Introduction to Mechanical Vibrations and Structural Dynamics

Transcription

1 Inroducion o Mechanical Viraions and Srucural Dynaics The one seeser schedule :. Viraion - classificaion. ree undaped single DO iraion, equaion of oion, soluion, inegraional consans, iniial condiions.. ree daped single DO iraion, equaion of oion, soluion, inegraional consans, iniial condiions. 3. The ending siffness, he spring asselies. 4. orced iraion, iraion under consan force. 5. orced iraion, haronically arying eernal force, paricular soluion, haronic response, apliude and phase characerisic. 6. orced iraion due o he cenrifugal force. 7. The roaional iraion, free and forced, orsional siffness. 8. The wo DO free iraion, he ode shape concep, he ode shape soluion, he ode shape orhogonaliy. 9. The wo DO free iraion - coplee soluion including inegraional consans.. The odal coordinaes, he odal ransforaion, he free (unconsrain syse), he syerical syse.. The odal analysis of he n-do syse, ari forulaion.. The haronic response of he n-do syse. 3. Viraion of recilinear eas.

2 . Viraion - classificaion. ree undaped single DO iraion, equaion of oion, soluion, inegraional consans, iniial condiions. The iraion classificaion. free (naural) iraion undaped iraion linear iraion iraion of discree ass (asses) iraion wih degree of freedo (DO) forced iraion daped iraion non-linear iraion iraion of coninuously disriued ass iraion wih ore DO ree undaped single DO iraion he spring he spring he ass iraion The spring siffness : he spring free lengh undefored spring he spring lenghening defored spring The ain spring paraeers : - he spring free lengh, he lengh of undefored spring (no lenghened nor pressed) [unis are,,... in square races he unis will e enioned in he whole e], - he spring lenghening (elongaion) [, ], - he lenghening force [N], - he spring siffness [N/, N/].

3 The eaple : The siffness of he spiral spring is : where : 4 G d 8 n D G - he elasic odulus (Young odulus) for shear deforaion [Pa N/, MPa N/ ], d - he spring wire diaeer [, ], D - he spiral diaeer [, ], n - nuer of spring coils [-]. 3 The spring poenial (deforaional) energy : he spring free lengh undefored spring he spring lenghening E P The acion - reacion principle : defored spring d d he spring free lengh undefored spring he spring lenghening S defored spring S - he eernal lenghening force, he acion (lac on figure), - he spring force, he inernal reacion force, caused y he spring deforaion, he spring resisance agains deforaion (red on figure). S 3

4 The equaion of oion he spring free lengh undefored spring he equiliriu posiion a defored spring S = where is he ass, a i S i a a is he acceleraion, is siffness, is displaceen. where : The soluion is : sin displaceen [] cos s deriaie, elociy [/s] sin a nd deriaie, acceleraion [/s ] placed ino he equaion of oion : sin sin sin sin sin sin 4

5 In he soluion : sin are : - ie, independen ariale [s], - he naural circular frequency [s - ], f - he naural frequency, nuer of cycles per second [s -, Hz], T - he period, he ie of one cycle [s], f, - inegraional consans. The soluion ie cure : T sin The inegraional consans physical eaning : - he apliude, he displaceen aiu alue, [, ] - he phase shif, he phase angle, [rad] diided y circular frequency gies he shif of he sinus cure o he lef on he ie ais (in seconds). The inegraional consans soluion The iniial condiions - he saus in he eginning of he iraion, in ie = =... = iniial displaceen, = iniial elociy. In he soluion : sin cos 5

6 6 use he iniial condiions : sin cos siply : sin cos he soluion of he wo equaions wih wo unnowns : arcan The apliude and phase shif are deerined y he iniial condiions. The alernaie epression of he displaceen soluion : cos sin sin cos sin where : cos sin are inegraional consans (in opposie o and, and hae no direc physical eaning) Inegraional consans soluion fro iniial condiions : cos sin sin cos The inegraional consans are hen : and : arcan arcan

7 The noe o he nuerical calculaion of he arcg funcion : The funcion arcg has always roos. or eaple : arcg.5 = 6.6, arcg,5 = 6,6. ecause oh g 6.6 = g 6,6 =.5 arcan II quadran III quadran I quadran IV quadran The coon calculaor always reurns he roo in he ineral -9, 9. u if <, he proper roo is shifed 8º or rad. arcan > 9, 8 (II. quadran) < arcan < > while calculaor reurns -6.6! -8, -9 (III. quadran) arcan 6. 6 or arcan while calculaor reurns 6.6!, 9 arcan , arcan 6. 6 (I. quadran) (IV. quadran) 7

8 . ree daped single DO iraion The daping is a physical phenoenon, which causes he decrease of iraion unil i copleely disappears. The causes of daping : - he enironenal resisance agains oion (in liquid, in air), - he inernal fricion inside aerial srucure - he aerial daping, - echnical deice - daper. he syol of daping The force resisance agains oion - he daping force : here : - coefficien of daping [N s - ], - elociy [/s]. The equaion of oion : s a a i S i a where : is he decay consan he soluion is : e sin 8

