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(1)Propagation of Impact Vibration in Capillary Tubes which takes place at the Opening of High Pressure Baal Valves (2nd Report) By. Fumie Institzate. NOGUCHI of Technology. .. (Received September 9, 1968). .. abstract After the experiments reported last time, which were made with the capillary tubes of various inner diameters, and miscellaneous lengths, the author continued the same experiments with longer tubes and at last terminated them unti! the pressure amplitude became zero. This time he made also some theoretical investigation about them.. S1. Recent experiments and their termination Whi!e we were making the experiments, reported in g 1. of last report with the tubes of various inner diameters and miscellaneous lengths, as in g3. of last report, and with longer tubes, we found that the pressure amplitude of the vibration successively diminished with the increasing of tube length. Then we chased this successive diminution and at last made the experiment with the tube lengths where the pressure amplitude became zero and terminated the experiment. e. The following table shows the tube lengths used this time: Tubes with inner diameter O.4mm........3, 5, 10, 20, 30, 40, 50, 60, 80, 100, 120,･140, 160, 180, 200,.220, 240, 260, 280, 300, 320, 340, 360, 380, 400, 440,. .. 480, 500, 540, 580, 600, 620 (32 pieces). Tubes with inner diameter O.6mm........3, 5, 10, 20, 30, 40, 50, 60, 80, 100, 130, 170, 200, 240, 280, 300, 400, 500, 600, 800, 1000, 1200, 1500, 1800, 2000 (25 pieces) Tubes with inner diameter O.7mm........5, 10, 20, 30, 40, 50, 60, 80, 110, 135, 170, 200, 225, 260, 300, 340, 380, 420, 460, 500, 560, 640, 700, 800, 1000, 1200,. 1500, 1800, 2200, 2400 (30 pieces) Tubes with inner diameter O.8mm........3, 5, 10, 20, 30, 40, 60, 80, 100, 140, 160, 180, 220, 260, 300, 340, 380, 420, 470, 520, 600, 700, 800, 1000, 1200, 1500,. 1800, 2100, 2400, 2700, 3000 (31 pieces).

(2) E NoGucHI. 26. Tubes with inner diameter O.9mm........5, 10, 20, 30, 40, 50, 60, 90, 110, 130, 150, 170, 210, 240, 260, 300, 340, 380, 420, 460, 500, 550, 600, 700, 800, 1000, ' 1200, 1500, 1800, 2100, 2400, 2700, 3100, 3700, 4000, 4400, 5000, 5300 (38 pieces). Tubes with inner diameter 1.0mm........5, 10, 20, 40, 60, 80, 100, 140, 180, 220, 270, 320, 380, 440, 500, 560, 620, 680, 740, 800, 1000, 1200, 1500, 1800, 2100, 2400, 2700, 3000, 3300, 3600, 3900, 4200, 4600, 5100, 5600, 6000, 6400 (37 pieces). After all, the experiments were made with the tubes of six sorts of diameters, with 193 pieces, and eight times with each piece, which amounted to 1544 times.. In addition, a great many experiments were made under same constant condition with the tubes of various diameters, to search for the maximum pressure amplitude by the same accumulated pressures. g2. Results of Experiinents. .. 1) Wave form of Impact Vibration As it is almost impossible under the circumstance of limited space to report all the photographs of wave forms of pressure vibration which appeared in oscilloscope in every one of the 1544 experiments mentioned above, this time we give only the results of experiments in Fig. 1 made with tubes with. inner diameter O.7mm as typical examples. In them, there are two sorts of photographs of wave forms taken a) at the open end, or Base Position of last report, of the test capillary tubes, and b) at the bottom end, or Top Position of last report, of the test capillary tubes.. a) The wave form taken at the open end of the test capillary tube. In either case the light point proceeds along the abscissa from point X to Xi in Fig. 2, at point a it bounds up in the direction ab, it reflects at point b, and. thereafter from this point it begins a damped vibration b-c-d-e-f-g ･･･, whose amplitude decreases rapidly and whose wave length (this time it coincides with the period, as the unit of abscissa is ip micro-second, the time) increases gradually. This phenomenon is thought to be as mentioned･in g 1 of last report, by after accumulating the pressure in the liquid in capillary tube and making in equi!ibrium the elastic forces of tube and liquid, and suddenly opening. the non-return ball valve, the elastic energy accumulated in the tube and liquid is altered into the kinetic energy of !iquid which escapes from the Neqative Pressure. (-) x. b i 1 l ' '. A.. f c. `. e. ao. g. xt. Time. d. Fig. 2.

