From linear mechanics to nonlinear mechanics


Material Information

From linear mechanics to nonlinear mechanics
Series Title:
Physical Description:
18 p. : ill. ; 27 cm.
Loeb, Julien
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Remote control -- Research   ( lcsh )
Nonlinear systems   ( lcsh )
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Consideration is first given to the technique used in telecommunication where a nonlinear system (the modulator) results in a linear transposition of a signal. It is then shown that a similar method permits linearization of electromechanical devices or nonlinear mechanical devices. A sweep function plays the same role as the carrier wave in radio-electricity. The linearizations of certain nonlinear functionals are presented.
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Julien Loeb.
General Note:
"Report date October 1955."
General Note:
"Translation of "De la mécanique linéaire a la mécanique non linéaire." From Annales des Télécommunications, vol. 5, no. 2, Feb. 1950."

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Full Text
N3tP4 M lr3L




By Julien Loeb2


It is first recalled how, in the technique of telecommunication, a
nonlinear system (the modulator) gives a linear transposition of a signal
and then it is shown that a similar method permits linearization of
electromechanical devices or nonlinear mechanical devices (relays, direc-
tional loops of radiogoniometry, etc.). The function of sweep plays
the same role as the carrier wave in radioelectricity.


For about 10

years, the introduction of the methods perfected by
engineers into the calculation of mechanical or
systems has constituted certain progress.

This is known to be due to the fact that the
since they are in first approximation of a linear
thoroughly studied by means of mathematical tools
during the last century, such as matrix calculus,

electrical systems,
nature, could be
already developed
symbolical analysis,

To the extent as they may be considered linear, the mechanical or
electromechanical systems can be treated by the same methods.

"De la mecanique lineaire a la mecanique non lineaire." Annales
des Tle4communications, vol. 5, no. 2, Feb. 1950, pp. 65-71.

1The author discussed the subject of this article orally on June 28,
1949 and October 18, 1949, during the Conferences of the Study Center of
Flight Mechanics, organized by the Special-Engines Section of the Aero-
nautical Technical Service. Moreover, the author had partially treated
this subject in his Preliminary Note No. 144 of the National Laboratory
of Radioelectricity.

2Chief Engineer at the C.N.E.T., Head of the Department Remote
Control and Counter Measures.

NACA TM 1596

Unfortunately, their characteristics are rarely linear. In partic-
ular, the mechanical systems can generally be considered linear only when
they are subject to small oscillations. This permits treatment of stabil-
ity problems, provided, however, that the magnitudes are not too large.

The methods taken over from the technique of electrical networks
do not permit, for instance, treating by means of relays, electro-
mechanical systems functioning in a discontinuous fashion.

This is very regrettable since relays constitute powerful and small
amplifiers which have only the one fault of not being linear.

The object of the present study is to show that in the arsenal of
telecommunications, one may yet find an entirely general method of
treating nonlinear problems or, more accurately, of transforming them
into linear problems.

The definition of this method is contained in five words: "Utili-
zation of a carrier frequency."



2.1. Linear Systems

Electromechanical systems are essentially conceived for transmitting
messages (telegraph, telephone, facsimile, television, etc.).

A message is characterized by a band width, that is, by the spectrum
of the frequencies it contains which must be passed.

Moreover, the "quantity of information" a channel can transmit is
limited by a "magnification" of the amplitude of the signal. Two signals
of the same frequency and different amplitudes are actually distinguish-
able only if the difference of their amplitudes exceeds a predetermined
magnitude. This magnitude stems principally from the noise always present
in a radio or wire connection. In mechanics, a system of remote control
also must convey a message. Here the band width is much smaller than that
of the telephone or even of the telegraph, because of the large inertia
of the mechanical components in motion. According to the utilization,
the band width varies between 1 cycle/second and 10 cycles/second, or
even, possibly, up to about 100 cycles/second.

