ACTS ELEMENTS OF FIELD GUN CARRIAGE DESIGN THESIS FOR DEGREE OF M. E. OPW MLO CLUES AB SD ore THESIS FL A LOCAL a ELD UPYTS Of FIELD GUY CARRIAGE DRECTC Thesis for Tesree of 1), F, vw (; : Tayton A ceney. THESI> FOREWORD, This paper, a compilation of the formulae now in use in the Ordnance Office, United States Army, is intended as a guide for future caloulations, The formulae have previous- ly been scattered throughout many sets of calculations on various carriages. They, together with their derivation and method of application, are here collected for the first time, D. A. G. 95433 LIST OF CONTENTS, Page 10. I. Velocity of projectile in the bore, 4 II, Velocity of free recoil. 9 III, Stability. 11 IV. Total pull. 12 V. Hydraulic resistance, } 17 VI. Velocity of retarded recoil, 18 VII. Throttling areas, ol VIII. Forces, 29 IX, Stresses, 34 X, Counter-recoil springs, 36 XI, General proceedure, 49 -4- I, VELOCITY OF PROJECTILE IN THE BORE, The formula used for the calculation of the curve of velocity of the projectile in the bore as a function of space is the binomial formula of Col, Ingalls published in the Journal of the United States Artillery of November—December, 1903. While these formulae probably do not as accurately represent the true action of the powder gases as do the trinomial formulae published by the same author at a later date, they are sufficiently accurate for calculating throt~ tling areas in gun carriages and possess the advantages of giving smooth pressure and velocity curves and of facility in use, The following elements must be known in order to cal- culate the pressure and velocity curves: Pnax=Normal maximum pressure in bore in pounds per square inoh. Vnuz.” Muzzle velocity of projectile in feet per second. u = Travel of projectile in inches, C = Capacity of powder chamber in cubic inches, d= Specific gravity of powder grains (about 1.56). w= Weight of projectile in pounds, a a Weight of charge in pounds, c¢ = Caliber in feet, From the above the following secondary constants can be calculated, Weight of charge weight of water required to fill chamber Called "density of loading." aB._1 12. i _A-~Jd = = ow Ads AS A= [1.44218] © = ~5~ Zo = (C F 1728 ) etree = [2.30955] 437 = reduced length of initial air space in chamber in feet, My Ow = Ww uo 144 7g oc” Zo ~ [3 J a’ u _ € uz. --Gr * number of exvansions,. The fundamental formulae for pressure and velocity are (1) P = M,X,(1 - NX,) (3) VWs MX, ( 1 = NX,) Col. Ingalls states that the pressure curve has a max- imum ordinate when€ = 00,4524, A large number of calculations with these formulae have shown that this is not strictly true for howitzers and field guns, The average of several calcou-~ lations made gives the maximum pressure to occur at € = .65 for field guns and at € = ,.6 for howitzers, The method of arriving at the true position of the maximum pressure is as follows: From the fundamental equations above we may write Voz. . UK (1 - NX, ) | MX, (1- NX) _ X, (1 - §X,) Pp - ~ - Bmax, My%g (L- NX,) AM X,(1 - NX.) ak, (1 - NX,) Vax, - YaX,NX,= PX, - PNX.X, PNX,X, - VaX,NX, = PX, - VX, (3) EX) - Vax, P ~- PX, vx XX, ~6- Take out X, and X, corresponding to E me from the tables of these functions, Assume & for maximum pressure in accordance with some Similar gun previously calculated. In the absence of any indication as to what this should be it should be assumed at about .6. Take out X., and X, corresponding to this 3 assumed value of € , Calculate N from the formula derived above, Calculate M from the following formula (4) Ma- Yr Xy - NXoNy To check: Calculate P from the formula P = M,X, (1 - NX), It should agree with the given maximum pressure, Now calculate P for the higher and lower