On the Calculation of Mechanisms of Certain Limit Pneumatic Regulators

V. K. NECHAEV

As is known, the characteristics of a limit pneumatic regulator (a maximum-speed limiter) of a carburetor automobile engine depend to a considerable degree on the laws governing the variation of the moment \(M_p\), developed by the spring on the regulator throttle, with respect to the angle of rotation \(\beta\) of this throttle.¹

Satisfactory operation of a pneumatic regulator can be ensured only if the actual law of variation of the moment \(M_p\) with respect to the angle \(\beta\) does not differ substantially from the required law established in the calculation of the regulator. The required curve \(M_p=f(\beta)\), in the coordinates \(M_p\) and \(\beta\), usually has a comparatively complex form and a curvature varying with the angle \(\beta\). In connection with the rapid increase of the aerodynamic moment \(M_c\) (created on the throttle by the flow of the working mixture) with the angle of rotation of the throttle, the curve \(M_p\) must have a steep rise in the region of large values of \(\beta\) (Fig. 1).

The initial point \(a\) of the curve \(M_p\) is strictly fixed. Its abscissa corresponds to the selected “setting” (initial) angle \(\beta_0\) of the regulator throttle. Its ordinate \(M_{p0}\) determines the position of the initial point of the regulator characteristic, that is, that engine speed \(n_a\) at which the regulator begins to come into action.

Fig. 1

This curve \(M_p=f(\beta)\) must be realized as accurately as possible by choosing the appropriate geometry of the linkage mechanism connecting the spring with the throttle and by selecting the corresponding characteristics of this spring. Here, however, a compromise is usually unavoidable: the required curve \(M_p\), obtained from the calculation of the regulator for a specified regulator characteristic, must be somewhat corrected so as to bring its form closer to the form of those curves \(M_p=f(\beta)\) that can be produced by the selected type of linkage mechanism connecting the regulator spring with the throttle. The limits of this correction are established according to the permissible deviations from the originally prescribed regulator characteristic of the engine.

In limit pneumatic regulators, cam and hinged-lever mechanisms are used for connecting the spring with the throttle. Cu-

A cam mechanism is used, for example, in the Handi limiting regulator. Here the required complex law of variation of the moment \(M_n\) with the angle \(\beta\) is ensured by selecting the appropriate profile of the cam fixed on the axis of the regulator throttle. Linkage mechanisms are used in regulators installed on domestic GAZ-51 and ZIS-120 engines. By selecting the parameters of such a mechanism, one can, with a greater or lesser degree of approximation, realize the required law of variation of the moment \(M_n\) with the angle \(\beta\). However, the method of such selection, as far as the author knows, has not been developed at all.

Below only certain linkage (camless) mechanisms connecting the regulating spring with the throttle are considered. The purpose of the study is to establish methods for calculating the parameters of such mechanisms for a prescribed curve \(M_n\).

Such a problem is also of known general interest, as one of the problems of designing a mechanism with a preassigned nonlinear characteristic of its elastic properties. In the general case, the question considered below reduces to the problem of designing a crank-slider mechanism with a linear (i.e., obeying Hooke’s law) spring that provides a prescribed nonlinear law of variation of the torque on the crankshaft with the angle of rotation of this shaft. To use the conclusions obtained below, it is sufficient here to replace the regulating throttle by a crank.

1. Direct connection of the spring with the throttle

In the simplest case (Fig. 2), the force \(F\) of the spring is applied to the throttle at a certain point \(A_0\), through a shackle \(A_0B\) hinged to the throttle and to the spring. Point \(A_0\) is located in the mid-plane of the throttle. The shackle \(A_0B\) is always set in the direction of the force \(F\), i.e., along the axis of the spring.

During operation of the regulator, the angle \(\beta\) may vary in the interval from \(\beta=\beta_0\) to some \(\beta=\beta_{\max}\), corresponding to full closing of the throttle. With a sufficiently great length of the spring (or, more precisely, with a sufficiently large distance of the point of attachment of the right end of the spring from the axis of rotation of the throttle), the change in the angle of inclination of the spring axis as the throttle closes may be neglected (see below, Sec. 3). Accordingly, here and in Sec. 2 it is assumed that the axis of the spring, and consequently also the shackle \(A_0B\), remains horizontal for all values of the angle \(\beta\), i.e., perpendicular to the axis of the regulator pipe.

