Dynamical analysis of hollow-shaft dual-rotor system with circular cracks:

Yang Yongfeng; Wang Jianjun; Wang Yanlin; Fu Chao; Zheng Qingyang; Lu Kuan

doi:10.1177/1461348420948287

Dynamical analysis of hollow-shaft dual-rotor system with circular cracks:

Yongfeng, Yang; Jianjun, Wang; Yanlin, Wang; Chao, Fu; Qingyang, Zheng; Kuan, Lu 2020-09-01 00:00:00 In this paper, we considered a dual-rotor system with crack in shaft. The influence of circular crack in hollow shaft on dynamical response was studied. The equations of motion of 12 elements dual-rotor system model were derived. Harmonic balance method was employed to solve the equations. The critical speed and sub-critical speed responses were investigated. It was found that the circular crack in hollow shaft had greater influence on the first-backward critical speed than the first-forward critical speed. Owing to the influence of crack, the vibration peaks occurred at the 1/2, 1/3 and 1/4 critical speeds of the rotor system, along with a reduction in sub-critical speeds and critical speeds. The deeper crack away from the bearing affected the rotor more significantly. The whirling orbits, the time-domain responses and the spectra were obtained to show the super-harmonic resonance phenomenon in hollow-shaft cracked rotor system. Keywords Rotor, hollow-shaft, circular cracks, critical speed, super-harmonic Introduction Rotating machines represent the maximal and most important class of machinery used for ﬂuid media transpor- 1–3 tation, metal working and forming, energy generation, providing aircraft propulsion and other purposes. High 4–6 speed and heavy power are the development directions of modern rotating machineries. In the past decades, 7 8 9 there are a lot of literatures that focus on the study of unbalance, clearance, base motions, damping ratio 10 11–13 identiﬁcation, rubbing and viscoelastic properties of rotor system. Especially, crack and misalignment 14–16 effects in rotor dynamic characteristics are frequently investigated. Fatigue crack of the rotor shaft observed in the rotating machinery should be avoided. It may lead to cata- strophic failure. In this situation, there are non-linear and non-stationary responses of the rotor system. However, the strong non-linearity can make the system possess characteristics that are substantially different from those of the linear system, such as self-excited oscillations and jump discontinuities. Detailed investigation into the non-linear dynamic response prediction of cracked shaft is very important for diagnosing and preventing 18–20 rotor cracks. The inﬂuence of transverse crack on a rotating shaft has been the attention of many researchers. Extensive 21 22 reviews of the dynamic response of cracked rotor systems were published by Dimarogonas and Wauer. Pennacchi et al. proposed a model-based transverse crack identiﬁcation method suitable for industrial machines. The excellent accuracy obtained at deﬁned position and depth of different cracks demonstrated the effectiveness and reliability of the proposed method. Patel and Darpe investigated the inﬂuence of the crack-breathing models Institute of Vibration Engineering, Northwestern Polytechnical University, Xi’an, China Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang, China CRRC Zhuzhou Electric Locomotive Institute Co., Ltd, Hunan, China Corresponding author: Yang Yongfeng, Northwestern Polytechnical University, P.O. Box 264 127, West Youyi Road, Xi’an, Shaanxi 710072, China. Email: yyf@nwpu.edu.cn 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). 2 1228 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) (switching-crack model and response-dependent breathing-crack model) on the non-linear vibration character- istics of the cracked rotor. Switching-crack modeling reveals chaotic, quasi-periodic and sub-harmonic vibration response for deeper cracks, and more realistic breathing crack model reveals no evidence of them. Rubio and Fernandez-Saez proposed a new procedure to analyze the non-linear dynamics of cracked rotors using an iterative technique that transformed the full non-linear problem into a succession of time-dependent linear ones. The calculations using the proposed method were over a 100 times faster than the corresponding to integrate the full non-linear problem, being very helpful in on-line crack-identiﬁcation procedures. Sinou adopted har- monic balance method to study the stability of the rotor system presenting a transverse breathing crack by considering the effects of crack depth, crack position and the rotating speed. The areas of instability will increase considerably when the crack deepened and that the crack’s position and depth were the main factors affecting not only the non-linear behaviour of the rotor system but also the different zones of dynamic instability in the periodic solution for the cracked rotor. Ishida and Inoue used harmonic excitation force to investigate the non-linear response of cracked rotor. The occurrence of various types of non-linear resonances due to crack was clariﬁed, and types of these resonances, their resonance points and dominant frequency component of these resonances were clariﬁed numerically and experimentally. Sawicki et al. investigated the modelling and analysis of machines with breathing cracks, which open and close due to the self-weight of the rotor, producing a parametric excitation. Penny and Friswell considered a cracked asymmetric Jeffcott rotor and studied the inﬂuence of small out-of- 29,30 balance forces and cracks. Al-Shudeifat proposed a new breathing function which could describe the breath- ing mechanism of the crack more precisely. The process of solving the stiffness matrix of the crack element is given in detail, and the dynamic characteristics of the rotor system with breathing crack or open crack are solved, respectively. These dynamic phenomena are veriﬁed by experiments. Moreover, Floquet theory is used to analyse the inﬂuence of crack and damping on the stability of the cracked rotor, and the changing law of the instability speed region is obtained. Cavalini et al. used a crack-identiﬁcation methodology based on a non-linear approach, which uses external applied diagnostic forces at certain frequencies, to estimate the location and depth of the crack. In the above two new diagnostic methods, the relationship between the response frequency and the crack is more explicit, which is expected to improve the accuracy of the crack identiﬁcation. In recent years, the hollow structures are widely used in aero-engine to improve efﬁciency. More attention is 32,33 needed about the dynamic characteristics of cracked rotor system with hollow shaft in the engineering rotor. In this work, the inﬂuence of circular crack in hollow shaft, the critical speed and sub-critical speed responses is studied. We are devoted to provide some guidance for the detection and identiﬁcation of hollow-shaft dual-rotor system crack faults. The modeling of the cracked rotor system The ﬁnite element modeling of open crack is addressed in this part. The crack leads to a reduction of stiffness where the synchronous breathing of the crack between compression and tension stress ﬁelds on the crack faces of contact may lead to a permanent plastic deformation by which the breathing mechanism becomes dominated by the permanently open crack state. Circular crack is a typical form of transverse crack models. As far as the author knows, fewer literatures study this crack model compared with the huge literatures of straight crack model. However, as a rotating shaft, it is easier for cracks to propagate along the circumference than along the axis. In this part, the crack stiffness model is 34–36 established by the neutral axis theory. When the shaft is subject to stress concentration, especially in the position where the cross-section suddenly changes or material defects exist, the fatigue cracks will propagate faster on both sides of the crack edge under long-term action of the alternating load. As a result, the area of the crack element section will become an annular one. To show this characteristic accurately, a ﬁnite element model of the rotor system with 12 elements and 13 nodes is established as shown in Figure 1. The bearing is located at nodes 1 and 13, and the disk is located at nodes 4 and 10. There is unbalance in disk 1, and the crack is located in element 4. For a cracked rotor system with hollow shaft, with the propagation of crack, penetration of the shaft’s inner wall may occur. The cross-section of the crack element before penetration is shown in Figure 2. The shadow area represents the crack. The initial angle of the crack is taken as 0. The o–xy is a ﬁxed coordinate system. The crack region is symmetric about the oy axis, and the depth of the cracks is same on each section. o is the centroid of cross-section when there is no crack, c is the centroid when the crack appears, e ¼ oc represents the change of centroid position and X is the rotational speed of the rotor. The outer radius and inner radius of the shaft are R Yongfeng et al. Yongfeng et al. 12293 Disk1 Disk2 1 2 3 4 5 ĂĂ 8 9 10 11 12 Bearing and supporting Figure 1. Finite element model of the cracked rotor system. (a) (b) Figure 2. The crack sketch before penetration. (a) Cross-section of the crack element; (b) Geometry of the crack area. and r, respectively. h is the crack angle, h is the depth of the crack and the non-dimensional crack depth is given by u ¼ h/R. – – The area moments of inertia of the cracked element about its centroidal x and y axes are constant quantities during the rotation of the shaft while the area moments of inertia of the cracked element about its ﬁxed x and y axes are time-varying quantities during the rotation of the shaft. The cracked element stiffness matrix in the – – 37 rotating x and y axes can be written in a form similar to that of the asymmetric rod in space in Pilkey. In the circular crack model, crack propagation appears as it increases in depth and crack angle, but h and h are relatively independent parameters. In order to study the crack propagation process, let cos(h) ¼ (R–h)/R. The crack depth is assumed to be constant. The moment of inertia of