International Journal of Structural Stability and Dynamics Vol. 12, No. 2 (2012) 251284 # .c World Scienti¯c Publishing Company DOI: 10.1142/S0219455412500034
PARAMETRIC INSTABILITY OF TWO-DISK ROTOR WITH TWO INERTIA ASYMMETRIES
Int. J. Str. Stab. Dyn. 2012.12:251-284. Downloaded from www.worldscientific.com by 166.111.55.40 on 03/13/13. For personal use only.
Q. K. HAN* and F. L. CHU† School of Precision Instruments and Mechanology Tsinghua University, Beijing 100084, China *
[email protected] †
[email protected] Received 1 November 2010 Accepted 5 January 2011 Determination of operating conditions of parametric instability is crucial to the design and usage of the inertia asymmetric rotor. Current research mostly focused on the rotor with single inertia asymmetric disk. There are few studies on the multi-disk rotors with multiple inertia asymmetries. In fact, the interaction between the multiple parametric excitations with various phasing and amplitude, which are induced by the multiple unsymmetrical disks, would make the instability behavior of the system di®er distinctly from that of the single-disk rotor system. Thus, the parametric instability of the two-disk rotor system with two inertia asymmetries is studied herein. Two important indicators for describing the unstable regions, namely the unstable rotating speed and width of the unstable region, are de¯ned and derived using the parametric instability theory and Taylor expansion technique. For a practical used two-disk unsymmetrical rotor, three design parameters (inertia excitation phasing, relative position of the disk, and inertia ratio) are discussed in detail for their e®ects on the two indicators. It is shown from the results that the dynamic instability of the two-disk unsymmetrical rotor system indeed has some unique features that di®er from that of the one-disk rotor system. The interaction of the two inertia parametric excitations could be utilized to control (or enhance) the unstable regions. Keywords: Parametric instability; rotor system; unsymmetrical inertias.
1. Introduction A rotor with rotary inertia inequality is known as a typical unsymmetrical rotor, whose dynamic behavior could be modeled by a coupled set of second-order linear di®erential equations with time-periodic coe±cients. This type of rotor system is of great theoretical and practical importance because of its many applications in various ¯elds of engineering. A two-blade propeller, a fan or pump impeller, a teetered wind turbine, a cam shaft, and a rotary plow all have unequal rotary inertia about two principal axes of the rotating disk. With the increasing demands of high speed, heavy load, and lightweight in modern rotor systems, the system operating speed may be * Corresponding
author. 251
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252
Q. K. Han & F. L. Chu
higher than the ¯rst-, second-, even multi-order critical speeds. Thus, the parametric excitation from the unsymmetrical inertia causes instability and severe vibration under certain operating conditions. Determination of operating conditions of parametric instability is crucial to the design and usage of the unsymmetrical rotor system. On the investigation of the inertia asymmetric rotors, many studies focused on the instability behavior of systems with single parametric excitation (induced by one unsymmetrical disk). In a remarkable paper presented in 1933, the asymmetric inertia properties in rotor system was ¯rst noticed by Smith.1 In 1961, Crandall2 conducted a quantitative analysis of instability problem for an asymmetric rotor with inertia inequality. Later, a series of in-depth theoretical and experimental studies on the unstable vibrations of a rotating shaft carrying an unsymmetrical rotor were carried out by Yamamoto.36 The e®ects of coupling between inertia and sti®ness inequalities, damping and °exible bearing pedestals were discussed in detail. Arday¯o and Frohrib7 extended the four degree-of-freedom rotor system of Yamamoto and Ota3 to include °exibility in the bearing supports, and obtained the unstable regions due to the system asymmetry parameters. Based upon the ¯nite element method and on the use of complex coordinates, the equations of motion for the study of the °exural dynamic behavior of an unsymmetrical rotor system were described by Genta.8 The model could take into account the presence of the nonaxisymmetrical stator and damping (both of viscous and hysteretic types). Recently, Nriot9 applied stochastic methods, dealing with uncertain inertia parametric excitation to rotating machines with constant rotating speed subjected to a base translational motion. For the asymmetric overhung rotor model of Crandall and Brosens2 with ¯nite displacement nonlinearities, the secondary whirling behavior was studied by Nandakumar and Chatterjee10 using multiple scales method. The usual phenomena of resonances, namely saddle-node bifurcations, jump phenomena, and hysteresis were observed in their study. Ishida and Lin11 proposed a simple passive control method utilizing discontinuous spring characteristics to eliminate the unstable ranges of an inertia asymmetrical rotor system. However, few studies have been made on the dynamic behavior of rotors with two (even multiple) asymmetric disks, which would induce multiple inertia parametric excitations. Wang et al.12 conducted the dynamic analysis for complex asymmetric rotor-bearing systems by the ¯nite element method. Kang et al.13 focused on the utilization of the ¯nite element technique to accommodate the e®ects of both deviatoric inertia and sti®ness due to the asymmetry of °exible shaft and multiple disks. The Timoshenko beam element is employed to simulate rotor-bearing systems by taking into account the gyroscopic moments, rotary inertia, transverse shear deformation of shaft, and asymmetry of disk and shaft. The steady-state analyses of asymmetric rotor systems for determining the synchronous critical speeds and subcritical speeds are estimated in conjunction with the harmonic balance method. Based upon these, Kang et al.14,15 proposed a modi¯ed in°uence coe±cient procedures for balancing the asymmetric rotor-bearing systems. Recently, Hsieh et al.16
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Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
253
developed an extended transfer matrix method for analyzing the coupled lateral and torsional vibration responses of asymmetric rotor-bearing system. As the literature shows, extensive e®orts have been devoted to study the steady-state response of rotors with multiple asymmetric disks, and only Ref. 17 has paid attention to the parametric instability. The in°uence of bearing damping on instability of the asymmetric rotor system was mainly investigated, while the interaction between the multiple inertia asymmetries was not considered in Ref. 17. Actually, the multiple parametric excitations with various phasing and amplitudes do in°uence each other. It has been shown theoretically18 and experimentally19 that the ranges of some unstable regions of a linear vibration system with two sti®ness parametric excitations would be attenuated and even vanished by adjusting the excitation phasing in a certain range. Thus, it is foreseeable that the multiple inertia (mass) parametric excitations would bring signi¯cant impacts to the instability of the unsymmetrical rotor system. The objective of this paper is to study the parametric instability of a two-disk rotor with two inertia asymmetries. The vibration model of the asymmetric rotor is established and written in a dimensionless form utilizing the Lagrange equation. The methods for analyzing the whirling properties and parametric instability are introduced. Then, the two important indicators for describing the unstable regions, namely the unstable rotating speed and width of unstable region, are derived using the instability theory and the Taylor expansion technique. By taking a practical used two-disk unsymmetrical rotor as an example, the whirling frequency and all of the unstable regions in excitation parameters' plane are computed numerically, and some typical features are obtained qualitatively. The e®ects of three design parameters (inertia excitation phasing, relative position of the disk and inertia ratio) on the two indictors are discussed in detail. Finally, some useful conclusions are presented. 2. Vibration Model of Two-Disk Rotor with Unsymmetrical Inertias A simply supported rotor system, as shown in Fig. 1, consists of a light shaft and two unsymmetrical disks without any static and dynamic unbalance. The angular rotating velocity is . Without considering the axial and torsional vibrations, the xi , yi and xi , yi (i ¼ 1; 2) are, respectively, utilized to describe the transversal and angle vibration displacements of the two rotating disks. Thus, the vibratory system has eight degrees of freedom. L, l1 , and l2 denote the rotor span and the distances to two disks from the left/right support points of the shaft. The masses of the two disks are m1 and m2 . Besides, three (one polar and two diametral) moments of inertia about the z, x, and y-coordinates are represented by Izi , Ixi and Iyi (i ¼ 1; 2), respectively. For the inertia unsymmetrical disks, Ixi 6¼ Iyi . Figure 2 shows an angle between the principal axes of inertia of the two disks due to the installation. In the following analysis, it will be shown that the angle characterizes the phasing of the two inertia parametric excitations, and would have an important impact on the dynamic instability of the rotor system.
Q. K. Han & F. L. Chu
Int. J. Str. Stab. Dyn. 2012.12:251-284. Downloaded from www.worldscientific.com by 166.111.55.40 on 03/13/13. For personal use only.
254
Fig. 1. A simply supported two-disk rotor system.
Fig. 2. Angle between the principal axes of inertia of the two disks.
