Send Orders for Reprints to reprints@benthamscience.ae 150 The Open Mechanica Engineering Journa, 2015, 9, 150-155 Open Access The Mechanica Properties Anaysis of Machine Too Spinde After Remanufacturing Based on Uncertain Constraints Ling Liu *,1,2 1 University of Xi an in Shaan Xi, Xi'an 710065, China 2 Schoo of Mechatronics, Northwestern Poytechnica University in Shanxi Xi'an 710072, China Abstract: In this paper, the CNC machine spinde after remanufacturing is researched as an object on uncertain constraints. At first, the equations of the machine spinde motion based on beam theory are estabished. This artice uses Finite Eement Anaysis (FEA) function to anayze the remanufacturing of machine spinde system in the free mode and whie static and the actua working conditions of muti-moda anaysis of the spinde s constraints state. By anaysis it is known that the spinde vibrates and deforms at high speeds, and some assumptions are used to improve the unreasonabe parameters, so that the spinde s dynamic performance is more stabe and reiabe in the conditions of the high speed and heavy oad operation. In addition, simpifying the cost and shortening the design cyce are the part of the anaysis. The resuts provides an optimized design and a basis for precision contro for the heavy-duty mechanica spinde system or machine spinde system. Keywords: Frequency domain anaysis, parametric modeing, spinde moda, static characteristics.. 1. INTRODUCTION The eectric spinde, magnetic suspension bearing, anguar contact ba bearer, iquid hydrostatic bearing and air bearing represents the deveopmenta trends of the spinde of the high-performance athe in China [1, 2]. Features of the spinde are mainy studied by using the methods such as FEA method, centraized parameter method, transfer matrix method and boundary eement method [3]. Boinger processes the radia anguar contact ba bearing as the radia spring and damper and estabishes the beam mode of the spinde of the athe [4]. KOSMATKA estabishes Timoshenko beam mode based on the Hamiton principe and anayzes the infuence of the axia force on stabiity of the beam [5]. Jorgensen and Shin estabish the bearing oaddistortion mode and studies infuence of the therma distortion on spinde rigidity and inherent frequency. CAO introduces centrifuga force, gyroscopic effect, pre-tightening force and rotor offset to Timoshenko beam mode and ays foundation for further research on the spinde of the athe [6-8]. In addition, experts and schoars have achieved some achievements in research on the spinde of the athe in China. Guan Xiyou summarizes the dynamic research methods on the spinde of the athe [9]. Xu Linfeng anayzes distortion of the spinde and guide of the athe [10] and deduces force features of the spinde and guide. Zhang Yaomai simpifies the bearing support as spring dampness unit and estabishes the dynamic finite eement mode of the spinde [11]. Yu Tianbiao anayzes high-speed spinde system of the grinding machine supported by the iquid hydrostatic bearing and studies the reation between the rotation rate of the spinde, system rigidity and inherence frequency [12]. Many unknown factors affect processing quaity in actua processing, e.g. casting quaity of semi-finished products and bearing ife. Based on the actua production conditions, this paper mainy anayzes the infuences of the bearing constraint position on the modes of the spinde. This paper first considers cutting infuences, estabishes the rigidity matrix of the spinde, and anayzes the vibration mode of the spinde by using the finite eement anaysis software. The anaysis resuts are vaidated via the hammer tria to prove correctness of the anaysis concusions. The research concusions can provide theoretica reference for the design of the spinde system of the athe. 2. ANALYSIS ON MILLING CUTTER FORCE The theoretica computing equation of the cutting force of the mier is described as: F t = cos(β γ ) = τ s ε n a p a w [ctgϕ + tg(ϕ + β γ )] (1) β friction ange; γ anterior ange; ϕ cutting ange; τ s cutting strength; ε distortion factor; a p cutting depth; a w cutting width; From the cutting force computing mode in the Fig. (1), the cutting force mainy incudes circumferentia cutting force F t, radia cutting force F r and radia cutting force F a. The circumferentia cutting force and radia cutting force wi make the spinde rotate. The axia cutting force wi make the spinde generate the bending moment [13]. 1874-155X/15 2015 Bentham Open
