Spring Pendulum with Dry and Viscous Damping

Transcription

1 Spring Pendulum with Dry and Viscus Damping Eugene I Butikv Saint Petersburg State University, Saint Petersburg, Russia Abstract Free and frced scillatins f a trsin spring pendulum damped by viscus and dry frictin are investigated analytically and with the help f numerical simulatins An idealized mathematical mdel is assumed (Culmb law) which nevertheless can explain many peculiarities in behavir f varius scillatry systems with dry frictin The amplitude f free scillatins diminishes under dry frictin linearly, and the mtin stps after a final number f cycles The amplitude f sinusidally driven pendulum with dry frictin grws at resnance withut limit if the threshld is exceeded At strng enugh nn-resnant sinusidal frcing dry frictin causes transients that typically lead t definite limit cycles peridic steady-state regimes f symmetric nn-sticking frced scillatins which are independent f initial cnditins Hwever, at the subharmnic sinusidal frcing interesting peculiarities f the steady-state respnse are revealed such as multiple cexisting regimes f asymmetric scillatins that depend n initial cnditins Under certain cnditins simple dry frictin pendulum shws cmplicated stick-slip mtins and chas Keywrds: dry frictin, dead zne, sinusidal frcing, resnance, threshld, steady-state regime, asymmetric scillatins 1 Intrductin Mechanical vibratin systems with cmbined viscus and dry (Culmb) frictin are f cnsiderable imprtance in numerus applicatins f dynamics in engineering When frictin is viscus, the spring scillatry systems are described by linear differential equatins This case allws an exhaustive explicit analytical slutin which is usually studied in undergraduate curses at universities and can be fund in mst textbks n general physics Hwever, the influence f dry frictin n scillatry systems remains as a rule beynd the scpe f the academic literature and traditinal physics curses Dry frictin results in a nnlinearity With dry frictin, the system acquires a nn-smth, discntinuus nnlinear character If the cefficient f dry frictin is sufficiently small, the scillating bdy slides under harmnic frcing and its velcity is zer nly fr the instants at which the directin f mtin reverses This kind f mtin f dry frictin scillatr with n stick phase is usually referred t as a pure slip (nn-sticking) mtin At strng enugh dry frictin sticking may ccur: the bdy remains at rest fr a finite time during the driving cycle after its velcity reaches zer A detailed histrical review n dry frictin and stick-slip phenmena can be fund in [1] Dry frictin as a nnlinearity is the current fcus f research activities Even the simplest dry frictin mdel, the Culmb frictin, can explain the principal peculiarities in mtin f dry frictin scillatr Damping f free scillatins under dry frictin is very clearly described in the textbk f Pippard [2] (see als [3]) Different appraches t the prblem are discussed in [4] [5] Den Hartg [6] was first t slve in 193 the peridic sliding respnse f a harmnically frced scillatr with bth viscus and dry-frictin damping Later n the analytical slutins f Preprint submitted t Internatinal Jurnal f Nn-Linear Mechanics Nvember 17, 213

2 nn-sticking respnses were widely discussed in the cntemprary scientific literature (see [7] [14] and references therein) The prblem was treated by using a number f varius analytical and numerical techniques In recent years, there has been an increasing interest in peridic and chatic mtins f discntinuus dynamical systems because f their imprtant rle in engineering (see, fr example, [15]) In the literature the analytical slutin t the prblem f scillatins in a system with dry frictin is usually btained by a simple methd f stage-by-stage integratin f the differential equatins which describe the system These equatins are linear fr the time intervals ccurring between cnsecutive turning pints, if the simplest (Culmb) mdel is assumed fr dry frictin The intervals are bunded by the instants at which the velcity is zer The cmplete slutin is btained by fitting these half-cycle slutins t ne anther fr adjining time intervals By virtue f the piecewise linear nature f the relevant differential equatins, explicit slutins can be fund fr the time intervals between the successive turnarunds In ur apprach t the prblem we try t rely primarily n the physics underlying the investigated phenmena In this paper we are cncerned with free scillatins f a trsin spring pendulum, and with frced scillatins f the pendulum kinematically driven by an external sinusidal frce, including cases f damping caused by dry (Culmb) frictin, and bth by viscus and dry frictin Mathematically, the pendulum driven by an external frce is equivalent t the spring-mass system with the bdy residing n the hrizntally scillating base The simple frmulae f analytical slutins are cnfirmed by graphs btained in cmputer simulatins New results are related with quantitative descriptin f the resnant grwth f scillatins under sinusidal frcing, and with clsed-frm analytical slutins at sub-resnant frequencies These slutins crrespnd t multiple asymmetric steady-state regimes cexisting at the same values f the system parameters Characteristics f such regimes depend n the initial cnditins Our analytical and numerical slutins are illustrated by a simplified versin f the relevant simulatin prgram (Java applet) available n the web [16] 2 The physical system The rtating cmpnent f the trsin spring scillatr investigated in the paper is a balanced flywheel whse center f mass lies n the axis f rtatin (Fig 1), similar t devices used in mechanical watches A spiral spring with ne end attached t the flywheel flexes when the flywheel is turned The ther end f the spring is attached t the exciter a driving rd, which can be turned by an external frce abut the axis cmmn with the flywheel axis The spring prvides a restring trque whse magnitude is prprtinal t the angular displacement f the flywheel relative t the driving rd In ther wrds, we assume that the flywheel is in equilibrium (the spring is unstrained) when the rd f the flywheel is parallel t the driving rd In the case f unfrced (free, r natural) scillatins in an islated system, the mtin is initiated by an external influence which ccurs befre a particular instant This influence determines the initial mechanical state f the system, that is, the displacement and the velcity f the scillatr at the initial instant These initial cnditins determine the amplitude and phase f subsequent free scillatins The frequency and damping rate f free scillatins are determined slely by the physical prperties f the system, and d nt depend n the initial cnditins Oscillatins are called frced if an scillatr is subjected t an external peridic influence whse effect n the system can be expressed by a separate term, a peridic functin f the time, in the differential equatin f mtin We are interested in the respnse f the system t the peridic external frce The behavir f scillatry systems under peridic external frces is ne f the mst imprtant issues in the thery f scillatins A ntewrthy distinctive 2

