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Investigation on the dynamic behaviors of a rod fastening rotor based on an analytical solution of the oil film force of the supporting bearing:

Investigation on the dynamic behaviors of a rod fastening rotor based on an analytical solution... The dynamic characteristics of rod fastening rotor supported by a finite journal bearing are investigated in this study. To model the dynamic behaviors of the bearing-rotor system, the oil film force of bearing is calculated by approximately solving the Reynolds equation with the variables separation method and Sturm–Liouville theory, and then a motion equation is developed with consideration of the contact and gyro effects of the disks of the rotor. To solve the motion equation with small error and excellent stability, an improved Newmark method is proposed. On this basis, the dynamics characteristics of the rod fastening rotor are analyzed for different rotor speeds, disk eccentricities, shaft bearing stiffness, and contact stiffness. And the orbits of the rod fastening rotor and integral rotor are compared. The numerical results indicate that the analytical solution of the oil film force has higher computational efficiency than the finite difference method. The rod fastening rotor shows higher stability than the integral rotor, and exhibits rich dynamic behaviors, such as periodic, qusi-periodic, period-2, period-4, and period-6. Keywords Rod fastening rotor, Newmark method, Sturm–Liouville theory, approximate solution, stability Introduction Rotor-bearing system is an important component in the power equipment, and its stability affects greatly the safety and normal service performance of the equipment. However, the oil film force of the bearing is usually nonlinear, and consequently causes complex and unstable dynamic behaviors of the rotor when the rotor is working under the conditions of high speed and overload. Therefore, the dynamic stability of the rotor-bearing 1–6 system caused by the nonlinear oil film force has investigated by many researchers. Yang et al. established the model of rotor with transverse crack by considering parametric uncertainties. And the equation of the system was solved by the Harmonic Balance method. Then, the influences of transverse crack and uncertain parameters on the responses of the rotor were studied. Wang et al. studied the stability of a flexible liquid-filled rotor based on Fourier series. The analytical expression of the dynamic pressure of the liquid was derived, and the dynamic stability of the rotor was analyzed by the numerical method. Desavale et al. established a comprehensive empirical model with dimensionless parameters. Department of Mechanical and Electrical Engineering, Shaanxi Railway Institute, Wei Nan, China Corresponding author: Di Hei, Department of Mechanical and Electrical Engineering, Shaanxi Railway Institute, Wei Nan 714000, China. Email: drillok@126.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 708 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 2 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Based on the dimensional analysis approach, the diagnosis of misalignment and bearing looseness were studied. The calculating results of the empirical model agreed well with the experimental results. Barbosa et al. predicted the rotor dynamics by using the Kriging surrogate models. The models can reduce the calcu- lation time effectively. Zhang et al. proposed an adjustable elliptical journal bearing to suppress the vibration of amplitude of the rotor, and the results indicated that the bearing can effectively reduce the vibration. Santos et al. studied the dynamic characteristic of a rigid-rotor system experimentally. The damping and natural frequencies were obtained, and the maximum servovalve voltage and radial injection pressure were found for the rotor system. In the studies above, the integral rotor was taken as the object. However, besides the integral rotor, the rod fastening rotor is also very commonly used in the heavy gas turbine and so on. For the rod fastening rotor, the rod bolts are used to press all disks tightly together. Therefore, the rod fastening rotor will show some different dynamic behaviors from the integral rotor. In this context, the dynamics characteristics of the rod fastening rotor has been studied by some researchers. Hei et al. established the model of rod fastening rotor supported by fixed- tilting pad journal bearing. The database method was employed to solve the oil film force of the fixed-tilting pad journal bearing. The dynamics behaviors of the rotor system were investigated by the orbit diagrams, the time series, the frequency spectrum diagrams. But, the gyro effects of the disks were not taken into account when modelling. Zhang studied the tangential contact stiffness of the disks of the rod fastening rotor based on the fractal contact theory. The correctness of the fractal contact theory was verified by experiment. At the same time, it was found that the fractal contact theory has its scope of application. The study only solved the tangential contact stiffness, but the dynamics behaviors of the rod fastening rotor were not analyzed. Li established the model of rod fastening rotor by considering the contact friction force and pre-tightening force of the disks. The effects of the contact friction force and pre-tightening force on the rotor dynamics were studied. The results shown that the contact stiffness and the natural frequency of the system increased with the increase of pre- tightening force. Hu et al derived the motion equation of the rod fastening rotor by Lagrange equation. The influence of the initial deflection caused by the unbalanced pre-tightening force on the dynamic behaviors of the rotor was studied. Their work showed that the initial deflection has a great effect on the rotor dynamics. Meanwhile, Hu et al. also studied the dynamic characteristics of the rub-impact rod fastening rotor. The bifurcation diagram, vibration waveform, frequency spectrum, shaft orbit, and Poincare map were used to analyze the dynamics responses of the rotor system. However, in the works of Hu et al. , the influence of the gyro effect on the dynamic behaviors of the rod fastening rotor was not considered, and a short bearing model was assumed. For the rod fastening rotor, the hydrodynamic sliding bearing is usually taken as the supporting part, and the oil film force of the bearing is very important for the analysis of the dynamic behaviors of the rotor. So, the investigation of the oil film force was implemented by many researches. In many studies, the model of infinite long 12–14 bearing and infinite short bearing were adopted widely. Considering journal bearing parametric uncertainties, Ramos et al. studied the dynamics of the rotor system, and the rotor is supported by the short fluid film bearings. Wei et al. studied the nonlinear dynamic behaviors of the multi-disk rotor-bearing-seal system which supported by short bearing. The response of the system was calculated by the fourth order Runge– Kutta method. Liu et al. studied the dynamics characteristics of the rod fastening rotor which supported by the infinite long bearing. The shooting method and path-following technique were adopted to calculate the dynamics responses of the system. The infinite long bearing and infinite short bearing models are very simple and idealized models. In practice, the slide bearings are finite long bearings. So, the two idealized models cannot describe the actual bearing. In order to solve the oil film force of finite long bearing accurately, numerical methods are proposed. Smolık et al. studied the shape of the threshold curve for a rigid rotor supported by journal slide bearing. In order to solve the oil film force of journal bearing, the four methods were proposed: the infinitely short approximation, the infinitely short approximation, the finite differences method, and the finite elements method. Tuckmantel investigated the vibration signature of rotor-coupling-bearing system by solving the oil film force of the bearing with finite volume method. Lu et al. calculated the oil film forces and their Jacobis by the variational constraint approach. However, the numerical methods (such as the Hei and Zheng 709 Hei and Zheng 3 Figure 1. Schematic diagram of the bearing-rod fastening rotor system. finite volume method, finite elements method, finite differences method, and variational constraint approach) have large calculation cost. In this study, an approximation analytical solution of oil film force is proposed for the supporting bearing of the rod fastening rotor by using the method of separation of variables and Sturm–Liouville theory. This method not only ensures the calculation accuracy, but also improves the calculation efficiency. Meanwhile, a model of the rod fastening rotor is established by considering the Gyro effect and contact effect of the disks, and an improved Newmark method is proposed to solve the dynamic responses of the rotor system. On this basis, the comparison of obit is implemented between the integral rotor and rod fastening rotor. The influence of rotating speed, eccentricity, bending stiffness, and contact stiffness on the dynamic behaviors of the rotor system is also investigated. Equation of the rod fastening rotor system The geometry model of the rod fastening rotor system supported by oil film hydrodynamic journal bearings is given in Figure 1. In this study, the contact effect and gyro effect of the disks are considered, and the contact effect of disks is taken as a bending spring with nonlinear stiffness. In Figure 1, O and O are the centers of the two disks, l is the length of the whole rotor, a and b are the length of 1 2 the two shafts, a is the distance between A and O , b is the distance between B and O , a is the distance between 1 1 1 1 2 A and O , b is the distance between B and O , m and m are the mass of disk O and O , m and m are the 2 2 2 O1 O2 1 2 A B mass of the two shafts (the length are a and b respectively), e and e are the eccentricities of the disk O and O , O1 O2 1 2 f and f are the nonlinear oil film forces of bearing in negative x and y directions, respectively. x y The dynamic equation of the rod fastening rotor system can be written as follows Mq € þ Gq_ þ Kq ¼ f þ Q þ W þ f (1) where M is the mass matrix, G is the gyroscopic matrix, K is the stiffness matrix, f is the vector of nonlinear oil film forces of bearing, Q is periodic exciting forces (i.e., unbalanced or steam excitation with the same phase as rotating speed) vector acting on rotor, W is the vector of gravity, f is the vector of nonlinear restoring force caused by the nonlinear stiffness, and q is the displacement vector of the rotor. The 12 degrees-of-freedom of the system are [x , y , x , y , h , h ,x , y , h , h , x , y ]. x and y are the displacements of the rotor at bearing station A in A O1 O1 x1 y1 O2 O2 x2 y2 B B A A x and y directions, x and y are the displacements of the center of disk O in x and y directions, h and h are O1 O1 1 x1 y1 the angles that disk O rotates around the x and y axes, x and y are the displacements of the center of disk O 1 O2 O2 2 in x and y directions, h and h are the angles that disk O rotates around the x and y axes, x and y are the x2 y2 2 B B displacements of the rotor at bearing station B in x and y directions. q ¼½x ; y ; x ; y ; h ; h ; x ; y ; h ; h ; x ; y � A A O1 O1 x1 y1 O2 O2 x2 y2 B B f ¼ ½� f ; � f ; 0; 0; 0; 0; 0; 0; 0; 0; � f ; � f � xA yA xB yB 710 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 4 Journal of Low Frequency Noise, Vibration and Active Control 0(0) W ¼ ½0; m g; 0; m g; 0; 0; 0; m g; 0; 0; 0; m g� A O1 O2 B 2 2 2 2 Q ¼ ½0; 0; m e x sinðxtÞ; m e x cosðxtÞ; 0; 0; m e x sinðxtÞ; m e x cosðxtÞ; 0; 0; 0; 0� O1 O1 O1 O1 O2 O2 O2 O2 3 3 3 3 T f ¼ ½0; 0; k ðx � x Þ ; k ðy � y Þ ; 0; 0; � k ðx � x Þ ; � k ðy � y Þ ; 0; 0; 0; 0� R O2 O1 R O2 O1 R O2 O1 R O2 O1 2 3 m 0 0 00 00 00 0 0 0 6 7 6 7 m 0 00 00 00 0 0 0 6 7 6 7 6 7 m 00 00 00 0 0 0 O1 6 7 6 7 6 m 00 0 0 00 0 0 7 O1 6 7 6 7 6 7 J 00 00 0 0 0 d1 6 7 6 7 6 7 J 0 0 00 0 0 d1 6 7 M ¼ 6 7 6 7 m 0 00 0 0 O2 6 7 6 7 6 7 m 00 0 0 6 O2 6 7 6 7 6 J 00 0 7 d2 6 7 6 7 6 7 J 00 d2 6 7 6 7 6 7 m 0 4 5 2 3 00 00 0 0 00 0 0 00 6 7 6 7 6 7 0 00 0 0 00 0 0 00 6 7 6 7 6 7 00 0 0 00 0 0 00 6 7 6 7 6 7 6 7 0 0 0 00 0 0 00 6 7 6 7 6 7 6 7 0 � J x 00 0 0 00 z1 6 7 6 7 6 7 6 J x 0 00 0 0 007 z1 6 7 G ¼ 6 7 6 7 00 0 0 00 6 7 6 7 6 7 6 7 0 0 0 00 6 7 6 7 6 7 6 7 0 � J x 00 z2 6 7 6 7 6 7 J x 0 00 6 7 z2 6 7 6 7 6 7 6 7 4 5 0 Hei and Zheng 711 Hei and Zheng 5 > > b k > 44 > > > k þ > > > 6 2 2 > > l l > > > > > > > > > > > 2k b > 14 1 > > > > � � �� < = 6 2 l k b k k k b 11 1 14 44 14 1 0 � þ 00 � 2 0 > > l l l l 6 > b k > > 0 2 44 > 6 > > þk þ > 11 > 2 2 6 > > l l > > 6 > > > � > 6 > > > > 2k b 6 > > > > 6 > � > : ; 8 9 > > 6 b k > 33 > > > > k þ > > 2 2 > > l l > > > > > �� �� > > < = 6 b k k k b 1 23 33 23 1 2k b b 23 1 0 2 0 0 �k þ � 0 � þ k 6 22 > 2 2 > l l l l l l > > > > > > 6 > > 0 0 > > 6 > k 2k b > 33 23 > > 6 > > þ � : ; 2 2 l l �� k k b 6 14 11 1 � 0 ðÞ k þ k 00 k 6 11 b 14 6 l l �� 6 k k b 23 22 1 6 0 � 0 ðÞ k þ k k 0 22 b 23 l l �� k k b 33 23 1 0 � 0 k k 0 23 33 l l 6 �� k k b 44 14 1 � 0 k 00 k 14 44 l l 6 � � 0 0 K ¼ k b k 6 11 14 � þ 0 �k 000 l l � � 0 0 k b k 6 2 22 23 0 � þ 0 �k 00 6 l l �� 0 0 6 k k b 33 23 0 � 0 000 l l �� 0 0 k k b 44 14 � 0 0 000 l l 8 � 9 6 k a b k > > 11 1 1 44 > > > � > > 2 2 > > > l l > > > > > > > > > > > k ðb � a Þ > 14 1 1 > > 6 � � � � > > l k a k k k a 11 1 14 44 14 1 0 � � 00 � � 0 0 > > l l l l 6 k a b k > > 2 2 11 44 > > > þ � > > > 6 2 2 > > l l > > > > �> > > > > 6 k ðb � a Þ > 2 2 > > > 6 þ : ; 8 9 6 � k a b k > 22 1 1 33 > > > > � > > > 2 2 > > l l > > > > > > > > > > > k ðb � a Þ > 23 1 1 > > 6 �� � � > > < = 6 2 k a k k k a l 22 1 23 33 23 1 0 0 � � � � 0 0 0 > > l l l l 6 k a b k > 2 2 > > 22 33 > > þ � > > > 2 2 > > l l > > > > > �> > 0 > > > k ðb � a Þ > 2 2 > > > : ; l 712 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 6 Journal of Low Frequency Noise, Vibration and Active Control 0(0) 8 9 k a b k > 11 1 1 44 > > > > > > � > > > > > 2 2 > l l > > > > > > > > > > > > k