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Analysis and synthesis of a high-speed pneumatic machine drive
Mechanismand MachineTheory,1978,VoL 13, pp. 293-300. PergamonPress. Printed in Great Britain
Analysis and Synthesis of a High-Speed Pneumatic Machine Drive I. I. Artobolevskiit and E. V. Hertz Received 1 June 1976; for publication 8 September 1977
Abstract The paper deals with the problem of increasing the velocity of actuating elements in a pneumatic machine drive. A rational structure of the drive is chosen by a comparative analysis of the dynamic characteristics of different drives. A dynamic analysis by computer of the most promising drive with different parameters is given. As a result the most important design parameters are established. The dynamic synthesis of this drive is made on the basis of reference data generated by the computer using a limited variation of essential parameters. The maximal piston velocity received at limited size and weight of the drive was taken as an optimal criterion. Accordingly, for this method, design parameters of the drive were chosen for which velocity is much higher than for common drives. Introduction PNEUMATIC machine drives have been widely adopted recently in the automation of technological processes in different branches of mechanical engineering, metallurgy, space-aeronautics and so on. They are used as elements in transporting and clamping mechanisms, for remote control of machines working in hazardous environments of dust, vibration, and radiation. Pneumatic drives are reliable, relatively inexpensive, have a very simple construction, and are easily maintained. Since pneumatic machine drives are used to initiate the motion of actuating elements, their working time is included into the total time of the cycle for a machine which utilizes these drives. The main design requirement for pneumatic machine drives and control systems is to increase their velocity since they are inferior in this respect to electrical drives. Experimental research studies of pneumatic machine drives used for the automation of labor-consuming operations such as metal cutting, stamping of different articles, and so on have shown that the most favorable velocity of actuating elements is about l0 m/see and higher. Pneumatic machine drives used in industry usually develop considerably lower velocities (1-2 m/see) when working from factory mains where the compressed air pressure is 4-5 kgJm 2. This paper deals with the problem of increasing pneumatic machine drive velocities with simultaneous limitation of their size by means of selecting the rational construction and optimal parameters [1]. The dynamics of different machine drive types is investigated, and a method of choosing the most promising optimal design parameters is worked out.
Dynamic Analysis of a Typical Machine Drive A typical pneumatic machine drive that has a drive containing no more than two chambers is shown in Fig. l(a). Compressed air from the main pressure equal to Pm enters chamber 1 of the working cylinder. Pressure in this chamber is thus increased. When this pressure reaches the value able to overcome the machine drive resistance equal to P - , the sum of all forces acting on piston I and on the actuating elements of the machine connected with it, the piston begins to move. The actuating element may be a clamping mechanism or a transport rmechanism. Chamber 2 is connected to the atmosphere whose pressure is given by/3A. tAcademician Institute for the Study of Machines, Academy of Sciences of the U.S.S.R., MoscowCenter, Ulitsa Griboedova4, K. 60, U.S.S.R. 293
294
(o) t
~ PA 2
I
I I I
i%
3
P
I
11
/7 ,¢1
•' ~.
%1 "v
, )
I //2 PA
~l
3
=
\\
11
¢--- p. ,,P
L ~
Rgure 1. Pneumatic machine drives. (a) Typical drive (b) High speed drive with a built-in reservoir; (c) High speed drive with a releasing mechanism. After thestroke of piston [ is over, the pressure Pt in the working chamber 1 increases up to pressure Pu of the main and drops to atmospheric pressure 13Ain the exhaust chamber. The distributor not shown!n_Fig. l(a) isthen switched over and compressed air enters from the main into chamber 2, while chamber 1 is connected to the atmosphere and the piston returns to its initial position. The initial conditions of a typical machine drive in general are: atmospheric pressure in the working chamber connected to the supply main and main pressure in the exhaust chamber connected to the atmosphere. Since the piston velocity depends on the pressure difference on both sides of the piston, it is advisable before the beginning of the piston movement to remove the compressed air from the exhaust chamber. Then the atmospheric pressure in both chambers will be the initial condition. In this case the piston velocity will be much ~eater. The dynamics of a typical pneumatic machine drive may be described by the following system of equations [2]. N 2 d2~
(I)
where
N = t~lf'EfmRTu~ u2 ~= ~ t Fl \PuFis] " ~; ~=--; tm '
~1 = ~ 1 .