tabular values of € next to that assumed, If this P is less in each case than Prax., the point assumed is the correct one for the max- imum P, If either is greater, calculate P for successive tabular values of € until P passes through a maximum, assume the value of € corresponding to this calculated EF as a new trial value and recalculate N, M, M,, and P as before, Proceed in this way until the true position of the maximun is obtained, The values of N and M thus derived are the true ones to be used in the equation v = MXo - MNX0oX, for plotting the velocity curve, The pressure curve may be calculated by the formula P = MX, - Ma NX,%, » if desired, but this curve is used in designing guns rather than carriages and is not required for our purpose, —7— To plot the curve of velocity of projectile as a funo~ tion of space, (Fig, 1). Having obtained M and N, V and consequently V may be obtained for any value of € oru by substitution fn the for- mula, The points may be taken rather far apart especially after passing the maximum pressure, The points may be plot- ted to any convenient scale and a smooth curve should be drawn through all of them, If any point is off the curve it should be recalculated, To calculate the curve of velocity of projectile as a function of time (F 3 A method of calculating this curve is given by Captain (now General) Crozier in Ordnance Construction Note No. 57. It consists in plotting the curve of reoiprocals of the velocity as a function of time|2 = f(t)] and of measuring the areas under successive portions of this curve. This area is fide = fdias = fat =. However, it is not necessary to plot the ourve of re- ciprocals, as the curve of velocity as a function of time may be obtained directly fram the curve of velocity as a function of space, The expression for the area under any portion of this curve is (Fig. 1) A= Axv, = ax os = 2° ys e Ax (5) oo At = “A The area under successive portions of the curve may be obtained by planimeter or by any other convenient method, and the successive times thus be calculated. The accuracy saat ams. LTA 0% -~3~ of this method increases, the smaller the increments of x become, Both of these methods of calculation fail near the origin. The first method fails because the Y-axis becomes tangent to the reciprocal curve at infinity and it is impossible to measure the area correctly or to plot the curve with any degree of accuracy through points near the Y-axis, The second method fails because the units of x and A become too small to be measured with accuracy near the origin, However, the true time for this portion of the curve may be approximated to any required degree of accuracy, Having calculated the time corresponding to a number of Spaces as near the origin of motion as practicable, plot the curve of velocity as a function of time to a large scale, preferably on cross section paper (See Fig. 3). One of the properties of this curve is that the area beneath any portim thereof represents the distance traveled over during that time. Therefore, if the area under the curve up to any point is unequal to the corresponding space traversed, the curve is incorrect and should be adjusted accordingly, In Fig. 3 the dotted line represents the preliminary or calculated curve while the full line represents the ad- justed or true curve, The whole of the curve of velocity of the projectile as a function of time may now be plotted to a convenient scale, The maximum abscissa of this curve is called t', -~9- II, VELOCITY OF FREE RECOIL (Fig, 3). The velocity of free recoil is the velocity which the recoiling parts would have if they were acted on only by the powder pressure and not retarded in any way. It attains its maximum