On the Problem of Balancing Piston Engines with Articulated Connecting Rods
Abstract Generated abstract
This paper develops a graphical and analytical method for evaluating the balance of single-shaft, multi-cylinder, multi-row piston engines, including engines with articulated connecting rods under specified geometric conditions. It represents first- and second-order inertia forces of reciprocating masses as sums of rotating vectors, allowing V-type, W-type, radial, and multi-throw crankshaft arrangements to be reduced to combinations of simpler vector schemes. The method gives constructions and formulas for resultant forces, elliptical force loci, and associated unbalanced moments, providing a practical framework for assessing balance without relying solely on cumbersome direct analytical calculations.
Full Text
On the Problem of Balancing Piston Engines with Articulated Connecting Rods
L. V. Nechaev
(Presented by the scientific seminar of the Department of “Internal Combustion Engines”)
In designing new internal combustion engines for automobiles, tractors, etc., one of the important factors ensuring minimum overall dimensions and the best arrangement of the engine is the rational choice of the number and disposition of the cylinders. As a rule, in making this choice an analysis is carried out of the balancing of the engine with respect to the inertia forces of the rotating and reciprocating masses.
The analytical method of balancing analysis usually used in practice, when applied to multi-cylinder multi-row engines, often proves to be very cumbersome and labor-intensive; and graphical methods, which are more visual and less labor-intensive, have been developed in a form convenient for practice only for in-line multi-cylinder and single-row radial engines.
In the present article a graphical-analytical method is proposed for analyzing the balancing of single-shaft multi-cylinder multi-row engines. This method is applicable both to engines with ordinary (central) connecting rods and to engines with articulated connecting rods, provided that
\[ \gamma_l=\gamma. \]
Here:
\(\gamma_l\)—the angle between the axis of the master connecting rod and the line joining the centers of the lower heads of the central and articulated connecting rods,
\(\gamma\)—the angle between the axes of the cylinders.
The method described below is based on the principle of graphically representing the inertia forces of the first order \(P_{\mathrm{I}}\) and of the second order \(P_{\mathrm{II}}\) of the reciprocating masses of the crank-connecting-rod mechanism of each cylinder as projections, on the cylinder axis, of vectors whose magnitudes are respectively equal to \(P_{\mathrm{I\,max}}\) or \(P_{\mathrm{II\,max}}\), and which rotate with angular velocity \(\omega\) (for \(P_{\mathrm{I\,max}}\)) and \(2\omega\) (for \(P_{\mathrm{II\,max}}\)). Here \(\omega\) is the angular velocity of rotation of the crankshaft.
In the general case, for a single-cylinder engine the magnitudes of the forces \(P_{\mathrm{I}}\) and \(P_{\mathrm{II}}\) are determined from the known expressions:
a) for the main cylinder and for any side cylinder with a central connecting rod
\[ P_{\mathrm{I\,cyl}}=P_{\mathrm{I\,cyl.\,max}}\cdot \cos \alpha; \]
\[ P_{\mathrm{II\,gl}}=P_{\mathrm{II\,gl\,max}}\cdot \cos 2\alpha, \]
where
\[ P_{\mathrm{I\,gl\,max}}=m_{2\mathrm{\,gl}}\cdot R\cdot \omega^2; \]
\[ P_{\mathrm{II\,gl\,max}}=\mu\cdot m_{2\mathrm{\,gl}}\cdot R\cdot \omega^2; \]
b) for a side cylinder with an articulated connecting rod, under the condition \(\gamma_i=\gamma\),
\[ P_{\mathrm{I\,pr}}=P_{\mathrm{I\,pr\,max}}\cdot \cos \alpha_i; \]
\[ P_{\mathrm{II\,pr}}=P_{\mathrm{II\,pr\,max}}\cdot \cos(2\alpha_i-\Theta), \]
where
\[ P_{\mathrm{I\,pr}}=m_{2\mathrm{\,pr}}\cdot R\omega^2; \]
\[ P_{\mathrm{II\,pr}}=F\cdot m_{2\mathrm{\,pr}}\cdot R\omega^2. \]
In these expressions, \(m_{2\mathrm{\,gl}}\) and \(m_{2\mathrm{\,pr}}\) are the masses, reduced to the axis of the piston pin, moving in translation in the crank-and-connecting-rod mechanisms of the main and side cylinders, respectively.