(3) i. 27. Impact Vibration of Ball Valves. valve, and the negative pressure head ob in Fig. 2, and when this negatiVe pressure head is changed into kinetic energy, it brings about the vibration b-c-d-e-f ･･･.. Judging from the common sense, ob must be taken downwards to show that it is a negative pressure, but by the circumstance of oscilloscope, we take it upward for convenieneei sake.. The height ob of the pressure head is in proportion with the pressure accumulated in the capillary tube.. When the accumulated pressure is 100kg/cm2, it amounts to 70.6kg/cm2, when 80kg/cm2 56.5kg/cm2, when 60 kg/cm2 42.4kg/cm2, and it always 70.6% of the accumulated pressure, but when 40 kg/cm2, it amounts to 26.75kg/cm2 .. and diminishes to 66.5%.. .. It was ascertained after a great many experiments that the heights of these pressure heads are always the same and constant with every tube inner diameter and length, when the accumulated pressure is once given under the same and constant condition of the experimental apparatus and by the measurement in a day.. b) The wave form taken at the bottom end of the test capillary tube. As far as the tube length remains short, this wave form much resembles. the wave form taken at the open end of test tube as shown in Fig. 2. But as with the increase of tube !ength, the amplitude diminishes and the wave. length increases. In general it takes the wave form asshown in Fig. 3. With the extension of tube length, there appears a roundness at point a, the lengths. ao and ac get longer, ob contracted, and the whole wave form gets the roundness. With a certain very long tube, the amplitude becomes at last zero, and the wave length very long at the same time. The diminution rate of the maximum amplitude ob is large as far as the tube is short, but is small when the tube is fairly long.. b. p. t 1 1. a. f c. o. ,. e. g. d Fig. 3. The length of the first half period ac of the wave form gets longer with the increase of the tube !ength. Once a set of tube inner diameter and tube length is set to a certain value,. this length of half period is always the same and independent of the accumulated pressure, but changes with the change of voltage to the oscilloscope. 2) Relation between the tube length and the maximum pressure amplitude. By changing the length of tubes with various inner diameters and plotting the maximum pressure amplitude which corresponds to that length, we can.

(4) 28 F. NoGucHi get the diagram shown in Fig. 4.. Fig. 4-1, Fig. 4-2, Fig. 4-3, Fig. 4-4, Fig. 4-5, Fig. 4-6 are the arrangements. of the results of experiments measured with tubes with inner diameters O.4 mm, O.6 mm, O.7 mm, O.8 mm, O.9 mm, l.O mm respectively. There are two sorts of lines in these figures; a) Maximum pressure ampli-. tude of wave form measured at the open end (= Base Position) of tubes. b) Maximum pressure amplitude measured at the bottom end (= Top Position) of tubes.. The lines of a), by the reason of g2. 1-a), independent with tube length, are four straight lines parallel to the abscissa.. The lines of b), four in each figures, by the reason of g2. 1-b), as far as the tube lengths are short, the maximum pressure amplitude rapidly reduces with the increase of tube length, all the lines in each figure in this region are the lines nearly like straight lines which descend steeply to the right. But this descent will be relaxed in due course and then followed,by curves which concave upwards, and then at last, as the rate of diminution of the maximum pressure amplitude is very small, they become lines almost nearly like straight lines which slopes slightly to the right. But by observing in more details, we find that this decreasing is by no means smooth, but poceeds gradually stepwise.. -･. .. These four lines of maximum pressure amplitude measured at the bottom end of tube meet at last at one point which corresponds to the tube length where the maximum pressure amplitude becomes zero. Moreover, in the case of tube with larger inner diameter, the rate of decreasing of the maximum pressure amplitude is smaller, and the lines incline. by more maximum Fig. and the. easy grade from the beginning, and the tube length at which the pressure amplitude becomes zero is much longer. 5 shows the relation between the tubes of various inner diameters tube lengths at which the maximum pressure amplitude at the bottom. end of tube becomes zero. ` lnner. Dia. O,4mm. O.6mm O.7mm. O,8mm. O.9mm LOmm' Io. 20. 50 40. 50. Fig. 5 *. ***. 60. TUBE LENGTH Meter. *.

