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A METHOD OF CALCULir PING THE INERTIA FORCES JJHILE, perhaps, a matter largely of academic interest to others than their designm, some understanding of

the method of calculating the inertia forces iii recifirocitt ilig, engines may not be out of place. In practice the usual method is graphic, e.g. Klein’s diagrams and polygons of forces. The formula for calculating the inertia forces at any selected

r.p.m. is : P.:1. war (cos + cos Si-i, ). For simplicity all higher harmonics than the second will be neglected. / P=force in lb. M=mass of piston and upper end of con.rod, etc., in lb. g=-32.2 (to bring poundals to pounds). Co= angular velocity in circular motion. [2ir radians =360°J. r= radius of crank throw in feet. 0=angle of crank to cylinder length of con.-rod. A usual ratio of axis at any moment. L -– — mums of crank throw’

con.-rod length to radius of crank throw is 4 to 1 approx. Let us first consider a single-cylinder engine, and from the results, work out the balance, or lack, thereof for multi-cylinder engines. As we are not concerned with the total inertia forces generated (which become enormous at really high r.p.m.) we m

will assume throughout that 7-g Or =4 lb. At top stroke = 0°, cos 0°=1, 20 = 0°, cos 0°= 1, … cos 0+ c°128 =1 +1.

Thus the formula gives 4(1+1)=5 lb. The piston is, thus, exerting a force of 5 lb. upwards at t.d.c., for a body continues in a state of rest or of uniform motion in a straight line unless acted upon by an external force. This force, stopping and reversing the motion of the piston, is exerted by the main bearings acting through the crank and con.-rod. At bottom d.c. 0= 180°, cos 180°=1, 20 =360°, cos 360°=1 Cos 180° is 1 acting downwards, cos 360° is 1 acting upwards. Thus P=4(1-1). 3 lb. acting downwards.

When the crank has turned 90°, 0=-,90°, cos 90°=0, 20= 180°, cos 180°=1, acting downwards. … P=4(1)=1. Thus at t.d.c. we have a force of 5 lb. acting away from crank „ b.d.c. I 9 3 lb. „ towards crank „ 900 or 270° , 1 lb. „ Maximum piston speed is reached when the con.-rod and crank are at right angles, so on the down stroke the piston is

We have sometimes been accused of not presenting sufficient matter of a technical nature, so we have much pleasure in

publishing this contribution on engine balance by Dr. R. •• ***** ••••••••• ****** ••• *********** ••••••••••••••••••••

decelerating below this point. Hence we find that when the crank has turned through 90° the inertia of piston, etc., exerts a force towards the crank. The forces exerted on the upward stroke are exactly the same, though the piston is first accelerated and then decelerated to t.d.c.

The forces at t. and b.d.c. may be counteracted by bobweights on the crank heavier than needed to balance the rotating weights, but, unfortunately, this excess weight is unbalanced at the 90° position and so helps to add to the vibration. Twin-cylinder engine.—(l) 360° twin: Double the number of power impulses, but the same lack of balance as a single. (Triumph Tiger.”)

(2) Side-by-side twin with cranks at 180°(e.g., Scott).—This is used only for 2 strokes: one piston at t.d.c., 5 lb. up ; one piston at b.d.c., 3 lb. down (see .fig. 1).

This introduces a couple in the axis of the cylinders. By the addition of bob-weights this couple is eliminated, but a new couple introduced at the 90° position. In addition, as a 4-stroke, this engine gives unequal firing intervals.

(3) Flat twin (e.g. Jowett).—Here we find that the movement of each piston assembly is exactly matched by its opposite number, but the necessity for the two crank throws at 180° means that the cylinders must be mutually offset and so a narrow couple is introduced. This is eliminated by the bob-weights, but they introduce a new couple at the 90° position. Neither in (2) nor in this case. does the elimination of the couple balance out the forces at the ends of the piston travel. (4) 90° twin (e.g. G.N.).—Here the best achievable balance is due to the bob-weights (see fig. 2). Piston A, 5 lb. away from the crank. Piston B, 1 lb. towards the crank (at rt. angles to A) Bob-weight 4 lb. in opposite direction to piston A. The resultant force is %/ ,lb., acting horizontally to left. After turning through 90°: B piston 5 lb. away from the crank, A piston 1 lb. to wards the crank. Bob-weight 4 lb. in opposition to B. vita Resultant %/A. horizontally to Rt, after turning another 90° A piston (b.d.c.) 3 lb. towards crank, B piston 1 lb. towards the crank. Bob-weight 4 lb. in opposition to A piston. S2

Resultant /21b. horizontally to L. In the same way, after a further turn through 90° (b.d.c. for

B piston) we find A/21b. acting horizontally to the Rt.

Thus we .find small unbalanced forces acting alternately to L and R., four times per revolution, and unequal fir:ng intervals. Flat twin with single. throw crank.—Here we have one piston at t.d.c. and the other at b.d.c., i.e., 5 lb. +3 lb., say, to the left. A bob-weight rxerting 8 lb. to the right exactly balances this at both ends of the stroke, in fact, the combined movement of the Wood, who is an acknowledged expert on this subject ; incidentally, he ran for a time a 6-cylinder Scott-engined

Aston-Martin, and now possesses a flat-four Jowett.—Ed. ••••••••••••••• +++++++++ •00•0••••••••••• ++++++ ++++++ ••••

two piston assemblies approximates very closely to S.ItM . At the 90 position, while the two pistons mutually balance out the 1 lb. towards the crank, the bob-weight is exerting a force of 8 lb. entirely unbalanced at right angles to the cylinder axis. Dr. Lanchester overcame this by using two bob-weighted crankshafts one al oVe the other, geared together and revolving in opposite directions. Thus, in the 90′ position the forces exerted by the two pistons balanced each other, as (lid the forces due to the bob-weights. This design necessitated two con.-rods from each piston. There was the additional disadvantage of unevenly spaced firing intervals. The 4-cylinder Brough Superior Golden Dream ” motor-cycle uses two engines, one above the other, with crankshafts geared together and so achieves real balance taus 4-cylinder torque. This layOut also allows efficient

Four-cylinder engines—Straight lour : Two pistons at t.d.c., 5+5 lb..10 lb. ; two pistons at b.d.c., 3+8 lb.= 6 lb.

