On the Stability of the Characteristics of Limit Pneumatic Controllers

V. K. Nechaev

Pneumatic throttle-type controllers (with a rotating throttle valve as the sensing and actuating elements of the control system) differ from ordinary centrifugal controllers not only in their design and principle of operation. Pneumatic controllers possess noticeable sensitivity to changes in the external atmospheric conditions under which the engine operates (the pressure and temperature of the outside air), and to changes in the resistances in the engine intake system ahead of the controller. In addition, because of the small magnitude of the driving forces in a pneumatic controller, produced by the aerodynamic action of the flow of the working mixture on the regulating valve, these controllers have increased sensitivity to the friction forces in the controller mechanism.

The sensitivity of pneumatic controllers of the type considered to external (atmospheric) conditions and to friction forces manifests itself in the not always sufficient stability of operation of such controllers. The position of the governing characteristic of an engine with a pneumatic throttle-controller proves to be not entirely definite, but variable. The maximum rotational speeds developed by the engine after a sudden removal of load, as limited by the limit controller, now depend not only on the position of the gas pedal (this dependence is natural for controllers of the K-49 and K-80 types, which have one “combined” regulating and throttle valve), but also on the magnitude of the friction forces in the controller and on atmospheric conditions.

Below we consider the question of the influence of changes in atmospheric pressure $p_0$, ambient temperature $T_0$, and (dry) friction in the mechanism of a controller of the K-49 or K-80 type.

1. Influence of Changes in Atmospheric Pressure $p_0$

As is known [1], the behavior of a pneumatic controller depends, first of all, on the relation between the driving forces or moments in the controller, on the one hand, and the elastic (restoring) moments, on the other.

The force of the controller spring gives on the axis of the regulating valve a certain “elastic” or “restoring moment” $M_n$, which depends only on the characteristics of the spring (its stiffness and initial tension) and on the geometric characteristics of the mechanism connecting the spring with the valve. This moment $M_n$ varies with the angle of rotation $\beta$ of the regulating valve and, with unchanged spring adjustment, is completely determined by this angle $\beta$.

The driving forces in the controller are created by the action on the regulating valve of the flow of working mixture passing from the carburetor through the controller into the engine. These forces give on the regulating valve a certain torq-

the driving moment \(M_c\), which is a complex function of a whole number of variables. Approximately [1]:

\[ M_c = k_1 q + k_2 \Delta p_{12}, \tag{1} \]

where \(q\) is the dynamic pressure of the flow of the working mixture ahead of the controller throttle:

\[ q=\frac{\gamma_1 V^2}{2g}; \]

\(\gamma_1\) is the specific weight of the working mixture ahead of the controller throttle;
\(V\) is the velocity of the mixture flow (at the same place);
\(k_1\) and \(k_2\) are quantities depending on the shape, dimensions, eccentricity of the throttle, and the angle \(\beta\) of its rotation from the axis of the controller branch pipe; in each particular controller \(k_1\) and \(k_2\) are functions only of the angle \(\beta\).

When the atmospheric pressure \(p_0\) decreases and the position of the throttle valve remains unchanged \((\beta=\mathrm{const})\), as does the engine speed, \(n=\mathrm{const}\) (which can be ensured, for example, by a corresponding change in the external load on the engine), the flow of mixture passing through the controller into the engine weakens; this leads to a decrease in the “driving” moment \(M_c\). The result of this is a delay in the action of the limiting pneumatic controller: the controller comes into operation later, that is, at a higher engine speed \(n\), the lower the pressure \(p_0\). The magnitude of this delay can be determined approximately as follows.

As is known [2], the filling coefficient of the engine \(\eta_v\), when the external pressure \(p_0\) changes (within limited ranges, of course), changes hardly at all. Therefore, if the changes in the relative magnitudes of the pressure losses in the carburetor and controller associated with the change in \(p_0\) are neglected, then it may be assumed that the velocity \(V\) of the mixture flow ahead of the controller at \(n=\mathrm{const}\) does not depend on the magnitude of \(p_0\) and remains constant during fluctuations of the pressure \(p_0\).