\(R\) is the crank radius,
\(L\) is the length of the main connecting rod,
\[ \mu=\frac{R}{L}, \]
\(\alpha\) and \(\alpha_i\) are the angles of rotation of the crank, measured respectively from the axes of the main and side cylinders,
\(F\) and \(\Theta\) are a dimensionless coefficient and an angle depending on the geometrical parameters of the main and articulated connecting rods and determined from formulas given, for example, in [1].
The formulas and propositions proposed below will be valid in any case if: a) the numbering of cylinders working on one crank is carried out in the direction of rotation of the crankshaft; b) the angle between the axes of the cylinders \(\gamma_i\) (\(\gamma_{12}\), \(\gamma_{13}\), etc.) is counted from the main cylinder in the direction of rotation of the crankshaft; c) the crank-and-connecting-rod mechanism has no offset.
If the first two conditions are not observed, the proposed formulas must be rearranged accordingly.
1. Inertia forces of the first order in multicylinder engines with a single-throw shaft
The end of the vector of the unbalanced inertia force of the first order (as, indeed, of any other order) for a multicylinder engine with a single-throw shaft describes an ellipse. This is not difficult to verify by considering, for simplicity, a two-cylinder engine with a single-throw shaft. Indeed, the total unbalanced inertia force of the first order is equal to (Fig. 1):
\[ P_1=\sqrt{x^2+y^2}, \]
where \(x\) is the sum of the projections of the first-order inertia forces of both cylinders onto the \(X\) axis,
\(y\) is the same onto the \(Y\) axis.
\[ y=P_{11y}+P_{12y}=P_{\mathrm{I\,max}}\cdot \cos\alpha\cdot \cos\frac{\gamma}{2} +P_{\mathrm{I\,max}}\cdot \cos(\alpha-\gamma)\cdot \cos\frac{\gamma}{2}; \]
\[ x=P_{1x}+P_{1x}=-P_{1\max}\cdot \cos \alpha \cdot \sin \frac{\gamma}{2} +P_{1\max}\cdot \cos(\alpha-\gamma)\cdot \sin \frac{\gamma}{2}. \]
Here \(P_{1y}, P_{1x}, P_{2y}, P_{2x}\) are the projections of the instantaneous values of the inertia forces of the translationally moving masses of both cylinders onto the coordinate axes.
After simple trigonometric transformations, \(x\) and \(y\) can be brought to the form
Fig. 1
\[ y=a\cdot \cos\left(\alpha-\frac{\gamma}{2}\right), \tag{1} \]
\[ x=b\cdot \sin\left(\alpha-\frac{\gamma}{2}\right). \tag{2} \]
Equations (1) and (2) represent the equation of an ellipse in parametric form. The semiaxes of the ellipse are equal to
\[ a=2P_{1\max}\cdot \cos^2 \frac{\gamma}{2}, \tag{3} \]
\[ b=2P_{1\max}\cdot \sin^2 \frac{\gamma}{2}. \tag{4} \]
In these derivations, as in the following ones, it is assumed as usual [2] that
\[ m_{2\text{ gl}}=m_{2\text{ pr}}=m_2. \]
V-shaped engines
The magnitude of the unbalanced inertia force of the first order is determined by equations (1, 2, 3, 4). Substituting in these expressions
\[ \cos^2 \frac{\gamma}{2}\ \text{and}\ \sin^2 \frac{\gamma}{2} \]
by
\[ \frac{1+\cos\alpha}{2}\ \text{and}\ \frac{1-\cos\alpha}{2}, \]
one may write
\[ a=P_{\mathrm{I}\max}+P_{\mathrm{I}\max}\cdot\cos\gamma;\qquad b=P_{\mathrm{I}\max}-P_{\mathrm{I}\max}\cdot\cos\gamma, \]
i.e.,
\[ a=A+B;\qquad b=A-B, \]
where
\[ A=P_{\mathrm{I}\max};\qquad B=P_{\mathrm{I}\max}\cdot\cos\gamma. \]
The quantities \(A\), \(B\), \(a\), and \(b\) can be determined graphically. To do this, the segment \(ON=P_{\mathrm{I}\max}\) (or \(OD=ON=P_{\mathrm{I}\max}\)), directed along the coordinate axis, must be projected onto the axis of either cylinder, and from the point of intersection \(M\) (or \(K\)) a straight line must be drawn parallel to the axis of the second cylinder until it intersects the given coordinate axis at point \(E\) (or \(C\)). Then (Fig. 2)
\[ ON=A,\qquad OK=B,\qquad OE=a,\qquad OC=b. \]
Thus, the resultant inertia force in a V-engine can be determined in the following way:
It is necessary to construct the ellipse of the unbalanced force from the known semiaxes \(a\) and \(b\), determined analytically or graphically. From point \(N\) on a circle of radius \(r=P_{\mathrm{I}\max}\cdot q\) (\(q\) is the scale), corresponding to the position of the crank, drop a perpendicular to the axis of either cylinder, and from the point \(T\) of their intersection draw a straight line parallel to the axis of the other cylinder until it intersects the ellipse at point \(D\). The vector \(OD\) will be the vector of the total unbalanced first-order inertia force at the given instant (Fig. 3). Conversely, for any point of the ellipse the corresponding position of the crank can be found.
