ACCUMULATION OF FLUID FLOW ENERGY BY VIBRATIONS EXCITATION IN SYSTEM WITH TWO DEGREE OF FREEDOM

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1 ENGINEERING FOR RURAL DEVELOPMENT Jelgava, ACCUMULATION OF FLUID FLOW ENERGY BY VIBRATION EXCITATION IN YTEM WITH TWO DEGREE OF FREEDOM Maris Eiduks, Janis Viba, Lauris tals Riga Technical University, Institute of Mechanics Abstract. Very large achieveents in echanics of the field of aerodynaics and hydrodynaics exist. In the sae tie there are any technical tasks to be solved. One of the tasks group is find out new kinds of energy transforation fro constant fluid flow to accuulators. Its needs optiization for and paraeters of technical objects. A criteria like energy saving or its useful utilization fro interaction with air or water flow ay be used. In the first part of report a otion of a vibrator with constant air or water flow velocity excitation is optiized. The ain task is to find out optial control law for variation of area of vibrating object within liits. It is shown that optial control action is on bounds of area liits. ynthesis of control law in phase coordinates shows that very effective real control law is change area of object when it stops right or left side. In the second part of report obtained optial control law is realized in real vibration syste. Keywords: otion control, air or water flow excitation, energy utilization, optial control, adaptive control, synthesis adaptive systes. Introduction Motion of a vibrator with two degree of freedo and constant air flow V excitation is investigated (Fig. ). yste consists of asses, with springs c, c and dapers b, b. The ain idea is to find out optial control law in tie for variation of additional area ( of vibrating ass within liits (): ( where lower level of additional area of ass ; upper level of additional area of ass ; t tie., () Fig.. chee of odel with area ( control:, asses of oving bodies; c, c stiffness of springs; b, b daping coefficients;.v air flow velocity;, lower and upper level of area; of ass ; ( area control in tie in phase plane (z n displaceent, velocity) The criterion of optiization K is tie T required to ove object fro initial position to end position. For syste excitation any tie ust be solved the high-speed proble () [ - ]: t K dt () = t To assue t = ; t =, we have K = T. In this investigation equation of otion for large flow velocity T V ay be described as (): && y= c y c && z = c ( y z) + b ( y z) b y& b ( y& ) u( ( V 9 ( y& ); + ), ()

2 ENGINEERING FOR RURAL DEVELOPMENT Jelgava, where y, y&, & y displaceent, velocity and acceleration of ass ; z,,& displaceent, velocity and acceleration of ass. To use new variables (phase coordinates) x y x y& x z x =, = =, =, = = the syste (6) ay be written in first order differential equation for (): = x ; = [ cx c = x ; = [ c x ) + b x ) b x b x ) u( ( V In this syste with two degree of freedo Hailtonian is [ -6]: here H H = ψ + ψ ( + ψ x ( c + ψ ( x ) + b = ψ X where () x )]; + x ( c x c x ) b x b x )) x ) u( ( V ψ ψ = ψ ; X ψ + x ) ). ) ] + ψ x + () (5) =. (6) calar ultiplication of two last vector functions ψ and X in any tie (function H) ust be axial. To have such axiu, control action u( ust be within liits u ( = u ; u( = u, depending only fro the sign of function ψ (5): H = ax H, if ψ ( u( ( V + x ) ) = ax The ain conclusion of optial control law (8, 9) for this syste with two degree of freedo is that value of area any tie ust be on the bounds (): ( = or ( =. ynthesis of real control action For realizing optial control actions (in general case) syste needs a feedback with two adapters: one for displaceent easureent and another for velocity easureent. x x + = x C C x x + (7) = x C, C Fig.. Control by two adapters and one separation line x = x C + C in phase plane: x, x phase coordinates of ass ; C, C integration constants;, lower and upper level of area x Fig.. Control by two adapters and one broken separation line in phase plane: x, x phase coordinates of ass ; C,, C integration constants;, - lower and upper level of area x 9

