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An Excerpt from an Unpublished Paper Intended to be Presented at the 74th Annual International Conference of the Society of Allied Weight Engineers, Inc. May 2015

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Society of Allied Weight Engineers, Inc. P.O. Box 60024, Terminal Annex Los Angeles, CA 90060

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RECIPROCATING PISTON ENGINE INERTIAL ENERGY LOSS Perhaps the most curious aspect of the modern piston engine is that utilizes a reciprocal motion more reminiscent of the inefficient reciprocating motion of nature (people, monkeys, birds, fish) than some of mankind’s more efficient creations (wheeled). The reciprocating motion characteristic of the piston engine has been dismissively referred to as “monkey motion”, and with good reason. This fact has long been recognized, and much effort has expended to find a rotary substitute for the reciprocating, such as the Wankle engine or the gas turbine, but as of this writing the reciprocating engine still reigns supreme for automotive propulsion. As noted in Chapter 2, the engine “rotating” (which includes the reciprocating pistons, valves, etc.) masses contribute tremendously to the automotive effective mass, especially in the lower gears, yet a simple minded reduction of those masses is not possible. The engine rotating mass term “I2” includes the flywheel, for which a certain amount of mass is essential to its function as an energy storage device, and the crankshaft, for which the inclusion of several large heavy counterweights is also essential to smooth out the inertia pulses of reciprocation. It is because of such complications that the reduction of the “I2” contribution to the effective mass was not considered in this paper, despite being very desirable. However, that does not mean that attempts to reduce the “I2” term have not been attempted in the past (which usually involved such things as “slipper” pistons, titanium con-rods, trimmed counterweights, light-weight flywheel, etc.), or that such efforts are not ongoing today. The October 2013 issue of the SAE journal Automotive Engineering reports that Chrysler has been engaged in a 3-year, $30 million dollar R&D program under United States Department of Energy contract to improve gasoline fueled, reciprocating piston engine efficiency by 25% over the present norm 1. Among such things as duel-stage turbochargers 2, multi-fuel operation, and cooled EGR with secondary air injection, the project also investigated the elimination of heavy balance shafts 3, and the consequent parasitic energy losses, through the use of a special crankshaft utilizing dynamic counterweights (“pendulums”) at each crankshaft “throw”; whether this approach will be successful enough to make it to production remains to be seen. Determination of the rotational inertia of a piston engine as commonly used for automotive propulsion is a variable inertia problem whose solution is complex but manageable. The variation in the rotational inertia with the angle of rotation is primarily the effect of the reciprocating motion of the piston and connecting rod. According to the Shock and Vibration 1

Reference [1], pg.16. The “norm” was represented by a 2009 Chrysler 4.0 L port-injected V6, which served as the project baseline engine. 2 The duel-stage turbocharger set-up minimizes what is known as “turbo-lag” through some clever mass properties engineering; the smaller first stage turbo has an especially low rotational-inertia compressor to minimize “spoolup” time. 3 The 1928-1934 Duesenberg J was powered by a 419.7 cid DOHC straight-eight “hemi” engine. To the crankshaft of this remarkable engine was bolted two containers, each partially filled with 16 oz (0.4536 kg) of mercury. The sloshing of the mercury within the containers provided significant vibration damping, although with a toxic risk that would not be allowed today.

Handbook (Harris and Crede) the rotational inertia “J” of a piston engine may be approximated by the following equation:


𝑱 = 𝑰𝒄𝒓𝒂𝒏𝒌𝒔𝒉𝒂𝒇𝒕 + 𝑵 �

𝑾𝒑 𝟐


+ 𝑾𝒄 �𝟏 − �� 𝑹𝟐 𝟐

(EQ. F.01)

J = piston engine rotational inertia (lb-in2). Icrankshaft = piston engine crankshaft rotational inertia (lb-in2). N = piston engine number of cylinders. Wp = weight of piston and wristpin with some allowance for oil (lb). Wc = weight of connecting rod (lb). h = con-rod C.G. location as the fraction “h′/l” of rod length (see Figure F.01).