The kinetic energy T of the rotor is: T ¼
: 1 1 m1 ðx_ 21 þ y_ 21 Þ þ Ix1 ð_x1 cos t þ y1 sin tÞ 2 2 2 : : : 1 1 þ Iy1 ð_x1 sin t y1 cos tÞ 2 þ Iz1 ðx1 y1 x1 y1 Þ 2 2
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
255
: 1 þ Ix2 ð_x2 cosðt þ Þ þ y2 sinðt þ ÞÞ 2 2 : 1 1 þ Iy2 ð_x2 sinðt þ Þ y2 cosðt þ ÞÞ 2 þ m2 ðx_ 22 þ y_ 22 Þ: 2 2
ð1Þ
By introducing the mean values of the diametral moments of inertia Ii ¼ ðIxi þ Iyi Þ=2, the inertia asymmetries of the two disks Ii ¼ ðIxi Iyi Þ=2 (i ¼ 1; 2) and the generalized freedom vector,
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¼ 1
2 T
¼ x1 y1
x2
y2
y1
x1
y2
x2 T ;
ð2Þ
one can rewrite Eq. (1) as: T ¼
1 :T : 1: ðM þ dMtÞ þ T G; 2 2
ð3Þ
in which the constant M, time-variant mass matrices dMt, and the gyroscopic matrix G are, respectively, derived as: " # M1 0 M¼ 0 M1 3 2 m1 7 6 7 6 I1 7 6 7 6 7 6 m2 7 6 7 6 7 6 7 6 I 2 7 6 ð4Þ ¼6 7; 7 6 m1 7 6 7 6 7 6 7 6 I1 7 6 7 6 7 6 m 2 5 4 I2 2
0 0 0 0 6 6 0 I1 cos 2t 0 0 6 6 60 0 0 0 6 6 6 0 0 I2 cos 2ðt þ Þ 60 dMt ¼ 6 6 0 0 0 60 6 6 6 0 I1 sin 2t 0 0 6 6 60 0 0 0 4 0 0 0 I2 sin 2ðt þ Þ
0
0
0
0 I1 sin 2t 0 0
0
0
0
0
0
0
0
0
0
I1 cos 2t
0
0
0
0
0
0
0
0
3
7 7 7 7 7 0 7 7 7 I2 sin 2ðt þ Þ 7 7; 7 0 7 7 7 7 0 7 7 7 0 5 I2 cos 2ðt þ Þ 0
ð5Þ
256
Q. K. Han & F. L. Chu
" G¼
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2
0
G1
G1
0
0
0
6 0 60 6 60 0 6 6 60 0 6 ¼6 60 0 6 6 6 0 Iz1 6 6 0 40 0
0
#
0
0
0
0
0
0
0
0 Iz1 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 Iz2 0
0
0
0
3
7 0 7 7 0 7 7 7 Iz2 7 7 7: 0 7 7 7 0 7 7 7 0 5
ð6Þ
0
The potential energy V of the lightly °exible shaft is 1 V ¼ T K; ð7Þ 2 in which K is the sti®ness matrix of the system. Due to material damping, the Rayleigh dissipation energy H could be expressed as: 1: : H ¼ T C; ð8Þ 2 where C denotes the damping matrix. By substituting Eqs. (3), (7), and (8) into the following Lagrange equation: d @ðT V Þ @ðT V Þ @H þ : ¼ 0; ð9Þ : dt @ @ @ one can obtain the vibration di®erential equations for the two-disk unsymmetrical rotor system without outer excitations as: :: : ~ ~ ðM þ MðtÞÞ þ ðC þ G þ GðtÞÞ þ K ¼ 0;
ð10Þ
~ ~ þ Te Þ ¼ in which MðtÞ ¼ dMt is the time-variable part of mass matrix, and Mðt ~ MðtÞ, where Te ¼ 2=!e is the parametric period and !e ¼ 2 is called the parametric (internal) frequency. Besides, the gyroscopic matrix has a time-variable part ~ ~ GðtÞ due to the inertia asymmetries. The expression of GðtÞ could be derived as: 2
0 0 6 0 I sin 2t 1 6 60 0 6 6 60 0 ~ 6 GðtÞ ¼ 26 0 0 6 6 6 0 I1 cos 2t 6 40 0 0 0
0 0 0 0 0 0 0 I2 sin 2ðt þ Þ 0 0 0 0 0 0 0 I2 cos 2ðt þ Þ
3 0 0 0 0 7 0 I1 cos 2t 0 0 7 7 0 0 0 0 7 7 0 0 0 I2 cos 2ðt þ Þ 7 7; 7 0 0 0 0 7 7 0 I1 sin 2t 0 0 7 7 5 0 0 0 0 0 0 0 I2 sin 2ðt þ Þ
ð11Þ
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Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
257
~ þ Te Þ ¼ GðtÞ ~ in which Gðt is also time-periodic. Obviously, Eq. (10) is a coupled set of second-order linear di®erential equations with periodically time-varying coe±cients, which is referred to as a parametrically excited system. Because of the two inertia asymmetric disks, there are two parametric excitations with various phasing angles pi (i ¼ 1; 2) in the range of ½0; 2. From the expressions ~ ~ of MðtÞ and GðtÞ, one can ¯nd that p1 ¼ 0 and p2 ¼ 2. Thus, 2 ½0; is closely related to the parametric excitation phasing, and it will be mainly considered for the e®ects upon the parametric instability of the two-disk rotor system. For the rigidly supported rotor, the compliance matrix F of the system is nonsingular. As long as the F is computed, the sti®ness matrix K could be easily obtained utilizing the relation K ¼ F 1 . For the engineering beam, the element fij of the compliance matrix is computed as: Z fij ¼ z
M 0j ðzÞM i0 ðzÞ dz; EJb
ð12Þ
where M i0 ðzÞ and M j0 ðzÞ are the bending moment distributions due to the unit load at the ith and jth nodes, respectively. The Young's modulus and moment of inertia of the shaft cross section are denoted by E and Jb . For the circular cross section with diameter d, the Jb ¼ d 4 =64. It is shown from Eq. (12) that fij ¼ fji and F is a symmetric matrix. The M i0 ðzÞ and M j0 ðzÞ for the simply supported beam under concentrated force or bending moment are easily obtained, and the elements of compliance matrix could be solved using Eq. (12). Then, the sti®ness matrix is given by: 2 3 K1 0 5 K ¼4 0 K1 3 2 k11 k12 k13 k14 0 0 0 0 7 6 7 6 7 6 k12 k22 k23 k24 0 0 0 0 7 6 7 6 7 6 6 k13 k23 k33 k34 0 0 0 0 7 7 6 7 6 7 6 0 0 0 7 6 k14 k24 k34 k44 0 7 6 ð13Þ ¼6 7: 6 0 0 0 0 k11 k12 k13 k14 7 7 6 7 6 7 6 6 0 0 0 0 k12 k22 k23 k24 7 7 6 7 6 7 6 6 0 0 0 0 k13 k23 k33 k34 7 7 6 5 4 0 0 0 0 k14 k24 k34 k44
258
Q. K. Han & F. L. Chu
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If the proportional viscous damping is considered and the coupling damping parts are ignored, then the damping matrix of the system could be expressed as: 3 2 c1 7 6 c3 7 6 7 6 c 2 7 6 7 6 c 4 7; 6 ð14Þ C¼6 7 c 1 7 6 7 6 c3 7 6 5 4 c2 c4 in which ci (i ¼ 1; 2; 3; 4) are the damping coe±cients. pffiffiffiffiffiffiffiffiffiffiffiffiffi A reference displacement and angular velocity are introduced as xst ¼ I1 =m1 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !st ¼ k11 =m1 . Then, the transversal displacements, time, and rotating velocity pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi could be written in nondimensional forms as: x 01 ¼ x1 = I1 =m1 , y 01 ¼ y1 = I1 =m1 , pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 02 ¼ x2 = I1 =m1 , y 02 ¼ y2 = I1 =m1 , t 0 ¼ t k11 =m1 , 0 ¼ = k11 =m1 . For convenience sake, the primes on the dimensionless quantities are omitted in the following. By substituting the above dimensionless quantities into Eq. (10), and de¯ning some parameters, including: the mass ratio ¼ m2 =m1 ; relative inertia asymmetries i ¼ Ii =Ii (i ¼ 1; 2); inertia ratios i ¼ Izi =Ii (i ¼ 1; 2); viscous damping ratios pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi g1 ¼ c1 = m1 k11 , g2 ¼ c2 =ðI1 k11 =m1 Þ, g3 ¼ c3 = m1 k11 , and g4 ¼ c4 =ðI2 k11 =m1 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffi dimensionless forms of the sti®ness matrix elements z12 ¼ k12 m1 =I1 =k11 , z13 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi k13 =k11 , z14 ¼ k14 m1 =I1 =k11 , z22 ¼ k22 m1 =ðk11 I1 Þ, z23 ¼ k23 m1 =I1 =k11 , z24 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi k24 m1 =ðk11 I1 Þ, z33 ¼ k33 =k11 , z34 ¼ k34 m1 =I1 =k11 , z41 ¼ k14 m1 I1 =ðk11 I2 Þ, pffiffiffiffiffiffiffiffiffiffiffi z42 ¼ k24 m1 =ðk11 I2 Þ, z43 ¼ k34 m1 I1 =ðk11 I2 Þ, and z44 ¼ k44 m1 =ðk11 I2 Þ, one gets the dimensionless form of the vibration di®erential equations for the system as: þG þG ~ ðtÞÞ: þ K þM ~ ðtÞÞ:: þ ðC ¼ 0; ðM where
" ¼ M
1 M
#
1 M
0 21
6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 4
0
ð15Þ
3 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 5
1 1 1 1 1
ð16Þ
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
2
~ MðtÞ
0 0 0 0 6 6 0 1 cos 2t 0 0 6 6 60 0 0 0 6 6 0 0 2 cos 2ðt þ Þ 60 ¼6 60 0 0 0 6 6 6 0 sin 2t 0 0 6 1 6 60 0 0 0 4 0 0 0 2 sin 2ðt þ Þ
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2 6 6 6 6 6 6 6 C ¼6 6 6 6 6 6 4