In the case of a regulator, the concept of band width appears less
clearly but still applies. Evidently the physical quantity which must


be maintained by the regulator and which is imposed from the outside
cannot vary. Nevertheless, the regulator must necessarily possess a
response sufficient to re-establish the desired condition when a pertur-
bation has momentarily disturbed it.

An automatic pilot for an airplane is at the same time something
of a remote controller and something of a regulator. It must follow
a course with sufficient rapidity, even if this course is modified by
the pilot, and in addition, it must return rapidly to its position
after gusts which play here the role of parasitic impulses.

2.2. Carrier Frequencies

Most frequently, the signal is not sent just as it is, but is used
for modulating an auxiliary current the frequency F of which is called
the carrier frequency. A modulator is essentially a nonlinear device
in which the carrier frequency and the signal are added. Let

P = Po sin 2tFt be the carrier frequency

S = SO sin 2tft the signal

Assume cp(a) to be the function representing the action of the
modulator. If one adds in the latter P and S, there results
c(P + S). Expanding in Taylor series, one obtains

p(P + S) = a + al(P + S) + a2(P + S)2 + a(P + S) + .

If one lets the modulator be followed by a filter which transmits
only a band centered on F, one obtains the term

2a2PS = 2a2(SO sin 2itft)P0 sin 2~Ft

Thus one has a carrier frequency P0 sin 2tFt the amplitude
SO sin 2tft of which represents the signal to be transmitted.

This makes sense only if f is much smaller than F. In practice,
f is at most of the order of one-third of F.

Thus it is seen that if one encounters in a chain of transmission
a nonlinear element (modulator), one finds a linear function of the
signal by operating as follows:

Add a carrier frequency.

Filter a band centered on this carrier frequency.

NACA TM 1596


We shall examine a certain number of mechanisms regarded as trans-
forming an input quantity into an output quantity. In the most general
case, this latter will be a continuous or discontinuous function or a

5.1. Continuous Function

3.11. Odd function.- Here is, for small amplitudes, a linear mech-
anism: the output quantity is, in magnitude and sign, simply propor-
tional to an input quantity. For instance, the servomechanisms in
which the geometrical position of an indicator is governed by the resist-
ance of a potentiometer belong to this category. (See fig. 1.)

The control in this case is effected by means of the slider C1
of the potentiometer Pl. The fed-back quantity is the position of the
slider C2 of the potentiometer P2. The input quantity e is the
geometrical displacement between the two sliders.

The signal furnished by the error-sensing device is the difference
in potential E between C1 and C2. E is proportional to e.

Another example is given by figure 2 which represents the classical
error-sensing device of an angular Selsyn control.

The control is effected by modification of the angle 8 which the
single-phase rotor (fed by 50 c/s) forms with a fixed reference. The
fed-back quantity is the angle 9'. The error (input quantity) is
e = e 9'.

The output quantity E is an alternating current having a fre-
quency of 50 c/s, the amplitude of which is in magnitude and sign
proportional to sin E.

One considers this system as linear for small values of e.

5.12. The output quantity is an even function of the input quantity.-
Let us examine, for instance, the case of radiogoniometry. It is well
known that there exist devices which permit automatically guiding an
indicator connected with the directional loop toward the source of electro-
magnetic radiation. They are the radio compasses.

Besides, even if the radiogoniometry is not automatic, the ensemble
formed by the directional loop and the operator constitutes a servo-
mechanism (and not one of the better ones, either).

NACA TM 1596

If one plots as abscissa the angular error and as ordinate the
detected value E of the current which leaves the' directional loop,
one obtains the curve of figure 3. Here one has the indication of the
absolute value of the angular deviation but one has no longer the sign.

A condition of the same kind was encountered when one attempted
to design a local oscillator producing a voltage the frequency of which
is to be made equal to a given frequency. Here again the frequency
of the beat current indicates the absolute value of the difference, but
the sign of the latter does not affect the output current.

