torsional vibrations in reciprocating engine shafts

Torsional Vibrations in Reciprocating Engine ShaftsAuthor(s): G. R. GoldsbroughSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 109, No. 749 (Sep. 1, 1925), pp. 99-119Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/94498 .

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99

Torsional Vibrations tn Recipr ocating Engine Shafts. By G. R. GOLDSBROUGH, D.Sc., Armstrong College, Newcastle-on-Tyrne.

(Communicated by Prof. T. H. Havelock, F.R.S.-Received May 12, 1925.)

? 1. Introduction. It is well known that, in the case of reciprocating engines, there are certain

critical speeds of running at which the torsional vibrations in the shaft become large in amplitude and introduce an element of danger into the system. Fairly simple methods have been devised for the practical calculation, from the constants of the machinery, of the location of these critical speeds. In these methods, the reciprocating parts of the engine are replaced by an " equivalent mass " which is assumed to contribute to the elastic vibrations of the shaft -in exactly the samie way as do the actual, rather complicated, system of crank, connecting-rod, piston and piston-rod.* It is the main purpose of this paper to examine the correctness of this equivalence.

In two particular cases examined by the author, the automatic records of the shaft vibraotions at about a critical speed showed a large amplitude at the expected point, but the period of the vibration was twice that anticipated. This anomaly is examined on p. 116 and the conditions of its existence exhibited.

The results obtained in this paper are applicable only to relatively low-speed engines. The anialysis would require sensible mnodification to cover speeds amounting to several thousands of revolutions per minute. A

? 2. Formation of the Equations. Consider a dynamical system composed of a crank, a

connecting-rod, and a piston. Let the axis of the crank be / through 0 perpendicular to the plane of the paper (fig. 1). G Let r be the radius of the crank OB, and I the length of the connecting-rod BA. The piston and piston-rod PA oscillate B along OA as OB rotates. Further, let r

ml bi- the mass of the piston and piston rod; ? m2 be the mass of the connecting rod; I' be the length GA, where G is the centre of mass of the connecting rod; k be the radius of gyration of I about an axis through G parallel to the

axis of rotation, and I be the moment of inertia of the crank about its axis of rotation.

* See, e.g. ' Proc. Inst. Civil Eng.,' vol. 162, p. 371. H 2



100 G. R. Goldsbrough.

Then if we take an origin in 0, put x for the length OA, 0, measured anti- clockwise for angle AOB, and qb for angle BAO, the kinetic energy of such a system is

T =2 [m1l2 + M2 (k2 + 1'2) b2 + M2d2 + 2m2d1' sin q . + 162].

From the simple relations x = I cos ' + r cos 0,

r sin 0 = 1 sin ',

we can express T in terms of 0 and 0 alone. It is convenient to expand the formula in powers of t, which equals r/l. Since r/1 is usually about 4 we find a series which converges fairly rapidly. Retaining the terms only as far as tX2, we have

T -- (in1 + n2) r202 [t - cos 20 + ,u (cos 0 .cos 30) + -,u2 (I1-cos 20) ..

+ ,12m 02 [(1 + cos 20) (k2 + 1'2)- rl' (cos 0 - cos 30) ...]

+ II62. (1)

It is worth noting that the radius of gyration of the connecting-rod and the position of its centre of mass only appear in terms factored by V2. So that, to the order used, approximate values of these quantities will suffice.

Next consider the thrust applied to the piston by the expanding gases in the, cylinder. This thrust is periodic and can be expressed in the form of a Fourier series. But the character of the series will depend upon the type of engine studied. In order to fix ideas we shall consider here only a six-cylinder four-stroke cycle, single-acting Diesel engine. The results obtained can easily be extended to other cases by simple modification of the constants involved. In such an engine the thrust on the piston may be expressed in the form

San sin nO/2 + Eb6, cos nO/2.

The period of this series is 4-x, corresponding to the two revelutions of the shaft in which the cycle of operations is completed.

The work done in a small displacement 3x of the piston is

- ax {Ea,, sin nO/2 +Ebn cos nO/2 }. But

Ax -r(sin 0 +I ,t sin 20 ...) 80,

the next higher term having the factor ~0.



Torsional Vibrations in Reciprocating Engine Shafts. 101

Hence the work done in the small displacement is

- a0 [an {cos (nf-2) 0/2 -cos (n+2) 0/2}

+bn {sin (n+2) 0/2+sin (n-2) 0/2}

+.lIan {cos (n -4) 0/2 -cos (n+4) 0/2} + bn {sin (n+4) 0/2+sin (n -4) 0/2}]. (2)

If now, we put H.-, and Hm as the torsional couples on either side of the rnth crank, we may write the equation of motion of the mth system as:

d aT aT _ + r/2E [an {cos (n-2) 0!2 -cos(n+2) 0/2} dt ao ao

+ bn {sin (n+2) 0/2+sin (n-2) 0/2}

+ -ta,ua { cos (n-4) 0/2-cos (n-4) 0/2}

+ 2 , {sin (n+4) 0/2+ sin (n-. 4) 0/2}] + Hqm--Hm. (3)

Consider now the arrangement of the separate systems along the shaft. First there are the six cranks of the six engine systems and the flywheel separated by short portions of shafting. Then there will be a longer length of shafting co(nnecting with another rotating mass-a propeller in a marine engine or an armature in the case of a dynamo driven by an engine, or perhaps only the rotor of a dynamometer; and this mass operates against a resistance, in overcoming which the energy of the engine is consumed.