Fig. 2.
Analyzing expressions (5) and (6), one can arrive at the following conclusion. The unbalanced inertia force, varying according to the law of an ellipse, can be represented as a geometric sum of two forces, \(P_{\mathrm{I}\max}\) and \(P_{\mathrm{I}\max}\cdot\cos\gamma\), the vectors of which \(A\) and \(B\) rotate in opposite directions with angular velocity \(\omega\), the vector
\[ A=P_{\mathrm{I}\max} \]
being directed along the crank axis and rotating together with it. At the initial instant \((\alpha=0)\), vector \(A\) coincides with the axis of the main cylinder, and vector \(B\) with the axis of the side cylinder. The direction from the crankshaft toward the engine head is taken as the positive direction of the vectors.
In fact (Fig. 4)
\[ ON=P_{\mathrm{I}\max}=A, \]
where the line \(NN'\) is parallel to the \(X\)-axis; from triangle \(ONC\),
\[ NC=P_{1\max}\cdot \sin(\gamma-\alpha), \]
from triangle \(OBN\),
\[ OB=DC=P_{1\max}\cdot \cos\alpha. \]
It is then not difficult to prove that
\[ \varphi=180-\varphi_2-\varphi_3. \]
\[ ND=\sqrt{NC^2+DC^2-2NC\cdot DC\cdot \cos\varphi_1} = P_{1\max}\cdot \cos\gamma, \]
where
\[ \varphi_1=180-\gamma-90=90-\gamma. \]
The angle \(\varphi_2\) is determined by the sine theorem
Fig. 3.
Fig. 4.
\[ \sin\varphi_2=\frac{NC}{ND}\cdot \sin\varphi_1, \]
whence, after substituting the values of \(ND\) and \(NC\),
\[ \varphi_2=\gamma-\alpha, \]
i.e.,
\[ \varphi=180-\gamma+\alpha-90+\frac{\gamma}{2} =90-\frac{\gamma}{2}+\alpha \]
or, for \(\alpha=0\),
\[ \varphi=90-\frac{\gamma}{2}, \]
i.e., the vector \(ND=B\) is directed parallel to the axis of the side cylinder. With an increase in the angle \(\alpha\), the vector \(B\) must rotate counterclockwise.
counterclockwise with the same angular velocity \(\omega\) with which the vector \(A\) rotates clockwise.
W-shaped engines
A W-shaped engine may be considered as a combination of a V-shaped engine with bank angle \(2\gamma\) and an in-line engine (Fig. 5). In this case the middle cylinder is usually the main one. The angle \(\alpha\) is measured from its axis, which is the axis of symmetry. The vector of the resultant unbalanced inertia force of the first order \(P_1\) is the geometric sum of the vectors of the inertia forces of all cylinders. In this case the vector of the unbalanced force may also be represented as the geometric sum of two vectors rotating in opposite directions with angular velocity \(\omega\).
Fig. 5.
Indeed, the unbalanced resultant \(P_1\) of the first-order inertia forces can be determined by summing the inertia force \(P_{1\,\mathrm{main}}\) in the main cylinder and \(P_{1V}\), the resultant force for the pair of cylinders forming a V-arrangement with bank angle \(2\gamma\), i.e.
\[ P_1=\sqrt{x^2+y^2}, \]
where
\[ y=y_V+y_1 \]
\[ x=x_V, \]
\(y_V\) is the projection of the first-order resultant force of the pair of cylinders forming the V-arrangement on the \(Y\)-axis;
\(y_1=P_{1\max}\cdot\cos\alpha\) is the inertia force of the main cylinder;
\(x_V\) is the projection of the first-order resultant force of the pair of cylinders forming the V-arrangement on the \(X\)-axis.
