Interactive Dynamics
Kinetic Analysis of a Connecting Rod and Piston in an Internal Combustion Engine

Two-Stroke Spark
Ignition Engine
Figure 1.
The figure at the left shows a cut-away view of a two-stroke, single-cylinder spark ignition engine. You can see the spark plug at the top, the piston below the spark plug, the connecting rod connected to the piston, and the crank connected to the bottom of the connecting rod. The large anvil-shaped thing on the crank is a counterweight to balance the crank.

Introduction

In the first part of this activity, you did a complete kinematic and limit-case analysis of the crank, connecting rod, and piston in a typical internal combustion engine. Your goal in this analysis was to:

Find the acceleration of the piston, acceleration of the mass center of the connecting rod, and the angular acceleration of the connection rod, all as a function of the crank angle.
Find the maximum angular acceleration of the connecting rod (and, hopefully, at what crank angle it occurs).
Find the maximum acceleration of the piston (and at what crank angle it occurs).
Find the maximum acceleration of the mass center of the connecting rod (and the corresponding crank angle).

In today's activity, you will continue your analysis of the slider-crank mechanism you began last time by finding all of the important dynamic loads on the piston and connecting rod.

Kinetic Analysis of the Slider-Crank

As with the last activity, all of the following discussion refers to those quantities defined in Figures 2 and 3.

Figure 2. Schematic of the slider-crank mechanism to be analyzed.

Figure 3. Another schematic of the slider-crank mechanism showing the mass centers of the crank and the connecting rod.

As we said last time, the slider-crank mechanism shown is driven by the combustion process that occurs above the piston at C. This combustion process generates a time-dependent force (one could also think of it as crank-angle dependent) P which drives the piston down. The motion of the piston drives the crankshaft at A around by way of the connecting rod BC. Unlike the last activity, you will now need to know the mass properties of each of the components in the slider-crank mechanism. You should use the physical parameters given in the following table for each of the three components:

Component	Mass	Centroidal Mass Moment of Inertia
Crank at A	2.5 kg	10,000 kg-mm²
Piston at C	0.5 kg	150 kg-mm²
Connecting Rod BC	0.4 kg	6,000 kg-mm²

Now that you have done the entire kinematic analysis and have the mass properties, you can go ahead and find the following forces acting on the system (not necessarily in any particular order):

The total force on the pin (B) connecting the crank to the connecting rod (as a function of crank angle).
The angle of the force on the pin at B relative to the orientation of the connecting rod (as a function of crank angle).
The total force on the pin (C) connecting the connecting rod to the piston (as a function of crank angle).
The angle of the force on the pin at C relative to the orientation of the connecting rod (as a function of crank angle).
The output torque delivered to the crank/crankshaft (again, as a function of crank angle).

Note that as part of the design process, you usually want to find the maximum forces that occur. Therefore, it would be nice to know the maximum value attained by each of these forces. Consequently, your next task is to use Mathematica to plot all of the forces/torques and angles you found above as a function of crank angle.

The force P on the top of the piston is assumed to be known. We will model P as a piecewise constant force that is equal to 3000 N for q between 0 and p (the explosion part) and is equal to 750 for q between p and 2p (the part of the cycle during which exhaust gases are ejected). At 2p, the force simply repeats. See Figure 4 for a plot of P versus q for q going from 0 to 2p.

Figure 4. The combustion force P.

A piecewise continuous force such as the one shown in Figure 4 can be difficult to deal with computationally so we will use the first 7 non-zero terms in the Fourier series representation for P as given by the Mathematica code below.

Needs["Calculus`FourierTransform`"] f = 3000 UnitStep[q] - 2250 UnitStep[q - p]; Plot[f, {q,0,2p}, PlotRange->{0,3000}] fs = FourierTrigSeries[f, {q,0,2p}, 11]
Plot[fs, {q,0,2p}, PlotRange->{0,3300}]

Figure 5 is plot of the first four non-zero terms in the Fourier series representation. Notice how the Fourier series repeats every 2p in the same way we want P to.

Figure 5. The combustion force P and the first four non-zero terms in its Fourier series.

Just for fun, we also show want to show you how the Fourier series converges (non-uniformly, we might add) to P as you add more terms in the series. See Figure 6.

Figure 6. The first 11 non-zero terms in P's Fourier series (along with P).

Activity Report

You should write one combined report for both parts of the slider-crank activity. In your report, you should comment on the following items (in no particular order):

Compare the maximum acceleration of the piston with the acceleration of gravity and with your estimate of the maximum acceleration of a fast automobile. Also compare with the maximum number of "g's" that a human can withstand.
Looking at your plot of output torque versus crank angle and given that we are assuming that the crank rotates at a constant angular velocity when in steady-state, why is the output torque not constant? In addition, what features might you add to the engine so that you obtain a nearly constant torque output for a constant angular velocity of the crank?
What is the area under your output torque versus crank angle curve for one complete cycle of the crank? Use that area and the time it takes for one rotation of the crank to compute the horsepower of your engine. Finally, compare the horsepower of your engine (see Section 14.5 of Pytel and Kiusalaas) with that of the 3 engines of the Titanic (you will find this on the web if you look hard). What is the horsepower of a typical automobile engine?
How does the maximum total force on each of the pins compare with the weight of a typical person?

This activity report will be due Tuesday, April 27.

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Prepared by Gary L. Gray and Francesco Costanzo.

This page was last modified on Thursday, April 22, 1999 at 18:20:58.