We shal1 assume that this latter resisting couple is uniform and of magnitude H, and that the terminating mass has a moment of inertia about the axis of rotation of magnitude I,. We shall also take the moment of inertia of the flywheel to be If.

All the parts of the mechanism are elastic. We shall, however, in order to reduce the work and secure a first approximation, assume that only the longer part of the shaft is flexible and that all the rest of the material is perfectly rigid. For the purposes of torsional vibrations, it is probable that the portions of the shaft between the various cranks must be considered as elastic. These portions a-re, however, usually short and stout, and, compared with the longer part of the shaft, may with sufficient accuracy for a first approximation be considered rigid. It is intended to examine this matter in detail in a later paper. We are now considering an engine and flywheel, rigidly connected, operating an elastic shaft with a load at the other end.

Let d be the diameter of this shaft, supposed uniform, L its length, and




,t' the modulus of rigidity of the material of the shaft; theDn if we neglect the mass of the shaft itself (this is usually quite small compared with the other masses), the torsional couple producing a difference in angular position of the two ends amounting to o is Ko, where K = -I -rV'd4/L.

The six cranks belonging to the separate cylinders are usually placed at 1200 difference in phase. As the crankshaft is assumed rigid, for our purpose it does not matter what the order of the cranks is. We can take it that at any instant the positions of the cranks will be

0, 0 + 1200, 0 + 2400, 0 + 3600, 0 + 4800, 0 + 6000.

Equation (3) will give the expressions for each system in turn if we replace 0 by 0 + 120, On making these substitutions in (3) and adding up the six equations, we find that all the terms disappear, except the constant terms and those which involve circular functions of multiples of 30. In particular,

T = M12 [3 (m1+ 2) r2 (1-, cos 30) + 3M2,2 (k2 + 1'2+ rl' cos 30) + 61].

The right-hand side of (3) gives

(r/2) [6a2 + 3taj] +3r [cos30 {a8-a4+ ip(ao0-a2)}

+ sin 30 {b4 - bs + 2 (b2 -blO)

+ cos 60 {- a10- 1ta8}

+ sin 60 {bl0 + ,pb8} + ...]-H6, since Hoz- 0.

The equation for the flywheel is

if; H6- -1',

where H' is the couple transmitted to the main shaft. On substituting tl-ese values in equations (3) and eliminating H6, we find,

0 [3r2 (inl + in + + 3m202 (k2 + i'2) + 61 + If - {3 pr2 (i.1 + M2)- 3m2 2rl'} cos 30]

+ 2 2 [P (Mi1 + i2) r2 - 2rl'] sin 3 0 - r (3a2 + 3/2 Va4)

+3r [cos 30 {a8 a4 + -1 (a10-a2)} + sin 30 {b - b8 + Rt (b? -b

+ cos 60 t- a'10 - 2

+ sin 60 {b;lo 1 p }+ ...2]-1 H'. (4)



Tortsional Vibrations in Reciproating Engi'ne Shafts. 103

0 is the angular displacement of the flexible shaft at the flywheel end. This will produce a torsional couple which will be transmitted to the opposite end. Let ~ be the, angular displacement at the latter. Then the couple transmidtted is

K (0-~)

This couple produces an acceleration in the mass of moment of inertia I, and also overcomes the resisting couple there, viz., H.

Henice we have

and (5) H'=K (0- 4~) J

These give the equations

0[3r2 (inl ? in2) + 3M2 ~L2 (k2 + 1'2) + 61 + If. -{3Vr2 (in + in2) - 3M2 ~t2rl'} cos 3 0]

+ 9O2 [p (in., +Mi2) r2 - ~t2m2rjl] sin 30 -r(3a2 + ~ta- 4 1t(a4) + 3r [cos 30 {a8 - 2a + (1-a2)}

+sin 30 {b4- b8 + ~t (b - b lo)} (6) +cos 60 {- al- 10- pa8}

+sin 60 {bj+10+tb8} ...1

I~y==K(0-q)-HK - (0 - )I These equations, within the small restrictions already stated, completely

express the motion of the engine and shaft. Inspection shows at once that these equations have not even a particular

solution representing uniform rotation, owing to the presence of the trigonometric terms. It is further clear that, since the mean applied couple must be equal to the, meani resistance,

II=r(3a2 + 3 ~a4)

and K (0 - $ must contain a constant part equal to H. This cons-tant part in the difference of 0 and i-I represents the mean twist of the shaft when under torsional stress.

