The Prediction of Dynamic Strain in Leaf-Type Compressor Valves With Variable Mass and Stiffness

Transcription

1 Purdue Uiversity Purdue e-pubs Iteratioal Compressor Egieerig Coferece School of Mechaical Egieerig 1972 The Predictio of Dyamic Strai i Leaf-Type Compressor Valves With Variable Mass ad Stiffess M. Moavei Detroit Ediso Compay R. Cohe Purdue Uiversity J. F. Hamilto Purdue Uiversity Follow this ad additioal works at: Moavei, M.; Cohe, R.; ad Hamilto, J. F., "The Predictio of Dyamic Strai i Leaf-Type Compressor Valves With Variable Mass ad Stiffess" (1972). Iteratioal Compressor Egieerig Coferece. Paper This documet has bee made available through Purdue e-pubs, a service of the Purdue Uiversity Libraries. Please cotact epubs@purdue.edu for additioal iformatio. Complete proceedigs may be acquired i prit ad o CD-ROM directly from the Ray W. Herrick Laboratories at Herrick/Evets/orderlit.html

2 THE PREDICTION OF DYNAMIC STRAIN IN LEAF-TYPE COMPRESSOR VALVES WITH VARIABLE MASS AND STIFFNESS Dr. M. Moavei, Seior Research Egieer Egieerig Research Departmet, Detroit Ediso, Detroit, Michiga Dr. R. Cohe, Professor of Mechaical Egieerig ad Director of the Ray W. Herrick Laboratories, Purdue Uiversity, West Lafayette, Idiaa Dr. J. F. Hamilto, Professor of Mechaical Egieerig Ray W. Herrick Laboratories, Purdue Uiversity, West Lafayette, Idiaa INTRODUCTION The cotiued operatio of a compressor with automatic valves is highly depedet o the lift of its suctio ad discharge valves. The fatigue life of the valves is govered by the maximum level of cyclic stress i the valves durig compressor operatio. The usual practice for the determiatio of valve stress is to experimetally measure the stress level for a give valve desig uder give operatig pressures. Predictio of the valve stress if the valve desig, such as the valve stop depth, or the operatig pressures are chaged ivolves remeasuremet of the valve strai uder the ew coditios. This paper discusses the developmet of a method for utilizig oe set of valve strai measuremets to: 1. Determie the maximum valve stress ad its positio o the valve, ad 2. Predict the value of maximum valve stress for a chage i valve stop positio ad/or operatig pressures. Valve strai eergy aalysis is used to estimate the valve respose before ad after the valve hits the stop. The dyamic strai modes of the valve are used to characterize the valve dyamics. I. GENERAL THEORY -- DYNAMICS OF NON-UNIFORM WIDTH BEAMS The differetial equatio of motio for the trasverse vibratio of beams i bedig is give by Where x is the positio coordiate alog the legth of the beam, y(x,t) is the absolute displacemet at ay poit x ad P(x,t) is the distributed load per uit legth. Usig the approach of modal expasio, the beam motio may be expressed by the superpositio of its atural modes of vibratio (1) y(x,t) = E (x)q (t) =l where (x) are the beam's eigefuctios (mode shapes) ad Q(t) are the participatio factors or time fuctios which deped o the forcig fuctio ivolved. The axial strai due to bedig is obtaied from the relatio or s(x,t) (x,t) ~ 2 Z A 2 ~ (x)q (t) =l Q (t) h 2 E ~ (x)q (t) =l where h is the thickess of the beam ad ~ (x) = (2) (3) (4) d2 > (x) 1 ( )(-) dx 2 A 2 (5) are odimesioal fuctios which represet the shape of the strai modes, ad where is a coveiet ormalizig factor which icreases with frequecy ad has the dimesio (l/i) 2, ad p is the mass desity of the beam material. For the case of a beam with uiform cross sectio both the displacemet modes, <P(x), ad the strai modes, ~(x), may be obtaied from the exact fuctioal relatios [1,5], but for the beams with ouiform cross sectios, these fuctioal relatios are ot usually kow. Researchers i the past [2,3,4] have tried to measure these modes i the laboratory or use equivalet uiform shapes for the shape of their valves. However, experimetal evaluatio of these modes is time cosumig ad most ofte does ot have the high degree of accuracy which is required for a (6) 156