the crack section in two directions is deﬁnite, but the crack will cause a certain offset in the sectional centroid position. The crack region in Figure 2(b) is divided into three parts, which is A ¼ A þ A – A , where A refers to the arcuate region between line y ¼ R crack 1 2 3 1 and line y ¼ R–h, A represents the trapezoidal area between line y ¼ R–h and line y ¼ (R–h)cos(h) and A means 2 3 the un-shaded arched area between line y ¼ R–h and line y ¼ (R–h)cos(h). The moment of inertia of the crack area on the axes ox I is given by ox A A A 1 2 3 I ¼ I þ I � I (1) ox ox ox ox where Z Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 1 2 2 I ¼ y dA ¼ 2y R � y dy ox A R�h Z Z R�h 2 3 I ¼ y dA ¼ 2y tanðhÞdy ox A ðR�hÞ� cosðhÞ Z Z qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R�h A 2 2 3 2 I ¼ y dA ¼ 2y ðR � hÞ � y dy ox A ðR�hÞ� cosðhÞ The moment of inertia of the crack area on the axes oy I is given by oy A A A 1 2 3 I ¼ I þ I � I (2) oy oy oy oy 4 1230 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Z Z R�sinðhÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 1 2 2 I ¼ x dA ¼ 2x ð R � x �ð R � hÞÞdx oy A 0 Z Z R�sinðhÞ A 2 2 I ¼ x dA ¼ 2x ðR � sinðhÞ� xÞ=tanðhÞdx oy A ðR� hÞ�sinðhÞ ðR� hÞ�sinðhÞ þ 2x ððR � hÞ�ð R � hÞ� cosðhÞÞdx Z Z qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðR� hÞ�sinðhÞ A 2 2 3 2 I ¼ x dA ¼ 2x ð ðR � hÞ � x �ð R � hÞ� cosðhÞÞdx oy A 0 The remaining area of the crack element A is given by ce 2 2 2 2 A ¼ A � A ¼ p �ðR � r Þ� ð h � R � h �ðR � hÞ Þ (3) ce crack The offset distance e of the centroid o is given by �� Z Z Z e ¼ ydA þ ydA � ydA =A (4) ce A A A 1 2 3 For the situation of crack penetration, conditions and parameter settings remain the same as those before penetration. The crack region in Figure 3(b) is divided into three parts, where A represents to the arcuate region between line y ¼ R and line y ¼ R–h, A represents to the trapezoidal area between line y ¼ R–h and line y ¼ r cos (h) and A means the unshaded arched area between line y ¼ r and line y ¼ r cos(h). The moment of inertia of the crack area on the axes ox I is given by ox A A A 1 2 3 I ¼ I þ I � I (5) ox ox ox ox where Z Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 1 2 2 I ¼ y dA ¼ 2y R � y dy ox A R� h Z Z R� h 2 3 I ¼ y dA ¼ 2y tanðhÞdy ox A r�cosðhÞ Z Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 A 2 2 I ¼ y dA ¼ 2y r � y dy ox A r�cosðhÞ (a) (b) Figure 3. The crack sketch after penetration. (a) Cross-section of the crack element; (b) Geometry of the crack area. Yongfeng et al. Yongfeng et al. 12315 The moment of inertia of the crack area on the axes oy is given by A A A 1 2 3 I ¼ I þ I � I (6) oy oy oy oy where Z Z R� sinðhÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 1 2 2 I ¼ x dA ¼ 2x ð R � x �ðR � hÞÞdx oy A 0 Z Z R� sinðhÞ A 2 2 I ¼ x dA ¼ 2x ðR �sinðhÞ� xÞ=tanðhÞdx oy A r�sinðhÞ r� sinðhÞ þ 2x ððR � hÞ� r �cosðhÞÞdx Z Z r�sinðhÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 3 2 2 I ¼ x dA ¼ 2x ð r � x � r �cosðhÞÞdx oy A 0 2 2 2 2 A ¼ A � A ¼ p �ð R � r Þ� ðh �R � h �r Þ (7) ce crack Z Z Z �� e ¼ ydA þ ydA � ydA =A (8) ce A A A 1 2 3 The decrease of the moment of inertia due to the crack can be deﬁned by I ¼ I þ A �e , I ¼ I . The 1 ox ce 2 oy moment of inertia of the crack element relative to the new centroid axis cx and cy can be obtained by ce I ¼ I � I cx (9) ce I ¼ I � I cy where I and I are very important parameters to calculate the stiffness reduction matrix. In order to show the 1 2 effect of different models, the moment of inertias for solid shaft and hollow shaft, circular crack, and straight crack are compared here. The physical parameters for a rotor system are shown in Table 1. Assume that the solid shaft has the same outer diameter as the hollow axis. Figures 4 and 5 show I , I and the relative 1 2 reduction of I with variety of crack depth. With the increasing of crack depth, the loss of moment of inertia in x and y directions increases. The effect of hollow shaft and circular crack is greater than the solid shaft and straight crack. Timoshenko beam-axis model is used to calculate the crack stiffness of the shaft. The coordinates of the ith element in rotor system can be expressed as ½x ; y ; h ; h ; x y ; h ; h � . x and y are the nodal displace- i i xi yi iþ1; iþ1 xðiþ1Þ yðiþ1Þ ments and h and h are the rotating angular displacements. When the crack is fully open, the stiffness reduction x y Table 1. Value of the physical parameters. Parameter Value Length of the rotor shaft, L 0.724 m Outer radius of rotor shaft, R 7.9 mm Inner radius of rotor shaft, r 4.74 mm Density, q 7800 kg/m Stiffness of bearing ðk :k Þ 5 � 10 N/m xx yy Damping of bearing ðc :c Þ 5 � 10 N s/m xx yy 1232 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Yongfeng et al. 7 12EI 12EI 1 2 where / ¼ , / ¼ and l is the shear coefﬁcient. 2 2 1 2 1 l A Gl l A Gl ce ce 1 1 The cosine switching function fðtÞ¼ð1 þ cosðXtÞÞ=2 for breathing cracks is used to describe the opening and closing of cracks during rotation. The dynamic equations of the cracked rotor can be given by € _ MqðtÞþ½C þ G�qðtÞ þðK � fðtÞK ÞqðtÞ¼ F cosðXtÞþ F sinðXtÞþ F (11) c 1 2 g where M, C, G and K are the mass matrix, damping matrix, gyro matrix and stiffness matrix of a non-crack rotor, respectively. For the crack-element stiffness matrix K ¼ K c crack For other elements, K equals 0 and F and F represent the unbalanced excitation vector. For the node 1 2 contains unbalance i 2 F ¼ m d � X ½cosb; sinb; 0; 0� i 2 F ¼ m d � X ½� sinb; cosb; 0; 0� where m is the eccentric mass, d is the eccentricity distance and b is the angle of unbalanced mass. For other nodes, F and F equal 0. F is the gravity force vector of the rotor system. The value of each node is given by 1 2 g Figure 4. I and I with variety of crack depth. F ¼ ½� m g; 0; 0; 0�, where m is the mass of node j. 1 2 j Harmonic balance method is used to solve equation (11), so the solution is assumed to be expressed in Fourier series as qðtÞ¼ A þ ½A cosðiXtÞþ B sinðiXtÞ� (12) 0 i i i¼1 where n represents the retained harmonic number. From our experience, n ¼ 4 is sufﬁcient to reveal the dynamic characteristics of cracked rotor system. By substituting equation (12) into equation (11), it is obtained that 2 3 hollow shaft with circular crack 2 3 2 3 K C K 0 A F 1 1 1 6 7 hollow shaft with straight crack 6 7 6 7 6 7 � C K � X M 0 K 6 7 6 7 B F 1 1 1 2 6 7 6 7 6 7 solid shaft with straight crack 6 7 6 7 6 7 6 K 0 K �ð 2XÞ M C . . . : 7 A 0 1 2 2 6 7 6 7 6 7 6 7 6 7 6 2 7 6 7 6 7 0 K � C K �ð 2XÞ M . . . : . . . : B 0 6 1 2 7 6 7 6 7 ¼ (13) 6 7 6 7 6 7 6 7 . . . . .. . . . . . . : K 0 : : 1 6 7 6 7 6 7 6 7 6 7 Figure 5. The relative reduction of I with variety of crack depth. 6 7 6 7 6 7 . . . . . . : . . . : 0 K : : 6 7 6 7 6 7 6 7 6 7 6 7 matrix K caused by the crack can be written as 6 � 7 crack K 0 K �ð 4XÞ MC A 0 4 4 5 4 5 1 4 4 5 2 3 B 0 0 K � C K �ð 4XÞ M 4 1 1 1 1 1 4 12I 0 06lI �12I 0 06lI 6 1 1 1 1 7 1 þ / 1 þ / 1 þ / 1 þ / 6 7 1 1 1 1 6 7 1 1 1 1 6 7 � 1 0 12I �6lI 00 �12I �6lI 0 � 1 � ^ � � � 6 2 2 2 2 7 where K ¼� K =4, F ¼ F þ K K F =2, K ¼ K � X M � K ðK ÞK =8, C ¼ sXC, s ¼ 1; 2; . . . ; n, 1 c 1 c g c c s 1 þ / 1 þ / 1 þ / 1 þ / 6 7 2 2 2 2 � 1 6 7 1 4 þ / 1 2 � / A ¼ K ðF þ K A =4Þ. By solving the linear equation (13), the steady-state response of the rotor system can 6 2 2 7 2 2 0 g c 1 0 �6lI l I 0 06lI l I 0 6 2 2 2 2 7 6 1 þ / 1 þ / 1 þ / 1 þ / 7 2 2 2 2 be obtained based on harmonic balance method. 6 7 1 4 þ / 1 2 � / 6 1 1 7 2 2 6lI 00 l I �6lI 00 l I 6 1 1 1 1 7 6 1 þ / 1 þ / 1 þ / 1 þ / 7 1 1 1 1 K ¼ 6 7 crack 1 1 1 1 6 7 �12I 00 �6lI 12I 00 �6lI 6 1 1 1 1 7 Numerical simulations 6 1 þ / 1 þ / 1 þ / 1 þ / 7 1 1 1 1 6 7 1 1 1 1 6 7 For the rotor system shown in Figure 1, the physical parameters are the same as shown in Table 1. Figure 6 shows 6 0 �12I 6lI 0 0 12I 6lI 0 7 2 2 2 2 6 7 1 þ / 1 þ / 1 þ / 1 þ / 2 2 2 2 the vibration amplitude of a non-crack rotor system with asymmetric stiffness and symmetric bearing stiffness. 6 7 1 2 � / 1 4 þ / 6 7 2 2 2 2 6 0 �6lI l I 0 06lI l I 0 7 2 2 2 2 The ﬁrst-forward critical speed is w ¼ 281.8 rad/s, and the ﬁrst-backward critical speed is w ¼ 272.9 rad/s. In f1 b1 6 7 1 þ / 1 þ / 1 þ / 1 þ / 2 2 2 2 6 7 addition, we can obtain the critical speed by the Eigenvalue Method, and the critical speed is w ¼ 281.8 rad/s and 4 1 2 � / 1 4 þ / 5 f1 2 1 2 1 6lI 00 l I �6lI 00 l I 1 1 1 1 1 þ / 1 þ / 1 þ / 1 þ / w ¼ 272.8 rad/s. For asymmetric-cracked shaft with isotropic bearing or symmetric intact shaft with anisotropic 1 1 1 1 b1 (10) Yongfeng et al. Yongfeng et al. 12337 12EI 12EI 1 2 where / ¼ , / ¼ and l is the shear coefﬁcient. 2 2 1 2 1 l A Gl l A Gl ce ce 1 1 The cosine switching function fðtÞ¼ð1 þ cosðXtÞÞ=2 for breathing cracks is used to describe the opening and closing of cracks during rotation. The dynamic equations of the cracked rotor can be given by € _ MqðtÞþ½C þ G�qðtÞ þðK � fðtÞK ÞqðtÞ¼ F cosðXtÞþ F sinðXtÞþ F (11) c 1 2 g where M, C, G and K are the mass matrix, damping matrix, gyro matrix and stiffness matrix of a non-crack rotor, respectively. For the crack-element stiffness matrix K ¼ K c crack For other elements, K equals 0 and F and F represent the unbalanced excitation vector. For the node 1 2 contains unbalance i 2 F ¼ m d � X ½cosb; sinb; 0; 0� i 2 F ¼ m d � X ½� sinb; cosb; 0; 0� where m is the eccentric mass, d is the eccentricity distance and b is the angle of unbalanced mass. For other nodes, F and F equal 0. F is the gravity force vector of the rotor system. The value of each node is given by 1 2 g F ¼ ½� m g; 0; 0; 0�, where m is the mass of node j. Harmonic balance method is used to solve equation (11), so the solution is assumed to be expressed in Fourier series as qðtÞ¼ A þ ½A cosðiXtÞþ B sinðiXtÞ� (12) 0 i i i¼1 where n represents the retained harmonic number. From our experience, n ¼ 4 is sufﬁcient to reveal the dynamic characteristics of cracked rotor system. By substituting equation (12) into equation (11), it is obtained that 2 3 2 3 2 3 ^ � K C K 0 A F 1 1 1 6 7 6 7 6 7 6 7 � C K � X M 0 K 6 7 6 7 B F 1 1 1 2 6 7 6 7 6 7 6 7 6 7 6 7 6 K 0 K �ð 2XÞ M C . . . : 7 A 0 1 2 2 6 7 6 7 6 7 6 7 6 7 6 2 7 6 7 6 7 0 K � C K �ð 