Development of Basic Conditions for Division of ... - Science Direct

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 150 (2016) 882 – 888

International Conference on Industrial Engineering, ICIE 2016

Development of Basic Conditions for Division of Mechanisms into Subfamilies L. Dvornikova, A. Fomina,b,* a

b

Siberian State Industrial University, Kirova Str. 42, Novokuznetsk 654007, The Russian Federation Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, Ac. Lavrentieva ave. 1, Novosibirsk 630090, The Russian Federation

Abstract This study provides the basics for structural organization and classification of mechanisms within their subfamilies. The division of all mechanisms into subfamilies is based on the application of various types of kinematic pairs with varied numbers of relative motions. The introduction of subfamilies gives the necessary base for completed synthesis of all mechanisms within their families. It has been analytically proved that all five families of mechanisms include fifty seven subfamilies, where thirty one are in the zero family, fifteen in the first family, seven in the second family, three in the third family and one in the fourth family. The study presents new foundations for the structural synthesis of the planar and spatial mechanisms which may be only developed with the approach of separation into subfamilies. This new approach gives necessary foundations to further synthesis of any kinds of kinematic chains, such as Assur groups with zero DOF or multi-mobility systems, as well as one DOF mechanisms. ©2016 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: kinematic pair; link; imposed constraint; DOF; subfamily.

1. Introduction We start the construction of any machine from the calculation of its structural parameters and projection of its kinematic scheme. This operation is named structural synthesis. It is the fundamental section in Mechanism and Machine Science (MMS) and it is much undeveloped today, compare with other MMS sections.

* Corresponding author. Tel.: +7-952-171-6434; fax: +7-384-346-5792. E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016

doi:10.1016/j.proeng.2016.07.038

883

L. Dvornikov and A. Fomin / Procedia Engineering 150 (2016) 882 – 888

New kinematic schemes of mechanisms have been constructed on a hunch with the help of former approaches or with the use of Assur method [1,2], in which structural groups of zero mobility [3,4] are added one by one beginning from a driving link. This solution is not always efficient as it does not include all diversity of presumable answers. The question of how to elaborate the structures of mechanisms including the preassigned number of links and the number of DOF (mobility) of the mechanical system is a multivariate problem. It is because of the situation when absolutely different schemes of mechanisms are constructed using identical number of links and pairs. These schemes are differentiated by the structure and types of kinematic pairs. This task might be solved through the analyzing of the mechanisms’ structure from the point of imposed constraints and application of different kinematic pairs with various numbers of their relative motions. Let’s consider the classification of families of mechanisms offered by Academician Artobolevsky I.I. [5]. It was proposed to divide all mechanical systems into families by the number of imposed constraints (m). Parameter m comes into the universal structural formula of mobility of kinematic chains which has been firstly derived by Professor Dobrovolsky V.V. [6]

Wm

k m 1

( 6  m) n  ¦ ( k  m) p k ,

(1)

k 5

where Wm – mobility of a kinematic chain, defining number of its DOF, n – number of movable links in a kinematic chain, pk – class of pairs from five to one. Mechanical systems with one DOF (Wm=1) are investigated in this paper. Parameter m accepts only positive integer values from 0 to 4. This allows separating all mechanisms into five various families beginning from the zero up to the fourth. 2. Division of mechanisms into families depending on the number of general imposed constraints Mechanisms of the zero family are also known as “classically” spatial mechanisms which have no any constraints imposed on the movements of their links [7-11]. This type of mechanisms includes kinematic pairs of five classes – from one-DOF pairs (ɪ5) up to five-DOF pairs (ɪ1). Introduction of m=0 into in the formula (1) will be give us the structural equation for zero family mechanisms looks as follows

W0

6n  5 p5  4 p 4  3 p3  2 p 2  p1 .

(2)

Fig. 1 presents the kinematic scheme of three-bar mechanism with osculating cylinders [12], which functions within zero family and has no imposed constraints. Mechanisms of the first family are spatial and have one general imposed constraint (m=1). If we input m=1 into the formula (1) we will get the equation shown below

W1

5n  4 p5  3 p4  2 p3  p2 .

(3)

These mechanisms might be organized by whole diversity of all kinematic pairs except pairs ɪ1. Number of relative motions of their links in the Cartesian space is five. The first family mechanism is shown in the Fig. 2.

