ENGINEERING FOR RURA DEEOPMENT Jelgava,.-.5.5. DOUBE PENDUUM IBRATION MOTION IN FUID FOW Janis iba, Maris Eiuks, Martins Irbe Riga Technical University, atvia janis.viba@rtu.lv, maris.eiuks@rtu.lv, martins.irbe@rtu.lv Abstract. The paper analyses the ouble penulum motion in the vertical plane interacting with the lui (air or water) low. The penulum consists o two plates, astene together with each other an with the ounation by the axes. The case was investigate where the lui lows in a horizontal irection but the plates move aroun the pivots in the vertical plane. It is assume that there is a laminar lui low. Double penulum interaction with the low is escribe as localize particles interaction with the plates, using the mass centre an angular momentum exchange theorems o mechanical systems. A system with two egrees o reeom is investigate. Interaction with lui is approximate by a square step epenence on the low relative velocity taking into account the irection o motion. The resulting system o equations is use to simulate the motion. The three cases o motion are analyse: ree oscillations against a stationary lui to etermine experimentally all physical parameters o the interaction; auto vibration motion at a constant low velocity; time variable low inuce oscillations. Some interesting results o the plate motion graphics are given. Keywors: ouble penulum, plate interaction with lui, low-inuce vibrations. Introuction Object vibrations in surrouning environment inclue motion insie lui at low spee region with a range when moving parts o elements change the velocity irection. Thereore, it is necessary to analyze the mechanical system motion as a ouble penulum () vibration in lui low () (Fig. ). Nature observation shows that some o the leaves o the trees start penulum motion (rattle) in winy weather. We can also observe water plant vibrations in the river by water stream lows. It is important to check up such vibrations or untraitional energy prouction (without propeller system) []. For that it is necessary investigating some simpler systems like a ouble penulum motion using the classical mechanics theory [-]. This report eals with analysis o two plates astene together with each other an with the ounation connecte to the pivots (that is, with the axis perpenicular to the irection o movement). The ouble penulum an lui interaction orces are analyze an their reuction to the penulum pivot points is oun out. Ater that, consiering win interaction orces ierential equations or the ouble penulum motion analysis are given. Some results o moelling are shown. Fig.. Double penulum in lui low: ouble penulum; horizontally low Moel o object The object consists o two rectangular plates an (Fig..). The plates are rotate aroun the horizontal axes Oz an Az. As generalize coorinates angles φ an φ are use. In the plates or the mass centers an vertically gravity orces mg an mg are applie respectively. The rotary axes Oz an Az can be place asymmetrically with istances an rom the plates ens. Flui low is horizontal with a spee. The analysis o the motion o the ouble penulum system by
ENGINEERING FOR RURA DEEOPMENT Jelgava,.-.5.5. 5 ierential equations requires ining the lui low interaction orces. They are obtaine in the next section. Fig.. Plane moel o penulum: penulum rotating part; penulum plane motion part; horizontal low; φ, φ generalize coorinates Reuction o lui low interaction orces Flui low interaction orces epen on the relative spee o the contacts points. The main interaction is the square unction o transverse spee components, perpenicular to the plate. Thereore, it is necessary to in out the point velocities projections on the perpenicular irections AO an AB (Fig..). It means that or the plate AO expression rom two graphs an (Fig. ) can be oun. Accoringly, or the plate AB all three graphs, an can be use. In this ouble penulum task or orces reuction it is convenient to use the points O an A as reuction centers. Then, the lui low principal vectors R, R an the principal moments M, M can be oun rom the ollowing integrals (or Fig. ): R M, Fin M R ; () R M, Fin M R. ()