But when the external pressure is reduced from \(p_0\) to some \(p_0'\), and the temperature \(T_0\) remains unchanged, the specific weight \(\gamma_1\) of the mixture ahead of the controller decreases to some \(\gamma_1'\). In the first approximation:

\[ \gamma_1'=\gamma_1\frac{p_0'}{p_0}, \]

if it is assumed that the change in external pressure does not affect the composition of the working mixture.^1) Accordingly, with the decrease in \(\gamma_1\), the dynamic pressure \(q\) and the static pressure drop \(\Delta p_{12}\) across the controller throttle decrease:

\[ q' = q\frac{p_0'}{p_0}, \]

\[ \Delta p'_{12}=\Delta p_{12}\frac{p_0'}{p_0}, \]

and likewise, according to formula (1), the moment \(M_c\) on the throttle also decreases. Consequently, for \(n=\mathrm{const}\) and \(\beta=\mathrm{const}\), approximately:

\[ M'_c=M_c\frac{p_0'}{p_0}. \tag{2} \]

^1) In fact, when the pressure \(p_0\) decreases, a certain enrichment of the mixture occurs: the excess-air coefficient decreases. This enrichment of the mixture somewhat compensates for the decrease in the specific weight of the air.

Now, as the external pressure \(p_0\) decreases, the curves \(M_c=f_2(n)\), plotted in the coordinates \(M_c\) and \(n\) for some fixed position of the regulating throttle valve \((\beta=\mathrm{const})\), approach the abscissa axis \(n\), and their slope decreases with increasing \(n\), i.e., the curves \(M_c\) become flatter (Fig. 1).

When the engine operates with the throttle fully open \((\beta=\beta_0)\), but without load, engagement of the regulator will occur only when the condition

\[ M'_c > M_{n0}. \]

is satisfied.

Here \(M_{n0}\) is the initial moment developed on the throttle valve by the regulator spring at the initial “setting” position \((\beta=\beta_0)\) of this valve. Here, because the curve \(M'_c=f'_2(n)\) for \(\beta=\beta_0\) runs very gently, the lag in regulator engagement, i.e., the difference \(n'_A-n_A\), may prove to be comparatively large.

Fig. 1

The law of variation of the moment \(M_c\) with the number of engine revolutions \(n\), for \(\beta=\mathrm{const}\), is fairly complicated and depends on a whole series of circumstances which, in a given engine, determine the values of \(q\) and \(\Delta p_{12}\) at each value of \(n\). Therefore, the most reliable method should be considered the experimental determination of the moments \(M_c\) as a function of \(n\) (for a number of fixed values of the angle \(\beta\)) by means of a special device, during engine tests on a stand at different values of \(p_0\).

From the data of such experiments, carried out, for example, at two values of atmospheric pressure \(p_0\) and \(p'_0\) for the case of full throttle opening \((\beta=\beta_0)\), one can construct, point by point, the curves \(M_c\) and \(M'_c\) (Fig. 1). Then the intersection of the horizontal through the initial point \(a\) of the curve \(M_n\) with the curves \(M_c\) and \(M'_c\) gives the points 1 and \(1'\), whose abscissas determine the engine rotational speeds \(n_A\) and \(n'_A\), at which the regulator will come into action (respectively at outside pressures \(p_0\) and \(p'_0\)).

Approximately, it may be assumed that, for small angles \(\beta\) and for small vacuums in the engine intake system, the pressures \(q\) and \(\Delta p_{12}\) increase in direct proportion to the square of the engine speed \(n\), that is, for \(p_0=\mathrm{const}\) and \(\beta=\mathrm{const}\):

\[ M_c=wn^2, \tag{3} \]

where \(w\) is a certain coefficient of proportionality, depending on the shape and dimensions of the throttle valve, the characteristics of the engine intake system, the angle \(\beta\), etc.

