On Criteria for the Imbalance of Reciprocating Internal Combustion Engines

V. K. Nechaev

The analytical and graphical methods currently known for analyzing the balancing of multicylinder reciprocating engines make it possible to determine the magnitudes and directions of the vectors of the resultants (sums) of the unbalanced inertia forces of the masses of the crank-and-connecting-rod mechanisms of the cylinders and of the moments of these forces in the engine.

However, knowledge of the absolute magnitude of these forces and moments does not yet provide sufficient grounds for a reliable judgment about the degree of imbalance of a given engine, or for a confident decision on whether or not it is necessary to develop design measures to improve the balancing of the engine.

In this connection, it would be desirable to have some criteria, as simple as possible, for a rapid but sufficiently reliable assessment of the degree of imbalance of an engine.

The literature gives the only numerical criteria for the imbalance of reciprocating engines known to us, proposed in their time by V. Ya. Klimov and B. S. Stechkin.

Here the imbalance of an engine in the plane of the cylinder axes (as applied to a multicylinder in-line engine) and in the horizontal plane perpendicular to it and passing through the crankshaft axis is characterized respectively by the coefficients \(\xi\) and \(\eta\), determined by the expressions:

\[ \xi = \frac{P_{\mathrm{I}} + \dfrac{1}{4}P_{\mathrm{II}}}{m_e \omega_b^2 D} + \frac{\left(M_{\mathrm{I}} + \dfrac{1}{4}M_{\mathrm{II}}\right)\cdot 6l} {m_e \omega_b^2(l^2+h^2)\cdot D}; \tag{1} \]

\[ \eta = \frac{P_n}{m_e \omega_b^2\cdot D} + \frac{M_n\cdot 6l}{m_e \omega_b^2(l^2+b^2)\cdot D}, \tag{2} \]

where \(P_{\mathrm{I}}, P_{\mathrm{II}}, M_{\mathrm{I}}, M_{\mathrm{II}}\) are the amplitudes of the resultants (sums) of the unbalanced inertia forces of the first and second orders and of the moments of these forces, acting in the plane of the cylinder axes;

\(P_n\) is the amplitude of the resultant unbalanced inertia force acting in the horizontalಿದ

(applicable to a vertical single-row engine) planes;

\(M_{\mathrm{II}}\) — moment of the forces \(P_{\mathrm{II}}\), likewise acting in this plane;

\(\omega_{\mathrm{в}}\) — angular speed of rotation of the engine shaft;

\(l, b, h\) — length, width, and height of the engine (taken without allowance for insignificant protruding parts of the engine);

\(D\) — diameter of the engine cylinders.

The engine is considered well balanced if

\[ \xi < 0.002;\qquad \eta < 0.002, \]

and poorly balanced if

\[ \xi > 0.01;\qquad \eta > 0.01. \]

Formulas (1) and (2), and the limiting values of \(\xi\) and \(\eta\) just indicated, are given in the handbook Diesels [1] without any qualifications or clarifications. It is only noted that these limits for \(\xi\) and \(\eta\) were obtained on the basis of calculations of a number of completed engines. Thus, the authors of the handbook, in essence, recommend these criteria for use in design practice in assessing the degree of imbalance of diesels of various types and purposes.

As follows from (1) and (2), the criteria \(\xi\) and \(\eta\) do not depend on the absolute value of the angular speed \(\omega_{\mathrm{в}}\), that is, the degree of imbalance of an engine, as assessed by these criteria, does not change when its speed regime changes. Thus two engines with different speed ratings, but having identical corresponding values of the coefficients \(\xi\) and \(\eta\), must be considered imbalanced to the same degree.

However, these latter conclusions cannot be regarded as correct. Rather, it should be considered that the independence of the criteria \(\xi\) and \(\eta\) from \(\omega_{\mathrm{в}}\) is their fundamental shortcoming, depriving these criteria of universality and severely limiting the sphere of their possible application in practical conditions.

Let us consider the simplest case of forced oscillations of a vertical engine under the action of the exciting forces \(P_{\mathrm{I}}\cos \omega_{\mathrm{в}} t\) and \(P_{\mathrm{II}}\cos 2\omega_{\mathrm{в}} t\) applied to it.

The engine is rigidly fastened to a massive foundation (or to a sub-engine frame) having mass \(m_{\phi}\). The foundation (frame) is installed on elastic elements (soil, springs, etc.), the total stiffness of which is equal to \(c\).

