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1. THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Series A F OF THE ROMANIAN ACADEMY Volume 9 Number 3 2008 pp 000 000 A NEW ANALYTICAL APPROACH TO NONLINEAR VIBRATION OF AN ELECTRICAL MACHINE Nicolae HERISANU Vasile MARINCA Toma DORDEA Gheorghe MADESCU Politehnica University of Timisoara Department of Mechanics and Vibration Timisoara Romania Center of Advanced Research in Engineering Sciences Romanian Academy Timisoara Branch Timisoara Romania Corresponding author Nicolae HERISANU E mail herisanu mec upt ro In this paper the nonlinear dynamic behaviour of an electrical machine exhibiting nonlinear vibration is investigated using a new analytical technique namely Optimal Homotopy Asymptotic Method This study provides an effective and easy to apply procedure which is independent on whether or not there exist small parameters in the considered nonlinear equation different from perturbation methods which require the existence of the small parameter The approximate analytic solution is in very good agreement with the numerical simulations results which prove the reliability of the method Key words Nonlinear vibration Optimal Homotopy Asymptotic Method Electrical machine 1 INTRODUCTION Electrical machines are widely used in engineering applications and industry due to their reliability They are dynamical systems encountering dynamical phenomena which can be detrimental to the system2. 0 OF 15Q 8Q0 0 C yQ in 4Q t QF 15Q 8N0 0 aioe The constants Qo Q C and C2 can be determined from Eqs 26 33 34 35 and by means of the residual which reads R t A Q Q C C o X axsinw t Px ysinw t 0 1 amp 2 2 Y 1 The last condition can be written with collocation method REA Oo CC 0 37 Finally five equations with five unknowns are obtained In the case when 1 1 2 1 5 1 58 a 2 75 B 12 5 y 0 2 we obtain A 1 1120201913228436 Qo 3 282055275793561 Q 3 39416043 17791465 C 0 0007293 171916262496 C 0 000714849572157505 Fig 1 shows the comparison between the approximate solution and the numerical solution obtained by a fourth order Runge Kutta method Figure 1 Comparison of the approximate solution with the numerical solution numerical solution approximate solution It can be seen that the solution obtained by our procedure is nearly identical with that given by the numerical method 7 A new analytical approach to nonlinear vibration of an electrical machine 4 CONCLUSIONS In the present study an analytical model for an electrical machine has been developed to obtain the nonlinear vibration response due to nonlinear stiffness The system is parametrically excited by an axial thrust and at the same time a forcing excitation caused by an unbalanced force of the rotor is acting on the system The mathematical model takes into account
3. From engineering point of view it is very important to predict the nonlinear dynamic behaviour of complex dynamical systems such as the electrical machines This is a significant stage in the design process before the machine is exploited in real conditions avoiding in this way undesired dynamical phenomena which could damage the system Basically the electric machines share the same dynamical problems with classical rotor systems having specific sources of excitation which lead to nonlinear vibration occurrence The main sources of dynamic problems are the unbalanced forces of the rotor 1 2 bad bearings or nonlinear bearings 3 4 mechanical looseness misalignments other electrical and mechanical faults which generate nonlinear vibration in the system These problems are usually solved by numerical simulations 5 experimental investigations 6 7 or by analytical developments 8 9 In general the nonlinear vibration problems are usually solved using perturbation methods which are the most used analytical techniques Some of the most used methods are the Lindstedt Poincare method 10 the Krylov Bogoliubov Mitropolsky method 11 12 the Adomian decomposition method 13 and other perturbation method 14 Unfortunately as it is well known the perturbation methods have their limitations since they are based on the existence of a small parameter and especially in strongly nonlinear systems these classical methods fail Theref