~2
P2.
Pro' E=
Pu' ,
n=F2;
X = .:-'
Fi t,,----,~.
N \pj:,
,
pmFl .
fl,/Lh area and flow coefficient of the inlet port; F~, F2, piston area on both tides.; m, s, piston mass and its stroke; R, gas constant; Tw, air temperature in the supply main; Pu,/~, air pressure in the supply main and in the i, the chamber, respectively; x, piston position coordinate; t, time; P, external force; g, gravity acceletation;/~, adiabatic index.
The equation for defining air pressure in the working chamber I has the following form k r "8" # dE1 +¢ " - 'T J'
dSi d-; =
(2)
where
.~Ol ,0(al) = ¢(a~) = X/(a~*
6o, =
T'
- g,.,/a),
when 0,528 < 8i < 1; ~o(8i)= 0,259, when
0 < 8; ~< 0,528; ~orreduced coordinate of the piston position which characterizes the initial volume of the working chamber. The equation for determining the pressure in the exhaust chamber 2 is
d8--2 d'r =
~o2+~; -~' [flH-'~°k-')/z*~° l (--~) - ~ d~]
~o2 = ~ .
f l = Iz,f, '
~p
P.
- ~(ai);
(3)
where Xoz-reduced coordinate of the piston position which characterizes the initial volume of the exhaust chamber; f2,/Lz-area and flow coefficient of the exhaust port. Equations (1)-(3) are solved by means of a computer for different values Some results are given in Fig. 2. The velocity ~ of the piston on time ~- with N = 0,4 (curve 1) is given in Fig. 2.
N= 0.4 25
/
//"
1.5
A
/
0.5~ 0
0.2
0.4
0.6
0.8
LO
1.2
"r
Rgure 2. Dependence of piston velocity on time. 1. Typical drive; 2. High speed drive with a built-in reservoir; 3. High speed drive with a releasing mechanism.
296
Dynamic Analysis of a Machine Drive with a Built-in Reservoir Pneumatic piston velocity is sometimes insufficient for a number of technological operations as for example in the case of a quick insertion of articles into a dangerous zone[3], the operations of marking and stamping[4], and so on. Since the piston velocity depends on the pressure difference on both sides of the piston it is expedient to increase this difference. To meet this requirement it is necessary to provide a faster supply of compressed air into the working chamber and to provide a faster exhaust into the atmosphere for the exhaust chamber. The size increase of inlet and outles devices results in an increase of size and weight of the machine drive and does not give a considerable increase of velocity. A more effective solution is the use of the so called high-speed machine drives (see Fig. lb)---having a third additional chamber--a built in reservoir 3 connected with chamber 1 by means of a large enough port in the diaphragm. When chamber 2 is connected with the atmosphere the pressure in the chamber decreases. When the pressure difference on both sides of the piston is sufficient to overcome the resistance forces, the piston begins to move. The equation of air pressure and air temperature in the built-in reservoir 3 are, in this case, added to eqns (I)--(3) for the pneumatic machine drive •
d83
dz - ~
\83/3
d--~-= ~¢~3 ~¢3~ $3 = P 3 .
~u'
(4)
- 83¢(83) + nl,353~(03)~0
bl3 = V
#F,'
(5)
~'~1.3= /Z3f3 83 = T3.
~,fl
Tu'
where P3, /'3, air pressure and temperature in reservoir 3; V, reservoir 3 volume; f3, ~3, area and flow coefficient of the diaphragm port. The corresponding corrections are also introduced into eqn (2) characterizing the pressure in the working chamber as the compressed air enters into it not from the main but from reservoir 3. Terms fl,.3 $3 V'(03) ¢($d$3) must be introduced into eqn (2) instead of the first term ~o($,) in straight brackets. The results of the solution of eqns (1)-(5) are given in the form of curve 2 in Fig. 2. As seen from this diagram the velocity of the high-speed machine drive is much higher than that of the typical one.
