Frequency response analysis of the gear box in a lathe machine using transfer functions

Transcription

1 International Journal of Scientific and Research Publications, Volume 4, Issue 11, November Frequency response analysis of the gear box in a lathe machine using transfer functions V Suma Priya 1a, B V S Raghu Vamsi 2b*, E Kavitha 1c, K. Srividya 1d 1a PG Student, Dept. of Mechanical Engineering, PVP Siddhartha Institute of Technology, Vijayawada 2b* Assistant Professor, Dept. of Mechanical Engineering, Gudlavalleru Engineering College, Gudlavalleru 1c Assistant Professor, Dept. of Mechanical Engineering, PVP Siddhartha Institute of Technology, Vijayawada 1d Assistant Professor, Dept. of Mechanical Engineering, PVP Siddhartha Institute of Technology, Vijayawada Abstract- In this research work, frequency response of the gear box in the medium duty lathe machine is studied using transfer functions. The effect of the torque acting at a particular rotor on the amplitude of vibration of the other rotors is studied. Initially, equation of motion is developed for the multi-rotor system in the gearbox and later, Laplace transforms are applied to find the transfer functions. Torque acting at various rotors is also calculated. The obtained characteristic equation and the transfer functions are solved for poles and eros (frequencies to attenuate the inputs at every rotor) by writing programming in MATLAB. Plots of the frequency response curves are plotted by writing the programming in MATLAB and the final conclusions are drawn. Index Terms- Frequency response curves, Laplace transforms, MATLAB, Poles, and Zeros Symbols and notations Den denominator of the transfer function G shear modulus of the shaft J mass moment of inertia Jeq equivalent mass moment of inertia for the mating gears Kt torsional stiffness of the shaft l length of the shaft N - speed of the shaft - torque acting at the rotor Z(s) Laplace transform Z11 (num) numerator of the transfer function (similar notation for the other functions) Angular deflection - angular acceleration T I. INTRODUCTION he study of Torsional vibrations in a multi-rotor system plays an important role while designing the power transmission systems in the machines and internal combustion engines. The torque acting at a particular rotor will make it to vibrate with large amplitudes and the effect of the same torque on the vibration characteristics of the other rotors in the transmission system is also significant. When the forcing frequency becomes equal to the natural frequency of any of the rotors, a state of resonance will occur. Apart from the torque acting on the rotors, the forces developed in the mating of two gears and forces during the operation of the corresponding machine also amplifies the magnitude of vibration to a much higher value. Hence, there is a need to study the effect of these torques and additional forces on the vibration characteristics of the entire power transmission system. In this research work, the effect of torques acting at the different rotors on the other rotors is studied. Guo Rui et. al [1] developed a 10-DOF lumped parameter model for the machine tool spindle system with geared transmission, for the purpose of analying the torsional vibrations caused by the gravitational torque arisen in a spindle system when machining a heavy work piece. By using the elementary method and Runge-kutte method in MATLAB, the eigen values problem was solved and the pure torsional vibration responses were obtained and examined. Wu Hao et. al [2] established a numerical model of the bending stiffness of the tapered roller bearing through mechanics and deformation analysis. On the base of the model, a new TMM (transfer matrix method) for bearing-rotor system was established; the new TMM considers the influences of the bearing structure on the vibration of the rotor system. The method is validated by analying the same problem (modal analysis of air-blower) using FEM. B.B. Maharathi et. al [3] presented a general formulation for the problem of the steady-state unbalance response of a dual rotor system with a flexible intershaft bearing using an extended transfer matrix method, where the transfer matrix assumes a dimension of (33x33) and the formulation is validated through a computer program. M. Aleyaasin et. al [4] considered the distributed-lumped model for the analysis of the flexural vibrations of a rotor-bearing system and derived a general formula for the determinant of the tri-diagonal partitioned matrix description of the system. The obtained results are compared to those acquired from the transfer matrix method. HU Qinghua et. al [5] developed a five degrees of freedom (5-DOF) model for aeroengine spindle dual-rotor system dynamic analysis. The proposed model mathematically formulates the nonlinear displacements, elastic deflections and contact forces of bearings with consideration of 5-DOF and coupling of dual rotors. The nonlinear equations of motions of dual rotors with 5-DOF are solved using Runge-Kutta-Fehlberg algorithm. Vishwajeet kushwaha and Prof. N.kavi [6] used finite element methods to find the natural frequencies for different possible cases of multi-rotor and gear-branched systems. The various mode shapes for several cases are also shown to illustrate the state of the system at natural frequencies. The results obtained have been compared with Holer s method and transfer matrix method to establish the effectiveness of finite element method for such systems.

