Dr R Tiwari, Associate Professor, Dept. of Mechanical Engg., IIT Guwahati,

Transcription

1 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i).3 Measuremet ad Sigal Proessig Whe we ivestigate the auses of vibratio, we first ivestigate the relatioship betwee the frequeies ad the rotatioal speed. We a do suh spetrum aalysis early usig FFT-aalyzer (Fast Fourier trasformatio) equipmets. Spetrum aalysers have various oveiet futios, suh as traig aalysis, Campbell diagram ad waterfall diagram. I traig aalysis, dyami harateristis of a rotatig mahie are ivestigated by hagig the rotatioal speed. A waterfall diagram is a 3-dimesioal plot of spetra at various speeds. Figure.9 shows these diagrams for illustratio. However, to use these futios orretly, we must have some bagroud owledge of sigal proessig. Pea veloity Magitude Frequey (Hz) (a) Spetrum diagram Spi Speed (rpm) (b) Spi Speed traig diagram Spi speed Frequey (Hz) Magitude Spi speed Frequey (Hz) ( ) Campbell diagram (d) Waterfall diagram Figure.9 Futios of spetrum aalyzer Further, if we have to ostrut a speifi data aalysis system that fits our researh, we must have suffiiet uderstadig of the fudametal of sigal proessig. The vibratio of the rotor is a whirlig motio ad therefore ot oly the frequeies but also the diretios of the whirlig motios are importat eough to pursue their auses. However, sie the usual fast Fourier trasform (FFT) theory gives iformatio about magitudes of frequeies ad phases oly, we aot ow the whirl diretio usig the ovetioal FFT-aalyser. For this purpose, Ishida (997) ad Lee () 447

2 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) proposed a sigal proessig method where the whirlig plae of a rotor is overlapped to the omplex plae. This method is alled the omplex-fft (or diretioal-fft) method, eables us to ow the diretios of whirlig motio besides the magitudes of the frequeies. They also used this method to extrat a ompoet form o-statioary time histories obtaied umeri simulatios ad experimeted data ad depited the amplitude variatio of the ompoet. We will disuss fudametal ideas eessary to uderstad sigal proessig by omputer. I additio, appliatios of the omplex-fft method of studyig statioary ad o-statioary vibratios are explaied..3. Measuremet ad Samplig Problem Measuremet system ad Digital Sigal: A measuremet system is show i Figure.a. Sesor (detetio) Iterfae (A/D-trasform) Persoal Computer (Proessig, display) Figure.(a) A measuremet system x(t) x t t Figure.(b) Aalog ad digital sigals Neessary data are deteted from the vibratig struture by sesors. I a rotatig mahie, rotor displaemets i two diretios formig a right agle ad a rotatig speed are deteted as voltage variatios. The output sigal x(t) from the sesor is a aalog sigal that is otiuous with time. But the sigal is disretised whe it is aquired by omputer through a iterfae. This digital sigal is a series of disrete data { x } obtaied by measurig (alled samplig) a aalog sigal istatly at every time iterval t ad is give as x( t), where is a iteger. This iterval t is alled x a samplig iterval. A digital sigal is desretised i both time ad magitude. Disretizatio i magitude is alled quatizatio, ad the magitude is represeted by biary umbers (uit: bits). 448

3 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) Digital data i a persoal omputer are proessed ito various forms usig software programs. I this operatio, two represetative proessig are performed. Oe is sigal extratio, where ueessary sigal ompoets are abadoed i the aquired data, ad the other is data trasformatio, where the data are overted to a oveiet form. Problems i Sigal Proessig: Whe a aalog sigal x(t) is haged ito a sequee of digital data { x } (,,,, N) a virtual (or imagiery) wave is obtaied if a fast sigal is sampled slowly. For example, whe a sigal illustrated by the full lie is sampled as show i Figure., a virtual sigal wave illustrated by the dashed lie appears, although it is ot otaied i the origial sigal. x(t) T t x Virtual wave t Figure. Aliasig This pheomeo is alled aliasig. It is obvious that we must sample with a smaller samplig iterval as the sigal frequey ireases. We a determie whether or ot we have this aliasig by followig the samplig theorem. It says: whe a sigal is omposed of the ompoets whose frequeies are all smaller tha f, we must sample it with a frequeies higher tha sae of ot losig the origial sigal s iformatio. The frequey f for the f is alled Nyquist frequey. For example, if a sie wave with period T is sampled wheever x ( t), that is, with samplig iterval T/, we have x. Therefore, two sampligs i a period are learly isuffiiet. However, 449

