How to play a disc brake

Ho w to pla y a d i sc brak e O.N. Kirillo v Institute of Mec h anics, Mosco w State Lomonosov Univers ity , Mic hurinskii pr. 1, 119192 Mosco w, Russia E-mail: kirillo v@imec.msu.ru Abstract W e consider a gy roscopic system with tw o degrees of freedom under th e action of small dissipativ e and non-conserv ative p ositional forces, which has its origin in the mod els of rotating b odies of revol u tion b eing in frictional contact. The s p ectru m of th e unp erturb ed gy roscopic system forms a ”spectral mes h ” in the plane ”frequency – gyroscopic parameter” with double semi-simple purely imaginary eigenv alues at zero v alue of the gyroscopic parameter. It is shown that dissipativ e forces lead to the splitting of the semi-simple eigen va lue with the creation of the so-called ”bub ble of instability” – a ring in th e three-dimensional sp ace of the gyroscopic parameter and real and imaginary parts of eigenv alues, which corresponds to complex eigen va lues. In case of full dissipation with a p ositiv e- d efinite damping matrix th e eigenv alues of th e ring hav e negative real parts makin g the bubble a latent source of instability b ecause it can ”emerge” to the region of eigen v alues with p ositive real parts du e to action of b oth indefinite damp ing and non-conserv ative p ositional forces. In the pap er, the instability mechanis m is analytically described with the use of th e p erturbation theory of multiple eigenv alues. Explicit conditions are established for th e origination of the bubble of instability and its transition from the latent to active p hase, clarifying th e key role of indefin ite damping and non-conserv ativ e p ositional forces in the dev elopment and localization of the sub critical flutter instabilit y . As an example stability of a rotating circular string constrained by a stationary load system is studied in d et ail. The theory develo p ed seems to give a first clear explanation of the mechanism of self- excited vibrations in the rotating structures in frictional contact, that is resp onsible for such well-kno wn phenomena of acoustics of friction as the squealing disc brake and the singing wine glass. Keywords: gyr osc opi c system, dissip ative and non-c onservative p ertur b ations, indefinite damping, ve ering and mer ging of mo des, sp e ctr al mesh, friction-induc e d oscil lations, disc br ake sque al, ac oustics of friction, singing wine glass 1 In tro d uction An axially symmetric shell, like a wine glass can ea sily pro duce sound when a wet finger is rubb ed ar ound its r im or w a ll, as it was observed alr eady in 1638 by Galileo Ga lilei in his Dialogues Co ncerning Two New 1 Sciences [1, 8, 3 4]. This principle is used in playing the g lass harmonica inv ented by Benjamin F ranklin in 1757, which he ca lled ”a rmonica”, where, in o rder to pro duce sound, one should touch by a moist finge r an edge of a glass b owl rotating around its axis o f symmetr y [15, 2 2, 34, 5 3]. This remar k able phenomenon ha s bee n studied ex per imentally; se e e.g. [40]. How e ver, an adequate analytica l theor y for its description seems to b e still mis s ing. Another closely related exa mple o f acoustics o f frictio n is the squea ling dis c brake [22, 2 6]. This mechanical system pr o duces so und due to transverse v ibr ations of a r o tating annular pla te caus ed by its int er action with the bra ke pads. Despite intensiv e exp erimental and theoretica l study , the pro blem o f predicting and controlling the squeal r emains an imp orta nt issue [4, 12, 19, 22 , 24, 26, 3 4, 47, 4 8, 51]. Sig nificant but still po orly under sto o d pheno mena a re squea ling and bar ring of calender rolls in pap er mills causing a n intensive noise a nd reducing the quality of the pap er [35]. The prese nc e of multiple eigenv alues in the sp ectr a o f free vibr a tions of axially symmetric shells and pla tes is well-known. Alr e ady Rayleigh, studying the acoustics of bells, reco gnized that, if the symmetry of a b ell were complete, the noda l mer idians of a transverse vibra tio n mo de would hav e no fixed p os ition but would trav el freely ar ound the b ell, a s do those in a wine glass dr iven by the moistened finger [2 2]. This is a reflection of the fact that sp ectra of fre e vibrations of a b ell, a wine glass, an annular plate a nd o ther b o dies o f r e volution contain double purely imag inary semi-s imple eigenv alues with tw o linearly indep endent eigenv ector s. Rotation ca uses the double eigenv a lues of an axially s ymmetric structure to split [2]. The ne w b o rn pair of simple eig env alues corresp onds to the for ward and backw ard tr av elling wav es, which propaga te alo ng the circumferential directio n [2, 3, 4, 9, 16, 19, 20]. Viewed from the rotationa l frame, the frequency of the forward trav elling wa ve app ears to decr ease a nd that of the backw a rd trav elling wa ve app ea r s to incr ease, as the spin increases. Due to this fact, double eigenv a lues originate aga in at non-zero angular velocities, forming the no des of the sp ectr al mesh in the plane ’e ig enfrequency’ versus ’angular velocity’. The spe c tr al meshes ar e characteristic fo r exa mple for the ro tating circular strings, r ing s, discs, and cylindrical a nd hemispherica l shells. The phenomeno n is a pparent als o in hydro dynamics, in the problem of sta bilit y of a vortex tub e [27] and in magnetohydro dynamics (MHD) in the problem of instability o f the s pherically symmetric MHD α 2 -dynamo [43]. It is known that str ik ing the wine glass excites a num b e r o f mo des, but rubbing the r im with a finger or bowing it r adially with a violin b ow gene r ally excites a single mo de [15]. The sa me is tr ue for the squealing disc br ake [26, 44, 45, 46, 48, 5 1]. F or this reason, we formulate the ma in pr oblem o f aco ustics of friction of rotating ela s tic b o dies of revolution as the description of the mechanism of ac tiv ating a pa rticular mo de of the contin uum by its contact with an exter nal b o dy . In ca se o f the disc brake, the frictional contact of the brake pa ds with the ro tating disc in tr o duces dissipative and non-c o nserv ative p ositional forces into the system [48, 51]. Since the no des of the sp ectral mesh corr e sp ond to the double eigenv alues , they a re mo st s ensitive to p er tur bations, esp ecially to those break ing the symmetries of the sy s tem. Consequen tly , the instability will most likely o ccur at the angular velocities close to that o f the no des of the sp ectral mesh and the unstable mo des