Crankshaft

Abstract

A crankshaft is used to convert reciprocating motion of the piston into rotary motion. The crankshaft in an engine is probably the most complex of all the shafts used in any machinery, and, as the name implies, it is far from being anywhere near a straight shaft. With the help of examples, crankshafts are classified depending on the type of supports as overhung and centre crankshaft or based on the number of throws as single throw or multi-throw shafts. The procedure for design the crankshafts is explained in detail using calculations of the crankshaft strength and stress. The factors affecting the fatigue strength are deliberated. Plots of oil film thickness explain the wear pattern. Inherently, single-, two-, three- and four- cylinder in line engines are not fully balanced for inertia forces or couples; if the cost permits, counter rotating balance shafts are designed to neutralize these forces. Otherwise, these forces are left unbalanced; they cause rigid body vibration of the engine and also are transmitted to the vehicle and to other parts. Vee engine shafts are treated slightly differently from the inline engine shafts. The designed shaft when made does not have mass distribution as per the design because of manufacturing tolerances to forging and machining. This imbalance is removed at a balancing machine within the limits specified in standards. The inertias of individual throws, piston, and connecting rod as well as the flywheel with the stiffness of the shaft result in multiple natural frequencies in the rotational direction. In case of long shafts as in the case of a six-cylinder engine, the torsional vibration can have a resonance frequency in the operating range of speeds and can induce fatigue usually starting from the oil hole in crank pin. Therefore, calculations of moments of inertia, equivalent stiffness and the natural frequencies as well as the amplitude of vibration are important. If the amplitude is sufficiently high, the torsional stress can exceed the fatigue strength limit leading to failure. To avoid the natural frequency in the speed range of operation an oscillator in the form of a torsional vibration damper is added. The new system not only shifts the frequencies but al-so reduces the dynamic magnifier to reduce the torsional stresses. Various parameters like characteristic frequency at fixed points, damping ratio, unit tuning ratio, optimum tuning ratio are introduced and with the aid of a characteristic help-graph, the tuning ratio of a rubber or spring damper can be selected. While rubber is relatively less expensive, heat dissipation in the damper must be carefully predicted to estimate the temperature of the rubber in the damper as the properties of rubber are highly dependent on temperature. The type of rubber is properly selected and manufactured with great care to avoid aging of rubber at the operating temperature. When the heat may not be easily dissipated fluid dampers are useful. Such dampers without a spring only dampen the vibration to save the shaft but do not play any major role in shifting the natural frequency of the shaft system. The engine load can be quickly simulated at specific rigs to estimate the bending fatigue or torsional fatigue. Finally, the importance of the design of bolts and the applied tightening torque are important to hold the flywheel, the crank pulley, connecting rods, bearing caps together, cannot be understated.

Annexure I

Mass elastic system as per BICERA (Payne and Scott 1958 )

Figure 15.38 shows the multi mass system for the crankshaft of a multiple cylinder engine crankshaft with the connected components like flywheel, gears, dampers etc.

Fig. 15.38

Multi-mass system

Moment of inertia per Cylinder Line, J_cyl

As described by BICERA (in Sect. 15.15) (Payne and Scott 1958), the recommended procedure for performing mass elastic calculations is to concentrate all the moments of inertia of each crank throw, including the connecting rod and piston, at the corresponding cylinder centre position on the equivalent shaft representing the crankshaft. The result is termed the moment of inertia per cylinder line. This method is typically used for an in line engine, but can be extended for Vee engines. In that case the inertia will include two connecting rod and piston assemblies, the moment of inertia per “line” instead of per “cylinder line.” The moment of inertia per cylinder line, J_cyl, is obtained as follows:

$$ \varvec{J}_{{\varvec{cyl}}} + \varvec{J}_{{\varvec{journal}}} + \varvec{J}_{{\varvec{crank}\,\varvec{pin}}} + \varvec{J}_{{{\mathbf{2}}\,\varvec{webs}}} + \varvec{J}_{{\varvec{counter}\,\varvec{weights}}} + \varvec{J}_{{\varvec{rod},\varvec{rotating}}} + \varvec{J}_{{\varvec{rod},\varvec{recip}}} + \varvec{J}_{{\varvec{piston}\,\varvec{assm} + \varvec{pin}}} $$

