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From: Dave Baker
Newsgroups: rec.crafts.metalworking
Subject: Re: Maximum piston acceleration? OT
Date: 30 Sep 1999 08:16:58 GMT

OK Peter - a while back you asked if there were any takers. Well I nearly
"took" but I wanted to be absolutely sure of my maths before I answered.
Several hours of calculations later I'm not only going to take but going to
prove what both you and 99.9% of the world's engineers believe is a given -
written in stone and unchallengeable - is wrong ! Not often someone like me
gets to do that sort of thing so I'll make the most of it :)

Let's assume for arguments sake that we classify acceleration away from TDC as
positive and deceleration towards BDC as negative. You believe that maximum
positive acceleration is at TDC and minimum acceleration (or maximum negative
acceleration) is at BDC. Your simplified formula you agree is not correct for
any angle other than 0 or 180 degrees ATDC but you believe that it is correct
at those points and that the results it gives at those points (as does the full
equation) are maxima.

They aren't - maximum acceleration is indeed at TDC and this applies to any
ratio of conrod to stroke length. Minimum acceleration IS NOT at BDC except in
the situation that conrod length is infinite and the piston motion becomes
truly sinusoidal at which point the values of acceleration (ignoring the sign)
are the same at TDC and BDC. For real world engines the minimum acceleration
comes well before BDC and in the case of the example given (stroke 3.236", rod
4.930") this point occurs at 138.96 degrees ATDC - i.e. over 40 degrees before

Many years ago I wanted to examine these sorts of issue. I wrote a spreadsheet
to calculate the position of the piston every degree after TDC. Over time I
added the calculations of piston velocity and acceleration, then included
piston and rod weights to enable me to see the forces acting on the rod bolt.
Later I included routines to calculate rod bolt size required at different rpm.

Finally I linked into cam profile data files and to see graphically the
relationship of the valve and piston positions to enable me to calculate
required valve cutout depths and the effects of advancing or retarding cam
timing on piston to valve clearance. In effect I can build a top end in a
computer simulation and check that the amounts to be skimmed off head or block,
cam timing , cam choice etc don't cause me a piston/valve clearance problem and
if so how much to machine to correct it. Anyway, not really relevant to this

As I used these routines more and more I added graphs, one of which is piston
acceleration. This is when I first became aware that the minimum acceleration
is not at BDC - this struck me as so unlikely based on accepted wisdom that I
ignored it for too long. Last night I checked through all of the maths in the
programme and reworked one example with trig manually to be sure. The programme
doesn't do pure calculus - to arrive at piston speed it just deducts two
adjacent piston positions and divides by time. This throws the true position of
the velocity shown out by one degree. A similar step to get acceleration
introduces another degree error. To overcome this I reworked the programme to
run not just every one degree but in steps I can pre select. By using steps of
0.1 or 0.01 degrees I can zero in on the true position of an event I want to
study more. I hope the above is clear.

For the example given the results are this:

Max acceleration      TDC                    29,331 m/s^2
Min acceleration    138.96 deg ATDC   -15,428 m/s^2
accel at BDC                                     -14,836 m/s^2
Peak piston velocity    73.4 deg ATDC

I suggest you check the following data - if you agree that it is right then the
rest follows. Here are the piston positions at various times in inches below
TDC every 30 degrees as shown by the programme.

0       0
30     0.284
60     1.012
90     1.891
120   2.630
150   3.086
180   3.236

Forget your equations - grab a pocket calculator and draw out the trig - it
only takes 3 calcs to arrive at each answer. If the positions are right then so
are the speeds and accelerations. If you graph out the acceleration over a
complete cycle the curve looks like a very slight "W" - peak positive values at
TDC on either side, minimum values somewhat either side of BDC and rising a bit
at BDC before falling again.

Now for the explanation - once you accept the facts and start to think about it
you can reason it out. Hint - consider as someone posted earlier a conrod the
same length as the crank throw - i.e. half the stroke length. If you retain the
assumption that the small end pin is constrained to the bore centreline then
what does this do to piston motion? Once you see what is happening here you can
extrapolate to a more real world length situation. So anyone care to explain
back to me why maximum negative piston acceleration does not occur at BDC?