2 International Journal of Scientific and Research Publications, Volume 4, Issue 11, November In section-i, literature review is presented. In section-ii, arrangement of gears in the gear box with the calculation of stiffness of the shaft and mass moment of inertia of the rotors is discusses. In section-iii, mathematical modeling of the rotor system is presented. In section-iv, the procedure of the transfer function analysis, involving the method to derive transfer functions is presented in detail. In section-v, the results are discussed in detail with the conclusion in section-vii.

3 International Journal of Scientific and Research Publications 3 II. GEAR BOX ARRANGEMENT IN THE LATHE MACHINE In this work, the gear box in a medium duty lathe machine is considered for the analysis. The frequency response of every rotor to the torque acting at the self as well as other rotors is studied, to understand the vibration characteristics of a gear box much better. The estimated details of the components in the gear box i.e. the stiffness of the shafts and moment of inertia of the rotors (gears) are presented in the table 1. The arrangement of the various rotors in the gear box adopted in this study is presented in the fig. 1. Stiffness of the shaft is calculated using the torsion equation: G t GJ J l R l (1) III. MATHEMATICAL MODELING OF MULTI ROTOR SYSTEM The spring-mass representation of the gear box is presented in fig. 2. Wherever two gears are mating, the equivalent mass moment of inertia is calculated and it is considered to be as an equivalent rotor. The equation of motion can be derived by Newton s law as:.. J K t (2) The equation of motion derived for the entire gear box is presented in as eq. (3) These equations are written in matrix form for simplicity:.. J K t Where J inertia matrix k stiffness matrix Torque matrix Angular Acceleration matrix: t T Angular displacement matrix: T (4)

4 International Journal of Scientific and Research Publications 4 J.. k eq J k k eq J k k eq J k k k Z k Z J J J eq Z6 Z6.. k eq k eq k eq Z 4 6 k Z6 J k Z k.. 4 eq Z6 J k k k Z k Z J J J J eq Z12 Z12.. eq eq eq12 k k.. Z k k Z eq13 13 k Z12 Z (3) Torque matrix: Inertia matrix: Jeq J eq 0 0 J eq Jeq J eq Jeq J J eq Jeq Jeq Jeq Jeq Jeq Jeq13 Stiffness matrix: T

5 International Journal of Scientific and Research Publications 5 K t K12 K K12 K12 K23 K K23 K23 K34 K K34 K K34 Z K45 0 K47 K K 47 K 48 Z K45 K K67 K Z K47 0 K67 K47 K Z6 Z K K48 K89 K Z6 K89 K K Z K 0 K K 10 K912 K913 Z K910 K K1112 K Z K K1112 K1112 K912 0 Z12 Z K K Z IV. TRANSFER FUNCTION ANALYSIS PROCEDURE These are the functions, which helps to study the effect of torque T i acting at the rotor R j. The procedure to find the transfer functions is given in detail below: Initially derive the expression for equation of motion for the each rotor. Apply Laplace transform to them and convert them into s-form. Find the transfer functions and characteristic equation for the each rotor, corresponding to each torque. Write the program to plot the frequency response curves in MATLAB. Step-1 presented in the above section i.e. eq. (3). Step-2: Laplace transforms: Applying Laplace transforms to the eq. (4), we get 2 Js K s s (5)

6 International Journal of Scientific and Research Publications 6 Where, 2 Jeq 1S K12 K K12 Jeq2S K12 K23 K K23 Jeq3S K23 K34 K Jeq4S K34 K K34 Z K K47 K K 47 K 48 Z K45 Jeq5S K Jeq6S K67 K Z K47 0 K67 Jeq7S K47 K Z6 2 JS K Z K Jeq8S K48 K89 K Z 6 2 Jeq9S K89 K K89 Z K K912 K913 K912 K 913 Z K910 Jeq10S K Jeq 11S K1112 K Z K912 0 K1112 Jeq12S K1112 K912 0 Z12 Z K Jeq 13S K 913 Z 12 T s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 s 10 s 11 s 12 s 13 s Z-matrix is given as T s Torque matrix is given as s s s s s s s s s s s s s