The Mechanica Properties Anaysis of Machine Too Spinde The Open Mechanica Engineering Journa, 2015, Voume 9 151 3. ESTABLISH RIGIDITY MODEL OF SPINDLE 3.1. Mode Simpification The spinde studied in this paper can be simpified as the structure in the Fig. (2). Based on the existing design parameters, it is known that D 1 =32 mm, D 2 =132 mm, D 3 =140 mm, D 4 =65 mm, L 1 =50 mm, L 2 =100 mm, L 3 =213 mm and L 4 =453 m. The spinde is made of 38CrMoAA. The main parameters of this materia are described as foows: moduus of easticity is 2.1e5 Mpa, yied strength is 835 MPa, Poisson ratio is 0.3 and shear strength is 765 MPa. Generay the bearing is fuy constrained in the traditiona anaysis on the spinde. This method ignores the infuence of the eastic support on the inherent frequency of the spinde. It is difficut for the existing FEA method to combine the bearing force and spinde vibration. The bearing is simpified as the spring unit with the axia and radia rigidity in anaysis. The spring rigidity can be seected by the type and size of the bearing, as shown in the Fig. (4). Based on the separate force anaysis of the bearing, the forces on the spring unit are different in four directions during rotation of the spinde and the force reduces from down to up, as shown in the Fig. (3). X The rotation ange θ is the derivative of the defection in the equation (3). Based on the geometric equation, we can get: = u x + u y = du θ (4) dx Stress component is: = u x = du dx y dθ dx The stress σ x inside the axis is: σ x = E (6) The stress component is =700/211 3.3. In addition, the shear stress τ xy =G shoud be considered. The factor =185/828 0.22. (5) Fn Ft Y Fig. (3). Whoe radia shift coud diagram of bearing. Fr Z P Fig. (1). Anaysis and computing of miing force. D3 D2 D1 D4 L1 L2 L4 L3 Fig. (4). Equivaent bearing distribution of spring. Fig. (2). Simpification mode of spinde. 3.2. Estabish Basic Mode For the shear axis, assuming the transverse fiber without extrusion, we can get ε y = 0, u = u = f ( x) (2) The radia shift at any point inside the axis is described as: u = u θy = f u x ( ) θ (3) 3.3. Infuences of Shear Based on the above anaysis, when the infuences of the transverse shear distortion are not considered, equation (5) shoud be changed as: ε = { } = u = du x dx y d 2 u dx 2 = B b a b (7) The stress matrix is: [ N 6 ] (8) B b = N 1 y N 3 y N 4 N 2 y N 5 y
152 The Open Mechanica Engineering Journa, 2015, Voume 9 Ling Liu = B b a b = [ B 1 B 2 B 3 B 4 ]a b, wherein (i=1,2,3, ) (9) b N i 0 ξ B i = 1 0 a N bξ i ab η = 1 i ( 1+ η o ) 0 0 aη i ( 1+ ξ 0 ) (10) a N i η b N 4ab aη i ( 1+ ξ 0 ) bξ i ( 1+ η 0 ) i ξ Assuming that the transverse shear distortion ony affects the defection, the additiona shift fied and additiona axis defection can be expressed as foows: u s = 0 υ s,a = s υ s1 υ s2 υ s can be expressed as: (11) υ s = N 1 υ s1 + N 2 υ s2 (12) To introduce the shift fied into the pane geometric equation matrix, ony the shear stress shown as the equation (13) is generated: = N 1 ε = [ N 2 ] υ s1 υ s2 Stress of axia unit is: = B ba b B s a s Unit stress is: E 0 σ = Dε = 0 G k = B sa s (13) (14) (15) G is the shear moduus in the equation (15). k is the coefficient for non-uniform distribution of the shear stress. By referring to reated information, k=10/9 for the round section. 3.4. Unit Rigidity Matrix For the unit body, to use the virtua shift equation, we can get: Ω ε δε T σ dω = δ a e F e (16) α e and F e are the shift vector and force vector of the unit node in the equation (16) when bending and shearing are considered and α e can be expressed as the equation (17). A is the sparse matrix, shown as the equation (18): a e = a b + Aa s (17) A = 0 1 0 0 0 0 0 0 0 0 1 0 T (18) To substitute the equation (14), (15) and (17) into the equation (15), we can get: δ a et ( K b a e K b Aa s ) + (19) δ a T s (K s a s + A T K b Aa s A T K b a e ) = δ a et F e where is K b = Ω e EB T b B b dω, K s = Ω e ( G k)b T s B s dω Based on the baance