Research Article  Open Access
On Structural Patterns of Mechanical Systems with Impacts with One and Two DegreesofFreedom
Abstract
Structural patterns of mechanical systems with impacts with one degreeoffreedom and two degreesoffreedom, with elastic connections, have been identified and described. For their identification, a general method proposed by the author has been applied. This method uses (i) a matrix representation of the system with impacts, (ii) procedures that enable generations of all combinations of such systems as well as their identification and elimination of redundant equivalent combinations, and (iii) a procedure for elimination of disconnected systems.
1. Introduction
Systems with one degreeoffreedom belong to the simplest mechanical systems. These systems have been analyzed by numerous authors (e.g., [1–44]), and their dynamics have been investigated in a wide range. The following issues have been considered in the abovementioned publications: steady periodic motions, subharmonic motions, stability of vibrations, elastic and plastic impacts, time and phase characteristics, behavior of the system under an influence of variations in the rigidity coefficient, the viscous damping coefficient, or the dry friction coefficient.
Special attention should be here drawn to works by Peterka and his coworkers, Nordmark, Chin et al., Ivanov, Shaw, and by Shaw and Holmes. Peterka et al. [32–34, 75] determine the regions of stable periodic motions and show a transition to chaotic motions both for the case of a linear, elasticdamping supporting structure, and for the case where dry friction occurs in the system. They present the results of numerical investigations in the form of time functions, phase trajectories, and Poincare maps. Nordmark [29] conducted the investigations on a motion of the onedegreeoffreedom oscillator in which impacts occur under the influence of external excitation. He undertook a trial to describe the moment of impact and showed that a change in external parameters could lead to qualitative changes in the motion character, from a periodic motion, through bifurcation cycles, up to chaotic motions. Moreover, using analytical methods, he presented a new impact phenomenon of the grazing type and a new type of bifurcation that follows from it. Various types of grazing bifurcations were discussed by Chin et al. [7]. The grazing type impacts were dealt with by Ivanov ([17, 18]) as well. Shaw and Holmes [37–39] discuss a simple conservative system with one limiting stop. On the assumption that the coefficient of restitution equals zero, the authors reduce the considerations to a onedimensional map whose analysis shows that stable orbits exist for nearly all frequencies of the excitation force. Shaw [36] observed harmonic, subharmonic, and chaotic motions while analyzing the linear model of an oscillator with two limiting stops subjected to harmonic excitation, in which the processes occurring in the system at the instant of impact are modeled with Newton’s hypothesis (Goldsmith [79]).
In many studies (e.g., [4, 19, 26, 49, 80–85]), the impact is modeled as an abrupt change in rigidity. It allows one to use various analytical methods in the dynamical analysis of the system. For instance, Blankenship and Kahraman [4] as well as Kahraman and Singh [19] analyzed a vibroimpact system with clearances subjected to parametric and periodic excitation. The results of the numerical analyses conducted on the basis of the methods of the harmonics balance were confirmed by the experimental results. Kim and Noah [80] employed the harmonics balance method connected with a Fourier transform procedure to analyze the system dynamics with linear characteristics in segments.
The literature devoted to mechanical systems with impacts on two degreesoffreedom is much less extensive. Although, in this case, the first studies were published slightly later, interesting applications were already mentioned in them. Special attention should be paid to Sadek [71], Masri [86], Masri and Caughey [70], Masri and Ibrahim [73], Araki et al. [87], Bapat and Sankar [88], Cempel [5], Peterka [32, 33], Chatterjee et al. [72], Bazhenov et al. [55], Peterka and BlazejczykOkolewska [75], including theoretical, experimental, and numerical analyses of impact vibration dampers, Karpenko et al. ([89–91]), providing extensive analytical, numerical, and experimental studies of a Jeffcott rotor with a snubber ring, and Koizumi [92], Park [93], and Luo et al. [60], describing the applications of impact oscillators as models of moulding machines. Numerous examples of other applications of systems with impacts can be found in A. E. Kobrinskii and A. Kobrinskii [20], Babickiĭ [47], and Jerrelind and Stensson [94].
Lately the efforts of researchers investigating the dynamics of vibroimpact systems have been focused on the theory of stability, bifurcations, the reasons for occurrence of chaos (e.g., Aidanpӓӓ and Gupta [51], Brach [95], Chen [96], Chin et al. [7], BlazejczykOkolewska and Kapitaniak [76, 77], Czolczynski [57], Czolczynski and Kapitaniakt [58], Di Bernardo et al. [9], Han et al. [67], Ivanov [18], Awrejcewicz and Lamarque [97], Bishop [98], BlazejczykOkolewska et al. [99], Brogliato [100], Guckenheimer and Holmes [101], Ibrahim [102], Rand and Moon [103], and Thompson and Bishop [104]), and control of such systems (e.g., Awrejcewicz et al. [2], Gutiérrez and Arrowsmith [105], de Souza and Caldas [66], de Souza et al. [106], Lee and Yan [23], Luo [107], Luo and Lv [62], Zhao and Wang [108], and Zinjade and Mallik [109]).
In the literature survey presented here, first of all the publications devoted to new dynamic behaviors have been mentioned. The occurrence of specific behaviors of the system is strictly related to the physical model assumed for its analysis. In BlazejczykOkolewska et al. [110], numerous schemes of models of mechanical systems with impacts were presented. For the initial analysis of the abovementioned systems, physical models in which all linear and nonlinear supporting structures are described with the most elementary rheological model, a spring with linear characteristics, can be used.
The comparative studies of physical models of vibroimpact systems used in scientific considerations have led to the determination of assumptions and principles of classification of mechanical systems with impacts in which models are rigid bodies that can move along a straight line without a possibility to rotate (BlazejczykOkolewska et al. [110]) and to the development of the method that allows one to determine structural patterns of all such systems with an arbitrary number of degreesoffreedom (BlazejczykOkolewska [111]). By a structural pattern of the mechanical system with impacts is understood a certain series of systems characterized by a specified structure of component elements (a definite configuration of fenders and connections). All structural patterns of mechanical systems constitute a set in which a kind of the connection (a spring or a damper), its character (linearity or nonlinearity), and the way the impact phenomenon is modeled are not differentiating parameters.
