Graphical Method for Determining Piston Acceleration with a Finite Connecting Rod Length
Abstract Generated abstract
This paper addresses the determination of piston acceleration in a steam engine when the connecting rod has finite length, a quantity needed for estimating inertia forces of reciprocating masses. Starting from the standard analytical expression for acceleration, it develops a geometric construction that represents the crank velocity by the crank radius and incorporates the ratio of crank radius to connecting rod length as an auxiliary offset. The proposed method allows the acceleration for any crank position to be found directly from a diagram, without first calculating tabulated values or constructing the full acceleration curve. The construction also accounts for crank angles beyond 90 degrees, showing how the sign change corresponds to the transition from acceleration to retardation.
Full Text
PROCEEDINGS
of the Tomsk Technological Institute
of Emperor Nicholas II
Vol. 19. 1910. No. 3.
II.
A. V. Ugarov.
Graphical Method for Determining Piston Acceleration with a Finite Connecting Rod Length.
1—4.
GRAPHICAL METHOD FOR DETERMINING PISTON ACCELERATION
WITH A FINITE CONNECTING ROD LENGTH
When taking into account the inertia forces of the masses of a steam engine moving together with the piston (the rod, crosshead, and part of the connecting rod), it is necessary to know the acceleration of these masses for a given angle of rotation of the crank from its dead-center position.
Analytically, the acceleration of the piston is expressed, as is known, by the formula:
\[ p=\frac{v^{2}}{R}\left(\cos\alpha+\frac{R}{L}\cos 2\alpha\right), \]
where \(p\) is the acceleration of the piston, \(L\) is the length of the connecting rod, \(R\) is the radius of the crank, \(\alpha\) is the angle of its rotation, and \(v\) is the uniform speed of rotation of the crank pin.
In order to judge the variation of acceleration over the entire path of the piston stroke between its dead-center positions, the so-called acceleration curve is usually drawn, by plotting, at each position of the piston, an ordinate proportional to its acceleration at that moment.
The acceleration curve may be obtained by the following methods*:
a) by means of a table of cosines, or else a table of piston paths;
b) by finding seven characteristic points of the curve;
c) by the method of drawing tangents; and
d) by the shortened method—the acceleration curve is taken to be a parabola**.
All these methods make it possible to find graphically the acceleration of the piston for an arbitrary angle of rotation of the crank only after the curve itself has been constructed.
* Radinger—Ueber Dampfmaschinen mit hoher Kolbengeschwindigkeit. 1892.
* Tolle—Regelung d-r Kraftmaschinen*. 1905.
Presented below is a method by which one can graphically find the acceleration for any position of the crank, without performing calculations or preliminarily drawing the entire curve.
Let us transform the formula given above, using the following notation: let the crank velocity \(v\), on some scale, be represented by a segment equal to the crank radius \(R\), and let the ratio
\[ \frac{R}{L}=\mu, \]
be some proper fraction.
Then, instead of the formula:
\[ p=\frac{v^{2}}{R}\left(\cos\alpha+\frac{R}{L}\cos2\alpha\right) \]
we obtain
\[ p=R(\cos\alpha+\mu\cos2\alpha). \]
Since \(\cos2\alpha=2\cos^{2}\alpha-1\), we may write
\[ p=R(\cos\alpha+2\mu\cos^{2}\alpha-\mu), \]
or finally
\[ p=R\cos\alpha+2\mu R\cos^{2}\alpha-\mu R. \]
This expression is easy to construct for any angle \(\alpha\).
Take (see the drawing) the crank circle described about the center \(O\), and lay off on its diameter from the center, in the direction opposite to the position of the slider (to the right in the drawing), the segment \(OO'=\mu R\)—for machines of the normal type \(\mu=\frac{1}{5}\). From the point \(O'\), as from a center, describe two circles: one with radius \(\mu R\), and the other with radius \(R\). This latter auxiliary circle, evidently, ob—
possesses the property that all its points are shifted parallel to the diameter by a distance \(\mu R\) from the crank circle to the right.
If now, for an arbitrary position of the crank \(OA\), we draw the line \(AD\) parallel to the line of dead points (the diameter of the crank circle) until it meets the perpendicular \(FD\), drawn through the point of intersection of the direction of the chosen crank radius with the circle of radius \(\mu R\), then the segment \(ED\) will represent the required acceleration of the piston on the scale in which the radius of the crank circle represents the speed of rotation \(v\) of the crank pin.
Let us drop from the point \(A\) a perpendicular to the diameter. From the drawing we have
\[ ED = AD - AE; \]
but
\[ AE = \mu R,\quad AD = BC = BO + OC, \]
where \(BO = R\cos\alpha\), and the segment \(OC = OF\cos\alpha\).
In turn, \(OF = OG\cos\alpha\); consequently
\[ OC = OG\cos^2\alpha = 2\mu R\cos^2\alpha, \]
whence finally
\[ ED = R\cos\alpha + 2\mu R\cos^2\alpha - \mu R, \]
which was required to be proved.
For angles of rotation greater than \(90^\circ\) we may take the designation \(\alpha = 180^\circ - \alpha_1\).
Since \(\cos(180^\circ-\alpha_1) = -\cos\alpha_1\), our basic formula takes the form:
\[ p = -R\cos\alpha_1 + 2\mu R\cos^2\alpha_1 - \mu R. \]
Taking the minus sign outside the parentheses, we have:
\[ p = -\left(R\cos\alpha_1 - 2\mu R\cos^2\alpha_1 + \mu R\right), \]
which in the drawing is expressed by the segment \(LN\).
Indeed:
\[ LN = LH + HN. \]
According to the drawing,
\[ LH = OQ - OP = R\cos\alpha_1 - 2\mu R\cos^2\alpha_1;\quad HN = \mu R, \]
whence
\[ LN = R\cos\alpha_1 - 2\mu R\cos^2\alpha_1 + \mu R. \]
The minus sign before the parentheses in the expression for \(p\) shows that the acceleration has changed into retardation, after passing through its zero value.