Technical Report - 1 Con Rod Length, Stroke, Piston Pin Offset, Piston Motion and Dwell in the Lotus-Ford Twin Cam Engine by T. L. Duell May 24 Terry Duell consulting 19 Rylandes Drive, Gladstone Park Victoria 343, Australia ph. +61 3 9338 464 tduell@planet.net.au http://www.planet.net.au/~tduell/cons.html
24 Terry Duell Published on a RISC OS system using!techwriter Pro+ Figures and diagrams prepared in Gnuplot,!Tau and!draw 2
1 Introduction A view has been expressed that Lotus-Ford Twin Cam engines built on the cross flow (Xflow) 711M engine block, using the 77.62mm stroke crankshaft and the 2737E con rod, do not breathe as well at high rpm, as the standard Lotus-Ford Twin Cam engine, which was built on the 12E engine block and used the 72.8mm stroke crankshaft and the 125E con rod. There are various opinions expressed about the importance of con rod length, and many can be found by using "piston dwell" as the search string of an internet search engine. Whilst there is a lot said about the effects of rod length and crank to rod ratios, very few of the opinions appear to be supported by firm data backed by analysis. There appears to be a general view that longer con rods (larger rod length to stroke ratio) will increase piston dwell time, which is said to increases time for combustion and lead to improved torque in the mid to upper rpm range. Some opinions also state that short rods will reduce dwell time and produce good intake and exhaust velocities at low to medium speeds but will cause reduced torque at high rpm This report details a limited investigation into the the geometry of the crank-rod combinations and the motion of the piston, in order to develop some understanding of the magnitude of the effects of changing the crank, rod and other basic engine geometry. The investigation does not aim to address any issues other than those that are directly affected by the crank, rod, and piston geometry. 2 Assumptions A number of assumptions are necessary to make the analysis of the crank, rod and piston geometry manageable. The basic assumptions are; Components are rigid Components have no mass There are no clearances in joints The constant π = 3.1415926 Crank angles are measured clockwise from 12 o'clock Clearly, some of these assumptions do not reflect real engines. The calculation of piston displacements, velocities and accelerations at high engine speeds is very complex as con rods stretch and bend, and crankshafts bend and twist. Accounting for all these real world effects would add significant complexity to the problem. The results obtained using these assumptions will indicate relative differences, all other things being equal, and should provide guidance on cause and effect trends. 3
3 Basic data Basic data is as shown in Table 1. Engine Bore (mm) Stroke (mm) Con Rod Length (mm) Twin Cam (standard) 711M (X-Flow std) Pin Offset (mm) Rod/Stroke ratio 82.55 72.8 121.95-1.16 1 1.6745 8.98 77.62 125.155-1.16 2 1.6124 Table 1. Basic engine data The Rod/Stroke ratio is defined as the ratio of the length of the con rod measured from the axis of the piston pin bush to the axis of the big end bearing, to the crankshaft stroke. The crankshaft stroke is twice the crankshaft radius, measured from the axis of the main bearing journal to the axis of the big end bearing journal. 4 The Crank-Rod Model Figure 1 shows the basic geometry for the crank-rod model with pin offset. Analysis of the model is as follows; Fig. 1 The basic crank-rod model, with pin offset 1 A positive offset is to the right of the cylinder axis, as shown in Fig.1. Ref.1 indicates that the Twin Cam piston pin offset is negative. 2 Assumed value 4