at the time the powder gases cease to act, i. 6., when the bore pressure becomes atmospheric pressure, = velocity of projectile in bore (previously calculated). Re ag = weight of charge, w = Weight of projectile. Y.. = weight of recoiling parts, Ve = velocity of free recoil. Assuming, as seems rational, that the average welocity of the powder is one-half that of the projectile, CO W. Vp= (w+ 2) vn volg. Wits (6) Ver = Vp W r The average velocity of the powder gases as they emerge from the bore after the projectile has left it is generally taken as 4700 ft./sec,. (Rausenberger gives 3.5 Vv. ). Y.. Vp max, = Wo Vuz. + 4700% , WV uz + 4700 (7) Ve max, W r The curve of velocity of projectile as a function of time can be used for velocity of free recoil as a function of time by simply changing the vertical scale in the ratio of w w+. . Then draw a horizontal line parallel to the axis ¥, of t at a distance of V, max therefrom and connect this line XX vas M " ae 89S OLS Oe s: Ms a an ae Sa ae aie and TOTAL SHADED AREA = E Te ARNT, ee py do SINGLE SHADED AREA = &- ee INR oo ada) - VT-€ at) af , wae -10- with the curve by means of a smooth curve, The exact position of the point of tangency of the curve and horizontal line can- not be accurately determined, but it makes very little differ- ence so far as the results of the calculations are concerned, The time to the point of tangency is termed T . It is equal approximately to 3t' for gun carriages and to est! for howitzer carriages, It will be generally sufficiently accurate to take it arbitrarily at some even thousandth of a second. +t! is the time elapsed when the projectile leaves the muzzle and T is the time elapsed when the powder gases cease to act, The area under this curve represents the distance tra- veled over in free recoil up to any time t for: Area = fVdt = fe dt =fdx =x The total area up to time 7 is the distance traveled in free recoil up to that time and is called. Eg, This area should be measured by any convenient method, preferably by planimeter, and determined, As a partial check measure the area under the curve up to time t', This area to the scale used for velocity of projectile in the bore should give the length of travel u, -ll- III, STABILITY. The stability of a carriage is that force which applied through the center of gravity of the recoiling parte in a direction parallel to the piston rod pull and having a lever arm to the bearing point of the spade is just sufficient to over- turn the carriage, It is denoted by R when in battery and by r when at full recoil (Fig. 4). W, = total weight Wy, = weight of recoiling parte ll Rz Wyx rz = W,x - Wb cos ¢d RZ — YZ = Wx ~ Meh, b cos ¢ = We b cos ¢, (8) R-gf Wr b cos g For 6 = 0; cos 6 = 1; andz=h W (9) R-r7rs b R=- r is known as the "loss of stability during recoil," It is used in derivine an expression for total resistance or "total pull," From figure 4 it is evident that as ¢ inoreases z passes through O and becomes negative. Rand r therefore, become very large and after the line of force passes through the bearing point of the spade no question of stability can enter, This increase in R with the increase in ¢ enables us to in- crease the piston rod pull and to shorten the recoil, as the gun is elevated, without endangering stability. N | nC eee = a ao ae nee | a oe oe ) AAEM ass A Ae ee ee A vf, i TT17 717777 m rs ie st ee Ss te == Fie. F x -12~ IV, TOTAL PULL (Fig. 5). We are now prepared to derive an expression for the total resistance tending to retard the recoiling parts and to bring then to rest; this is called the "total pull." For convenience the characters used are here again set down and defined, R = stability at beginning of recoil