2XÞ M . . . : . . . : B 0 6 1 2 7 6 7 6 7 ¼ (13) 6 7 6 7 6 7 6 7 . . . . .. . . . . . . : K 0 : : 1 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 . . . . . . : . . . : 0 K : : 6 7 6 7 6 7 6 7 6 7 6 7 6 � 7 K 0 K �ð 4XÞ MC A 0 4 4 5 4 5 1 4 4 5 B 0 0 K � C K �ð 4XÞ M 4 1 4 � 1 � 1 � ^ � � � where K ¼� K =4, F ¼ F þ K K F =2, K ¼ K � X M � K ðK ÞK =8, C ¼ sXC, s ¼ 1; 2; . . . ; n, 1 c 1 c g c c s � 1 A ¼ K ðF þ K A =4Þ. By solving the linear equation (13), the steady-state response of the rotor system can 0 g c 1 be obtained based on harmonic balance method. Numerical simulations For the rotor system shown in Figure 1, the physical parameters are the same as shown in Table 1. Figure 6 shows the vibration amplitude of a non-crack rotor system with asymmetric stiffness and symmetric bearing stiffness. The ﬁrst-forward critical speed is w ¼ 281.8 rad/s, and the ﬁrst-backward critical speed is w ¼ 272.9 rad/s. In f1 b1 addition, we can obtain the critical speed by the Eigenvalue Method, and the critical speed is w ¼ 281.8 rad/s and f1 w ¼ 272.8 rad/s. For asymmetric-cracked shaft with isotropic bearing or symmetric intact shaft with anisotropic b1 8 1234 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Yongfeng et al. 9 w =281.8rad/s f1 w =272.9rad/s b1 Non-dimensional crack depth Rotation speed (rad/s) Rotation speed (rad/s) Figure 9. Rotation speed–crack depth–vibration amplitude waterfall at node 4. Figure 6. Vibration amplitude at node 4 for non-crack rotor with asymmetric/symmetric stiffness. f1 b1 1/2w f1 1/2w b1 Rotation speed (rad/s) f1 1/3w b1 1/2w b1 b1 1/2w f1 1/4w 1/3w 1/4w f1 b1 f1 Rotation speed (rad/s) w f1 b1 Non-dimensional crack depth 1/2w 1/3w b1 b1 Figure 7. The first critical speed with variety of crack depth. 1/2w 1/4w f1 1/3w b1 f1 (a) (b) Rotation speed (rad/s) w w f1 b1 1/2w b1 1/3w b1 1/2w f1 1/4w 1/3w b1 1/4w f1 f1 Rotation speed (rad/s) Figure 10. Rotation speed–vibration amplitude diagram of rotor system at node 4 with crack located in different elements when crack depth u¼ 0.5. bearings, the ﬁrst-forward and backward whirl speeds could be excited by unbalance force. As shown in Figure 6, the results obtained by two methods are in good agreement with each other. Non-dimensional crack depth Non-dimensional crack depth The ﬁrst critical rotational speed of a hollow-shaft rotor system with a circular crack is shown in Figure 7. It can be seen that the ﬁrst critical speed of the cracked rotor decreases with the crack depth going deeper. Figure 8. The first critical speed with crack position in different elements. Vibration amplitude (m) Rotation speed (rad/s) Rotation speed (rad/s) Rotation speed (rad/s) Mean of vertical Mean of vertical Mean of vertical Mean of vertical Vibration amplitude (m) amplitude (m) amplitude (m) amplitude (m) amplitude (m) Yongfeng et al. Yongfeng et al. 12359 Non-dimensional crack depth Rotation speed (rad/s) Figure 9. Rotation speed–crack depth–vibration amplitude waterfall at node 4. f1 b1 1/2w f1 1/2w b1 Rotation speed (rad/s) f1 1/3w b1 1/2w b1 b1 1/2w f1 1/4w 1/3w 1/4w f1 b1 f1 Rotation speed (rad/s) w f1 b1 1/2w 1/3w b1 b1 1/2w 1/4w f1 1/3w b1 f1 Rotation speed (rad/s) w w f1 b1 1/2w b1 1/3w b1 1/2w f1 1/4w 1/3w b1 1/4w f1 f1 Rotation speed (rad/s) Figure 10. Rotation speed–vibration amplitude diagram of rotor system at node 4 with crack located in different elements when crack depth u¼ 0.5. bearings, the ﬁrst-forward and backward whirl speeds could be excited by unbalance force. As shown in Figure 6, the results obtained by two methods are in good agreement with each other. The ﬁrst critical rotational speed of a hollow-shaft rotor system with a circular crack is shown in Figure 7. It can be seen that the ﬁrst critical speed of the cracked rotor decreases with the crack depth going deeper. Mean of vertical Mean of vertical Mean of vertical Mean of vertical Vibration amplitude (m) amplitude (m) amplitude (m) amplitude (m) amplitude (m) Mean of vertical Mean of vertical Mean of vertical Mean of vertical amplitude(m) amplitude amplitude amplitude(m) 10 1236 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Yongfeng et al. 11 (a) (b) (c) f1 f1 f1 x(m) x(m) x(m) Figure 13. Phase orbits near 92.1 rad/s of node 4 when crack depth u¼ 0.4. (a) (b) (c) Figure 11. Rotation speed–vibration amplitude diagram of rotor system at node 4 with no crack. (a) (b) (c) x(m) x(m) x(m) Figure 14. Phase orbits near 69 rad/s of node 4 when crack depth u¼ 0.4. (a) (b) (c) x(m) x(m) x(m) Figure 12. Phase orbits near 138.3 rad/s of node 4 when crack depth u¼ 0.4. Figure 15. Power spectrum of node 4 when crack depth u¼ 0.4. y(m) y(m) y(m) y(m) y(m) y(m) y(m) y(m) y(m) Yongfeng et al. Yongfeng et al. 1237 11 (a) (b) (c) x(m) x(m) x(m) Figure 13. Phase orbits near 92.1 rad/s of node 4 when crack depth u¼ 0.4. (a) (b) (c) x(m) x(m) x(m) Figure 14. Phase orbits near 69 rad/s of node 4 when crack depth u¼ 0.4. (a) (b) (c) Figure 15. Power spectrum of node 4 when crack depth u¼ 0.4. y(m) y(m) y(m) y(m) y(m) y(m) 12 1238 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Yongfeng et al. 13 In particular, the ﬁrst-backward critical speed is very sensitive to the crack depth while the ﬁrst-forward critical Funding speed is not. The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this For the above study, the crack position is constant in element 4. Here, we investigate the effect of crack article: This study was funded by National Natural Science Foundation of China (grant number 11972295), the Key position. Figure 8 shows the ﬁrst critical speeds at different element locations. When the crack position is close Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University (grant to the bearing position (Element 1), the ﬁrst critical speed decreases slowly with increasing crack depth. When the number VCAME201803), and Graduate Innovation Fund of Northwestern Polytechnical University (grant number crack position is far away from the bearing, especially for the element near the middle of the shaft (Element 4, CX2020124). Element 5, Element 6), and the ﬁrst-order critical speed drops faster. Element 6 is the fastest one. Figure 9 shows the rotation speed–crack depth–vibration amplitude waterfall at node 4 when the circular crack ORCID iD is located in element 4. Due to the circular crack, vibration peaks appear at the 1/2, 1/3 and 1/4 ﬁrst-forward and Yang Yongfeng https://orcid.org/0000-0003-0402-4440 backward critical rotational speeds. Since we study the response of a cracked rotor system, the basic character- istics of response are the same as given in Ishida and Yamamoto. When the crack depth is small (u¼ 0.1 and References u¼ 0.2), the vibration peaks near 1/3 and 1/4 critical speeds are insigniﬁcant. With the crack expansion and 1. Li CF, She HX, Tang QS, et al. The effect of blade vibration on the nonlinear characteristics of rotor-bearing system deepening, the vibration peaks near 1/3 and 1/4 critical speeds become very obvious. At the same time, the sub- supported by nonlinear suspension. Nonlin Dyn 2017; 89: 987–1110. critical rotational speeds of the cracked rotor system also decrease with the increasing of the crack depth. In short, 2. Hou L, Chen YS, Fu YQ, et al. Application of the HB-AFT method to the primary resonance analysis of a dual-rotor the crack will cause the rotor system to have vibration peaks near the ﬁrst critical speed and 1/n (n¼ 2, 3, 4) ﬁrst system. Nonlin Dyn 2017; 88: 2531–2551. critical speed. The amplitude of vibration peak near 1/4 critical speed is very sensitive to the crack depth and 3. Luo Z, Zhu YP, Zhao XY, et al. Determining dynamic scaling laws of geometrically distorted scaled models of a cantilever position. There are vibration peaks between the 1/2 ﬁrst critical speed and the ﬁrst critical speed. The rotation plate. J Eng Mech 2016; 142: 04015108. speed is near 215 rad/s. With the crack depth going deeper and the element closer to crack position, the rotation 4. Zhang GH and Ehmann KF. Dynamic design methodology of high speed micro- spindles for micro/meso-scale machine tools. Int J Adv Manuf Technol 2015; 76: 229–246. speed will go down and the amplitude will go up. This is an important dynamic characteristic to detect the crack 5. Dai HH, Jing XJ, Wang Y, et al. Post-capture vibration suppression of spacecraft via a bio-inspired isolation system. Mech fault in rotor system. Syst Signal Process 2018; 105: 214–240. Figure 10 shows the rotation speed–vibration amplitude diagram of node 4 with the crack locating in different 6. Qin ZY, Yang ZB, Zu J, et al. Free vibration analysis of rotating cylindrical shells coupled with moderately thick annular elements when the circular crack depth u¼ 0.5. It can be seen that when the crack is located near the support plate. Int J Mech Sci 2018; 142-143: 127–139. position of element 1, the vibration peaks near 1/3 and 1/4 critical speeds are not obvious. While the crack is far 7. Fu C, Xu YD, Yang YF, et al. Response analysis of an accelerating unbalanced rotating system with both random and away from the bearing support, the vibration peaks near 1/3 and 1/4 critical speed become obvious. There is a interval variables. J Sound Vib 2020; 466: 115047. peak near 215 rad/s. With the crack depth going deeper and the measurement point closer to crack position, the 8. Li HG, Meng G, Meng ZQ, et al. Effects of boundary conditions on a self-excited vibration system with clearance. Int J speed will go down and the amplitude will go up. As a result, the crack located in the middle shaft of the rotor has Nonlin Sci Num 2007; 8: 571–579. a greater inﬂuence on the dynamic characteristics than in the supporting position. Figure 11 shows the rotation 9. Han QK and Chu FL. Parametric instability of ﬂexible rotor-bearing system under time-periodic base angular motions. Appl Math Model 2015; 39: 4511–4522. speed–vibration amplitude diagram of rotor system at node 4 with no crack. It can be seen that the vibration peak 10. Wang WM, Li QH, Gao JJ, et al. An identiﬁcation method for damping ratio in rotor systems. Mech Syst Signal Process only occurred at the critical speed of the rotor system, and there is no super-harmonic resonance phenomenon in 2016; 68-69: 536–554. no-cracked rotor systems. 