(5) lmpact Vibration of Ball Valves 29 By a great many experiment made afterwards, it was verified, as in g2. 1-a), the maximum pressure amplitude at the open end of capillary tubes does not vary with the inner diameter and length of tubes, and we meet the result that four each line takes the same value in every figure respectively.. And besides, by the experiments with the tubes shorter than 5cm, it was also verified that the maximum pressure amplitudes measured at the bottom end of tubes get very close to those measured at the open end when the lengths of tubes are extremely short, and we understood that when we extend the curves b) to the direction where the tube length is zero, this may coincide with the. maximum pressure amplitude measured at the open end of tubes at the point where the tube length is zero. Then, it needs correction to Fig. 10-15 in last report.. .. 3) Relation between the tube length and the first half period of vibration. wave. The length of first half period ab of the vibration measured at the open. .. end of capillary tubes is a!ways the same in every case. But the length of first ha!f period ab of the vibration measured at the bottom end of capillary tubes changes with the inner diameter and length of. '. a. But they are invariable in spite of the change of the accumulated pressure, when the inner diameter and length of tube remain the same, and the diagram of relation between the tube length and the first half period is only a line with respect to the tubes with same inner diameter. But as they vary with the change of voltage to the oscilloscope, we converted them in terms of 5 volts by multiplying the conversion rate every time when the voltage is altered. Fig. 6 is the diagram showing this relation, and in it the results measured with the six sorts of tubes of different inner diameters are gathered and arranged in one chart. In this diagram, it is shown that every line starts from the same point,. as the first ha!f period of every sort of tubes is the same when the tube length is zero, and every line terminates at points of same height of ordinate, `. as the first half period ab of every tube is the same at the tube !ength where !. the maximum pressure amplitude becomes zero. These six lines passes stepwise between these points, these may sometimes depend on errors of the experiment, but by observing them on every line, we can not think that they are caused only by errors. g3. Theoreticai consideration. : : 1 1. 1. l. 1. 1) The wave form of the impact pressure vibration at the open end of capillary tube.. The initial negative pressure head of the wave form of pressure vibration at the open end of tube is thought to depend on the fact that the elastic energy.

(6) 30 F. NoGucHI. by the stresses caused by the expansion of tube wall and the compression of liquid by accumulating pressure in the liquid in tube is altered into kinetic energy by the sudden throw open of valve and this energy is again changed into the negative pressure head and the kinetic energy of liquid which escape$. fromthevalve. ･ . '. Let. L: Length of tube D: Inner diameter of tube. b: Thickness of tube wall '. p: Pressure accumulated in the tube and liquid . A: Interna! sectional area of tube={i-D2. .. E: Modulus of longitudinal elasticity of tube wall material'. * Energy accumulated in the liquid column (=The work necessary to compress the !iquid) Since the pressure in tube ascends from zero to p Mean ascension of pressure in the tube==-li-P. '. Changes of volume per unit volume of liquid==-ft'. The work necessary to compress the liquid w,==->-･-il-Il ･AL= A2LKPLiL. * Energy accumulated in the tube wal! (=The work necessary to expaRd the tube) E!ongation per unit length of tube= ll}-･DLP' 2blL =-2Db-PE-. ' . Length increased at the circumference of tube==zD.-2-DbPI-=-Zi//i-IPWork necessary to expand the tube, as the e!ongation of tube increases. a. evenly from zero. 1 PDL 2AP-AP2DL. W2=-2-'-2nd' DIE-) --2JETE. ･･. 'This accumulated energy w,+w, is changed into kinetic energy. The tube expanded as shown by the dotted line in Fig. 7 by the accumulation of pressure into the tube conbt,ra fiS, a,&d,.dr?;.:O,r, e.S.tdO.t,h,ehglifi..",",a.i.,St.atie,7,,ia,S.fih,O.W.",. li.,I,./1l･l,ll･i･lili･ll'itl,l.is,ltl,tji:,:'i.,･t,:,-.itri:.i:･i',i-tr'.l,il･l･1/r･li,,iIP. of liquid whose volume is As (s is the displacement. of the liquid) dashes out from the tube, and the ･pressure restores to zero, the original state. Fig. 7. Assuming that the pressure changes linearly the mean pressure -l}-p, and. 'from P to zero, we can say that there acts always.

(7) Impact Vtbration of BaH Va!ves 31 the amount of kinetic energy is. -g-.A,- pes ･････････a) and at the same time there happens an impact caused by the sudden movement of liquid whose amount is. g,As .. p,A,s ..(2) t: Time during which the liquid has moved. Then. PSS =.,+.,= A2LKP2+ A2LbPiD .........(3) +. By this kinetic energy the liquid moves one more distance s, and the energy changes into the negative pressure head P,/r and the kinetic energy of liquid escaping from the valve v2/2g, and after all. .. Air( ;O + 2V.2 )= A2.iP2 + "2ibPiD. eo + 8,2 -S(k+ bDE) ･･･････-(4) Fig. 8 represents the wave form of the pressure vibration of liquid felt by the strain gauge pressure indicator at the open end of tube as shown in Fig. 2.. bx % lXXxx f(t). lcSs-.---.e-----f g.-..-.--. a lo l. 1. :. lb. l. l. Fig. 8 In Fig. 8 it is thought that the amount of P, is indicated by the height. l. l. 1 ,/z. L･ i<<S<. i. .. ob, as the statical pressure p is not felt by pressure indicator and the abscissa indicates the pressure zero, a is the point when the impact strikes, ab is the path to drop down to the negative pressure P,, ao is the time of descent to P,.. After the point b, there follows a damped vibration b-c-d-e-f･･･, whose amp!itude diminishes rapidly and whose wave length increases gradually. Then it is thought that it is not a damped vibration of simple harmonic. motion but a damped vibration of the sum of great many simple harmonic I. motlons. The vibration after the point b, taking P as pressure at an arbitrary time.. P= P,f(t) (cos Pt+cos 2Pt+ cos 3Pt ･･･) ･････････( 5) f(t) is the function of time t and decreases rapidly with the progress of time.. l l l.

