A Method to Determine Structural Patterns of Mechanical Systems with Impacts

Abstract

A structural classification of vibroimpact systems based on the principles given by Blazejczyk-Okolewska et al. (2004) has been proposed for an arbitrary finite number of degrees-of-freedom. A new matrix representation to formulate the notation of the relations occurring in the system has been introduced. The developed identification and elimination procedures of equivalent systems and identification procedures of connected systems enable the determination of a set of structural patterns of systems with impacts.

1. Introduction

In the theory of vibrations of mechanical systems, systems with impacts preoccupy an important place. They started to be investigated already in the mid 1950s, and since then the interest in them has been growing and growing continuously (see, e.g., references [1–51]). Examples of their application include physical models of buildings that are used to predict effects of earthquakes (e.g., Natsiavas [39] and Nigm and Shabana [41]), pile drivers for piles or pipes in oil mining, rammers for moulding mixes, crushers, riveting presses, hammer drills (e.g., Ajibose et al. [1], Babickiĭ [3], Bajkowski [5], A. E. Kobrinskii and A. Kobrinskii [30], Krivtsov and Wiercigroch [31], and Pavlovskaia and Wiercigroch [43]), vibration dampers (e.g., Bajkowski [5], Bapat [6], Masri and Ibrahim [37], Peterka [44], and Peterka and Blazejczyk-Okolewska [46]), low-loaded toothed and cam gears (e.g., Kahraman and Singh [29], Lin and Bapat [33], Natsiavas [39], and Nguyen et al. [40]), vibrating conveyors, bar screens, gun lock mechanisms, electric automatic cutouts (e.g., Nguyen et al. [40]), printing heads in needle printers (e.g., Babickiĭ [3], Bapat [6], A. E. Kobrinskii and A. Kobrinskii [30], and Tung and Shaw [50]), and heat exchangers (e.g., Blazejczyk-Okolewska et al. [13] and Lin and Bapat [33]). In numerous cases, for example, in impact machines, vibration dampers, or any other type shakers, this phenomenon (the phenomenon of impact) plays a very useful role. On the other hand, however, its occurrence is very undesirable, as it causes, for example, additional dynamic loads, as well as faulty operation of machines and devices.

Intensive development of investigations on nonlinear phenomena comprises more and more complex vibroimpact systems. They differ as far as the design of their components is concerned, which results in various dynamical behaviors. While analyzing the studies devoted to mechanical systems with impacts, one can state that the researchers’ attention has been drawn to systems that differ in (cf. [11]) (i) a number 𝑛 of degrees-of-freedom (e.g., [1, 16, 18, 31, 33, 40, 42, 43, 47–49, 51] for 𝑛=1, [3, 6–10, 16, 19, 32, 33] for 𝑛=2, [22, 33, 35] for 𝑛=3, [17, 35, 41] for arbitrary 𝑛), (ii) a number 𝑧 of fenders (e.g., [19, 20, 27, 33, 48, 49] for 𝑧=1 and [9, 10, 21, 33, 34, 38, 44, 46, 51] for 𝑧=2), (iii) the way the limiting stops are arranged (e.g., [7, 9, 45, 46], (iv) designs of the supporting structure (e.g., [7, 9, 33, 51], and (v) a number 𝑒 of excitations applied (e.g., [3, 6, 24, 26, 33, 37, 39–41, 46, 47, 50, 51] for 𝑒=1, [35, 45] for 𝑒>1).

Due to the fact that the majority of vibroimpact systems is characterized by several above-mentioned properties at the same time, a question arises which of these properties should decide about a type of the system and in what way. During the research on dynamics of various mechanical systems, the author asked herself the following questions for many times: how a type of the system with impacts should be defined, how many such types can be differentiated, and what their properties are. The comparative investigations of physical models of vibroimpact systems analyzed in scientific studies have led to a presentation of assumptions and development of principles for the classification method of mechanical systems which models are rigid bodies that can move along a straight line without possibility to rotate (see Blazejczyk-Okolewska et al. [11]), shortly recalled in Section 2, and to provide the geometrical conditions for assembly and impacts of such systems with one and two degrees-of-freedom (see Blazejczyk-Okolewska et al. [12]).

In the present study, a method for determination of structural patterns of systems with impacts with an arbitrary finite number of degrees-of-freedom, based on the principles given by Blazejczyk-Okolewska et al. [11], is discussed. The systems differ as far as the following issues are concerned: a number of degrees-of-freedom, a number and a configuration of fenders, and a number and a configuration of connections. Without loss of generality (see Section 4.3), it has been assumed that the possible connections are springs. To develop the method proposed in the study, the following has been required: (i) to use a matrix representation of the system with impacts, (ii) to provide a characterization of equivalent systems according to the rules given in Blazejczyk-Okolewska et al. [11], (iii) to develop procedures for generation of all possible combinations of these systems and to identify and eliminate unnecessary equivalent combinations, and (iv) to eliminate disconnected systems. The approach leads to a division of all considered systems with impacts into disconnected subsets characterized by the fact that the behavior of systems of the same type (elements of one subset) can be described with equations of motion of the same structure.