3 -d + d -q + q q(t) j(t) Figure 1: Schematic diagram f the driven trsin scillatr with dry frictin characteristic f frced scillatins is the phenmenn f resnance, in which a small peridic disturbing frce can prduce an extrardinarily large respnse in the scillatr Resnance is fund everywhere in physics and s a basic understanding f this fundamental prblem has wide and varius applicatins The phenmenn f resnance depends upn the whle functinal frm f the driving frce and ccurs ver an extended interval f time rather than at sme particular instant In this paper we draw attentin t peculiarities f resnance in an scillatr with dry frictin In ur mdel f an scillatry system, free scillatins f the flywheel ccur when the driving rd is immvable (θ = ) Frced scillatins are excited when the driving rd rtates back and frth sinusidally abut its middle psitin θ = between the angles θ and θ (see Fig 1): θ(t) = θ sin ωt This mde differs frm the dynamical mde cnsidered usually in textbks, accrding t which scillatins are excited by a given external frce exerted n the system Our mde can be called kinematical, because in this mde scillatins are excited by frcing ne part f the system (the driving rd) t execute a given mtin (in ur case a simple harmnic mtin) This kinematical mde is especially cnvenient fr bservatin, because the mtin f the exciter can be seen simultaneusly with scillatins f the flywheel In the Culmb mdel f dry frictin, as lng as the system is mving, the magnitude f dry frictin is assumed t be cnstant, and its directin is ppsite that f the velcity, that is, its directin changes each time the directin f the velcity changes When the system is at rest, the frce f static dry frictin takes n any value frm sme interval F max t F max The actual value f static frictinal frce can be fund frm the requirement f balancing the ther frces exerted n the system In ther wrds, the frce f static frictin adjusts itself t make equilibrium with ther external frces acting n the bdy The magnitude f the frce f kinetic dry frictin is assumed in this mdel t be equal t the limiting frce F max f static frictin In real physical systems dry frictin is characterized by mre cmplicated dependencies n the relative velcity (see, fr example, [17]) The limiting frce f static frictin is usually greater than the frce f kinetic frictin When the speed f a system increases frm zer, kinetic frictin at first decreases, reaches a minimum at sme speed, and then gradually increases with a further increase in speed These peculiarities are ignred in the idealized z-characteristic f dry frictin Nevertheless, this idealizatin allws us t understand many imprtant features f scillatins in real physical systems In ur mdel f a trsin scillatr sme amunt f dry frictin can exist in the bearings f the flywheel axis Because the magnitude f static frictinal trque can assume any value up t N max, there is a range f values f angular displacement called the stagnatin interval r dead zne in which static frictin can balance the restring elastic trque f the strained spring 3

4 At any pint within this interval the system can be at rest in a state f neutral equilibrium, in cntrast t a single psitin f stable equilibrium prvided by the spring in the case f scillatr with viscus frictin The stagnatin interval extends equally t either side f the pint at which the spring is unstrained The strnger the dry frictin in the system, the mre extended the stagnatin interval The bundaries f the interval ±d are determined by the limiting trque N max f static frictin In Fig 1 these bundaries b and +b are shwn fr the case in which the driving rd is in its middle psitin θ = 3 The differential equatin f the scillatr The rtating flywheel f the trsin scillatr is simultaneusly subjected t the restring trque D(φ θ) prduced by the spring, the trque B φ f viscus frictin which is prprtinal t the angular velcity, and the trque N fr f kinetic dry frictin When the exciter is frced t mve peridically accrding t θ(t) = θ sin ωt, the differential equatin describing the rtatinal mtin f the flywheel with the mment f inertia J is thus J φ = D(φ θ sin ωt) B φ + N fr (1) The trque N fr is directed ppsitely t angular velcity φ, and is cnstant in magnitude while the flywheel is mving, but may have any value in the interval frm N max up t N max while the flywheel is at rest: N fr ( φ) = N max sign φ = { Nmax fr φ >, N max fr φ < (2) Here N max is the limiting value f the static frictinal trque It is cnvenient t express the value N max in terms f the maximal pssible deflectin angle d f the flywheel at rest, when the driving rd (see Fig 1) is immvable at its middle psitin θ = : N max = Dd The angle d crrespnds t the bundary f the stagnatin interval Dividing all terms f equatin (1) by J, we get φ + 2γ φ + ω 2 d sign φ + ω 2 φ = ω 2 θ sin ωt (3) The damping cnstant γ is a measure f the intensity f viscus frictin It is intrduced here by the relatin 2γ = B/J The frequency ω = D/J characterizes undamped natural scillatins The sign φ functin is meant t take the undetermined values between 1 and 1 at zer argument, which crrespnds t stick phase The actual value f the static dry frictin trque is such that the system is in equilibrium The differential equatin (1) fr an scillatr with dry frictin, as well as equatin (3), is nnlinear because the trque N fr ( φ) abruptly changes when the sign f φ changes at the extreme pints f scillatin This is the s-called Filippv system [18] In the idealized case f the z-characteristic this is a piecewise smth system, and we may cnsider the fllwing tw linear equatins instead f Eq (3): φ + 2γ φ + ω 2 (φ + d) = ω 2 θ sin ωt fr φ >, (4) φ + 2γ φ + ω 2 (φ d) = ω 2 θ sin ωt fr φ < (5) Whenever the sign f the angular velcity φ changes, the pertinent equatin f mtin als changes The nnlinear character f the prblem reveals itself in alternate transitins frm ne f the linear equatins (4) (5) t the ther 4