7 International Journal of Scientific and Research Publications 7 Step-3: Deriving for transfer functions. 2 JS K From matrix, we will get transfer functions by applying below procedure and transfer functions are arranged in a matrix as Transfer function matrix shown below. Z 11 represents the effect of torque τ 1 acting at rotor 1 on the rotor 1 itself. Similarly, Z 119 represents the effect of torque τ 9 acting at rotor 9 on the rotor Let us consider det Z11 det 1313 det 1212 = By elimination of 1 st row & 1 st column of JS 2 K matrix. Z 12 det 1212 det 1313 det 1212 = By elimination of 1 st row & 2 nd column of JS 2 K matrix. Similar procedure is followed for the remaining transfer functions. det 1212 = denominator = den det 1313 = numerator = num Sample calculations for transfer functions: 1 11num Z num Z 1 den 11 s den s Z num Z num e s e s e s e s s e s e s 6.73e s 3.26e s 4.610e s 2.24e s 2.32e s 7.43e

8 International Journal of Scientific and Research Publications 8 Den 5.02e s 2.48e s 1.12e s 1.910e s 0.02s 7.06e s 1.53e s e s e s e s e s e s e s e s Step-4: Plotting Frequency response curves Once the transfer functions are derived, then the programming is written to plot the frequency domain behavior of each transfer function, using MATLAB. Frequency domain behavior means identifying the magnitude and phase characteristics of each transfer function, showing how they change as the frequency of the forcing function is varied over a frequency range. Each transfer function is evaluated in the frequency domain by evaluating it as s j, where w is the frequency of the forcing function, radians /sec. For plotting the frequency response, different methods are available. In this work, transfer functions are given as input, in the form of num and den for the each and bode command with no left hand arguments is used by choosing the frequency range and the graphs plotting magnitude and phase generating automatically. On x-axis, range of frequencies is taken and on the y-axis, magnitude of vibration and phase angles are plotted. The obtained response curves are tabulated below with the titles as their transfer functions. V. ANALYSIS OF FREQUENCY RESPONSE CURVES From the results it has been observed that The peak points on the response curves indicate the poles and the valley points on the curves indicates the eros. The poles are the roots of the characteristic equation. They show the frequencies where the system will amplify the inputs. All the transfer functions will have same characteristic equation i.e. denominator ( den ). The poles depend only on the distribution of mass and stiffness throughout the system under analysis. But not on where the toques are applied or where the displacements are measured. The eros are the roots of the numerator of a given transfer function. Zeros show the frequencies where the system will attenuate inputs. They are different for different transfer functions. Some transfer functions may have no eros. Z ij i j In the low frequency range (< 1000 rad / sec), the response of all transfer functions,i=j=1, 2, 3.13, the response curve profiles are similar and presents higher gain. This is because of all the 1 rotors acted upon by the torque directly and the rigid body motion is falling off at the rate. In that operating range, all the rotors are running out of phase with the applied torque. Z ij i j In the high frequency range (> 1000 rad / sec), the response of all transfer functions, i=j=1, 2, 3.13, the response curve profiles are similar and presents lesser gain. This is because of all the rotors acted upon by the torque directly and the rigid body motion is observed to be falling off at the 1 3 rate. In that operating range, all the rotors are running out of phase with the applied torque. R j 1 When the effect of the torques acting on the rotors, on the rotor R 1, is considered, all the response curves are looking alike for the transfer functions, At the higher frequencies, the rigid body motion is falling off at the rate with different gains. In that operating range, the rotor R 1 is running in phase with some torques and out of phase with remaining torques. At the lower frequencies, the effect of all the torques (except 1 ) is 1 similar and the rigid motion is falling off at the rate with same gains. R j, on the rotor R2, is considered, all the response curves are looking alike for the transfer functions, and for the 2 j, When the effect of the torques acting on the rotors transfer functions, At the higher frequencies, the rigid body motion is falling