5.2. Discontinuous Functions

In mechanisms with "on or off" operation, the output signal increases
abruptly from 0 to a given constant value when the input quantity is
positive, and from 0 to an opposite value, likewise constant, when
that quantity is negative. The diagram of figure 4 which describes the
same device as that of figure 1, with one addition, a polarized relay,
gives an example for this. In this case, the function is discontinuous
and odd (in general, such systems present a sensitivity threshold which
the error must exceed for the signal to exist).

Figure 5 gives the curve of E as a function of E.

5.5. Functionals

It happens quite frequently that the output signal is not a function
of the error (case where the output voltage depends only on the actual
value of the input quantity) but a functional (case where the output
voltage depends not only on the actual value of the variable but also on
its prior values).

5.51. Continuous functional.- Figure 6 shows the diagram of a system
derived from that in figure 1 in which the input signal is amplified by
means of an electromechanical amplifier, an "amplidyne" for instance.

Figure 7 gives the curve of E as a function of e.

3.52. Discontinuous functional.- The example for this is furnished
by a phase-sensing device used at the Centre National d'Etudes des Tel4-
communications; its principle is as follows:

An output signal is to be obtained which is a function of the phase
displacement E of two currents of the impulses D and G (fig. 8),
taken positively if G lags behind D, and negatively in the opposite
case (only small values of e are dealt with here).

NACA TM 1596

A "seesaw" or "flip-flop" is actuated by these two currents: D acts
on the tube at right,. G on the tube at left (fig. 9).

The signal E will be the mean value of the voltage between the
plates. If E is positive, the flip-flop is in its position G during
almost the entire period. If E is negative, it remains in D during
the same time.

There occurs therefore an abrupt jump when E passes through zero.
In fact, E must slightly exceed the value of zero in order to make the
flip-flop operate. One then obtains the diagram represented in fig-
ure 10.


One now has to find out whether the systems described above can be
transformed into linear systems. Section 2.2 furnishes us the method.

It has been seen that the modulation of a carrier wave by the signal
requires the use of a nonlinear network. One then obtains a modulated
carrier the amplitude of which is a linear function of the input signal.
Here we shall accept our nonlinear system such as it is; we shall super-
impose on its circuits the input quantity to be transmitted and a peri-
odic function, of time frequency F sinusoidall, for instance). A
periodic function of the frequency F modulated by the input signal
results. Furthermore, after detection it will be possible to obtain
a linear function of the input signal. We shall see how this general
idea is applied to the concrete cases.

4.1. Method Followed by an Operator

As happens very frequently when one attempts to invent a new
automatic device, one must begin by analyzing what motions an operator
performs who actuates a known manual device. We shall consider for this
analysis the operation of a directional loop in radiogoniometry. The
operator ignores the actual direction of the wave which reaches him.
He orients his directional loop at random, and generally perceives a
signal. This fact informs him that his direction loop does not occupy
the desired position but it cannot tell him in what direction to turn
the directional loop in order to nullify the signal. Reasoning is sub-
stituted for lacking indications. The operator displaces the directional
loop in a certain manner; if this displacement produces a reinforcement
of the signal, the operator has made the displacement in the wrong direc-
tion; therefore, he will displace the directional loop in the opposite
direction which will bring it closer to the correct position.

NACA TM 1596

It will frequently happen that the motion imparted to the directional
loop will overshoot the mark, and the error will have changed in sign.
If the operator does not have good reflexes, it might even happen that
every motion intended to bring the directional loop into the desired posi-
tion overshoots its mark, and the combination of directional loop plus
operator will get into a self-sustained oscillation.

There will be the more chances for the appearance of this oscilla-
tion, for the same operator, the greater the required accuracy.

In the language of servomechanisms, one may say that the more the
error is amplified, the more the system tends to oscillate by itself.

4.2. General Procedure of Linearization

The above example shows that it might be of advantage to set up
mechanisms which reproduce the alternating motion effected by the radio-
goniometry operator in search of his mark. That is precisely where the
carrier frequency mentioned before comes in. One adds to the input
quantity an arbitrary sinusoidal function of time, of the form
EO sin 2iFt. Other periodic functions may be used, notably "saw-tooth"

We shall call this function the "sweep function." As in the domain
of telecommunications, the frequency F must be placed far beyond the
band width of the system.

