a coupled dynamic valve spring and engine performance simulation
TRANSCRIPT
A Coupled Dynamic Valve Spring and Engine Performance Simulation
Nigel Fleming, Richard Pearson, and Mike Bassett, Lotus Engineering Software, Lotus Engineering, UK
AbstractNow that the use of simulation is an established part of the engine design and development process much current activity is directedtoward the coupling of codes to obtain greater model refinement. In this paper results from an engine cycle simulation code coupledwith a valve-spring dynamics code, with the facility to model a hydraulic lash adjuster, are presented. The coupled simulation enablesthe examination of the influence of the force generated by the cylinder gas pressure on the valve motion and this modified behaviouris then fed back in to the engine cycle simulation code to assess the impact on the engine performance. The most significant departureof the valve motion from that generated by a kinematic analysis was found to be due to the valve-train flexibility. For both anautomotive engine and a high-speed racing engine the dynamically modified valve motion gives rise to only small differences inpredicted engine performance.
A model of a gas spring of the type used in some racing engines is also described in the paper. The model includes the effects of heatand mass transfer and gas properties.
1. IntroductionIn the past 30 years computer simulation techniques for mechanical analysis and thermodynamic cycle simulation have become firmlyestablished as an inextricable part of the powertrain design process. Engine performance simulation codes have evolved from thestatus of research tools to fully supported commercial software packages. These thermodynamic cycle simulations, which includenon-steady gas dynamics, are used routinely to predict engine performance [1,2], thereby assisting in the design of the intake andexhaust manifolds (and silencers), the specification of cam profiles, the prediction of the gas pressure forces and thermal loading onthe engine. Valve-train and crank-train models of varying levels of complexity are then used to perform the mechanical and thermaldesign analysis of the base-engine components. Classical calculations for crank train components are performed at the concept stageand increasingly sophisticated modelling techniques are applied, leading to full finite element models of the engine structure. Initialdesign of valve-train systems is often performed using simple kinematic models of individual valve-train units that can then be refinedby conducting dynamic analyses and, ultimately, building a full model of an engine valve train.
Engine performance simulation codes and cam profile design / valve-train analysis codes have a symbiotic relationship with eachother within the engine concept design process. The results of parametric analyses using the former programs specify the lift andduration targets for the profile design carried out using the latter. The resulting cam profiles may then be used as input to the enginesimulation program in a latter stage of the engine design to assess the effect of mechanical constraints of the valve-train upon theperformance of the engine.
Valve-train analysis has become increasingly sophisticated as the engine design process has evolved [3-6]. Models of componentssuch as hydraulic lash adjusters have been introduced [6] and the effects of gas pressure forces on valve motion have been modelled[6]. This paper describes a cam profile design and analysis tool that can be used to rapidly generate profiles using a kinematicapproach. The resulting model of the valve-train system can then be used as the basis of a 1-D dynamic model that can be coupled atrun-time to an engine performance simulation code. This enables the effects on the valve motion of the varying gas pressure forces inthe engine cylinder to be accounted for whilst providing the engine simulation with a more realistic valve motion.
A brief description of the engine performance simulation code is given in Section 2, followed by a more detailed account of the valve-train analysis software in Section 3, including a description of the gas spring model used in the simulation of the racing engine.Section 4 describes the coupling of the two programs whilst Section 5 presents the results and discussion of the work.
2. The Engine Simulation ModelThe engine cycle simulation program used in the present work was Lotus Engine Simulation. The governing equations of one-dimensional unsteady gas flow in ducts are solved using a shock-capturing finite volume scheme [2] in order to capture the effects ofintake and exhaust manifold geometry on engine performance. Boundary models are available for poppet, reed, piston-ported, or diskvalves, pipe junctions, superchargers, turbocharger compressors and turbines, charge-coolers and other major engine components.Heat release models are used to characterise the combustion event and a choice of in-cylinder heat transfer models is available.
