“IMT Oradea” - 2021
P ROCEEDINGS
OF THE
A NNUAL S ESSION
OF
S CIENTIFIC P APERS
Volume XX (XXX)
27
th– 28
thMay 2021 Baile Felix SPA, Oradea
ROMÂNIA
ISSN 2457- 8347 ISSN-L 2285-3278
U NIVERSITY OF O RADEA P UBLISHING H OUSE Universitatii street, Nr.1
Oradea, Cod.410087, Jud.Bihor, Romania Tel/Fax.+40 259 408627
E-mail: [email protected]
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PROCEEDINGS OF THE ANNUAL SESSION OF SCIENTIFIC PAPERS
“IMT ORADEA - 2021”
27th – 28th May, Baile Felix SPA, Oradea, Romania
Preface
The International Conference, ANNUAL INTERNATIONAL SESSION OF SCIENTIFIC PAPERS - IMT Oradea 2021, with topics in the field of industrial engineering, took place in the last week of May 2021, for two days, Thursday 27 May, respectively Friday 28 May, in the virtual environment, as well as in 2020, given the travel restrictions decided by authorities around the world. According to the preliminary schedule, all participants presented their work as a result of their research.
The organizers, editors and members of the technical staff continuously assisted, from a logistical point of view, the entire development of the scientific event.
The Conference was opened by its Chairman, Assoc. Prof. PhD Gavril Grebenişan. He, as every year, invited Prof. PhD Alexandru Viorel Pele, as dean of the Faculty of Managerial and Technological Engineering of the University of Oradea (FIMT-UO), as the main sponsor of the scientific event, to address, to the participants and guests, a word of welcome. In his speech, the dean of FIMT-UO, wanted to express his enthusiasm generated by the possibility of holding the Conference, even if the conditions are so unfavourable. The speaker also showed that the joy of meeting on the occasion of the scientific event, known as IMT Oradea 2021, can not be overshadowed by the fact that it takes place virtually and not in such a friendly environment in Oradea-Baile Felix SPA, where we meet, with great pleasure, every year, in the last week of May. After expressing, with deep respect, thanks, both in his name and in the name of the faculty of FIMT-UO, which he leads, to those present, the dean wanted to emphasize that he wants the scientific meeting IMT Oradea 2021 to represent a great success, and at the next IMT Oradea Conference 2022, to meet in Oradea, as we were used to, before the outbreak of the Corona Virus pandemic.
After the welcome speech of the dean of FIMT-UO, the Chairman of the Conference, he invited the Coordinator of the Presentations, Prof. PhD Vesselenyi Tiberiu, to take control of the presentation program.
The first presentation was that of Prof. PhD Florin Blaga and Assist. Prof. PhD Alin Pop, as Keynote Speakers, who presented the paper "The efficiency of modelling and simulation of manufacturing systems using Petri nets". Also, as Keynote Speaker, the following day, Friday 28 May, presented works of great interest, and Prof. PhD Erdei Timotei István and Prof. PhD Husi Géza, who presented the work
"Incorporation of Industry 4.0 into education at the Faculty of Engineering of the University of Debrecen", respectively Prof. PhD Vesselenyi Tiberiu who presented the paper entitled "Artificial Neural Networks Image Processing in industrial machining applications".
The conference program was strictly observed by the organizers and coordinated by the Editorial Committee, in each of the two days, Thursday 27 May, respectively Friday 28 May, and works are scheduled between 10 AM and 4 PM.
Discussions followed the presentation of each participant's paper, comments and clarifications requested by participants. In general, 15 minutes were allocated for each article, of which five minutes for discussions. It has rarely happened that these time intervals are not observed.
Two independent reviewers reviewed all papers, unknown to the authors, and one editor for each article.
The acceptance rate of the works was 67%, which means that a third of the works were rejected.
The Editorial Board thanks, in this way, Mrs Lucy Evans, Coordinator of IOP Conference Series:
Materials Science and Engineering, who was kind enough to participate in the work of the IMT Oradea 2021 Conference.
Editorial Board Baile Felix SPA, Oradea, May 2021
THE CONFERENCE’S COMMITTEES
Proceedings’ Editors :
o Gavril GREBENIȘAN, University of Oradea, Romania o Alexandru-Viorel PELE, University of Oradea, Romania o Florin Sandu BLAGA, University of Oradea, Romania o Horia BELEŞ, University of Oradea, Romania
Scientific Committee :
o Stelian ALACI, Stefan cel Mare University, Suceava
o Catalin ALEXANDRU, Transilvania University Brasov, Romania
o Silvia AVASILCĂI, Technical University Gheorghe Asachi of Iaşi, Romania o Calin BABAN, University of Oradea, Romania
o Marius BABAN, University of Oradea, Romania
o Miroslav BADIDA, Technical University of Košice, Slovakia o Pouya DERAKHSHAN-BARJOEI, Islamic Azad University, Iran o Horia BELEŞ, University of Oradea, Romania
o Florin BLAGA, University of Oradea, Romania o Sanda BOGDAN, University of Oradea, Romania
o Ioan BONDREA, Lucian Blaga University of Sibiu, Romania o Dan BRINDASU, Lucian Blaga University of Sibiu, Romania
o Adriana BUJOR, Technical University Gheorghe Asachi of Iaşi, Romania o Constantin BUNGAU, University of Oradea, Romania
o Nicolae BURNETE, Technical University of Cluj-Napoca, Romania
o Samuel SÁNCHEZ CABALLERO - Univeristat Politecnica de Valencia, Spain o Miguel ANGEL SELLES CANTO - Univeristat Politecnica de Valencia, Spain o Marco CECCARELLI, University of Cassino, Italy
o Florina-Carmen CIORNEI, Stefan cel Mare University, Suceava o Ciprian CRISTEA, Technical University of Cluj-Napoca, Romania o Predrag DASIC, High Technical Mechanical School, Trstenik, Serbia o Izabela DEMBINSKA, University of Szczecin, Poland