It is known from experience th~at uniform angular velocity is an approximation to the motion defined by the equations (6). We shall therefore assuLme that the motion can be represented by fluctuations about steady motion and seek to determine the nature of these fluctuations.




? 3. Solution of the Equations. In the left-hand member of the first of equations (6) the trigonometric terms

represent the effects of the inertia of the reciprocating parts. The usual plan of replacing these by a single equivalent mass leads only to the term

0 [3M2(i1 2) + 3Mi2 2 (k2+ 2) + 6I + If].

The term in the squared brackets is the " equivalent mass " of the engine system, and this is the only term generally used. But it is evident that the term factored by 02 is too important to be rejected without scrutiny. Especi- ally in the cases of higher velocities this term .must increase in importance.

We shall assume the mean angular velocity to be to, and, to find the oscillations about this mean value, we shall assume

coxt+, (7) -cot +JT 7

The substitution of these values in the left-hand members of (6) takes an approximately full account of the inertia of the system. In the trigonometric terms of the right-hand member we shall omit the variation i. Its inclusion would represent the effects of the oscillations in modifying the compression of the gases in the cylinder, and these we omit in order to keep the equations of manageable size.

Further to simplify the work, we shall retain only linear terms in > and .

The quantity + is the mean difference between 0 and 6, and is such that

K = H. Now divide the first of equations (6) through by

I' - 3r2 (,i1 + Mn2) + 3iM2 2 (k2 + 1'2) + 6I + If; put

{3Ft2 (mi1 + 2) - 3[2m2rl'} --- I' ?;

change the independent variable from t to x, 3 cot, and indicate differentia- tion with regard to X by dashes. Then

0 (1- E cos 30) + 30'2E sin 30

-H/9I'co2 +fi cos 30 +f2 cos 60 * (8)

+ g1 sin 30+ g2sin 60 - K (0 -)/9I'c, 2

ff= K (O - )/9IPco2 - H/9I1Co2 J




In these equations we have written,

fi ~ 3r {as -a4 + 2 (alo -a2) 921

A 2 3r (-a10 - aO 91-3r {b4 -b8 + e t (b2 - b10)}j 9W2I',

g 2-3r (b10 + I V-b8) 9 2J'.

Now substitiute (7) in (8) except in the trigonometric terms of the right-hand member, and putting for short

I1 K/9I'co2, l2 = K/9IJ02, we have

0" (1 -?~ Cos r) + Cl ( -) + e (0' sin r + -1 cos r) --l sin r + f1 cos r +If2 cos 2r + g, sin -r+ -g2sin 2-c,

<'+ KC (2 i) 0.

The quantity e is small compared with unity and we shall treat it as of the first order o f small quantities. The quantities f and g are also of the first order of smallness. Hence working to this order only,

i + KC (j -) (1 + e cos r) + e (0' sin r + j cos r) -6 sin r + fi cos r + f2 cos 2x + g, sin C + g2 si (9)

<+ KI 2 (9)

Or, finally, on eliminating i,

<"'+ (tC1 + - K2) ? 1{K 1 " COS -" + K271') sin C ( + K2TI) COS x} e-s sin C +ff cos C +f 2 cos 2x + g, sin C + g2 sin 2-. (10)

If we remove the pressures of the gases on the pistons, that is, in equation (6) put H, and all the a's and b's zero, and as a consequence put @ in (7) also zero, equation (10) would still have the same form except that fl, f2, g, and g2 would be zero. The term- le sin C is still outstanding in the right member and introduces something of the nature of a forced oscillation. It arises from the incomplete balance of the reciprocating parts.

Omitting completely the right member of (10), we have a linear homogeneous equation with periodic coefficients. The roots of the characteristic equation are readily found to be ? /C, 0, 0, where IC2 -C1 + K2. We shall proceed to the solution on the supposition that Ic is not an integer. Take first the solution in the vicinity of the zero roots. Theory shows that this has the form

- eCT

Y a io + d 1 + s 2 + p d e CS .C1+ FCG2 + * s ;

aUnd y is purely periodic of period 27r.




On substituting in (10) the given value for 1, we find

y"" + 4ay"' + 6a2y" + 4a3y' + a4y + /C2 (yt + 2ay' + a2y)

+ e {Kj cos X (y" + 2ay' + a2y) + sin X (y"' + 3ay" + 3a2y' + ,3y

+K2y + IC2GY) + 2 COS - (y" + 2ay' + a2y+ ECY)}-O

Now introduce the values for y and a, and we have for the terms independent of e,

'"No + KC2%<1 _0.

The solution of this of period 27t, is

= a constant = c0.

To make the solution quite complete we require certain initial conditions. We shall assume that when - 0, c0. It follows that at X = 0,

1-2-*.-?0.

The terms in ? give

'h11-- + 4Gp%0"'+ K2 (n1"t + 2ap-%') 0.

This equation leaves a, indeterminate and requires

11 = a. cos cr + b1 sin Kc- + cl.