Now, according to formula (2), for the same \(\beta\), but with a new value of the external pressure \(p_0'\), the torque on the throttle plate is determined by the expression:

\[ M_c'=-\frac{p_0'}{p_0}\,wn^2. \]

At the instant when the regulator is engaged, respectively for points 1 and \(1'\) (Fig. 1), we have:

\[ M_{n0}=wn_A^2; \]

\[ M_{n0}'=-\frac{p_0'}{p_0}\,wn_A'^2 \]

whence

\[ n_A'=n_A\sqrt{-\frac{p_0}{p_0'}} \]

and the delay in regulator engagement when the atmospheric pressure is lowered from \(p_0\) to \(p_0'\) is:

\[ \Delta n_A=n_A'-n_A=n_A\left(\sqrt{-\frac{p_0}{p_0'}}-1\right). \tag{4} \]

For \(-\dfrac{p_0}{p_0'}=1.1\) (which cannot be regarded as exceptional even when an engine is operated in lowland areas),

\[ \Delta n_A=0.05\,n_A. \]

In reality this delay may have a considerably greater value, especially at large angles \(\beta\) (a nearly closed throttle), since here the parabolic character of the dependence of the torque \(M_c\) on \(n\), adopted in formula (3), is plainly violated, and the torque \(M_c\) increases with the number of engine revolutions \(n\) more slowly than is represented by formula (3). In Fig. 2, as an example, one of the experimental curves \(M_c=f_2(n)\) is shown, obtained in tests of the GAZ-51 engine with a K-49 carburetor in the laboratory of the Internal-Combustion Engines Department of the TPI.

Fig. 2

But a decrease in the external pressure \(p_0\) leads not only to a decrease in the quantities \(q\), \(\Delta p_{12}\), and \(M_c\) (for \(\beta=\mathrm{const}\) and \(n=\mathrm{const}\)), but also to a decrease in the amount of working mixture entering the engine. The result of this is a reduction in the indicated and effective power of the engine. Now the external,

and also the partial (with respect to the angle \(\beta\)) engine characteristics \(N_e=f_1'(n)\) in the coordinates \(N_e\) and \(n\) are located closer to the abscissa axis \(n\).

A consequence of the two circumstances noted above—the decrease in the torque \(M_c\) and in the effective power of the engine \(N_e\) as the outside pressure \(p_0\) decreases (at \(\beta=\mathrm{const}\))—is not only a displacement of the initial point \(A\) of the governor characteristic, but also a shift of this entire characteristic toward higher values of \(n\) and lower values of \(N_e\).

Let us assume approximately that the indicated power \(N_i\) of a carburetor engine at \(n=\mathrm{const}\) and \(\beta=\mathrm{const}\) varies in direct proportion to the outside pressure \(p_0\), while the friction power \(N_{mp}\) does not depend on the pressure \(p_0\) [2]. Consequently, the effective power of the engine at a reduced outside pressure \(p_0'\) and \(\beta=\mathrm{const}\), \(n=\mathrm{const}\), may be determined by the expression:

\[ N_e' = N_i \frac{p_0'}{p_0} - N_{mp}, \]

where \(N_i\) is the indicated power of the engine at outside pressure \(p_0\) and the same values of \(\beta\) and \(n\). The point of intersection of the curve \(N_i\) with the curve \(N_{mp}\) (Fig. 3) determines the maximum (runaway) speed of the engine at this \(p_0'\) and \(\beta=\mathrm{const}\), while the vertical segments between the curves \(N_i'\) and \(N_{mp}\) represent the magnitude of the effective power of the engine for the given \(\beta\), \(p_0'\), and \(n\).

Fig. 3

In this way one can construct the external (at \(\beta=\beta_0\)) and partial (for a number of definite fixed values of the angle \(\beta\)) characteristics of the effective power of the engine for different values of the outside pressure \(p_0\). Now, in order to analyze the influence of a change in the pressure \(p_0\) on the governor characteristic \(AB\) of the engine, one may use the graphical method proposed by the author [1].

Figure 4 presents the external characteristics of the engine \(N_e, N_e', N_e''\) (quadrant I) and the curves of the aerodynamic torque on the throttle valve \(M_c, M_c', M_c''\) (quadrant IV), constructed respectively for three values of atmospheric pressure \(p_0, p_0', p_0''\) (where \(p_0>p_0'>p_0''\)) with the unchanged “set” position of the governor throttle valve \(\beta=\beta_0\). In quadrant III, as usual, the curve of the spring torque \(M_n=f_3(\beta)\) is shown.