in lbs, = stability at end of recoil in lbs, total pull at beginning of recoil in lbs, ll total pull at end of recoil in lbs, length of recoil in feet, = length of free recoil at time T. y mo ids wv & ll = time elapsed when powder gases cease to act, W M = mass of recoiling parts = — g For convenience in making the calculations it is assumed lst that P is constant up to time T, end that the slope of curve of total pull from P to p shall be parallel to that from R tor, This gives the greatest possible stability consistent with the lst assunmp- tion, Very little would be gained by letting the slope of the curve from P to p extend over the entire length of b. The retarding effect of P on the velocity of free recoil is Vs fadt ft at = fat = 2 for t = T the retardation is pt ~13- The velocity of recoil at this time is therefore Ve max, Pr = V.. The effect of P on the distance traveled is S = fvat = fF tat = (tat = Bt for t= T the effect is st The distance recoiled at time T is therefore PT® o - From the diagran (Fig. 5) we may now write —+=-P_, oO = R-Fr b- &+ P— b oM B (10) psp-R=f(p~ €+ pl) ell o Ptpwip. -R=2 . pt = P 2b ( E Po? . MM T\2 FSP (db - Ee + fr) = 3 ( v,- 22) a (R-7r) (b= & + Po) 200 oy Pp -~ _ Tt”) _©€ _ o FF 3b eu l(b - E+ Par) = 3 (Ve pt ) 242 22 5 P(p -#) + Ft ~ (R=) (p ~ g+ PED = Xv, - pt)? 2 P(b -£) a (R = x) (> -£)7 — (R= x) a(> -€) pL? BD Bd R-3r p'rt wee 3 8 2 02CtCiCAE - erp 7+ Beg R-r peTt4 4 The term 3D “a3 contains T* and may be neglected -14~ in comparison with the other terms, R~ 2 8 _ (11) P [» - E+ vypeT- SS (bv -§) T- We + R SF (b _¢#)2 This is readily solved for P, For the condition of theoretical stability at 0° elevation W Wy Wy & W " R- t€& (12) P = 7 2 , max, 1. Jr. T h BM If therefore P as derived fron equation (11) is greater than Prox,, stability will not obtain. Pp is obtained from the equation (10) p =P ~ ReE (yp ~ E+ Ph) Checks: t\2 (13) PSL (p ~ e+ po = Me “ Py)“ For uniform pull P is constant and P =p -E+ VT This might have been written by inspection from equation (11) by placing R- r= 0, This formula is useful in obtaining a preliminary estimate 2 MV. of P; for —— is easily calculated; b may be assumed and & and ¥,T estimated from similar carriages, -15= Check; 2 (15) P(o- £ +P To) a (vp- pt)® For carriages in which variable recoil is used it is customary to.use the formula for long recoil (11) for eleva- tions up to about 5°, For elevations above the line where the Over-turning force passes through the bearing point of the spade, 17° approximately, the formula for short recoil (14) should be used, If we attempt to use the fornula for long recoil (11) — for all intermediate elevations it is found that R —- r and consequently P — p becomesvery large for the higher elevations, 1, @, as h becomes small in the formula R- 1r = “2 ° The slope of the curve of pull increases very rapidly therefore with the elevation. As the slope of the curve of pull for short recoil is 0, the transition between the two formulae is irregular, Since stability at these intermediate elevations is gen~ erally ample it is customary to ignore equation (11) and to make the slope of the curve of pull vary uniformly with the elevation from that calculated at 5° (approximately) to zero at the lst position where the short recoil formula (14) is used. Pp. This slope is(.——>~-b72—) be Et on b, €& , T? and M are known for all elevations, P can be estimated roughly as beine inversely propor= tional to b. We can then make a table like the following: Elev. b- & + pt Slope P-—p 0° 8,677 117 313 5° 2,512 151 380 10° 2.08 100 808 15° 1.923 50 96 20° 1.600 O 0 Instead of formula (11) we derive the following: let P-~p=C! p =P -—C! P+tp— . &! 2 PS > .