Fig. 2

Let us denote:

\(F_n\) — the initial tension of the regulating spring when the throttle is in the vertical (in Fig. 2) position \(\beta=0\);

\(F_0\) — the spring force at the “set” position of the regulating throttle \(\beta=\beta_0\);

\(F\) — the spring force at an arbitrary position of the throttle (at any angle \(\beta\) to the vertical);

\(c\) — the stiffness of the regulating spring;

\(z\) — the elongation deformation of the spring compared with its length at \(\beta=0\);

\(h\) is the arm of the force \(F\) relative to the axis \(O\) of rotation of the flap;

\(a\) is the distance from the point \(A_0\) at which the shackle \(AB\) is attached to the flap to the axis of rotation of the latter.

According to Fig. 2,

\[ z = a \sin \beta; \qquad h = a \cos \beta . \]

Now:

\[ F = F_n + cz = F_n + ca \sin \beta, \tag{1} \]

\[ F_0 = F_n + ca \sin \beta_0 \]

and the moment developed by the force \(F\) about the axis of the flap is:

\[ M_n = Fh = (F_n + ca \sin \beta)a \cos \beta \]

or

\[ M_n = ca^2(\lambda + \sin \beta)\cos \beta, \tag{2} \]

where the following notation has been introduced:

\[ \lambda = \frac{F_n}{ca^2}. \tag{3} \]

The curves of variation of the “specific” moment \(\dfrac{M_n}{ca^2}\) as a function of the angle \(\beta\), calculated from expression (2) for various values of the coefficient of initial tightening \(\lambda\), are presented in Fig. 3 (curves 1, 2, 3). It is easy to see that the form of these curves by no means satisfies the requirements imposed on it: the steepness of the curves here decreases with increasing \(\beta\), whereas the specified regulating characteristic of the engine dictates exactly the opposite. The maximum of the curves \(M_n\) is reached here before the value \(\beta = 45^\circ\), and the earlier the greater \(\lambda\) is. Consequently, the interval of possible working angles \(\beta_{\max} - \beta_0\) is very small here and does not even reach \(35 \div 40^\circ\).

Fig. 3

To improve the character of the curves \(M_n\), it is expedient to move the point \(A\) out of the mid-plane of the flap. In Fig. 4, the swing axis of the shackle is fixed at a certain point \(A\) below the plane of the flap, so that the line \(OA\) makes a certain angle \(\gamma\) with the mid-plane of the flap. The point \(A\) is located on the normal \(A_0A\) to the plane of the flap.

According to Fig. 4:

\[ z = a_1 \sin \gamma + a_1 \sin(\beta-\gamma), \]

\[ F = F_n + ca_1[\sin \gamma + \sin(\beta-\gamma)] \tag{5} \]

and the spring moment:

\[ M_n = [F_n + ca_1 \sin \gamma + ca_1 \sin(\beta-\gamma)]a_1 \cos(\beta-\gamma) \]

or

\[ M_n = ca_1^2[\lambda_1 + \sin(\beta-\gamma)]\cos(\beta-\gamma), \tag{6} \]

where

\[ \lambda_1 = \frac{F_n}{ca_1} + \sin \gamma = \lambda \cos \gamma + \sin \gamma. \tag{7} \]

Here, as before, \(F_n\) is the initial tension of the spring at \(\beta = 0\).

Fig. 4

In Fig. 5 (curves 4–7) are shown graphs of the variation of the ratio \(\dfrac{M_n}{ca^2}\) as a function of the angle \(\beta\), calculated from expression (6) for a series of values of \(\lambda\) and \(\gamma\). It follows from Fig. 5 that moving the point \(A\) at which the shackle \(AB\) is attached out of the middle plane of the damper leads to a substantial change in the curves \(M_n = f(\beta)\). The interval of angles \(\beta\) of rotation of the regulating damper in which a continuous increase of the moment \(M_n\) with respect to the angle \(\beta\) occurs is considerably widened and, for example, at \(\gamma = 60^\circ\) already amounts to more than \(90^\circ\). But the wider this interval, the easier it is to select on the curves \(\dfrac{M_n}{ca^2}\) a segment satisfying the requirements of the form and character of the necessary (specified) curve \(M_n = f(\beta)\).