Dynamic Analysis of s Machine Drive with a Releasing Mechanism The further increase of velocity can be achieved at the expense of the maximum pressure difference by using a high-speed machine drive with a releasing mechanism [4] as Shown in Fig. l(c). Piston I is retained in a fixed position by means of the piston rod II up to the moment when pressure in the working chamber will be the same as in the main, and the pressure in the exhaust chamber is the same as the atmospheric pressure. After this the compressed air is supplied to the bottom chamber of the releasing mechanism and piston II moves upward releasing piston rod I which moves rapidly forward. During this process the working chamber is constantly receiving air from the reservoir. The dynamics of this machine drive may be also described by a system of equations given above (1)-(5), but the initial conditions of m6vement for this case are not the same as previously described. These equations were solved by a computer according to a general purpose program for designing complex pneumatic systems [4]. The algorithm is based on the idea of formalized composition of equations by the machine itself, according to basic equations taken as a model and their consequent solution foreseen in the same program. When deriving the basic equations, analytical descriptions of the motion of the actuating elements and thermodynamic descriptions of the processes taking place in the drive chambers were used. The basic equations are 1. The equation of p/ston motion under the influence of all the forces applied is analogous to eqn (1). 2. The equation of conservation of compressed air
297
energy in the chamber which is the basic one for receiving eqns (2)-(4). 3. the equation of the conservation of the compressed air mass in the chamber used for deriving eqn (5). All these equations are derived with the following assumptions: Thermodynamic processes are considered as quasi-stationary ones and the heat exchange with the surrounding environment is not taken into account. The equations are given in a dimensionless form since this allows the solution results to be applied to a whole range of drives of the same type. The derivation of these equations is given in the book by Hertz[2]. For the solution of eqns (1)-(5) the information concerning the machine drive construction (the number of pistons and chambers) in a formalized form of special matrices, as well as reference data and initial conditions are introduced beforehand into a computer. Curve 3 in Fig. 2 characterizes the change of the dimensionless velocity ~ of the piston from time ~- given in the dimensionless form. The analysis of the computation results carred out with different values of parameters N, fl and load X has shown that the most effective design is a high-speed machine drive with a releasing mechanism (Fig. lc) whose velocity is 30-50% higher than in common drives. The dynamic analysis of this machine drive was carried out to determine the influence of different physical parameters (f, F, m, s, V) as well as dimensionless values (N, II, ~3, II) on its dynamics. The analysis showed that the dimensionless parameters ~:3 and fl have the greatest influences. These parameters give the characteristics of the built-in reservoir volume and the area ratios of the outlet and inlet ports. The volume of the built-in reservoir has the greatest influence on the terminal piston velocity ~k at small values of N (0.1-0.3).
Dynamic Syntheses of s Machine Drive with s Releasing Mechanism The synthesis of the high-speed machine drive was carried out by the computer using reference data. In this case a definite class of pneumatic machines driven with a releasing mechanism limited by a range of parameter changes: 0.1 ~
MMT V~J 1~ Nn
298 G,,O,
8 6
~x
005
2
0
0.1
0.2
0.3
0,4
0,6
~
0,8
1.0 N
Rgure 3. Regions of optimal Paramete-rs: I, area ratio £ of outlet and inlet; II, total volumes (6+ 6o2) of the reservoir and exhaust chamber; Ill, reservoir volume 63. Figure 3 is a diagram showing the dependence of dimensionless optimal values of the reservoir volume 63 and the parameter II on the ratio N of the machine drive constructive parameters. From the diagram it is seen that the increase of the volume 63 more than 1.5-2.0 times is inexpedient with relatively high values of N (0.6-1.0). In this range of the change of N the influence of the initial volume 60: of the exhaust chamber 2 is not high. However, for machine drives with a small constructive parameter N, the influence of 6o~ on the terminal velocity of the piston may be neglected only at values 1~ close to optimal ones (see zone 1 Fig. 3). Zones II and III are used to chose optimal parameters 6.o:and 63. The upper boundary line of the zones is circumscribed by a curve corresponding to a load X = 0.05 and the lower boundary to a load X = 0.15. In the process of optimization we have taken as the optimal criterion[l] the ratio of the energy of the moving parts of the machine drive to its weight
2H'
(6)