The frequency of free vertical oscillations of such a system is

\[ \omega_0=\sqrt{\frac{c}{m_{\mathrm{d}}+m_{\phi}}}. \tag{3} \]

Let us assume, to simplify the exposition, that there are no damping forces in the system. Then the amplitudes \(Z_{\mathrm{I}}\) and \(Z_{\mathrm{II}}\) of the forced vertical oscillations of the engine with the foundation, caused respectively by forces of the first and second orders, are determined by the expressions:

\[ Z_{\mathrm{I}}= \frac{1}{1-\dfrac{\omega_{\mathrm{в}}^2}{\omega_0^2}}\cdot\frac{P_{\mathrm{I}}}{c} = \frac{1}{\left(\omega_0^2-\omega_{\mathrm{в}}^2\right)} \cdot \frac{P_{\mathrm{I}}}{\left(m_{\mathrm{d}}+m_{\phi}\right)}; } \tag{4} \]

\[ Z_{\mathrm{II}}= \frac{1}{1-\dfrac{4\omega_v^2}{\omega_o^2}}\cdot \frac{P_{\mathrm{II}}}{c} = \frac{1}{\left(\omega_o^2-4\omega_v^2\right)} \cdot \frac{P_{\mathrm{II}}}{\left(m_d+m_f\right)} . \tag{5} \]

In order to make it possible to evaluate the degree of engine imbalance by dimensionless coefficients, let us refer \(Z_{\mathrm{I}}\) and \(Z_{\mathrm{II}}\) to some basic dimension of the engine, which, following Klimov and Stechkin, we shall conventionally take to be the cylinder diameter \(D\). Then the relative amplitudes of the vertical oscillations of the engine are

\[ \xi_{\mathrm{I}}'=\frac{Z_{\mathrm{I}}}{D}; \qquad \xi_{\mathrm{II}}'=\frac{Z_{\mathrm{II}}}{D}. \tag{6} \]

The graph of the dependence of the amplitudes \(Z_{\mathrm{I}}\) and \(Z_{\mathrm{II}}\) on the ratio \(\dfrac{\omega_v}{\omega_o}\) is shown in Fig. 1.

Similarly, the curves for \(\xi_{\mathrm{I}}'\) and \(\xi_{\mathrm{II}}'\) will evidently have the same character.

Now the absolute and relative amplitudes of the total oscillations of the engine under the simultaneous action in it of the unbalanced forces \(P_{\mathrm{I}}\cos \omega_v t\) and \(P_{\mathrm{II}}\cos 2\omega_v t\) are:

Fig. 1.

\[ Z_{\mathrm{I}}+Z_{\mathrm{II}} = \frac{1}{\left(\omega_o^2-\omega_v^2\right)} \cdot \frac{P_{\mathrm{I}}+\sigma P_{\mathrm{II}}} {m_d\left(1+\dfrac{m_f}{m_d}\right)}; \tag{7} \]

\[ \xi_{\mathrm{I}}'+\xi_{\mathrm{II}}' = \frac{1}{\left(\omega_o^2-\omega_v^2\right)} \cdot \frac{P_{\mathrm{I}}+\sigma P_{\mathrm{II}}} {m_d\left(1+\dfrac{m_f}{m_d}\right)}, \tag{8} \]

where the following notation has been adopted:

\[ \sigma= \frac{1-\dfrac{\omega_v^2}{\omega_o^2}} {1-\dfrac{4\omega_v^2}{\omega_o^2}} . \tag{9} \]

As follows from (7) and (8), when \(\omega_v\to\infty\) or when \(\omega_o\to 0\), that is, when the stiffness \(c\) of the foundation support is reduced to zero, the amplitude of the forced oscillations of the engine tends to the limiting value

\[ Z_{\mathrm{I}\,\mathrm{lim}}+Z_{\mathrm{II}\,\mathrm{lim}} = \frac{P_{\mathrm{I}}+\dfrac{1}{4}P_{\mathrm{II}}} {m_d\left(1+\dfrac{m_f}{m_d}\right)\omega_v^2}, \tag{10} \]

since here, according to (9), \(\sigma=1/4\).

If the engine has no foundation \((m_f=0)\), then the limiting relative amplitude of its oscillations is determined, according to (8), by the formula

\[ \xi_{\mathrm{I}}+\xi_{\mathrm{II}}= \frac{P_{\mathrm{I}}+\dfrac{1}{4}P_{\mathrm{II}}}{m_{\partial}\omega_b^2D}, \tag{11} \]

which fully coincides with the first half of the right-hand side of expression (1) for the Klimov–Stechkin criterion.

