Analysis of dynamical behaviors of a double belt friction-oscillator model

Transcription

1 Analysis of dynamical behaviors of a double bel fricion-oscillaor model Ge Chen Shandong Normal Universiy School of Mahemaical Sciences Ji nan, PR China @qq.com Jinjun Fan Shandong Normal Universiy School of Mahemaical Sciences Ji nan, PR China fjj18@126.com Absrac: In his paper, analyical resuls of complex moions of a double bel fricion-oscillaor are invesigaed using he flow swichabiliy heory of he disconinuous dynamical sysems. The fricion-oscillaor is composed of a mass conneced by viscoelasic elemen and linear spring-loading and ineracing wih wo moving bels by means of dry fricion. Differen domains and boundaries for such sysem are defined according o he fricion disconinuiy, which exhibis muliple disconinuous boundaries in he phase space. The necessary and sufficien condiions of he sick moions, non-sick moions and grazing moions of such sysem are given in he form of heorem mahemaically. The swiching planes and basic mappings will be defined o sudy grazing moions and periodic moions. The resuls of compuer simulaion of he sick moions and grazing moions for differen parameers are submied in he presen paper. Wih appropriae mapping srucures, he simulaion of he sick and non-sick periodic moions for such an oscillaor are also given. Key Words: double bel fricion-oscillaor; disconinuous dynamical sysem; swichabiliy; sick moion; grazing moion 1 Inroducion Disconinuous dynamical sysems exis everywhere in engineering [1-13]. In mechanical engineering, mos of he dynamical sysems are disconinuous. One used o adop coninuous models for approximae descripions of disconinuous dynamical sysems. However, such coninuous modeling canno provide adequae predicions of disconinuous dynamical sysems, and also makes he problems solving be more complicaed and inaccurae. To beer describe he real world, one should realize ha disconinuous models can provide adequae and real predicaions of engineering sysems. Therefore, a heory applicable o disconinuous dynamical sysems should be buil. The early sudy of disconinuous dynamical sysems goes back o Den Harog [1] in Den Harog considered a forced oscillaor wih Coulomb and viscous damping. In 1960, Levian [2] invesigaed a fricion oscillaor wih he periodically driven base, and also discussed he sabiliy of he periodic moion. In 1966, Masri and Caughey [3] discussed a disconinuous impac damper, and obained he sabiliy of he symmerical period-1 moion of he impac damper. More deailed discussions on he general moion of impac dampers were also developed in Masri [4]. In 1976, Ukin [5] firs invesigaed he conrolled dynamical sysems in view of disconinuiy. This mehod is called sliding mode conrol. Ukin [6] applied he sliding mode conrol in variable srucure sysems, and more deailed heory of his mehod was also developed in [7] by Ukin. In 1986, Shaw [8] invesigaed he non-sick periodic moion of a dry-fricion oscillaor, and discussed he sabiliy of his moion hrough he Poincare mapping. In 1988, Filippov [9] invesigaed he dynamic behaviors of a Coulomb fricion oscillaor and developed differenial equaions wih disconinuous righ-hand sides. The analyical condiions of sliding moion along he disconinuous boundary were developed hrough differenial inclusion, and he exisence and uniqueness of he soluion were also discussed. Leine eal. [10] invesigaed he sick-slip vibraion induced by an alernae fricion models hrough he shooing mehod in In 1999, Galvaneo and Bishop [11] discussed dynamics of a simple dynamical sysem subjeced o an elasic resoring force, viscous damping and dry fricion forces and sudied he non-sandard bifurcaions wih analyical and numerical ools. Pilipchuk and Tan [12] sudied he fricion induced vibraion of a wo-degree-of-freedom fricion oscillaor in In 2005, Casini and Vesroni [13] invesigaed dynamics of wo double-bel fricion oscillaors by means of E-ISSN: Volume 15, 2016

2 analyical and numerical ools. However, he dynamical behaviors of disconinuous dynamical sysem is silled difficul o invesigae. Luo [14-19] developed a general heory for disconinuous dynamical sysems and gave is applicaions in engineering. The G-funcions for disconinuous dynamical sysems are he mail ools o invesigae singulariy in disconinuous dynamical sysems and inroduced by Luo [17] in 2008 and furher developed on ime-varying domains by Luo [18] in Based on he G-funcions, he full and half sink and source, non-passable flows o he separaion boundary in disconinuous dynamical sysems were discussed, and he necessary and sufficien condiions of he swiching bifurcaions beween he passable and non-passable flows were presened. The deailed discussion can be referred o Luo [19]. Based on his heory, los of disconinuous models can be invesigaed easily, for example [20 26]. In his paper, analyical condiions for sick, non-sick and grazing moions of he double-bel fricion oscillaor will be developed using he flow swichabiliy heory of he disconinuous dynamical sysems. Differen domains and boundaries for such sysem are defined according o he fricion disconinuiy, which exhibis muliple disconinuous boundaries in he phase space. Based on he above domains and boundaries, he analyical condiions of he sick moions and grazing moions are obained mahemaically. The swiching planes and basic mappings will be defined o sudy grazing moions and periodic moions. For a beer undersanding of he dynamical behaviors, he numerical simulaions are given o illusrae he analyical resuls of he complex moions. o a fixed wall, as shown in Fig. 1. This fricioninduced oscillaor includes a mass m, a spring of siffness k and a damper of viscous damping coefficien c. In his configuraion, he mass m is coninuously in conac wih boh bels which are pushed ono he mass wih a consan forced F N and possess he same fricion characerisics. The periodic driving force A 0 + B 0 cos Ω exers on he mass, where A 0, B 0 and Ω are he consan force, exciaion srengh and frequency raio, respecively. Since he mass conacs he moving bels wih fricion, he mass can move along or res on he bel 1 or bel 2 surface. Furher, a kineic fricion force shown in Fig. 2 is described as = (µ 1 + µ 2 )F N, ẋ [ v 2, + ), [(µ 1 µ 2 )F N, (µ 1 + µ 2 )F N ], ẋ = v 2, F f (ẋ) = (µ 1 µ 2 )F N, ẋ [ v 1, v 2 ], [ (µ 1 + µ 2 )F N, (µ 1 µ 2 )F N ], ẋ = v 1, = (µ 1 + µ 2 )F N, ẋ (, v 1 ], (1) where ẋ := dx/d,f N and µ k (k = 1, 2) are a normal force o he conac surface and fricion coefficiens beween he mass m and he bel k (k = 1, 2), respecively. Here we assume ha v 2 > v 1 and µ 1 µ 2. 2 Physical Model Figure 2: Fricion force The moions of he mass in a double-bel fricion oscillaor can be divided ino wo cases. If he mass moves along bel 1 and bel 2, he corresponding moion is called he non-sick moion. If he mass moves ogeher wih bel 1 or bel 2, he corresponding moion is called he sick moion. For he mass moving wih he same speed of he bel 1 surface, he force acing on he mass in he x- direcion is defined as Figure 1: Physical model Consider a periodically forced oscillaor, aached F s1 = A 0 + B 0 cos Ω kx cẋ + µ 2 F N for ẋ = v 1. (2) If his force canno overcome he fricion force µ 1 F N (i.e., F s1 µ 1 F N ), he mass does no have any rel- E-ISSN: Volume 15, 2016