where ~k is the dimensionless terminal velocity and H, the dimensionless weight of the machine drive. Numerous calculations have shown that the terminal velocity of the piston increases with the increase of the reservoir volume. The weight of the drive increases with the increase of velocity (mainly due to the reservoir) and there arises a moment when even a considerable increase of volume does not give a considerable increase of velocity increment; coefficient K reaches at that moment its maximum value-Kin,. A further increase of 63 results in the decrease o f / ~ as the increase of the machine drive weight considerably outstrips the velocity increase. The value of Km~xat this moment corresponds to an optimal value 63° of the reservoir volume (see Fig. 4). The analysis of calculations carried out and the analysis of reference data based on these calculations shows that the value K,~, increases when the constructive parameter N is decreased. This corresponds to a more effective influence of 63 on the terminal velocity ~k. There is a definite zone of values of K in the interval K~b ~ K ~ Km~ where the minimum allowable energy chosen on the base of technological consiJeratio~s K ~ . = a Km~ is taken as the lower boundary. Figure 5 gives dependencies for choosing the optimal reservoir volume according to velocity and weight data. Curve I-I represents the upper boundary of optimal dimensionless volume reservoir 63 corresponding to the maximum velocity & received on the condition that this velocity is increasing to the end of the piston stroke. Curve II-II gives the dependence of the
299 K N=0.3 x =0.15 2.3 K
f
I
2.2
I
2.1
\6,
I I I
I
I 2.0
i.o
1.5
I
2.o ¢;
63
Figure 4. Dependence of the coefficient K and piston velocity on the reservoir volume. so3°, optimum value of the reservoir volume.
3.0
L
2.5
2.0 Krnin 1.5
0.2
04
0.6
I 0.8
1.0 N
Figure 5. Regions of optimal design fora stroke drive (upper region) and a transport one (lower region). I-I, curve of maximum terminal velocity; I1-11,curve of maximum coefficient K; II1-111,curve of minimum coefficient K. maximum coefficient K on value ~. Curve III-IlI is the lower boundary of optimal values of the reservoir volume corresponding to the minimum allowable energy level that is necessary for the given technological process. Two zones of optimal parameters are given in this Figure. It is advisable to choose constructive parameters from the upper zone for the case when the terminal velocity ~k of the piston is of great importance as for example for the shock pneumatic machine drive. But if it is necessary to receive a minimum weight or size, thereby achieving high enough piston velocities, it is recommended to choose drive parameters from the lower zone. This may be the case for transport machine drives, manual devices, emergency systems, and so on. In all cases it is recommended that the designer avoid choosing values st3 and N lying outside the indicated optimization zones since for this case there will be a sharp increase of the machine drive sizes or a decrease of the kinetic energy of the moving parts. The diagram of Fig. 5 is given for a load X = 0.05. Analogous diagrams where obtained for other values of the load.
Concluelon The results of the dynamic analysis of typical pneumatic and high speed drives have been discussed. The dynamics of these drives were described by means of a system of non-linear high order differential equations and computers were widely used in solving these systems. A universal computer program for dynamic calculation has been worked out. As a result of calculations carried out, it has been stated that drives with built-in reservoirs develop con-
300 siderably higher speeds than typical drives of a more simple construction. The highest efficiency has a drive with a realeasing mechanism the speed of which is higher by 30-50%. The influence of different design parameters such as the built in reservoir volume, the inlet and outlet ports areas, the weights of the elements, and so on have been investigated. The regions of existence of design parameters, close to optimal ones have been determined. The authors offer a technique for choosing the most essential parameters of drives with the aim of determining their maximum speed. This technique can be used for shock drives as well as for transport ones. In the last case it is important to receive a relatively high drive speed with limited size and weight. The above mentioned calculation methods are used in practical design of a number of drives which at present are in use in industry.
References 1. I. I. Artobolevskii,The choice of optimum machine parameters. Machinovedenje,No. 4, p. 3 (1973). 2. E. V. Hertz, Pneumatic machine drives. M., Machinostro]enie, p. 359 (1969). 3. K. H. Larson, Computer sizes parameters of a pneumatic scram system. Hydraulic and Pneumatics, VoL 17, No. l0 (1%4). 4. E. V. Hertz and V. S. Doldzenkov, Investigation of the dynamics of a high speed pneumatic machine drive with a releasing mechanism. Mavhinovedenje,No. 2, p. 29 (1974).
Analysis and Synthesis of a High-Speed Pneumatic Machine Drive I. I. Artobolevskiit and E. V. Hertz Received 1 June 1976; for publication 8 September 1977
Abstract The paper deals with the problem of increasing the velocity of actuating elements in a pneumatic machine drive. A rational structure of the drive is chosen by a comparative analysis of the dynamic characteristics of different drives. A dynamic analysis by computer of the most promising drive with different parameters is given. As a result the most important design parameters are established. The dynamic synthesis of this drive is made on the basis of reference data generated by the computer using a limited variation of essential parameters. The maximal piston velocity received at limited size and weight of the drive was taken as an optimal criterion. Accordingly, for this method, design parameters of the drive were chosen for which velocity is much higher than for common drives. Introduction PNEUMATIC machine drives have been widely adopted recently in the automation of technological processes in different branches of mechanical engineering, metallurgy, space-aeronautics and so on. They are used as elements in transporting and clamping mechanisms, for remote control of machines working in hazardous environments of dust, vibration, and radiation. Pneumatic drives are reliable, relatively inexpensive, have a very simple construction, and are easily maintained. Since pneumatic machine drives are used to initiate the motion of actuating elements, their working time is included into the total time of the cycle for a machine which utilizes these drives. The main design requirement for pneumatic machine drives and control systems is to increase their velocity since they are inferior in this respect to electrical drives. Experimental research studies of pneumatic machine drives used for the automation of labor-consuming operations such as metal cutting, stamping of different articles, and so on have shown that the most favorable velocity of actuating elements is about l0 m/see and higher. Pneumatic machine drives used in industry usually develop considerably lower velocities (1-2 m/see) when working from factory mains where the compressed air pressure is 4-5 kgJm 2. This paper deals with the problem of increasing pneumatic machine drive velocities with simultaneous limitation of their size by means of selecting the rational construction and optimal parameters [1]. The dynamics of different machine drive types is investigated, and a method of choosing the most promising optimal design parameters is worked out.
Dynamic Analysis of a Typical Machine Drive A typical pneumatic machine drive that has a drive containing no more than two chambers is shown in Fig. l(a). Compressed air from the main pressure equal to Pm enters chamber 1 of the working cylinder. Pressure in this chamber is thus increased. When this pressure reaches the value able to overcome the machine drive resistance equal to P - , the sum of all forces acting on piston I and on the actuating elements of the machine connected with it, the piston begins to move. The actuating element may be a clamping mechanism or a transport rmechanism. Chamber 2 is connected to the atmosphere whose pressure is given by/3A. tAcademician Institute for the Study of Machines, Academy of Sciences of the U.S.S.R., MoscowCenter, Ulitsa Griboedova4, K. 60, U.S.S.R. 293
294
(o) t
~ PA 2
I
I I I
i%
3
P
I
11
/7 ,¢1
•' ~.
%1 "v
, )
I //2 PA
~l
3
=
\\
11
¢--- p. ,,P
L ~
Rgure 1. Pneumatic machine drives. (a) Typical drive (b) High speed drive with a built-in reservoir; (c) High speed drive with a releasing mechanism. After thestroke of piston [ is over, the pressure Pt in the working chamber 1 increases up to pressure Pu of the main and drops to atmospheric pressure 13Ain the exhaust chamber. The distributor not shown!n_Fig. l(a) isthen switched over and compressed air enters from the main into chamber 2, while chamber 1 is connected to the atmosphere and the piston returns to its initial position. The initial conditions of a typical machine drive in general are: atmospheric pressure in the working chamber connected to the supply main and main pressure in the exhaust chamber connected to the atmosphere. Since the piston velocity depends on the pressure difference on both sides of the piston, it is advisable before the beginning of the piston movement to remove the compressed air from the exhaust chamber. Then the atmospheric pressure in both chambers will be the initial condition. In this case the piston velocity will be much ~eater. The dynamics of a typical pneumatic machine drive may be described by the following system of equations [2]. N 2 d2~
(I)
where
N = t~lf'EfmRTu~ u2 ~= ~ t Fl \PuFis] " ~; ~=--; tm '
~1 = ~ 1 .
