SURFACE PROFILOGRAPHY BY THE METHOD OF INTERMITTENT PROBING

A. N. Eremin.

Under the influence of various cutting factors involved in the chip-formation process, the finish of the machined surface of a product may have varying quality.

On the basis of experimental investigations carried out in recent years abroad and in the USSR, it has been established that the most realistic and objective indicator of the quality of surface finish should be considered the magnitude of the roughnesses (irregularities) formed on the surface as a result of machining.

The smaller the height of the irregularities, the cleaner and smoother the machined surface, the higher the quality of its finish, and conversely.

Consequently, in order to characterize the external condition of a given machined surface, it is sufficient to measure the heights of its roughnesses and, from the magnitude of the latter, to assess the quality of the machining finish.

For this purpose there now exists a whole series of instruments whose operation is based on various measurement principles.

Among them, the most widespread are instruments that make it possible to measure roughness by directly probing the surface profile with the aid of a needle (the probing method).

The essence of this measurement method is very simple. A sharply pointed needle is moved continuously over the surface being tested with light contact; under the influence of the irregularities it performs oscillatory motions.

Depending on the principle of operation of the instrument, these oscillations of the needle are either recorded, with the aid of an optical device, on an enlarged scale in the form of a curve corresponding to the profile of the given surface (instruments of Abbott and Firestone (1); Berndt and Wallichs (2); Ammon (3), etc.), or are indicated by the readings of an electrical instrument (the capacitor instruments of Voronov (4) and Berman (5)).

Despite the relative differences in construction and in the methods of recording the measurement results (optical or electrical), all the instruments mentioned are completely similar in that:

1. Measurement of surface roughness is carried out by the method of continuous probing by means of a sliding needle.

2. The action of the recording devices (the instrument readings) is continuous during the process of measuring (probing) the surface.

This similarity also contains the principal shortcoming common to all instruments of this type, since with the continuous method of probing roughness by a sliding needle and with continuous recording of the measurement results, distortions inevitably arise in the readings of the instruments:

a) due to the phenomenon of abrasion and blunting of the tip of the needle against the surface;

b) due to jumps of the needle over the irregularities and destruction of the latter (by the needle), which is especially observed when probing surfaces with strongly

with a discontinuous profile (surface tears), i.e., when there is no smooth transition between individual roughness ridges;

c) due to the influence of dynamic and elastic vibrations and shocks arising during operation of the instrument (influence from the hand or the motor) in individual couplings.

To eliminate the influence of the listed distortions on the accuracy of measurement results, we propose a profilograph instrument in which, unlike existing ones, all operations associated with measuring (and reading the indications of) surface irregularities are intermittent in character (the method of intermittent probing).

At present the instrument has been built and installed in the metal-cutting laboratory of the Tomsk Industrial Institute¹).

Principle of Operation of the Profilograph.

Figs. 1 and 2 show the general view of the experimental profilographic installation with the specimen (article) secured for measurement.

In general, the operation of the profilograph is based on the optico-mechanical principle of measurement, similar to that used in the instruments of Abbott, Berndt, and Ammon.

The roughness is probed by a needle; the measurement results are recorded on photographic film, on an enlarged scale, by a light beam in the form of the profile curve of the surface.

The operating scheme of the optico-mechanical part of the instrument is sufficiently clear from Fig. 3.

The characteristic features of this profilographic installation consist in the following.

First, unlike existing instruments operating by the probing method²), when roughness is measured the needle moves (together with the instrument) not over the surface being tested, but in a raised (initial) position above the surface and is periodically lowered to record the ordinates of points of the surface profile, located at a very small distance from one another. As a result, the image of the profile curve is obtained in the form of a dashed line (Figs. 4a and 4v).

During automatic operation of the instrument, the contact points of the needle with the surface can be produced one after another at distances of: 0.007 mm, 0.014 mm; 0.028 mm; with manual control of the instrument—at any horizontal distance.

As is evident from the profilograms presented, the dashed recording gives a sufficiently clear representation of the nature of the surface profile. The clarity of the image can be increased if all points of the curve are joined by a continuous line (Figs. 4-b and 4-g).

When taking profilograms, the instrument performs a number of operations (automatically or by hand), repeated cyclically with each recorded point of the profile curve.

The cycle of instrument operations associated with recording one point has the following sequence:

1. Stopping the motion of the instrument along the length of the surface. At this time the needle is raised to the upper initial position.
2. Lowering the needle until it touches the surface (contact).
3. Switching on the light source (lighting the lamp).

¹) The material part of the instrument was made by the laboratory mechanics, comrades A. A. Chernykh and K. G. Gyngazov.

²) With the exception of the instrument designed by the National Physical Laboratory in England. D. Clayton—“An instrument for measuring surface cleanliness,” Engineering, 1935, No. 3611, p. 325.

Fig. 1.

Fig. 2.

Fig. 6.

Visible annotations in the drawing: “Light source”; “Diaphragm”; “Lens”; “Needle”; “Frosted glass”; “Scale.”

Fig. 7.

Visible annotations in the drawing: “EMB”; “EMB”; “Insul.” / “Insulation.”

Lettered labels visible in the drawings are preserved as in the original: \(a, b, c, d, e, f, g, h, i, k, l, m, n, o, p, r, s, t, u, v, x, y, z\), and capital labels \(A, B, C, D, E, F, G, J, K, M, N, P, X, Y\).

1. Fixing the point by a light beam (exposure).
2. Switching off the light source (extinction of the lamp).
3. Lifting the needle to the upper initial position.
   Simultaneously, rotation of the drum with the film.
4. Movement of the instrument along the surface to the next stop (point).

In accordance with what is set out in Fig. 5, a diagram is given of the motion of the needle in the horizontal and vertical directions when measuring the ordinates of points of the surface profile.

Since in this instrument there is absolutely no continuous sliding of the needle over the surface roughnesses, therefore:

a) the phenomenon of abrasion (wear) and blunting of the needle is eliminated, regardless of the hardness and cleanliness of the surface.

b) distortions in the profile curve are eliminated, as are errors in the instrument readings, which inevitably result from jumps of the needle over irregularities and from the destruction of the latter in the case of measurement by other instruments using the method of continuous probing (Abbott, Berndt, Ammon, Berman, etc.).

Secondly, all actions of the instrument connected with measuring the profile of a given surface section are fully automated. The participation of the experimenter at the moment of recording the profilogram is eliminated.

The continuously periodic movement of the carriage, on which the measuring chamber of the profilograph with needle 1 is mounted, is produced by motor 2 with reducer 3, installed on a separate support from the profilograph plate (Fig. 1).

At the moment of fixing each point of the profile curve, the motor with reducer is automatically disconnected by electromagnetic clutch 4, thereby eliminating the influence of vibrations on the instrument readings.

Thirdly, the light source for recording the measurement results is switched on only for the duration of the exposure of the ordinate of the given point, at the moment of the static state of the instrument. This achieves:

a) sharpness of the image of the point on the film (it is not blurred).

b) complete elimination of the influence of random shocks and oscillations on the accuracy of the instrument readings, which in profilographs existing up to now does not occur, and therefore constitutes a significant shortcoming in their operation.

Design of the profilographic setup.

The experimental profilographic setup (Figs. 1 and 2) consists of three main parts:

1. The profilograph proper, which is a measuring chamber 1, in the lower part of which is located a mechanism with a needle for probing surface roughnesses.

2. Stand 5 with a bracket (overhang) for fastening and moving the profilograph in the horizontal and vertical directions.

3. Power unit for automatic control of the profilograph during its operation.

Figure 6 shows the structural design of the photo-optical part of the instrument, mounted inside measuring chamber 1.

From a light source located outside the chamber in a separate casing, the light beam passes through the aperture of a diaphragm and a lens, strikes the movable mirror $b$, is reflected onto the fixed mirror $q$, and from there falls on the film fastened to the rotating drum $d$.

Outside, in the lower part of the chamber, there is placed a mechanism for probing roughnesses, consisting of a lever $c$, carrying on arm $a$ the needle. The latter can be moved in the slot of the lever and fixed with a screw.

![Fig. 3 diagram]

Light source
Diaphragm
Mirror
Ground glass and scale
Film drum
Rotating mirror
Lens
Movable mirror-prism
Spring hinge
Needle
Workpiece

Fig. 3.

![Fig. 4 profilograms]

a) \(v = 60;\ t = 0.50;\ S = 0.136;\ M = 1150\)

b)

c) \(v = 15;\ t = 2.1;\ S = 0.055;\ M = 1150\)

d)

Fig. 4.