A method was already indicated above for determining the positions of points 1 and \(1'\), whose abscissas \(n_A\) and \(n_A'\) give the engine speeds, respectively, for the pressures \(p_0\) and \(p_0'\), at which the governor comes into action. The vertical through point 1 intersects the curve \(N_e\) at point \(A\), which is the initial point of the governor characteristic of the engine at outside pressure

$p_0$. Similarly, by means of the vertical line drawn through point $1'$, the initial point $A'$ of the governing characteristic of the engine at the external pressure $p_0'$ is determined. As the pressure $p_0$ decreases, point $A$ moves on the diagram to the right and downward; the delay in regulator engagement increases.

But this delay in regulator engagement $\Delta n_A$ cannot be unlimited. With a gradual decrease in the external pressure, this delay finally approaches the limit $\Delta n_{A\max}$, when a further increase in the engine speed upon removal of the load from it with the throttle fully open $(\beta=\beta_0)$ will already be limited not by the regulator, but by the engine itself. Let us consider this case, which, of course, under ordinary operating conditions of engines is only of theoretical interest. ^1)

Fig. 4

Let us carry the end points of the characteristics $N_e$, $N_e'$ and $N_e''$, corresponding to the no-load operation of the engine and lying on the abscissa axis, down to the corresponding curves $M_c$, $M_c'$, $M_c''$. Through the points thus obtained we draw a curve. This curve $Z_2$ may be called the no-load curve of the engine according to its external characteristic, that is, for $\beta=\beta_0$, with a variable value of the external pressure $p_0$. A horizontal line drawn through point $a$ intersects this curve at a certain point $y$. It is obvious that the abscissa of point $y$ determines the limiting speed that the engine can develop with the given limiting throttle-regulator after the removal from the engine of the entire

^1) However, the method of analysis described here may prove useful for solving already practical questions about the maximum possible delay in regulator engagement when the engine is operating with a partially closed throttle. Here it is necessary only to replace the external characteristics of the engine by the corresponding partial characteristics (for the given angle $\beta$) and to construct the curves $M_c$, $M_c'$, $M_c''$ for this same angle $\beta$.

load at the maximum opening \((\beta=\beta_0)\) of the throttle valve. Now the segment \(n_y-n_A\) gives the maximum possible delay in the engagement of the limit pneumatic governor.

To determine the position of the end point \(B\) of the governor characteristic, it is necessary to have the partial characteristics of the engine and the torque curves \(M_c\) for a number of values of the angle \(\beta\), at different atmospheric pressures.

In Fig. 4, the partial characteristics of the engine and the torque curves \(M_c\) are shown by dashed lines for some fixed position of the throttle valve \((\beta=\beta_1)\) at two values of the external pressure: \(p_0\) and \(p_0'\). We transfer the end points of the external and partial power characteristics of the engine at pressure \(p_0\) onto the torque curves \(M_c\) corresponding to this pressure (constructed for the angles \(\beta=\beta_0\) and \(\beta=\beta_1\)). Through the points thus obtained we draw the no-load curve \(Z\), which, transferred horizontally into quadrant III, gives the curve \(Z_1\). The intersection of the curve \(Z_1\) with the spring-torque curve \(M_n\) gives the point \(b\), by which, using the construction indicated in the drawing, we find the required end point \(B\) of the governor characteristic \(AB\) at the external (atmospheric) pressure \(p_0\). When the atmospheric pressure is reduced to \(p_0'\), the curves \(Z\) and \(Z_1\) are transformed, respectively, into the curves \(Z'\) and \(Z_1'\), and the point \(b\) moves to the point \(b'\), which then determines the end point \(B'\) of the governor characteristic \(A'B'\) at pressure \(p_0'\).

It is easy to see that a reduction in atmospheric pressure leads not only to a delay in the engagement of the governor, but also to a decrease in its degree of nonuniformity.

An increase in the resistance of the air cleaner (as it becomes contaminated) has, in general, a similar effect on the operation of the throttle governor. However, the limiting delay \(\Delta n_{A\max}\) will here be different, since the decrease in pressure ahead of the carburetor due to an increase in the resistance of the air cleaner, with unchanged exhaust back pressure, entails a decrease in the volumetric efficiency of the engine.

This circumstance must be taken into account when constructing the calculated curves \(N_e\) and \(M_c\) for the specified reduced pressure ahead of the carburetor.