The discussed 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.

2. Fundamental Assumptions and Principles

Let us recall the idea of the classification method proposed in [11]. Assume that the models of systems are rigid bodies with the masses 𝑚_𝑗 (𝑗=1,2,…,𝑛), connected by, for example, springs that can move along a straight line without a possibility to rotate. We say that a system has 𝑛 degrees-of-freedom if it is composed of 𝑛 bodies (referred to as subsystems further on), and it is not subdivided into independent systems.

For a fixed number of degrees-of-freedom 𝑛, we build the basic spring system (with 𝑠=[𝑛(𝑛+1)]/2 springs), that is, the system in which each subsystem (mass) is connected with another one and the frame by a spring, as well as the basic impact system (with 𝑧=𝑛(𝑛+1) fenders), that is, the system in which each subsystem impacts on any other subsystem and the frame at both possible senses of the relative velocity. If we remove even one spring from the basic spring system, we obtain a system with another combination of spring arrangements. These systems will be referred to as spring combinations. The number of all possible spring combinations is as follows: 𝑖_𝑠=2^𝑠. Analogously, if we remove even one fender from the basic impact system, we obtain a system with another combination of fender arrangements. These systems will be referred to as impact combinations. The number of all impact combinations 𝑖_𝑧=2^𝑧. The basic spring systems and basic impact systems for 𝑛=1,2,3 are given in Blazejczyk-Okolewska et al. [11].

Combining a basic spring system with a basic impact system, we get a basic spring-impact system, in which every subsystem is connected with any other subsystem and the frame, and each subsystem impacts on any other subsystem and the frame at both possible senses of the relative velocity. The basic spring-impact systems for one, two, and three degrees-of-freedom are given in Blazejczyk-Okolewska et al. [11]. If even one spring or even one fender is removed from the basic spring-fender system, we obtain a system with another combination of arrangements of springs or fenders. They will be referred to as spring-impact combinations.

All spring-impact systems are obtained as a result of the two-phase procedure. Phase I, referred to as a generation phase, consists in matching each case of the spring combination with each case of the impact combination. It should be noticed here that the number of all possible spring-impact systems is considerably lower than the number 𝑖_𝑠𝑧=2^𝑠⋅2^𝑧=2^𝑠+𝑧 of all possible spring-impact combinations. Unfortunately, the applied method to differentiate spring-impact combinations has a certain fault. It turns out that combinations of models of spring-impact systems (with a different configuration of springs and fenders) looking apparently different can be assigned to the same physical model. Some examples of equivalent pairs of spring-impact systems are given in Blazejczyk-Okolewska et al. [11]. It turns out that even for 𝑛=2 there are numerous spring-impact combinations which are equivalent to other combinations. An identification of the sets of equivalent combinations among spring-impact combinations (referred to as classes of relations) and a selection of their representatives for any 𝑛 will be considered in Section 4.1.

All the systems in which a subdivision into two or more independent subsystems that are not connected either by a spring or by an impact occurs will be refered to as disconnected systems. The systems in which a division into independent systems does not occur will be referred to as connected systems. Let us notice that already for systems with 𝑛=3, matching a disconnected spring combination with a disconnected impact combination can lead to a connected spring-impact combination (see, e.g., the system analyzed by Dabrowski and Kapitaniak [22]). The identification of connected systems will be considered in Section 4.2.

The above-mentioned considerations lead to the second phase in determination of all spring-impact systems. Phase II is an elimination phase, and it consists in elimination of redundant equivalent spring-impact combinations that correspond to one physical system (subphase I) and elimination of combinations that are faulty due to their disconnectedness (subphase II).

Here, the author would like to point out that already for systems with 𝑛=2, identification of equivalent combinations is not a trivial task. If we consider a system with three, four, or more degrees-of-freedom, we can state that it is difficult to control even the number of “subdivisions” into subsystems, not to mention the identification of equivalent combinations. An application of a matrix representation of the physical model of the mechanical system with impacts, proposed in Section 3, has contributed greatly to solving the above-mentioned problems.