![Fig. 5 principle of intermittent probing]

Horizontal movement of the instrument needle.

Upper initial position of the needle above the surface.

Vertical movement of the needle.

Surface profile.

Fig. 5.

![Fig. 8 instrument diagram]

Section of the disk

3 mm

to the tracking probe

Fig. 8.

At one end, lever \(c\) is attached to a support on two thin and narrow springs \(O\) (the length of the free part is 1 mm), on which it can turn through a small angle, as on a hinge with no play along its axis. At the other end, lever \(c\) has a prism \(f\), against which the movable mirror \(b\) is lightly pressed by means of a spring.

With this interconnection between the mechanical and optical parts of the kinematic chain of the instrument, every vertical oscillation of the needle inevitably also causes an oscillation of the light beam along the generatrix of drum \(d\).

The forced raising of the needle into the upper initial position and its lowering onto the surface during the measurement of roughnesses is carried out by lever \(k\), attached to the armature of electromagnet E. M. I. (needle electromagnet, Fig. 7).

To eliminate the influence of the weight of lever \(c\) on the pressure of the needle, the lever is balanced by the action of spring \(i\), the tension of which can be adjusted by resetting screw \(h\) (Fig. 6).

To record each point of the profile curve on the film, drum \(d\) is automatically turned through a certain angle by a worm mechanism actuated by electromagnet E. M. B. (drum electromagnet, Fig. 7).

The operation of the mechanism is as follows. Under the action of an electric current and spring \(l\), the armature of the electromagnet performs oscillatory motions and, by means of rocker \(m\) and pawls \(n\), turns ratchet wheel \(s\), seated immovably on the axis of the worm, in one direction. The rotation of the latter is transmitted to the worm wheel and to the drum axle, which are connected to one another by friction. To make the connection reliable, the end of the drum axle is split and acts as a spring (Figs. 6 and 4).

The angle of rotation of the drum can be set in accordance with the feed (rotation) of the ratchet wheel by 1 or by 2 teeth. In this case the distance between the points of the curve on the film along the abscissa axis (Fig. 4) will be equal to either 0.35 mm or 0.7 mm. As can be seen from Fig. 7, in order to change the angle of rotation of the drum it is sufficient to turn eccentric \(t\) by 180° in one direction or the other by pin \(p\); this changes the magnitude of the oscillation amplitude of the electromagnet armature and, consequently, the feed of the ratchet wheel.

Handle \(r\), seated at the opposite end of the worm, serves for turning the drum manually through any desired angle.

Drum \(d\) (Fig. 6) is placed in a closed cylinder (cassette), which has along its upper generatrix a narrow slit (\(2 \times 90\) mm) for admitting the light beam onto the film.

The slit is covered by shutter \(u\), which also has the form of a cylinder, by turning it by the head of pin \(z\). Thanks to this, the cassette can be removed from the chamber and inserted into it under full external illumination.

The cassette is reloaded with film in a dark room. The largest size of the film fastened to the drum is \(10 \times 15\) cm; the greatest usable area for recording the profilogram is \(9 \times 14\) cm.

On the front side of the measuring chamber, facing the experimenter (Figs. 6 and 7), a ground glass is inserted, onto which the light beam can be directed by means of folding mirror \(x\). On the outside of the glass a scale is fixed, with divisions every 1 mm, for taking readings of the instrument indications (by eye).

The purpose of the ground glass is twofold.

First, the glass serves for observing the oscillations of the beam (the light “spot”) during adjustment of the profilograph before the start of measuring the roughness of the given surface, i.e. for coarse setting of the limiting deviations of the beam. The latter is necessary in order to be certain that, during the photographing of the profilogram, all oscillations of the light beam will be recorded within the field of the film.

Secondly, the frosted glass is needed for taking readings of the magnitudes of beam deflections according to the scale indications, in the case where the profilogram is constructed from direct-observation data (not on film).

Regardless of the external illumination, the visibility of the light “spot” on the frosted glass is distinct.

During the recording of the profilogram on film, mirror \(x\) must be turned toward the wall of the instrument, and the frosted glass must be covered from the outside with a lid.

The light source is a 25-watt electric lamp having a straight filament.

Adjustment of the sharpness of the light point (the “spot”) is achieved by changing the length of the tube and the diameter of the diaphragm aperture (the lens is fixed). The smallest diameter of the point on the film, achieved by the indicated adjustment, is \(0.2 \div 0.3\) mm.

The mirrors are made of stainless steel.

The construction of the stand is shown in Fig. 7. By means of the stand, the profilograph (camera) is moved in the horizontal direction during measurement of surface roughnesses and in the vertical direction at the moment of the preparatory adjustment of the instrument for measurement.

With the aid of bracket (overhang) \(M\) and column \(N\), the profilograph is directly connected with carriage \(E\), which can move along the horizontally arranged guides \(B\). The carriage, and together with it the profilograph, receive translational motion from the rotation of worm gear \(D\) through lead screw \(F\), the nut of which \(P\) is connected with the carriage by means of strip \(Q\), seated freely on pins.

The arrangement of such a connection is caused by the need to eliminate the harmful influence of runout of screw \(F\) (owing to manufacturing inaccuracy) on the accuracy of the instrument reading.

For manual horizontal feed of the profilograph (needle) during the experiment, one should rotate the worm shaft by the flywheel “\(L\)”; in automatic operation of the instrument, the worm is rotated through cardan shaft 6 from the power unit (Figs. 1 and 7).

For rapid movement of the carriage with the profilograph, it is necessary first to disengage cam clutch \(H\) from the worm wheel, and then to rotate handle “\(T\),” mounted directly on the lead screw (Fig. 7).

One revolution of the worm corresponds to a horizontal feed of the profilograph of \(0.028\) mm.

In manual control of the instrument, the amount of feed can be made greater or smaller than \(0.028\) mm by changing the angle of rotation of the worm (flywheel) within any desired limits.

Only three automatic feeds are implemented in this profilograph: \(0.007\) mm, \(0.014\) mm, and \(0.028\) mm, the magnitudes of which correspond to \(1/4\), \(1/2\), and 1 revolution of the worm.

For rough positioning of the needle above the surface, bracket “\(M\),” together with the instrument, is moved along column “\(N\)” and secured with stop “\(G\).” Smooth lowering of the needle (with the instrument) onto the surface being tested at the moment of adjustment for measurement is performed by rotating nut “\(W\),” which imparts translational motion to screw \(v\) owing to the presence of key \(y\).

In order to reduce oscillations of the entire system when nut \(W\) is turned, the latter has a double conical surface of rotation, whereby both centering (lower cone) and support (upper cone) are achieved simultaneously.

During measurement of roughnesses, the vertical-feed screw \(v\) must be fixed immovably with locknut \(I\) and stop \(R\). By means of the locknut the clearance in the thread of the screw and nut \(W\) is eliminated, while by the stop, compressing split bushing \(X\) (fixed immovably to the body of the stand),

the play on the key \(Y\) of screw \(v\) is eliminated. Such a method of fastening quite reliably ensures the rigidity of the entire upper part of the stand with the profilograph.

By means of the limb made integrally with nut \(W\), one can observe the magnitude of the vertical displacement of the needle (of the instrument); by means of limb \(I\)—the magnitude of the horizontal displacement of the needle.

Microadjustment, i.e., setting the light spot on any division of the scale, is most simply achieved by additional adjustment of the pressure by stop \(R\).

The arrangement of the power unit is shown in Fig. 8.

The unit comprises: motor 2 (not indicated in Fig. 8), reduction gear 3, ring current interrupter \(P_1\), electromagnetic clutch E. M. M., and automatic switch \(P_2\). By means of the unit, automatic control is effected of all actions of the instrument mechanisms during the process of recording the profilogram on film, and automatic switching of the electrical connections circuit to manual control at the moment when recording is completed.

The interaction of the individual kinematic links of the unit occurs as follows.

From motor 2, mounted on the same stand as reduction gear 3 (see also Fig. 1), through a friction transmission, motion is transmitted to worm wheel \(e\) and to ring current interrupter \(P_1\), seated on the same hollow shaft \(B_1\). Simultaneously, from pin \(f\) on the worm wheel, a four-ended Maltese cross \(g\) on shaft \(B_2\) is set in motion.

Owing to this, during one revolution of the worm wheel, or, what is the same, during one revolution of the ring current interrupter \(P_1\), shaft \(B_2\) makes only \(1/4\) revolution.