We shall demonstrate the two following results, valid for small
values of the input quantity e:

(1) If the system is odd (representative curve symmetrical with
respect to the origin), the mean value of the signal given at its out-
put is proportional in magnitude and in sign to e.

(2) If the system is even, the component of the output current
which is at the sweep frequency has an amplitude which is proportional
in magnitude and in sign to e.


The above theorem is evident when th'e output voltage of the error-
sensing device is an odd and continuous function of the error.

8 NACA TM 1596

5.1. Odd and Discontinuous Function

When the output voltage of the error-sensing device is an odd and
discontinuous function of the error, this result continues to exist as
is demonstrated in the report by Mac Coll.3

Figure 11 brings, in substance, this demonstration which applies
to an "on or off" system.

This system gives a signal:

+E as long as e > 0

-E as long as E < 0

The difference between the durations of the applications of the
signals +E and -E is equal to 4r, with

cO sin -T=


E << EC


and the mean value E of the signal is

T = E x" = 2C E
T A 0

This proportionality continues to exist if the system gives a curve
analogous to that of figure 5. We shall not give here the demonstration
of this result because this demonstration is entirely similar to the one
that will be given further on, with regard to the discontinuous functional.

5.2. Odd and Continuous Functional

We may extend this result to include the case where the output
current is an odd functional.

In fact, let F(U) be this functional. In the cases we are
studying (curve of hysteresis), it will lead to a function of (U) which
may have two values.

3Mac Coll (L. A.), Elements de la theorie des servomecanismes
(FuLndamental theory of servomechanisms). New York, D. Van Nostrand Co.,
Inc., 1945.

NACA TM 1396

For instance, in the case of figure 7 one has F(U) = F1(U) when
(U) varies from -U0 to +U0, and one has F(U) = F2(U) when U comes
back from +UO to -UO.

When one has U = E + eO sin at, the functional F(U) becomes a
function of t and of .

In the most general case, the instants when the function changes
its value will be continuous functions of e. Consequently, the integral
J F(cE co sinj t)dt likewise will be a continuous function of e.

If the origin of the coordinates is chosen in such a manner that,
for E = 0, this integral becomes zero (symmetry of the hysteresis
curve with respect to the origin) and if the first derivative of this
integral with respect to E does not become zero, one thus has a signal
which is proportional in magnitude and in sign to e.

5.5. Odd and Discontinuous Functional

We shall obtain the same result with the system of figure 10.
(See fig. 12.)

'The curve I represents the variation of 60 sin at as a function
of time. This function reaches the magnitude +8 at the instant t;
then, when it attains in descending the magnitude +8 by hypotheses,
nothing passes because of the hysteresis. At this moment the signal
has the value +CO. Only when the function e0 sin at will have
attained the value -8 (instant tl), the signal will be reversed so
as to attain -C1. Likewise, it will become positive again only at the
instant t2.

Since the time intervals t0, t1, t2 are equal, the mean value
of the signal is zero.

This is no longer the case when one has, instead of the signal
CO sin wt

E + CO sin at (curve II)

At this moment, the instant t where the function attains +8
comes before to, the instant tl' where the function attains -8 comes

NACA TM 1596

after tl, and the instant t2' where the function exceeds +6 (exactly
one period after t0') is likewise ahead of t2.

As a result, the signal is equal to +CO during a longer time
than it is equal to -CO.

The mean value of the signal is therefore no longer zero, and it
can easily be seen that, for e < 6, with eO > 8, it is given by the
following formula:

1 -2

Thus one sees that one does not only obtain an output signal which
is a linear function of e but that one has found a method which permits
modifying the coefficient of proportionality at will.

For instance, the following experimental curves (fig. 15) were
drawn up for various values of e in the phase-sensing device described
in section 5.32.

The Sperry compass, with its oscillating "hunting" device, likewise
illustrates this method of linearization in the case of an odd functional.
In this case, the sweep is "saw-tooth" type.


