Technical Vibration - text 2 - forced vibration, rotational vibration
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1 Technicl Viion - e - foced viion, oionl viion 4. oced viion, viion unde he consn eenl foce The viion unde he eenl foce. eenl The quesion is if he eenl foce e is consn o vying. If vying, wh is he foce funcion. The consn foce nd he honiclly vying foce will e nlyzed. The viion unde he consn eenl foce he fee lengh undefoed sping he equiliiu posiion s v e = cons The equion of oion : i e s e
2 Technicl Viion - e - foced viion, oionl viion The heicl ppoch : ho Ce hoogenous picul sin p p p consn inlly : consn consn e s e whee Ce sin s s is so clled sic defoion (he defoion of he sping unde he consn foce ). e [] s Ce sin s [s]
3 Technicl Viion - e - foced viion, oionl viion The echnicl ppoch : Le us define he sic defoion s : e s Le us hen ove he oigin of he coodine syse y s (see figue). ol = s + () he fee lengh undefoed sping he equiliiu posiion s s v e = cons In he equiliiu posiion he sping foce, coesponding o he sic defoion (so clled sic pe-sess), is in he equiliiu wih he eenl consn foce e. s _ s s e The equion of oion is : e i e ol s v The ol defoion ol cn e epessed s he su of sic defoion s nd vying displceen () : The equion of oion is hen : nd : Becuse (see ove) : ol s e v s v e e s s 3
4 Technicl Viion - e - foced viion, oionl viion he igh side of he equion is hen equl o zeo : v In he end he equion of oion is he se s his fo fee (nul) viion. The soluion is lso he se : Ce sin The conclusion :. The viion unde consn eenl foce is nlyzed he se s fee (nul) viion. Bu :. The equiliiu posiion - he oigin of he coodine syse, is deeined no y he sping fee lengh u y he sic defoion s. 3. The ol loding of he sping is given y wo coniuions, sic nd dynic : s _ ol s _ s s _ dyn s 4
5 Technicl Viion - e - foced viion, oionl viion 5. The viion unde he honiclly vying eenl foce The honiclly vying eenl foce (eciing foce) ens foce, vying ccoding o he sinus funcion : sin hee - he pliude of he eciing foce (iu vlue) [N], - he cicul fequency of he eciing foce [s - ], f - he fequency of he eciing foce [Hz], T - he peiod of he eciing foce [s]. f s sin () The equion of oion is : v sin The soluion hs wo ps - hoogenous soluion nd picul soluion. hoogenous picul T f The hoogenous soluion is he soluion fo zeo igh side of he equion of oion, epesens fee (nul) viion (ed line see dig elow) : ho Ce sin T ho Ce sin o he pees see fee (nul) viion. 5
6 Technicl Viion - e - foced viion, oionl viion The picul soluion eflecs non-zeo igh side, epesens foced viion (lue line see dig elow) : p sin hee - he pliude of he picul soluion [, ], - he cicul fequency of he eciing foce [s - ], - phse shif [d]. T sin p The soluion of he equion of oion is he supeposiion of oh hoogenous nd picul soluion : hoogenous picul Ce sin sin Ce sin sin ho Ce sin sin p coplee soluion hoogenous soluion picul soluion nsien pocess sedy se The coplee soluion cn e spli ino wo phses.. phse - nsien pocess The soluion is deeined s supeposiion of oh hoogenous soluion (fee viion) nd picul soluion (foced viion). The ie ehvio is he copliced ecuse of he coinion of wo diffeen fequencies. Tes liied ie.. phse - sedy se Ss he oen he fee viion disppes (due o dping). Sedy se is epesened y he only picul soluion. Tes infinie ie. 6
7 Technicl Viion - e - foced viion, oionl viion The pliude nd phse shif clculion The picul soluion nd is deivives : p p p us sisfy he equion of oion : sin cos sin sin sin cos sin sin Using he igonoeic fouls : sin cos he equion cn e epessed s : sin cos cos sin cos cos sin sin cos sin cos sin sin cos sin cos sin Coping of he ces wih sin nd cos ee leds o : o he second equion diecly : o n cos sin The noe :, 8º o, d. sin cos sin n cos cn cos sin o he fis equion, if sin n nd n cos n hen : 7
8 Technicl Viion - e - foced viion, oionl viion o susequen nlysis i is useful o define he diensionless pees : he uning coefficien, he dping io (he elive dping). Hee is he nul cicul fequency of he fee undped viion. The cicul fequency of he eciing foce cn e epessed s : he decy consn cn e epessed s : Puing hese epessions ino he foul fo pliude of he foced viion : fcoizing : if hen siilly : s cn cn The io (pliude of he eciing foce) nd (siffness) is so clled sic defoion. This epesens he sic defoion of he sping unde he consn foce. s The physicl ening of he pliude is siply iu displceen. 8
9 Technicl Viion - e - foced viion, oionl viion The physicl ening of he phse shif is he ie dely of he displceen esponse () fe he foce ()., sin sin The iu vlue of displceen (lue cuve on he dig ove) ppes cein ie dely le hn he iu vlue of he eciing foce (ed cuve). This ie dely is deeined s whee - phse shif [d], - cicul fequency of he eciing foce [s - ]. The noe : Be ceful,, 8º o, d. If <, he clculo euns he vlue -9º,. Bu he coec esul is 9º, 8º. The eseche is esponsile o dd 8 º. The pliude nd phse shif soluion fo undped viion -,. s s s, cn cn cn = º if o <, = 8º if o >. 9
10 Technicl Viion - e - foced viion, oionl viion The pliude chceisic nd he phse chceisic These wo e he pliude s funcion of (cicul fequency of he eciing foce) nd phse shif s funcion of. The pliude chceisic s pliude ) = =, =, - dping io s ) - uning coefficien 3) es = = = On he dig he ed cuve efes o he undped foced viion, while lue nd geen cuves efe o he dped foced viion. Poin ) coesponds o he consn foce (zeo fequency, infinie long peiod), pliude is equl o he sic displceen = s. Poin ) coesponds o so clled esonnce. The esonnce ppes if o. In esonnce he pliude inceses o high vlues. Poin 3) coesponds o he eeely high fequency of he eciing foce. In his cse he sque oo in denoino of he epession fo inceses infiniely, he whole fcion nd lso he pliude deceses o zeo.
11 Technicl Viion - e - foced viion, oionl viion The phse chceisic cn cn phse shif = = / =, =, - uning coefficien 3 Wih incesing fequency of he eciing foce nd incesing uning coefficien he phse shif inceses fo = o = 8º ( d). o lge dping (ed cuve on dig) he cuve is sooh, fo slle dping (lue cuve) he funcion ends o sep chnge. In esonnce (see ove) he phse shif hs he vlue close o = 9º (/ d).
12 Technicl Viion - e - foced viion, oionl viion 6. The foced viion due o cenifugl foce The viing syse is epesened y he ss, siffness nd coefficien of dping. The p of he viing ody is oing ss o. y cen = o e,, e = cen sin( ) v o e - he oing ss [g], - he ngul velociy of he oing ss [d/s], - ecceniciy, he disnce of he cene of ss fo he is of oion []. The oing poduces he consn cenifugl foce : cen o e The cenifugl foce oes wih he oing ss wih he ngul velociy. The ngle eween he cenifugl foce nd y is (pependicul o he is, he diecion of viion) inceses s : The coponen of he cenifugl foce (he coponen pesen in he equion of oion) is : cen sin cen sin The equion of oion is hen : cen sin nd he picul soluion (see ove) is : p sin
13 Technicl Viion - e - foced viion, oionl viion The susequen soluion hs wo levels. Level A, he soluion fo given oionl speed. If he ngul velociy is defined s consn, he coesponding cenifugl foce cn e clculed : cen o e The pliude of foced viion is : cen cen 3
