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.
Systems with one degreeoffreedom belong to the simplest mechanical systems. These systems have been analyzed by numerous authors (e.g., [
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. [
In many studies (e.g., [
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 [
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., Aidanp
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. [
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. [
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 [
Systems with two degreesoffreedom (
The maximal number of springs (spring connections)
Connected spring combinations for
Disconnected spring combinations for
The spring connectedness zone is located between the subsystems with the masses
Similarly, the number
In the case of connected spring combinations (Figure
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
We will conduct a similar analysis for impact connections. The maximal number of fenders (impact connections) is equal to
Connected impact combinations (fenders
Connected impact combinations (fender
Connected impact combinations (fender
Disconnected impact combinations (lack of the fenders
The impact connectedness zone is located between the subsystems of masses
Let us notice that the elements and indices of some impact adjacency matrices, similarly as spring adjacency matrices, have been bolded (in Figure
In the case of fortyeight connected impact combinations (Figures
According to the assumed principle (
The way connected and disconnected springimpact combinations are generated (before elimination) for



 



 



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
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)
As an example, let us consider the springimpact system
Table
Selected classes of springimpact relations and their representatives.
Classes of springimpact relations 


Representatives of the class of relations  Remaining systems from the class of relations  Representatives of the class of relations  Remaining systems from the class of relations 



































































































































Below, in the context of the twodegreesoffreedom system, an application of the procedures developed by BlazejczykOkolewska [
Table
Table of spring relations for
Fields of spring information  





2  5 


4  7 
5  2 


7  4 


Table
Table of impact relations for
Fields of impact information  






1 


2  9  5  17 

33  33 

4  41  37  19 
5  17  2  9 

25 

25 
7  49  34  11 
8  57  38  27 
9  2  17  5 
10  10  21  21 
11  34  49  7 
12  42  53  23 

18  18 

14  26  22  29 

50  50 

16  58  54  31 
17  5  9  2 

13  13 

19  37  41  4 

45  45 

21  21  10  10 
22  29  14  26 
23  53  42  12 
24  61  46  28 

6 

6 
26  14  29  22 
27  38  57  8 
28  46  61  24 
29  22  26  14 

30 


31  54  58  16 

62  62 


3  3 

34  11  7  49 

35 



43  39  51 

19  4  41 

27  8  57 

51  36  43 

59 

59 
41  4  19  37 
42  12  23  53 
43  36  51  39 
44  44  55  55 

20  20 

46  28  24  61 

52  52 

48  60  56  63 
49  7  11  34 

15  15 

51  39  43  36 

47  47 

53  23  12  42 
54  31  16  58 
55  55  44  44 
56  63  48  60 
57  8  27  38 
58  16  31  54 

40 

40 
60  48  63  56 
61  24  28  46 

32  32 

63  56  60  48 

64 


List of connected and disconnected representatives of classes of connectionimpact (springimpact) relations.
Connected representatives of classes of relations  





























 




















 







































 




























 



























 
 




















 
Disconnected representatives of classes of relations  
 









 












 








*Intensively shaded.
^{ #}Slightly shaded.
The remaining two spring systems (
The total number of springimpact systems which remain after the elimination of equivalent systems is thus
Table
The list has been obtained with the identity procedures (BlazejczykOkolewska [
Table
The springimpact systems (structural patterns) presented in Table
One of the springimpact systems described in Table
The analysis of springimpact systems is definitely simplified for
The maximum number of springs for the system with one degreeoffreedom (
Spring combinations for
In this case it is difficult to talk about the spring connectedness (subphase II of
The maximum number of fenders (impact connections) for
Impact combinations for
According to the assumed principle (
Connectionimpact (springimpact) combinations for
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
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
Examples from the references (
No.  Model of the system  Considered by, for example,  Structural pattern ( 

1 

Cempel [ 

 
2 

Krivtsov and Wiercigroch [ 

 
3 

Pavlovskaia et al. [ 

 
4 

Ajibose et al. [ 

 
5 

Babickiĭ [ 

 
6 

Chin et al. [ 

 
7 

de Weger et al. [ 

 
8 

Hinrichs et al. [ 

 
9 

Bichri et al. [ 

 
10 

Di Bernardo et al. [ 

 
11 

Stefanski and Kapitaniak [ 

 
12 

Awrejcewicz et al. [ 

 
13 

Ma et al. [ 

 
14 

Nordmark [ 

 
15 

Ing et al. [ 

 
16 

Ma et al. [ 

 
17 

Andreaus et al. [ 

 
18 

Půst and Peterka [ 

 
19 

Ma et al. [ 

 
20 

Babickiĭ [ 

 
21 

Luo and Menon [ 

 
22 

Blankenship and Kahraman [ 

 
23 

Lin and Bapat [ 

 
24 

Leine and van Campen [ 

Examples from the references (
No.  Model of the system  Considered by, for example,  Structural pattern ( 

1 

Aidanpää and Gupta [ 

 
2 

Wiercigroch et al. [ 

 
3 

Valente et al. [ 

 
4 

Peterka [ 

 
5 

Bazhenov et al. [ 

 
6 

Ho et al. [ 

 
7 

Czolczynski [ 

 
8 

Czolczynski and Kapitaniakt [ 

 
9 

BlazejczykOkolewska et al. [ 

 
10 

Luo [ 

 
11 

Luo and Lv [ 

 
12 

Luo et al. [ 

 
13 

Wagg [ 

 
14 

de Souza et al. [ 

 
15 

De Souza and Caldas [ 

 
16 

Chatterjee et al. [ 

 
17 

Masri and Ibrahim [ 

 
18 

BlazejczykOkolewska [ 

 
19 

BlazejczykOkolewska and Kapitaniak [ 

 
20 

Yue and Xie [ 

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 the study, the classification method developed by BlazejczykOkolewska [
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.