3 aive moion o he bel 1. The equaion of he moion for he mass in such sae is described as ẋ = v 1, ẍ = 0. (3) For he mass moving wih he same speed of he bel 2 surface, we can also obain he equaion for he mass as follows ẋ = v 2, ẍ = 0. (4) For he non-sick moions of he fricion-induced oscillaor, we can obain he equaions of he moions as follows mẍ = A 0 + B 0 cos Ω kx cẋ + (µ 1 + µ 2 )F N for ẋ < v 1, mẍ = A 0 + B 0 cos Ω kx cẋ (µ 1 µ 2 )F N for v 1 < ẋ < v 2, mẍ = A 0 + B 0 cos Ω kx cẋ (µ 1 + µ 2 )F N for ẋ > v 2. (5) 3 Domains and Boundaries From he previous discussion, here are five moion saes including hree non-sick moions in he hree regions and wo sick moions on he boundaries. The phase plane can be pariioned ino hree domains and wo boundaries, as shown in Fig. 3. In each domain, he moion can be described hrough a coninuous dynamical sysem. The corresponding boundaries are defined as: } = Ω 21 = {(x, ẋ) x (, + ), ẋ = v 1, } Ω 23 = Ω 32 = {(x, ẋ) x (, + ), ẋ = v 2. (7) Based on he above domains and boundaries, he vecors for moions of he mass in he domains can be inroduced as follows x (λ) = (x (λ), ẋ (λ) ) T, F (λ) = (ẋ (λ), F (λ) ) T, (8) where λ = 1, 2, 3 and F (1) (x (1), ) = c mẋ(1) k m x (1) + B 0 m cos Ω + 1 m [A 0 + (µ 1 + µ 2 )F N ], F (2) (x (2), ) = c mẋ(2) k m x (2) + B 0 m cos Ω + 1 m [A 0 (µ 1 µ 2 )F N ], F (3) (x (3), ) = c mẋ(3) k m x (3) + B 0 m cos Ω + 1 m [A 0 (µ 1 + µ 2 )F N ]. (9) From (5), he equaions of he non-sick moions for he mass are rewrien in he vecor form of ẋ (λ) = F (λ) (x (λ), ) for λ {1, 2, 3}. (10) For he sick moion, he equaions of he moion for he mass are rewrien in he vecor form of ẋ (0) (λ) = F(0) (λ) (x (λ), ) for λ {1, 2} (11) and where F (0) (λ) (x(0) (λ), ) = 0, (12) Figure 3: Domains and boundaries The hree domains are expressed by Ω α ( α = 1, 2, 3 ): { } Ω 1 = (x, ẋ) x (, + ), ẋ (, v 1 ), { } Ω 2 = (x, ẋ) x (, + ), ẋ (v 1, v 2 ), { } Ω 3 = (x, ẋ) x (, + ), ẋ (v 2, + ). (6) x (0) (λ) = (x(0) (λ), ẋ(0) (λ) )T, F (0) (λ) = (v λ, F (0) (λ) )T. 4 Analyical Condiions By he heory of he flow swichabiliy o a specific boundary in disconinuous dynamical sysem in [17], he swiching condiions of he passabiliy, sick moions and grazing flows of he double-bel fricion oscillaor will be developed in his secion. For convenience, we firs inroduce some conceps and several lemmas in flow swiching heory. E-ISSN: Volume 15, 2016

4 Consider a disconinuous dynamical sysem ẋ (α) F (α) (x (α),, P α ) R n (13) in domain Ω α (α = i, j) which has a flow x (α) = Φ( 0, x (α) 0, P α, ) wih an iniial condiion ( 0, x (α) 0 ), and on he boundary { = x ϕ ij (x,, λ) = 0, } ϕ ij is C r coninuous (r 1) R n 1, (14) here is a flow x (0) = Φ( 0, x (0) 0, λ, ) wih an iniial condiion ( 0, x (0) 0 ). The 0-order G-funcions of he flow x (α) o he flow x (0) on he boundary in he normal direcion of he boundary are defined as G (α) (x (0) G (0,α) (x (0), ±, x (α) ±, P α, λ), ±, x (α) ±, P α, λ) = D x (0) n T (x (α) ± x (0) ) + n T (ẋ (α) ± ẋ (0) ). (15) The 1-order G-funcions for a flow x (α) o a boundary flow x (0) in he normal direcion of he boundary are defined as G (1,α) (x (0), (α) ±, x(α) ±, P α, λ) = D x 2 n T (0) (x (α) ± x (0) ) +2D x (0 ) n T (ẋ (α) ± ẋ (0) ) + n T (ẍ (α) ± ẍ (0) ), where he oal derivaive D x (0) ( ) := ( ) x (0) ẋ (0) + ( ), (16) he normal vecor of he boundary surface a poin x (0) () is given by n T (x (0),, λ) = ϕ ij (x (0),, λ) = ( ϕ ij and ± = ± 0., ϕ ij x (0) 1 x (0) 2,, ϕ ij ) T (, x (0) n x, (0) ) (17) If he flow x (α) conacs wih he boundary a he ime m, ha is x (α) m = x m = x (0) m, and he boundary is linear, independen of ime, we have G (0,α) (x m, m, P α, λ) := G (0,α) (x (0) m, m±, x (α) m±, P α, λ) = n T ẋ (α) G (1,α) (x m, m, P α, λ), (x m, m± ) (18) := G (1,α) (x (0) m, m±, x (α) m±, P α, λ) = n T ẍ (α). (x m, m± ) (19) Here m+ and m are he ime before approaching and afer deparing he corresponding boundary, respecively. Lemma 1 [17] For a disconinuous dynamical sysem ẋ (α) = F (α) (x (α),, P α ) R n, x( m ) = x m a ime m. For an arbirarily small ε > 0, here is a ime inerval [ m ε, m ). Suppose x (i) ( m ) = x m = x (j) ( m ). Boh flows x (i) () and x (j) () are C[ r m ε, m )-coninuous (r 1) for ime, and d r+1 x (α) /d r+1 < (α {i, j}). The necessary and sufficien condiions for a sliding moion on Ω αβ are G (0,α) (x m, m, P α, λ) < 0 G (0,β) (x m, m, P β, λ) > 0 where α, β {i, j} and α β. for n Ω αβ Ω α, (20) Lemma 2 [17] For a disconinuous dynamical sysem ẋ (α) = F (α) (x (α),, P α ) R n, x( m ) = x m a ime m. For an arbirarily small ε > 0, here are wo ime inervals [ m ε, m ) and ( m, m+ε ]. Suppose x (i) ( m ) = x m = x (j) ( m+ ). Boh flows x (i) () and x (j) () are C[ r m ε, m ) and Cr ( m, m+ε ] - coninuous (r 1) for ime, respecively, and d r+1 x (α) /d r+1 < (α {i, j}). The flow x (i) () and x (j) () o he boundary is semipassable from domain Ω i o Ω j iff eiher (x m, m, P i, λ) > 0 (x m, m+, P j, λ) > 0 for n Ω αβ Ω j, (21) G (0,i) G (0,j) E-ISSN: Volume 15, 2016