In an analogous way one could obtain an expression for the limiting (as \(\dfrac{\omega_b}{\omega_0}\to\infty\) and \(m_{\phi}=0\)) relative amplitude of the total vertical oscillations of the end points of the engine casing (along the crankshaft axis) under the action of the unbalanced moments \(M_{\mathrm{I}}\cos\omega_b t\) and \(M_{\mathrm{II}}\cos 2\omega_b t\), applied in the plane of the cylinder axes:

\[ \xi_{\mathrm{I}}+\xi_{\mathrm{II}}= \frac{M_{\mathrm{I}}+\dfrac{1}{4}M_{\mathrm{II}}}{\Theta_y\omega_b^2D}\cdot\frac{l}{2}. \tag{12} \]

Here \(\Theta_y\) is the moment of inertia of the engine mass with respect to the horizontal axis of angular oscillations of the latter.

If the engine is taken approximately as a parallelepiped with mass \(m_{\partial}\) uniformly distributed over its volume, then

\[ \Theta_y=m_{\partial}\frac{l^2+h^2}{12}. \]

Now (12) fully coincides with the second half of the right-hand side of expression (1).

The fundamental similarity of expressions (11) and (12) allows us, in the subsequent analysis of the forced oscillations of the engine (and foundation) under the action only of the forces \(P_{\mathrm{I}}\) and \(P_{\mathrm{II}}\), with \(M_{\mathrm{I}}=0\) and \(M_{\mathrm{II}}=0\), to confine ourselves to this case. The conclusions obtained in this way are readily extended to the case of the action in the engine of the moments \(M_{\mathrm{I}}\) and \(M_{\mathrm{II}}\), and to the case of the presence in the engine of unbalance in the horizontal plane.

As is evident from the preceding, the Klimov–Stechkin criteria estimate engine unbalance by those limiting amplitudes of forced oscillations, \(Z_{\mathrm{I\,pred}}\) and \(Z_{\mathrm{II\,pred}}\), which the engine would have if, having no foundation (sub-engine frame), it were suspended in free space.

But according to (7) and (10) and Fig. 1, the amplitudes \(Z_{\mathrm{I}}\) and \(Z_{\mathrm{II}}\) of the engine with a foundation (frame), for \(c>0\), may differ greatly from \(Z_{\mathrm{I\,pred}}\) and \(Z_{\mathrm{II\,pred}}\). Introduce the notation:

\[ \varepsilon_{\mathrm{I}}=\frac{Z_{\mathrm{I}}}{D};\qquad \varepsilon_{\mathrm{II}}=\frac{Z_{\mathrm{II}}}{D}. \]

Then the actual values \(\xi'_{\mathrm{I}}\) and \(\xi'_{\mathrm{II}}\) of the criteria of engine unbalance (separately for the inertia forces of the first and second orders) are determined by the expressions

\[ \xi'_{\mathrm{I}}=\varepsilon_{\mathrm{I}}\cdot\xi_{\mathrm{I}}; \tag{13} \]

\[ \xi'_{\mathrm{II}}=\varepsilon_{\mathrm{II}}\cdot\xi_{\mathrm{II}}, \tag{14} \]

where

\[ \xi_{\mathrm{I}}=\frac{P_{\mathrm{I}}}{m_{\mathrm{д}}\omega_{\mathrm{в}}^{2}D}; \tag{15} \]

\[ \xi_{\mathrm{II}}=\frac{1}{4}\cdot\frac{P_{\mathrm{II}}}{m_{\mathrm{д}}\omega_{\mathrm{в}}^{2}D} \tag{16} \]

— partial criteria of imbalance according to Klimov–Stechkin.

The coefficients \(\varepsilon_{\mathrm{I}}\) and \(\varepsilon_{\mathrm{II}}\) may be regarded as necessary correction factors to the Klimov–Stechkin criteria.