4 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) this theorem teahes us that digital data with more tha two poits durig oe period a express the origial sigal orretly. For example, if we sample the sigal havig ompoets of, ad 6 Hz with a samplig frequey of Hz, we have a imagiary spetrum of 4 Hz, whih does ot exist pratially. But, if we sample it with a frequey of more tha Hz ( 6 Hz), suh a alisig problem does ot our. I pratial measuremets, we do ot ommoly determie the samplig frequey by trial measuremet. Istead, we use a low-pass filter to elimiate the ueessary highfrequey ompoets i the sigal ad sample with the frequey higher tha twie the utoff frequey. By suh a proedure, we a prevet aliasig..3. Fourier Series I data proessig, we must first ow the frequey ompoets otaied i a sigal. The fudametal owledge eessary for it is the Fourier series. We will briefly summaries it from the poit of view of sigal proessig. O type of Fourier series is expressed by real umbers, while the other is by omplex umber. (i) Real Fourier Series: A periodi futio x (t) with period T a be expaded by trigoometri futios whih belog to the orthogoal futio systems as follows x a ( t) + ( a os t + b si ωt) ω (.) where give by ω / T. This series is alled the Fourier series or real Fourier series. Its oeffiiets are T / a x( t)os ω tdt, b x( t)si ω tdt (.3) T T T / T / T / (ii) Complex Fourier Series: Fourier series a be expressed by omplex umbers usig Euler s formula θ e j osθ + j siθ. Complex umbers mae it easier to treat the expressios. As will be metioed later, omplex represetatio maes it possible to represet a whirlig motio o the omplex plae. Substitutig Euler s formula ito equatio (.), we have 45

5 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) jωt x( t) e (.4) where the omplex oeffiiets are give by T T / T / x( t) e ω dt (, ±, ±, ) (.5) j t Equatio (.4) is alled the omplex Fourier series. Betwee the real ad omplex Fourier oeffiiets, the relatioships a jb, a, a + jb (.6) hold, where >. from this we ow the followig relatioship (.7) Therefore, if these quatities are illustrated i the figure taig the order (, ±, ±, ) as the absissa, the real part is symmetri about the ordiate axis, ad the imagiary part is sew-symmetri about the origi. These omplex Fourier oeffiiets a also be represeted by jθ e (.8) where the absolute value a + b is alled a amplitude spetrum, the agle ( b a ) θ ta a phase spetrum ad a power spetrum. As a example, the omplex oeffiiets of the square wave with period T 8 are show i Figure.. This wave is defied as x ( t) for t ad 7 t 8 for t 7 45

6 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) T ω (a) Time History (b) Spetrum Figure. Spetrum of a square wave (omplex form) For this square wave, we have the followig from equatio (.5) /T, si ω + j for (.9) Tω Sie x (t) is a ever futio i.e. f ( x) f ( x), the imagiary part of is zero. siθ si( θ ) siθ os( θ ) osθ Figure.3 (a) A odd futio (b) A eve futio.3.3 Fourier Trasform Whe x (t) is a isolated pulse, it aot be overted to a disrete spetrum sie it is ot periodi. However, let us osider that this iterval is exteded to ifiity. The the spetra obtaied will represet the spetra of the isolated pulse. Substitutig equatio (.5) ito equatio (.4), we get x( t) x( t) e dt e T / jωt jω t π T (.) T / where the frequey ω / T of the fudametal wave is deoted by ω. Here we represet of the th order by ω ω ad the differee i frequeies betwee the adjaet ompoets by ω ω ω ω. If we mae T, we have + / T 45