3 successful stress predictio. I this work a mathematical model was selected so that the displacemet ad strai modes of ay leaf-type valve (modeled as beams) ca be evaluated aalytically. The model is geeral ad it represets the dyamics of cotiuous beams with variable mass ad stiffess. I the model the differetial equatios of motio are coverted ito itegral equatios ad these equatios are solved by umerical itegratio o a digital computer. Strai Modes Substitutio of Equatio (2) ito Equatio (1) sults i az [EI(x) 2: d 2 (x) ax 2 =l dx 2 _ Q(t)] d 2 Q (t) + m(x) [ 2: (x) "'l For free vibratio ad _] dt 2 P(x,t) y(x,t) = 2: (x) =l substitutig i Equatio (7), oo d 2 (x) 2: ] =l dx 2 P(x,t) si w t yields m(x) 2: =l (x)w 2 re- itegratig Equatio (9) two times with respect to x, yields EI(x) 2: d 2 (x) (7) (8) (9) = f f m(x) 2: (x) w 2 dx (1) =l dx 2 =l Dividig by EI(x) ad otig that Wu 2 is a costat, with two more itegratios the fial form of Equatio (1) may be writte as (x) = w 2 1 ff EI(x) ff m(x) (x) dx (11) This equatio idicates that for a cotiuous system there are ifiite umbers of modes that will satisfy this equatio, each oe correspodig to a particular frequecy. Equatio (11) may be solved o a digital computer to obtai as may modes ad frequecies desired for ay compressor leaf-type valve with variable mass ad stiffess. For the details of the process ivolved ad compariso of the experimetal ad aalytical results o displacemet ad strai modes, see published iformatio i refereces [6,7]. Time Fuctios The time fuctios or the factors Q(t) i Equatio (2) determie how much each vibratio mode co~ribu~es to the valve motio at ay particular momet i time. For accuracy i the stress predictio, it is also importat that the values of the time fuc ~ios be kow ~o the highes~ possible accuracy. Through a geeral derivatio, it ca be show [7) tha~ the relatio betwee these factors ad the expressios which cotai the ivolved forcig fuctio ad the atural modes of vibratio may be writte as t f (x)p(x,t)dx o m ' f m(x) 2 (x) dx o m m=-=1,2,... The right had side of this equatio ivolves the itegratio of the product of the displacemet modes, <P(-x), ad the forcig fuctio, P(x,t). (12) Ufortuately, i most cases adequate mathematical expressios for the forcig fuctios i the compressor are ot available because the equatios of the gas flow aroud the valve ad through the valve ports ad ~heir iteractio with the valve dyamics is ot kow. To overcome the complexity of Equatio (12), the followig alterative approach was cosidered for the evaluatio of these fuc~ios. The approach used.here calls f.or the direct recordig of the s~rai time history of N strai gages applied to the valve. The umber of gages (N) applied depeds upo the decisio or assumptio of the umber of modes ecessary to describe the valve respose such that the predic~ed results would be withi a reasoable accuracy. I this study from the aalysis of the strai sigal, Figure 3, a three mode approximatio to the valve dyamics was cosidered ~o be sufficiet. However, after obtaiig the ~hree participatio factors ad comparig their magitudes, it was foud that the cotributio of the third mode was egligible. For a value of N=3, the expaded form of Equatio (4) becomes where the umber 1, 2, ad 3 idicate the first, secod, ad third strai modes ad the letters K, L, M correspod to three differet gage locatios (13) o the valve. From this equatio ~he time factors ql(t), qz(t), ad q 3 (t) may be obtaied for ay time, durig valve opeig ad closig, provided that the strai time histories at three locatios K, L, ad M o the valve are kow. These time histories, for ay compressor operatig coditio, may be obtaied by the applicatio of three strai gages at these locatios ad the recordig their correspodig strai sigal o film (oscillograms), Figure 3. The selectio of the locatios K, L, ad M o the valve where strai gages are to be mouted is ot completely arbitrary, as these locatios have to be selected such that the determiat of the strai mode matrix i Equatio (13) is ot sigular or geerally ill-coditioed. The locatio of the highest strai o the valve thus does ot have to be kow or ivestigated by the applicatio of strai gages as this locatio will be predicted whe the etire strai distributio is obtaied by 157