~ £! - Ty) tT 2 2 2 3 2 ct tT? otlsowv (b - £) au a (b -£) -P5- & — i = PVpT + (16) P(b - E+ yet TO!) WVe? fp - £) 4M 2 2 from which we get P and therefore p. Check; P + 2 (13) SSP ip- eg +PL je Uy, ~ p TH -17- V. HYDRAULIC RESISTANCE (Fig. 5). The hydraulic resistance is the net resistance to motion to be taken up in the hydraulic cylinder after subtracting all other forms of resistance from the total pull and allowing for the component of the weight of the recoiling parts parallel to the piston rod, It is denoted by Rie Let S m load on the springs at any length of recoil. F = friction on recoil guides, B = stuffine box friction, P_ = total pull at any distance x from origin, f = coefficient of friction, (17) Then R, = Py + Wy sin, 6 - S-F -B, In general R, can be expressed a a linear function of x for; Py = f(x) fram diagram, It is, of course, a different function before and after time T , but linear in both cases, W, sin, ¢ is constant for any elevation, (Note that this term is negative when the gun is in depression), S = f(x) (linear), F = Wf cos ¢ is accurate enough in general, although the friction due to the pinching action on the recoil guides is neglected, f should be taken small, .1, for this calculation, B= 100 lbs. per in, of diameter of piston rod, a con= stant, Let A = effective area of piston then 7h = pressure in recoil cylinder, -18- VI, VELOCITY OF RETARDED RECOIL (Figs. 3 and 6). The velocity of retarded recoil is the actual velocity of the recoiling parts with respect to the carriage, For times prior to T it is solved for as follows: 2 =v,-p¢ = Prt Vy Ve - Py x= €&- BM Draw ordinates of the curve of velocity as a function of time about .001 to .002 sec. apart anc measure the areas beneath the curve between these successive ordinates, then construct a table like the following: t Pé Vp V;, Pte E. x oid ~001 © 008 003 » 005 «007 » 009 »013 ~017 2019 =T a Slide rule results for Pi and rt are sufficiently accurate, Vp and € are obtained from the curve, As a check we may plot Vy as a function of *, The result should be a smooth curve, The point of Vy max will occur somewhat before Vy and it is at this point that the maximum opening of the throttling orifice occurs, For times after T the following equation holds VesAG iE fo ete a eae a > vi ne ~19—~ P + P (——*) x= ( ¢- Pkp/ =3 ( v3 = 0,2) Ae Py is linear it may be expressed as P, = c — dx where c and d are numérical and known (see paze 17), P +o - dx tS, Maw 2 we EH EHP oD & Vy" ~ V,") 2 (P+) x- (P +0)( €- PED ~ ax? + ax (¢ - PL ll u(v,? = v,*) [Pp + ta ( re] 8 _ (p Pr C &- ay JX 7 OX - (P+) ( E-5) a 2 2 UV? — uv, 2 PT? PT 2 oe (18) v;° = We tt o)( & - a) F tota (gE - BM Mu M Check; Place x = b 2 Vy must = 0, The equation is of tne form 2 (19) V,° =A-Bx + 6,° As a large number of points are necessary to be found on the curve it is less laborious to solve it by the method of differences, On account of the form of the equation, the second difference ie constant. We write; 2 V7 = A - Bx, + Cx, Vh5* = A- BX9 + Cx” 20+ tC @ ct m4 00 lj xX, + a where a =Zix ve a A — Bx, ~ Ba + Cx B+ 20X,.a + Ca® xo SAT HAD 1 1 V.. eX A - BX, + Cx, Veo” = Ve” = Ca® + 2 XB ‘X1 = ~ Ra Cax, Similarly <3" - V9" = Ca - Bu. + 2Caxs Vig? - Veo" = Ca’ — Ba + 2Cax, + 2Ca? ee ond difference = 2ca® at origin x = x, = 0, . first 2 es let/difference = - (Ba - Ca”) Having selected "a" and made the other necessary substi- tutions prepare a table of lst differences, Subtract these successively from A, The result is a from which we can obtain Vy. e For the condition of uniform pull P, =P, a constant, and the formula becones P(x — e+ PD) _u OM 2 (V,.