STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE AFDELING MACHINEBOUW - DROOGBAK la - AMSTERDAM

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1 REPORT N. 31 M APRIL 1960 STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE AFDELING MACHINEBOUW - DROOGBAK la - AMSTERDAM NETHERLANDS RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION ENGINEERING DEPARTMENT - DROOGBAK la - AMSTERDAM MODEL TESTS CONCERNING THE DAMPING COEFFICIENT AND THE INCREASE IN THE MOMENT OF INERTIA DUE TO ENTRAINED WATER OF SHIP'S PROPELLERS MODELONDERZOEK NAAR DE DEMPINGSCOEFFICIENT EN DE VERGROTING VAN HET MASSATRAAGHEIDSMOMENT TENGEVOLGE VAN HET MEETRILLENDE WATER VAN SCHEEPSSCHROEVEN BY N.J. VISSER (HIEF ENGINEER - WERKSPOOR N.y. AMSTERDAM JL1O Issued b the Cuncil THIS REPORT IS NOT TO BE PUBLISHED UNLESS VERBATIM AND UNABRIDGED

2 RESEARCH COMMITTEE Prf. Dr. Ir. J. J. Kcn PrL Dr. Ir. W. P. A. VAN LAMMEREN Dr. Ir. W. J. MULLER Ir. W. H. C. E. RÖSINGH Ir. E. STRUYK Ir. A. DE Mr (ex ffici)

3 PART I. CONTENTS FESTS WITH NON-ROTATING PROPELLERS Summary 4 Symbls 5 Purpse f the measurements 5 Test prcedure 6 Mdel prpellers used 7 Evaluatin f the test data 7 Effect f the specific gravity f the prpeller 8 Accuracy f the results 8 Crrectin f the results 8 Results and cnclusins 9 Increase in the mment f inertia 9 Damping factr i Water height abve the prpeller 10 page PART II. TESTS WITH ROTATING PROPELLERS I. Purpse f the measurements 12 II. Test prcedure Experimental arrangement and methd Mdel prpellers used Evaluatin f the test data 13 Increase in the mment f inertia 13 Damping Accuracy f the measurements 14 III. Measurement f the mment f inertia 14 i. Results Cnclusins 14 IV. Damping 15 I. Results Cnclusins 16 V. Discussin 17 VI. Appendix I 19 Cmparisn with the results btained in part I 19 VII. Appendix II Determinatin f the damping

4 SUMMARY The experiments were perfrmed in a twing tank with mdel prpellers. The experiments with nn-rtating prpellers were carried ut using 36 mdel prpellers f the B-3 and B-4 series f the Netherlands Shipbuilding Experiment Tank at Wageningen. Mdel prpellers f the B-4 series nly were used in the rtating and twed prpeller tests. In the first case the trsinal vibratins were generated by tw Philips exciters cupled tgether. In the secnd case the excitatin was generated by the driving mtr with the aid f an alternating current adjustable in regard t frequency and vltage. The increase in the mment f inertia f the prpeller due t entrained water is calculated in bth cases frm the decrease in the natural frequency f the trsinal vibratin system. It is shwn that the mass f the entrained water increases linearly with the pitch. The nn-rtating prpellers indicated an increased mass cupling cmpared t the rtating prpeller. The mass increase appears t be independent f the r.p.m., the twing speed and the slip. The investigatin des nt give infrmatin abut the influence f scale effect, flw inhmgenity and flw restrictin. Cmparisn with shipbard measurements is desirable. In the first case the mean increase in the mment f inertia is apprximately 38%. In the secnd case the crrespnding increase is apprximately 35%. It is nrmal practice t cnsider a 25/ mean increase in the mment f inertia. In the first case the damping is determined frm the decrease in the angular amplitudes during the decay f the vibratins. In the secnd case the damping is determined frm the phase angle between the angular amplitudes f tw pints f the system. The methds used t determine the damping cefficients were nt particularly accurate but enabled the effect f varius factrs n the damping cefficient t be established.