The conditions to be fulfilled show that

al b C- 0. The terms in e give

_2" + 4G210" + 4f111" + 6ca2no + K (12 + 2G2%' + 2alNi + a1 0) = 2 C2Co COS X-,

Since the right-hand side has no constant term, therefore a1, 0. From this

2 2 qi2c 2cos 2

+ a2 coS Kcr + b2 sin Kc- + C2. 1 - K,2

From the conditions to be fulfilled,

a2 =b2 ? , and 2 = 2_

From the terms in e3/2 we have

3"+ K2 '3 = 0,

as the other terms cut out. Hence - 0 as before




From the terms in el,

N 4 + 4G2i 2""+ K2 (n + 2a2yA2' + a2 20)

--?- + 4 1 2 82 + 8- 2 4 +

+ - 2cos r]. To avoid the explicit appearance of X we must have

-2 2

- 2 (2X1- 1 + K2), (11)

and then Co (4Q1l2 + 3K2 2- cos2) s2 - Ac0(C22

4 (1 _

K2) (4 _

2) (1 2) 2cs. (2 It is not necessarv to proceed to higher terms of this series. We pass next to the solutions associated with the roots i LK of the

characteristic equation. In this case the solution has the form

-ey Yo+ (Th+rTh+

Y LIC + e1 + Z2a2 +...:

and y, as before, has the period 27r. On substituting, we have, for the terms independent of e,

"o + 4K!/// 5- 2o - 2LK370o' - 0.

The only solution of this which is periodic, of period 2r, is

= = a constant - c0. The terms in c give

rn""1 - 4li"- 5K - - 2L, - 2tic3cy 611 - 12 210 cos t.

Therefore ai6 0, and

= - K2170 coS (1 + 5K2 l,22 (4LK - 2t/c3) o sin No

On applying the condition that 1 0, when X 0, we have finally,

=(1 2 2 {1 + 5K, - (1 + 5K,) cos X - (4L'c - 2LKc) sin x}. (13) (-K2)2 (i - 2K)

The terms in s2 give the equation

Y~2 H 4LK7 2" 5K,2"21' - 2Wcv2' -K -

(1 _C-K2 (1 + 5K2)

Kl+2 2) + periodic terms. (1 - K 2)2 (1 -2K -2

2




Hence - 2CK52 42 (1 + 5K2 (g + 'K2t - 2L~~3a2 -

It is unnecessary to carry the calculation of these quantities any further. By changing the sign of fC in (13) and (14), we can obtain the solution corre-

sponding to the other root,- LK, of the characteristic equation. Collecting up the results we see that the integral sought has the form

= cle (') y1 + c2e (2)Ty2 + C384(3I y3 + )

where c(1) and a(2) are given by (12) and ((3) and a(4 can be inferred from (14). Inspection shows that in general the characteristic exponents a are distinct and none of them are congruent to zero, mod. V(- 1).

To complete the solution of equation (10) we must take account of the right- hand member. Without working out the result in detail we can infer the nature of this particular integral. Since the right-hand member of (10) is purely periodic, with period 27, and the solutions already found of the equation without this member have their characteristic exponents unequal and none of them congruent to zero, mod. V( 1), except, it may be, in special cases, it follows that the particular integral sought will also be purely periodic, and will contain no real exponents and no terms involving the time explicitly.

? 4. Examination of the Solutions Found. It is necessary now to examine the solutions found in the last section in

order to determine under what conditions, if any, they are valid, and also to see what types of motion they indicate. We have determined the solution in the form of a power series in the quantity s. The general theory of linear equations of this type shows that the series will be convergent for all time for value of X sufficiently. small, except at, and in the neighbourhood of, critical points. Though we have determined very few of the terms of the various series, it is clear from inspection of the results (11) to (14) that critical points occur at K2 - 1, 4, ... ; that is, whenever Kc is an integer. In these. cases the solutions already given will fail; and in the vicinity of these critical points the results will be of doubtful value. We must, therefore, construct fresh solutions valid in the neighbourhood of, and at, these critical points.

Keeping clear of their particular values, we can examine the nature of the motion given by the solutions.

Referring to the characteristic exponents, that determined by (14) will always be imaginary within the region of its validity as also is its conjugate



Torsional Vibrations in Reciprocating Engine Shaft8. 109

term. These, therefore, will not give rise to anything exceptional in the motion. The exponents determined by (11), however, seem to be almost entirely real. Since KC1 + 2 = K2, we may write this expression

,2a2 -yk2(

So that except for values of K1 > 1 - C2, a 2 is a real positive quantity. This is the more emphasised in that K1 is usually small in comparison with KC2.

It would, then, appear as though the vibrations given by this expression would increase slowly in amplitude, for a large range of values in the constants, in the entire absence of friction.

The vibrations included in the particular integral are everywhere finite in amplitude, except in the vicinity of the critical points mentioned.

? 5. Solution of the Equations in Critical Cases. I. K = 1.--Returning now to equation (10) we shall find the solution valid

for the critical case K = 1. In order, however, to cover more than the exact critical point, we shall put KC2 1 + oCe2, where a is a small variable quantity. The equation then becomes

n" + <' (1 -4- oCe2) e { COS T - (Y" + K(jY2) sin r + 2 ( + K2 ) COS r}

+ (g1 - -) sin XK+ g2 sin 2xr +fi cos X +f2 cos 2X. (15)

We shall take the equation first without the last member, and so study the homogeneous part.