The code has a built-in ‘drag-and-drop’ model builder. A screen shot of a typical model is shown in Figure 1. This model includes‘Sensor and Actuator’ elements that one used to vary the timing of the intake camshaft and control the operation of a variablegeometry intake manifold. Sensors and actuators can make use of more sophisticated signal-operator functions which are built in tothe code. As the software can operate in the Component Object Model (COM) environment, the input and output signals associatedwith the sensors and actuators can also be passed externally so that the model is driven by, for example, a Simulink control model.
Fig. 1: Screen shot of model in Lotus Engine Simulation
3. The Valve-Train Simulation ModelThe simulation program used in this study, Lotus Concept Valve Train, provides the facility to build valve-train models of variousdegrees of complexity. Initial cam profile design can be executed using a wholly kinematic analysis with an empirical factor that isused in the determination of float speed limits. The cam profile may be designed using a segmented-polynomial or Bezier-curveapproach, as described below. Later in the design process a dynamic model can be generated directly from the kinematic model. Amulti-mass representation of a single or double wire spring can be constructed or a gas spring of the type used in motorsportapplications may be modelled. It is also possible to include either solid or hydraulic lash adjusters in the dynamic model.
3.1. Cam Profile GenerationCam profiles can be defined using a segmented polynomial approach. The particular polynomial form for each segment is dependenton the number of points defined and the boundary conditions imposed on each segment. A typical polynomial for a profile segment is
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The user constructs the polynomial curve to define the valve/cam motion. A segmented polynomial curve is pre-defined which theuser can modify and impose additional constraints upon. The default curve is based on an eleven-point, six-segment, definition. Eachsegment is represented by a polynomial having the required order to suit the number of points and boundary conditions specified. Thecomplete displacement curve and the first three derivatives (velocity, acceleration and jerk) are simultaneously displayed. Featuressuch as the ramp heights and velocities, the symmetry of the profile, the maximum lift, and the profile period may be modified.
‘Clipped velocity’ profiles can be created using 15 points and 10 segments, whilst ‘clipped acceleration’ profiles are based on 17points with 12 segments. Clipped velocity or acceleration profiles enable the user to limit the maximum and minimum values of theseparameters to address problems such as eccentricity limitations. The points of extrema can be coupled so that, when modified, theirmagnitudes are identical.
Existing profiles can be imported in to the software and analysed. During this import process Chebyshev polynomials [7] may be usedto interpolate the data on to a regular crank-angle basis, smooth the data if required, or produce undefined derivatives. Importedprofiles can be fitted to the polynomial format described above using a ‘feature-identification’ approach where the matching process isperformed according to user-defined criteria
An additional option for designing cam profiles is the use of piecewise Bezier curves [8] which are constructed as a sequence of cubicsegments [9]. The construction of Bezier curves uses a special case of cubic Hermite interpolation based on a construction due toBernstein [10] in which the interpolating polynomials depend upon certain control points. Figure 2 shows a Bezier curve defined by aset of n+1 control points Z0, Z1,…,Zn. according to the equation
)()( ,0
tBt nin
ti�
�
� ZC , (2)
where )(, tB ni is a Bernstein polynomial of the form
inini tt
iinntB �
�
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!)(, , (3)
and ]1,0[�t . The Bezier curve passes through the first and last control points and is tangential at its end points to Z1-Z0 and Zn-Zn-1.Performing translations and rotations on the control points gives rise to these effects on the Bezier curve.
Fig. 2: Bezier curve defined by control points Z0-Z4
The cam profile can be designed by using Bezier curves to define the acceleration, the velocity, or the lift. Six segments, each with aminimum of four points, are used to define an acceleration diagram, with the ‘smoothness’ of the global curve being conditioned bygradient continuity across the transitions. An additional requirement that the distance is identical between the last and first two pointson contiguous segments is imposed. Conventional ramps are added to the first and last points of the Bezier curve to create a completecam profile. The curves for velocity and lift are calculated via differentiation, and jerk from integration, of the global curve. Thisdense but unevenly spaced data is then interpolated back onto the required regular cam angle increment. A number of constraints arethen applied to ensure that the basic shape of the piecewise curve follows that required for a coherent cam profile.
