MODEL OF THE PNEUMATIC DOUBLE ACTING CYLINDER COMPILED BY RHD RESISTANCES

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1 Journal of alied science in the thermodynamics and fluid mechanics Vol. 3, No. 1/009, ISSN MODEL OF THE PNEUMATIC DOUBLE ACTING CYLINDER COMPILED BY RHD RESISTANCES *Lukáš DVOŘÁK * Deartment of Hydrodynamics and Hydraulic Equiment, VŠB - Technical University of Ostrava, Faculty of Mechanical Engineering 17. listoadu 15, , Ostrava-Poruba, Czech Reublic Phone:+ (40) (59) Fax: + (40) (59) [email protected] This aer deals with mathematical modelling of neumatic double acting cylinder and also of other comonents by means of RHD resistances. Resistances are resistance to motion R, resistance to acceleration H and deformation resistance D. In the aer formulas for calculation of RHD resistances and comonents mathematical models are described. Results of simulation by this method of modelling are comared with exerimental results in the end. Keywords: modelling; simulation; neumatic cylinder 1 INTRODUCTION - MODELLING BY MEANS OF RHD RESISTANCES This way of modelling is based on electro-neumatic analogy. This method is used with the good results in the field of hydraulic systems [1] however in the area of neumatic systems is not common. The reason can be in roerties of the comressed air what makes difficult also much more simle calculations. This method was more recisely described in my doctoral thesis []. Program which comutes neumatic system characteristic by RHD resistances method in the software Matlab Simulink was created. It consists of three arts i.e. model of comressed air source, model of valve and iing and model of neumatic cylinder. MODEL OF PRESSURE SOURCE, PIPELINE AND DIRECTIONAL VALVE Air tank is considered as a ideal ressure source in the model. It means that the ressure is constant all-time of neumatic cylinder oeration. In to the model of source it is necessary to enter the value of working ressure, which is inut to the model of directional control valve and iing. Directional control valve can be described as a resistance to motion R. Resistance is caused by restriction when air flows through the valve. Quantity of resistance can be calculated from the flow coefficient Kv and also deend on the ressure ratio. Similarly the iing causes a ressure looses by friction when air flows through it. Flow caacity of iing can be defined by flow coefficients for examle Kv. Equivalent flow coefficient Kv of both elements can be then comuted by methods described in literature [3], [4], [5]. By the hel of equivalent flow coefficient Kv it is ossible to comute the total resistance of valve and iing, equation (1). R m 9 = 10 Kv ρ a 1a 8 [ s ] m N (1) In equation ρ is air density at working ressure, 1a is absolute ustream ressure (source ressure) and a is absolute downstream ressure (ressure in working chamber). It alies to air flow from ressure source to working chamber. In the case of air flow from exhaust chamber to atmoshere 1a is ressure in exhaust chamber and a is atmosheric ressure. The formula was verified by exeriment and it is ossible to use it for subsonic and for choked (sonic) flow rate too. Volume of air in ies resents deformation resistance D. It is imossible to ignore it. Volume of ieline can be added to dead volume of neumatic cylinder. This modification does not influence results but makes the calculation simler. Outut value of valve and ieline model is volumetric flow rate which is calculated from equation (). The flow rate is inut to neumatic cylinder model. 1

2 Journal of alied science in the thermodynamics and fluid mechanics Vol. 3, No. 1/009, ISSN Q1 = () R m 3 MODEL OF PNEUMATIC CYLINDER Model of neumatic cylinder is more comlicated than models of other neumatic elements. It can be described as a combination of all resistance tyes. Resistance to motion R is caused by friction of iston and iston rod and its quantity can be calculated from efficiency of cylinder η, theoretical force F theor, cylinder diameter D and iston velocity v. R = loss Q = S F loss v Ftheor = 1, 61 4 D v ( 1 η ) [N s m 5 ] (3) Resistance to acceleration H is caused by inertia of iston and iston rod mass and moving mass connected with iston rod. H = m red S =m red h =1,61 m red 4 [N s m 5 ] (4) V 1 D Deformation resistance D is caused by air comressibility. From exeriment aears that it is ossible to consider air comression in working chamber (chamber which is sulied with comressed air) as a isothermal rocess. Then the simly equation for resistance calculation can be used. D su = a V su [N m 5 ] (5) In the equation a is absolute ressure in the end of comression and V su is variable working chamber volume. It can be calculated with relation (6) where h act is actual iston osition and S is its area, constant 0,003 resents dead volume of cylinder and V h is ieline (hose) volume. V su =S h act 0,003 V h [m 3 ] (6) Actual quantity of exhaust chamber deformation resistance D can be calculated by the relation (7) D ex = n V ex act n 1 [N m 5 ] (7) where act is actual ressure in exhaust chamber, n is normal ressure and V ex is half of exhaust chamber volume. Working chamber can be described as a net of RHD resistances, see fig. 1. In comarison with model listed in my thesis [1] this version is modified. RHD resistance net in the icture 1 is simler but the results of simulation are the same as results obtained by original model. Q 1 Q mov R H Q mov Q Q Dsu Q Dex su D su F i D ex ex Figure 1: Model of working chamber Figure : Model of exhaust chamber Oeration of neumatic cylinder can be distributed to three time arts. The first art concerns fulfilment of dead volume. Dead volume is added to volume of working chamber (equation (6)) and by the hel of this volume the deformation resistance D su is calculated, equation (5). During the first time art the ressure

