ n4 p4 ÞF 11 ðT 2 Þ þ 2n2 p2 I 2 Xbf IF 21 ðT 2 Þ ¼ G3 ðT 2 Þ; ðb2f  n2 p2 I 2 b2f þ n4 p4 ÞF 21 ðT 2 Þ  2n2 p2 bf I 2 XIF 11 ðT 2 Þ ¼ G4 ðT 2 Þ:

ð34Þ

Similarly, for coefficient of ebb T 0 I it is obtained ðb2b  n2 p2 I 2 b2b þ n4 p4 ÞF 12 ðT 2 Þ þ 2n2 p2 I 2 Xbb IF 22 ðT 2 Þ ¼ H 3 ðT 2 Þ; ðb2b  n2 p2 I 2 b2b þ n4 p4 ÞF 22 ðT 2 Þ  2n2 p2 bb I 2 XIF 12 ðT 2 Þ ¼ H 4 ðT 2 Þ:

ð35Þ

Eqs. (34) and (35) constitute systems of two inhomogeneous algebraic for F11(T2), F21(T2), and F12(T2), F22(T2), respectively. Their homogeneous parts have a nontrivial solution. So, their solvability conditions can be written as [30]   n4 p4  b2  n2 p2 I b2  2 f f   2n2 p2 I 2 Xbf I

 G3 ðT 2 Þ   ¼ 0; G4 ðT 2 Þ 

  n4 p4  b2  n2 p2 I b2  2 b b   2n2 p2 bb XIF 12 ðT 2 Þ

 H 3 ðT 2 Þ   ¼ 0: H 4 ðT 2 Þ 

ð36Þ

After simplification, the solvability conditions are reduced to expressions same as Eq. (23). Therefore, application of multiple scales method to the original gyroscopic partial differential equations (Eq. (13)) and to the discretized equations (Eq. (27)) gives the same results.

4. Free vibration analysis Expressing F1 and F2 in a polar form 1 F 1 ðT 2 Þ ¼ a1 ðT 2 ÞeIh1 ðT 2 Þ ; 2

1 F 2 ðT 2 Þ ¼ a2 ðT 2 ÞeIh2 ðT 2 Þ ; 2

ð37Þ

where ai(T2) (i = 1, 2) and hi(T2) (i = 1, 2) are amplitudes and phase angles of the response, respectively. Substituting Eq. (37) into Eq. (23), and separating real and imaginary parts, the modulation equations are obtained: 1 1  K1 D2 a1 ðT 2 Þ þ ca1 ðT 2 Þbf ¼ 0;  K2 D2 a2 ðT 2 Þ þ ca2 ðT 2 Þbb ¼ 0; 2 2 1 1 2  K1 a1 ðT 2 ÞD2 h1 ðT 2 Þ þ Ca1 ðT 2 Þa2 ðT 2 Þ  n6 p6 a31 ðT 2 Þ ¼ 0; 2 8 1 1  K2 a2 ðT 2 ÞD2 h2 ðT 2 Þ þ Ca2 ðT 2 Þa21 ðT 2 Þ  n6 p6 a32 ðT 2 Þ ¼ 0: 2 8

ð38Þ

ð39Þ

Solving Eq. (38) for a1(T2) and a2(T2) a1 ðT 2 Þ ¼ C 1 e2cbf T 2 =K1 ;

a2 ðT 2 Þ ¼ C 2 e2cbb T 2 =K2 :

ð40Þ

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Substituting Eq. (40) into (39), and solving for h1(T2) and h2(T2) h1 ðT 2 Þ ¼

