ANALYSIS OF LINEAR AND NONLINEAR EQUATION FOR OSCILLATING MOVEMENT

Transcription

1 УД Anna Macurová arol Vasilko Faculy of Manufacuring Tecnologies Tecnical Universiy of ošice Prešov Slovakia ANALYSIS OF LINEAR AND NONLINEAR EQUATION FOR OSCILLATING MOVEMENT Macurová Anna Vasilko arol 7 Моделювання процесу вібрації машин вимагає знання характеру диференціального рівняння вібраційного руху Проаналізовано лінійне та нелінійне диференціальні рівняння які моделюють процес виникнення стійких та нестійких вібрацій у вібраційних системах Modelling of oscillaing processes requires one o know e caracer and properies of a differenial equaion for oscillaing oveen Tis paper presens an analysis of e soluions for linear and nonlinear equaions of oscillaing oveen aied a solving oscillaions in ecnological syses of vibraing acines Proble definiion Te applicaion of a nonlinear oscillaing syse is deonsraed o be suiable for e analysis of vibraing acines for ecnological processing of ecanical pars For e realisaion i is necessary o asseble a aeaical apparaus suiable for odelling of is process We aep for is purpose a soluion of a se of linear and nonlinear differenial equaions for an oscillaing syse Analysis of soluions of equaion & + ω ε f ( were f ( u F cos( ω u A siple eaple of oscillaing oion is provided by a solid objec wi weig wic is deviaed fro is seady sae posiion According o Hooke s Law ere is a force F (elasic wic acs on e deviaed objec wic is proporional o e deviaion in e for of F k were k k > is a caracerisic consan Te objec will ove (oscillae under e influence of is force Te equaion in e for & k d d were is e weig of e objec and ( & & epresses e oscillaory oion on d d assupion a e graviaion forces and resisance of e environen are disregarded Wen oscillaing objec is oving in e environen wic presens e oving objec wi resisance and e resisance is proporional o speed & us e resisance is q& were q > is e consan of is proporion Equaion epressing e oveen of an oscillaing objec becoes & + q& + k Equaion above represens oion wic is called an independen oscillaion Afer rearranging Equaion becoes q k & + & + Le s denoe q p and k ω Differenial equaion can be en rewrien as & + p& + ω (3 96

2 Equaion (3 is a linear differenial equaion wi consan coefficiens and wi e rig side equal o zero Is caracerisic equaion is λ + p λ + ω (4 and is roos are λ p + p ω λ p p ω Tree scenarios are possible ere a Te roos of Equaion (4 are cople Le s denoe p ω Ω were Ω > Equaion (4 will ave cople roos wen p ω < wic is wen q < k In is scenario e roos of Equaion (4 are λ p + iω λ p iω Ten a general soluion of differenial Equaion is e funcion p p ce cos( p ω + ce sin( p ω or in e for of p p c e cos( Ω + ce sin( Ω (5 were cc are real consans Funcion (5 represens e posiion of oscillaing poin in ie For e oveen o occur i is necessary a for consans cc i olds a c i a leas for one i I is proven a e oveen wic occurs in is case as e following caracerisic propery: Te oscillaing objec crosses e seady sae posiion (null posiion indefinie nuber of ies over equal ie inerval T Tis is called a alf-period of oscillaion wile ie T is called a period of oscillaion Le s subsiue e following for consans i c i c r sinα c r cosα were r > Ten i becoes possible o rearrange Funcion (5 as follows p e ( r sin α cos( Ω + r cosαsin( Ω fro wic p e r sin ( Ω +α (6 A grapical represenaion of is funcion for one paricular eaple is sown in Fig Fig Grapical represenaion of funcion e sin 97

3 Funcion (6 as a rivial soluion of wen Ω + α is an ineger uliple of π ie Ω k + α kπ were k Tis eans a oscillaing objec crosses e seady posiion infinie nuber of ies in ie k Te ie period beween wo subsequen crossings of e seady sae posiion can be derived fro e following equaions + α ( k π fro wic we can derive Ω k + + Ω k + α nπ ( π Ω k + k Hence for a alf period of oscillaion T π k + k Ω As q Ω ω p p en π π T ω p 4 ω q Paraeer Ω represens circular frequency of e observed oion Suc a oion for wic e objec crosses e seady sae posiion a leas wice we call oscillaory oion oerwise we refer o non-oscillaory or non-vibraing oion b Te second scenario for e caracerisic Equaion (4 wen e roos are bo real and differen provides for a generic soluion of differenial Equaion in e for λ c e λ + c e c c R λ p + p ω λ p p ω were p ω > q as p en q > 4ω and q > ω In is case e objec crosses e seady sae posiion a os once wen ie equaion λ λ c e + ce wen c i i as a os one soluion In is case e ype of oion is non-oscillaory (nonvibraing Te corresponding grapical represenaion of is funcion for one paricular eaple is sown in Fig c Te ird scenario for e roos of caracerisic Equaion wen λ p were p ω ie p ω p ω ω > p > A generic soluion of differenial Equaion is a funcion p p c e + c e p e ( + or e ( + λ c c ω c c us e oion is non-oscillaory Is grapical represenaion for one paricular eaple is sown in Fig 3 98