3. Matrix Representation of the Physical Model

Below, a new way of representation of a physical model of the mechanical system with impacts is proposed. We will employ the graph terminology proposed by Deo [23]. It has been assumed for the needs of the present study that the subsequent vertices will be the bodies of masses 𝑚_𝑗, that is, the subsequent subsystems 1,2,…,𝑛 up to the frame marked as 𝑛+1, and the edges—the segments that describe the connections between the subsystems, that is, spring or impact connections. An undirected graph which describes spring connections occurring in a system will be called a spring graph. Figure 1(b) illustrates the spring graph of the system shown in Figure 1(a) (hereafter, we follow the notation of springs and fenders used in Blazejczyk-Okolewska et al. [11]). One can read from it that there is a spring connection of subsystems 1 with 3 (subsystem 1 with the frame) and subsystem 2 with 3 (subsystem 2 with the frame) and that there is no spring connection between subsystems of the masses 𝑚₁ and 𝑚₂ (there is no edge connecting vertices 1 with 2). This graph describes a disconnected spring combination.

(a)

(b)

(c)

(d)

Figure 1 

Example of a disconnected spring-impact combination: (a) scheme of the system, (b) spring graph, (c) impact graph, and (d) spring-impact graph.

A description of impact connections requires a sense of displacements of subsystems and the frame (although a displacement of the frame is not possible, we can imagine it for a while) to be accounted for. It leads to assigning proper directions (orientations) to the impact graph edges. While displacing each subsystem and the frame upwards (matter of convention), we encounter a fender of another subsystem or a fender of the frame on the way; then we can talk about an impact connection, and we mark the edge orientation. Otherwise, there is no impact connection (there is no edge). A directed graph that describes impact connections (impact relations) occurring in a system will be referred to as an impact graph. As an example, let us analyze the impact graph from Figure 1(c). There is an impact connection of frame 3 with the subsystem 1 (the edge orientation informs us about it) in the graph, but not otherwise. There are also impact connections of 2 with 3 and of 3 with 2 (again, the respective orientation of the edge manifests it), and there are no impact connections of 1 with 2 and of 2 with 1. This graph describes a disconnected impact combination.

Although a pictorial representation of the graph is very convenient and clear, a matrix representation is sometimes more suitable. The adjacency matrix of an undirected graph with 𝑛+1 vertices and without parallel edges is the symmetric binary matrix 𝐒=[𝑠_𝑖𝑗] of the dimensions 𝑛+1×𝑛+1 defined in such a way that 𝑠_𝑖𝑗=1, if there is an edge between the 𝑖th and the 𝑗th vertex, and 𝑠_𝑖𝑗=0, if there is no edge between them. At the spring graph in Figure 1(b), its spring adjacency matrix 𝐒=[𝑠_𝑖𝑗], 𝑖,𝑗=1,2,3 has been written.

The adjacency matrix of a directed graph with 𝑛+1 vertices and without parallel edges is the 0-1 matrix 𝐙=[𝑧_𝑖𝑗] of the dimensions 𝑛+1×𝑛+1 defined in such a way that 𝑧_𝑖𝑗=1, if there is an edge directed from the 𝑖th vertex to the 𝑗th vertex, and 𝑧_𝑖𝑗=0 otherwise. At the impact graph in Figure 2(c), there is an impact adjacency matrix 𝐙.

(a)

(b)

Figure 2 

Example of equivalent spring systems.

By analogy, for the need of the present study, we can construct a spring-impact graph and introduce a notion of the adjacency matrix of the spring-impact system. A spring-impact system (e.g., this from Figure 1(a)) can be shown with two graphs, that is, a spring graph (Figure 1(b)) and an impact graph (Figure 1(c)). The adjacency matrix of the spring-impact system with the spring adjacency matrix 𝐒 and the impact adjacency matrix 𝐙 is the block matrix 𝐒𝐙=[𝐒𝐙].

The spring and impact graphs that describe spring-impact systems can be treated as one spring-impact graph and can be written on one drawing. The spring-impact graph for the system from Figure 1(a) has been shown in Figure 1(d).

The above-described notions of the connectedness and the graph adjacency matrix can be a helpful tool to identify systems in which a subdivision into independent systems occurs, disconnected systems (subphase II of Phase II), and to identify the combinations equivalent to another combination (subphase I of Phase II).

4. Determination of Structural Patterns of Vibroimpact Systems

For one and two degrees-of-freedom, it is possible to draw all spring-impact graphs and state their properties. However, for higher 𝑛, the corresponding investigations are not that easy. Firstly, a way in which subsequent spring-impact systems are generated should be determined in order not to omit or multiply any of them. The method that enables such generation of systems is presented in Section 4.1. In this subsection, a characterization of equivalent systems is introduced, and identification and elimination procedures of equivalent systems are given. In Section 4.2, a standard graph theory algorithm has been implemented for identification of connected systems. Section 4.3 includes the final procedure to determine structural patterns of mechanical systems with impacts.