5 I. SYMBOLS I. Mment f inertia f the prpeller '1 Mment f inertia f the prpeller in air '2 Mment f inertia f the prpeller in water I Mment f inertia at the prpeller due t the shaft etc. I, '2 Mments f inertia befre deducting It 1m. Mment f inertia f the driving mtr rtr Mment f inertia f the prpeller in air with Vi J Mment f inertia f the prpeller in water with y, I,, Mment f inertia f the prpeller in air with V2 ISO Mment f inertia f the prpeller in water with y, y1 Resnance frequency f the trsinal vibratins in air y, Resnance frequency f the C C2 trsinal vibratins in water Trsinal rigidity f the shaft between the mtr and the prpeller Trsinal rigidity between the first trsigraph and the prpeller kgcmsec' kgcmsec' kgcmsec2 kgcmsec' kgcmsec' kgcmsec' kgcmsec' kgcmsec' kgcmsec' sec sec-' kgcmrad' kgcmrad' Lgarithmic decrement n Number f vibratins sec-' D Damping rati with respect t critical damping Damping cefficient per unit angular velcity V Prpeller vlume cm3 r1 Inertia radius cm a Vi, V2 e Amplificatin factr Specific gravities f the prpeller Natural lgarithm base kgcmsecrad kgcm3 d Shaft diameter cm g Acceleratin due t gravity cmsec' M0 Excitatin cuple (maximum value) kgcm 0, 0m, & Angular amplitudes at the prpeller, mtr and first trsigraph rad, cpj, Phase angle between O and M0 and between 0s0m and M0 rad w Circular frequency radsec ß Phase angle between 0, and eilt Pitch-diameter rati f the prpeller rad II. PURPOSE OF THE MEASUREMENTS This investigatin has been made t determine the increase in the mment f inertia f a prpeller submerged in water resulting frm the vibratin f the surrunding water. This increase is dependerit n a number f factrs, f which the pitchdiameter rati f the prpeller, the number f blades and the blade area rati are the mst imprtant. At the same time, the effects due t the frequency, the amplitude and the height f water abve the prpeller were examined. Damping measurements were als carried ut. 5

6 III. TEST PROCEDURE Fig. 1. The increase in the mment f inertia f ship's prpellers was determined by means f mdel prpellers in a simple vibrating system. The decrease in the natural frequency f the trsinal vibratin f the first degree f this system, after submerging the prpeller, is a measure f the increase. The system was cnstructed as fllws: a small shaft is clamped in a blck, the mass f which is assumed t be infinitely large. A mdel prpeller was attached t the free end f the shaft. The blck rests n a stand (see Figure 1). The shaft with the mdel prpeller is set int trsinal vibratin by means f a butterfly-ring by tw exciters which are attached t the supprt. A cuple is applied thrugh bth exciters s that n bending vibratins are induced. The exciters are cnnected in series and suitably scillatr are p life r dlsplacemenr scillscpe Fig. 2. blck exciter pick p Experimental arrangement height 6

7 matched in regard t amplificatin and phase displacement. A displacement recrder, fixed t the blck, registers the trsinal vibratins via the butterflyring. These vibratins are cnverted int an alternating vltage and are made visible n an scillscpe (see Figure 2). The damping rati is determined by setting the prpeller in resnance by means f the exciters. The exciters are then suddenly switched ff, after which the vibratin dies away due t damping. If the signal is shwn with a sufficient lw timebase n the scillscpe, the amplitude behaviur f the phenmena can be recrded phtgraphically. IV. MODEL PROPELLERS USED The prpellers used in these tests cmprised six series each f six prpellers, 36 in all. These series are referred t by the letter-figure cmbinatin: B-3-35 B-3-50 B-3-65 B-4-40 B-4-55 B-4-70 The first number indicates the number f prpeller blades, the secnd the blade area rati. Each series cnsisted f prpellers with different speeddiameter ratis. The crrespnding ratis fr each series were: V. EVALUATION OF THE TEST DATA The vibratin system cmprises ne mass and ne degree f freedm. The knwn frmulae fr this system are: 42v12 = - and 4t2v2.... (1) which give: '2 y2 (2) '1 22 The increase in the mment f inertia due t the entrained water is usually expressed as a percentage f the mment f inertia f the prpeller in air. This is the reasn fr determining the rati The percentage increase is: x 100% J()2 i x 100% (3) '1 I'2 J T measure the damping cefficient, tw amplitudes are measured n the phtgraphic recrding at a distance f n vibratins. The rati f the tw cnsecutive amplitudes, fr the case f pure fluid damping, which is here assumed, is equal t e, where 5 is the lgarithmic decrement. The amplitude rati after n vibratins is therefre e1ò. if this is determined by the abve-mentined methd, 6 can then be calculated. The damping rati D is given by: 2rD 6= (4) ViD2 Since D is very small and therefre D «i (see Table 2) we get: 2r Fr the derivatin see TIM05HENK0, Vibratin prblems in Engineering third editin page 72, which gives: D=2 (6) hence T determine the increase in the mment f inertia as a functin f the height f water abve the prpeller the same system was used. Hwever, tw ther prpellers were used as mdel prpellers, f which the mments f inertia were calculated frm a pendulum test as 39.5 x l0 kgcmsec2 and 28.3 x l0 kgcmsec2 respectively. The calculatins were carried ut as befre. The height f the water was measured as the distance between the lwer plane f the prpeller nrmal t the shaft, and the water surface (see Figure 2). The increase in the mment f inertia as a functin f the water height is again btained frm the decrease in frequency with increasing water height. (5) (7)