9 International Journal of Scientific and Research Publications off at the 9 15 rate and at the 5 9 rate with lesser but different gains. In that operating range, the rotor R 2 is running in phase with some torques and out of phase with remaining torques. At the lower frequencies, the effect of all the torques (except 2 ) is similar 1 and the rigid motion is falling off at the rate with same gains. R j, on the rotor R3, is considered, 3 j, When the effect of the torques acting on the rotors all the response curves are looking alike for the transfer functions , 37 and for and for. At the higher frequencies, the rigid body motion is falling off at 1 1 the rate and with the rate with lesser but different gains. In that operating range, the rotor R 3 is running in phase with some torques and out of phase with remaining torques. These many variations are because of mating of two gears, which develops forces affecting the vibrating characteristics. This increases the amplitude of vibration also. At the lower frequencies, the effect of all the torques (except 3 ) is similar and the rigid motion is 1 falling off at the rate with same gains. 4 j, When the effect of the torques acting on the rotors R j, on the rotor R4, is considered, all the response curves are looking alike for the transfer functions , 44 and for 45 48,, and for , and for.the magnitude of vibration is high for the transfer functions 43 44, which is because of the mating of gears resulting in the increasing of forces and amplitudes. At the higher frequencies, the rigid body motion is falling 1 1 off at the 5 9 rate with lesser but different gains. In that operating range, the rotor R 4 is running in phase with some torques and out of phase with remaining torques. At the 3, 4 lower frequencies, the effect of all the torques (except ) is similar and the rigid motion 1 is falling off at the rate with same gains. R j, on the rotor R5, is considered, 5 j, When the effect of the torques acting on the rotors all the response curves are looking alike for the transfer functions and for , and for At the higher frequencies, the rigid body motion is falling off at the rate with lesser but different gains. In that operating range, the rotor R 5 is running in phase with some torques and out of phase with remaining torques. At lower frequencies, the effect of all the torques (except 5 ) is similar and the rigid motion is falling 1 off at the rate with same gains. R j, on the rotor R6, is considered, 6 j, When the effect of the torques acting on the rotors all the response curves are looking alike for the transfer functions and for , and for At the higher frequencies, the rigid body motion is falling off at the rate with lesser but different gains. In that operating range, the rotor R 6 is

10 International Journal of Scientific and Research Publications 10 running in phase with some torques and out of phase with remaining torques. At lower frequencies, the effect of all the torques (except 6 ) is similar and the rigid motion is falling 1 off at the rate with same gains. R j, on the rotor R7, is considered, 7 j, When the effect of the torques acting on the rotors all the response curves are looking alike for the transfer functions 71 73,, and for , and for and for At the higher frequencies, the rigid body motion is 1 1 falling off at the 5 11 rate with lesser but different gains. In that operating range, the rotor R 7 is running in phase with some torques and out of phase with remaining torques. At lower frequencies, the effect of all the torques (except 7 ) is similar and the rigid motion is 1 falling off at the rate with same gains. R j, on the rotor R8, is considered, 8 j, When the effect of the torques acting on the rotors all the response curves are looking alike for the transfer functions and for and for At the higher frequencies, the rigid body motion is falling 1 1 off at the 5 11 rate with lesser but different gains. In that operating range, the rotor R 8 is running in phase with all the torques. At lower frequencies, the effect of all the torques (except 8 ) is similar and the rigid motion is falling off at the 1 rate with same gains. Due to the generation of forces developed during mating, at rotor R 9, it has a significant effect in amplifying the amplitudes at rotors R 8, R 9, R 12, and R 13. R j, on the rotor R9, is considered, 9 j, When the effect of the torques acting on the rotors all the response curves are looking alike for the transfer functions and for and for At the higher frequencies, the rigid body motion is falling 1 1 off at the 5 13 different rates and with lesser but different gains. In that operating range, the rotor R 9 is running in phase with all the torques. At lower frequencies, the effect of all the torques (except 9 ) is similar and the rigid motion is falling off at the 1 rate with same gains. Due to the generation of forces developed during mating, at rotor R 9, it has a significant effect in amplifying the amplitudes at rotors R 8, R 9, R 12, and R 13. R j j, on the rotor R 10, is considered, When the effect of the torques acting on the rotors, 10 all the response curves are looking alike for the transfer functions and for and for At the higher frequencies, the rigid body motion is 1 1 falling off at the 7 13 different rates and with lesser but different gains. In that operating range, the rotor R 10 is running in phase with all the torques. At lower frequencies, the effect of all the torques (except 10 ) is similar and the rigid motion is falling off at the 1 rate with same gains.