Fig. 1. Three-bar mechanism with osculating cylinders (m=0)

884

L. Dvornikov and A. Fomin / Procedia Engineering 150 (2016) 882 – 888

Fig. 2. Angled transmission mechanism (m=1).

Two general constraints (m=2) are imposed on the motion of mechanisms of the second family, this also means that these mechanisms can be organized by pairs ɪ5, ɪ4 and ɪ3. Fig. 3 shows the mechanism for reproduction of spatial curves [13] of the second family (m=2). The structural formula connects number of movable links, kinematic pairs and mobility is written as

W2

4n  3 p5  2 p4  p3 .

(4)

The third family includes planar and spherical mechanisms with the number of general imposed constraints of three. The mobility equation (1) with m=3 includes only one- and two-DOF pairs

W3

3n  2 p5  p 4 .

(5)

Four-bar cam mechanism with variable intermediate link [14] is in the Fig. 4. It has three imposed constraints and works within third family.

Fig. 3. Mechanism for reproduction of spatial curves (m=2).

Fig. 4. Four-bar mechanism with variable link (m=3).

It is possible to synthesize only screw and wedge mechanisms within the fourth family where four general constraints (m=4) are imposed. These mechanisms include kinematic pairs only of the fifth class

W4

2n  p5 .

(6)

885

L. Dvornikov and A. Fomin / Procedia Engineering 150 (2016) 882 – 888

Fig. 5 presents known three-bar screw mechanism, it reproduces two relative motions in the Cartesian space. So, the number of general imposed constraints (m) gives an opportunity to create mechanisms in five various families depending on the mobility equations (2)-(6) which include such parameters as the numbers of classes of kinematic pairs, the number of DOF for a constructed mechanism and the number of movable links. 3. Multi-family mechanisms with fractional number of general imposed constraints

It should be noticed that there is a special type of mechanisms which cannot be referred to any of the above mentioned families when parameter m takes fractional values beginning from 0.5 up to 3.5. Such mechanisms are multi-family, they contemporaneously contain mechanisms or Assur groups of various families [15]. There is a kinematic scheme of the multi-family five-bar screw linkage which is presented in the Fig. 6 [16]. It simultaneously includes the planar four-bar mechanism of the third family (W3=1) composed from the links 1, 2, 3 and 4, and onebar group of zero DOF of the fourth family (W4=0) formed from links 1 and 5.

Fig. 5. Three-bar screw mechanism (m=4).

Fig. 6. Multi-family five-bar screw linkage (m=3,5).

The mobility of this multi-family mechanism (Fig. 6) cannot be calculated by only one of the equations (2)-(6). It should be calculated in conformity with all chains involved in the mechanism. So, number of DOF for the part of the third family is W3 3 ˜ 3  2 ˜ 4 1 . For the part belonging to the fourth family number of DOF is W4 2 ˜ 1  2 0 . The overall number of DOF for this multi-family mechanism will be W3,4=W3+W4=1, it means input motion is set for one link in the mechanism (in this case - for crank 2). 4. Subfamilies of mechanisms, their structural formulas of mobility

Within families mechanisms might have more special differences in their structure. One of these differences is the composition of the used kinematic pairs of various classes. Thereby subfamilies of mechanisms were introduced within families to consider more detailed distinctions among structures. Let’s look to the equations (2)-(6) to extract possible subfamilies of mechanisms. It is feasible to write thirty one equations for subfamilies of mechanisms in the zero family which have totally various compositions of kinematic pairs [17,18]. The most complex subfamily includes contain of pairs from fiveup to one-DOF and describes by the formula W0 (1)

6 n  5 p 5  4 p 4  3 p 3  2 p 2  p1 ,

(7)

886

L. Dvornikov and A. Fomin / Procedia Engineering 150 (2016) 882 – 888

when the simplest one subfamily, thirty first, includes only one-DOF pairs W0 ( 31)

6 n  p1 .