1 C 22 CK2 e4cbb T 2 =K2 1 C 21 n6 p6 e4cbf T 2 =K1  þ C4; K1 cbb cbf 16 2

h2 ðT 2 Þ ¼

1 C 21 CK1 e4cbf T 2 =K1 1 C 22 n6 p6 e4cbb T 2 =K2  þ C3; K2 cbf cbb 16 2

ð41Þ

where Ci (i = 1–4) are constants determined by initial conditions. Substituting Eqs. (40) and (41) into Eq. (37), and then substituting outcomes in Eq. (31), one may obtain v ¼ /n ðxÞ½Af cosðF1 A2b  F2 A2f þ bf T 0 þ C 4 Þ þ Ab cosðF4 A2b  F3 A2f þ bb T 0 þ C 3 Þ; w ¼ /n ðxÞ½Af sinðF1 A2b  F2 A2f þ bf T 0 þ C 4 Þ þ Ab sinðF4 A2b  F3 A2f þ bb T 0 þ C 3 Þ;

ð42Þ

where Af ¼ C 1 eD1 T 0 ;

A b ¼ C 2 eD 2 T 0

ð43Þ

and parameters Fi (i = 1–4), Dj (j = 1, 2) are defined as 2cbf 2cbb CK2 n6 p6 D1 ¼ ; D2 ¼ ; F1 ¼ ; F2 ¼ ; K1 K2 16K1 cbb 2cbf

F3 ¼

CK1 ; 16K2 cbf

F4 ¼

n6 p6 : 2cbb

ð44Þ

These analytical expressions describe the free vibration of a rotating shaft with nonlinearities in curvature and inertia in two transverse planes, where only one mode (nth mode) has been excited. Eq. (42) shows that both forward and backward natural frequencies are excited (generally, C2 – 0). Also, due to the nonlinearity, the frequency of the vibration is dependent on the amplitude of the motion. 5. Numerical example In this section, numerical examples are considered to examine the free vibrations of a rotating shaft with nonlinearities in curvature and inertia. We should note that in this section, all quantities are in nondimensional form. Time histories of the displacement of the rotating shaft midpoint are presented in Figs. 3 and 4, where only the first mode is excited and X = 10, c = 0.05. In Fig. 3, only the plane v has been excited by the initial dis_ placement ðvð0Þ ¼ 0:01; v_ ð0Þ ¼ 0; wð0Þ ¼ 0; wð0Þ ¼ 0Þ. But, there exist vibrations in plane w. The reason is

Amplitude v

0.01 Numerical Simulation Perturbation

0.005 0 -0.005 -0.01

0

10

20

30

40

50

60

70

80

90

100

Amplitude w

0.01 Perturbation Numerical Simulation

0.005 0 -0.005 -0.01

0

20

40

60

80

100

Time _ Fig. 3. Time history of the shaft midpoint displacement; vð0Þ ¼ 0:01; v_ ð0Þ ¼ 0; wð0Þ ¼ 0; wð0Þ ¼ 0; I 2 ¼ 0:000625.

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281

Amplitude v

0.01 Numerical Simulation Perturbation

0.005 0 -0.005 -0.01

0

10

20

30

40

50

60

70

80

90

100

-6

x 10

Amplitude w

2

Numerical Simulation Perturbation

1 0 -1 -2

0

10

20

30

40

50

60

70

80

90

100

Time _ Fig. 4. Time history of the shaft midpoint displacement; vð0Þ ¼ 0:01; v_ ð0Þ ¼ 0; wð0Þ ¼ 0; wð0Þ ¼ 0; I 2 ¼ 0.

due to the presence of the gyroscopic effect. Eq. (13) are linearly coupled by the gyroscopic terms. When one plane is excited, the other plane due to the gyroscopic effect oscillates, too. Because the absolute values of the forward and backward natural frequencies are close, a beat phenomenon is observed. The data in Fig. 4 is the same as Fig. 3 except the rotary inertia and consequently, the gyroscopic terms are neglected (I2 = 0). Fig. 4 shows that the plane w does not oscillate, and the beat phenomenon does not occur. To validate the results of perturbation method, numerical simulations have been used in Figs. 3 and 4. Both methods agree very well. Eq. (43) shows that the amplitudes Af and Ab are exponential functions of parameters D1 and D2. Also, D1 and D2 are functions of rotating speed (X), damping coefficient (c) and diametrical mass moment of inertia (I2). The effects of rotating speed on the parameters D1 and D2 are shown in Fig. 5, for different values of parameter I2. Only first mode is considered. Parameter D1 is descending but parameter D2 is ascending with respect to the