5 or (x m, m, P i, λ) < 0 (x m, m+, P j, λ) < 0 for n Ω αβ Ω i. (22) G (0,i) G (0,j) Lemma 3 [17] For a disconinuous dynamical sysem ẋ (α) = F (α) (x (α),, P α ) R n, x( m ) = x m a ime m. For an arbirarily small ε > 0, here is a ime inerval [ m ε, m+ε ]. Suppose x (α) ( m± ) = x m. The flow x (α) () is C[ r m ε, m+ε ] -coninuous (r α 2) for ime, and d r+1 x (α) /d r+1 < (α {i, j}). A flow x (α) () in Ω α is angenial o he boundary iff G (0,α) (x m, m, P α, λ) = 0 for α {i, j }; (23) eiher G (1,α) (x m, m, P α, λ) < 0 for n Ωαβ Ω β, or WSEAS TRANSACTIONS on MATHEMATICS (x m, m, P α, λ) > 0 for n Ωαβ Ω α, (24) G (1,α) where α, β {i, j} and α β. More deailed heory on he flow swichabiliy such as high-order G-funcions, he definiions or heorems abou various flow passabiliy in disconinuous dynamical sysems can be referred o [17] and [19]. From he aforemenioned definiions and lemmas, we give he analyical condiions for he flow swiching in he double-bel fricion oscillaor. For he double-bel fricion oscillaor in Secion 2, he normal vecors of he boundaries and Ω 23 are given as n Ω12 = n Ω21 = (0, 1) T, n Ω23 = n Ω32 = (0, 1) T. (25) The G-funcions for such fricion oscillaor are simplified as G (0,α) (x (α), m± ) or G (1,α) (x (α), m± ). Theorem 4 For he double-bel fricion oscillaor described in Secion 2, we have he following resuls: (i) The sick moion on x m a ime m appears iff he following condiions can be obained: F (1) (x m, m ) > 0 and F (2) (x m, m ) < 0. (26) (ii) The sick moion on x m Ω 23 a ime m appears iff he following condiions can be obained: F (2) (x m, m ) > 0 and F (3) (x m, m ) < 0. (27) Proof: From he aforemenioned definiions, he 0- order G-funcions for he sick boundaries and Ω 23 in he double-bel fricion oscillaor are and G (0,1) (x m, m± ) = n T F (1) (x m, m± ), G (0,2) (x m, m± ) = n T F (2) (x m, m± ), G (0,2) Ω 23 (x m, m± ) = n T Ω 23 F (2) (x m, m± ), G (0,3) Ω 23 (x m, m± ) = n T Ω 23 F (3) (x m, m± ). (28) (29) From (25), he Eqs. (28) and (29) can be compued as: and G (0,1) (x m, m ) = F (1) (x m, m ), G (0,2) (x m, m ) = F (2) (x m, m ), G (0,2) Ω 23 (x m, m ) = F (2) (x m, m ), G (0,3) Ω 23 (x m, m ) = F (3) (x m, m ). (30) (31) By Lemma 1, he sick moion on x m a ime m appears iff G (0,1) (x (m), m ) > 0 and G (0,2) (x (m), m ) < 0, (32) i.e. F (1) (x m, m ) > 0 and F (2) (x m, m ) < 0. (33) Therefore, (i) holds. Similarly, (ii) holds. Theorem 5 For he double-bel fricion oscillaor described in Secion 2, we have he following resuls: (i) The non-sick moion (or called passable moion o boundary) on x m a ime m appears iff he following condiion can be obained: F (1) (x m, m± ) F (2) (x m, m ) > 0. (34) (ii) The non-sick moion on x m Ω 23 a ime m appears iff he following condiion can be obained: F (2) (x m, m± ) F (3) (x m, m ) > 0. (35) E-ISSN: Volume 15, 2016

