MANUFACTURING TOLERANCES AS A CAUSE FOR AUDIBLE NOISE OF INDUCTION MOTORS

Transcription

1 MANUFACTURING TOLERANCES AS A CAUSE FOR AUDIBLE NOISE OF INDUCTION MOTORS Delaere K., Franen J., Hameyer K., Belman R. Katholieke Univeriteit Leuven, De. EE (ESAT) Div. ELEN, Kardinaal Mercierlaan 94, B-300 Leuven, Belgium. INTRODUCTION In deigning and manufacturing induction motor, it i neceary to allow dimenional tolerance on every art, e.g. on tator and rotor taming, tator frame, rotor aembly, bearing and endhield. Generally, the larger the tolerance, the lower the manufacturing cot []. On the other hand, the tolerance are often related to erformance of the machine, and increae intolerance caue a deterioration in the erformance; e.g. lower efficiency, torional vibration, radial ocillation, horter lifetime due to increaed overall temerature or hot ot. In everal aer (e.g. [,3,4]) it ha been reorted that audible noie data vary coniderably between nominably identical mall machine, and thi variation i mot likely to be caued by the variation of the dimenion of art between the tolerance limit in ma roduction. For electric machine with mall airga, uch light variation in the dimenion of the tator, rotor, end hield and bearing may lead to a coniderable variation of the airga at different angular oition between tator and rotor. Thi nonuniform airga give rie in oeration to unbalanced magnetic ull. The u.m.. due to a tatic eccentricity (incorrect oitioning of the rotor with reect to the tator bore) i a tatic force, in mot cae not cauing any roblem. In two-ole motor, and to a minor extend alo in more ole machine, vibrating comonent at double uly frequency are reent that may caue detrimental effect [5]. In a dynamically eccentric machine, i.e. a machine with a bended haft due to mechanical unbalance, the unbalanced magnetic ull lead to a decreae of the critical eed of the machine [6]. Again a two ole machine give a ecial force comonent, having double li frequency.. ECCENTRICITY FIELDS-UNDAMPED When the rotor i eccentrically oitioned with reect to the tator bore, the airga length i a function of time and ace: { } (, t) co( t ) α α ϕ () 0 ( ) for a tatic eccentricity for a dynamic eccentricity () (: number of oleair; :li; : uly ulation). i the airga length, increaed for the lotting and aturation. The relative eccentricity i (e: eccentricity) e (3) The airga ermeance i the invere of the airga length: (, t) λα Λ λ co[ λα ( t ϕ) ] (4) α (, t) λ 0

2 Λ λ For a mall relative eccentricity: λ ( λ 0) λ ( > 0) { } 0 (, t) + co( α t ϕ) λα (6) During tranient, inufficient time i available to generate the large vibration amlitude required for increaed and utained audible noie. Therefore, with regard to audible noie, the teady tate condition i imortant and therefore, the li i mall ( 0 rated ). If daming by rotor current i neglected, the reulting airga flux denity i ( α, ) co( α ϕm) + B{ co[ ( + ) α ( + ) t ( ϕm + ϕ) ] + co ( ) α ( ) t ( ϕ ϕ ) b t B t [ m ]} with B the almot contant, undamed amlitude of the fundamental field in the machine. Thi ituation i definitevely true when the machine i ulied directly from the grid, where a contant voltage i alied. When the induction motor i ulied from a frequency inverter, field weakening i alied in the high eed range (above bae eed given by the motor rated voltage and frequency and the inverter dc link voltage). Thi clearly indicate that the electromagnetically generated audible noie comonent get le ronounced at high eed. Furthermore, at high eed the airborne audible noie dominate. Thee ulementary comonent are the bai of the extra force and audible noie comonent that are oberved in ractice [7-0]. (5) (7) 3. INFLUENCE OF DAMPING The eccentricity field are damed by current in the rotor bar, both in amlitude and in hae. The hae hift i not conidered here, a it doe not influence the force comonent to a major extend. Therefore, the imaginary art i neglected and the factor i []: k ξ α (8) β + + σ with the li m gr tatic eccentricity dynamic eccentricity At ynchronou eed ( 0), the tatic eccentricity i damed while the dynamic eccentricity remain undamed. The amlitude of the eccentricity field are B, Bα (0) with number of oleair, ulation and hae angle: (9)