of the p ertur bed sys tem will have the frequencies close to 2 Figure 1 : The sp ectra l mesh of system (1) when δ = ν = κ = 0. that of the double eigenv alue s at the no des. This picture qualitatively a grees with the existing exp erimental data [1 5, 22, 26, 4 4, 45, 46]. In the following with the use of the p er turbation theory of m ultiple eigen v alues [3 3, 38, 39, 43] we will show that independently on the definiteness of the damping matrix, there exist co mbinations o f dissipa tive and non-c o nserv ative p ositiona l force s causing the flutter insta bilit y in the v icinit y of the no des of the sp ectral mesh for the angular velocities fro m the sub critica l range. W e will show that zero and nega tive eigenv alues in the sp ectrum of the damping matr ix encoura ge the development of the lo c a lized sub critical flutter instability while zero eigenv alues in the matrix of non-conserv ative p ositiona l forces suppress it. Explicit expres s ions describing the movemen ts of eigenv a lues due to change of the system parameters will b e obtained. Conditions will b e deriv ed fo r the eigenv alues to mo ve to the right pa rt of t he complex plane. Appro ximatio ns o f the domain o f as ymptotic sta bility will be found and s ingularities o f the stabilit y boundar y resp o ns ible for the developmen t o f instability will b e describ ed and cla ssified. The metho dology develop ed for the study of the t wo-dimensional system will finally be employ e d to the detailed in vestigation of the stabilit y o f a rotating circular string c o nstrained by a stationar y load system. 2 The sp ectral mesh of a t w o-dimensional gyroscopic system Consider an autono mous non-cons erv ative sy stem descr ib ed by a linea r differe nt ia l equatio n of se cond or der ¨ x + (2Ω G + δ D ) ˙ x + (( β 2 − Ω 2 ) I + κ K + ν N ) x = 0 , (1) where a dot ov er a sy m b o l denotes time differentiation, x ∈ R 2 , and I is the identit y matrix. The real matrices D = D T , G = − G T , K = K T , and N = − N T are related to diss ipa tive (damping), gyrosco pic, po tent ia l, and no n-conserv ative po sitional (circulator y) forces with magnitudes co n tr olled by scaling factor s δ , 3 Ω, κ , and ν resp ectively; β > 0 is the fr e q uency of free vibr a tions o f the po tent ia l system corres po nding to δ = Ω = κ = ν = 0. The parameter s and v ar ia bles o f the system are assumed to b e non-dimensional quantities. Without loss in generality we ass ume det G = det N = 1 . Equation (1) a pp ea rs a s a tw o-mo de a pproximation of the mo dels of rotating b o dies of revolution in frictional contact after their lineariza tion a nd discretization [19, 2 6, 5 1]. Separating time by setting x ( t ) = u exp( λt ) we arrive at the eigenv alue problem Lu = 0 , L = I λ 2 + (2Ω G + δ D ) λ + ( β 2 − Ω 2 ) I + κ K + ν N . (2) Applying the Le verrier-Barnett a lgorithm [11] to the matrix p olynomia l L we find the characteristic poly no mial of the sy s tem P ( λ ) = λ 4 + δ tr D λ 3 + (2( β 2 + Ω 2 ) + δ 2 det D + κ tr K ) λ 2 + (4Ω ν + δ ( β 2 − Ω 2 )tr D + δ κ (tr K tr D − tr KD )) λ + κ 2 det K + κ tr K ( β 2 − Ω 2 ) + ( β 2 − Ω 2 ) 2 + ν 2 . (3) In the absence of dissipative, external p otential, and non- conserv ative p ositiona l forces ( δ = κ = ν = 0) the characteristic p olynomial (3) corresp o nding to the o p e r ator L 0 (Ω) = I λ 2 + 2 λ Ω G + ( β 2 − Ω 2 ) I , which belong s to the class of matrix po lynomials cons ider ed, e.g., in [21], has four purely imagina ry ro ots λ + p = iβ + i Ω , λ + n = − iβ + i Ω λ − p = iβ − i Ω , λ − n = − iβ − i Ω . (4) In the plane (Ω , Im λ ) these ro ots considered as functions of Ω orig ina te a co llection of straight lines intersecting with each other (Fig. 1) – the sp e ctr al mesh [43]. Two no des o f the mesh at Ω = 0 cor resp ond to the double semi-simple eige n v alues λ = ± iβ . At the o ther t wo no des at Ω = ± Ω d there exist double semi-simple e ig env alues λ = 0. The range | Ω | < Ω d = β is called sub critic al for the gy roscopic par ameter Ω since the zero eig env alue corres p o nds to the divergence b oundary [21]. The double semi-simple eigenv alue iβ at Ω = Ω 0 = 0 ha s tw o linea rly-indep endent eigenv ec to rs u 1 and u 2 u 1 = 1 √ 2 β   0 1   , u 2 = 1 √ 2 β   1 0   . (5) An y linea r co mbination of these vectors is an eigenv ector to o . The eigenv ecto rs are orthogo nal u T 2 u 1 = u T 1 u 2 = 0 and satis fy the norma lization condition u T 1 u 1 = u T 2 u 2 = (2 β ) − 1 . Under p er tur bation of the gyro scopic parameter Ω = Ω 0 + ∆Ω, the double e ig env alue iβ into tw o s imple ones bifurca tes. The asymptotic formula for the p erturb ed eigenv alues is [38] λ ± p = iβ + i ∆Ω f 11 + f 22 2 ± i ∆Ω r ( f 11 − f 22 ) 2 4 + f 12 f 21 (6) where the quantities f ij are f ij = u T j ∂ L 0 (Ω) ∂ Ω u i     Ω=0 ,λ = iβ = 2 iβ u T j Gu i . (7) 4 Since G is a s kew-symmetric matrix, we find f 11 = f 22 = 0 , f 12 = − f 21 = i, (8) and λ ± p = iβ ± i ∆Ω = iβ ± i Ω in agreement with (4). 3 Conserv ativ e, dissipativ e, and n on-conserv ativ e deformation of the sp ectral mesh F ollowing the approa ch of [38], we consider a p ertur bation of the gyr o scopic system L 0 (Ω)+ ∆ L (Ω) b y dissipative, po tent ia l, and non-co nserv ative p ositiona l forces breaking the symmetries of the initial sys tem. A s s uming that the size of the p er turbation ∆ L (Ω) = δ λ D + κ K + ν N ∼ ε is sma ll, where ε = k ∆ L (0) k is the F rob enius norm of the p er turbation at Ω = 0 corr esp onding to the double eig env alue iβ , the b ehavior of the pe r turb ed eigenv alues for sma ll Ω and s mall ε is describ ed by the following a symptotic for m ula [38] λ ± p = iβ + i Ω f 11 + f 22 2 + i ǫ 11 + ǫ 22 2 ± i r (Ω( f 11 − f 22 ) + ǫ 11 − ǫ 22 ) 2 4 + (Ω f 12 + ǫ 12 )(Ω f 21 + ǫ 21 ) , (9) where the co efficients f ij are given by the expres sions (7) and ǫ ij are small complex num b ers of orde r ε ǫ ij = u T j ∆ L (0) u i = iβ δ u T j Du i + κ u T j Ku i + ν u T j Nu i . (10) With the vectors (5) we find ǫ 11 = i δ 2 d 22 + κ 2 β k 22 , ǫ 12 = i δ 2 d 12 + κ 2 β k 12 + ν 2 β ǫ 22 = i δ 2 d 11 + κ 2 β k 11 , ǫ 21 = i δ 2 d 12 + κ 2 β k 12 − ν 2 β , (11) where d ij and k ij denote the entries o f the matrices D and K . Substituting the quantities (8) and (11) into equation (9 ) we obtain λ ± p = iβ + i ρ 1 + ρ 2 4 β κ − µ 1 + µ 2 4 δ ± √ c, (12) where c =  µ 1 − µ 2 4  2 δ 2 −  ρ 1 − ρ 2 4 β  2 κ 2 +  i Ω + ν 2 β  2 − iδ κ 2tr KD − tr K tr D 8 β , (13) and µ 1 , µ 2 and ρ 1 , ρ 2 are eigenv a lues o f the matrices D and K , resp ectively , and thus sa tisfy the equations µ 2 − µ tr D + det D = 0 , ρ 2 − ρ tr K + det K = 0 . (14) Separation o f real and imaginary parts in equation (12) yields the rela tions 4  Re λ + tr D 4 δ  4 − 4  Re λ + tr D 4 δ  2 Re c − (Im c ) 2 = 0 , 4  Im λ − β − tr K 4 β κ  4 + 4  Im λ − β − tr K 4 β κ  2 Re c − (Im c ) 2 = 0 , (15) 5 where Re c =  µ 1 − µ 2 4  2 δ 2 −  ρ 1 − ρ 2 4 β  2 κ 2 − Ω 2 + ν 2 4 β 2 , Im c = Ω ν β − δ κ 2tr KD − tr K tr D 8 β . (16) F rom the equations (15) explicit express ions follo w for the r eal and imaginary parts of the eigenv alue s originated a fter the splitting of the do uble semi-simple eig env alue iβ Re λ = − µ 1 + µ 2 4 δ ± s Re c + p (Re c ) 2 + (Im c ) 2 2 , Im λ = β + ρ 1 + ρ 2 4 β κ ± s − Re c + p (Re c ) 2 + (Im c ) 2 2 . (17) The for m ula s (1 2)-(17) desc r ib e splitting o f the do uble eig env alues at the no des of the sp ectra l mesh due to v aria tio n of parameters, including those corres po nding to diss ipative and non-co nserv ative p ositional forces. Such splitting obviously le a ds to the deformatio n o f the mesh, in pa r ticular, to the ve ering and mer ging of eigenv alue br anches. Although the veering phenomenon in the systems with g yrosco pic co upling was studied bo th numerically a nd analytica lly , e.g in [4 , 5, 9, 1 0, 16, 20, 37], the explicit expr essions (12)-(17) for the splitting of the double eigenv a lues due to action of for ces of all types were not previously derived. Note that the approa ch used in our pap er differs from that of the cited w o rks b ecause it is ba sed on the per tur bation theor y of multiple eigenv alues [3 3, 38, 39]. The adv an ta g e o f our a pproach is the description of the sp ectrum of the p erturb ed system by means of only the deriv atives of the op era tor with res pe c t to parameters and the eigenv e c to rs o f the m ultiple eige nv alue calculated directly at a no de of the sp ectr al mesh. 3.1 Conserv ativ e deformation of the sp ectral mesh W e first study the influence of a conserv a tive p erturbation w ith the matrix K on the s pec tral m e s h of the gyrosc opic system. Substitution of δ = ν = 0 into the formulas (15) and (1 6) yields  Im λ − β − ρ 1 + ρ 2 4 β κ  2 − Ω 2 =  ρ 1 − ρ 2 4 β  2 κ 2 , Re λ = 0 . (18) When κ 6 = 0, equa tion (18) describ es a h yp erb o la in the plane Im λ versus Ω. Lo ca tion o f the br anches of the hyperb ola with resp ect to the lines of the unp erturb ed sp ectral mesh Im λ = β ± Ω dep ends on the definiteness of the matr ix K , see Fig. 2. Indeed, accor ding to equation (1 8) the hyperb ola ha s the a symptotes Im λ = β + ρ 1 + ρ 2 4 β κ ± Ω . (19) The asy mptotes cross each other ab ov e the no de (0 , β ) of the non-defor med sp ectral mesh for tr K > 0, exactly at the no de for ρ 1 = − ρ 2 , and b elow the no de for tr K < 0. The bra nches o f the hype r b ola intersect the axis Ω = 0 a t the p oints β 1 = β + ρ 1 2 β κ, β 2 = β + ρ 2 2 β κ. (20) If the eig env alues ρ 1 , 2 hav e the same sign, the intersection p o ints are a lwa ys lo cated ab ove or b elow the no de of the sp ectral mes h for K > 0 or K < 0, resp ectively , see Fig. 2(a). In case when one of the eigenv alue s ρ 1 , 2 is 6 Figure 2: Conser v ative defor ma tion of t he sp ectral mesh ( κ > 0): K is p ositive-definite (a); K is p o sitive semi-definite (b); K is indefinite (c). zero implying semi-definiteness of the matrix K , one of the bra nches of the hyper b o la go es thro ugh the node o f the sp ectral mesh and the other crosses the axis Ω = 0 a b ove the no de, if K is p ositive semi-definite, or b elow it, if K is neg ative semi-definite, Fig . 2(b). If K is indefinite, one of the intersection p oints β 1 , 2 is loca ted ab ov e the no de a nd another one below the no de, as indicated in Fig. 2(c). Therefore, the conserv ative deformation of the sp ectra l mesh do es no t shift the eigenv alues from the imagi- nary a xis near the no des (0 , ± β ) and thus preserves the marg inal stability . How ever, the deforma tion pattern depe nds on the definitenes s of the p erturbatio n matr ix K . In particular, in the degenerate case when det K = 0, one o f the eigenv a lue br anches, origina ted after the splitting of the double eig env alue, alwa y s pa sses through the p oint co rresp onding to the no de of the unp erturb ed sp ectral mesh, Fig. 2(b). 3.2 Creation a nd activ ation of the la t en t sources of instabil ity by dissipation Now we consider the effect of dissipative forces on the stability o f the gyro scopic sy stem and study its dep endence on the prop er ties of the matrix D . Assuming ν = κ = 0 in expre ssion (13) we rewrite for m ula (12) in the for m λ = iβ − µ 1 + µ 2 4 δ ± s  µ 1 − µ 2 4  2 δ 2 − Ω 2 . (21) Since Im c = 0, equa tions (15) can be tra nsformed into  Re λ + µ 1 + µ 2 4 δ  2 + Ω 2 = ( µ 1 − µ 2 ) 2 16 δ 2 , Im λ = β (22) when Ω 2 − ( µ 1 − µ 2 ) 2 16 δ 2 < 0 , (23) and into Ω 2 − (Im λ − β ) 2 = ( µ 1 − µ 2 ) 2 16 δ 2 , Re λ = − µ 1 + µ 2 4 δ, (24) when the sign in inequalit y (23) is opposite. F or a given δ eq uation (2 4) defines a hyperb o la in t he plane (Ω , Im λ ), while (22) is the equa tion of a circle in the plane (Ω , Re λ ), a s shown in Fig. 3(a,c). F or tra cking the 7 Figure 3: Originatio n of a laten t source of the sub critical flutter instabilit y in presence of full dissipation: Submerged bubble of instability (a); co alescence of eigenv alues in the complex plane at tw o exceptional p oints (b); hyper bo lic tra jector ies of imag inary par ts (c). complex eigenv alues due to change of the gyr oscopic parameter Ω, it is conv e nie nt to consider the eig env alue branches in the three - dimensional space (Ω , Im λ, Re λ ). In this space the circle b e lo ngs to the plane Im λ = β and the h yp er b o la lies in the plane Re λ = − δ ( µ 1 + µ 2 ) / 4, see Fig. 4(a,c). Belo w we show tha t the circle of the complex eigenv alue s – the bubble of instability [23] – plays the cr ucial role in the developmen t and lo calizatio n of the sub critica l flutter instability and that its pro p erties dep end on whether the matrix D is definite or indefinite. 