In the present study, the abovementioned method has been illustrated on the example of one and twodegreesoffreedom systems. All structural patterns of mechanical one and twodegreesoffreedom systems with impacts, with arbitrary connections, have been identified and described with this method. Next, structural patterns have been assigned to vibroimpact systems selected from the literature. The knowledge of all systems with impacts of a given number of degreesoffreedom allows us to state which types have been already analyzed and which have not been investigated yet. The proposed classification of mechanical systems with impacts according to their structure allows us to rearrange the knowledge on systems with impacts and is the basis for understanding the sources of their diversity. Providing a full set of objects to be analyzed, it gives hints for new ideas and directions in designing technical devices.
In the considerations below, the terminology, concepts, and notations introduced by BlazejczykOkolewska [111] have been used.
2. Systems with Two DegreesofFreedom
Systems with two degreesoffreedom () have one connectedness zone. Numerous equivalent combinations and many disconnected systems can be identified among them.
The maximal number of springs (spring connections) ( is the spring connecting the subsystem of a mass with the frame, the spring connecting the subsystem of a mass with the subsystem of the mass , and the spring connecting the subsystem of a mass with the frame). The number of possible spring combinations is . All connected and disconnected combinations are presented in Figures 1 and 2.
The spring connectedness zone is located between the subsystems with the masses and . This zone consists of just one spring , which decides whether the basic spring system will be divided into two subsystems or not (in the case of a system with a higher number of degreesoffreedom, a division into two or more subsystems can occur, of course). The number of all connected spring combinations is equal to a product of the number of possible arrangements of the springs that decide about connectedness (one) and the number of possible arrangements beyond the spring connectedness zone of the remaining 0, 1, or 2 springs (four); that is to say, (see Figure 1). The notations of connected spring combinations (, , , and ) have been introduced according to the generation procedure presented by BlazejczykOkolewska ([111, Subsection ]). One of these systems () is the basic spring system.
Similarly, the number of all disconnected spring combinations can be defined. This number is equal to a product of the number of possible arrangements of the springs that decide about the connectedness (one) and the number of possible arrangements beyond the spring connectedness zone of the remaining 0, 1, or 2 springs (four); that is to say, (see Figure 2). The notations of disconnected spring combinations (, , , and ) have been introduced according to the abovementioned generation procedure. The total number of connected and disconnected spring combinations is of course equal to the number of all spring combinations; that is, .
In the case of connected spring combinations (Figure 1), the body of a mass acts on the body of a mass (and vice versa, of course), due to an occurrence of the spring connection between these bodies (the spring ). On the other hand, in the case of disconnected spring combinations (Figure 2), the body of a mass will never affect the behavior of the body of a mass (and vice versa, of course). A lack of the spring connection results in the fact that the bodies “will not know” about each other.
Spring adjacency matrices are attributed to all spring combinations. To list them, also the generation procedure has been used. Let us pay attention to the fact that elements and indices of some spring adjacency matrices have been bolded (in Figure 1: , ; in Figure 2: , ). This bolding refers to spring systems that are equivalent to each other by inversion (the definition of equivalency has been given by BlazejczykOkolewska [111]).
We will conduct a similar analysis for impact connections. The maximal number of fenders (impact connections) is equal to . These are the following connections: is the upper impact connection of the subsystem of a mass with the frame, the lower impact connection of the subsystem of a mass with the frame, the upper impact connection of the subsystem of a mass with the subsystem of a mass , the lower impact connection of the subsystem of a mass with the subsystem of a mass , —upper impact connection of the subsystem of a mass with the frame, and the lower impact connection of the subsystem of a mass with the frame. The number of impact connections is equal to . All connected and disconnected impact combinations are presented in Figures 3, 4, 5, and 6.
The impact connectedness zone is located between the subsystems of masses and . This zone consists of two fenders and that decide whether a division of the basic impact system into two subsystems will occur or not. The number of all connected impact combinations is equal to a product of the number of possible arrangements of the impact connections that decide about the connectedness (three) and the number of possible arrangements, beyond the impact adjacency zone, of the remaining impact connections (); that is to say, . Figure 3 shows the systems: , , , , , , , , , , , , , , , and , that is, all such connected impact combinations that the connection between subsystems occurs both via the upper fender and via the lower fender . One of these systems () is the basic impact system. In Figure 4 the following systems are marked: , , , , , , , , , , , , , , , and , that is, all connected impact combinations in which the connection between subsystems occurs only via the lower fender . Figure 5 presents the systems: , , , , , , , , , , , , , , , and , that is, all connected impact combinations in which the connection between subsystems occurs only via the upper fender . The notations of connected impact combinations have been introduced according to the generation procedure discussed by BlazejczykOkolewska ([111, Section ]). The number of all disconnected impact combinations is equal to a product of the number of possible arrangements of the impact connections that decide about the disconnectedness (one) and the number of possible arrangements, beyond the impact connectedness zone, of the remaining impact connections (); that is to say, . In Figure 6, the following systems are depicted: , , , , , , , , , , , , , , , and , that is, all disconnected impact combinations. The total number of connected and disconnected impact combinations is equal to the number of all impact combinations . The impact adjacency matrices are attributed to all impact combinations.
Let us notice that the elements and indices of some impact adjacency matrices, similarly as spring adjacency matrices, have been bolded (in Figure 3: , , , , , and ; in Figure 4: , , , and ; in Figure 5: , , , and ; in Figure 6: , , , , , and ). These are all impact systems equivalent to each other via inversion or translocation.
In the case of fortyeight connected impact combinations (Figures 3, 4, and 5), the body of a mass acts on the body of a mass due to an occurrence of the impact connection between the subsystems (fenders and in Figure 3, one of these fenders in Figures 4 and 5). On the other hand, in the case of sixteen disconnected impact combinations (Figure 6), the body of a mass will never affect the behavior of the body of a mass (and vice versa, of course). A lack of the impact connection (a lack of both fenders and ) will cause that the bodies “will not know” about each other.