11. Ma H, Shi CY, Han QK, et al. Fixed-point rubbing fault characteristic analysis of a rotor system based on contact theory. Figures 12 to 15 show the phase orbits and the power spectrum of node 4 near the sub-critical speeds when the Mech Syst Signal Process 2013; 38: 137–153. circular crack depth is 0.4. It can be seen that when the rotating speed is close to the 1/2 ﬁrst-forward critical 12. Chu FL and Lu WX. Experimental observation of nonlinear vibrations in a rub-impact rotor system. J Sound Vib 2005; speed, it appears as two overlapping ellipses in the phase orbits and 2� component in the frequency domain. 283: 621–643. Actually, the 2� component is the ﬁrst-forward critical speed. In the same way, for 1/3 and 1/4 critical speeds, the 13. Zhang WM and Meng G. Stability, bifurcation and chaos analyses of a high-speed micro-rotor system with rub-impact. phase orbits are three and four overlapping ellipses, respectively. Similarly, the frequency components are dom- Sensor Actuat A-Phys 2006; 127: 163–178. inated by the 3� component or 4� component. For the backward critical speed, we can observe the same 14. Yang YF, Chen H and Jiang TD. Nonlinear response prediction of cracked rotor based on EMD. J Franklin Inst 2015; 352: 3378–3393. phenomenon. It shows the super-harmonic resonance phenomenon in cracked rotor systems. 15. Yang YF, Wu QY, Wang YL, et al. Dynamic characteristics of cracked uncertain hollow-shaft. Mech Syst Signal Process 2019; 124: 36–48. 16. Ma H, Zeng J, Feng RJ, et al. Review on dynamics of cracked gear systems. Eng Fail Anal 2015; 55: 224–245. Conclusion 17. Patel TH and Darpe AK. Inﬂuence of crack breathing model on nonlinear dynamics of a cracked rotor. J Sound Vib 2008; A hollow-shaft rotor system with circular cracks is studied in this paper. The time-varying stiffness matrix of the 311: 953–972. crack element is deduced. The inﬂuence of the crack on the critical speed and sub-critical speed is shown. It is 18. Qin WY, Meng G and Zhang T. The swing vibration, transverse oscillation of cracked rotor and the intermittence chaos. J Sound Vib 2003; 259: 571–583. found that the circular cracks could reduce both the ﬁrst-forward and backward critical rotational speeds of the 19. Lu YJ, Zhang YF, Shi XL, et al. Nonlinear dynamic analysis of a rotor system with ﬁxed-tilting-pad self-acting gas- rotor system, especially the latter one. Owing to the inﬂuences, the vibration peaks occur near 1/n (n¼ 1, 2, 3, and lubricated bearings support. Nonlin Dyn 2012; 69: 877–890. 4) critical rotational speed. As the crack increases, the peaks become more prominent. When the crack is located 20. Meng G, Zhang WM, Huang H, et al. Micro-rotor dynamics for micro-electro-mechanical systems (MEMS). Chaos in the middle of shaft, the effect will be greatest. Super-harmonic resonance phenomena can be observed in the Soliton Fract 2009; 40: 538–562. cracked rotor system. The results of this paper can provide some guidance for detection and identiﬁcation of crack 21. Dimarogonas AD. Vibration of cracked structures: a state of the art review. Eng Fract Mech 1996; 55: 831–857. fault in hollow-shaft dual-rotor system. 22. Wauer J. Dynamics of cracked rotors: literature survey. Appl Mech Rev 1990; 43: 13–17. 23. Pennacchi P, Bachschmid N and Vania A. A model-based identiﬁcation method of transverse cracks in rotating shafts suitable for industrial machines. Mech Syst Signal Process 2006; 20: 2112–2147. Declaration of conflicting interests 24. Rubio L and Fernandez-Saez J. A new efﬁcient procedure to solve the nonlinear dynamics of a cracked rotor. Nonlin Dyn The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this 2012; 70: 1731–1745. article. 25. Sinou JJ. Effects of a crack on the stability of a non-linear rotor system. Int J Nonlin Mech 2007; 42: 959–972. Yongfeng et al. Yongfeng et al. 1239 13 Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: This study was funded by National Natural Science Foundation of China (grant number 11972295), the Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University (grant number VCAME201803), and Graduate Innovation Fund of Northwestern Polytechnical University (grant number CX2020124). ORCID iD Yang Yongfeng https://orcid.org/0000-0003-0402-4440 References 1. Li CF, She HX, Tang QS, et al. The effect of blade vibration on the nonlinear characteristics of rotor-bearing system supported by nonlinear suspension. Nonlin Dyn 2017; 89: 987–1110. 2. Hou L, Chen YS, Fu YQ, et al. Application of the HB-AFT method to the primary resonance analysis of a dual-rotor system. Nonlin Dyn 2017; 88: 2531–2551. 3. Luo Z, Zhu YP, Zhao XY, et al. Determining dynamic scaling laws of geometrically distorted scaled models of a cantilever plate. J Eng Mech 2016; 142: 04015108. 4. Zhang GH and Ehmann KF. Dynamic design methodology of high speed micro- spindles for micro/meso-scale machine tools. Int J Adv Manuf Technol 2015; 76: 229–246. 5. Dai HH, Jing XJ, Wang Y, et al. Post-capture vibration suppression of spacecraft via a bio-inspired isolation system. Mech Syst Signal Process 2018; 105: 214–240. 6. Qin ZY, Yang ZB, Zu J, et al. Free vibration analysis of rotating cylindrical shells coupled with moderately thick annular plate. Int J Mech Sci 2018; 142-143: 127–139. 7. Fu C, Xu YD, Yang YF, et al. Response analysis of an accelerating unbalanced rotating system with both random and interval variables. J Sound Vib 2020; 466: 115047. 8. Li HG, Meng G, Meng ZQ, et al. Effects of boundary conditions on a self-excited vibration system with clearance. Int J Nonlin Sci Num 2007; 8: 571–579. 9. Han QK and Chu FL. Parametric instability of ﬂexible rotor-bearing system under time-periodic base angular motions. Appl Math Model 2015; 39: 4511–4522. 10. Wang WM, Li QH, Gao JJ, et al. An identiﬁcation method for damping ratio in rotor systems. Mech Syst Signal Process 2016; 68-69: 536–554. 11. Ma H, Shi CY, Han QK, et al. Fixed-point rubbing fault characteristic analysis of a rotor system based on contact theory. Mech Syst Signal Process 2013; 38: 137–153. 12. Chu FL and Lu WX. Experimental observation of nonlinear vibrations in a rub-impact rotor system. J Sound Vib 2005; 283: 621–643. 13. Zhang WM and Meng G. Stability, bifurcation and chaos analyses of a high-speed micro-rotor system with rub-impact. Sensor Actuat A-Phys 2006; 127: 163–178. 14. Yang YF, Chen H and Jiang TD. Nonlinear response prediction of cracked rotor based on EMD. J Franklin Inst 2015; 352: 3378–3393. 15. Yang YF, Wu QY, Wang YL, et al. Dynamic characteristics of cracked uncertain hollow-shaft. Mech Syst Signal Process 2019; 124: 36–48. 16. Ma H, Zeng J, Feng RJ, et al. Review on dynamics of cracked gear systems. Eng Fail Anal 2015; 55: 224–245. 17. Patel TH and Darpe AK. Inﬂuence of crack breathing model on nonlinear dynamics of a cracked rotor. J Sound Vib 2008; 311: 953–972. 18. Qin WY, Meng G and Zhang T. The swing vibration, transverse oscillation of cracked rotor and the intermittence chaos. J Sound Vib 2003; 259: 571–583. 19. Lu YJ, Zhang YF, Shi XL, et al. Nonlinear dynamic analysis of a rotor system with ﬁxed-tilting-pad self-acting gas- lubricated bearings support. Nonlin Dyn 2012; 69: 877–890. 20. Meng G, Zhang WM, Huang H, et al. Micro-rotor dynamics for micro-electro-mechanical systems (MEMS). Chaos Soliton Fract 2009; 40: 538–562. 21. Dimarogonas AD. Vibration of cracked structures: a state of the art review. Eng Fract Mech 1996; 55: 831–857. 22. Wauer J. Dynamics of cracked rotors: literature survey. Appl Mech Rev 1990; 43: 13–17. 23. Pennacchi P, Bachschmid N and Vania A. A model-based identiﬁcation method of transverse cracks in rotating shafts suitable for industrial machines. Mech Syst Signal Process 2006; 20: 2112–2147. 24. Rubio L and Fernandez-Saez J. A new efﬁcient procedure to solve the nonlinear dynamics of a cracked rotor. Nonlin Dyn 2012; 70: 1731–1745. 25. Sinou JJ. Effects of a crack on the stability of a non-linear rotor system. Int J Nonlin Mech 2007; 42: 959–972. 14 1240 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) 26. Ishida Y and Inoue T. Detection of a rotor crack using a harmonic excitation and nonlinear vibration analysis. J Vib Acoust 2006; 128: 741–749. 27. Sawicki JT, Friswell MI, Kulesza Z, et al. Detecting cracked rotors using auxiliary harmonic excitation. J Sound Vib 2011; 330: 1365–1381. 28. Penny JET and Friswell MI. The dynamics of rotating machines with cracks. MSF 2003; 440–444: 311–318. 29. Al-Shudeifat MA. On the ﬁnite element modeling of the asymmetric cracked rotor. J Sound Vib 2013; 332: 2795–2807. 30. Al-Shudeifat MA. Stability analysis and backward whirl investigation of cracked rotors with time-varying stiffness. J Sound Vib 2015; 348: 365–380. 31. Cavalini AA Jr, Sanches L, Bachschmid N, et al. Crack identiﬁcation for rotating machines based on a nonlinear approach. Mech Syst Signal Process 2016; 79: 72–85. 32. Lu ZY, Hou L, Chen YS, et al. Nonlinear response analysis for a dual-rotor system with a breathing transverse crack in the hollow shaft. Nonlin Dyn 2016; 83: 169–185. 33. Guo CZ, Al-Shudeifat MA, Vakakis AF, et al. Vibration reduction in unbalanced hollow rotor systems with nonlinear energy sinks. Nonlin Dyn 2015; 79: 527–538. 34. Du Y, Zhou SX, Jing XJ, et al. Damage detection techniques for wind turbine blades: a review. Mech Syst Signal Process 2020; 141: 106445. 35. Liu Y, Zhao YL, Li JT, et al. Application of weighted contribution rate of nonlinear output frequency response functions to rotor rub-impact. Mech Syst Signal Process 2020; 136: 106518. 36. Yang YF, Zheng QY, Wang JJ, et al. Dynamics response analysis of airborne external storage system with clearance between missile-frame. Chin J Aeronaut. Epub ahead of print 20 June 2020. DOI: 10.1016/j.cja.2020.06.008. 37. Pilkey WD. Analysis and design of elastic beams. 1st ed. New York: John Wiley and Sons, 2012. 38. Ishida Y and Yamamoto T. Linear and nonlinear rotor dynamics. New York: Wiley, 2012. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE http://www.deepdyve.com/lp/sage/dynamical-analysis-of-hollow-shaft-dual-rotor-system-with-circular-BQDgRFNjBV

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Abstract