0
0
0
0 1 sin 2t 0 0
0
0
0
0
0
0
0
0
0
1 cos 2t
0
0
0
0
0
0
0
0
259
3
7 7 7 7 7 0 7 7 2 sin 2ðt þ Þ 7 7; 7 0 7 7 7 0 7 7 7 0 5 2 cos 2ðt þ Þ 0
ð17Þ
3
g1
7 7 7 7 7 7 7 7; 7 7 7 7 7 5
g3 g2 g4 g1 g3 g2
ð18Þ
g4 " ¼ G 2
0
1 G
1 G
0
0
#
0
6 0 60 6 60 0 6 6 60 0 6 ¼6 60 0 6 6 6 0 1 6 6 0 40 0 2
0 0 6 6 0 1 sin 2t 6 6 60 0 6 6 6 0 0 ~ ðtÞ ¼ 26 G 60 0 6 6 6 0 cos 2t 6 1 6 60 0 4 0 0
0
0
0
0
0
0
0
0 1 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 2 0
0
0
0
0
0
3
7 0 7 7 0 7 7 7 2 7 7 7; 0 7 7 7 0 7 7 7 0 5
ð19Þ
0
0
0
0
0
0
0
0 1 cos 2t 0
0
0
0
0
0
0
2 sin 2ðt þ Þ
0
0
0
0
0
0
0
0
0
0
0 1 sin 2t 0
0
0
0
0
0
0 2 cos 2ðt þ Þ 0
0
0
0
3
7 7 7 7 7 0 7 7 2 cos 2ðt þ Þ 7 7; 7 0 7 7 7 0 7 7 7 0 5 2 sin 2ðt þ Þ 0
ð20Þ
260
Q. K. Han & F. L. Chu
" ¼ K
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2
1 K 0 1
0
#
1 K z12 z13 z14
0
0
0
0
3
7 6 0 0 0 7 6 z12 z22 z23 z24 0 7 6 6z 0 0 0 7 7 6 13 z23 z33 z34 0 7 6 7 6z z z z 0 0 0 0 7 6 41 42 43 44 ¼6 7: 7 6 0 0 0 0 1 z z z 12 13 14 7 6 7 6 6 0 0 0 0 z12 z22 z23 z24 7 7 6 7 6 0 0 0 z13 z23 z33 z34 5 4 0 0
0
0
0
ð21Þ
z41 z42 z43 z44
3. Numerical Methods for Whirling Properties and Parametric Instability The dynamic model of the two-disk rotor system has been established and written in dimensionless form in Sec. 2. Here, the numerical methods for analyzing the whirling properties and parametric instability will be introduced in brief. 3.1. Whirling theory for multi-disks rotor system Without considering the inertia asymmetries (1 ¼ 2 ¼ 0) and viscous damping ~ ðtÞ ¼ 0 and C ¼ 0. Thus, the vibration ~ (g1 ¼ g2 ¼ g3 ¼ g4 ¼ 0), we have MðtÞ ¼G model of the rotor system (shown as Eq. (15)) could be expressed as: ( 1 : 2 þ K 1 :: 1 1 ¼ 0 M 1 þ G : ð22Þ :: : 1 2 G 1 1 þ K 1 2 ¼ 0 M In dynamic analysis of the multi-disks rotor system, a complex vector is utilized to describe thepdisplacements for the axial centers of rotating disks, i.e., z ¼ 1 þ j2 , in ffiffiffiffiffiffiffi which j ¼ 1. Then, Eq. (22) is rewritten as: 1 z:: þ K 1 z:: jG 1 z ¼ 0: M
ð23Þ
By substituting the solution form z ¼ z0 e j!t into the above equation, we obtain the eigenvalue problem of the system 1þK 1 z0 ¼ 0: 1 þ !G ½! 2 M
ð24Þ
As mentioned above, the two-disk rotor system has eight degrees of freedom. Given one rotating angular velocity , eight whirling frequencies !i , (i.e., four forward whirling frequencies (i ¼ 1; 2; 3; 4) and four backward whirling frequencies (i ¼ 1; 2; 3; 4)), could be obtained by solving the eigenvalue problem in Eq. (24). Here, the polyeig function in MATLABr is used in the solution. The
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
261
variations of the eight whirling frequencies with rotating angular velocity are gained by repeating the above process. In addition, if the rotating angular velocity equals to one of the four forward whirling frequencies, it is also called as synchronous forward whirling velocity. There are four of such velocities, denoted by si (i ¼ 1; 2; 3; 4).
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3.2. Numerical method for determining parametric unstable regions The parametric excitation from the unsymmetrical inertia causes instability and severe vibration under certain operating conditions. The aim of the parametric instability analysis is to determine the unstable ranges in the plane of parametric frequency and amplitude. The discrete state transition matrix (DSTM) method is commonly used in the parametric instability analysis. For given parameters, whether the parametric system is unstable or not could be judged by estimating the modulus of the complex eigenvalues of the DSTM. Thus, the unstable range is determined point by point in the parameters' plane. Obviously, the key point for the method is, how to obtain the DSTM of the system. Presently, many numerical methodologies, such as the direct integration technique and approximation technique, have been developed for computing the DSTM. Detailed research progress about this area is available in Ref. 20. Here, a numerical method presented by Friedmann21 is utilized to estimate the DSTM. By taking yðtÞ ¼ ½_ T , Eq. (15) could be transformed into the state space as: ::
y ¼ AðtÞy; in which the coe±cient matrix AðtÞ is expressed as: " # 1 " # þM ~ ðtÞ M ~ ðtÞ 0 0 M M AðtÞ ¼ : þG þG ~ ðtÞ þM ~ ðtÞ C 0 K M
ð25Þ
ð26Þ
Obviously, Aðt þ Te Þ ¼ AðtÞ. One parametric period ð0; Te Þ is divided into N equal parts and every part is k (k ¼ 1; 2; . . . ; N). The kth time interval ð k1 ; k Þ is denoted as k , and the length is %k ¼ ð k k1 Þ. Then, the time-periodic coe±cient matrix AðtÞ in the kth time interval could be approximately replaced by a timeconstant matrix Ck , which is computed numerically as: Z k 1 Ck ¼ AðÞd 2 k : ð27Þ %k k1 Thus, the approximate DSTM A ðT ; NÞ in one period could be gained as: A ðT ; NÞ ¼
N Y
e %k C k :
ð28Þ
k¼1
According to the research results by Hsu,22A ðT ; NÞ will approach the exact solution as N ! 1. Thus, as long as the parametric period is divided into small enough, the
262
Q. K. Han & F. L. Chu
obtained DSTM would meet the accuracy requirements. The characteristic multipliers i , which are just the eigenvalues of the DSTM, are utilized to determine whether the parametric system is unstable or not. For an unforced periodic system to be stable, it is su±cient that
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ji j 1
8i:
ð29Þ
If ji j > 1 for any i, then the system is to be unstable and parametric resonance occurs. Thus, the process for the stability analysis of parametric system by DSTM method is summarized as: ¯rst, for selected system parameters, the DSTM is computed numerically using Eq. (28); then, the characteristic multipliers are obtained by solving the eigenvalue problem of the matrix; ¯nally, whether the system is stable or not with given system parameters is judged by Eq. (29). For other system parameters, the instability would be determined by repeating the above-mentioned process. Thus, the unstable regions of the system are scanned one by one in the parameter's plane. 4. Two Indicators for Unstable Region For the inertia unsymmetrical rotor system, the parametric frequency !e is twice the rotating angular velocity , and the amplitudes of the parametric excitation are controlled by the relative inertia asymmetries 1 and 2 . Two types of instability are of interest: the primary and combination instability regions, which are denoted pðcÞ by U ijp (i ¼ j) and U ijc (i 6¼ j), respectively. A typical unstable region U ij without damping in the plane 1ð2Þ is given in Fig. 3. When the relative inertia asympðcÞ metries equal to zero (1ð2Þ ¼ 0), the range of U ij is just a point. Then, with the pðcÞ increasing of 1ð2Þ , the range of U ij is enlarging gradually, indicating that the parametric excitation becomes more and more intensive. Here, the unstable rotating pðcÞ pðcÞ speed ij and the width of unstable region ij , as shown in the ¯gure, are utilized as two indicators in order to describe the unstable region more conveniently. The unstable rotating speed is just the angular velocity corresponding to the unstable
Fig. 3. A typical parametric unstable region without damping.
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
263
starting point. Clearly, the width of unstable region is closely related to the unstable rotating speed and the inertia asymmetries. In the following, the methods for solving the two indicators will be introduced based upon the parametric instability theory and the Taylor expansion technique.
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4.1. Unstable rotating speed According to the parametric instability theory, if the parametric excitation frequencies are equal to (or close to) particular combinations of the natural frequencies, then the system would become unstable and resonant. For the rotor system, the natural frequencies are referred to as the forward whirling frequencies. Thus, when the system is at the starting point of primary instability region U ijp , the parametric excitation frequency !e and the forward whirling frequency ! pi should satisfy !e ¼ 2! pi
ði ¼ 1; 2; 3; 4Þ:
ð30Þ
By substituting !e ¼ 2 pij into Eq. (30), one can obtain the primary unstable rotating speed pij as: pij ¼ ! pi
ði ¼ j ¼ 1; 2; 3; 4Þ:
ð31Þ
It is shown that the ith primary unstable rotating speed is such a rotating angular velocity that has the equal values with the ith forward whirling frequency. Obviously, the primary unstable rotating speed is just the synchronous forward whirling velocity si . For the system at the starting point of the combination instability region U ijc , the relation between the excitation frequency and two orders of forward whirling frequency is !e ¼ ! ci þ ! cj
ði 6¼ j ¼ 1; 2; 3; 4Þ:
ð32Þ
Similarly, the combined unstable rotating speed cij could be gained as: cij ¼ ð! ci þ ! cj Þ=2 ði 6¼ j ¼ 1; 2; 3; 4Þ:
ð33Þ
One can ¯nd that the cij is such a rotating angular velocity that equals to the half of the sum of the ith and jth forward whirling frequencies. In whirling analysis, for a certain rotating velocity, the whirling frequencies could be solved. Then, in virtue of the polynomial ¯tting technique, the variations of the four forward whirling frequencies with rotating velocity are obtained, respectively. Then, according to the relationship in Eqs. (31) and (33), the four primary unstable rotating speeds ( p11 , p22 , p33 and p44 ) and six combined unstable rotating speeds ( c12 , c13 , c14 , c23 , c24 and c34 ) could be computed one by one. 4.2. Width of unstable region Based upon the parametric instability theory and the Taylor expansion technique, an pðcÞ approximate method is introduced to compute the width of the unstable region ij .
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264
Q. K. Han & F. L. Chu
Without considering system damping, there are two frequency components ! and $ ($ ¼ 2 !) in the free response of the rotor system due to the inertia asymmetries. Thus, the free response of the system could be expressed as follows: 8 x1 ¼ A1 cos !t E1 sin !t þ B1 cos $t F1 sin $t > > > > y1 ¼ A1 sin !t þ E1 cos !t þ B1 sin $t þ F1 cos $t > > > > > > x2 ¼ A2 cos !t E2 sin !t þ B2 cos $t F2 sin $t > > > < y ¼ A sin !t þ E cos !t þ B sin $t þ F cos $t 2 2 2 2 2 ; ð34Þ > ¼ C sin !t þ G cos !t þ D sin $t þ H 1 1 1 1 cos $t > > x1 > > > y1 ¼ C1 cos !t G1 sin !t þ D1 cos $t H1 sin $t > > > > > x2 ¼ C2 sin !t þ G2 cos !t þ D2 sin $t þ H2 cos $t > > : y2 ¼ C2 cos !t G2 sin !t þ D2 cos $t H2 sin $t in which Ai , Bi , Ci , Di , Ei , Fi , Gi and Hi (i ¼ 1; 2) are the unknown constant coef¯cients. Substituting Eq. (34) into Eq. (15) and taking the coe±cients of the harmonic function to be zero, one can obtain the linear algebraic equations about the unknown constant coe±cients 3 2 A1 6A 7 6 27 7 6 6 C1 7 7 6 6 C2 7 7 6 6B 7 6 17 7 6 6B 7 2 36 2 7 7 A11 A12 A13 A14 6 6 D1 7 7 6 6A 7 6 21 A22 A23 A24 76 D2 7 ð35Þ 7 ¼ 0; 6 76 4 A31 A32 A33 A34 56 E1 7 7 6 7 A41 A42 A43 A44 6 6 E2 7 6G 7 6 17 7 6 6 G2 7 7 6 6 F1 7 7 6 6F 7 6 27 7 6 4 H1 5 H2 in which A11 ¼ A33 2 ð1 ! 2 Þ z13 z12 6 6 z13 ðz33 ! 2 Þ z23 6 ¼6 2 6 z12 z23 ð! þ 1 ! þ z22 Þ 4 z41
z43
z42
z14 z34 z24 ð! 2 þ 2 ! þ z44 Þ
3 7 7 7 7; ð36Þ 7 5
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
A22 ¼ A44 2 ð1 $ 2 Þ z13 z12 6 6 z13 ðz33 $ 2 Þ z23 6 ¼6 6 z12 z23 ð$ 2 þ 1 $ þ z22 Þ 4 z41
z43
Int. J. Str. Stab. Dyn. 2012.12:251-284. Downloaded from www.worldscientific.com by 166.111.55.40 on 03/13/13. For personal use only.
A12 ¼ A21 ¼ A34 ¼ A43
0 0
z24
2
ð37Þ
3
0
6 0 60 0 ¼6 6 0 0 !$ 4 1 0
7 7 7 7; 7 5
z34
0
0 0
3
z14
ð$ 2 þ 2 $ þ z44 Þ
z42
2
265
7 7 7; 7 5
0 0
ð38Þ
2 !$ cos 2
0 0 0
6 60 0 0 A14 ¼ A23 ¼ A32 ¼ A41 ¼ 6 60 0 0 4
3
0
7 7 7; 7 5
0 0
ð39Þ
0 0 0 2 !$ sin 2 A13 ¼ A31 ¼ A24 ¼ A42 ¼ 0:
ð40Þ
Thus, one can obtain the frequency equation by taking the determinant of Eq. (35) to be zero as: 02 31 A11 A12 A13 A14 B6 A 7C B6 21 A22 A23 A24 7C detB6 ð41Þ 7C ¼ 2 ð!; Þ ¼ ðf f þ Þ 2 ¼ 0; @4 A31 A32 A33 A34 5A A41 A42 A43 A44 in which f ¼ detðA11 Þ, f ¼ detðA22 Þ, ¼ 1 21 þ 2 1 2 þ 3 22 þ 4 21 22 , where i (i ¼ 1; 2; 3; 4) are the sub-determinants, which will not be expressed here due to the space limitation. In fact, f and f are just the frequency equations of the system without taking into account the inertia asymmetries, that is, the determinant of the coe±cient matrix in Eq. (24) 1þK 1 j; 1 þ !G f ¼ j !2 M 1þK 1 j: 1 þ $G f ¼ j $ 2 M pðcÞ !i
ð42Þ ð43Þ
pðcÞ ij
and are the forward whirling frequency and unstable rotating speed As the pðcÞ pðcÞ of the rotor system with zero inertia asymmetries, so fð! i ; ij Þ ¼ 0, pðcÞ pðcÞ pðcÞ pðcÞ fð! ; Þ ¼ 0, and ð! ; Þ ¼ . When the inertia asymmetries are greater i
ij
i
ij
pðcÞ
pðcÞ
than zero (1ð2Þ > 0), the unstable region is around the point ð! i ; ij Þ, as shown in Fig. 3. Thus, the width of unstable region could be obtained through the Taylor pðcÞ pðcÞ expansion of at the point (! ¼ ! i þ &, ¼ ij þ ). Assuming &, and 1ð2Þ to
266
Q. K. Han & F. L. Chu
be small, one have pðcÞ
ð! i
pðcÞ
þ &; ij þ Þ
@ @ 1 @ @ 2 : pðcÞ pðcÞ þ
& þ
¼ ð! i ; ij Þ þ & ¼ 0: þ @! @ 2 @! @
ð44Þ
@ f @ f @ @ @f @f @! ¼ f @! þ f @! ¼ 0, @ ¼ f @ þ f @ ¼ 0, @f @ f @f @ f @ 2 @!@ ¼ @! @ þ @ @! . By substituting these
From Eq. (41), one can obtain:
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@f @ f @f @ f @ ¼ 2 @! @! , @ 2 ¼ 2 @ @ , and relations into Eq. (44) and simplifying, we have @f @ f 2 @f @ f @f @ f @f @ f 2
þ
&þ
þ ¼ 0: & þ @! @! @! @ @ @! @ @ @ 2 @! 2
2
Solving the quadratic equation about &, one can obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @f @ f @f @ f @f @ f @f @ f @f @ f 2 ð @!
þ
Þ ð @! @ @ @! @ @ @! Þ 4 @! @! : &¼ @f @ f 2ð @! @! Þ
ð45Þ
ð46Þ
When the system is in the unstable region, the & is a complex number, whose imaginary part characterizes the intensity of the parametric instability. The absolute value of the imaginary part of & is denoted by , and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi 1 @f @ f @f @f @f @f ¼
2: ð47Þ 4 2 @! @! @ @! @ @! When ¼ 0, the system is at the unstable boundary, and we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @f @f @ f @f @f @f
0 ¼ 4 @ @! @ @! : @! @! pðcÞ
Thus, the width ij pðcÞ
ij
ð48Þ
pðcÞ
of the unstable region U ij could be obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @f @f @ f @f @f @f ¼ 2 0 ¼ 4 @ @! @ @! : @! @!
ð49Þ
Based upon the parametric instability theory and Taylor expansion technique, the width of the unstable region could be solved approximately utilizing Eq. (49). In the following section, a practical used two-disk rotor system is taken as an example to study the whirling properties and parametric instability using the methods in the above sections. 5. Computation and Discussions The actual parameters and dimensions of a two-disk rotor system are5: m1 ¼ m2 ¼ 10:43 kg, I1 ¼ I2 ¼ 0:1397 kgm2, L ¼ 0:5045 m, and l1 ¼ 0:168 m. The Young's modulus of the shaft is E ¼ 2:058 10 11 N/m2. As the cross-sectional diameter
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
267
is d ¼ 0:01188 m, so moment of inertia of the shaft cross section is Jb ¼ 9:778 10 10 m4. A relative distance of disk 2 from the right support point is de¯ned as lr ¼ l2 =l1 . When lr ¼ 1, the two disks are symmetrically mounted on the shaft about the midpoint. For the case of lr > 1 (or lr < 1), disk 2 is more close to (or further from) the midpoint compared with disk 1. The dimensionless mass ratio ¼ 1 and inertia ratios in the range of 0 < 1 ; 2 < 2 will be considered in the analysis. In addition, the parametric excitation is assumed to be relative small, i.e., the relative inertia asymmetries varying in the range of 0 1 and 2 0:4.
Here, the parametric excitations are neglected (1 ¼ 2 ¼ 0). For 1 ¼ 2 ¼ 0:7 and lr ¼ 1, Fig. 4 gives all of the eight whirling frequencies !i varying with the rotating angular velocity in the range of 05. The velocities corresponding to the intersection points of the angular velocity line with the four forward whirling frequency lines are just the synchronous forward whirling velocities (also the primary unstable rotating speeds), as shown in the ¯gure. With the increasing of the rotating angular velocity, except for the third and fourth forward whirling frequencies, the other whirling frequencies seems to be little a®ected. In addition, as the rotating angular velocity increasing, the !3 line is ¯rst approaching then maintains parallel with the !4 line. With the same values of the above parameters, the variations of $i ¼ 2 !i and !i (i ¼ 1; 2; 3; 4) with the rotating angular velocity are plotted in Fig. 5. According to Eq. (33), one can ¯nd that the angular velocities corresponding to the intersection points are the six combined unstable rotating speeds, as marked in the ¯gure. 5
ω1 ω2 ω3 ω4 ω1 ω2 ω3 ω4 ω=Ω
4
3 Whirling frequency ωi
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5.1. Whirling properties
2
1
0
–1
–2
s
s
0
0.5
p
s p Ω2 (Ω22) Ω3 (Ω33)
p
Ω1 (Ω11)
1
1.5
2
Ωs4 (Ωp44)
2.5
3
3.5
4
4.5
5
Rotating angular velocity Ω
Fig. 4. Whirling frequencies of rotor system with 1 ¼ 2 ¼ 0:7 and lr ¼ 1.
268
Q. K. Han & F. L. Chu 4
ω1 ω2 ω3 ω4 ϖ1 ϖ2 ϖ3 ϖ4
3
Whirling frequency
2
1
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0
–1 c Ω12
–2
0
0.5
c
Ω14
Ωc13
1
Ωc
23
c
Ω24
1.5
c
Ω34
2
2.5
3
Rotating angular velocity Ω
Fig. 5. Combined unstable rotating speeds of rotor system with 1 ¼ 2 ¼ 0:7 and lr ¼ 1.
5.2. E®ects of design parameters on unstable rotating speed In the section, some of the design parameters, involving the inertia ratios (1 ¼ 2 ¼ ) and relative distance lr of disk 2, are studied for their e®ects upon the pðcÞ unstable rotating speeds ij . Three values of lr (0.81, 1, and 1.19) are considered in the computation, and the variations of the four primary and six combination unstable rotating speeds with in the range of 02 are plotted in Figs. 6 and 7, respectively. The p11 and p22 have been increasing for varying from 0 to 2, as shown in Figs. 6(a) and 6(b). However, p33 and p44 are only existing for in the range of 01. When is around or equal to 1, the values of the two primary unstable rotating speeds approaches in¯nity, which is shown in Figs. 6(c) and 6(d). As long as is greater than 1, there are no unstable speeds existing. This means that the p p system does not have the two primary instability regions U 33 and U 44 for 1 < < 2. Adding the value of lr with certain would reduce the unstable rotating speeds. As the is changing from 0 to 2, the combination unstable rotating speeds except for c34 are increasing with continuously, as shown in Figs. 7(a)7(e). Especially for c13 , c14 , c23 and c24 , the values of these unstable speeds increase rapidly when is approaching 2. The variation of c34 with is similar with that of the p33 and p44 , which is shown in Fig. 7(f). Thus, one can say that the combination c would not exist if the is designed in the range of 12. In instability region U 34 addition, increasing the value of lr makes the combined unstable rotating speeds decrease to some extent.
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries 0.5
1.4 lr=1 lr=1.19 lr=0.81
0.45 0.4 Ωp
1.2 Ωp
0.35
11
1
22
0.3
0.8
lr=1 lr=1.19 lr=0.81
0.25 0.6
0.2
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0.15
0
1 β
0.4
2
0
(a)
2
10 lr=1 lr=1.19 lr=0.81
6 33
1 β
(b)
8
Ωp
269
lr=1 lr=1.19 lr=0.81
8 Ωp
6
44
4
4 2
0
2
0
1 β
(c)
2
0
0
1 β
2
(d)
Fig. 6. Primary unstable rotating speeds varying with inertia ratio in the range of 02: (a) p11 ; (b) p22 ; (c) p33 ; and (d) p44 .
5.3. Unstable regions By utilizing the DSTM method, the unstable regions could be numerically obtained. Then, some qualitative properties for the parametric instability of the unsymmetrical rotor system will be discussed in the section. One parametric period ð0; Te Þ should be divided into N equal parts before the computation of DSTM. The greater the value of N taken, the higher the accuracy of the DSTM obtained. However, the greater value of N also means higher calculation cost. In the following, the value of N is taken to be 600 to meet the accuracy requirements. For 1 ¼ 2 ¼ 0:5 and lr ¼ 1, ten unstable rotating speeds could be, respectively, obtained after the whirling analysis: p11 ¼ 0:2716, p22 ¼ 0:9404, p33 ¼ 1:2412, p44 ¼ 1:8604, c12 ¼ 0:5847, c13 ¼ 0:6832, c14 ¼ 0:9910, c23 ¼ 1:0759, c24 ¼ 1:3780, and c34 ¼ 1:5710. The damping coe±cients are assumed to be equal, and g1 ¼ g2 ¼ g3 ¼ g4 ¼ g. The inertia asymmetry of disk 2 is neglected, i.e., 2 ¼ 0. When g ¼ 0, all of the unstable regions varying with the 1 in the range of 00.4 are
270
Q. K. Han & F. L. Chu 1
20
20 lr=1 lr=1.19 lr=0.81
0.9 15
0.8 Ωc 12
Ωc
0.7
13
lr=1 lr=1.19 lr=0.81
15 Ωc
14
10
10
0.6 0.5
l =1 r l =1.19 r l =0.81
0.4
5
5
r
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0.3
0
1 β
0
2
0
(a)
lr=1 lr=1.19 l =0.81
20
0
r
2
10 lr=1 lr=1.19 l =0.81
20
8
r
Ωc 15
Ωc
24
15
1 β
(c)
25
25
23
0
2
(b)
30
Ωc
1 β
6
34
10
4
5
5
2
0
0
0
10 l =1 r lr=1.19 l =0.81 r
0
1 β
(d)
2
0
1 β
(e)
2
0
1 β
2
(f)
Fig. 7. Combined unstable rotating speeds varying with inertia ratio in the range of 0-2: (a) c12 ; (b) c13 ; (c) c14 ; (d) c23 ; (e) c24 ; and (f) c34 .
obtained and shown in Fig. 8. In virtue of the unstable rotating speeds, each of the ten unstable regions are identi¯ed and marked in the Fig. 8. Obviously, points X1 (coordinates: 1 ¼ 0:16, ¼ 1:6) and X3 (coordinates: 1 ¼ 0:1, ¼ 1:45) are in unstable region, and points X2 (coordinates: 1 ¼ 0:27, ¼ 2) and X4 (coordinates: 1 ¼ 0:14, ¼ 2:25) are in stable region. Utilizing the numerical integration technique, the free responses of the asymmetric rotor system : at the above four points under initial conditions (ð0Þ ¼ ½0:01 0 0 0 0 0 0 0) are solved and plotted in Fig. 9. The free responses of unstable points X1 and X3 are all enlarging with time rapidly, while the maximum response amplitude of the system at points X2 and X4 seems to be little changed with time. Thus, one can say that the instability regions obtained by DSTM method are believable. With the increasing of 1 , the ranges of every unstable region are enlarging pðcÞ gradually. For g ¼ 0 and 1 ¼ 0:05; 0:1, the width of the unstable regions ij are computed, respectively, utilizing the approximate method in Sec. 4.2, and compared
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
c U23
0.7 1.1
Ω
0.6
1.05
c U12
0.5
c U13
0.95 p U11
0.3
0
p U22
0.9 0.1
0.2
0.3
0.4
0.85
0
0.1
0.2
(a)
0.3
0.4
0.3
0.4
(b) 2.5
1.7
2.4
c U34
X
1.6
2.3
1
X4
2.2
1.5
c U24
Ω
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0.2
c U14
1
0.4
271
X2
2.1
1.4
p U44
2 1.3
X
3
1.9
1.2
p U33
0
1.8 0.1
0.2 ∆1
(c)
0.3
0.4
0
0.1
0.2 ∆ 1
(d)
Fig. 8. Ten unstable regions for g ¼ 0: (a) in [0.2, 0.75]; (b) in [0.85, 1.15]; (c) in [1.12, 1.78]; and (d) in [1.74, 2.5].
with the ones obtained by the DSTM method. The results are listed in Table 1. pðcÞ There is little di®erence between the results of ij obtained by the two methods, and the maximal relative di®erence is less than 10%. Thus, the approximate method derived in the paper for computing the width of the unstable region is accurate and reasonable for small inertia parametric excitations. When the damping is considered (g ¼ 0:005; 0:03), similar computations are carried out and the instability results are presented in Figs. 10 and 11, respectively. For small damping, i.e., in Fig. 10, one can ¯nd that the lower orders of unstable p c c region (U 11 , U 12 and U 13 ) are reduced, while the higher orders of unstable region seems to be little a®ect by the damping. Continuing to increase the damping coefp c c ¯cient g ¼ 0:03, the U 11 and U 12 have been disappeared and U 13 has been reduced signi¯cantly in the considered range of inertia asymmetry (00.4), as shown in Fig. 11(a). The other unstable regions also have been narrowed with di®erent degrees, as plotted in Figs. 11(b)11(d). It is shown from the above investigations that the damping has certain weaken e®ect upon the unstable regions, especially for the lower orders of unstable region.
272
Q. K. Han & F. L. Chu 200
0.02
150 0.01 Response
Response
100 50 0 –50
0
–0.01
–100 0
50
100 time
–0.02
150
0
50
(a)
100 time
150
200
(b)
9
6
x 10
0.02
4 0.01 Response
2 Response
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–150
0 –2 –4
0
–0.01
–6 –8
0
50
100
150
–0.02
0
50
time
100 time
(c)
(d)
Fig. 9. Free responses of the asymmetric rotor system at four points of Fig. 8: (a) point X1 ; (b) point X2 ; (c) point X3 ; and (d) point X4 .
Table 1. Comparisons for the widths of unstable regions obtained by approximate and DSTM methods. 1 ¼ 0:05
p U 11 p U 22 p U 33 p U 44 c U 12 c U 13 c U 14 c U 23 c U 24 c U 34
1 ¼ 0:1
DSTM
Approx.
Di®erence (%)
DSTM
Approx.
Di®erence (%)
0.0011 0.0162 0.0550 0.0600 0.0043 0.0065 0.0058 0.0285 0.0255 0.0581
0.0012 0.0167 0.0566 0.0598 0.0044 0.0067 0.0060 0.0288 0.0258 0.0579
9.1 3.1 2.9 0.3 2.3 3.1 3.4 1.1 1.2 0.3
0.0023 0.0333 0.1125 0.1273 0.0088 0.0132 0.0118 0.0560 0.0497 0.1162
0.0024 0.0334 0.1133 0.1195 0.0088 0.0134 0.0120 0.0576 0.0516 0.1158
4.3 0.3 0.7 6.1 0.0 1.5 1.7 2.8 3.8 0.3
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
0.7
c U13
Ω
1.05 c U12
0.5
0.95 p U11
0.3
0
0.1
0.2
p U22
0.9
0.3
0.4
0.85
0
0.1
(a)
0.2
0.3
0.4
(b) 2.5
1.7
c U34
2.4
2.3
1.6
2.2
1.5
c U24
Ω
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c U14
1
0.4
0.2
c U23
1.1
0.6
273
1.4
2.1 p U44
2
1.3 1.9
1.2 U p 33
0
1.8
0.1
0.2 ∆1
(c)
0.3
0.4
0
0.1
0.2 ∆1
0.3
0.4
(d)
Fig. 10. Ten unstable regions for g ¼ 0:005: (a) in [0.2, 0.75]; (b) in [0.85, 1.15]; (c) in [1.12, 1.78]; and (d) in [1.74, 2.5].
The parametric instability of the system simultaneously excited by the two inertia asymmetries of the disk 1 and 2 will be investigated in the following. In analysis, the inertia asymmetry of disk 1 1 is ¯xed to be invariable, and the inertia asymmetry of disk 2 2 is varying in a range with di®erent angles , in order to ¯nd whether the interaction between the two inertia asymmetries would in°uence the system instability or not. Given g ¼ 0 and 1 ¼ 0:1, all of the unstable regions varying with 2 in the range of 00.4 for two values of inertia angle ( ¼ 0; =2) are computed and plotted in Figs. 12 and 13, respectively. Due to the operation of 1 , every unstable regions are not a point even for 2 ¼ 0. With the increasing of 2 , the variations of di®erent unstable regions are p p p p also di®erent. For ¼ 0, as shown in Fig. 12, four primary (U 11 , U 22 , U 33 , and U 44 ) c c and two combination (U 13 and U 24 ) instability regions are all enhancing continually with 2 , while the remaining four combination instability regions c c c c (U 12 , U 14 , U 23 , and U 34 ) are reduced gradually. In 2 ¼ 0:1, the four unstable regions have been disappeared. For the range of 2 > 0:1, the four unstable
274
Q. K. Han & F. L. Chu
0.7
Ω
0.6
1
0.4
0.95
0.3
0.9 0
0.1
0.2
0.3
0.4
0.85
p U22
0
0.1
(a)
0.2
0.3
0.4
0.3
0.4
(b) 2.5
1.7
2.4
c U34
1.6
2.3 2.2
1.5 Ω
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1.05
0.5
0.2
c U23
1.1
c U13
2.1
c U24
1.4
p U44
2
1.3 1.9
p U33
1.2
1.8 0
0.1
0.2 ∆1
(c)
0.3
0.4
0
0.1
0.2 ∆1
(d)
Fig. 11. Ten unstable regions for g ¼ 0:03: (a) in [0.2, 0.75]; (b) in [0.85, 1.15]; (c) in [1.12, 1.78]; and (d) in [1.74, 2.5].
regions are reappearing and enlarging with 2 . When the angle is adjusted to ¼ =2, as shown in Fig. 13, the variations of the unstable regions with 2 are opposite with that of the ¼ 0. In this case, the four primary and two combination instability regions ¯rst decrease to zero, then reappear and amplify with the increasing of 2 . The other four combination instability regions are all enlarging with 2 . The above phenomena indicate that when the system is excited by the two inertia asymmetries, the widths of unstable regions would be signi¯cantly in°uenced by the inertia angle. Research in this area will be further expanded in Sec. 5.4. 5.4. E®ects of design parameters on width of unstable region The design parameters, involving the inertia angle , relative distance lr of disk 2 and inertia ratio , are, respectively, analyzed to show their e®ects upon the width of the unstable region of the unsymmetrical rotor system.
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
c U23
0.7 1.1
Ω
0.6
1.05
0.5
c U13
c U12
0.95 p U11
0.3
0
0.9 0.1
0.4
0.85
p U22
0
0.1
(a)
0.4
(b) 2.5
c U34
1.7
2.4 2.3
1.6 1.5
2.2
c U24
Ω
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c U14
1
0.4
0.2
275
2.1
1.4
p U44
2 1.3 1.9 1.2
p U33
0
1.8 0.1
∆2
(c)
0.4
0
0.1
∆
0.4
2
(d)
Fig. 12. All of the unstable regions for 1 ¼ 0:1 and ¼ 0: (a) in [0.2, 0.75]; (b) in [0.85, 1.15]; (c) in [1.12, 1.78]; and (d) in [1.74, 2.5].
5.4.1. Inertia angle The relative inertia asymmetry of disk 1, inertia ratio, and relative distance are ¯xed pðcÞ to be 1 ¼ 0:1, ¼ 0:5, and lr ¼ 1, respectively. The variations of the ij with in ½0; and 2 in ½0; 0:3 are considered in the analysis. From the discussions for Figs. 12 and 13 in the above section, it is indicated that the 10 unstable regions might p p p p c be divided into two groups (A and B). Group A includes U 11 , U 22 , U 33 , U 44 , U 13 , c c c c and U 24 . Group B contains the remaining unstable regions, i.e., U 12 , U 14 , U 23 , and p c U 34 . Here, the U 11 is chosen to represent the unstable regions of group A, and the three-dimensional (3D) and contour plots of the width p11 are given in Fig. 14, respectively. With a certain value of 2 , both the 3D and contour plots of Fig. 14 show that increasing from 0 to =2 would reduce the p11 . When ¼ =2, the p11 reaches a local minimum value. Instead, the p11 would increase if the varies from =2 to . : When 2 ¼ 0, the p11 ¼0:002. If the inertia asymmetry 2 of disk 2 is utilized to attenuate the unstable region, one can ¯nd from Fig. 14(b) that the and 2 should
276
Q. K. Han & F. L. Chu
c U23
0.7 1.1
Ω
0.6
1.05 c U12
0.5
c U13
1
0.4
0.95 p U11
0.3
0
0.1
0.4
0.85
0
c U14
0.1
(a)
0.4
(b) 2.5
c U34
1.7
2.4 2.3
1.6
2.2
1.5
c U24
Ω
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0.2
p U22
0.9
1.4
2.1 p U44
2 1.3 1.9
p U33
1.2 0
1.8
0.1
∆2
(c)
0.4
0
0.1
∆2
0.4
(d)
Fig. 13. All of the unstable regions for 1 ¼ 0:1 and ¼ =2: (a) in [0.2, 0.75]; (b) in [0.85, 1.15]; (c) in [1.12, 1.78]; and (d) in [1.74, 2.5].
be set within the plane surrounded by the curve 0.002. Especially for ¼ =2 and 2 ¼ 1 ¼ 0:1, the p11 is close to zero, as shown in Fig. 14(a), indicating that the p U 11 has been eliminated in this case. The variations of the widths of the other unstable regions in group A with the inertia angle are the same with that of the p11 with . Thus, if the is near =2, the interaction between the two inertia asymmetry would make the unstable regions of group A have relative lower widths. In order to control the unstable regions of group A, besides the should be set around =2, the 2 of disk 2 should also be set around a certain value, which is closely related to the inertia asymmetry of disk 1, inertia ratio, and relative position of disk 2. c The U 34 is taken as an example to show the widths of unstable regions in group B varying with inertia angle . Results for the width c34 are presented by both 3D and contour plots in Fig. 15. It is shown that the c34 would ¯rst increase and then decrease as the changing from 0 to . When equals to 0 or with certain 2 , the c34 obtains the minimum value. For the ¼ =2, the value of c34 is locally maximal. : Without considering 2 , Table 1 shows that c34 ¼0:1. If the operation of disk 2 is
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
277
0.01 0.008 p
λ 11
0.006 0.004 0.002 0 0.4 pi 3pi/4
0.2
pi/2
0.1
pi/4
∆2
0
0
φ
(a)
0.0
09
0.0
09
06 05
0.0
0.0
7
03
00
8
0.
00
7
0.0
04
05
00
0.0
0.
0.2
0.
06
0.25
04
0.0
0 0.
0.0
0.0
08
0.0
02
03 0.0
0.001
0.0
06
0.0
05
03
0.0
2
0.0
0.00
05
0.0
06
0.0
0.15
04
∆2
0.1
04
0.0
0.00
0.05
0.
0.002
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0.3
3 .00
4
00
1
0
0.003
2
0.00
0 0
\pi/4
pi/2 φ
3pi/4
pi
(b) Fig. 14. Width
p11
varying with in ½0; and 2 in ½0; 0:3: (a) (3D) plot and (b) the contour plot.
used to reduce the unstable width, then the values of and 2 should be given within the ranges surrounded by the two curves 0.1 in Fig. 15(b). Especially for ¼ 0 or and 2 ¼ 0:1, the c34 almost equal to zero and the unstable region is disappeared, as shown in Fig. 15(a). The variations of the widths of the other unstable regions in group B are similar with that of c34 with . Thus, the should be set around 0 or to attenuate the unstable regions of group B. Besides, the value of 2 should also be taken as a certain value, which could be ascertained as long as the inertia asymmetry of disk 1, inertia ratio, and relative position of disk 2 are given. Obviously, the unstable regions of both group A and B could not be weakened simultaneously through adjusting the inertia angle in the range of ½0; . In other words, the
278
Q. K. Han & F. L. Chu
0.5 0.4
λ
0.3
c 34
0.2 0.1
0.3
pi 3pi/4
0.2 pi/2
0.1
∆2
pi/4 0
φ
0
(a)
5
0.3
0.3
0.
4
4
0.
5
0.3
2
0.2
5
5
0.1
0.
0.3
0.2
0.45
0.2
0.25
0.2
0.1 0.2
5 0.2
0.1
5
0.05
25
0. 25
0.
0.1
0.1
0.1
0.1
3
0.
3
0.
0.15
0.2
∆2
5
0.35
5
0.0
5
0.05 0.1
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0 0.4
0.15 0.1
0
0
\pi/4
pi/2
3pi/4
pi
φ
(b) Fig. 15. Width c34 varying with in ½0; and 2 in ½0; 0:3: (a) (3D) plot and (b) the contour plot.
repression of the unstable regions of group A would result in the enhancement of the unstable regions of group B. 5.4.2. Relative distance lr The relative distance lr of disk 2 not only in°uences the unstable rotating speeds (it has been discussed in Sec. 5.2), but also a®ects the widths of unstable regions, which will be investigated in the section. The 1 and are set to be 0.1 and 0.5 in the analysis. For three values of lr (0.81, 1, and 1.19), the widths of the unstable regions of both the group A and B varying with 2 are solved and plotted in Fig. 16 and 17, respectively. The inertia angle is taken as 0 and =2.
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries 0.02
0.2 lr=0.81 l =1 r lr=1.19
0.015
0.6 lr=0.81 l =1 r lr=1.19
0.15
p λ11
p λ22
0.01
279
lr=0.81 l =1 r lr=1.19
0.5 p λ330.4
0.1
0.3 0.005
0
0.05
0
0.1
0.2
0
0.3
0.2
0
0.1
0.2
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∆2
0.3
0
0.1
∆2
(a)
0.3
(c)
0.1 lr=0.81 l =1 r l =1.19
0.2 ∆2
(b)
0.8
0.6
0.1
0.25 lr=0.81 l =1 r l =1.19
0.08
r
0.2
r
λp44 0.4
0.2
λc130.06
λc240.15
0.04
0.1
0.02
0.05
0
0
lr=0.81 l =1 r l =1.19 r
0
0
0.1
0.2
0.3
0
0.1
∆
(d)
0.2
0.3
0
0.1
∆
2
(e)
0.2
0.3
∆
2
2
(f)
Fig. 16. E®ect of lr on widths of unstable regions of group A for ¼ 0: (a) p11 ; (b) p22 ; (c) p33 ; (d) p44 ; (e) c13 ; and (f) c24 .
For ¼ 0, Fig. 16 shows that the widths of unstable regions of group A are all linearly increasing with 2 . Except for p22 (Fig. 16(b)) and c24 (Fig.16(f)), increasing the value of lr would not only decrease the width values, but also reduce the slopes of the lines, as shown in Figs. 16(a), 16(c), 16(d), and 16(e). When ¼ =2, except for c23 , increasing the value of lr would also reduce the widths of unstable regions of group B, which is shown in Fig. 17. Thus, one can say that increasing the relative position of disk 2 would have suppression e®ects upon most of the unstable regions in the unsymmetrical rotor system. 5.4.3. Inertia ratio Inertia ratio has signi¯cant e®ects on the unstable rotating speeds, and some of the unstable rotating speeds would not exist if the varies from 1 to 2. Besides, the widths of unstable regions will also be in°uenced by the . The 1 and lr are given to be 0.1 and 1. For three values of (0.3, 0.5, and 0.7), the variations of the every unstable widths with 2 are solved, respectively. The widths of unstable regions of group A with ¼ 0 are plotted in Fig. 18, while Fig. 19 gives the widths of unstable
280
Q. K. Han & F. L. Chu 0.04
0.1
0.035
lr=0.81 l =1 r l =1.19
0.08
r
0.03 c 12
λ
c 14
λ
0.025 0.02
0.06 0.04
0.015
l =0.81 r l =1 r l =1.19
0.01
0.02
r
0.005
0
0.1
0.2
0
0.3
0
0.1
0.2
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∆2
0.3
∆2
(a)
(b)
0.25
0.7 0.6
0.2
0.5 c λ23
0.15
c λ34
0.4 0.3
0.1
0.2
lr=0.81 lr=1 l =1.19
0.05
lr=0.81 lr=1 l =1.19
0.1
r
0
0
0.1
0.2
r
0
0.3
0
0.1
0.2
∆2
0.3
∆2
(c)
(d)
Fig. 17. E®ect of lr on widths of unstable regions of group B for ¼ =2: (a) c12 ; (b) c14 ; (c) c23 ; and (d) c34 .
3
10
x 10
0.14
1
0.12
0.8
8 λp 11
0.1 λp 22 0.08
6
β=0.3 β=0.5 β=0.7
p λ330.6
0.4 0.06 4
2
β=0.3 β=0.5 β=0.7
0
0.1
0.2 ∆2
(a)
β=0.3 β=0.5 β=0.7
0.04 0.3
0.02
0
0.1
0.2
0.2
0.3
0
0
0.1
∆
0.3
2
(b)
Fig. 18. E®ect of on widths of unstable regions of group A for ¼ 0: (a) (e) c13 ; and (f) c24 .
0.2 ∆
2
(c) p11 ;
(b)
p22 ;
(c) p33 ; (d) p44 ;
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries 1.4
0.07 β=0.3 β=0.5 β=0.7
1.2 1 λp
44
0.35 β=0.3 β=0.5 β=0.7
0.06 λc
0.8
0.25
0.05
c λ240.2
13
0.04
0.15 0.03
0.4
0.1
0.02
0.2 0
0.1
0.2
0.01
0.3
0.05 0
0.1
∆
0.2
0
0.3
0
0.1
∆
2
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β=0.3 β=0.5 β=0.7
0.3
0.6
0
281
(d)
0.2
0.3
∆
2
2
(e)
(f)
Fig. 18. (Continued )
0.04
0.06
0.035
β=0.3 β=0.5 β=0.7
0.05
0.03 λc 12
0.025
λc 0.04
0.02
0.03
14
0.015
β =0.3 β =0.5 β =0.7
0.01 0.005
0
0.1
0.02
0.2
0.01
0.3
0
0.1
∆2
0.2
0.3
∆2
(a)
(b) 0.35 β=0.3 β=0.5 β=0.7
0.3 0.25 λc 23
0.2 0.15 0.1 0.05 0
0
0.1
0.2
0.3
∆2
(c)
(d)
Fig. 19. E®ect of on widths of unstable regions of group B for ¼ =2: (a) c12 ; (b) c14 ; (c) c23 ; and (d) c34 .
282
Q. K. Han & F. L. Chu
regions of group B with ¼ =2. From the two ¯gures, one can ¯nd that for the unstable regions of both group A and B, their widths are all reduced with di®erent degrees by decreasing the value of . Except for p11 (Fig. 18(a)), p22 (Fig. 18(b)), and c12 (Fig. 19(a)), the reductions are signi¯cant. This means that most of the unstable regions in inertia asymmetrical rotor system could be attenuated by decreasing the inertia ratio.
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6. Conclusions The whirling properties and parametric instability of a two-disk rotor system with unsymmetrical inertias has been studied numerically in the paper. Two important indicators for describing the unstable regions, namely the unstable rotating speed and width of unstable region, have been derived using the instability theory and Taylor expansion technique. For a practical used two-disk unsymmetrical rotor, the e®ects of three design parameters (inertia excitation phasing, relative position of the disk, and inertia ratio) on the two indictors have been discussed in detail. Some conclusions can be drawn from this study: (1) For the inertia ratio varying from 0 to 1, all of the unstable rotating speeds are increasing continually. However, some unstable rotating speeds, involving: p33 , p44 , and c34 , have disappeared for in the range of 1–2. This indicates that these unstable regions could not exist in the system. Moreover, adding the values of lr makes the unstable rotating speeds decrease with a certain value of . pðcÞ (2) According to the variations of the widths ij of the unstable regions with p inertia angle , the unstable regions are divided into two groups: group A (U 11 , p p p c c c c c c U 22 , U 33 , U 44 , U 13 , and U 24 ) and group B (U 12 , U 14 , U 23 , and U 34 ). If is around =2, the unstable regions of group A would obtain the local minimum values. When the is approaching to 0 or , the widths of the unstable regions of group B would have local minimum values. Besides, in order to achieve the optimal control of the unstable regions of group A (or B), the relative inertia asymmetry of one disk should also be set a certain value, which is closely related to the inertia asymmetry of the other disk, inertia ratio, and the relative position of the disk. However, the unstable regions of both group A and B could not be weakened simultaneously through adjusting the inertia angle in the range of ½0; . (3) Increasing the relative position of disk 2 or decreasing the inertia ratio would all have suppressive e®ects on most of the unstable regions in the unsymmetrical rotor system.
Acknowledgments The research work described in the paper was supported by the National Science Foundation of China under Grant No. 10732060 and 51075224.
Parametric Instability of a Two-Disk Rotor with Two Inertia Asymmetries
283
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References 1. D. M. Smith, The motion of a rotor carried by a °exible shaft in °exible bearings, Proceedings of the Royal Society, Series A 142 (1933) 92118. 2. S. H. Crandall and P. J. Brosens, On the stability of rotation of a rotor with rotationally unsymmetric inertia and sti®ness properties, Transactions of the ASME, Journal of Applied Mechanics 83 (1961) 567570. 3. T. Yamamoto and H. Ota, On the unstable vibrations of a shaft carrying an unsymmetrical rotor, Transactions of the ASME, Journal of Applied Mechanics 86 (1964) 515522. 4. T. Yamamoto and H. Ota, The damping e®ect on unstable whirlings of a shaft carrying an unsymmetrical rotor, Memoirs of the Faculty of Engineering, Nagoya University 19(2) (1967) 197215. 5. T. Yamamoto, H. Ota and K. Kono, On the unstable vibrations of a shaft with unsymmetrical sti®ness carrying an unsymmetrical rotor, Transactions of the ASME, Journal of Applied Mechanics 90 (1968) 313321. 6. T. Yamamoto and K. Yasuda, Unstable vibrations of an unsymmetrical rotor supported by °exible bearing pedestals, Bulletin of the JSME 15(87) (1972) 10631073. 7. D. Arday¯o and D. A. Frohrib, Instabilities of an asymmetric rotor with asymmetric shaft mounted on symmetric elastic supports, Transactions of the ASME, Journal of Engineering for Industry 98(4) (1976) 11611165. 8. G. Genta, Whirling of unsymmetrical rotors: A ¯nite element approach based on complex co-ordinates, Journal of Sound and Vibration 124(1) (1988) 2753. 9. N. Nriot, C. H. Lamarque and A. Berlioz, Dynamics of a rotor subjected to a base translational motion and an uncertain parametric excitation, 12th IFToMM World Congress (Besancon, France, 2007). 10. K. Nandakumar and A. Chatterjee, Nonlinear secondary whirl of an overhung rotor, Proceedings of the Royal Society, Series A 466 (2010) 283301. 11. Y. Ishida and J. Liu, Elimination of unstable ranges of rotors utilizing discontinuous spring characteristics: An asymmetrical shaft system, an asymmetrical rotor system, and a rotor system with liquid, Transactions of the ASME, Journal of Vibration and Acoustics 132 (2010) 011011011018. 12. P. J. Wang, H. F. Teng, D. J. Liu, et al., The dynamic analysis for complex asymmetric rotor-bearing systems by ¯nite element method, Acta Mechanica Solida Sinica 12(4) (1991) 370376 (In Chinese). 13. Y. Kang, Y. P. Shih and A. C. Lee, Investigation on the steady-state responses of asymmetric rotors, Transactions of the ASME, Journal of Vibration and Acoustics 114 (1992) 194208. 14. Y. Kang, G. J. Sheen and S. M. Wang, Development and modi¯cation of a uni¯ed balancing method for unsymmetrical rotor-bearing systems, Journal of Sound and Vibration 119(3) (1997) 349368. 15. Y. Kang, C. P. Chiang, C. C. Wang, et al., The minimization method of measuring errors for balancing asymmetrical rotors, JSME International Journal, Series C 46(3) (2003) 10171025. 16. S. C. Hsieh, J. H. Chen and A. C. Lee, A modi¯ed transfer matrix method for the coupled lateral and torsional vibration of asymmetric rotor-bearing systems, Journal of Sound and Vibration 312 (2008) 563571. 17. Y. Kang and Y. G. Lee, In°uence of bearing damping on instability of asymmetric shafts, part III: disk e®ects, International Journal of Mechanical Sciences 39(9) (1997) 10551065. 18. J. Lin and R. G. Parker, Mesh sti®ness variation instabilities in two-stage gear systems, Transactions of the ASME, Journal of Vibration and Acoustics 124 (2002) 6876.
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19. Q. K. Han, J. J. Wang and Q. H. Li, Parametric instability of a cantilever beam subjected to two electromagnetic excitations: Experiments and analytical validation, Journal of Sound and Vibration (2010) under review. 20. Q. K. Han, Theory and experiment study on dynamic characteristics of parametrically excited system, Ph.D. dissertation (BeiHang University, Beijing, 2010). 21. P. Friedmann, E. Hammond and T. H. Woo, E±cient numerical treatment of periodic systems with application to stability problems, International Journal for Numerical Methods in Engineering 11 (1977) 11171136. 22. C. S. Hsu, On approximating a general linear periodic system, Journal of Mathematical Analysis and Applications 45 (1974) 234251.