6 Proof: By Lemma 2, passable moion on he boundary x m a ime m appears iff G (0,1) (x m, m± ) G (0,2) (x m, m ) > 0. (36) By (25), we obain G (0,1) (x m, m± ) = F (1) (x m, m± ), G (0,2) (x m, m ) = F (2) (x m, m ). (37) Then (36) and (37) implies ha (i) holds. The proof for (ii) is similar. Theorem 6 For he double-bel fricion oscillaor described in Secion 2, we have he following resuls: (i) The grazing moion on x m a ime m appears iff he following condiions can be obained: if α = 1, F (α) (x m, m± ) = 0 for α {1, 2}, (38) F (α) (x m, m± ) F (α) (x m, m± )+ F (α)(x m, m± ) m <0, if α = 2, (39) F (α) (x m, m± ) F (α) (x m, m± )+ F (α)(x m, m± ) m >0. (40) (ii) The grazing moion on x m Ω 23 a ime m appears iff he following condiions can be obained: F (α) (x m, m± ) = 0 for α {2, 3}, (41) From (25), (28) and (29), we have G (0,α) (x m, m± ) = n T F (α) (x m, m± ) = F (α) (x m, m± ) for α {1, 2}. (46) From (19), we obain (x m, m± ) = n T D x (0) F (1) (x m, m± ) m ( T = (0, 1) D x (0) ẋ (1), F (1) (x m, )) m (x m, m± ) = F (1) (x m, m± ) F (1) (x m, m± )+ F (1)(x m, m± ). m G (1,1) Similarly, G (1,2) (x m, m± ) (47) = F (2) (x m, m± ) F (2) (x m, m± )+ F (2)(x m, m± ) m. (48) From (46),(47) and (48), (i) holds. In a similar manner, (ii) holds. 5 Swiching Planes and Mappings The swiching planes are inroduced as (λ = 1, 2): Σ 0 (λ) = {(x i, ẋ i, Ω i ) ẋ i = v λ }, Σ 1 (λ) = {(x i, ẋ i, Ω i ) ẋ i = v λ }, Σ 2 (λ) = {(x i, ẋ i, Ω i ) ẋ i = v + λ }, (49) if α = 2, F (α) (x m, m± ) F (α) (x m, m± )+ F (α)(x m, m± ) m <0, if α = 3, (42) F (α) (x m, m± ) F (α) (x m, m± )+ F (α)(x m, m± ) m >0. (43) Proof: By Lemma 3, he sufficien and necessary condiions for he grazing flows on he boundary are G (0,α) (x m, m± ) = 0 for α {1, 2}, (44) G (1,1) (x m, m± ) < 0, G (1,2) (x m, m± ) > 0. (45) where v λ = lim δ 0(v λ δ) and v + λ = lim δ 0(v λ + δ) for arbirary small δ > 0. Therefore, eigh basic mappings will be defined as: P 1 : Σ 0 (1) Σ0 (1), P 2 : Σ 1 (1) Σ1 (1), P 3 : Σ 2 (1) Σ2 (1), P 4 : Σ 0 (2) Σ0 (2), P 5 : Σ 1 (2) Σ1 (2), P 6 : Σ 2 (2) Σ2 (2), P 7 : Σ 1 (2) Σ2 (1), P 8 : Σ 2 (1) Σ1 (2). (50) In phase plane, he rajecories of mappings P λ (λ {2, 3, 5, 6, 7, 8}) in Ω α (α {1, 2, 3}) s- aring and ending a he velociy boundaries and sick mappings P λ (λ = 1, 4) are illusraed in Fig. 4. E-ISSN: Volume 15, 2016

7 Figure 4: Basic mappings From foregoing (49) and (50), we obain P 1 : (x i, v 1, Ω i ) (x i+1, v 1, Ω i+1 ), P 2 : (x i, v1, Ω i) (x i+1, v1, Ω i+1), P 3 : (x i, v 1 +, Ω i) (x i+1, v 1 +, Ω i+1), P 4 : (x i, v 2, Ω i ) (x i+1, v 2, Ω i+1 ), P 5 : (x i, v2, Ω i) (x i+1, v2, Ω i+1), P 6 : P 7 : P 8 : (x i, v + 2, Ω i) (x i+1, v + 2, Ω i+1), (x i, v 2, Ω i) (x i+1, v + 1, Ω i+1), (x i, v + 1, Ω i) (x i+1, v 2, Ω i+1). (51) Wih (11) and (12), he governing equaions for P λ (λ = 1, 4) can be described as x i+1 = v 1 ( i+1 i ) + x i, A 0 +B 0 cos Ω i+1 kx i+1 cv 1 +µ 2 F N =µ 1 F N, (52) x i+1 = v 2 ( i+1 i ) + x i, A 0 +B 0 cos Ω i+1 kx i+1 cv 2 µ 1 F N =µ 2 F N, (53) respecively. For he double-bel fricion oscillaor, he domains Ω α (α {1, 2, 3}) are unboubded. From he basic heorems of disconinuous dynamical sysem, only hree possible bounded moions exis in he hree domains. In domain Ω α (α {1, 2, 3}), he displacemen expressions and velociy expressions of he mass can be solved from (9) and (10). Using he displacemen expressions and velociy expressions of he mass in domain Ω α (α {1, 2, 3}), he governing equaions of mapping P λ (λ {2, 3, 5, 6, 7, 8}) are obained. Wih (51), he governing equaions of each mapping P λ (λ {2, 3, 5, 6, 7, 8}) can be expressed as f (λ) 1 (x i, Ω i, x i+1, Ω i+1 ) = 0, f (λ) 2 (x i, Ω i, x i+1, Ω i+1 ) = 0. (54) The grazing moion occurs when a flow in a domain is angenial o he boundary and hen reurns back o his domain. The analyical condiions for he grazing moion in he double-bel fricion oscillaor were described as Lemma 3 and Theorem 6. If he grazing moion occurs a (x m, m ) Ω αβ (α, β {1, 2} or {2, 3}, α β), more deailed heorem on he grazing moions will be developed. For he double bel fricion oscillaor described in Secion 2, here are four cases of grazing moions on he boundaries: he flow in domain Ω 1 angenial o he boundary, he flow in domain Ω 2 angenial o he boundary Ω 21, he flow in domain Ω 2 angenial o he boundary Ω 23, and he flow in domain Ω 3 angenial o he boundary Ω 32, corresponding o he mapping P 2, P 3, P 5 and P 6, respecively. Wih (54), we can obain he following heorem. Theorem 7 For he double-bel fricion oscillaor described in Secion 2, here are four kinds of grazing moions: (i) Suppose he flow in domain Ω 1 reaches x m a ime m, he grazing moion on he boundary appears (i.e. he mapping P 2 is angenial o he boundary ) iff mod(ω m, 2π) [ 0, π + Θ cr 2 ) ( 2π Θcr 2, 2π ] for 0 < γ 2 < B 0 m Ω; mod(ω m, 2π) [ 0, 3 2 π ) ( 3 2π, 2π ] for 0 < γ 2 = B 0 m Ω; mod(ω m, 2π) [ 0, 2π ] for 0 < B 0 m Ω < γ 2; mod(ω m, 2π) ( 0, π ) for γ 2 = 0; mod(ω m, 2π) ( Θ cr 2, π Θcr 2 ) ( 0, π ) for γ 2 < 0 and B 0 m Ω > γ 2 ; mod(ω m, 2π) {Ø} for γ 2 < 0 and B 0 m Ω < γ 2, (55) where and Θ cr 2 = arcsin( γ 2m ), γ 2 = c mẍ(1)( m ) + k mẋ(1)( m ). (ii) Suppose he flow in domain Ω 2 reaches x m Ω 21 a ime m, he grazing moion on he boundary Ω 21 appears (i.e. he mapping P 3 is E-ISSN: Volume 15, 2016

