Finite Element Vibration Analysis of a Rotating shaft System with an Open Crack by the harmonic excitation

Finite Element Vibration Analysis of a Rotating shaft System with an Open Crack by the harmonic excitation Nobuhiro NAGATA, Tsuyoshi INOUE, Yukio ISHIDA Deptartment of Mechanical Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8603 Japan ABSTRACT The vibration of the rotating shaft with an open crack under the harmonic excitation is investigated. The concise FEM rotating shaft model for the quantitative analysis, which was developed in the previous studies, is utilized, and the fundamental equations governing the vibration of the cracked shaft with the harmonic excitation are deduced. Furthermore, the experimental system using the active magnetic bearing for the harmonic excitation is developed, and the validity of the obtained theoretical results are confirmed experimentally. 1. INTRODUCTION Health monitoring technique for the rotating machinery has been developed and the vibration diagnosis of a rotor crack has been investigated in last 40 years (1)(2). There are many studies to investigate the influence of the crack by using the finite element method (FEM) for the modeling of the cracked shaft (3) (8). In these conventional studies, some of them which compared the influence of the crack quantitatively with experiment by using the commercial 3D-FEM software in the process of the modeling of the crack (8). In the detail vibration analysis or fault diagnosis of the practical rotating shaft, it will be generally required to perform the time-domain numerical analysis or the frequency-domain theoretical analysis recursively and evaluate the influence of various parameters. Therefore, it seems complex and costly to apply these conventional quantitative modeling techniques directly to such vibration analysis or fault diagnosis. In this point, the author has developed the concise and general purpose oriented models of the open crack in the rotor system which is available for the quantitative vibration analysis, and confirmed the validity of them experimentally (9) (10). In this paper, the vibration of the rotating shaft with an open crack under the harmonic excitation is investigated. The concise FEM rotating shaft model for the quantitative analysis, which was developed in the previous studies, is utilized, and the fundamental equations governing the vibration characteristics of resonances which are caused by the coexistence of the rotor crack and the harmonic excitation are deduced. Furthermore, the experimental system using the active magnetic bearing for the harmonic excitation is developed, and the validity of the obtained theoretical results are confirmed experimentally. 2. MODELING OF THE ROTATING SHAFT WITH AN OPEN CRACK 2 1 Modeling of the rotating shaft system The rotor system shown in Fig.1(a) is considered. An elastic shaft is simply supported by ball bearings. The length l and the radius R of the shaft are l900 mm and R10 mm. Three disks of disk 1, disk 2, and disk 3 are mounted on the shaft. Disk 1 is mounted at the position l d1 100 mm from bearing 1. Its diameter, thickness, and mass are D 1 135 mm, H 1 15.5 mm, and m d1 1.73 kg, respectively. Disk 2 and Disk 3 are mounted at the positions of l d2 400 mm and l d3 550 mm from bearing 1, respectively. Notation l c is the distance from bearing 1 to the position of the crack. The finite element metnod (FEM) is used in the modelling. The crack is modelled using the element at the crack position and the neighboring elements (9) (12) The displacements of translation and rotation at nodes i 1,...,n 1 are represented as the state vector q {x 1 y 1 φ x1 φ y1... x n1 y n1 φ x(n1) φ y(n1) } T. The equation of motion for the rotor 1