4.1. Generation of Spring-Impact Systems

4.1.1. Generation of All Adjacency Matrices

Spring and impact adjacency matrices are binary matrices with elements equal to “0” or “1.” To determine a set of all such matrices, we will use a representation of natural numbers in the binary system. Let us remind here that in this system the expression 𝑐_𝑛𝑐_𝑛−1⋯𝑐₀, where 𝑐_𝑛,𝑐_𝑛−1,…,𝑐₀ are 0’s or 1’s, denotes the number 𝑐_𝑛2^𝑛+𝑐_𝑛−12^𝑛−1+⋯+𝑐₀2⁰.

Constructing adjacency matrices of spring and impact graphs consists in generation of respective binary series and proper arrangements of their terms in the matrice tables. A simple way to find successive digits of the binary notation of the number presented in the decimal notation is finding remainders of subsequent divisions by two for a series of numbers where the first term is the number whose binary expression we seek for, and the next terms are integral parts from the previous divisions. Reversing the sequence of the terms in the obtained series of remainders, we obtain the binary expansion sought. A number of series corresponding to all spring adjacency matrices for 𝑛 degrees-of-freedom are equal to 2^𝑠. As each adjacency matrix of the spring graph is symmetric, it is enough to generate the respective triangular matrix. A number of series corresponding to all impact adjacency matrices for 𝑛 degrees-of-freedom are equal to 2^𝑧. The adjacency matrix of the impact graph does not have to be symmetric, thus we have to have 2^𝑧 of binary series. In the sequel, we will write S_𝑘=0𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₁𝑐₀𝑐_𝑛−10𝑐_2𝑛−2⋯𝑐_𝑛+1𝑐_𝑛𝑐_𝑛−2𝑐_2𝑛−20⋯𝑐_2𝑛𝑐_2𝑛−1𝑐⋯⋯⋯⋯⋯⋯₁𝑐_𝑛+1𝑐_2𝑛⋯0𝑐_𝑠−1𝑐₀𝑐_𝑛𝑐_2𝑛−1⋯𝑐_𝑠−10,Z_𝑙=0𝑑_𝑛−1𝑑_𝑛−2⋯𝑑₁𝑑₀𝑑_𝑧−10𝑑_2𝑛−2⋯𝑑_𝑛+1𝑑_𝑛𝑑_𝑧−3𝑑_𝑧−20⋯𝑑_2𝑛𝑑_2𝑛−1𝑑⋯⋯⋯⋯⋯⋯_𝑠+𝑛𝑑_𝑠+𝑛+1𝑑_𝑠+𝑛+2⋯0𝑑_𝑠−1𝑑_𝑠𝑑_𝑠+1𝑑_𝑠+2⋯𝑑_{𝑠+𝑛−1}0, (1) for 𝑘=1,2,…,2^𝑠 and 𝑙=1,2,…,2^𝑧, where 𝑐_𝑠−1,…,𝑐₁,𝑐₀ and 𝑑_𝑧−1,…,𝑑₁,𝑑₀ are the binary notations of 2^𝑠−𝑘 and 2^𝑧−𝑙, respectively.

A disadvantage of the matrix representation proposed in this study is that various adjacency matrices can correspond to one physical system. This fault can be overcome via identification and elimination of unnecessary matrices.

4.1.2. Characterization and Identification of Equivalent Systems

Subphase I of Phase II (Section 2) comprises elimination of spring-impact systems equivalent to another system. Having in mind convenience of this presentation, we will refer to spring, impact, and spring-impact combinations as spring, impact, and spring-impact systems to the end of this subsection.

To characterize and identify equivalent systems, let us introduce the following notions. The transpose of the adjacency matrix 𝐀 is denoted with the symbol 𝐀^𝑇. The operation due to which from the system 𝐀 we obtain the system described by 𝐀^𝑇 will be referred to as the transposition of the system 𝐀. The transposition of the system can be treated as a change in the orientation of the frame of reference introduced during the investigations of the system dynamics.

Let 𝐀 be an adjacency matrix of the type 𝑛+1×𝑛+1. Inversing rows 𝑤_𝑖 with rows 𝑤_{𝑛−𝑖+1} for 𝑖=1,2,…,[(𝑛+1)/2] and then inversing columns 𝑘_𝑗 with columns 𝑘_{𝑛−𝑗+1} for 𝑗=1,2,…,[(𝑛+1)/2], we obtain a new adjacency matrix 𝐀^𝑃, which will be called the inverse of adjacency matrix 𝐀. The symbol [𝑥] denotes the integral part of the number 𝑥. The operation due to which from the system 𝐀 we obtain the system described by 𝐀^𝑃 will be called the inversion of the system 𝐀. The inversion causes a change in the arrangement of vertices (subsystems).