8 angenial o he boundary Ω 21 ) iff mod(ω m, 2π) ( π + Θ cr 3, 2π Θcr 3 ) ( π, 2π) for 0 < γ 3 < B 0 m Ω; mod(ω m, 2π) {Ø} for 0 < B 0 m Ω γ 3; mod(ω m, 2π) ( π, 2π ) for γ 3 = 0; mod(ω m, 2π) [ 0, Θ cr 3 ) ( π Θcr 3, 2π ] for γ 3 < 0 and B 0 m Ω > γ 3 ; mod(ω m, 2π) [ 0, π 2 ) ( π 2, 2π ] for γ 3 < 0 and B 0 m Ω = γ 3 ; mod(ω m, 2π) [ 0, 2π ] for γ 3 < 0 and B 0 m Ω < γ 3, (56) where and Θ cr 3 = arcsin( γ 3m ), γ 3 = c mẍ(2)( m ) + k mẋ(2)( m ). (iii) Suppose he flow in domain Ω 2 reaches x m Ω 23 a ime m, he grazing moion on he boundary Ω 23 appears (i.e. he mapping P 5 is angenial o he boundary Ω 23 ) iff mod(ω m, 2π) [ 0, π + Θ cr 5 ) ( 2π Θcr 5, 2π ] for 0 < γ 5 < B 0 m Ω; mod(ω m, 2π) [ 0, 3 2 π ) ( 3 2π, 2π ] for 0 < γ 5 = B 0 m Ω; mod(ω m, 2π) [ 0, 2π ] for 0 < B 0 m Ω < γ 5; mod(ω m, 2π) ( 0, π ) for γ 5 = 0; mod(ω m, 2π) ( Θ cr 5, π Θcr 5 ) ( 0, π ) for γ 5 < 0 and B 0 m Ω > γ 5 ; mod(ω m, 2π) {Ø} for γ 5 < 0 and B 0 m Ω < γ 5, (57) where and WSEAS TRANSACTIONS on MATHEMATICS Θ cr 5 = arcsin( γ 5m ), γ 5 = c mẍ(2)( m ) + k mẋ(2)( m ). (iv) Suppose he flow in domain Ω 3 reaches x m Ω 32 a ime m, he grazing moion on he boundary Ω 32 appears (i.e. he mapping P 6 is angenial o he boundary Ω 32 ) iff mod(ω m, 2π) ( π + Θ cr 6, 2π Θcr 6 ) ( π, 2π) for 0 < γ 6 < B 0 m Ω; mod(ω m, 2π) {Ø} for 0 < B 0 m Ω γ 6; mod(ω m, 2π) ( π, 2π ) for γ 6 = 0; mod(ω m, 2π) [ 0, Θ cr 6 ) ( π Θcr 6, 2π ] for γ 6 < 0 and B 0 m Ω > γ 6 ; mod(ω m, 2π) [ 0, π 2 ) ( π 2, 2π ] for γ 6 < 0 and B 0 m Ω = γ 6 ; mod(ω m, 2π) [ 0, 2π ] for γ 6 < 0 and B 0 m Ω < γ 6, (58) where and Θ cr 6 = arcsin( γ 6m ), γ 6 = c mẍ(3)( m ) + k mẋ(3)( m ). Proof: (i) For he double-bel fricion oscillaor described in Secion 2, by Theorem 6, he grazing moion condiions for he flow x (1) () in domain Ω 1 on he boundary a ime m are given as F (1) (x m, m± ) = 0, (59) F (1) (x m, m± ) F (1) (x m, m± )+ F (1)(x m, m± ) <0. m (60) Wih (9), he Eqs. (59) and (60) can be compued as c mẋ(1)( m ) k m x (1)( m ) + B 0 m cos Ω m + 1 m [A 0 + (µ 1 + µ 2 )F N ] = 0, (61) c mẍ(1)( m ) k mẋ(1)( m ) B 0Ω m sin Ω m < 0. (62) The grazing condiions are compued hrough (54), (61) and (62). Three equaions and an inequaliy have four unknowns, hen one unknown mus be given. From (62), he criical value for mod(ω m, 2π) is inroduced hrough Θ cr 2 = arcsin( γ 2m ), where γ 2 = c mẍ(1)( m ) + k mẋ(1)( m ), and he superscrip cr represens a criical value relaive o grazing. E-ISSN: Volume 15, 2016

9 If 0 < γ 2 < B 0 m Ω, hen 1 < γ 2m have < 0, we mod(ω m, 2π) [ 0, π + Θ cr 2 ) ( 2π Θ cr 2, 2π ]. If 0 < γ 2 = B 0 m Ω, hen γ 2m = 1, we have mod(ω m, 2π) [ 0, 3 2 π ) ( 3 π, 2π ]. 2 If 0 < B 0 m Ω < γ 2, hen γ 2m < 1, we have mod(ω m, 2π) [ 0, 2π ]. If γ 2 = 0, hen γ 2m = 0, we have mod(ω m, 2π) ( 0, π ). If γ 2 < 0 and B 0 m Ω > γ 2, hen 0 < γ 2m < 1, we have mod(ω m, 2π) ( Θ cr 2, π Θ cr 2 ) ( 0, π ). If γ 2 < 0 and B 0 m Ω < γ 2, hen γ 2m > 1, we have mod(ω m, 2π) {Ø}. Therefore (i) holds; (ii) Similarly, for he double-bel fricion oscillaor described in Secion 2, by Theorem 6, he grazing moion condiions for he flow x (2) () in domain Ω 2 on he boundary Ω 21 a ime m are given as F (2) (x m, m± ) = 0, (63) F (2) (x m, m± ) F (2) (x m, m± )+ F (2)(x m, m± ) >0. m (64) Wih (9), he Eqs. (63) and (64) can be compued as c mẋ(2)( m ) k m x (2)( m ) + B 0 m cos Ω m + 1 m [A 0 (µ 1 µ 2 )F N ] = 0, (65) c mẍ(2)( m ) k mẋ(2)( m ) B 0Ω m sin Ω m > 0. (66) The grazing condiions are compued hrough (54), (65) and (66). Three equaions and an inequaliy have four unknowns, hen one unknown mus be given. From (66), he criical value for mod(ω m, 2π) is inroduced hrough Θ cr 3 = arcsin( γ 3m ), where γ 3 = mẍ(2)( c m ) + mẋ(2)( k m ), and he superscrip cr represens a criical value relaive o grazing. If 0 < γ 3 < B 0 m Ω, hen 1 < γ 3m < 0, we have mod(ω m, 2π) ( π + Θ cr 3, 2π Θ cr 3 ) ( π, 2π). If 0 < B 0 m Ω γ 3, hen γ 3m 1, we have mod(ω m, 2π) {Ø}. If γ 3 = 0, hen γ 3m = 0, we have mod(ω m, 2π) ( π, 2π ). If γ 3 < 0 and B 0 m Ω > γ 3, hen 0 < γ 3m < 1, we have mod(ω m, 2π) [ 0, Θ cr 3 ) ( π Θ cr 3, 2π ]. If γ 3 < 0 and B 0 m Ω = γ 3, hen γ 3m = 1, we have mod(ω m, 2π) [ 0, π 2 ) ( π, 2π ]. 2 If γ 3 < 0 and B 0 m Ω < γ 3, hen γ 3m > 1, we have mod(ω m, 2π) [ 0, 2π ]. Therefore (ii) holds. Similarly we can prove ha (iii) and (iv) hold. 6 Numerical Simulaions To verify he analyical condiions of he sick moions and grazing moions obained in Secion 4 and Secion 5, he moions of he mass in he double-bel fricion oscillaor will be demonsraed hrough he ime hisories of displacemen and velociy, he corresponding rajecory of he mass in phase space. In his secion, he force produc will be presened o illusrae he s- aring and vanishing poins of sick moions. The displacemens or he velociies of he bel 1 and bel 2 are presened by black curves, and he displacemen or he velociy of he mass, he corresponding rajecories of he moions of he mass in phase space are presened by red curves. The onse poins of sick or periodic moions are marked by blue-solid circular symbols, he vanishing poins of sick or periodic moions are marked by green-solid circular symbols, he swiching poins are depiced by red-solid circular symbols. Consider he sysem parameers as A 0 = 15, B 0 = 10, Ω = 1, c = 1, k = 1, m = 1, g = 9.81, v 1 = 1, v 2 = 5, µ 1 = 0.8, µ 2 = 0.6, F N = 9.81 o demonsrae a sick moion of he mass on E-ISSN: Volume 15, 2016

