On Some Characteristics of Centrifugal Governors with Ball Weights

V. K. Nechaev

Centrifugal governors with ball weights have become widespread in modern engines and are used, in particular, in domestic high-speed diesels D-6 and D-12.

Below we consider the question of the frictional resistances that arise during oscillations of the sleeve of such a governor.

Knowledge of the magnitude and nature of the resistances in the control system and, in particular, in the governor is necessary for concrete calculations of the transient processes of engine control.

Fig. 1

Let us denote (Fig. 1)

\(\omega\) — the angular velocity of rotation of the governor shaft, which in the subsequent exposition is assumed to be constant,

\(x\) — the instantaneous axial displacement of the sleeve from its equilibrium position (corresponding to the given \(\omega\)),

\(c\) — the stiffness of the governor springs reduced to the sleeve,

\(E\) — the total normal pressure of all the balls on the flat (movable) plate.

The force \(E\) depends on the magnitude of the total centrifugal force \(C_x\) of all the balls, their total additional radial (perpendicular to the axis of rotation of the governor shaft) inertial force \(Q_x\), the angle \(\beta\) of inclination of the generatrix of the fixed conical plate to the shaft axis, and the friction forces \(T_1\) and

\(T_2\), arising at the points of contact of the balls with the plates during radial displacements of these balls.

In addition, the magnitude \(E\) depends on the direction of motion of the balls at the given instant of time (\(E'\)—when the balls move apart, when \(\dfrac{dy}{dt} > 0\), and \(E''\)—when the balls move together, when \(\dfrac{dy}{dt} < 0\)).

For the case in which the balls move apart (move away from the axis of the governor shaft, \(\dfrac{dy}{dt} > 0\)) we have (Fig. 1a)

\[ \begin{aligned} T'_1 + T'_2 \sin \beta + N' \cos \beta &= C_x + Q_x,\\ E' - N' \sin \beta + T'_2 \cos \beta &= 0, \end{aligned} \tag{1} \]

where \(N'\) is the total normal pressure of all the balls on the conical (fixed) plate.

Let

\[ \left. \begin{aligned} T'_1 &= \mu_1 E',\\ T'_2 &= \mu_2 N', \end{aligned} \right\} \tag{2} \]

where \(\mu_1\) and \(\mu_2\) are, respectively, the coefficients of friction between the balls and the flat and conical plates.

Now from (2)

\[ N'=\frac{E'}{\sin\beta-\mu_2\cos\beta} \]

and

\[ E'=\frac{C_x+Q_x}{\dfrac{\cos\beta+\mu_1\sin\beta}{\sin\beta-\mu_2\cos\beta}+\mu_1}. \]

In an analogous manner, for the case in which the balls move together \(\left(\dfrac{dy}{dt}<0\right)\), one can obtain (Fig. 1b)

\[ E''=\frac{C_x+Q_x}{\dfrac{\cos\beta-\mu_2\sin\beta}{\sin\beta+\mu_2\cos\beta}-\mu_1}. \]

In the absence of friction \((\mu_1=\mu_2=0)\), the total pressure of all the balls on the flat plate is determined by the expression (Fig. 2)

\[ E_0=(C_x+Q_x)\operatorname{tg}\beta \tag{3} \]

independently of the direction of the radial motion of the balls.

Now, taking (3) into account, we have

\[ E'=\frac{1}{\left[\dfrac{\cos\beta+\mu_2\sin\beta}{\sin\beta-\mu_2\cos\beta}+\mu_1\right]\cdot \operatorname{tg}\beta}\cdot E_0 \]

\[ E''=\frac{1}{\left[\dfrac{\cos\beta-\mu_2\sin\beta}{\sin\beta+\mu_2\cos\beta}-\mu_1\right]\cdot \operatorname{tg}\beta}\cdot E_0 \]

or, neglecting terms containing products \(\mu_1,\ \mu_2\), approximately:

\[ E' = E_0 \frac{\sin \beta - \mu_2 \cos \beta}{\cos \beta + (\mu_1 + \mu_2)\sin \beta}\cdot \frac{1}{\operatorname{tg}\beta}, \tag{4} \]

\[ E'' = E_0 \frac{\sin \beta + \mu_2 \cos \beta}{\cos \beta - (\mu_1 + \mu_2)\sin \beta}\cdot \frac{1}{\operatorname{tg}\beta}. \tag{5} \]

Whence it follows that

\[ E' < E_0 < E''. \]

It is obvious that the differences

\[ R' = E_0 - E', \tag{6} \]

\[ R'' = E'' - E_0 \tag{7} \]

are the total friction forces of the balls reduced to the sleeve, respectively for the cases of divergence \((R')\) and convergence \((R'')\) of these balls.