Through gear wheels with the transmission ratio

\[ \frac{Z_1}{Z_2}=\frac{4}{1};\quad \frac{Z_3}{Z_4}=\frac{2}{1}\ \text{and}\ \frac{Z_5}{Z_6}=\frac{1}{1} \]

motion is transmitted to shaft \(B_3\) with the electromagnetic clutch—E. M. M.—fastened at its end, which, by attracting armature \(k\), rotates the cardan shaft of the profilograph.

With successive engagement of gears \(Z_1\), \(Z_3\), and \(Z_5\) by means of the movable key \(h\), the cardan shaft makes \(1\), \(1/2\), and \(1/4\) revolutions per one revolution of current interrupter \(P_1\), which corresponds to a feed of the profilograph (needle) in the horizontal direction of \(0.028\), \(0.014\), or \(0.007\) mm (see above).

Ring current interrupter \(P_1\) consists of three parts connected together: a middle part, in the form of a complete ring, to which current from the external electrical network is continuously supplied (through a brush), and two outer half-rings arranged mutually opposite one another.

From one half-ring I, current is taken into the circuit for simultaneously supplying the electromagnetic clutch E. M. M. (Fig. 8) and the electromagnets for the needle and drum, i.e., EMI and EMB (Fig. 7); from the other half-ring II, current is taken into the circuit supplying the light source (lamp) of the profilograph.

When the ring interrupter rotates, electric current is taken from both half-rings (I and II) alternately.

Consequently, during one revolution of the current interrupter either all the electromagnets (EMM, EMI, EMB) operate, or only the light source (lamp) operates; in other words, during one revolution of the current interrupter a cycle of all actions of the instrument is performed, necessary for fixing one point of the curve on the profilogram (Fig. 4).

In Fig. 9 a diagram is given of the relative interaction of all mechanisms of the profilograph during one revolution of the ring current interrupter.

Here: $\alpha$ is the angle of action of semicircular ring I, corresponding to the period of current supply to the electromagnets EMM, EMI, EMB (the armatures are attracted).

$\gamma$ is the angle of rotation of the E. M. M. together with the cardan shaft, corresponding to the period of feed of the profilograph (needle) in the horizontal direction.

$\beta$ is the angle of action of semicircular ring II, corresponding to the period of current supply to the light source. (Exposure time of the point.)

$\delta$ is the angle of rotation of the current interrupter corresponding to the period of inactivity of all the mechanisms of the profilograph. This is provided in order to achieve damping of oscillations in all parts of the instrument by the moment the point is fixed on the film (angle $\beta$), i.e., by the moment of automatic reading of the measured quantity.

The symmetrical arrangement of the angles is adopted for operation with the reverse direction of rotation of the annular current interrupter.

Depending on the sensitivity of the film, the time of one cycle of actions of the instrument, performed during one revolution of the current interrupter, ranges from 1 sec. and upward.

The duration of the cycle is varied by regulating the number of revolutions of the motor.

From the bevel gear transmission $Z_7$ and $Z_8$ (Fig. 8) on the shaft $B_2$ of the reducer, motion is transmitted to screw $i$ (pitch 0.9 mm) with a split nut $m$, which has translational motion along guide rods $l$, at the ends of which a switch $P_2$ of special design is mounted.

By means of this switch, at the moment the recording of the profilogram (on the film) is completed, the entire installation of the instrument is switched off (stopped), and at the same time the circuit of electrical connections for automatic control of the profilograph is switched over to the circuit for manual control of the actions of the E. M. I. and of the light source (see below, Fig. 10).

The switch is actuated by pressing the slide $n$, either by hand or automatically, by the pressure of nut $m$ at the end of its travel along the screw. The length of the full travel of the nut along the screw corresponds to a complete revolution of the drum $d$ with the film (Fig. 7), i.e., to the length of the profilogram—about 140 mm.

By setting nut $m$ in the initial position at any place along the length of screw $i$, it is possible to obtain any length of the surface-profile curve, recorded automatically on the film, of any magnitude (within the limits up to 140 mm).

The rotating electromagnetic coupling—E. M. M.—is supplied with electric current through ring contacts $P$, over which the brushes slide (Fig. 8).

From the general view of the instrument installation (Figs. 1 and 2) it is seen that the stand with the profilograph and the driving cardan shaft are mounted on one plate, installed on a massive pedestal, separately from the support of the power unit.

Such an arrangement was made in order to eliminate the influence of motor vibrations on the accuracy of the instrument readings during the recording of the profilogram.

For the same reasons, the installation of an electromagnetic coupling is provided, making it possible to introduce a break between the cardan shaft and the power unit.

From Fig. 8 it is seen that the armature $k$ of the electromagnetic coupling E. M. M., under the action of the spring, always tends to move away from the coupling as soon as the attractive force ceases. The resulting gap between the armature and the electromagnetic coupling ($\sim 0.55$ mm) is quite sufficient to isolate the transmission of vibration of the unit to the profilograph at the moment when points are fixed on the profilogram.

Fig. 9.

Visible annotations in the figure: “Direction of motion”; “reading”; “brush”; “brushes”; “to the light source”; “to the electromagnets”; \( \alpha = 140^\circ \), \( \gamma = 90^\circ \), \( \delta = 30^\circ \), \( \beta = 160^\circ \); I; II.

Fig. 10.

Visible annotations in the circuit diagram include: \(220\,\mathrm{V}\), \(V\), \(M\), \(R\), \(A\), \(C_1\), \(R_2\), \(K\), \(B\), \(P_1\), \(P_4\), \(P_8\), \(P_9\), \(P_{12}\), \(P_{14}\), \(P_2\), \(P_1\), \(P\), \(EMB\), \(ENM\), \(EM1\).

Fig. 11.

Visible annotations in the figure include: “scale of the ruler”; \(H\); \(A'\), \(B'\), \(B\), \(C\), \(D\), \(R\), \(R_{\min}=L\), \(\Delta R_{\min}\), \(q\), \(a\), \(b\), \(\Delta b\), \(x\), \(20^\circ\), \(30^\circ\), \(90^\circ\), \(\alpha\), \(\delta\); “needle”; “prism”; [[unclear: small label near the right-hand ray]].

Fig. 12.

a)

Visible annotations: \(c\), \(c_1\), \(b_1\), \(O\), \(O'\), \(a\), \(\beta\), \(\gamma\), \(e\), \(f\).

b)

Visible annotations: \(c\), \(O\), \(O'\), \(\alpha\), \(\beta\), \(\gamma\), \(x\), \(f\), \(b\), \(b_1\), \(b_2\), \(z_c\).

The cardan shaft 6 (Fig. 1) has a telescopic joint in its middle part, which makes it possible, without disturbing the kinematic interaction between the power unit and the profilograph stand, to carry out vertical and horizontal displacement of the latter.

Diagram of the electrical connections of the installation.

Figure 10 presents the diagram of the electrical connections of the profilographic installation for automatic and manual control of the instrument.

Designations:

$M$ — series electric motor for 110 V. Power 44 W. It has 5 speed stages, regulated by a starting rheostat.

$R_1$ — starting rheostat of the motor.

EMM — electromagnetic clutch with ring collecting rings.

EMB — electromagnet for rotating the drum with the film.

EMI — electromagnet for raising and lowering the needle.

$L_1$ — lamp (light source) for fixing points on the film.

$R_2$ — rheostat for regulating the light intensity of lamp $L_1$.

$LR_1$—$LR_4$ — lamp rheostats.

$P_1$ — ring current interrupter.

$P_2$ — switch of the electrical-connection circuit from automatic control of the instrument to the circuit with manual control (by button $K_1$).

$K_1$ — button for manual control of the lighting of lamp $L_1$ and the actions of E.M.I.

$B$ — switch.

$L_2$ — signal lamp (or bell) indicating the moment when recording of the profilogram is completed.

$C$ — spark suppressor on the brushes.

The electrical-connection circuit for automatic operation of the instrument corresponds to the position of switch $P_2$ (1—1) and button $K_1$, as indicated in Fig. 10, with switch $B$ in the on state.

To start the instrument during recording of a profilogram on film, it is sufficient to switch on the starting rheostat $R_1$.

At the moment when recording of the profilogram is completed, switch $P_2$ automatically moves to position (2—2); at the same time the motor stops and the signal lamp lights up (or the bell begins to operate).

By opening switch $B$, the operator stops the signaling action and at the same time obtains the electrical-connection circuit for manual control (by button $K_1$) of the actions of lamp $L_1$ (light source) and E.M.I.