In Fig. 1, b = R Cos (θ) a = R Sin (θ) d = a o c = L 2 d 2 The piston position = R Cos (θ) + L 2 d 2 = R Cos (θ) + L 2 (a o) 2 = R Cos (θ) + L 2 (R Sin (θ) o) 2 Note that the piston is at top dead center (TDC) position when the crank and rod are in line. The distance moved by the piston from TDC is; g = (L + R) 2 o 2 [R Cos (θ) + L 2 (R Sin (θ) o) 2 ] where R = crank radius = stroke / 2. L = con rod length θ = crank angle af ter vertical o = piston pin of f set The volume, due to piston movement from TDC, can be calculated from g and the cylinder diameter. If the angular velocity of the crank is ω rads/sec, then the linear velocity of the crank pin is Rω, which has a direction normal to the crank. The instantaneous centre of rotation of the con rod at the crankshaft (big-end) is on a line projected along the crank. The velocity of the piston V p will be directed along the axis of the cylinder, and the instantaneous centre of rotation of the small end of the con rod will on a line normal to the axis of the bore through the axis of the piston pin. The center of rotation of the con rod will be at the intersection of those lines projected normal to the direction of the linear velocities, at that instantaneous position. Clearly, the angular velocity of the con rod will be same at both the small and the big-ends, and the velocity of the small end of the con rod will be the same of the piston. Referring to Fig.1, Rω Angular velocity of con rod = e R Rω V p = (f o) e R and e = (c + b) Cos (θ) f = e Sin (θ) Rω Hence V p = e Sin (θ) o e R Note that if N is the engine speed (rpm), then ω = 2πN (rads/sec) 6 5
Piston dwell at TDC and at bottom dead centre (BDC) are often mentioned. It should be noted that strictly, there is no dwell period. The piston comes to rest at precisely the crank angle that the crank and rod are in line (TDC and BDC), and is moving at all other crank angles. At crank angles which are very close to the TDC and BDC angles, the piston is moving slowly. It is this slow movement in the vicinity of TDC and BDC that give rise to the term piston dwell. In order that the effects of varying con rod length, stroke and piston pin offset can be assessed, it is necessary to introduce a small piston displacement δ, and then calculate the crank angle range over which the piston remains within that small distance of TDC and BDC. These angle ranges will be the dwell angle. The value of δ used in this investigation is.254 mm (.1"). The basic geometry for the dwell model is shown in Fig. 2. Fig. 2 Basic geometry of the dwell model In Fig. 2a, the crank angle for TDC is established as follows, of f θ = arcsin ( L + R ) 6
and the angle for BDC (Fig. 2b) is, of f β = arcsin ( L R ) θ = β + 18 In Fig. 2c the maximum angle at the limit of δ, from TDC, is θ + α + β, and these angles are found as follows, OB 2 = of f 2 + ( L 2 + R 2 of f 2 δ) 2 α = arccos ( R2 + OB 2 L 2 2 R OB ) β = arccos ( (L + R)2 + OB 2 δ 2 2 (L + R) OB ) The minimum angle at the limit of δ from TDC, in Fig. 2c, is θ + β γ. The total dwell angle at TDC, for a piston movement of δ, is, Total TDC angle = θ + α + β + θ + β γ γ = arccos ( R2 + OB 2 L 2 2 R OB ) To find the angles defining the dwell limits at BDC, we use Fig. 2d. of f β = arcsin (L R) 2 of f 2 OD = ( (L R) 2 of f 2 + δ) 2 + of f 2 µ = arccos ( R2 + OD 2 L 2 2 R OD ) β = arccos ( (L R)2 + OD 2 δ 2 2 (L R) OD ) θ = µ β 9 and the max angle θ = 27 θ ϕ = arccos ( R2 + OD 2 L 2 2 R OD ) and the min angle = β + ϕ and the total dwell angle, at BDC, for a piston movement of δ, is (27 θ) β + ϕ The crank stroke, rod length and piston pin offset will each have some effect on the forces and moments on the piston and the axial force on the rod which results in torque on the crank. The geometry for the analysis of the crank moment arm is shown in Fig. 3(a), and the free body diagrams of the piston showing the forces and moments acting on the piston are Fig. 3(b) and Fig. 3(c). 7