(8) 32 . 'F.NoGucHi Taking the whole progress inc!uding ab of Fig. 8 as a Fourier Series of the sum of great many simple harmonic motions, (5) can be re-written as. follows: '. P== Pi sin Pt+ P, sin 2Pt+ P, sin 3pt ････････････( 6). 2) Vibration of liquid in an one end closed tube. The vibrations of liquid in the tube which take p!ace at the sudden opening of valve at the open end of tube after accumulating pressure into the tube are. thought to be the following two sorts and others. a) Vibration of liquid particles in the tube caused by the contraction of tube wall and the expansion of liquid.. b) Vibration by the impact of negative pressure head caused by ･the sudden opening of valve at the open end of tube. The vibration in a) is thought to be nearly equal to the free vibration, but it damps rapidly by the friction by the movement of liquid particles in the tube.. In this vibration, the liquid particles in the tube are thought to move in miscellaneous directions with various velocities. But analysing them in the radial direction and the axial direction of tube, the former may be very small and neg!isible, and it is thought that this is a composition of a lot of longit i+['i.il･`Illt'･i':..li:.i':1.:":･":-',lii,::j:-:';.l'.isr'.il:Ei. -::;. 'f."I,:1:i:igg･L.::':r-i:t.:t's. :･ :･･. cc-----ldxP-. tudinal vibrations in axial direction with various amplitudes and periods. In this case, it is the vibration in a one end closed pipe as shown above in Fig. 9, and resembles the case of free longitudinal vibration in a one end fixed elastic bar as shown beneath in Fig. 9.. . This time, each liquid particle in the tube has. t' =T --- r= theirinertiaandelasticity,andmakesitsownarbitrary vibration. This phenomenon is thought to be the same as the composition of many mass particles and elastic springs as shown below in Fig. 9.. 1. Fig.g Putting the frictional resistance of liquid out of consideration, and /J. Let u: Displacement of liquid partic!e in axial direction x: Position of an arbitrary section dx from the closed end. t: Time From the relation Inertia force of dx oo Elastic force of liquid. Oo2t#=a2aO.2% a2=CiS. --･--(7>. The general solution of this equation of motion with respect to the section dx IS. '.

(9) Impact Vibration of Ball Valves 33 u=sin -4tlitx-(Ai cos zT2alt+Bi sinit zn2alt). The longitudinal vibration of the whole liquid column is the sum up of these equatlons. u=,=,?i,1,,...sinZ2ZIX(AicosZT2alt+BisinM2alt) .........(s). ' Putting x=:O in this equation (8), we get u=O that means the liquid particles do not make'vibration at the closed end.. Put x=l, (u)=ex, (a)= O, we get the maximum value. t==o t=o. ' '(.u=),-9El(i+-,'-+,i,+･･･)--9LZ-i･-Z,iil-Ei ･････････(g). '. t=o From these result, we understand that there is a loop of vibration at the open end of the tube, where the liquid particles make the most violent vibration, and the vibration more diminishes at deeper places, and at the innermost place no vibration exists, and there is a node of vibration there. But it is generally. .. guessed that there exist maximum pressure at the node of vibration, and therefore a pressure vibration whose phase and period differs to that of the impact vibration. But this vibration is thought to be very smaller than the impact vibration. The vibration in b) corresponds to the case of giving a sudden impact on the liquid fi11ed in a one end closed tube at its open end, as shown above in Fig. 10, and resembles the case hammering an elastic' L bar whose other end is built-in a wall, at its free end.. In this paper, the impact acts in reality in;-+-.: nega- Fo --u:--;-:'. :-.:s--;:t. tive sense, but in anyway in this case as the liquid in. 1tt t----tt. i':･l･'i,//;ill,/i.}'-itagSl-'. ---E---. x-N. thetuberesistsagainst the impact force with its i!i= ---i-inertia force as one body, the remainder force of the substruction of inertia force from impact force acts upon every particles of liquid and raises the pressure. -- -. p. i l t. -Fo Fo. Fig. 10 by compression or causes a negative pressure by tension. This matter may be represented by the mass body and spring shown beneath in Fig. 10.. The repulsion of this inertia force is more large or violent. with more rapid and severe impact force, and when we fit up a pressure i. 1 !. indicator at the bottom end of the tube, it feels the pressure, positive or negative, caused by the'residual force of subtraction ofelastic force from impact force.. Put the friction between the mass particles out of consideration and take the amount (impact force--inertia force) as m times of inertia force,. ･t .t. - mrSA O,2,¥ -=AKi3i", This can be re-written in the same form as (7) [. 1. i. l. i. ･････････(10).