The transpose of the inverse of the adjacency matrix 𝐀 will be called the translocate of adjacency matrix 𝐀 and denoted by 𝐀^𝑃𝑇. The operation due to which from the system 𝐀 we obtain the system described by 𝐀^𝑃 is called the translocation of the system 𝐀. The translocation causes a change in the arrangement of vertices and then a change in the orientation of edges.

The above-mentioned definitions concern the spring systems (𝐒^𝑇,𝐒^𝑃,𝐒^𝑃𝑇), the impact systems (𝐙^𝑇,𝐙^𝑃,𝐙^𝑃𝑇), and the spring-impact systems (𝐒𝐙^𝑇=[𝐒^𝑇𝐙^𝑇], 𝐒𝐙^𝑃=[𝐒^𝑃𝐙^𝑃], and 𝐒𝐙^𝑃𝑇=[𝐒^𝑃𝑇𝐙^𝑃𝑇]).

Let 𝐀 and 𝐀^′ be spring or impact adjacency matrices, respectively. We say that the system 𝐀 is equivalent via transposition, inversion, or translocation with the system 𝐀^′, when 𝐀^𝑇=𝐀^′, 𝐀^𝑃=𝐀^′, and 𝐀^𝑃𝑇=𝐀^′, correspondingly. If at least one of these equivalencies holds, we can say that the systems 𝐀 are equivalent to 𝐀^′.

The identification of spring-impact systems equivalent to other spring-impact systems is conducted in three different ways. We say that a spring-impact system [𝐒𝐙]  is equivalent via transposition (way I) to a spring-impact [𝐒^′𝐙^′] if 𝐒=𝐒^′∧𝐙^𝑇=Z^′. (2) The symbol ∧ denotes a conjunction.

We say that a spring-impact system  [𝐒𝐙]  is equivalent via inversion (way II) to a spring-impact system [𝐒^′𝐙^′] if 𝐒^𝑃=𝐒^′∧𝐙^𝑃=𝐙^′or𝐒^𝑃=𝐒^′∧𝐙^𝑃=𝐙=𝐙^′. (3)

We say that a spring-impact system [𝐒𝐙]  is equivalent via translocation (way III) to a spring impact system [𝐒^′𝐙^′] if 𝐒^𝑃=𝐒^′∧𝐙^𝑃𝑇=𝐙^′or𝐒^𝑃=𝐒^′∧𝐙^𝑃𝑇=𝐙=𝐙^′. (4)

We say that a spring-impact system [𝐒𝐙]  is equivalent to a spring-impact system [𝐒^′𝐙^′] if [𝐒𝐙] and [𝐒^′𝐙^′] are equivalent via transposition, inversion, or translocation. Then, we write: [𝐒𝐙]∼[𝐒^′𝐙^′]. The author would like to point out that the notion of equivalent mechanical systems is not identical with the notion of isomorphic systems (i.e., systems whose graphs are isomorphic), and therefore, the standard methods for determining isomorphic graphs are not applicable here. For example, the impact systems with adjacency matrices 𝐙=[𝑧_𝑖𝑗] and 𝐙^′=[𝑧^′_𝑖𝑗],  𝑖,𝑗=1,2,3, where 𝑧_𝑖𝑗=0 if (𝑖=𝑗) or (𝑖=1 and 𝑗=2) and 𝑧_𝑖𝑗=1 otherwise, and 𝑧^′_𝑖𝑗=0 if (𝑖=𝑗) or (𝑖=3 and 𝑗=1) and 𝑧^′_𝑖𝑗=1 otherwise, have isomorphic graphs, but they are not equivalent.

The following conclusions result from the above-mentioned definitions, namely, the following.(1)We tend to identify spring-impact systems that are assigned to one model, but while generating all possible combinations (Phase I—generation phase), they were treated as different models. It has been observed that the system in which the following was altered: (a) the orientation of the frame of reference, (b) the sequence of subsystem numeration, and (c) both the sequence of subsystem numeration as well as the orientation of the frame of reference became a new model in an artificial way.(2)Transposition of the system can be treated as a change in the orientation of the frame of the reference introduced during the investigations of the system dynamics. As the spring adjacency matrix is symmetric, thus it is equivalent (via transposition) to itself only. (3)Inversion of the system causes a change in the arrangement of vertices (subsystems). However, it should be noticed that rows and columns are to be arranged in the same order. Thus, if two matrix rows are interchanged (e.g., 𝐒 or 𝐙), then the columns corresponding to them should be interchanged as well. (4)Translocation of the system causes a change in the arrangement of vertices (subsystems) and then a change in the orientation of edges (change in the orientation of the frame of reference). The procedure of translocation of the given spring system is identical to the procedure of inversion of this system. This is not always the case for impact systems, however.