In this paper, we considered a dual-rotor system with crack in shaft. The influence of circular crack in hollow shaft on dynamical response was studied. The equations of motion of 12 elements dual-rotor system model were derived. Harmonic balance method was employed to solve the equations. The critical speed and sub-critical speed responses were investigated. It was found that the circular crack in hollow shaft had greater influence on the first-backward critical speed than the first-forward critical speed. Owing to the influence of crack, the vibration peaks occurred at the 1/2, 1/3 and 1/4 critical speeds of the rotor system, along with a reduction in sub-critical speeds and critical speeds. The deeper crack away from the bearing affected the rotor more significantly. The whirling orbits, the time-domain responses and the spectra were obtained to show the super-harmonic resonance phenomenon in hollow-shaft cracked rotor system. Keywords Rotor, hollow-shaft, circular cracks, critical speed, super-harmonic Introduction Rotating machines represent the maximal and most important class of machinery used for ﬂuid media transpor- 1–3 tation, metal working and forming, energy generation, providing aircraft propulsion and other purposes. High 4–6 speed and heavy power are the development directions of modern rotating machineries. In the past decades, 7 8 9 there are a lot of literatures that focus on the study of unbalance, clearance, base motions, damping ratio 10 11–13 identiﬁcation, rubbing and viscoelastic properties of rotor system. Especially, crack and misalignment 14–16 effects in rotor dynamic characteristics are frequently investigated. Fatigue crack of the rotor shaft observed in the rotating machinery should be avoided. It may lead to cata- strophic failure. In this situation, there are non-linear and non-stationary responses of the rotor system. However, the strong non-linearity can make the system possess characteristics that are substantially different from those of the linear system, such as self-excited oscillations and jump discontinuities. Detailed investigation into the non-linear dynamic response prediction of cracked shaft is very important for diagnosing and preventing 18–20 rotor cracks. The inﬂuence of transverse crack on a rotating shaft has been the attention of many researchers. Extensive 21 22 reviews of the dynamic response of cracked rotor systems were published by Dimarogonas and Wauer. Pennacchi et al. proposed a model-based transverse crack identiﬁcation method suitable for industrial machines. The excellent accuracy obtained at deﬁned position and depth of different cracks demonstrated the effectiveness and reliability of the proposed method. Patel and Darpe investigated the inﬂuence of the crack-breathing models Institute of Vibration Engineering, Northwestern Polytechnical University, Xi’an, China Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang, China CRRC Zhuzhou Electric Locomotive Institute Co., Ltd, Hunan, China Corresponding author: Yang Yongfeng, Northwestern Polytechnical University, P.O. Box 264 127, West Youyi Road, Xi’an, Shaanxi 710072, China. Email: yyf@nwpu.edu.cn 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). 2 1228 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) (switching-crack model and response-dependent breathing-crack model) on the non-linear vibration character- istics of the cracked rotor. Switching-crack modeling reveals chaotic, quasi-periodic and sub-harmonic vibration response for deeper cracks, and more realistic breathing crack model reveals no evidence of them. Rubio and Fernandez-Saez proposed a new procedure to analyze the non-linear dynamics of cracked rotors using an iterative technique that transformed the full non-linear problem into a succession of time-dependent linear ones. The calculations using the proposed method were over a 100 times faster than the corresponding to integrate the full non-linear problem, being very helpful in on-line crack-identiﬁcation procedures. Sinou adopted har- monic balance method to study the stability of the rotor system presenting a transverse breathing crack by considering the effects of crack depth, crack position and the rotating speed. The areas of instability will increase considerably when the crack deepened and that the crack’s position and depth were the main factors affecting not only the non-linear behaviour of the rotor system but also the different zones of dynamic instability in the periodic solution for the cracked rotor. Ishida and Inoue used harmonic excitation force to investigate the non-linear response of cracked rotor. The occurrence of various types of non-linear resonances due to crack was clariﬁed, and types of these resonances, their resonance points and dominant frequency component of these resonances were clariﬁed numerically and experimentally. Sawicki et al. investigated the modelling and analysis of machines with breathing cracks, which open and close due to the self-weight of the rotor, producing a parametric excitation. Penny and Friswell considered a cracked asymmetric Jeffcott rotor and studied the inﬂuence of small out-of- 29,30 balance forces and cracks. Al-Shudeifat proposed a new breathing function which could describe the breath- ing mechanism of the crack more precisely. The process of solving the stiffness matrix of the crack element is given in detail, and the dynamic characteristics of the rotor system with breathing crack or open crack are solved, respectively. These dynamic phenomena are veriﬁed by experiments. Moreover, Floquet theory is used to analyse the inﬂuence of crack and damping on the stability of the cracked rotor, and the changing law of the instability speed region is obtained. Cavalini et al. used a crack-identiﬁcation methodology based on a non-linear approach, which uses external applied diagnostic forces at certain frequencies, to estimate the location and depth of the crack. In the above two new diagnostic methods, the relationship between the response frequency and the crack is more explicit, which is expected to improve the accuracy of the crack identiﬁcation. In recent years, the hollow structures are widely used in aero-engine to improve efﬁciency. More attention is 32,33 needed about the dynamic characteristics of cracked rotor system with hollow shaft in the engineering rotor. In this work, the inﬂuence of circular crack in hollow shaft, the critical speed and sub-critical speed responses is studied. We are devoted to provide some guidance for the detection and identiﬁcation of hollow-shaft dual-rotor system crack faults. The modeling of the cracked rotor system The ﬁnite element modeling of open crack is addressed in this part. The crack leads to a reduction of stiffness where the synchronous breathing of the crack between compression and tension stress ﬁelds on the crack faces of contact may lead to a permanent plastic deformation by which the breathing mechanism becomes dominated by the permanently open crack state. Circular crack is a typical form of transverse crack models. As far as the author knows, fewer literatures study this crack model compared with the huge literatures of straight crack model. However, as a rotating shaft, it is easier for cracks to propagate along the circumference than along the axis. In this part, the crack stiffness model is 34–36 established by the neutral axis theory. When the shaft is subject to stress concentration, especially in the position where the cross-section suddenly changes or material defects exist, the fatigue cracks will propagate faster on both sides of the crack edge under long-term action of the alternating load. As a result, the area of the crack element section will become an annular one. To show this characteristic accurately, a ﬁnite element model of the rotor system with 12 elements and 13 nodes is established as shown in Figure 1. The bearing is located at nodes 1 and 13, and the disk is located at nodes 4 and 10. There is unbalance in disk 1, and the crack is located in element 4. For a cracked rotor system with hollow shaft, with the propagation of crack, penetration of the shaft’s inner wall may occur. The cross-section of the crack element before penetration is shown in Figure 2. The shadow area represents the crack. The initial angle of the crack is taken as 0. The o–xy is a ﬁxed coordinate system. The crack region is symmetric about the oy axis, and the depth of the cracks is same on each section. o is the centroid of cross-section when there is no crack, c is the centroid when the crack appears, e ¼ oc represents the change of centroid position and X is the rotational speed of the rotor. The outer radius and inner radius of the shaft are R Yongfeng et al. Yongfeng et al. 12293 Disk1 Disk2 1 2 3 4 5 ĂĂ 8 9 10 11 12 Bearing and supporting Figure 1. Finite element model of the cracked rotor system. (a) (b) Figure 2. The crack sketch before penetration. (a) Cross-section of the crack element; (b) Geometry of the crack area. and r, respectively. h is the crack angle, h is the depth of the crack and the non-dimensional crack depth is given by u ¼ h/R. – – The area moments of inertia of the cracked element about its centroidal x and y axes are constant quantities during the rotation of the shaft while the area moments of inertia of the cracked element about its ﬁxed x and y axes are time-varying quantities during the rotation of the shaft. The cracked element stiffness matrix in the – – 37 rotating x and y axes can be written in a form similar to that of the asymmetric rod in space in Pilkey. In the circular crack model, crack propagation appears as it increases in depth and crack angle, but h and h are relatively independent parameters. In order to study the crack propagation process, let cos(h) ¼ (R–h)/R. The crack depth is assumed to be constant. The moment of inertia of the crack section in two directions is deﬁnite, but the crack will cause a certain offset in the sectional centroid position. The crack region in Figure 2(b) is divided into three parts, which is A ¼ A þ A – A , where A refers to the arcuate region between line y ¼ R crack 1 2 3 1 and line y ¼ R–h, A represents the trapezoidal area between line y ¼ R–h and line y ¼ (R–h)cos(h) and