It now remains to choose the corresponding values of the quantities \(\gamma, a_1, c\), and \(F_n\) in order to ensure coincidence of the actual curve \(M_n\) with the necessary curve—at least at three specified points, provided only that the character of the necessary curve \(M_n\) is similar to the character of the curves presented in Fig. 5.

Let these three specified points of the curve \(M_n\) (Fig. 1) have, respectively, abscissas \(\beta_0, \beta_1, \beta_2\) and ordinates \(M_{n0}, M_{n1}, M_{n2}\). Then, using expression (6), one may write:

\[ \begin{aligned} M_{n0} &= ca_1^2[\lambda_1+\sin(\beta_0-\gamma)]\cos(\beta_0-\gamma),\\ M_{n1} &= ca_1^2[\lambda_1+\sin(\beta_1-\gamma)]\cos(\beta_1-\gamma),\\ M_{n2} &= ca_1^2[\lambda_1+\sin(\beta_2-\gamma)]\cos(\beta_2-\gamma). \end{aligned} \tag{8} \]

Fig. 5

We form the ratios:

\[ a_1=\frac{M_{n1}}{M_{n0}} = \frac{[\lambda_1+\sin(\beta_1-\gamma)]\cos(\beta_1-\gamma)} {[\lambda_1+\sin(\beta_0-\gamma)]\cos(\beta_0-\gamma)}, \tag{9} \]

\[ a_2=\frac{M_{n2}}{M_{n0}} = \frac{[\lambda_1+\sin(\beta_2-\gamma)]\cos(\beta_2-\gamma)} {[\lambda_1+\sin(\beta_0-\gamma)]\cos(\beta_0-\gamma)}. \tag{10} \]

From (9):

\[ \lambda_1= \frac{1}{2}\, \frac{\sin 2(\beta_1-\gamma)-a_1\sin 2(\beta_0-\gamma)} {a_1\cos(\beta_0-\gamma)-\cos(\beta_1-\gamma)}, \tag{11} \]

and, analogously, from (10):

\[ \lambda_1= \frac{1}{2}\, \frac{\sin 2(\beta_2-\gamma)-a_2\sin 2(\beta_0-\gamma)} {a_2\cos(\beta_0-\gamma)-\cos(\beta_2-\gamma)}. \tag{12} \]

Equating the right-hand sides of expressions (11) and (12), we obtain an equation in which the only unknown quantity is the angle \(\gamma\). But a solution of this equation with respect to \(\gamma\) is hardly possible in general form. In practice, the required value of the angle \(\gamma\) can be found from relations (11) and (12) by trial selection or graphically, as the abscissa of the point of intersection of two curves \(\lambda_1=f_1(\gamma)\) and \(\lambda_1=f_2(\gamma)\), constructed point by point, respectively from expressions (11) and (12), for a number of arbitrary values of the angle \(\gamma\).

Now, from design considerations, we accept the value of the radius \(a_1\) and, for example, from the first of expressions (8) we find the required stiffness of the regulating spring:

\[ c=\frac{M_{n0}}{a_1^2[\lambda_1+\sin(\beta_0-\gamma)]\cos(\beta_0-\gamma)} ; \tag{13} \]

the quantity \(\lambda_1\) entering here, with \(\gamma\) already known, is determined from expressions (11) or (12).

Next, from formula (7):

\[ F_n=ca_1(\lambda_1-\sin\gamma). \tag{14} \]

Whence the initial preload of the spring \(F_n\) is determined. Finally, from the expression:

\[ F_0=F_n+ca_1[\sin\gamma+\sin(\beta_0-\gamma)] \tag{15} \]

we find the preload of the spring at \(\beta=\beta_0\), whose value is needed during assembly and adjustment of the regulator.

At this point the principal calculation of the regulator mechanism may be considered complete. The values of the parameters \(\gamma\), \(c\), and \(F_0\) calculated above, for the chosen \(a_1\), ensure the coincidence of the actual curve of the moment \(M_n\) with the specified (required) one at three preselected points of the latter.