The pairs of spring-impact systems [𝐒₂𝐙₂] and [𝐒₂𝐙₉], [𝐒₂𝐙₉] and [𝐒₅𝐙₁₇], and [𝐒₂𝐙₁₇] and [𝐒₅𝐙₉], where the corresponding spring systems 𝐒_𝑘 and impact systems 𝐙_𝑗 are given in Figures 2 and 3, are examples of systems in which equivalency, respectively, via transposition (way I), via inversion (way II), and via translocation (way III) takes place.

(a)

(b)

(c)

(d)

Figure 3 

Example of equivalent impact systems.

If two spring-impact systems are equivalent, then we say that they belong to the same class of relations. Elements of the class of relations have to be identified with an arbitrary chosen representative of this class.

Description of the Procedure of Identification of Equivalent Systems. We assume that for the given degree-of-freedom 𝑛, we have all adjacency matrices of spring and impact systems (generated according to the procedure described in Section 4.1) at our disposal. In the first phase, we will deal with impact systems. For each system, we find systems equivalent to it. The information on the kind of equivalency is recorded in the respective impact information fields: 𝐙∣𝐙^𝑇∣𝐙^𝑃∣𝐙^𝑃𝑇, where 𝐙 denotes the number of the matrix 𝐙 of the given system and 𝐙^𝑇, 𝐙^𝑃, 𝐙^𝑃𝑇 the numbers of the matrices 𝐙^𝑇, 𝐙^𝑃, 𝐙^𝑃𝑇 of the systems equivalent to the given system via transposition, inversion, and translocation, respectively. Thus, a table of impact relations, which includes full information on equivalence between impact systems, will be obtained.

In the second phase, we deal with spring systems. As each spring adjacency matrices 𝐒 is symmetric, thus 𝐒^𝑇=𝐒 and 𝐒^𝑃𝑇=𝐒^𝑃. Hence, for each spring system, it is enough to find a system equivalent to it via inversion. The information on equivalency is written in two fields of spring information: 𝐒∣𝐒^𝑃, where 𝐒 denotes the number of matrix 𝐒 of the given system and 𝐒^𝑃—the number of matrix 𝐒^𝑃 of the system equivalent to the given system via inversion. As a result, we will obtain a table of spring relations including information on equivalencies between spring systems.

In the last stage, we deal with spring-impact systems. We generate such systems by matching spring and impact systems with each other. Next, using the table of spring relations and the table of impact relations, we identify equivalent systems, according to principles (1), (2), and (3). Thus, obtained sets of equivalent systems will be referred to as classes of relations or equivalency classes. Let us notice that undistinguishable systems as they correspond to one physical system, which has specified spring and impact connections of subsystems, belong to the same class of relations. Hence, it is necessary not only to identify all classes of relations but also to select representatives of classes and to eliminate the systems that are not representatives as well.

The set of all spring-impact combinations is the sum of disjoint sets of the form 𝐶={𝐒𝐙,𝐒𝐙^𝑇,𝐒𝐙^𝑃,𝐒𝐙^𝑃𝑇,𝐒^𝑃𝐙,𝐒^𝑃𝐙^𝑇,𝐒^𝑃𝐙^𝑃,𝐒^𝑃𝐙^𝑃𝑇}. The latter sets can be seen as the unions of the sets 𝐶₁={𝐒𝐙,𝐒𝐙^𝑇,𝐒^𝑃𝐙^𝑃,𝐒^𝑃𝐙^𝑃𝑇} and 𝐶₂={𝐒𝐙^𝑃,𝐒𝐙^𝑃𝑇,𝐒^𝑃𝐙,𝐒^𝑃𝐙^𝑇}. According to rules (1), (2), and (3), the possible cases are as follows.(1)If 𝐒=𝐒^𝑃, then 𝐶₁=𝐶₂=𝐶, and therefore 𝐶 constitutes an equivalency class which consists of(i)one element when 𝐙=𝐙^𝑇=𝐙^𝑃,(ii)two elements when (𝐙=𝐙^𝑇 and 𝐙≠𝐙^𝑃) or (𝐙=𝐙^𝑃 and 𝐙≠𝐙^𝑇),(iii)four elements when 𝐙, 𝐙^𝑇, 𝐙^𝑃 and 𝐙^𝑃𝑇 are four different numbers.(2)If 𝐒≠𝐒^𝑃 and 𝐙=𝐙^𝑃, then 𝐶₁=𝐶₂ and 𝐶 is an equivalency class that consists of(i)two elements when 𝐙=𝐙^𝑇,(ii)four elements when 𝐙≠𝐙^𝑇.(3)If 𝐒≠𝐒^𝑃 and 𝐙≠𝐙^𝑃, then 𝐶₁ and 𝐶₂ are two different equivalency classes and both consist of (i)two elements when 𝐙=𝐙^𝑇,(ii)four elements when 𝐙≠𝐙^𝑇.