2. Influence of changes in the temperature of the outside air \(T_0\)

An increase in the ambient temperature from \(T_0\) to some \(T_0'\) leads to a corresponding decrease in the specific weight of the air and of the working mixture; the volumetric efficiency of the engine remains almost constant. Consequently, the pressures \(q,\Delta p_{12}\) and the torque \(M_c\) (with unchanged \(\beta\) and \(n\)) decrease in the ratio

\[ \frac{T_0}{T_0'}. \]

In connection with the decrease in the weight filling of the engine, the effective power of the engine decreases.

Obviously, the influence of an increase in temperature \(T_0\) on the operation of the limit pneumatic governor is, in general, analogous to the influence of a reduction in the external pressure \(p_0\). Therefore, the delay in the engagement of the governor when the air temperature is increased from \(T_0\) to \(T_0'\) may be approximately determined by an expression similar to (4):

\[ \Delta n_A=n_A\left(\sqrt{\frac{T_0'}{T_0}}-1\right). \]

Thus, for example, at \(T_0'=300^\circ\) and \(T_0=240^\circ\),

\[ \Delta n_A=0.12\,n_A. \]

A detailed account of the influence of a change in temperature \(T_0\) on the operation of the limit governor may be carried out graphically, by means of constructions similar to those described above in item 1.

3. Effect of Friction in the Governor Mechanism

Above, in considering the question of the delay in the engagement of the governor when the external atmospheric conditions change, and also in the author’s previously mentioned work [1], friction in the governor mechanism was not taken into account.

In connection with the generally small magnitudes of the “driving” moments \(M_c\) on the governor throttle, especially at small angles \(\beta\), this friction (primarily friction on the axis of the governor throttle) may lead to the appearance of appreciable insensitivity of the governor. This insensitivity of the governor will manifest itself, first of all, in a delay in its engagement, in an increase of \(n_A\). Oscillations of the friction forces, inevitable under conditions of poor lubrication, will lead to oscillations of \(n_A\), i.e., to a disturbance of the stability of the governor action.

The insensitivity of the throttle-governor depends not only on the magnitude of the friction forces (or, more precisely, on the moment of the friction forces \(M_{mp}\) relative to the axis of rotation of the governor throttle) and on the magnitude of the “driving” moments \(M_c\), but also on the laws governing the variation of these moments with the engine speed \(n\) and the angle \(\beta\).

A reliable estimate and allowance for the friction forces in a pneumatic governor are hardly possible. The reasons for this lie not only in the nonconstancy of the friction coefficients in the governor mechanism, but also in the difficulties of taking into account such circumstances as oscillations of the throttle during engine operation (owing to pulsations of the working-mixture flow), engine vibrations, etc.

Below, the question of allowing for friction forces in the mechanism of a pneumatic governor is considered in the most simplified form. The purpose of the investigation is not so much to obtain a numerical estimate of the insensitivity of the throttle-governor as to clarify the nature of certain general regularities that are useful to take into account when developing the governor design.

For the friction moment on the axis of the governor throttle, neglecting, as an approximation, the friction in the connection between the spring and the throttle, one may write [4]:

\[ M_{mp}=2M_0+k_1R, \]

where \(k_1\) is the reduced coefficient of friction in the bearings,
\(M_0\) is the friction moment of an unloaded ball bearing,
\(R\) is the radial force acting on the bearings.

The magnitude \(R\) is determined as the resultant of the force of the governor spring \(F\) and the pressure force \(Q\) of the mixture flow on the throttle.

The force \(F\) is directed along the spring axis and is a function of the angle \(\beta\); for each angle \(\beta\) the magnitude of the force \(F\) is readily determined from consideration of the spring deformations.

The force \(Q\), which we take to be normal to the plane of the throttle, may be approximately determined from the expression:

\[ Q=(c_1q+\Delta p_{12})F_z, \]

where \(F_z\) is the throttle area;
\(c_1\) is a coefficient depending on the shape of the throttle and on the angle \(\beta\) (the magnitude of the coefficient \(c\) is determined by blowing the throttle on a special test rig);
\(q\) and \(\Delta p_{12}\) have the same meanings as above, in Sec. 1.