5 4 Damping f free scillatins under dry frictin Fr the case f free (unfrced) scillatins the right-hand side f equatins (4) (5) is zer In case the dry frictin is absent (the dead zne vanishes: d = ), damping f free scillatins ccurs slely due t viscus frictin Fr this idealized case the differential equatin f mtin becmes linear It has a well knwn analytical slutin, accrding t which the amplitude f free scillatins under viscus frictin decreases expnentially with time That is, the cnsecutive maximal deflectins f the scillatr frm its equilibrium psitin frm a diminishing gemetric prgressin because their rati is cnstant In an idealized linear system such scillatins cntinue indefinitely, their amplitude asympttically appraching zer The duratin f expnential damping can be characterized by a cnventinal decay time τ = 1/γ The expnential character f damping caused by viscus frictin fllws frm the prprtinality f frictin t velcity Sme ther relatinship between frictin and velcity prduces damping with different characteristics The slutin t equatins (4) (5) fr nn-zer dry frictin (d ) can be fund by using the methd f the stage-by-stage integratin f each f the linear equatins fr the half-cycle during which the directin f mtin is unchanged These slutins are then jined tgether at the instants f transitin frm ne equatin t the ther in such a way that the displacement at the end pint f ne half-cycle becmes the initial displacement at the beginning f the next half-cycle This array f slutins cntinues until the end pint f a half-cycle lies within the dead zne An imprtant feature f free scillatins damped by dry frictin is that the mtin cmpletely ceases after a finite number f cycles As the system scillates, each subsequent change f its velcity ccurs at a smaller displacement frm the mid-pint f the stagnatin interval Eventually the turning pint f the mtin ccurs within the stagnatin interval, where static frictin can balance the restring trque f the spring, and s the mtin abruptly stps At which pint f the interval this event ccurs, depends n the initial cnditins, which may vary frm ne situatin t the next These characteristics are typical f varius mechanical systems with dry frictin Fr example, dry frictin may be encuntered in measuring instruments, such as a mving-cil galvanmeter, in which readings are taken with a needle The needle f the cil may cme t rest at any pint f the stagnatin interval n either side f the dial pint which gives the true value f the measured quantity This circumstance explains the rigin f randm errrs inevitably ccurring in the readings f mving-cil measuring instruments The larger the dry frictin, the larger the errrs f measurement In rder t find the fundamental characteristics f scillatins which are damped under the actin f dry frictin, next we assume that viscus frictin is absent (γ = ) ϕ( t) ϕ( ) t T Dead zne 12 degrees, initial deflectin 175 degrees, initial angular velcity Figure 2: Phase trajectry (left) and graphs f φ(t) and φ(t) (right) fr scillatins whse damping ccurs under dry frictin 5

6 At the initial instant t =, let the flywheel be displaced t the right (clckwise) frm the equilibrium psitin s that φ() >, and then released withut a push In the phase plane shwn in the left-hand part f Fig 2 this initial state is represented by the pint which lies t the extreme right n the hrizntal axis, the φ-axis If this displacement exceeds the bundary f the stagnatin interval, ie, if φ() > d, the flywheel begins mving t the left ( φ < ) and its mtin is described by equatin φ + ω(φ 2 d) = The slutin t this equatin with the given initial cnditins (φ() = φ, φ() = ) is simple harmnic mtin whse frequency is ω The midpint f the mtin is d This pint cincides with the right-hand bundary f the stagnatin interval The displacement d f the midpint frm zer is caused by the cnstant trque f kinetic frictin This trque is directed t the right (clckwise) while the flywheel is mving t the left The initial segment f the φ(t) graph in the right-hand part f Fig 2 is the first half-cycle f the csine curve, whse midpint is at a height f d abve the abscissa axis The amplitude f this scillatin abut the midpint d is φ d The crrespnding prtin f the phase trajectry lies belw the hrizntal axis This curve is the lwer half f an ellipse whse center is at the pint d n the hrizntal axis This pint crrespnds t the right-hand bundary f the stagnatin interval Since the amplitude f the first half-cycle is φ b, the extreme left psitin f the flywheel at the end f the half-cycle is (φ 2b) When the flywheel reaches this psitin, its velcity is mmentarily zer, and it starts t mve t the right Since its angular velcity φ is subsequently psitive, we must nw cnsider equatin φ + ω(φ 2 + d) = The values f φ and φ at the end f the preceding half-cycle are taken as the initial cnditins fr this half-cycle Thus the subsequent mtin is again a half-cycle f harmnic scillatin with the same frequency ω as befre but with the midpint d displaced t the left, ie, with the midpint at the left-hand bundary f the stagnatin interval This displacement is caused by the cnstant trque f kinetic frictin, whse directin was reversed when the directin f mtin was reversed The amplitude f the crrespnding segment f the sine curve is φ 3d In the phase plane this stage f the mtin is represented by half an ellipse lying abve the φ-axis The center f this secnd semi-ellipse is at the pint d n the φ-axis Cntinuing this analysis half-cycle by half-cycle, we see that the flywheel executes harmnic scillatins abut the midpints alternately lcated at d and d The frequency f each cycle is the natural frequency ω, and s the duratin f each full cycle equals the perid T = 2π/ω f free scillatins in the absence f frictin The cmplete phase trajectry is frmed by such increasingly smaller semi-ellipses, alternately centered at d and d The diameters f these cnsecutive semi-ellipses lie alng the φ-axis and decrease each half-cycle by 2d The lps f the phase curve are equidistant The phase trajectry terminates n the φ-axis at the pint at which the curve meets the φ-axis inside the dead zne (between d and d) The jining tgether f these sinusidal segments, whse midpints alternate between the bundaries f the stagnatin interval, prduces the curve that describes scillatry mtin damped by dry frictin (Fig 2) The maximal deflectin decreases after each full-cycle f these scillatins by a cnstant value equal t the dubled width f the stagnatin interval (ie, by the value 4d) The scillatin cntinues until the end pint f sme next in turn segment f the sine curve ccurs within the dead zne ( d, d) Thus, in the case f dry frictin, cnsecutive maximal deflectins diminish linearly in a decreasing arithmetic prgressin, and the mtin stps after a final number f cycles, in cntrast t the case f viscus frictin, fr which the maximal displacements decrease expnentially in a gemetric prgressin, and fr which the mtin cntinues indefinitely Energy transfrmatins in free scillatins damped by dry frictin are shwn in Fig 3 While the flywheel is rtating in ne directin, the trque N max f kinetic frictin is cnstant, and the ttal energy E tt (φ) f the scillatr decreases linearly with the angular displacement, 6

7 E tt E pt E tt E kin E pt -j j T Figure 3: Energy transfrmatins in free scillatins damped by dry frictin φ, f the flywheel This linear dependence f the ttal energy n φ is clearly indicated in the left-hand part f Fig 3, where the parablic ptential well f the elastic spring is shwn A representing pint whse rdinate gives the ttal energy E tt (φ) and whse abscissa gives the angular displacement f the flywheel, scillates with time between the slpes f this well, gradually descending t the bttm f the well The trajectry f this pint cnsists f rectilinear segments jining the slpes f the well These segments are straight because the negative wrk dne by the frce f dry frictin is prprtinal t the angle f rtatin, φ The amunt f this wrk N max φ equals the decrease E tt f ttal energy The time rate f dissipatin f the ttal energy, de tt /dt, is prprtinal t the magnitude f the angular velcity, φ(t) Thus, the greatest rate f dissipatin f mechanical energy thrugh frictin ccurs when the magnitude f the angular velcity, φ, is greatest, that is, when the flywheel crsses the bundaries f the dead zne Near the pints f extreme deflectin, where the angular velcity is near zer, the time rate f dissipatin f mechanical energy is smallest (right-hand part f Fig 3) Unlike the case f viscus frictin, the scillatr with dry frictin may retain sme mechanical energy E fin at the terminatin f the mtin Such ccurs if the final angular displacement (within the dead zne) is nt at the midpint f the stagnatin interval Then the spring remains strained, and its elastic ptential energy is nt zer The remaining energy des nt exceed the value Dd 2 /2 = N max d/2 When the initial excitatin is large enugh, that is, when the initial energy is much greater than Dd 2 /2, the scillatr executes a large number f cycles befre the scillatins cease In this case it is reasnable t cnsider the ttal energy averaged ver a perid f the scillatin, E tt (t) The decrease f E tt (t) during a large number f cycles depends quadratically n the lapse f time because the amplitude f scillatin decreases linearly with time and because the averaged ttal energy is prprtinal t the square f the amplitude If t fin is the final mment when scillatins cease, then at the time t the averaged ttal energy E tt (t) is prprtinal t (t t fin ) 2 This statement (which clearly applies nly fr t < t fin ) is exactly true nly when the flywheel cmes t rest at the center f the stagnatin interval Hwever, even if such is nt the case and there is a residual ptential energy stred in the spring after the mtin ceases, the statement is apprximately true In systems with bth dry and viscus frictin the damping f scillatins can als be investigated by the stage-by-stage slving f the equatins f mtin and by using the mechanical state at the end f the previus half-cycle as the initial cnditins fr the next in turn half-cycle The phase trajectry cnsists in this case f the shrinking alternating halves f spiral lps that are characteristic f a linear damped scillatr The fcal pints f these spirals alternate between the bundaries f the stagnatin interval The lps f the phase trajectry are n lnger equidistant Nevertheless their shrinking des nt last indefinitely: the phase trajectry in this case als terminates after sme finite number f turns arund the rigin when it reaches the stagnatin interval n the φ-axis 7

8 T figure ut the relative imprtance f viscus versus dry frictin, we can cmpare the decrease in amplitude caused by each f these effects during ne cmplete cycle It was established abve that under the actin f dry frictin this decrease equals the cnstant value f the dubled width f the stagnatin interval 4d On the ther hand, viscus frictin decreases the amplitude f the scillatin during a cmplete cycle by an amunt which is prprtinal t the amplitude Indeed, fr γt 1, ie, fr rather large values f the quality factr Q = 2ω /γ, expressin fr the decrease a during ne perid T in the mmentary amplitude a due t viscus frictin can be written apprximately as fllws: a = a(1 e γt ) aγt = aγ 2π ω = πa Q (6) Equating a t the dubled width 4d f the stagnatin interval, we find the amplitude ã which delimits the predminance f ne type f frictin ver the ther: ã = 4d = 4 Qd Qd (7) γt π If the actual amplitude is greater than ã, the effect f viscus frictin dminates Cnversely, if the actual amplitude is less than ã, the effect f dry frictin dminates When the initial excitatin f the scillatr is great enugh, the amplitude may exceed the value ã Qd In this instance, the initial damping f the scillatins is influenced mainly by viscus frictin, and the decrement in the width f several initial lps f the phase trajectry (caused by viscus frictin) is greater than the separatin f the centers f adjining half-lps (ie, the decrement exceeds the width f the stagnatin interval) It is clear that in this case the shrinking f the spiral caused by viscus frictin is mre influential in shwing the effects f damping than is the alternatin f the centers f half-lps caused by dry frictin When the value f a falls belw that f ã (when a < ã Qd), the effects f dry frictin dminate In the phase plane this dminance prduces a trajectry f cnsecutive half-lps whse centers alternately jump between the ends f the stagnatin interval, d and d, until the phase trajectry reaches the segment f the φ-axis in the stagnatin interval If viscus frictin is strng, that is, if γ > ω, and if the initial displacement f the flywheel φ() lies beynd the bundaries f the stagnatin interval, φ() > d, the released flywheel mves withut scillating tward the nearest bundary f the stagnatin interval At this pint the flywheel stps turning 5 Resnance in the scillatr with dry frictin under sinusidal excitatin In this sectin we analyze frced scillatins f the trsin spring pendulum in cnditins f resnance, that is, when the frequency f excitatin ω equals natural frequency ω f the scillatr (T = T = 2π/ω ) Generally at large enugh dry frictin sticking may ccur: the flywheel remains at rest fr a finite time after the velcity reaches zer Hwever, if the amplitude f excitatin θ in equatins (4) (5) exceeds sme threshld value, the mtin f the flywheel is purely sliding (nn-sticking), and in the absence f viscus frictin the amplitude f scillatins grws indefinitely An example f such resnant scillatins is shwn in Fig 4 The phase trajectry and the graphs f φ(t) and φ(t) are btained by cmputer simulatin which is based n numeric integratin f equatins (4) (5) We nte the linear grwth f the amplitude: the successin f maximal deflectins f the flywheel frms an arithmetic prgressin Next we find analytically the threshld fr excitatin f such grwing scillatins, and calculate the increment f the amplitude after each driving cycle 8