° ” v,.") Px - P( €- pre , u 2 Uwe 2 an.’ 73. Yr ~ 3 Yx MV," + SP ( pr® v,° - “Yr - €- on ) T. Sx. This is the equation of a simple parabola, in which the firet difference is constant. It may be solved directly Or, more easily, by differences, Check; for x =b; Vx° must = 0, VII.THROTTLING AREAS, (a) for Straicht throttling through an orifice between the piston and cylinder, The theoretical velocity of efflux from any orifice is v( theoretical) = (Zen. Denoting the pressure in pounds per square inch by p and the weight of a cubic foot of the liquid by d we may write p ad orh = ar aa v( theoretical) =/2888 For hydroline, J = 53, and substituting this value and also the numerical value of g we have (288 - 32,8 (21) v(theoretical) =H 53 “=p = 13,28 /p Let v(actual) = ~ theoretical) = © were c and K are constants to be determined experimentally, If a is the area of orifice, V the velocity of the piston, (the Vx previously calculated) and A the effective piston area, VA = va= vi theoretical) a- Ca a = KVA _ - KVA v(theoretical) -— C 13,60 fP - ¢ p= Eve? 13.22 /Rra- C For purposes of analysis we can neglect C, it being small in comparison with the other term of the denoninator, ~2o— Then a = xva/2 13,82 Ry, From which we see that "a" varies directly as the Velocity of recoil and inversely as the square root of the hydraulic pull, If the latter ie conetantjet Q=h, v,° = QR, (b~x) = Q(b~x) z = Q2 fo-x a® = constant (b-x) This is the equation of a parabola and under these con- ditions "a" is independent of the velocity and depends only on x, This is very useful property as it means that with & properly constructed throttling orifice we get substan-~ tially correct recoil no matter what the initial velocity, As R, is not quite constant due to springs, piston clearance, etc., the actual recoil will vary slightly under certain con- ditions but the variation is not sufficient to cause ser- ious mechanical difficulty, For straight throttling through an orifice between the piston and cylinder the values determined for the con- etants in the area formula are K = 2,28; C == 96, In order to avoid the danger of over cutting the bars, K should be taken about 90% of 2.22 or = 2, for the initial trial value, If firines show it to be too emall the area can then be enlarged, For convenience in handling, it is sometimes desirable to have the coefficients of contraction expressed directly in terms of p, ~23~ Place v(actual) = v( theoretical) Ci + pCos VA = va = v(theoretical) a Cy + pls (23) a = £1 + pCg) VA 13.28 Jp By makine proper substitutions we obtain C,; = 1.697; Co = 0001081, (33a) 9 a(2.697 + 0001082 p) VA 13.28 /p or using the safe value of 90% "a" as before (23b) (1,527 + ,0000973 p) VA ~ 13,22 p From the value of "a" derived from either formula 22 of fornula 23 it is necessary to subtract the clearance area between the piston and cylinder, a,. The amount of area to remove from the throttling bars is therefore a - a,, (4) For the condition of throttling through an orifice from the front of the piston into a bypass and from the bypass _ through another orifice into the space behind the piston, We will first derive a method for determining the coef~ ficients of contraction under this peculiar condition, az KVA as before, 13,22 /p-C The area of orifice is made up of three parts &4o = clearance, a constant as = area in front of piston (toward which it is roving) a, = area in rear of piston (opposite to aj) Let K = total coefficient of contraction corresponding to ao; (Cc = 0) 24m Let K, = total coefficient of contraction corresponding to &) and a> p = pressure in front of piston, Py = pressure in bypass, Pressure in rear of piston = 0, Vo = velocity of oil through ao. ~ il Rn = pA; Ry = (p - Pi) A; Ro =P, A VA = Voa, + Vy a, = Vo a, + Voas (21) wtheoretical) = 13.223 (Dp 