According to (7), (15), (16),

\[ \varepsilon_{\mathrm{I}}= \frac{1}{\dfrac{\omega_{\mathrm{o}}^{2}}{\omega_{\mathrm{в}}^{2}}-1} \cdot \frac{1}{1+\dfrac{m_{\mathrm{ф}}}{m_{\mathrm{д}}}}; \tag{17} \]

\[ \varepsilon_{\mathrm{II}}= \frac{1}{\dfrac{\omega_{\mathrm{o}}^{2}}{\omega_{\mathrm{в}}^{2}}-1} \cdot \frac{4\sigma}{1+\dfrac{m_{\mathrm{ф}}}{m_{\mathrm{д}}}}. \tag{18} \]

Graphs of the dependence of these coefficients on the ratio \(\dfrac{\omega_{\mathrm{в}}}{\omega_{\mathrm{o}}}\) for three particular cases of the ratio \(\dfrac{m_{\mathrm{ф}}}{m_{\mathrm{д}}}\) are shown in Fig. 2.

For \(m_{\mathrm{ф}}=0\) and \(\dfrac{\omega_{\mathrm{в}}}{\omega_{\mathrm{o}}}\to\infty\), that is, for \(\omega_{\mathrm{в}}\to\infty\) or \(\omega_{\mathrm{o}}\to0\), both coefficients \(\varepsilon_{\mathrm{I}}\) and \(\varepsilon_{\mathrm{II}}\) tend to the limiting value 1. Only here do the Klimov–Stechkin criteria \(\xi_{\mathrm{I}}\) and \(\xi_{\mathrm{II}}\) coincide with the actual values of the relative amplitudes \(\xi'_{\mathrm{I}}\) and \(\xi'_{\mathrm{II}}\).

Fig. 2.

In Fig. 3, constructed on the basis of the same expressions (17) and (18), the shaded areas indicate the regions where the coefficients \(\varepsilon_{\mathrm{I}}\) and \(\varepsilon_{\mathrm{II}}\) turn out to be greater than unity, that is, where the criteria \(\xi_{\mathrm{I}}\) and \(\xi_{\mathrm{II}}\) underestimate the actual imbalance of the engine. These regions are bounded by the curves \(\varepsilon_{\mathrm{I}}=1\) and \(\varepsilon_{\mathrm{II}}=1\), respectively, for the cases of the presence in the engine of first- and second-order forces and moments.

According to Fig. 3, in the region

\[ \frac{\omega_v}{\omega_o}<\frac{\sqrt{2}}{2} \]

— for forces and moments of the first order, and

\[ \frac{\omega_v}{\omega_o}<\frac{\sqrt{2}}{4} \]

for the second order, the coefficients \(\varepsilon_I\) and \(\varepsilon_{II}\) remain less than unity at all times. Here the amplitudes \(Z_I\) and \(Z_{II}\) increase with increasing \(\omega_v\), remaining, however, always smaller than \(Z_{I,\mathrm{lim}}\) and \(Z_{II,\mathrm{lim}}\).

Fig. 3

But these regions are the usual operating regions of the foundations of stationary installations. Consequently, as applied to stationary engines, the Klimov–Stechkin criteria always give an overestimated (that is, worsened) estimate of the actual imbalance of the engine. This error increases as the ratio

\[ \frac{\omega_v}{\omega_o} \]

decreases.

Automobile engines (on elastic supports), as a rule, operate in the beyond-resonance region:

\[ \frac{\omega_v}{\omega_o}\gg 1, \]

where the amplitudes of the forced vibrations of the engine decrease with increasing \(\omega_v\). It is easy to see that here the Klimov–Stechkin criteria assess the imbalance of the engine with sufficient reliability. Under these conditions the permissible values of the criteria \(\xi\) and \(\eta\) can indeed be established independently of the engine speed.

However, as applied to stationary engines these standards may prove excessively stringent. Here we can allow larger values of the criteria \(\xi\) and \(\eta\), the larger the mass of the foundation \(m_f\), the smaller the cylinder diameter \(D\), and the lower the engine speed \(\omega_v\).

Indeed, substituting into (4) the expression for \(P_1\) in terms of \(\xi_1\) from (15), we obtain

\[ Z_I=\frac{1}{\dfrac{\omega_o^2}{\omega_v^2}-1}\cdot \xi_1\,\frac{m_d}{m_d+m_f}\cdot D . \tag{19} \]

Let \(Z_{\mathrm{perm}}\) be the permissible value of the amplitude of forced vibra-

of the engine and foundation. Then the corresponding permissible value of the criterion \(\xi_{\mathrm{I}}\) (in the case where only the force \(P_{\mathrm{I}}\) is present in the engine) is:

\[ \xi_{\mathrm{I}dop}=\frac{Z_{dop}}{D}\left(1+\frac{m_{\phi}}{m_{\partial}}\right)\left(\frac{\omega_o^2}{\omega_v^2}-1\right). \tag{20} \]

Similarly, when only the force \(P_{\mathrm{II}}\) acts in the engine:

\[ \xi_{\mathrm{II}dop}=\frac{Z_{dop}}{D}\left(1+\frac{m_{\phi}}{m_{\partial}}\right)\left(\frac{\omega_o^2}{4\omega_v^2}-1\right). \tag{21} \]

The current standards for the design of foundations for stationary engines [3] allow, as the limiting amplitude of forced vibrations of the foundation, \(0.2\) mm. Obviously, in the case of a tenfold safety margin with respect to this figure, i.e., at \(Z=0.02\) mm, we shall have every reason to regard the engine as well balanced.