Manual control (of the actions of $L_1$ and E.M.I.) is used either at the moment of setting up the instrument for recording a profilogram from a given section of the surface under test, or in the process of constructing a profilogram on the basis of observation data from the instrument readings on the scale (with manual horizontal feed).

The lamp rheostats $LR_1 \div LR_4$ are selected so that each electrical circuit has the proper current strength.

All rheostats, except the starting rheostat $R_1$, are mounted in one cabinet (Figs. 1 and 2).

The diagram shown (Fig. 10) demonstrates that all control of this profilographic installation consists of very simple operations by the operator, requiring an expenditure of time of no more than 3–5 seconds.

Devices for securing the specimens under test

By its design the instrument is intended for measuring roughness only on external surfaces of cylindrical and flat types.

For securing specimens of circular and rectangular cross-section there are two interchangeable fixtures—tables, which are installed on the plate of the instrument in a special slot.

As an example, Figs. 1 and 2 show a fixture for securing circular-section specimens between centers.

The design of the fixture makes it possible to measure surface roughness in two mutually perpendicular directions, i.e., along and across the axis of the article.

To measure one or another section of the surface of the article, the fixture is moved by hand along the slot and is firmly fastened to the plate by bolts 7.

When measuring surface roughness in a direction perpendicular to the axis of the article, the latter is slowly rotated in the centers of the fixture by means of handle 8. The angle of rotation is read from limb 9.

For one and the same angle of rotation of the limb, the length of the arc on the circumference of the specimen (article) may vary with a change in the machining diameter. To simplify the calculation of the distances traversed over the surface of the specimen, there is a chart expressing the dependence among the angle of rotation of the limb, the diameter of the specimen, and the arc of the circumference.

The limiting dimensions of the test specimens secured in the fixtures of this profilograph installation are:

a) Circular cross-section. Length 240–260 mm
        Diameter up to 200 mm

b) Rectangular cross-section. Length up to 300 mm
        Width and height up to 150 mm

If, for testing, the specimens are installed directly on the instrument plate on prisms, then in this case the length of the specified specimens may be increased to 1000 mm and more.

It should be noted that for articles of great length, roughness can be measured only in the longitudinal direction.

Mathematical investigation of the influence of the instrument’s kinematic scheme on the accuracy of its readings

Figure 11 presents the kinematic scheme of the measuring part of the profilograph, in which, for greater clarity of the mathematical investigation, the intermediate mirror changing the direction of the light ray is not shown, as indicated in Fig. 3.

Adopted notation:

\(\delta\)—vertical displacement of the stylus (lifting, lowering), equal in magnitude to the measured height in mm.

\(h\)—displacement of the light ray (“spot”) along the scale of the ground glass or along the film, corresponding, at scale \(\delta\), in mm.

\(H\)—maximum travel of the light ray in mm.

\(x\)—displacement of the prism of lever \(c\) in mm.

\(f\)—height of the prism in mm (not shown in the diagram).

\(\alpha\)—angle of rotation of levers \(a\) and \(c\).

\(\alpha_1\)— ” ” ” \(b\) of the movable mirror.

\(2\alpha_1\)— ” ” of the light ray.

\(L = R_{\min}\)—distance from the movable mirror to the scale or film in mm.

\(R\)—length of the light ray in mm.

$e$ — the length of the working part of the needle (not indicated in the diagram) from the line passing through lever $c$ in the horizontal position and its point of rotation $O$.

$a$, $b$, and $c$ — the lengths of the levers in mm.

To what extent the accuracy of the profilograph readings depends on the combination of the selected dimensions and structural forms of all elements of the kinematic scheme is set forth in the analysis given below.

1. Mathematical relationship between the quantities

In deriving this relationship in general form, it is conventionally assumed that

\[ e = 0\ \text{mm} \]

\[ f = 0\ \text{mm} \]

According to the diagram in Fig. 11, the magnitude of the prism displacement (at $f=0$), equal to the chord $x$, is common to the arcs described by radii $c$ and $b$; therefore:

\[ x = 2c \sin \frac{\alpha}{2} \tag{1} \]

and

\[ x = 2b \sin \frac{\alpha_1}{2} \tag{2} \]

From the equality of equations (1) and (2),

\[ \frac{b}{c} = \frac{\sin \dfrac{\alpha}{2}} {\sin \dfrac{\alpha_1}{2}} \tag{3} \]

From the geometry of the rotation of lever $c$, which carries on arm $a$ the needle (at $e=0$), the magnitude of the vertical displacement of the latter is determined as

\[ \delta_{e=0} = 2a \sin \frac{\alpha}{2} \tag{4} \]

Substituting the value of $\sin \dfrac{\alpha}{2}$ from equation (3), we find

\[ \delta_{e=0} = 2 \frac{ab}{c} \cdot \sin \frac{\alpha_1}{2} \tag{5} \]

The greatest value of the travel of the light beam along the scale is

\[ H = 2 R_{\max} \sin \alpha_1 \tag{6} \]

or

\[ H = 4 R_{\max} \sin \frac{\alpha_1}{2} \cos \frac{\alpha_1}{2} \tag{7} \]

The ratio of the quantities $H$ and $\delta_{e=0}$ represents the transmission ratio of the instrument kinematics, or, what is the same, its magnification scale of the measured heights along the vertical of the profilogram.

\[ M = \frac{H}{\delta_{e=0}} \]

Replacing $H$ and $\delta_{e=0}$ by their values from equations (7) and (4), we determine the general mathematical relationship between the displacement of the beam $H$ and the lifting of the needle $\delta_{e=0}$, expressed through the magnitudes of all elements of the instrument kinematics.

\[ M = \frac{H}{\delta_{e=0}} = 2 \frac{c}{ab} R_{\max} \cos \frac{\alpha_1}{2}. \tag{8} \]

Relation (8) shows that with a change in the arm \(a\), i.e., the distance at which the needle is fastened, the magnification scale of the instrument \(M\) also changes, and vice versa. For a given \(M\)

\[ a=2\frac{c}{b}\cdot \frac{R_{\max}}{M}\cdot \cos \frac{\alpha_1}{2} \]

or

\[ a=2\frac{c}{b}\cdot \frac{\delta_{e=0}}{H} R_{\max}\cdot \cos \frac{\alpha_1}{2} \tag{9} \]

Since equation (8) expresses the mathematical interrelation between the elements of the instrument kinematics only at the extreme positions of the light beam and under the condition that \(f=0\) and \(e=0\), it is therefore necessary to prove whether this relation will remain constant for any position of the beam along the scale \(H\), and also in the case where \(e\ne 0\). It must be remembered that, for accurate operation of the instrument, this relation must always remain constant.

2. Influence of changes in \(R\) on the accuracy of the instrument readings.

With uniform rotation of the movable mirror (Fig. 11), a light beam of length \(L=R_{\max}\) gives segments that are uniform and proportional to the angle of rotation only along the arc \(AKB=l\), which is, as it were, a fictitious scale of the readings measured by the profilograph.

The actual reading scale is arranged in the instrument along the straight line \(A^1B^1=H\) (the length of the generatrix of the drum with film) or the scale of the ground glass; therefore, when the light beam is deflected to one side or the other from the middle position \(O^1K\), its length varies within the limits

\[ \text{from } L=R_{\min} \text{ to } R_{\max}. \]

Consequently, on the actual scale \(A^1B^1\) the light beam marks off segments that are nonuniform and not proportional to the angle of rotation, as on the arc \(AKB\), but that gradually increase in magnitude from the line \(O^1K\) to both sides.

Comparison of the length of the arc \(l\) with the straight line \(H\) for the full angle of rotation of the beam \(2\alpha_1\) gives the total discrepancy (i.e., the error) in the profilograph readings due to the kinematics of its optical part.

The magnitude of this error, in % of \(l\), is:

\[ \Delta l=\left(\frac{H}{l}-1\right)100 \tag{10} \]

where: \(l\) is the length of the arc at radius \(R_{\min}=\text{const.}\) and angle \(2\alpha_1\), determined by the formula

\[ l=0.01745\,R_{\min}\,2\alpha_1 \tag{11} \]

\[ H=90 \text{ mm, according to the size of the working part of the film,} \]

\[ R_{\min}=L=500 \text{ mm, according to the optics of the instrument.} \]

From the relation:

\[ \frac{H}{2R_{\min}}=\operatorname{tg}\alpha_1=0.090 \tag{12} \]

\[ \text{angle } \alpha_1=5^\circ 8'34''=5.143^\circ;\qquad \sin \frac{\alpha_1}{2}=0.04486 \tag{12'} \]

Determining, by formula (11), the length of the fictitious scale \(l\) and substituting its value into formula (10), we find the greatest magnitude of the error in the readings of the actual scale of the instrument due to the optics:

\[ \Delta l_{\max}=0.284\% \text{ of } l \quad \text{(at the extreme positions of the beam)} \]

On average, at the position of the beam this error is equal to zero. Owing to the small magnitude of \(\Delta l_{\max}\), the nonuniformity of the scale \(H\) may be neglected; therefore, a scale with uniform divisions at intervals of \(1\) mm is installed in the instrument.

