Techno’s

# And Piston Velocity

Introduction

This 3-part article is about the piston and crankshaft movements in the Mazda MX-5 1.6 and 1.8 litre engines. The purpose is to expose the stresses and strains upon the piston, connecting rod and crankshaft during acceleration and at particular rpm's.

The principles and algorithms used in this article can be applied to any 4-stroke engine. To assist the reader to understand the calculations I have included the more complex algorithms for reference.

Part 1 - The Basics

Mazda has made two basic engines for the MX-5; the 1.6 litre and the 1.8 litre. In Australia the 1.6 litre engine was used in the NA6A model cars (1989 to 1993). The 1.8 litre is stock for all Australian delivered cars since 1994 (NA8A, NB8A and NB8B).

Table 1 contains the base data used in the article. The values in the shaded cells are the primary data used in the calculations for the other values.

Table 1 MX-5 Engine Specifications

 Model Description Ref. Value MX-5 1.6 litre MX-5 1.8 litre Stroke mm S 83.6 85.0 Bore mm B 78.0 83.0 Connecting Rod Length (+/- 0.05mm) R 132.90 132.90 Number of Cylinders N 4 4 Bore Stroke ratio B/S 0.933014 0.976471 Stroke Bore ratio S/B 1.071795 1.024096 Rod Stroke ratio R/S 1.589713 1.563529 Displacement per cylinder (cc) D[i] 399.47 459.90 Engine Displacement (cc) D*N 1597.88 1839.61 Crank Arm mm S/2 41.8 42.5 Other Variables Used in the Algorithms RPM T Revolutions per minute of the Crank Picton Velocity V This is the speed of the Piston's movement measured in meters per second Piston Acceleration A The rate of change in velocity, measured in meters per second per second Crank Angle K Crank angle in degrees after TDC

Table 1 contains all the data used in the calculations. The column titled "Ref Value" contains the variables assigned various components in the algorithms. In the worked examples I have used the 1.8 litre engine and 6,000 rpm as a benchmarks for illustrations. To enable the formula to be Excel computable I have used Excel functions such as "SQRT", "ATAN" and "^", etc.

Understanding the Piston, Rod and Crank Arrangement

To understand the data contained in this article it is useful to be able to identify the parts of the engine being discussed. Figures 1,2 and 3 display the movement and relationships of the components.

Reference to TDC and BDC are explained in relation to the piston's location and the stages in the rotation of the crank.

 Figure1 Piston at Top Dead Centre  (TDC) Figures 1 shows the piston (red) at top dead centre (TDC). This is the highest position of the piston and occurs when the con-rod (green) and crank arm (blue) both fully extended. The crankshaft (white) is at zero degrees and ready to start a 360-degree revolution. Figure 2 Piston at Bottom Dead Centre (BDC) Figure 2 shows the piston at bottom dead centre. The crank has turned 180 degrees the piston has travelled one stroke and is at its lowest point, called bottom dead centre (BDC).   When the piston is in this position the con-rod is vertical and the crank arm is facing downward. Figure 3 Piston Approaching TDC Figure 3 Shows 1.      The con-rod links the piston to the crank arm. 2.      The con-rod and crank arm geometry translates the vertical movement of the piston into a rotating motion of the crankshaft. 3.      The length of the con-rod and crank arm determine the stroke travel of the piston.

Figure 3, while demonstrating the relationships of the components in motion also shows the location of the piston when passing a timing point, in this case 14 degrees before TDC.

There are 360 degrees in one rotation of the crankshaft.  When the timing is at 14 degrees before TDC the spark is fired at 346 degrees of rotation of the crank. Imagine the end of the crank being a clock dial with 12 o'clock being 0 degrees. At 14 degrees BTDC the clock would be at 11.57 and 20 seconds.

Part 2 - The Movement of The Piston

During one 360 revolution of the crank in a MX-5 1.8 litre engine the piston:-

·         travels down 85.0 mm (TDC to BDC); stops

·         travels up 85.0mm (BDC to TDC); and stops

When the engine is turning over at 6,000 rpm the piston travels down 6,000 times per minute and up 6,000 times per minute. Therefore, at 6,000 rpm piston travels 12,000 x 0.85m = 1,020 meters per minute, or as is more widely use, 17.0 meters per second  (1,020/60). Since the piston stops at each 180-degree rotation of the crank then 17.0m per second is the average, or mean, speed of the piston. So what is the velocity of the piston at during its movement?

Graph 1 plots the velocity of the piston at 10-degree intervals over one revolution or the crank. Notice how the maximum velocity is reached when the crank angle is around 74 degrees ATDC. This is very close to the position when the crank arm and the con-rod form a 90-degree angle. The maximum piston velocity, in the 1.8 litre is reached at when the crank angle is 73.715 degrees after TDC.