According to the assumed principle (Phase I, BlazejczykOkolewska [111]), we will match now each spring combination with each impact combination. The total number of connected springimpact combinations (for before elimination) is equal to , whereas the total number of disconnected springimpact combinations (before elimination) is equal to . Table 1 shows Phase I for the system of . As a result of matching all connected spring combinations (Figure 1) with all connected impact combinations (Figures 3, 4, and 5), connected springimpact combinations arise. Their total number equals . As a result of matching all connected spring combinations (Figure 1) with all disconnected impact combinations (Figure 6), connected springimpact combinations arise. Their total number is . As a result of matching all disconnected spring combinations (Figure 2) with all connected impact combinations (Figures 3, 4, and 5), connected springimpact combinations arise. Their number is equal to . As a result of matching all disconnected spring combinations (Figure 2) with all disconnected impact combinations (Figure 6), disconnected springimpact combinations arise. Their number equals . The number of connected springimpact combinations added to the number of disconnected springimpact combinations is equal to the number of all springimpact combinations; that is to say, .

Due to the convenience of presentation, spring, impact and springimpact combinations will be referred to as spring, impact, and springimpact systems up to the end of this section.
The next stage consists in the elimination of equivalent combinations (subphase I of Phase II, BlazejczykOkolewska [111]). For , it can be conducted without a computer, but it is a very timeconsuming procedure.
Let us begin with the generation of the adjacency matrix of all springimpact systems. We take the first one, and we generate a transposed, inverted, and translocated matrix to it. Next, we check the equivalency of the taken matrix to itself and to the adjacency matrices of the subsequent springimpact systems. If any equivalency is identified (see equations (2) (4), BlazejczykOkolewska [111]), we make a class of relations, and we choose the system with the highest member as its representative (see the principles of selection, BlazejczykOkolewska [111]). The abovementioned activities should be repeated for each springimpact system which has not been included in any class of relations yet.
As an example, let us consider the springimpact system . It is equivalent to other systems and . The class of relations in this case has four elements. We choose as its representative as it is the system with the highest number, and the systems S_{2}Z_{23}, S_{2}Z_{53}, and S_{5}Z_{12} are thus eliminated.
Table 2 presents a list of selected classes of relations and their representatives. The first two columns include all springimpact systems that arise as a result of matching the spring system with all impact systems . An analogous list can be given for the remaining spring systems equivalent to one another via inversion, that is, , , and (in Table 2, we replace by the subsequent systems , , and ). The last two columns in the table include all springimpact systems that result from matching the spring system with all impact systems . An analogous list can be prepared for the spring system (in Table 2, we replace by the spring system and by the spring system ). For , there are 168 classes of relations. Among them, we can distinguish 24 classes of relations in which springimpact systems are disconnected (see Table 5 discussed further in this section as well). Classes of springimpact relations can be oneelement (one springimpact system which is simultaneously the representative of the class of relations), two elements (two equivalent springimpact systems and one of them is the representative of the class of relations), or four elements (four equivalent springimpact systems and one of them is the representative of the class of relations).

Below, in the context of the twodegreesoffreedom system, an application of the procedures developed by BlazejczykOkolewska [111] will be discussed. It is worth noticing that, whenever we take the subsequent adjacency matrix, we will have the full information on equivalencies to other adjacency matrices (both for spring and impact systems).
Table 3 is a table of spring relations for , that is to say, a list of all spring systems (the first column ) together with the information on their inversed equivalency (the second column ). This table has been made on the basis of the spring table generation procedure (BlazejczykOkolewska [111, Section ]). Whenever we take the subsequent spring adjacency matrix, we have the full information on it. It follows from Table 3 that four spring systems are equivalent to one another via inversion (, , , and ), and there are two pairs of spring systems equivalent via inversion: ~ and ~ (, ).

Table 4 is a table of impact relations for , that is to say, a list of all impact systems (the first column ) with the information on their transposed (the second column ), inversed (the third column ), and translocated (the fourth column ) equivalency. This table has been made according to the impact table generation procedure (BlazejczykOkolewska [111, Section ]). Whenever we take the next impact adjacency matrix, we have the full information on it. We can read from Table 4 that for twentyfour impact systems, the number of the matrix (it refers to the lower index) fulfils condition (6) described by BlazejczykOkolewska [111]. All the systems (, , , , , , , , , , , , , , , , , , , , , , , and ) will be matched to those spring systems which are equivalent to one another via inversion (, , , and ). After matching, we will obtain 96 springimpact systems that are the representatives of classes of relations. Among them, 48 systems are equivalent to one another via inversion or translocation. Their lower indices will be bolded.

 
*Intensively shaded. ^{ #}Slightly shaded. 
The remaining two spring systems ( and ) should be matched to impact systems fulfilling condition (6) and, additionally, to impact systems from the classes of relations that fulfill condition (7) of the numbers equal to the higher number out of the two numbers and . All the impact systems satisfying these conditions, and we have 36 such systems altogether (, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and ), after matching to the spring systems ( and ), yield 72 springimpact systems that make another group of the representatives of classes of relations.
The total number of springimpact systems which remain after the elimination of equivalent systems is thus .
Table 5 shows all springimpact systems that have not been eliminated. They are divided into two groups: connected systems and disconnected systems.
The list has been obtained with the identity procedures (BlazejczykOkolewska [111, Subsections 4.1 and 4.2]), which eliminate equivalent systems and investigate their connectedness. In the first column of Table 5, there are spring systems. In the remaining part of the table, there are impact systems. The digit written under the symbol of the impact system corresponds to the number of systems in the given class of springimpact relations (including the representative of this class). The bolding of this digit and of the index of the adjacency matrix means that the given springimpact system is equivalent to itself via inversion of translocation. On such a list as the one prepared, an actual springimpact system can be found easily. There are altogether 144 connected springimpact systems (including 3 combinations without impacts), whereas the number of all disconnected springimpact systems is equal to 24. It means that, as a result of the elimination of equivalent and disconnected systems, the original number of systems (512) has decreased more than three times (144).
Table 5 presents these springimpact systems that have been selected as the representatives of classes of relations. Its comparison to Tables 3 and 4 provides the information on all the systems which belong to the given class of relations.