Let us choose, as illustrative, the following ranges of possible values of the parameters of stationary installations:

\[ D=100\div250\ \text{mm}; \qquad \frac{m_{\phi}}{m_{\partial}}=0\div5; \qquad \frac{\omega_v}{\omega_o}=\frac{1}{4}\div\frac{1}{2}. \]

Now, for \(Z_{dop}=0.02\) mm, from (20) and (21) we obtain the corresponding intervals of permissible values of the criteria \(\xi_{\mathrm{I}}\) and \(\xi_{\mathrm{II}}\):

\[ \xi_{\mathrm{I}dop}=0.0002\div0.018; \]

\[ \xi_{\mathrm{II}dop}=0\div0.076. \]

However, if we are guided by the standards cited above on p. 4, we would have to classify a considerable portion of our engines as poorly balanced.

In calculating the intervals \(\xi_{\mathrm{I}dop}\), \(\xi_{\mathrm{II}dop}\), we took the quantity \(Z_{dop}\) to be constant, independent of the engine speed, i.e., of the value \(\omega_v\). It would be more correct to reduce the value of \(Z_{dop}\) somewhat as \(\omega_v\) increases [3]. It is easy to see that in this case the intervals for \(\xi_{\mathrm{I}}\) and \(\xi_{\mathrm{II}}\) would be still wider.

Let us note that the approximate interval computed above for \(\xi_{\mathrm{II}dop}\) turned out to be considerably wider than the interval for \(\xi_{\mathrm{I}dop}\). This is due to the fact that the Klimov–Stechkin criteria seriously underestimate the influence on engine imbalance of second-order inertia forces and moments of forces in comparison with first-order forces and moments.

In the numerator of expression (1) for the criterion \(\xi\), before \(P_{\mathrm{II}}\) and \(M_{\mathrm{II}}\) there is a factor \(1/4\). Thus it is as though it were assumed that the influence of second-order forces and moments on engine imbalance is, in general, four times less significant than the influence of first-order forces and moments.

However, according to (7) and (8), in the general case (for \(c>0\)), before \(P_{\mathrm{II}}\) and \(M_{\mathrm{II}}\), instead of the factor \(1/4\), there should be placed the factor \(\sigma\), whose value is determined by expression (9).

As can be seen from the graph (Fig. 4), only when $\dfrac{\omega_v}{\omega_o} > \sqrt{0.4}$ does the coefficient $\sigma$ prove to be less than unity, tending as $\dfrac{\omega_v}{\omega_o} \to \infty$ to the limiting value $1/4$. Throughout the entire subresonance region $\sigma > 1$.

Thus, as applied to stationary engines (and also to other motor installations operating under subresonance conditions), the Klimov–Stechkin criteria give an error in estimating the influence of second-order forces and moments of at least a factor of 2.5 (when $\dfrac{\omega_v}{\omega_o} \to 0$). Further, with increasing $\dfrac{\omega_v}{\omega_o}$ this error rapidly increases and, for example, when $\dfrac{\omega_v}{\omega_o} = 0.4$, it is more than sixfold.

Fig. 4.

All that has been said above leads to the conclusion that the Klimov–Stechkin criteria are highly conditional, that their mechanical application to estimating the degree of imbalance of engines of various types is inadmissible, and that it is necessary to take into account the specific installation and operating conditions of the engine when assessing its imbalance.

These criteria, without major reservations and limitations, are suitable only for automobile and aircraft (piston) engines installed on an elastic suspension.

REFERENCES

1. Diesels. A Design Engineer’s Reference Manual, ed. Prof. V. A. Vansheidt, Mashgiz, 1957.

2. Savinov O. A. Foundations for Machines, Stroiizdat, 1955.

3. Technical Specifications for the Design of Foundations for Machines with Dynamic Loads (TU—60—49), Stroiizdat, 1949.