7 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) ( ) ( ) jωt jωt x t x t e dt e dt (.) where ω, forms as follows ω ad are replaed by ω, dω ad, respetively. This a be expressed i separate with jωt x t ω e dt ( ) ( ) (.) jωt ( ω) x( t) e dt (.3) Equatio (.) is alled the Fourier trasform of x (t) ad equatio (.3) is alled the iverse Fourier trasform of (ω ). As a example, Figure.4 shows a square pulse defied as x(t) for t for all other t ad its otiuous spetrum is obtaied by equatio (.3) i.e. by the Fourier trasformatio siω ( ω) (.4) πω Now, let us ompare the spetrum of a square wave of period T as show i Figure. ad that of a square pulse show i Figure.4. From equatio (.9) ad (.4) that the Fourier oeffiiets i Figure 4 have the followig relatioship to (ω ). T ( ω ) (.5) where ω is the fudametal frequey. Therefore, the evelope of the quatities obtaied by multiplyig T / to the lie spetra of the Fourier oeffiiets of the square wave gives the otiuous spetra of the Fourier trasform (ω ) of the square pulse. As show i Figure.5, 453

8 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) multiplyig the spetrum for the square wave with period T 8 i Figure.4 gives the spetrum for the square pulse i Figure.5. x(t) (ω) /π t 4π 3π π π 3π 4π (a) Square pulse i time domai (b) Fourier trasform of the square pulse Figure.4. Spetrum of a square pulse x(t) (ω) /π 8.5 π ( ω ) T t 4π 3π π π 3π 4π (a) Square wave i time domai (b) Fourier trasform of the square wave Figure.5. Compariso betwee spetra for a square wave ad square pulse..3.4 Disrete Fourier Trasform So far, we have disussed the Fourier series ad Fourier trasform o the assumptio that we ow a otiuous sigal wave i the ifiite time domai. However, i pratial experimets, the data aquired, overted from the data measured by a aalog-to-digital overter, are sequees of data { x } (,,,, -, ) that are disrete ad with fiite umber. To perform spetrum aalysis usig these fiite umbers of disrete data, we must use the disrete Fourier trasform (DFT). This DFT is defied as follows: Give N data sampled with the iterval t, the DFT is defied as a series expasio o the assumptio that the origial sigal is periodi futio with the period (although the origial sigal is ot eessary periodi). However, various problems our i the ourse of this proessig. The first is the aliasig problem. Whe the sigal is sampled with iterval t, iformatio about the ompoets with frequeies higher tha t N t is lost. Therefore, we must 454

9 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) pay attetio to valid rage of the spetra obtaied. The seod is the problem of the oiidee of periods. It is impossible to ow the orret period of the origial sigal before the measuremet. Therefore, the period of the origial sigal ad the period of DFT do ot oiide, ad this differee produes the leaage error. We will disuss this leaage error ad its outermeasure later. The third is the problem about the legth of measuremet. I the ase of a isolated sigal x(t), we aot have data i a ifiite time rage. However, sie the Fourier oeffiiets ad Fourier trasform ( ω) by oetig the values of smoothly. I the followig, we explai how to ompute DFT. Let us assume that we obtaied data sequee x, x, x,, x N by samplig. These data are exteded periodially, as show by the dashed urve i Figure.6. x x x x x x N N x N x x x N t t t N t ( N ) t ( N ) t t Figure.6. Sampled sigal sequees The fudametal period is T N t ad the fudametal frequey is ω ω T. If this dashed urve is give as a otiuous time futio, its Fourier series expasio is give by the expressios obtaied by replaig ω with ω i equatio (.4) ad (.5). However, i the ase of a disrete sigal, the itegral of equatio (.6) must be alulated by replaig t, T, x(t) ad with t, N t, x ad respetively. By suh replaemets, we have x e t x e N N j ω t j ( π / T ) ( T / N ) T N (.6) We represet the right-had side of this expressio by x, that is N j π / N xe N ; (,,,, N-) (.7) 455

10 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) ad all this desrete Fourier trasform of the disrete sigal x, x,, x N. Paired with this is the followig expressio, alled the iverse disrete Fourier trasform (IDFT) N e j / N ; (,,,, (N-)) (.8) These trasformatios map the disrete sigal of a fiite umber o the time axis to the disrete spetra of a fiite umber o the frequey axis, or vie versa. These expressios usig omplex umbers are alled omplex disrete Fourier trasform ad the omplex iverse disrete Fourier trasform. We also have trasformatio usig oly real umbers. Oe is the real disrete Fourier trasform, give by A B N x os N N x si N π N π N (,,,, N-) (.9) where A ad Fourier trasform is give by B are quatities defied by A + jb. Further, the iverse real disrete x N A os N B si ; (,,,, N-) (.3) N We explai the harateristis of the spetra obtai by DFT usig a example i the followig. Figure.7(a) shows a square wave with period T 8 ad sixtee sampled data: x x ad 4 x x obtaied by samplig with iterval t. 5. I this example, the sigal is 5 5 sampled itetioally i the rage that oiides with the period of the origial square wave to avoid the leaage error. Figure.7(b)-(e) shows spetra represetig the real part of part of harateristis:, the amplitude The spetra is periodi with period N., ad the phase, the imagiary. These spetra have the followig The same spetra as those of the egative order -N/,, - also appear i the rage N/,, (N-). 456