Because it is possible to drag the points of one curve independent of the others, (except for enforced continuity), the overall curve willnot normally meet the fundamental requirements required by a particular profile and additional operations are necessary. For asymmetrical profile it is sufficient to ensure that the zero velocity point at maximum lift occurs at the centre of the profile – this willalso ensure that the velocity and lift at the top of the closing ramp are correctly matched. For an asymmetrical profile, once theposition of maximum lift has been set, a further two matching operations must be carried out, ensuring the compatibility of bothvelocity and lift at the top of the closing ramp. Two points in the closing segments of the profile are manipulated to give theappropriate ramp velocity and the lift is then checked against the specified lift at the top of the closing ramp. If the latter parameter isnot correct the acceleration curve is re-manipulated to achieve the closing ramp velocity and the top-of-ramp lift is checked again.This process is iterated automatically by the software according to prescribed criteria.
Four segments are used if the velocity curve is used to manipulate the cam profile and two segments are used if the user wishes todefine the lift curve. Additional points may be added within each Bezier segment up to a maximum of 20.
Figure 3a shows a screen-shot from Lotus Concept Valve Train where a control point has been selected in the acceleration diagram –in this case the selected point can be freely manipulated in the vertical direction (see yellow buttons at top of control panel) in order tomodify the cam profile. An example of the output results from a kinematic analysis is shown in Figure 3b.
Z4
Z3
Z2
Z1
Z0
Fig. 3a: Manipulation of acceleration diagram during cam profile design
Fig. 3b: Example of output from valve-train static analysis.
3.2. Dynamic Model of Valve SpringOnce a cam profile has been designed it can be analysed in more detail by converting the kinematic model of the valve-train system into a dynamic model. Only single valve actuation assemblies can be modelled in Lotus Concept Valve Train and hence no camshafttorsional vibration effects are included. The stiffness of the camshaft bearings, the cam lobe, the tappet, the valve stem, and the valveseat-to-cylinder head contact region are used in the dynamic model. The component mass data is carried over from the kinematicmodel and the valve spring element can be resolved in greater detail by sub-dividing it into multiple discrete masses, each with theirown stiffness and damping values, as shown in Figure 4. The simulation calculates the motion of the individual masses (shown to theleft of the mass elements in Figure 4) and is capable of modelling the ability of the spring to control the motion of the valve across theengine speed range.
3.3. Model of Hydraulic Lash AdjusterHydraulic lash adjusters (HLA’s) are now common in modern automotive engines. As part of the valve-train dynamic modellingfacility Lotus Concept Valve Train contains a means of representing these devices which is based on that described in reference [11].A screen-shot of an HLA model in Lotus Concept Valve Train is shown in Figure 4. It can be seen that the ball-check valve and itsspring, which control the flow of fluid between the high- and low-pressure reservoirs, is included in the model. This mass transfer andthe effect of aeration levels on the bulk modulus of elasticity of the oil are accounted for in the simulation. The results of incorporatingsuch a model into a coupled analysis with an engine performance simulation will be described later in the paper.
Fig. 4: Model of hydraulic lash adjuster
3.3. Gas Spring ModelCurrent F1 racing engines and some GP1 motorcycle engines employ small chambers filled with a gas, usually nitrogen, whichserve the same function as a mechanical spring in a conventional valve train. These ‘gas springs’ circumvent the problem of springsurge by eliminating the coil spring entirely. A schematic of such a system is shown in Figure 5. The gas spring itself isrepresented by a chamber, 1, which is fed by the system supply connections, 2. The chamber also experiences mass transferthrough a calibrated orifice, 3, and via leakage of the gas to the ambient, 4, via a sealing ring. The volume of the chamber varies asthe valve is moved. The system is shown as part of a ‘finger-follower’ actuation mechanism which are common in highperformance engines due to the lower frictional penalty they impose and the greater freedom of valve lift range and profile shapethey facilitate.
