Kinematics of a slider-crank mechanism

Michael Kearney40274982

Group Members:D PotterA Pirlo

M Bonaventura

Date of Experiment:15th August 2005

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Abstract

This report presents the results of a theoretical and experimental inves-tigation of a single-cylinder, four-stroke internal combustion engine. Atheoretical expression for the piston displacement as a function of crankangle is developed. Results from this expression agree with experimentalmeasurements to within 3.3% of the stroke of the piston. The timing ofthe opening of the inlet and exhaust valves relative to the piston motionwere also measured. Both valves are open at the same time over a 20◦

angular displacement of the crankshaft when the piston is near top deadcentre between the inlet and exhaust strokes.

2

Contents

1 Introduction 4

2 Theory 42.1 Four-stroke cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Kinematics of the slider-crank mechanism . . . . . . . . . . . . . 5

3 Apparatus 6

4 Procedure 7

5 Results and Discussion 9

6 Conclusions 13

Appendices 14

A Apparatus measurements 14

B Raw experimental data 15

C Derived experimental data 20

3

1 Introduction

The purpose of the slider-crank mechanism is to convert the linear motion ofthe piston to rotational motion of the crankshaft. One common application ofthis mechanism is in internal combustion engines.

The first aim of this experiment is to investigate and compare the theoreticalkinematic relationship between the displacement of the piston and the angle ofthe crankshaft with that measured for a single-cylinder engine. The other aimis to investigate the four-stroke cycle by simultaneously observing the motionof the piston and valves.

2 Theory

2.1 Four-stroke cycle

An internal combustion engine operates by burning a small amount of a high-energy content fuel, such as petroleum, and using the energy released to drivea shaft. The four-stroke combustion cycle, developed by Nikolaus Otto in1867, is commonly used in petrol-driven internal combustion engines.

Figure 1: Four-stroke engine cycle. Reproduced from the engine’s manual [1]

The four strokes in the Otto cycle are shown in Figure 1. These are:

Intake: The inlet valve is open and the piston moves downwards, drawing in amixture of fuel and air into the cylinder.

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Compression: Both valves are shut and the piston moves upwards to compress the fuel-air mix. The spark plug fires just before the piston reaches its top deadcentre postion (the position where the piston reaches its maximum ver-tical location). This initiates the combustion of the mixture.

Power: Again both valves are closed. The hot gases due to the combustion of thefuel air mix drive the cylinder down. The connecting rod transfers thislinear motion of the piston to rotational motion of the crankshaft. Thetorque thus applied to the crankshaft can be used to drive a mechanism,such as the blades of a lawn mower.

Exhaust: The exhaust valve opens and the upward motion of the piston drives theexhaust gasses out of the cylinder.

Note: The terminology used to describe of the ”four strokes” varies in differentsources [1] [3].

2.2 Kinematics of the slider-crank mechanism

The slider crank mechanism, shown in Figure 2, is a kinematic mechanism. Thepiston displacement from top dead centre, x, can be determined from the geom-etry of the mechanism, in terms of the lengths of the conrod, L, and crank, R,and the crank angle, θ. From the geometry and noting that θ = ϕ = 0 whenx = 0, x can be expressed as

x = R−R cos(θ) + L− L cos(ϕ). (1)

Figure 2: Slider-crank mechanism.

Also from the geometry, it can be seen that

L sin(ϕ) = R sin(θ) (2)

and

[L cos(ϕ)]2 = L2 − [L sin(ϕ)]2 . (3)

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Substituting for L sin(ϕ) from Equation 2 in Equation 3, leaves θ as the onlyvariable on the right hand side of the expression,

[L cos(ϕ)]2 = L2 − [R sin(θ)]2 . (4)

Equation 4 can be substituted into Equation 1 to obtain the kinematic equationfor the slider crank mechanism (Equation 5),

x = R−R cos(θ) + L−√

L2 − [R sin(θ)]2. (5)

Equation 5 can then be rearranged by introducing another parameter, n, theratio of the length of the conrod, L, to the radius of crankshaft, R, as

x = R

1− cos θ + n

1−

√1−

(sin θ

n

)2

, (6)

wheren =

L

R. (7)

Equation 6 is the kinematic equation for the slider-crank mechanism given inthe practical handout [2]. The values of parameters R and n are determined bymeasurement of the Briggs & Stratton engine.