On the basis of equations (1) and (2), with the angle \(\alpha\) measured from the axis coinciding with the axis of the main cylinder,
\[ y_V=2P_{1\max}\cdot\cos^2\gamma\cdot\cos\alpha, \]
\[ x_V=2P_{1\max}\cdot\sin^2\gamma\cdot\sin\alpha, \]
then
\[ y=2P_{1\max}\cdot\cos\alpha+P_{1\max}\cdot\cos 2\gamma\cdot\cos\alpha. \]
The expressions for \(x\) and \(y\) may be written in the form
\[ x=b\cdot\sin\alpha;\qquad y=a\cdot\cos\alpha; \]
here
\[ b=2P_{1\max}\cdot\sin^2\gamma;\qquad a=2P_{1\max}+P_{1\max}\cos 2\gamma. \]
Taking, as before,
\[ a=A+B\quad\text{and}\quad b=A-B, \]
we obtain
\[ a=A+B=2P_{1\max}+P_{1\max}\cdot \cos 2\gamma, \]
\[ b=A-B=2P_{1\max}\cdot \sin^2\gamma-P_{1\max}-P_{1\max}\cdot \cos 2\gamma, \]
whence
\[ A=1.5P_{1\max}; \tag{5} \]
\[ B=0.5P_{1\max}+P_{1\max}\cdot \cos 2\gamma. \tag{6} \]
Consequently, the total unbalanced inertia force for a W-type engine may be represented as the geometric sum of two forces: \(1.5P_{1\max}\) and \(0.5P_{1\max}+P_{1\max}\cdot \cos 2\gamma\), whose vectors \(A\) and \(B\) rotate in opposite directions with angular velocity \(\omega\). At the initial instant \((\alpha=0)\) both vectors are directed along the axis of the main cylinder. For any angle of rotation, vector \(A\) is directed along the crank.
The graphical determination of the magnitudes of vectors \(A\) and \(B\) is shown in Fig. 5. Along the coordinate axes the segments are laid off
\[ OF=OP=P_{1\max}; \]
\[ ON=OC=1.5P_{1\max}. \]
From point \(P\) a perpendicular is dropped to the axis of any side cylinder, and from the point of intersection \(K\) a line is drawn parallel to the axis of the second side cylinder until it intersects the \(X\)-axis at point \(M\). Then the segment \(MC\) is equal to
\[ MC=0.5P_{1\max}+P_{1\max}\cdot \cos 2\gamma=B. \]
Indeed,
\[ OK=P_{1\max}\cdot \sin\gamma; \]
\[ OM=2\cdot OK\cdot \cos(90-\gamma)=2P_{1\max}\cdot \sin^2\gamma; \]
\[ PM=OM-OP=2P_{1\max}\cdot \sin^2\gamma-P_{1\max}=-P_{1\max}\cdot \cos 2\gamma; \]
\[ MC=PC-PM=0.5P_{1\max}+P_{1\max}\cdot \cos 2\gamma=B. \]
Representing the unbalanced inertia force in a multicylinder engine as the geometric sum of two forces, whose vectors rotate in different directions, makes it possible to solve complex balancing problems by reducing a complicated engine scheme to a combination of several V- or V- and W-schemes.
Radial Engines
When analyzing the balance of a radial engine, it should be remembered that, depending on whether the number of cylinders in the star is even or odd, the initial assumptions must be different.
An engine with an odd number of cylinders may be represented as a series of V-type engines and one W-type engine. In this case, each V pair is formed by cylinders arranged symmetrically with respect to the axis of the main cylinder, which is the axis of symmetry of the engine. In this case, for each V pair and for the cylinders forming the W-scheme, the vectors \(A\) and \(B\) should be determined analytically or graphically. It is clear that all vectors \(A_1, A_2\), etc. are directed along the crankpin of the shaft in one direction, while the vectors \(B_1, B_2\), etc. are also along one straight—
...but in different directions (the direction is determined by the sign before \(B\) of each pair in the \(V\)- and \(W\)-schemes).
In the case of an odd number of cylinders in a star, the unbalanced inertia force of the first order is constant in magnitude and is equal to [1, 2]
\[ P_1=\frac{i}{2}P_{1\max}. \]
Geometrically, the force \(P_1\) can be represented in the form of a vector \(A\), directed along the crank and rotating with angular velocity \(\omega\) in the direction of rotation of the crankshaft.
