Experimental Optimization According to Pareto of

Proceedings of IEEE Instrumentation & Measurement Technology Conference, IMTC/97 Sensing, Processing, Networking, Volume 2, pp. 936-941, May 19-21, 1997, Ottawa, Canada

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Experimental Optimization According to Pareto of Proportional Pneumatronics Diaphragm Transducers with Variable Entrance Throttles V. B. Bokov Vicerrectorado de Investigaciones, ESPE, Ave. El Progreso S/N, Sangolqui, Quito, ECUADOR Phone (593-2) 334-960 Fax (593-2)-334-952 E-mail: [email protected]

Abstract - Experimental investigation of pneumatronic diaphragmatic quick-acting transducers is presented as a result of parametrical and structural optimization according to Pareto of the transducers’ time - response and sensitivity. The linear regression model of response of the initial transducer allowed an optimization of its parameters and definition the ways of design improvement with the goal of minimization of time – response. Optimization by the Box - Wilson method was used together with optimization according to Pareto for the creation of principally novel pneumatronic transducers with variable entrance throttles and improved static and dynamical performance. Keywords: experimental optimization, pneumatronic transducers. I. INTRODUCTION Proportional pneumatronic diaphragmatic differential transducers are the most important elements of pneumatronic measuring and control systems. In the measurers, these transducers are located between a pneumatic sensor (nozzle) and an electronic system. Often this transducer is the single mechanical link of transmission in the systems of jet sensors. They have long time - response. This changes for the worse the dynamics of measuring system. Moreover the sensitivity of these transducers is insufficient in some cases of applications. The analysis of linear regression model is one of the most efficient ways of pneumatronic analog device development for desirable dynamical and static properties. Therefore, taking this as a basis, it is possible to improve characteristics of pneumatronic devices and to develop an advanced technology of the transducers’ design. II. DESCRIPTION OF INITIAL PNEUMATRONIC TRANSDUCER Proportional pneumatronic transducers are designed according to the principle of flow rate compensation. The flow rate through the outlet variable throttle of a working chamber compensates the flow rate through the inlet variable throttle of the working chamber. In this case, pressure in the working chamber can be constant or equal to pressure in the back pressure chamber. This device works by the principle of force compensation (Fig. 1).

Fig. 1 Prototype of pneumatronic transducer: 1 - base, 2 - interlayer, 3 - cover, 4 - inserted seat, 5 - stem (rod), 6 - armature stock, 7 - axial armature, 8 - air bearing, 9 - case of inductive transducer, 10 - nut, 11 - conical grip, 12 - constant throttle, 13 - adjustable throttle, 14 - hard center, 15 diaphragm, 16 - stock nut, 17 - spiral spring, 18 - pipe connection

Pneuma-mechanical transducer is the most inertial part of the pneumatronic transducer. An initial analysis [1] has shown that the pneumamechanical transducer, shown in Fig. 2, has the best dynamical and good static performance.