Any takers? <g>

Dave Baker at Puma Race Engines (London - England)  - specialist cylinder head
work, flow development and engine blueprinting. Web page at

From: Dave Baker
Newsgroups: rec.crafts.metalworking
Subject: Re: Maximum piston acceleration? OT
Date: 30 Sep 1999 23:48:03 GMT

>From: Robert Bissett
>At TDC when the crank starts to rotate the the throw distance relative to
>the crank centerline starts to get smaller. At the same time the crank
>end of the rod is moved sideways making it relatively shorter. Both
>actions add in the same direction to get the high acceleration of the
>piston. Nothing new here.
>Now consider what happens as the crank passes BDC. (It's easier to
>visualize after BDC, but it is the same approaching BDC). As the crank
>rotates from BDC the throw distance gets shorter at exactly the same rate
>as leaving TDC and starts the upward acceleration of the piston. Again
>the crank end of the rod is moved sideways making it relatively shorter,
>but this time the relative shortening of the rod subtracts from the total
>upward motion reducing the acceleration.
>As stated by Dave an infinitely long rod would have equal piston
>acceleration at TDC and BDC because the shortening of the rod would not
>have any effect. So for any real world engine there is some point after
>BDC where the shortening throw distance overcomes the opposite shortening
>effect of the rod and achieves the max acceleration. A longer rod moves
>that point towards BDC, a shorter rod moves it away.
> OK?

It is quite difficult to mentally consider motions in terms of acceleration
being as it is a second derivative of distance (well it is difficult for me
anyway). I think I understand what you are saying here - you are using the term
"throw" to mean vertical distance from crank centreline to big end centre if I
follow correctly - normally this term is used for the radius of rotation of the
crank pin. However, your explanation, I believe,  applies to piston velocity
not piston acceleration. The explanation of the changing position of maximum
negative acceleration is complex but here's one way of considering it.

Imagine, as I previously posted, the theoretical situation where rod length is
equal to crank radius. It is actually possible to construct an engine like this
although it wouldn't run too well :)

The piston moves away from TDC normally, starting at maximum acceleration which
falls as the crank rotates. At 90 degrees past TDC the rod lies horizontally on
the same plane as the crank arm. The piston pin centre, being constrained to
the bore centreline now lies in the same position as the crank centreline. The
piston has stopped moving and from this position until 270 degrees ATDC the rod
and crank arm move as a unit with no further piston motion. Given that the
piston has stopped moving by 90 degrees ATDC it is a given that maximum
deceleration has already occured so proving that this point cannot be BDC for
all crank geometries.

In fact the motion in the above situation is very anomalous. The piston is
still acelerating in the downward plane right up to 90 degrees ATDC. It reaches
both maximum speed and zero speed at the same time with deceleration being
infinitely large. A tad hard on big end bearings I suspect. Any increase in rod
length, however small, to a value greater than 0.5 x stroke and the point of
maximum piston speed drops back to around the 70 degree ATDC mark -  from which
point it gradually moves round to 90 degrees ATDC again as rod length increases
towards infinity.

Now consider the situation as rod length is gradually increased. The piston
will now continue to move downwards right up until BDC although the movement
after 90 degrees ATDC is still very slight - but increasing as rod length
increases. So gradually the point of maximum deceleration must occur later in
the cycle and eventually tend towards 180 degrees ATDC.

To understand how this position of maximum deceleration changes with changing
rod/stroke ratio I ran a series of calculations though the computer. Here are
the results to the nearest degree ATDC.

RATIO                 DEGREES ATDC
0.5                          90
0.6                          94
0.7                          98
0.8                          103
0.9                          108
1.0                          113
1.1                          117
1.2                          122
1.3                          127
1.4                          132
1.5                          138
1.6                          144
1.7                          151
1.8                          160

Now I have to use finer increments as the motion becomes more exaggerated.

1.85                         166
1.86                         168
1.87                         170
1.88                         172
1.89                         175
1.90                         180

The numbers are behaving strangly here - most functions which approach a limit
do so asymptotically. In this geometry the function literally crashes into the
180 degree barrier at the point at which rod/stroke ratio reaches 1.90. From
then on there can be no further change. The function behaves normally close to
the 90 degrees ATDC point when the ratio is 0.5 - in other words, as rod length
increases, the point of max deceleration moves slowly away from 90 degrees ATDC
but with increasing rapidity - a normal asymptotic function.

So what is special about rod/stroke ratios in the region of 1.85 to 1.90? That
I still don't know but if you examine the consensus of experiment as to the
best ratio for optimum engine performance what ratio do you think you find ????
Any takers :)

I wonder if anyone has really studied the effects of changes in rod length on
engine performance in terms of the changing position of maximum deceleration? I
suspect not. There seems to be more here than meets the eye though.

I hope the original poster is appropriately sorry for opening up what turned
out to be such a major can of worms <g>

So Pete - what do you have to say to all that then ?

Dave Baker at Puma Race Engines (London - England)  - specialist cylinder head
work, flow development and engine blueprinting. Web page at

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