3 Apparatus

A single-cylinder, four-stroke Briggs and Stratton Engine was studied in thisexperiment. A side view of the engine is shown in Figure 3. A top view of theengine, which shows a close up of the cylinder, piston and valves, is shown inFigure 4. The head has been removed and a dial gauges have been attachedto allow measurement displacement of the inlet and exhaust valves. The pistonand cylinder are also accessible for measurement. A protractor is attached tothe crankshaft to facilitate measurement of the crank angle.

The engine specifications are:

• 3 HP maximum power

• 127 cc capacity

• Model No: 81232

• Type: 0209-01

• Code: 79042603

• Mechanical Engineering Catalog No.: M2820

• Modifications:

– The head has been removed allowing access to the piston and valves.

– A dial gauge has been installed to measure the displacement of thevalves.

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Figure 3: Briggs & Stratton engine that was used for the practical. Modifica-tions are shown.

– A 360◦ protractor has been installed on the crank shaft to allowmeasurement of the crank angle

The measurements were made using these apparatus

• Dial Gauge:Manufactured by Mercer, EnglandResolution: 0.01 mm

• Vernier Calipers:Manufactured by TricleModel Number: P02270108ID Number: 4051904Resolution: 0.02 mm

• 360◦ Protractor:Generic School ProtractorResolution: 0.5◦

4 Procedure

The experiment was completed in the following sequence:

1. The cylinder diameter was measured using the vernier calipers.

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Figure 4: Top view of the engine, clearly showing the piston, cylinder and valves.

2. The inner diameter of the large and small ends of the conrod (Dlarge

and Dsmall in Figure 5 respectively), and dimension F (also shown inFigure 5) were measured using vernier calipers. These numerical valueswere substituted into Equation 8 to determine the kinematic length of theconrod.

L = F − 12

(Dlarge + Dsmall) (8)

3. The crank was positioned such that the piston is at top dead centre,between the exhaust and intake strokes. Then the displacement of thepiston from the top of the cylinder was measured using vernier calipers.

4. The crank was rotated 15◦ anti-clockwise, and the new piston displacementwas measured using the vernier calipers and recorded in the logbook. Thisstep was repeated for 15◦ increments until one complete cycle (360◦ ofrotation) was completed.

5. Steps 3 and 4 were repeated twice and averages of these measurementswere calculated. The kinematic length of the crank, R, was then deter-mined from the average measurements using Equation 9.

R = 12 (x|BDC − x|TDC) (9)

Here

xTDC is the piston extension at top dead centre,

xBDC is the piston extension at bottom dead centre, and

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Figure 5: Dimensioned sketch of conrod.

R is the kinematic length of crank.

6. The crank was returned to the top dead centre position between exhaustand intake stokes. The dial gauge was positioned over the inlet valve andthe reading on the dial gauge was recorded into the logbook.

7. The crank was turned 10◦ anti-clockwise and the the measurement on thedial gauge was recorded. (Note: It is necessary to correct the readingsfrom the dial gauge for the initial offset. This was done by subtractingthe reading on the dial when the valve was fully closed from the otherreadings (See Tables 4 and 5). This step was repeated for 10◦ incrementsuntil one complete cycle (720◦ of rotation) was completed.

8. Steps 6 and 7 were then repeated for the exhaust valve.

5 Results and Discussion

Figure 6 shows the displacements of the piston and the valves as a function ofthe angular displacement of the crankshaft from top dead centre (the raw dataused to generate this graph is given in Appendix B). The piston displacement ismeasured from the top dead centre position and valve displacement is measuredfrom the fully-closed position. It shows that the inlet valve is open primarilyfor every second downward stroke of the piston (increasing piston displacement)and that the exhaust valve is open primarily for every second upwards strokeof the piston. The upwards stroke when the exhaust valve is open precedes thedownwards stroke when the inlet valve is open. It is noted that the valves open

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just before and close just after the piston changes direction. The intake valveopens 20◦ before the intake stroke and closes 40◦ into the subsequent upwardsstroke. The exhaust valve opens 40◦ before the exhaust stroke, and closes justafter top dead centre between the exhaust and intake strokes (exhaust valvedisplacement is 0.07 mm at top dead centre). Both the inlet and exhaust valvesare open for approximately 20◦, near and including top dead centre.