l 3

2

8

P1 B

1

V4

4

9

10

β

5

S23

A V2

C

7

11

6

Figure 2. Principal design scheme of initial pneumamechanical transducer


It is comprised of two pipelines 1 and 2 supplied in parallel from the same compressed gas source 3, through one calibrated inlet orifice 4 and one variable inlet throttle 5. One of the two pipelines, pipeline 1, is the regulating pipeline in the example shown in Fig. 2, and has a regulating outlet orifice 6 to the atmosphere. The other pipeline, or measuring pipeline 2, is fitted with an outlet orifice to the atmosphere by a measuring jet sensor 7 (nozzle-flapper sensor), while the regulating pipeline 1 is fitted with another outlet orifice 6 to the atmosphere. Here the outlet orifice 6 of the regulating pipeline 1 is of constant cross-section, and only one measuring orifice, i.e. 7 of pipeline 2 is used with the device. The measuring pipeline 2 has an entrance throttle 5 in which is mounted a tapered portion 8 axially movable in the throttle so as to modify the effective area of the latter, and forming a part of a rod integral with a very flexible diaphragm 9. The diaphragm is capable of being deformed, without strain, in a chamber 10. The diaphragm 9 divides the interior of the chamber 10 into two chambers B and C linked respectively to the two pipelines 2 and 1. When the flapper 11 is removed, air passes freely through the measuring jet sensor 7, and the tapered portion 8 is in a top position. This leads to a maximum flow rate of air through the device. When the flapper 11 appears in front of the measuring sensor 7, it increases pressure in the chamber B and causes the diaphragm 9 and the tapered portion 8 to move down. The movement will be continued as far as a new balance will be settled between differential pressure, weight of movable parts, and effort of the pressing spring. The moving rod is located in the aerostatic guides (Fig. 1.) for elimination of influence of dry friction. In this case, the measurement is read on the indicating instrument joined to the tapered portion 8. The tapered portion is moved by the diaphragm 9 under the influence of the differential pressure to which it is subjected before attaining its position of equilibrium. The pressure is nearly same for both chambers B and C. Because a very flexible diaphragm is used here and the pressure in the chamber C is nearly constant mass of air in the chamber B is changed very little and the time - response is very short. III. OPTIMAL EXPERIMENT WITH INITIAL PNEUMATRONIC TRANSDUCER Experimental investigation of the initial pneumatronic transducer with the volume of chamber B equal to 45 cm 3 and the weight of moving parts equal to 15 g, has shown that time - response was 60 - 70 mc. These results have been obtained by means of one - factor experiments. In this case only one parameter was changed. Changing the corner β of the tapered portion, an area f 23 of the cross - section of measuring orifice (jet sensor) and stiffness K of the spring did the experiments.

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For influence definition of all essential technological and design parameters of the transducer to its dynamical properties, more investigations were carried out. The first step in creation of a pneumatronic transducer was a theoretical investigation. Then its construction, optimum parameters and conditions of work were chosen. Many parameters have an influence on the dynamic performance of proportional pneumatronic transducer and therefore the best method of investigation is the Design of Experiments. During an active experiment preparation, the prototype of transducer was projected so that its parameters of design and conditions of work could be possible to alter independently according to a matrix of an experimental design. The design of the prototype is shown in Fig. 1. In the active experiment a time - response was chosen as a response characteristic. The input factors of the experiment were: F24 ( X 1 ) - effective area of the diaphragm; f 2 3 ( X 2 ) - area of the cross - section of measuring orifice; f 1 4 ( X 3 ) - cross - section area of inlet orifice of the back pressure chamber C; V 2 ( X 4 ) - volume of the working chamber B; K ( X 5 ) -stiffness of the elastic system;

V 4 ( X 6 ) - volume of the back pressure chamber C and P1 ( X 7 ) - inlet pressure.

The other factors, which have an influence on the timeresponse, were supported constant in the active experiment. In chamber B the parameters of inlet variable throttle were chosen as follows: diameter of entrance orifice d 1 2 = 3 mm, diameter of the rod d 12 r =2.5 mm, corner of the tapered 0

portion β = 35 . These dimensions of the throttle guaranty the ordinary manufacturing and the small expenditure of compressed air. The displacement of the moving rod was equal to +/- 0.3 mm. The mass of the moving details in all tests was maintained equal to 19.8g. Rubber diaphragm was used with thickness equal to 0.35 mm. The jet sensor was the measuring orifice of 2 mm diameter (nozzle) with flapper. An inductive sensor of 234 model (mass of moving details equals to 15.4 g) of the measuring electronic system (217 model of "Caliber" firm, Russia) was used as a second electronic transducer. The LVDT Dimensional Gage Heads of Schaevitz™ Sensors and similar ones may be used as well. The response characteristics of the transducer were obtained using a special stand equipped with a fast-response recorder. In compliance with the preliminary results [1] of investigations values of basic levels and intervals of variation


of the factors were elected (lines 1 - 5 of Table I) with a purpose of creation a linear model of time - response of the transducer. For minimization of the number of tests a 27−4 replica was used in the design matrix [2]. It has the ability of low solution and all linear effects are mixed with interactions of couple. With purpose of removing the coefficients’ influence of the couple effects additional replica was used in

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which all signs of matrix members were changed by opposed ones.