Fig. 3 Crank moment arm and piston free body diagram Referring to Fig. 3(a), we can establish the crank moment arm (OE), at crank angle θ, as follows; AC = RSin (θ) BC = AC of f Sinβ = BC L RSin (θ) of f β = arcsin ( ) L φ = 9 β θ Cos (φ) = OE R OE = RCos (9 β θ) Fig. 3(b) shows the forces acting on the piston. The gas force ( F g ) acts along the centreline of the cylinder. The thrust force ( F T ) acts normal to the piston, and the friction force ( F ) acts on the thrust face of the piston, parallel to the cylinder axis, and in a direction opposite to the direction of travel of the piston. If we assume that the thrust force acts through the axis of the piston pin, and we transfer the line of action of the gas force and the friction force to the axis of the piston pin, we have the forces and couples as shown in Fig.3(c). 8
F R F F = F T = F g F F Cos (β) F g (Tan (µ) Tan (β)) (1 Tan (µ) Tan (β)) = (F g F F ) Tan (β) The couple on the piston due to f riction f orce = M F = F F ( b 2 of f ) The couple on the piston due to gas f orce = M g = F g of f where F g = Gas f orce on piston F = Friction f orce F T = Thrust f orce F R = Axial f orce on rod M g = Couple due to gas f orce M F = Couple due to f riction f orce β = Rod angle to cylinder axis µ = Angle of f riction (Tan (µ) = mu) R = crank radius L = rod length of f = piston pin of f set and the net couple on the piston = M F M g For the purposes of looking at the effect of stroke, rod length and piston pin offset on the torque developed at the crank, we can assume a unit value for the net force on the piston (F g F ), and hence arrive at a corresponding value for F R which is applied to the crank through the distance OE. This analysis provides an insight into the geometric effects without the requirement to know anything about the complexities of the variation of F g with crank angle, or the requirement to quantify the friction between the piston, rings and cylinder wall. If a suitable value of µ and a suitable function or lookup table for F g can be determined for a particular engine, then the more detailed analysis developed above can be used to derive the crank torque-crank angle function for a cycle. 9
5 Results Using the relations derived above, the crank angles for TDC and BDC are as shown in table 2. Engine Crank angle TDC (deg) Crank angle BDC (deg) Std TC -.3677 179.3192 X-Flow -.355 179.3258 Table 2. TDC and BDC crank angles for Std TC and X-Flow The dwell angles are as shown in table 3. Engine TDC dwell angle (deg) BDC dwell angle (deg) Std TC 3.757 5.111 X-Flow 3.622 4.991 Table 3. Crankshaft angular displacement for piston dwell The effects of con rod length to stroke ratio and piston pin offset on piston dwell are shown in Figs. 4 and 5. Contours of constant dwell angle are shown on the base of the plots. Piston dwell (deg) TDC dwell angle 3.7 3.65 3.6 3.55 3.5 3.75 3.7 3.65 3.6 3.55 3.5 3.45 1.2 1.3 1.4 1.5 1.6 Rod/Stroke ratio 1.7 1.8 1.9 2 2.1-1 -.5-1.5-2 -2.5.5 1 1.5 2 2.5 Piston pin offset (mm) Fig. 4 Piston dwell (TDC), for varying rod/stroke ratios and pin offsets 1
Piston dwell (deg) 5.5 5.4 5.3 5.2 5.1 5 4.9 4.8 4.7 1.2 1.3 1.4 1.5 1.6 Rod/Stroke ratio 1.7 1.8 1.9 2 2.1-1 -.5-1.5-2 -2.5 BDC dwell angle 5.4 5.3 5.2 5.1 5 4.9 4.8.5 1 1.5 2 2.5 Piston pin offset (mm) Fig. 5 Piston dwell (BDC), for varying rod/stroke ratios and pin offsets The effects of con rod length to crank stroke ratio and piston pin offset on the crank torque, with a unit net piston force, for varying crank angle, are shown in Figs 6 and 7. Contours of constant crank torque are shown on the base of the plots. Crank torque (Nmm) Crank torque for 1 N piston force, pin offset=1.16mm 4 3 2 1 45 4 35 3 25 2 15 1 5-5 -1-2 2 4 6 8 1 12 14 16 2.1 2 1.9 1.8 1.7 1.6 1.5 Rod/crank ratio 1.4 1.3 18 2 1.2 Fig. 6 Crank torque for rod/stroke ratio 11