As before,* put = e0fTy,

Y -O+ 1+ F2+" 2

a ea, + e2a2 +

On substituting the terms without ? give

= ao + bo cos -c + co sin -c,

where ao, bo and c0 are arbitrary constants.

The terms in e give

" + 4orjY%"' + ?J1" + 2a1Yj0=' - ET), where we have written

E) (*) = v cos -r + (" + K2 ) sin 2 + 2 ( + CS) COS T.

* The tacit assumption that y and a are expansible in simple power series in e can be justified by suitable modifications of the usual existence theorems.




To avoid the constant terms appearing in 0 we must have

bo = 0.

Also, to remove the term in cos x.,

2alco =K2a0; (16) and, finally,

e1=-al+b1cos-r+clsinT+ (5-3K2) co sin 2-. (17)

Up to this stage, the quantities ai and one of the two quantities a0, c0 remain

undetermined. The terms of the original equation factored by e2 give

2" + 4crpm,"' + 4c2%'ff"' + 6g1210" + 12 ? 2a1jN1 + 2a2%''

+ a12yj0 ? n = 0 d1). (18)

On picking out from 0 (n1) the constant terms and those involving cos X

and sin -r, we have the equations

a12 a0 = b,/4 (5 - 3C2), (19)

2a bi - ac- 5a1 c0 T-- (5- 3K2) ( 4 + 3K2) co, (20)

- 2c1 - 272c0- 2 c2a1. (21 )

On eliminating ao, c0 and b, from the equations (16), (19) and (20) we arrive at the equLation for a1, which is

32a14 - 5K2 (5 - 3C2) a2 + K2/192 (5 - 3Kc2)2 (4 - 3K 2)

-o c2 (5-31C2)=-0. (22)

When a1 has been determined from this equation, (20) determines c0 in terms of ao, which we may regard as arbitrary, and (19) determines b1 in terms of ao. The relation (21) gives one equation for determining a2 corresponding to (16) for a1. For the complete evaluation of C2 and cl, we must proceed to the equation formed by the terms factored by ?3.

So far as the work has been carried, the exponent a is known up to the first power of ?. This is sufficient to give a clear idea of the motion, and the approximations are carried no further.

II. Kf = 2.-We proceed next to the case where the quantity K approaches the next critical value and, in equation (10) we put K2 = 4 + M2. The solution follows the same lines as before, and it is only necessary to write down the final results.



Torsional Vibrations in Reciprocating Engine Shafts. It 1

The values of the constants fall into two separate groups:

(a) 12 =9K6 2 (1 + 2), (23)

ao is arbitrary,

bo=-o = 0,

b 1 0,

el - -%L2 (5 -3K)

1l - -?- 384 al

a, - C 2aO,

1=a, + cl sin 2-r + laO2 COS T.

(b) a1 _o, 256a22 + {16oc + T (110KC2 2 -130c2- 456)}

{16oc + 7 2w (- 25/C2 + 2752- 756)} 0, (24)

b _ 2K2 a

? 4 -3K ?'

{- 16. + 7Tl (- 25K22 + 275K2-756)} bo. 16a2

Equations (23) and (24) show four values of the exponent a, thus com- pleting the necessary number. In a similar fashion we could determine the solutions in the vicinity of fC _ 3, 4,.

? 6. Exarmination of the Solutions Found in ? 5. Before proceeding to find the particular integrals of the equation (10) under

different conditions, it is of service to examiine the partial solutions already found. The character of the solution is fixed by the nature of the exponent a, as the co-factor of the exponent is purely periodic in each case.

(1) Case when K 1. --The index a is in the nature of a series, CS = ?Sa1 + ?262 "' . As ? is known to be a very small quantity, we have merely determined the first term. The four values of a1 are given by equation (22), which we repeat:-

32a14 -5c52 (5 - 3c2) a12 + -s2- (5 - 3K2)2 (4 - 3K2) 192 - ocK2 (5- 3K2) -O (25)

In this equation fC2 is a constant. depending upon the masses of the reciprocating machine and the other constants of the motion. The quantity ca is a variable parameter, introduced to give an idea of the character of the motion near to the critical point c = 1.




If the equation (25) has as a solution for a1 a positive real value, then the motion tends to depart largely from uniform angular velocity, though it is clear, of course, that with lapse of time the original equations, framed on the assumption of small values of 0 and -, break down. On the other hand, purely imaginary values of a1 mean that the motion consists of small variations about unifornm angular velocity. We therefore proceed to examine the nature of the roots of this bi-quadratic equation.

Since K1 + KC2 1 + 20C, and the last term is small, K2 < 1. Hence the factors (5 - 3K2) and (4 - 3KC2) are positive.