Discrete massrepresentingpart of thevalve spring
Hydraulic lashadjuster
Fig. 5. Schematic of gas spring model
The gas spring is modelled in Lotus Concept Valve Train by considering the change in properties of the gas in the spring chamber dueto its change in volume and the mass and heat transfers between the various connected reservoirs. An ideal gas is assumed, thecomposition of which is defined by specifying the mole fractions of each constituent from a choice of 13 species. The variations of theproperties of the gas as a function of temperature are also included. For each species the enthalpy is given by
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where � is the Universal Gas Constant and the coefficients a1 to a5 are taken from Benson [12]. The enthalpy of the mixture is thengiven by
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N
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where Nspec is the number of species in the gas mixture and xj is the mole fraction of specie j.
The specific internal energy and mass of the gas in the spring chamber is integrated using a fourth-order Runge-Kutta scheme. Theinternal energy at time level n+1 is given by
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where
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The chamber mass is integrated in the same way. The rate of change of internal energy with time is obtained from the First Law ofThermodynamics for an unsteady, open system in the form
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where h0 is the specific stagnation enthalpy of the gas entering or leaving the chamber, and Q represents the heat transfer rate throughthe gas chamber walls. The mass flow rates ( m� ) are obtained by solving the equations of compressible flow through an orifice [13].
Figure 6 shows a screen-shot from Lotus Concept Valve Train which depicts a model of a direct-acting valve mechanism with a gasspring. The results of incorporating such a model into a coupled analysis with an engine performance simulation will be describedlater in the paper.
5
32 2
1
41. Gas spring.2. System supply reservoir.3. Calibrated orifice.4. Ambient.5. Finger follower.
4. Coupled Analysis of VaIt is possible to link Lotus Engine Senvironment. In the present work, hdynamic link library (DLL) which scoupled simulation significantly and
The purpose of coupling the codes isthe simulation of the valve-train dynlift curve. The dynamics of each valvtheir property sheets, as shown in FiConcept Valve Train, producing a wmodel can be loaded. After loadingusual way.
r
SpringchambeFig. 6. Screen-shot of gas spring model
lve Train Dynamics and Engine Performanceimulation externally to any other computer program which is capable of operating in the COM
owever, a facility to link Lotus Engine Simulation directly with Lotus Concept Valve Train via ahares dynamic memory was developed. This approach simplifies the procedure for setting up a reduces the computational overhead associated with the COM environment.
to enable the force generated by the pressure differential across the valve head to be included inamics and, conversely, to provide the engine performance simulation with a more realistic valvee in the engine simulation model can be modelled simply by selecting the ‘Dynamic’ option from
gure 7 (inside dashed line). Clicking the option labelled ‘Dynamic Model Data’ then opens Lotusindow similar to that shown in Figure 4, so that the appropriate data can be entered or an existing or setting up the appropriate valve-train data the engine simulation code can then be run in the
Fig. 7. Screen-shot of model used for analysis
5. Results and DiscussionFigure 8 shows a comparison of measured and predicted valve spring strain for the direct-acting valve train. The measured resultswere obtained using a test rig on which the cylinder-head of a two-litre, four-cylinder, 16-valve engine was motored at 6000 rev/min.(3000 rev/min. camshaft speed). The valve-train system has single wire springs, and hydraulic tappets. The correlation betweenmeasured and predicted test results is very good.