~2
P2.
Pro' E=
Pu' ,
n=F2;
X = .:-'
Fi t,,----,~.
N \pj:,
,
pmFl .
fl,/Lh area and flow coefficient of the inlet port; F~, F2, piston area on both tides.; m, s, piston mass and its stroke; R, gas constant; Tw, air temperature in the supply main; Pu,/~, air pressure in the supply main and in the i, the chamber, respectively; x, piston position coordinate; t, time; P, external force; g, gravity acceletation;/~, adiabatic index.
The equation for defining air pressure in the working chamber I has the following form k r "8" # dE1 +¢ " - 'T J'
dSi d-; =
(2)
where
.~Ol ,0(al) = ¢(a~) = X/(a~*
6o, =
T'
- g,.,/a),
when 0,528 < 8i < 1; ~o(8i)= 0,259, when
0 < 8; ~< 0,528; ~orreduced coordinate of the piston position which characterizes the initial volume of the working chamber. The equation for determining the pressure in the exhaust chamber 2 is
d8--2 d'r =
~o2+~; -~' [flH-'~°k-')/z*~° l (--~) - ~ d~]
~o2 = ~ .
f l = Iz,f, '
~p
P.
- ~(ai);
(3)
where Xoz-reduced coordinate of the piston position which characterizes the initial volume of the exhaust chamber; f2,/Lz-area and flow coefficient of the exhaust port. Equations (1)-(3) are solved by means of a computer for different values Some results are given in Fig. 2. The velocity ~ of the piston on time ~- with N = 0,4 (curve 1) is given in Fig. 2.
N= 0.4 25
/
//"
1.5
A
/
0.5~ 0
0.2
0.4
0.6
0.8
LO
1.2
"r
Rgure 2. Dependence of piston velocity on time. 1. Typical drive; 2. High speed drive with a built-in reservoir; 3. High speed drive with a releasing mechanism.
296
Dynamic Analysis of a Machine Drive with a Built-in Reservoir Pneumatic piston velocity is sometimes insufficient for a number of technological operations as for example in the case of a quick insertion of articles into a dangerous zone[3], the operations of marking and stamping[4], and so on. Since the piston velocity depends on the pressure difference on both sides of the piston it is expedient to increase this difference. To meet this requirement it is necessary to provide a faster supply of compressed air into the working chamber and to provide a faster exhaust into the atmosphere for the exhaust chamber. The size increase of inlet and outles devices results in an increase of size and weight of the machine drive and does not give a considerable increase of velocity. A more effective solution is the use of the so called high-speed machine drives (see Fig. lb)---having a third additional chamber--a built in reservoir 3 connected with chamber 1 by means of a large enough port in the diaphragm. When chamber 2 is connected with the atmosphere the pressure in the chamber decreases. When the pressure difference on both sides of the piston is sufficient to overcome the resistance forces, the piston begins to move. The equation of air pressure and air temperature in the built-in reservoir 3 are, in this case, added to eqns (I)--(3) for the pneumatic machine drive •
d83
dz - ~
\83/3
d--~-= ~¢~3 ~¢3~ $3 = P 3 .
~u'
(4)
- 83¢(83) + nl,353~(03)~0
bl3 = V
#F,'
(5)
~'~1.3= /Z3f3 83 = T3.
~,fl
Tu'
where P3, /'3, air pressure and temperature in reservoir 3; V, reservoir 3 volume; f3, ~3, area and flow coefficient of the diaphragm port. The corresponding corrections are also introduced into eqn (2) characterizing the pressure in the working chamber as the compressed air enters into it not from the main but from reservoir 3. Terms fl,.3 $3 V'(03) ¢($d$3) must be introduced into eqn (2) instead of the first term ~o($,) in straight brackets. The results of the solution of eqns (1)-(5) are given in the form of curve 2 in Fig. 2. As seen from this diagram the velocity of the high-speed machine drive is much higher than that of the typical one.
