Exercise 8 CRITICA SEEDS OF TE ROTATING SAFT. Ai of the exercise Observatio ad easureet of three cosecutive critical speeds ad correspodig odes of the actual rotatig shaft. Copariso of aalytically coputed ad easured critical speed values.. Theoretical itroductio The critical speed otio is associated with rotary achies ad is used to describe the rotatioal speed at which a excessive icrease i shaft deflectios ad resultig vibratios of the rotor housig ca be observed. The critical speed occurs whe the speed of rotatios is equal to the atural frequecy of the shaft. Two ethods for deteriig atural frequecies of the shaft are preseted i the theoretical itroductio... Rotatig shaft as a cotiuous syste To deterie atural frequecies of the uifor shaft with the ass distributed alog its legth, oe ca use the bea equatio of lateral vibratios i the followig for: y y a, x t (8.) where: µ a, (8.) µ ass per uit legth, flexural stiffess of the bea. If we use the ethod of separatio of variables, the foral solutio to bea equatio (8.) takes the for: y ( x, U ( x) T (. (8.) The substitutio of this solutio i Eq. (8.) gives: U I ( x) T( a U ( x) T& (, (8.) thus: U I ( x) T& ( a. (8.5) U ( x) T ( To satisfy Eq. (8.5) for arbitrary x ad t values, both sides of this equatio have to be equal to a costat value deoted as k : U I ( x) T&& ( a k U ( x) T(. (8.) Thus, oe obtais two idepedet differetial equatios: U I ( x) k U ( x) ; (8.7) T& ( T(, (8.8) where: k a. (8.9) The hoogeous solutio to Eq. (8.7) has the followig for:
U(x) A si kx B cos kx C sih kx D cosh kx, (8.) where: A, B, C, D costat quatities deteried fro the boudary coditios. For siply supported eds, the boudary coditios have the followig for: y(, ; y( l, ; y( x, t ; y( x, t x x i. (8.) Substitutig solutio (8.) i boudary coditios (8.), oe receives the eigefuctio sequece i the for: π x U ( x) si k x si ; l,,... (8.) The solutio to Eq. (8.8) is as follows: T ( K cos t si t, (8.) where K ad are costat quatities deteried fro the iitial coditios. O the basis of Eqs. (8.9) ad (8.), the atural frequecies, related to successive eigefuctios of the bea, ca be deteried fro the followig equatio: k π ;,,... (8.) a l µ Fially, the geeral solutio to the bea equatio takes the for: y( x, ( K πx cost si si l (8.5).. Rotatig shaft as a discrete syste. yklestad ethod A coplex or cotiuous syste ca be divided ito fiite eleets or segets ad thus it ca be approxiated with a equivalet discrete syste. To explai this techique, oe ought to itroduce the cocept of a state vector ad a trasfer atrix. A state vector is a colu of ubers, whose values represet variables at a give statio i the syste. The ubers describe the state of the proble. ece, each eleet of the state vector is called a state variable. Typical variables ad their correspodig state vectors are show i Fig. 8.. a) x b) ϕ c) θ T d) { Z} x F F { Z} { Z} T θ { Z} Fig. 8.. State vectors, geeralized forces ad displaceets for: a) tesio, b) bedig, c) torsio, d) shear. For exaple, if a cobiatio of shear ad bedig occurs at the statio, the correspodig state vector is:
T { Z} [ ]. (8.) A trasfer atrix trasfers state variables fro oe statio to aother. et us take ito accout a bea structure divided ito segets. A typical seget of the bea, as show i Fig. 8., cosists of a assless spa which is characterized by the bedig stiffess ad the poit ass. The superscripts ad deote the left- ad the right-had side of the ass, respectively. The flexural properties of the seget are described by the field trasfer atrix of the spa ad the iertial effects of the seget are described by the poit trasfer atrix of the ass. To describe the field trasfer atrix, cosider a free-body sketch of a uifor bea of the legth as show i Fig. 8.(b). I this case, the field trasfer atrix trasfers the state variables of the state vector {Z} - i the left ed to the state vector {Z} i the right ed of the spa. a) - spa b) seget c) X X Fig. 8.. Derivatio of the trasfer atrix of the bea For the equilibriu state, we require that: ;, (8.7) where: - bedig oet, - shear force. Accordig to Fig. 8.(b), the chage i the slope Φ of the spa is equal to:. (8.8) After the substitutio of Eq. (8.7) i (8.8) ad rearrageet, we obtai:. (8.9) The chage i the deflectio of the spa is as follows:. (8.) After the substitutio of ad fro Eq. (8.7) i (8.) ad rearrageet, get: we
. (8.) The field trasfer atrix is obtaied by writig Eqs. (8.7), (8.9) ad (8.) i the atrix for:. (8.) To derive the poit trasfer atrix, cosider a free-body sketch of (Fig. 8.c). The equatios for the shear ad the oet are as follows: ad, (8.) where - ass oet of iertia of about its axis oral to the (x, y) plae. For the rigid body otio of, the followig relatios are fulfilled: ;. (8.) The poit trasfer atrix is obtaied fro Eqs. (8.) ad (8.):. (8.5) Substitutig {Z} fro Eq. (8.) i (8.5), oe receives the trasfer atrix:. (8.) Equatio (8.) ca be writte i the followig for: { } { } Z Z, (8.7) where is the trasfer atrix of the -th eleet. Usig the recurrece forula, the state vector Z} { at a typical statio ca be related to the state vector Z } { at the boudary of the syste: { } { }{ } Z Z.... (8.8) Equatio (8.8) presets the recurrece forula which is used i the yklestad ethod for the atural frequecy calculatio. The coo boudary coditios for the bea proble are preseted i Table 8..