The springimpact systems (structural patterns) presented in Table 5 are written in such a way that the given spring connection and the first line starting with correspond to the connections of the impact systems having both fenders in the connectedness zone, the second line beginning with to the connections of the impact systems that have one fender in the connectedness zone, and the third line starting with to the connections of the impact system that do not have fenders in the connectedness zone. The shaded cells describe systems in which there is no impact connection (the cell intensively shaded), they have no spring connection (the cell lightly shaded), or there is no impact or spring connection (connection of the cell slightly shaded to the cell very intensively shaded). Such systems will be considered in Sections 3 and 4.
One of the springimpact systems described in Table 5, that is, , includes the full configuration of springs and fenders (the maximum number of springs and the maximum number of fenders). It is thus the basic springimpact system for . All the remaining springimpact systems can be made out of it via removing a certain number of springs or fenders.
The analysis of springimpact systems is definitely simplified for and becomes much more difficult for . A classification of springimpact systems with one degreeoffreedom is presented below.
3. Systems with One DegreeofFreedom
The maximum number of springs for the system with one degreeoffreedom () is equal to ( is the spring connecting the body of a mass with the frame, see the first system in Figure 7). The number of possible spring combinations . In Figure 7, all combinations with their adjacency matrices are presented.
In this case it is difficult to talk about the spring connectedness (subphase II of Phase II): either the system has no spring connecting it to the frame ( in Figure 7), or has such a spring ( in Figure 7), but it does not belong to the spring connectedness zone.
The maximum number of fenders (impact connections) for is ( is the upper impact connection of the body of a mass with the frame and and the lower impact connection of the body of a mass with the frame). Both impact connections and their notations are marked in Figure 8. The number of possible impact combinations . Figure 8 shows all such combinations along with their adjacency matrices. In the case of impact combinations of , it is also difficult to talk about impact connectedness (subphase of Phase II): either the system does not have any fenders connecting it to the frame ( in Figure 8), has one fender (lower in Figure 8 or upper in Figure 8) or has two such fenders ( in Figure 8), but they do not belong to the impact connectedness zone. The following relations take place in them: the impact combination is equivalent to itself, the combination is equivalent to itself, and the combination is equivalent to the combination via transposition.
According to the assumed principle (Phase I), we match now each spring combination with each impact combination. As a result of the elimination of redundant equivalent combinations (via transposition), we obtain the springimpact systems presented in Figure 9. As all spring and impact combinations with one degreeoffreedom are connected; thus all springimpact systems formed as a result of matching such combinations are connected. In Figure 9 the systems in which there are no impact connections (), spring connections ( and ), impact connections, and spring connections () are marked.
Let us notice that systems with one degreeoffreedom can be treated as subsystems of disconnected systems with two degreesoffreedom. The subsystems of matching spring systems would correspond to the systems shown in Figure 9: (the spring connection of the system of a mass with the frame) and (there is no spring connection of the subsystem of a mass with the frame), with the following disconnected impact systems: (the impact connection via the fenders and of the subsystem of a mass with the frame), (the impact connection via the fender of the subsystem of a mass with the frame), and (there is no impact connection of the subsystem of a mass with the frame).
4. Examples of Systems Considered in the Literature and Their Structural Patterns
Let is notice that instead of a spring connection, we can introduce any other connection that describes the action of at least one force (linear or nonlinear) that depends on displacement or velocity in the system. It cannot only be an elasticity force, but also a viscous damping force, a friction force, or an elasticdamping force or even a triple combination of these forces. As the occurrence of a fender does not determine the way the impact is modeled, thus the approach described here enables identification of all structural patterns of connectionimpact systems for the given . Let us draw attention to the fact that their number does not grow with an appearance of subsequent qualitatively different forces (dependent on displacement or velocity) or with a change in the impact modeling method. All the considerations conducted for springimpact systems can be thus repeated without any obstacles for connectionimpact systems. The obtained adjacency matrices of PZ connectionimpact systems are identical to the adjacency matrices of SZ springimpact systems. Hence, we can state that for (, , , and in Figure 9) and (all connected representatives of classes of relations without , , and ; see Table 5); we have identified all structural patterns of connectionimpact systems. Tables 6 and 7 include selected examples of mechanical one and twodegreesoffreedom systems chosen from the literature. for all systems, their structural patterns have been assigned according to the classification method proposed in [111]. Similarly, structural patterns for other examples from the literature can be defined.


An introduction of arbitrary connections instead of spring connections also leads to the determination of all structural patterns for mechanical systems without impacts (referred to as oscillators) of ( in Figure 9) and (, , and in Table 5). Such systems are commonly analyzed. As an example, we can take the pattern, which is considered to be a dynamic vibration damper in the majority of books on theory of vibrations of machines and devices (e.g., Den Hartog [112], Timoshenko [113]). On the other hand, with the (Figure 9) pattern, all systems with one degreeoffreedom can be described. The analysis of various oscillators of (linear and nonlinear) can be found, for example, in monographs by Den Hartog [112], Timoshenko [113], Kapitaniak [114], and Guckenheimer and Holmes [101].
5. Conclusions
In the study, the classification method developed by BlazejczykOkolewska [111] has been illustrated in the example of mechanical systems with one and two degreesoffreedom. The relations occurring in the system have been described in terms of the matrix representation. The identification and the elimination of redundant adjacency matrices and the identification of connected systems have been conducted with the previously developed procedures. According to the principles of selection, the connected representatives of connectionimpact classes of relations have been determined. They constitute a set of all structural patterns of vibroimpact systems. Four patterns of impact oscillators with one degreeoffreedom and 141 patterns of impact oscillators with two degreesoffreedom have been distinguished. The knowledge of classification principles enables classification of each mechanical system with impacts with arbitrary connections. The procedure for classification of the given system has been presented in numerous examples taken from the literature devoted to the subject.
The considerations conducted for systems with impacts have allowed for defining the number of patterns of systems without impacts, commonly referred to as oscillators. In this case, one pattern with one degreeoffreedom and three patterns with two degreesoffreedom have been distinguished.
The proposed classification of mechanical systems with impacts according to characteristic properties of their structure seems to be a natural classification. It reflects the relationships between the system structures, tells us about their way of evolution, and presents their genesis. It allows us to rearrange the knowledge on systems with impacts and is the basis for understanding the sources of their diversity. Providing a full set of objects to be analyzed, it gives hints for new ideas and directions in designing technical devices.