Now consider the case when oc 0. It is easily found that if K2> about 0 1, the values of a12 are both real and positive. Even if this condition regarding Kc2 is not fulfilled, the real part of the value of a,2 will always exist and will be positive. Hence in both cases we shall have one pair of values of a1 with their real parts positive. The introduction of a positive value for oc makes the values of a12 more certainly real, one value of a12 becoming greater and the other less. Ultimately, when

(5 - 3K2) (4 - 3'C2)

192

we have one pair of values of a1 zero. This is an important point. When a negative value of oc is taken, the values of a12 tend to be more certainly

complex; but the real parts remain positive and the results are as before. There is clearly no value of oc that would make the values of a1 purely imaginary. Hence it can be said definitely that at every point in the vicinity of Kt- 1 the vibrations increase in amplitude with the time.

(ii) Case when - 2.-The four values of the exponent are given by (23) and (24). The first is always real and positive, the second real and negative. Both are independent of a, and the former indicates, in the vicinity of K = 2, vibrations of increasing amplitude. The third and fourth indices begin with S2a2, and hence are extremely small. In the expression (29) the quantity (- 25K22 + 275K2- 756) is negative for 0 < K2 < 4; the quantity (110 K222-13OK2-456) is negative for values of K2 between 0 and about 2 7, but for higher values of K2 is positive. Hence, when oc- 0 and K2 < 2'7, a2 is real and negative. The motion then shows no tendency to exaggerated amplitudes of vibration. If, on the other hand, K2> 2'7, as is the case in many types of engine, a2 is real and positive and the motion will exhibit vibrations of increasing amplitude at the critical point.




A pair of zero values of a2 occurs at

and = TTI2 (456 + 13OK2 - 110 K22)( andi (26)

X TT-I (756 - 275Kc2 + 25K22) J

These indicate two points very close together, which may or may not include the point K 2 between them, though it must always be near them. Wheni x has a value intermediate to these two values, a22 is real and positive and the vibrations have increasing amplitudes. Outside this range the motion shows no such increase in amplitudes.

? 7. The Nature of the particulatr Integrals in the Critical Cases.

Having found the solutions of equation (10) without the right-hand member, we now proceed to discuss the nature of the particular integrals when that member is included. We can infer a good deal about these integrals from the general theory of such equations already quoted. The right-hand member is purely periodic of period 2 r. If, then, the characteristic exponents already found are all unequal and none of them congruent to zero mod. V/( - 1), then it is known that the particular integrals will also be purely periodic. But if either of these conditions is unfulfilled, the particular integrals will involve the time explicitly. From the nature of the case, none of the characteristic exponents can be equal to an imaginary integer. But we do have certain pairs of exponents equal to zero. WhenK2 --1 + e2%C, this occurs, as already shown, at

(5- 3Kc2) (4-3K 2)

192

Owing to the smallness of e, this occurs very close to the point te = 1. But it is important to notice that this occurs independently of the type of term in the right-hand member. All that is requisite is that the term should be. periodic.

Again, in the vicinity of K = 2, the exponents given by (23) lead to no terms in the particular integral other than periodic terms.

The exponents given by (24) produce terms involving the time explicitly in the particular integrals at the points indicated by (26). In practice these points are close together and very close to the point K _ 2.

The right-hand member of equation (10) has the value

- C sin X +fi cos X +f2cos 2 X + g_ sin C + g2sin 2 x. VOL. CIX.-A. I




Of these terms, the first arises from the inertia of the reciprocating parts, while the others arise from the thrust on the piston. From what has been found above, it is clear that the first term alone is capable of producing terms increasing linearly with the time in the vicinity of the points Kc 1, K 2, and, by induction, K equal to any integer. In other words, quite apart from the character of the indicator diagram altogether, the system of the reciprocating parts produces vibrations,.increasing in amplitude with the time, at each of the critical points defined by Kc equal to an integer. This point is of such importance that it is worth examining in greater detail.

? 8. Further Development of the particular Integral. In order to avoid critical points, the best method of constructing a particular

integral is to apply the method of arbitrary constants to the solutions with the characteristic exponents already found. A readier method, if less exact in certain places, is to follow the well-known process of assuming a solution and finding the sequence relation between the coefficients. We shall adopt the latter plan. We require a solution of the equation

" + K2 m" + p {C, lcos T + + K2 ') sin - + - (<' + I2C) COS '} ~~~~ (27)~~~~~~~~~~

-(g, 6 C) sin Xr + 92 sin 2xr + fi cos X d-49 cos 2,r (27) and we shall take, more particularly,.the cases of Kt approximating to integers.

Consider only the sine terms of the right-hand member first, and assume 00

E B,, sin nr.