(8)

There are fifteen subfamilies of mechanisms within the first family. The first subfamily is the most complex as it involves kinematic pairs of the fifth (p5), fourth (p4), third (p3) and second (p2) classes. The fifteenth subfamily is the simplest one. It includes one class of pairs - p2

­W1(1) 5n  4 p5  3 p 4  2 p3  p 2 ; ° °W1( 2 ) 5n  4 p 5  3 p 4  2 p 3 ; °W 5n  4 p 5  3 p 4  p 2 ; ° 1( 3 ) °°W1( 4 ) 5n  4 p 5  3 p 4 ; ® °W1( 5 ) 5n  4 p5  2 p 3  p 2 ; °W 5n  4 p 5  2 p 3 ; ° 1( 6 ) °W1( 7 ) 5n  4 p 5  p 2 ; ° °¯W1( 8 ) 5n  4 p 5 ;

W1( 9 )

5n  3 p 4  2 p 3  p 2 ;

W1(10 )

5n  3 p 4  2 p 3 ;

W1(11)

5n  3 p 4  p 2 ;

W1(12 )

5n  3 p 4 ;

W1(13 )

5n  2 p 3  p 2 ;

W1(14 )

5n  2 p 3 ;

W1(15 )

5n  2 p 2 .

(9)

Mechanisms of the second family are organized by three- (p3), two- (p4) and one-DOF (p5) kinematic pairs. So, seven various subfamilies have been extracted [19, 20], their analytical equations are shown in the system (10)

­W2 (1) 4 n  3 p 5  2 p 4  p 3 ; ° °W2 ( 2 ) 4 n  3 p 5  2 p 4 ; ® °W2 ( 3 ) 4 n  3 p 5  p 3 ; °W ¯ 2 ( 4 ) 4n  3 p5 ;

W2 ( 5 )

4n  2 p 4  p3 ;

W2 ( 6 )

4n  2 p 4 ;

W2 ( 7 )

4n  p3 .

(10)

The third family has only three subfamilies. The first one is the most complex as it simultaneously includes one(p5) and two-DOF (p4) pairs, while the second and third subfamilies involve either pairs p5 or p4 separately

­°W3 (1) ® °¯W3 ( 2 )

3n  2 p 5  p 4 ; 3n  2 p 5 ;

W3 ( 3 )

3n  p 4 .

(11)

Pairs of the fifth class (p5) connect links in the mechanisms of the fourth family, so a singular subfamily has been picked out and its analytical equation is totally repeated formula (6). It should be noted that to create a mechanism with one DOF, i.e. to provide condition Wm=1, is not feasible for each subfamily out of 57 ones. For example, the thirtieth subfamily of the zero family where only kinematic pairs of the second class are used and is described by the equation (12) W0( 30 )

6n  2 p 2

(12)

887

L. Dvornikov and A. Fomin / Procedia Engineering 150 (2016) 882 – 888

does not give an opportunity to synthesize a mechanism because the coefficients before n and p2 in are even numbers, so condition W0(30)=1 is not satisfied. There might be constructed mechanical systems with zero or gerade numbers of DOF in this subfamily. 5. Mechanisms synthesis within subfamilies on the example of elementary mechanisms of the third family

Let’s turn to the subfamilies of mechanisms of the third family. The necessary condition for the first subfamily is the simultaneous use of kinematic pairs p5 and p4. Writing the equation for the first subfamily from the system (11) with W3(1)=1 we will get n

( 2 p5  p 4  1) / 3 ,

(13)

from which minimal value of the parameter n will be equal to two (n=2), then minimal numbers of kinematic pairs p5 and p4 are 2 and 1 (p5=2, p4=1). Show the cam mechanism with higher kinematic pair p4 in the Fig. 7a. Second subfamily involves only one-DOF pairs. Introduce W3(2)=1 in the mobility formula for the second subfamily from the system (11) and then express parameter n like n

( 2 p 5  1) / 3 ,

(14)

where minimum n is 3 and p5 is 4. There is a four-bar jointed mechanism shown in the Fig. 7b, which is constructed with the found parameters of n and p5. The third subfamily is organized only by pairs of the fourth class p4. Let’s express parameter n from the equation from the system (11) in the following form n

( p 4  1) / 3 ,

(15)

where minimum values are n=1, p4=2. Show in the Fig. 7c two-bar mechanism of the third subfamily [21] according to the calculated parameters.

Fig. 7. Mechanisms of the third family.