Fig. 6

However, such coincidence can by no means always be obtained, since the assortment of possible forms of the curves in Fig. 3 is limited. Here there are no, for example, curves like those in Fig. 9, which, over a small interval of angles \(\beta_2-\beta_0\), have a slight rise at the beginning and a rapid rise at the end. Analytically this is expressed in the fact that, for a sufficiently large difference \(a_2-a_1\) and a small interval \(\beta_2-\beta_0\), equations (11) and (12) for \(\lambda_1\) and \(\gamma\) prove incompatible.

In Fig. 6, as an example, a family of curves \(M_n=f(\beta)\) is presented, which is given by the mechanism of Fig. 4, between two specific prescribed points—

by points 1 and 2. Curves passing through these points, but situated in the interval 1–2 below a certain limiting line (corresponding in the present example to the value \(\gamma \simeq 65^\circ\)), cannot be provided by the mechanism of Fig. 4.

2. Mechanism with limited motion of the shackle \(AB\)

In domestic designs of limiting regulators (carburetors K—49, K—80, etc.) a somewhat more complex design is used for the mechanism connecting the regulating spring with the throttle. Here the angular displacements of the shackle \(AB\) relative to the throttle are limited by a special projection (stop) on the latter (Fig. 7).

Such an arrangement has certain advantages in comparison with the mechanism considered above (Fig. 4), where the shackle \(AB\) was always set in the direction of the spring force. The mechanism of Fig. 7 provides the possibility:

Fig. 7

a) of obtaining a steep rise of the curve \(M_n\) in the region of large values of the angle \(\beta\), with a comparatively slow rise of it at small \(\beta\);

b) of obtaining a curve \(M_n\) with a sharp break at some prescribed point of it, with abscissa \(\beta_y\);

c) of satisfying the required curve at four or five selected points (provided the form of this curve is suitable).

Here it is necessary to consider two different periods of operation of the mechanism of Fig. 7: before and after engagement of the stop.

a) Before engagement of the stop \((\beta \leqslant \beta_y)\)

The limiting stop (in Figs. 4 and 8, shown conventionally in the form of a certain post \(E\), fixed on the throttle below its plane) permits the shackle \(AB\), rocking about point \(A\), to deviate only through a certain angle \(\varphi_y\) from the normal to the radius \(OA\).

For positions of the throttle at angles \(\beta\) to the vertical smaller than \(\varphi_y + \gamma\), and with the horizontal position of the axis of the spring, the shackle \(AB\) is also situated horizontally (see Fig. 4; here the stop \(E\) is shown by a dashed line).

It is obvious that up to the instant of engagement of the stop (that is, up to the instant of contact of the shackle with the projection on the throttle, which occurs at \(\beta_y=\varphi_y+\gamma\)) the regulator mechanism remains completely identical with the mechanism considered above in Fig. 4 (without a stop). Consequently, here the same—

the expressions (5) and (6) obtained above are valid. The law of variation of the ratio \(\dfrac{M_n}{c a^2}\) for angles \(\beta \leq \beta_y\) will be represented by the same curves shown in Fig. 3.

b) After engagement of the stop \((\beta > \beta_y)\)

Now (for \(\beta > \varphi_y+\gamma\)) the link \(AB\) no longer remains horizontal: as the throttle closes it is deflected downward (Fig. 7); the triangle \(AOB\) becomes rigid. At the instant the stop engages, this triangle occupies the position \(A_yOB_y\), shown by dashed lines in Fig. 8, with the horizontal side \(A_yB_y\). Now the spring force \(F_y\) acts at the point \(B_y\), the magnitude of which, according to (5), is determined by the expression:

\[ F_y = F_{\mathrm{н}} + c a_1[\sin \gamma + \sin \varphi_y], \tag{16} \]

since

\[ \beta_y = \varphi_y + \gamma. \]

Fig. 8.