3 ν, ± ± (), ± () ϕ, ϕm ± ϕ (3) When calculating the u.m.., the damed amlitude have to be ued. For the radial force other than eccentricity generated, the daming may be neglected, a it concern motly high ole field yielding α. The amlitude of the comonent are the roduct of the tator harmonic, damed by the rotor and accounting for the addition uing the aroriate hae angle for the lot and lotting harmonic. When an eccentricity i reent, the eccentricity harmonic have to be taken into conideration. The undamed eccentricity harmonic (amlitude B ) induce at rated eed in the rotor bar current having the r.m..-value: I j, R, kξ ( β + j ) ( + σ gr,, ± ) B 0 u (4) with ulation, ± ( )( ) Thee current generate rotor harmonic, having the amlitude: 0 B ± kξ ± I, λ, R, with the number of oleair ( ) λ ± gn +, ξλ ξ, k Bu β + + σ gr,, (5) (6) (7) and with the ulation a een by a tator oberver: g N, ( + ) + ( ) (8) When the teeth and the yoke are highly aturated, the amlitude B, ± may differ a lot from the reult a hown here with an overall correction to the airga. Due to aturation, an extra reluctance can be introduced, being high where the flux denity i high and vice vera [] [ ] (, t) co( t ) λ α λ λ α ϕ (9) at average aturation m m In the machine dicued further on, aturation i not critical. 4. STATOR FLUX DENSITY In a three hae machine with q lot er ole and er hae, the flux denity i comoed of a number of wave (eccentricity and aturation neglected): b( α, t) Bν co( να t ϕν) (0) ν

4 with ν 6g + with g 0 () In general the lot and lotting harmonic are ufficient: ( ) B K ξ B (lotting) () n c n u B Wn ξn B ν ξ u I I m α (lot) (3) n with ( 6 ) n gq + g ±, ± (4) α n k ξn + (5) σ grn The reulting amlitude of the undamed tator field are ( in ϕ co ϕ, coϕ ower factor): B B + B + B B inϕ (6) rnu n wn n wn 5. FORCE CALCULATION The Maxwell tre from the combination of eccentricity and lotting i: B Brna σ 0 They have the number of oleair: λ ± + ( ± ) + ( ) r, n g N n gn+ ± n± r± (8) [ ] They differ from the normal one by ±. An a uch ilent machine can become loud due to the fact that the frequency of the force generated by the eccentricity are cloe to the natural frequencie. The frequencie of the radial force are g N f f ( ) r± u ly ± + 0 (7) (9) The ± i dealing with the ± choice, while 0 deal with the um a difference in the combination of trigonometric function. For the tatic eccentricity, 0, yielding a frequencie g N f f ( ) r± u ly + 0 (30) 6. PRACTICAL EXAMPLE The following data are available of a ractical motor that howed eriou audible noie roblem. Number of oleair 3. Number of tator lot N 54. Number of rotor lot N 68. Number of lot er ole and er hae in the tator q3. Coil itch in the tator w/τ 8/9. At rated eed, the li i mall comared to ( ), leading to

5 g 68 fr± fu ly The reult of thi analyi are hown in the following table for a uly frequency of 40 Hz. g When thee reult are comared to the value a found in the meaured curve, the evidence i obviou. A main audible noie frequency wa exerimentally reent at 365 Hz with two ide band at ±80 Hz. Thi i exactly redicted by the theory (both main frequency and ide band). Alo the 800 Hz value i correctly redicted. Alo the lower audible noie value around 800 Hz robably coincide with the theoretical value, but are not oible to ae in great detail on the meaured curve. 7. CONCLUSIONS It i demontrated how a tatic eccentricity can contribute to the generation of audible noie. The influence of the uly frequency and the number of rotor bar i analyed. When the frequency of the generate force coincide with a bending natural frequency of the tator aembly, roblem may be exected. A the tatic eccentricity i not checked in the manufacturing roce, the oibility i clearly reent. It hould be treed that chooing a different number of rotor bar will not hel, a the roblem will occur at a different uly frequency. Only a reduction of the caue, i.e. the eccentricity will cure the roblem. A the roblem i due to the fundamental uly frequency, higher inverter witching frequencie will not influence the henomenon.