(10) 34 ENoGucHi. 02u 02u ,., ai Ot2 Ox2. .........(11). Put some conditions at the instant of impact, we get the general so!ution. u =f(at- x) +f, (at+ x). f(at-x) means that a curve represented byf proceeds with its form unchanged and with a constant speed. a-V I;g ･････････(i2) from the open end to the closed end of tube, and f,(at+x) proceeds in inverse direction. In this case we may consider only about f(at-x). 3) Pressure vibration at the bottom end of tube. The vibration in this paper is the overlap of these in a) and b) in g3. 2.) The wave form of pressure vibration felt by the pressure indicator at the bottom end of tube is thought usually that every point on the wave form of pressure vibration at the open end of tube is transmitted to the bottom end and reformed into it, it can be also thought that many pressure simple harmonic vibrations of (6) in g3.1) are transmitted with the velocity a. There can be thought two cases about the inertia force of liquid: -of the mass of the whole liquid acts as one body, -of the mass of each liquid particle or their group. The former is the case in which the impact is rapid and strong, the latter is the case of s!ow or weak impact. The weaker ones of the pressure sirnple harmonic vibrations are gradually diminished by the inertia force of the liquid mass particles on their ways of propagation. . Hitherto we have not considered about the frictional resistance caused by the mutual collision and rubbing when the liquid particles in the tube make movements, but in fact we can by no means neglect them in the case of capillary tubes. But it is generally diflicult to solve the equations of motion containing the frictional resistance, and in this occasion we suppose only that they will be a rapid damped vibration. It is thought that the pressure transmitted to the bottom end of tube will be more decreased than before by the addition of frictional resistance after the off-set of inertia force from the impact force.. The diminished pressure vibration moreover may interfere with the pressure vibration at the bottom end of tube, as their phases and period generally differ from each other, their pressures once are added up, and at another time substructed. 4) Hypothesis about the distribution of moving velocity of liquid particle in a one end closed tube. We make now consideration about the moving velecity of liquid particle, as the friction of liquid in g3. is more increased with the augmentation of moving velocity of !iquid particle. As the vibration in this experiment is thought to be the overlap of the vibrations a) and b) in g3. 2), the movement. ,. -.

(11) Impact Vibration of Ball Valves 35 of liquid particle in the tube is also thought to be the overlap of movements of these two. The vibration of a) is most active at the open end of tube and zero at the bottom end, then the liquid particle moves violently at the open S,".d.,Oi, 2",b,e,.a,n,9,,IPeiE.,Ve,i,Of,it,Y,.ifi,,Ma,,Xi.m,:r?fi,b.",t,,gr,a.d",a,ii,y,.b,e.;o. mes gentie. ' A more violent movement of liquid particlbs can be supposed in the vibra-. '. '. tion in b), in this case this movement is most violent at the open end but becomes more gentle towards the interior, but this movement is suppoSed to be sudden!y damped at a certain depth, and thereafter there exists' only a weak movement and this continues to the bottom end, as in the case of waves on the sea surface or stone throwing in the pond. Therefore, in the case where these two movements a) and b) are overlapped, the moving velocity of liquid particle is maximum at the open end of tube, and diminished towards the interior, but suddenly damped at a certain depth, and thereafter there continues a weak moving velocity to the bottom end.. 5) Hypothesis about the frictional resistance for the liquid vibration in tube.. Much is not yet known about the frictional resistance for the liquid vibration in tube, and we take it after the frictional resistance of water flow. h=2 l: Length of tube d: Inner diameter of tube. l v2. 72g ,.･. ･････････(12). '. v: Mean velocity of liquid particles in tube ,. Z: Frictional coefficient. v is thought to be the mean of the moving velocity of many liquid partic!es in tube in g3. 4). o. From the view in 4), the frictiona! resistance is very large at the open end of tube, but diminishes toward the interior as is said in 4). When the tube is very short, the liquid particle in tube is thought to move very lively almost over the whole length, and the frictional resistance per unit length of tube is very large.. On the contrary, when the tube is fair!y long, from the view in 4), the frictional resistance is very large at the open end of tube, but much reduced eoVebreas&earlti in length, the frictional resistance per unit length of tube is thought. About the difference of frictional resistance of liquid by the cases when the inner diameter of tube is larger or smallerj it is thought as follows: The frictional resistance is caused by the moving velocity of liquid particles, gfndtutbhgSaveOiginbgy VtehieOCggYseiSotfhOfiUoee.XgOwbaetger?ximum at the center of the section. In the case of the tube with larger inner diameter, gradient of moving.