Crank torque (Nmm) Crank torque for 1 N piston force, L/Stroke=1.6124 4 3 2 1 45 4 35 3 25 2 15 1 5-5 -1-2 2 4 6 8 1 12 14 16 18 2-1 -.5-1.5-2 -2.5 Fig. 7 Crank torque for pin offset Piston accelerations, for TC and XF, at 6 rpm are shown in Fig. 8. 3.5 1 1.5 2 2.5 XF piston acceleration (m/sec/sec) TC piston acceleration (m/sec/sec) Pin offset (mm) 2 1 Piston acceleration (m/sec/sec) -1-2 -3-4 -.6 -.4 -.2.2.4.6.8 1 Fig. 8 Piston accelerations TC and XF at 6 rpm Con rod angles, for varying con rod length/stroke ratio, and for varying piston pin offset, are shown in Figs. 9 and 1. Contours of constant rod angle are shown on the base of the plots. 12
Rod angle (deg) Con Rod angle, piston pin offset=1.16mm 2 15 1 5 25 2 15 1 5-5 -2 2.1 2 1.9 1.8 1.7 2 1.6 4 6 8 1.5 Rod/crank ratio 1 12 1.4 14 16 1.3 18 2 1.2 Fig. 9 Con Rod angle for crank angle and rod/crank ratio Rod angle (deg) Con Rod angle, L/Stroke=1.612 15 1 5 2 15 1 5-5 -2 2 4 6 8 1 12 14 16 18 2-1 -.5-1.5-2 -2.5 Fig. 1 Rod angle for crank angle and piston pin offset.5 1 1.5 2 2.5 Piston pin offset (mm) Piston volume and velocity for crank angle, for the TC and XF are shown in Figs. 11 and 12. 13
.3 XF piston displacement (ml) from TDC TC piston displacement (ml) from TDC.25.2 Piston displacement (ml).15.1.5 -.6 -.4 -.2.2.4.6.8 1 Fig. 11 Volume change for TC and XF.4.3 XF piston velocity (mm/sec) TC piston velocity (mm/sec).2.1 Piston velocity (mm/sec) -.1 -.2 -.3 -.4 -.5 -.6 -.4 -.2.2.4.6.8 1 Fig. 12 Piston velocity for TC and XF 14
6 Discussion The dwell angles (table 3), for a piston displacement of.254mm, are not insignificant, and the X-Flow has slightly smaller dwell angles than the TC. Figs. 4 and 5 show that piston pin offset has a very small effect on dwell angle, relative to the effects of stroke and con rod length. Contours of constant dwell angle are shown on the base of the plot. The detailed data shows that the variation in dwell angle over the range of piston pin offsets used, (-2.32 to 2.32mm) is only approx..7 deg. The maximum dwell angles, for both TDC and BDC, for all rod/stroke ratios examined, occur when the piston pin offset is zero. Piston accelerations near TDC are not noticeably different for the TC and XF (Fig. 8). Where the acceleration is not zero, the acceleration of the XF piston is always higher than that of the TC, for the crank angles shown. Con rod angle (Figs. 9, 1) is affected by both rod/stroke ratio and piston pin offset. Decreasing rod/stroke ratio increases the rod angle, particularly in the region of crank angles of about 4 to 14 degrees. Negative piston pin offsets increase the rod angle, and this effect is also more marked at crank angles nearer 9 deg than nearer 18 degrees. Cylinder volume change with variation in crank angle (Fig. 11), shows the TC and XF very much the same approaching TDC, with the XF showing larger volume change after TDC. Note that the TDC crank angles are not the same (Table 2). Piston velocities near TDC (Fig. 12), shows the same trend as piston accelerations. Where the velocity is not zero, the XF piston acceleration is always higher than that of the TC. The effects of rod length to stroke ratio and piston pin offset on the crank torque for a unit net piston (gas) force, at small crank angles, are shown in Figs. 6 and 7. Contours of crank torque are shown on the base of the plot. Negative piston pin offsets have only a small effect on the