8 VI. EFFECT OF THE SPECIFIC GRAVITY OF THE PROPELLERS The material f the mdel prpellers is nt always the same. If it is desired t cmpare the results f the varius tests, this factr must be taken int accunt. The increase in the mment f inertia due t the vibrating water is usually expressed as a percentage f the mment f inertia f the prpeller in air. In the case f a prpeller f sme material, different frm that f the mdel prpellers emplyed, the increase calculated as a percentage f its wn mment f inertia in the absence f water is Y2 times greater than that fr an identical prpeller made f the same material as the mdel prpellers. A cmparisn f tw similar prpellers f different materials gives: '21 = 111+a11, 122 I,2+a111 = V = 112+ayi rt2 g vi '22 = Ii2+_cc_y2rt2 V2 g = I,2+a112 = 12(1 ' V +a >'2 >2 VII. ACCURACY OF THE RESULTS The nly errr affecting the calculatin f the increase in the mment f inertia, is due t the inaccuracy f the exciter. The maximum errr f this apparatus is 0.25% f the induced frequency. With vibratin in air the resnance is very sharp, s that the maximum adjusting and reading errr is abut 0.1 c.p.s. and therefre amunts fr the lwest frequency measured 41.1 c.p.s. in air and 29.9 c.p.s. in water with the B-3-65; = 1.4 t 0.244% f the natural frequency in air. With vibratin in water the resnance peak is less sharp, s that 0.4 c.p.s. r 1.35% f the natural frequency in water must be accepted as the adjusting and reading errr. Accrding t frmula (2): '2 r2 Il r2 In accrdance with the methd f calculating the errrs, the errr in is twice the maximum errr f r1 plus twice the maximum errr f r2. The ttal errr is therefre: 2( )% + 2( )% = 4.168% Fr the highest frequency measured, 58 c.p.s. in air and 55 c.p.s. in water, with the B 4 40; = 0.5 the ttal errr is: 2( )% + 2( )% = 2.8% The damping determinatin is inaccurate. The recrdings f the damping in air d nt indicate an expnential but a linear decay f the vibratins. This is prbably due t the fact that the damping is nt viscus. The damping in water als frequently shws deviatins and side effects. The results can nly be taken as a rugh determinatin f the damping cefficient. VIII. CORRECTION OF THE RESULTS Sme preliminary experiments were made with three calibratin discs instead f mdel prpellers. The dimensins f these discs were knwn accurately, s that the mments f inertia culd be calculated. These values were , and kgcmsec2 respectively. The shaft diameter was 13 mm and the length 900 mm. Fr the purpse f accurate calculatin, the mments f inertia f the shaft, the washer and the nut were taken int accunt. 8 Fr the trsined sectin f the rd: ly -. - d41 kgcmsec2 = 3 g kgcmsec2... (10) The mment f inertia f the threaded sectin, tgether with the washer and nut was estimated at kgcmsec2. Tgether, this prduces what will later always be called the,,added" mment f inertia: = kgcmsec2.