Graph 1 Piston Velocity Graph 2 shows the (Maximum) Piston Velocity of a 1.8 litre MX-5 piston as it passes the 73.715-degree mark ATDC, in 500 RPM increments.

Graph 2 Maximum Piston Velocity Graph 3 shows the acceleration rate of the piston over one full crank rotation. The velocity of the piston is a consequence of the rate and duration of acceleration of the piston. Acceleration (A) is measured in meters per second per second.

Graph 3 Piston Acceleration Note from the graph

·         the maximum acceleration rate is achieved immediately before and after TDC, and

·         the acceleration rate goes from positive to negative at around 74 degrees after TDC. This is immediately after the highest velocity reading is obtained and the piston is now slowing (and reverses at 286 degrees (ie 74 Before TDC)).

Table 2 shows the piston acceleration (A) for the 1.6 litre and 1.8 litre engines at 500 RPM intervals and compares that to the 1,000 RPM index value. The acceleration rate is recorded at the commencement of movement at TDC, which is the greatest acceleration rate at any time in the crank rotation. The piston acceleration of a 1.8 litre engine is a constant 1.02082 times the 1.6 litre because of the increased stroke length. Notice how a doubling of the RPM creates a squaring of the acceleration rate. The piston of a 1.6 litre Mazda MX-5 engine accelerated at 603 meters per second per second at 1,000 rpm. This acceleration rate is 4 times that at 2,000 RPM, 16 times at 4,000 RPM and 36 times at 6,000 RPM. In the 1.8 litre the g forces at 7,200 RPM are 64 times that at idle (900 RPM).

Table 2 Maximum Piston Acceleration Rate at Modelled RPM in Meters Per Second Per Second

 RPM Mazda MX-5 1.6 Litre 83.6 mm Stroke Mazda MX-5 1.8 Litre 85 mm Stroke Relationship  to Acceleration at 1,000 RPM 1,000 603 615 1.00 1,500 1,356 1,384 2.25 2,000 2,410 2,460 4.00 2,500 3,766 3,844 6.25 3,000 5,423 5,536 9.00 3,500 7,381 7,535 12.25 4,000 9,641 9,842 16.00 4,500 12,202 12,456 20.25 5,000 15,064 15,378 25.00 5,500 18,227 18,607 30.25 6,000 21,692 22,144 36.00 6,500 25,458 25,988 42.25 7,000 29,526 30,140 49.00 7,500 33,894 34,600 56.25 8,000 38,564 39,367 64.00

Note that the acceleration rate at 6,000 rpm is 4 times that at 3,000 RPM

and a 20% increase in RPM from 6,000 to 7,200 results in a 44% increase in the acceleration rate of the piston.

There are 101.9716 gravitational acceleration units "g forces" per 1,000 meters per second per second acceleration. Therefore, at 6,000 RPM, when the 1.8 litre engine's piston has a maximum acceleration rate of 22,144 m/s/s there are 2,258.06 "g forces" exerted. Small wonder that lightweight pistons and con-rods are the order of the day on high performance engines.

Part 3 - Location and Displacement of the Piston at Crank Angles.

Each piston in the 1.8 litre engine strokes 85mm, and with an 82mm bore, displaces a volume of 459.902 cc's. Because of the geometry of the rod and crank arm the movement of the piston in the cylinder is bell shaped when plotted on a graph.

Graph 4 shows the graph shows the location of the piston in mm from TDC at various crank angles (scaled to the left side of the graph) and the displacement at crank angles (scaled to the right side).

Graph 4 Piston Location and Displacement Table 3 shows the location of the top of the piston in the bore for the 20 degrees of crank rotation approaching Top Dead Centre (TDC).

Table 3 Location of Piston Top for 20 Degrees of Rotation Before TDC

 Crank Angle Before TDC MX-5 1.6 litre piston location in mm from TDC MX-5 1.8 litre piston location in mm from TDC Crank Angle Before TDC MX-5 1.6 litre piston location in mm from TDC MX-5 1.8 litre piston location in mm from TDC -20 3.2920 3.3604 -10 0.8334 0.8507 -19 2.9759 3.0377 -9 0.6756 0.6896 -18 2.6750 2.7306 -8 0.5342 0.5453 -17 2.3896 2.4392 -7 0.4092 0.4178 -16 2.1196 2.1637 -6 0.3008 0.3071 -15 1.8654 1.9041 -5 0.2090 0.2134 -14 1.6269 1.6607 -4 0.1338 0.1366 -13 1.4044 1.4336 -3 0.0753 0.0769 -12 1.1979 1.2228 -2 0.0335 0.0342 -11 1.0075 1.0285 -1 0.0084 0.0085 TDC 0.0000 0.0000

Notice the difference between the 1.6 litre and 1.8 litre models. This is because the 1.8 litre (85mm) has a longer stroke than the 1.6 litre (83.6mm).