Below, the principles for selection of representatives of classes of relations are given; thus, the criteria for elimination of equivalent spring-impact systems are specified.

A selection of representatives of classes of relations of all spring-impact systems is conducted according to the following rules.(1)The representatives of classes of relations obtained by matching spring systems 𝐒,𝐒^𝑃 equivalent to themselves (𝐒=𝐒^𝑃) with equivalent impact systems 𝐙, 𝐙^𝑇, 𝐙^𝑃, 𝐙^𝑃𝑇 will become the systems 𝐒𝐙^∗ fulfilling condition 𝐙^∗=max𝐙,𝐙^𝑇,𝐙^𝑃,𝐙^𝑃𝑇. (5)(2)The representatives of classes of relations obtained as a result of matching equivalent spring systems 𝐒,𝐒^𝑃 satisfy condition 𝐒≠𝐒^𝑃 with(a)equivalent impact systems 𝐙, 𝐙^𝑇, 𝐙^𝑃, 𝐙^𝑃𝑇 for which 𝐙=𝐙^𝑃 are the systems 𝐒^∗𝐙^∗ that fulfill the relation 𝐒^∗=max𝐒,𝐒^𝑃∧𝐙^∗=max𝐙,𝐙^𝑇,𝐙^𝑃,𝐙^𝑃𝑇, (6)(b)equivalent impact systems 𝐙, 𝐙^𝑇, 𝐙^𝑃, 𝐙^𝑃𝑇 such that 𝐙≠𝐙^𝑃 are the systems 𝐒^∗𝐙^∗ and 𝐒𝐙^∗∗ fulfilling condition 𝐒^∗=max𝐒,𝐒^𝑃∧𝐙^∗=max𝐙,𝐙^𝑇,𝐙^𝑃,𝐙^𝑃𝑇∧𝐙^∗∗=max𝐙^∗𝑃,𝐙^∗𝑃𝑇. (7)

Let us consider the two spring systems shown in Figures 2(a) and 2(b). The spring adjacency matrices of these systems will be denoted as 𝐒₂ and 𝐒₅. Let us see that 𝐒₂^𝑃=𝐒₅, and thus the systems 𝐒₂ and 𝐒₅ are equivalent via inversion. As 2=𝐒₂<𝐒₂^𝑃=5, thus the system 𝐒₅ has a higher numeration, and it is the representative of the two-element class of spring relations {𝐒₂,𝐒₅} (classes of spring relations can have one or two elements).

Now, let us consider the four impact systems shown in Figures 3(a), 3(b), 3(c), and 3(d). The adjacency matrices of these systems will be denoted as 𝐙₂, 𝐙₅, 𝐙₉, and 𝐙₁₇, correspondingly. The systems are equivalent, and they form a four-element class of impact relations {𝐙₂,𝐙₅,𝐙₉,𝐙₁₇}. Matching the system 𝐒₂ (the system 𝐒₅) with all impact systems from Figure 3, we will obtain four cases. Having in mind the fact that each spring adjacency matrix is symmetric, we should eliminate, via equivalency and via transposition, the following spring-impact systems (leaving the systems of a higher numeration): 𝐒₂𝐙₂(𝐒₂𝐙₂^𝑇∼𝐒₂𝐙₉) and 𝐒₂𝐙₅(𝐒₂𝐙₅^𝑇∼𝐒₂𝐙₁₇) and 𝐒₅𝐙₂(𝐒₅𝐙₂^𝑇∼𝐒₅𝐙₉) and 𝐒₅𝐙₅(𝐒₅𝐙₅^𝑇∼𝐒₅𝐙₁₇).

It can be stated that having applied the equivalency via transposition, eight equivalent spring-impact systems are reduced to four systems, which are equivalent to the systems eliminated. By using the equivalency via inversion and via translocation, four noneliminated systems can be reduced to two. In such a situation, we leave the spring-impact systems with the highest numeration: 𝐒₅𝐙₁₇(𝐒₂𝐙₉^𝑃∼𝐒₅𝐙₁₇) and 𝐒₅𝐙₉(𝐒₂𝐙^𝑃₁₇∼𝐒₅𝐙₉). Let us notice that the first system 𝐒₅𝐙₁₇ is the case described by principle (2(a)), that is, a matching of the spring system that fulfills condition 𝐒>𝐒^𝑃 with the impact system from the table of impact relations that fulfills condition (6). The second system 𝐒₅𝐙₉ is the case described with principle (2(b)), that is, a matching of the spring system fulfilling condition 𝐒>𝐒^𝑃 with the impact system from the class of relations of the system fulfilling condition (7) of the number higher out of two numbers 𝐙^𝑃 and 𝐙^𝑃𝑇.