B1, B2, B1, B2, B1, B2,

-0.01

Β1,Β2

-0.015

I2=0.0000625 I2=0.0000625 I2=0.000625 I2=0.000625 I2=0.00625 I2=0.00625

-0.02

-0.025

-0.03

-0.035 0

10

20

30

40

50

60

70

80

Ω Fig. 5. Parameters D1 and D2 versus rotating speed; c = 0.05.

90

282

S.A.A. Hosseini, S.E. Khadem / Mechanism and Machine Theory 44 (2009) 272–288 First Mode 20

I2=0.00625 I2=0.000625

15

I2=0.0000625

βf , -βb

10

5

0

-5

-10 0

20

40

60

80

100

Ω Fig. 6. Parameters bf and bb versus rotating speed.

rotating speed and diametrical mass moment of inertia. These parameters are linear functions of damping and descending (Eq. (44)). The figure shows that for the same data, the inequality D2 P D1 is always satisfied. In Fig. 6, forward and backward linear natural frequencies bf and bb are plotted versus the rotating speed for different values of parameter I2. The values of bf and bb increases as the parameter X increases. This increase rate becomes higher for larger values of the parameter I2. Eq. (42) shows that the natural frequencies have two parts. The first, constant parts bf and bb which are linear natural frequencies. The second, parts that depend on the amplitudes, and consequently on time: ðF1 A2b þ F2 A2f Þ=T 0 and ðF4 A2b þ F3 A2f Þ=T 0 . Therefore, the natural frequency is functions of I2,c and X as well

First Mode 16

I2=0.00625 I2=0.000625

FNNF, BNNF

14

I2=0.0000625

12

FNNF

10

8

BNNF

6

10

20

30

40

50

60

70

80

90

Ω Fig. 7. Parameters FNNF and BNNF versus rotating speed; c = 0.05.

100

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283

Second Mode

28

FNNF

26

FNNF, BNNF

BNNF 24

22

20

I2=0.00625

18

I2=0.000625 I2=0.0000625

16 0

10

20

30

40

50

60

70

80

Ω Fig. 8. Parameters FNNF and BNNF versus rotating speed; c = 0.05.

as time. So, to investigate the effects of parameters I2, c and X on the nonlinear natural frequencies, one should set the time on a specified value (e.g. T0 = 1) and then examine the effects of diametrical mass moment of inertia (I2), damping coefficient (c) and rotating speed (X). Here, Forward and Backward Nonlinear Natural Frequencies are defined as FNNF ¼ F1 A2b þ F2 A2f  bf ;

BNNF ¼ F3 A2f þ F4 A2b  bb ;

where Af ¼ C 1 eD1 ; Ab ¼ C 2 eD2 and Ci (i = 1, 2) are determined from the initial conditions vð0Þ ¼ _ 0:01; v_ ð0Þ ¼ 0; wð0Þ ¼ 0; wð0Þ ¼ 0. First Mode 60

I2=0.00625 I2=0.000625

50

I2=0.0000625

FNNF

40

30

20

10

0 10 -3

10 -2

10 -1

c Fig. 9. Parameter FNNF versus damping coefficient; X = 50.

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First Mode 25

I2=0.00625 I2=0.000625

BNNF

20

I2=0.0000625

15

10

5

10

-3

10

-2

10

-1

c Fig. 10. Parameter BNNF versus damping coefficient; X = 50.