(12) 36 ENoGucHi. velocity from the center to the circumference may be gentle, but in the case of tube with smaller inner diameter, this gradient may be steep and the zone of' small velocity may be narrow. Then, on the average, in the case of tube with smaller inner diameter, the moving velocity of liquid is thoght to be larger,. and the frictional resistance is' larger. ' On the other hand, consideration about the friction of liquid particle based on the tube wall, this is thought to be proportional with inner circumference of tube, consequently inner diameter d.. The movement of liquid particle is caused by the impact, and the impact is, by eq. (2) of g3, proportional with the internal sectional area A,-consequently the square of inner diameter d2. Accordingly, the frictional resistance per impact is d/d2=1/d, that is, inverse-. ly proportional to inner diameter. Then it is thought when the inner diameter of tube is large, the frictional resistance is small, and when the diameter is･ small the resistance is great. Frictional resistance is the addition or multiplication of these two. But in actuality, it is not always so simple as mentioned above, there may also act more factors other than those forementioned, in reality, with a very thin tube such as with inner diameter O.4mm, it is proved by experiment that the resistance is far larger than with the tube with slightly larger diameter.. e. .. 6) Relation between the tube length and the maximum amplitude of pressure vibration at the bottom end of tube. As the elastic force of the liquid in the tube is the remainder of the sub-. struction of inertia force and friction resistance from impact force when we take into account the frictional resistance, this force will be less than in the case of taking no account of the friction resistance, consequently reduction of pressure felt by the pressure indicator at the bottom end of tube. Final!y the maximum amplitude of pressure vibration felt by the pressure. indicator at the bottom end of tube becomes smaller than that at the open end of tube.. ". a) As far as the tube length is short. In an extreme case that the tube length is zero, as both the inertia force and the frictional resistance of liquid are zero at this time, the impact force is. directly transmitted to the pressure indicator at the bottom end of tube as it. is, the wave form felt by it must coincide with the wave form at the open end of tube. When the tube length is slightly longer, inertia force only increases in proportion to the increment of tube length, but the frictional resistance, whose value is large as the movement of !iquid particle is very active all over the length, increases with this large value in proportion to the increment of tube. !ength, and the maximum amplitude of wave form of pressure vibration fe!t by the pressure indicator at the bottom end of the tube is rapidly diminished, and the lines of maximum amplitude in the region of short length in fig. 4. F. l. i 1. !. I.

(13) " tr･. Impact Vibration of Ball Valves 37 rnake steep descent to the right.. b) When the tube length is medium long. When the tube length is beyond a certain length, there broken out an inactive part of the movement of liquid particle at the innermost of tube, and ･the fractional resistance per unit length of tube decreases, and the rate of .increase of the fractional resistance becomes dull, but on the other hand the inertia force of liquid reaches to an effective amount at such a length, and increases in proportion to the increase of tube length, the result of the sum of these two is that the lines in Fig. 4 gradually lie to the right side and make ･curves concave upward. c) When the tube length is fairly long. In this case, inertia force of liquid has reached to a great amount but yet iincreases in proportion to the increment of tube length, frictional resistance also has reached to a big amount but its rate of increase is very smal! compared with the case of short length, and makes little increase with the increase. .. ef tube length. - .. .. The rate of increase of the sum of these two is therefore very little. In Fig. 4, all the lines in this region slope very gentle to the right nearly to the horizon, as the decreasing of pressure amplitude is small when the tube length is enough long. But by observing carefully these slQping lines, we find that decreasing is 'by no means smooth, but progresses graduallyi'stepwise, this may be thought. by the reason in next 7). When the tube length is extremely long, the .maximum amplitude felt by the pressure indicator at the bottom end of tube iought to be zero, when the sum of the inertia force and the frictional resistance. iis equal to the impact force. ･ But in reality, there appear and disappear intermittent a very slight pres:sure wave, this may be because of the remaining of pressure vibration at the. ,bottom end of tube. ･ d) About the inner diameter of tube. '. i. The amount of inertia force felt by the pressure indicator is independent -of the change of inner diameter of tube, as it is the inertia force per unit. ". L. '. sectional area of tube. . '. In the case of tube with larger inner diameter, by the reason above, the. -frictional resistance may diminish not only inversely proportional to the inher. L. tdiameter of tube, but also proportional to the square of the reduction of. i. velocity, as the moving velocity of liquid particle is more slow, and the rate. i. ef reduction of maximum pressure amplitude by the case of tube with larger .inner diameter is smaller than in the case of smaller inner diameter, declination of lines in diagram is more gentle than the case of smaller diameter. In the case of the tube with smaller diameter, inverse phenomenon is to be thought, the descent of lines is steeper and the tube length where the maxi-. 1. r. f i /l. 1 i '. mum pressure amplitude becomes zero is shorter than in the case of larger.