ðb � a Þ > 14 1 1 � � � � > > > < = > 0 0 0 0 þ k b k k k b 2 2 2 > 11 14 44 14 � þ 00 � 0 > 0 0 k a b k > > > l l l l 2 2 11 44 > > > > þ � > > > 2 2 > > > > > l l > �> > > 0 > > > > > k ðb � a Þ > 2 2 > > > > > : ; > 2 > l > 8 9 � > k a b k > 22 1 1 33 > > > > > > > 2 2 > > > l l > > > > > > > > > k ðb � a Þ > 23 1 1 � �� � > > < = 0 0 0 þ > b k k k b > 2 2 2 23 33 23 > 0 l 0 � k þ � 0 0 0 0 > k a b k l l l l > > 2 2 > > 22 33 > > þ � > > > 2 2 > > > l l > > �> > 0 > > > k ðb � a Þ > 2 2 > 23 >> > > þ ; : > l > � � > k a k 11 1 14 �k 0 00 � � 0 > l l � � k a k > 22 1 23 > 0 �k 00 0 � � b > l l > � � k k a 33 23 1 > 000 0 0 � � l l � � > k k a 44 14 1 000 0 � � 0 > l l � � 0 0 k a k 11 14 0 0 ðk þ k Þ 00 k � � 0 11 14 > l l � � > 0 0 > k a k 2 > 22 23 0 0 0 ðk þ k Þ k 0 0 � � > 22 23 l l > � � 0 0 k k a > 33 23 > 0 0 0 k k 0 0 � � 23 33 l l � � > 0 0 k k a 2 > 44 14 0 0 k 00 k � � 0 > 14 44 > l l > 8 9 2 > > > > k a k > 11 44 1 > > > > þ > > > 2 2 > > > l l > > > > > > > > 2k a > 14 1 > > > � � � � < = 0 0 0 0 þ þ > k a k k k a 2 2 2 11 14 44 14 l > � � 00 � � 0 0 2 0 > k a k l l l l > > > 11 2 44 > > þ > > > 2 2 > > > l l > � > > > 0 > > > 2k a > > > 2 > > > > þ > : 2 ; l > 8 9 > > > k a k > > > 22 33 > 1 > > > þ > > > > > 2 2 > > l l > > > > > > > > > > > 2k a 23 1 � � � � > > > < = > 0 0 0 0 þ k a k k k a 2 > 2 2 22 23 33 23 l > 0 � � � � 0 0 > 0 2 0 > > k a k > > l l l l > 22 2 33 > > > > þ þ > > 2 2 > > > l l > > � > > > > > 2k a > > 2 > > > > > þ > : ; > where f and f are the nonlinear oil film forces of bearing at the station A in negative x and y directions, xA yA respectively. f and f are the nonlinear oil film forces of bearing at the station B in negative x and y directions, xB yB respectively. J and J are the polar moments of the inertia for disks O and O . J and J are the equatorial d1 d2 1 2 z1 z2 moments of the inertia for disks O and O . k is the bending stiffness of the rod. k is the nonlinear 1 2 b R stiffness of contact interface. x is the rotating speed of the rotor. k and k are the bending stiffness of shaft 11 22 AO in the x and y directions. k and k are the stiffness caused by the swing angle which disk O oscillated 33 44 1 0 0 around the x and y axes, k and k are the cross stiffness of the shaft AO. k and k are the bending stiffness of 14 23 11 22 0 0 shaft BO in the x and y directions. k and k are the stiffness caused by the swing angle which disk O oscillated 33 44 2 0 0 around the x and y axes. k and k are the cross stiffness of the shaft BO. The rod fastening rotor is a symmetric 14 23 0 0 0 0 rotor in this paper, so the relationship of the stiffness are k ¼ k ¼ k ¼ k , k ¼ k ¼ k ¼ k , 11 22 11 22 33 44 33 44 0 0 19 k ¼ k ¼ k ¼ k . 14 23 14 23 Hei and Zheng 713 Hei and Zheng 7 For the convenience of calculating and derivation, the following dimensionless variables are introduced. pffiffiffiffiffiffiffi X ¼ x =c; Y ¼ y =c; m ¼ m =m ði ¼ A; O ; O ; BÞ; l ¼ l=c; x ¼ x c=g; s ¼ xt; i i i i i i 1 2 3 2 e ¼ e =c ðj ¼ O ; O Þ; r ¼ lxBr =ðmgc Þ j j 1 2 F ¼ f=ðrmgÞ; K ¼ K ¼ k c=ðmgÞ; K ¼ K ¼ k =ðmgcÞ; 11 22 11 33 44 33 K ¼ K ¼ k =ðmgÞ; J ¼ J=ðmc Þ 23 14 23 where c is radius clearance, r is the radius of the shaft journal, B is the width of the bearing, and r is Sommerfeld number. Nonlinear oil film force Solution of nonlinear oil film force The geometry of the journal bearing is shown in Figure 2, and the cross section of the bearing is shown in Figure 3. In the two figures, O is the center of the bearing, O is the center of the shaft journal, h is the deviation b j angle, U is the angle goes from negative y-axis direction along the clockwise direction, u is the angle which goes from extension of O O along the clockwise direction, f and f are oil film forces in the negative radial and the j b r t Figure 2. Coordinate of finite length journal bearing. Figure 3. The cross section of the finite length journal bearing. 714 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 8 Journal of Low Frequency Noise, Vibration and Active Control 0(0) tangential directions, f and f are oil film forces in negative x and y directions, h is the thickness of oil film, R is x y the radius of the bearing, and W is the load of the bearing. By assuming that the lubricant is in-compressible, the Reynolds equation for the lubrication of the journal bearing can be written as �� 3 3 1 @ h @pðu; zÞ @ h @pðu; zÞ @h @h þ ¼ x þ 2 (2) R @u 6 @u @z 6 @z @u @t where l is the viscosity of the lubricating oil film. p is the distribution of pressure of oil film, h is the thickness of oil film (h ¼ c þ ecosu), R is the radius of bearing, and z is the axial coordinate. For the convenience of derivation and calculation, the dimensionless variables are introduced as follows �� k ¼ z ; H ¼ 1 þ ecosu; e ¼ e=c; s ¼ xt; e ¼ de=ds; h ¼ dh=ds; (3) w ¼ c=R; P ¼ p=p ðp ¼ 2xl=w Þ 0 0 where k is the dimensionless axial coordinate, H is the dimensionless thickness of the oil film, e is the dimensionless eccentricity, P is the dimensionless oil film pressure distribution, w is clearance ratio, and e is the radial velocity. Substituting the dimensionless variables into equation (2), the dimensionless Reynolds equation can be written as �� �� �� @ @P D @ @P 3 3 0 H þ H ¼ �3esinu þ 6e cosu þ 6eh sinu (4) @u @u B @k @k where eh is the dimensionless tangential velocity. To solve equation (4), the method of separation of variables and Sturm–Liouville theory are used. Based on the method of separation of variables, the dimensionless oil film pressure P can written as Pðu; kÞ¼ P ðu; kÞþ P ðu; kÞ (5) p h where P (u, k) is the special solution, P (u, k) is the general solution, and their expressions are given as p h P ðu; kÞ¼ P ðuÞþ P ðkÞ (6) p p1 p2 P ðu; kÞ¼ P ðuÞP ðkÞ (7) h h1 h2 In equation (5), it can be seen that P (u, k) and P (u, k) determine the oil film pressure distribution P. p h Therefore, P (u, k) and P (u, k) need to be solved respectively, and their solutions are introduced in the next section. p h Calculation of special solution. Substituting equation (6) into equation (4), the dimensionless Reynolds equation can be written as �� 2 2 0 d P ðkÞ B d P ðuÞ 3esinu dP ðuÞ 3esinu 6e cosu 6eh sinu p2 p1 p1 ¼ � þ � þ þ (8) 2 3 3 3 D du H du H H H dk The expression for the left item of equation (8) is a function of k, and the expression for the right item of equation (8) is a function of u. Let d P ðkÞ p2 ¼ C (9) dk Hei and Zheng 715 Hei and Zheng 9 �� 2 0 d P ðuÞ dP ðuÞ B 3esinu 3esinu 6e cosu 6eh sinu p1 p1 � þ � � ¼ C (10) 2 3 3 3 D du H du H H H P (k) can be obtained by integrating equation (9) twice. p2 P ðkÞ¼ Ck =2 þ c k þ c (11) p2 1 2 In equation (11), C is arbitrary constant, c and c are integral constants. Let C ¼ 0, c and c can be 1 2 1 2 obtained by solving the boundary conditions. The boundary conditions of the special solution in the axial direc- tion are as follows P j ¼ 0 (12) p2 k¼�1 Based on equation (12), the constant c and c are obtained, and c ¼ c ¼ 0. 1 2 1 2 Let dP ðuÞ p1 ¼ P ðuÞ (13) p1 du Substituting equation (13) into equation (10), P ðuÞ can be written as p1 0 0 P ðuÞ¼ ðc þ 6e sinu þ 3ecosu � 6eh cosuÞ (14) p1 3 ð1 þ ecosuÞ where c is undetermined constant. By integrating equation (13), P (u) can be expressed as 3 p1 P ðuÞ¼ P ðuÞ du þ c (15) p1 p1 4 where c is undetermined constant. In order to obtain the expression of P (u), Sommerfeld transformation is introduced as follows p1 1=2 1=2 2 2 2 1 � e ð1 � e Þ sinuð1 � e Þ e þ cosu 1 þ ecosu ¼ ; du ¼ da; sina ¼ ; cosa ¼ ; 1 � ecosa 1 � ecosa 1 þ ecosu 1 þ ecosu rffiffiffiffiffiffiffiffiffiffiffi (16) 1 � e u a ¼ 2 arctan tan 1 þ e 2 Based on Sommerfeld transformation, expression of P (u) can be written as follows p1 2 2 0 a � 2esina þ 0:5e sinacosa þ 0:5e a 3e P ðuÞ¼ c þ p1 3 5=2 2 ð1 � e Þ eð1 þ ecosuÞ �� (17) 3e �6eh sinacosa 3ea þ þ e sina þ sina � e � þ c 5=2 5=2 2 2 2 2 ð1 � e Þ ð1 � e Þ The boundary conditions of the special solution in the circumferential direction are as follows @P p1 u ¼ u ¼ 0; P ðu Þ¼ 0; u ¼ u ¼ p; j ¼ 0 (18) p1 a a c u¼u @u 716 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 10 Journal of Low Frequency Noise, Vibration and Active Control 0(0) where u is the starting position of the oil film pressure and u is the termination position of the oil film. a c Based on the boundary conditions (i.e., equation (18)), c and c can be expressed as follows 3 4 3e c ¼� (19) eð1 þ eÞ �� �� 0 0 5=2 6eh 3e 2 esina cosa 3ea 3e 3e 2 c c c � e sina þ sina � � þ � ð1 � e Þ c c 2 2 5=2 5=2 2 2 2 2 ðÞ ðÞ eð1þeÞ eð1þecosu Þ 1�e 1�e c ¼ c (20) 2 2 a � 2esina þ 0:5e sina cosa þ 0:5e a c c c c c The special solutions of the oil film force in the radial and tangential directions can be obtained by integrating the P (u, k), and they can be written as follows Z Z 1 u F ¼� P ðuÞcosu dudk; p1 �1 u (21) Z Z 1 u > p > F ¼� P ðuÞsinu dudk p1 : t �1 u Calculation of general solution. By substituting P (u, k) into equation (4), then the equation (4) can be written as follows �� 2 2 @ P ðu; kÞ @P ðu; kÞ D @ P ðu; kÞ h h h 3 2 3 H � 3H esinu þ H ¼ 0 (22) 2 2 @u @u B @k Substituting equation (7) into equation (22), the following expression can be obtained �� �� 2 2 00 00 0 P ðkÞ B P ðuÞ B 3 P ðuÞ h2 h1 h1 � ¼ � esinu (23) P ðkÞ D P ðuÞ D H P ðuÞ h2 h1 h1 In order to solve the equation (23), the following expressions should be satisfied P ðkÞ h2 � ¼j (24) P ðkÞ h2 �� �� 2 2 00 0 B P ðuÞ B 3 P ðuÞ h1 h1 � esinu ¼j (25) D P ðuÞ D H P ðuÞ h1 h1 For the journal bearing, the following boundary conditions should be satisfied P ðu; �1Þ¼ P ðuÞþ P ð�1Þ¼�P ðu; �1Þ¼�P ðuÞP ð�1Þ (26) p p1 p2 h h1 h2 Based on the boundary conditions above, the axial boundary conditions for the general solution is obtained P j ¼ 1; h2 k¼1 (27) P j ¼ 1 h2 k¼�1 Hei and Zheng 717 Hei and Zheng 11 In order to solve equation (24), three situations of j need to be considered (i.e., j > 0, j ¼ 0, and j < 0). 1. when j ¼ 0 Equation (24) can be written as P ðkÞ¼ 0 (28) h2 By integrating the equation (28) twice, the expression of P (k) can be obtained as follows h2 P ðkÞ¼ c k þ c (29) h2 5 6 where c and c are integral constants. c and c can be obtained by the axial boundary conditions (i.e., equation 5 6 5 6 (27)), and c ¼ 0, c ¼ 1. Substituting c and c into equation (29), it is found that the function of P (k) is not 5 6 5 6 h2 related to k. So, the situation of j ¼ 0 is not suitable. 2. when j > 0 Let j ¼ k , the solution of equation (24) is as follows P ðkÞ¼ c cosðkkÞþ c sinðkkÞ (30) h2 7 8 where c and c are constant. There is no solution to the equation (24) combining with the boundary conditions, so 7 8 the equation (30) is also not suitable. 3. when j < 0 Let j ¼�k , the solution of equation (24) can be expressed as follows kk �kk P ðkÞ¼ c e þ c e (31) h2 9 10 where c and c are constant, they can be obtained by using the axial boundary conditions of the general solution, 9 10 k 2k and c ¼ c ¼ e /(1 þ e ). The equation (31) is the solution of equation (24). The final expression of P (k) can be 9 10 h2 obtained kk �kk P ðkÞ¼ ðe þ e Þ (32) h2 2k 1 þ e To solve the equation (25), the following transformations are adopted 2 3 H ¼ c u; P ðuÞ¼ fðuÞ=H ; H ¼ H (33) h h h1 h where H is an approximation function of the thickness of the oil film, c is the coefficient of the approximation h h function, and f(u) is the transformation function of P (u). h1 Equation (25) can be expressed as Strum–Liouville equation based on equation (33), and then the equation (25) can be written as follows f ðuÞ� k fðuÞ¼ 0 (34) The boundary conditions of the equation (34) are as follows fðu Þ¼ 0; fðu Þ¼ 0 (35) a c 718 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 12 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Equations (34) and (35) are taken as Strum–Liouville equation, and the equation (34) can be expressed as follows �� � � pðuÞf ðuÞ � qðuÞfðuÞ� k qðuÞfðuÞ¼ 0 (36) where p(u) is the coefficient function of Strum-Liouville equation, q(u) is the potential function of Strum- Liouville equation, q(u) is the weight function of Strum–Liouville equation, and p(u) ¼ 1, q(u) ¼ 0, q(u) ¼ 1. The eigenvalue and eigenfunction of equation (36) can be obtained directly as follows D ip k ¼ ; i ¼ 1; 2 . . . (37) B ðu � u Þ c a �� �� D ipðu � u Þ f ðuÞ¼ sin k ¼ u � u ¼ sin (38) i i ð Þ B ðu � u Þ c a where i is the number of eigenvalue, k is the ith eigenvalue, and f (u) is the ith eigenfunction. Because of the i i orthogonality of eigenfunction, the following expression can be obtained 0 1 f ðuÞ¼ g f ðuÞ g ¼ Z (39) i B u C i i i @ 2 A qðuÞf ðuÞ du Then, general solution can be expressed as follows P ðu; kÞ¼ r f ðuÞP ðkÞ (40) h i i h2 i¼1 where u k k k �k k i i ðÞ r ¼ g � �qðuÞ�p ðuÞ�f ðuÞ du P ðkÞ¼ e þ e i p1 i h2 i i 2k 1 þ e The general solutions of the oil film force in the radial and tangential directions can be obtained by integrating P (u, k), and they are expressed as Z Z 1 u > c > h F ¼� P ðu; kÞcosu du dk; < h �1 u Z Z (41) 1 u F ¼� P ðu; kÞsinu du dk > h �1 u Nonlinear oil film force. According to the equations (21) and (41), the nonlinear oil film forces in the radial and tangential directions can be expressed as follows P h F ¼ F þ F r r (42) P h F ¼ F þ F t t The expressions of the nonlinear oil film force in the x and y directions are as follows F ¼ F cosh þ F sinh x t r (43) F ¼�F sinh þ F cosh y t r Hei and Zheng 719 Hei and Zheng 13 Verification of the proposed method To verify the proposed solution method of nonlinear oil film force of the bearing. The nonlinear oil film forces of bearing calculated by the proposed method, the finite difference method, and the infinite long bearing model are 0 0 compared. In the verification, the parameter of the bearing is listed as follows: B/d ¼ 1, e ¼ 0.5, x ¼ 0.1, y ¼ 0.2, and i ¼ 30. The comparison result is shown in Figure 4. From Figure 4, it can be seen that the oil film force calculated by the proposed method is in good agreement with the result calculated by the finite difference method. Meanwhile, the calculation time of the proposed method and the finite difference method are also compared, and shown in Figure 5. From Figure 5, it can be seen that the calculation time of the proposed method is smaller than that of the finite difference method. The results of Figures 4 and 5 