References
 U. Andreaus, L. Placidi, and G. Rega, “Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2603–2616, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. Awrejcewicz, K. Tomczak, and C. H. Lamarque, “Controlling systems with impacts,” International Journal of Bifurcation and Chaos, vol. 9, no. 3, pp. 547–553, 1999. View at: Google Scholar
 A. Bichri, M. Belhaq, and J. PerretLiaudet, “Control of vibroimpact dynamics of a singlesided Hertzian contact forced oscillator,” Nonlinear Dynamics, vol. 63, no. 12, pp. 51–60, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 G. W. Blankenship and A. Kahraman, “Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type nonlinearity,” Journal of Sound and Vibration, vol. 185, no. 5, pp. 743–765, 1995. View at: Publisher Site  Google Scholar
 C. Cempel, The Periodical Vibration with Impacts in Mechanical Discreet Systems, Dissertation Series no. 44, Technical University of Poznan, Poznan, Poland, 1970.
 W. Chin, E. Ott, H. E. Nusse, and C. Grebogi, “Grazing bifurcations in impact oscillators,” Physical Review E, vol. 50, no. 6, pp. 4427–4444, 1994. View at: Publisher Site  Google Scholar  MathSciNet
 W. Chin, E. Ott, H. E. Nusse, and C. Grebogi, “Universal behavior of impact oscillators near grazing incidence,” Physics Letters A, vol. 201, no. 23, pp. 197–204, 1995. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. de Weger, W. van de Water, and J. Molenaar, “Grazing impact oscillations,” Physical Review E, vol. 62, no. 2, pp. 2030–2041, 2000. View at: Google Scholar
 M. Di Bernardo, M. I. Feigin, S. J. Hogan, and M. E. Homer, “Local analysis of Cbifurcations in ndimensional piecewisesmooth dynamical systems,” Chaos, Solitons & Fractals, vol. 10, no. 11, pp. 1881–1908, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 W. Fang and J. A. Wickert, “Response of a periodically driven impact oscillator,” Journal of Sound and Vibration, vol. 170, no. 3, pp. 397–409, 1994. View at: Publisher Site  Google Scholar
 Y. S. Fedosenko and M. I. Feigin, “Periodic motion of vibrating hammer including the presence of sliding regime,” Prikladnaja Matematika i Mekhanika, vol. 35, no. 5, pp. 892–898, 1971 (Russian). View at: Google Scholar
 M. I. Feigin, “Perioddoubling at Cbifurcation in the piecewisecontinuous systems,” Prikladnaja Matematika i Mekhanika, vol. 34, pp. 861–869, 1970 (Russian). View at: Google Scholar
 S. Foale and S. R. Bishop, “Bifurcations in impact oscillations,” Nonlinear Dynamics, vol. 6, no. 3, pp. 285–299, 1994. View at: Publisher Site  Google Scholar
 N. Hinrichs, M. Oestreich, and K. Popp, “Dynamics of oscillators with impact and friction,” Chaos, Solitons & Fractals, vol. 8, no. 4, pp. 535–558, 1997. View at: Google Scholar
 K. H. Hunt and F. R. E. Crossley, “Coefficient of restitution interpreted as damping in vibroimpact,” Journal of Applied Mechanics, vol. 42, no. 2, pp. 440–445, 1975. View at: Google Scholar
 H. M. Isomäki, J. Von Boehm, and R. Räty, “Devil's attractors and chaos of a driven impact oscillator,” Physics Letters A, vol. 107, no. 8, pp. 343–346, 1985. View at: Publisher Site  Google Scholar  MathSciNet
 A. P. Ivanov, “Stabilization of an impact oscillator near grazing incidence owing to resonance,” Journal of Sound and Vibration, vol. 162, no. 3, pp. 562–565, 1993. View at: Publisher Site  Google Scholar
 A. P. Ivanov, “Impact oscillations: linear theory of stability and bifurcations,” Journal of Sound and Vibration, vol. 178, no. 3, pp. 361–378, 1994. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. Kahraman and R. Singh, “Dynamics of an oscillator with both clearance and continuous nonlinearities,” Journal of Sound and Vibration, vol. 153, no. 1, pp. 180–185, 1992. View at: Google Scholar
 A. E. Kobrinskii and A. Kobrinskii, Vibroimpact Systems, Nauka Press, Moscow, Russia, 1973.
 M. Z. Kolovskij, Nonlinear Theory of Impact Systems, Nauka Press, Moscow, Russia, 1966.
 S. Kundu, S. Banerjee, J. Ing, E. Pavlovskaia, and M. Wiercigroch, “Singularities in softimpacting systems,” Physica D, vol. 231, no. 5, pp. 553–565, 2011. View at: Google Scholar
 J. Y. Lee and J. J. Yan, “Position control of doubleside impact oscillator,” Mechanical Systems and Signal Processing, vol. 21, no. 2, pp. 1076–1083, 2007. View at: Publisher Site  Google Scholar
 R. I. Leine and D. H. van Campen, “Discontinuous bifurcations of periodic solutions,” Mathematical and Computer Modelling, vol. 36, no. 3, pp. 259–273, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Q. H. Li and Q. S. Lu, “Coexisting periodic orbits in vibroimpacting dynamical systems,” Applied Mathematics and Mechanics, vol. 24, no. 3, pp. 261–273, 2003. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 S. Q. Lin and C. N. Bapat, “Estimation of clearances and impact forces using vibroimpact response: random excitation,” Journal of Sound and Vibration, vol. 163, no. 3, pp. 407–421, 1993. View at: Publisher Site  Google Scholar
 A. C. J. Luo and S. Menon, “Global chaos in a periodically forced, linear system with a deadzone restoring force,” Chaos, Solitons & Fractals, vol. 19, no. 5, pp. 1189–1199, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. C. Moon and S. W. Shaw, “Chaotic vibrations of a beam with nonlinear boundary conditions,” International Journal of NonLinear Mechanics, vol. 18, no. 6, pp. 465–477, 1983. View at: Publisher Site  Google Scholar  MathSciNet
 A. B. Nordmark, “Nonperiodic motion caused by grazing incidence in an impact oscillator,” Journal of Sound and Vibration, vol. 145, no. 2, pp. 279–297, 1991. View at: Google Scholar
 A. B. Nordmark, “Effects due to low velocity impact in mechanical oscillators,” International Journal of Bifurcation and Chaos, vol. 2, no. 3, pp. 597–605, 1992. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. Pavlovskaia, M. Wiercigroch, and C. Grebogi, “Modeling of an impact system with a drift,” Physical Review E, vol. 64, no. 5, Article ID 056224, 9 pages, 2001. View at: Google Scholar
 F. Peterka, “An investigation of the motion of impact dampers, paper I, II, III,” StrojniCku Casopis, vol. 21, no. 5, 1971 (Czech). View at: Google Scholar
 F. Peterka, Introduction to Vibration of Mechanical Systems with Internal Impacts, Academia, Praha, Czech Republic, 1981.