o Nebojsa DENIC, Alfa University Belgrade, Serbia
o Cristian Vasile DOICIN, Polytechnic University of Bucharest, Romania o George DRAGHICI, Polytechnic University of Timisoara, Romania o Marek DYLEWSKI, University of Szczecin, Poland
o Catalin FETECAU, Dunărea de Jos University, Romania o Beata FILIPIAK, University of Szczecin, Poland
o Cornel Cătălin GAVRILĂ, Transilvania University Brasov, Romania o Gavril GREBENIȘAN, University of Oradea, Romania
o Mikulas HAJDUK, Technical University Kosice, Slovakia o Voichiţa HULE, University of Oradea, Romania
o Geza HUSI, University of Debrecen, Hungary
o Nicolae ISPAS, Transilvania University Brasov, Romania
o Monica IZVERCEANU, Polytechnic University of Timisoara, Romania
o Sinisa KUZMANOVIC, University of Novi Sad, Serbia o Amit KUMAR, Nanjing Forestry University, Nanjing, China o Mihai -Tiberiu LATEŞ, Transilvania University Brasov, Romania o Aleksandar MAKEDONSKI, Technical University of Sofia, Bulgaria o Juan Lopez MARTINEZ, Polytechnic University of Valencia, Spain o Sergiu MAZURU, Technical University of Moldova, R. Moldova o Tudor MITRAN, University of Oradea, Romania
o Gina Maria MORARU, Lucian Blaga University of Sibiu, Romania o Doina MORTOIU, Aurel Vlaicu University of Arad, Romania o Sorin PATER, University of Oradea, Romania
o Alexandru-Viorel PELE, University of Oradea, Romania o Delia POP, University of Oradea, Romania
o Mircea Teodor POP, University of Oradea, Romania o Mariana Adriana PRICHICI, University of Oradea, Romania
o Miguel Ángel Peydro RASERO, Universitat Politècnica De València, Spain o Mariana RATIU, University of Oradea, Romania
o Sergio ROSSETTO, Polytechnic University of Torino, Italy o Alexandru RUS, University of Oradea, Romania
o Nazzal SALEM, Zarqa University, Jordan
o Dan SAVESCU, Transilvania University Brasov, Romania
o Laurentiu SLATINEANU, Gheorghe Asachi Technical University of Iasi, Romania o Radu STANCIU, Polytechnic University of Bucharest, Romania
o Ivan NAGY SZOKOLAY, Széchenyi István University, Hungary o Edit SZUCS, University of Debrecen, Hungary
o Ioan Constantin TARCA, University of Oradea, Romania o Radu Catalin TARCA, University of Oradea, Romania o Imre TIMAR, University of Pannonia, Hungary
o Marian TOLNAY, Slovak University of Technology, Slovakia o Stefan VALCUHA, Slovak University of Technology, Slovakia o Radu VELICU, Transilvania University Brasov, Romania o Tiberiu VESSELENYI, University of Oradea, Romania o Agostino VILLA, Polytechnic University of Torino, Italy
o Marian ZAHARIA, Petroleum- Gas University of Ploiesti, Romania o Máté ZÖLDY, Budapest University of Technology and Economics, Hungary o Sahin YILDIRIM, Erciyes University, Turkey
Steering Committee :
o Dan CHIRA, University of Oradea, Romania o Nicolae CRECAN, University of Oradea, Romania
o Lehel Szabolcs CSOKMAI, University of Oradea, Romania o Ovidiu MOLDOVAN, University of Oradea, Romania o Georgeta NICHITA, University of Oradea, Romania o Alin POP, University of Oradea, Romania
o Pavel Dan TOCUT, University of Oradea, Romania
IN PARTNERSHIP WITH:
o EMSIL GROUP
o COMAU ROMANIA -ORADEA
o RESEARCH CENTER "PRODUCTICA IMT"– ORADEA
o RESEARCH CENTER IN MECHANICAL ENGINEERING AND AUTOMOTIVE-"IMA"- ORADEA
o GENERAL ASSOCIATION OF ROMANIAN ENGINEERS -BRANCH BIHOR o SOCIETY OF AUTOMOTIVE ENGINEERS OF ROMANIA
o ROMANIAN ASSOCIATION FOR NONCONVENTIONAL TECHNOLOGIES -BRANCH BIHOR
o ASSOCIATION OF MANAGERS AND ECONOMIC
o ASSOCIATION FOR INTEGRATED ENGINEERING AND INDUSTRIAL MANAGEMENT o ROBOTICS SOCIETY OF ROMANIA
o ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING
SPONSORS
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Copyright 2021
PROCEEDINGS OF THE ANNUAL SESSION OF SCIENTIFIC PAPERS is printed edition of on-line Journal “ANNALS OF THE UNIVERSITY
OFORADEA. FASCICLE OF MANAGEMENT AND TECHNOLOGICAL ENGINEERING”, ISSN 1583 - 0691, CNCSIS "Clasa B+", Issue
#1. The responsibility for the content of each paper is solely upon the authors. Accordingly, neither the University of Oradea, nor their officers, members of the editorial board, are responsible for the accuracy or authenticity of any published paper.
INDEX OF PAPERS AND AUTHORS
C Alexandru
Transilvania University of Braşov, Romania
A STUDY ON THE EFFECT OF THE DEFORMATIONS IN FLEXIBLOCKS ON THE BEHAVIOR OF THE VEHICLE AXLE SUSPENSION MECHANISMS... 1 C Alexandru
Transilvania University of Braşov, Romania
DESIGN SENSITIVITY ANALYSIS IN THE KINEMATICS OF THE 4SS-AXLE GUIDING MECHANISM WITH
PANHARD BAR ... 7 S Alaci, F-C Ciornei, C-C Suciu and I-C Romanu
„Stefan cel Mare” University of Suceava, Suceava, Romania
THE EFFECT OF STRUCTURAL ASPECT FOR PLANAR SYSTEMS WITH 2DOF UPON THE STABILITY OF
MOTION... 12 Gy Korsoveczki, G Husi
University of Debrecen, Debrecen, Hungary
THE MODEL OF A PERMANENT MAGNET DC MOTOR IN TIME DOMAIN AND FREQUENCY DOMAIN BASED ON BOND GRAPH MODELLING AND DESIGN OF POSITION CONTROL USING PID CONTROLLER... 19 F S Blaga, A Pop, V Hule and C I Indre
University of Oradea, Oradea, Romania
THE EFFICIENCY OF MODELING AND SIMULATION OF MANUFACTURING SYSTEMS USING PETRI NETS... 24 C C Gavrila and M T Lates
“Transilvania” University of Brasov, Brasov, Romania
3D MODELLING AND FEM ANALYSIS ON METAL COIN EDGE PUNCHING ERROR... 51 N. Salem
Zarqa University, Jordan
OIL CONDITION MONITORING AND PREDICTING ACTIONS USING AN ARTIFICIAL INTELLIGENCE TECHNIQUE:
PRINCIPAL COMPONENTS ANALYSIS ALGORITHM... 61 M Ratiu, M A Prichici, D M Anton and D C Negrau
University of Oradea, Oradea, Romania
COMPRESSION TESTING OF SAMPLES PRINTED ON DELTA AND CARTESIAN 3D PRINTER... 70 A M Prada, F S Blaga and M Ursu
Universitatea din Oradea, Oradea, Romania
CHOICE OF OFFERS BY MEANS OF MULTI-CRITERIA DECISION SYSTEMS, USING FUZZY LOGIC. CASE STUDY... 74 M F Popa, D M S Capătă and N Burnete