So far we did not encounter error-sensing devices which were at the
same time even and discontinuous; their treatment would be the same. The
curves given by those one may encounter have, in general, the appearance
of figure 14.

Since here the first term of the expansion of F in terms of the
increasing powers of its argument is of the form K(e + 0 sin wt)2,
one obtains, as frequency term w, an expression of the form Kee0 sin ut.

Therefore, the signal has, at the frequency cu, an amplitude propor-
tional to the error e in magnitude and sign, that is to say, its phase
changes by 1800 when the error changes in sign. It is analogous to the
signal one receives in position-controlling servomechanisms operating
on alternating current (section 5.11, fig. 2). This result continues
to be valid in the case of an angular curve represented in figure 15.

NACA TM 1596

This form of signal is advantageous when the desired power exceeds
the power the system can furnish directly (error-sensing device in a
servomechanism, for instance) because an alternating voltage can be
amplified much more easily than a continuous one.

This procedure appears, therefore, to be better than the "false-
zero" methods usually employed for avoiding the difficulty presented by
even systems, with the possibility of instability of the zero point
which characterizes these methods.

An example of these systems (in the present case, an error-sensing
device in a servomechanism) has been given above; it is the directional
loop in radiogoniometry, with all its possible variants one of the best
known of which is the automatic tracking device of radar; here again the
angle-sensing device, that is to say, the antenna placed in the focus
of a mirror, has a symmetrical diagram; one adds to the angular deviation
to be sensed an angular deviation which is a sinusoidal function of
time, by making the antenna rotate eccentrically.

A method of the same character is described by Pierre Debraine and
Cestmir Simane4 for the regulation of the magnetic field of a cyclotron.

There one has to control the magnetic field with respect to the
ionic output current in such a manner as to maintain it at a value pre-
cisely equal to that required by the other characteristics of the appa-
ratus (notably the frequency of the voltage applied to the "Dees")., The
curve giving the ionic current as a function of the error E committed
in the magnetic field is even, at least in first approximation. The
authors applied the principle of linearization by sweep, adding to the
magnetic field a component alternating at 5 cycles/second. The ionic
current involves an error signal at 3 cycles/second the amplitude of
which is proportional to the error, and the phase of which is 00 or 1800
according to whether the error is positive or negative. This signal
at 5 cycles/second is filtered (in particular, it is separated from the
parasitic signal of the frequency 6 cycles/second) and after amplification
and detection it is applied to the excitation of the generator which
feeds the electromagnet.


An arbitrary function or functional may always be considered as the
sum of two functions (or functionals) of which one is even and the other

2F(U) = (F(U)) + F(-U) + (F(U)) F(-U)

4Dispositif de synchronisation automatique du cyclotron. C.R. Acad.
Sci. Fr., (25 fevr. 1948) 226, no. 8, pp. 648-650.

NACA TM 1596

The procedure of linearization will generally give, if E is a
small constant:

(a) A continuous signal proportional to E.

(b) An alternating signal of the form KE sin wt.

Both may be used.


We have thus shown that the technique of telecommunications can
furnish solutions in the field of mechanics, even in the case of non-
linear systems.

It is evident that mechanical and electrical engineers, struggling
with their problems, have thought of solutions of the type which were
indicated above without waiting for the telecommunication engineers.
However, the above outline constitutes an attempt at generalization
and classification of these methods.

The analogy thus established between the problems of mechanics and
the problems of telecommunications permits, besides, recognizing a
limitation imposed on this type of solution by the existence of the
allowable frequency bands.

Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics

NACA TM 1596

Figure 1.-

Figure 2.-

Figure 3.-


NACA TM 1396


Figure 4.-

Figure 5.-

NACA TM 1596

Figure 6.-


Figure 7.-

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Current G


Figure 8.-






Figure 9.-

Figure 10.-

NACA TM 1596

Duration of the positive signal
I --- _

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Figure 11.-

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b,' _-. T

IACA TM 1596



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NACA Langley Field, Va.



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