3 increases (equation (8)) but the iston does not move yet. Journal of alied science in the thermodynamics and fluid mechanics Vol. 3, No. 1/009, ISSN Δ su = D su Q 1 dt Δ F (8) Δ F = F ex S S (9) When the ressure reaches Δ F what is determined by force on iston rod and ressure in exhaust chamber the first time art is finished and starts second art, i.e. movement of iston. During second time art inlet flow rate is shared into two branches. The flow rate called flow rate to motion Q mov which flows through R and H resistances causes movement of iston. Piston velocity and osition can be calculated from this flow rate, equation (11), (1). h 0; h max Q mov = D su H [ Q 1 dt ] dt D su H [ Q mov dt ] dt R H Q mov dt (10) v = Q mov S (11) h act = v dt (1) Flow rate Q Dsu influence ressure in working chamber which can be solved by relation (14). Q Dsu =Q 1 Q mov (13) Δ su =Δ F D su Q Dsu dt (14) When the iston reaches end osition the second time art is finished. All inlet flow rate causes growing of ressure in chamber (16). h act =h max Q 1 =Q D su (15) Δ su =Δ F D su Q 1 dt (16) Exhaust chamber can be described as a one deformation resistance, see fig.. Equation (17) allows to comute ressure during all time arts of cylinder oeration. Δ ex = i D ex S v dt D ex Q dt (17) In relation i is initial ressure in chamber, S is annulus area, v iston velocity and Q is outut flow which is calculated by the hel of valve and ie model. 4 SIMULATION AND MEASUREMENT RESULTS Results of the mentioned neumatic system model i.e. curves of osition, iston velocity and ressure were comared with results of exeriment. The real mechanism consists of neumatic cylinder C9SDB (SMC), directional valve SV5-M5-B (Q n = 95 dm n 3. min -1 Kv = 0,0855 m 3. h -1 ) from FESTO and lastic hoses with inside diameter d = 4 mm and length L = 0,7 m. Scheme of mechanism is in fig. 3. Load mass on the iston rod was m = 11,5 kg and inclination angle of cylinder was α = 90 deg. 3

4 Journal of alied science in the thermodynamics and fluid mechanics Vol. 3, No. 1/009, ISSN Figure 3: Simle neumatic system Into the models comiled by means of RHD resistances it is necessary to enter arameters resented in Tab.1. Elements arameters as dimensions of cylinder, efficiency and valve flow coefficient can be find out in catalogues of neumatic comonents. Table 1: Inut arameters directional valve SV5-M5-B (FESTO) Kv 0,0855 m 3. h -1 olyurethane hoses L 0,7 m d 0,004 m D 0,04 m d c 0,016 m h 0,5 m cylinder F 0 N C9SDB (SMC) m 11,5 kg η 0,93 - α 90 deg working ressure 5, Pa In the following ictures there are results of simulation and results of measurement. From the comarison of simulation and exerimental results aears quite good corresondence of curves of iston osition and velocity. Movement of iston calculated by model starts earlier contrary of the exeriment results but stroke time is almost identical. The simulated curves of ressures in chambers differ from exeriment results however it does not influence dynamics of mechanism. Figure 4: Curves of iston ositio 4

5 Journal of alied science in the thermodynamics and fluid mechanics Vol. 3, No. 1/009, ISSN Figure 5: Curves of iston velocity Figure 6: Curves of ressure in chambers 5 CONCLUSIONS Method of modelling of neumatic elements and systems by means of RHD resistances was more recisely described in my doctoral thesis []. In this aer the modified model of neumatic cylinder was mentioned. Method of neumatic systems modelling based on RHD models rovides quite good results. However in the meantime this method is verified only for the double-acting cylinders with the iston diameter u to D = 63 mm and with stroke u to h = 500 mm. Another limit of the model is the size of directional control valve. The good results were obtained with the models where the directional control valve with flow coefficient u to Kv = 0,14 m 3. h -1 was used. REFERENCES [1] HRUŽÍK, L. KOZUBKOVÁ, M. Dynamika tekutinových mechanizmů návody do cvičení. Ostrava: FS VŠB-TU Ostrava, [cit ] Available from: <htt:// [] DVOŘÁK, L. Metodika návrhu, simulace a exerimentální ověření neumatických systémů. Ph.D. Thesis. Ostrava: VŠB-TU Ostrava, [3] KOPÁČEK, J. Pneumatické mechanismy díl 1. Pneumatické rvky a systémy. Ostrava: VŠB-TU Ostrava, ISBN [4] NEVRLÝ, J.: Modelování neumatických systémů. Brno: Akademické nakladatelství CERM Brno, ISBN [5] SMC. Basic of neumatics Technical course [CD-ROM]