9 The ttal mments f inertia determined frm the measurements and calculated frm C C eni2= (2rvi)2 (2v2)2 with C = 2586 kgcmrad1 must nw be reduced by the added mments f inertia t btain the values 1 and 12, thus: 1=1It and 12=12t1t... (11) The experiments with the calibratin discs gave the fllwing results: disc N. calculated frequency measured frequency I II III sec' sec' The errr is f rder f 1% I. RESULTS AND CONCLUSIONS 1. Increase in the mment f inertia The results f the experiments are cntained in Table 1. The results f each series are summarised graphically (see Figures 3 and 4) Hf D e 40 B i 4 H/D 0 80 e , Fig Fig. 4. 9

10 Cnclusins The graphs fr each prpeller series, shw with very fair apprximatin a straight line. Fr bth the 3-bladed and 4-bladed series, the entrained water increases with increasing blade area rati and HiD rati, nly the values f all series fr H/D-0.5, shw clse agreement. prpeller series prpeller n. Table I 's/hi. 100% Ii B B B B B B Damping rati Table 2 shws the value btained fr the damping rati D. The side effects already mentined were in sme cases s disturbing that n results culd be btained This is the reasn fr the gaps in the table. prpeller series prpeller n. Table 2 D in air D in water B B B B B B Cnclusins Because f the inaccuracy f these measurements definite cnclusins cannt be made. The values given in the fllwing tables must be cnsidered as nly apprximate. 3. Water height abve the prpeller The experiments were perfrmed with three different shaft diameters and tw different prpellers 10 with mments f inertia f 39.5 x I0 kgcmsec2 and 28.3 x 10-e kgcmsec2 respectively, (prpellers ns 930 and 1022). The results are summarised in Figures 5 and 6. The mst marked change in frequency ccurs with water heights between O and 30 mm, while with water heights exceeding 80 mm the variatin is practically nil. The maximum variatin in the mment f inertia therefre als ccurs between O and 30 mm water

11 60 t prpeller n. 930 shaft diameter 15.5 height, while with a height f water greater than 80 mm practically n variatin ccurs. In additin the excitatin frce was varied, by varying the current strength, which crrespnds t variatin in the amplitude f the trsinal vibratins. These data were used t calculate the average value f the frequency crrespnding t a given water height. The frequency variatin as a functin f the impulsive frce is shwn in Figure 7 and appears t be small, i.e. abut 0.6% which lies inside the limits f accuracy f the measurements shalt diameter shaft diameter 1Z water height in mm Fig.. Relatinship between frequency and water height. i E prpeller n. 930 E shaft diameter shaft diameter shaft diameter 'IP!_ 100 Fig Current in amps Variatin in frequency with current variatin increase in mment f inertia Fig. 6. Relatinship between the mment f inertia increase and the water height li

12 PART II TESTS WITH ROTATING PROPELLERS I. PURPOSE OF THE MEASUREMENTS The measurements described in this sectin can be cnsidered as supplementary t these in Part I. Whereas previusly the mdel prpeller did nt rtate, it was nw tested while rtating in undisturbed flwing water. The effect f the vibrating water is determined as befre i.e., by determining the increase in the mment f inertia f the prpefler and the damping effect due t the surrunding water. II. TEST PROCEDURE 1. Experimental arrangement and methd An attempt was made s far as pssible, t reprduce a tw mass system. This was achieved by means f a system cmprising the mtr, which als served as the exciter, the prpeller shaft, which frmed the elastic element, and the mdel prpel- 1er (see Figure 9). This system was accmmdated in a twing mdel which was twed thrugh the water (see Figure 8). Fig. 8. The principal reasn why this system des nt cmpletely satisfy the requirements was the presence f the heavy shaft emerging frm the tube f the twing mdel. Althugh it was sturdily made it was nt entirely rigid, and because f its relatively great length, it displayed a certain elasticity. The reasn why this length had t be s great was that the prpeller had t be twed. This was necessary t allw the prpeller t rtate in a flw field as hmgeneus and undisturbed as pssible. If the prpeller were pushed the flw field wuld be f such a disturbed character that reprducible measurements wuld be very difficult. The shaft was driven by an adjustable speed electric mtr; this mtr als served as the vibratry trque exciter, the intensity and frequency f the trque being adjustable. A three phase mtr was used with a slip ring 12