A means 2 3 the un-shaded arched area between line y ¼ R–h and line y ¼ (R–h)cos(h). The moment of inertia of the crack area on the axes ox I is given by ox A A A 1 2 3 I ¼ I þ I � I (1) ox ox ox ox where Z Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 1 2 2 I ¼ y dA ¼ 2y R � y dy ox A R�h Z Z R�h 2 3 I ¼ y dA ¼ 2y tanðhÞdy ox A ðR�hÞ� cosðhÞ Z Z qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R�h A 2 2 3 2 I ¼ y dA ¼ 2y ðR � hÞ � y dy ox A ðR�hÞ� cosðhÞ The moment of inertia of the crack area on the axes oy I is given by oy A A A 1 2 3 I ¼ I þ I � I (2) oy oy oy oy 4 1230 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Z Z R�sinðhÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 1 2 2 I ¼ x dA ¼ 2x ð R � x �ð R � hÞÞdx oy A 0 Z Z R�sinðhÞ A 2 2 I ¼ x dA ¼ 2x ðR � sinðhÞ� xÞ=tanðhÞdx oy A ðR� hÞ�sinðhÞ ðR� hÞ�sinðhÞ þ 2x ððR � hÞ�ð R � hÞ� cosðhÞÞdx Z Z qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðR� hÞ�sinðhÞ A 2 2 3 2 I ¼ x dA ¼ 2x ð ðR � hÞ � x �ð R � hÞ� cosðhÞÞdx oy A 0 The remaining area of the crack element A is given by ce 2 2 2 2 A ¼ A � A ¼ p �ðR � r Þ� ð h � R � h �ðR � hÞ Þ (3) ce crack The offset distance e of the centroid o is given by �� Z Z Z e ¼ ydA þ ydA � ydA =A (4) ce A A A 1 2 3 For the situation of crack penetration, conditions and parameter settings remain the same as those before penetration. The crack region in Figure 3(b) is divided into three parts, where A represents to the arcuate region between line y ¼ R and line y ¼ R–h, A represents to the trapezoidal area between line y ¼ R–h and line y ¼ r cos (h) and A means the unshaded arched area between line y ¼ r and line y ¼ r cos(h). The moment of inertia of the crack area on the axes ox I is given by ox A A A 1 2 3 I ¼ I þ I � I (5) ox ox ox ox where Z Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 1 2 2 I ¼ y dA ¼ 2y R � y dy ox A R� h Z Z R� h 2 3 I ¼ y dA ¼ 2y tanðhÞdy ox A r�cosðhÞ Z Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 A 2 2 I ¼ y dA ¼ 2y r � y dy ox A r�cosðhÞ (a) (b) Figure 3. The crack sketch after penetration. (a) Cross-section of the crack element; (b) Geometry of the crack area. Yongfeng et al. Yongfeng et al. 12315 The moment of inertia of the crack area on the axes oy is given by A A A 1 2 3 I ¼ I þ I � I (6) oy oy oy oy where Z Z R� sinðhÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 1 2 2 I ¼ x dA ¼ 2x ð R � x �ðR � hÞÞdx oy A 0 Z Z R� sinðhÞ A 2 2 I ¼ x dA ¼ 2x ðR �sinðhÞ� xÞ=tanðhÞdx oy A r�sinðhÞ r� sinðhÞ þ 2x ððR � hÞ� r �cosðhÞÞdx Z Z r�sinðhÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 2 2 3 2 2 I ¼ x dA ¼ 2x ð r � x � r �cosðhÞÞdx oy A 0 2 2 2 2 A ¼ A � A ¼ p �ð R � r Þ� ðh �R � h �r Þ (7) ce crack Z Z Z �� e ¼ ydA þ ydA � ydA =A (8) ce A A A 1 2 3 The decrease of the moment of inertia due to the crack can be deﬁned by I ¼ I þ A �e , I ¼ I . The 1 ox ce 2 oy moment of inertia of the crack element relative to the new centroid axis cx and cy can be obtained by ce I ¼ I � I cx (9) ce I ¼ I � I cy where I and I are very important parameters to calculate the stiffness reduction matrix. In order to show the 1 2 effect of different models, the moment of inertias for solid shaft and hollow shaft, circular crack, and straight crack are compared here. The physical parameters for a rotor system are shown in Table 1. Assume that the solid shaft has the same outer diameter as the hollow axis. Figures 4 and 5 show I , I and the relative 1 2 reduction of I with variety of crack depth. With the increasing of crack depth, the loss of moment of inertia in x and y directions increases. The effect of hollow shaft and circular crack is greater than the solid shaft and straight crack. Timoshenko beam-axis model is used to calculate the crack stiffness of the shaft. The coordinates of the ith element in rotor system can be expressed as ½x ; y ; h ; h ; x y ; h ; h � . x and y are the nodal displace- i i xi yi iþ1; iþ1 xðiþ1Þ yðiþ1Þ ments and h and h are the rotating angular displacements. When the crack is fully open, the stiffness reduction x y Table 1. Value of the physical parameters. Parameter Value Length of the rotor shaft, L 0.724 m Outer radius of rotor shaft, R 7.9 mm Inner radius of rotor shaft, r 4.74 mm Density, q 7800 kg/m Stiffness of bearing ðk :k Þ 5 � 10 N/m xx yy Damping of bearing ðc :c Þ 5 � 10 N s/m xx yy 1232 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Yongfeng et al. 7 12EI 12EI 1 2 where / ¼ , / ¼ and l is the shear coefﬁcient. 2 2 1 2 1 l A Gl l A Gl ce ce 1 1 The cosine switching function fðtÞ¼ð1 þ cosðXtÞÞ=2 for breathing cracks is used to describe the opening and closing of cracks during rotation. The dynamic equations of the cracked rotor can be given by € _ MqðtÞþ½C þ G�qðtÞ þðK � fðtÞK ÞqðtÞ¼ F cosðXtÞþ F sinðXtÞþ F (11) c 1 2 g where M, C, G and K are the mass matrix, damping matrix, gyro matrix and stiffness matrix of a non-crack rotor, respectively. For the crack-element stiffness matrix K ¼ K c crack For other elements, K equals 0 and F and F represent the unbalanced excitation vector. For the node 1 2 contains unbalance i 2 F ¼ m d � X ½cosb; sinb; 0; 0� i 2 F ¼ m d � X ½� sinb; cosb; 0; 0� where m is the eccentric mass, d is the eccentricity distance and b is the angle of unbalanced mass. For other nodes, F and F equal 0. F is the gravity force vector of the rotor system. The value of each node is given by 1 2 g Figure 4. I and I with variety of crack depth. F ¼ ½� m g; 0; 0; 0�, where m is the mass of node j. 1 2 j Harmonic balance method is used to solve equation (11), so the solution is assumed to be expressed in Fourier series as qðtÞ¼ A þ ½A cosðiXtÞþ B sinðiXtÞ� (12) 0 i i i¼1 where n represents the retained harmonic number. From our experience, n ¼ 4 is sufﬁcient to reveal the dynamic characteristics of cracked rotor system. By substituting equation (12) into equation (11), it is obtained that 2 3 hollow shaft with circular crack 2 3 2 3 K C K 0 A F 1 1 1 6 7 hollow shaft with straight crack 6 7 6 7 6 7 � C K � X M 0 K 6 7 6 7 B F 1 1 1 2 6 7 6 7 6 7 solid shaft with straight crack 6 7 6 7 6 7 6 K 0 K �ð 2XÞ M C . . . : 7 A 0 1 2 2 6 7 6 7 6 7 6 7 6 7 6 2 7 6 7 6 7 0 K � C K �ð 2XÞ M . . . : . . . : B 0 6 1 2 7 6 7 6 7 ¼ (13) 6 7 6 7 6 7 6 7 . . . . .. . . . . . . : K 0 : : 1 6 7 6 7 6 7 6 7 6 7 Figure 5. The relative reduction of I with variety of crack depth. 6 7 6 7 6 7 . . . . . . : . . . : 0 K : : 6 7 6 7 6 7 6 7 6 7 6 7 matrix K caused by the crack can be written as 6 � 7 crack K 0 K �ð 4XÞ MC A 0 4 4 5 4 5 1 4 4 5 2 3 B 0 0 K � C K �ð 4XÞ M 4 1 1 1 1 1 4 12I 0 06lI �12I 0 06lI 6 1 1 1 1 7 1 þ / 1 þ / 1 þ / 1 þ / 6 7 1 1 1 1 6 7 1 1 1 1 6 7 � 1 0 12I �6lI 00 �12I �6lI 0 � 1 � ^ � � � 6 2 2 2 2 7 where K ¼� K =4, F ¼ F þ K K F =2, K ¼ K � X M � K ðK ÞK =8, C ¼ sXC, s ¼ 1; 2; . . . ; n, 1 c 1 c g c c s 1 þ / 1 þ / 1 þ / 1 þ / 6 7 2 2 2 2 � 1 6 7 1 4 þ / 1 2 � / A ¼ K ðF þ K A =4Þ. By solving the linear equation (13), the steady-state response of the rotor system can 6 2 2 7 2 2 0 g c 1 0 �6lI l I 0 06lI l I 0 6 2 2 2 2 7 6 1 þ / 1 þ / 1 þ / 1 þ / 7 2 2 2 2 be obtained based on harmonic balance method. 6 7 1 4 þ / 1 2 � / 6 1 1 7 2 2 6lI 00 l I �6lI 00 l I 6 1 1 1 1 7 6 1 þ / 1 þ / 1 þ / 1 þ / 7 1 1 1 1 K ¼ 6 7 crack 1 1 1 1 6 7 �12I 00 �6lI 12I 00 �6lI 6 1 1 1 1 7 Numerical simulations 6 1 þ / 1 þ / 1 þ / 1 þ / 7 1 1 1 1 6 7 1 1 1 1 6 7 For the rotor system shown in Figure 1, the physical parameters are the same as shown in Table 1. Figure 6 shows 6 0 �12I 6lI 0 0 12I 6lI 0 7 2 2 2 2 6 7 1 þ / 1 þ / 1 þ / 1 þ / 2 2 2 2 the vibration amplitude of a non-crack rotor system with asymmetric stiffness and symmetric bearing stiffness. 6 7 1 2 � / 1 4 þ / 6 7 2 2 2 2 6 0 �6lI l I 0 06lI l I 0 7 2 2 2 2 The ﬁrst-forward critical speed is w ¼ 281.8 rad/s, and the ﬁrst-backward critical speed is w ¼ 272.9 rad/s. In f1 b1 6 7 1 þ / 1 þ / 1 þ / 1 þ / 2 2 2 2 6 7 addition, we can obtain the critical speed by the Eigenvalue Method, and the critical speed is w ¼ 281.8 rad/s and 4 1 2 � / 1 4 þ / 5 f1 2 1 2 1 6lI 00 l I �6lI 00 l I 1 1 1 1 1 þ / 1 þ / 1 þ / 1 þ / w ¼ 272.8 rad/s. For asymmetric-cracked shaft with isotropic bearing or symmetric intact shaft with anisotropic 1 1 1 1 b1 (10) Yongfeng et al. Yongfeng et al. 12337 12EI 12EI 1 2 where / ¼ , / ¼ and l is the shear coefﬁcient. 2 2 1 2 1 l A Gl l A Gl ce ce 1 1 The cosine switching function fðtÞ¼ð1 þ cosðXtÞÞ=2 for breathing cracks is used to describe the opening and closing of cracks during rotation. The dynamic equations of the cracked rotor can be given by € _ MqðtÞþ½C þ G�qðtÞ þðK � fðtÞK ÞqðtÞ¼ F cosðXtÞþ F sinðXtÞþ F (11) c 1 2 g where M, C, G and K are the mass matrix, damping matrix, gyro matrix and stiffness matrix of a non-crack rotor, respectively. For the crack-element stiffness matrix K ¼ K c crack For other elements, K equals 0 and F and F represent the unbalanced excitation vector. For the node 1 2 contains unbalance i 2 F ¼ m d � X ½cosb; sinb; 0; 0� i 2 F ¼ m d � X ½� sinb; cosb; 0; 0� where m is the eccentric mass, d is the eccentricity distance and b is the angle of unbalanced mass. For other nodes, F and F equal 0. F is the gravity force vector of the rotor system. The value of each node is given by 1 2 g F ¼ ½� m g; 0; 0; 0�, where m is the mass of node j. Harmonic balance method is used to solve equation (11), so the solution is assumed to be expressed in Fourier series as qðtÞ¼ A þ ½A cosðiXtÞþ B sinðiXtÞ� (12) 0 i i i¼1 where n represents the retained harmonic number. From our experience, n ¼ 4 is sufﬁcient to reveal the dynamic characteristics of cracked rotor system. By substituting equation (12) into equation (11), it is obtained that 2 3 2 3 2 3 ^ � K C K 0 A F 1 1 1 6 7 6 7 6 7 6 7 � C K � X M 0 K 6 7 6 7 B F 1 1 1 2 6 7 6 7 6 7 6 7 6 7 6 7 6 K 0 K �ð 2XÞ M C . . . : 7 A 0 1 2 2 6 7 6 7 6 7 6 7 6 7 6 2 7 6 7 6 7 0 K � C K �ð 2XÞ M . . . : . . . : B 0 6 1 2 7 6 7 6 7 ¼ (13) 6 7 6 7 6 7 6 7 . . . . .. . . . . . . : K 0 : : 1 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 . . . . . . : . . . : 0 K : : 6 7 6 7 6 7 6 7 6 7 6 7 6 � 7 K 0 K �ð 4XÞ MC A 0 4 4 5 4 5 1 4 4 5 B 0 0 K � C K �ð 4XÞ M 4 1 4 � 1 � 1 � ^ � � � where K ¼� K =4, F ¼ F þ K K F =2, K ¼ K � X M � K ðK ÞK =8, C ¼ sXC, s ¼ 1; 2; . . . ; n, 1 c 1 c g c c s � 1 A ¼ K ðF þ K A =4Þ. By solving the linear equation (13), the steady-state response of the rotor system can 0 g c 1 be obtained based on harmonic balance method. Numerical simulations For the rotor system shown in Figure 1, the physical parameters are the same as shown in Table 1. Figure 6 shows the vibration amplitude of a non-crack rotor system with asymmetric stiffness and symmetric bearing stiffness. The ﬁrst-forward critical speed is w ¼ 281.8 rad/s, and the ﬁrst-backward critical speed is w ¼ 272.9 rad/s. In f1 b1 addition, we can obtain the critical speed by the Eigenvalue Method, and the critical speed is w ¼ 281.8 rad/s and f1 w ¼ 272.8 rad/s. For asymmetric-cracked shaft with isotropic bearing or symmetric intact shaft with anisotropic b1 8 1234 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Yongfeng et al. 9 w =281.8rad/s f1 w =272.9rad/s b1 Non-dimensional crack depth Rotation speed (rad/s) Rotation speed (rad/s) Figure 9. Rotation speed–crack depth–vibration amplitude waterfall at node 4. Figure 6. Vibration amplitude at node 4 for non-crack rotor with asymmetric/symmetric stiffness. f1 b1 1/2w f1 1/2w b1 Rotation speed (rad/s) f1 1/3w b1 1/2w b1 b1 1/2w f1 1/4w 1/3w 1/4w f1 b1 f1 Rotation speed (rad/s) w f1 b1 Non-dimensional crack depth 1/2w 1/3w b1 b1 Figure 7. The first critical speed with variety of crack depth. 