Technical University of Cluj-Napoca, Cluj Napoca, Romania
ANALYSIS OF DEFECTS IN THE BRAKING SYSTEMS OF COMMERCIAL VEHICLES... 89 R L Păcurariu1,2 , E S Lakatos1,2, L M Nan1,2, L Bacali 1,2 and D Seitoar2
1Technical University of Cluj-Napoca, Cluj-Napoca, Romania
2Institute for Research in Circular Economy and Environment “Ernest Lupan”, Cluj-Napoca, Romania
AN ANALYSIS OF EUROPEAN UNION’S CIRCULAR ECONOMY INDICATORS WITH FOCUS ON MATERIALS:
IMPLICATIONS FOR THE MANUFACTURING INDUSTRY... 96 G Grebenişan1, N Salem2, S Bogdan1, D C Negrău1
1University of Oradea, Romania
2Zarqa University, Jordan
OIL CONDITION MONITORING, AN AI APPLICATION STUDY USING THE CLASSIFICATION LEARNER TECHNICS... 105
LI Culda, ES Muncut (Wisznovszky), GM Erdodi University “Aurel Vlaicu” of Arad, România
TRANSFORMING 3D PRINTED CNC PROTOTYPE INTO A PLOTTER... 114 D M S Capătă, M F Popa, T Dan, N Burnete
Technical University of Cluj Napoca, Romania
EXPERIMENTAL STUDIES REGARDING THE THERMAL OPERATING REGIME FOR SCR CATALYSTS OF
HEAVY-DUTY COMMERCIAL VEHICLES... 121 C F Baban1, M Baban1, I S Popi2
1University of Oradea, Oradea, Romania
2S.C. Top Metal Factory SRL Oradea, Romania
Quality improvement of the primary cable of the handbrake lever: a QFD approach... 126 M B Tudose and S Avasilcai
Gheorghe Asachi Technical University, Iași, Romania
FINANCIAL PERFORMANCE MANAGEMENT AND ECONOMIC CYCLE VARIATIONS. EVIDENCE FOR TEXTILE
INDUSTRY... ... 131 O Popa, C. Mihele, C Făgărășan and A. Pîslă
Technical University of Cluj-Napoca, Cluj-Napoca, Romania
LEADERSHIP APPROACH TOWARDS AGILE, WATERFALL AND ITERATIVE IMPLEMENTATION OF THE SOFTWARE DEVELOPMENT PRODUCTS... 141 C Cristea, M Cristea, F M Șerban, C Făgărășan and C E Stoenoiu
Technical University of Cluj-Napoca, Cluj-Napoca, Romania
PRODUCTIVITY ASSESSMENT OF THE ROMANIAN CONSTRUCTION INDUSTRY USING MALMQUIST
PRODUCTIVITY INDEX... 151 A Vîlcu, I Herghiligiu, I Verzea, M Pîslaru
”Gheorghe Asachi” Technical University of Iasi, Romania
INSULAR GENETIC ALGORITHM FOR OPERATIONAL MANAGEMENT... 158 F-C Ciornei, S Alaci, P Bulai and I-C Romanu
„Stefan cel Mare” University of Suceava, Suceava, Romania
A COMPARATIVE EXPERIMENTAL STUDY FOR DRY AND WET COLLISIONS... 166 A I Radu
University “TRANSILVANIA” of Brasov, Brasov, Romania
ANALYSIS OF DRIVER’S SEAT RIGIDITY IN THE CASE OF REAR-END COLLISIONS USING A VIRTUAL
MULTIBODY MODEL... 174 P D Tocuț, I Stanasel, F T Avram
University of Oradea, Oradea, Romania
STUDY OF THE CLAMPING FORCE IN A PNEUMATIC WEDGE DEVICE... 181 D.I. Țarcă, R.V. Ghincu, D. Crăciun
University of Oradea, Oradea, Romania
DESIGNING A VACUUM CHAMBER AND SUBSTRATE POSITIONING SYSTEM FOR MAGNETRON SPUTTERING
DEPOSITION APPLICATIONS... 186 R. V. Ghincu, D.I. Țarcă, O. A. Moldovan
University of Oradea, Oradea, Romania
DEVELOPMENT OF A DATA ACQUISITION SYSTEM FOR A VACUUM THIN FILM DEPOSITION EQUIPMENT... 191 C Fagarasan, O Popa, A Pisla and C Cristea
Technical University of Cluj-Napoca, Cluj-Napoca, Romania
AGILE, WATERFALL AND ITERATIVE APPROACH IN INFORMATION TECHNOLOGY PROJECTS...197
M Ratiu, D M Anton and D C Negrau University of Oradea, Oradea, Romania
EXPERIMENTAL STUDY ON THE SETTINGS OF DELTA AND CARTESIAN 3D PRINTERS FOR SAMPLES PRINTING... 207 NV Burnete, F Mariasiu, D Moldovanu, N Burnete, D Capata, B Jurchis
Technical University of Cluj-Napoca, Cluj-Napoca, Romania
PARAMETRIC STUDY OF AIR-COOLED TEG HEAT EXCHANGER DESIGN FOR WASTE HEAT RECOVERY
IN HEAVY-DUTY VEHICLE... 212 D C Negrau, G Grebenișan, T Vesselenyi, D M Anton, C I Indre
University of Oradea, Oradea, Romania
MODELING AND BUILDING A 3D PRINT HEAD... 218 D C Negrau, G Grebenișan, M Ratiu, D M Anton
University of Oradea, Oradea, Romania
CONTROL SYSTEM FOR FDA DEPOSITION USING A CNC MILLING MACHINE... ... 224 L Andrei1, V Chindea2 and D L Băldean2
1Infectious Disease Hospital of Cluj-Napoca, Cluj-Napoca, Romania
2Technical University of Cluj--Napoca, Cluj-Napoca, Romania
RESEARCHING DIESEL CAR CO2 EMISSIONS IN COLD AND WARM-UP TRANSIENT TEST CYCLE... 230 L Andrei1, V Chindea2 and D L Băldean2
1Infectious Disease Hospital of Cluj-Napoca, Cluj-Napoca, Romania
2Technical University of Cluj--Napoca, Cluj-Napoca, Romania
RESEARCHING PARTICULATE MATTER CHARACTERISTICS FOR MITIGATING HEALTH RISKS GENERATED
BY ROAD VEHICLES... 235 F M Serban, C Cristea, C E Stoenoiu
Technical University of Cluj-Napoca, Cluj-Napoca, Romania
MARKET CONCENTRATION IN ROMANIAN INSURANCE INDUSTRY... 240 F B Scurt, T Vesselenyi, R C Țarcă, H Beleș, G Dragomir
University of Oradea, Oradea, Romania
AUTONOMOUS VEHICLES: CLASSIFICATION, TECHNOLOGY AND EVOLUTION... 247 C E Stoenoiu, F M Serban, C Cristea
Technical University of Cluj-Napoca, Cluj-Napoca, Romania
THE VARIATION OF THE TURNOVER AND ITS EFFECT ON THE EXPLOITATION RISK... 255 G G L Mitrea- Curpanaru
Gheorghe Asachi Technical University, Iași, Romania
MODELS FOR MEASURING THE PERFORMANCE OF AN ORGANIZATION... 260 I C Gherghea, C Bungau, C I Indre, and D C Negrau
University of Oradea, Oradea, Romania
ENHANCING PRODUCTIVITY OF CNC MACHINES BY TOTAL PRODUCTIVE MAINTENANCE (TPM)
IMPLEMENTATION. A CASE STUDY... 269 I G Radu, E Vultur and C N Drugă
University of Brasov, Brasov, Romania
FINITE ELEMENT ANALYSIS OF THE MAIN STRESSES IN GAMMA ROD OSTEOSYNTHESIS... 279 A Saadah, G Husi
University of Debrecen, Debrecen, Hungary
KUKA KR5 ARC WELDING INDUSTRIAL MANIPULATOR WORKSPACE MODELLING BASED ON KINEMATICS STUDY... 285 M Török, T I Erdei, Sz Tóth, G Husi
University of Debrecen, Debrecen, Hungary
DESIGNING AND BUILDING A REMOTE-CONTROLLED 3D PRINTED PROTOTYPE ROBOT ARM IMPLANT... 292
G G L Mitrea- Curpanaru
Gheorghe Asachi Technical University, Iași, Romania