13 armature by means f which it was pssible t cmbine the driving mtr and vibratry trque exciter in a single machine, thus prducing a very simple vibrating system. The statr winding f the mtr was supplied with a variable frequency three phase alternating current. The rtr was supplied with direct current and in additin, with an alternating current which culd be adjusted bth in regard t frequency and phase. This alternating current prvided the trsinal excitatin f the system. As described in the experiments in Part I, the increase in the mment f inertia f the prpeller in water is calculated frm the natural frequency f the system. The results are determined by means f tw identical trsigraphs, which prduce a true phase recrd f the trsinal vibratins. The utputs frm the trsigraph were registered n a pen recrder after amplificatin. In rder t determine their phase - relatin the cmbined signal was recrded in the same way. The frequency f the vibratins was read directly frm the scillatr, the prpeller revlutins were measured by means f an ptical system and an electrnic cunter. The twing speed f the ship's mdel and therefre f the prpeller was als measured with an electrnic cunter. directin f travel Fig. 9. Mdel prpellers used Of the prpellers used in Part I nly the series B-4-40, B-4-55 and B-4-70 were used fr these experiments i.e., three series f six prpellers. As previusly explained the figure 4 indicates the number f blades f the prpeller, the figures 40, 55 and 70 the blade area ratis. Each series cnsists f prpellers with different ratis, i.e., = Evaluatin f the test data a. Increase in tile mment f inertia Fr purpses f calculatin the vibratin system was cnsidered as an ideal tw mass system. The natural frequency was: ==j/1, (1) These values are fund by cmparing the mments f inertia f the prpellers in air with the values determined in Part I (see Appendix I). The maximum errr fr these experiments in air is 2.1%; the mean errr is less than I %. b. Damping The damping was determined by a methd entirely different t that used in the experiment with a 13

14 nn-rtating prpeller, being calculated n the basis f prpeller equilibrium. The theretical derivatin is given in Part II f the Appendix. The damping cefficient g in kgcmsecrad' is calculated frm the fllwing frmula (see Appendix II): g Jw = tan (q22-ç1) (2) Table 3 prpeller series r1 c.p.s. I kgcmsec2 r2 c.p.s. I kgcmsec2 ( 100% B B B Accuracy f the measurements The precisin f this methd is nt great. The signals have been recrded simultaneusly. One f the signals was very small s that its errr cannt be less than 5%. Althugh fr this reasn cnsiderable scatter f the results was t be expected and did indeed ccur, nevertheless imprtant cnclusins can be derived. III. MEASUREMENT OF THE MOMENT OF INERTIA 1. Results The natural frequency was determined fr each prpeller fr different twing speeds and prpeller r.p.m. The mment f inertia f the prpeller in water '2 was then calculated with the aid f frmula I. Table 3 shws the prpeller series, the rati, the natural frequency r1 f the system with the prpeller in air, the mment f inertia '1 f the prpeller in air, the natural frequency r2 f the system with the prpeller in water, the mment f inertia '2 f the prpeller in water and the percentage increase in the mment f inertia. The natural frequencies were determined as the mean f a few experiments with different r.p.m. and twing speeds. The scatter was nt mre than 0.2 c.p.s. The results fr the three series f pr- pellers are given in Figure 10. Fr cmparisn the results f the previus experiments with nn-rtating prpellers are reprduced. 2. Cnclusins The increase in the mment f inertia appears t be independent f the prpeller revlutins, the twing speed and the slip in the regin f maximum utput. The time available in the tank did nt permit this t be established fr all values f prpeller revlutins, twing speeds and slip. The scatter f the results was as already stated 0.2 c.p.s. The curves clearly indicate that the increase in the mment f inertia increases linearly with the rati, in agreement with the results btained with nnrtating prpellers. 14