11 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) The spetra of the real part ad those of the amplitude are both symmetri about N/. The spetra of the imagiary part ad those of the phase are sew symmetri about N/. The spetra i the left half of the zoe,, (N-) are valid. The spetra i the right half are virtual ad are too high ompared to the samplig frequey. If the samplig iterval is arrowed the umber of spetra ireases, ad therefore suh a spetra diagram writte i the iterval ω / T exteds to the right. For ompariso, the spetrum of N 3 is depited i Figure.7(f). If the samplig frequey is shorteed, the sampled data beome substatially equal to the otiuous wave, ad therefore its spetra will approah those of the Fourier series show i Figure 4. The magitude of is.33 i Figure.7(a) ad is.8 i Figure.7(f). This value approahes C.5 i Figure.3 as the umber of data sampled ireases. Note : Differet types of defiitio of DFT ad IDFT are used, depedig upo persoal preferee. Some use the followig defiitios, i whih the magitudes of trasform i Figure.4. oiide with that of Fourier ad N j π / N (,,, N-) (.3) t x e x N j π / N e T (,,, N-) (.33) Some use followig expressios, whih have the oeffiiet /N i the outer-part expressio: (MATLAB uses this) ad N j π / N xe (,,, N-) (.34) x N j π / N e N (,,, N-) (.35) Of ourse, every defiitio has the same futio as mappig. However, we must be areful whe we iterpret the physial meaig of the magitude of the spetra. For example, for equatio (.7) that gives a spetrum with magitude. x( t) si t, it is 457

12 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) x x4 x x5 x x5 (a) Square wave ad 6 sampled data t A Period N6 B (b) Real part of N 8 5 N/ N Sew symmetri about N/ ( ) Imagiary part of Symmetri about N/ N/ 5 6 (d) Amplitude (e) phase Sew-symmetri about N/ Period N 3 Symmetri (f) Case of N 3 Figure.7 Digital sigal i the time ad frequey domais 458

13 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i).3.5 First Fourier Trasform The vast omputatioal tas eessary for DFT preveted its pratial utilizatio. I 965, Cooley ad Tuey proposed a algorithm that eabled the fast omputatio of DFT. The algorithm is alled Fast Fourier Trasform (FFT), has made real-time spetrum aalysis a pratial tool. I the alulatio of DFT give by equatio (6), we must perform may multipliatios ad additios. However, the same alulatio appears repeatedly sie the futio e j / N os { ( N )} j si{ ( N )} has a periodi harateristi. The FFT algorithm elimiated suh repetitio ad allowed the DFT to be omputed with sigifiatly fewer multipliatios tha diret evaluatio of DFT. For further details refer to boo by Newlad (99) Radom Vibratio ad Spetral Aalysis. The FFT algorithm has the restritio that the umber of data must be (,,, ). Whe the umber of data N is N, DFT eeds multipliatios ad FFT eeds N multipliatios. For example, whe 4, about,5, multipliatios are eessary i DFT ad about,48 i FFT. If N ireases this differee ireases extremely large. MATLAB has FFT futio ame fft x ) where x x}. ( N {.3.6 Leaage Error ad Coutermeasures (i) Leaage Error: I FFT or DFT, omputatios are based o the assumptio that the data sampled over a time period are repeated before ad after data measuremet. Figure.8 shows the assumed sigals ad their spetra for two types of measuremet of a siusoidal sigal x( t) si t. Both ases have 3 sampled data, but their samplig itervals are differet. I ase A, the samplig iterval is t π /6.396 ad the rage measured is exatly twie the fudametal period. The omputatio of FFT or DFT is performed for the wave as show by the dotted lie. I this ase the assumed wave is same as the origial sigal ad therefore we get a orret sigal spetrum. I ase B, the samplig iterval is t. 36, ad the rage measured is about.5 times the period of the origial sigal. I this ase, the assumed wave show i Figure.8() is ot smooth at the jutio ad differs fro the origial sigal i time domai. As a result, the magitude of the orret spetrum dereases ad spetra that do ot exist i the origial sigal appear. As see i this example, if the time duratio measured ad the period of the origial sigal do ot oiide, the magitude of the orret spetrum dereases ad spetra that do ot exist i the origial sigal appear o both sides of the orret spetrum. This pheomeo is alled leaage error. 459