10 (a) (b) (c) (d) Figure 5: Numerical simulaion of a sick moion on he boundary : (a) displacemen-ime hisory, (b) velociy-ime hisory, (c) phase rajecory, (d) force-ime hisory. (A 0 = 15, B 0 = 10, Ω = 1, c = 1, k = 1, m = 1, g = 9.81, v 1 = 1, v 2 = 5, µ 1 = 0.8, µ 2 = 0.6, F N = 9.81, 0 = 0, x 0 = 0, ẋ 0 = 0). he boundary. The iniial condiions are 0 = 0, x 0 = 0, ẋ 0 = 0. For a beer undersanding of mechanism of sick moions, he ime-hisory responses for displacemen, velociy, he corresponding rajecory and force response will be illusraed. The ime hisories of displacemen and velociy are shown in Fig. 5(a) and (b), respecively. Wih he iniial condiions 0 = 0, x 0 = 0, ẋ 0 = 0, he sick moion of he bel 1 appears afer a period of ime. The blue-solid circular symbol represens he onse poin of he sick moion. I can be seen ha he velociy of he mass is equal o he speed of he bel 1, and hen he mass and he bel 1 move ogeher for some ime. During his ime, he mass and he bel 1 have he same displacemen increase. The green-solid circular symbol represens he vanishing poin of he sick moion. Afer vanishing of sick moion on he boundary, he mass moves under he influence of fricion force. In Fig. 5(c), he corresponding rajecory of he sick moion on he boundary is ploed. Furher, he force-ime hisories are also presened in Fig. 5(d). The blue curve sands for F (1) and he green curve sands for F (2). Beween he blue-solid circular symbol and he green-solid circular symbol, i is observed ha F (1) > 0 and F (2) < 0. This saisfy he analyical condiions of he sick moion on he boundary described in Theorem 4. Afer vanishing of he sick moion, F (1) F (2) > 0. Consider a sick moion on he boundary Ω 23 wih he sysem parameers A 0 = 5, B 0 = 10, Ω = 0.1, c = 1, k = 1, m = 1, g = 9.81, v 1 = 1, v 2 = 5, µ 1 = 0.8, µ 2 = 0.6, F N = The ime-hisory responses for displacemen, velociy, he corresponding rajecory and force response of such a sick moion are illusraed in Fig. 6. The iniial condiions 0 = 0, x 0 = 0, ẋ 0 = 5 are seleced o make he sick E-ISSN: Volume 15, 2016

11 (a) (b) (c) (d) Figure 6: Numerical simulaion of a sick moion on he boundary Ω 23 : (a) displacemen-ime hisory, (b) velociy-ime hisory, (c) phase rajecory, (d) force-ime hisory. (A 0 = 5, B 0 = 10, Ω = 0.1, c = 1, k = 1, m = 1, g = 9.81, v 1 = 1, v 2 = 5, µ 1 = 0.8, µ 2 = 0.6, F N = 9.81, 0 = 0, x 0 = 0, ẋ 0 = 5). moion on he boundary Ω 23 appear in such iniial condiions. The blue-solid circular symbol and he green-solid circular symbol represen he onse and vanishing poins of he sick moion, respecively. In Fig. 6(a) and (b), he curves beween he blue-solid circular symbol and he green-solid circular symbol are he inersecion beween he black curve and he red one. In his period, he mass and he bel 2 moves ogeher and have he same velociy and displacemen. The corresponding rajecory of he sick moion on he boundary Ω 23 is presened in Fig. 6(c). Afer he vanishing of he sick moion, he rajecory of he mass in he phase plane exiss in domain Ω 2 for some ime. The force-ime hisories are also ploed in In Fig. 6(d). The blue curve sands for F (2) and he green curve sands for F (3). I is observed ha he sick moion on he boundary Ω 23 saisfy F (2) > 0 and F (3) < 0. A he green-solid circular symbol, F (2) = 0 and F (3) < 0. Then he sick moion vanishes. The sysem parameers A 0 = 5, B 0 = 10, Ω = 1, c = 2, k = 2, m = 1, g = 9.81, v 1 = 5, v 2 = 10, µ 1 = 0.7, µ 2 = 0.2, F N = 9.81 are chosen o illusrae a grazing moion of mapping P 2 in domain Ω 1. The iniial condiions are 0 = 2.5, x 0 = , ẋ 0 = In Fig. 7(b) and (c), he ime hisory responses for velociy of such a grazing moion are presened. In Fig. 7(b), i is clear ha he velociy of he mass is equal o he bel 1 a he red poin only. I means ha he moion is angenial o he velociy boundary and hen leaves his boundary again, so he red poin is grazing poin. The blue poin sands for he sar poin of a simple periodic moion. A he green poin, he firs period finishes and he nex period begins. The periodiciy of his simple periodic moion can be seen clearly in Fig. E-ISSN: Volume 15, 2016