Now, under the condition that the spring axis is perpendicular to the axis of the governor pipe, we have:

\[ R=\sqrt{(F+Q\cos\beta)^2+(Q\sin\beta)^2}. \]

For \(\beta=\mathrm{const}\), the force \(F\) is constant and the friction moment \(M_{mp}\) depends only on the force \(Q\), that is, ultimately, under constant atmospheric

conditions—on the engine speed \(n\). But for \(\beta=\mathrm{const}\) the moment \(M_c\) is also a function of \(n\). Consequently, for a fixed position of the regulating throttle \((\beta=\mathrm{const})\), the friction moment \(M_{tr}\) may be regarded as a function of the moment \(M_c\).

In Fig. 5, by the line \(CD\), in coordinates \(M_c\) (ordinate axis) and \(M_{tr}\) (abscissa axis), the law of variation of the friction moment \(M_{tr}\) on the axis of the regulating throttle is shown for \(\beta=\beta_0\).

Now the driving moment for the throttle will be the moment \(M'\), determined by the difference:

\[ M'=M_c-M_{tr}. \]

Construction of the curve \(M'\) from the curves \(M_c\) and \(M_{tr}\) can be carried out by the simple graphical method shown in Fig. 5: any point 1 on the curve \(M_c\) is projected horizontally to point 2 on the line \(CD\); from point 2 we draw the ray \(2\!-\!3\) at an angle of \(45^\circ\) (in the case of equal scales adopted for \(M_c\) and \(M_{tr}\); with different scales this angle, of course, will be different) to the horizontal, until it intersects the ordinate axis \(M_c\) at point 3; the horizontal through point 3 gives, on the vertical through point 1, point 4 belonging to the curve \(M'\). It follows from Fig. 5 that the presence of friction in the regulator mechanism leads to a decrease of the already small (at small angles \(\beta\)) angle of inclination of the curve of driving moments (now the curve \(M'\)) to the abscissa axis and to a displacement of this curve upward.

Fig. 5

Now, in the presence of friction in the regulator, the initial point \(A\) of the regulator characteristic \(AB\) (Fig. 5) will move from \(A\) (where it would have been in the absence of friction) to some new point \(A_1\). The position of this point \(A_1\) is determined by means of a construction similar to that just described. For this purpose, from point 5 of intersection of the horizontal through the initial point \(a\) of the curve \(M_n\) of the spring moment with the ordinate axis \(M_n, M_c\), we draw the line \(5\!-\!6\) at an angle of \(45^\circ\) (for equal scales adopted for \(M_c\) and \(M_{tr}\)) to the horizontal, until it intersects the curve \(M_{tr}\) at some point 6. The horizontal \(6\!-\!7\) intersects the curve \(M_c\), constructed for \(\beta=\beta_0\), at point 7.

It is evident that the vertical drawn through this point 7 gives, on the external characteristic of the engine, the desired point \(A_1\), and on the abscissa axis—the engine speed at which the controller will come into action. The segment \(\Delta n_A\) will characterize the lag in the engagement of the controller in the presence of friction forces in it, characterized by the curve \(CD\).

But the influence of friction in the controller is not limited to shifting the initial point \(A\) of the controller characteristic to the right, toward higher engine speeds. Owing to friction forces, the entire controller characteristic is shifted to the right, and its end point \(B\) is shifted by a value \(\Delta n_B\), different from \(\Delta n_A\).

To reduce this lag in the engagement of the controller and to diminish the sensitivity of the controller to fluctuations of friction forces in it, it is necessary:

a) to reduce friction in the controller mechanism (by mounting the controller throttle valve on rolling bearings, etc.);

b) to increase the distance from the point of attachment of the spring to the throttle valve to the latter’s axis of rotation (in order to reduce the force \(F\));

c) to increase the eccentricity of the throttle valve (which leads to a decrease in the pressure \(Q\) for the same \(M_c\), especially at large values of the angle \(\beta\));

d) to choose an initial, “setting” angle of the controller throttle valve that is not too small.

REFERENCES

1. V. K. Nechaev. On limiting pneumatic controllers of carburetor automobile engines. Izvestiya TPI, vol. 75.

2. A. S. Orlin et al. Internal Combustion Engines, vol. 1, Moscow, 1951.

3. M. M. Maslennikov and M. S. Rapoport. Aircraft Piston Engines. Moscow, 1951.

4. I. Levin (ed.). Handbook for the Designer of Precision Instruments, Moscow, 1953.