14 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i). Measured Rage ase A A Measured Rage ase B B -. 4 π 6π 8π π t (a) Origial sigal ad measured rage T 4 π 6π 8π π (b) Assumed sigal ad its spetrum (ase A) T 4π 6π 8π π T ( ) Assumed sigal ad its spetrum (ase B) Figure.8. Leaage error (ii) Coutermeasures for leaage error (Widow Futio): To derease the leaage error due to disrepay betwee the time duratio measured ad the period of the origial sigal, we must oet the repeated wave smoothly. For this purpose we multiply a weightig futio that derease gradually at both sides. This weightig futio is alled time widow. Represetative time widows: the Haig widow, Hammig widow ad Blama-Harris widow are show i Figure.9. These widows are defied i the rage: N as Haig widow w( ).5.5os ( π / N ) Hammig widow w( ) os( π / N ) Blama-Harris widow w( ) os( π / N ) +.79 os( 4 π / N ) ad outside < N ω ( ) for all three ases. (.37) 46

15 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) o Blama- Harris widow Hammig widow Haig widow N- Figure.9 Widow futio For a disussio of their harateristis ad the effets of these widow futios, refer to some boos o sigal proessig. (iii) Prevetatio of leaage by oiidig periods: As disussed above, we a obtai the orret result if the time duratio measured oiides with the iteger multiple of the period of the origial sigal. If we a attai this by some meas, it is better tha the use of widow futios, whih distorts the origial sigal. For example, for umerial alulatios that a be repeated i exatly the same way ad whose samplig iterval a be adjusted freely, we a determie the measuremet duratio after we ow the period of the origial sigal by trial simulatio, ad the exeute the atual umerial simulatio. O the otrary, for experimets, fie adjustmet of samplig itervals is geerally impossible usig pratial measurig istrumets. However, if the pheomeo appears withi a speed rage, we a hage the rotatioal speed little by little ad adopt the best result where the period, ofte determied by the rotatioal speed, ad the samplig iterval fit..3.7 Appliatios of FFT to Rotor Vibratios I the ivestigatio of rotor vibratio, we must ow the diretio of a whirlig motio as well as its agular veloity. I FFT (or DFT), elemets of data sequee { x } obtaied by samplig are osidered as real umbers ad those of data sequee { } obtaied by disrete Fourier trasform are osidered as omplex umbers. I the followig, we itrodue a method that a distiguish betwee whirlig diretios utilizig the revised FFT. I this FFT, rotor whirlig motio is represeted by a omplex umber by overlappig the whirlig plae o the omplex plae ad applyig FFT to these omplex sampled data. Let us assume that a dis mouted o a elasti shaft is whirlig i the y-z-plae. We get sampled data { y } ad { } z by measurig the defletios y (t) ad z (t) i the y ad z diretios respetively. Taig the y-axis as real axis ad the z-axis as 46

16 Dr R Tiwari, Assoiate Professor, Dept. of Mehaial Egg., IIT Guwahati, (rtiwari@iitg.eret.i) imagiary axis, we overlap the whirlig plae o the omplex plae. Usig sampled data we defie the omplex umbers as follows: y ad z, S y + jz (,,,., N-) (.38) ad apply FFT (DFT) to them. We all suh a method the omplex-fft method. Example: Subharmoi resoae of order ½ of a forward +ω / +ω +ω / +ω - - Baward Forward (a) Spetrum without widow - - ( ) Spetrum obtaied by adjustig samplig iterval +ω / +ω - - (b) Spetrum with widow Figure.3. Spetra of the sub-harmoi resoae of ½ order of a forward whirlig mode ω b ω f ω - - Baward Forward Figure.3. Spetrum of the ombiatio resoae (omplex FFT method) 46