When the light beam moves along a uniform and rectilinear scale, the opposite phenomenon is observed; namely, as the beam is deflected from the mean position, the increment of the angle of rotation \(\alpha_1\) decreases.

Setting the condition of equality

\[ l = H = 90\ \text{mm}, \]

we determine the total decrease in the increment in the angle of rotation of the beam

\[ \Delta 2\alpha_{1\max} = -0.28\% . \tag{13} \]

However, the total decrease in the increment of the angle \(2\alpha_1\) (or \(\alpha_1\)) does not yet give a clear representation of the degree of nonuniformity of this increment during uniform motion of the light beam along the scale \(H\) (from the mean position to the extreme positions).

To analyze the nonuniformity it is necessary to use formula (12)

\[ \frac{H}{2}\cdot \frac{1}{R_{\min}} = \operatorname{tg}\alpha_1 , \]

Since \(R_{\min}=L=500\ \text{mm}=\text{const}\), then, assigning to \(\frac{H}{2}\) different values from \(0\) to \(45\) mm, we assume that every \(1\) mm it is possible to determine the corresponding magnitudes of the increment of the angle of rotation of the beam \(\Delta\alpha_1\) and the degree of nonuniformity of this increment.

The results of the calculations are given in Table 1, where the quantities \(\Delta\alpha_1\) are indicated as averages of five values, since when \(\frac{H}{2}\) changes by \(1\) mm within the limits of \(5\) mm, no appreciable regularity is observed in the change of \(\Delta\alpha_1\) (the calculations were made using five-place tables of tangents).

Table 1.

\(\dfrac{H}{2}\), mm	\(\operatorname{tg}\alpha_1\)	Increment of the angle of rotation \(\Delta\alpha_1\), for beam motion by 1 mm of the scale, on average	% decrease of the increment \(\Delta\alpha_1\)	Note
0	0	0	0.000	Beam in the mean position
1–5	0.002–0.010	\(6' 52.56''\)	0.008
6–10	0.012–0.020	\(6' 52.54''\)	0.103
11–15	0.022–0.030	\(6' 52.14''\)	0.107
16–20	0.032–0.040	\(6' 52.12''\)	0.200
21–25	0.042–0.050	\(6' 51.74''\)	0.279
26–30	0.052–0.060	\(6' 51.42''\)	0.533
31–35	0.062–0.070	\(6' 50.40''\)	0.558
36–40	0.072–0.080	\(6' 50.30''\)	0.728
41–45	0.082–0.090	\(6' 49.60''\)

The error is insignificant in magnitude; therefore it is entirely possible not to take it into account and not to introduce corrections for it into the general equation (8).

At any magnification scale of the instrument, \(\Delta l_{\max}\), \(\Delta 2\alpha_{1\max}\), and the % decrease of the increment \(\Delta\alpha_1\) (degree of nonuniformity) are constant quantities.

The greatest length of the light ray \(R_{\max}\) from formula (6)

\[ R_{\max}=502\ \text{mm} \]

Increase of the ray

\[ \Delta R=R_{\max}-R_{\min}=2\ \text{mm} \]

With a focal distance from the lens to the reading scale (or to the drum) of \(600\ \text{mm}\), an elongation by \(2\ \text{mm}\) does not cause a noticeable loss in the sharpness of the image of the light “spot.”

3. Influence of the kinematics of levers \(c\) and \(b\) on the instrument readings.

The interaction of levers \(c\) and \(b\), indicated in the diagram of Fig. 11, corresponds to the theoretical case of their arrangement when the prism height is \(f=0\). In this case the ratio of \(b\) to \(c\), according to formula (3), is

\[ \frac{b}{c}=\frac{\sin \dfrac{\alpha}{2}}{\sin \dfrac{\alpha_1}{2}}=0.10 \tag{14} \]

since the structural dimensions of the levers have been taken as

\[ b=3.25\ \text{mm};\quad c=32.5\ \text{mm}. \]

The actual arrangement and interaction of levers \(c\) and \(b\) at \(f>0\) is shown in the diagram of Fig. 12 a) and b). The line \(OO'\), connecting the centers of rotation of the levers, is not horizontal, as in the diagram of Fig. 11, but is inclined at an angle \(\gamma\). The arm of lever \(c\) is the distance from the vertex of the prism to the center of rotation \(O\).

Substituting the value of \(\sin \dfrac{\alpha_1}{2}\) from formula (12) into formula (14), we find

\[ \sin \frac{\alpha}{2}=\frac{b}{c}\sin\frac{\alpha_1}{2}=0.004486 \tag{15} \]

Whence the greatest angle of rotation of lever \(c\) is

\[ \alpha=0^\circ 30'54'' \tag{15'} \]

When lever \(c\) rotates, its length, determined by the position of the prism, remains constant, while the length of lever \(b\) varies within the limits

\[ \text{from } b \text{ to } b_1;\ \text{where } b>b_1. \]

The magnitude of the shortening of lever \(b\), denoted by \(\Delta b\), reaches its greatest value at the middle position of \(b\) and \(c\) (Fig. 12 b)

\[ \Delta b_{\max}=Z_c+Z_b \tag{16} \]

where: \(Z_c\) and \(Z_b\) are the arrows of arcs corresponding to radii \(c\) and \(b\) and angles \(\alpha\) and \(\alpha_1\),

\[ Z_c=c\left(1-\cos\frac{\alpha}{2}\right). \]

\[ Z_b=b\left(1-\cos\frac{\alpha_1}{2}\right) \]

By substituting the calculated values of \(Z_c\) and \(Z_b\) into formula (16), we obtain:

\[ \Delta b_{\max}=0.00358\ \text{mm} \]

\[ \text{or } \Delta b_{\max}=0.11\% \text{ of } b \tag{16'} \]

\[ \text{and } b_1=3.2464\ \text{mm}. \]

The segments enclosed between the two arcs of radii $c$ and $b$ (Fig. 12b) graphically depict the oscillations $\Delta b$.

With uniform rotation of lever $c$ within the limits of angle $\alpha$, the oscillation of the length of radius $b$ gives a nonuniform change in angle $\alpha_1$.

The nonuniformity of $\alpha_1$ can be traced by formula (15)

\[ \sin \frac{\alpha_1}{2}=\frac{c}{b_x}\cdot \sin \frac{\alpha}{2} \tag{17} \]

where: $b_x$ is the current value of radius $b$ within the limits from $b_1$ to $b$.

Since the limits of variation of the angles $\dfrac{\alpha}{2}$ and $\dfrac{\alpha_1}{2}$ are small:

\[ \frac{\alpha}{2}=0^\circ 15' 27'' \quad \text{by formula (15)} \]

\[ \frac{\alpha_1}{2}=2^\circ 34' 17'' \quad \text{by formula (12')} \]

the values of the sines of these angles vary proportionally to the angles (Table 2)

Table 2.

$\dfrac{\alpha}{2}$	$\sin \dfrac{\alpha}{2}$	Increment $\sin \dfrac{\alpha}{2}$
$0^\circ$	0,00000	0,00000
$0^\circ30'$	0,00873	0,00873
$1^\circ00'$	0,01745	0,00872
$1^\circ30'$	0,02618	0,00873
$2^\circ00'$	0,03490	0,00872
$2^\circ30'$	0,04362	0,00872
$3^\circ00'$	0,05234	0,00872

On the basis of this dependence, (17) takes the following form

\[ \frac{\alpha_1}{2}=\frac{c}{b_x}\cdot \frac{\alpha}{2} \tag{17'} \]

Hence it is clear that, with uniform increase of $\dfrac{\alpha}{2}$, at $c=\text{const.}$, the angle $\dfrac{\alpha_1}{2}$ increases somewhat more slowly owing to the increase of $b_x$.