(15.1)

Moment of Inertia of Main Bearing Journal, J _journal

J_journalcan be obtained via CAD packages. For a solid journal J_journal can be estimated by:

$$ J_{journal} = \pi \rho \left( {D_{main}^{4} .L_{main\,total} } \right)/32.g $$

(15.2)

where

$$ \begin{aligned} & g = gravitational\,acceleration, \\ & \rho = the\,density\,of\,the\,journal, \\ & D_{main} = the\,diameter\,of\,the\,main\,bearing\,journal\,and \\ & L_{main total} = the length\,of\,the\,main\,bearing\,journal \\ \end{aligned} $$

Moment of Inertia of Crankpin, J _crankpin

J_crankpinalso can be obtained via CAD packages. For details, see Sect. 15.15. For a solid journal Jcrankpin can be estimated by:

$$ J_{crankpin} = \pi \rho /4.g.L_{{pin {-} total}} .\left\{ {\left( {d_{pin}^{4} /8} \right) + R^{2} d_{pin}^{2} } \right\} $$

(15.3)

where

$$ \begin{aligned} & g = gravitational\,acceleration, \\ & \rho = the\,density\,of\,the\,crankpin, \\ & R = the\,crankshaft\,radius, \\ & d_{pin} = diameter\,of\,the\,crankpin,and \\ & L_{pin total} = length\,of\,the\,crankpin \\ & Moment\,of\,inertia\,of\,Webs\,and\,Counterweights = J_{2webs} \,and\,J_{counterweights} \\ \end{aligned} $$

CAD packages are typically employed to calculate this moment of inertia. Numerous graphical and tabulation methods also exist, but can be cumbersome to use.

Reciprocating and Rotating Weights of Connecting Rod

The motion of the crankpin end of the connecting rod assembly describes a circle. Whereas the small end motion is reciprocating. The mass moment of inertia of the connecting rod assembly is determined by assuming that the total weight is composed of two concentrated weights. A rotating weight is concentrated at the big end centre, and a reciprocating weight is concentrated at the small end centre. Therefore, the sum of the rotating and reciprocating assembly weights are equal to the total connecting rod weight. Although the reciprocating and rotating weights of a connecting rod assembly can be easily measured (see Sect. 15.15), prototype connecting rod assemblies often are not available until later in the engine design stage. In this case, the typical procedure in obtaining concentrated weight data is to employ CAD packages to calculate the location of the centre of gravity and the total weight of the connecting rod assembly. The connecting rod assembly includes the connecting rod and cap, cap screws, washers (if applicable) connecting rod big end bearings, and connecting rod small end bushing. Once the total weight and centre of gravity are determined, then the reciprocating and rotating weights are determined as follows:

$$ W_{recip} = \left( {W_{total} .L_{BE} } \right)/L $$

(15.4)

$$ W_{rot} = \left( {W_{total} .L_{SE} } \right)/L $$

(15.5)

where

$$ \begin{aligned} W_{recip} & = reciprocating\,weight\,of\,connecting\,rod\,assembly \\ W_{rot} & = rotating\,weight of connecting rod assembly \\ W_{total} & = total\,weight\,of\,connecting\,rod\,assembly \\ L_{BE} & = distance\,from\,the\,connecting\,rod\,big\,end\,centre\,to\,CG\,of\,rod \\ L_{SE} & = distance\,from\,the\,connecting\,rod\,small\,end\,centre\,to\,CG\,of\,rod \\ L & = connecting\,rod\,length,centre\,to\,centre \\ \end{aligned} $$

Moment of inertia of Rotating Portion of Connecting Rod, J _{rod, rot}

The mass moment of inertia of the rotating portion of the connecting rod, Jrod, rot can be calculates as follows:

$$ J_{rod,rot} = R^{2} .W_{rot} /g $$

(15.6)

where

$$ \begin{aligned} & W_{rot} = rotating\,weight\,of\,connecting\,rod\,assembly\,from\,Equation\,5 \\ & g = gravitational\,acceleration \\ & R = crankshaft\,radius \\ \end{aligned} $$

Moment of Inertia of Reciprocating Parts, W _{rod, recip} and W _{pistonassm + pin}

If the reciprocating weight of the connecting rod, W_{rod, recip}, is added to the weight of the piston assembly plus the piston pin, W_{pistonassm+pin}, then the total mass moment of inertia of the reciprocating parts can be calculated as follows:

$$ J_{tot\,recip} = R^{2} .W_{totrecip} /2g $$

(15.7)

where

$$ \begin{aligned} W_{tot.recip} & =\Sigma \,weights:reciprocating\,rod,piston\,assembly,piston\,pin \\ g & = gravitational\,acceleration \\ R & = the\, crank\,shaft\,radius \\ \end{aligned} $$

Moment of Inertia of the Gear Train

The front and/or rear gear train and components are modelled by adding the rotating inertia with the inertia of the crankshaft gear. Gear and gear train component inertial can be obtained through CAD packages, by hand calculations, or through experimentation. These values should be adjusted to take into account the components’ respective speed ratios with respect to the crankshaft (and thus crankshaft gear) speed by the following equation:

$$ J_{eff} = \left( {N_{g} /N_{c} } \right)^{2} .\left( {J_{comp} } \right) $$

(15.8)

where

$$ \begin{aligned} J_{eff} & = effective\,inertia\,of\,the\,gear\,train \\ N_{g} & = speed\,of\,the\,individual\,component \\ N_{c} & = speed\,of\,the\,crankshaft \\ J_{comp} & = inertia\,of\,the\,individual\,component \\ \end{aligned} $$

The adjusted inertia can then be summed and added to the inertia of the crankshaft gear.

Stiffness

The stiffness of the crank-throw and the coupling could be calculated based on the B. I. C. E. R. A. For example, the stiffness of the crank of the engine under study by the first method was 2. 08 MNm/rad and by the second 2. 01 MNm/rad and hence there is no difference practically between them.

Shaft stiffness, K, can be calculated by

$$ K = GI_{p} L $$

where

$$ \begin{aligned} & G = modulus\,of\,rigidity \\ & L = shaft\,length \\ & I_{p} = \pi \,D^{4} /32,for\,a\,solid\,shaft \\ & D = shaft\,diameter \\ \end{aligned} $$

It can also be measured experimentally:

$$ K = \frac{T}{\theta } $$

T is the applied torque and

$\theta$ is the resulting torsional deflection (in radians)

$$ Inertia\,for\,a\,disc = 0.5\,r^{2} \,M $$

where

$$ \begin{aligned} r & = disc\,radius \\ M & = mass \\ Inertia\,for\,a\,ring & = M\left( {R^{2} + r^{2} } \right)/2 \\ \end{aligned} $$

where

$$ \begin{aligned} M & = mass \\ R & = outer\,ring\,radius \\ R & = the\,inner\,ring\,radius \\ \end{aligned} $$

Equivalent stiffness

Stiffness, k = T/θ is determined using the concept of equivalent lengths. It is generally convenient for evaluating complicated shaft portions, such as crank-throws. Any shaft portion could be accounted for either by means of its stiffness k or by indicating its equivalent length, Le with reference to a fixed value chosen as the equivalent diameter, De. The stiffness of the complicated shaft between the two masses is determined from the sum of the contributions of the individual lengths.

Crank-throw Stiffness

During the design stage of an engine crankshaft, the torsional stiffness of each crankshaft portion between cylinder centres (or centres of the crankpin for Vee engines) is determined. This requires a value for the equivalent length of the crank throw. The typical method to determine the equivalent length of the crank throw uses the BICERA provisional formula, as taken from Sect. 15.15.

$$ L_{e} = D_{e}^{4} \left[ {\frac{{L_{j} }}{{D_{j}^{4} - d_{j}^{4} }} + \frac{{L_{c} }}{{D_{c}^{4} - d_{c}^{4} }} + \frac{{0. 07L_{e}^{3} }}{{R_{0}^{2} \left( {D_{c}^{4} - d_{c}^{4} } \right)}} + \frac{{kR_{0} }}{{L_{w} B^{3} }}} \right] $$