11 International Journal of Scientific and Research Publications 11 R j 11 j, When the effect of the torques acting on the rotors, on the rotor R 11, is considered, all the response curves are looking alike for the transfer functions for At the higher frequencies, the rigid body motion is falling off at the rate and at the 5 9 rate with different rates respectively and with lesser but different gains. In that operating range, the rotor R 11 is running in phase with some of the torques and out of phase with the remaining torques. At lower frequencies, the effect of all the torques (except 11 ) is similar and the rigid motion is falling off at the 1 rate with same gains. R j 12 j, When the effect of the torques acting on the rotors and, on the rotor R 12, is considered, all the response curves are looking alike for the transfer functions for and for At the higher frequencies, the rigid body motion is falling off at the 7 11 and at the 5 7 different rates respectively and with lesser but different gains. In that operating range, the rotor R 12 is running in phase with the torques and out of phase with the remaining torques. At lower frequencies, the effect of all the torques (except 12 ) is similar and the rigid motion is falling off at the 1 rate with same gains. R j j, on the rotor R 13, is considered, When the effect of the torques acting on the rotors, 13 all the response curves are looking alike for the transfer functions for and for At the higher frequencies, the rigid body motion is falling off at the 9 15 and at the 5 7 different rates respectively and with lesser but different gains. In that operating range, the rotor R 13 is running in out of phase with the torques and in phase with the remaining torques. At lower frequencies, the effect of all the torques (except 13 ) is similar and the rigid motion is falling off at the 1 rate with same gains. and and VI. CONCLUSION This study reveals that there is a need to study the effect of torques acting at a particular rotor on another rotor and the affect is significant. At lower forcing frequencies, all the rotors are exhibiting same behavior irrespective of the magnitude of torques acting at different rotors and the rate of fall of rigid body motion is 1. At higher forcing frequencies, rotors are exhibiting different behaviors with the fall in rigid body motion at the rate. The gain in amplitude is also varying from rotor to rotor and the effect of forces generated in the mating of gears is also a reason for that. REFERENCES [1] Guo Rui, Jang Sung-Hyun, Choi Young-Hyu, Torsional vibration analysis of lathe spindle system with unbalanced workpiece, J. Cent. South Univ. Technol. (2011) 18: , Springerlink

12 International Journal of Scientific and Research Publications 12 [2] Wu Hao, Zhou Qiong1, Zhang Zhiming and An Qi, Vibration analysis on the rolling element bearing-rotor system of an air blower, Journal of Mechanical Science and Technology 26 (3) (2012) 653~659, Springerlink [3] B.B. Maharathi, P.R. Dash, A.K. Behera, Dynamic Behaviour Analysis of a Dual-Rotor System Using the Transfer Matrix Method, International Journal of Acoustics and Vibration, Vol. 9, No. 3, [4] M. Aleyaasin, M. Ebrahimi, R. Whalley, Vibration analysis of distributed-lumped rotor systems, Comput. Methods Appl. Mech. Engrg. 189 (2000) 545±558, Elsevier [5] Hu Qinghuaa, Deng Sierb, Teng Hongfeia, A 5-DOF Model for Aeroengine Spindle Dual-rotor System Analysis, Chinese Journal of Aeronautics 24 (2011) , Elsevier [6] Vishwajeet Kushwaha, Prof. N.Kavi, Professor, Analysis of torsional vibration characteristics for multi-rotor and gear-branched systems using finite element method, A Thesis, National Institute of Technology, Rourkela, [7] Dr. Rajiv Tiwari, A short term course on Theory and Practice of Rotor Dynamics, Department of Mechanical Engineering, Indian Institute of Technology Guwahati, (15-19 Dec 2008) S.No Shaft Table. 1 Stiffness of the shaft K, in N-m/rad S.No Moment of inertia, J in kg-m² 1 S J₀ = S J₁ = S J₂ = S J₃ = S J₄ = S J₅ = S J₆ = S J₇ = S J₈ = S J₉ = S J₁₀ = S J₁₁ = J₁₂ = J₁₃ = J₁₄ = J₁₅ =