reation in the equation (20): Q = GA k γ = GA du s xy k dx = GA k we can get: Q = dm dx = d dx EI d 2 u dx 2 = EI d 3 u b dx 3 = 6EI 3 ( ), (20) u s2 u s1 (21) 2( u b2 u b1 ) ( θ 1 +θ 2 ) b=12k*e*i/(g*a* 2 ), E is the eastic moduus and E=2.11e. I is the sectiona inertia moment, G is the shearing moment and G=8.28e. A is the section area. A = 1 2πr 2 = 0.5 3.14 64 2 = 6430mm 2. To substitute vaues, we can get: ( ) b = 12 10 9 211 0.05 652 32 2 = 0.03 (22) 828 0.5 π 64 2 373 2 u 2 -u 1 =u b2 -u b1 +u s2 -u s1, so we can get: u s2 u s1 = b 1+ b u 2 u 1 ( ) b ( 2( 1+ b) θ 1 +θ 2 ) (23) If the transverse shear distortion is considered, the unit rigidity matrix is: EA 0 0 EA 0 0 12EI 6EI 0 0 12EI 6EI ( 1+ b) 3 ( 1+ b) 2 ( 1+ b) 3 ( 1+ b) 2 6EI ( 4 + b)ei 6EI ( 2 b)ei 0 0 ( 1+ b) 2 ( 1+ b) ( 1+ b) 2 ( 1+ b) K e = EA EA 0 0 0 0 0 12EI 6EI 12EI 6EI 0 ( 1+ b) 3 ( 1+ b) 2 ( 1+ b) 3 ( 1+ b) 2 6EI ( 2 b)ei 6EI ( 4 + b)ei 0 0 ( 1+ b) 2 ( 1+ b) ( 1+ b) 2 ( 1+ b) After computing, we can get: 0.4 0 0 0.4 0 0 0 0.2 1 0 0.2 1 0 1 0.2 0 1 0.6 K e = 0.4 0 0 0.4 0 0 0 0.2 1 0 0.2 1 0 0.2 0.3 0 1 0.6 (24) (25) The whoe rigidity matrix of the axis unit is computed as foows:
The Mechanica Properties Anaysis of Machine Too Spinde The Open Mechanica Engineering Journa, 2015, Voume 9 153 1.4 0 1 0.2 EA 1.92 0 1.4 0.4 0.4 1 0.4 1.8 0.6 0.2 0.4 0.62 0.4 u 1 v 1 u 2 v 2 1 0 = σ 0 0 The whoe node shift of the unknown variant is (26) (a) first-order frequency a = [ u 1 v 1 u 2 v 2 ] T = σ [ 0.000539 0.0005292 0 0 ] T. By comparing the resuts of the soution of the variants obtained by using this mode with resuts of the FEA, the resuts are found to be simiar, so the scae can be used as the basis for preiminary design. (b) second-order frequency 4. MODE ANALYSIS Based on the Timoshenko beam mode mq + k q = 0, m is the mass matrix and k is the rigidity matrix. By substituting the easy-to-know mass matrix and rigidity matrix estabished in the above section into the Timoshenko beam mode, the vibration frequency of the spinde of the athe can be obtained [14]. The anaysis of the vibration of the simpified spinde mode under free state was performed by using an FEA software. The first-order and second-order vibration is bending and third-order and fourth-order vibration is torsion, as shown in Fig. (5). The so-caed free state indicates to the constrain freedom at one end of the spinde in five directions. Another end is the open end. The anaysis resuts indicate that the third-order and fourth-order torsion ampitude is arger and is about 15 times of first-order and second-order torsion ampitude. The third-order and fourthorder vibration frequency is about 4 times of the first-order and second-order vibration frequency, so the spinde of the athe cannot meet the usage requirement under free state. To constrain the bearing at another end, generay the anguar contact ba bearing is used as the spinde of the athe. Ony rotation in Y and Z axis is restricted in the simuation anaysis. The anaysis resuts indicate that the vibrationa ampitude of the spinde wi reduce after the bearing is constrained, as shown in Fig. (6). The bearing constraint is ocated at 30 mm inward at the constraint section and 100 mm inward at the free end. Based on the anaysis of the constrained spinde, first-order and secondorder vibration ampitude is simiar to 3-order and fourthorder vibration ampitude and reduces itte compared to without the constraint, but the third-order and fourth-order vibration frequency speeds up. The increased frequency is very obvious. It indicates that the od vibration frequency of the spinde is increased after the bearing constraint is imposed, so the probabiity of sef-excitation and forced vibration wi reduce consideraby. (c) Third-order frequency (d) Fourth-order frequency 5. EXPERIMENT The dynamic features of the spinde are mainy studied by the hammering experiment method. The factor speed is not considered in our research, so the spinde can be fixed at the bearing constraint and one part of the spinde is Fig. (5). Vibration frequency of spinde under free state. hammered by using an iron hammer. The vibration frequency of the spinde is detected by using an advanced test instrument, as shown in Fig. (7).