 F. Peterka and J. Vacík, “Transition to chaotic motion in mechanical systems with impacts,” Journal of Sound and Vibration, vol. 154, no. 1, pp. 95–115, 1992. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Půst and F. Peterka, “Impact oscillator with Hertz's model of contact,” Meccanica, vol. 38, no. 1, pp. 99–114, 2003. View at: Publisher Site  Google Scholar
 S. W. Shaw, “The dynamics of a harmonically excited systems having rigid amplitude constraints—part I, part II,” Journal of Applied Mechanics, vol. 52, no. 2, pp. 459–464, 1985. View at: Publisher Site  Google Scholar  MathSciNet
 S. W. Shaw and P. J. Holmes, “A periodically forced piecewise linear oscillator,” Journal of Sound and Vibration, vol. 90, no. 1, pp. 129–155, 1983. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. W. Shaw and P. J. Holmes, “A periodically forced impact oscillator with large dissipation,” Journal of Applied Mechanics, vol. 50, no. 4, pp. 849–857, 1983. View at: Google Scholar
 S. W. Shaw and P. Holmes, “Periodically forced linear oscillator with impacts: chaos and longperiod motions,” Physical Review Letters, vol. 51, no. 8, pp. 623–626, 1983. View at: Publisher Site  Google Scholar  MathSciNet
 A. Stefanski and T. Kapitaniak, “Using chaos synchronization to estimate the largest Lyapunov exponent of nonsmooth systems,” Discrete Dynamics in Nature and Society, vol. 4, no. 3, pp. 207–215, 2000. View at: Publisher Site  Google Scholar
 A. Stefanski and T. Kapitaniak, “Estimation of the dominant Lyapunov exponent of nonsmooth systems on the basis of maps synchronization,” Chaos, Solitons & Fractals, vol. 15, no. 2, pp. 233–244, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. M. T. Thompson and R. Ghaffari, “Chaos after perioddoubling bifurcations in the resonance of an impact oscillator,” Physics Letters A, vol. 91, no. 1, pp. 5–8, 1982. View at: Publisher Site  Google Scholar  MathSciNet
 P. C. Tung, “The dynamics of a nonharmonically excited system having rigid amplitude constraints,” Journal of Applied Mechanics, vol. 59, no. 3, pp. 693–695, 1992. View at: Publisher Site  Google Scholar
 G. S. Whiston, “Global dynamics of a vibroimpacting linear oscillator,” Journal of Sound and Vibration, vol. 118, no. 3, pp. 395–424, 1987. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. M. Krivtsov and M. Wiercigroch, “Dry friction model of percussive drilling,” Meccanica, vol. 34, no. 6, pp. 425–435, 1999. View at: Publisher Site  Google Scholar
 O. K. Ajibose, M. Wiercigroch, E. Pavlovskaia, and A. R. Akisanya, “Global and local dynamics of drifting oscillator for different contact force models,” International Journal of NonLinear Mechanics, vol. 45, no. 9, pp. 850–858, 2010. View at: Publisher Site  Google Scholar
 V. I. Babickiĭ, Theory of Vibroimpact Systems, Nauka, Moscow, Russia, 1978. View at: MathSciNet
 Y. Ma, J. Ing, S. Banerjee, M. Wiercigroch, and E. Pavlovskaia, “The nature of the normal form map for soft impacting systems,” International Journal of NonLinear Mechanics, vol. 43, no. 6, pp. 504–513, 2008. View at: Publisher Site  Google Scholar
 J. Ing, E. Pavlovskaia, M. Wiercigroch, and S. Banerjee, “Bifurcation analysis of an impact oscillator with a onesided elastic constraint near grazing,” Physica D, vol. 239, no. 6, pp. 312–321, 2010. View at: Publisher Site  Google Scholar
 M. Wiercigroch and V. W. T. Sin, “Experimental study of a symmetrical piecewise baseexcited oscillator,” Journal of Applied Mechanics, vol. 65, no. 3, pp. 657–663, 1998. View at: Google Scholar
 J. O. Aidanpää and R. B. Gupta, “Periodic and chaotic behaviour of a thresholdlimited twodegreeoffreedom system,” Journal of Sound and Vibration, vol. 165, no. 2, pp. 305–327, 1993. View at: Publisher Site  Google Scholar
 G. W. Luo and J. H. Xie, “Hopf bifurcations and chaos of a twodegreeoffreedom vibroimpact system in two strong resonance cases,” International Journal of NonLinear Mechanics, vol. 37, no. 1, pp. 19–34, 2002. View at: Publisher Site  Google Scholar
 M. Wiercigroch, R. D. Neilson, and M. A. Player, “Material removal rate prediction for ultrasonic drilling of hard materials using an impact oscillator approach,” Physics Letters A, vol. 259, no. 2, pp. 91–96, 1999. View at: Google Scholar
 A. X. C. N. Valente, N. H. McClamroch, and I. Mezić, “Hybrid dynamics of two coupled oscillators that can impact a fixed stop,” International Journal of NonLinear Mechanics, vol. 38, no. 5, pp. 677–689, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 V. A. Bazhenov, O. S. Pogorelova, T. G. Postnikova, and O. A. Luk'yanchenko, “Numerical investigations of the dynamic processes in vibroimpact systems in modeling impacts by a force of contact interaction,” Strength of Materials, vol. 40, no. 6, pp. 656–662, 2008. View at: Publisher Site  Google Scholar
 J. H. Ho, V. D. Nguyen, and K. C. Woo, “Nonlinear dynamics of a new electrovibroimpact system,” Nonlinear Dynamics, vol. 63, no. 12, pp. 35–49, 2011. View at: Publisher Site  Google Scholar
 K. Czolczynski, “On the existence of a stable periodic motion of two impacting oscillators,” Chaos, Solitons & Fractals, vol. 15, no. 2, pp. 371–379, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 K. Czolczynski and T. Kapitaniakt, “On the existence of a stable periodic solution of two impacting oscillators with damping,” International Journal of Bifurcation and Chaos, vol. 14, no. 11, pp. 3931–3947, 2004. View at: Publisher Site  Google Scholar