3488 13.322 v, 2 228 7D VV. a 13.28 Se 13,22 ° K, jr 3 L kK, Pp Pi; Vo = Ki [P+ -25— 1 C = 13,22 py - Kv A— . L a A = ec ~ KV C= 15.22 e ae PO RV ay In any round for which both the pressures and velocities have been measured irstrumentally’V, p, &@,, and a, are known at any point. Consequently a straight line represent-— ing the equation C = f(K) can be plotted for any point of the curve, Pilot several such curves, The average of the in- tersections should give the correct values of K and C, From a certain round for 3.8inch Howitzer Carriage so plotted these values were found to be K = 1,75; C = = 50, These values have since been tried on a number of car- riages and have given fairly good results on all. To calculate the throttling area knowing the constants K_ and C. On account cf the complexity cf the formula we will neglect the enall amourt of oil which passes through a, in our calculations, Then when the areas have been computed we will deduct &, therefrom, 13,22 [=e + 50 1.75 VA - 50 a @ —> — ples Rn Ry a , = > Ra = ®h_ (1.75 VA - 50 ap )2 A UA 13.22 as a. = ‘1,75 VA i“ 13,22 ant + 50 a, = 1,75_VA 13 ewr ao ee EA 1 = 13.22 (1, 75 vae/* — 50 Afas 2 (k Rr 1, = ee ee, ee eee ee. on ee oe ae es oe ee oe +. 13.68 ao 50 (24) 75. V 21 = 13,22 1.75 473 Vo. s50k2 \2 Kk Rn - ( - )" + 5 13,22 an 13,00) ° Check; Substitute a,a, for ao, Vnax, and the corresponding R,. &@; should = a5, At the point of rnaxinun velocity the area aleo must be naxinum, ee ay = a2, and Py = 5 R | a [ty > {05 li (25) . 1,75 V _ @«¢ a... = UaAx. 13,22/22 + OA 50 The method of using the formulae for throttlins areas is as follows: ist; Determine V, and R;, for values of x whose distance @part is ecual to the thickness cf the piston. (For points prior to maxinum velocity it is tetter that they should be sep- arated by only one-half or one-quarter this arount.) énd; Solve for a,,,. (formula 25). -27~ Srd; Substitutin: a... for a> in formulae (34) and the oroper vaiues cf Ry and Vy, solve for aj, ahd area for the point next nearest the naxinum Now (a4, - this a;) is the area covered br the piston in movin: frou the 2nd position to the lst, ”. it must be added to anax, to set the new ao, Solve for each successive value of a, in a sinilar nanner, In formula (24), a, and ao may be interchanged and the Bane equation be used for solvins for as to set the points on the curve after the waxirun in a sinilar manner, Havinz solved for area on vcotk sides of the viston at all necessary points subtract fron each the clearance area a,,. The greater portion cf the area is generally nade up of drilled holes throuzh the recoil valve. Havins deter- mined the size of hole which is to te used find its area, a =~ &o area noie The numoer cf holes at any point of recoil is then l! = It is customary to driil the holes for trial one or two twist drill sizes smaller than calculated, ard to sradually enlarge them in orcer to cet correct recoil and good pres- sure and velocity curves, Length of ports in the liner, If tne rows cf holes in the valve are a cert:in nunber of decrees apart it would at first appear that to have row No, 2 besin te open at the sane tine thst row No. l begins tc close, the ports in tke liner should be te sanie number of decrees lons, and that to cet the correct theoret- ieal openins at elevations interrediate to those fcr which the rows cf holes in the valve were calculated the oorts in the liner should have square ends i. e., the’ should be this ~38~ —/0°— Pe ol~ 4 : 4 - shape: . |, C) ‘ \o Experience has shown, however, ‘Ie o Ww: - that with ports of this shape too much oil escapes through the throttling orifices at the intermediate elevations and too great recoil and too hich end pressures