l y y l d 3 l d1 l c l d 2 d o x φωt x crack disk 1 disk 2 disk 3 (a) FEM rotor model with open crack (b) Coordinate systems Fig. 1 FEM rotor model and cordinate systems system is obtained by the finite element method as follows (13) (15) M q (C G) q K(t)q f(t) (1) Here, M is the mass matrix, G is the gyroscopic matrix, C is the damping matrix, K(t) is the stiffness matrix. The force vector f(t) includes the gravitational force, unbalance force, and support reaction forces from the bearings. The stiffness of the shaft is asymmetric due to the open crack, and its value varies in the inertia coordinate system with the shaft rotation angle ωt as shown in Fig.1(b). Thus, the stiffness matrix K(t) is the function of time t. Here, ω is the shaft s rotational speed. Bearing support system is represented as linear springs by using spring constants k b. In our experimental setup, the Young s modulus was estimated as E 1.876 10 11 N/m 2 and the bearing stiffness were derived as k b 1.36 10 6 N/m respectively (10). 2 2 Modeling of the Open-crack This paper considers a transverse crack. The inertia coordinate system O-xy and the rotating coordinate system O-x y, which rotates with the shaft rotation angle ωt, are used as shown in Fig.1(b). The positive direction of the y axis is taken to the crack direction, and the x axis is taken perpendicular to it. Variables in the rotating coordinate system are denoted with prime notation. In this paper, the symmetrical shaft with a circular cross section is considered. Therefore, when the element of the healthy un-cracked part of the shaft is considered, its area moments of inertia for the x axis and the y axis are the same, and these are identical to the area moments of inertia for the x axis and the y axis in the inertia coordinate system O- xy, respectively. Their values are designated as I 0. While, when the elements representing the effect of the crack are considered, the area moments of inertia for the x axis and the y axis are different from each other, and they vary depending on the shaft s rotating angle. Hence, in such a case, the area moment of inertia in a rotating coordinate system O-x y is firstly obtained. The area moments of inertia of the element representing the effect of crack for both x and y directions decrease due to the open crack. Their values for x axis and y axis at the crack position are denoted as I x c and I y c. Figure 2 shows the cross section of the shaft at the crack position. Notation d is the depth of the crack, h is the distance between the un-cracked end and the neutral axis. 3. FINITE ELEMENT MODEL OF THE SHAFT WITH AN OPEN CRACK 3 1 The Modeling of Crack In this paper, the concise FEM rotating shaft model, which was developed in the previous studies, is utilized. (9) (10) The area moment of inertia I varies linearly from the area moment of inertia of cracked part I x (z) and I y (z) to the area moment of inertia of normal part I 0 as shown in Fig.3. The shaft position z which is apart from the crack position z c is considered, and the area moment of inertia I x (z) and I y (z) is assumed to be influenced by the crack at shaft position z c 2

2 0 0 2 y d h x neutral axis Area moment of inertia (mm 4 ) 8000 7000 6000 5000 4000 3000 2000 1000 I y'c I x'c y' x' 2l wcx 2l wcy l c 400mm D20mm depth50% 300 350 400 450 500 Position from the left end of the shaft z (mm) Fig. 2 Cross section view of the shaft Fig. 3 Variation of the area moment of inertia in the open crack models I c Normal rotor I 0 I y c I x c 0 90 180 ΔI c ωt Fig. 4 The area moment of inertia at crack position (I c ) and represented as: I x (z) I 0 I x c l wcx I y (z) I 0 I y c l wcy z z c I x c (z c l wcx < z < z c l wcx ) z z c I y c (z c l wcy < z < z c l wcy ) (2) Here, l wcx and l wcy represent the range of decrease of the area moment of inertia from crack position. Then, l wcx and l wcy are proportional to depth of crack d and represented