14 Technicl Viion - e - foced viion, oionl viion Level B, he pliude chceisic If we suppose vying oionl speed, hen he cenifugl foce depends on he ngul velociy nd he pliude of foced viion is : cen o e e o e o The funcion is diffeen fo he one if he pliude of he eciing foce is supposed o e consn. ) = =, - dping io =,35 e ) 3) 3 4 es = = = 3 - uning coefficien Poin ) coesponds o he zeo oionl speed, zeo cenifugl foce nd susequenly zeo displceen. Poin ) coesponds o so clled esonnce (see ove). Poin 3) coesponds o he eeely high oionl speed. Then he lii gives he pliude. o li e e o 4
15 Technicl Viion - e - foced viion, oionl viion 7. The oionl viion, fee nd foced, osionl siffness The fee undped oionl viion Suppose he igid ody suppoed y he pin join nd fleile sping. In he equiliiu posiion he ody - he hin od is in he hoizonl posiion, he sping is in he veicl posiion, pependicul o he od. The ody will oe., I oion y = p sin p S p I - he ss oen of inei of he ody [g ], p - he sping siffness [N/], - he, he pependicul disnce of he pin-join o he sping is [], - he oionl ngle, he ngul coodine [d], - he ngul velociy [d/s], - he ngul cceleion [d/s ]. The echnicl unis : - he oionl ngle [º, evolue], d = (8/)º 57,3º, evolue = 36º = d 6,8 d. n - he oionl speed [ev/in], n 3, ev/in = /3 d/s,5 d/s. 5
16 Technicl Viion - e - foced viion, oionl viion Duing oion he sping is lenghened nd he sping foce S ppes. If he oionl ngle is he sll, he soluion cn e lineized. The equion of oion : I M i p S The sping foce is : he sping lenghening is : S p p sin p The noe : The siplificion sin is used. The eo depends on he vlue of he ngle sin eo sin sin [ ] [d] [-] [%],7453,745,5 % 5,877,876,3 %,7453,7365,5 % 5,6,59, %,349,34 % 3,54,5 5 % 6,47,866 % 9,57 57 % The siplificion sin is used usully if <5º. The equion of oion hen is : The susiuion = p oligoy). The equion of oion is hen : I p I p (oionl siffness [N /d]) cn e used (voluny, no I I The equion of oion cn e coped wih he equion of oion of he nslionl viion of he ss picle. 6
17 Technicl Viion - e - foced viion, oionl viion The pllel (nlogy) eween oh cn e seen. he nslionl viion of he ss picle equion of oion he ss he siffness he coodine nd i s second deivive he soluion C sin he nul cicul fequency he iniil condiions =... =, v=v - iniil displceen v - iniil velociy C pliude v phse shif cn v he oionl viion of he ss ody equion of oion I insed of he ss he ss oen of inei I is used insed of he siffness he oionl siffness = p is used insed of he coodine oionl coodine nd i s second deivive is used he soluion C sin he nul cicul fequency I p I he iniil condiions =... =, = - iniil oionl ngle - iniil ngul velociy pliude C phse shif cn 7
18 Technicl Viion - e - foced viion, oionl viion The fee dped oionl viion Suppose he igid ody suppoed y he pin join, fleile sping nd dpe. In he equiliiu posiion he ody - he hin od is in he hoizonl posiion, oh sping nd dpe e in he veicl posiion, pependicul o he od. I, oion y p v = q S q p Ecep of I,, p,, nd (see ove) : - he coefficien of dping [N - s], q - he, he pependicul disnce of he pin-join o he dpe is [], Duing oion he sping is lenghened nd he sping foce S ppes. The dpe pison oves nd he dping foce ppes. The sping foce is : The dping foce is : S p v q q The cicufeenil velociy is : whee is he ngul velociy. The equion of oion : v q I M i p q p p q q S I q I q p p Afe he susiuion = p s he oionl siffness [N /d] nd = q s he oionl coefficien of dping [N s] : I I The equion of oion is nlogous o he one of nslionl viion of he ss picle : 8