FNNF and BNNF as a function of rotating speed (X) are shown in Figs. 7 and 8 for the first two modes. In the first mode, FNNF curves increase as the rotating speed increases, for all values of I2. But in the second mode, if I2 = 0.00625, the value of FNNF increases first, and then decreases as the rotating speed increases. In both first and second modes, BNNF curves are descending with respect to the rotating speed, for all values of I2. For the same data, the values of FNNF are larger than BNNF. It is seen from the figures that due to the gyroscopic effects, for a rotating speed change, the change rate of FNNF and BNNF are higher for the larger values of I2. In Figs. 9–12, FNNF and BNNF are plotted versus damping coefficient c for different values of I2. Only first two modes are considered. It is observed from figures that the general characteristics of the curves are

Second Mode I2=0.00625 1000

I2=0.000625 I2=0.0000625

FNNF

800

600

400

200

0 10

-3

10

-2

10

-1

c Fig. 11. Parameter FNNF versus damping coefficient; X = 50.

S.A.A. Hosseini, S.E. Khadem / Mechanism and Machine Theory 44 (2009) 272–288

285

Second Mode 500

I2=0.00625 I2=0.000625 I2=0.0000625

BNNF

400

300

200

100

0 -3 10

10

-2

10

-1

c Fig. 12. Parameters BNNF versus damping coefficient; X = 50.

similar. For small c, FNNF and BNNF have high values and have sharp changes as c changes. For large values of c, FNNF and BNNF change slowly. In each curve, there exists a point that is local minimum. For the second mode, this point has moved toward larger values of c. Curves for the first mode, changes sharper than the second mode. 6. Summary and conclusions Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia is investigated analytically here. Rotary inertia and gyroscopic effects are included but, shear deformation is neglected. To analyze the free vibrations of the shaft, the multiple scales method is used. An expression is derived which describes the nonlinear free vibration of the rotating shaft in two transverse planes. The effects of diametrical mass moment of inertia, damping coefficient and rotating speed on the FNNF and BNNF are considered. The most important results of the paper can be expressed as  In the nonlinear free vibration analysis, both forward and backward nonlinear natural frequencies are excited.  The results from direct and discretized perturbation analysis are the same.  When one plane is excited, the other plane due to the gyroscopic effect oscillates, too.  In free vibrations, the beat phenomenon is observed.  The results of numerical simulation and perturbation solution agree very well.  For the same data, the values of FNNF are larger than BNNF.  In the first mode, FNNF curves increase as the rotating speed increases, for all values of I2. Also, in both modes, BNNF curves are descending with respect to the rotating speed, for all values of I2.  For small c, FNNF and BNNF have high values and have sharp changes as c changes. For large values of c, FNNF and BNNF change slowly. FNNF and BNNF curves for the first mode, changes sharper than the second mode. Acknowledgement The authors would like to thank the Amir Kabir University Energy and Control Center of Excellence for her support.