as: (16) l wcx 2.10 d l wcy 1.32 d (3) 3 2 Equation of motion for the FEM model of the rotating shaft The area moment of inertia of the crack element is expressed in the inertia coordinate system O-xy. The difference of the area moments of inertia between the crack direction, I x c, and its perpendicular direction, I y c, is represented as 2 I c, and their area moments of inertia are related as I c (I y c I x c)/2. When the shaft rotates with the rotational speed ω, the area moments of inertia I xc and I yc in the inertia coordinate system O-xy varies between the values of I y c and I x c as the function of time t as shown in Fig.4. The area moment of inertia of the element at the crack position for x direction is represented in the inertia coordinate system as: I c (t) I y c I c I c cos2ωt (4) 3

The stiffness matrix of the rotating shaft with the open crack, K(t), is derived using this representation as follows. First, the stiffness matrix of the element representing the effect of the crack is obtained in the rotating coordinate system O-x y, and it is represented as k e c. Here, superscript notation e designates the element matrix, and subscript notation c designates the element representing the effect of the crack. Then, the element s stiffness matrix k e c is transformed into the inertial coordinate system O-xy and represented as k e c. This element s stiffness matrix k e c in the inertial coordinate system is divided in to three components corresponding the terms in Eq.(4), namely, the constant matrix k e c0 concerning I y c, constant matrix k e c0 concerning I c, and the matrix k e c2ω concerning I c cos2ωt. The derived element s stiffness matrices representing the effect of the crack, k e c, are put in to the corresponding parts of the total stiffness matrix K(t) in Eq.(1). As a result, three total matrices K 0, K 0, and K 2ω are obtained. The equation of motion of Eq.(1) is transformed to the style in which the state variables of the x and y directions are separated (13)(14). The total state variables vector {q} is re-arranged as { q} { q T x qt y } T where q x {x 1 φ y1... x n1 φ y(n1) } T and q y {y 1 φ x1... y n1 φ x(n1) } T. Each matrix in Eq.(1) is also re-arranged corresponding the state variables vector q. The transformed equation of motion is obtained as (9) : [ ]{ } [ ]{ } ([ ] [ ]){ } M 0 q x C ωḡ q x K const 0 K 2ω C 2ω K 2ω S 2ω qx 0 M q y ωḡ C q y 0 K const K 2ω S 2ω K 2ω C 2ω q y Here, M is the total mass matrix with the mass value of each element in the diagonal component, Ḡ is the total gyromatrix with the element gyro-matrices (13). Furthermore, notations K const K 0 K 0, C 2ω cos2ωt and S 2ω sin2ωt are introduced. The force vectors f xall and f yall include the support forces of both bearings in x and y directions, the unbalance force, the gravitational force, and so on. The contributions of the bearing support forces in these force vectors f xall and f yall are extracted and represented as K b q x and K b q y. Matrices K b is the stiffness matrices concerning the bearing support stiffness. As a result, the equation of { f xall motion of a rotating shaft with an open crack is represented as: [ ]{ } [ ]{ } [ ]{ } [ ]{ } } M 0 q x C ωḡ q x K constb 0 qx K 2ω C 2ω K 2ω S 2ω qx { f x 0 M q y ωḡ C q y 0 K constb q y K 2ω S 2ω K 2ω C 2ω q y f y Here, K constb K const K b, force vectors f x and f y do not include the bearing support force, but include the unbalance force, gravitational force, and so on. 4. ANALYSIS OF FORCED VIBRATION BY HARMONIC EXCITATION The analytical method of the vibration of the cracked shaft with harmonic excitation is developed. The equation of motion (6) for the shaft is used, and the notations f x FcosΩt and f y FsinΩt are introduced as harmonic excitation. In that case, the equation of motion (6) is represented as: [ ]{ } [ ]{ } [ ]{ } [ ]{ } { } M 0 q x C ωḡ q x K constb 0 qx K 2ω C 2ω K 2ω S 2ω qx FcosΩt 0 M q y ωḡ C q y 0 K constb q y K 2ω S 2ω K 2ω C 2ω q y FsinΩt (7) f yall } (5) (6) 4 1 Assumption of the solution In the case of the orbit is considered to be a circle, the term of the fundamental whirling motion with excitation frequency Ω causes the additional whirling motion of 2ω Ω through the parametric term of K 2ω due to crack. Then, Ω component and 2ω Ω component have the closed relationship. Therefore, the vibration solution with hormonic excitation is assumed as: { } { } qx q(2ω Ω) cos((2ω Ω)t δ (2ω Ω) ) q Ω cos(ωt δ Ω ) (8) q y q (2ω Ω) sin((2ω Ω)t δ (2ω Ω) ) q Ω sin(ωt δ Ω ) 4

Furthermore, q (2ω Ω) cosδ (2ω Ω) q (2ω Ω)c. q (2ω Ω) sinδ (2ω Ω) q (2ω Ω)s, q Ω cosδ Ω q Ωc, q Ω sinδ Ω q Ωs, cos(2ω Ω)tC (2ω Ω), sin(2ω Ω)tS (2ω Ω), cosωtc Ω and sinωts Ω are introduced, the vibration solution is assumed as: { } { } qx q(2ω Ω)c C (2ω Ω) q (2ω Ω)s S (2ω Ω) q Ωc C Ω q Ωs S Ω (9) q y q (2ω Ω)s C (2ω Ω) q (2ω Ω)c S (2ω Ω) q Ωs C Ω q Ωc S Ω The assumed solution of Eq.(9) is substituted in Eq.(7), and the constant term and the coefficients of cosωt, sinωt, cos(2ω Ω)t and sin(2ω Ω)t are equated for both x and y directions. The obtained equations are re-arranged in terms of vector { q Ωc q Ωs q (2ω Ω)c q (2ω Ω)s } as: Ā Ω ΩωḠ Ω C K 2ω 0 q Ωc F Ω C Ā Ω ΩωḠ 0 K 2ω q Ωs F K 2ω 0 Ā (2ω Ω) (2ω Ω)ωḠ (2ω Ω) C q (2ω Ω)c 0 0 K 2ω (2ω Ω) C Ā (2ω Ω) (2ω Ω)ωḠ 0 q (2ω Ω)s Here, the notations Ā Ω Ω 2 M ( K 0all K b ) Ā (2ω Ω) (2ω Ω) 2 M ( K 0all K b ) are introduced. Eq.(10) is solved for { q Ωc q Ωs q (2ω Ω)c q (2ω Ω)s }. For example, the 2ω Ω vibration component at the node n is obtained by R (2ω Ω)(n) q 2 (2ω Ω)c(2n 1) q2 (2ω Ω)s(2n 1). In the case of rotational speed ω500(rpm) and the magnitude of excitation F5.0(N), the result of theoretical analysis of the amplitude of 2ω Ω component is shown in Fig.5. The lower figure of Fig.5 shows the theoretical results of the natural frequency (10) and sub-resonance point of 2ω Ω component. The upper figure of Fig.5 shows the theoretical results of the amplitude of 2ω Ω component. As shown in Fig.5, sub-resonance peaks of 2ω Ω component (in upper figure) agree with sub-resonance points of 2ω Ω (in lower figure), and the validity of the obtained equation (10) is confirmed. 5 1 Experimental setup 5. EXPERIMENTAL SYSTEM The experimental system is shown in Fig.6(a). The horizontal shaft, whose length is 900mm and diameter is 20mm. Three disks are mounted to the shaft, and the motor drives the shaft through the pulley and spring. Here, disk2 is excitated by magnetic bearing(amb) as shown in Fig.6(b), and the shaft displacements of the disk 2 in both x and y directions were measured using the displacement sensors (Baumer, IWRM 18U9511). An open crack, whose width is about 0.2m1m and depth is 10mm (50% for the shaft s diameter), is made on the shaft by electrospark machining. (10) 5 2 Control of harmonic excitation Control circuit of harmonic excitation is shown in Fig.7. The voltage of harmonic signal from Function Generator and the output voltage of displacement from sensor is transfered to the signal voltage through DSP. Here, current values in coils of magnetic bearing are formulated to generate proper harmonic excitation in DSP. Electromagnetic forces in coils of magnetic bearing are generated by the signal voltages through the pawer amplifer. In this paper, horizontal harmonic excitation F 0 cosωt is caused. Considering the characteristic of magnetic bearing, the harmonic excitation is controled by calculating the proper value of current with DSP. The excitation force F 0 cosωt is represented as: I 2 2 I1 2 F 0 cosωt k 1 (R 1 x δ 1 ) 2 k 2 (R 2 x δ 2 ) 2 (11) Here notations R 1 and R 2 is distance between magnetic bearing and disk2, x is displacement of disk2, and k 1, k 2, δ 1 and δ 2 are the characteristic parameters of magnets of the magnetic bearing. Each parameters (R 1, R 2, k 1, k 2, δ 1, δ 2 ) are estimated experimentally in advance and harmonic excitation is generated by setting up with MATLAB/Simulink with DSP. 5