4 substitutig the results of Equatio (13) ito Equatio (4). II. EXPERIMENTAL MEASUREMENTS The two compressors used for the experimetal work of this study had cosiderable differece i their valve geometry, capacity ad applicatio. I geeral, they may be classified i the category of sigle actig, reciprocatig, hermetic compressors usig a halocarbo for a refrigerat. Figures 1 ad 2 show the suctio valve ad valve plate assembly of the two compressors. These valves will be refered to as the AU1Pl2 ad 14 suctio valves respectively. FIGURE 3. Simultaeous Strai ad Valve Tip Deflectio of 14 Suctio Valve. Strai at x/ =.5 i. Showig i Sequece the Momet of Stop Cotact ad Maximum Strai. (Ps,Pd)=(25,65) psig FIGURE 1. AU1Pl2 Valve With Valve Plate FIGURE Compressor Block With Valve Plate ad Suctio Valve Removed Strai gages used i this study were of MA Series, miiature trasducer gages, supplied by Micromeasuremet Ic. These are self temperature compesatio gages with highest level of accuracy ad stability. The experiece ad techiques developed by the Ray W. Herrick Laboratories [3,5] were used for the successful applicatio of these gages for measurig the dyamic strai of the valves durig the actual operatio of the compressor. The top trace of Figure 3 shows a typical strai-time history of the 14 suctio valve. For better uderstadig of the valve motio ad aa~ysis of strai sigal, a variable reluctace proximity trasducer was istalled i the vaive plate to measure the valve tip displacemet [7]. The locatio of the trasducer was primarily govered by where it could be coveietly istalled without iterruptig the ormal gas flow. The bottom trace i Figure 3 is the output of this trasducer which represets the motio ear the tip of the 14 suctio valve. The istrumetatio ad collectio of the strai data for the AU1Pl2 valve show i Figure 1, was origially doe by Alle G. Doige who used the uit for his experimetal model. The details of the istrumetatio ad the parts ivolved are described i his Ph.D. Thesis!3]. I this work his data are used to evaluate the applicability of the mathematical model, ad the eergy cocepts developed here, by compariso of the theoretical ad experimetal results. III. STRAIN SIGNAL ANALYSIS As idicated by Equatio (4) the strai is a fuctio of the positio x alog the valve ad the time t. At the momet whe the tip of the valve first cotacts the stop, t=tc, the strai i the valve is called the cotact strai. The momet of cotact may be idetified by otig the disturbace i the strai sigal whe valve strikes the stop. It was foud that this disturbace was a reliable idicator for cotact strai, I3,7], ad the use of other kids of idicators, like a electrical cotactor i the stop positio of the valve was ot ecessary. The level of the highest strai i the valve occurs at t=~, soo after the valve strikes stop, is defied here as the maximum strai. momets of stop cotact ad maximum strai are show i sequece i Figure 3. which the The I the method of modal aalysis, cosideratio of the correct boudary coditios of the valve is of utmost importace for a successful predictio of strai ad deflectio alog the valve. Experimetal time histories idicate that the tip of the valve, immediately after strikig the stop positio rebouds from the stop ad does ot remai i cotact with it whe maximum strai occurs [7). Thus, the clamped-free boudary coditios were used for both cotact ad maximum strai predictios alog the valve. Furthermore, the chage i strai which occured i the valve durig the time ~t=tm-tc was called the "reboud" strai. The relatio betwee maximum, cotact, ad reboud strai may be writte as 158

