Torsional Vibrations in Reciprocating Engine Shafts

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  • 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 .Accessed: 07/05/2014 02:00

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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

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  • 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.

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  • 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

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  • 102 G. R. Goldsbrough.

    ,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)

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  • 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 trigono- metric 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 approxima- tion 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.

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  • 104 G. R. Goldsbrough.

    ? 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 oscilla- tions 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

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  • Torsional Vibrations in Reciprocating Engine Shafts. 105

    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,

  • 106 G. R. Goldsbrough.

    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%

  • Torsional Vibrations in Reciprocating Engine Shafts. 107

    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...

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