4. 0 x 0 A x 0 0 18 LX LU xa Cf t C O Fee CHOW pal Hp Mis Oy DA AI O i 12 m x 0 0 x 0 0 19 Note that can be determined avoiding the presence of secular terms in the left hand side of Eq 19 The frequency Q depends on the arbitrary parameter and we can apply the so called principle of minimal sensitivity 30 in order to fix the value of We do this by imposing that 2o dr At this moment m th order approximation given by Eq 14 depends on the parameters C1 C2 Cm The constants C can be identified via various ways for example collocation method Galerkin method least square method etc It must be highlighted that our procedure contains the auxiliary function h t p which provides us with a simple way to adjust and optimally control the convergence region and rate of solution series Note that instead of an infinite series the OHAM searches for only few terms mostly three terms 20 Nicolae HERISANU Vasile MARINCA Toma DORDEA Gheorghe MADESCU 3 APPLICATION OF OHAM TO THE INVESTIGATION OF NONLINEAR VIBRATION OF THE CONSIDERED ELECTRICAL MACHINE The validity of the proposed procedure is illustrated for the electrical machine whose dynamic behaviour is governed by Eq 1 Form Eq 18 it is obtained the following solution Xy t Acost For i 1 into Eqs 17 and 19 we obtain Die 2 0 3 ON N X 0A Qo X 0 A xX AX si S Bx yon Substituting Eq 2
5. Q Q 1 Q 9 3 A new analytical approach to nonlinear vibration of an electrical machine where X t is an initial guess of x t Therefore as the embedding parameter p increases from 0 to 1 o t p varies from the initial guess xo t to the solution x t so does Q p from the initial guess Qo to the exact frequency Q Expanding t p Q p in series with respect to the parameter p one has respectively A T p Xp T px T p x T 10 Q p Q pQ p Q 11 If the initial guess Xo t and the auxiliary function h t p are properly chosen so that the above series converges at p 1 one has XM T X_ T X T X5 T 4 12 O O O Q 13 Notice that series 10 and 11 contain the auxiliary function h t p which determines their convergence regions The results at the m th order approximations are given by X T x9 T 2 T 4 T 14 Q Q Q O 15 We propose that the auxiliary function h t p to be of the form h t p PIC f t Cr falt CfE 16 where Cy C C are constants and f t fo t f t are functions depending on variable t k being a fixed arbitrary number Substituting Eqs 12 and 13 into Eq 7 yields N 9 Q No Xp Q9 4 PNI Xp X1 Q9 Q 0 2 17 te IN Ke XX2 Q0 QQ Po Fc If we substitute Eqs 17 and 16 into Eq 5 and equate the coefficients of various powers of p equal to zero we obtain the following linear equations L x
6. in x t requires that o 3C C pea Ay Ce eae 25C BA 25699 2O 404 4Q 192Q C C 30 T C BA 320 C ip Arc 192Q03 C C 512Q3 C C R 0 31 From Eqs 26 and 30 we obtain the frequency in the form Q 0 Q 32 The parameter can be determined applying the principle of minimal sensitivity From Eq 20 we obtain the following condition 3B A 6C 14C C 5C3 256 3C C a Q C3RA 29Q4 30 30 _ Y4Q 3 96 By means of Eqs 30 and 32 Eq 31 becomes 23C2BA 30 2880 C C 1Q7 C C 33 0 Q Q 34 The Eq 31 can be written as C C aA 20 02 C aAo 07 269 Gwo o 02 49 57624 52007 0 Q o n 2C yo 59 0 2C yo 0 179 994 102a o 22501 34070 0 35 The first order approximate solution is X T X9 T 2x T or by means of Eqs 21 27 and 3 Nicolae HERISANU Vasile MARINCA Toma DORDEA Gheorghe MADESCU 6 3 3 3 3 3 X t A al oe cos Qt CBA cos 30t CPA os 5Ot ne cos 70t 3202 1920 3207 962 1922 C C aAQ C C JaA a A ae ee sin Q t sin Q t 200 2Q 200 29 Q C AQ T z sin 59 t F 5x sin 5Q r 205 2407 1090 0 205 249 10Q0 07 36 2 2 a O gage A sin 2Q t OF Q OF BO 40 0 2 2 gt a lt sin w 2Q r ae z Sin 4Q r Qf 3Q 4Qe
7. 1 into Eq 22 it is obtained A Ny o QA al 0 o h pa ose H cos3t sli sin amp 1 ht sin amp 1 h ysin r 2 Q Q Q If we choose k 3 and fi 1 1 fi 1 2cos2t f t 2c0s 4t then Eq 19 becomes for i 1 OF x x A C C 0 BA cos T Topa cos3t G CBA C A Q 0 A BA Jcos 5t TOBA cos7t 5C C aA sin 2 i sin 22 1 nite G oA amp i5 lr sin amp 5 t Cysin r C y sin BiG Q 2 Q Q Q Q sin 2 sin amp 4 lr sin amp 4 1 Q Q Q Avoiding the presence of a secular term needs oF oF SRA With this requirement the solution of Eq 25 is x t M cost Ncos3t Pcos5t Qcos7t Rsint where m CPA OBA y CBA CBA g CBA 320 19202 3205 9603 1920 paler t amp JaAQ 207 07 f C aAQa 269 Co Q4 5 427 Jo Q4 576Q 520705 05 QUQ a 2C Q0 59 07 2C yQ0 0 179 05 92 1007 a 0f 05 225Q 3407 07 21 22 23 24 25 26 27 28 5 A new analytical approach to nonlinear vibration of an electrical machine Substituting Eqs 21 and 27 into Eq 19 we obtain the following equation Q x x cost 1C C BA 2M N A C C 2Q Q 4 C C a Q A A A 3 3 12 T OLLCP C o A P gt CBA N 0 2P 29 29007 49 2 0 2l 4 2B Q BA C Rsint NT where N T means the other nonresonant terms No secular term