9 ϕ( t) ϕ( ) t Drive amplitude 25 degrees, dead zne 15 degrees, initial displacement -15 degrees T Figure 4: Phase trajectry with Pincaré sectins (left) and graphs f φ(t) and φ(t) (right) fr scillatins at resnance with dry frictin We chse fr simplicity the initial deflectin φ() f the flywheel cinciding with the left bundary f the dead zne, that is, φ() = d, and initial angular velcity zer: φ() = Such initial cnditins prvide the sliding (nn-sticking) mtin frm the very beginning with tw turnarunds during each cycle f excitatin During the first half f the excitatin perid ( < t < T /2) the angular velcity is psitive ( φ(t) > ), and we shuld use equatin (4) The slutin t this equatin (with γ = ), satisfying the abve indicated initial cnditins, can be written as fllws: φ(t) = 1 2 θ (ω t cs ω t sin ω t) d, φ(t) = 1 2 θ ω 2 t sin ω t, < t < T /2 (8) Accrding t (8), next maximal elngatin t the right side ccurs at t = T /2 and equals 1 2 πθ d This elngatin is greater in magnitude than the preceding (initial) elngatin d t the left side by 1 2 πθ 2d T find the increment in the amplitude during the secnd half-cycle f excitatin, when the flywheel rtates in the ppsite directin, we shuld use equatin (5) An analytical slutin t this equatin is given in the Appendix It ccurs that the increment in amplitude during the secnd half-cycle is the same as during the first half-cycle Therefre during the whle cycle the increment in amplitude equals πθ 4d Specifically, fr θ = 25 and d = 15 (the values crrespnding t the simulatin shwn in Fig 4) the amplitude shuld increase during each cycle by 1854 The simulatin in Fig 4 shws that during the first six cycles the amplitude increased by = 111, which gives fr increment during ne cycle the value 185 in a gd agreement with the theretical predictin The grwth f scillatins amplitude ccurs if the value f increment πθ 4d during a cycle f excitatin is psitive Hence the threshld f resnance (θ ) min fr the scillatr with dry frictin is given by the fllwing cnditin: θ > 4 π d, (θ ) min = 4 d (9) π Fr given width d f the dead zne (fr given dry frictin) equatin (9) defines the critical (minimal) value (θ ) min f the drive amplitude which prvides nn-sticking frced scillatins f the flywheel after a rather shrt transient During the transient, depending n the initial cnditins, sticking is pssible Fr θ > (θ ) min, after the transient is ver, at the initial mment t n = nt = nt f each in turn cycle f excitatin angular velcity φ(t n ) f the flywheel is zer: φ(nt ) = This means that Pincaré sectins in the phase plane (crrespnding t time mments t n = nt ) apprach the abscissa axis during the transient and remain n its negative side further n Since the increment in the elngatin is the same fr each cycle, the pints f Pincaré sectins n the axis are equidistant (see Fig 4) 9

10 46 25 ϕ( t) ϕ( ) t θ ( t ) Drive amplitude 191 degrees, dead zne 15 degrees, initial displacement -25 degrees T Figure 5: Oscillatins at the threshld cnditins at resnance with dry frictin Statinary peridic scillatins at the threshld cnditins are shwn in Fig 5 At arbitrary initial values f φ and φ the phase trajectry appraches eventually a limit cycle similar t the cycle shwn in the left-hand side f Fig 5 The amplitude f steady-state frced scillatins at the threshld depends n initial cnditins If initial velcity is zer ( φ() = ), steady-state scillatins ccur frm the very beginning, withut any transient, in case initial displacement φ() is negative and lies beynd the dead zne, that is, if φ() <, φ() d The amplitude f these scillatins equals φ() This mde f scillatins is unstable with respect t variatins in parameters θ and d: a slight increment f the drive amplitude r decrement in the dead zne width causes an indefinite grwth f the amplitude If the amplitude f the exciter θ is smaller than the critical value (θ ) min given by equatin (9), but greater than the dead zne width d, a steady-state regime with tw sliding phases and tw sticking phases establishes after the transient is ver Fr θ smaller than the dead zne width d, the flywheel, depending n the initial cnditins, either remains immvable frm the very beginning, r makes several mvements with sticking and then finally stps at sme pint f the dead zne The resnant grwth f amplitude ver the threshld is restricted if sme amunt f viscus frictin is present in the system In a dual-damped system steady-state scillatins with a cnstant amplitude eventually establish fr arbitrary initial cnditins An example f resnant scillatins in the system with bth dry and viscus frictin is shwn in Fig 6 Equatins (4) (5) allw us t calculate the amplitude a f such resnant symmetric steady-state scillatins We chse the time rigin t = at the beginning f the next in turn drive cycle At this mment ϕ( t) Drive amplitude 15 degrees, dead zne 5 degrees, quality 1, initial deflectin -15 degrees T Figure 6: Oscillatins at resnance with dry and viscus frictin the flywheel ccurs at the extreme displacement t the left side (φ() = a) and has the angular velcity zer ( φ() = ) During the first half-cycle f the drive ( < t < T /2) it mves t the right, s that φ is psitive during this interval Therefre we shuld use equatin (4) with ω = ω in its right-hand part It is cnvenient t intrduce instead f φ(t) a new unknwn 1

11 functin ψ(t) = φ(t) + d, which, accrding t (4), satisfies the fllwing equatin: ψ + 2γ ψ + ω 2 ψ = ω 2 θ sin ω t (1) We can search fr its peridic partial slutin in the frm ψ(t) = A cs ω t This functin satisfies equatin (1), if A = (ω /2γ)θ = Qθ Next we add t this partial slutin the general slutin f the crrespnding hmgeneus equatin: ψ(t) = Qθ cs ω t + (C cs ω t + S sin ω t) exp( γt) (11) It fllws frm the initial cnditin ψ() = that in (11) S = (γ/ω )C T find C, we require that in the steady-state symmetric regime elngatins t bth sides shuld be equal: φ() = φ(t /2) Frm this cnditin we get C = 2d 1 exp( γt /2) = 2d 1 exp( π/2q) (12) Substituting these C and S values in equatin (11), we btain the time dependence f the angular displacement φ(t) = ψ(t) d fr the first half-cycle f excitatin The desired amplitude a f this steady-state resnant scillatin is given by φ(): ( a = Qθ d 2 1 exp( π/2q) 1 ) ( Q θ 4d ) (13) π The latter apprximate expressin is valid in case f rather weak viscus frictin, when Q 1 In the absence f dry frictin (at d = ) the grwth f amplitude at resnance is restricted due t viscus frictin by the value Qθ, which is Q times greater than the amplitude f the driving rd θ, in accrdance with the first term in equatin (13) With dry frictin, the steadystate amplitude is apprximately Q times greater than the excess f the drive amplitude θ ver the threshld 4d/π We emphasize that dry frictin alne is unable t restrict the grwth f amplitude ver the threshld at ω = ω Nevertheless, equatin (13) shws that when dry frictin is added t the system with viscus frictin, the steady-state amplitude at resnance is smaller than Qθ Frm the numerical simulatin (Fig 6) we see that with θ = 15 and Q = 1 the resnant amplitude equals nly 863, if the dead zne d equals 5 (cmpare with Qθ = 15 at d = ) This experimental value 863 is in a gd agreement with the theretical result expressed by (13), accrding t which the steady-state amplitude shuld be 862 In cnditins f exact tuning t resnance (at ω = ω ) the energy is transferred t the scillatr frm the external surce (frm the exciter) with maximal efficiency, if at the beginning f each excitatin cycle the flywheel ccurs at an extreme elngatin t the left-hand side Indeed, in this case the sinusidally varying external trque exerted n the flywheel by the exciter acts during the whle cycle in the directin f the flywheel rtatin, and ver the threshld (at πθ > 4d) vercmes the trque f dry frictin: the amplitude grws linearly (see Fig 4) increasing during a cycle by πθ 4d On the cntrary, if at the beginning f the excitatin cycle the flywheel ccurs at an extreme elngatin t the right-hand side, the external trque f the spring during the whle cycle is directed against the flywheel s angular velcity tgether with the frictinal trque In this case the amplitude reduces during each cycle by the amunt πθ + 4d After the amplitude reduces t zer, the phase relatins between the flywheel and exciter change t the ppsite and becme favrable fr the transfer f energy t the scillatr: the amplitude begins t grw An example f such behavir is shwn in Fig 7 At the drive amplitude θ = 6366 and the dead zne 25, the amplitude linearly reduces during each cycle f the initial stage f the prcess by πθ +4d = 3 After 3 full driving cycles the amplitude diminishes frm initial 9 t zer During the further resnant grwth the amplitude linearly increases during each cycle by πθ 4d = 1, and after next 9 cycles becmes 9 11