Substituting the limiting values $b_x$ ($b_1$ and $b$), we determine the magnitude of the greatest decrease in the increment of the angle $\dfrac{\alpha_1}{2}$ when lever $b$ is rotated from the middle position to the extreme positions.

\[ \Delta \frac{\alpha_1}{2}_{\max.}=-0{,}11\% \text{ of } \frac{\alpha}{2} \tag{18} \]

This circumstance is very favorable for ensuring a slowed increase of the angle $\alpha_1$ of rotation of the light beam, as it moves (from the middle to the extreme positions) along a uniform and rectilinear scale $H$.

Consequently, the inevitable decrease in the increment of the rotation angle of the beam $\alpha_1$ by $\Delta \alpha_{1\max}=-0{,}28\%$ (13) is somewhat compensated by the decrease in the rotation angle $\dfrac{\alpha_1}{2}$ of lever $b$, by the amount $\Delta \dfrac{\alpha_1}{2}_{\max}=-0{,}11\%$ (18).

Thus, the greatest possible error in the instrument readings (at the extreme positions of the beam on the scale) due to the optics will be approximately 0.17% of the true value.

The analysis presented indicates that the kinematics of levers \(c\) and \(b\) has a positive effect on the accuracy of the instrument readings; therefore there is no need to introduce a correction into equation (8) to account for the magnitude \(\Delta \dfrac{\alpha_1}{2}_{\max}\).

4. Influence of the Needle Length on the Instrument Readings

The actual length of the working part of the needle \(e>0\); therefore, when lever \(c\) is rotated through the angle \(\alpha\), its end moves from point \(K\) (Fig. 13) to point \(m\) along an arc of radius:

\[ a_1>a \]

From the right triangle \(\Delta OPK\),

\[ a_1=\frac{a}{\cos\beta}=\frac{e}{\sin\beta} \tag{19} \]

\[ \frac{e}{a}=\operatorname{tg}\beta \tag{20} \]

For the relatively small value of the angle of rotation \(\alpha=0^\circ30'54''\) (15′), the length of the arc \(\overset{\frown}{km}\) of radius \(a_1\) may quite well be replaced by the chord \(\overline{km}\):

\[ \overset{\frown}{km}=\overline{km}=S \]

Under this condition, from the isosceles triangles \(\Delta kom\) and \(\Delta ouk\), the angle

\[ \psi=\beta \]

The horizontal displacement of the needle during measurement is

\[ y=\delta_{e>0}\operatorname{tg}\beta \tag{21} \]

where: \(\delta_{e>0}\) is the greatest rise of the needle corresponding to the measured height at the given magnification scale.

Equation (8), expressing the general mathematical dependence between the displacement of the light beam \(H\) and the rise of the needle \(\delta_{e=0}\), was derived under the condition \(e=0\).

From comparison of the diagrams in Fig. 11 and Fig. 13 it is seen that the displacement of the point \(p\) at which the needle is fastened to lever \(c\) was taken as the magnitude of the needle rise \(\delta_{e=0}\).

In reality, when the needle length \(e>0\), the magnitude of the needle rise \(\delta_{e>0}\) corresponds to the length of the leg \(nm\) in \(\Delta kmn\) (Fig. 13).

In order that equation (8) also be valid for the case \(\delta_{e>0}\), it is necessary to show that

\[ \delta_{e=0}=\delta_{e>0} \]

On the one hand, from \(\Delta kmn\),

\[ \delta_{e>0}=S\cos\beta \tag{22} \]

where: \(S=\overline{km}\) is the chord; from \(\Delta kom\) it is equal to

\[ S=2a_1\sin\frac{\alpha}{2} \]

Substituting the value of \(S\) into equation (22) and replacing \(a_1\) by its value from (19), we find:

\[ \delta_{e>0}=2a\sin\frac{\alpha}{2} \tag{23} \]

Fig. 13.

Direction of needle motion

Actual profile

Represented profile

Fig. 14.

Direction of the light beam

Cardan-shaft rotations

Magnification:
vertical \(M = 23000\)
horizontal \(M = 35.6\)

\(\Delta h\)

Fig. 15.

On the other hand, in \(\Delta opp_1\) the quantity \(\delta_{e=0}=pp_1\) is also equal to

\[ \delta_{e=0}=2a\sin \frac{\alpha}{2} \tag{24} \]

Thus, the equality of (23) and (24) proves that

\[ \delta_{e>0}=\delta_{e=0}=\delta . \tag{25} \]

Consequently, the needle length \(e\), all other conditions being equal, does not affect the instrument readings and does not change the magnification scale \(M\).

Changing the length \(e\), as equations (20) and (21) indicate, changes only the magnitude of the horizontal displacement of the needle—\(y\), owing to which a shift is introduced into the profile of the irregularities being recorded (Fig. 14).

In the diagram, the profile curve is depicted with its peaks shifted in the direction of motion of the needle over the surface. This shift is the greater, the greater \(y\) is, i.e., the greater the value of \(e\), or the smaller the arm \(a\) (equations 20 and 21).

At a magnification scale \(M=1000\), from equation (9) \(a=10\) mm. The working length of the needle in this instrument is always set to

\[ e \approx 2\ \text{mm}. \]

Under the given conditions, from equations (20) and (21)

\[ \frac{e}{a}=\operatorname{tg}\beta=0.2 \]

the angle

\[ \beta=11^\circ 20' \]

\[ y=\delta \operatorname{tg}\beta=0.2\delta, \]

i.e., the horizontal displacement of the needle amounts to 20% of the measured heights \(\delta\).

Such a percentage of displacement of the depicted profile can occur only when the profilogram is recorded at natural size (Fig. 14), i.e., when the vertical magnification \(M\) and horizontal magnification—\(m\)—are equal to

\[ M=m=1. \]

In reality, the profilogram obtained in the instrument has:

\[ M>m>1, \]

therefore, in the depicted profile the ratio

\[ \frac{y}{\delta}\ne \operatorname{tg}\beta=\operatorname{tg}\beta \frac{m}{M} \tag{26} \]

From this it is evident that the larger \(M\) and the smaller \(m\) (with \(\operatorname{tg}\beta=\mathrm{const.}\)), the smaller the shift of the profilogram.

Table 3 gives the values of \(\dfrac{y}{\delta}\) in percent, calculated from equation (26) for different magnification scales of the instrument.

Table 3.

Vertical magnification \(M\)	Horizontal magnification \(m=12.5\)	Horizontal magnification \(m=25\)	Horizontal magnification \(m=50\)	Horizontal magnification \(m=100\)
1000	0.25	0.5	1.0	2.0
2000	0.5	1.0	2.0	4.0

The shifts in the profilograms are insignificant and therefore are completely imperceptible to the eye. This is also confirmed by the examples of the above-mentioned profilograms (Figs. 4a and c).

Does the angle \(\alpha\) change uniformly with the rise of the needle?

For given values of \(a\) and \(e\), the lower limiting position of the needle tip (point \(k\), Fig. 13) is characterized by the angle \(-\varphi\); the upper limiting position (point \(m\)) by the angle \(\varphi_1\).

\[ \varphi=\beta+\frac{\alpha}{4}\quad \text{for } \delta=0 \]

\[ \varphi_1=\beta-\frac{\alpha}{2}\quad \text{for } \delta=\delta_{\max}. \]

The difference of the limiting angles is equal to

\[ \varphi-\varphi_1=\alpha \tag{27} \]

i.e., to the full angle of rotation of the levers \(a\) and \(c\).

Let us denote:

\(\varphi_x\)—the current value of the angle of rotation of the needle corresponding at the given moment to the measured height \(\delta_x\).

From the geometry of the scheme, Fig. 13,

\[ \delta_x=\overline{ku}-\overline{k_1u_1} \tag{a} \]

where the legs are:

\[ \overline{ku}=a_1\sin\varphi \tag{b} \]

\[ \overline{k_1u_1}=a_1\sin\varphi_x \tag{c} \]

from equations (a), (b), and (c) we find

\[ \sin\varphi_x=\sin\varphi-\frac{\delta_x}{a_1} \]

Replacing \(a_1\) by its value (19), we finally have:

\[ \sin\varphi_x=\sin\varphi-\delta_x\frac{\cos\beta}{a} \tag{28} \]

Equation (28) expresses the dependence between the magnitude of the needle rise \(\delta_x\) and the angle of rotation \(\alpha\) (through \(\varphi_x\)), since on the right-hand side of the equation, for a given magnification scale \(M\), all quantities are constant except \(\delta_x\).