Resultant, 4 lb. upwards.

90′ position, four pistons, 1+1+1+1 =4 lb. down. Resultant, 4 lb. downwards.

180° position, two pistons t.d.c., 5+5 lb.—10 lb. (low-tints ; two pistons b.d.c., 3+3 lb.-6 lb. downwards.

Resultant, 4 lb. upwards.

‘270° position, four pistons, 1+1+1-1-1=4 lb. downwards.

Thus we see in each revolution we have two ” kicks ” of 4 lb. upwards and two ” kicks ” of 4 lb. downwards, upwards and downwards alternating. This is rather poor and hence we have rubber mountings I The Lanchester harmonic balancer counteracts these unbalanced forces and was used on one Vturdiall model. Bob-weights in this case do not affect the balance, as they cancel each other’s forces out, lad they ease the load on the main bearings very considerably.

Rat four (e.g. jowett): With the two front pistons at t.d.e. and the two back pistons at b.d.c., we have 5 lb. to 11, 5 lb. to L, but rather further back, 8 lb. to L, further back still, and 3 lb. to right at the rear (see fig. 3). The resultants of these give 8 lb. to R and 8 lb. to L, but nOt quite in line, giving a very narrow couple, which alternates every 1800. At 90 position the balance is perfect. Jowetts mount their flat-four on rubber to absorb t his narrow couple, resulting in a very smooth engine.

Six-cy/imh.r in line engines : The crank throws are set with No. 1 at (t.d.c.). No. 2 at 240′ to it. No. 3 at I 20to it, No. 4 Iii line with No. 3, No. 5 in line with No. 2, and No. 6 in line with No. I (see Jig. 4).

No. 1, 5 lb. upwards. No. 2-244r, cos 2400 —-1, cos 4800= 1, both act ing downwards, i.e., cos 8 4_cos. 20 .” • downwards.

No. 3-120° in the same way exerts force of”; lb. (10 WIIW 11’4 s.

Thus the force due to No. 1 is exactly equalled by that due to Nos. 2 and 3, and in the same way Nos. 4 and 5 equal force due to No. 6. This holds good for all positions of the crank and the engine is in balance. In passing, it may be noted that the engine is out of balance for the third harmonic. The long crank and the layout of the ” throws ” plus the had gas distribution of an unblown 6-cylinder accounts for their liability to break their cranks at high r.p.m. Alllilcar overcatuue t his by to reeoi induction and Freddie Dixon by fitting six carburetters to his Itileys.

Eight-cylinder engines—Straight eight : All these engines now have a crank consisting of a 4-cylinder crank between and at. right angles to the four end throws of another 4-cylinder crank (i.e., two throws at each end).

t.d.c. b.d.c. 90° 90′ 90′ 90° b.d.c. 1 2 3 4 3 6 7 8

(1) 5 lb. upwards, (2) 3 lb. downwards, (3). (4), (5), (6) each 1 lb. downwards.* lb. downwards ; (7) 3 lb. downwards, (8) 5 lb. upwards. Combining these we get resultants of 10 lb. upwards halfway along the engine, and 10 lb. downwards halfway along the engine, i.e., two equal and opposite forces acting at the same point.. This engine is, thus, in balance, and this holds good for the four positions considered in this article. The disadvantage of such engines is the great length of the crankshaft.

V-eight : With No. I throw throw is 90′ behind it, No. 3 throw opposite No. I. The behind the other. at t.d.e. for No. 1 cylinder, No. 2 throw opposite No. 2. and No. 4 engine is four 90 V-twins, one

This gives 2 v’.01). to L and 2 ?. ‘,lb. to R, and acting at the same point, thus cancelling each other out. This holds good for all the positions considered and the engine balance is very good indeed. Ittpis noteworthy that the 14-litre Al(sreedes, which won the G.P. of Tripoli just before the war at a record speed had a V-eight engine.

Two-crank engines (e.g., Junkers, Troubling, etc.): In Yorkshire there is a design ready for production of a 4-cylinder 4stroke engine on the lines of the Junkers diesel 2-stroke. with a crank at the top and another at the bottom of the cylinder block. This engine has eight pistons and con.-rods, two to each cylinder. Each piston moves in an exact mirror image of its fellow in the same bore, and therefore each pair is in balance. which means that the whole engine is in balance. Each crank is of a normal 4-cylinder layout. The stroke of each piston is very Short, giving low piston speeds at high r.p.m., but the effect of the ” explosion ” occurring I wtween two pistons gives the benefit of a long stroke, and so will allow the engine to pull a high top gear. There is, further, very little waste heat as there is no cylinder-head in the accepted sense of that term. Great things are hoped for from this engine.

Good balance takes a great load off flic chassis-frame and allows it to be lightened, but it should lit stressed that though an engine is in perfect I, lance and so vibrationless,ithe inertia forces are just as gnat as in an unbalanced engine. and e011.-rods just as likely to appear (luring over-revving !

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