4 Fig Grapical represenaion of funcion e + e Fig 3 Grapical represenaion of funcion e + e By using e lii of li we will sow a for all ree scenarios wi increasing e deviaion of converges o ie funcion ( represens daped oveen (daped vibraion daped non-oscillaory oveen If p > ie q > e following relaions old: p a lie ( c cos( Ω + c sin( Ω li λ λ b li li( c e + c e li c e were p + p ω < p p ω < p+ p ω p p ω + ce p p c li li( c e + ce If ere is no resisance fro environen ie wen q us p Te oveen can be described by e equaion & ω or & + ω ω & + 99

5 If we consider a uni weig ( en we will analyse only a siplified equaion & + ω If ere is an eernal force P wic is only a funcion of ie and wic acs on a oving syse us e resuling oion is described by e following differenial equaion & + q& + k P( (7 and is is an equaion of inernal oscillaion One of e siples fors of is funcion is P F cos( ω ω > ie funcion P is periodic Non-linear second order differenial equaion Le s consider e soluion properies of e following differenial equaion M + B + u + d u (8 ( ( ( ( were M ( u u( P ( ε f ( u are coninuous funcions I is possible o epress Equaion (8 using a syse of differenial equaions Le ( T ( ( ( B ( u M M M + (9 is a general soluion of Syse (9 For eac soluion ( J we assue a i eiss wiin inerval J Le s denoe e rig boundary posiion of inerval J as > > us J Le s in Equaion (8 se ( following syse were ( C M B u en differenial Equaion (8 is epressed by e ( D R C ( D J R R ( M a Le s assue Syse suc a ( are so a c c is a consan and ( c( c R en all soluions of J of Syse R For becoes -rivial soluion and -consan soluion a Le s assue Syse suc a ( Le for eac non-rivial soluion J of Syse eis a funcion r > and funcion u suc a u C ( J R u were C ( J R c is a space of real funcions wic can be derived and ese funcions are of one real variable defined over e inerval J Le e following old for soluions J i i r cosv r( sin v( r is called a polar funcion and funcion Funcion v is called an angular funcion Syse is epressed using equaions in e for r v r sin v v r sin v cos ( ( (

6 were ( ( ( ( r cosv r sin v + r cosv( v ( M r cosv Afer rearranging Syse we derive equaions r ( ( r( cosv( sin v cosv cosv sin v r M (3 v ( sin v ( r cosv ( cosv π v ( k + k Z M For reasons of breviy we denoe I i i inegrals as follows I sin y cos y I cos ( r cos y M sin y cos y d is an ineger ( ( r cos y π sin y ( y d v ( k + k Z M (4 y J denoes a coninuous funcion Le for all coninuous funcions y J eis inegrals I I as defined in e following saeens a o f Ten all non-rivial soluions J of Syse (9 are: a unbounded wen I I ± b kπ unbounded wen I I were R is a consan + u( k Z is an ineger c bounded so a wen I I ± d bounded so a wen I I were R is a consan kπ + u( k Z is an ineger e bounded wen I L L > I ± f bounded wen I L L > I R ineger Inegraing Syse (3 over e inerval r ( r( ep sin y cos y ( v v( + ( y( kπ is a consan + u( k Z yields ( r cos y M sin y cos y d is an ( r cos y sin ( cos y d (5 M Considering e assupion a a cerain value of r ( eiss suc a r ( > ( r ( r is bounded (unbounded en eac non-rivial soluion ( J bounded (unbounded Te siuaion wen r ( if a polar funcion In our fuure work we will consider Equaion (8 in e following scenarios: wen funcion P ε f ; P as values ( ie of Syse (9 is occurs eacly wen I or if u

7 wen ere is no resisance fro environen ie for q us p ie e oveen is described by equaion & ω or & + ω ω & + If we consider a uni weig ( en we will analyse only a siplified equaion & + ω Conclusions In e firs par of is paper we ave analysed a linear differenial equaion of an oscillaing oveen In e following paper we will coninue eplore e soluions of is non-linear alernaive ursweil J Obyčajné diferenciálne rovnice Praa 978 Ponrjagin LS Prosyje differencialnie urovnenija Moscow 98 3 Socko Z Maeaical Design of Vibraory-Cenrifugal Hardening of Surface of Cylindric 4 Long-Sized Macine Pars In:Conference: New ways in anufacuring ecnologies 6 Prešov 366 S