12 (a) (b) (c) (d) Figure 7: Numerical simulaion of a grazing moion of mapping P 2 in domain Ω 1 : (a) displacemen-ime hisory, (b) velociy-ime hisory, (c) velociy-ime hisory for muliple period, (d) phase rajecory. (A 0 = 5, B 0 = 10, Ω = 1, c = 2, k = 2, m = 1, g = 9.81, v 1 = 5, v 2 = 10, µ 1 = 0.7, µ 2 = 0.2, F N = 9.81, 0 = 2.5, x 0 = , ẋ 0 = ). 7(c). The ime hisory response for displacemen of such a grazing moion is ploed in Fig. 7(a). Three black curves in Fig. 7(a) represen he displacemen of special poins on he bel 1, such as he grazing poin, he sar and end poins of he firs period. I can be observed ha he red poin is he inersecion beween he black curve and he red one, so a he red poin, he mass and he grazing poin have he same displacemen and velociy. However, a he blue and green poins, he mass and he blue or green poin only have he same displacemen, he velociies of hem are no equal, so he blue and green poins are no grazing poin. In phase plane, he rajecory of he grazing moion is angenial o he velociy boundary in domain Ω 1, as shown in Fig. 7(d). In phase plane, he blue poin and he green poin are he same poin. The sysem parameers A 0 = 10, B 0 = 16, Ω = 2, c = 4, k = 8, m = 2, g = 9.81, v 1 = 4, v 2 = 4, µ 1 = 0.8, µ 2 = 0.5, F N = and he iniial condiions 0 = 2.5, x 0 = , ẋ 0 = are given o demonsrae he grazing moion of mapping P 3 in domain Ω 2 in Fig. 8. The hisory of velociy of he grazing moion is shown in Fig. 8(b). I can be seen ha he velociy of he mass is equal o he bel 1 a he red poin and hen leaves his velociy boundary again. I means ha he moion is angenial o he velociy boundary Ω 21, so he red poin is grazing poin. Afer he grazing moion, he velociy of he mass reaches he velociy boundary again a he blue poin. More deailed illusraion is shown in Fig. 8(c). A he poins beween he blue poin and he green poin, he force response saisfy F (1) > 0 and E-ISSN: Volume 15, 2016

13 (a) (b) (c) (d) Figure 8: Numerical simulaion of a grazing moion of mapping P 3 in domain Ω 2 : (a) displacemen-ime hisory, (b) velociy-ime hisory, (c) velociy-ime hisory, (d) phase rajecory. (A 0 = 10, B 0 = 16, Ω = 2, c = 4, k = 8, m = 2, g = 9.81, v 1 = 4, v 2 = 4, µ 1 = 0.8, µ 2 = 0.5, F N = 19.62, 0 = 2.5, x 0 = , ẋ 0 = ). F (2) < 0. Then he sick moion appears in such condiions, and he mass moves ogeher wih he bel 1. The blue and he green poins sand for he onse and he vanishing poins of he sick moion, respecively. Afer he vanishing of sick moion on he bel 1, he velociy of he mass leaves his velociy boundary again. In Fig. 8(a), wo black curves represen he hisory of displacemen of he grazing poin(or called he red poin) and he onse poin of sick moion(or called he blue poin). The inersecion of he red curve and black curves is he red poin and he curve beween he blue and he green poin, sand for he grazing moion and he sick moion, respecively. The rajecory of he grazing moion in domain Ω 2 is shown in Fig. 8(d). I can be observed ha he rajecory of he grazing moion in domain Ω 2 is angenial o he velociy boundary Ω 21, and he rajecory of he sick moion is a par of he velociy boundary Ω 21. From he basic mappings in Secion 5, non-sick and sick periodic moions of he double-bel fricion oscillaor can be obained. The following Fig. 9 and Fig. 10 illusrae he non-sick and sick periodic moions, respecively. Also, he ime-hisory responses of displacemen, velociy and he corresponding rajecory will be illusraed. Besides, for a beer undersanding of he mechanisms of non-sick and sick periodic moions, he force responses relaive o ime, displacemen and velociy will be presened, oo. The sysem parameers A 0 = 2, B 0 = 9, Ω = 4, c = 0.2, k = 3, m = 0.1, g = 9.81, v 1 = 1, v 2 = 25, µ 1 = 0.3, µ 2 = 0.2, F N = 10 are used o illusrae a non-sick periodic moion of mapping P 32 = P 3 P 2. Wih he iniial condiions 0 = , x 0 = , ẋ 0 = 1, numerical simulaions can be obained, as shown in Fig. 9. The ime-hisory responses of displacemen, velociy and E-ISSN: Volume 15, 2016

14 (a) (b) (c) (d) (e) (f) Figure 9: Numerical simulaion of a non-sick periodic moion relaive o mapping P 32 = P 3 P 2 : (a) displacemen-ime hisory, (b) velociy-ime hisory, (c) phase rajecory, (d) force-ime hisory, (e) forcedisplacemen hisory, (f) force-velociy hisory (A 0 = 2, B 0 = 9, Ω = 4, c = 0.2, k = 3, m = 0.1, g = 9.81, v 1 = 1, v 2 = 25, µ 1 = 0.3, µ 2 = 0.2, F N = 10, 0 = , x 0 = , ẋ 0 = 1). E-ISSN: Volume 15, 2016

15 (a) (b) (c) (d) (e) (f) Figure 10: Numerical simulaion of a sick periodic moion relaive o mapping P 4645 = P 4 P 6 P 4 P 5 : (a) displacemen-ime hisory, (b) velociy-ime hisory, (c) phase rajecory, (d) force-ime hisory, (e) forcedisplacemen hisory, (f) force-velociy hisory (A 0 = 5, B 0 = 9, Ω = 1.75, c = 0.2, k = 3, m = 0.1, g = 9.81, v 1 = 12, v 2 = 1, µ 1 = 0.5, µ 2 = 0.3, F N = 10, 0 = , x 0 = , ẋ 0 = 1). E-ISSN: Volume 15, 2016