−100 0 100 200 300 400 500 600 700 800−10

0

10

20

30

40

50Four−stroke engine cycle − valve and piston displacements

Crank angle from TDC (°)Theoretical piston displacement Measured piston displacement Inlet valve displacement Exhaust valve displacement

Figure 6: Piston and valve displacements over an entire four stroke cycle.

The theoretical curve for piston displacement is also shown in Figure 6. Theparameters R and n, in Equation 6, were determined from measurements (seeAppendices A and C for more details) to be 22.45 mm and 3.54, respectively.When these values are substituted into Equation 6 the theoretical piston dis-placement can be expressed as

x = 22.45

1− cos θ + 3.54

1−

√1−

(sin θ

3.54

)2

. (10)

Figure 6 shows that the measured piston displacement follows the theoreticalcurve very well. The maximum difference between theory and measurement is1.5 mm. This equates to 3.3% of the piston’s stroke. There is an apparent biasin the data, as most of the measurements fall to the right of the theoreticalpiston displacement curve. Closer inspection of Fig. 6 allows this bias to beestimated to be approximately 4◦ − 5◦. The most likely cause of the bias is anoffset in the zero angle of the protractor that is used to measure the angulardisplacement of the crank from top dead centre.

The engine capacity (displacement) is defined as the swept volume of the cylin-der [3]. The length of the sweep is the stoke of piston. This is the difference

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between the piston extension from top dead centre (0◦ and 360◦ crank angle)to bottom dead centre (180◦ crank angle). From this and the diameter of thecylinder, the engine capacity was calculated to be 0.126 l. This is close to thedisplacement quoted for this engine by the manufacturer (0.127 l). Figure 7shows the cylinder head, which sits on top of the cylinder. Note that the extravolume under the head is not included in the engine’s capacity [3].

Figure 7: Head of engine showing the internal volume. This volume is notincluded in the engine’s capacity.

The motion of the piston (for a constant crank angular velocity) is close tosimple harmonic. This allows one to estimate the crank angle at which themaximum speed of the piston is obtained (for a constant crank angular veloc-ity). The maximum downward speed would occur at a crank angle of 90◦ andthe maximum upward speed would occur at a crank angle of 270◦ and the mea-surements indicate that the piston displacements at these crank angles are 24.39mm and 26.69 mm, respectively.

The camshaft (used to open and close the valves at the appropriate times) turnsat half the speed of the crankshaft. This can be seen from Fig. 6 where it isapparent that the piston completes two cycles over the same angular displace-ment of the crank for which the cams complete a single cycle.

If it is assumed that the motion of the inlet valve can be approximated assimple harmonic motion, then the velocity and acceleration of the valves can

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be estimated for a given engine speed. Such calculations indicate that for acrankshaft rotational speed of 3000 rpm, the maximum acceleration experiencedby the valve is 112.4 ms−2. This corresponds to approcimately 11.5 g.

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6 Conclusions

The kinematic motion of the slider in the slider-crank mechanism can be ex-pressed in terms of the lengths of the crank and the conrod, and the angulardisplacement of the crankshaft. The experimental measurements of piston dis-placement agree with the predictions of a theoretical model of the piston motionto within 3.3% of the stroke of the piston. In the present experiment, an offsetbetween the theoretical and experimental values for piston displacement wasalso observed. This was attributed to the incorrect setting of the zero angle ofthe protractor that measures crank angle. This offset is estimated as 4◦ − 5◦.

The inlet valve was open during the intake stroke and the exhaust valve wasopen during the exhaust stroke. The opening range of both valves extended pastthe top-dead-centre postions for their respective strokes. Near top dead centrebetween the exhaust and intake strokes, both valves were open for approximately20◦ and angular rotation of the crankshaft. The increased range of valve openingallows more air to move in during the intake stroke and out during the exhauststroke.

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Appendices

A Apparatus measurements

The cylinder diameter is required to determine the volume of the cylinder andthe capacity of the engine. It has been measured by taking the average of anumber of readings of the diameter at different angles around the cylinder. Re-sults are shown in Table 1. The average value of the diameter is 59.8 mm.