PERFORMANCE MANAGEMENT – A STRATEGIC AND INTEGRATED APPROACH TO ENSURING THE SUCCESS OF ORGANIZATIONS... 296 B Sovilj1, I Sovilj-Nikić1, S Sovilj-Nikić2, V Blanuša3
1University of Novi Sad, Novi Sad, Serbia
2Iritel a.d. Beograd, Belgrade, Serbia
3High Technical School of Professional Studies, Novi Sad, Serbia
RESEARCH OF THE INFLUENCE OF AXIAL DISPLACEMENT ON THE WEAR OF HOB MILLING TOOLS... 301 R Veres, S Ilea, B C Feier, G Veres, F Corb
University of Oradea, Oradea, Romania
APPLICATIONS OF NONCONVENTIONAL TECHNOLOGIES IN THE CURRENT GLOBAL PANDEMIC... 311 A Rohan and M Baritz
University Transilvania from Brasov, Brasov, Romania
RESEARCH ON THE BIOMECHANICAL BEHAVIOR OF DENTAL IMPLANTS... 316 D M Anton, S E Szabo, C A Mociar and A F Iuhas
University of Oradea, Oradea, Romania
LINE FOLLOWER MOBILE ROBOTS, PROTOTYPES OF ROBOTS AND FUNCTIONAL OF MOBILE ROBOTS... 324 M G Statache, E Vultur and C N Drugă
“Transilvania” University of Brasov, Brasov, Romania
BUILDING A CONTINUOUS PASSIVE MOBILIZATION MACHINE (CPM) FOR THE INFERIOR LIMB RECUPERATION... 329 D M Anton, R C Milas, A M Stiube and L E Stiube
University of Oradea, Oradea, Romania
AREA MAPPING FOR AUTONOMOUS DISINFECTION ROBOT... 337 S Ilea and R Veres
University of Oradea, Oradea, Romania
STRIPPING OF PLATED PLASTIC MATERIALS... 341
A study on the effect of the deformations in flexiblocks on the behavior of the vehicle axle suspension mechanisms
C Alexandru
Transilvania University of Braşov, Romania E-mail: [email protected]
Abstract. This work deals with a study on the effect of the deformations in flexiblocks (bushings) on the behavior of the vehicle axle suspension mechanisms. The mechanism in study is a representatiove one for the structural group/class of mono-mobile axle suspension systems, namely the guiding mechanism by five points - on five spheres (so called 5SS). In a previous work, it was established that the rigid joint model for this class of mechanisms does not provide results close to the real flexible bushing model, reason for which the research in this paper is directed towards the identification of a theoretical model of joint which, at a lower complexity than that of the real model, would still provide viable results, in terms of axle guidance accuracy in the spatial movement relative to car body. In this regard, static and kinematic models were developed and analyzed by using the MBS (Multi-Body Systems) software solution ADAMS.
1. Introduction
The suspension mechanisms of vehicles’ rear (beam) axles perform two functions, namely the function of guiding the beam axle in movement relative to car body, respectively the function of transmitting the contact forces from wheels. Regarding the guiding function, this is defined by the spatial movement (position and orientation) of the axle, which is established by the positional analysis of the guiding mechanism. The guidance function is studied on models in which the car body is considered fixed (namely, kinematic and static models). In the case of the dynamic model, where the car body is a mobile part, the axle is forced, by wheels, to follow the road profile (more or less accurately, depending on the elasticity of the tires).
Therefore, the analysis of the guidance function can be performed on the kinematic model, considering the loading of the mechanism through vertical positions imposed on the wheels, respectively the static model, where the external loading is performed by forces applied to the wheels.
Obviously, the determination of the guidance function on the static model is closer to reality, because the elastic elements of the suspension are also taken into account.
The (quasi)static analysis of the axle guiding mechanisms can be performed on the basis of two models, in correlation with the modelling of the connections between the guiding bars and the adjacent parts (car body and axle), which are actually made by flexiblocks/bushings (compliant joints that support linear and angular deformations), as follows: rigid models, in which the bushings are modelled by spherical couplings/joints, with 3 degrees of freedom (DOF) corresponding to the rotational movements (these models are also the basis of the structural synthesis and kinematic study of the axle guiding mechanisms [1]); flexible/compliant models, with 6 degrees of freedom restricted elastically, which considers both linear and angular movements (deformations).
Opting for one or the other model is, first of all, a choice dictated by the available computational tool. If the rigid models, due to the low degree of mobility they have (DOM = 1 or DOM = 2), are suitable for analysis by specialized methods and in-house programs, the elastic models, with a large number of degrees of mobility, require the use of MBS (Multi-Body Systems) commercial software solutions, which are actually used in a large variety of applications [2-6]. Approaching the compliant bushing model through classical formalisms, in addition to being extremely difficult, involves the risk of obtaining significant computation errors, a consequence of the particularly vast mathematical apparatus that should be managed. For the positional analysis of the axle guiding mechanisms, there is frequently considered the simplifying assumption that the rigid models approximate quite well the behaviour of the real models with compliant bushings. Obviously, the following problem/question can be raised: what is the deviation/error generated by the symbolic representation of bushing by spherical coupling regarding the guiding function of the mechanism? It is interesting, in fact, for which category of axle guiding mechanisms the rigid model hypothesis can be accepted, regarding the accuracy of the results provided for the axle movement, compared to the flexible model.