Figure F.01 – SCHEMATIC DIAGRAM OF CONNECTING ROD The calculation of “Icrankshaft” may be accomplished by the usual methods of “weight accounting”. Such “usual methods” may constitute the traditional but tedious “hand calc”

technique of breaking down the crankshaft into standard volumes 4, multiplying by the material density, and “summing”; or the utilization of the “mass properties analysis” function of CATIA or whatever 3D CAD/CAM system the crankshaft may be modeled in. The example engine for this exposition is the Jaguar XK150S 3.4L straight-six engine of 1958, so of necessity the traditional and tedious “hand calc” method was employed 5, resulting in the following values:

Icrankshaft = 151.11 lb-in2 Wp = 2.52 lb Wc = 1.52 lb h = 0.272 R = 2.1 in

“Plugging” these values into Equation F.01 produces the following result for the rotational inertia “J” of the crankshaft/con-rods/piston assembly:

𝑱 = 𝟏𝟓𝟏. 𝟏𝟏 + 𝟔 �

𝟎. 𝟐𝟕𝟐 𝟐. 𝟓𝟐 + 𝟏. 𝟓𝟐 �𝟏 − �� 𝟐. 𝟏𝟐 𝟐 𝟐

= 𝟐𝟏𝟗. 𝟐𝟎 𝒍𝒃 − 𝒊𝒏𝟐

This approach obscures the variable inertia nature of the piston engine rotating mass, which is of considerable significance with regard to the engine induced sprung mass vibration problem, and with respect to the problem of energy loss (vibration, sound, light, and heat are all forms of energy loss) decrementing engine efficiency. To account for the inertia variation with rotation this author developed an appropriate equation based on a consideration of the inertia forces resulting from the angular acceleration “α1” as per the following single cylinder free-body diagrams:


The Shock and Vibration Handbook presents some specialized formulae for the calculation of the inertia of the crankshaft webs which may be used instead, and it is stated that the formulae are also applicable to marine propellers with blades of “ogival” section. 5 Jaguar XK150S 3.4L engine “engineering” drawings may be found in Jaguar by Lord Montagu of Beaulieu, Haynes Publishing, 1979.

Figure F.02 – SINGLE CYLINDER INERTIA ABOUT CRANKSHAFT AXIS The torque “T” about the crankshaft axis is equal to the sum of all the inertial resistances:

𝒍 − 𝒉′ 𝟐 𝑰𝟐 𝜶𝟐 𝑻 = 𝑰𝟏 𝜶𝟏 + 𝒎𝟐 � � 𝑹 𝜶𝟏 + �𝒎𝟑 𝒂𝟑 + � � 𝐬𝐢𝐧 𝜽� 𝑹 𝒍 𝒍

Substitute “α1 R sin(θ)/l” for “α2” and substitute “α1 R” for “α3”:

𝒍 − 𝒉′ 𝟐 𝑹 𝟐 𝟐 𝑻 = 𝑰𝟏 𝜶𝟏 + 𝒎𝟐 � � 𝑹 𝜶𝟏 + 𝒎𝟑 𝜶𝟏 𝑹 + 𝑰𝟐 𝜶𝟏 � � 𝒔𝒊𝒏𝟐 𝜽 𝒍 𝒍

Divide through by “α1”:

𝟐 ′ 𝑻� = 𝑰 + 𝒎 �𝒍 − 𝒉 � 𝑹𝟐 + 𝒎 𝑹𝟐 + 𝑰 �𝑹� 𝒔𝒊𝒏𝟐 𝜽 𝟏 𝟐 𝟑 𝟐 𝜶𝟏 𝒍 𝒍

This is the effective rotational inertia (“Ieff = T/α1”) in terms of “θ”, but calculation is more convenient in terms of crankshaft angle “ψ” as it is the crankshaft rotation which causes the inertial flux. The relationship of the crankshaft angle “ψ” to connecting rod angle “θ” and how the effective “R’” varies with “ψ” may be determined from the following diagram:

Figure F.03 – CRANKSHAFT ANGLE vs. CONNECTING ROD ANGLE So now we may substitute “R cos(ψ)” for “R” and “sin-1(R/l) cos(ψ)” for “θ”, also we may change “T/α1” to “Ieff” or “J” (for better unity in symbolism with Equation F.01):