3.2.1 F ull dis sipation and p erv asiv e damping: A latent state of the bubble of instabili ty F rom the formulas (22) a nd (24) we see that the basic geometric characteristics o f the tra jectories of eigenv a lues of the dissipatively p erturb ed gyros copic system ar e defined b y the eigenv alues µ 1 and µ 2 of the matrix D . F or example, the ra dius of the bubble of instability r b and the distance d b of its center from the plane Re λ = 0 are r b = | ( µ 1 − µ 2 ) δ | 4 , d b = | ( µ 1 + µ 2 ) δ | 4 . (25) As a cons equence, the bubble do es not intersect the plane Re λ = 0 under the condition d b ≥ r b , which is equiv alent to the inequality µ 1 µ 2 = det D ≥ 0 . (26) Moreov e r , the bubble of instability is ” submerged” under the surface Re λ = 0 in the space (Ω , Im λ, Re λ ), if δ ( µ 1 + µ 2 ) = δ tr D > 0 . (27) Since the inequalities (2 6) and (2 7) imply p ositive se mi-definiteness of the matrix δ D , we conclude that the ro le of full dissipation or p erv asive da mping is to defor m the s pe c tral mesh in such a wa y that the double 8 Figure 4: The mech a nism o f subc r itical flutter instability (b old lines): The r ing (bubble) of complex eigenv alues submerged under the surface Re λ = 0 due to action of dis sipation with det D ≥ 0 - a latent source o f ins ta bilit y (a); repulsion o f eig env alue branches of the sp ectral mesh due to action of non-conser v ative p ositional fo r ces (b); emersion of the bubble of instabilit y due to indefinite damping with det D < 0 (c); colla pse of the bubble of instability and immersion and emer sion of its pa rts due to c o mbined action o f dissipative and non-conser v ative po sitional forces (d). 9 semi-simple eigenv alue is inflated to the bubble of complex eigenv alues (2 2) connected with the tw o br anches of the hyper b ola (2 4) at the p o int s Im λ = β , Re λ = − δ ( µ 1 + µ 2 ) / 4 , Ω = ± δ ( µ 1 − µ 2 ) / 4 , (28) and to plunge all the eig env alue cur ves into the region Re λ ≤ 0. The eigenv alues at the p o ints (28) are double and hav e a Jo rdan chain of generalized eigenv ector s of order 2. In the co mplex plane the eigenv alues of the per turb ed system move with the v ariation of Ω alo ng the lines Re λ = − d b un til they meet at the p oints (28) and then split in the orthog onal direction; how ever, they never cross the imag inary axis, se e Fig. 3 (b). Note that a s imila r pro cess o f unfolding the semi-simple eigenv a lue (diab olica l p o int ) into tw o double eigenv alues with the Jor dan blo cks (exceptional p oints) for Hermitian matrices under arbitrar y co mplex per turbation and its a pplication to crysta l optics hav e b een describ ed in [30, 3 8 ]. The bubble of instability has tw o remark able prop erties importa nt for the ex planation of the pheno menon of squeal. Fir st, there can exist p erturbations causing its growth and e mersion ab ov e the sur face Re λ = 0 (flutter); second, for small δ the instabilit y is lo calized in the vicinity of the frequency Im λ = β and the v alue o f the gyrosc opic par a meter Ω = 0. The na rrow frequency band is a characteris tic prop erty of squeal. W e conclude that the bubble is a la tent sour c e of sub critical flutter instability lo ca lized in a nar row ra nge of change o f the gyrosc opic parameter with | Ω | < Ω d at the frequency corresp o nding to the double eig env alue of the non-rotating system. 3.2.2 Indefinite dampi ng: An activ e s tate of the bubble of i nstability As it is see n from equa tions (25), the radius of the bubble of instability is g reater then the depth of its submer sion under the surface Re λ = 0 only if the eigenv alues µ 1 and µ 2 of the damping ma tr ix hav e different signs , i.e. if the damping is indefinite . The damping with the indefinite matrix app ears for example in the systems with frictional contact when the friction co efficient is dec r easing with relative sliding velocity as well as in the pro blems of propaga tion of w aves in diss ipa tive media [17, 24, 26]. Indefinite da mping is known to b e a destabilizing factor [18, 49, 50]. In our c ase it leads to the emersion o f the bubble of instability meaning that the eigenv alues o f the bubble have pos itive real parts in the ra nge Ω 2 < Ω 2 cr , where Ω cr = δ 2 √ − det D (29) is found from the equatio n (22) after assuming there Re λ = 0. The r ight side of the formula (29) is real o nly for det D < 0, i.e. for the indefinite matrix D . W e see that the do main of as ymptotic stabilit y is defined by the c onstraints δ tr D > 0 and Ω 2 > Ω 2 cr . In the plane of the parameter s δ and Ω for det D < 0 it ha s a form shown in Fig. 5(a ). Due to the singularity existing at the o rigin, an uns ta ble system with indefinite damping ca n b e stabilized by sufficiently strong gy roscopic forces, as shown by the da shed line in Fig. 5(a). With the incr ease of det D the stability domain is getting wider and fo r det D > 0 it is defined by the c o ndition δ tr D > 0 , Fig . 5(c). At det D = 0 the line Ω = 0 do e s not belo ng to the domain of asy mptotic stability , Fig. 5(b). 10 Figure 5: Approximation based o n the first-o r der p ertur bation theory o f double e ig env alues to the do main of asymptotic stability (white) and its b oundar y (b old lines) for the diss ipa tively p erturb ed gyrosc opic system (1) when ν = κ = 0, tr D > 0 and det D < 0 (a); det D = 0 (b); de t D > 0 (c). Therefore, changing the da mping matrix δ D from p ositive definite to indefinite we trigge r the state of the bubble o f insta bility fr om latent (Re λ < 0 ) to active (Re λ > 0), se e Fig. 4(a,c). Since fo r small δ w e have Ω cr < Ω d , the flutter instability is lo calized in the vicinity of Ω = 0. As we show b elow, the non-co ns erv ative po sitional forces play the similar role. 3.3 Activ ation of the bu bble of instab ility by non-conserv ativ e p ositional forces In th e absence of dissipa tio n non-conserv ative p ositional forces destr oy t he marginal stabilit y of g y roscopic systems [6, 7]. One ca n ea sily chec k that this prop erty is v alid for system (1) b y assuming δ = κ = 0 in the formulas (12) a nd (13), which yield λ ± p = iβ ± i Ω ± ν 2 β , λ ± n = − iβ ± i Ω ∓ ν 2 β . (30) Equations (30) show that the eigenv alues of the branches iβ + i Ω and − i β − i Ω o f the spe ctral mesh get p ositive real parts due to p erturbation by the no n- conserv ative p ositiona l forces. The eig env alues of the other tw o bra nches a re shifted to the left from the imagina r y a xis, s e e Fig. 4(b). Note that the r eal part of the p ertur b ed eigenv alue is pro po rtional to 1 /β , which mea ns that the des tabilizing effect o f non-conser v ative po sitional forces is less pronounced for higher frequencies. This is imp orta n t in the study of systems with many degrees of fr eedom. Thus, the ins tability induced by the non-conser v ative forces only is not lo calized near the no des of the sp ectral mesh, in contrast to the effect of indefinite damping. W e s how now that in co m bina tion with the dissipa tive forces , bo th definite and indefinite, the non-conserv ative forc e s can create sub c r