In the literature such a problem is addressed mainly for front wheel guiding mechanisms (four-bar, McPherson and multi-link configurations) [7-13].The conclusions of these works converge on the idea that, although there are some differences between the behaviour of rigid and flexible joint models in terms of guiding accuracy, the rigid model is a viable alternative for the positional analysis of the wheel guiding mechanisms. These approaches are especially kinematic, as it is not possible to make a quantitative assessment of the dimensional deviations in bushing (i.e. the deformations as a function of forces), much less a correlation between the deformations for all bushings in the guiding mechanism.
In a previous work by the author [14], a comparative analysis between rigid and flexible joint models for the two main categories of axle guiding mechanisms (in terms of the number of degrees of mobility, DOM = 1 and DOM = 2) was carried out. A similar study was performed in [15], by considering other structural variants of axle guiding mechanisms and respectively numerical values of the specific geometric parameters. The conclusions resulting from these papers can be summarized as follows: the hypothesis of the rigid coupling model is valid only in the case of bi-mobile (DOM = 2) axle guidance mechanisms, while in the case of mono-mobile (DOM = 1) mechanisms, it is necessary to consider models of couplings/joints closer to reality (i.e. compliant bushings), which must allow linear deformations.
Starting from the previously mentioned research, the present work deals with a more detailed study on the influence of the joints’ deformations on the guidance function of the mono-mobile axle suspension mechanisms (for which the rigid joint model hypothesis is invalid), in order to identify the simplest models that ensure an appropriate behavior, close to that of the real model. The study is conducted by developing and testing MBS models made by using the virtual prototyping environment ADAMS of MSC.Software.
2. The MBS model of the suspension system
To study the influence of elasticities in bushings on the guiding function of the axle suspension mechanisms, the static model of the mechanism was initially developed, considering that the car body is fixed and the external loading of the guiding mechanism is realized by forces applied to the wheels.
The suspension system in study is based on an axle guiding mechanism by five points - on five spheres (5SS), whose equivalent structural model has a single degree of mobility, corresponding to the vertical movement of one of the wheels (left or right).
The static model of the suspension system based on 5SS axle guiding mechanism, which is shown in Figure 1, contains the bodies/parts in the suspension system (axle & wheels, guiding links/bars, car body), the connections between bodies (which in the real case are made by bushings), the spring &
damper assemblies (which are arranged between the axle and the car body), the compression &
extension bumper stops (which are arranged inside the shock absorbers / dampers, thus limiting the relative movement between their pistons and cylinders).
PROCEEDINGS of the ANNUAL SESSION OF SCIENTIFIC PAPERS
“IMT ORADEA - 2021”
27th – 28th May, Oradea, Romania
Figure 1. The static model of the 5SS suspension system.
For this type of axle suspension systems, there were modeled and analyzed rigid (where the connections of the guiding bars to axle and car body are made by spherical couplings/joints) and flexible (couplings by bushings) models. By taking into account the bushings’ elasticity, all the parameters that describe the guiding function of the mechanism (i.e. the spatial movement of the axle) will obviously be influenced. The guiding function of the mechanism is reported/defined by the roll angle and the vertical coordinate of the axle’s center, because these are the parameters that actually describe the global character of the spatial movement of the axle relative to the car body.
3. Results and conclusions
The comparative analysis of the rigid and flexible models was carried out for several functional situations determined by the nature and values of the wheel contact forces. In this work, the test model considered corresponds to the situation where the vehicle is suspended, with the car body locked, and it is operated on the left wheel by a variable vertical force (Fz), in the range of values Fz[0, 800] daN.
The results of these analyzes led to conclusions similar to those of previous researches [14, 15]
confirming the invalidity of the rigid joint model in the case of the mono-mobile suspension system.
Further, it was started from the premise that it might be possible that for certain types of bushings, where due to high stiffnesses there are practically no linear deformations, the behavior of the flexible model is close to that of the rigid model. From the point of view of the linear (radial and axial) deformations that occur in bushings, the flexible model for the 5SS guiding mechanism led to the results shown in Figure 1, for the connections on axle of the lower longitudinal bars (1 - left, 2 - right), the upper longitudinal bars (3 - left, 4 - right), and the transversal bar (5). Similar deformations were obtained for the bushings through which the guiding bars are connected to car body. According to these diagrams, the linear deformations in bushings are very small, of the order of tenths-hundredths of a millimeter. So, there are enough small deformations in bushings for the law of motion through the flexible model to be very different from that of the rigid model. The question then arises: what should a bushing look like from a geometric point of view in which not even such small deformations appear?
According to the relations in [16], the linear rigidities in bushings are mainly influenced by the thickness of the rubber intermediate layer. High rigidities, which determine even the proximity of the condition of linear non-deformability of the elastic element (a necessary condition for the validation of the rigid model), involve an extremely small thickness of the rubber between the outer and inner metallic rings of the bushing. However, such a constructive solution would have effects on the necessary angular mobility, on the sound and anti-vibration insulation capacity of the bushing.
Moreover, the symbolic representation bushing spherical coupling, which is the base of the structural synthesis [1], would no longer be valid either, because in fact the spherical rotations could no longer occur. The analysis of a large number of constructive solutions showed that no constructive variant of bushing, at least the known ones, does not provide the necessary conditions for the validation of the rigid model for the mono-mobile axle guiding mechanisms.
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a. b.
c.
Figure 2. The radial and axial deformations in bushings for the 5SS axle guiding mechanism.
On the other hand, the analysis of the guidance function of the axle guiding mechanisms can also be performed on the kinematic model. For this, the elastic and damping elements (springs, dampers, bump stops) are removed from the suspension system, and the input (actuating) is made through vertical positions imposed on the left wheel (ZGs [-80, 80] mm, from the static/rest position when ZGs = -77 mm). Because the kinematic model does not contain the elastic suspension elements, which would generate internal reaction forces, the linear deformations in bushings are much smaller than in the static model. Even under these conditions, according to the diagrams in Figure 2, the rigid model for the guiding mechanisms with DOM = 1 behaves differently from the flexible model, referring to the law of motion by the guiding mechanism, which strengthens the conclusion regarding the invalidation of the rigid joint model for this structural group of mechanisms.
Since in bushings there are, theoretically, linear deformations in all directions (axial and radial), the influence of each linear elasticity characteristic on the guiding function of the axle suspension mechanism will be further identified. In this regard, the bushings were modeled by connections with 4 degrees of freedom (DOF), considering in addition to the spherical rotations one of the linear deformations (along X, Y, Z, by case), the 4-DOF coupling models being defined by compound sphere-translation joints. Such an approach allows the identification of the simplest model for bushing, which is able to ensure a behavior close to that of the real 6-DOF bushing. The results obtained by analyzing the 4-DOF joint models for the parameters of interest that define the spatial position of the axle are shown in Figures 3 and 4 (a - static analysis, b - kinematic analysis).
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a. b.
Figure 3. The results of the comparative analysis for the kinematic models.
a. b.
Figure 4. The roll angle of the axle for the 4-DOF joint models.
a. b.
Figure 5. The vertical displacement of the axle’s centre for the 4-DOF joint models.
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According to these diagrams, the 4-DOF joint model "X", which allows radial linear deformation along X, leads to results close to the real compliant model (with 6-DOF bushings), while the 4-DOF joint
"Y", which allows axial linear deformation along Y, is very close to the rigid model (with 3-DOF spherical joints). Therefore, the simplest viable model for rear axle bushings is the composed sphere- translation joint on the longitudinal axis, with 4 elastically restricted degrees of freedom. Consequently, the main reaction forces in bushings are the longitudinal ones, while and lateral reaction forces (along the bushing’s axis) are practically insignificant. This information is also important for the proper sizing of the bushings in the 5SS axle guiding mechanism.
References
[1] Alexandru C 2009 The kinematic optimization of the multi-link suspension mechanism used for rear axle of the motor vehicle Proceedings of the Romanian Academy - A 10(3) pp 244-253 [2] Alexandru C and Comşiț M 2007 Virtual prototyping of the solar tracking systems Renewable
Energy and Power Quality Journal 1(5) pp 105-110
[3] Alexandru C and Pozna C 2008 Virtual prototype of a dual-axis tracking system used for photovoltaic panels Proceedings of the IEEE International Symposium on Industrial Electronics - ISIE pp 1598-1603
[4] Alexandru C and Pozna C 2009 Dynamic modeling and control of the windshield wiper mechanisms WSEAS Transactions on Systems 8(7) pp 825–834
[5] Alexandru P, Macaveiu D and Alexandru C 2012 A gear with translational wheel for a variable transmission ratio and applications to steering box Mechanism and Machine Theory 52, pp 267-276
[6] Geonea ID, Alexandru C, Margine A and Ungureanu A 2013 Design and simulation of a single DOF human-like leg mechanism Applied Mechanics and Materials 332 pp 491-496
[7] Attia HA 2003 Kinematic analysis of the multi-link five-point suspension system in point coordinates Journal of Mechanical Science and Technology 17(8) pp 1133-1139
[8] Balike KP, Rakheja S and Stiharu I 2008 Kinematic analysis and parameter sensitivity to hard points of five-link rear suspension mechanism of passenger car Proceedings of the Design Engineering Technical Conference pp 755-764
[9] Hiller M and Woernle C 1985 Kinematical analysis of a five point wheel suspension ATZ 87 pp 59-64 [10] Knapczyk J and Maniowski M 2002 Selected effects of bushings characteristics on five-link
suspension elastokinematics Mobility and Vehicle Mechanics 3(2) pp 107-121
[11] Knapczyk J and Maniowski M 2006 Elastokinematic modeling and study of five-rod suspension with subframe Mechanism and Machine Theory 41(9) pp 1031-1047
[12] Simionescu PA and Beale D 2002 Synthesis and analysis of the five-link rear suspension system used in automobile Mechanism and Machine Theory 37(9) pp 815-832
[13] Tică M, Dobre G and Mateescu V 2014 Influence of compliance for an elastokinematic model of a proposed rear suspension International Journal of Automotive Technology 15(6) pp 885-891 [14] Țoțu V and Alexandru C (2013) Study concerning the effect of the bushings’ deformability on the
static behavior of the rear axle guiding linkages Applied Mechanics and Materials 245 pp 132- 137
[15] Țoțu V (2014) A comparative analysis between the rigid and compliant joint models for the guiding system of the cars axles Annals of the Oradea University, Fascicle of Management and Technological Engineering XXIII pp 131-134
[16] Alexandru C (2019) Method for the quasi-static analysis of beam axle suspension systems used for road vehicles Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 233(7) pp 1818-1833
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Design sensitivity analysis in the kinematics of the 4SS-axle guiding mechanism with Panhard bar
C Alexandru
Transilvania University of Braşov, Romania E-mail: [email protected]
Abstract. This work deals with the design sensitivity analysis in the kinematics of the vehicle rear axle guided by four points - on four spheres (so called 4SS), with Panhard bar. The geometric parameters that define the kinematic scheme of the axle guiding mechanism are considered in this study, which aims to identify the influence of these parameters on the specific kinematic functions. The purpose is to achieve a separation of geometric parameters depending on their influences on the objective functions to be optimized from a kinematic point of view, so as to simplify the process of effective optimization, by taking into account (as design variables) only the main parameters.
1. Introduction
Relative to car body, the vehicle wheels can be guided independently - by means of a guiding mechanism for each wheel, or dependently - by a guiding mechanism of the rigid axle. The first solution is frequently used for the front & rear wheels of the passenger cars, while the second one is mainly used for the rear axles of larger gauge cars. For the rear axle guidance, spatial mechanisms formed by a number of binary links (bars) are interposed between axle and car body. The bars’
connections to axle and car body are made by using compliant joints (i.e. bushings) [1-3]. Usually, for the kinematic study (where the car body is fixed), the bushings are modelled as spherical joints, the corresponding models having a low number of degree of mobility (DOM=1 or DOM=2) [4, 5].
The guidance of the rear axle is made by driving a number of its points on suitably chosen surfaces and curves (sphere, circle or coupler curve). By the guidance of four axle points on four spheres with centres on car body (4SS), bi-mobile (DOM=2) guiding mechanisms are obtained. According to the study carried out in [6], for this structural group/class of axle guiding mechanisms, the spherical joint model assumption can be accepted because the behaviour of these mechanisms is closer to that of the real model with compliant joints (bushings). The structural variants of 4SS axle guiding mechanism are presented in Figure 1. For the scheme shown in Figure 1.a, all the guiding bars of the mechanism are arranged in the longitudinal direction, while for the mechanism shown in Figure 1.b one of the bars (4) is arranged transversely (so called Panhard bar), which ensures a better takeover of the transversal forces from the wheels, the latest solution being addressed in this work.
The kinematic optimization of the axle guiding mechanisms is a rather complex problem, considering both the variety of kinematic parameters that need to be optimized and the multitude of geometric parameters that define the mechanism. For this reason, it would be preferable to achieve a separation of geometric parameters depending on the influence they have on the objective functions to be optimized. Such a study will allow that in the effective optimization only the main parameters, which significantly influence the behaviour of the axle guiding mechanism, to be taken into account.
a. b.
Figure 1. Configurations of 4SS-axle guiding mechanism.
Under these terms, the present work deals with the design sensitivity analysis in the kinematics of the 4SS axle guiding mechanism with Panhard bar. The kinematic analysis is carried out through the characteristic points’s method, which was depicted in [7] as part of a more detailed numerical algorithm for determining the equilibrium position of the axle suspension system. The kinematic analysis method was algorithmized using the C ++ programming environment.
2. Geometric and kinematic parameters
Its position and orientation, in relation to the global reference frame OXYZ that is fixed in car body, describe the spatial movement of the rear axle. The global coordinates XP0, ZP0 determine the initial position of the axle in OXYZ reference frame. Due to the symmetry of the guiding linkage relative to the longitudinal plane of the car, there are the following relationships between the geometrical parameters: XM01= XM02, |YM01|= YM02, ZM01= ZM02, YM03= 0, XM1(P)= XM2(P), |YM1(P)|=YM2(P), ZM1(P)= ZM2(P), YM3(P)= 0. Therefore, the geometrical model of the 4SS guiding mechanisms is defined by 19 geometrical parameters, as follows (Figure 2): XM01,2, YM01,2, ZM01,2, XM03, ZM03, XM04, YM04, ZM04, XM1,2(P), YM1,2(P), ZM1,2(P), XM3(P), ZM3(P), XM4(P), M4(P), ZM4(P), l1,2, l3, l4.
Figure 2. The geometrical model of the 4SS guiding mechanism with Panhard bar.
Relative to the car body, the rear axle must have the possibility of vertical displacement and rotation around the longitudinal axis of the vehicle (X). When the vehicle is moving, besides the above mentioned necessary motions, secondary undesirable motions can occur, as follows: displacements of the axle centre P along the longitudinal and transversal axes, axle pivoting movement, and axle rotation around its own axis). The minimization of the undesirable motions represents the goal of the kinematic optimization [4, 5].
The spatial position & orientation of the axle are defined by the following kinematic parameters:
the linear displacements of the axle’s centre:
XP = XP - XP0, YP = YP - YP0, ZP = ZP - ZP0; (1)
the rotations of the rear axle in the global coordinate system’s plans:
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Gs Gd
Gs Gd
xy Y Y
X arctgX
,
Gs Gd
Gs Gd
yz Y Y
Z arctgZ
,
) P ( G
P P G G
xz X
Z Z Z arcsinZ
0
0
. (2)
The global coordinates of the characteristic points (Gs, Gd, G) and of the axle’s center (P) are determined in accordance with the numerical method depicted in [**]. The vertical coordinates of the wheels’ centers / axle’s ends (ZGs, ZGd) are independent kinematic parameters.
Usually, the coordinates of the guiding points on axle (X, Y, Z)Mi(P) are established by constructive criteria. Afterwards, it will be analyzed the influence of the other parameters on the kinematic behavior of the guiding mechanisms (in fact, on the undesirable motions). The spatial configuration of the mechanism is defined by taking into account the disposing of the guiding bars, in accordance with the schemes shown in Figure 3 (the longitudinal bars - 1, 2, 3) and Figure 4 (the transversal bar - 4).
Figure 3. The positioning of the longitudinal bars.
Figure 4. The positioning of the transversal bar.
The unitary position vectors u1-3 (for the longitudinal bars) and u4 (for the transversal bar) are defined by the following equations:
, sin
cos cos
cos sin
u u u u , ...
i , sin
cos sin
cos cos
u u u u
' x
' x z
' x z
z y x
' iy
' iy iz
' iy iz
z i
y i
x i
i
4 4 4
4 4
4 4 4
3 4
1
(3)
where:
'iy = arctg(tgiy cosiz), i = 1...3, '4x = arctg(tg4x cos4z). (4) The global coordinates of points/joints on car body (XM0, YM0, ZM0) will be:
XM0i = XMi – li cosiz cos'iy, YM0i = YMi – li siniz cos'iy, ZM0i = ZMi + li sin'iy, i = 1…3, (5) XM04 = XM4 – l4 sin4z cos'4x, YM04 = YM4 + l4 cos4z cos'4x, ZM04 = ZM4 + l4 sin'4x, where the global coordinates of the guiding points on axle (XM, YM, ZM) correspond to the known initial position of the mechanism.
By noting k = l3/l1(2) (the ratio between the lengths of the upper and lower longitudinal bars), there are obtained the following geometrical parameters (whose influence on the kinematic behaviour of the axle guiding mechanism will be analysed): k, l1, l4, 1y, 1z, 3y, 4x, 4z.
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3. Results and conclusions
The results of the design sensitivity study are presented in the diagrams shown in Figures 5-9, corresponding to the functional case ZGs=ZGd, considering the vertical travel of the wheels in the interval Z[-80, 80]mm relative to the initial position (vehicle in rest). The influence of parameters l4, 1z and 4z on the undesirable motions XP, YP, xy and xz is insignificant; on the other hand, all geometrical parameters have a small influence on the axle pivoting motion xy (for this reason, the corresponding variation diagrams are no longer presented). The following values of the geometrical parameters were considered: 1y = {-6, 0, +6}, 3y = {-6, 0, +6}, l1 = {400, 500, 600} mm, k = {0.4, 0.6, 0.8}, 4x = {-6, 0, +6}.
Figure 5. The influence of parameter 1y.
Figure 6. The influence of parameter 3y.
Figure 7. The influence of parameter l1.
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Figure 8. The influence of parameter k.
Figure 9. The influence of parameter 4x.
According to these results, a separation of the geometrical parameters was obtained, as follows:
main parameters, with great influence on kinematic behaviour of the axle guiding mechanism: k, 1y,
3y; secondary parameters, with small influence: l1(2), l4, 1(2)z, 4x, 4z. There can be considered that the undesirable motions are given by the following functions, in pairs [kinematic parameters;
(geometric parameters)]: XP F1[ZGs,d; (k, 1y, 3y)], YP F2[ZGs,d; (4x)], xy F3[ZGs,d], xz F4[ZGs,d; (k, 1y, 3y)]. Afterwards, the kinematic synthesis of the axle guiding mechanism can be carried out on the basis of the main geometrical parameters, by neglecting the secondary parameters, which simplifies the optimal design process, with beneficial effects on the allotted time.
References
[1] Knapczyk J and Maniowski M 2002 Selected effects of bushings characteristics on five-link suspension elastokinematics Mobility and Vehicle Mechanics 3(2) pp 107-121
[2] Knapczyk J and Maniowski M 2006 Elastokinematic modeling and study of five-rod suspension with subframe Mechanism and Machine Theory 41(9) pp 1031-1047
[3] Tică M, Dobre G and Mateescu V 2014 Influence of compliance for an elastokinematic model of a proposed rear suspension International Journal of Automotive Technology 15(6) pp 885-891 [4] Alexandru C 2009 The kinematic optimization of the multi-link suspension mechanism used for
rear axle of the motor vehicle Proceedings of the Romanian Academy - A 10(3) pp 244-253 [5] Simionescu PA and Beale D 2002 Synthesis and analysis of the five-link rear suspension system
used in automobile Mechanism and Machine Theory 37(9) pp 815-832
[6] Țoțu V and Alexandru C (2013) Study concerning the effect of the bushings’ deformability on the static behavior of the rear axle guiding linkages Applied Mechanics and Materials 245 pp 132-137 [7] Alexandru C 2019 Method for the quasi-static analysis of beam axle suspension systems used for road vehicles Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 233(7) pp 1818-1833
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The effect of structural aspect for planar systems with 2DOF upon the stability of motion
S Alaci1, F-C Ciornei1, C-C Suciu1 and I-C Romanu1
1Mechanics and Technologies Department, „Stefan cel Mare” University of Suceava, Suceava, Romania
E-mail: [email protected]
Abstract. The paper analyses all planar chains with 2DOF from the point of view of stability of motion. For the rotation-prismatic structural solution first there are obtained the equations of motion and then, the numerical integration procedure is applied. A strong instability of the system can be noticed. The same dynamical system is modelled using dynamical analysis software and the instability is confirmed.
1. Introduction
There are uncommon the cases when a part of a mechanical structure moves unrestrained, without interacting with other parts from its proximity. The majority of the mobile mechanical structures are made of solid elements that come into contact to each other. When two of the components of a system make contact the number of degrees of freedom of the two elements reduces. One of the main classification criteria of the constraints which take place between two parts, named kinematical joints, is the class of the pair, defined as the number of degrees of freedom taken for one of the element of the pairs, when the other element is considered fixed, [1]. Considering the fact that a rigid body has 6 degrees of freedom (DOF) it results that the class of the pair can have values from 1 to 5, the limit case of 0 representing the lack of linkage and the pair of class 6 represents the relative rigidity of the two parts that behave as a single part.
The second especially important notion when analyzing the mobility of a structure is the family, concept that illustrates the number of common restraints imposed to each element of the structure. The monographs of mechanisms theory consider as main categories of mechanisms: spatial mechanisms, in the general significance, spherical mechanisms that have as characteristic the existence of the same fixed point for all the elements of the mechanism and planar mechanisms that have as characteristic the fact that all the elements move parallel to the same plane. Both spherical and planar mechanisms are mechanisms of family 3, [2]. Since a kinematical chain of a certain family cannot contain pairs of a class inferior or equal to the family of the chain, for a planar kinematical chain - that will be the referred next, there is possible the existence of pairs of class 5 - rotation R and prismatic T and also of superior pairs of class 4. The dynamical systems with two degrees of freedom are the simplest systems where the chaos phenomenon can appear.
2. Possible structural solutions for planar kinematical chains with 2 DOF
The effect of chaos for a double physical pendulum, figure 1, is presented in [3] for the situation when the launch position differs substantially from the position of static equilibrium, that corresponds to the vertical orientation of both rods and with the center of mass in the lowest position. To illustrate the
effect of chaos that occurs in the system, two identical pendula were considered, positioned symmetrically with respect to a vertical plane. The motion of the pendula was modeled using dynamical simulation software. It was observed that for a relatively short time the motions of the pendula were identical, but after that, the oscillations of the pendula differed. From experiments performed with the double pendulum, figure 1, it was concluded that the attempt to obtain identical motions from two similar launches, with the same angular amplitudes, 1090o, 2090o, is unsuccessful. The explanation of the phenomenon resides in the strong dependency of the evolution of the system on the initial conditions. But, in spite of the efforts, the precision of obtaining the initial position cannot be superior to the precision of the instruments of measure used in the estimation of the positional parameters of the system.
In figure 2 is presented a system having in structure a prismatic pair and a rotation pair for which Voinea [4] presents the differential equations of motion. In order to appreciate how stable the system is, the manner used for the double pendulum is applied. For that reason, using dynamical simulation software, two identical systems were modeled, positioned in mirror, with the same initial position, figure 4. The angle characterizing the initial position of the rod is 0135o. Unlike the double pendulum from figure 1, where after a few seconds the motions of the pendula are completely different, in this case, the motions of the two systems are rigorously identical, as shown in figure 5 where the variations of velocities of the prismatic bodies were plotted. To validate this affirmation, in figure 6 is presented the dependency of the velocity of prism 2 as a function of the velocity of the prism 1. The straight line plot from figure 6 shows the equality of the two velocities.
Figure 1. Kinematical chain 2 DOF with two rotation pairs
Figure 2. Kinematical chain 2 DOF with one rotation pair
and one prismatic pair
Figure 3. Kinematical chain 2 DOF with higher pair
Figure 4. Mirrored identical systems
1
2
G g m
2 1
v2 v1
g m
G r
C N
T
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Figure 5. Variation of the velocities of the prisms Figure 6. The dependency v2 vs v1 A kinematical chain that contains a higher kinematic pair is presented in figure 3. The system has two degrees of freedom , because when the friction is absent, the fundamental theorems of dynamics provide three scalar equations (the momentum theorem gives two scalar equations and the moment of momentum theorem gives a single equation) which allow for finding three unknowns, here the two positional parameters and and the magnitude of the normal reaction N. But particularly interesting is the case when friction occurs in the higher pair. The magnitude of the friction force T is added besides the three unknowns , and N . To solve the problem, a supplementary equation is required. Two cases can be distinguished, depending on the friction type:
a) when rolling friction is present, the supplementary equation is r and the system has 1 DOF, and in fact it is a cycloidal pendulum with the dynamics presented in [5] for which there is no risk of chaos phenomenon occurrence.
b) when dry friction is present, the fourth equation is given by the relation between the magnitude of friction force and the magnitude of normal reaction, TN according to Coulomb’s law. In this case the system has again 2 DOF.
3. The 2DOF linkage with rotation and prismatic pairs
Next it is presented the effect produced by reciprocate changing the pairs from the kinematical chain from figure 2, as in figure 7, problem proposed by Spiegel [6]. It is considered the particular case when the homogenous rod is jointed to the ground in the center of mass. A sleeve with negligible dimensions slides without friction on the rod.
Figure 7. System with 2 DOF
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