The matrix of the experimental plan and the results of tests are tabulated in Table I (lines 6 - 23). To take this as the basis and with aid of known equations [2] the coefficients of regression were calculated (Table I, line 25). Significance of the coefficients of regression was evaluated by the Student’ s criterion. TABLE I

EXTREMAL EXPERIMENT WITH INITIAL PNEUMATRONIC TRANSDUCER N o1

Factors of Investigation

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Basic levels Intervals of Variation Upper levels Lower levels Codes of Factors Main/Additional Replicas Tests - 1/9 Tests - 2/10 Tests - 3/11 Tests - 4/12 Tests - 5/13 Tests - 6/14 Tests - 7/15 Tests - 8/16 Steep ascent Coefficients of Regression Intervals of Variation Steps Test No 113 Test No 209 Test No 302 Test No 415 Test No 614

F24 , mm2

f 23 , mm2

f 14 , mm2

1232 61 1293 1171 X1

2.327 0.154 2.481 2.173 X2

0.747 0.147 0.905 0.610 X3

-/+ +/-/+ +/-/+ +/-/+ +/-

-/+ -/+ +/+/-/+ -/+ +/+/-

-0.0146 -0.8855 -0.89 1333 1418 1501 1602 1779

0.0134 0.0021 0.002 2.101 1.909 1.723 1.497 1.099

V2 , cm3

K, N/m

26.44 3.14 29.58 23.30 X4

161 52 211 107 X5

-/+ -/+ -/+ -/+ +/+/+/+/-

+/-/+ -/+ +/+/-/+ -/+ +/-

-0.0095 -0.0014 -0.0014 0.905 1.039 1.169 1.327 1.606

0.0040 0.0126 0.013 24.97 23.72 22.52 21.04 18.44

The obtained equation of regression is t = 0.1181 X 0 - 0.0146 X 1 + 0.0134 X 2 - 0.0095 X 3 + 0.0040 X 4 + 0.0021 X 5 - 0.0072 X 7 , (1) where t is a time - response of the transducer. Adequacy of this equation was examined by the Fisher’ s criterion. With the aid of the linear model, the steepest descent was realized (Table I, lines 24 - 37). From Table I it follows the time - response of the transducer was decreased 1.5 times. The linear dependencies of the rod’ s displacements l from variations of a measuring circular clearance S 2 3 (Fig. 3.) show that an absolute sensitivity (the amplification factor) of the transducer is increased with decrease of its time response.

P1 ,

V4 , cm3 8.09 1.36 9.35 6.83

kPa

Tests’ results t, c

X6

49.0 9.8 58.8 39.2 X7

+/-/+ +/-/+ -/+ +/-/+ +/-

+/+/-/+ -/+ -/+ -/+ +/+/-

-/+ +/+/-/+ +/-/+ -/+ +/-

0.143/0.095 0.088/0.151 0.148/0.088 0.137/0.099 0.098/0.122 0.088/0.150 0.140/0.101 0.109/0.132

0.0021 0.109 0.11 149 138 128 115 94

-0.0006 -0.0008 -0.001 8.20 8.30 8.32 8.51 8.70

-0.0072 -0.0706 -0.07 56.9 63.6 70.1 78.0 92.0

0.071 0.058 0.051 0.049 0.043

means that the design of initial pneumatronic transducer is not optimal according to Pareto because it is possible to improve both these characteristics simultaneously. l , mm 0.15 Test No 614

0.10

Test No 302

0.05 0.00

0.20 0.25

0.30

-0.05 -0.10

0.35 0.40 S23 , mm

Test No 113

-0.15

Fig. 3 Static characteristics of initial pneumatronic transducer

So, in this case, it is possibly to improve simultaneously dynamical and static characteristics of the transducer. This