crank torque. Rod to stroke ratio has a more noticeable effect. Increasing the crank moment arm and rod angle at low crank angles has potential to improve engine torque, as the cylinder pressure, and hence piston force is very high over the initial stages of the power stroke. The net effect will depend on the rate of increase in the friction force. We can get some idea of the difference that the engine geometry makes to the friction and thrust forces by looking at the ratio of friction force to gas force and the ratio of thrust force to gas force, versus crank angle, for some values of the coefficient of friction. F F F g = F T F g mu tan (β) 1 + mu tan (β) = (1 mu tan (β) ) tan (β) 1 + mu tan (β) where, as before, mu = coef f icient of f riction, β = Rod angle to cylinder axis This approach does not require that we know anything about the magnitude of the gas force, or its variation with crank angle. Results of this analysis, for the TC and XF engine geometries, are shown in Figs. 13 to 16. 15
Piston Friction Force/Gas Force Ratio Ratio of Friction Force to Gas Force, TC.3.2.1.4.35.3.25.2.15.1.5 -.5-2 2 4 6 8 1 12 14 16 18 2.9 1.8.7.6.5.4 Piston/bore friction coefficient.3.2.1 Fig. 13 Ratio of Friction force to Gas Force, TC geometry Piston Friction Force/Gas Force Ratio Ratio of Friction Force to Gas Force, XF.3.2.1.4.35.3.25.2.15.1.5 -.5-2 2 4 6 8 1 12 14 16 18 2.9 1.8.7.6.5.4 Piston/bore friction coefficient.3.2.1 Fig. 14 Ratio of Friction Force to Gas Force, XF Geometry 16
Piston Thrust Force/Gas Force Ratio Ratio of Piston Thrust Force to Gas Force, TC.3.2.1.35.3.25.2.15.1.5 -.5-2 2 4 6 8 1 12 14 16 18 2 1.9.8.7.6.5.4Piston/bore friction coefficient.3.2.1 Fig. 15 Ratio of Thrust Force to gas Force, TC Geometry Piston Thrust Force/Gas Force Ratio Ratio of Piston Thrust Force to Gas Force, XF.3.2.1.35.3.25.2.15.1.5 -.5-2 2 4 6 8 1 12 14 16 18 2.9 1.8.7.6.5.4Piston/bore friction coefficient.3.2.1 Fig. 16 Ratio of Thrust Force to gas Force, XF Geometry The indications are that the TC friction/gas force ratio is lightly lower than that of the XF, for the same coefficient of friction, and the TC thrust/gas force ratio is also slightly lower than that of the XF, for the same coefficient of friction. This is because of the lower rod angle of the TC, resulting from the larger rod/stroke ratio. 17
7 Conclusions A few basic facts can be derived from the analysis, and these are summarised in Table 4. Twin Cam X-Flow Rod/Stroke ratio 1.6745 1.6124 TDC Dwell 3 (deg) 3.757 3.622 BDC Dwell (deg) 5.111 4.991 TDC Dwell time at 6rpm (sec) BDC Dwell time at 6 rpm (sec).144.16.142.1386 Table 4. Basic data from analysis The effects of changing basic geometric variables, all other things equal, are summarised in Table 5. Change Reduce rod/stroke ratio Reduce piston pin offset 4 Effect Reduces TDC dwell Increases BDC dwell Increases rod angle Increases friction/gas and thrust/gas force ratios Increases crank torque minor increase in crank torque Increases rod angle Reduces piston couple due to gas force (couple= at offset=) Reduces piston couple due to piston friction Maximum dwell angles at piston pin offset= Table 5. Effect of changing basic geometry Note that 'all other things equal' does not always apply. The effects in table 5 are indicative 3 Assuming that the dwell angle is defined by the crank angle range for which a piston remains within.254mm of the TDC or the BDC position. 4 Offset is positive as shown in Fig. 1, reducing includes going from + to - offset 18
of general trends only, and in practice may not be achievable due to 'other things no longer being equal'. 19
8 References 1. Ford Motor Company of Australia Limited, "Escort Workshop Manual 197-1975", March 197 2