In the case of an even number of cylinders in a star, for the analysis it is convenient to make one of the coordinate axes coincide with the axis of the master cylinder. Such a radial engine can be represented as a series of \(V\)-type engines and one opposed engine. Each \(V\)-pair is formed by cylinders symmetrically located with respect to the axis of the master cylinder. In this case the vectors \(A_1, A_2\), etc. are also directed along the crank in one direction, while the vectors \(B_1, B_2\), etc. lie on one straight line, but in opposite directions, depending on the sign before \(B\). It can be shown that
\[ A=2P_{1\max}+\frac{i-2}{2}P_{1\max}=\left(1+\frac{i}{2}\right)P_{1\max}, \tag{7} \]
\[ B=-P_{1\max}. \tag{8} \]
Here \(i\) is the number of cylinders.
Consequently, for a radial engine with an even number of cylinders the resultant unbalanced inertia force may be represented by the geometric sum of two forces \(\left(1+\frac{i}{2}\right)P_{1\max}\) and \(-P_{1\max}\), whose vectors \(A\) and \(B\) rotate in opposite directions with angular velocity \(\omega\). At the initial instant \((\alpha=0)\) the vectors are oppositely directed, the vector \(A\) coinciding with the direction of the crank.
The coordinates of the end of the vector may be found from the equations
\[ x=b\cdot\sin\alpha, \]
\[ y=a\cdot\cos\alpha, \]
where
\[ b=A-B;\quad a=A+B. \]
\(A\) and \(B\) are determined from expressions (7) and (8).
2. First-order inertia force in multi-row engines with a multi-throw crankshaft
In view of the fact that the unbalanced inertia force for a group of cylinders operating on one crank can be found by the geometric summation of two forces whose vectors are equal to \(A\) and \(B\), the unbalanced force in an engine with a multi-throw crankshaft can be represented as the geometric sum of the forces \(R_A\) and \(R_B\), i.e.,
\[ R_1=R_A+R_B, \]
where \(R_A\) is the geometric sum of the vectors \(A\) of all cranks,
\(R_B\) is the geometric sum of the vectors \(B\) of all cranks.
The vectors \(R_A\) and \(R_B\) rotate in opposite directions with angular velocity \(\omega\). The unbalanced inertia force in a multi-row, multi-cylinder engine also varies according to the law of an ellipse whose semiaxes are equal to
\[ a=R_A+R_B,\quad b=R_A-R_B. \]
It is not difficult to see that if \(R_A=0\), then also \(R_B=0\), i.e., the engine is balanced with respect to first-order inertia forces.
When \(R_A=\sum A_n\ne 0\), evidently \(R_B=\sum B_n\ne 0\) as well, and, consequently, for the analysis of balance it is necessary to determine the magnitudes and directions of the vectors \(R_A\) and \(R_B\) at \(\alpha=0\). In this case the vectors \(R_A\) and \(R_B\) are rotated with respect to the \(Y\)-axis by the angle
\[ \beta=\sum_{k=1}^{n}2^{k-n-1}\cdot \gamma_k . \]
Here: \(k\) is the ordinal number of the crank,
\[ \begin{aligned} n&\text{ is the number of cranks,}\\ \gamma_k&\text{ is the angle between the first crank and the given crank.} \end{aligned} \]
For example, for a four-crank shaft,
\[ \beta=\frac{\gamma_1}{16}+\frac{\gamma_2}{8}+\frac{\gamma_3}{4}+\frac{\gamma_4}{2}. \]
For the vector \(R_A\), the angle \(\beta\) is measured from the first crank in the direction of rotation, and for the vector \(R_B\), opposite to the rotation of the crankshaft. The method for determining the resultant force \(R_A\) or \(R_B\) is completely analogous to the determination of the resultant centrifugal force in a single-row multi-cylinder engine and presents no serious difficulties.