IV. ANALYSIS OF EXPERIMENTAL RESULTS AND SYNTHESIS OF NOVEL DEVICES The tendencies of variation of the factors f 1 4 , V 2 , K, and V 4 were analyzed. The linear model of time - response and the optimal experiment have demonstrated that it is possible to decrease the time - response of the transducer by means of its design improvement. So, a direction of compressed spring’ s force action is directed to meet mobile details’ transference in a moment of transducer’ s operation and this action coincides with the direction of an inertial force’ s action. As a result, the inertia of the transducer is increased. That is why, to decrease the time - response, it is necessary to place the spiral spring in the working chamber B [3], as shown in Fig. 4. This spring’ s placement creates a coincident direction of en elastic forces’ action with a direction of mobile details’ motion. It allows decreasing the influence of inertial force and a timeresponse shortening. In this case the value of the factor V 2 will be decreased and the value of the factor V 4 will be increased by the value of the spring’ s volume. According to equation (1), it will also assist in increasing the speed of transducer’ s response. β

V2

V4

area provides a minimum flow rate, but at the moment of operation the throttle’ s flow area is increased. It provides an increase of pressure in the back - pressure chamber C. At this moment, an acceleration of transition’ s completion is reached in the back - pressure chamber. It also decreases the time response of the transducer. The transducer reaches a maximum increase of a quick-acting work, if the spring is installed in the working chamber and the entrance constant throttle of the back - pressure chamber is replaced by a changeable one (Fig. 5). This device works on the both back-pressure and differential schemes.

l P1

β

V2 V4 P1

A

S23

B C

β

Fig. 5 Principal scheme of novel transducer

l P1

939

A

S23

B C

Fig. 4 Novel design of transducer with spring in working chamber

The results of the transducer tests with the placement of spring in the working chamber and the tests with spring in the back pressure chamber had shown that the transducer in the first case has the speed of response increased 1.5 times than in the second case. These tests were conducted with the spring rates equal to 110 N/m and 210 N/m of the spiral springs, that were situated by turn in the working chamber and in the back pressure chamber. The other very interesting factor for improvement of the scheme is the factor f 1 4 . The increase in value of this factor, according to equation (1), leads to an increase of a flow rate through the transducer. For elimination of this defect, it is possible at the entrance of the back - pressure chamber to install a throttle with alternated geometric flow area during the working process. So, in the initial state the throttle’ s flow

V. OPTIMAL EXPERIMENT WITH NOVEL TRANSDUCER For definition of optimum parameters of the transducer shown in Fig. 5 with minimum time – response an optimal experiment was employed although. The same procedure of optimal experiment was used as for the transducer on the scheme of Fig. 1. Although a fractional saturated plan in assembly with fold-over method has been used. A total list of the active experiment’ s entrance factors was: V 2 - volume of the working chamber B; F 2 4 - effective area of the diaphragm; β - corner of the tapered portions; f 23 - area of cross - section of the measuring orifice; K - stiffness of elastic system; P1 - inlet absolute pressure; V 4 - volume of the back pressure chamber C. Values of basic levels of the factors, intervals of variation and the values of the upper and lower levels were tabulated in Table II. The experimental runs were fulfilled according to the design matrix and randomization. All these tests were repeated twice.


The known equations [2] were used for coefficients of regression calculation and an equation of regression was obtained as t = 0.0610 X 0 + 0.0035 X 1 + 0.0070 X 2 - 0.0020 X 3 0.0030 X 7 , (2) where t is the time - response. Also, using the equation of regression, the steepest descent was realized. As a result, the time - response of the transducer was decreased approximately in five times more than the initial optimal transducer shown in Fig. 1 and 10 ms time response was obtained. Static characteristics of the novel transducer in the tests #700 and #1500 are shown in Figure 6. The absolute sensitivity of the transducer is decreased with a decrease of its time response. So, in this case, it is impossible to improve simultaneously the dynamical and static characteristics of the transducer. It means the scheme of the novel pneumatronic transducer is optimal according to Pareto on the criterion of time response and on the criterion of sensitivity. 0.15

l, mm Test No 700

0.10

0.00 0.10

0.15

0.20

0.25

0.30 S ,mm 23

-0.05 -0.10 Fig. 6 Static characteristics of novel pneumatronic transducer

l P1

β

A

V2 B

γ

V4 P1

C A

Fig. 7 Principal scheme of transducer with increased sensitivity.