It comes from the Fig. 7a,b,c that planar mechanisms of the third family have principal differences in structure, function features and compositions of kinematic pairs while all these mechanisms have three imposed constraints (m=3) and their links work in a plane. 6. Conclusions

The idea of separation of mechanisms into subfamilies substantially simplifies the problem of structural synthesis of mechanisms and their classification, gives an opportunity to input more exact parameters for the designed mechanisms.

888

L. Dvornikov and A. Fomin / Procedia Engineering 150 (2016) 882 – 888

Theory of mechanisms subfamilies, developed in the current paper, might be elaborated in the direction of the designing of new mechanisms where kinematic pairs might be put in various positioning with the possibility to change pairs within their classes, i.e. to change relative movements of such pairs. For instance, if the system includes three-DOF pairs we could introduce spherical, planar or other pair of the third class. This approach gives an opportunity to design new mechanisms within their families and subfamilies. Acknowledgements

The reported study was funded by the Ministry of Education and Science of the Russian Federation (registration number RFMEF160714X0106). References [1] L.V. Assur, The investigation of planar rod mechanisms with lower pairs from the view of their structure and classification, Acad. USSR Publishing. 1952 (in Russ.). [2] M. Ceccarelli, Distinguished Figures in Mechanism and Machine Science: Their Contributions and Legacies, Part 3, Springer, 2007. [3] E.E. Peisakh, Classification of the planar Assur groups, J. Mech. Mach. Theory. 9 (2007) 5௅17. (in Russ.). [4] L.T. Dvornikov, L.N. Gudimova, The problem of diversity searching of eight-bar planar jointed Assur groups, J. Mech. Mach. Theory. 1 (2008) 15௅29. (in Russ.). [5] I.I. Artobolevski, Theory of Mechanisms and Machines, Moscow, 1965. (in Russ.). [6] V.V. Dobrovolski, Basic principles of rational classification of mechanisms, Moscow, 1936. (in Russ.). [7] L.W. Tsai, Mechanism Design: Enumeration of kinematic structures according to function, CRC Press LLC, Boca Raton, 2000. [8] R.I. Alizade, J. Duffy, E.T. Hajiyev, Mathematical models for analysis and synthesis of spatial mechanisms: Four-link spatial mechanisms, J. Mech. Mach. Theory. 18 (1983) 301–307. [9] T.S. Mruthyunjaya, Kinematic structure of mechanisms revisited, J. Mech. Mach. Theory. 38 (2003) 279–320. [10] H. Guangbo, K. Xianwen, M. Qiaoling, Design and modelling of spatial compliant parallel mechanisms for large range of translation, in: Proceeding of ASME 2010 international design engineering technical conferences and computers and information in engineering conference. (2010) 329௅340. [11] J. Angeles, Spatial Kinematic Chains: Analysis-Synthesis-Optimization, Springer-Verlag, 2012. [12] I.I. Artobolevski, Mechanisms in the current technique, Vol. 1, Moscow, 1979. (in Russ.). [13] L.T. Dvornikov, RU Patent 2309051. (2007). [14] L.T. Dvornikov, N.Yu. Ermolaeva, RU Patent 2533369. (2014). [15] L.T. Dvornikov, The problems of the investigation of multi-family mechanisms, in: Proceeding of Materials of the VI scientific and practical conference of the problems of metallurgical and mining machines, Novokuznetsk. (1997) 15௅23. (in Russ.). [16] A.S. Fomin, M.E. Paramonov, M.S. Fomin, RU Patent 141622. (2014). [17] L.T. Dvornikov, The bases of general (universal) classification of mechanisms, J. Mech. Mach. Theory. 9 (2011) 18௅29. (in Russ.). [18] L.T. Dvornikov, A.V. Stepanov, To the question of classification of mechanisms, Bulletin of the Tomsk Polytechnic University. 314 (2009) 31௅34. (in Russ.). [19] A.S. Fomin, Substantiation of the methods of structural synthesis, kinematic and kinetostatic analysis of mechanisms of the second family, Dr. diss., Omsk, 2013. (in Russ.). [20] A.S. Fomin, L.T. Dvornikov, The development of the fundamental conditions of structural synthesis of mechanisms of the second family, J. Fundamental Research. 10 (2013) 2188௅2192. (in Russ.). [21] N.O. Adamovich, Structural synthesis, kinematics and statics of planar non-Assur mechanisms, Dr. diss., Novosibirsk, 1998. (in Russ.).