1/2w 1/4w f1 1/3w b1 f1 (a) (b) Rotation speed (rad/s) w w f1 b1 1/2w b1 1/3w b1 1/2w f1 1/4w 1/3w b1 1/4w f1 f1 Rotation speed (rad/s) Figure 10. Rotation speed–vibration amplitude diagram of rotor system at node 4 with crack located in different elements when crack depth u¼ 0.5. bearings, the ﬁrst-forward and backward whirl speeds could be excited by unbalance force. As shown in Figure 6, the results obtained by two methods are in good agreement with each other. Non-dimensional crack depth Non-dimensional crack depth The ﬁrst critical rotational speed of a hollow-shaft rotor system with a circular crack is shown in Figure 7. It can be seen that the ﬁrst critical speed of the cracked rotor decreases with the crack depth going deeper. Figure 8. The first critical speed with crack position in different elements. Vibration amplitude (m) Rotation speed (rad/s) Rotation speed (rad/s) Rotation speed (rad/s) Mean of vertical Mean of vertical Mean of vertical Mean of vertical Vibration amplitude (m) amplitude (m) amplitude (m) amplitude (m) amplitude (m) Yongfeng et al. Yongfeng et al. 12359 Non-dimensional crack depth Rotation speed (rad/s) Figure 9. Rotation speed–crack depth–vibration amplitude waterfall at node 4. f1 b1 1/2w f1 1/2w b1 Rotation speed (rad/s) f1 1/3w b1 1/2w b1 b1 1/2w f1 1/4w 1/3w 1/4w f1 b1 f1 Rotation speed (rad/s) w f1 b1 1/2w 1/3w b1 b1 1/2w 1/4w f1 1/3w b1 f1 Rotation speed (rad/s) w w f1 b1 1/2w b1 1/3w b1 1/2w f1 1/4w 1/3w b1 1/4w f1 f1 Rotation speed (rad/s) Figure 10. Rotation speed–vibration amplitude diagram of rotor system at node 4 with crack located in different elements when crack depth u¼ 0.5. bearings, the ﬁrst-forward and backward whirl speeds could be excited by unbalance force. As shown in Figure 6, the results obtained by two methods are in good agreement with each other. The ﬁrst critical rotational speed of a hollow-shaft rotor system with a circular crack is shown in Figure 7. It can be seen that the ﬁrst critical speed of the cracked rotor decreases with the crack depth going deeper. Mean of vertical Mean of vertical Mean of vertical Mean of vertical Vibration amplitude (m) amplitude (m) amplitude (m) amplitude (m) amplitude (m) Mean of vertical Mean of vertical Mean of vertical Mean of vertical amplitude(m) amplitude amplitude amplitude(m) 10 1236 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Yongfeng et al. 11 (a) (b) (c) f1 f1 f1 x(m) x(m) x(m) Figure 13. Phase orbits near 92.1 rad/s of node 4 when crack depth u¼ 0.4. (a) (b) (c) Figure 11. Rotation speed–vibration amplitude diagram of rotor system at node 4 with no crack. (a) (b) (c) x(m) x(m) x(m) Figure 14. Phase orbits near 69 rad/s of node 4 when crack depth u¼ 0.4. (a) (b) (c) x(m) x(m) x(m) Figure 12. Phase orbits near 138.3 rad/s of node 4 when crack depth u¼ 0.4. Figure 15. Power spectrum of node 4 when crack depth u¼ 0.4. y(m) y(m) y(m) y(m) y(m) y(m) y(m) y(m) y(m) Yongfeng et al. Yongfeng et al. 1237 11 (a) (b) (c) x(m) x(m) x(m) Figure 13. Phase orbits near 92.1 rad/s of node 4 when crack depth u¼ 0.4. (a) (b) (c) x(m) x(m) x(m) Figure 14. Phase orbits near 69 rad/s of node 4 when crack depth u¼ 0.4. (a) (b) (c) Figure 15. Power spectrum of node 4 when crack depth u¼ 0.4. y(m) y(m) y(m) y(m) y(m) y(m) 12 1238 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Yongfeng et al. 13 In particular, the ﬁrst-backward critical speed is very sensitive to the crack depth while the ﬁrst-forward critical Funding speed is not. The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this For the above study, the crack position is constant in element 4. Here, we investigate the effect of crack article: This study was funded by National Natural Science Foundation of China (grant number 11972295), the Key position. Figure 8 shows the ﬁrst critical speeds at different element locations. When the crack position is close Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University (grant to the bearing position (Element 1), the ﬁrst critical speed decreases slowly with increasing crack depth. When the number VCAME201803), and Graduate Innovation Fund of Northwestern Polytechnical University (grant number crack position is far away from the bearing, especially for the element near the middle of the shaft (Element 4, CX2020124). Element 5, Element 6), and the ﬁrst-order critical speed drops faster. Element 6 is the fastest one. Figure 9 shows the rotation speed–crack depth–vibration amplitude waterfall at node 4 when the circular crack ORCID iD is located in element 4. Due to the circular crack, vibration peaks appear at the 1/2, 1/3 and 1/4 ﬁrst-forward and Yang Yongfeng https://orcid.org/0000-0003-0402-4440 backward critical rotational speeds. Since we study the response of a cracked rotor system, the basic character- istics of response are the same as given in Ishida and Yamamoto. When the crack depth is small (u¼ 0.1 and References u¼ 0.2), the vibration peaks near 1/3 and 1/4 critical speeds are insigniﬁcant. With the crack expansion and 1. Li CF, She HX, Tang QS, et al. The effect of blade vibration on the nonlinear characteristics of rotor-bearing system deepening, the vibration peaks near 1/3 and 1/4 critical speeds become very obvious. At the same time, the sub- supported by nonlinear suspension. Nonlin Dyn 2017; 89: 987–1110. critical rotational speeds of the cracked rotor system also decrease with the increasing of the crack depth. In short, 2. Hou L, Chen YS, Fu YQ, et al. Application of the HB-AFT method to the primary resonance analysis of a dual-rotor the crack will cause the rotor system to have vibration peaks near the ﬁrst critical speed and 1/n (n¼ 2, 3, 4) ﬁrst system. Nonlin Dyn 2017; 88: 2531–2551. critical speed. The amplitude of vibration peak near 1/4 critical speed is very sensitive to the crack depth and 3. Luo Z, Zhu YP, Zhao XY, et al. Determining dynamic scaling laws of geometrically distorted scaled models of a cantilever position. There are vibration peaks between the 1/2 ﬁrst critical speed and the ﬁrst critical speed. The rotation plate. J Eng Mech 2016; 142: 04015108. speed is near 215 rad/s. With the crack depth going deeper and the element closer to crack position, the rotation 4. Zhang GH and Ehmann KF. Dynamic design methodology of high speed micro- spindles for micro/meso-scale machine tools. Int J Adv Manuf Technol 2015; 76: 229–246. speed will go down and the amplitude will go up. This is an important dynamic characteristic to detect the crack 5. Dai HH, Jing XJ, Wang Y, et al. Post-capture vibration suppression of spacecraft via a bio-inspired isolation system. Mech fault in rotor system. Syst Signal Process 2018; 105: 214–240. Figure 10 shows the rotation speed–vibration amplitude diagram of node 4 with the crack locating in different 6. Qin ZY, Yang ZB, Zu J, et al. Free vibration analysis of rotating cylindrical shells coupled with moderately thick annular elements when the circular crack depth u¼ 0.5. It can be seen that when the crack is located near the support plate. Int J Mech Sci 2018; 142-143: 127–139. position of element 1, the vibration peaks near 1/3 and 1/4 critical speeds are not obvious. While the crack is far 7. Fu C, Xu YD, Yang YF, et al. Response analysis of an accelerating unbalanced rotating system with both random and away from the bearing support, the vibration peaks near 1/3 and 1/4 critical speed become obvious. There is a interval variables. J Sound Vib 2020; 466: 115047. peak near 215 rad/s. With the crack depth going deeper and the measurement point closer to crack position, the 8. Li HG, Meng G, Meng ZQ, et al. Effects of boundary conditions on a self-excited vibration system with clearance. Int J speed will go down and the amplitude will go up. As a result, the crack located in the middle shaft of the rotor has Nonlin Sci Num 2007; 8: 571–579. a greater inﬂuence on the dynamic characteristics than in the supporting position. Figure 11 shows the rotation 9. Han QK and Chu FL. Parametric instability of ﬂexible rotor-bearing system under time-periodic base angular motions. Appl Math Model 2015; 39: 4511–4522. speed–vibration amplitude diagram of rotor system at node 4 with no crack. It can be seen that the vibration peak 10. Wang WM, Li QH, Gao JJ, et al. An identiﬁcation method for damping ratio in rotor systems. Mech Syst Signal Process only occurred at the critical speed of the rotor system, and there is no super-harmonic resonance phenomenon in 2016; 68-69: 536–554. no-cracked rotor systems. 11. Ma H, Shi CY, Han QK, et al. Fixed-point rubbing fault characteristic analysis of a rotor system based on contact theory. Figures 12 to 15 show the phase orbits and the power spectrum of node 4 near the sub-critical speeds when the Mech Syst Signal Process 2013; 38: 137–153. circular crack depth is 0.4. It can be seen that when the rotating speed is close to the 1/2 ﬁrst-forward critical 12. Chu FL and Lu WX. Experimental observation of nonlinear vibrations in a rub-impact rotor system. J Sound Vib 2005; speed, it appears as two overlapping ellipses in the phase orbits and 2� component in the frequency domain. 