12 T Drive frequency 1 w, drive amplitude 6366 degrees, dead zne 25 degrees, initial angle 9 degr, init velcity Figure 7: Phase diagram with Pincaré sectins and graph f φ(t) f scillatins with dry frictin at resnance with initial deflectin φ() = +9 6 Nn-resnant frced scillatins In the case f exact tuning t resnance, in cntrast t the scillatr with viscus damping, dry frictin alne is unable t restrict the grwth f the amplitude f frced scillatins ver the threshld In nn-resnant cases (ω ω ) f harmnic excitatin, after a transient f a finite duratin, steady-state scillatins f cnstant amplitude can establish due t dry frictin even in the absence f viscus frictin Nn-resnant frced scillatins in the scillatr with dry frictin received significant attentin in the literature Since the pineer s wrk f Den Hartg [6] in 193, several researchers [7] [11] investigated the system analytically and numerically, and btained exact slutins, describing the steady-state nn-sticking mtin with tw turnarunds f per cycle fr a harmnically excited dry frictin scillatr T Drive amplitude 45 degrees, drive frequency 7w, dead zne 2 degrees Figure 8: Phase diagram and graphs f φ(t) and φ(t) f nn-sticking steady-state scillatins with dry frictin at ω = 7ω An example f such nn-resnant scillatins in the system with cnsiderable amunt f dry frictin is shwn in Fig 8 The peridic mtin cnsists f tw nn-sticking phases f equal duratin T/2 The angular velcity is negative in ne phase and psitive in the ther Unfrtunately, it is impssible t express φ(t) in a clsed analytical frm, because the turnarund pints dividing the tw phases are determined by a transcendental equatin T find the amplitude a(ω) f this symmetric scillatin, it is sufficient t cnsider nly ne phase between successive turnarunds, which is described by differential equatin (5) The calculatins are similar t thse abve described fr the resnant case (thugh mre cmplicated) Using peridicity and symmetry f the desired slutin, we find the fllwing dependence f the steady-state amplitude n the driving frequency ω and amplitude θ = 45 f the excitatin: 1 a(ω) = θ (1 ω 2 /ω) d2 (ω /ω) 2 sin 2 π(ω /ω) 2 θ(cs 2 π(ω /ω) + 1) (14) 2 12

13 Fr the frequency f excitatin ω = 7ω, drive amplitude θ = 45, and dead zne d = 2 we get frm (14) fr the steady-state amplitude the value 863, which is in perfect agreement with the numerical simulatin illustrated by Fig 8 a /q d = w/w 1 - d =, 2 - /q = 35, 3 - /q = 5, 4 - /q = 6 d d d Figure 9: Amplitude frequency characteristics f sinusidally driven spring scillatr with dry frictin damping Expressin (14) fr the steady-state amplitude f nn-sticking scillatins cincides (in smewhat different ntatins) with results published earlier in the literature [7], [11] Frequencyrespnse resnant curves (amplitude-frequency characteristics) given by (14) fr the scillatr with dry frictin are shwn in Fig 9 fr several values f relative width d/θ f the dead zne (in the frequency regin ω > 5ω ) We emphasize that expressin (14) is valid nly fr sliding (nn-sticking) symmetric mtins f the scillatr Such mtins are pssible if the fllwing simple implicit cnditin (see [7]) n the parameters is fulfilled: a(ω, θ, d) d θ ( ω ω ) 2 (15) Slving equatin (15) numerically fr the unknwn d at ω = 7ω and θ = 45 (these values were used fr the simulatin shwn in Fig 8), we find that the maximal width d max f the dead zne fr which steady-state nn-sticking symmetric mtins are pssible equals 325 A clsed-frm frmula fr the dmain f steady-state symmetric nn-sticking scillatry respnses was btained in [8] It prvides the minimum driving trque amplitude (θ ) min required t prevent sticking fr given width d f the dead zne and given drive frequency ω: (θ ) min = d (ω 2 ω 2 1 ) 2 [ 1 + (ω/ω ) 2 sin 2 (πω /ω) (1 + cs(πω /ω) 2 ] (16) Certainly, this equatin can be used als t find the maximal width d max f the dead zne fr which steady-state nn-sticking symmetric mtins are pssible at given frequency ω and amplitude θ f the driving trque Substituting ω = 7ω and θ = 45 in (16), we get d max = 325, in accrdance with the abve estimate (15) The upper part f Fig 1 illustrates scillatins ccurring n this edge f such nn-sticking regime (dead zne 325 ) Fr initial cnditins φ() =, φ() =, sticking ccurs several times during a shrt transient which ends with a nn-sticking symmetric steady-state scillatins Accrding t (14), their amplitude must equal 663, in gd agreement with the simulatin Fr cmparisn, the lwer part f Fig 1 shws the steady-state scillatins at the same values f the frequency and amplitude f the exciter (ω = 7ω and θ = 45 ), but fr a smewhat greater dry frictin (dead zne 37 ) In this case sticking ccurs twice during each cycle f excitatin Nt surprisingly that the amplitude f steady-state symmetric frced scillatins with sticking bserved in the simulatin is smaller than equatin (14) predicts (55 against 62 ) 13