Fig. 2

From (4) and (6)

\[ R' = E_0 \frac{\mu_1 \sin \beta + \dfrac{\mu_2}{\sin \beta}} {\cos \beta + (\mu_1 + \mu_2)\sin \beta} \tag{8} \]

and analogously, from (5) and (7),

\[ R'' = E_0 \frac{\mu_1 \sin \beta + \dfrac{\mu_2}{\sin \beta}} {\cos \beta - (\mu_1 + \mu_2)\sin \beta}. \tag{9} \]

From expressions (8) and (9) it follows that

\[ R' < R''. \]

This “asymmetry” of the friction forces increases with increasing coefficients of friction \(\mu_1\) and \(\mu_2\) and angle \(\beta\). But for small \(\mu_1\) and \(\mu_2\), and not too large \(\beta\), the difference between \(R'\) and \(R''\) is insignificant. Here, according to (8) and (9), for both travels of the sleeve one may take the same value of the friction force,

\[ R = E_o \left( \mu_1 \operatorname{tg}\beta + \frac{2\mu_2}{\sin 2\beta} \right). \tag{10} \]

Dependences (8)—(10) for the force \(R\) were obtained on the basis of relation (2), i.e., under the assumption that the friction forces \(T_1\) and \(T_2\) at the points of contact of the balls with the disks obey Coulomb’s law. But the actual laws of friction of the balls in the governor under consideration are considerably more complex and require special consideration, taking into account the influence of lubrication, the dimensions of the balls, the complex kinematics of their motion during oscillations of the governor sleeve, etc. An analysis of these laws is not part of the purpose of this article.

Experimental investigations of the resistances in the governors of D-6 and D-12 engines, carried out by E. V. Kovalenok in the internal-combustion-engine laboratory of the Tomsk Polytechnic Institute, showed that for \(\omega = \mathrm{const}\) the friction force \(R\), reduced to the sleeve, is practically directly proportional to the instantaneous velocity of the axial motion of the sleeve.\(^1\)

This gives us grounds to assume that the coefficients of friction \(\mu_1\) and \(\mu_2\) in relations (2) are directly proportional to the velocities of radial displacement, along the disks, of the points of contact of the balls with these disks,

\[ \mu_1 = \Psi \frac{dy}{dt}, \]

\[ \mu_2 = \frac{\Psi}{\sin \beta}\frac{dy}{dt}, \]

where \(\Psi\) is a certain coefficient depending on the characteristics of the lubricating oil, the quality of machining of the surfaces of the balls and disks, etc.

Consequently, instead of (10) one may write

\[ R = - \Psi E_o \left( \mu_1 \operatorname{tg}\beta + \frac{2\mu_2}{\sin 2\beta} \right)\cdot \frac{dy}{dt} \tag{11} \]

or, in general form,

\[ R = - k \frac{dx}{dt}, \tag{12} \]

where

\[ k \]
— is the coefficient of linear (viscous) friction in the governor, reduced to the sleeve,

\[ \frac{dx}{dt} \]
— is the instantaneous velocity of displacement of the sleeve along the axis of the governor shaft.

Now, for free oscillations of the governor sleeve, disconnected from the engine and the fuel pump and rotating with constant angular

\(^1\) The investigations were carried out on a pendulum apparatus similar to the apparatus developed by V. Krutov at NAMI.

with speed \(\omega\) (for example, from an electric motor—when testing the governor on a pendulum installation), one may write

\[ (M+m_{\sh}\operatorname{tg}^2\beta)\frac{d^2x}{dt^2} +k\frac{dx}{dt} +(c-m_{\sh}\omega^2\operatorname{tg}^2\beta)x=0, \]

where

\[ \begin{aligned} M&\text{ is the mass, reduced to the sleeve, of all governor parts moving during axial}\\ &\text{ displacements of the sleeve (excluding the balls),}\\ m_{\sh}&\text{ is the total mass of all the balls,}\\ m_{\sh}\operatorname{tg}^2\beta&\text{ is the mass of the balls reduced to the sleeve,}\\ c-m_{\sh}\omega^2\operatorname{tg}^2\beta&\text{ is the so-called stability factor of the governor.} \end{aligned} \]

From (11) and (12)

\[ k=\Psi E_0\left(\operatorname{tg}^2\beta+\frac{1}{\sin\beta\cdot \cos^2\beta}\right) \tag{13} \]

and

\[ k=\Psi(C_x+Q_x)\left(\operatorname{tg}^3\beta+\frac{1}{\cos^3\beta}\right) \]

or, for \(Q_x\ll C_x\),

\[ k\simeq \Psi C_x\left(\operatorname{tg}^3\beta+\frac{1}{\cos^3\beta}\right). \]

Fig. 3

Thus, the coefficient \(k\), for \(E_0=\mathrm{const}\), increases rapidly with an increase in the angle \(\beta\) of inclination of the generatrix of the dish cone to the axis of the governor shaft. For example, increasing the angle \(\beta\) from \(60\) to \(70^\circ\) leads to an increase of the coefficient \(k\) by \(104\%\) (Fig. 3).

Therefore, experimental data on the coefficient \(k\), obtained when testing one governor, cannot, without the appropriate recalculations, be trans-

to the same governor, differing from the first only by the angle \(\beta\). This recalculation should be carried out on the basis of expression (13).

Finally, the strong dependence of the coefficient \(k\) on the angle \(\beta\) gives the designer certain possibilities for controlling the resistances in the engine-control system in order to obtain the desired quality characteristics of regulation. By adopting different values of the angle \(\beta\), one can obtain different values of \(k\) with unchanged quality of lubrication, quality of machining of the balls and rolling surfaces, etc.