154 The Open Mechanica Engineering Journa, 2015, Voume 9 Ling Liu (a) First-order frequency This experiment is mainy performed to detect the frequency of the spinde under fourth-order simuated vibration. The simuation data are simiar to the experimenta data, which indicates that the stabished mode and moda anaysis method are correct, but about 8% difference exist in the simuation data and experimenta data due to the simpified mode and simpe test means. This error is within the acceptabe range, as shown in Fig. (8). (b) Second-order frequency (c) Third-order frequency Fig. (8). Comparison of spinde frequency. CONCLUSION (d) Fourth-order frequency (1) This paper estabishes the rigidity matrix equation of the spinde and provides the basis for manua computing of the spinde frequency; (2) This paper anayzes the moduus of the spinde by using the FEA method, embodies the roe of the bearing constraint intuitivey, and further studies the infuence factors of the spinde frequency. (3) The simuation mode is vaidated via the hammering experiment to prove vaidity of the mode and provide references for design in future. CONFLICT OF INTEREST Fig. (6). Vibration frequency of spinde under bearing constraint state. The author confirms that this artice content has no confict of interest. Fig. (7). LMS vibration test instrument.
The Mechanica Properties Anaysis of Machine Too Spinde The Open Mechanica Engineering Journa, 2015, Voume 9 ACKNOWLEDGEMENTS [7] Thanks to the strong support of the Science and Technoogy Department of Shaanxi Province Project No. 2014K06-43! Thanks to the strong support of the Xi'an Science and Technoogy Project No. CXY1443WL24! [8] [9] REFERENCES [1] [2] [3] [4] [5] [6] Q. He, Research on cutting sef-exited vibration contro based on TK401 digita controed athe, Mechanica and Eectric Engineering, vo. 24, no. 3, pp. 55-56, 2007. W. Xiong, Anaysis on current technica conditions of spinde of high-performance athe in China, Meta Processing (Cod Processing), no. 18, pp. 6-11, 2011. Avaiabe from: www.metaworking1950.com S. Gao, and G. Meng, Research progress on dynamic features of athe s spinde system, Vibration and Impact, vo. 26, no. 6, pp. 103-109, 2007. S.J. Zhang, and S. To, The effects of spinde vibration on surface generation in utra-precision raster miing internationa, Journa of Machine Toos & Manufacture, no. 71, pp. 52-56, 2013. L. Yan, Research on on-site measurement method and key technoogy of high-precision arge bearing parts, Doctora Dissertation of Shanghai University, Apri, 2008. U. V. Kumara, and T. L. Schmitzb, Spinde dynamics identification for Receptance Couping Substructure Anaysis, Precision Engineering, no. 36, pp. 435-443, 2012. Received: January 8, 2015 [10] [11] [12] [13] [14] Revised: January 15, 2015 155 Z. Cui, Optimization techniques and appications on the highspeed grinder design, Ph.d thesis, Hunan University, China, November 2010. Y. Ma, The energy characteristic state modes for mechanica system conceptua design, Ph.d thesis, Daian University of Technoogy, Septmber 2011. X. Guan, and W. Sun, Research on dynamic feature anaysis method of spinde system of numerica contro machine, Combined Lathe and Automated Processing Technoogy, vo. 56, no. 4, pp. 1-5, 2010. L. Wu, F. Guo, and L. Shang, Anaysis on distortion of athe s spinde and guide, Journa of North China University of Water Resources and Eectric Power, vo. 27, no. 4, pp. 52-54, 2006. Y. Zhang, C. Lui, and Z. Xie, Einite eement anaysis of spinde component of high-speed athe, Journa of Northeastern University (Natura Science Version), vo. 29, no. 10, pp. 14741477, 2008. T. Yu, X. Wang, and P. Guan, Moduus anaysis of spinde system of utra-high-speed grinding and cutting athe, Journa of Mechanica Engineering, vo. 48, no. 17, pp. 183-188, 2012. Y. Wang, Research on Optima Design of Lathe s Spinde, Master Dissertation of University of Eectronic Science and Technoogy of China, 2009. L. Can, S. Ma, and Y. Zhao, Finite eement modeing and moduus anaysis of spinde of heavy machine under muti-constraint state, Journa of Mechanica Engineering, vo. 48, no. 3, pp. 165-172, 2012. Accepted: January 16, 2015 Ling Liu; Licensee Bentham Open. This is an open access artice icensed under the terms of the Creative Commons Attribution Non-Commercia License (http://creativecommons.org/icenses/by-nc/3.0/) which permits unrestricted, non-commercia use, distribution and reproduction in any medium, provided the work is propery cited.