8. 330 1334 2005 12 YAMGOUE S B KOFANE T C Application of the Krylov Bogoliubov Mitropolsky method to weakly damped strongly non linear planar Hamiltonian systems Int J Non Linear Mechanics 42 10 pp 1240 1247 2007 13 ADOMIAN G A review of the decomposition method in applied mathematics J Math Annal amp Appl 135 pp 501 544 1998 14 NAYFEH A H Introduction to perturbation techniques Wiley New York 1981 15 HE J H Variational iteration method a kind of nonlinear analytical technique Some examples Int J Nonlinear Mech 34 pp 699 708 1999 16 MARINCA V HERISANU N Periodic solutions for some strongly nonlinear oscillations by He s variational iteration method Computers and Math with Applications 54 7 8 pp 1188 1196 2007 17 HERISANU N MARINCA V Solution of a nonlinear oscillator using an iteration procedure WSEAS Transaction on Systems 6 1 pp 156 161 2007 18 ODIBAT Z M MOMANI S Application of variational iteration method to Nonlinear differential equations of fractional order Int J Nonlinear Sci Num Simul 7 1 pp 27 34 2006 19 RAMOS J I On the variational iteration method and other iterative techniques for nonlinear differential equations Applied Mathematics and Computation 199 1 pp 39 69 2008 20 MARINCA V HERISANU N Periodic solutions of Duffing equation with strong non linearity Chaos Solitons and Fractals 37 1 pp 144 149 2008 21 LIAO S J Homotopy analysis met
9. chinery 6 pp 191 200 2000 4 HARSHA S P SANDEEP K PRAKASH R The effect of speed of balanced rotor on nonlinear vibrations associated with ball bearings International Journal of Mechanical Sciences 45 pp 725 740 2003 5 LEE D S CHOI D H A dynamic analysis of a flexible rotor in ball bearings with nonlinear stiffness characteristics International Journal of Rotating Machinery 3 pp 73 80 1997 6 SINOU J J VILLA C THOUVEREZ F Experimental and Numerical Investigations of a Flexible Rotor on Flexible Bearing Supports International Journal of Rotating Machinery 11 pp 179 189 2005 7 CRISTALLI C PAONE N RODRIGUEZ R M Mechanical fault detection of electric motors by laser vibrometer and accelerometer measurements Mechanical Systems and Signal Processing 20 pp 1350 1361 2006 8 FINLEY W R HODOWANEC M M HOLTER W G An analytical approach to solving motor vibration problems IEEE Transactions on Industry Applications 36 pp 1467 1480 2000 9 HERISANU N MARINCA V MARINCA B An analytic solution of some rotating electric machines vibration Int Review of Mech Eng IREME 1 5 pp 559 564 2007 10 CHEUNG Y K CHEN S H LAU S L A modified Lindstedt Poincare method for certain strongly nonlinear oscillators Int J Non Linear Mech 26 3 4 pp 367 378 1991 11 CVETICANIN L Free vibration of a Jeffcott rotor with pure cubic non linear elastic property of the shaft Mechanism and Machine Theory 40 pp 1
10. convenient form X x axsin f Bx ysinat 0 x 0 A x 0 0 2 where a 4 B k a the dot denotes derivative with respect to time and A is the amplitude of the oscillations Note that it is unnecessary to assume the existence of any small or large parameter in Eq 2 The main purpose of the present paper is to use the Optimal Homotopy Asymptotic Method OHAM for obtaining solutions of strongly nonlinear vibration of the electrical rotating machinery under study 2 BASIC IDEA OF OHAM 28 29 The Eq 2 describes a system oscillating with an unknown period T We switch to a scalar time t 2nt T Qt Under the transformation t Or 3 the original Eq 2 becomes Va a arsin 1 Br ysin t 0 4 where the prime denotes the derivative with respect to T By the homotopy technique we construct a homotopy in a more general form A ot p h t p 1 p JL 0 1 p U t p NI 1 p Q X p 9 5 where L is a linear operator o T uyn pno ERa p while N is a nonlinear operator 2 GH b 2 gt 2 NLM P 20 p Pp AEP e 296 p alt p sin 2r 7 B6 c p ysin Et pO p where pe 0 1 is the embedding parameter h t p is an auxiliary function such as h t 0 0 h t p 0 for p 0 is an arbitrary parameter From Eqs 2 and 3 we obtain the initial conditions T p 0 pJ A 0 p ee 0 8 Obviously when p 0 and p 1 it holds O 7 0 X9 T HtL 2 7 O 0 J