Amplitude of (2ω-Ω) conponent (mm) 1.4-4.42 37.70 1.2 1.0 0.8 0.6 0.4 0.2 0.0-30 -20-10 0 10 20 30 40 800 Rotational speed ω(rpm) 700 600 500 400 300 p b -4.41 2ω-Ωp f p f 37.74 2ω-Ωp b 200-30 -25-20 -15-10 -5 0 5 10 15 20 25 30 35 40 Excitation frequaency Ω(Hz) Fig. 5 Analysis of excitation 4 AMB y motor bearing1 crack & sensor bearing2 x 1 spring 2 100mm 400mm 550mm 900mm 3 (a) rotor system Fig. 6 Experimental system (b) magnetic bearing Function generator Harmonic signal(v) DSP Power amp. Current(A) Magnetic bearing & sensor Displacement x,y Fig. 7 control of external force 6

6. COMPARISON WITH THE EXPERIMENT The experimental system shown in Fig.6 is used, and the rotating test of the shaft with an open crack with the depth of 50% was performed in the case of rotating speed ω500 rpm. Here, excitation force F is 5N in x direction only. The shaft displacements at the disk 2 in both x and y directions were measured for each excitation frequency using the displacement sensors. Figure 8 shows resonance curve of 2ω Ω component, and the theoretical result is confirmed by the experiment (the symbols ). Therefore, the validity of the developed FEM rotating shaft model with the open crack were clarified experimentally. Amplitude of (2ω-Ω) conponent (mm) 0.5 0.4 0.3 0.2 0.1 2ω-Ωp f theory experiment 4.42 4.63 4.4 4.7 0.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Excitation frequency (Hz) (a) Results of analysis and experiment ( 2ω Ωp f ) Fig. 8 Amplitude of (2ω-Ω) conponent (mm) 0.10 0.08 0.06 0.04 0.02 2ω-Ωp b theory experiment 37.70 37.92 37.6 37.9 0.00 35.5 36.0 36.5 37.0 37.5 38.0 38.5 39.0 Excitation frequency (Hz) (b) Results of analysis and experiment ( 2ω Ωp b ) Results of analysis and experiment of 2ω Ω component 7. CONCLUSION The analytical method of vibration of a rotating shaft with an open crack under the harmonic excitation is developed by utilizing our concise FEM rotating shaft model for the quantitative analysis, which was developed in the previous studies. Furthermore, the fundamental equations governing the vibration of the cracked shaft with the harmonic excitation are deduced. The experimental system using the active magnetic bearing for the harmonic excitation is developed, and the validity of the obtained theoretical results are confirmed experimentally. 8. ACKNOWLEDGEMENT This work was supported by Grant-in-Aid for Scientific Research C 20560214 from Japan Society for the Promotion of Science (JSPS). REFERENCES LIST (1) J.Wauer, On the dynamics of cracked rotors:a literature survey, Transaction of the American Society Mechanical Engineers, Applied Mechanics Review, vol43,(1990),pp.13-17. (2) R.Gasch, A Survey of Dynamic Behaviour of a Simple Rotating Shaft with a Transverse Crack, Japan of Sound and Vibration, Vol.160, No.2, (1993), pp.313-332. (3) Inagaki,T., Kanki,H., Shiraki,K., 1982, Transverse Vibration of a General Cracked-Rotor Bearing System, Transactions of the American Society Mechanical Engineers, Journal of Mechanical Design, Vol.104, pp.345-355. (4) Mayes,I.W., and Davies,W.G.R., 1984, Analysis of the Response of a Multi-Rotor-Bearing System Containing a Transverse Crack in a Rotor, Transactions of the American Society Mechanical Engineers, Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol.106, pp.139-145. (5) Nelson,H.D., and Nataraj,C., 1986, The Dynamics of a Rotor System with a Cracked Shaft, Transactions of the American Society Mechanical Engineers, Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol.108, pp.189-197. 7

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