5 E; = E: + E; m c r Strai ad deflectio distributio alog the valve for ay momet i time durig opeig ad closig of the valve, may be obtaied from Equatios (4) ad (2) respectively. The two momets cosidered were the momet of valve tip cotact with the stop, t=tc, ad the momet of maximum strai, t=t The values for the strai modes, ~(x), ad themdisplacemet modes, (x), used i the above equatios, were the oes predicted by the mathematical model. The measured strai values at cotact time ad maximum time were substituted ito the right side of Equatio (13) to compute the time fuctios at t=tc ad t=tm respectively. Results are show i Figures 4 ad 5. 1 zooo E Mll FIGURE ZOOD 6 E:xperiJ]le:t e DATA POINT used i the JUOd!!!l - 'l'h<!oory (predicted by the model) AU1Pl2 VALVJ.l 5 l.o.1.1 Theoretical Distributio of Dyamic Strai ad Deflectio at Cotact Time, t=tc' for AU1Pl2 Valve -----LOJ Experi.et ::".::r... Ul Theory (l'4iauol by the -el) AU1Pl2 VALVJ! FIGURE 5. Theoretical Distributio of Dyamic Strai ad Deflectio at Maximum Time t=t, (P,Pd)=(8,3) psig ' m s IV. COMPRESSOR DESIGN CHANGES AND MAXIMUM STRAIN Chage i Stop Positio ~(i.) The depth of the stop positio i the compressor block is a importat factor to cotrol the maximum strai i the valve. The mai reaso for providig a stop is to limit the maximum strai by restrictig the valve motio. The values of the modal participatio factors, Q(t), i Equatios (2) ad (4) o.z. y are fuctios of time ad they will chage whe cotact ad maximum times are chaged, due to the chages i the stop positio. To this ed, the relatios betwee these factors ad the valve stop positio had to be developed. For a two mode approximatio to the valve respose (foud to be sufficiet to predict stress alog the valve) equatio for the valve tip positio 8 ad the valve strai eergy are ad the values of the time fuctios are foud by solvig these equatios as 1 Ql (t) = 2!2 {-Al6 ± [(Al8)2-4A2(A362-2~(t))2]Z} 1 Q2(t) = (2) {6-1(2)Ql(t)} 2 where the costats A's are fuctios of valve geometry ad boudary coditios!7]. (14) From these equatios the time factors Ql(t) ad Q 2 (t) ca be determied for ay momet i time provided that the expressio for the strai eergy, V(t) ad the valve tip deflectio, 6, is kow. I this aalysis, 6 is regarded as a idepedet parameter ad at cotact time it correspods to the stop depth. At this poit it remais to fid a expressio for the strai eergy i the valve as a fuctio of time or valve tip deflectio. Mathematical Expressio for Valve Strai Eergy The valve strai may be characterized by the amout of strai eergy i the valve at ay momet i time. This strai eergy is give by V(t) = 2! EI(x)[h E:(x,t)] dx (15) where EI(x) is the local bedig stiffess alog the valve. By substitutio of Equatio (2) ad (4) ito Equatio (15) it ca be show [7] that the expressio for the strai eergy i the valve up to cotact may be writte as E q2(t) 2 q3(t) 2 2 V(t) = 2 [Cl + C2(q (t)) + C3(ql(t)) +... ]6 1. (16) where the costat C's are fuctios of the valve respose to gas forces, valve geometry ad its boudary coditio. Assumig that the ratios of the relative cotributio of differet modes to the valve respose remai substatially costat as log as the iitial coditios or the valve boudary coditios are ot chaged; the the expressio i the bracket i Equatio (16) remais a costat {7) ad the equatio may be writte as V(t) = ~ K (17) The value of K 2 may be determied from Equatio (17) if a kow valve tip deflectio ad the correspodig valv~ strai eergy is kow. This iformatio 159

6 ca be obtaied from the momet of valve tip cotact with the stop for which the experimetal strai data has bee collected. Desigatig this stop depth 6 (usually o is the stadard stop depth) ad the correspodig cotact strai eergy as V (tc), the from Equatio (17) 2V (t ) c K2 = ;sz ad substitutig back ito Equatio (17) 2V(t) = 2Vo(t) 62 E Ea 2 for ay tip positio up to cotact with the stop. To evaluate the cotact strai eergy V (tc), the (18) (19) values of strai distributio s(x,tc), show i Figure 4 for the AU1Pl2 valve, are substituted i Equatio (15). The result is substituted i Equatio (19) to obtai the valve strai eergy as a fuctio of tip deflectio. Approximate Mathematical Expressios For Maximum Eergy The maximum eergy is defied to be the valve strai or potetial eergy at t=tm, whe maximum strai occurs. At this momet the tip of the valve (o=om) is somewhere betwee the seat ad stop depth ad the rate or slope of the strai sigal is zero, Figure 3. Thus, the kietic eergy for the momet of maximum strai is zero ad all the eergy i the valve is its strai or potetial eergy, {5,7]. Figure 6 shows the valve deflectio curve (No. 3) for the momet of maximum strai. The deflectio curve No. 2, is for the same valve tip deflectio, o=om, usig the precotact deflectio curve. FIGURE 6. Coordiate Set-up ad Valve Deflectio at: Positio 1. t"'o Tip Deflectio '"' Positio 2. t'"'t Tip Deflectio o=om Positio 3. t'"'tml Tip Deflectio o=om The expressio for the valve strai eergy as a fuctio of tip deflectio developed prior is for ay momet i time durig valve operatio up to cotact. Thus, a expressio for the maximum eergy at t=tm may be writte as m v(tm) = v(t)o=o + ~v where 6V represets the et eergy added to the valve by the gas forces durig the time iterval betwee deflectio curves 2 ad 3. Substitutig m (2) the expressios give by Equatio (19) for V(t), whe o=om, ito Equatio (2) results i; 2V(t ) 2V (t) m_ = _o 2 + 2~V E Eo 2 m E The amout of eergy desigated by ~V above, is called the reboud eergy. This is the et eergy (21) added to the valve durig the time iterval ~tm=tmtml where t=tm1 correspods to the momet whe the valve tip is at o=om before cotact with the stop. I high speed compressors, this time iterval is very short. I fact, the total time from t=o.o up to t=tm was about oe millisecod i the AU1Pl2 ad 1.5 millisecods i the 14 compressor; also, the measured time betwee cotact ad maximum' strai 6t=tm-tc, was about.25 millisecods i AU1Pl2 ~d.5 millisecods i the 14 compressor ad remaied almost costat whe stop depth was chaged [5,7]. For a steady state operatio of the compressor durig this very short time, ~tm, the et eergy, ~V, added to the valve durig this time iterval is assumed to be idepedet of stop depth. This eergy may be determied from Equatio (21) uder a kow set of operatig coditios ad stop depth. For this evaluatio the predicted values of strai distributio s(x,tm),(show i Figure 5 for the AU1Pl2 valve) are substituted i Equatio (15) to compute V(tm). Substitutio of V(tm) with the correspodig value of Om (which is predicted by the model at the time whe E(x,tm) is obtaied) ito Equatio (21) results i the evaluatio of the reboud eergy, ~V. It was metioed that the valve tip deflectio om at the momet of maximum strai may ot be the same as the stop depth. Accordig to Figure 6, the rel~tio betwee valve tip reboud, re, tip deflect~o, Om ad stop positio, o, may be writte as 8 = o + r m e The valve reboud was assumed to be a fuctio of the reboud eergy, ad the assumed fuctio was writte as where R 1 is a Equatio (23) taied for a as discussed Total Tip Excursio (22) (23) costat which ca be evaluated from whe the values of re ad ~V are obset of compressor operatig coditios earlier i this paper. The total tip excursio is defied as the maximum valve tip deflectio uder a set of operatig coditios with o stop i the compressor. This total excursio desigated here by ot, is a fuctio of valve ad compressor desig ad operatig coditio. The operatig coditio determies the mass rate of flow which is primarily a fuctio of suctio or ilet pressure. Chages i discharge pressure do ot have sigificat affect o the level of maximum strai which is proportioal to valve deflectio ad thus they are eglected here [7,5]. 16

7 A study of Doige's axperimetal data o the determiatio of valve cotact with the stop ad observatio of the cotactor sigal i the 14 compressor, resulted i the derivatio!7] of a expressio for the total tip excursio as a fuctio of suctio pressure i the form of p 1/4 = (~) t to P so (24) where Oto ad Pso are the correspodig kow valve tip cotact ad suctio pressure respectively. A kowledge of the total tip excursio is ecessary for the determiatio of the maximum level of the eergy stored or give to the valve by the gas forces up to the time t=~, uder ay operatig coditio. Whe the depth of the stop positio is chaged, it will fially reach a poit where