Typical Valve Spring Strain Correlation
-2000
-1000
0
1000
2000
3000
4000
5000
6000
Mic
rost
rain
TestPredicted
Time
Fig. 8. Comparison of measured and predicted valve spring strain
Figure 9 shows the difference in valve lift predicted by a dynamic model of a direct-acting valve train with a mechanical tappetcompared with the kinematically generated profile [(dynamic lift)-(kinematic lift)]. No tappet-to-valve clearance was used in this caseand the system has been set up to ensure that there is no valve float. Results for both 2000 rev/min and 6500 rev/min are shown. In thevalve-closed period there is a slight recession of the valve caused by the seating force applied by the spring. The two large peaks ineach curve are caused by the system flexibility due to loading from the two main acceleration events (see Figure 3a for the generalform of the acceleration curve). The effects caused by these main acceleration events are larger at 6500 rev/min. than at 2000 rev/min..The higher momentum of the valve in the deceleration event at 6500 rev/min. causes the departure of the profile from the kinematicprofile to reduce below that of the level found at 2000 rev/min.
-3.50E-02
-3.00E-02
-2.50E-02
-2.00E-02
-1.50E-02
-1.00E-02
-5.00E-03
0.00E+000 90 180 270 360
Cam angle / [deg]
Cha
nge
in v
alve
lift
/ [m
m]
mechanical tappet: 2000 rev/min
mechanical tappet: 6500 rev/min
Fig. 9. Effects of valve-train dynamics on valve lift for direct-acting system with mechanical tappet
Figure 10 shows results produced by modelling a system with a hydraulic tappet. In this case the change in valve lift is relative to akinematic profile which has not been modified in any way. The presence of the hydraulic tappet introduces additional flexibility intothe system due to the compressibility of the oil. Further compliance is caused by the flow of oil through the ball-check valve before itseats and the leakage of oil past the tappet seals. The differential lift does not recover to the extent that it does with a mechanicaltappet as the spring force, which retains control of the profile at all times, causes leakage of oil out of the tappet.
-1.20E-01
-1.00E-01
-8.00E-02
-6.00E-02
-4.00E-02
-2.00E-02
0.00E+000 90 180 270 360
Cam angle / [deg]
Cha
nge
in v
alve
lift
/ [m
m]
Hydraulic tappet: 2000 rev/min
Hydraulic tappet: 6500 rev/min
Fig. 10. Effect of valve-train dynamics on valve lift for direct-acting system with mechanical tappet
Figure 11 shows the effects of introducing increasing levels of complexity to a dynamic model of a valve-train system with ahydraulic tappet. An inlet valve is considered operating at an engine speed of 6500 rev/min.. The kinematic profile against which thedynamic profiles were compared had 0.05mm at the valve opening point and 0.06mm at the valve closing point (varying linearlyacross the profile duration) subtracted from the profile in order to account for the effects of tappet collapse described above. Theresults obtained using a rigid valve show that the flexibility of this element contributes significantly to the departure of the dynamicvalve motion from the kinematic profile. Changing other system stiffness values for components such as the cam lobe and shaft andvalve seat also have a large effect and the sum of these accounts for most of the change in valve lift.
-6.00E-02
-5.00E-02
-4.00E-02
-3.00E-02
-2.00E-02
-1.00E-02
0.00E+00
1.00E-02
2.00E-02
3.00E-02
0 90 180 270 360
Cam angle / [deg.]
Cha
nge
in v
alve
lift
/ [m
m]
Basic dynamic model (no dynamic spring)Basic dynamic model with rigid valveBasic dynamic model with gas pressureDynamic spring modelDynamic spring model + gas pressureBasic dynamic model with rigid valve, seat, and camshaft
Fig. 11. Effect of valve-train dynamics on valve lift for direct-acting system with mechanical tappet
The large dip in the change-in-valve-lift curve is due to the inclusion of the gas pressure forces in the cylinder which cause significantvalve recession in to the seat when the cylinder pressure is large (during the valve-closed period). The cylinder pressure makes only asmall difference to the intake valve in the valve-open period as the cylinder pressure is low during this time. Adding the dynamicspring model to the system has the effect of slightly increasing the change in valve lift.