On substituting we find the coefficient of sin n-c in the left-hand side is

n2 (n2 - K) Bn + le {-(Kl + -1) (n- 1)2 (n 1)3? K2+(n -)}B 1

+ 2? {(- Ct + 1) (n + 1)2 + (n + 1)3 K2(- + -2)} B 1 jf, for convenience, we write

pn=le{-n2(K1+ 2) n3F n 2(fl+)}K

and qn=2{n2(- K+ 2)+ 3 K2(n 2)}

the resulting equations are

(12 - K2)Bl+q2B2 -1 - -UEI

pBj+ 22(22 -2)B3 IC2I

P2B2+ 32 (32 - tC2)B3 -+ q4B4 - 0 S (28)

pn-l Bn_] + n2 (n2 - K2) Bn + q.+, Bn,j ? 0, *

** . ***** .......... J




The general relation between the coefficients is of a well-known form.* It is readily shown that the sequence of coefficients forms an absolutely convergent series, provided we add the further condition that

L B- +l.. n-+0 B17b

In practice, the application of this condition amounts to. the replacement of a finite series for the infinite series, the finite series stopping at some term Bn such that B,+1/B. would be less than a certain quantity which is regarded as the lower limit of magnitudes to be considered.

Working, then, with the finite series, and writing

A 2- K2, q2 0 ?0 0

P1 22(22- 2), q3 0 O

0 , P2 , 32 (32 i2), 0

@-2 I@@ 2(n-1)2{(n 1)2- K2} 9n

... ,0 Pn-I X n2(n2_-C2)

and A, the same determinant in which the rth column has its first and second

menmbers replaced by g, - Is and g2, respectively, the remaining members of that column being all zero. Then the solution of the equations (28) is

Br = Ar .

for all values of r from 1 to n. And the particular integral of equation (27) is

n A= Er sin rtjA. (29)

It is clear from this solution that in those positions where A is small the -amplitudes of the harmonic terms increase; and when A vanishes the solution as it stands fails. But it is readily shown that in the latter case, by the device commonly used for this type of equation, there appears a factor C

explicitly attached to the coefficient of each term. This corresponds closely with what we found in an entirely different way

in the last section. It is to be noticed that as the p's and q's all have the factor S, which is small, the value of A is small whenever K is an integer. But A

* CJf. Lamb: 'Hydrodynamics' (Fourth Edition), p. 323; and 'Proc. Lond. Math. Soc.,' Ser. 2, vol. 14, p. 38.

i 2




does not rigorously vanish for any integral value of K. We may put it that the necessary value of K for that purpose has the form K2 r2 + OE2, where r is an integer and oc is small. In fact, the solution of the equiation A 0 should give the same values of oc in the vicinity of r 1 and r = 2, as we have already found by the other method.

Now let A() be the determinalnt formed by deleting fronm the determinant A the first (r - 1) rows and columns. Then

B1 {(g1 -?) A\2) - g2q2(3A} ? A,

B2 {(12 - K2)g2A(3) - pl (gl - s) A(3)} ' A

and so on.

Now, suppose g -- 92= 0. That is, we have in the right-hand member of equation (27) onily the term - le sin x, which arises from the inertia of the reciprocating machinery, and no actual external force acting on the system. Suppose, further, that Kc 2. We shall avoid the complication of A = 0 by considering what happens when A is small, because Kc-- 2. Then

B1 - 1 A(2 I1--- A 1A,

B2 6 - 6p1A(3)/A. But, as K 2,

(2) -_p2q3A A4)" Hence

B2/Bj - P1A(3) + p2q3 A (4).

Now the first terms respectively of A3) and A(4) are independent of s, and that of A(3) is greater (numerically) than that of Y'). So that, for sufficiently small values of e it is evident that B2 may be made a large multiple of B1. In the series for -, in such a case, the term in sin 2x will preponderate, although the motion is excited by a term involving sin Xr only. We may then expect to find, arising purely from the inertia of the system itself, and apart from any external forces, large vibrations produced in the vicinity of KC

equal to an integer. Again, restoring g, and .92, it may happen that the constants of the system

are such that, in the vicinity of K = 2,

(12 - K2) q2 -P (i -c)

or, at any rate, the values may be such that the difference of the two members is small compared with

1 C.) A g2q) A U} /\ In such a case the term of n in sin 2-6 will be small compared with the term

in sin T. We should then find that though the vibrations were large they




had the period 2-r, and not -n, as would be anticipated from the fact that K 2.

This departure from anticipation is exaggerated in calculating the value of i. From equation (9)

i ! |K2 +T Now in a large number of cases of a certain type Kc2 is large compared with

KI. Therefore, since K1 + K2 K2 = 4, approximately, it is clear that K2 is also approximately 4. Now, remembering that = B1 sin Tr + B2 sin 2T + and already B2 is smaller than B1, we find

i K2 21sin X + - B sin2t+ .... K2 K2

So that, at this point, the coefficient of sin 2t is still further reduced. An instrument, therefore, put to register the motion at the engine end of

the shaft, would find the vibrations about uniform rotation almost purely of period 27z.*

In the general case, with no particular relations among the terms, the solution given by (34) would show exaggeration cf amplitudes in the vicinity of the critical points, but would not show a vibration of simple form; it would be compounded of all the terms of the series, none of which would markedly predominate, but the whole would have period 27.