the valve tip does ot cotact ay more. If the suctio pressure is icreased, the cotact occurs agai. Therefore, for ay operatig coditio, i.e., suctio pressure, there is a limit o the positio of the stop depth for which the valve tip just cotacts the stop ad does ot reboud. I this case tc=tm ad om=oc=ot Thus Equatios (19) ad (21) become 2V(t) 2 Vo(t) 2 Ps 1/2 -E~ = E --2- to (P) 8 so t=t =t (25) c m This idicates that for the momet of t=tc=tm the valve cotact strai eergy ad maximum eergy are equal ad their values caot exceed the limit give by Equatio (25). This limit is a fuctio of the compressor operatig coditio which is represeted here by the suctio pressure, Ps. V. PREDICTION OF MAXIMUM STRAIN WHEN STOP POSITION IS CHANGED The eergy expressios ad other correspodig relatios which have bee developed so far were applied to the 14 ad AU1Pl2 suctio valves with the oly iput beig measured values of the cotact ad maximum strai from three gages o the valve ad the value of a suctio pressure Pso with its correspodig tip excursio. Figure 7 shows the results of the applicatio of the eergy expressios to the AU1Pl2 valve. This figure represets the cotact ad maximum eergy of the valve as a fuctio of valve tip deflectio. For the momet of cotact, the tip deflectio is the same as the stop depth, but for the momet of maximum strai, the relatio betwee tip deflectio ad stop depth is determied from Equatio (22). The level of the maximum total eergy for the particular operatig coditio show i this figure was obtaied from Equatio (25) with the values of Ps=8 psig ad ot=-.35 iches. Aother way to obtai this level is to compute the total excursio for ay operatig ' coditio from Equatio (24) ad locate it o the horizotal axis of Figure 7. The correspodig value of cotact strai eergy would be the desired level....,.,. c.,..!: QIO NI&U ><,.-,5" ::1:, ::: c ,-- --r-----r-----,---:-:--r---:-.,---,,.--:-:----, Experimet -Theory (model)' :Y ~~~~~-+~ ~- ~OL.. ~~~~~~~~_L~~~~~ ~.1.2 FIGURE 7. Cotact ad Maximum Strai Eergy as a Fuctio of Tip Deflectio for AU1Pl2 Valve. (Ps,P )=(8,3) psig To predict the value of cotact strai whe stop depth is chaged, the correspodig values of cotact strai eergy ad tip deflectio (Figure 7 or Equatio 19) are substituted i Equatio (14) to obtai the correspodig values of Ql(t) ad Q2(t). These values are the used i Equatios (2) ad (4) to evaluate the cotact strai ad deflectio distributio for the particular stop depth or tip deflectio. I a similar maer the maximum strai ad deflectio distributios for the momet of t=tm are obtaied; except that i this case the maximum eergy curve i Figure 7 or Equatio (21) is used. Figure 8 shows the predicted cotact ad maximum strai for the locatio x=.31 iches from the clamped ed, as the stop depth is icreased. Experimetal data are also show i the figure for the purpose of compariso. The dotted lie show i Figure 8 is the result of the predictio of maximum strai by Doige, usig the theory he developed i his study [3]. Figures 9 ad 1 show the etire strai ad deflectio distributio as stop depth is chaged for the momets of t=tc ad t=tm respectively. The icremet i the stop depth was ~o=-.3 iches. Figure 11 shows similar results for the 14 suctio valve at t=tm. 1 Maximum Doige m Ml.l - I g o.a & (~) FIGURE 8. Theoretical Cotact ad Maximum Strai.2. c (sc ad Em) as a Fuctio of Stop Depth Compared to Experimetal Data, (Gage Locatio x=.31 i). AU1Pl2 Valve Experimet e AD Experimet sm --Theory (Model) 161