Table 1: Measurements and estimated values of the cylinder’s diameter.Diameter of Cylinder Diameter (mm)Measurement 1 60Measurement 2 59.65Measurement 3 60.02Measurement 4 59.6Average Measurement 59.8Variation 0.42

The determination of the kinematic length of the connecting rod, L, requiresmeasurement of the dimensions SmallD, LargeD and F of the conrod (seeFig. 5). These measurements are given in Table 2.

Table 2: Measurements made to determine the length of the conrod.Total Length (mm) Smaller Diameter(mm) Larger Diameter (mm)

Measurement 1 98.04 25.1 11.88Measurement 2 97.98 25.14 11.88Measurement 3 97.96 25.16 11.62Measurement 4 98.02 25.26 12.3

Average Measurement 98 25.16 11.92Variation 0.04 0.1 0.38

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B Raw experimental data

The raw and corrected measurements of the piston displacement are shown inTable 3 and the raw and corrected measurements of the displacement of theinlet and exhaust valves are shown in Tables 4 and 5, respectively.

Table 3: Piston DisplacementCrank Angle Piston Extension (mm)

(◦) M1 M2 M3 Average Max Variation0 0 0 0 0 0

15 0.86 0.80 0.90 0.85 0.0530 3.12 3.24 3.28 3.21 0.0945 7.58 7.32 7.36 7.42 0.1660 12.72 12.46 12.46 12.55 0.1775 18.28 18.08 18.20 18.19 0.1190 24.7 24.26 24.22 24.39 0.31

105 30.00 29.94 30.40 30.11 0.29120 35.20 34.96 35.24 35.13 0.17135 39.28 39.12 39.32 39.24 0.12150 41.90 42.08 42.16 42.05 0.15165 44.14 43.94 43.78 43.95 0.19180 44.98 45.00 44.72 44.90 0.18195 44.34 44.30 44.36 44.33 0.03210 42.86 42.66 42.72 42.75 0.11225 40.30 40.10 40.40 40.27 0.17240 36.46 36.56 36.56 36.52 0.06255 32.32 32.08 32.10 32.16 0.16270 26.68 26.72 26.68 26.69 0.03285 20.06 20.84 20.84 20.58 0.52300 14.80 14.82 15.20 14.94 0.26315 9.78 9.60 9.70 9.69 0.09330 4.80 5.04 4.82 4.89 0.15345 1.68 1.88 1.58 1.71 0.17360 0 0 0 0 0

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Table 4: Inlet valve extension over a full four stroke cycle

Crank angle raw measurement(mm) corrected measurement(mm)0 1.46 0.8110 2.06 1.4120 2.68 2.0330 3.21 2.5640 3.66 3.0150 4.04 3.3960 4.3 3.6570 4.51 3.8680 4.64 3.9990 4.69 4.04100 4.7 4.05110 4.69 4.04120 4.67 4.02130 4.59 3.94140 4.4 3.75150 4.15 3.5160 3.79 3.14170 3.38 2.73180 2.87 2.22190 2.29 1.64200 1.66 1.01210 1.21 0.56220 0.96 0.31230 0.9 0.25240 0.9 0.25250 0.9 0.25260 0.92 0.27270 0.82 0.17280 0.65 0290 0.65 0300 0.65 0310 0.65 0320 0.65 0330 0.65 0340 0.65 0350 0.65 0360 0.65 0370 0.65 0380 0.65 0390 0.65 0400 0.65 0410 0.65 0420 0.65 0

Continued on next page

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Crank angle raw measurement(mm) corrected measurement(mm)430 0.65 0440 0.65 0450 0.65 0460 0.65 0470 0.65 0480 0.65 0490 0.65 0500 0.65 0510 0.65 0520 0.65 0530 0.65 0540 0.65 0550 0.65 0560 0.65 0570 0.65 0580 0.65 0590 0.65 0600 0.65 0610 0.65 0620 0.65 0630 0.65 0640 0.65 0650 0.65 0660 0.65 0670 0.65 0680 0.65 0690 0.69 0.04700 0.75 0.1710 1.01 0.36720 1.44 0.79

