On the Question of Resistances in the Control System of a High-Speed Diesel
Abstract Generated abstract
The paper examines resistance forces in the control system of a high-speed diesel engine, focusing on friction in plunger-type fuel pumps reduced to an equivalent resistance at the fuel rack. It first derives the conventional viscous-friction model, then extends it by including a velocity-independent friction component on the plunger surface and the combined effects of rotational and translational plunger motion. The resulting expressions show that the effective coefficient of resistance can decrease as plunger translational speed, and thus pump camshaft speed, increases, in agreement with reported experiments. The analysis also suggests that at small rotational slip velocities the constant friction component behaves like an additional viscous term and does not introduce nonlinearity into the control system.
Full Text
ON RESISTANCES IN THE CONTROL SYSTEM OF A HIGH-SPEED DIESEL ENGINE
V. K. NECHAEV
The fuel pump is one of the main sources of resistances in the regulating system of a diesel engine.
The friction arising during the mutual displacement of the elements of the fuel pump in the process of regulation is usually taken into account in aggregate form as a certain resistance, reduced to the rack, to the displacement of the rack \(R\). The magnitude \(R\) includes, in a certain way, the friction forces in the rack guides, in the gear supports, in the engagement of the gears with the rack, and the friction forces between the plungers and the sleeves, arising during the processes of changing the fuel delivery. These latter resistances (to the rotation of the plungers in the sleeves) are usually the principal components in the magnitude \(R\), and only they are considered and taken into account in the present article.
Figure 1 shows an elementary kinematic diagram of an ordinary plunger-type fuel pump.
Fig. 1
Here \(r_1\) and \(r_2\) are, respectively, the radius of the plunger and the initial radius of the gear,
\(v_p\) — the instantaneous velocity of displacement of the rack in the process of regulation,
\(v_n\) — the corresponding circumferential velocity on the lateral (cylindrical) surface of the plunger;
\(T\) — the tangential friction force on the surface of the plunger, arising during rotation of the plunger and perpendicular to its axis,
\(F = 2\pi r_1 l\) — the lateral surface of the plunger in contact with the sleeve,
\(i\) — the number of plunger pairs in the fuel pump under consideration.
If it is assumed that between the plunger and the sleeve there is only purely fluid friction, then one may write
\[ T=\eta \frac{F}{\delta}v_n, \]
where \(\eta\) is the coefficient of viscosity of the fuel in the clearance between the plunger and the sleeve,
\(\delta\) is the magnitude of this (radial) clearance.
Now the resistance force \(R\), reduced to the rack, opposing rotation of the plungers will be determined by the expression
\[ R=iT\frac{r_1}{r_2}=i\eta\frac{F}{\delta}\frac{r_1}{r_2}v_n =i\eta\frac{F}{\delta}\frac{r_1^2}{r_2^2}v_p \]
or
\[ R=k_p v_p, \]
where
\[ k_p=i\eta\frac{F}{\delta}\frac{r_1^2}{r_2^2}. \tag{1} \]
The quantity \(k_p\) may be regarded as a certain conventional coefficient of viscous friction in the fuel pump, reduced to the rack (that is, proportional to the velocity of displacement of the rack).
But experiments do not confirm dependence (1).
Experimental investigations of resistances in plunger-type fuel pumps, carried out by V. Krutov at NAMI [1] and by E. V. Kovalenok in the internal-combustion-engine laboratory of the Tomsk Polytechnic Institute [2], showed that the coefficient \(k_p\) decreases with increasing rotational speed of the pump shaft, that is, with increasing translational velocity of the plunger \(v_0\) ^1).
Let us consider the question of the coefficient \(k_p\) in a somewhat more general form (Fig. 2). We shall assume that the friction force \(dQ\) on an elementary area \(dF\) of the lateral surface of the plunger contains a certain constant component \(dQ_0\) (independent of the sliding velocity) and obeys the two-term dependence
\[ dQ=\frac{\eta w}{\delta}\,dF+dQ_0 \]
or
\[ dQ=w\,dq+\lambda_0 dF, \]
where it is denoted
\[ q=\eta\frac{F}{\delta}, \]
\[ \lambda_0=\frac{dQ_0}{dF}. \]
Fig. 2
^1) At a constant engine speed \(n\), the velocity \(v_0\) varies continuously in accordance with the profile of the fuel-pump cams. Below, \(v_0\) means the average value of this velocity over one revolution of the pump shaft.