Dynamic Analysis of s Machine Drive with a Releasing Mechanism The further increase of velocity can be achieved at the expense of the maximum pressure difference by using a high-speed machine drive with a releasing mechanism [4] as Shown in Fig. l(c). Piston I is retained in a fixed position by means of the piston rod II up to the moment when pressure in the working chamber will be the same as in the main, and the pressure in the exhaust chamber is the same as the atmospheric pressure. After this the compressed air is supplied to the bottom chamber of the releasing mechanism and piston II moves upward releasing piston rod I which moves rapidly forward. During this process the working chamber is constantly receiving air from the reservoir. The dynamics of this machine drive may be also described by a system of equations given above (1)-(5), but the initial conditions of m6vement for this case are not the same as previously described. These equations were solved by a computer according to a general purpose program for designing complex pneumatic systems [4]. The algorithm is based on the idea of formalized composition of equations by the machine itself, according to basic equations taken as a model and their consequent solution foreseen in the same program. When deriving the basic equations, analytical descriptions of the motion of the actuating elements and thermodynamic descriptions of the processes taking place in the drive chambers were used. The basic equations are 1. The equation of p/ston motion under the influence of all the forces applied is analogous to eqn (1). 2. The equation of conservation of compressed air
297
energy in the chamber which is the basic one for receiving eqns (2)-(4). 3. the equation of the conservation of the compressed air mass in the chamber used for deriving eqn (5). All these equations are derived with the following assumptions: Thermodynamic processes are considered as quasi-stationary ones and the heat exchange with the surrounding environment is not taken into account. The equations are given in a dimensionless form since this allows the solution results to be applied to a whole range of drives of the same type. The derivation of these equations is given in the book by Hertz[2]. For the solution of eqns (1)-(5) the information concerning the machine drive construction (the number of pistons and chambers) in a formalized form of special matrices, as well as reference data and initial conditions are introduced beforehand into a computer. Curve 3 in Fig. 2 characterizes the change of the dimensionless velocity ~ of the piston from time ~- given in the dimensionless form. The analysis of the computation results carred out with different values of parameters N, fl and load X has shown that the most effective design is a high-speed machine drive with a releasing mechanism (Fig. lc) whose velocity is 30-50% higher than in common drives. The dynamic analysis of this machine drive was carried out to determine the influence of different physical parameters (f, F, m, s, V) as well as dimensionless values (N, II, ~3, II) on its dynamics. The analysis showed that the dimensionless parameters ~:3 and fl have the greatest influences. These parameters give the characteristics of the built-in reservoir volume and the area ratios of the outlet and inlet ports. The volume of the built-in reservoir has the greatest influence on the terminal piston velocity ~k at small values of N (0.1-0.3).
Dynamic Syntheses of s Machine Drive with s Releasing Mechanism The synthesis of the high-speed machine drive was carried out by the computer using reference data. In this case a definite class of pneumatic machines driven with a releasing mechanism limited by a range of parameter changes: 0.1 ~
MMT V~J 1~ Nn
298 G,,O,
8 6
~x
005
2
0
0.1
0.2
0.3
0,4
0,6
~
0,8
1.0 N
Rgure 3. Regions of optimal Paramete-rs: I, area ratio £ of outlet and inlet; II, total volumes (6+ 6o2) of the reservoir and exhaust chamber; Ill, reservoir volume 63. Figure 3 is a diagram showing the dependence of dimensionless optimal values of the reservoir volume 63 and the parameter II on the ratio N of the machine drive constructive parameters. From the diagram it is seen that the increase of the volume 63 more than 1.5-2.0 times is inexpedient with relatively high values of N (0.6-1.0). In this range of the change of N the influence of the initial volume 60: of the exhaust chamber 2 is not high. However, for machine drives with a small constructive parameter N, the influence of 6o~ on the terminal velocity of the piston may be neglected only at values 1~ close to optimal ones (see zone 1 Fig. 3). Zones II and III are used to chose optimal parameters 6.o:and 63. The upper boundary line of the zones is circumscribed by a curve corresponding to a load X = 0.05 and the lower boundary to a load X = 0.15. In the process of optimization we have taken as the optimal criterion[l] the ratio of the energy of the moving parts of the machine drive to its weight
2H'
(6)