itical flutter instability in the vicinity of diab olical p o int s . Recall that the real and imaginary parts of the eigenv a lue s or iginated a fter the spitting of the double eig e n- v alues due to combined action of dissipative and no n-conserv ative po sitional fo r ces are g iven b y the express io ns (15) and (17), where we assume κ = 0 . Multiplying equatio ns (17) we find that the tra jectories of the eigenv alues 11 Figure 6: Sub critical flutter instability due to combined action o f dissipa tive a nd non-c o nserv ative po sitional forces: Collapse and emersio n of the bubble of instability (a); excur sions of e ig env alues to the r ight side of the complex pla ne when Ω go es from nega tive v alues to po sitive (b); crossing of imaginar y parts (c). in the complex plane are des crib ed by the formula  Re λ + tr D 4 δ  (Im λ − β ) = Ω ν 2 β . (31) When ν = 0 a nd δ 6 = 0 is given, the e igenv alues mov e with the v ariation o f the gyro scopic parameter Ω alo ng the lines Re λ = − tr D / 4 and Im λ = β and merg e at the p o ints (2 8), see Fig. 3(b). Non-conserv ative p ositiona l forces with ν 6 = 0 destroy the merging of mo des. As a conse q uence, the eigen- v alues mov e a long the s eparated tra jectories. Accor ding to r elations (30) the e igenv alues with | Im λ | incre a sing due to an increase in | Ω | mov e closer to the imaginary axis then the others, as shown in Fig 6(b). Therefore, in the space (Ω , Im λ, Re λ ) the action of the non- c o nserv ative p o sitional forces s e parates the bubble of insta bilit y and the adjacent h yp er bo lic eige nv alue branches int o tw o no n-intersecting c urves, see Fig 4(d). It is rema rk able that the for m of each of the new eigenv alue cur ves carries the memor y ab out the o riginal bubble of instability . Due to this sp e ctral pec ulia rity the real parts of the eigenv alues can b e p ositive for the v alues of the gyr oscopic parameter lo calized nea r Ω = 0 in the range Ω 2 < Ω 2 cr . T ak ing int o a ccount that Re λ = 0 at the critical v alues of the g y roscopic para meter we find Ω cr from the e q uations (17) Ω cr = δ tr D 4 s − ν 2 − δ 2 β 2 det D ν 2 − δ 2 β 2 (tr D / 2) 2 . (32) Additionally , it follows from (17) that the eig enfrequencies of the unsta ble mo des fro m the interv al Ω 2 < Ω 2 cr are lo ca lized near the frequency o f the double semi-simple e ig env alue at the no de of the undeformed sp ectral mesh: ω − cr < ω < ω + cr ω ± cr = β ± ν 2 β s − ν 2 − δ 2 β 2 det D ν 2 − δ 2 β 2 (tr D / 2) 2 . (33) When the radicand in formulas (32) and (33) is real, the e ig env alues move in the complex plane making the excursion to its r ight side, as shown in Fig. 6(b). As it follows fro m (32), in pr esence of no n-conserv ative 12 po sitional forces such ex c ur sions b ehind the stabilit y bounda r y a re possible for the eigenv alues , ev en when dissipation is full (det D > 0 ). One can s ay that similarly t o the indefinite damping t he non- c onservative p ositional for c es activate the latent sour c es of flut t er instability cr e ate d by the ful l dissip ation . The equation (32) describ es the surfac e in the space o f the pa r ameters δ , ν , and Ω, which is an appr oximation to the stability b o undary sepa rating the do mains o f asymptotic stability and flutter. F or b etter under s tanding the s ha p e of this surface we extract the para meter ν in (3 2) ν = ± δ β tr D s δ 2 det D + 4Ω 2 δ 2 (tr D ) 2 + 16Ω 2 . (34) If det D ≥ 0 and Ω is fixed, the formula (34) desc r ib es tw o independent curves in the plane ( δ, ν ) intersecting with ea ch other at the o rigin a long the straig ht lines g iven b y the expression ν = ± β tr D 2 δ. (35) How ever, in case when damping is indefinite and det D < 0, th e ra dical in the fo r mula (34) is real only for δ 2 < − 4Ω 2 / det D meaning that (34) de s crib es tw o branches o f a clos ed lo op in the plane o f the pa rameters δ and ν . The lo op is smo oth at every p oint ex c e pt at the orig in, where it is self-intersecting with the tangents given by the expression (35). Therefor e , for a given Ω this curve lo oks like figur e ” 8”. When Ω go es to zero, the size of the self-int e r secting curve tends to zero to o . W e conclude that in case det D < 0 the sha pe of the surface describ ed by equatio n (32) or (34) is a cone with the ”8 ” -shap ed lo op in a cross -section, see Fig. 7 (a). Due to self-intersections the cone co nsists of four p o ck ets . The asy mptotic stability doma in is inside the tw o of them, s elected by the inequality (27), as shown in Fig. 7(a). The s ingularity of the stability domain at the origin is the degenera tion of a mor e ge ne r al configur ation describ ed first in [50]. The doma in of a symptotic stability bifurc a tes when det D changes from nega tive to po sitive v alues. This pro cess is shown in Fig. 7. Indeed, according to formula (3 4 ) with increasing det D < 0 tw o po ck ets of the domain of asymptotic sta bilit y mov e to wards eac h other until they have a common tangen t line ν = 0 at det D = 0, see Fig. 7(b). When det D is p os itive, this temp or arily g lued co nfiguration unfolds to the unique domain of a symptotic stability b ounded b y the tw o surfaces intersecting along the Ω-a xis, as shown in Fig. 7(c). In Fig. 7(a) we see that in case of indefinite da mping there always ex ists a n instability gap due to the singularity at the origin. Sta rting in the flutter domain a t Ω = 0 for any combination o f the parameters δ and ν one ca n r each the doma in of a symptotic stability at higher v alues of | Ω | (g y roscopic stabiliza tio n), as shown in Fig. 7(a) b y the dashed line. The gap is resp onsible for the sub critical flutter instability lo calized in the vicinity of the no de o f the s pe c tr al mesh of the unp erturb ed gyro scopic system. When det D = 0, the gap v anis he s in the direc tion ν = 0. In case of full dissipa tion (det D > 0) the singularity a t the orig in unfolds. Howev er, the memory ab out it is preserved in the tw o instability gaps lo cated in the folds of the stability b ounda ry with the lo cally strong curv ature , Fig. 7(c). W e see that in case of full dissipation for some combinations o f the parameters δ a nd ν the system is asymptotically stable a t any Ω. There exist, howev er, the v alues of δ and ν for which o ne ca n p enetra te the 13 Figure 7: Do mains of asymptotic stability in the space ( δ, ν , Ω) for different t yp es of da mping: Indefinite damping det D < 0 (a); semi-definite (p erv asive) damping det D = 0 (b); full dissipa tion det D > 0 (c). fold of the s