indicate that the proposed method not only ensures the calculation accuracy, but also improves the calculation efficiency. Solving method of nonlinear dynamics Improved Newmark method The Newmark method is usually based on an assumption of constant acceleration. It is also called constant average acceleration hypothesis. The solution process of the Newmark method is as follows At time t þDt, the equation (1) can be written as follows € _ Mq þ Gq þ Kq ¼ P (44) tþDt tþDt tþDt tþDt where P ¼f þQ þWþf . According to the constant average acceleration hypothesis, the following equations can be obtained q_ ¼ q_ þ a q € þ a q € (45) tþDt t 6 t 7 tþDt where a and a are integral parameters of the Newmark method, and a ¼ Dtð1 � cÞ, a ¼ cDt. Dt is the step of 6 7 6 7 time, c is the control parameter of the Newmark method, and c � 0.5. q ¼ q þ q_ Dt þ ½ð0:5 � aÞq € þ aq € �Dt (46) tþDt t t t tþDt Figure 4. Nonlinear oil film forces versus eccentric angle h. (a) F versus h; (b) F versus h. x y 720 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 14 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 5. Comparison of calculation efficiency of two method. where a is the control parameter of the Newmark method, and a � 0.25(0.5 þ c). Based on constant average acceleration hypothesis, q € and q_ can be expressed by q , q € , q_ , and q in tþDt tþDt tþDt t t t the following forms € _ € q ¼ a ðq � q Þ� a q � a q (47) 0 2 3 tþDt tþDt t t t q_ ¼ a ðq � q Þ� a q_ � a q € (48) 1 4 5 tþDt tþDt t t t where a , a , a , a , a , and a are integral parameters of the Newmark method, and 0 1 2 3 4 5 1 c 1 1 c c a ¼ ; a ¼ ; a ¼ ; a ¼ � 1; a ¼ � 1; a ¼ � 1 Dt 0 1 2 3 4 5 aDt aDt aDt 2a a 2a By substituting the equations (47) and (48) into equation (44), and then the equation (44) can be written as ~ ~ Kq ¼ P (49) tþDt tþDt where the equivalent stiffness matrix (K ¼ K þ a M þ a G) is irrelevant to time, and the equivalent load vector 0 1 P ¼ P þ M½a q þ a q_ þ a q € �þG½a q þ a q_ þ a q € � tþDt tþDt 0 2 3 1 4 5 t t t t t t is relevant to time. q , q_ , q € can be obtained by the equations (47) to (49). q , q_ , q € satisfy the tþDt tþDt tþDt tþDt tþDt tþDt constrained condition of motion, but cannot satisfy the dynamic equation simultaneously. _ € Substituting the values of q , q into the equation (44), q can be obtained tþDt tþDt tþDt �1 € _ q ¼ M ðP � Gq � Kq Þ (50) tþDt tþDt tþDt tþDt _ € It should be noted that q , q , q satisfy the dynamic equation, but they do not satisfy the motion tþDt tþDt tþDt constrained condition simultaneously. In order to made the displacement, velocity, and acceleration satisfy the motion constrained condition and dynamic equation simultaneously, the Newmark method is improved as follows. From the equations (47) and (50), it can be seen that Newmark method will produce an imbalance acceleration € € (dq ) at each calculation step, and dq can be obtained by equations (47) and (50). tþDt tþDt € € € dq ¼ q � q (51) tþDt tþDt tþDt Hei and Zheng 721 Hei and Zheng 15 By substituting the equation (51) into equation (50), the following equation can be obtained € _ Mq þ Gq þ Kq ¼ P þ dP (52) tþDt tþDt tþDt tþDt tþDt � � where dP is the increment of dynamic load, and dP ¼ Mdq € . tþDt tþDt tþDt In order to eliminate the increment of dynamic load in equation (52), an increment equation of dynamics can be obtained, and it is as follows � � � � Mdq € þ Gdq_ þ Kdq ¼� dP (53) tþDt tþDt tþDt tþDt � � where dq € , dq_ , and dq are the increments of displacement, velocity, and acceleration respectively, which tþDt tþDt tþDt are caused by � dP . They can be calculated by the following equations tþDt � � 1 � dq ¼ � K dP (54) tþDt tþDt � � dq_ ¼ a dq (55) tþDt tþDt � � dq € ¼ a dq (56) tþDt tþDt � � � � € _ dq , dq , and dq satisfy the motion constrained condition. Because the calculate disturbance � dP is tþDt tþDt tþDt tþDt � � � � � very small, dq € , dq_ , and dq will also satisfy the equation (53). This means that the dq € , dq_ , and tþDt tþDt tþDt tþDt tþDt dq satisfy the motion constrained condition and the dynamic equation simultaneously. tþDt By adding equation (52) to equation (53), and the increment of dynamic load is eliminated. And then the displacement, velocity and acceleration at the time (tþDt) can be written as follows � � q ¼ q þ dq (57) tþDt tþDt tþDt � � q_ ¼ q_ þ dq_ (58) tþDt tþDt tþDt � � q € ¼ q € þ dq € (59) tþDt tþDt tþDt � � q , q_ , q € are taken as the true solution to equation (44). tþDt tþDt tþDt From the analysis above, it can be seen that the solution of the improved Newmark method not only satisfies the motion constrained condition, but also satisfies dynamics equation basically. Verification of the improved Newmark method In order to verify the improved Newmark method, a model of the rotor bearing system is adopted. The com- parison of the calculating results among Newmark method, improved Newmark method and Runge–Kutta method is implemented. In the verification, the parameters of the rotor-bearing system are as follows: the width of the bearing B¼ 0.16 m, the width to diameter ratio B/d¼ 1, the viscosity of lubricating oil l¼ 0.02626 Pa s, the radius clearance c¼ 0.000288 m, the parameter of system s¼ 0.2662. The equivalent dimensionless mass of the rotor at the two bearing stations are m � ¼ 0.3825 and m � ¼ 0.3825, the dimensionless mass of the disk a b � � are m � ¼ 1.235, the dimensionless stiffness of two shafts are k ¼ k ¼ 4.3, and the eccentricity of the two disks O 1 2 is e ¼ 0.2. Figure 6 shows the bifurcation diagrams which calculated by the proposed method (i.e., improved Newmark method) and Newmark method. From Figure 6, it can be seen that the dynamic behaviors of the rotor can be investigated for a wide range of speeds when the improved Newmark method is adopted. Figure 7(a) shows the orbit of the rotor calculated by the improved Newmark method, Newmark method and Runge–Kutta method when x � ¼ 1. From Figure 7(a), it can be seen that the orbit calculated by the improved Newmark method is very closed to the calculating result of the Runge–Kutta method. Because the error of each step is compensated, 722 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 16 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 6. The bifurcation diagram calculated by improved Newmark method and Newmark method. (a) Bifurcation diagram calculated by the proposed method; (b) Bifurcation diagram calculated by Newmark. Figure 7. The comparison of obits and calculation efficiency between Runge–Kutta method and the improved Newmark method. (a) Obits calculated by the three method for x  ¼ 1 (b) Comparison on the calculation efficiency between the Runge-Kutta method and the improved Newmark method. the improved Newmark method has higher precision than the Newmark method. The comparison of calculation efficiency between the Runge-Kutta method and the improved Newmark method has been implemented in Figure 7(b). From Figure 7(b), it can be seen that the improved Newmark method has higher computational efficiency than the Runge–Kutta method. Hei and Zheng 723 Hei and Zheng 17 Figure 8. The results of the two models for x ¼ 1; 1:1. (a) The results of the two models for x ¼ 1; (b) The results of the two models for x  ¼ 1:1. Numerical examples and results The rod fastening rotor is adopted, and the oil film force is calculated by the proposed method. The improved Newmark method is adopted to calculated the dynamic response of the rod fastening rotor-bearing system. The parameter of the rotor system are as follows: r ¼ 0.075 m, l ¼ 3 m, a ¼ 1.5 m, b ¼ 1.5 m, a ¼ 1.3 m, b ¼ 1.7 m, 1 1 a ¼ 1.7 m, b ¼ 1.3 m, m ¼ m ¼ 206.75 kg, h ¼ 0.4 m (the width of disks), R ¼ 0.16 m, e e ¼ 0.3c (i.e., 2 2 A B O1 ¼ O2 2 2 e ¼ 0.3), m ¼ m ¼ 250.93 kg, J ¼ J ¼ 3.2118 kg�m , J ¼ J ¼ 1.6059 kg�m ,2 m ¼ m þ m þ m þ m , o1 o2 z1 z2 d1 d2 A B o1 o2 m ¼ 457.668 kg, B ¼ 0.15 m, c ¼ 0.000195 m, l ¼ 0.02 N·s/m . The stiffness of the shafts are as follows: 7 6 6 0 0 0 0 k ¼ k ¼ k ¼ k ¼ 1.0116 � 10 N/m, k ¼ k ¼� 3.7733 � 10 N, k ¼ k ¼ 3.7733 � 10 N, k ¼ k ¼ 11 22 11 22 14 23 14 23 33 44 0 0 k ¼ k ¼ 2.0847 � 10 N·m, k ¼ k ¼ 5k /3. The number of eigenvalue i ¼ 30. 33 44 b R 11 The comparison of the dynamic behaviors is implemented between the integral rotor and rod fastening rotor. Figure 8(a) and (b) show the orbits of the rotor at bearing station for x  ¼ 1 and x  ¼ 1:1. From Figure 8(a), it can be seen that the orbits are all periodic orbits, and the orbits amplitude of the rod fastening rotor is smaller than that of integral rotor. In Figure 8(b), it can be seen that the motion of the rod fastening rotor is periodic motion, but the motion of the integral rotor is not periodic motion. The result in Figure 8 demonstrated that the rod fastening rotor is more stable. Taking the dimensionless speed as control parameter Taking the dimensionless speed x as control parameter, and the dimensionless speed range changes from 0.9 to 1.35. Figure 9 shows the bifurcation diagram of the rotor at bearing station and disk 1 station. From Figure 9, the rotor shows rich dynamic behaviors when the dimensionless speed of the rotor changes. Specially, the rotor is stable when the rotating speed is low, and the rotor becomes unstable when the dimensionless speed of the rotor increases. When x  ¼ 0:9, Figure 10 shows the orbit, time series and spectrum of the rotor at the bearing station. From Figure 10(a), it can be seen that the orbit of the rotor is a closed curve, which is similar to a circle. It means that the motion of the rotor is periodic and the motion of the rotor is stable. Figure 11 shows the swing angles of the two disks. In Figure 11, it can be seen that the swing angles of the disks also change periodically. With the increase of the rotating speed, the stable periodic response of the rotor system changes to period-2 response. Figure 12 shows the orbit, Poincare map, time series and spectrum of the rotor at bearing station when x  ¼ 1:1. From Figure 12(a), it can be seen that the motions of the rotor are typical half- frequency whirl motions, and there are two fixed point when the orbit pass through the Poincare section. 724 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 18 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 9. Bifurcation diagram of y versus x at bearing station and disk stations. (a) Bifurcation diagram of y at bearing station; (b) Bifurcation diagram of y at disk 1. Figure 10. The orbit, time series and spectrum of the rotor at bearing station for x  ¼ 0:9. (a) The orbit of the journal at bearing station A; (b) Time series; (c) The spectrum. Hei and Zheng 725 Hei and Zheng 19 Figure 11. The swing angle of disks 1 and 2 around the x and y axis for x ¼ 0:9. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Figure 12. The orbit, Poincare map, time series and spectrum of the rotor at bearing station for x  ¼ 1:1. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series; (d) The spectrum. 726 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 20 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Therefore, the motion of the rotor is period-2 motion. The swing angles of the disks are shown in Figure 13, and it can be seen that the swing of the disks is consistent with the motion of the rotor. With the increase of the rotating speed, the period-2 motion of the rotor turns to period-4 motion. Figure 14 shows the orbit, Poincare map, time series and spectrum of the rotor at bearing station when x  ¼ 1:18. It can be seen that there are four closed curves in Figure 14(a), and there are four fixed points in the Poincare section in Figure 14(b). Figure 15 shows the swing angles of disks. In Figure 15, it is demonstrated that the swing of the disks is consistent with the motion of the rotor. For x ¼ 1:26, Figure 16 shows the orbit, Poincare map, time series and spectrum of the rotor at bearing station. From Figure 16(a), it can be seen that the orbit of the rotor is disarray, and the fixed point of the Poincare section is also disarray in Figure 6(b). The frequency spectrum is continuous spectrum in Figure 16(d). Figure 17 shows the swing angles of the disks. From Figure 17, it can be seen that the swing angles of the disks are irregular. From the describe above, it can be seen that the chaos phenomenon is exhibited (i.e., the periodic-4 motion of the rotor turns to chaos motion) with the rotating speed increase continuously. Taking the dimensionless bending stiffness of the shafts as control parameter 0 0 The dimensionless bending stiffness of the shafts contain K , K , K , K . In this section, 11 22 11 22 0 0 K ¼ K ¼ K ¼ K ¼ K. Taking the dimensionless bending stiffness K as control parameter, the dynamics 11 22 11 22 responses of the rod fastening rotor system are investigated when x  ¼ 1:18. When the dimensionless stiffness K changes from 3.0 to 6.0, Figure 18 shows the bifurcation diagram of the rotor at bearing station and disk 1station respectively. From Figure 18, it can be seen that the motion types of the rotor system are period-4 motion, period-2 motion, periodic motion, quasi-periodic motion, period-6 motion, quasi-periodic motion, and periodic motion when the dimensionless stiffness K are 3.0, 3.5, 3.8, 4.3, 4.5, 4.7, and 6.0. When the dimensionless bending stiffness of the shaft is low, the motion of the rotor is period-4 motion. When K ¼ 3.0, the orbit, Poincare map, time series and spectrum of period-4 motion of the rotor are shown in Figure 14. With the increase of the dimensionless bending stiffness of the shaft, period-4 motion of the rotor turns to a period-2 motion. When K ¼ 3.5, Figure 19 shows the orbit and Poincare map of the period-2 motion of the rotor. Figure 13. The swing angle of disks 1 and 2 around the x and y axis for x  ¼ 1:1. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Hei and Zheng 727 Hei and Zheng 21 Figure 14. The orbit of rotor, Poincare map, time series and spectrum of the rotor at bearing station for x ¼ 1:18. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series; (d) The spectrum. Figure 15. The swing angle of disks 1 and 2 around the x and y axis for x  ¼ 1:18. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. 