For the transducers with variable entrance throttles, it is easy to reach the improvement of static characteristic [4]. In case if it is necessary to increase the sensitivity the needle of cone of variable entrance throttle must be installed in the back pressure chamber as shown in Fig. 7. Here the cone β of the throttle of the working chamber B must be greater than the cone γ of the throttle of the back pressure chamber C in order to insure the stability of the transducer. VI. CONCLUSIONS An adequate linear regression model of the initial transducer’ s time - response was founded, which allowed the optimization of its parameters and definition of the ways for design improvement with the aim of achievement the minimum time – response. For the pneumatronic analog diaphragmatic transducers the best effect of the quick-acting rise is reached with synchronous optimization of parameters and scheme according to Pareto, that is based on the linear model of response of the transducer, on the analysis of factors of regression and on the structural analysis. Optimization of the initial transducer’ s construction was used for creation of the novel pneumatronic transducer with variable entrance throttles and improved static and dynamical properties.

Test No 1500

0.05

940

S23

REFERENCES [1] O. B. Balakshin, V. S. Pogoreliy and B. I. Bokov. "Investigation of quick-acting pneumatic transducer for dimensional measurements". - Transactions: Computer Simulation Tasks of the Machine-Building, Nauka, Moscow, 1976, pp.187-196 (in Russian). [2] V. V. Nalimov and N. A. Chernova. Statistical methods of optimal experiments planning, Moscow, 1965 (in Russian). [3] B. I. Bokov and V. B. Bokov "Pneumatic differential transducer for dimensional measurements", USSR Patent No 991163, filed January 23, 1982, Int. Cl. G 01 B 13/02 (in Russian). [4] O. B. Balakshin, B. I. Bokov, V. B. Bokov and V. A. Serebrennikov "Pneumatic differential transducer for dimensional measurements", USSR Patent No 1116314, filed September 30, 1984, Int. Cl. G 01 B 13/02 (in Russian).


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TABLE II.

EXTREMAL EXPERIMENT WITH NEW PNEUMATRONIC TRANSDUCER No1

Factors of Investigation

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Basic levels Intervals of Variation Upper levels Lower levels Codes of Factors Main/Additional Replicas Tests - 1/9 Tests - 2/10 Tests - 3/11 Tests - 4/12 Tests - 5/13 Tests - 6/14 Tests - 7/15 Tests - 8/16 Steep rising Coefficients of Regression Intervals of Variation Steps Test No 143 Test No 300 Test No 500 Test No 700 Test No 1100 Test No 1500

V2 ,

F24 , mm2

βo

f 23 , mm2

K, N/m

P1 , kPa

V 4 , cm 3

23.43 1.89 25.32 21.54 X1

3340 133 3473 3207 X2

32.5 2.5 35 30

X3

1.492 0.079 1.571 1.413 X4

161 52 211 107

55.0 5.0 60.0 50.0

27.06 2.51 29.57 24.55

+/+/-/+ -/+ -/+ -/+ +/+/-

+/-/+ -/+ +/-/+ +/+/-/+

+/-/+ +/-/+ -/+ +/-/+ +/-

+/-/+ +/-/+ +/-/+ +/-/+

+/-/+ -/+ +/+/-/+ -/+ +/-

+/+/-/+ -/+ +/+/-/+ -/+

+/+/+/+/-/+ -/+ -/+ -/+

0.076/0.058 0.060/0.070 0.044/0.076 0.060/0.058 0.060/0.064 0.066/0.058 0.074/0.044 0.054/0.064

0.0035 0.0066 0.007 22.43 21.33 19.93 18.53 15.73 12.93

0.0070 0.9310 0.93 3207 3061 2875 2689 2317 1945

-0.0020 -0.0050 -0.005 33.2 34.0 35.0 36.0 38.0 40.0

0.0013 0.0001 0.0001 1.478 1.462 1.442 1.422 1.382 1.342

0.0008 0.0416 0.04 155 149 141 133 117 101

0.0013 0.0065 0.007 54.0 52.9 51.5 50.1 47.3 44.5

-0.0030 -0.0075 -0.008 28.20 29.46 31.06 32.66 35.86 39.06

0.058 0.044 0.028 0.020 0.014 0.010

cm3

X5

X6

Results of Tests t, c

X7