As a result of matching two spring systems from Figure 2 with four impact systems from Figure 3, two classes of spring-impact relations arise—they both have four elements. There are the following systems in one class of spring-impact relations: 𝐒₅𝐙₁₇ (the representative of the class of relations) and 𝐒₂𝐙₂, 𝐒₂𝐙₉, 𝐒₅𝐙₅. The second class of spring-impact relations comprises the following systems: 𝐒₅𝐙₉ (the representative of the class of relations) and 𝐒₅𝐙₂, 𝐒₂𝐙₅, 𝐒₂𝐙₁₇.

Further on, we will consider representatives of spring-impact classes of relations only. Connected and disconnected systems are among them.

4.2. Identification of Connected Systems

Subphase II of Phase II (Section 2) comprises elimination of all systems in which a subdivision of the spring-impact system into two or more independent systems, which are not connected either by a spring or a fender, has occurred. Let us notice that in the spring-impact system that represents a mechanical system with impacts, a subdivision into at least two independent systems will occur when the graph formed from the spring-impact graph as a result of neglecting the vertex corresponding to the frame and all the edges incidental to it will not be connected. The analysis of the graph connectedness can be conducted with the algorithm for integration of vertices (see Deo [23]) applied to the matrix 𝐌=[𝑚_𝑖𝑗], 𝑖,𝑗<𝑛+1, obtained as a result of the logical sum of the spring adjacency matrix 𝐒 and the symmetrized impact adjacency matrix 𝐙 (𝑚_𝑖𝑗=𝑠_𝑖𝑗 OR 𝑧_𝑖𝑗 OR 𝑧_𝑗𝑖 for all 𝑖,𝑗).

4.3. Structural Patterns of Vibroimpact Systems

Employing all the above-described procedures, we identify equivalent combinations, select representatives of classes of relations, and identify connected spring-impact systems. The so-obtained representatives of classes of spring-impact relations form a set of all structural patterns of vibroimpact systems with an arbitrary number of degrees-of-freedom. This is a consequence of the fact 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 can be an elasticity force, but also a viscous damping force, a friction force or an elastic-damping force, or even a triple combination of these forces. Each structural pattern of mechanical systems with impacts corresponds to a class of systems characterized by a specific structure of component elements (a definite configuration of fenders and connections). All structural patterns of mechanical systems constitute a set in which the kind of connection (a spring or a damper) and its character (linearity or nonlinearity) and the way the impact phenomenon is modeled are not differentiating parameters.

The presented method of identification and description of structural patterns has been discussed on the example of systems with one and two degrees-of-freedom in the author’s study [8].

5. Conclusions

A remarkable increase in the interest in investigations of more and more complex mechanical systems with impacts, as well as a multitude and a diversity of such systems, imposes a need to classify them. Taking advantage of simplicity of the spring connection that commonly occurs in mechanical systems, a structural classification method of systems with impacts has been proposed. The essence of the method consists in a proper matching of spring and impact systems. The obtained spring-impact systems can be connected or disconnected. In the case of systems with two degrees-of-freedom, the matchings of disconnected spring and impact systems lead to disconnected spring-impact systems. However, for systems with three or more degrees-of-freedom, the situation does not have to be the same. Therefore, while building more complex systems, disconnected spring and impact systems should be accounted for.

In the notation of relations occurring in vibroimpact systems, a matrix representation that allows to determine all systems has been introduced. Its disadvantage lies in the fact that it is possible to assign various adjacency matrices to the same physical system. To overcome this discrepancy, procedures for identification and elimination of unnecessary adjacency matrices have been developed. In the further considerations, only representatives of spring-impact class of relations are analyzed. The criteria for their selection are given in the selection principles developed. Next, a procedure for identification of connected and disconnected systems, that is, systems in which a subdivision into independent systems does not occur or occurs, respectively, is introduced. The developed method enables structural classification of systems with impacts with an arbitrary number of degrees-of-freedom.

The proposed classification of mechanical systems with impacts according to the characteristic properties of 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 mechanical devices. Obviously, it will not always satisfy fully designers for whom a functional classification allowing for a selection of the proper system functionally indispensable in the given device would be equally important. However, it has not been possible to combine the properties of structure and function in any existing classification yet. Besides, trials to develop a functional classification would be unsatisfactory due to two reasons. Firstly, one system can belong simultaneously to a few different classes considered in functional terms. Secondly, as the progress in technology goes further and further, new functions of systems with impacts can appear. Thus, in principle, the functional classification would not fulfill the condition of exclusiveness and full completeness, which is satisfied by the structural classification presented herein.