tability boundar y with the change of Ω, as shown in Fig. 7(c) by the da shed line. F o r such δ and ν the flutter instability is lo ca liz ed in the vicinity of Ω = 0. It is remar k able that in presence o f non-co ns erv ative po sitional force s the system with full dissipa tion can suffer from the subcr itical flutter lo calized near the no des of the s pe ctral mesh. A go o d illustratio n of this fact is the formula for the maximal real part of the unstable eigenv alue attained at Ω = 0 (see Fig. 6(a)) (Re λ ) max = − µ 1 + µ 2 4 δ + s  µ 1 − µ 2 4  2 δ 2 + ν 2 4 β 2 . (36) F rom our previous consider a tions it follows that the phenomenon of the lo ca l sub critica l flutter instability is controlled b y the eigenv a lues of the matrix D . When bo th of them are p ositive, the folds of the stability bo undary are mor e prono unced if one of the eigenv alues is close to z e ro. If one o f the eigenv a lues is nega tive and the other is po sitive, the lo cal subc r itical flutter instability is p oss ible for any combination of δ and ν including the ca se when the non-conserv ative po sitional forces are absent ( ν = 0). Even if the structure of the da mping ma tr ix D is unknown, we realize tha t the main ro le of diss ipation of any kind is the creation o f the bubble of insta bilit y . It is submerged below the s urface Re λ = 0 in the spa ce (Ω , Im λ, Re λ ) in case of full diss ipa tion and partially lies in the doma in Re λ > 0 when damping is indefinite. Non-conserv ative po sitional forces destroy the bubble of instability in to tw o branches and shift one of them to the reg ion of p os itive real par ts even in case o f full dissipation. Since the bra nch remembers the existence of the bubble, the instability is developing lo cally near the no des of the sp ectra l mesh. Ther efor e, the instability me chanism b ehind t he sque al is the emersion (or activation) due to indefinite damping and non-c onservative p ositional for c es of the bubbles of instability cr e ate d by the ful l dissip ation in the vicinity of the no des of the sp e ctr al mesh . 14  Figure 8: A ro ta ting circula r string and its ” keyboar d” constituted by the no des (marked by white and grey) of the sp ectr al mesh (only 30 mo des ar e shown). 4 Example. A rotating circular string constrained b y a stationary load system F ollowing Y ang a nd Hutton [16], we consider a rotating cir cular string of displa cement W ( ϕ, τ ), constr a ined at ϕ = 0 b y a stationary loa d s y stem consis ting of a spring, a damp er, a nd a massless eyelet g enerating a constant frictional follow er force F [19] on the string, a s shown in Fig. 8. The par ameters r and ρ are the radius and mass per unit length of the string. The ro tational speed o f the str ing is γ . The following as sumptions are adopted in developing the gov er ning equa tion of the proble m: the circumfer e nt ia l tension P in the string is co nstant; the stiffness o f the spr ing supp orting the string is K and the damping co efficient of the visco us damp er is D ; the velocity of the string in the ϕ direction has constant v a lue γ r [1 6]. Int r o ducing the following non-dimensiona l v ariables a nd para meters t = τ r s P ρ , w = W r , Ω = γ r r ρ P , k = K r P , µ = F P , d = D √ ρP , (37) we arrive a t the non-dimensiona l governing equatio n and bo undary conditions [1 6]: w tt + 2Ω w tϕ − (1 − Ω 2 ) w ϕϕ = 0 , (38) w (0 , t ) − w (2 π , t ) = 0 , (1 − Ω 2 )[ w ϕ (2 π , t ) − w ϕ (0 , t )] + k w (0 , t ) + dw t (0 , t ) + µw ϕ (0 , t ) = 0 . (39) The b o undary c o nditions (39) reflect the co n tinuit y of the string displacement a nd the discontin uity of its slop e following from the force ba lance at the eyelet in the ass umption of smallness of the norm of w and w ϕ with resp ect to r . The inclusion in Fig. 8 shows the m utua l c o nfiguration of the vectors of the frictional follow er force F and of the circ umfer ent ia l tension P . 15 Separation o f time by the subs titution w ( ϕ, t ) = u ( ϕ ) exp( λt ) yields the b oundary v alue problem Lu = λ 2 u + 2 Ω λu ′ − (1 − Ω 2 ) u ′′ = 0 , (40) u (0) − u (2 π ) = 0 , u ′ (0) − u ′ (2 π ) = λd + k 1 − Ω 2 u (0) + µ 1 − Ω 2 u ′ (0) , (41) where prime denotes partia l different ia tion with resp ect to ϕ . The non-self-adjoint b ounda ry eige nv alue problem depe nds on four para meters Ω, d , k , and µ express ing the sp eed of ro tation, and damping, stiffness, a nd friction co efficients of the eyelet, resp ectively . T aking the s c alar pro duct ( L u, v ) = R 2 π 0 ¯ v L u dϕ , where the bar over a symbol deno tes complex conjugate, then in teg rating it b y parts and emplo ying the bounda r y conditions (41) w e arrive at the b oundary v a lue problem a djoint to (40) and (41) L ∗ v = ¯ λ 2 v − 2Ω ¯ λv ′ − (1 − Ω 2 ) v ′′ = 0 , (42) v (0) − v (2 π ) = − µ 1 − Ω 2 v (2 π ) , v ′ (0) − v ′ (2 π ) = ¯ λd + k 1 − Ω 2 v (2 π ) + 2Ω µ (1 − Ω 2 ) 2 v (2 π ) . (43) Let us first cons ider the string without co nstraints ( d = k = µ = 0 ). Then, the s y stem is gy r oscopic so that the eigenfunctions of the a djoint b oundary v alue pr oblems corresp o nding to a purely imaginary eigenv a lue λ coincide, i.e. v = u . Assuming the s o lution o f the equation (40) in the form u = C 1 exp  λ 1 − Ω ϕ  + C 2 exp  − λ 1+Ω ϕ  and substituting it into the bo undary conditions (4 1) we obtain the ch a racteris tic eq uation 8 λ sin π λ i (1 − Ω) sin π λ i (1 + Ω) e − 2 πλ Ω Ω 2 − 1 Ω 2 − 1 = 0 . (44) The ro ots of the equation (44) ar e λ + n = in (1 + Ω) , λ − n = in (1 − Ω) , (45) where n ∈ Z . They are the eigenv alues o f the b oundary eigenv a lue problem (4 0), (41) with the eigenfunctions u + n = cos( nϕ ) − i sin( nϕ ) , u − n = cos( nϕ ) + i sin( nϕ ) , (46) resp ectively . The eigenv alues are purely imag ina ry and form a mesh of lines intersecting with each other in the plane Im λ versus Ω, as shown in Fig. 8. The eigenv a lues (45) are simple almost at all v alues of Ω exce pting those corres po nding to the no des o f the sp ectral mesh. Indeed, tw o eigenv a lue br anches λ ε n = in (1 + ε Ω) and λ δ m = im (1 + δ Ω), where ε, δ = ± , intersect each other at Ω = Ω εδ mn Ω εδ mn = n − m mδ − nε (47) and or iginate the double eigenv alue λ εδ mn = inm ( δ − ε ) mδ − nε , (48) which has tw o linearly indep endent eigenfunctions u ε n = cos( nϕ ) − ε i sin( nϕ ) , u δ m = cos( mϕ ) − δ i sin( mϕ ) . (49) 16 The no des (47), (48) of the spectr al mes h of the r otating circular string in the absence o f the external loading are marked by white and grey dots in Fig. 8. At Ω = 0 the sp ectr um of the non-rotating