16 force response for he non-sick periodic moion are ploed in Fig. 9(a), (b) and (d). The blue curve s- ands for F (1) and he green curve sands for F (2). The swiching poins are marked by red-solid circular symbols. From Fig. 9(d), i can be observed ha F (1) F (2) > 0 on he red poins a he same ime. By Theorem 5, he moion will pass hrough he velociy boundary. Furher, he corresponding rajecory in phase plane is presened in Fig. 9(c). Obviously, here is no sick moion exiss in Fig. 9(c). The mapping swiching from P α o P β (α, β {2, 3} and α β) is coninuous. The relaion beween he displacemen and force in Fig. 9(e) and he relaion beween he velociy and force in Fig. 9(f) verifies he crieria in Theorem 5 a he velociy boundary. The sysem parameers A 0 = 5, B 0 = 9, Ω = 1.75, c = 0.2, k = 3, m = 0.1, g = 9.81, v 1 = 12, v 2 = 1, µ 1 = 0.5, µ 2 = 0.3, F N = 10 are given o illusrae a sick periodic moion for mapping srucures P The iniial condiions 0 = , x 0 = , ẋ 0 = 1 are seleced o make he sick moion on he boundary Ω 23 appear in such iniial condiions. The ime-hisory responses of displacemen, velociy and he corresponding rajecory in phase plane are shown in Fig. 10(a), (b) and (c). The blue-solid circular symbol and he green-solid circular symbol represen he onse and vanishing poins of he sick moion, respecively. I is clearly seen ha wo sick pars( i.e. P 4 ) exis in his periodic moion. The ime-hisory responses of forces for he sick periodic moion are ploed in Fig. 10(d), he blue curve sands for F (2) and he green curve sands for F (3). Obviously, on he blue poins a he same ime we have F (2) F (3) < 0, hen he sick moion on he boundary Ω 23 appears. However, a he green poins a he same ime we have F (2) F (3) = 0. I means he vanishing of he sick moion and he beginning of non-sick moion. In Fig. 10(e) and (f), he relaions beween he force and displacemen or velociy are presened. These can help us o undersand he mechanisms of sick periodic moion relaive o displacemen and velociy. 7 Conclusion This paper was concerned wih he dynamics of a double bel fricion oscillaor which was subjeced o periodic exciaion, linear spring-loading, damping force and wo fricion forces using he flow swichabiliy heory of he disconinuous dynamical sysems. D- ifferen domains and boundaries for such sysem were defined according o he fricion disconinuiy, which exhibied muliple disconinuous boundaries in he phase space. Based on he above domains and boundaries, he analyical condiions of he sick moions, grazing moions and periodic moions were obained mahemaically. The numerical simulaions were given o illusrae he analyical resuls of hese moions. Acknowledgemens: This research was suppored by he Naional Naural Science Foundaion of China(No , No ) and Naural Science Foundaion of Shandong Province(No. ZR2013AM005). Corresponding auhor: Jinjun Fan. References: fjj18@126.com(j.fan). [1] J.P. Den Harog, Forced vibraions wih Coulomb and viscous damping, Transacions of he American Sociey of Mechanical Engineers 53, 1930, pp [2] E.S. Levian, Forced oscillaion of a spring-mass sysem having combined Coulomb and viscous damping, Journal of he Acousical Sociey of America 32, 1960, pp [3] S.F. Masri, T.D. Caughey, On he sabiliy of he impac damper, ASME Journal of Applied Mechanics 33, 1966, pp [4] S.F. Masri, General moion of impac dampers, Journal of he Acousical Sociey of America 47, 1970, pp [5] V.I. Ukin, Variable srucure sysems wih s- liding modes, IEEE Transacions on Auomaic Conrol 22, 1976, pp [6] V.I. Ukin, Sliding modes and heir applicaion in variable srucure sysems, Moscow:Mir 1978 [7] V.I. Ukin, Sliding regimes in opimizaion and conrol problem, Moscow: Nauka 1981 [8] S.W. Shaw, On he dynamical response of a sysem wih dry-fricion, Journal of Sound and Vibraion 108, 1986, pp [9] A.F. Filippov, Differenial Equaions wih Disconinuous Righhand Sides, Kluwer Academic Publishers, Dordrech Boson London [10] R.I. Leine, D.H. Van Campen, A.De. Kraker, L. Van Den Seen, Sick-slip vibraion induced by alernae fricion models, Nonlinear Dynamics 16, 1998, pp [11] U. Galvaneo, S.R. Bishop, Dynamics of a simple damped oscillaor undergoing sick-slip vibraions, Meccanica 34, 1999, pp [12] V.N. Pilipchuk, C.A. Tan, Creep-slip capure as a possible source of squeal during deceleraing sliding, Nonlinear Dynamics 35, 2004, pp E-ISSN: Volume 15, 2016

17 [13] P. Casini, F. Vesroni, Nonsmooh dynamics of a double-bel fricion oscillaor, IUTAM Symposium on Chaoic Dynamics and Conrol of Sysems and Processes in Mechanics, 2005, pp [14] A.C.J. Luo, A heory for non-smooh dynamical sysems on connecable domains, Communicaion in Nonlinear Science and Numerical Simulaion 10, 2005, pp [15] A.C.J. Luo, Imaginary, sink and source flows in he viciniy of he separarix of nonsmooh dynamic sysem, Journal of Sound and Vibraion 285, 2005, pp [16] A.C.J. Luo, Singulariy and Dynamics on Disconinuous Vecor Fields, Amserdam: Elsevier 2006 [17] A.C.J. Luo, A heory for flow swichabiliy in disconinuous dynamical sysems, Nonlinear Analysis: Hybrid Sysems 2, 2008, pp [18] A.C.J. Luo, Disconinuous Dynamical Sysems on Time-varying Domains, Higher Educaion Press, Beijing China [19] A.C.J. Luo, Disconinuous Dynamical Sysems, Higher Educaion Press, Beijing and Springer- Verlag Berlin Heidelberg [20] A.C.J. Luo, B.C. Gegg, On he mechanism of sick and non-sick periodic moion in a forced oscillaor including dry-fricion, ASME Journal of Vibraion and Acousics 128, 2006, pp [21] A.C.J. Luo, B.C. Gegg, Sick and non-sick periodic moions in a periodically forced,linear oscillaor wih dry fricion, Journal of Sound and Vibraion 291, 2006, pp [22] A.C.J. Luo, S. Thapa, Periodic moions in a simplified brake dynamical sysem wih a periodic exciaion, Communicaion in Nonlinear Science and Numerical Simulaion 14, 2008, pp [23] A.C.J. Luo, Fuhong Min, Synchronizaion of a periodically forced Duffing oscillaor wih a periodically excied pendulum, Nonlinear Analysis: Real World Applicaions 12, 2011, pp [24] A.C.J. Luo, Jianzhe Huang, Disconinuous dynamics of a non-linear, self-excied, fricioninduced, periodically forced oscillaor, Nonlinear Analysis: Real World Applicaions 13, 2012, pp [25] Yanyan Zhang, Xilin Fu, On periodic moions of an inclined impac pair, Commun Nonlinear Sci Numer Simula 20, 2015, pp [26] Xilin Fu, Yanyan Zhang, Sick moions and grazing flows in an inclined impac oscillaor, Chaos, Solions & Fracals 76, 2015, pp E-ISSN: Volume 15, 2016