The limit of oscillation of the angle \(\varphi_x\) according to equation (27) is equal to the angle \(\alpha\), i.e., it is a relatively small quantity (\(\alpha_{\max}=0^\circ 30'54''\)); therefore \(\sin\varphi_x\) varies proportionally to \(\varphi_x\).

Consequently, the angle \(\varphi_x\), or equivalently the angle \(\alpha\), varies proportionally to \(\delta_x\) (28).

This proportionality is preserved for any \(M\), since \(\alpha_{\max}=\mathrm{const}\) for the given instrument.

5. Conclusions.

On the basis of all the data of the mathematical investigation, the following conclusion may be drawn.

a) With the selected relationship of the elements of the kinematics (\(H, L, c\), and \(b\)), the opto-mechanical part of the instrument has practically no influence on the accuracy of the profilograph readings, since the greatest (total) discrepancy in the readings from the true value is reached at the extreme positions of the light ray and amounts to fractions of a percent (\(\simeq 0.17\%\)).

b) Equation (8) is theoretical and computational, and therefore may have the following form:

\[ M=\frac{h}{\delta}=2\,\frac{c}{ab}\cdot R\cos\frac{\alpha_1}{2} \tag{29} \]

Limits of measurement and accuracy of the profilograph readings.

As applied to particular cases of measuring the surface finish of surfaces of various quality, the profilograph permits changes in the magnification scales along the coordinates of the profilogram within the following limits:

a) for the height of the irregularities (ordinates), magnification

\[ M = 1000 \div 2000 \]

b) for the length of the surface (abscissa), with horizontal feed of the needle by: 0.028 mm; 0.014 mm; 0.007 mm, the magnification is

\[ m = 12.5;\,25 \qquad 25;\,50 \qquad 50;\,100 \]

Vertical magnification is achieved by resetting the needle on lever \(c\) (Figs. 3 and 6); horizontal magnification—by changing the angle of rotation of the drum by means of electromagnet EMB (Fig. 7) and by changing the horizontal feed of the needle.

Since the greatest length of the profile curve imaged on the film is 140 mm, the actual length of the measured section of the surface, depending on the horizontal magnification, varies from 1.4 mm to 11.2 mm.

When constructing the profilogram from direct-observation data on the scale of readings (on the ground glass), the length of the measured section is unlimited.

The limiting height of the measured irregularities allowed by the instrument along the ordinate axis of the profilogram (90 mm) is equal to

\[ \begin{aligned} \text{for } M &= 1000 && 90 \text{ microns}\\ \text{for } M &= 2000 && 45 \text{ microns} \end{aligned} \]

The possible accuracy of readings (by eye) of the instrument indications on the scale (with divisions every 1 mm) is

\[ \begin{aligned} 0.2 \text{ micron} &\qquad \text{for } M = 1000\\ 0.1 \text{ ”} &\qquad \text{ ” } M = 2000 \end{aligned} \]

The technically possible accuracy of measurement provided by the profilograph (within the limits of the scale of readings) depends:

1) on the ratio of the dimensions and the designs of all elements of the kinematic scheme of the optico-mechanical part of the instruments (length of the light beam, ratio of the lever arms, dimensions and shape of the scale of readings—straight or concave, etc.), Fig. 11.

2) On the accuracy of manufacture of the parts of the entire system and of the kinematics of the instrument (for example: accuracy of parallel planes, mirrors, etc.).

On the basis of the mathematical investigation carried out, it was established that the selected ratio and design of the kinematic elements of the optico-mechanical part of this profilograph have practically no influence on the accuracy of measurement, since the greatest possible error in the instrument reading (at the extreme positions of the light beam on the scale) is \(\simeq 0.17\%\).

To determine the total influence of the unavoidable errors (inaccuracies) in the manufacture of the instrument parts on the accuracy of the readings of the latter within the limits of the entire scale, measurement (by probing) was carried out on a smooth and even surface.

If all parts of the profilograph had been manufactured accurately, then the light beam should in this case have depicted on the profilogram a line all points of which lie on a straight line without scattering, and conversely.

The surfaces subjected to measurement were those of two plane-parallel plates (Johansson type) and the surface of a glass polished plate. The accuracy and finish of the indicated surfaces are sufficiently high,

in comparison with the surfaces for whose measurement the instrument is intended, so as not to doubt their selection as a standard of a smooth and even surface.

The measurement was carried out by the method of probing with a sliding needle with manual feed. Readings were taken on the scale every \(0.056\) mm and \(0.028\) mm, i.e., after 2 or 1 revolution of the cardan shaft. The vertical magnification was \(M = 1150\). To eliminate oscillations of the light beam (scale readings) due to random scratches on the reference surfaces, the needle had a somewhat blunted tip. The standards were placed on the measuring table with a slight inclination so that, as the needle moved, the light “spot” would move within the limits of the entire reading scale.

On the basis of the measurement results, for each surface enlarged curves of the instrument readings were plotted. In Fig. 15, a part of one of the profilograms is shown with a magnification along the ordinate axis of 23000 (a 20-fold magnification of the instrument readings at \(M = 1150\)) and along the abscissa axis with a magnification of 35.6.

For clarity in representing the amplitude of oscillation of the light beam, the curves are brought to the horizontal position.

As can be seen from Fig. 15, the instrument readings are arranged not along a straight line, but along a wave-shaped line. The length and height of the principal waves (the small oscillations are explained by inaccuracy of reading) are not the same in different sections.

From comparison of the profilograms it became clear that all the curves, in their character, turned out to be of the same type, and when superposed one on another their contours coincide (according to the principal waves).

The identity of the curves proves that the cause of the wave-shaped scattering of the instrument readings (points) should be considered to be not roughness on the reference surfaces (in which it is difficult to assume coincidence for all surfaces), but errors inherent in the mechanism of the opto-mechanical part of this profilograph, due to the influence of inaccuracies in the manufacture of the instrument parts.

The greatest value of the scattering (error) of the instrument readings, \(\Delta h\), determined by the vertical segment between parallel straight lines drawn tangent to the crests of the waves on both sides of the curves (Fig. 15), is measured on the scale of readings as 1 mm.

This means that if the contour of any profile curve recorded by the profilograph from some machined surface (for example, Fig. 4) is outlined by a line 1 mm thick, then all errors in the instrument readings will fall within the limits of its thickness.

The uniform arrangement of the points of the instrument readings relative to the mean line \(xx\) (Fig. 15) permits one to consider that the greatest absolute error in readings on the scale of readings is

\[ \Delta h_{\max} = \pm 0.5 \text{ mm}. \]

For a given vertical magnification of the instrument \(M\), this error is equal to a constant value, i.e.,

\[ \begin{aligned} \text{at } M &= 1000; && \Delta h_{\max} = \pm 0.5 \text{ micron} = \text{const} \\ \text{” } M &= 2000; && \Delta h_{\max} = \pm 0.25 \text{ ” } \quad \text{”} \end{aligned} \]

When measuring different heights of irregularities, \(h\) (at \(M = \text{const}\)), the relative error in the readings is a variable quantity

\[ \frac{\Delta h_{\max}}{h} \ne \text{const}. \]

In the graph of Fig. 16, the values of the relative errors in %, at different magnifications \(M\), are given.

![Figure 16]

Graph labels:

• Vertical axis: Relative measurement error, \(\Delta \delta / \delta\), in %
• Horizontal axis: Height of irregularities \(R\), in microns
• Inset: Relative accuracy of measurement at various magnifications of the instrument \(M\)

Fig. 16.

![Figure 17]

Graph labels:

• Title inside plot: Calibration curve of the profilograph
• Vertical axis: Needle displacement \(h\), in mm
• Horizontal axis: Vertical displacement of the needle \(\delta\), in microns
• Legend:
• \(\odot\) — needle raising
• \(\times\) — lowering
• \(\operatorname{tg}\alpha = 1.15\)
• \(M = 1150\)

Fig. 17.

![Figure 18]

Diagram labels:

• Lever
• Needle
• Capacitor
• Surface profile
• Direction of motion of the workpiece
• \(x\), \(y\)
• \(B'\), \(A\), \(B\)
• \(\varphi\)
• \(\Delta h\)
• \(h\)
• \(h_{\max}\)
• \(\ell\)

Fig. 18.

The graph shows that, with an allowable measurement error within \(\pm 10\%\), this profilograph can measure irregularities up to 2.5 microns high at \(M = 2000\), and up to 5 microns at \(M = 1000\).

When measuring heights \(h = 1\) micron, the relative error reaches \(\pm 50\%\) at \(M = 1000\) and \(\pm 25\%\) at \(M = 2000\).