(14) 38 ENoGucHi. inner diameter. In the case of the tube with extremely large inner diameter, as frictional resistance is very small, and the factor which diminishes the impact is only the inertia of liquid co!umn, gentle is the rate of decrease of maximum pressure amplitude, the lines in diagram descend extremely slowly to the right and almost less concaved part in the middle. 7) Wave form of pressure vibration felt by the pressure indicator at the. bottom end of tube. The wave form of pressure vibration at the open end of tube is transmitted to the pressure indicator at the bottom end of tube via the liquid column in the tube with the velocity a of formula (12) of g3. 2-b). This ve!ocity a is said to be constant and near the acoustic velocity.. The wave form of pressure vibration at the open end of tube is thought to be the sum of many simple harmonic vibrations as shown by the formula. p. (6) of g3. 1), and each simple harmonic vibration is transmitted to the bottom. end of tube with the velocity a. But these many.simple harmonic vibrations are the collection of those with different amplitude and period. When these vibr,ations travel from open end to the innermost, they are reduced at their amplitude, and the vibrations with shorter period among them are extinguished by the inertia of liquid particles or their collective body and the friction resistance on their way, as they are usually with very small amplitude, and those which gan reach to the bottom end of tube are only the gathering of vibrations with sma!1 amplitude and long period. The longer the tube, the less the vibrations with short period, so appears the vibration with longer period. But as this disappearing may .be. done step-. -. w.ise.with respect to tube'length, it could be explained that the increasing of. the first half period is made stepwise with respect to the tube length in Fig. 6.. With this theory above we could also explain the reason of stepwise reduction of maximum amplitude of fair!y long tube in g3. 6-c). wavLt fgSrmthjl:.tghhtrokhnadtnetshs9'gathering of those resiguai vi,brations becomes a. N. I :. i. /.e '' *****. References :-. S.TIMosHENKo VibrationProblemsiriEngineering. t. l. J.SHOGENJI WaterTurbinep.385 C.TsuBoi Sindo-ron Postscript:rOn ,the way of this experiment we often met troubles of apparatus especially that of pressure indicator at top position, and were obliged to alter the reading of ordinate scale of images on account of change of tone by the. l i. trouble, and last year there built a factory in the neighbourhood of our. ; 1. I.

(15) Impact Vibration of Ball Valves. 39. laboratory and from there current an electric noise vibration on wire we were ' disturbed by the addition of noisy vibration on oscillograph images.. Acknowledgements Author is very much grateful to Mr. MAEDA who gave him precious teaching, and Mr. IsHIKAwA, Mr. KAwAGucHI, Mr. KoBAyAsHI, Mr. SuzuKI and Mr. SuGiyAMA who assisted this experiment, and Mr. NABEsHiMA who made many capillary tube joints.. '. .. 8'v. T. l. gE mo. 'i. si. .. - op :. //.. l. d. ie. .b'. 't "Vlpt. ptS ... '. ,,. .1. t. Y- O 50. Has .･=. M8 v atv-. i･. IX. 1. l. l' .p .1".".=. .r"'i''. "-.F'. -'-'-' '-". t' "'. jl. L"r"x'. ---- r .y- E. ""･t=--=-. P- q. l/ 11 ' il'. t. g. ･r--･. Ia]4s fi Ts. 10 ll. is'. LO iO. ,2". 21 30. 33 eb.. 40. f6. '. so. TubeLength Meter. Fig. 6.' Relationship between the first half period of wave･form of,.pressure vibration and the tube'length. ' ' ''. '' ' ' tt. tt. .ttt '. '. ' lt-.,-.,, -.t t. ･//. e･. v. ･t t. sa.

(16) 40 F．NOGUCHI. O．7T. p＝100kq！℃m 80. ヴ。魍瞳＿輌5瞬駆旧田圃薗塑二二田田国国圏園. 四団国璽響唖聾雷響盟盟齢樋薗㌔哩．’讐灘翼翼“富. 曝囲囲. 墜. 閣類騨獺鰯， i騒騒膨照’囲騰置醗琶藍碧瞳璽日置置. 匪圏国田. 闇闇6・匿醗盟. t＝0 50 100 150 200 250 300 5ンμS. Fig．1−a． Photographs of oscillograph images of tube with inner dia．0．7 mm，. at Top Position， as typical examples．. 句.

(17) Impact Vibration of Ball Valves 41. 0．7T. ，. o. 200. 専. ＿一225国. 団団四囲嗣睡劉腿囲畷国園圃鰯鯛26。隅覇覇照鰯職幽幽幽囲. 薩幽囲闘’韻.