circular string consists of the double semi-simple purely imaginary eigenv alues in , n ∈ Z , each of which splits into tw o simple purely imaginary eigenv alues due to change in the angular velocity [9, 10, 16, 20]. At Ω = ± 1 all the eigenv alue branches cr oss the a xis Im λ = 0, see Fig. 8. In the following we will consider the spe ctrum for the angula r velocities from the sub critical range when Ω ∈ ( − 1 , 1). Now we consider the deformation of the spe c tr al mesh ca used by the interaction of the rotating str ing with the external loading sy s tem. W e pro ceed analo gously to our in vestigation of a tw o-dimensiona l system a nd study the splitting of the double eig env alues a t the no des o f the s pe ctral mesh. F or this purp os e we use the per turbation theor y of multiple eigenv alue s of non-self-a djoint differential o p e rators developed in [33, 39, 43]. According to this theory the per turb ed eigenv alues a re expr essed by the fo llowing asymptotic for mula λ = λ εδ nm − f εε nn + f δδ mm 2 − ǫ εε nn + ǫ δδ mm 2 ± s  f εε nn − f δδ 22 + ǫ εε nn − ǫ δδ 22  2 4 − ( f εδ nm + ǫ εδ nm )( f δε mn + ǫ δε mn ) . (50) The co efficients f εδ nm are defined by f εδ nm = 2 λ εδ nm R 2 π 0 u ε n ′ ¯ u δ m dϕ + 2 Ω εδ nm R 2 π 0 u ε n ′′ ¯ u δ m dϕ 2 q R 2 π 0 ( λ εδ nm u ε n + Ω εδ nm u ε n ′ ) ¯ u ε n dϕ R 2 π 0 ( λ εδ nm u δ m + Ω εδ nm u δ m ′ ) ¯ u δ m dϕ ∆Ω , (51) while the qua nt ities ǫ εδ nm are ǫ εδ nm = ( dλ εδ nm + k ) u ε n (0) ¯ u δ m (0) + µu ε n ′ (0) ¯ u δ m (0) 2 q R 2 π 0 ( λ εδ nm u ε n + Ω εδ nm u ε n ′ ) ¯ u ε n dϕ R 2 π 0 ( λ εδ nm u δ m + Ω εδ nm u δ m ′ ) ¯ u δ m dϕ . (52) where ∆Ω = Ω − Ω εδ nm . Calculating the integrals in (51) and (52) a nd taking into account expr essions (47) a nd (48), we o btain f εε nn = − iεn ∆Ω , f εδ nm = f δε mn = 0 , f δδ mm = − iδ m ∆Ω , ( 5 3) and ǫ εε nn = dλ εδ nm + k − ε inµ 4 π ni , ǫ εδ nm = dλ εδ nm + k − ε inµ 4 π i √ nm , ǫ δε mn = dλ εδ nm + k − δ imµ 4 π i √ nm , ǫ δδ mm = dλ εδ nm + k − δ imµ 4 π mi . (54) T aking into acco unt the results of o ur ca lculations (53) and (54) we find λ = λ εδ nm + i εn + δ m 2 ∆Ω + i n + m 8 π nm ( dλ εδ nm + k ) + ε + δ 8 π µ ± √ c (55) where c =  i εn − δ m 2 ∆Ω + i m − n 8 π mn ( dλ εδ nm + k ) + ε − δ 8 π µ  2 − ( dλ εδ nm + k − iεnµ )( dλ εδ nm + k − i δ mµ ) 16 π 2 nm . (56) 17 According to the exp erimental data [22, 26, 44, 45, 46] the frequency of sound emitted by a singing wine glass a nd a squealing lab orator y brake at lo w spins is close to a do uble eig enfrequency of the non-r otating bo dies. F or this reaso n it is imp ortant to study first the influence of exter nal stiffness, damping and frictio n on the defor mation o f the sp ectral mesh near the no des corr e sp onding to Ω = 0. A t Ω = 0 the do uble eigenv alue λ εδ nm originates due to intersection of the eig env alue branches λ ε n and λ δ m , where m = n a nd ε = − δ . T ak ing this in to account, we find from (55) and (56) the expres sions describing splitting of the double eigenv alue in due to action of gyrosco pic forces and an ex ternal s pr ing λ = in + i k 4 π n ± i r n 2 Ω 2 + k 2 16 π 2 n 2 . (57) Equation (5 7) des crib es a hyper bo la in the plane I m λ versus Ω, which intersects the axis Ω = 0 at the v alues λ = n and λ = n + k 2 π n , see F ig . 9 (a ). The g ap b etw een the branches decrease s with the increase of th e nu mber n o f a mo de. It is r emark able that the low er branch pa sses throug h the p oint corres p o nding to the no de o f the s pec tral mesh of the non-p er turb e d gy roscopic system, which a grees with the numerical re s ults of [16]. Remember that in case of tw o-dimensio na l systems the rea son for such a degenerate b ehavior is a zero eigenv alue in the matrix K o f e x ternal p otential fore s . Analogously , fro m express io ns (55) and (56) follows the asy mptotic formula for the e igenv alues or iginating after the s plitting o f the double eigenv alue in at Ω = 0 due to p erturba tion by the gyr oscopic fo rces and a n external dampe r λ = in − d 4 π ± r d 2 16 π 2 − n 2 Ω 2 . (58) According to (58) the real parts o f the eigenv alues as functions of Ω orig inate a bubble of instability in the plane Re λ versus Ω  Re λ + d 4 π  2 + n 2 Ω 2 = d 2 16 π 2 , Im λ = n . (59) The ellipse (59) is submerged under the plane Re λ = 0 in the space (Ω , Im λ, Re λ ) so that it touches the plane at the origin, a s shown in Fig. 9(b). The ellipse is connected w ith the branches of the hyperb ola of complex eigenv alues n 2 Ω 2 − (Im λ − n ) 2 = d 2 16 π 2 , Re λ = − d 4 π . (60) The horizontal diameter of the ellipse decrea ses with the increase in the mo de num b er n , while the vertical one do es not change. As we see, the external damp er cr eates a latent source of lo cal sub c ritical flutter instability exactly as it happ ens in tw o dimensio ns when the matrix of dissipa tive forces D is semi-definite, i.e. when it has one zer o eigenv alue. The range o f change of the gy r oscopic parameter co rresp onding to the latent instability is lo cated co mpactly ar ound the orig in and decreases with the increa se in n . The deformation of the sp ectral mesh near the double eig e nv alue in at Ω = 0 due to combined action of gyrosc opic forces a nd externa l friction is de s crib ed by the expre s sion λ = in ± r  in Ω + µ 4 π  2 − µ 2 16 π 2 , ( 6 1) 18 Figure 9: Deformation of the sp ectr al mesh of the rotating string near the no des (0 , 3 ), (0 , 2), and (0 , 1) caused by the actio n of the exter nal spring with k = 0 . 3 (a), damp er with d = 0 . 3 (b), and friction with µ = 0 . 3 (c), resp ectively . 