15 The increase in the mment f inertia is greater with a nn-rtating prpeller than with a rtating ne; this difference decreases as the blade area rati increases. When applying the results f this investigatin t actual prpellers the scale effect must be taken int accunt, tgether with the effects due t rtatin in an inhmgeneus and restricted flw regin. The mean value f the increase in the mment f inertia used in practical calculatin is 25%. In sme cases the increase appears t be less and did nt vary in accrdance with increase in the rati and blade area rati. With adjustable prpellers hwever, sme experiences shwed a strng dependence n the rati. It wuld be mst useful if actual shipbard measurements f the increase in the mments f inertia f prpellers due t the vibratin f the surrunding water, culd be assembled, classified and cmpared with the results f the present investigatin. e B-4-40I Fig. IO. Rtating prpeller - -: Nn-rtating prpeller B III IV. MEASUREMENTS OF THE DAMPING COEFFICIENT I. Results Table 4 shws the prpeller series, the H/D rati, the prpeller revlutins n, the mdel twing speed r and the damping cëfficiënt. The prpeller vibratin amplitude was apprximately 2 fr all the measurements, except the ne series B The damping cefficient was always determined at the resnance frequency. The prpeller r.p.m. value n is always the same as the mean nrmal r.p.m. f the actual prpeller (in tw cases f prpeller B-4-70 this did nt apply because measurement did nt appear pssible at this prpeller speed). The twing speed r is always taken as O r "nrmal", i.e. just equivalent t r less than the maximum prpeller efficiency. The results f the 3 series are shwn in Figures 11 and 12. Because the scatter f the pints is large the curve has nt been drawn. A few measurements were made with prpeller B-4-55 with = 0,8 with varying r.p.m., speed and frequency. Table 5 and Figure 13 give the results fr different r.p.m. but always fr the same pint f prpeller peratin, i.e. in crrespndence 15

16 prpeller series prpeller revlutins per sec Table 4 e rn/sec kgcrnsec rad-1 M kgcm B ,5 7, ,6 7, B B with the twing speed f the mdel. Table 6 and Figure 14 give the results btained with different amplitudes, btained by nt adjusting the frequency t the resnance frequency; table 7 gives the results btained with different twing speeds r, cnstant prpeller r.p.m. n and cnstant vibratin amplitude f 2. prpelier series prpeller n. 2. Cnclusins Table 5 prpeller revlutins rn/sec pr sec kgcmsec rad B Althugh the methd used did nt pssess a high degree f accuracy and the results indicate cnsiderable scatter, it is pssible t draw sme cnclusins cncerning the behaviur f the damping cefficient as a functin f varius quantities. Figures 11 and 12 shw that the damping cefficient increases with increase in, therefre with increase in the cuple M. Figure 13 indicates that the damping cefficient increases with increasing prpeller r.p.m. if the cnditins therwise remain unaltered. Since the prpeller mment increases with the prpeller this phenmenn is t be expected. Figure 14 gives an unexpected picture f the dependence f the damping cefficient n frequency and amplitude. At frequencies belw the resnance frequency the damping cefficient decreases with increase infrequency. Under cnditins f resnance, apparently because f the increase in displacement, the damping cefficient decreases at a slwer rate. It appears therefre that the increase in frequency has a diminishing influence n the damping cefficient and increase in displacement the ppsite effect. Table 7 gives the values f the damping cefficient as a functin f the twing speed fr cnstant prpeller revlutins. The values are such that n cnclusins can be drawn. 16

17 E 0.8 B-4-40 E Mfr V= z E, 1.2 B ? Fig. 12. ' E 0.6 prpeller n. 156 F/F = 0.55 H/D : Mfr V= revs/sec rs O e rn/sec Fig. 11. Fig. 13. V. DISCUSSION As in the case f the increase f the mment f inertia little is knwn cncerning the influence f the scale effect n damping. It may be expected that this effect will nt be great, since in the case f prpeller mment and utput the effect is very small. Special experiments wuld give mre pre- cise data n this pint. The surface rughness f the prpeller mdels naturally has a very marked influence n the damping. A clser investigatin int the effect f the amplitude wuld als prvide useful infrmatin. 17