(18) F．NOGUCHI O．7T. ま. 匪駆罵廻田田隈翌翌田圃思あ国国誕. 圏圏墨圏量感露塵謹白腫里目麗麗 ’∫■曜四囲四国囲囲曜劉旧圏. ，総躍塵櫻野圏. 哩開脚圏璽. P.

(19) Impact Vibration of Ball Valves 43. 0．7T. IOO 80. 9粥盟野露幽麺四壁. 雇. 蕪。＼. 明目溺盤.

(20) 嶋. 、. F．NOGUCHI. 0．7T. ら . ドぐレ B●四四璽1＿璽三層回賑三噛. 四劃幽雌三三．凹凹凹凹四四. ：腰縣三三願一. ’. 鎖幽・、. タヰココ. ●.

(21) Impact Vibration of Ball Valves. O．7T ニぐ、。圏囲田臣團圏L．園圃囲圏1 自由幽出盧幽・1麓萱箇猶臨麟圃. 田図：・1国囲国. 囲囲層幽幽溢 ■圏‘r引田薗田田口. 幽幽画調圏圃. ρ. ．口固鞘噸醐幽幽髄墜1題回国園圃1．：幽. づ50. H臨灘謡講. 、幽幽囲凹田囲. 1，幽幽囲園圃囲. ・開顕圏温国願翻圏.

(22) 46. F．NOGUCHI. αIT. 8。闘. 嚇. 圏 4. 4. 口口目騒羅懸＿纏. 塵温鑑田圃口匹雪目覆. 温照懸鰻一圏目引置 26。甲種園田 . 瞳. ・.

(23) Impact Vibration of B211 Valves 47. 0．7T. こマ. r塵. 4．○. 箇幽3qo 騒騒猶. 目■目 r. 1. 歯一Z…醐420灘鱒額難．タ. 鞭階置謹460膿m唾灘一. ■屡翅㌦口開調田田. 田躍邑露盤萱盟サ. 胴置目遡謹圃 ■■摺目■圏. 盈躍溢襲一戸困盈蹴.

(24) F．NOGUCHI. 0．7T ヰ . 醒劉囲囲．、、 8堅塁塁壁墨型園圃隔：園圃闘＿騒騒旧圓. 0園圃圃置薗嗣640薗醒幽薗興置. 融融翻 ’1．回国四囲. 顯顯懸. o. 盤幽幽醤脚脚ヒ’’”魑謹幽，麺貧. 幽． 1 1 1 ’ 1. ノコ. 屋騒騒野田開縣響層’口匪翅「■…■饗、. 屠留置圏橿田露顕. 一，、コ、幽・’・ ’ 宇. 鷹裂麗麗隔 ’ ｝ F塑圏酉肖國幽幽翌日1：”昌. ゼー. ：囲国璽劉暉暉∫：∫圏圏唾日日日. 溜騒騒圏温田翻瀟瀟のロミ. 囲翌翌姻囲国融融囲題聞醐，・＾墜劉闇閣鵠凹墜幽瞳塑甑臨1800三囲囲囲醗，. ，.

(25) Impact Vibration of Ball Valves. d't"J."T:t,.':'-t'/,'. 491F. '. ,r.. '. ,'v.. ... .,.,,,.,111,lt,. ''v ･,. 1,･,.). O･7T ".. 60. ' x '',. , x.. 't' ,r. ". 8 /. '. t/. o･. 2200. '. vl. .. ;"tt. 't :l'waee. ). 2400 t''' .. s,･ ,1. ,v'. t-' Y.. 'i' tt/. ',I. '. r. tt t-. 1. 'L ,. t. -t ･1. l. '. 40. d.

(26) 50 F．NOGUCHI. O。7B ザへ . ．購田圃薩颯騙ン田園園田劇団。幽幽堕璽四国3。。 ■醒自画翻■．、幽幽幽幽幽幽. 340. 曜劉唖聾. り. 9. 38。1国画顯一、，一。、. 420. 460. 町. 500. 560. Fig．1−b． Photographs of oscillograph images of tubes with inner dia．0．7 mm，. at Base Position， as typical examples．.

(27) ノ . 、！ Impact Vibratlon of B∂11 Valves 51. 0。7B． . も. 園圃藤覇願嗣』騒園圃園圃覇た. 麟躍謹響響騒騒. 陰. 懸欝響曹㍑一一熟』黛 340 ｝｛｛許｛！此. 国騒騒騒騒塑. 耀醗騨一ガ躍田温潤阻隔盟隔. 願醗睡醤圏題葭. 囲闘圏温田園田団団題圏. 麗麗懸幽謄圏躍圏圏闘囲畷翻旧団団閣. 璽謄岡岬鑑躍.

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