With a further increase in the angle \(\beta\), the triangle \(AOB\) rotates about the point \(O\) together with the throttle; the point \(B\) moves along an arc of radius \(a_2 = OB\) from the center \(O\). In this process the spring receives an additional elongation \(\Delta z\); correspondingly, the spring force also increases:

\[ F = F_y + c\Delta z. \tag{17} \]

According to Fig. 8,

\[ \Delta z = a_2\sin(\gamma' - \varphi_y) - a_2\sin\psi \]

or

\[ \Delta z = a_2\sin(\gamma' - \varphi_y) + a_2\sin(\beta - \gamma''), \]

where \(\gamma'\) is the angle at the vertex \(O\) of the rigid (for \(\beta > \beta_y\)) triangle \(AOB\), and

\[ \psi = \gamma'' - \beta. \]

Here \(\gamma''\) is the angle between the radius \(OB\) and the plane \(OA_0\) of the shutter:

\[ \gamma''=\gamma+\gamma'. \tag{18} \]

Now, instead of expression (17), one may write:

\[ F=F_{\mathrm{н}}+ca_1(\sin\gamma+\sin\varphi_y)+ca_2[\sin(\gamma'-\varphi_y)+\sin(\beta-\gamma'')] \]

or

\[ F=ca_2[\lambda_2+\sin(\beta-\gamma'')], \]

where it is denoted that

\[ \lambda_2=\lambda_1\,\frac{a_1}{a_2}+\frac{a_1}{a_2}(\sin\gamma+\sin\varphi_y)+\sin(\gamma'-\varphi_y). \tag{19} \]

Here, as before:

\[ \lambda_1=\frac{F_{\mathrm{н}}}{ca_1}, \]

and \(F_{\mathrm{н}}\) is the spring preload at \(\beta=0\).

The arm of the force \(F\):

\[ h=a_2\cos\psi=a_2\cos(\beta-\gamma''). \]

The moment developed by the spring on the shutter at \(\beta\geqslant\beta_y\):

\[ M_{\mathrm{п}}=Fh=ca_2^2[\lambda_2+\sin(\beta-\gamma'')]\cdot\cos(\beta-\gamma''). \tag{20} \]

The quantities \(\gamma'\) and \(a_2\) entering expressions (18) and (20) can be determined as follows. From triangle \(AOB\):

\[ \frac{l}{a_1}=\frac{\sin\gamma'}{\cos(\gamma'-\varphi_y)}, \tag{21} \]

\[ \frac{a_2}{a_1}=\frac{\cos\varphi_y}{\cos(\gamma'-\varphi_y)}. \]

Hence

\[ \tan\gamma'=\frac{l\cos\varphi_y}{a_1-l\sin\varphi_y}, \tag{22} \]

\[ a_2=a_1\frac{\cos\varphi_y}{\cos(\gamma'-\varphi_y)} \tag{23} \]

where \(l\) is the length of the shackle \(AB\).

Depending on the magnitude of \(l\) and of the other mechanism parameters entering formulas (6) and (12), the curve \(M_{\mathrm{п}}=f(\beta)\) will have a different form and even a sharp break at the point corresponding to the instant at which the stop is engaged (Fig. 9).

The formulas obtained above may be used for designing the regulator mechanism (Fig. 7) according to a prescribed (required) curve \(M_{\mathrm{п}}=f(\beta)\). At present, in these formulas there are nine unknowns: \(c, F_{\mathrm{н}}, \gamma, \gamma'\) (or \(\gamma''\)), \(\varphi_y, a, a_1, a_2\), and \(l\). But the quantities \(l, a_1, a_2, \varphi_y\), and \(\gamma'\) are related to one another by relations (21) and (23); in addition,

\[ a=a_1\cos\gamma. \]

The radius \(a_1\) (or \(a\)) should be chosen on the basis of design considerations,\(^1\) since its absolute value does not affect the character of the course of the curve

\(^1\) Here it is necessary to take into account that a decrease in the radius \(a_1\) leads to a corresponding increase in the spring forces \(F\), that is, to an increase in the loads on the axis of the regulating shutter and to an increase in the friction forces on it. The result of this (for a small value of \(a_1\)) may be the appearance of appreciable insensitivity of the regulator.

\(M_n=f(\beta)\) and determines only the absolute values of the required stiffness \(c\) of the regulating spring and its initial preload \(F_n\). As a result, five unknowns remain: \(c, F_n, \gamma, \gamma'\), and \(\varphi_y\), by the selection of which the required curve \(M_n=f(\beta)\) can be ensured at four or five points (provided, of course, that this curve has a suitable form).

