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Does Length Matter? Part Deux
By Khiem Dinh
Khiem Dinh is an engineer for Honeywell Turbo Technologies at the time of this writing. All statements and opinions expressed by Khiem Dinh are solely those of Khiem Dinh and not reflective of Honeywell Turbo Technologies.
In part I, we looked at the effect of connecting rod length on dwell time of the piston around top dead center. This was determined by plotting the position of the piston in relation to crank angle. Using some physics and calculus, we can also determine the velocity and acceleration of the piston in crank angle. If we assume an angular velocity (or revolutions per minute), we can determine the velocity and acceleration in the time domain which is what everyone is more familiar with.
The original equation I derived was not so friendly for taking a derivative in order to calculate velocity and acceleration. While browsing around the web, I can upon an equation for piston position and its derivates (velocity and acceleration) explaining piston motion on Wikipedia.
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| Here’s the new equation to determine piston position. It has the exact same result as my previous equation, but it is easier to take derivatives of this one. |
Taking the derivative of an equation for variable X gives you a new equation that tells you the rate of change of variable X. In our first case, variable X is position. What is the rate of change of position? Velocity. For example, let’s say you’re on the football field and doing a 40 yard dash. You start at the goal line and get to the 40 yard line in 5.0 seconds. That gives you an average velocity of 0.8 yards per second. If your velocity is zero, that means you are standing still and not moving anywhere, therefore your change in position is also zero. So the first derivative of a position equation gives you the velocity equation.
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| Taken from Wikipedia, here are the equations for piston position, velocity, and acceleration with respect to crank angle. |
What is the rate of change of velocity? Acceleration. If you’re driving at 100mph towards a cliff with zero acceleration (in this case, you’d probably want negative acceleration), then you will continue to go 100mph until you fly off the cliff. If you do not want to end up like Thelma and Louise, you will probably want to hit the brakes so that you have negative acceleration which will change you velocity from 100mph to zero and keep you alive. So taking the first derivative of the velocity equation gives you the acceleration equation. Consequently, the acceleration equation is the second derivative of the position equation. Cool how math works out right?
So far, our equations have been with respect to crank angle. To make it more useful to us, we want to change it to the time domain. If we assume an engine rpm, we can calculate an angular velocity in radians per second and calculate velocity and acceleration in meter/second and meters/second^2 respectively.
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| The funny looking ‘w’ is omega and stands for angular velocity in radians per second. There are 2 x pi radians per revolution, dividing by 60 is to convert revolutions per minute to revolutions per second. |