728 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 22 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 16. The orbit, Poincare map, time series and spectrum of rotor at bearing station for x  ¼ 1:26. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series (d) The spectrum. Figure 17. The swing angle of disks 1 and 2 around the x and y axis for x  ¼ 1:26. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Hei and Zheng 729 Hei and Zheng 23 Figure 18. Bifurcation diagram of y versus K at bearing station and disk stations. (a) Bifurcation diagram of y at bearing station; (b) Bifurcation diagram of y at disk 1 station. Figure 19. The orbit, Poincare map of rotor at bearing station for K¼ 3.5. (a) The orbit of the journal at bearing station A; (b) Poincare map. 730 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 24 Journal of Low Frequency Noise, Vibration and Active Control 0(0) With the dimensionless bending stiffness of the shaft increase continuously, the period-2 motion bifurcates to stable periodic motion. When K¼ 3.8, Figure 20 shows the orbit of periodic motion of the rotor. Then, the quasi- periodic bifurcation (i.e., the periodic motion turns to quasi-periodic motion) appears with the increase of the dimensionless bending stiffness of the shaft. When K¼ 4.3, Figure 21 shows the orbit, Poincare map, time series and spectrum of the rotor. From Figure 21(a), it can be seen that the quasi-periodic orbit is in a bounded region, and the projection of the Poincare map is a closed curve which is shown in Figure 21(b). From Figure 21(c) and (d), it can be seen that the amplitude of the orbit is irregular, and frequency spectrum contains multiple frequency components. Figure 22 shows the swing angle of the disks, and the swing angle of the disks are irregular. Figure 20. The orbit of rotor at bearing station for K¼ 3.8. Figure 21. The orbit, Poincare map, time series and spectrum of the rotor at bearing station for K¼ 4.3. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series (d) The spectrum. Hei and Zheng 731 Hei and Zheng 25 Figure 22. The swing angle of disks 1 and 2 around the x and y axis for K¼ 4.3. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Figure 23. The orbit, Poincare map, time series and spectrum of the rotor at bearing station for K¼ 4.5. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series; (d) The spectrum. 732 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 26 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 24. The swing angle of disks 1 and 2 around the x and y axis for K¼ 4.5. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Figure 25. The orbit, Poincare map of rotor at bearing station for K¼ 4.7. (a) The orbit of the journal at bearing station A; (b) Poincare map. Hei and Zheng 733 Hei and Zheng 27 When K¼ 4.5, Figure 23 shows the orbit, Poincare map, time series and spectrum of the rotor. From Figure 23 (a), it can be seen that a period-6 motion is appeared for the rotor, and the orbit contains six closed curves. Moreover, there are six fixed points in the Poincare section, as shown in Figure 23(b). From Figure 23(c) and (d), it can be seen that there are multiple amplitudes and frequency components. Figure 24 shows the swing angle of the disks. With the increase of dimensionless bending stiffness of the shaft, the motion of the rotor changed from period-6 motion to quasi-periodic motion, and then, the quasi-periodic motion turns to stable periodic motion. When K¼ 4.7, Figure 25 shows the orbit, Poincare map of the rotor. In Figure 25, it can be seen that the orbit of the rotor is in a bounded area, and projection of the Poincare map is a closed curve. Figure 26 shows the orbit of the rotor when K¼ 6, and the orbit of the rotor is a closed curve. Figure 26. The orbit of rotor at bearing station for K¼ 6.0. Figure 27. Bifurcation diagram of y versus e at bearing station and disk stations. (a) Bifurcation diagram of y at bearing station; (b) Bifurcation diagram of y at disk 1 station. 734 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 28 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Taking the dimensionless eccentricity of the disks as control parameter Taking the dimensionless eccentricity of the disks as control parameter, the dynamic behaviors of the rotor system are investigated. In the investigation, the dimensionless rotating speed x ¼ 1:18, and the range of dimensionless eccentricity of the disks changes from 0.01 to 0.47. Figure 27 shows the bifurcation of the rotor at the bearing and disk 1 station. From Figure 27, it can be seen that the rotor will exhibits quasi-periodic motion, periodic motion, period-2 motion, period-4 motion, period-2 motion, and periodic motion with the increase of the dimensionless eccentricity. Figure 28. The orbit, Poincare map of rotor at bearing station for e¼ 0.01. (a) The orbit of the journal at bearing station A; (b) Poincare map. Figure 29. The orbit of rotor at bearing station for e¼ 0.08. Hei and Zheng 735 Hei and Zheng 29 Figure 30. The orbit, Poincare map of rotor at bearing station for e¼ 0.15. (a) The orbit of the journal at bearing station A; (b) Poincare map. Figure 31. The orbit, Poincare map of rotor at bearing station for e¼ 0.42. (a) The orbit of the journal at bearing station A; (b) Poincare map. 736 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 30 Journal of Low Frequency Noise, Vibration and Active Control 0(0) When the dimensionless eccentricity of the disks is very low, the motion of the rotor is quasi-periodic motion. Figure 28 shows the orbit and Poincare map of the rotor at the bearing station for e¼ 0.01. Then, quasi-periodic motion will turn to periodic motion. When e¼ 0.08, Figure 29 shows the periodic orbit of the rotor at the bearing station. With the increase of the dimensionless eccentricity of the disks, the periodic motion bifurcates to a period- 2 motion. When e¼ 0.15, Figure 30 shows the orbit and Poincare map of the rotor at the bearing station. The bifurcation phenomenon occurs again when the dimensionless eccentricity increases continuously, and the Figure 32. The orbit of rotor at bearing station for e¼ 0.46. Figure 33. Bifurcation diagram of y versus x at bearing station and disk stations for K ¼K ¼0.1,0.7,2,20. (a) Bifurcation diagram of y b R for K ¼ K ¼ 0.1; (b) Bifurcation diagram of y for K ¼ K ¼ 0.7; (c) Bifurcation diagram of y for K ¼ K ¼ 2; (d) Bifurcation diagram of b R b R b R y for K ¼ K ¼ 20. b R Hei and Zheng 737 Hei and Zheng 31 period-2 motion will turn to period-4 motion. When e¼ 0.3, Figure 14 shows the calculating results. The period-4 motion bifurcates to period-2 motion when the dimensionless eccentricity increase continuously. Figure 31 shows the orbit and Poincare map of the rotor at the bearing station when e¼ 0.42. Then, the motion of the rotor will turn to stable periodic motion with the increase of the dimensionless eccentricity of the disks. Figure 32 shows the orbit of the rotor at bearing station when e¼ 0.46. Taking the dimensionless contact stiffness of the disks as control parameter Letting K K , the influence of the dimensionless contact stiffness K and K on the dynamics of the rod fastening b¼ R b R rotor bearing system is studied. The parameters are the same as the section “Taking the dimensionless speed as control parameter.” For K K ¼0.1, 0.7, 2, 20, Figure33(a) to (d) show the bifurcation diagram at bearing b¼ R station. From Figure 33, it can be seen that the types of the motions of the rotor system are similar when x  < 1:195, and the motions of the rotor system are periodic motion, period-2 motion, and period-4 motion. The motion of the rotor system turns to chaos motion when x  � 1:195. From Figure 33(a), it can be seen that the chaos motion of the rotor system turns to period-4 motion, and then, the period-4 motion turns to period-2 motion with the increase of the rotating speed. Finally, the period-2 motion turns to chaos motion. The range of rotating speed in which the motions of the rotor system are period-4 motion and period-2 motion is from 1.25 to 1.29, and this range of rotating speed shrinks gradually with the increase of the contact stiffness which are shown in Figure 33(b) and (c). At the same time, it is found that the rotor collides with the wall of the bearing in the speed range of 1.21–1.245 when K K ¼0.1. The speed range shrinks gradually with the increase of the dimensionless contact stiffness, and b¼ R the range disappears completely when K ¼K ¼20. For K ¼K ¼0.1 and x  ¼ 1:24, Figure 34 shows the orbit and b R b R time series of the rotor at the bearing station and swing angle of the disk 1. From Figure 34(a) and (b), it can be seen that the orbit of the rotor at the bearing station is regular, and the orbit becomes disordered with the increase of the time. The corresponding time series is shown in Figure 34(c). Figure 34(d) shows the swing angle of the Figure 34. The orbit, time series of the rotor at bearing station and the swing angle of disks 1 around the x and y axis for K ¼K ¼0.1, x  ¼ 1:24. (a) The orbit of the journal for 25–0 circle; (b) The orbit of the journal for 25-last; (c) Time series; (d) The b R swing angle of disks 1 around the x and y. 738 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 32 Journal of Low Frequency Noise, Vibration and Active Control 0(0) disk 1. From Figure 34(d), it can be seen that the swing angle becomes large abruptly. The instability of the rotor is due to the sudden increase of the swing angle of the disk 1. Therefore, it is very important to choose the appropriated contact stiffness. Conclusions In this study, to investigate the dynamic behaviors of a rod fastening rotor. A motion model of rod fastening rotor is developed by considering the contact effect and gyro effect of the disks. An approximate analytical solution of the oil film force for the supporting bearing of the rotor is proposed based on the method of separation of variables and Sturm–Liouville theory, and the Newmark method is improved to solve the motion model of the rotor bearing system. The main results are as follows: 1. The proposed approximate analytical solution can solve the oil film force of the bearing fast, and the results calculated by the proposed approximate analytical solution agree well with the results calculated by the finite difference method. Moreover, compared with the Newmark method, the improved Newmark method has high precision by compensating the disturbance error, and the improved Newmark method has higher calculation efficiency than Runge–Kutta method. 2. Based on the approximate analytical solution of the oil film force and the improved Newmark method, the dynamics behaviors of the rod fastening rotor system and the integral rotor system are studied. The results indicate that the rod fastening rotor system is more stable than the integral rotor system. 3. Taking the dimensionless rotating speed, bending stiffness of the shaft and the eccentricity of the disks as control parameter, the rod fastening rotor system exhibits rich and complex nonlinear dynamic behaviors. Furthermore, the influence of the contact stiffness of the disks on the dynamic motion of the rotor system is analyzed. The result indicates that the contact stiffness of the disks has great effect on the dynamic behaviors of the rotor system when the rotor operates under non-periodic motion state. Meanwhile, a large swing angle will be observed for the disks when the contact stiffness is inappropriate, and it will consequently result in collision between the rotor and bearing. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Scientific Research Program of Shaanxi Province of China (No. 2019JQ-928), Research Program of Weinan of Shaanxi Province of China (Nos.2016KYJ-1–2, 2018-ZDYF-JCYJ-60), Research Program of Shaanxi Railway Institute of China (No. KY2018-53). ORCID iD Di Hei https://orcid.org/0000-0002-4609-1679 References 1. Yongfeng Y, Qinyu W, Yanlin W, et al. Dynamic characteristics of cracked uncertain hollow-shaft. Mech Syst Signal Process 2019; 124: 36–48. 2. Wang G and Yuan H. Dynamic stability analysis for a flexible rotor filled with liquid based on three-dimensional flow. ASME J Fluids Eng 2018; 20: 2253–2267. 3. Desavale RG. Dynamics characteristics and diagnosis of a rotor-bearing’s system through a dimensional analysis approach: an experimental study. ASME J Comput Nonlinear Dyn 2019; 14: 014501–014501. 4. Barbosa MPF and Alves WM. Kriging-based surrogate modeling for rotordynamics prediction in rotor-bearing system[C]. In: Cavalca K, Weber H (eds). Proceedings of the 10th international conference on rotor dynamics-IFToMM. Cham: Springer, 2019. doi: 10.1007/978-3-319-99270-9_22. 5. Zhang S, Xu H, Zhang L, et al. Vibration suppression mechanism research of adjustable elliptical journal bearing under synchronous unbalance load. Tribol Int 2018; 132: 1–286. doi: https://doi.org/10.1016/j.triboint.2018.10.043. Hei and Zheng 739 Hei and Zheng 33 6. Santos IF and Svendsen PK. Noninvasive parameter identification in rotordynamics via fluid film bearings-linking active lubrication and operational modal analysis. ASME J Eng Gas Turbines Power 2017; 139: 062507. 7. Hei D, Lu Y, Zhang Y, et al. Nonlinear dynamic behaviors of a rod fastening rotor supported by fixed-tilting pad journal bearings. Chaos Solitons Fractals 2014; 69: 129–150. 8. Zhang S, Lv C and Li J. Experimental study on tangential contact stiffness of plate joint surface of the rod fastening rotor. DTEEES 2018; 10: 12783. dio: 10.12783/dteees/epme2018/24631. 9. Li Y. Research on the dynamic characteristics of the rod fastening rotor. In: Proceedings of the 2018 7th international conference on energy and environmental protection. Paris: Atlantis Press, 2018. dio: 10.2991/iceep-18.2018.173. 10. Hu L, Liu YB, Zhao L, et al. Nonlinear dynamic behaviors of circumferential rod fastening rotor under unbalanced pre- tightening force. Arch Appl Mech 2016; 86: 1621–1631. 11. Hu L, Liu YB, Zhao L, et al. Nonlinear dynamic response of a rub-impact rod fastening rotor considering nonlinear contact characteristic. Arch Appl Mech 2016; 86: 1869–1886. 12. Ramos DJ, Ferraz AR, Daniel GB, et al. Dynamic analysis of rotating systems considering uncertainties in the bearings’ parameters. In: Proceedings of the 10th international conference on rotor dynamics-IFToMM. Cham: Springer, 2019. doi: 10.1007/978-3-319-99272-3_32. 13. Wei Y, Chen Z, Jiao Y, et al. Computational analysis of nonlinear dynamics of a multi-disk rotor-bearing-brush seal system. Int J Str Stab Dyn 2018; 18: 350–362. 14. Liu Y, Liu H, Wang X, et al. Nonlinear dynamic characteristics of a three-dimensional rod-fastening rotor bearing system. Proc IMechE Part C: J Mechanical Engineering Science 2015; 229: 882–894. 15. Smolık L, Rendl J, Dyk S, et al. Threshold stability curves for a nonlinear rotor-bearing system. J Sound Vibrat 2019; 442: 698–713. 16. Wenzel da Silva Tuckmantel F and Cavalca KL. Vibration signatures of a rotor-coupling-bearing system under angular misalignment. Mech Machine Theor 2019; 133: 559–583. 17. Lu Y, Ji L, Zhang Y, et al. Dynamic behaviours of the rotor non-linear system with fixed-tilting-pad journal bearings support. IMechE Part J: J Engineering Tribology. 2010; 224: 1037–1047. 18. Qian Z, Cheng L, Chen W, et al. Analysis on bistable response of a disk-rod-fastening rotor. J Aerospace Power 2011; 26: 1563–1568. (In Chinese). 19. Zhong YE, He YZ, et al. Dynamics of the rotor. Beijing: Tsinghua University Press, 1987. (in Chinese). 20. Feng L. The dynamic characteristics of tilting-pad journal bearing. Xi’an: Xi’an University of Technology, 2011. (in Chinese). 21. Hei D and Zheng M. Dynamics analysis of flexible rotor system based on Wilson-h method. China Mech Eng 2011; 22: 2918–2922. (in Chinese). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE

Investigation on the dynamic behaviors of a rod fastening rotor based on an analytical solution of the oil film force of the supporting bearing:

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Abstract

The dynamic characteristics of rod fastening rotor supported by a finite journal bearing are investigated in this study. To model the dynamic behaviors of the bearing-rotor system, the oil film force of bearing is calculated by approximately solving the Reynolds equation with the variables separation method and Sturm–Liouville theory, and then a motion equation is developed with consideration of the contact and gyro effects of the disks of the rotor. To solve the motion equation with small error and excellent stability, an improved Newmark method is proposed. On this basis, the dynamics characteristics of the rod fastening rotor are analyzed for different rotor speeds, disk eccentricities, shaft bearing stiffness, and contact stiffness. And the orbits of the rod fastening rotor and integral rotor are compared. The numerical results indicate that the analytical solution of the oil film force has higher computational efficiency than the finite difference method. The rod fastening rotor shows higher stability than the integral rotor, and exhibits rich dynamic behaviors, such as periodic, qusi-periodic, period-2, period-4, and period-6. Keywords Rod fastening rotor, Newmark method, Sturm–Liouville theory, approximate solution, stability Introduction Rotor-bearing system is an important component in the power equipment, and its stability affects greatly the safety and normal service performance of the equipment. However, the oil film force of the bearing is usually nonlinear, and consequently causes complex and unstable dynamic behaviors of the rotor when the rotor is working under the conditions of high speed and overload. Therefore, the dynamic stability of the rotor-bearing 1–6 system caused by the nonlinear oil film force has investigated by many researchers. Yang et al. established the model of rotor with transverse crack by considering parametric uncertainties. And the equation of the system was solved by the Harmonic Balance method. Then, the influences of transverse crack and uncertain parameters on the responses of the rotor were studied. Wang et al. studied the stability of a flexible liquid-filled rotor based on Fourier series. The analytical expression of the dynamic pressure of the liquid was derived, and the dynamic stability of the rotor was analyzed by the numerical method. Desavale et al. established a comprehensive empirical model with dimensionless parameters. Department of Mechanical and Electrical Engineering, Shaanxi Railway Institute, Wei Nan, China Corresponding author: Di Hei, Department of Mechanical and Electrical Engineering, Shaanxi Railway Institute, Wei Nan 714000, China. Email: drillok@126.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 708 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 2 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Based on the dimensional analysis approach, the diagnosis of misalignment and bearing looseness were studied. The calculating results of the empirical model agreed well with the experimental results. Barbosa et al. predicted the rotor dynamics by using the Kriging surrogate models. The models can reduce the calcu- lation time effectively. Zhang et al. proposed an adjustable elliptical journal bearing to suppress the vibration of amplitude of the rotor, and the results indicated that the bearing can effectively reduce the vibration. Santos et al. studied the dynamic characteristic of a rigid-rotor system experimentally. The damping and natural frequencies were obtained, and the maximum servovalve voltage and radial injection pressure were found for the rotor system. In the studies above, the integral rotor was taken as the object. However, besides the integral rotor, the rod fastening rotor is also very commonly used in the heavy gas turbine and so on. For the rod fastening rotor, the rod bolts are used to press all disks tightly together. Therefore, the rod fastening rotor will show some different dynamic behaviors from the integral rotor. In this context, the dynamics characteristics of the rod fastening rotor has been studied by some researchers. Hei et al. established the model of rod fastening rotor supported by fixed- tilting pad journal bearing. The database method was employed to solve the oil film force of the fixed-tilting pad journal bearing. The dynamics behaviors of the rotor system were investigated by the orbit diagrams, the time series, the frequency spectrum diagrams. But, the gyro effects of the disks were not taken into account when modelling. Zhang studied the tangential contact stiffness of the disks of the rod fastening rotor based on the fractal contact theory. The correctness of the fractal contact theory was verified by experiment. At the same time, it was found that the fractal contact theory has its scope of application. The study only solved the tangential contact stiffness, but the dynamics behaviors of the rod fastening rotor were not analyzed. Li established the model of rod fastening rotor by considering the contact friction force and pre-tightening force of the disks. The effects of the contact friction force and pre-tightening force on the rotor dynamics were studied. The results shown that the contact stiffness and the natural frequency of the system increased with the increase of pre- tightening force. Hu et al derived the motion equation of the rod fastening rotor by Lagrange equation. The influence of the initial deflection caused by the unbalanced pre-tightening force on the dynamic behaviors of the rotor was studied. Their work showed that the initial deflection has a great effect on the rotor dynamics. Meanwhile, Hu et al. also studied the dynamic characteristics of the rub-impact rod fastening rotor. The bifurcation diagram, vibration waveform, frequency spectrum, shaft orbit, and Poincare map were used to analyze the dynamics responses of the rotor system. However, in the works of Hu et al. , the influence of the gyro effect on the dynamic behaviors of the rod fastening rotor was not considered, and a short bearing model was assumed. For the rod fastening rotor, the hydrodynamic sliding bearing is usually taken as the supporting part, and the oil film force of the bearing is very important for the analysis of the dynamic behaviors of the rotor. So, the investigation of the oil film force was implemented by many researches. In many studies, the model of infinite long 12–14 bearing and infinite short bearing were adopted widely. Considering journal bearing parametric uncertainties, Ramos et al. studied the dynamics of the rotor system, and the rotor is supported by the short fluid film bearings. Wei et al. studied the nonlinear dynamic behaviors of the multi-disk rotor-bearing-seal system which supported by short bearing. The response of the system was calculated by the fourth order Runge– Kutta method. Liu et al. studied the dynamics characteristics of the rod fastening rotor which supported by the infinite long bearing. The shooting method and path-following technique were adopted to calculate the dynamics responses of the system. The infinite long bearing and infinite short bearing models are very simple and idealized models. In practice, the slide bearings are finite long bearings. So, the two idealized models cannot describe the actual bearing. In order to solve the oil film force of finite long bearing accurately, numerical methods are proposed. Smolık et al. studied the shape of the threshold curve for a rigid rotor supported by journal slide bearing. In order to solve the oil film force of journal bearing, the four methods were proposed: the infinitely short approximation, the infinitely short approximation, the finite differences method, and the finite elements method. Tuckmantel investigated the vibration signature of rotor-coupling-bearing system by solving the oil film force of the bearing with finite volume method. Lu et al. calculated the oil film forces and their Jacobis by the variational constraint approach. However, the numerical methods (such as the Hei and Zheng 709 Hei and Zheng 3 Figure 1. Schematic diagram of the bearing-rod fastening rotor system. finite volume method, finite elements method, finite differences method, and variational constraint approach) have large calculation cost. In this study, an approximation analytical solution of oil film force is proposed for the supporting bearing of the rod fastening rotor by using the method of separation of variables and Sturm–Liouville theory. This method not only ensures the calculation accuracy, but also improves the calculation efficiency. Meanwhile, a model of the rod fastening rotor is established by considering the Gyro effect and contact effect of the disks, and an improved Newmark method is proposed to solve the dynamic responses of the rotor system. On this basis, the comparison of obit is implemented between the integral rotor and rod fastening rotor. The influence of rotating speed, eccentricity, bending stiffness, and contact stiffness on the dynamic behaviors of the rotor system is also investigated. Equation of the rod fastening rotor system The geometry model of the rod fastening rotor system supported by oil film hydrodynamic journal bearings is given in Figure 1. In this study, the contact effect and gyro effect of the disks are considered, and the contact effect of disks is taken as a bending spring with nonlinear stiffness. In Figure 1, O and O are the centers of the two disks, l is the length of the whole rotor, a and b are the length of 1 2 the two shafts, a is the distance between A and O , b is the distance between B and O , a is the distance between 1 1 1 1 2 A and O , b is the distance between B and O , m and m are the mass of disk O and O , m and m are the 2 2 2 O1 O2 1 2 A B mass of the two shafts (the length are a and b respectively), e and e are the eccentricities of the disk O and O , O1 O2 1 2 f and f are the nonlinear oil film forces of bearing in negative x and y directions, respectively. x y The dynamic equation of the rod fastening rotor system can be written as follows Mq € þ Gq_ þ Kq ¼ f þ Q þ W þ f (1) where M is the mass matrix, G is the gyroscopic matrix, K is the stiffness matrix, f is the vector of nonlinear oil film forces of bearing, Q is periodic exciting forces (i.e., unbalanced or steam excitation with the same phase as rotating speed) vector acting on rotor, W is the vector of gravity, f is the vector of nonlinear restoring force caused by the nonlinear stiffness, and q is the displacement vector of the rotor. The 12 degrees-of-freedom of the system are [x , y , x , y , h , h ,x , y , h , h , x , y ]. x and y are the displacements of the rotor at bearing station A in A O1 O1 x1 y1 O2 O2 x2 y2 B B A A x and y directions, x and y are the displacements of the center of disk O in x and y directions, h and h are O1 O1 1 x1 y1 the angles that disk O rotates around the x and y axes, x and y are the displacements of the center of disk O 1 O2 O2 2 in x and y directions, h and h are the angles that disk O rotates around the x and y axes, x and y are the x2 y2 2 B B displacements of the rotor at bearing station B in x and y directions. q ¼½x ; y ; x ; y ; h ; h ; x ; y ; h ; h ; x ; y � A A O1 O1 x1 y1 O2 O2 x2 y2 B B f ¼ ½� f ; � f ; 0; 0; 0; 0; 0; 0; 0; 0; � f ; � f � xA yA xB yB 710 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 4 Journal of Low Frequency Noise, Vibration and Active Control 0(0) W ¼ ½0; m g; 0; m g; 0; 0; 0; m g; 0; 0; 0; m g� A O1 O2 B 2 2 2 2 Q ¼ ½0; 0; m e x sinðxtÞ; m e x cosðxtÞ; 0; 0; m e x sinðxtÞ; m e x cosðxtÞ; 0; 0; 0; 0� O1 O1 O1 O1 O2 O2 O2 O2 3 3 3 3 T f ¼ ½0; 0; k ðx � x Þ ; k ðy � y Þ ; 0; 0; � k ðx � x Þ ; � k ðy � y Þ ; 0; 0; 0; 0� R O2 O1 R O2 O1 R O2 O1 R O2 O1 2 3 m 0 0 00 00 00 0 0 0 6 7 6 7 m 0 00 00 00 0 0 0 6 7 6 7 6 7 m 00 00 00 0 0 0 O1 6 7 6 7 6 m 00 0 0 00 0 0 7 O1 6 7 6 7 6 7 J 00 00 0 0 0 d1 6 7 6 7 6 7 J 0 0 00 0 0 d1 6 7 M ¼ 6 7 6 7 m 0 00 0 0 O2 6 7 6 7 6 7 m 00 0 0 6 O2 6 7 6 7 6 J 00 0 7 d2 6 7 6 7 6 7 J 00 d2 6 7 6 7 6 7 m 0 4 5 2 3 00 00 0 0 00 0 0 00 6 7 6 7 6 7 0 00 0 0 00 0 0 00 6 7 6 7 6 7 00 0 0 00 0 0 00 6 7 6 7 6 7 6 7 0 0 0 00 0 0 00 6 7 6 7 6 7 6 7 0 � J x 00 0 0 00 z1 6 7 6 7 6 7 6 J x 0 00 0 0 007 z1 6 7 G ¼ 6 7 6 7 00 0 0 00 6 7 6 7 6 7 6 7 0 0 0 00 6 7 6 7 6 7 6 7 0 � J x 00 z2 6 7 6 7 6 7 J x 0 00 6 7 z2 6 7 6 7 6 7 6 7 4 5 0 Hei and Zheng 711 Hei and Zheng 5 > > b k > 44 > > > k þ > > > 6 2 2 > > l l > > > > > > > > > > > 2k b > 14 1 > > > > � � �� < = 6 2 l k b k k k b 11 1 