283: 621–643. Actually, the 2� component is the ﬁrst-forward critical speed. In the same way, for 1/3 and 1/4 critical speeds, the 13. Zhang WM and Meng G. Stability, bifurcation and chaos analyses of a high-speed micro-rotor system with rub-impact. phase orbits are three and four overlapping ellipses, respectively. Similarly, the frequency components are dom- Sensor Actuat A-Phys 2006; 127: 163–178. inated by the 3� component or 4� component. For the backward critical speed, we can observe the same 14. Yang YF, Chen H and Jiang TD. Nonlinear response prediction of cracked rotor based on EMD. J Franklin Inst 2015; 352: 3378–3393. phenomenon. It shows the super-harmonic resonance phenomenon in cracked rotor systems. 15. Yang YF, Wu QY, Wang YL, et al. Dynamic characteristics of cracked uncertain hollow-shaft. Mech Syst Signal Process 2019; 124: 36–48. 16. Ma H, Zeng J, Feng RJ, et al. Review on dynamics of cracked gear systems. Eng Fail Anal 2015; 55: 224–245. Conclusion 17. Patel TH and Darpe AK. Inﬂuence of crack breathing model on nonlinear dynamics of a cracked rotor. J Sound Vib 2008; A hollow-shaft rotor system with circular cracks is studied in this paper. The time-varying stiffness matrix of the 311: 953–972. crack element is deduced. The inﬂuence of the crack on the critical speed and sub-critical speed is shown. It is 18. Qin WY, Meng G and Zhang T. The swing vibration, transverse oscillation of cracked rotor and the intermittence chaos. J Sound Vib 2003; 259: 571–583. found that the circular cracks could reduce both the ﬁrst-forward and backward critical rotational speeds of the 19. Lu YJ, Zhang YF, Shi XL, et al. Nonlinear dynamic analysis of a rotor system with ﬁxed-tilting-pad self-acting gas- rotor system, especially the latter one. Owing to the inﬂuences, the vibration peaks occur near 1/n (n¼ 1, 2, 3, and lubricated bearings support. Nonlin Dyn 2012; 69: 877–890. 4) critical rotational speed. As the crack increases, the peaks become more prominent. When the crack is located 20. Meng G, Zhang WM, Huang H, et al. Micro-rotor dynamics for micro-electro-mechanical systems (MEMS). Chaos in the middle of shaft, the effect will be greatest. Super-harmonic resonance phenomena can be observed in the Soliton Fract 2009; 40: 538–562. cracked rotor system. The results of this paper can provide some guidance for detection and identiﬁcation of crack 21. Dimarogonas AD. Vibration of cracked structures: a state of the art review. Eng Fract Mech 1996; 55: 831–857. fault in hollow-shaft dual-rotor system. 22. Wauer J. Dynamics of cracked rotors: literature survey. Appl Mech Rev 1990; 43: 13–17. 23. Pennacchi P, Bachschmid N and Vania A. A model-based identiﬁcation method of transverse cracks in rotating shafts suitable for industrial machines. Mech Syst Signal Process 2006; 20: 2112–2147. Declaration of conflicting interests 24. Rubio L and Fernandez-Saez J. A new efﬁcient procedure to solve the nonlinear dynamics of a cracked rotor. Nonlin Dyn The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this 2012; 70: 1731–1745. article. 25. Sinou JJ. Effects of a crack on the stability of a non-linear rotor system. Int J Nonlin Mech 2007; 42: 959–972. Yongfeng et al. Yongfeng et al. 1239 13 Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: This study was funded by National Natural Science Foundation of China (grant number 11972295), the Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University (grant number VCAME201803), and Graduate Innovation Fund of Northwestern Polytechnical University (grant number CX2020124). ORCID iD Yang Yongfeng https://orcid.org/0000-0003-0402-4440 References 1. Li CF, She HX, Tang QS, et al. The effect of blade vibration on the nonlinear characteristics of rotor-bearing system supported by nonlinear suspension. Nonlin Dyn 2017; 89: 987–1110. 2. Hou L, Chen YS, Fu YQ, et al. Application of the HB-AFT method to the primary resonance analysis of a dual-rotor system. Nonlin Dyn 2017; 88: 2531–2551. 3. Luo Z, Zhu YP, Zhao XY, et al. Determining dynamic scaling laws of geometrically distorted scaled models of a cantilever plate. J Eng Mech 2016; 142: 04015108. 4. Zhang GH and Ehmann KF. Dynamic design methodology of high speed micro- spindles for micro/meso-scale machine tools. Int J Adv Manuf Technol 2015; 76: 229–246. 5. Dai HH, Jing XJ, Wang Y, et al. Post-capture vibration suppression of spacecraft via a bio-inspired isolation system. Mech Syst Signal Process 2018; 105: 214–240. 6. Qin ZY, Yang ZB, Zu J, et al. Free vibration analysis of rotating cylindrical shells coupled with moderately thick annular plate. Int J Mech Sci 2018; 142-143: 127–139. 7. Fu C, Xu YD, Yang YF, et al. Response analysis of an accelerating unbalanced rotating system with both random and interval variables. J Sound Vib 2020; 466: 115047. 8. Li HG, Meng G, Meng ZQ, et al. Effects of boundary conditions on a self-excited vibration system with clearance. Int J Nonlin Sci Num 2007; 8: 571–579. 9. Han QK and Chu FL. Parametric instability of ﬂexible rotor-bearing system under time-periodic base angular motions. Appl Math Model 2015; 39: 4511–4522. 10. Wang WM, Li QH, Gao JJ, et al. An identiﬁcation method for damping ratio in rotor systems. Mech Syst Signal Process 2016; 68-69: 536–554. 11. Ma H, Shi CY, Han QK, et al. Fixed-point rubbing fault characteristic analysis of a rotor system based on contact theory. Mech Syst Signal Process 2013; 38: 137–153. 12. Chu FL and Lu WX. Experimental observation of nonlinear vibrations in a rub-impact rotor system. J Sound Vib 2005; 283: 621–643. 13. Zhang WM and Meng G. Stability, bifurcation and chaos analyses of a high-speed micro-rotor system with rub-impact. Sensor Actuat A-Phys 2006; 127: 163–178. 14. Yang YF, Chen H and Jiang TD. Nonlinear response prediction of cracked rotor based on EMD. J Franklin Inst 2015; 352: 3378–3393. 15. Yang YF, Wu QY, Wang YL, et al. Dynamic characteristics of cracked uncertain hollow-shaft. Mech Syst Signal Process 2019; 124: 36–48. 16. Ma H, Zeng J, Feng RJ, et al. Review on dynamics of cracked gear systems. Eng Fail Anal 2015; 55: 224–245. 17. Patel TH and Darpe AK. Inﬂuence of crack breathing model on nonlinear dynamics of a cracked rotor. J Sound Vib 2008; 311: 953–972. 18. Qin WY, Meng G and Zhang T. The swing vibration, transverse oscillation of cracked rotor and the intermittence chaos. J Sound Vib 2003; 259: 571–583. 19. Lu YJ, Zhang YF, Shi XL, et al. Nonlinear dynamic analysis of a rotor system with ﬁxed-tilting-pad self-acting gas- lubricated bearings support. Nonlin Dyn 2012; 69: 877–890. 20. Meng G, Zhang WM, Huang H, et al. Micro-rotor dynamics for micro-electro-mechanical systems (MEMS). Chaos Soliton Fract 2009; 40: 538–562. 21. Dimarogonas AD. Vibration of cracked structures: a state of the art review. Eng Fract Mech 1996; 55: 831–857. 22. Wauer J. Dynamics of cracked rotors: literature survey. Appl Mech Rev 1990; 43: 13–17. 23. Pennacchi P, Bachschmid N and Vania A. A model-based identiﬁcation method of transverse cracks in rotating shafts suitable for industrial machines. Mech Syst Signal Process 2006; 20: 2112–2147. 24. Rubio L and Fernandez-Saez J. A new efﬁcient procedure to solve the nonlinear dynamics of a cracked rotor. Nonlin Dyn 2012; 70: 1731–1745. 25. Sinou JJ. Effects of a crack on the stability of a non-linear rotor system. Int J Nonlin Mech 2007; 42: 959–972. 14 1240 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) 26. Ishida Y and Inoue T. Detection of a rotor crack using a harmonic excitation and nonlinear vibration analysis. J Vib Acoust 2006; 128: 741–749. 27. Sawicki JT, Friswell MI, Kulesza Z, et al. Detecting cracked rotors using auxiliary harmonic excitation. J Sound Vib 2011; 330: 1365–1381. 28. Penny JET and Friswell MI. The dynamics of rotating machines with cracks. MSF 2003; 440–444: 311–318. 29. Al-Shudeifat MA. On the ﬁnite element modeling of the asymmetric cracked rotor. J Sound Vib 2013; 332: 2795–2807. 30. Al-Shudeifat MA. Stability analysis and backward whirl investigation of cracked rotors with time-varying stiffness. J Sound Vib 2015; 348: 365–380. 31. Cavalini AA Jr, Sanches L, Bachschmid N, et al. Crack identiﬁcation for rotating machines based on a nonlinear approach. Mech Syst Signal Process 2016; 79: 72–85. 32. Lu ZY, Hou L, Chen YS, et al. Nonlinear response analysis for a dual-rotor system with a breathing transverse crack in the hollow shaft. Nonlin Dyn 2016; 83: 169–185. 33. Guo CZ, Al-Shudeifat MA, Vakakis AF, et al. Vibration reduction in unbalanced hollow rotor systems with nonlinear energy sinks. Nonlin Dyn 2015; 79: 527–538. 34. Du Y, Zhou SX, Jing XJ, et al. Damage detection techniques for wind turbine blades: a review. Mech Syst Signal Process 2020; 141: 106445. 35. Liu Y, Zhao YL, Li JT, et al. Application of weighted contribution rate of nonlinear output frequency response functions to rotor rub-impact. Mech Syst Signal Process 2020; 136: 106518. 36. Yang YF, Zheng QY, Wang JJ, et al. Dynamics response analysis of airborne external storage system with clearance between missile-frame. Chin J Aeronaut. Epub ahead of print 20 June 2020. DOI: 10.1016/j.cja.2020.06.008. 37. Pilkey WD. Analysis and design of elastic beams. 1st ed. New York: John Wiley and Sons, 2012. 38. Ishida Y and Yamamoto T. Linear and nonlinear rotor dynamics. New York: Wiley, 2012.

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

Published: Sep 1, 2020

Keywords: Rotor; hollow-shaft; circular cracks; critical speed; super-harmonic

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Dynamical analysis of hollow-shaft dual-rotor system with circular cracks:

Dynamical analysis of hollow-shaft dual-rotor system with circular cracks:

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Dynamical analysis of hollow-shaft dual-rotor system with circular cracks:

Dynamical analysis of hollow-shaft dual-rotor system with circular cracks:

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