11. hod a new analytical method for nonlinear problems Appl Math And Mech 19 10 pp 957 962 1998 22 LIAO S J Beyond perturbation Introduction to the Homotopy Analysis Method Chapman amp Hall CRC 2003 23 HE J H Homotopy perturbation technique Comp Methods in Appl Mech and Eng 178 pp 257 262 1999 24 CVETICANIN L Homotopy perturbation method for pure nonlinear differential equation Chaos Solitons amp Fractals 30 5 pp 1221 1230 2006 25 GORJI M GANJI D D SOLEIMANI S New application of He s homotopy perturbation method Int J of Nonlinear Sci Num Simul 8 3 pp 319 328 2007 Nicolae HERISANU Vasile MARINCA Toma DORDEA Gheorghe MADESCU 8 26 RAMOS J I Series approach to the Lane Emden equation and comparison with the homotopy perturbation method Chaos Solitons and Fractals 38 2 pp 400 408 2008 27 SHOU D H HE J H Application of parameter expanding method to strongly nonlinear oscillators Int J Nonlinear Sci Numer Simul 8 1 pp 121 124 2007 28 MARINCA V HERISANU N Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer Int Communications in Heat and Mass Transfer 35 6 pp 710 715 2008 29 MARINCA V HERISANU N NEMES I Optimal homotopy asymptotic method with application to thin film flow Central European J of Physics 6 3 pp 648 653 2008 30 AMORE P ARANDA A Improved Lindstedt Poincare method for the solut
12. ion of nonlinear problems J Sound Vibr 283 pp 1115 1136 2005 Received September 10 2008
13. ore scientists are continuously concerned in developing new analytical techniques which aim at surmounting these limitations Recently new powerful analytical tools were developed such as the Variational Iteration Method 15 16 17 18 19 20 Homotopy Analysis Method 21 22 Homotopy Perturbation Method 23 24 25 26 the parameter expanding method 27 in an attempt to obtain effective analytical tools valid for any strongly nonlinear problems In this paper a new analytical procedure namely Optimal Homotopy Asymptotic Method is employed in order to study the problem of nonlinear vibrations of an electric machine The investigated electrical machine is considered to be supported by nonlinear bearings and the assumption made in development of the mathematical model is that these bearings are characterised by nonlinear stiffness of Duffing type In the Member of the Romanian Academy Nicolae HERISANU Vasile MARINCA Toma DORDEA Gheorghe MADESCU 2 same time the entire dynamical system is subjected to a parametric excitation caused by an axial thrust and a forcing excitation caused by an unbalanced force of the rotor which is obviously harmonically shaped In these conditions the dynamical behaviour of the investigated electrical machine will be governed by the following second order strongly nonlinear differential equation mi k qsing t x k x fsinot x 0 A 0 0 1 which can be written in the more
14. the sources of nonlinearity and the corresponding equation of motion is solved using the Optimal Homotopy Asymptotic Method to graphically obtain the time history of nonlinear response The proposed procedure is valid even if the nonlinear equation does not contain any small or large parameter The OHAM provide us with a simple way to optimally control and adjust the convergence of the solution series and can give good approximations in few terms The convergence of the approximate solution series given by OHAM is determined by the auxiliary function h t p The obtained approximate analytical solution is in very good agreement with the numerical simulation results which proves the validity of the method This paper shows one step in the attempt to develop a new nonlinear analytical technique which is valid in the absence of a small or large parameter REFERENCES 1 FLEMING P POPLAWSKI J V Unbalance Response Prediction for Rotors on Ball Bearings Using Speed and Load Dependent Nonlinear Bearing Stiffness International Journal of Rotating Machinery 11 pp 53 59 2005 2 DEMAILLY D THOUVEREZ F JEZEQUEL L Unbalance Responses of Rotor Stator Systems with Nonlinear Bearings by the Time Finite Element Method International Journal of Rotating Machinery 10 pp 155 162 2004 3 CHANG Y P JEN S C TU S H SHYR S S KANG Y Mode locking quasi period and chaos of rotors mounted on nonlinear bearings International Journal of Rotating Ma
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