14 44 14 1 0 � þ 00 � 2 0 > > l l l l 6 > b k > > 0 2 44 > 6 > > þk þ > 11 > 2 2 6 > > l l > > 6 > > > � > 6 > > > > 2k b 6 > > > > 6 > � > : ; 8 9 > > 6 b k > 33 > > > > k þ > > 2 2 > > l l > > > > > �� �� > > < = 6 b k k k b 1 23 33 23 1 2k b b 23 1 0 2 0 0 �k þ � 0 � þ k 6 22 > 2 2 > l l l l l l > > > > > > 6 > > 0 0 > > 6 > k 2k b > 33 23 > > 6 > > þ � : ; 2 2 l l �� k k b 6 14 11 1 � 0 ðÞ k þ k 00 k 6 11 b 14 6 l l �� 6 k k b 23 22 1 6 0 � 0 ðÞ k þ k k 0 22 b 23 l l �� k k b 33 23 1 0 � 0 k k 0 23 33 l l 6 �� k k b 44 14 1 � 0 k 00 k 14 44 l l 6 � � 0 0 K ¼ k b k 6 11 14 � þ 0 �k 000 l l � � 0 0 k b k 6 2 22 23 0 � þ 0 �k 00 6 l l �� 0 0 6 k k b 33 23 0 � 0 000 l l �� 0 0 k k b 44 14 � 0 0 000 l l 8 � 9 6 k a b k > > 11 1 1 44 > > > � > > 2 2 > > > l l > > > > > > > > > > > k ðb � a Þ > 14 1 1 > > 6 � � � � > > l k a k k k a 11 1 14 44 14 1 0 � � 00 � � 0 0 > > l l l l 6 k a b k > > 2 2 11 44 > > > þ � > > > 6 2 2 > > l l > > > > �> > > > > 6 k ðb � a Þ > 2 2 > > > 6 þ : ; 8 9 6 � k a b k > 22 1 1 33 > > > > � > > > 2 2 > > l l > > > > > > > > > > > k ðb � a Þ > 23 1 1 > > 6 �� � � > > < = 6 2 k a k k k a l 22 1 23 33 23 1 0 0 � � � � 0 0 0 > > l l l l 6 k a b k > 2 2 > > 22 33 > > þ � > > > 2 2 > > l l > > > > > �> > 0 > > > k ðb � a Þ > 2 2 > > > : ; l 712 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 6 Journal of Low Frequency Noise, Vibration and Active Control 0(0) 8 9 k a b k > 11 1 1 44 > > > > > > � > > > > > 2 2 > l l > > > > > > > > > > > > k ðb � a Þ > 14 1 1 � � � � > > > < = > 0 0 0 0 þ k b k k k b 2 2 2 > 11 14 44 14 � þ 00 � 0 > 0 0 k a b k > > > l l l l 2 2 11 44 > > > > þ � > > > 2 2 > > > > > l l > �> > > 0 > > > > > k ðb � a Þ > 2 2 > > > > > : ; > 2 > l > 8 9 � > k a b k > 22 1 1 33 > > > > > > > 2 2 > > > l l > > > > > > > > > k ðb � a Þ > 23 1 1 � �� � > > < = 0 0 0 þ > b k k k b > 2 2 2 23 33 23 > 0 l 0 � k þ � 0 0 0 0 > k a b k l l l l > > 2 2 > > 22 33 > > þ � > > > 2 2 > > > l l > > �> > 0 > > > k ðb � a Þ > 2 2 > 23 >> > > þ ; : > l > � � > k a k 11 1 14 �k 0 00 � � 0 > l l � � k a k > 22 1 23 > 0 �k 00 0 � � b > l l > � � k k a 33 23 1 > 000 0 0 � � l l � � > k k a 44 14 1 000 0 � � 0 > l l � � 0 0 k a k 11 14 0 0 ðk þ k Þ 00 k � � 0 11 14 > l l � � > 0 0 > k a k 2 > 22 23 0 0 0 ðk þ k Þ k 0 0 � � > 22 23 l l > � � 0 0 k k a > 33 23 > 0 0 0 k k 0 0 � � 23 33 l l � � > 0 0 k k a 2 > 44 14 0 0 k 00 k � � 0 > 14 44 > l l > 8 9 2 > > > > k a k > 11 44 1 > > > > þ > > > 2 2 > > > l l > > > > > > > > 2k a > 14 1 > > > � � � � < = 0 0 0 0 þ þ > k a k k k a 2 2 2 11 14 44 14 l > � � 00 � � 0 0 2 0 > k a k l l l l > > > 11 2 44 > > þ > > > 2 2 > > > l l > � > > > 0 > > > 2k a > > > 2 > > > > þ > : 2 ; l > 8 9 > > > k a k > > > 22 33 > 1 > > > þ > > > > > 2 2 > > l l > > > > > > > > > > > 2k a 23 1 � � � � > > > < = > 0 0 0 0 þ k a k k k a 2 > 2 2 22 23 33 23 l > 0 � � � � 0 0 > 0 2 0 > > k a k > > l l l l > 22 2 33 > > > > þ þ > > 2 2 > > > l l > > � > > > > > 2k a > > 2 > > > > > þ > : ; > where f and f are the nonlinear oil film forces of bearing at the station A in negative x and y directions, xA yA respectively. f and f are the nonlinear oil film forces of bearing at the station B in negative x and y directions, xB yB respectively. J and J are the polar moments of the inertia for disks O and O . J and J are the equatorial d1 d2 1 2 z1 z2 moments of the inertia for disks O and O . k is the bending stiffness of the rod. k is the nonlinear 1 2 b R stiffness of contact interface. x is the rotating speed of the rotor. k and k are the bending stiffness of shaft 11 22 AO in the x and y directions. k and k are the stiffness caused by the swing angle which disk O oscillated 33 44 1 0 0 around the x and y axes, k and k are the cross stiffness of the shaft AO. k and k are the bending stiffness of 14 23 11 22 0 0 shaft BO in the x and y directions. k and k are the stiffness caused by the swing angle which disk O oscillated 33 44 2 0 0 around the x and y axes. k and k are the cross stiffness of the shaft BO. The rod fastening rotor is a symmetric 14 23 0 0 0 0 rotor in this paper, so the relationship of the stiffness are k ¼ k ¼ k ¼ k , k ¼ k ¼ k ¼ k , 11 22 11 22 33 44 33 44 0 0 19 k ¼ k ¼ k ¼ k . 14 23 14 23 Hei and Zheng 713 Hei and Zheng 7 For the convenience of calculating and derivation, the following dimensionless variables are introduced. pffiffiffiffiffiffiffi X ¼ x =c; Y ¼ y =c; m ¼ m =m ði ¼ A; O ; O ; BÞ; l ¼ l=c; x ¼ x c=g; s ¼ xt; i i i i i i 1 2 3 2 e ¼ e =c ðj ¼ O ; O Þ; r ¼ lxBr =ðmgc Þ j j 1 2 F ¼ f=ðrmgÞ; K ¼ K ¼ k c=ðmgÞ; K ¼ K ¼ k =ðmgcÞ; 11 22 11 33 44 33 K ¼ K ¼ k =ðmgÞ; J ¼ J=ðmc Þ 23 14 23 where c is radius clearance, r is the radius of the shaft journal, B is the width of the bearing, and r is Sommerfeld number. Nonlinear oil film force Solution of nonlinear oil film force The geometry of the journal bearing is shown in Figure 2, and the cross section of the bearing is shown in Figure 3. In the two figures, O is the center of the bearing, O is the center of the shaft journal, h is the deviation b j angle, U is the angle goes from negative y-axis direction along the clockwise direction, u is the angle which goes from extension of O O along the clockwise direction, f and f are oil film forces in the negative radial and the j b r t Figure 2. Coordinate of finite length journal bearing. Figure 3. The cross section of the finite length journal bearing. 714 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 8 Journal of Low Frequency Noise, Vibration and Active Control 0(0) tangential directions, f and f are oil film forces in negative x and y directions, h is the thickness of oil film, R is x y the radius of the bearing, and W is the load of the bearing. By assuming that the lubricant is in-compressible, the Reynolds equation for the lubrication of the journal bearing can be written as �� 3 3 1 @ h @pðu; zÞ @ h @pðu; zÞ @h @h þ ¼ x þ 2 (2) R @u 6 @u @z 6 @z @u @t where l is the viscosity of the lubricating oil film. p is the distribution of pressure of oil film, h is the thickness of oil film (h ¼ c þ ecosu), R is the radius of bearing, and z is the axial coordinate. For the convenience of derivation and calculation, the dimensionless variables are introduced as follows �� k ¼ z ; H ¼ 1 þ ecosu; e ¼ e=c; s ¼ xt; e ¼ de=ds; h ¼ dh=ds; (3) w ¼ c=R; P ¼ p=p ðp ¼ 2xl=w Þ 0 0 where k is the dimensionless axial coordinate, H is the dimensionless thickness of the oil film, e is the dimensionless eccentricity, P is the dimensionless oil film pressure distribution, w is clearance ratio, and e is the radial velocity. Substituting the dimensionless variables into equation (2), the dimensionless Reynolds equation can be written as �� �� �� @ @P D @ @P 3 3 0 H þ H ¼ �3esinu þ 6e cosu þ 6eh sinu (4) @u @u B @k @k where eh is the dimensionless tangential velocity. To solve equation (4), the method of separation of variables and Sturm–Liouville theory are used. Based on the method of separation of variables, the dimensionless oil film pressure P can written as Pðu; kÞ¼ P ðu; kÞþ P ðu; kÞ (5) p h where P (u, k) is the special solution, P (u, k) is the general solution, and their expressions are given as p h P ðu; kÞ¼ P ðuÞþ P ðkÞ (6) p p1 p2 P ðu; kÞ¼ P ðuÞP ðkÞ (7) h h1 h2 In equation (5), it can be seen that P (u, k) and P (u, k) determine the oil film pressure distribution P. p h Therefore, P (u, k) and P (u, k) need to be solved respectively, and their solutions are introduced in the next section. p h Calculation of special solution. Substituting equation (6) into equation (4), the dimensionless Reynolds equation can be written as �� 2 2 0 d P ðkÞ B d P ðuÞ 3esinu dP ðuÞ 3esinu 6e cosu 6eh sinu p2 p1 p1 ¼ � þ � þ þ (8) 2 3 3 3 D du H du H H H dk The expression for the left item of equation (8) is a function of k, and the expression for the right item of equation (8) is a function of u. Let d P ðkÞ p2 ¼ C (9) dk Hei and Zheng 715 Hei and Zheng 9 �� 2 0 d P ðuÞ dP ðuÞ B 3esinu 3esinu 6e cosu 6eh sinu p1 p1 � þ � � ¼ C (10) 2 3 3 3 D du H du H H H P (k) can be obtained by integrating equation (9) twice. p2 P ðkÞ¼ Ck =2 þ c k þ c (11) p2 1 2 In equation (11), C is arbitrary constant, c and c are integral constants. Let C ¼ 0, c and c can be 1 2 1 2 obtained by solving the boundary conditions. The boundary conditions of the special solution in the axial direc- tion are as follows P j ¼ 0 (12) p2 k¼�1 Based on equation (12), the constant c and c are obtained, and c ¼ c ¼ 0. 1 2 1 2 Let dP ðuÞ p1 ¼ P ðuÞ (13) p1 du Substituting equation (13) into equation (10), P ðuÞ can be written as p1 0 0 P ðuÞ¼ ðc þ 6e sinu þ 3ecosu � 6eh cosuÞ (14) p1 3 ð1 þ ecosuÞ where c is undetermined constant. By integrating equation (13), P (u) can be expressed as 3 p1 P ðuÞ¼ P ðuÞ du þ c (15) p1 p1 4 where c is undetermined constant. In order to obtain the expression of P (u), Sommerfeld transformation is introduced as follows p1 1=2 1=2 2 2 2 1 � e ð1 � e Þ sinuð1 � e Þ e þ cosu 1 þ ecosu ¼ ; du ¼ da; sina ¼ ; cosa ¼ ; 1 � ecosa 1 � ecosa 1 þ ecosu 1 þ ecosu rffiffiffiffiffiffiffiffiffiffiffi (16) 1 � e u a ¼ 2 arctan tan 1 þ e 2 Based on Sommerfeld transformation, expression of P (u) can be written as follows p1 2 2 0 a � 2esina þ 0:5e sinacosa þ 0:5e a 3e P ðuÞ¼ c þ p1 3 5=2 2 ð1 � e Þ eð1 þ ecosuÞ �� (17) 3e �6eh sinacosa 3ea þ þ e sina þ sina � e � þ c 5=2 5=2 2 2 2 2 ð1 � e Þ ð1 � e Þ The boundary conditions of the special solution in the circumferential direction are as follows @P p1 u ¼ u ¼ 0; P ðu Þ¼ 0; u ¼ u ¼ p; j ¼ 0 (18) p1 a a c u¼u @u 716 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 10 Journal of Low Frequency Noise, Vibration and Active Control 0(0) where u is the starting position of the oil film pressure and u is the termination position of the oil film. a c Based on the boundary conditions (i.e., equation (18)), c and c can be expressed as follows 3 4 3e c ¼� (19) eð1 þ eÞ �� �� 0 0 5=2 6eh 3e 2 esina cosa 3ea 3e 3e 2 c c c � e sina þ sina � � þ � ð1 � e Þ c c 2 2 5=2 5=2 2 2 2 2 ðÞ ðÞ eð1þeÞ eð1þecosu Þ 1�e 1�e c ¼ c (20) 2 2 a � 2esina þ 0:5e sina cosa þ 0:5e a c c c c c The special solutions of the oil film force in the radial and tangential directions can be obtained by integrating the P (u, k), and they can be written as follows Z Z 1 u F ¼� P ðuÞcosu dudk; p1 �1 u (21) Z Z 1 u > p > F ¼� P ðuÞsinu dudk p1 : t �1 u Calculation of general solution. By substituting P (u, k) into equation (4), then the equation (4) can be written as follows �� 2 2 @ P ðu; kÞ @P ðu; kÞ D @ P ðu; kÞ h h h 3 2 3 H � 3H esinu þ H ¼ 0 (22) 2 2 @u @u B @k Substituting equation (7) into equation (22), the following expression can be obtained �� �� 2 2 00 00 0 P ðkÞ B P ðuÞ B 3 P ðuÞ h2 h1 h1 � ¼ � esinu (23) P ðkÞ D P ðuÞ D H P ðuÞ h2 h1 h1 In order to solve the equation (23), the following expressions should be satisfied P ðkÞ h2 � ¼j (24) P ðkÞ h2 �� �� 2 2 00 0 B P ðuÞ B 3 P ðuÞ h1 h1 � esinu ¼j (25) D P ðuÞ D H P ðuÞ h1 h1 For the journal bearing, the following boundary conditions should be satisfied P ðu; �1Þ¼ P ðuÞþ P ð�1Þ¼�P ðu; �1Þ¼�P ðuÞP ð�1Þ (26) p p1 p2 h h1 h2 Based on the boundary conditions above, the axial boundary conditions for the general solution is obtained P j ¼ 1; h2 k¼1 (27) P j ¼ 1 h2 k¼�1 Hei and Zheng 717 Hei and Zheng 11 In order to solve equation (24), three situations of j need to be considered (i.e., j > 0, j ¼ 0, and j < 0). 1. when j ¼ 0 Equation (24) can be written as P ðkÞ¼ 0 (28) h2 By integrating the equation (28) twice, the expression of P (k) can be obtained as follows h2 P ðkÞ¼ c k þ c (29) h2 5 6 where c and c are integral constants. c and c can be obtained by the axial boundary conditions (i.e., equation 5 6 5 6 (27)), and c ¼ 0, c ¼ 1. Substituting c and c into equation (29), it is found that the function of P (k) is not 5 6 5 6 h2 related to k. So, the situation of j ¼ 0 is not suitable. 2. when j > 0 Let j ¼ k , the solution of equation (24) is as follows P ðkÞ¼ c cosðkkÞþ c sinðkkÞ (30) h2 7 8 where c and c are constant. There is no solution to the equation (24) combining with the boundary conditions, so 7 8 the equation (30) is also not suitable. 3. when j < 0 Let j ¼�k , the solution of equation (24) can be expressed as follows kk �kk P ðkÞ¼ c e þ c e (31) h2 9 10 where c and c are constant, they can be obtained by using the axial boundary conditions of the general solution, 9 10 k 2k and c ¼ c ¼ e /(1 þ e ). The equation (31) is the solution of equation (24). The final expression of P (k) can be 9 10 h2 obtained kk �kk P ðkÞ¼ ðe þ e Þ (32) h2 2k 1 þ e To solve the equation (25), the following transformations are adopted 2 3 H ¼ c u; P ðuÞ¼ fðuÞ=H ; H ¼ H (33) h h h1 h where H is an approximation function of the thickness of the oil film, c is the coefficient of the approximation h h function, and f(u) is the transformation function of P (u). h1 Equation (25) can be expressed as Strum–Liouville equation based on equation (33), and then the equation (25) can be written as follows f ðuÞ� k fðuÞ¼ 0 (34) The boundary conditions of the equation (34) are as follows fðu Þ¼ 0; fðu Þ¼ 0 (35) a c 718 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 12 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Equations (34) and (35) are taken as Strum–Liouville equation, and the equation (34) can be expressed as follows �� � � pðuÞf ðuÞ � qðuÞfðuÞ� k qðuÞfðuÞ¼ 0 (36) where p(u) is the coefficient function of Strum-Liouville equation, q(u) is the potential function of Strum- Liouville equation, q(u) is the weight function of Strum–Liouville equation, and p(u) ¼ 1, q(u) ¼ 0, q(u) ¼ 1. The eigenvalue and eigenfunction of equation (36) can be obtained directly as follows D ip k ¼ ; i ¼ 1; 2 . . . (37) B ðu � u Þ c a �� �� D ipðu � u Þ f ðuÞ¼ sin k ¼ u � u ¼ sin (38) i i ð Þ B ðu � u Þ c a where i is the number of eigenvalue, k is the ith eigenvalue, and f (u) is the ith eigenfunction. Because of the i i orthogonality of eigenfunction, the following expression can be obtained 0 1 f ðuÞ¼ g f ðuÞ g ¼ Z (39) i B u C i i i @ 2 A qðuÞf ðuÞ du Then, general solution can be expressed as follows P ðu; kÞ¼ r f ðuÞP ðkÞ (40) h i i h2 i¼1 where u k k k �k k i i ðÞ r ¼ g � �qðuÞ�p ðuÞ�f ðuÞ du P ðkÞ¼ e þ e i p1 i h2 i i 2k 1 þ e The general solutions of the oil film force in the radial and tangential directions can be obtained by integrating P (u, k), and they are expressed as Z Z 1 u > c > h F ¼� P ðu; kÞcosu du dk; < h �1 u Z Z (41) 1 u F ¼� P ðu; kÞsinu du dk > h �1 u Nonlinear oil film force. According to the equations (21) and (41), the nonlinear oil film forces in the radial and tangential directions can be expressed as follows P h F ¼ F þ F r r (42) P h F ¼ F þ F t t The expressions of the nonlinear oil film force in the x and y directions are as follows F ¼ F cosh þ F sinh x t r (43) F ¼�F sinh þ F cosh y t r Hei and Zheng 719 Hei and Zheng 13 Verification of the proposed method To verify the proposed solution method of nonlinear oil film force of the bearing. The nonlinear oil film forces of bearing calculated by the proposed method, the finite difference method, and the infinite long bearing model are 0 0 compared. In the verification, the parameter of the bearing is listed as follows: B/d ¼ 1, e ¼ 0.5, x ¼ 0.1, y ¼ 0.2, and i ¼ 30. The comparison result is shown in Figure 4. From Figure 4, it can be seen that the oil film force calculated by the proposed method is in good agreement with the result calculated by the finite difference method. Meanwhile, the calculation time of the proposed method and the finite difference method are also compared, and shown in Figure 5. From Figure 5, it can be seen that the calculation time of the proposed method is smaller than that of the finite difference method. The results of Figures 4 and 5 indicate that the proposed method not only ensures the calculation accuracy, but also improves the calculation efficiency. Solving method of nonlinear dynamics Improved Newmark method The Newmark method is usually based on an assumption of constant acceleration. It is also called constant average acceleration hypothesis. The solution process of the Newmark method is as follows At time t þDt, the equation (1) can be written as follows € _ Mq þ Gq þ Kq ¼ P (44) tþDt tþDt tþDt tþDt where P ¼f þQ þWþf . According to the constant average acceleration hypothesis, the following equations can be obtained q_ ¼ q_ þ a q € þ a q € (45) tþDt t 6 t 7 tþDt where a and a are integral parameters of the Newmark method, and a ¼ Dtð1 � cÞ, a ¼ cDt. Dt is the step of 6 7 6 7 time, c is the control parameter of the Newmark method, and c � 0.5. q ¼ q þ q_ Dt þ ½ð0:5 � aÞq € þ aq € �Dt (46) tþDt t t t tþDt Figure 4. Nonlinear oil film forces versus eccentric angle h. (a) F versus h; (b) F versus h. x y 720 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 14 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 5. Comparison of calculation efficiency of two method. where a is the control parameter of the Newmark method, and a � 0.25(0.5 þ c). Based on constant average acceleration hypothesis, q € and q_ can be expressed by q , q € , q_ , and q in tþDt tþDt tþDt t t t the following forms € _ € q ¼ a ðq � q Þ� a q � a q (47) 0 2 3 tþDt tþDt t t t q_ ¼ a ðq � q Þ� a q_ � a q € (48) 1 4 5 tþDt tþDt t t t where a , a , a , a , a , and a are integral parameters of the Newmark method, and 0 1 2 3 4 5 1 c 1 1 c c a ¼ ; a ¼ ; a ¼ ; a ¼ � 1; a ¼ � 1; a ¼ � 1 Dt 0 1 2 3 4 5 aDt aDt aDt 2a a 2a By substituting the equations (47) and (48) into equation (44), and then the equation (44) can be written as ~ ~ Kq ¼ P (49) tþDt tþDt where the equivalent stiffness matrix (K ¼ K þ a M þ a G) is irrelevant to time, and the equivalent load vector 0 1 P ¼ P þ M½a q þ a q_ þ a q € �þG½a q þ a q_ þ a q € � tþDt tþDt 0 2 3 1 4 5 t t t t t t is relevant to time. q , q_ , q € can be obtained by the equations (47) to (49). q , q_ , q € satisfy the tþDt tþDt tþDt tþDt tþDt tþDt constrained condition of motion, but cannot satisfy the dynamic equation simultaneously. _ € Substituting the values of q , q into the equation (44), q can be obtained tþDt tþDt tþDt �1 € _ q ¼ M ðP � Gq � Kq Þ (50) tþDt tþDt tþDt tþDt _ € It should be noted that q , q , q satisfy the dynamic equation, but they do not satisfy the motion tþDt tþDt tþDt constrained condition simultaneously. In order to made the displacement, velocity, and acceleration satisfy the motion constrained condition and dynamic equation simultaneously, the Newmark method is improved as follows. From the equations (47) and (50), it can be seen that Newmark method will produce an imbalance acceleration € € (dq ) at each calculation step, and dq can be obtained by equations (47) and (50). tþDt tþDt € € € dq ¼ q � q (51) tþDt tþDt tþDt Hei and Zheng 721 Hei and Zheng 15 By substituting the equation (51) into equation (50), the following equation can be obtained € _ Mq þ Gq þ Kq ¼ P þ dP (52) tþDt tþDt tþDt tþDt tþDt � � where dP is the increment of dynamic load, and dP ¼ Mdq € . tþDt tþDt tþDt In order to eliminate the increment of dynamic load in equation (52), an increment equation of dynamics can be obtained, and it is as follows � � � � Mdq € þ Gdq_ þ Kdq ¼� dP (53) tþDt tþDt tþDt tþDt � � where dq € , dq_ , and dq are the increments of displacement, velocity, and acceleration respectively, which tþDt tþDt tþDt are caused by � dP . They can be calculated by the following equations tþDt � � 1 � dq ¼ � K dP (54) tþDt tþDt � � dq_ ¼ a dq (55) tþDt tþDt � � dq € ¼ a dq (56) tþDt tþDt � � � � € _ dq , dq , and dq satisfy the motion constrained condition. Because the calculate disturbance � dP is tþDt tþDt tþDt tþDt � � � � � very small, dq € , dq_ , and dq will also satisfy the equation (53). This means that the dq € , dq_ , and tþDt tþDt tþDt tþDt tþDt dq satisfy the motion constrained condition and the dynamic equation simultaneously. tþDt By adding equation (52) to equation (53), and the increment of dynamic load is eliminated. And then the displacement, velocity and acceleration at the time (tþDt) can be written as follows � � q ¼ q þ dq (57) tþDt tþDt tþDt � � q_ ¼ q_ þ dq_ (58) tþDt tþDt tþDt � � q € ¼ q € þ dq € (59) tþDt tþDt tþDt � � q , q_ , q € are taken as the true solution to equation (44). tþDt tþDt tþDt From the analysis above, it can be seen that the solution of the improved Newmark method not only satisfies the motion constrained condition, but also satisfies dynamics equation basically. Verification of the improved Newmark method In order to verify the improved Newmark method, a model of the rotor bearing system is adopted. The com- parison of the calculating results among Newmark method, improved Newmark method and Runge–Kutta method is implemented. In the verification, the parameters of the rotor-bearing system are as follows: the width of the bearing B¼ 0.16 m, the width to diameter ratio B/d¼ 1, the viscosity of lubricating oil l¼ 0.02626 Pa s, the radius clearance c¼ 0.000288 m, the parameter of system s¼ 0.2662. The equivalent dimensionless mass of the rotor at the two bearing stations are m � ¼ 0.3825 and m � ¼ 0.3825, the dimensionless mass of the disk a b � � are m � ¼ 1.235, the dimensionless stiffness of two shafts are k ¼ k ¼ 4.3, and the eccentricity of the two disks O 1 2 is e ¼ 0.2. Figure 6 shows the bifurcation diagrams which calculated by the proposed method (i.e., improved Newmark method) and Newmark method. From Figure 6, it can be seen that the dynamic behaviors of the rotor can be investigated for a wide range of speeds when the improved Newmark method is adopted. Figure 7(a) shows the orbit of the rotor calculated by the improved Newmark method, Newmark method and Runge–Kutta method when x � ¼ 1. From Figure 7(a), it can be seen that the orbit calculated by the improved Newmark method is very closed to the calculating result of the Runge–Kutta method. Because the error of each step is compensated, 722 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 16 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 6. The bifurcation diagram calculated by improved Newmark method and Newmark method. (a) Bifurcation diagram calculated by the proposed method; (b) Bifurcation diagram calculated by Newmark. Figure 7. The comparison of obits and calculation efficiency between Runge–Kutta method and the improved Newmark method. (a) Obits calculated by the three method for x  ¼ 1 (b) Comparison on the calculation efficiency between the Runge-Kutta method and the improved Newmark method. the improved Newmark method has higher precision than the Newmark method. The comparison of calculation efficiency between the Runge-Kutta method and the improved Newmark method has been implemented in Figure 7(b). From Figure 7(b), it can be seen that the improved Newmark method has higher computational efficiency than the Runge–Kutta method. Hei and Zheng 723 Hei and Zheng 17 Figure 8. The results of the two models for x ¼ 1; 1:1. (a) The results of the two models for x ¼ 1; (b) The results of the two models for x  ¼ 1:1. Numerical examples and results The rod fastening rotor is adopted, and the oil film force is calculated by the proposed method. The improved Newmark method is adopted to calculated the dynamic response of the rod fastening rotor-bearing system. The parameter of the rotor system are as follows: r ¼ 0.075 m, l ¼ 3 m, a ¼ 1.5 m, b ¼ 1.5 m, a ¼ 1.3 m, b ¼ 1.7 m, 1 1 a ¼ 1.7 m, b ¼ 1.3 m, m ¼ m ¼ 206.75 kg, h ¼ 0.4 m (the width of disks), R ¼ 0.16 m, e e ¼ 0.3c (i.e., 2 2 A B O1 ¼ O2 2 2 e ¼ 0.3), m ¼ m ¼ 250.93 kg, J ¼ J ¼ 3.2118 kg�m , J ¼ J ¼ 1.6059 kg�m ,2 m ¼ m þ m þ m þ m , o1 o2 z1 z2 d1 d2 A B o1 o2 m ¼ 457.668 kg, B ¼ 0.15 m, c ¼ 0.000195 m, l ¼ 0.02 N·s/m . The stiffness of the shafts are as follows: 7 6 6 0 0 0 0 k ¼ k ¼ k ¼ k ¼ 1.0116 � 10 N/m, k ¼ k ¼� 3.7733 � 10 N, k ¼ k ¼ 3.7733 � 10 N, k ¼ k ¼ 11 22 11 22 14 23 14 23 33 44 0 0 k ¼ k ¼ 2.0847 � 10 N·m, k ¼ k ¼ 5k /3. The number of eigenvalue i ¼ 30. 33 44 b R 11 The comparison of the dynamic behaviors is implemented between the integral rotor and rod fastening rotor. Figure 8(a) and (b) show the orbits of the rotor at bearing station for x  ¼ 1 and x  ¼ 1:1. From Figure 8(a), it can be seen that the orbits are all periodic orbits, and the orbits amplitude of the rod fastening rotor is smaller than that of integral rotor. In Figure 8(b), it can be seen that the motion of the rod fastening rotor is periodic motion, but the motion of the integral rotor is not periodic motion. The result in Figure 8 demonstrated that the rod fastening rotor is more stable. Taking the dimensionless speed as control parameter Taking the dimensionless speed x as control parameter, and the dimensionless speed range changes from 0.9 to 1.35. Figure 9 shows the bifurcation diagram of the rotor at bearing station and disk 1 station. From Figure 9, the rotor shows rich dynamic behaviors when the dimensionless speed of the rotor changes. Specially, the rotor is stable when the rotating speed is low, and the rotor becomes unstable when the dimensionless speed of the rotor increases. When x  ¼ 0:9, Figure 10 shows the orbit, time series and spectrum of the rotor at the bearing station. From Figure 10(a), it can be seen that the orbit of the rotor is a closed curve, which is similar to a circle. It means that the motion of the rotor is periodic and the motion of the rotor is stable. Figure 11 shows the swing angles of the two disks. In Figure 11, it can be seen that the swing angles of the disks also change periodically. With the increase of the rotating speed, the stable periodic response of the rotor system changes to period-2 response. Figure 12 shows the orbit, Poincare map, time series and spectrum of the rotor at bearing station when x  ¼ 1:1. From Figure 12(a), it can be seen that the motions of the rotor are typical half- frequency whirl motions, and there are two fixed point when the orbit pass through the Poincare section. 724 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 18 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 9. Bifurcation diagram of y versus x at bearing station and disk stations. (a) Bifurcation diagram of y at bearing station; (b) Bifurcation diagram of y at disk 1. Figure 10. The orbit, time series and spectrum of the rotor at bearing station for x  ¼ 0:9. (a) The orbit of the journal at bearing station A; (b) Time series; (c) The spectrum. Hei and Zheng 725 Hei and Zheng 19 Figure 11. The swing angle of disks 1 and 2 around the x and y axis for x ¼ 0:9. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Figure 12. The orbit, Poincare map, time series and spectrum of the rotor at bearing station for x  ¼ 1:1. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series; (d) The spectrum. 726 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 20 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Therefore, the motion of the rotor is period-2 motion. The swing angles of the disks are shown in Figure 13, and it can be seen that the swing of the disks is consistent with the motion of the rotor. With the increase of the rotating speed, the period-2 motion of the rotor turns to period-4 motion. Figure 14 shows the orbit, Poincare map, time series and spectrum of the rotor at bearing station when x  ¼ 1:18. It can be seen that there are four closed curves in Figure 14(a), and there are four fixed points in the Poincare section in Figure 14(b). Figure 15 shows the swing angles of disks. In Figure 15, it is demonstrated that the swing of the disks is consistent with the motion of the rotor. For x ¼ 1:26, Figure 16 shows the orbit, Poincare map, time series and spectrum of the rotor at bearing station. From Figure 16(a), it can be seen that the orbit of the rotor is disarray, and the fixed point of the Poincare section is also disarray in Figure 6(b). The frequency spectrum is continuous spectrum in Figure 16(d). Figure 17 shows the swing angles of the disks. From Figure 17, it can be seen that the swing angles of the disks are irregular. From the describe above, it can be seen that the chaos phenomenon is exhibited (i.e., the periodic-4 motion of the rotor turns to chaos motion) with the rotating speed increase continuously. Taking the dimensionless bending stiffness of the shafts as control parameter 0 0 The dimensionless bending stiffness of the shafts contain K , K , K , K . In this section, 11 22 11 22 0 0 K ¼ K ¼ K ¼ K ¼ K. Taking the dimensionless bending stiffness K as control parameter, the dynamics 11 22 11 22 responses of the rod fastening rotor system are investigated when x  ¼ 1:18. When the dimensionless stiffness K changes from 3.0 to 6.0, Figure 18 shows the bifurcation diagram of the rotor at bearing station and disk 1station respectively. From Figure 18, it can be seen that the motion types of the rotor system are period-4 motion, period-2 motion, periodic motion, quasi-periodic motion, period-6 motion, quasi-periodic motion, and periodic motion when the dimensionless stiffness K are 3.0, 3.5, 3.8, 4.3, 4.5, 4.7, and 6.0. When the dimensionless bending stiffness of the shaft is low, the motion of the rotor is period-4 motion. When K ¼ 3.0, the orbit, Poincare map, time series and spectrum of period-4 motion of the rotor are shown in Figure 14. With the increase of the dimensionless bending stiffness of the shaft, period-4 motion of the rotor turns to a period-2 motion. When K ¼ 3.5, Figure 19 shows the orbit and Poincare map of the period-2 motion of the rotor. Figure 13. The swing angle of disks 1 and 2 around the x and y axis for x  ¼ 1:1. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Hei and Zheng 727 Hei and Zheng 21 Figure 14. The orbit of rotor, Poincare map, time series and spectrum of the rotor at bearing station for x ¼ 1:18. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series; (d) The spectrum. Figure 15. The swing angle of disks 1 and 2 around the x and y axis for x  ¼ 1:18. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. 728 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 22 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 16. The orbit, Poincare map, time series and spectrum of rotor at bearing station for x  ¼ 1:26. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series (d) The spectrum. Figure 17. The swing angle of disks 1 and 2 around the x and y axis for x  ¼ 1:26. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Hei and Zheng 729 Hei and Zheng 23 Figure 18. Bifurcation diagram of y versus K at bearing station and disk stations. (a) Bifurcation diagram of y at bearing station; (b) Bifurcation diagram of y at disk 1 station. Figure 19. The orbit, Poincare map of rotor at bearing station for K¼ 3.5. (a) The orbit of the journal at bearing station A; (b) Poincare map. 730 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 24 Journal of Low Frequency Noise, Vibration and Active Control 0(0) With the dimensionless bending stiffness of the shaft increase continuously, the period-2 motion bifurcates to stable periodic motion. When K¼ 3.8, Figure 20 shows the orbit of periodic motion of the rotor. Then, the quasi- periodic bifurcation (i.e., the periodic motion turns to quasi-periodic motion) appears with the increase of the dimensionless bending stiffness of the shaft. When K¼ 4.3, Figure 21 shows the orbit, Poincare map, time series and spectrum of the rotor. From Figure 21(a), it can be seen that the quasi-periodic orbit is in a bounded region, and the projection of the Poincare map is a closed curve which is shown in Figure 21(b). From Figure 21(c) and (d), it can be seen that the amplitude of the orbit is irregular, and frequency spectrum contains multiple frequency components. Figure 22 shows the swing angle of the disks, and the swing angle of the disks are irregular. Figure 20. The orbit of rotor at bearing station for K¼ 3.8. Figure 21. The orbit, Poincare map, time series and spectrum of the rotor at bearing station for K¼ 4.3. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series (d) The spectrum. Hei and Zheng 731 Hei and Zheng 25 Figure 22. The swing angle of disks 1 and 2 around the x and y axis for K¼ 4.3. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Figure 23. The orbit, Poincare map, time series and spectrum of the rotor at bearing station for K¼ 4.5. (a) The orbit of the journal at bearing station A; (b) Poincare map; (c) Time series; (d) The spectrum. 732 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 26 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 24. The swing angle of disks 1 and 2 around the x and y axis for K¼ 4.5. (a) The swing angle of disks 1 around the x and y axis; (b) The swing angle of disks 2 around the x and y axis. Figure 25. The orbit, Poincare map of rotor at bearing station for K¼ 4.7. (a) The orbit of the journal at bearing station A; (b) Poincare map. Hei and Zheng 733 Hei and Zheng 27 When K¼ 4.5, Figure 23 shows the orbit, Poincare map, time series and spectrum of the rotor. From Figure 23 (a), it can be seen that a period-6 motion is appeared for the rotor, and the orbit contains six closed curves. Moreover, there are six fixed points in the Poincare section, as shown in Figure 23(b). From Figure 23(c) and (d), it can be seen that there are multiple amplitudes and frequency components. Figure 24 shows the swing angle of the disks. With the increase of dimensionless bending stiffness of the shaft, the motion of the rotor changed from period-6 motion to quasi-periodic motion, and then, the quasi-periodic motion turns to stable periodic motion. When K¼ 4.7, Figure 25 shows the orbit, Poincare map of the rotor. In Figure 25, it can be seen that the orbit of the rotor is in a bounded area, and projection of the Poincare map is a closed curve. Figure 26 shows the orbit of the rotor when K¼ 6, and the orbit of the rotor is a closed curve. Figure 26. The orbit of rotor at bearing station for K¼ 6.0. Figure 27. Bifurcation diagram of y versus e at bearing station and disk stations. (a) Bifurcation diagram of y at bearing station; (b) Bifurcation diagram of y at disk 1 station. 734 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 28 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Taking the dimensionless eccentricity of the disks as control parameter Taking the dimensionless eccentricity of the disks as control parameter, the dynamic behaviors of the rotor system are investigated. In the investigation, the dimensionless rotating speed x ¼ 1:18, and the range of dimensionless eccentricity of the disks changes from 0.01 to 0.47. Figure 27 shows the bifurcation of the rotor at the bearing and disk 1 station. From Figure 27, it can be seen that the rotor will exhibits quasi-periodic motion, periodic motion, period-2 motion, period-4 motion, period-2 motion, and periodic motion with the increase of the dimensionless eccentricity. Figure 28. The orbit, Poincare map of rotor at bearing station for e¼ 0.01. (a) The orbit of the journal at bearing station A; (b) Poincare map. Figure 29. The orbit of rotor at bearing station for e¼ 0.08. Hei and Zheng 735 Hei and Zheng 29 Figure 30. The orbit, Poincare map of rotor at bearing station for e¼ 0.15. (a) The orbit of the journal at bearing station A; (b) Poincare map. Figure 31. The orbit, Poincare map of rotor at bearing station for e¼ 0.42. (a) The orbit of the journal at bearing station A; (b) Poincare map. 736 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 30 Journal of Low Frequency Noise, Vibration and Active Control 0(0) When the dimensionless eccentricity of the disks is very low, the motion of the rotor is quasi-periodic motion. Figure 28 shows the orbit and Poincare map of the rotor at the bearing station for e¼ 0.01. Then, quasi-periodic motion will turn to periodic motion. When e¼ 0.08, Figure 29 shows the periodic orbit of the rotor at the bearing station. With the increase of the dimensionless eccentricity of the disks, the periodic motion bifurcates to a period- 2 motion. When e¼ 0.15, Figure 30 shows the orbit and Poincare map of the rotor at the bearing station. The bifurcation phenomenon occurs again when the dimensionless eccentricity increases continuously, and the Figure 32. The orbit of rotor at bearing station for e¼ 0.46. Figure 33. Bifurcation diagram of y versus x at bearing station and disk stations for K ¼K ¼0.1,0.7,2,20. (a) Bifurcation diagram of y b R for K ¼ K ¼ 0.1; (b) Bifurcation diagram of y for K ¼ K ¼ 0.7; (c) Bifurcation diagram of y for K ¼ K ¼ 2; (d) Bifurcation diagram of b R b R b R y for K ¼ K ¼ 20. b R Hei and Zheng 737 Hei and Zheng 31 period-2 motion will turn to period-4 motion. When e¼ 0.3, Figure 14 shows the calculating results. The period-4 motion bifurcates to period-2 motion when the dimensionless eccentricity increase continuously. Figure 31 shows the orbit and Poincare map of the rotor at the bearing station when e¼ 0.42. Then, the motion of the rotor will turn to stable periodic motion with the increase of the dimensionless eccentricity of the disks. Figure 32 shows the orbit of the rotor at bearing station when e¼ 0.46. Taking the dimensionless contact stiffness of the disks as control parameter Letting K K , the influence of the dimensionless contact stiffness K and K on the dynamics of the rod fastening b¼ R b R rotor bearing system is studied. The parameters are the same as the section “Taking the dimensionless speed as control parameter.” For K K ¼0.1, 0.7, 2, 20, Figure33(a) to (d) show the bifurcation diagram at bearing b¼ R station. From Figure 33, it can be seen that the types of the motions of the rotor system are similar when x  < 1:195, and the motions of the rotor system are periodic motion, period-2 motion, and period-4 motion. The motion of the rotor system turns to chaos motion when x  � 1:195. From Figure 33(a), it can be seen that the chaos motion of the rotor system turns to period-4 motion, and then, the period-4 motion turns to period-2 motion with the increase of the rotating speed. Finally, the period-2 motion turns to chaos motion. The range of rotating speed in which the motions of the rotor system are period-4 motion and period-2 motion is from 1.25 to 1.29, and this range of rotating speed shrinks gradually with the increase of the contact stiffness which are shown in Figure 33(b) and (c). At the same time, it is found that the rotor collides with the wall of the bearing in the speed range of 1.21–1.245 when K K ¼0.1. The speed range shrinks gradually with the increase of the dimensionless contact stiffness, and b¼ R the range disappears completely when K ¼K ¼20. For K ¼K ¼0.1 and x  ¼ 1:24, Figure 34 shows the orbit and b R b R time series of the rotor at the bearing station and swing angle of the disk 1. From Figure 34(a) and (b), it can be seen that the orbit of the rotor at the bearing station is regular, and the orbit becomes disordered with the increase of the time. The corresponding time series is shown in Figure 34(c). Figure 34(d) shows the swing angle of the Figure 34. The orbit, time series of the rotor at bearing station and the swing angle of disks 1 around the x and y axis for K ¼K ¼0.1, x  ¼ 1:24. (a) The orbit of the journal for 25–0 circle; (b) The orbit of the journal for 25-last; (c) Time series; (d) The b R swing angle of disks 1 around the x and y. 738 Journal of Low Frequency Noise, Vibration and Active Control 40(2) 32 Journal of Low Frequency Noise, Vibration and Active Control 0(0) disk 1. From Figure 34(d), it can be seen that the swing angle becomes large abruptly. The instability of the rotor is due to the sudden increase of the swing angle of the disk 1. Therefore, it is very important to choose the appropriated contact stiffness. Conclusions In this study, to investigate the dynamic behaviors of a rod fastening rotor. A motion model of rod fastening rotor is developed by considering the contact effect and gyro effect of the disks. An approximate analytical solution of the oil film force for the supporting bearing of the rotor is proposed based on the method of separation of variables and Sturm–Liouville theory, and the Newmark method is improved to solve the motion model of the rotor bearing system. The main results are as follows: 1. The proposed approximate analytical solution can solve the oil film force of the bearing fast, and the results calculated by the proposed approximate analytical solution agree well with the results calculated by the finite difference method. Moreover, compared with the Newmark method, the improved Newmark method has high precision by compensating the disturbance error, and the improved Newmark method has higher calculation efficiency than Runge–Kutta method. 2. Based on the approximate analytical solution of the oil film force and the improved Newmark method, the dynamics behaviors of the rod fastening rotor system and the integral rotor system are studied. The results indicate that the rod fastening rotor system is more stable than the integral rotor system. 3. Taking the dimensionless rotating speed, bending stiffness of the shaft and the eccentricity of the disks as control parameter, the rod fastening rotor system exhibits rich and complex nonlinear dynamic behaviors. Furthermore, the influence of the contact stiffness of the disks on the dynamic motion of the rotor system is analyzed. The result indicates that the contact stiffness of the disks has great effect on the dynamic behaviors of the rotor system when the rotor operates under non-periodic motion state. Meanwhile, a large swing angle will be observed for the disks when the contact stiffness is inappropriate, and it will consequently result in collision between the rotor and bearing. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Scientific Research Program of Shaanxi Province of China (No. 2019JQ-928), Research Program of Weinan of Shaanxi Province of China (Nos.2016KYJ-1–2, 2018-ZDYF-JCYJ-60), Research Program of Shaanxi Railway Institute of China (No. KY2018-53). ORCID iD Di Hei https://orcid.org/0000-0002-4609-1679 References 1. Yongfeng Y, Qinyu W, Yanlin W, et al. Dynamic characteristics of cracked uncertain hollow-shaft. Mech Syst Signal Process 2019; 124: 36–48. 2. Wang G and Yuan H. Dynamic stability analysis for a flexible rotor filled with liquid based on three-dimensional flow. ASME J Fluids Eng 2018; 20: 2253–2267. 3. Desavale RG. Dynamics characteristics and diagnosis of a rotor-bearing’s system through a dimensional analysis approach: an experimental study. ASME J Comput Nonlinear Dyn 2019; 14: 014501–014501. 4. 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Journal

"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: May 31, 2020

Keywords: Rod fastening rotor; Newmark method; Sturm–Liouville theory; approximate solution; stability

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