14 T Drive frequency 7w, drive amplitude 45 degrees, dead zne 325 degrees T Drive frequency 7w, drive amplitude 45 degrees, dead zne 37 degrees Figure 1: Phase diagram and graphs f φ(t) and φ(t) f the transient that leads t nn-sticking steady-state scillatins at ω = 7ω and critical width d = 325 f the dead zne (upper part), and steady-state symmetric scillatins with sticking at d = 37 (lwer part) 7 Harmnic excitatin at sub-resnant frequencies Generally characteristics f steady-state behavir f the peridically frced scillatr with dry frictin, as well as f the scillatr with viscus frictin, are uniquely defined by the system parameters, and by the frequency and amplitude f the excitatin Certain exceptins are revealed if the frequency ω f sinusidal excitatin cincides with ne f subharmnics f the natural frequency: ω = ω /n, where n is an integer number Analytical steady-state slutins at sub-harmnic excitatin were cnsidered fr the first time in [12] In the present paper we suggest a simpler and physically mre transparent apprach t the prblem, and discuss peculiarities f such scillatins in mre detail Simple clsed-frm slutins are illustrated by time-dependent graphs and phase rbits btained with the help f cmputer simulatins Accrding t equatin (14), at frequencies f excitatin ω = ω /2, ω = ω /4, the amplitude f steady-state nn-sticking symmetric scillatin, independently f the dead zne width d, shuld be equal t the amplitude f frced steady-state scillatin in the absence f frictin (that is, at d = ): a(ω) = θ /(1 ω 2 /ω 2 ) (Certainly, this arbitrary value f d shuld satisfy the cnditin (16) fr nn-sticking mtins, which at ω = ω /n gives d θ /3) Figure 11 shws that frequency-respnse curves fr different d values at ω = ω /2 graze the curve fr d = Cmputer simulatins testify that in these cases steady-state scillatins are generally asymmetric: the angular excursin t ne side is greater than t the ther This means that at ω = ω /n ccurrence f special analytical slutins can be expected Belw we shw that in cntrast with the general case f frced scillatins, fr which the steady-state regime is described by the unique slutin which is independent f the initial cnditins, a cntinuum f asymmetric nn-sticking slutins exists at ω = ω /2n Each slutin gives an asymmetric limit cycle (attractr) that crrespnds t initial cnditins frm a certain basin f attractin T study the excitatin f scillatr at ω = ω /n, it is mre cnvenient t chse further n 14

15 14 a/q 12 d = d/q = w/w 14 a/q 12 d = 1 8 d/q = a/q 12 d = w/w 1 8 d/q = w/w Figure 11: Frequency respnse curves fr the scillatr with dry frictin given by equatin (14) at excitatin frequencies ω < ω /2 fr several values f the dead zne width (d/θ = 1, d/θ = 3, d/θ = 1) fr the time rigin t = the mment at which the exciter reaches its maximal deflectin θ, that is, t assume fr θ(t) the fllwing time dependence: θ(t) = θ cs ωt With this chice, as we will see later, the turnarunds in the steady-state mtin f the scillatr ccur apprximately at t = and t = T/2 This simplifies the frm f analytical slutins Fr definiteness we restrict further discussin t the case n = 2 If ω = ω /2, the spectrum f steady-state asymmetric scillatins at sufficiently small dry frictin (narrw dead zne) cnsists primarily f the principal harmnic with the frequency f excitatin ω and its secnd harmnic, whse frequency 2ω equals the natural frequency ω f the scillatr (see Fig 12) A small admixture f the third harmnic is als nticeable Mathematically, the principal harmnic crrespnds t the frced peridic partial slutin f the nnhmgeneus differential equatin f mtin (3) with γ = and with the sinusidal frcing term ω 2 θ cs ωt whse frequency ω equals ω /2 This partial slutin is 4 3 θ cs ωt The trque f dry frictin mves the mid-pint f this scillatin t d (left bundary d the dead zne) if φ > and t d if φ < This peridic (square-wave) displacement f the mid-pint caused by dry frictin explains the appearance f the third harmnic in the slutin shwn in Fig 12 The secnd harmnic with the frequency 2ω = ω is the general slutin f the hmgeneus equatin that crrespnds t (3) This general slutin describes natural scillatins with the frequency ω = 2ω, and can be represented (at γ = ) as A cs 2ωt + B sin 2ωt, where A and B are arbitrary cnstants In cntrast t a system with viscus frictin, 15

16 Drive frequency 5w, drive amplitude 6, dead zne 5, initial angle 65, initial angular velcity w, maximum elngatin 65, minimum elngatin T Figure 12: Phase diagram and time-dependent graphs f φ(t) and φ(t) with the graphs f their harmnics fr nn-sticking asymmetric steady-state scillatins at ω = ω /2 and small width f the dead zne (d = 5 ) nw this general slutin scillatin with the natural frequency des nt damp ut in the curse f time during the transient The simulatin shws (in accrdance with the requirement φ() = φ(t/2) = ) that in the steady-state regime the phase f this secnd harmnic is such that B = (see Fig 12) Hence the asymmetric steady-state mtin at ω = ω /2 can be apprximately described by the fllwing equatins: and φ + (t) = 4 3 θ cs ωt + A + cs 2ωt d, φ >, (17) φ (t) = 4 3 θ cs ωt + A cs 2ωt + d, φ <, (18) φ + (t) = 4 3 ωθ sin ωt 2ωA + sin 2ωt, φ >, (19) φ (t) = 4 3 ωθ sin ωt 2ωA sin 2ωt, φ < (2) One cnditin n cnstants A + and A fllws frm the requirement f cntinuity f φ(t) at the turnarund pints, when the sign f velcity reverses These are the mments t = and t = T/2 (see Fig 12) Frm φ + () = φ () we get A + A = 2d, r A + = A + 2d (cnditin φ (T/2) = φ + (T/2) yields the same relatin between A + and A ) Therefre nly ne f these cnstants remains arbitrary The steady-state regime described by equatins (17) (2) ccurs frm the very beginning (that is, withut any transient) if the initial cnditins are chsen prperly At t = we get frm (19) r (2) that the required initial angular velcity equals zer: φ() = This value is independent f the dead zne width d and the drive amplitude θ Since during the first halfcycle φ() is negative, fr the required initial displacement we shuld use equatin (18), which yields φ = φ() = 4θ 3 + A + d We see that the arbitrary cnstants A + and A, which determine the cntributin f the secnd harmnic int the steady-state mtin, depend n an arbitrary initial displacement φ : A + = φ 4 3 θ + d, A = φ 4 3 θ d (21) This means that in the system with dry frictin, in cntrast t the scillatr with viscus frictin, different initial displacements φ lead generally t different regimes f steady-state scillatins (t different limit cycles) Substituting these values f A + and A in (17) (2), we get the 16