3 ENGINEERING FOR RURAL DEVELOPMENT Jelgava, x x x x Fig.. Control by two adapters with fixing only levels of phase coordinates: x, x phase coordinates of ass ;, - lower and upper level of area Fig. 5. Control by only one velocity adapter which fix zero level: x, x phase coordinates of ass ;, - lower and upper level of area Modeling equation for control action with fixing levels of both phase coordinates (Fig..) is (8): [ k ( V + ) ( F( z, ))] [ k ( V + ) (,5,5 ))] U = (8) where (see Equation ) U = u( ( V + ) ; + z+ C) F( z, ) = (,5,5 ) ); k, C constants. Modeling equation for control action with one zero level of phase coordinate - velocity (Fig..) is (9): [ k ( V + ) (,5+,5 ))] [ k ( V + ) (,5,5 ))]. U = (9) Motion odeling with constant excitation paraeters Results of odeling are shown in Fig Exaples of otion for control (8) by two adapters with fixing only levels of phase coordinates are shown in Fig Motion is very stable because trajectories in phase plane for ass practically do not crosse (Fig. 7.). Exaples of otion in phase plane for second ore efficiency control (9) (by only one velocity adapter which fix zero level) is shown in Fig vy n y n z n Fig. 6. Motion of ass in phase plane with control (8) in the syste I Fig. 7. Motion of ass in phase plane with control (8) in the syste I 95

4 ENGINEERING FOR RURAL DEVELOPMENT Jelgava, t n z n Fig. 8. Control action (8) in tie t n doain in the I syste Fig. 9. Control action (8) as function of displaceent z n in the I syste vy n y n z n Fig.. Motion of ass in phase plane (in I syste) with control (9). Motion is very stable Fig.. Motion of ass in phase plane (in I syste) with control (9). Motion is very stable t n z n Fig.. Control action (9) in tie t n doain (in I syste) Fig.. Control action (9) as function of displaceent z (in I syste) Motion odeling with haronica and rando paraeters excitation To check up stability of otion ass was investigated two kinds of wind velocity exchange: by haronica and rando paraeters. Investigation shows that otion is very stable (Fig. -7). 96

5 ENGINEERING FOR RURAL DEVELOPMENT Jelgava, V.5 sin n s.75 π Fig.. Control action (9) and wind velocity haronica exchange in tie t n doain (in I syste) t n...5 vy n y n z n Fig. 5. Motion of ass in phase plane (y n displaceent, vy n velocity) with control Fig. (in I syste) Fig. 6. Motion of ass in phase plane (z n displaceent, velocity) with control Fig. (in I syste).5.5 Fig. 7. Control action (9) and wind velocity ixed exchange in tie t n doain Fig. 8. Motion of ass in phase plane (z n displaceent, velocity) with control (9) Analyses of controls (8) and (9) show that these adaptive systes have very stable otion (Fig. - 8). It eans that trajectories of ain ass in phase plane practically dos not crossed and periodical cycle is achieved after sall tie. At the end of investigations soe experiental works inside wind tunnel are analyzed. Investigated syste includes console spring and plane laina at the end (Fig. 9). Experients z n 97

6 ENGINEERING FOR RURAL DEVELOPMENT Jelgava, confir that airflow excitation is very efficient and vibration was excited in large region of air flow velocity. Fig. 9. Wind tunnel and vibrating syste inside (righ with console spring and plane laina Conclusions Air or water flow ay be used for excitation objects otion in vibration technique. Control of object area allows finding very efficient echatronic systes for energy conservation. Use of new vibration systes with air or water flow is in starting position and needs ore fundaental investigations. References. L.. Pontryagin. Математическая теория оптимальных процессов. М.: Физматгиз, с., Совместно с В. Г. Болтянским, Р. В. Гамкрелидзе и Е. Ф. Мищенко.. Pontryagin Maxiu Principle [online] [viewed..8]. Available at: Iyanaga, Y. Kawada, (Eds.). "Pontrjagin's [sic] Maxiu Principle." 88C in Encyclopedic Dictionary of Matheatics. Cabridge, MA: MIT Press, pp , 98.. E. Lavendelis. ynthesis of optial vibro achines. Zinatne, Riga, (97). (in Russian). 5. Viba J. Optiization and synthesis of vibro ipact systes. Zinatne, Riga, (988). (in Russian). 6. E.Lavendelis, J. Viba Methods of optial synthesis of strongly non linear (ipac systes. cientific Proceedings of Riga Technical University. Mechanics. Volue. Riga (7). pp