19 following from (55) and (5 6). The imagina ry parts of the eigenv alues of the deformed sp e ctral mes h Im λ = n ± 1 2 π q 2 π 2 n 2 Ω 2 ± π n Ω p 4 π 2 n 2 Ω 2 + µ 2 (62) cross at the no de (0 , n ), as in the non-p er turb ed case. Howev er, the cro ssing is degenera te b ecaus e the eigenv alue branches touch each o ther at the node, s ee Fig. 9(c). Indeed, expanding expressio n (62) in the vicinity of Ω = 0 we find that Im λ = n ± 1 2 π p π nµ | Ω | + O (Ω 3 / 2 ) . (63) Clearly , at Ω = 0 the imaginary parts do not split due to non- conserv ative per turbation from the eyelet. It is instructive to note that for Ω → ∞ the imaginary parts asymptotica lly tend to n (1 ± Ω). By this reaso n for small p erturba tions the sp e ctral mesh depicted in the plane Im λ versus Ω lo oks non- defo rmed at the fir st glance. The rea l parts of the deformed sp ectr a l mesh describ ed by the expr ession Re λ = ± 1 2 π q − 2 π 2 n 2 Ω 2 ± π n Ω p 4 π 2 n 2 Ω 2 + µ 2 (64) cross at Ω = 0, so that Re λ = 0. The c rossing o f the r e al parts is dege ne r ate Fig. 9(c), which is confirmed by the a symptotic expr ession Re λ = ± 1 2 π p π nµ | Ω | + O (Ω 3 / 2 ) . (65) F or Ω → ∞ the real pa rts follow the asymptotic law Re λ = ± µ 4 π ∓ µ 3 128 π 3 n 2 Ω 2 + o (Ω − 2 ) (66) As is seen in Fig. 9 (c), the r e al parts almo st always ar e close to the lines ± µ/ (4 π ), except for the vicinity of the no de o f the sp ectral mesh, where the r e al parts r a pidly tend to zero. This b ehavior agr ees with the results of nu mer ical ca lculations of [16]. W e see that the double semi-simple eige n v alue in do es not split due to v aria tion o f only the parameter of non- conserv ative po sitional for ces µ . In the tw o-dimensional case this would corresp o nd to the degenerate matrix N , det N = 0 . Since N is skew-symmetric, the degener acy mea ns N ≡ 0, i.e. the absence o f non- c onserv ative per turbation. In case of more then tw o degree s of freedom the degenera c y of the op era tor o f non-conse rv ative po sitional forces le ads to the cuspidal devia tion of the generic splitting picture r egistered in Fig. 9(c). W e hav e co nsidered lo cal deformation of the sp ectral mesh of the ro tating string nea r the no des a t Ω = 0. How ever, the formulae (55) and (56) enable us to inv estiga te similar ly the deformation o f the mesh in the vicinity of all the other no de s . Concluding, we note that the source of p erturbation of the rotating str ing concentrated at o ne p oint lea ds to the de fo rmations of its sp ectral mesh, which cor r esp ond to that c aused by the semi-definite matr ices of co n- serv ative, dissipative and non-conser v ative fo r ces in the tw o- dimensional case. Zero eigenv a lues of the damping op erator encour age the a ctiv ation of the latent bubble o f instability , while the zero eig env alue o f the o pe r ator of 20 non-conser v ativ e p ositional for ces suppresses this pro cess. T o get lo cal sub critical flutter instability , describ ed in the previous sections for the finite-dimensional mo del, the op erator s of dissipativ e and non-conser v ative per turbations must b e ge neric, which excludes their semi-definiteness. W e notice that the s ingular sourc e s o f momentum and energy in the pro blem of stability of a r otating string as a reaso n for deg e ne r acies were dis- cussed also in [28, 31]. One o f the ways to av oid this degener acy is to co nsider not p oint wise but distributed contact with the dissipative, stiffness and friction c har acteristics dep ending on the material coo rdinates. A step in this direction is taken in [51], where a mo del of distr ibuted pads was developed. F o r simplicity , in [51] the characteristics o f the pins cons tituting the pads were assumed not dep ending on the co ordinates, which led to the same semi- de finite degeneracy . This could b e a reflection o f the so-c alled Her rmann-Smith pa r adox of a bea m r esting on a Winkler-type ela stic fo undation and loaded by a follow e r force [25]. The degener a cy in the Herrmann-Smith problem is r emov ed by as s uming a non-uniform mo dulus o f elasticity . Similar mo dification of the mo del of the distributed brake pads of [51] could give g eneric per turbation op erato rs a nd o p en the wa y to the mo deling of the disk br a ke squeal catching its most sig nificant features. 5 Conclusion: Disc brak e as a m usical instrumen t As we alr e a dy mentioned, the principle of a ctiv ating sound by friction is the sa me for a wine glass, a disc bra ke, and the g lass har monica. The latter is an ancient m usica l instrument for which the famo us ”Dance of the Sugar Plum F airy” in the first e ditio n o f ”The Nutcracker” ballet was comp osed by P .I. Tchaiko wsky in 1891 [53]. The results obtained in the present pap er s how that the ”keyboa rd” of ro ta ting elastic b o die s of revolution, among which are the g lass harmonica and the disc of a br ake, is formed by the no des of the sp e ctral mes h, corres p o nding to angular velocities in the sub critica l r ange. The frictiona l co nt a ct is the s o urce o f dissipative and non-co nserv ative forces, which make the sys tem unstable in the vicinity of the no des and force a rotating structure to vibr ate at a frequency close to the do uble frequency of the no de and at the ang ular velo city close to that of the no de. These co nclusions agr ee with the results of recent exp eriments with the lab or atory brake [44, 45, 46]. The higher mo des ar e more efficiently damp ed than the lower ones, and a particular freq ue nc y is selected by the sp eed of rotation and the loading conditions, including such para meters a s the siz e o f the fr iction pads a nd their placement with resp ect to the disc. It is known that dissipative and non-conserv ative forces may influence the sta bility in a non-intuitiv e manner [6, 7, 13, 14, 29, 32, 41, 42, 50, 52]. W e hav e shown that the former crea te the latent lo cal sour ces of insta bility around the no des of the sp ectral mesh (bubbles of instabilit y), while the latter a ctiv a te these sour ces by inflating and des tructing the bubbles . It turns out that the eig env alues o f the damping matrix control the development of instability . F or b etter stabilit y b oth of them should be p o sitive and s tay far from zero. If one of the eigenv alues of the damping matrix is close to zero or b eco mes negative, the instabilit y can o ccur with the weaker no n- conserv ative pos itional for ces or even without them. With the use of the p er turbation theor y of multiple eigenv alues w e hav e obtained ex plicit for m ula s describing 21 the defor mation o f the sp ectral mesh by diss ipa tive a nd non-co nserv ative p erturba tions. The tra jectories of eigenv alues are a nalytically describ ed and cla s sified. The approximations of the domain o f asymptotic stability are obta ined with the use of the deriv atives of the op erator and the eigenvectors of the double eig env alues calculated at the no des of the sp ectra l mes h. Singular ities of the stability b o unda ry of a new type were found and their ro le in the development o f ins tability was c la rified. The theory develope d seems to b e the first analytica l explanation of the basic mechanism of friction-induced instabilities in ro tating elastic bo dies of revolution. The application of the theory to mor e complicated discrete and contin uous mo dels of sque a ling disc bra kes, singing wine glasse s , and ca lender rolls will b e published elsewhere. 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