13 International Journal of Scientific and Research Publications 13 Fig 1: Layout of gear box in all geared lathe Fig 2. Spring-mass representation of the gear box

14 International Journal of Scientific and Research Publications 14 Frequency response curves for various rotors corresponding to different transfer functions Z11 Z12 Z13 Z14 Z15 Z16

15 International Journal of Scientific and Research Publications 15 Z17 Z18 Z19 Z110 Z111 Z112

16 International Journal of Scientific and Research Publications 16 Z113 Z21 Z22 Z23 Z24 Z25

17 International Journal of Scientific and Research Publications 17 Z26 Z27 Z28 Z29 Z210 Z211

18 International Journal of Scientific and Research Publications 18 Z212 Z213 Z31 Z32 Z33 Z34

19 International Journal of Scientific and Research Publications 19 Z35 Z36 Z37 Z38 Z39 Z310

20 International Journal of Scientific and Research Publications 20 Z311 Z312 Z313 Z41 Z42 Z43

21 International Journal of Scientific and Research Publications 21 Z44 Z45 Z46 Z47 Z48 Z49

22 International Journal of Scientific and Research Publications 22 Z410 Z411 Z412 Z413 Z51 Z52

23 International Journal of Scientific and Research Publications 23 Z53 Z54 Z55 Z56 Z57 Z58

24 International Journal of Scientific and Research Publications 24 Z59 Z510 Z511 Z512 Z513 Z61

25 International Journal of Scientific and Research Publications 25 Z62 Z63 Z64 Z65 Z66 Z67

26 International Journal of Scientific and Research Publications 26 Z68 Z69 Z610 Z611 Z612 Z613

27 International Journal of Scientific and Research Publications 27 Z71 Z72 Z73 Z74 Z75 Z76

28 International Journal of Scientific and Research Publications 28 Z77 Z78 Z79 Z710 Z711 Z712

29 International Journal of Scientific and Research Publications 29 Z713 Z81 Z82 Z83 Z84 Z85

30 International Journal of Scientific and Research Publications 30 Z86 Z87 Z88 Z89 Z810 Z811

31 International Journal of Scientific and Research Publications 31 Z812 Z813 Z91 Z92 Z93 Z94

32 International Journal of Scientific and Research Publications 32 Z95 Z96 Z97 Z98 Z99 Z910

33 International Journal of Scientific and Research Publications 33 Z911 Z912 Z913 Z101 Z102 Z103

34 International Journal of Scientific and Research Publications 34 Z104 Z105 Z106 Z107 Z108 Z109

35 International Journal of Scientific and Research Publications 35 Z1010 Z1011 Z1012 Z1013 Z111 Z112

36 International Journal of Scientific and Research Publications 36 Z113 Z114 Z115 Z116 Z117 Z118`

37 International Journal of Scientific and Research Publications 37 Z119 Z1110 Z1111 Z1112 Z1113 Z121

38 International Journal of Scientific and Research Publications 38 Z122 Z123 Z124 Z125 Z126 Z127

39 International Journal of Scientific and Research Publications 39 Z128 Z129 Z1210 Z1211 Z1212 Z1213

40 International Journal of Scientific and Research Publications 40 Z131 Z132 Z133 Z134 Z135 Z136

41 International Journal of Scientific and Research Publications 41 Z137 Z138 Z139 Z1310 Z1311 Z1312

42 International Journal of Scientific and Research Publications 42 Z1313 AUTHORS First Author V Suma Priya, PG Student, PVP Siddhartha Institute of Technology, Vijayawada, sumapriya995@gmail.com Second Author B V S Raghu Vamsi, M.tech, Assistant Professor, Gudlavalleru Engineering College, Gudlavalleru bvsraghuvamsi@hotmail.com Third Author E Kavitha, M.Tech, Assistant Professor, PVP Siddhartha Institute of Technology, Vijayawada, kavithavarikola@gmail.com Fourth Author K Srividya, M.Tech, Assistant Professor, PVP Siddhartha Institute of Technology, Vijayawada, srividya_kode@yahoo.com Correspondence Author B V S Raghu Vamsi, bvsraghuvamsi@hotmail.com, bvsraghuvamsi@gmail.com, Contact number: