ENGINEERING FOR RURAL DEVELOPMENT Jelgava, MECHANISM MOTION STUDIES WITH COLLISIONS AT SEVERAL POINTS

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1 EGIEERIG FOR RURAL DEVELOPMET Jelgava, MECHAISM MOTIO STUDIES WITH COLLISIOS AT SEVERAL POITS Edgars Kovals 1, Janis Viba 1, Lauris Sals 1, Svelana Sokolova 1, Vialy Krupenin 1 Riga Technical Universiy, Lavia; Russian Academy o Sciences IMASH edgars.kovals@gmail.com, janis.viba@ru.lv, lauris.sals@gmail.com, svelana.sokolova@ru.lv, krupeninser@gmail.com Absrac. The paper analyses he moion o mechanical sysems, which causes simulaneous collisions (shocks) in heir elemens a more han wo conac poins. The main hypohesis, which is used in he heoreical calculaions, is ha he menioned impacs begin and end a he same ime a all impac conac poins. Impac ineracion o conac poins includes normal reacion and dry ricion orce wih correlaions under sopping areas. Descripion o he relaionship impacs a he same ime a wo, hree or more poins is given. For he descripion o he plane moion or mechanical sysems he mass cener moion heorem and he heorem o kineic momenum changes are used. Acquired correlaions are used in he mechanical sysems plane moion analysis o ransiion and saionary moion regimes under a number o collision poins. The modelling wih MahCAD program is carried ou or sysems wih one, wo and hree degrees o reedom, which akes place in collisions in wo and more poins. The inspecion o heory has been perormed by he Working Model D Program or mechanical sysems wih one, wo, hree degrees o reedom wih collisions a hree or more poins. Validaion o sudies is signiican. In addiion he demonsraion resuls or experimenal sudies or vibro drive sysem are given. Accordingly, he vibro engine is creaed by disbalance moors o he pendulum plaorm in plane moion. Impacs are generaed in wo la springs aer roaion abrupion rom undamen. The resuls o he work may be used in calculaions o equipmen in he mechanical posiioning sysem, as well as in he design o new mechanisms or machines, such as vibro conveyors or vibro engines. Keywords: shocks, impacs in mechanical sysem, simulaneous collision modelling. Inroducion Example o collision ineracion in one poin can be described by normal reacion and dry ricion orce ±F (Fig. 1). Graphics o normal reacion and one dry ricion orce (or ull slip F) in ime domain is shown in Fig.. K ±F y v c ω, ω 1, ω C (xc;yc) () F() SI SFI 1 τ SII SFII Fig. 1. Model o collisions in poin K: normal reacion; ±F dry ricion orce wih variable direcion (±) along angen in conac poin K, ω, ω 1, ω angular velciy o body in hree ime momens (iniial, in he middle, a he end) Here F( ) = ( ); Fig.. Graphics o normal reacion and dry ricion orce (or ull slip F) in ime domain: ime inerval o he irs phase o collision; τ ime inerval o he second phase o collision SI = ( ) d; SII = τ ( ) d; where dry ricion coeicien; SI, SII impulses o normal reacion (); SFI = F( ) d; τ SFII = F( ) d, (1) 114

2 EGIEERIG FOR RURAL DEVELOPMET Jelgava, SFI, SFII impulses o dry ricion orce F() in he irs and second impac phase. According o he heory o body plane collisions wih obsacle in one poin, here or collisions a several poins he exisence o one o seven dieren impac ineracion cases can be checked as ollows [1-4]. 1. Full slips depending on he iniial conac poin K suicien angenial velociy componen.. Full slip in one direcion depending o special body geomery and wih special moion iniial condiion (when he conac poin K has zero angenial velociy componen). 3. Parial slip wih end a he irs phase o impac ( ). 4. Parial slip wih end a he second phase o impac ( τ). 5. Full slip in wo direcions wih dry ricion orce reverse in he irs phase o impac ( ). 6. Full slip in wo direcions wih dry ricion orce reverse in he second phase o impac ( τ). 7. o slip in he conac poin. Four o hese seven ineracion cases are shown in Fig. -4. These seven dieren impac ineracion cases [5] can be used or he invesigaion o collisions a several poins [4]. SI 1 SI SI SII 1 SII SII SFI 1 F 1 1 SFI τ SFII F SFI 1 τ SFII 1 SFII τ1 Fig. 3. Graphic o normal reacion and dry ricion orce (ull slip in wo direcions) in a case 5: SFI 1 dry ricion orce impulse beore reverse in he irs phase o impac (1 ); SFI negaive dry ricion orce impulse aer reverse ill ; SI 1, SI pars o normal reacion impulse in he irs phase o impac Fig. 4. Graphic o normal reacion and dry ricion orce (ull slip in wo direcions) in a case 6: SFII 1 dry ricion orce impulse beore reverse in he second phase o impac ( τ1 τ); SFII negaive dry ricion orce impulse aer reverse ill τ; SII 1, SII pars o normal reacion impulse in he second phase o impac Simulaneous collisions o roaing body in wo poins Collisions in wo poins in he body wih one degree o reedom will be observed. Assume ha he roaing body collides wih a rigid obsacle in he poin K (Fig. 5). According o he collision heory he weigh and oher large disance ineracions can be negleced [1-6]. Then impac orces will be: in he axis O orces wih componens XO, YO; reacions in he conac poin K like normal reacion and dry ricion orce F. The idea o calculaion includes a possibiliy o wrie equaions o mechanics in special ime inervals: in he ime momen when he conac poin K velociy normal componen is zero; he ime momen when he conac poin K angenial velociy componen is changing direcion (e.g., dry ricion orce changes direcion, oo). In a case o ull slipping a he conac poin K (when aer collisions he body springs a an obsacle) normal reacion () and dry ricion orce F() in ime domain are shown in Fig

3 EGIEERIG FOR RURAL DEVELOPMET Jelgava, For he given model using classical mechanics equaions abou exchange o linear momenum or cener mass C and angular momenum agains he roaion axis O in ime momens and τ can be wrien ormulas (-4), [3; 6; 7]: d SI K ±F y YO v c C(xc;yc) h x () F() 1 SII SFI SFII O XO τ ω, ω 1, ω Fig. 5. Model o collision o roaing body Fig. 6. Diagrams o normal reacion () and dry ricion orces F() in ime domain ( m ω yc) = SXI SI; m ω xc= SYI SFI; JO ω= SI h SFI d; m ω = SXII SII; m ω xc = SYII SFII; JO ω = SII h SFII d; () (3) SII = R SI; SFI = SI; SFII = SII, (4) where m mass o he body; JO momen ineria body agains perpendicular axis in poin O; xc, yc coordinaes o cener mass; ω iniial angular velociy o he body; R coeicien o normal impulse resiuion; dry ricion coeicien; SI, SII, SFI, SFII impulses o normal reacion and dry ricion orce in he irs and second impac inervals and τ; SXI, SXII, SYI, SYII impulses o reacion orce componens XO and YO a poin O (Fig. 5). Eigh unknowns can be ound rom equaions () (4): R ω ( d h) JO ω ω= ; SI = ; h h JO ω R JO ω SFI = ; SFII = ; h h R JO ω SII = ; h (5) ( h SXI = m ω JO ω ( h ; SXII = R h m ω JO ω ; h ( h SYI = m ω xc h JO ω ; (6) ( h SYII = R m ω xc h JO ω. Generally known resul abou jamming in dry ricion mechanisms ollows rom he irs ormula (3), e.g.: 116

4 EGIEERIG FOR RURAL DEVELOPMET Jelgava, i h > d, hen ω < and he body springs back rom he obsacle. Oherwise, i h < d jamming will be and he body will sick o he obsacle. I addiionally he ime o collision τ is given, middle values o impac orces, F, XO, YO (Fig., 5, 6.) can be calculaed rom ormulas (-4). For opimizaions o he sysem parameers h, d, xc, yc, m and JO when a crierion is given (or example, minimum o impac impulses in roaing axe) ormula (6) can be used. Simulaneous collisions o roaing body in our poins The model o collision is shown in Fig. 7, where such collision coniguraion is invesigaed, in which all impac poins are placed in one line O, K1, K and K3. In hese poins here are no dry ricion ineracions, because velociies o all conac poins are perpendicular o he obsacle. v c y C (xc;yc) YO K3,3, L3 K,, L K1,1,L ω, ω 1, ω Fig. 7. Model o simulaneous collisions o roaing body in our poins For a roaing body when collisions ake place simulaneously in our poins (hree poins K1, K, K3 o he obsacle and one in he axis O) he above described heory and recommendaions are used. For roaing moion a he end o he irs impac phase and a he end o ull impac can be ound [4,6]: JO ω= S1I L1 SI L S3I L3; (7) JO ω = R1 S1I L1 R SI L R3 S3I L3, where R1, R and R3 coeiciens o resiuions in dieren poins K1, K and K3; L1, L and L3 disances rom he axe O ill poins K1, K and K3; S1I, SI, S3I normal impulses a he end o he irs phase. For moion o cener C mass can be calculaed: m ω m ω yc= SXO; m ω xc m ω xc= (1 R1) S1I (1 R) SI (1 R3) S3I SYO, where SXO, SYO reacion impulse componens in he axis O; xc, yc coordinaes o cener mass C. According o equaions o consrain proporionaliy can be used: L1 L S1I = S3I ; SI = S3I. (9) L3 L3 From equaions (7-9) all six unknowns ω, SXO, SYO, S1I, SI, and S3I can be ound. For example: R1 ω L1 R ω L R3 ω L3 ω =. L1 L L3 O XO x (8) 117