3. Second-order inertia forces in multi-row engines with a single-crank shaft
First let us consider the balance of engines with a single-crank shaft. The second-order inertia force for the main cylinder is equal to
\[ P_{II\,\mathrm{main}}=\psi m_{2\,\mathrm{main}}\cdot R\omega^2\cos 2\alpha, \]
for the articulated cylinder,
\[ P_{II\,\mathrm{tr}}=F\cdot m_{2\,\mathrm{tr}}\cdot R\omega^2\cos(2\alpha_1-\Theta), \]
where
\[ \alpha_1=\alpha-\gamma . \]
Denoting
\[ \psi m_{2\,\mathrm{main}}\cdot R\omega^2=P_{II\,\mathrm{main\,max}} \]
and
\[ Fm_{2\,\mathrm{tr}}\cdot R\omega^2=P_{II\,\mathrm{tr,max}}, \]
we have
\[ P_{II\,\mathrm{main}}=P_{II\,\mathrm{main\,max}}\cdot \cos 2\alpha;\quad P_{II\,\mathrm{tr}}=P_{II\,\mathrm{tr\,max}}\cdot \cos\left[2\alpha-(2\gamma+\Theta)\right]. \]
In Fig. 6 the vector \(OA\) represents the force \(P_{II\,\mathrm{main\,max}}\), and the vector \(OB\) represents the force \(P_{II\,\mathrm{tr\,max}}\); the latter is located at the angle
\[ 2\alpha_1-\Theta=2\alpha-(2\gamma+\Theta) \]
to the axis of the articulated cylinder. Both vectors rotate in the direction of rotation of the crankshaft with double angular velocity \(2\omega\). Consequently, the inertia force of the second order in the articulated cylinder can be represented graphically by the projection of the vector \(OB=P_{\mathrm{II\,pr\,max}}\) onto the axis of the articulated cylinder. The vector \(OB\) for the \(i\)-th cylinder rotates with double angular velocity \(2\omega\) in the direction of rotation of the crankshaft and lags behind the vector \(P_{\mathrm{II\,gl\,max}}\) by the angle \(\gamma_i+\Theta_i\). At the initial instant the vector \(OB\) is located at the angle \(2\gamma_i+\Theta_i\) to the axis of the articulated cylinder, measured opposite to the rotation of the crankshaft. The resultant inertia force can be found by the geometric summation of the inertia forces in the main and articulated cylinders (Fig. 6).
Fig. 6.
The resultant force of the second order \(R_{\mathrm{II}}\) may be represented in the form
\[ R_{\mathrm{II}}=\sqrt{x^2+y^2}, \]
where \(x\) is the sum of the projections of all inertia forces on the \(X\) axis,
\(y\) is the sum of the projections of all inertia forces on the \(Y\) axis.
For a two-cylinder engine
\[ x=OT+OM;\qquad y=OS+OK \]
\[ OT=P_{\mathrm{II\,gl\,max}}\cdot \cos 2\alpha \cdot \sin \frac{\gamma}{2} \]
\[ OS=P_{\mathrm{II\,gl\,max}}\cdot \cos 2\alpha \cdot \cos \frac{\gamma}{2}. \]
Let us denote:
\[ P_{\mathrm{II\,gl\,max}}\cdot \sin \frac{\gamma}{2}=A_1;\qquad P_{\mathrm{II\,pr\,max}}\cdot \cos \frac{\gamma}{2}=B_1. \]
Then
\[ OT=A_1\cdot \cos 2\alpha, \]
\[ OS=B_1\cdot \cos 2\alpha. \]
Similarly, for the articulated cylinder
\[ OM=P_{\mathrm{II\,pr\,max}}\cdot \cos(2\gamma+\Theta-2\alpha)\cdot \sin \frac{\gamma}{2} =A_2\cos(2\gamma+\Theta-2\alpha). \]
\[ OK=P_{\mathrm{II\,pr\,max}}\cdot \cos(2\gamma+\Theta-2\alpha)\cdot \cos \frac{\gamma}{2} =B_2\cdot \cos(2\gamma+\Theta-2\alpha), \]
i.e., the inertia force of the second order in any cylinder of the engine can be represented as the geometric sum of the projections of the vector \(B_i\)
onto the \(Y\) axis and vector \(A_i\) onto the \(X\) axis. For the main cylinder, vector \(A_1\) at the initial instant is directed along the \(X\) axis, and vector \(B_1\) along the \(Y\) axis. For any articulated cylinder, vector \(A_i\) is turned through an angle \(2\gamma_i+\Theta_i\) from the \(X\) axis, and \(B_i\) through the same angle from the \(Y\) axis, opposite to the rotation of the crankshaft; i.e., the angle between the vectors \(A_i\) and \(B_i\) is \(90^\circ\). For the main cylinder, vector \(A_1\) lags behind vector \(B_1\); for all articulated cylinders, on the contrary, vector \(A_i\) leads vector \(B_i\) (Fig. 7).
Fig. 7.
When the crankshaft rotates, the mutual arrangement of the vectors \(A_i\) and \(B_i\) does not change, and the whole system of force vectors rotates with angular velocity \(2\omega\) in the direction of rotation of the crankshaft.