(15.9)

where

$$ \begin{aligned} & L_{e} = equivalent\,length\,of\,the\,shaft, \\ & k = 4.559x + 0.439\,{\text{when all linear dimensions are in inches or}} \\ & k = 11.58x + 0.439\,{\text{when all linear dimensions are in cm}}; \\ & x = L_{w} B^{3} /R\,D_{j}^{4} \,{\text{in}}\,{\text{both}}\,{\text{cases}} \\ \end{aligned} $$

$$ \begin{aligned} L & = length, \\ D & = outer\,diameter, \\ d & = inner\,diameter \\ D_{e} & = equivalent\,diameter\left( {usually\,crankpin\,or\,main\,journal\,diameter} \right) \\ R & = crankshaft\,radius \\ B & = web\,width\,or\,equivalent\,web\,width\left( {Be} \right),and \\ L_{w} & = web\,thickness. \\ e\, & indicates\,an\,equivalent\,value, \\ j\, & indicates\,the\,main\,bearing\,journal, \\ c\, & indicates crankpin, and \\ w\, & Indicates\,the\,crank\,web. \\ \end{aligned} $$

Since many webs have a curvilinear profile, it is often necessary to calculate an equivalent web width, B_e:

$$ \frac{1}{{B_{e}^{3} }} = \frac{1}{2}\left( {\frac{1}{{B_{max}^{3} }} + \frac{1}{{B_{min}^{3} }}} \right) $$

(15.10)

where

$$ \begin{aligned} & Bmax = maximum\,web\,width\,and \\ & Bmin = minimum\,web\,width. \\ \end{aligned} $$

See Fig. 15.39: Web Width Illustrations below.

Fig. 15.39

Web Width Illustrations (Payne and Scott 1958)

This value for L_e can then be used in the calculation to determine the torsional stiffness between each cylinder centre along the crankshaft. If the main bearing length and web widths are constant for a crankshaft, the torsional stiffness between cylinder centres will also be constant. Minor web differences can usually be neglected in the early stages of the crankshaft design. The crank throw stiffness is given by:

$$ K = \frac{{\pi D_{e}^{4} G}}{{32\,L_{e} }} $$

(15.11)

where

$$ \begin{aligned} K & = crank\,throw\,stiffness, \\ D_{e} & = equivalent\,diameter\,\left( {crankpin\,or\,main\,journal\,diameter} \right), \\ G & = modulus\,of\,rigidity,and \\ L_{e} & = equivalent\,crank\,throw\,length,as\,calculated\,previously \\ \end{aligned} $$

Alternatively, finite element techniques may be used to calculate stiffness.

Front and Rear Crankshaft End Stiffness

The typical method is to calculate the stiffness of each constant diameter section, and then sum the individual stiffness in series at each end. The torsional stiffness of the first half throw (from the first main bearing centre to the first cylinder centre) is twice the torsional stiffness of the full throw geometry as calculated by the BICERA formula (Eq. 11). This stiffness will be added in series with the stiffness of the front crank nose. The rear most half throw should be treated in a similar manner.

For each section examined, the torsional stiffness can be calculated as follows:

$$ K = \frac{{\pi \left( {D^{4} - d^{4} } \right)}}{32L} $$

(15.12)

where

$$ \begin{aligned} & D = outside\,diameter\,of\,section, \\ & d = inside\,diameter\,of\,section, \\ & G = modulus\,of\,rigidity,and \\ & L = section\,length. \\ \end{aligned} $$

When summing in series,

$$ \frac{1}{{K_{tot} }} = \sum {\frac{1}{K}} $$

When a front and/or rear crankshaft gear is utilized, stiffness should be calculated for the crankshaft end sections from the mounting face to half the gear length by summing in series. The next series of sections should be summed starting at the gear half-length to the end main bearing mid length, and the end half throw should then be added in series. Unknown source.

Natural Frequency Calculation

Holzer’s tabular method is used in the present work to calculate the natural frequencies. It is useful for determination of natural frequencies and relative amplitudes required for the study of torsional vibration.

$$ {\text{J}}_{\text{cyl}} + {\text{J}}_{\text{journal}} + {\text{J}}_{\text{crankpin}} + {\text{J}}_{{2\,{\text{webs}}}} + {\text{J}}_{\text{counterweights}} + {\text{J}}_{{{\text{rod}},{\text{rotating}}}} + {\text{J}}_{{{\text{rod}},{\text{recip}}}} + {\text{J}}_{{{\text{piston}}\,{\text{assm}} + {\text{pin}}}} $$