EGIEERIG FOR RURAL DEVELOPMET Jelgava, 5.-7.5.16. For opimizaion o he sysem parameers ormulas (7-9) can be used when a crierion and limis are given.

The sysem is excied by he roaing eccenric 5 when he acuaor 4 sars o draw away rom he main collision elemen. This ime momen impacs in conac poins K1-K4 occur.

An example o plaorm velociy obained rom modelling is shown in Fig. 9. According o he given resuls a real driver was made (Fig. 1). 5 4 1 3 Fig. 8.

5 EGIEERIG FOR RURAL DEVELOPMET Jelgava, For opimizaion o he sysem parameers ormulas (7-9) can be used when a crierion and limis are given. Simulaneous collisions in vibro driver sysem The vibro impac driver model or horizonal moion is shown in Fig. 8. The sysem is excied by he roaing eccenric 5 when he acuaor 4 sars o draw away rom he main collision elemen. This ime momen impacs in conac poins K1-K4 occur. Using he heory above, he sysem parameers or moion o he righ side was ound. This complicaed mechanism was invesigaed by Working Model D program. An example o plaorm velociy obained rom modelling is shown in Fig. 9. According o he given resuls a real driver was made (Fig. 1) Fig. 8. Horizonal vibro driver sysem: 1 undamen; moving plaorm; 3 main collision elemen; 4 vibro impac acuaor; 5 eccenric, which is driven by elecro moor; K1; K; K3 and K4 impacs poins; O roaion axis o he eccenric Fig. 9. Horizonal velociy graphics or moving plaorm Fig. 1. Experimenal model wih eccenric 118

6 EGIEERIG FOR RURAL DEVELOPMET Jelgava, Resuls and discussion 1. I is possible o calculae simulaneous impacs a several poins using coeiciens o resiuions and dry ricion coeiciens.. Given calculaion mehod allows o ind analyical ormulas or simulaneous collisions a several poins. 3. Analyical ormulas mus be used or he sysem parameer opimizaion. 4. Resuls o modelling wih Working Model were used or design o he experimenal model. Conclusions 1. Here given mehod or calculaion o impacs a several poins is applicable o shock calculaions, mosly or one solid body: in plane or roaion moions by means o he classical mechanics relaionships.. For calculaions o sysems wih impacs a more han hree poins compuer programs can be used [7]. 3. The analysis o impacs in complicaed sysems by compuer programs needs checking o addiional resuls, which akes ino accoun seven dieren impac cases [7]. Reerences 1. Plavnieks V. The calculaion o oblique impac agains an obsacle. Problems o dynamics and srengh. Proceeding 18: Рига: Зинатне pp (in Russian).. Kepe O., Viba J. Theoreical Mechanics. Riga: Zvaigzne, Lavian. 578 p. 3. Viba J. Opimizaion and synhesis o vibro impac sysems. Riga: Zinane, Russian. 53 p. 4. Lavendelis, E.; Viba, J.; Grasmanis, B Collision o he rigid body wih obsacle a more han one poin, in nd Inernaional Conerence o Mechanical Engineering Mechanics 97 Proceedings, 3 5 Sepember, 1997, Vilnius, Par 1: Machine Dynamics and Diagnosics MDD. Machine Design, Compuaion and Opimizaion MDCO, pp Anhony Bedord and Wallace Fowler. Engineering Mechanics; Saics & Dynamics. 4h ed. Pearson Educaion, Inc. USA p. 6. Goldsein H., Poole C., Sako J. Classical Mechanics, hird ediion, Addison-Wesley,, 647 p. 7. MSC. Soware Corporaion. Working Model Tuorial, 4,. [online] [ ] Available a: hp:// 119

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