 B. BlazejczykOkolewska, J. Brindley, K. Czolczynski, and T. Kapitaniak, “Antiphase synchronization of chaos by noncontinuous coupling: two impacting oscillators,” Chaos, Solitons & Fractals, vol. 12, no. 10, pp. 1823–1826, 2001. View at: Publisher Site  Google Scholar
 G. W. Luo, Y. L. Zhang, and J. N. Yu, “Dynamical behavior of vibroimpact machinery near a point of codimension two bifurcation,” Journal of Sound and Vibration, vol. 292, no. 12, pp. 242–278, 2006. View at: Publisher Site  Google Scholar
 G. W. Luo, “Dynamics of an impactforming machine,” International Journal of Mechanical Sciences, vol. 48, no. 11, pp. 1295–1313, 2006. View at: Publisher Site  Google Scholar
 G. W. Luo and X. H. Lv, “Controlling bifurcation and chaos of a plastic impact oscillator,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2047–2061, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 G. Luo, L. Ma, and X. Lv, “Dynamic analysis and suppressing chaotic impacts of a twodegreeoffreedom oscillator with a clearance,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 756–778, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. J. Wagg, “Periodic sticking motion in a twodegreeoffreedom impact oscillator,” International Journal of NonLinear Mechanics, vol. 40, no. 8, pp. 1076–1087, 2005. View at: Publisher Site  Google Scholar
 S. L. T. de Souza, A. M. Batista, I. L. Caldas, R. L. Viana, and T. Kapitaniak, “Noiseinduced basin hopping in a vibroimpact system,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 758–767, 2007. View at: Publisher Site  Google Scholar
 S. L. T. de Souza and I. L. Caldas, “Controlling chaotic orbits in mechanical systems with impacts,” Chaos, Solitons & Fractals, vol. 19, no. 1, pp. 171–178, 2004. View at: Publisher Site  Google Scholar
 R. P. S. Han, A. C. J. Luo, and W. Deng, “Chaotic motion of a horizontal impact pair,” Journal of Sound and Vibration, vol. 181, no. 2, pp. 231–250, 1995. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. C. J. Luo, “Perioddoubling induced chaotic motion in the LR model of a horizontal impact oscillator,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 823–839, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. L. T. de Souza and I. L. Caldas, “Calculation of Lyapunov exponents in systems with impacts,” Chaos, Solitons & Fractals, vol. 19, no. 3, pp. 569–579, 2004. View at: Publisher Site  Google Scholar
 S. F. Masri and T. K. Caughey, “On the stability of the impact damper,” Journal of Applied Mechanics, vol. 33, pp. 586–592, 1966. View at: Google Scholar  MathSciNet
 M. M. Sadek, “The behaviour of the impact damper,” Proceedings of the Institution of Mechanical Engineers, vol. 180, pp. 895–906, 1965. View at: Google Scholar
 S. Chatterjee, A. K. Mallik, and A. Ghosh, “On impact dampers for nonlinear vibrating sytsems,” Journal of Sound and Vibration, vol. 187, no. 3, pp. 403–420, 1995. View at: Publisher Site  Google Scholar
 S. F. Masri and A. M. Ibrahim, “Stochastic excitation of a simple system with impact damper,” Earthquake Engineering and Structural Dynamics, vol. 1, pp. 337–346, 1973. View at: Google Scholar
 B. BlazejczykOkolewska, “Analysis of an impact damper of vibrations,” Chaos, Solitons & Fractals, vol. 12, no. 11, pp. 1983–1988, 2001. View at: Publisher Site  Google Scholar
 F. Peterka and B. BlazejczykOkolewska, “Some aspects of the dynamical behavior of the impact damper,” Journal of Vibration and Control, vol. 11, no. 4, pp. 459–479, 2005. View at: Publisher Site  Google Scholar
 B. BlazejczykOkolewska and T. Kapitaniak, “Dynamics of impact oscillator with dry friction,” Chaos, Solitons & Fractals, vol. 7, no. 9, pp. 1455–1459, 1996. View at: Publisher Site  Google Scholar
 B. BlazejczykOkolewska and T. Kapitaniak, “Coexisting attractors of impact oscillator,” Chaos, Solitons & Fractals, vol. 9, no. 8, pp. 1439–1443, 1998. View at: Google Scholar
 Y. Yue and J. H. Xie, “Symmetry and bifurcations of a twodegreeoffreedom vibroimpact system,” Journal of Sound and Vibration, vol. 314, no. 12, pp. 228–245, 2008. View at: Publisher Site  Google Scholar
 W. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids, Edward Arnold Publishers Ltd., London, UK, 1960. View at: MathSciNet
 Y. B. Kim and S. T. Noah, “Stability and bifurcation analysis of oscillators with piecewiselinear characteristics. A general approach,” Journal of Applied Mechanics, vol. 58, no. 2, pp. 545–553, 1991. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. L. Lau and W. S. Zhang, “Nonlinear vibrations of piecewiselinear systems by incremental harmonic balance method,” Journal of Applied Mechanics, vol. 59, no. 1, pp. 153–160, 1992. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Narayanan and P. Sekar, “Periodic and chaotic responses of an SDF system with piecewise linear stiffness subjected to combined harmonic and flow induced excitations,” Journal of Sound and Vibration, vol. 184, no. 2, pp. 281–298, 1995. View at: Publisher Site  Google Scholar
 S. Natsiavas, “Dynamics of piecewise linear oscillators with van der Pol type damping,” International Journal of NonLinear Mechanics, vol. 26, no. 34, pp. 349–366, 1991. View at: Google Scholar
 Y. Wang, “Dynamics of unsymmetric piecewiselinear/nonlinear systems using finite elements in time,” Journal of Sound and Vibration, vol. 185, no. 1, pp. 155–170, 1995. View at: Publisher Site  Google Scholar