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Appendix A As explained in Section 2, one can use Eqs. (2), (4), (5), (8) and (9) to derive the differential equations of motion governing the nonlinear bending–bending–torsional vibration of a rotating shaft. These equations are as follows:   Z xZ x Z x 00 02 0 0 02 0 0 0 02 0 0 02 0 0 € w Þdx dx  v € w Þdx m €v  v ð_v þ €v v þ w_ þ w ð_v þ €v v þ w_ þ w  l 0 0 1 1 € 00 þ 2w_ 0 v_ 0 w00 þ I 1 X w02 w_ 00 þ w0 w00 w_ 0 þ w_ 00 þ v0 w_ 0 v00 þ v02 w_ 00 þ I 1 ð/_ w_ 00 þ /_ 0 w_ 0 þ w02€v00 þ /w 2 2 þ 2w0 v_ 00 w_ 0 þ 2w0€v0 w00 þ /_ 00 w0 þ 2w0 v_ 0 w_ 00 Þ þ I 2 ð€v00 v02 þ €v00 þ v0 w0 w€00 þ v00 w_ 02 þ 2€v0 v0 v00 € 0 þ 2_v0 v0 v_ 00 þ v00 w0 w € 0 þ v_ 02 v00 þ 2v0 w_ 00 w_ 0 Þ  D11 ð/000 w0 þ 2/00 w00 þ /0 w000 þ 4w0 v000 w00 þ v0 w00 w þ w02 vðIV Þ þ 2w002 v00 þ 2w0 v00 w000 Þ  D22 ð4v00 v0 v000 þ wðIV Þ w0 v0 þ w0 v00 w000 þ 3w00 v0 w000 þ w002 v00 þ v003 þ vðIV Þ v02 þ vðIV Þ Þ ¼ 0;   Z x Z xZ x 0 02 0 0 02 0 0 00 02 0 0 02 0 0 € w Þdx  w € w Þdx dx m € ww ð_v þ €v v þ w_ þ w ð_v þ €v v þ w_ þ w l 0  0  1 1 _ v00 þ 2w0 v_ 0 v_ 00 þ /_ 0 v_ 0 þ w00 v_ 02 Þ  I 1 X v_ 00 v02 þ v_ 00 þ v_ 0 w0 w00 þ v_ 0 v0 v00 þ v_ 00 w02  I 1 ð/_ 2 2 w0 w0 w00 þ 2w_ 0 w0 w_ 00 þ w_ 02 w00 þ w0 v00€v0 þ v0€v0 w00 þ w€00 w02 þ w00 v_ 02 þ 2_v0 v_ 00 w0 þ w€00 þ v0€v00 w0 Þ þ I 2 ð2€

ðA:1Þ

þ D11 ð2w0 v00 v000 þ /0 v000 þ w00 v002 þ /00 v00 Þ  D22 ð4w00 w0 w000 þ w00 v0 v000 þ w003 þ wðIV Þ w02 þ w00 v002 þ wðIV Þ þ v0 vðIV Þ w0 þ 3w0 v00 v000 Þ ¼ 0; €  €v0 w0  w_ 0 v_ 0 Þ þ D11 ðv000 w0 þ v00 w00 þ /00 Þ ¼ 0: I 1 ð/

ðA:2Þ ðA:3Þ

The associated boundary conditions at x = 0 and x = l are D11 ð/0 w0 þ w02 v00 Þ þ D22 v00 ¼ 0;

v ¼ 0;

w ¼ 0; D22 w00 ¼ 0;

/ ¼ 0:

ðA:4Þ

It was noted in Section 2 that the torsional inertia terms can be neglected in comparison with the flexural inertia and stiffness terms. Using this assumption, Eqs. (A.3) and (A.4) give Z x v00 w0 dx þ    : ðA:5Þ /¼ 0

As mentioned earlier, because the shaft is slender, rotary inertia is small. Therefore, nonlinear terms that involve the rotary inertia in Eqs. (A.1) and (A.2) are negligible. Now, substituting Eq. (A.5) into Eqs. (A.1) and (A.2), and using nondimensional quantities (12) one may obtain Eq. (13). Appendix B B.1. Terms W1 and W2 used in Eq. (18) are defined as Z x Z 0 0 2 0 0 2 0 0 2 0 2 00 W1 ¼ v1 ðw1 D0 w1 þ v1 D0 v1 þ ðD0 w1 Þ þ ðD0 v1 Þ Þdx  v1 0

x 1

Z

x

ðw01 D20 w01 þ v01 D20 v01 þ ðD0 w01 Þ

2

00 þ ðD0 v01 Þ Þdx dx þ 2I 2 ðD0 D2 v001 þ XD2 w001 Þ  ðcD0 v1 þ 4v001 v01 v000 1 þ 2D0 D2 v1 þ v1 2