𝒍 − 𝒉′ 𝟐 𝑱 = 𝑰𝟏 + 𝒎𝟐 � � 𝑹 + 𝒎𝟑 (𝑹 𝐜𝐨𝐬 𝝍)𝟐 𝒍

𝟐 (𝑹 𝐜𝐨𝐬 𝝍)𝟐 𝑹 −𝟏 + 𝑰𝟐 �𝒔𝒊𝒏 �𝒔𝒊𝒏 � � 𝐜𝐨𝐬 𝝍�� 𝒍 𝒍𝟐

This is valid for only a single cylinder engine, for an engine of “N” cylinders the equation becomes:


𝑱 = 𝑰𝟏 + 𝑵𝒎𝟐 �

𝟐𝝅𝑲 𝑵


𝟐 𝟐 � 𝑹𝟐 + ∑𝑲=𝑵 𝑲=𝟏 �𝒎𝟑 𝑹 𝒄𝒐𝒔 �𝝍 −


� 𝒔𝒊𝒏𝟐 �𝒔𝒊𝒏−𝟏 � 𝒍 � 𝐜𝐨𝐬 �𝝍 −

𝟐𝝅𝑲 𝑵


𝟐𝝅𝑲 𝑵

𝑹 𝟐

� + 𝑰𝟐 � 𝒍 � 𝒄𝒐𝒔𝟐 �𝝍 −

(EQ. F.02)

Where: J = rotational inertia of crankshaft/con-rods/pistons assembly (lb-in2). I1 = crankshaft rotational inertia (lb-in2). N = number of cylinders. m2 = weight of connecting rod (lb). l = distance along connecting rod between centers of rotation (in). h′ = distance along connecting rod between crankshaft center and CG of connecting rod (in). R = crankshaft “throw” distance (in). K = counter variable. m3 = weight of piston, piston rings, wristpin, plus “h′/l” fraction of “m2” (lb). ψ = crankshaft angle (radians).

I2 = rotational inertia of con-rod about the wristpin center (lb-in2). An equation such as Equation F.02 is best evaluated by means of a computer program, hence the following listing of the Commodore BASIC language program (c. 1982) “CYLINERT.BAS” (“N” is input at line 10, plotting info is required lines 100 to 150, all other input values are embedded in lines 2000 to 2005)…


The output from this program, using values appropriate for the Jaguar XK150S 3.4L straight-six engine of 1958, is as follows… 6 CYLINERT.BAS OUTPUT

The inertial flux with rotation can clearly be seen for “N = 1, 2, 4, 5”. It is the combination of these inertial pulses plus the power pulses resulting from the ignition cycle that 6

Note this analysis concerns itself only with what may be termed “primary” forces and couples; there are “secondary” forces and couples of a lesser magnitude not addressed; see Reference [2] pp. 25-39.

compose most of the piston ICE vibration output (exhaust pulsations running through the exhaust system are also significant). Note that for “N = 6” the inertial flux seems completely nonexistent, which explains the traditional popularity of big straight-six engines such as those found in early XK and E-Type Jaguars. Lastly, interpretation of the “N = 6” plot indicates that the internal rotational inertia of the 1958 Jaguar 3.4L engine is about 220 lb-in2, which is in close agreement with the 219.2 lb-in2 obtained by use of Equation F.01. Unfortunately the determination of the rotational inertia of the crankshaft/conrod/piston assembly is only the first step in obtaining the total engine rotational inertia about the crankshaft axis “I2”. To this first step value must be added the inertial contribution of the camshafts and valve train 7, flywheel, and various pulleys, sprockets, belts, chains, engine accessories. For the 1958 Jaguar 3.4L engine the inertia total was determined to be:







Crankshaft Pulley



Pulley Nut



Drive Sprocket



Idler Sprockets



Fan Belts



Camshaft Chain





Water Pump









“I2” Total


Crankshaft Camshafts/Valve Train Flywheel



lb-in2, or

6.78 lb-ft2

Thomson, William T.; Vibration Theory and Applications, Prentice-Hall Inc., Englewood Cliffs, NJ; 1965, page 16 may be helpful in the evaluation of the valve train inertia.



Brooke, Lindsay; “Chrysler Sees the ICE Future”, Automotive Engineering, pp. 16-19, SAE, October 2013.


Newton, K.; W. Steeds, and T.K. Garrett, The Motor Vehicle; Jordan Hill, UK; ButterworthHeinemann, SAE R-298, 1996.

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