Table 8.. Boudary coditio for the bea proble Siple support Free ed Fixed ed For exaple, the deflectio ad the oet at a siple support have to be zero, whereas the slope ad the shear force are ukow ad o-zero. At the begiig poit or statio of the bea, there are two o-zero boudary coditios, dictated by the type of support. Siilarly, there are two o-zero boudary coditios at the other ed of the bea. The procedure of the yklestad ethod for the atural frequecy calculatio cosists i assuig the frequecy ad proceedig with the coputatio. The process repeats util the value that satisfies siultaeously the boudary coditios at both eds of the bea is foud. This value is the atural frequecy. Exaple: Usig the yklestad ethod, fid atural frequecies of the bea show i Fig. 8.. Data:. kg, l µ,,d l l l, d., πd /. -, E. N/, µ /l. kg/. Fig. 8.. uped-ass represetatio of the bea Solutio The bea was divided ito two segets of the legths: l.5 ad l.75 (Fig. 8.). The ass of the first seget is cocetrated at its right ed. The ass of the secod seget has bee divided i two equal parts ad located at its both eds. The laped-ass represetatio of the bea:.875 kg ad.85 kg. I this case, recurrece forula (8.8) for the atural frequecy calculatio has the followig for: { Z} { Z}, (a) Z, (b) ad are the ukow oet ad shear force at the fixed ed. where: { } { } T { } T Applyig Eq. (8.), oe receives the trasfer atrices for both segets:,5,,95,87,,5 5,8,7,5,7 (c)
,75, 5, 5,,,75 8,,,75,8 Substitutig (c) ad (d) i (a) i the geeral descriptio, we obtai: { Z} { Z} [ ]{ Z}. (e) (d) The oet ad the shear force have to be zero at the free ed of the bea (Table 8.). Usig the otatios fro (e), oe ca write these coditios i the followig for: ; (f). For a otrivial solutio to the siultaeous hoogeeous equatios, the characteristic deteriat of (f) has to vaish, that is:. (g) O the basis of Eq. (g), atural frequecies of the bea ca be deteried. For the give data, the correspodig equatio is as follows: 5,,8 α 7 8,,,,8 (h) or, i the developed for: 7 5,97,9. (i) The atural frequecies are equal to:.8 rad/s 88. rad/s f.7 z, f z.. easureet device A schee of the easureet device is show i Fig. 8.. Fig. 8.. easureet device
Shaft () i the for of a steel rod is supported i self-aligig ball bearigs (). It is drive by electric otor () with flexible couplig (). Both the otor ad bearig housigs are outed o the coo base. A potetioeter is used to vary the speed of the otor. A photoelectric trasducer is used to easure the agular velocity of the shaft. The shaft properties are as follows: ass. kg, diaeter d, legth: l. I the workig rage of the drivig otor ( rev/i), oe ca observe three cosecutive critical speeds of the shaft.. Course of the exercise Calculate aalytically three lowest values of the critical speed of the shaft treated as a cotiuous syste usig the real syste data preseted earlier. Record the results i Table 8. i the colu. Next, calculate the critical speeds of the shaft treated as a discrete syste, usig the yklestad ethod. I the uerical calculatios, use the DERIE package. Calculate the critical speeds of the ivestigated shaft for differet segetatios. Record the calculatio results i the colu of Table 8.. I the experietal part of the exercise, easure ad observe the critical speeds of the shaft. The critical speed occurs whe the lateral deflectios of the rotatig shaft have extree values. The easureets are to be coducted both at speeds lower tha the critical oe ad higher tha the critical oe, but such at which o ipacts agaist the liiters occur. Record the easureet results i Table 8. ad calculate the ea values. Table 8.. Ivestigatio results easured frequecy p [rad/s] ea value p [rad/s] [rad/s] / p % / p % I critical speed II critical speed III critical speed After copletig the easureets, copare the easured ad calculated values of critical speeds. Calculate the ratio p / ad write it i the last colu of Table 8.. ake drawigs of the observed deflectio shapes of the shaft. 5. aboratory report should cotai: ) Ai of the exercise. ) Experietal ad calculatio results i the for of Table 8.. ) Drawigs of the observed deflectio shapes of the shaft correspodig to the cosecutive critical speeds.
) Coclusios ad rearks. Refereces. Tse F.S., orse I.E., ikle R.T.: echaical ibratios - Theory ad Applicatios, Ally ad Baco Ic., Bosto, odo, Sydey, Toroto, 978.. Kapitaiak T.: Wstęp do teorii drgań. Wydawictwo olitechiki Łódzkiej, Łódź 99.. arszewski Z.: Drgaia i dyaika aszy. WN, Warszawa 98.