19 Technicl Viion - e - foced viion, oionl viion The pllel (nlogy) eween oh cn e seen. he nslionl viion of he ss picle equion of oion ecep of he ss, he siffness nd coodine nd i s deivives, he oionl viion of he ss ody equion of oion I ecep of he ss oen of inei I, he oionl siffness = p nd he oionl coodine nd i s deivives, he coefficien of dping Ce he soluion sin he nul cicul fequency of undped viion he decy consn he nul cicul fequency of dped viion insed of he coefficien of dping he oionl coefficien of dping Ce = q is used he soluion sin he nul cicul fequency of undped viion I p I he decy consn q I I he nul cicul fequency of dped viion he iniil condiions =... =, v=v - iniil displceen v - iniil velociy pliude C v phse shif cn v he iniil condiions =... =, = - iniil oionl ngle - iniil ngul velociy C pliude phse shif cn 9
20 Technicl Viion - e - foced viion, oionl viion The foced oionl viion Suppose he igid ody suppoed y he pin join, fleile sping nd dpe. In he equiliiu posiion he ody - he hin od is in he hoizonl posiion, oh sping nd dpe e in he veicl posiion, pependicul o he od. The honiclly vying eenl foce () ipcs on he ody.,, I oion S () = sin( ) q p Ecep of I,,, p, q,, nd (see ove) : () = sin( ) - he honiclly vying eenl foce [N], - he pliude of he foce [N], = f - he cicul fequency of he foce [s - ], f - he fequency of he foce [Hz], - he, he pependicul disnce of he pin-join o he foce []. Noe : The suden us PERECTELY diffeenie eween - he cicul fequency of he eenl foce nd - he ngul velociy of oion. The equion of oion is : I q I q p p sin sin Afe he susiuion = p s he oionl siffness [N /d] nd = q s he oionl coefficien of dping [N s] nd M = s he pliude of he eenl oen [N ] : I M sin
21 Technicl Viion - e - foced viion, oionl viion The equion of oion is nlogous o he one of nslionl viion of he ss picle : sin The pllel (nlogy) eween oh cn e seen. he nslionl viion he oionl viion of he ss ody of he ss picle equion of oion equion of oion sin I M sin ecep of he ss, he coefficien of dping, he siffness nd coodine nd i s deivives, ecep of he ss oen of inei I, he oionl coefficien of dping = q, he oionl siffness = p nd he oionl coodine nd i s deivives, he picul soluion sin s pliude phse shif cn cn whee : s - he sic defoion - he uning coefficien - he dping io M I I he picul soluion sin s pliude phse shif cn M cn whee : s - he sic p defoion - he uning coefficien - he dping io
22 Technicl Viion - e - foced viion, oionl viion The osionl siffness Suppose he long, hin od, fied on one end (op end on figue), eposed o he osionl oen (he oque). This esuls o he osionl defoion of he od, he osionl ngle. G, J p T - he lengh of he od [, ], G - he she elsic odulus (Young odulus) of he od eil [P, MP], J p - he coss secion pol qudic oen of inei [ 4, 4 ], e.g. fo cicul coss secion of he diee d, he pol oen of inei is : M T J p 3 d 4 M T - osionl oen (he oque) [N ], - he osionl ngle [d]. The osionl defoion of he od, epessed s he osionl ngle, is : M T G J The io of he osionl oen nd he osionl ngle cn e epessed s osionl siffness [N /d] : T M T P G J P Once, when he osionl defoion ppes, he ecion oen M R cs gins defoion. M T is he eenl oen s he cuse of defoion, M R is he secondy ecion oen s he esisnce gins defoion (nlogous o he sping foce). M T M R T
23 Technicl Viion - e - foced viion, oionl viion Suppose fhe he syeicl dis on he fee end of he osionl od. T I M R I - ss oen of inei of he dis [g ], T M R = T f - he osionl siffness of he od [N /d] - he ecion oen [N ], - he osionl ngle [d], - he ngul velociy [d/s], - he ngul cceleion [d/s ], The noe : Suppose he fee end of he od is suppoed y he eing o void he ending defoion. The equion of oion of fee undped osionl viion is : o including dping : I I T M i I T M R T T o fo foced viion : I T T M sin The soluion see ove.... 3
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