9 where : - naural circular frequency of daped iraion [s - ], The soluion ie cure : f - he naural frequency (he nu. of cycles per sec.) [Hz], T - he period (he ie of one cycle) [s], f, - inegraional consans. T () (+T) e T T 3 T 4 T 5 T 6 T 7 T e sin The inegraional consans soluion The iniial condiions - he saus in he eginning of he iraion, in ie = =... = iniial displaceen, = iniial elociy. used in he soluion, (alernaie epression) : e e e e sin e cos sin cos sin cos sin cos sin arcan arcan 9

10 urher : is he daping raio, ln T is he logarihic decreen. T 4 The eponenial funcion e deerines he ie of he iraion disappearing : [] a e a = 37 % [s] ie consan he angenial line Le us define so called ie consan : The funcion is hen : e e The nuerical alues of he funcion can y calculaed as follows : = = = 3 = 4 = 5 a_ e e e, 37 a_ e e e, a_ e e e, 5 4 a_ e e e, a_ e e e, 7 a = 37% a = 4% a = 5% a = % a =,7% Tha can e seen ha in he ie = 5 he res of he iraion apliude is less han % of he iniial alue. This ie can e undersood as he iraion disappearing ie.

11 3. The ending siffness, he ending iraion, he spring asselies The ending siffness Suppose he ass paricle on he free end of he fleile ea (of he negligile ass). The paricle can irae. y Le us epress so called ending siffness of he fied ea. E J y The ending of he fied ea under he force can e epressed as : 3 y 3 E J where - he ending force [N], - he ea lengh [], E - he ea aerial Young odulus [Pa N/ ], J - he ea cross secion quadraic oen of ineria [ 4 ]. The ending siffness can e epressed as : 3 E J in ers [N/] 3 y The ea cross secion quadraic oen of ineria can e calculaed as : d J d 4 64 full circle J 3 recangle

12 ll cases of iraion can e susequenly soled as he ass-spring syse. Of course he ea-suppor syse can e differen : / E J / 48 E J 3 a E J 3 E J a E J a 3 E J

13 The siffness of he hydraulic circui Suppose he hydraulic cylinder, eposed o he force. he force he pison V y he hydraulic cylinder V he hydraulic liquid - he force, pressing he hydraulic liquid [N], V - he iniial olue of he liquid [ 3 ], V - he olue change (decrease), he copressed olue [ 3 ], K p - he odule of he liquid olue copressiiliy [Pa] (appro. K = GPa), - he hydraulic pressure [Pa], - he pison cross secion area [ ], y - he pison displaceen []. The pressure - relaie copression forula (analogous o he Hoo s law for solid aerials) V p K V here V V represens he relaie olue change (decrease). Pressure can e epressed as : The copressed olue can e epressed as : p V y 3

14 Then : V p K V is : y K V and finally : hyd y K V can e undersood as hydraulic siffness in ers [N/]. The following soluion is equal o he ass - spring syse. S V y V hyd 4

15 The hydraulic circui can conain ore han only he hydraulic cylinder. S V y V In his case V is no only he olue under he cylinder pison, u olue of he hydraulic liquid in he whole circui. 5

16 The spring asselies The wo springs can e coined in wo ways. The parallel assely : The wo springs wih siffnesses and are coined side y side as shown on figure. s s The eernal force drags oh springs hrough coon ase. There are wo iporan eys :. he deforaion of oh springs is he sae,. he spring forces s and s in oh springs (red on figure) are suarized and in equiliriu wih he eernal force (lac on figure). s s s The su of siffneses ( + ) can e inerpreed as so called oal siffness : and he syse ehaes as wih one spring oal s real syse susiuional syse oal 6

17 The serial assely : The wo springs wih siffneses and are coined in one line as shown on figure. oal = / oal s s s = s / = s / The eernal force drags oh springs in poin. There are wo iporan eys :. he oal deforaion oal is he su of oh parial deforaions of oh springs, oal. he spring forces s and s (red on figure) are in equiliriu in poin and are equal, he spring force s and he eernal force (lac on figure) are in equiliriu in poin and are equal. s s The parial deforaions can e epressed as : liewise : s oal oal oal s oal In he noinaor = s = s can e canceled and finally : oal oal s s oal real syse susiuional syse oal 7

18 The classificaion of he spring assely can e confusing due o he isual for. The assely sech on fig. loos siilar o he assely sech of he serial assely. u only loos siilar. u... s s. The ody displaceen represens deforaion of oh springs and, hen :. The su of spring forces us e in he equiliriu wih eernal force : s s Tha is clear, ha he eaple represens he parallel assely and oal siffness is : oal real syse oal susiuional syse... 8