In this case
\[ A_i = F_i m_2 R \omega^2 \sin\left(\gamma_i-\frac{\gamma_1}{2}\right); \]
\[ B_i = F_i m_2 R \cdot \omega^2 \cos\left(\gamma_i-\frac{\gamma_1}{2}\right). \]
In the case of an even number of cylinders, when the axis of the main cylinder coincides with the \(Y\) axis,
\[ A_1 = \mu m_2 \cdot R \omega^2, \]
\[ B_1 = 0, \]
\[ A_i = F_i \cdot m_2 R \omega^2 \cdot \sin \gamma_i, \]
\[ B_i = F_i \cdot m_2 \cdot R \omega^2 \cdot \cos \gamma_i. \]
Denoting the resultants of all vectors \(A_i\) by \(A\), and \(B_i\) by \(B\), for all cylinders operating on one crank, we can determine the unbalanced inertia force as the geometric sum of the projections of vectors \(A\) onto the \(X\) axis and \(B\) onto the \(Y\) axis. Both vectors rotate in the direction of rotation of the crankshaft. Figure 8 shows
graphical method for determining the magnitudes of the vectors \(A_i\) and \(B_i\) from the known \(P_{\mathrm{II\,gl\,max}}\) and \(P_{\mathrm{II\,pr\,max}}\); in this case the vector \(P_{\mathrm{II\,pr\,max}}=ON\) lags behind \(P_{\mathrm{II\,gl\,max}}=OM\) by the angle \(\gamma_i+\Theta_i\).
To determine the magnitude and direction of the vectors \(A_1, A_2, B_1, B_2\), etc., for any position of the crankshaft, it is necessary first to determine the magnitudes \(P_{\mathrm{II\,gl\,max}}\) and \(P_{\mathrm{II\,pr\,max}}\) for all cylinders. Then, at an angle \(2\alpha\) (laying it off in the direction of rotation of the crankshaft) to the axis of the main cylinder, direct the vector \(P_{\mathrm{II\,gl\,max}}\), and at the angles \(\gamma_i+\Theta_i\) (laying them off opposite to the direction of rotation of the crankshaft) lay off, on the same scale, the vectors \(P_{\mathrm{II\,pr\,max}}\) of all the articulated cylinders. Having brought the vector \(P_{\mathrm{II\,gl\,max}}\) into coincidence with the axis of the main cylinder and then projected it onto the axes \(X\) and \(Y\), draw two arcs, one with radius \(OS\), the other with radius \(OT\), to their intersections with the lines \(CS\) or \(CE\) (or their extensions). The points \(F\) and \(J\) will be the ends of the vectors \(B\) and \(A\), respectively, i.e., \(OF=B_1\), \(OJ=A_1\).
Fig. 8.
Similarly, one may determine the magnitude and direction of the vectors \(A_i\) and \(B_i\) of any articulated cylinder. In doing so, it should not be forgotten that the angle between the vectors \(A_i\) and \(B_i\) is \(90^\circ\). For the main cylinder the vector \(A_1\) lags behind the vector \(B_1\); for all articulated cylinders it leads. For any articulated connecting rod, the vector \(A_i\) leads the vector \(A_1\) by the angle \(180-2\gamma_i-\Theta_i\).
The resultant inertia force, as is known, varies according to the law of an ellipse whose axes are inclined to the coordinate axes by the angle \(\Theta_0\) (Fig. 10), where
\[ \tan 2\Theta_0=\frac{2AB\sin\varepsilon}{B^2-A^2}. \tag{9} \]
Here \(\varepsilon\) is the angle between the vectors \(A\) and \(B\) (Fig. 9).
If the right-hand side of expression (9) has the sign \(+\), then the angle \(\Theta'\) is reckoned in the direction of rotation of the crankshaft.
The semiaxes of the ellipse are determined from the expressions:
\[ a=\frac{\sqrt{2AB\cos\varepsilon}} {\sqrt{A^2+B^2+\sqrt{(A^2+B^2)^2-4A^2B^2\cos\varepsilon}}}\,. \]
\[ b=\frac{\sqrt{2AB\cos\varepsilon}} {\sqrt{A^2+B^2-\sqrt{(A^2+B^2)^2-4A^2B^2\cos\varepsilon}}}\,. \]
Just as we represented the resultant force of the first order as the geometric sum of two forces whose vectors are equal to \(A\) and \(B\), so in this case the unbalanced inertia force of the second order can be represented as the geometric sum of two forces.
the vectors of which, \(C\) and \(D\), rotate in opposite directions with angular velocity \(2\omega\), the vector \(C\) rotating in the same direction as the crankshaft.