In some cases it is expedient to choose in advance the moment at which the stop is engaged (if, for example, the required curve \(M_n\) has a clearly expressed break point, after which it rises especially steeply). Here a new condition appears:

\[ \varphi_y=\beta_y-\gamma \]

and the required curve \(M_n\) can be ensured at four points (including the break point). Let us consider this case as an example.

Fig. 9

Let four points of the curve \(M_n=f(\beta)\) (Fig. 10) be given, with fixed abscissas \(\beta_0, \beta_1, \beta_2, \beta_3\) and the corresponding ordinates \(M_{n0}, M_{n1}, M_{n2}, M_{n3}\). Let us set the moment at which the stop is engaged at the point with abscissa \(\beta=\beta_2\), that is, let us take

\[ \beta_y=\beta_2 . \]

On the basis of expressions (16) and (20), we now have:

\[ M_{n0}=c a_1^2[\lambda_1+\sin(\beta_0-\gamma)]\cos(\beta_0-\gamma), \tag{24} \]

\[ M_{n1}=c a_1^2[\lambda_1+\sin(\beta_1-\gamma)]\cos(\beta_1-\gamma), \tag{25} \]

\[ M_{n2}=Ca_1^2[\lambda_1+\sin(\beta_2-\gamma)]\cos(\beta_2-\gamma), \tag{26} \]

\[ M_{n3}=Ca_2^2[\lambda_2+\sin(\beta_3-\gamma'')]\cos(\beta_3-\gamma''). \tag{27} \]

For the point with abscissa \(\beta=\beta_2\), one could also write the relation

\[ M_{n2}=Ca_2^2[\lambda_2+\sin(\beta_2-\gamma'')]\cos(\beta_2-\gamma''), \]

which, however, in the presence of the already existing relations (22) and (23), is equivalent to dependence (26).

We choose the value of the radius \(a_1\). Then, by jointly solving equations (24), (25), and (26) in the manner indicated above on p. 270, we find the sought quantities \(\gamma\), \(C\), \(F_n\), as well as \(a\), \(\lambda_1\), \(\lambda_2\), and the angle \(\varphi_y=\beta_y-\gamma\).

Fig. 10

It now remains to determine the required value of the radius \(a_2\) and the closely related values of the angle \(\gamma''\) and the length of the shackle \(l\). For this purpose one should use expression (27) for the ordinate of the extreme right fixed point \((\beta=\beta_3)\) of the moment curve \(M_n\) in Fig. 10. But the value \(a_2\) can be found from this only by trial: we assign the expected value of the angle \(\gamma''\) and, from formula (23), find the corresponding value of \(a_2\), and from expression (21) the length of the shackle \(l\). Next, by formula (19) we calculate the coefficient \(\lambda_2\) and, finally, by expression (27) we compute the moment \(M_{n3}\). This value of the moment \(M_{n3}\) is compared with the required value of the spring moment at the point \(\beta=\beta_3\). If, for example, the value of the moment \(M_{n3}\) found from expression (27) proves to be greater than the required one, it is necessary to recalculate, taking in relations (19) and (23) a new, somewhat smaller value of the angle \(\gamma''\), etc.

To check the results obtained, it is advisable to compute, by formula (20), several intermediate points of the moment curve \(M_n\) on the interval between \(\beta=\beta_2\) and \(\beta=\beta_3\). In the case of a significant deviation of these points from the prescribed curve \(M_n=f(\beta)\), a recalculation must be performed, proceeding, for example, from some new value of the angle \(\beta_y\) corresponding to the moment of engagement of the stop.

3. Allowance for the influence of the finite length of the regulator spring

Above, in subsections 1 and 2, it was assumed that at any angles of rotation \(\beta\) of the regulator flap, the direction of the force \(F\) of the spring remains unchanged (horizontal in Figs. 4 and 7). But such a picture can occur only

in the case where the fastening of the right end of the spring is placed very far from the axis of rotation of the flap. Only under this condition are all the preceding layouts and formulas valid, and the segment \(z\) represents the absolute extension of the spring in comparison with its length when the flap is in the vertical position.