18 Table 6 prpeller series prpeller n. n revs/sec u rn/sec vibratins per sec amplitude in degrees kgcmsecrad-1 B F prpeller n. 156 FI F 0.55 = 0.8 r = 6.3 revs/sec 0.98 rn/sec /.5, 42 Fig. 14. J / resnance O / / cl 00 D D cpn 3 3 O prpeller series prpeller n. Table 7 ' revs/sec rn/sec kgcmsec rad-' B f) amplitude

19 APPENDI I COMPARISON WITH THE RESULTS OBTAINED IN PART I The vibrating system is cnceived as an ideal tw mass system, with a natural frequency: c V1=-r=--- 2r 2r 1/ ' t I The mment f inertia f the prpellers is knwn frm previus experiments. The circular frequency w f the system is measured. If the experiments had been cmpletely free frm errr and if the system was in fact ideal then: w2_11 =0 (2) 't, trn and C were determined by reducing the functin: C C2 (3) Ii+It I) t a minimum. Putting m (1) = q, frm the methd f least squares: f(c, It, q) = (w2 The minimum cnditins fr f (C, I, q) are: f OC ' OI 'ôq This results in the fllwing three equatins: I1i(c2 (Il+l)2(w2 (& - Il+CI, C q)= _jj C )2 'i't q)= q) = whereby the w and I measurements fr the mdel prpellers are summated, i.e. the results fr the 3 series f 6 prpellers (18 prpellers). An exact slutin f the equatins is very cmplicated and labrius; the slutin is therefre btained by "trial and errr", which cmes dwn t "trying and interplating". The results thus btained are: C = 1349 kgcmrad' It = kgcmsec2 q = 5771 and therefre Im = kgcmsec2 Frm this crrelatin it appears that: 1) the values f C agrees well with the calculated value 1/C = 742 x 10-6 therefre C = 1347 (see Table 8); the values f I lie higher than expected frm the calculated value; the value f im is lwer than the value given by the makers. Table 9 shws the measured frequencies r and the crrespnding values f I, als the values f I fund in the earlier experiments with the difference expressed as a percentage. The maximum errr is nly 2.1%; the mean errr 0.92%. Table 8 sectin I 1 kgcmsec2 prpeller (+ entrained water) Tabel 9 1/C rad/kgcm t be measured - streamlined nut and streamlined cnnecting piece n the shaft shaft sectin between the prpeller and the first trsigraph l29.l0 first trsigraph+pulley shaft sectin between the first and secnd trsigraph _6 secnd trsigraph+pulley shaft sectin between the secnd trsigraph and the mtr _6 mtr (accrding t the maker's data ttal _6 prpeller series c.p.s. I kgcmsec2 ' previusly determined deviatin % B B B

20 APPENDI II DETERMINATION OF THE DAMPING Figure 1 5a is a diagramatic representatin f the tw mass system with the tw trsigraphs drive arrangement. The determinatin f the prpeller damping and the mment f inertia f the prpeller in water is perfrmed with the aid f a vectr diagram (see Figure 15b). The fllwing equatins f mtin can be derived fr the tw mass system, where the excitatin is prvided by the mtr and the damping by the prpeller: 0m2+C(9m08) = M0 C(OsOm) -08I50)2+O3eic = 0 91w2 - + Oeiw = M0 In the case f resnance the fllwing cnditins apply. The amplitudes f the tw trsigraphs and the difference in _the signals frm a vectr triangle with the and a. The defrmatin f the sectin L1 can nw be calculated frm the elasticity rati f L2 t L1. The side a f this triangle can nw be extended by the calculated sectin b and gives directly the magnitude f 9. Determinatin f this value permits the cuple 0snIsnO2 t be set ut, and since the directin f O5ew and 0Ic2 is knwn, the entire diagram can be cmpleted n the basis f the previusly mentined equatins f mtin. q1 is nw the phase angle between 9 and A1' = 90e, while 2 is the phase angle between s0m and M0. The damping is given by tan (92l) = 0513w2 18w Figure 1 5c gives a three dimensinal representatin f the phase difference f the prpeller mass as cmpared with the mtr mass. dann p ingk T r '2x2 LI ea t ra sg raph i9n, (.)f9,nc excitatin Fig. 15. Tw mass system with and withut phase displacement with trsigraph drive and vectr diagrams