are obtained. Con- sequently it has been found neceesary not only to provide the ports with circular ends, but also to make them slightly shorter than 10°, No entirely satisfactory explanation of this phenon- enon has been advanced, The following empirical formula for length of port is based on experiments performed on 4 differ- ent calibers, viz: 3", 3.8", 4,7" and 6", and may te taken as correct within these limits only. Length = ,O811-°(Inside diameter of liner), VIII, FORCES, The esterrnal forces on a nobvile carria-e are: (a) Powder pressure, (c) Force cr rifling, (c) Teicht, (d) Vertical reaction under wheels, (e) Vertical reaction on float. (f) Knowins the first 3, it is possible to obtain the re- 4 7 iorizontal reaction on sxvade, t mainder fren the equations, 2X = 9; =Y = 03 aPa=0, It is, however, necessary to obtain also the forces at all connections throusnout the carriaz-e in crder to determine the strensth cf the various parts, "e wili take as typical the case of a howitzer in battery at 0° elevation with powder gases acting, Forces on recoilirs parts, ; . A C LK ‘a S bl ns — 7: =, 2-0 0 lf oe! ew cee enue » ase wr Fila * £*R. 2-22 We have here as knowns A, Fy, and V 9 r a°x ° Unknowns i aie? B, and C, The unknowns may readily be determined fron the three T° fundamental equations, =X = 0; a>Y = 0; > Pa = 0, -30— A, B, and C are transmitted to the cradle, Forces on tipping parts (except rocker) Knowns Unknowns A, B, C, Wy D, Et, r Transmitted to rocker F, Transmitted to top carriage D', Et, Forces on rocker Knowns Unknowns Transnitted to top carriage Dy, E,, G iy OE Knowns Unknowns D = D, + Dy (algebraic) L E = EF, + E, (algebraic) lt W, N Transmitted to trail L, i, N. Forces on trail Knowns Unknowns L, My N, Wey T, P, R ——— eee ~* Bon If T — h l a) 1.6 A's T= Bly == £ poanrpt ’ 797°? @ 7g DAB (c) Each Fornulae, Ey a similar method of reasoninzs we find,usins the Pach fornulae b e = -~ .2 4 I = ( 1 - 1,3130% + 1,1750% ~ 44704) 1 1 _ ole Let = A" 1 = 1,313¢" + 1.175¢e% - .447c* = BN" h 1 hé esas I= =—— n 1.79 #AN ? 8.05 E ZT ~2 79 an on -=% h° A" Pp ~45~ ~.OW arrance the formulae in parallel colums as bvefore: Case IV. (Square bar with roun (a) Theoretical ¢ = V2rl , § . 2 Uren S h G h 2 f = Sr, 1 2 h BE b BG V = h°1(1 - .8580¥) _.21. g2 ,p8p Y= vT— 5 G ~ “E550?” 2 APB Let C = (7 - “B5eee ? vsor 24 An analysis of the above formu (1) (2) (3) (4) varies as Fiber stress being Fiber stress being equal, P varies Load being equal, f varies as &, 2,0 1 2 Gog oF Be . > *, and 4 three sets of formulae civen above, A, AB that equ2l,f varies as A, A', or A", a8 AE, A'B! or A"B" i Bf ° Fiber stress beins equal, V (,°, weicrht of spring) are tabulated below for each of the ~46~ page, These constants are plotted to scale on the following Theoretical Haitte 0 A AB - 3 An oatpr 2, od, 0, l. 1. 1. 1. 1, 1. 1. 1. 005 1,030 1,064 1,006 .947] 1.020 1.014 1,003 .966 wi 1,062 1,036 1.025 .902 11.042 1.025 1,016 .928 »15 1,096 1,041 1.053 .861 | 1.064 1.025 1.037 .900 08 1.133 1,036 1,094 ,823 | 1,086 1,019 1,064 ,873 0-65 1,172 1.082 1.147 .791 | 1.111 1,010 1.099 .843 00 1.813 .999 1.215 .762]1.137 .994 1.143 .818 4 1,306 .931 1.403 .710|1.191 ,.950 1.254 ,764 25 1.414 .833 1.698 ,66711.250 .884 1,414 ,718 Bach °C A" A™B" A, on 0, 1. 1. 1. 1. 05 1,011 1,007 1,003 .980 eL 1,081 1,007 1,018 ,.966 15 1,032 1.005 1.028 .947 08 1.043 .996 1.047 .931 65 1,056 .986 1,071 ,911 20 1,067 .970 1.099 .895 0-4 $1,098 ,934 1.178 ,846 25 1.118 ,884 1.26 ,.795 a 4 ys a ~S 3 a r ’ 4,6 ar 2 4 ee / } 9 | \ f ij } ee s . a ro | | \ re i in j i > | | \ * | \ - is | | sy ry, 1 oy } \ = a! | r } Ps A “a | | a 4 te | | | | | Ma Maat eae. AIL Nee wT) Oka VALUE of Cc" lal aaa Le a Peano t