Here \(w\) is the total instantaneous slip velocity on the side surface of the plunger
\[ w=\sqrt{v_n^2+v_0^2}. \]
The force \(Q\) is directed opposite to the velocity vector \(w\). Now, according to Fig. 2,
\[ T=\int_F dQ \sin \alpha=Q_0\sin\alpha+qw\sin\alpha=Q_0\sin\alpha+qv_n, \]
where
\[ \sin\alpha= \frac{\dfrac{v_n}{v_0}} {\sqrt{1+\left(\dfrac{v_n}{v_0}\right)^2}}. \]
Therefore
\[ T=qv_n+ \frac{Q_0\dfrac{v_n}{v_0}} {\sqrt{1+\left(\dfrac{v_n}{v_0}\right)^2}}. \tag{2} \]
Now
\[ R=i\,\frac{r_1}{r_2} \left( q+\frac{Q_0} {v_0\sqrt{1+\left(\dfrac{v_n}{v_0}\right)^2}} \right)\cdot v_n \]
or
\[ R=iF\,\frac{r_1^2}{r_2^2} \left( \frac{\eta}{\delta}+ \frac{\lambda_0} {v_0\sqrt{1+\left(\dfrac{v_n}{v_0}\right)^2}} \right)\cdot v_p . \tag{3} \]
Hence
\[ R=k_p v_p, \]
where
\[ k_p=iF\,\frac{r_1^2}{r_2^2} \left( \frac{\eta}{\delta}+ \frac{\lambda_0} {v_0\sqrt{1+\left(\dfrac{v_n}{v_0}\right)^2}} \right). \tag{4} \]
Here the value of the coefficient \(k_p\) depends not only on the viscosity \(\eta\) of the fuel and the clearance \(\delta\), but also on the absolute value of the velocity \(v_0\) and on the ratio \(v_n/v_0\). For a small value of this ratio, instead of (3) and (4) we approximately have
\[ T=i\left(q+\frac{Q_0}{v_0}\right)\cdot v_n \tag{5} \]
and
\[ k_p=iF\,\frac{r_1^2}{r_2^2} \left( \frac{\eta}{\delta}+\frac{\lambda_0}{v_0} \right). \tag{6} \]
Consequently, when the velocity \(v_n\) is small in comparison with the velocity \(v_0\), the entire tangential friction force \(T\) proves to be directly proportional to \(v_n\). Thus, in plunger pairs, during transient control processes, there occurs as it were a transformation of the constant component \(Q_0\) into fric-
…viscous. This transformation is due to the influence of the translational velocity of the plunger \(v_0\). As a result, the presence of the constant \(Q_0\) in the friction forces on the lateral surface of the plunger does not introduce nonlinearity into the control system.
The coefficient \(k_p\) decreases with increasing velocity \(v_0\), that is, ultimately, with an increase in the number of revolutions \(n\) of the camshaft of the fuel pump per minute (Fig. 3). However, this dependence of \(k_p\) on \(n\) weakens as the velocity \(v_n\) increases. For
\[ \frac{v_n}{v_0} \gg 1 \]
instead of (3) and (4) we have
\[ T = qv_n + Q_0 \]
and
\[ k_p = iF \frac{r_1^2}{r_2^2}\left(\frac{\eta}{\delta} + \frac{\lambda_0}{v_n}\right). \]
Fig. 3
Here the “transformation” of the constant friction \(Q_0\) into viscous friction already ceases, and the coefficient \(k_p\) falls with increasing velocity \(v_n\).
For
\[ \frac{v_n}{v_0} \ll 1 \]
the decrease of the coefficient \(k_p\) with increasing \(n\) may be still more rapid than follows from formula (4), owing to a decrease in the viscosity of the fuel \(\eta\), caused by increased heating of the fuel in the gap \(\delta\) at high mean velocities of the translational motion of the plungers \(v_0\).
REFERENCES
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Krutov V. I. Braking factor of the fuel-supply system of a diesel engine, Proceedings of NAMI, issue 57, 1949.
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Kovalenko E. V. Investigation of resistances in the control system of high-speed compression-ignition engines (types D-6 and D-12), Candidate’s dissertation, Tomsk Polytechnic Institute, 1954.