Let the right end of the spring be fastened at some point \(D\) at a comparatively small distance from the axis \(O\) (Fig. 11); \(l_1\) and \(l_2\) are the coordinates of this point. Now the axial line of the spring is inclined, at a certain angle \(\alpha\) to the horizontal (variable as a function of \(\beta\)). The spring receives an additional extension, as a result of which the spring force increases from the value \(F\) (which would be in the spring at \(\alpha=0\)) to some new value \(F'\). At the same time the arm of the spring force relative to the axis of rotation of the flap decreases from the magnitude \(h\) (when the spring axis remained horizontal all the time) to some new value \(h'\).

Fig. 11

The force \(F'\) gives the moment on the flap:

\[ M'_n = F'h'. \]

The ratio:

\[ q=\frac{M'_n}{M_n}=\frac{F'h'}{Fh} \]

may be regarded as a quantitative characteristic of the influence of the finite length of the spring (or, more precisely, of the magnitude of the segment \(OD\)) on the curve \(M_n=f(\beta)\).

Below the question of the magnitude of \(q\) is considered for the case of operation of the mechanism according to Figs. 7, 8 up to engagement of the stop \((\beta \leq \beta_y)\). According to Fig. 11:

\[ h' = h\,\frac{\cos(\beta' + \alpha)}{\cos\beta'}, \]

where

\[ \beta'=\beta-\gamma, \]

\[ \tg \alpha=\frac{a_1\cos\beta' - l_2}{a_1\sin\beta' + l_1}. \]

The total absolute elongation of the spring (as compared with its length at \(\beta=0\)):

\[ z' = AD - A'D = \frac{l_1+a_1\sin\beta'}{\cos\alpha} - \frac{l_1-a_1\sin\gamma}{\cos\alpha_0} \]

or

\[ z'=\frac{z}{\cos\alpha}+k, \]

where the following notation has been adopted:

\[ k=(a_1\sin\gamma-l_1)\left(\frac{1}{\cos\alpha_0}-\frac{1}{\cos\alpha}\right). \]

Here \(\alpha_0\) is the angle of inclination of the spring axis to the horizontal for the vertical \((\beta=0)\) position of the regulating damper, determined from the obvious relation:

\[ \tan\alpha_0=\frac{a_1\cos\gamma-l_2}{l_1-a_1\sin\gamma}, \]

and \(z\), as before, is the elongation of the spring in the case \(l_1=\infty\):

\[ z=a_1[\sin\gamma+\sin(\beta-\gamma)]. \]

Now the force in the spring is

\[ F'=F_{\mathrm{н}}+cz'=F_{\mathrm{н}}+c\left(\frac{z}{\cos\alpha}+k\right) \]

and the moment developed by the spring on the axis of rotation of the damper is

\[ M'_n=\left[F_{\mathrm{н}}+c\left(\frac{z}{\cos\alpha}+k\right)\right]\cdot h\, \frac{\cos(\beta'+\alpha)}{\cos\beta_1}. \]

The required ratio is

\[ q=\frac{M'_n}{M_n}= \frac{\left[F_{\mathrm{н}}+c\left(\frac{z}{\cos\alpha}+k\right)\right]\cdot\cos(\beta'+\alpha)} {(F_{\mathrm{н}}+cz)\cos\beta'} \]

or, on the basis of relation (3),

\[ q= \frac{\lambda+\dfrac{1}{a}\left(\dfrac{z}{\cos\alpha}+k\right)} {\lambda+\dfrac{z}{a}} \cdot \frac{\cos(\beta'+\alpha)}{\cos\beta'}. \]

An analogous expression can also be obtained for the period of operation of the mechanism after the stop has been engaged.

Specific calculations carried out for the mechanism of regulator K—49 show that here the coefficient \(q\) does not exceed the value \(1.005\), which indicates the sufficient practical accuracy of the relations given above in §§ 1 and 2.

But the deviation of the axis of the regulating spring from the horizontal (in Figs. 4, 7, 8) may have a substantial influence on the moment at which the stop is engaged. To compensate for this influence, it is necessary to increase the previously calculated value of the angle \(\varphi_y\) to \(\varphi_y+\alpha_y\), where \(\alpha_y\) is the angle of inclination of the spring axis to the horizontal at the moment the stop is engaged \((\beta=\beta_y)\).

1. See the author’s paper “On Limit Pneumatic Regulators of Carburetor Automobile Engines,” published in the present volume of the Proceedings of TPI. ↩