The vectors \(C\) and \(D\) are determined from the expressions:
Fig. 9. Fig. 10.
\[ C=\sqrt{\frac{A^2+B^2+2AB\cos\varepsilon}{4}}, \]
\[ D=\sqrt{\frac{A^2+B^2-2AB\cos\varepsilon}{4}}. \]
For a certain angle of rotation of the crankshaft \(\alpha_1\), which is determined by expression (10), both vectors \(C\) and \(D\) are directed along the semiaxis of the ellipse \(b\):
\[ \tg^2\alpha_1=\frac{z-\tg\psi}{P\tg\psi+1}, \tag{10} \]
where \(\psi\) is the initial position of the vector \(A\) (see Fig. 9),
\[ z=\frac{B}{A}\left(\frac{1-\cos 2\Theta}{\sin 2\Theta\cdot \cos\varepsilon}\right)-\tg\varepsilon, \]
\[ \cos 2\Theta= \frac{B^2-A^2}{\sqrt{(B^2-A^2)^2-4A^2B^2\sin^2\varepsilon}}, \]
\[ \sin 2\Theta= \frac{2AB\cdot \sin\varepsilon}{\sqrt{(B^2-A^2)^2-4A^2B^2\cdot \sin^2\varepsilon}}. \]
At the crankshaft rotation angle \(\alpha=0\), the vectors \(C\) and \(D\) are positioned relative to the semiaxis \(b\) at a certain angle \(2\alpha'_0\), which is determined by the expression:
\[ \sin 2\alpha'_0= \frac{A\cdot \sin(\psi+\varepsilon)\cdot \cos\Theta - B\cos\psi\cdot \sin\Theta}{a}. \]
4. Second-Order Inertia Forces in a Multi-Row Engine with a Multi-Crank Shaft
To determine whether the engine is balanced with respect to second-order inertia forces, it is necessary to determine the magnitude and direction of the resultant vector \(R_C\) of all vectors \(C\), and of the resultant vector \(R_D\) of all vectors \(D\).
If \(R_C \ne 0\), then, evidently, \(R_D \ne 0\) as well; if \(R_C = 0\), then likewise \(R_D = 0\). Both vectors are rotated relative to the semiaxis \(e\) of the ellipse of forces by the same angle, which is determined by a geometrical construction. The total unbalanced force will vary according to the law of an ellipse whose semiaxes are
\[ a = R_C - R_D,\qquad b = R_C + R_D, \]
and the semiaxis \(e\) is inclined to the \(Y\)-axis at an angle \(\Theta'_0\), determined from equation (9).
5. Moments of the First- and Second-Order Inertia Forces in a Multi-Row Engine
Representing the unbalanced inertia force of the first or second order as the geometrical sum of two vectors rotating in opposite directions makes it possible to analyze the balancing of the engine with respect to the moments of the inertia forces of the reciprocating masses analogously to the analysis of the balancing of the engine with respect to the moments of the centrifugal forces. In doing so, one should determine separately the magnitude and direction of the vector of the total moment of the forces \(A_I\) and \(B_{II}\) (for the first order), or \(C_I\) and \(D_{II}\) (for the second order). The unbalanced moment will be the geometrical sum of the moments of the forces
\[ A_I \text{ and } B_{II}\quad (\text{or } C_m \text{ and } D_m) \]
\[ M_I = M_A + M_B,\qquad M_{II} = M_C + M_D . \]
Here \(M_A, M_B, M_C, M_D\) are the geometrical sums of the moments respectively of the forces \(A, B, C\), and \(D\) of each crank.
The vectors of the total moments \(M_A\) and \(M_C\) rotate in the direction of rotation of the crankshaft with angular velocities \(\omega\) and \(2\omega\), respectively, while the vectors of the moments \(M_B\) and \(M_D\) rotate in the opposite direction, with the same velocities.
It is convenient to determine all moments relative to the midpoint of the crankshaft (the middle main journal), taken as the center of gravity of the engine, and the direction of all vectors according to the general rule for determining the direction of the moment vector of a force.
References
- M. A. Levin and G. V. Senechkin, eds. Aircraft Engines. Mashgiz, 1951.
- Aircraft Piston Engines. Calculation Handbook, edited by I. Sh. Neiman. Oborongiz, 1950.
- I. I. Kholmakov. Dynamics of Aircraft Engines. Oborongiz, 1938.