Unfortunately, the indicated measurement accuracy of this profilograph cannot be compared with the measurement accuracy of other existing instruments, since the authors of the latter are silent on this point. Usually only the accuracy with which the instrument readings are read is given, which by no means characterizes the attainable measurement accuracy.

With each repositioning of the needle along the length of lever \(c\) (Figs. 3 and 6), the vertical magnification (ordinate) of the instrument \(M\) is determined by calibration. During calibration the measuring chamber is in a stationary state. Raising (or lowering) of the needle is effected by measuring plane-parallel gauge blocks of various lengths. The instrument readings (by the light “spot”) are observed either on the frosted-glass scale or directly on the drum. In the latter case a scale grid is fastened to the drum, and, for observation of the beam motion, the side wall of the measuring chamber is removed.

On the graph (Fig. 17) two calibration “curves” (straight lines), shifted for clarity, are drawn; they correspond to one and the same setting of the needle on lever \(c\) (Figs. 3 and 6).

The scale of vertical magnification is

\[ M = \frac{h}{\delta} = \tg \alpha, \]

where: \(\alpha\) is the angle of inclination of the line to the axis of abscissae—\(\delta\).

For the case of needle setting shown in Fig. 17, \(\tg \alpha = 1.15\), which corresponds to \(M = 1150\).

The rectilinearity of the arrangement of the points indicates that the law by which the instrument readings change with the measured quantities remains constant over the entire reading scale.

The parallelism of both calibration “curves” indicates that the optico-mechanical part of the profilograph operates quite reliably.

A certain (but uniform) scatter of the points relative to the calibration lines is explained by the influence of the following factors:

1. Inaccuracy in the manufacture of the parts of the optico-mechanical part of the instrument and of the plane measuring gauge blocks (in length).

2. Unequal degree of tightness of the seating of the gauge blocks against the surface of the measuring table during calibration.

3. Inaccuracy in reading the scale (observer errors).

On the basis of repeated calibration runs it was established that the greatest amplitude of fluctuation (scatter) of the instrument readings reaches, in absolute value, \(\pm 0.4\) mm on the instrument scale, which corresponds to

\[ \begin{aligned} &\text{at } M = 1000; \ \pm 0.4 \text{ micron} \\ &\text{at } M = 2000; \ \pm 0.2 \text{ micron}. \end{aligned} \]

The smaller magnitude of the light-beam fluctuations during calibration, in comparison with the values established above (\(\pm 0.5\) and 0.25 micron), is explained by the fact that, during calibration, errors due to the influence of the instrument feed mechanism are excluded.

It is very important to point out that, with the optical method of recording measurement results by a light beam (as is adopted in this profilograph), the accuracy of measuring roughnesses does not depend on the inclination of the surface under test relative to the line of horizontal displacement of the needle.

(or of the article). Only the position of the profile curve on the field of the profilogram depends on the inclination of the measured surface segment; for example, horizontally, as indicated in Fig. 4, or somewhat inclined to the axis of abscissas.

This is one of the advantages (in addition to those already indicated above) of the given profilograph in comparison with instruments using an electrical method of reading the indications. (Instruments with capacitors: those of Berman (5) and Voronov (4)).

For example, in Berman’s instrument the zero reading corresponds to the position of the needle tip on the so-called line of the “dominant heights” of the irregularities on the measured surface segment (Fig. 18).

If the line of motion of the needle (or of the article) \(xx\) is parallel to the line of “dominant heights” \(yy\), then every lowering of the needle from the zero position (the line \(yy\)) until it contacts the surface at a given point to the depth of the depression \(h\) is marked by the galvanometer reading corresponding to the magnitude \(h\).

In most cases the line of “dominant heights” \(yy\) may be inclined to the line of motion of the needle (or of the article) \(xx\) at some angle \(\varphi\) (Fig. 18).

When the article moves to the left from point \(A\) (point \(A\) corresponds to the position of the needle when the instrument readings are set to zero), the needle will be lowered until contact with the surface at each given point by the height

\[ h + \Delta h \]

and, conversely, when it moves to the right—by the height

\[ h - \Delta h. \]

Consequently, the galvanometer readings will likewise be greater or less than the true value \(h\).

How large the error may be when measuring with Berman’s instrument (and similar ones) is shown quite convincingly by the following example.

Let the length of the measured surface segment \(AB\) be \(l = 1\) mm. The magnification scale of the instrument is \(M = 2000\). Let us assume that, practically always, an inclination of the line of “dominant heights” to the line of motion of the article is possible at an angle \(\varphi = 0^\circ 5'\).

Then

\[ \frac{\Delta h}{l} = \operatorname{tg}\varphi = 0.00145 \]

and

\[ \Delta h = 0.00145 \text{ mm at point } B. \]

According to the readings of the instrument, at \(M = 2000\) this value will be equal to \(\Delta h = 2.9\) microns.

If the measured height of the depression (at point \(B\)) is \(h = 4\text{–}5\) microns, then the relative error in the instrument reading is

\[ \frac{\Delta h}{h} = 72.5 \div 58\% \text{ of } h, \]

i.e., a value that is completely inadmissible.

Since, in order to characterize the smoothness of the surface, Berman takes the mean value of the measured quantities \(h_1, h_2, h_3\), etc., the instrument errors \(\Delta h_1, \Delta h_2, \Delta h_3\), etc., are distributed more uniformly, but even in this case they will influence the accuracy of the quantity \(h_{\mathrm{av}}\).

As for the determination (according to Berman) of the maximum height of the irregularities \(h_{\max}\) on the given surface segment \(AB\) (or \(AB'\)), the entire error \(\Delta h\) (at the given point) may enter completely into the value of \(h_{\max}\).

If to these errors in the instrument readings one also adds distortions due to electrical errors (which Berman himself notes [5]) and distortions from a continuously sliding needle, then in general one may conclude that existing instruments of the capacitance type (Voronov’s and Berman’s) give very inaccurate measurements and therefore are unsuitable for operational purposes, despite their main advantage over other instruments—a rapid (but incorrect) reading of the measured quantities \(h\), \(h_{\mathrm{av}}\), and \(h_{\max}\).

Conclusion

On the basis of the foregoing, the following conclusions may be drawn:

1. The advantage of the present experimental profilographic installation consists in the fact that, owing to the use of the intermittent (point) method of probing the surface and intermittent recording of the measurement results at the moment when the instrument is in a static state:

a) the influence of measurement errors inherent in all instruments operating by the method of continuous probing and with continuous recording (optical or electrical) of the measurement results has been completely eliminated,

b) complete automation of the process of taking profilograms has been achieved, without distortion due to the influence of vibrations and shocks.

1. The use of the optical method of recording (intermittent) measurement results eliminates the influence of the inclination of the measured surface on the accuracy of measurement. This is one of the advantages over instruments of the capacitance type with an electrical method of reading the instrument indications.

2. The design of the present profilograph installation makes it possible to measure the roughness of the surface of specimens (articles):

a) in two mutually perpendicular directions,

b) having considerable dimensions in length (over \(1000\) mm) and in cross-section (\(150 \times 150\) mm),

c) obtained as a result of finishing on lathes, planers, milling machines, and grinding machines,

d) over a section of any length, in the case of constructing the profilogram from readings on the scale, and over a length of up to \(11\) mm when taking it on film,

e) with a measurement accuracy of up to \(\pm 0.5\) micron at \(M = 1000\) and \(\pm 0.25\) micron at \(M = 2000\).

1. The disadvantages of the profilograph should include:

a) profile measurement is possible only on the external surfaces of specimens,

b) the limits for detecting microcracks and tears on the measured surface are limited by the radius of the probing needle (5–6 microns), which is in general characteristic of all probing methods without exception,

c) the time for taking a profilogram is 6–12 minutes.

Literature

1. Abbott and Firestone.—A profilograph for measuring irregularities of machined surfaces. Automotive Industries No. 8 (1933), p. 2040.

2. Schumacher H.—The present state of the technique of measuring surface finish. Maschinenbau/Betrieb, vol. 14 (1935), p. 379.

3. Ammon K. L.—Methods for determining the quality of a machined surface. Collection of abbreviated reports of the conference on metal cutting. Publishing House of the Academy of Sciences, 1937, p. 365.

4. Domoratskii A. E.—Modern methods for studying the cutting process and apparatus. Collection of abbreviated reports of the conference on metal cutting. Publishing House of the Academy of Sciences, 1937, p. 352.

5. Berman S. R.—Determination of surface quality. Mashinostroitel’, No. 8, 1938, p. 15.