Figure 12a shows the kinematic profile generated for use on the direct-acting, four-cylinder, 16-valve automotive engine for which thecorrelation between measured and predicted results shown in Figure 8 was produced. Values of 0.05mm and 0.1mm (varying linearlyacross the profile duration) have been subtracted from the original profile in order to account for the hydraulic tappet collapse. Figure12b shows that the major influence on the dynamic valve-lift curve is from the system stiffness and the collapse of the hydraulictappet. The rise in the change-in-valve-lift curve in the later part of the opening period is due to the over-compensation in thekinematic profile of subtracting the predefined tappet collapse from the kinematic profile. The engine speed at which the results wereobtained was 6500 rev/min..
0
2
4
6
8
10
0 90 180 270 360 450 540 630 720Crank angle / [deg.]
Valv
e lif
t / [m
m]
LES - Fixed lift
Fig. 12a. Kinematic valve lift for direct-acting system with hydraulic tappet tappet – automotive engine
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0 90 180 270 360 450 540 630 720Crank angle / [deg.]
Cha
nge
in v
alve
lift
/ [m
m]
LES + Hydraulic tappet modelLES + Tappet + SpringLES + Tappet + Spring + Gas loads
7000 rev/min
Fig. 12b. Effect of valve-train dynamics on valve lift for direct-acting system with hydraulic tappet. Automotive engine – 6500rev/min.
A comparison of measured and predicted brake torque for the engine is shown in Figure 13. This prediction was performed using thekinematic profile modified as described above. Figures 14 and 15 show the effects of introducing various degrees of sophisticationinto the valve-train model, running Lotus Engine Simulation coupled with Lotus Concept Valve Train. In Figure 15 the change intorque relative to that predicted using the kinematic profile is shown. It is clear that the largest effect is caused by the assumptionregarding the tappet collapse used in the kinematic curve.
150
160
170
180
190
200
5000 5500 6000 6500 7000 7500Engine speed/ [rev/min]
Torq
ue
Measured
Predicted - LES - fixed valve event
10 Nm
Fig. 13. Comparison of measured and predicted bake torque
150
160
170
180
190
200
5000 5500 6000 6500 7000 7500Engine speed / [rev/min]
Torq
ue
MeasuredLES - fixed valve eventLES + Hydraulic tappet modelLES + Tappet + SpringLES + Tappet + Spring + Gas loads
10 Nm
Fig. 14. Comparison of predicted bake torque for various valve-train models
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.05000 5500 6000 6500 7000 7500
Engine speed / [rev/min]
Torq
ue c
hang
e (N
m)
LES + Hydraulic tappet modelLES + Tappet + SpringLES + Tappet + Spring + Gas loads
Fig. 15. Comparison break torque differences due to various valve-train models
Figure 16 shows a simulation model of a high-speed five-cylinder racing engine. The engine performance was modelled using a fixedkinematic valve lift profile and by coupling the engine and valve-train simulation models. The valve-train-dynamic models included agas spring element of the type described in Section 3.3; the effect on the valve motion of the variation of cylinder pressure was alsoincluded in one of the simulation runs.
Fig. 16. Simulation model of high-speed racing engine
The effects on the valve motion, at an engine speed of 16000 rev/min., of introducing the gas spring model and the variation ofcylinder gas pressure are shown in Figure 17. Again, the major contribution to the change in valve lift if due to the elasticity of thevalve-train components. It can be seen that, due to the very high engine speed, the effect of the valve acceleration is very largecompared with the valve head and seat deflection due to the gas pressure forces.
-3.00E-01
-2.50E-01
-2.00E-01
-1.50E-01
-1.00E-01
-5.00E-02
0.00E+00
5.00E-02
0 90 180 270 360
Cam angle / [deg]
Cha
nge
in v
alve
lift
/ [m
m]
Air spring modelAir spring + gas loads
Fig. 17. Effect of valve-train dynamics on valve lift for direct-acting system with mechanical tappet and gas spring. Racing engine– 16000 rev/min.