ðIV Þ

2

0 3

ðIV Þ

00 0 000 02 0 0 þ w001 v001 þ v001 w01 w000 1 þ 3w1 v1 w1 þ v1 v1 þ w1 v1 w1 Þ; Z xZ x ðw01 D20 w0 þ v01 D20 v01 þ ðD0 w01 Þ2 þ ðD0 v01 Þ2 Þdx dx W2 ¼ w001 1 0 Z x 2 2 ðw01 D20 w01 þ v01 D20 v01 þ ðD0 w01 Þ þ ðD0 v01 Þ Þdx þ 2I 2 ðD0 D2 w001  XD2 v001 Þ  w01

ðB:1Þ

0 ðIV Þ

ðIV Þ

00 0 000 003 002 00 00 0 000 0 0 02  ðcD0 w1 þ w001 v01 v000 1 þ 4w1 w1 w1 þ w1 þ 2D0 D2 w1 þ v1 w1 þ 3v1 w1 v1 þ v1 v1 w1 þ w1 w1 Þ:

ðB:2Þ

S.A.A. Hosseini, S.E. Khadem / Mechanism and Machine Theory 44 (2009) 272–288

287

B.2. Terms W3 and W4 used in Eq. (30) are defined as W3 ¼ cD0 V 1  2n2 p2 I 2 XD2 W 1  n6 p6 ðV 1 W 21 þ V 31 Þ  2D0 D2 V 1  2n2 p2 I 2 D0 D2 V 1   3 2 2 1 4 4 2 2 þ n p  n p ðV 21 D20 V 1 þ V 1 ðD0 V 1 Þ þ V 1 ðD0 W 1 Þ þ V 1 W 1 D20 W 1 Þ; 8 3 W4 ¼ cD0 W 1 þ 2n2 p2 I 2 XD2 V 1  n6 p6 ðW 31 þ V 21 W 1 Þ  2D0 D2 W 1  2n2 p2 I 2 D0 D2 W 1   3 2 2 1 4 4 2 2 n p  n p ðW 21 D20 W 1 þ W 1 ðD0 V 1 Þ þ W 1 ðD0 W 1 Þ þ V 1 W 1 D20 V 1 Þ: þ 8 3

ðB:3Þ

ðB:4Þ

B.3. Terms Gi (i = 3, 4) and Hi (i = 3, 4) used in Eq. (32) are defined as G3 ðT 2 Þ ¼ 2Iðn2 p2 I 2 bf  n2 p2 I 2 X þ bf ÞD2 F 1 ðT 2 Þ 1 þ CF 1 ðT 2 ÞF 2 ðT 2 ÞF 2 ðT 2 Þ  4n6 p6 F 1 ðT 2 ÞF 21 ðT 2 Þ  cbf F 1 ðT 2 ÞI; 2

ðB:5Þ

G4 ðT 2 Þ ¼ 2ðn2 p2 I 2 bf þ n2 p2 I 2 X  bf ÞD2 F 1 ðT 2 Þ 1  CIF 1 ðT 2 ÞF 2 ðT 2 ÞF 2 ðT 2 Þ þ 4n6 p6 IF 1 ðT 2 ÞF 21 ðT 2 Þ  cbf F 1 ðT 2 Þ; 2

ðB:6Þ

H 3 ðT 2 Þ ¼ 2Iðn2 p2 I 2 bb þ n2 p2 I 2 X þ bb ÞD2 F 2 ðT 2 Þ 1 þ CF 1 ðT 2 ÞF 1 ðT 2 ÞF 2 ðT 2 Þ  4n6 p6 F 2 ðT 2 ÞF 22 ðT 2 Þ  cbb F 2 ðT 2 ÞI; 2

ðB:7Þ

H 4 ðT 2 Þ ¼ 2ðn2 p2 I 2 bb þ n2 p2 I 2 X þ bb ÞD2 F 2 ðT 2 Þ 1 þ CIF 1 ðT 2 ÞF 1 ðT 2 ÞF 2 ðT 2 Þ  4n6 p6 F 2 ðT 2 ÞF 22 ðT 2 ÞI þ cbb F 2 ðT 2 Þ; 2

ðB:8Þ

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