8 FIGURE 9. 6 MH I l.o '1->/D. Theoretical Distributio Strai ad Deflectio at t=tc, for Differet Stop AU1Pl2 Valve y' ;,_ of Dyamic Cotact Time, PositioEJ. 2. ().2.5" -x.jp... FIGURE 1. Theoretical Distributio of Dyamic MH I 1. Strai ad Deflectio at Maximum Time, t=tm for Differet Valve Tip Deflectios. AU1Pl2 Valve. re=-.91 i, (Ps,Pd)=(8,3) psig,.,. Chages I Compressor Operatig Coditios For the evaluatio of maximum strai as a fuctio of operatig coditios, the followig relatios had to be foud r: b) c) a) The relatio betwee the total tip excursio ad suctio pressure which was show i Equatio (24). The relatio betwee the reboud eergy ad suctio pressure which was foud to be approximately!j.v = 1~ (PS)l/2 E P 1/2 so where!j.v ad Pso correspod to the particular operatig coditio for which the strai data is obtaied. The approximate relatio betwee the suctio pressure ad valve tip reboud which was writte i the form of r eo re = (;")114 so (P )1/4 where reo is the correspodig valve tip reboud to the kow suctio pressure Pso s (26) (27) Equatio (26) was applied to the AU1Pl2 valve to obtai the values of the reboud eergy for differet compressor operatig coditios (suctio pressure). These values were the used i Equatio (21) to evaluate the maximum eergy i the valve as a fuctio of valve tip deflectio or stop depth for ay particular operatig coditio. Equatios (24) ad (25) were also used to determie the correspodig total tip excursio ad the level of maximum total eergy respectively. The results for two differet operatig coditios are show i Figure '-... "' 1 NJl.l.l )( r--.j >...-.~,>..1 o.~."3 he bm (i.) I FIGURE 12. Theoretical Cotact ad Maximum Strai Eergy as a Fuctio of Valve Tip Deflectio ad Operatig Coditios. AU1Pl2 Valve. FIGURE 11. Theoretical Distrihtitio of Dyamic Strai ad Deflectio at Maximum Time, t=tm, for Differet Valve Tip Deflectios. 14 Valve, re=-.54 i, (Ps,Pd)=(25,65) psig 162

9 These eergy curves or their correspodig equatios were the used, as described earlier i this paper, to predict the time factors ad cosequetly the maximum strai ad deflectio distributio alog the valve as a fuctio of stop depth for a ew compressor operatig coditio. A typical result is show i Figure 13. VIII. REFERENCES 1. Bishop, R.E.D. ad Johso, D.C., The Mechaics of Vibratio, Cambridge Uiversity Press, Lodo, Lowery, R.L. ad Cohe, R. "Experimetal Determiatio of Natural Frequecies ad Modes ---, of Compressor Reed Valves", ASHRA.E Joural Yol. 5, No. 2, pp. 9~98, ~ebruary MII { y, ~. 3. Doige, A. G., "A Stress ad Vibratio Aalysis of a Leaf-Type Compressor Valve", Ph.D. Thesis, Purdue Uiversity, Cadma, R.V., "Measuremet Techiques for Measurig Reed Valves Oscillatios", M.S. Thesis, Purdue Uiversity, Meirovitch, L., "Aalytical Methods i Vibratios", The MacMilla Compay, New York, o.s -x.;;.. VI. SUMMARY 1. FIGURE 13. Results of Maximum Strai ad Deflectio Distributio as a Fuctio of Valve Tip Deflectio. AU1Pl2 Valve (Ps,Pd)~(3,1) psig The results of this paper may be used to predict the dyamic stress distributio alog the leaf-type compressor valves whe chages i compressor desig, operatig coditios or small chages i the valve geometry are made. 6. Cohe, R., "The Stodola Method for Computig Frequecies ad Modes of Beams", Mimeographed Notes, Purdue Uiversity, July Moavei, M., "Predictio of Dyamic Strai i Leaf-Type Compressor Valves with Variable Mass ad Stiffess", Ph.D. Thesis, Purdue Uiversity, February Beski ad Roseberg, "Higher Modes of Vibratio by a Method of Sweepig", Joural of the Aeroautical Scieces, Vol. 13, No. 11,-pp: , November, (Primary source for referece 6 above) The mathematical model for ouiform beams is a useful tool for predictig lower frequecies, strai ad displacemet modes of leaf-type compressor valves with variable mass ad stiffess. By measurig the strai time histories o a few locatios alog the valve, the omial strai ad deflectio distributio alog the valve ca be predicted usig the approach of modal aalysis ad the techiques show i this paper. The cotact ad maximum eergy i the valve as a fuctio of compressor desig chages ad/or operatig coditios ca be evaluated usig the results of the eergy expressios ad cocepts discussed here. The kowledge of valve strai eergy ca be effectively used to predict valve maximum stress as a fuctio~ of the above chages. VII. ACKNOWLEDGMENTS The authors would like to thak the staff, techicias, ad the use of the excellet facilities of the Ray W. Herrick Laboratories where the etire project was coducted, ad the Dafoss A/S of Demark for their partial sposorship ad iterest i the project. Note: = d2mx~' lj>"(x) ~ dx 2 163