ENGINEERING FOR RURA DEEOPMENT Jelgava,.-.5.5. Here, & ; & ; where, length o plates;, angular velocities;, constants, incluing rag coeicients, ensity o lui; with o plates. I the rotation axis o the plates are not at the ens, the low principal vectors R, R an the principal moments M, M can be oun by separate length (- till, an - till ) rom ormulas () an (), taking into account eviations an (see Fig..). Fig.. Graphics o ouble penulum points velocities components: graphic o irst penulum rotation velocity component; secon penulum relative rotation velocity component; horizontal low component; R, R principal vectors; M, M principal moments Dierential equations o penulum motion Dierential equations o motion can be oun, or example, by the D Alembert s principle o the given mechanical system (see Fig..). Sum o orces moments against the point A or only the secon plate gives: M φaon rsin ( J m r ) && m && r mg rsin. Sum o orces moment against the point O or both plates gives: () 6
ENGINEERING FOR RURA DEEOPMENT Jelgava,.-.5.5. Here (see Fig..): M M R ( J m r ) && ( J m r m r ) && φan sin m( r ) φaon rsin φaon mg ( r sin Mφ J && ; Mφ J && ; φaon m & ; mg r sin rsin( ). φot m r && ; φat m r&& ; φaot m &&, where J, J central moments o inertia mass m, m; r, r istance rom center masses till pins; length o irst plate; &, & angular accelerations o plates. φon m r & ; && φan m r & ; () Fig.. D Alembert s principle orces: (mg, mg, R, R, M, M) active orces an their moments;, M components o orces o inertia an their moments From two equations () an () the angular accelerations&, & can be oun. Then, by integrations &, & the motion parameters can be calculate, using numerical computer calculation methos (or example, Euler metho). There exists an aitional possibility o analysis o this motion by special lui low analysis computer programs, or example, Working Moel. 7
ENGINEERING FOR RURA DEEOPMENT Jelgava,.-.5.5. Motion moelling In this research the investigation was mae by soli boies plane motion calculation computer program Working Moel D. Some interesting graphics o motion are shown in Fig. 5-7. omments are given below or each igure. 5 Fig. 5. Transitional ampe vibration motion o penulum: angular velocity o irst plate as time unction; initial position o ouble penulum; mile position o penulum; close en rest position o penulum; 5 angular velocity o secon plate (see Fig. ) 5 Fig. 6. Stationary vibration motion o penulum with large amplitue: angular velocity o irst plate as time unction; angular velocity o secon plate as time unction; initial position o ouble penulum; positions o rotation axis are with eccentricities an (see Fig. ); 5 experiment in win tunnel with low velocity about 6 m sec - Fig. 7. Biurcation o vibration motion (with small amplitue): angular velocities; initial position o ouble penulum 8
ENGINEERING FOR RURA DEEOPMENT Jelgava,.-.5.5. Results an iscussion Analysis o the moeling results shows that it is possible to obtain a stationary oscillations motion in the ouble penulum system. For that the axis o rotation can be isplace by eccentricity, against the en points o the plates. This means that lui lows at a constant spee in the observe system generating perioically vibration motion. In this system the existence o vibration biurcations shoul be checke out. This question requires aitional investigations an is not covere by the given article. onclusions. Double penulum vibration system by using lui low interaction can be use or untraitional energy prouction (without propeller system).. Stationary vibration motion epens on special penulum rotation axis isplacements.. Theory valiation was checke by the win tunnel. Reerences. iba J., itols D., Grischenko M., Eiuks M., Kulikovskis G. Investigation o unerwater robotic system or object motion propulsion an energy generation. Proceeings o International conerence Engineering or rural evelopment, 9-.5., Jelgava, atvia, pp. 6-65.. Martynyuk A. A. an Nikitina N.. The Theory o Motion o a Double Mathematical Penulum, International Applie Mechanics, ol. 6, No. 9,, pp. 5-58.. Double Penulum rom Eric Weisstein s. Worl o Physics scienceworl.wolram.com Mechanics Penula [online][..] Available at: http://scienceworl.wolram.com/ physics/doublepenulum.html. Double Penulum. [online][..] Available at: http://www.myphysicslab.com/bl_ penulum.html 5. Arnol,. I. Problem in Mathematical Methos o lassical Mechanics, n e. New York: Springer-erlag, p. 9, 989. 6. Wells, D. A. Theory an Problems o agrangian Dynamics. New York: McGraw-Hill, pp. -,, an -, 967. 7. arry M. Bates; James M. Nester. On D Alembert s Principle. ommunications in Mathematics (). olume: 9, Issue:, page 57-7. ISSN: 8-88. 9