The shaded lines at -10 and -14 degrees before TDC show the location of the piston's top at the popular timing point alternatives for the 1.6 litre models. A 4-degree advance in the timing is a 40% change in degrees BTDC and a 95.21% increase in the distance (in mm) travelled by the piston from the timing point and TDC.

The effect this has on the engines' performance is discussed in Techno's “Know your car Series #10: What Happens When the Timing Is Advanced".

The Definitive Guide to Piston and Crank Movement.

Table 4 shows the comparisons between the 1.6 litre and the 1.8 litre engines running at 6,000 rpm.

Table 4 Comparison of 1.6 litre and 1.8 Litre Engines at 6,000 RPM

 Data 1.6 Litre Ref Value 1.8 Litre Revolutions per minute at 6,000 RPM 6,000 T 6,000 Revolutions Per Second at 6,000 RPM 100.00 T/60 100.00 Degrees rotation per second at 6,000 RPM 36,000 360*(T/60) 36,000 Milliseconds Per Revolution at 6,000 RPM 10.00 1/(T/60)*1000 10.00 Milliseconds per degree at 6,000 RPM 0.02778 1/(360*T/60)*1000 0.02778 Maximum Piston Acceleration (Positive) (Gmax meters per second squared) at 6,000 RPM 21,692 ((T^2*S)/182.38)*(1+1/(2*(R/S)))/1000 22,144 Maximum Piston Deceleration in meters per second squared at 6,000 RPM - 11,597 A Plotted - 11,761 Time for Piston to Move from Timing mark (10) to TDC (Milliseconds) at 6,000 RPM 0.2778 10/(360*T/60)*1000 0.2778 Piston travel from Timing Point to TDC (mm) at 6,000 RPM 1.62692 W when K = 360 minus W when K = 350 1.66074 Mean Piston Speed Meters Per Second at 6,000 RPM 16.720 ((S/1000)*T*2)/60 17.000 Mean Piston Speed (Kph) at 6,000 RPM 60.192 ((S/1000)*T*2)*60 /1000 61.200 Maximum Piston Speed at 6,000 RPM (Meters per Second) 27.542 V 28.047 Maximum Piston Speed at 6,000 RPM (Kms per hr) 99.15 V*60*60 /1000 100.97

Summary

Congratulations, you have made it this far through this article, well done. Now you should have an understanding of the movements of a piston, con-rod and crank. The forces that act on the components are indeed impressive and help us to understand just how complex engines are, and the engineering developments required to design and build reliable engines that last for 300,000 Kms or more.

Safe journey

Rob (Techno) Spargo

January 2004

Formulae for Calculations in Excel Format

The tables above provide the explanations for the lettered variables

To calculate the velocity (V) of the piston when the crank is at a particular degree of rotation (K) ATDC and at a given RPM (T) the formula is:

Piston Velocity at a given crank angle V=(T*ATAN(1)/7.5)*((S/2)/1000)*SIN(((ATAN(1)/7.5)*K/6))*(1+COS(((ATAN(1)/7.5)*K/6))/(SQRT(((R/(S/2))^2)-SIN(((ATAN(1)/7.5)*K/6))*SIN(((ATAN(1)/7.5)*K/6)))))

The formula for calculating the acceleration rate (A) of a piston at a given crank angle after TDC (K) is

Acceleration in Meters per second per second at K =((T*ATAN(1)/7.5)^2)*((S/2)/1000)* ((1-COS(4*((ATAN(1)/7.5)*K/6)))/(8*(SQRT(((R/(S/2))^2)-SIN(((ATAN(1)/7.5)*K/6))*SIN(((ATAN(1)/7.5)*K/6)) ))^ 3)+COS(2*((ATAN(1)/7.5)*K/6))/(SQRT(((R/(S/2))^2)-SIN(((ATAN(1)/7.5)*K/6))*SIN(((ATAN(1)/7.5)*K/6)) ))+COS(((ATAN(1)/7.5)*K/6)))

The formula for the location (W) and displacement (D) of the piston for a given crank angle (K) after TDC is:

Displacement at K = D = (+PI()*(B^2)*W/1000)/4

[i] Displacement in cc's = D = PI()*(B^2)*S/1000