Figure 18 shows the impact of the dynamic valve motion on the results from engine performance simulation. At the peak-power speedof 16000 rev/min. there is a reduction of 7.5 hp compared with the prediction obtained with the kinematic profile – this is consistentwith the reduction in valve lift across the profile caused by the dynamic effects in the valve train.
5
6
7
8
9
10
11
5000 7500 10000 12500 15000 17500 20000Engine speed / [rev/min]
BMEP
Fixed valve eventAir spring modelAir spring + gas loads
1 bar
Fig. 18. Effect of valve-train dynamics on engine performance simulation.
Figure 19 shows that the inclusion of the gas pressure forces in the valve-train model make a considerable difference to the calculationof valve seat forces for the five-cylinder racing engine. The peak seat force is approximately five times higher than the case when thegas pressure forces are not included. The valve seat force gives and indication of the quality of the valve seating and affects valve seatrecession.
0
1000
2000
3000
4000
5000
6000
Time
Valv
e se
at fo
rce
/ [N
]
Air spring model + gas loadsAir spring model
Fig. 19. Effect of valve-train dynamics on engine performance simulation
6. ConclusionsThe coupled engine and valve-train simulation enables the examination of the effects of the force generated by the cylinder gaspressure on the valve motion. For both an automotive engine and a high-speed racing engine the dynamically modified valve motiongives rise to only small differences in engine performance. The most significant departure of the valve motion from that generated bya kinematic analysis was found to be due to the valve-train flexibility, with smaller effects being attributable to the collapse of thehydraulic lash adjuster, when included. Omitting the gas pressure forces causes a considerable under-estimate of the peak valve seatforce.
7. References
1. Winterbone, D.E. and Pearson, R.J., Design techniques for engine manifolds – Wave action methods for IC engines. Professional EngineeringPublications, London, 1999.
2. Winterbone, D.E. and Pearson, R.J., Theory of engine manifold design – Wave action methods for IC engines. Professional EngineeringPublications, London, 2000.
3. Turlay, J.D., Smooth acceleration cam development.Trans. SAE, p.715, 61, 1953.4. Kosugi, T., and Seino, T., Valve motion simulation method for high-speed internal combustion engine. SAE paper no. 850179, 1985.5. Nishiura, H., and Akahane, H., Valve gear movement simulation. I.Mech.E. paper no. C430/006, International Conference: Computers in Engine
Technology, Robinson College, University of Cambridge, 10-12 September, 1991.6. Watson, H.C., and Chow, C.H., Inclusion of hydraulic lash adjuster behaviour and gas dynamic forces in camshaft design. I.Mech.E. paper no.
C430/064, International Conference: Computers in Engine Technology, Robinson College, University of Cambridge, 10-12 September, 19917. Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P., Numerical Recipes in FORTRAN. 2nd Edition. Cambridge University Press,
Cambridge, 1992.8. Chiorescu, C., Cristea, G., Oprean, M., and Jebelean, P., Computer analysis of the automotive valve gear. 3rd ATA International Cnference on
Innovation and Reliability in Automotive Design and Testing, Firenze, Italy, April 8-10, 1992.9. Piegl, L., Fundamental Developments of Computer Aided Geometric Design. Academic Press, San Diego, 1993.10. Lorentz, G.G., Bernstein polynomials. University of Toronto Press, Toronto, 1953.11. ADAMS/Engine Product Guide, v12, MSC Software, 2002.12. Benson, R.S., Advanced engineering thermodynamics. 2nd Edition. Pergamon Press, Oxford, 1977.13. Benson, R.S., The thermodynamics and gas dynamics of internal combustion engines. Clarendon Press, Oxford, 1982.