The cosine terms of the right-hand member of equation (27), which were omitted, can be handled in exactly the same way as the sine terms and produce similar results. The final value of n is obtained by adding together the two series.

? 9. Summnary. In this paper we have investigated analytically the vibrations about steady

motion of an elastic shaft having an engine at one end and a mass operating against a resistance at the other end. The engine is assumed to be of the reciprocating type, and to fix ideas we have taken it to be of the six-cylinder variety of four-stroke, single-acting Diesel engine. To simplify the work we have assumed the engine parts to be rigid. We have taken fully into account the inertia of the reciprocating parts of the engine, but we have omitted entirely any account of the internal resistance to vibration in the flexible shafting and the resistance to vibration which must also occur at the load end of the shaft. The omission is due to our ignorance of the character of these resistances.

* Cf. Rayleigh: ' Phil. Mag.,' vol. 24, 1887, p. 145. A. Stephenson: ' Quarterly Journal of Pure and Appliedi Mathematics,' vol. 37, p. 353 (1906).



118 Torsional Vibrations in Reciprocating Engine Shafts.

The oscillations about steady rotation are found to be of two types, analogous to free and forced vibrations respectively. The forced v7ibrations do not arise entirely from the pressure of the expanding gases on the pistons. They are also produced by the incomplete balance of the reciprocating parts. Practically all the phenomena associated with dangerous critical speeds would appear if the fuel were cut off and the enigine made to run against no resistance with the requisite speed.

The positions of the critical speeds are approximately those found by the usual methods. Slight differences appear, but in practice they would be probably unimportant.

The vibrations, analogous to free vibrations, appear to be of the form eot multiplied bv periodic terms, where a is a small constant and t is the time. Throughout the whole range of possible speeds of rotation at least one of the quantities C appears to have a real part, existing and positive. This would point to the fact that, apart from frictional resistances, the vibrations everywhere tend to increase in amplitude with time. As this is contrary to experience, it is probable that the various kinds of frictional resistance that are possible have the effect of cutting out the real and positive part of a, or, if not exactly cancelling it, at any rate of leaving the real part negative. This would cause any such vibrations, if at any time excited, to be damped out.

Such frictional resistances would not produce, in the case of vibrations analogous to forced vibrations, any marked change of clharacter. Such change as occurs would be one of degree and not kind.. We may then interpret the vibrations of this type as having a direct physical significance.

As in the last case, we find the critical speeds are moved slightly from the values obtained by the approximate theory, but not so mLuch as to be of practical importance. Between the critical speeds the vibrations are small. As already remarked, however, these critical speeds do not arise only from the periodic firing in the cylinders, but also from the incomplete balance of the reciprocating parts. Moreover, the latter is equally effective in point of magnitude in producing large vibrations at certain velocities.

The type of vibration produced is in general very complicated and cannot be definitely said to consist at any speed of a simple vibration of definite period. Two very important facts, however, appear:-

(a) Suppose that the applied force has a term of the form P sin 3 &t, where co is the mean angular velocity of the shaft and t is the time. This would corre- spond to three vibrations in the period of one rotation of the shaft. According to the usual theory, a critical speed would occur whenever the natural frequency



High-Frequency Fatigue Tests. 119

of elastic vibrations of the system was three per revolution of the shaft. It is shown, however, that the same term can produce a critical speed whenever the natural frequency of vibration of the shaft is 3, 6, 9, . per revolution of the shaft. This means that the existence of terms such as sin 6wt, sin 9ct in the applied force is not essential, and their absence cannot be taken to imply absence of corresponding critical speeds.

(b) Under certain circumstances, depending upon the magnitudes of the parts of the systeim, in the vicinity of a critical speed corresponding to six vibrations of the shaft per revolution we may find that the curve is not at all like the curve of sin 6cot, but is almost exactly sin 3 &t. The critical speed is right and the vibrations large in amplitude, but the period of the curve of vibrations is twice that anticipated. This explains the phenomenon found in an actual case and automatically registered on a torsiograph.

High-Frequency Fatigue Tests. By C. F. JENKIN.

(Communicated by Sir Alfred Ewing, F.R.S.-Received May 18, 1925.)

PART I.

The experiments on high-frequency fatigue in copper, Armco iron, and mild steel described in the following paper were carried out in the Engineering Laboratory, Oxford, for the Fatigue Panel of the Aeronautical Research Committee. The cost of the apparatus was defrayed by a grant from the Engineering Research Board of the Department of Scientific and Industrial Researcb.

In 1911 Prof. B. Hopkinson* called attention to the importance of ascer- taining whether the fatigue limit of metals was dependent on the rate of alternation of stress. He designed and made an electric alternating direct- stress machine, and published the results of tests on mild steel carried out at about 7,000 periods per. minute (116 per second), which was more than three times as fast as any tests made up to that time. The results at this speed were compared with those made by Dr. Stanton at the National Physical Laboratory on the same material at 2,000 periods per minute (33 per second).

* ' Roy. Soc. Proc.,' A, vol. 86 (November, 1911).



torsional vibrations in reciprocating engine shafts

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