17 clsed-frm analytical slutins fr asymmetric steady-state subresnant regimes f the dryfrictin scillatr at ω = ω /2 Belw we shw that such steady-state regimes ccur frm the very beginning if the value f φ belngs t a certain interval Nw we can derive sme interesting prperties f the discussed steady-state slutins Extreme elngatins crrespnd t the turnarund pints and hence ccur at t and at t T/2 Maximum displacement t the right-hand side ccurs at t and, accrding t equatin (17) r (18), equals φ max = 4 3 θ +A +d (r, equivalently, φ max = 4 3 θ +A + d) Extreme elngatin t the left-hand side ccurs t T/2 and equals φ min = 4 3 θ A d = 4 3 θ A + + d We get that the ttal angular excursin φ max + φ min between the extreme pints fr all pssible slutins (17) (18) equals 8 3 θ : φ max + φ min = φ max φ min = 8 3 θ (22) It depends slely n the drive amplitude θ, and des nt depend n the intensity f dry frictin (n the width d f the dead zne) The difference between the extreme elngatins characterizes the asymmetry f this steadystate regime: φ max φ min = 2(A + d) = 2(A + d) = 2(φ 4 3 θ ) (23) The extreme elngatins t bth sides are equal t ne anther if φ 4 3 θ = In this case A + = d and A = d, and the secnd harmnic in the scillatin described by equatins (17) (18) vanishes Such symmetric steady-state scillatin with the amplitude 4 3 θ ccurs nly if the initial displacement φ equals 4 3 θ and the initial velcity zer Drive frequency 5w, drive amplitude 6, dead zne 5, initial angle 8, initial angular velcity w, maximum elngatin 8, minimum elngatin T Figure 13: Phase diagram and time-dependent graphs f φ(t) and φ(t) with the graphs f their harmnics fr nn-sticking symmetric steady-state scillatins at ω = ω /2 and small width f the dead zne (d = 5 ) The phase trajectry and time-dependent graphs f φ(t) and φ(t) (tgether with the graphs f their harmnics) fr such symmetric scillatin are shwn in Fig 13 (These graphs at d θ almst merge with the graphs f their principal harmnics) The initial cnditins that lead t the greatest asymmetry f the limit cycle can be fund as fllws We are interested in the unsticking regime with tw turnarunds per ne excitatin cycle These turnarunds ccur near t = and t = T/2, when the angular velcity φ changes sign We can rely n physical cnsideratins in finding the cnditin fr a turnarund ccurring withut sticking fr a finite time interval Indeed, t avid sticking at this pint, the restring trque f the spring exerted n the flywheel shuld be greater r at least equal t the greatest 17

18 pssible trque f dry frictin The desired cnditin crrespnds t the equality f these tw trques When the trque exerted by the spring equals the trque f static frictin in magnitude, the angular acceleratin φ f the flywheel equals zer Next we cnsider this cnditin fr each f the turnarunds, ccurring at t = and t = T/2 Fr the first turnarund at t = we shuld require φ() = using equatin (18) that crrespnds t negative angular velcity φ(t) < (see Fig 23) Frm this requirement we find immediately A = 1 3 θ Substituting this value t equatin (21), we get the first (lwer) bundary φ (lwer) f admissible initial deflectins: φ (lwer) = θ + d (24) With this initial displacement ne f the tw pssible mst asymmetric steady-state scillatins ccurs, in which the extreme deflectins are: φ max = θ + d, φ min = 5 3 θ + d (25) T find the ther bundary f the admissible initial deflectins φ (upper), we shuld require φ(t/2) = using equatin (17) This yields A + = 1θ 3, and fr the secnd (upper) bundary f admissible initial deflectins we get: φ (upper) = 5 3 θ d (26) With this initial displacement the ther f the tw pssible mst asymmetric steady-state scillatins ccurs, in which the extreme deflectins are: φ max = 5 3 θ d, φ min = θ d (27) T verify these theretical predictins with the help f numerical simulatins, we chse the drive amplitude θ = 6, which means that in the steady-state regime the ttal angular excursin between the extreme pints φ max + φ min shuld equal 8θ 3 = 16 independently f the intensity f dry frictin Width d f the dead zne in this simulatin equals 5 One f the tw pssible steady-state scillatins with the greatest asymmetry (see Fig 12) ccurs frm the very beginning (withut any transient), accrding t (24), at the initial cnditins φ() = θ + d = 65, φ() = The extreme elngatins shuld be φ max = θ + d = 65 and φ min = 5θ 3 +d = 95 These theretical predictins agree perfectly well with the cmputer simulatin (see Fig 12) Steady-state scillatin with equal elngatins t bth sides (φ max = φ min = 8 ) at the same values f the system parameters (θ = 6, d = 5 ) ccurs if the initial displacement φ = 4θ 3 = 8 This case is illustrated by the simulatin shwn in Fig 13 The secnd case f the greatest asymmetry ( φ(t/2) = ) ccurs, accrding t (26), at the initial cnditins 5θ 3 d = 95, φ() = Extreme elngatins in this case, accrding t (27), are φ max = 5θ 3 d = 95 and φ min = θ d = 65 These asymmetric scillatins are illustrated by the simulatin shwn in Fig 14 Three different limit cycles f nn-sticking scillatins shwn in Figs crrespnd t the same values f the system parameters ω = ω /2, θ = 6, and d = 5 Actually, at ω = ω /2 there exists a cntinuum f different steady-state nn-sticking mtins with the same ttal angular excursin 8θ 3, prprtinal t the drive amplitude θ This is a manifestatin f multistability a typical feature f nnlinear systems If the initial angular velcity φ() 18