where ~k is the dimensionless terminal velocity and H, the dimensionless weight of the machine drive. Numerous calculations have shown that the terminal velocity of the piston increases with the increase of the reservoir volume. The weight of the drive increases with the increase of velocity (mainly due to the reservoir) and there arises a moment when even a considerable increase of volume does not give a considerable increase of velocity increment; coefficient K reaches at that moment its maximum value-Kin,. A further increase of 63 results in the decrease o f / ~ as the increase of the machine drive weight considerably outstrips the velocity increase. The value of Km~xat this moment corresponds to an optimal value 63° of the reservoir volume (see Fig. 4). The analysis of calculations carried out and the analysis of reference data based on these calculations shows that the value K,~, increases when the constructive parameter N is decreased. This corresponds to a more effective influence of 63 on the terminal velocity ~k. There is a definite zone of values of K in the interval K~b ~ K ~ Km~ where the minimum allowable energy chosen on the base of technological consiJeratio~s K ~ . = a Km~ is taken as the lower boundary. Figure 5 gives dependencies for choosing the optimal reservoir volume according to velocity and weight data. Curve I-I represents the upper boundary of optimal dimensionless volume reservoir 63 corresponding to the maximum velocity & received on the condition that this velocity is increasing to the end of the piston stroke. Curve II-II gives the dependence of the
299 K N=0.3 x =0.15 2.3 K
f
I
2.2
I
2.1
\6,
I I I
I
I 2.0
i.o
1.5
I
2.o ¢;
63
Figure 4. Dependence of the coefficient K and piston velocity on the reservoir volume. so3°, optimum value of the reservoir volume.
3.0
L
2.5
2.0 Krnin 1.5
0.2
04
0.6
I 0.8
1.0 N
Figure 5. Regions of optimal design fora stroke drive (upper region) and a transport one (lower region). I-I, curve of maximum terminal velocity; I1-11,curve of maximum coefficient K; II1-111,curve of minimum coefficient K. maximum coefficient K on value ~. Curve III-IlI is the lower boundary of optimal values of the reservoir volume corresponding to the minimum allowable energy level that is necessary for the given technological process. Two zones of optimal parameters are given in this Figure. It is advisable to choose constructive parameters from the upper zone for the case when the terminal velocity ~k of the piston is of great importance as for example for the shock pneumatic machine drive. But if it is necessary to receive a minimum weight or size, thereby achieving high enough piston velocities, it is recommended to choose drive parameters from the lower zone. This may be the case for transport machine drives, manual devices, emergency systems, and so on. In all cases it is recommended that the designer avoid choosing values st3 and N lying outside the indicated optimization zones since for this case there will be a sharp increase of the machine drive sizes or a decrease of the kinetic energy of the moving parts. The diagram of Fig. 5 is given for a load X = 0.05. Analogous diagrams where obtained for other values of the load.
Concluelon The results of the dynamic analysis of typical pneumatic and high speed drives have been discussed. The dynamics of these drives were described by means of a system of non-linear high order differential equations and computers were widely used in solving these systems. A universal computer program for dynamic calculation has been worked out. As a result of calculations carried out, it has been stated that drives with built-in reservoirs develop con-
300 siderably higher speeds than typical drives of a more simple construction. The highest efficiency has a drive with a realeasing mechanism the speed of which is higher by 30-50%. The influence of different design parameters such as the built in reservoir volume, the inlet and outlet ports areas, the weights of the elements, and so on have been investigated. The regions of existence of design parameters, close to optimal ones have been determined. The authors offer a technique for choosing the most essential parameters of drives with the aim of determining their maximum speed. This technique can be used for shock drives as well as for transport ones. In the last case it is important to receive a relatively high drive speed with limited size and weight. The above mentioned calculation methods are used in practical design of a number of drives which at present are in use in industry.
References 1. I. I. Artobolevskii,The choice of optimum machine parameters. Machinovedenje,No. 4, p. 3 (1973). 2. E. V. Hertz, Pneumatic machine drives. M., Machinostro]enie, p. 359 (1969). 3. K. H. Larson, Computer sizes parameters of a pneumatic scram system. Hydraulic and Pneumatics, VoL 17, No. l0 (1%4). 4. E. V. Hertz and V. S. Doldzenkov, Investigation of the dynamics of a high speed pneumatic machine drive with a releasing mechanism. Mavhinovedenje,No. 2, p. 29 (1974).