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Table 5: Exhaust valve extension over a full four stroke cycle

crank angle raw measurement(mm) corrected measurement(mm)0 0.55 0.0710 0.48 020 0.48 030 0.48 040 0.48 050 0.48 060 0.48 070 0.48 080 0.48 090 0.48 0100 0.48 0110 0.48 0120 0.48 0130 0.48 0140 0.48 0150 0.48 0160 0.48 0170 0.48 0180 0.48 0190 0.48 0200 0.48 0210 0.48 0220 0.48 0230 0.48 0240 0.48 0250 0.48 0260 0.48 0270 0.48 0280 0.48 0290 0.48 0300 0.48 0310 0.48 0320 0.48 0330 0.48 0340 0.48 0350 0.48 0360 0.48 0370 0.48 0380 0.48 0390 0.48 0400 0.48 0410 0.48 0420 0.48 0

Continued on next page

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crank angle raw measurement(mm) corrected measurement(mm)430 0.48 0440 0.48 0450 0.48 0460 0.48 0470 0.48 0480 0.48 0490 0.48 0500 0.61 0.13510 1.09 0.61520 1.71 1.23530 2.32 1.84540 2.85 2.37550 3.28 2.8560 3.64 3.16570 3.88 3.4580 4.07 3.59590 4.17 3.69600 4.2 3.72610 4.2 3.72620 4.2 3.72630 4.14 3.66640 4.03 3.55650 3.82 3.34660 3.53 3.05670 3.13 2.65680 2.72 2.24690 2.16 1.68700 1.57 1.09710 0.97 0.49720 0.55 0.07

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C Derived experimental data

The length of the conrod, L, is determined using Equation 8 and the averagemeasurements in Table 2. The conrod length is calculated as 79.46 mm.

The kinematic radius of the crank, R, is determined by halving the displacementof the piston from its top dead centre to its bottom dead centre positions.Table 3 shows that this distance is 44.90 mm. Therefore the kinematic radiusof the crank is 22.45 mm.

The ratio of conrod length to crank radius, n(= L/R = 79.46/22.45), is calcu-lated to be 3.54.

The capacity of the cylinder is calculated using the formula for the volume of acylinder,

V =πD2h

4. (11)

The height of the cylinder, h, is determined by the difference in piston dis-placement between top dead centre and bottom dead centre postions. As above,this was measured to be 44.90 mm. The measurement of the piston diameteris recorded in Appendix A as 59.8 mm. Using Equation 11, the capacity of thecylinder is then calculated to be 126 cm3 or 0.126 l.

Table 6 shows the crank angles at which the inlet and exhaust valves open andclose.

Table 6: Crank angle where valves open and closeValve Opens Closes

INLET −20◦ (700◦) 220◦

EXHAUST 500◦ 720◦ (0◦)”

To a first approximation, the motion of the opening and closing of the inletvalve can be written as

y =12ymax

[1− cos

(πt

T

)], (12)

where

y is the displacement of valve from the closed postion,

ymax is the maximum displacement of the valve,

t is time, and

T is the period of time for which the inlet valve is open.

Equation 12 can be differentiated with respect to time to obtain an expres-sion for the velocity of the inlet valve and then differentiated with respect totime again to obtain an expression for the acceleration of the valve. The velocity

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and acceleration of the valve are thus given respectively by Equations 13 and 14.

y =12ymax

[sin

(πt

T

)] ( π

T

)(13)

y =12ymax

[cos

(πt

T

)] ( π

T

)2

(14)

Table 6 shows that the inlet valve is open for an angular displacement ofthe crankshaft of 240◦. The period of time that this corresponds to can then becalculated from Equation 15.

T =60N

240◦

360◦(15)

where N is the rotational speed of the crankshaft in rpm.

If rotational speed of the crankshaft is 3000 rpm, then N = 3000 and, fromEq. 15, the inlet valve will be open for T = 13.3 ms. The maximum displacementof the inlet valve can be found from Table 4 to be ymax = 4.05 mm. Themaximum acceleration can be calculated from Eq. 14 to be

ymax =ymaxπ

2

2T 2. (16)

Substituting these values for T and ymax into Eq. 16 gives the estimated maxi-mum valve of inlet valve acceleration of 112.4 m/s (11.5g).

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References

[1] Briggs & Stratton Service and Repair Instructions. 1976

[2] Mee, D, Laboratory Sheet: Kinematic of Mechanisms. Mechanical Engineer-ing Division University of Queensland. 1997.

[3] Pulkrabek, W. W, Engineering fundamentals of the internal combustionengine, Prentice Hall, New Jersey 1997.

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