 C. W. Wong, W. S. Zhang, and S. L. Lau, “Periodic forced vibration of unsymmetrical piecewiselinear systems by incremental harmonic balance method,” Journal of Sound and Vibration, vol. 149, no. 1, pp. 91–105, 1991. View at: Google Scholar
 S. F. Masri, “Analytical and experimental studies of multiunit impact dampers,” The Journal of the Acoustical Society of America, vol. 45, no. 5, pp. 1111–1117, 1964. View at: Google Scholar
 Y. Araki, I. Yokomichi, and Y. Jinnouchi, “Impact damper with granular materials: 4th report frequency response in a horizontal system,” Bulletin of the Japanese Society of Mechanical Engineers, vol. 29, no. 258, pp. 4334–4338, 1986. View at: Google Scholar
 C. N. Bapat and S. Sankar, “Multiunit impact damperreexamined,” Journal of Sound and Vibration, vol. 103, no. 4, pp. 457–469, 1985. View at: Google Scholar
 E. V. Karpenko, E. E. Pavlovskaia, and M. Wiercigroch, “Bifurcation analysis of a preloaded Jeffcott rotor,” Chaos, Solitons & Fractals, vol. 15, no. 2, pp. 407–416, 2003. View at: Publisher Site  Google Scholar
 E. V. Karpenko, M. Wiercigroch, E. E. Pavlovskaia, and M. P. Cartmell, “Piecewise approximate analytical solutions for a Jeffcott rotor with a snubber ring,” International Journal of Mechanical Sciences, vol. 44, no. 3, pp. 475–488, 2002. View at: Publisher Site  Google Scholar
 E. V. Karpenko, M. Wiercigroch, E. E. Pavlovskaia, and R. D. Neilson, “Experimental verification of Jeffcott rotor model with preloaded snubber ring,” Journal of Sound and Vibration, vol. 298, no. 45, pp. 907–917, 2006. View at: Publisher Site  Google Scholar
 K. Koizumi, Analysis of vibration with collision and applications to leaf beating machine [Doctor thesis], University of Tokyo, Precision Mechanical Engineering, 1980.
 W. H. Park, “Massspringdamper response to repetitive impacts,” Journal of Engineering for Industry, vol. 89, no. 4, pp. 587–596, 1967. View at: Publisher Site  Google Scholar
 J. Jerrelind and A. Stensson, “Nonlinear dynamics of parts in engineering systems,” Chaos, Solitons & Fractals, vol. 11, no. 13, pp. 2413–2428, 2000. View at: Google Scholar
 R. M. Brach, Mechanical Impact Dynamics: Rigid Body Collisions, John Wiley and Sons, 1991.
 G. Chen, Controlling Chaos and Bifurcations in Engineering Systems, CRC Press, Boca Raton, Fla, USA, 2000. View at: MathSciNet
 J. Awrejcewicz and C.H. Lamarque, Bifurcation and Chaos in Nonsmooth Mechanical Systems, vol. 45 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing, River Edge, NJ, USA, 2003. View at: MathSciNet
 S. R. Bishop, “Impact oscillators,” Philosophical Transactions of the Royal Society A, vol. 347, no. 1683, pp. 347–351, 1994. View at: Publisher Site  Google Scholar
 B. BlazejczykOkolewska, K. Czolczynski, T. Kapitaniak, and J. Wojewoda, Chaotic Mechanics in Systems with Impacts and Friction, vol. 36 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing, Singapore, 1999. View at: Publisher Site  MathSciNet
 B. Brogliato, Nonsmooth Mechanics, Springer, 1999.
 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. View at: MathSciNet
 R. A. Ibrahim, VibroImpact Dynamics: Modeling, Mapping and Applications, vol. 43 of Lecture Notes in Applied and Computational Mechanics, Springer, Berlin, Germany, 2009. View at: Publisher Site  MathSciNet
 G. X. Li, R. H. Rand, and F. C. Moon, “Bifurcations and chaos in a forced zerostiffness impact oscillator,” International Journal of NonLinear Mechanics, vol. 25, no. 4, pp. 417–432, 1990. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. H. T. Thompson and S. R. Bishop, Nonlinearity and Chaos in Engineering Dynamics, John Wiley and Sons, 1994.
 E. Gutiérrez and D. K. Arrowsmith, “Control of a double impacting mechanical oscillator using displacement feedback,” International Journal of Bifurcation and Chaos, vol. 14, no. 9, pp. 3095–3113, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. L. T. de Souza, M. Wiercigroch, I. L. Caldas, and J. M. Balthazar, “Suppressing grazing chaos in impacting system by structural nonlinearity,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 864–869, 2008. View at: Publisher Site  Google Scholar
 A. C. J. Luo, “On flow switching bifurcations in discontinuous dynamical systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 1, pp. 100–116, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 W. L. Zhao and L. Z. Wang, Random Fatigue and Clearance Nonlinearity of Mechanical Vibrating Systems, Science Publishing House, Beijing, China, 2007.
 P. B. Zinjade and A. K. Mallik, “Impact damper for controlling frictiondriven oscillations,” Journal of Sound and Vibration, vol. 306, no. 12, pp. 238–251, 2007. View at: Publisher Site  Google Scholar
 B. BlazejczykOkolewska, K. Czolczynski, and T. Kapitaniak, “Classification principles of types of mechanical systems withimpacts—fundamental assumptions and rules,” European Journal of Mechanics A/Solids, vol. 23, no. 3, pp. 517–537, 2004. View at: Publisher Site  Google Scholar
 B. BlazejczykOkolewska, “A method to determine structural patterns of mechanical systems with impacts,” http://arxiv.org/abs/1212.0193. View at: Google Scholar
 J. P. Den Hartog, Mechanical Vibration, McGrawHill Book Company, 1956.
 S. Timoshenko, Vibration Problems in Engineering, D. Van Nostrand Company, Inc., 1956.
 T. Kapitaniak, Chaotic Oscillations in Mechanical Systems, Nonlinear Science: Theory and Applications, Manchester University Press, Manchester, UK, 1991. View at: MathSciNet
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