A New Harmonic Analyser.
Year: 1893

page_text: Date of Application, 30th Aug., 1893
Complete Specification Left, 30th May, 1894-Accepted, 25th Aug., 1894
PROVISIONAL SPECIFICATION.
A New Harmonic Analyser.
I, ARCHIBALD SHARP, Engineer, 13 Fairlawn Grove, Chiswick, in the County
of Middlesex, do hereby declare the nature of this invention to be as follows:-
In the Fourier expansion
the coefficients being connected by the relations
or
anc Cos an
-b
= Ca
sin a
•
c³₁ = a² + b²
tan an=
-
=
Let the curve A₁ A A, (Fig. 1) be that represented by the equation y = f(x),
the scale of abscissæ being such that A, A, 2. Suppose a wheel W to roll on
the paper, and to be connected with a tracing point P in such a manner that as P
moves uniformly in the direction OX, the axis of the wheel W turns uniformly in
a horizontal plane, and the amount of rolling of the wheel at any instant is equal
to the corresponding displacement of P in the direction OY; then pp¹ (Fig. 2) is
= cos x dy.
equal to dy (Fig. 1), p¹g sin x dy and p q
=
If the axis of the rolling wheel makes one complete turn while P moves over the
curve from A, to A, the point of contact of the rolling wheel will describe a
ה

page_text: If the axis of the rolling wheel makes n revolutions while P moves from A₁
to A2, the corresponding values of P₁ ? and <P P₁ C will be nπ c and a
Various arrangements are suggested for connecting the rolling wheel with the
tracer, the following seems the most suitable, as it is adapted for an instrument to
give more than one simple harmonic constituent term for one tracing of the
curve.
The motion of the rolling wheel relative to the paper is compounded of two
movements (a). a pure rolling, the distance rolled being equal to dy the element
traced out by the tracer P, (b) a motion of rotation, the point of contact of the
wheel and paper being the centre of rotation, the angle turned through being
proportional to the abscissa of P. The wheel may therefore be rolled along a
fixed straight line, while the paper is made to turn; the centre of rotation of the
paper being the point of contact of the wheel with it, which is continually varying
in position.
Fig. 3 represents diagramatically the mechanism. The carriage F F which forms
the base of the instrument is supported by an axle with two equal wheels wi, and a
third wheel v, which roll on the paper on which the curve A, A A, is drawn, the
direction of motion of the carriage being OX. The axle drives by bevil gearing
the disc d. A long key on the upper surface of this disc fits into a groove on the
under surface of disc d, which is thus free to move in a straight line relative to
disc d₁. A groove on the upper surface of disc d, at right angles to that on its lower
surface has a key from the lower surface of disc d. resting in it. Thus the disc d₂
always turns with disc d₁. Any point on disc da may be made the centre of
rotation, the three discs being kinematically equivalent to Oldham's coupling for the
transmission of motion between two parallel shafts. The keys and grooves would
be replaced in the actual instrument by wheels and rails, in order to diminish
frictional resistance.
The tracing point P is mounted on a smaller carriage f, which is free to run in
the direction OY relative to main carriage. This smaller carriage carries also the
rolling wheel W, which rolls on the disc d..
The rolling wheel W should be spherical in form, and of as small diameter as
possible, so that its surface of contact with the paper on disc d, approximates
to a point. The friction between the wheel W and disc d, is great enough to
prevent any sliding. As the tracer P moves over the curve A, A A₂, the point of
contact of the wheel W will describe on the disc d, the curve p₁ P. p¹ P₂ (Fig. 2).
A needle or pencil n carried by the carriage f, is made to mark on the disc d.
when the tracer P is in its initial and final positions A₁, and A.. The direction
of the initial line O R will be recorded by the pencil n on the disc ds by making the
tracer P move along the line O Y.
The instrument gives by measurement the amplitude and epoch of each separate
simple harmonic constituent. The gearing must be such that the disc dz turns once
while the tracer P moves from A₁ to A,, the equal wheels wi w rolling on the paper.
If the wheels w, w, be half the diameter of w₁ and roll on flat rails lying on the
paper, (wheels w₁ w, being clear) the disc d3 will turn twice while the tracer moves over
the curve f₂ and a2 will be obtained. Similarly by having the diameters wз w
respectively, t, . . . of w₁ the constants of the 3rd, 4th, terms of the
expansion may be obtained. To ensure that as the tracer P moves in the
direction (Y the wheel W will roll on the disc d, and not cause the said disc dz to
slide relatively to dises d, and d, a second rolling wheel W₁ is fixed on the same
•

page_text: axle as the rolling wheel W, the wheel W₁ rolling on a fixed position of the
carriage F.
An analyser to give a number of pairs of coefficients C and a₁ may be constructed
by having a number of sets of discs turning with the velocity ratios 1, 2, 3, 4, . . .
being driven by bevil gearing from a pair of axles, on each of which is fixed a pair
of equal wheels w₁ w₁ which will roll on the paper.
It will be noticed that with a small disc d, the rolling wheel W may run off
before the whole of the curve pi p p¹ p₁ (Fig. 2.) is described. In this case let the
circle (Fig. 4) represent the out-line of the disc ds and p₁ R the initial line, suppose
the pencil n to have drawn the portion p of a curve when there occurs the danger of
the wheel W rolling off the disc. Keeping the tracing point P fixed the discs d
may be moved relative to d₁ so that the pencil n takes up any convenient position a
the portion a¹ of the curve is then described, at 6 the tracer P is again fixed and
the disc d, moved so that the pencil n takes up the position b₁ the remainder 6' P,
of the curve is then described.
From A draw a B equal and parallel to a' b and from B draw B P, equal and
parallel to b¹ Pa P₁ P, and P, p₁ R are the required length and angle respectively.
Dated this Thirtieth day of August 1893.
0
COMPLETE SPECIFICATION.
A New Harmonic Analyser.
I, ARCHIBALD SHARP, Engineer, 13, Fairlawn Grove, Chiswick, in the County
of Middlesex, do hereby declare the nature of this invention and in what manner
the same is to be performed, to be particularly described and ascertained in and by
5 the following statement:-
T

page_text: In the following description Figs. 1, 2, 3 and 4 refer to the sheet of drawings
filed with the Provisional Specification; Figs. 5-11 to the sheet of drawings filed
with the Complete Specification.
Let the curve A, A A, (Fig. 1) be that represented by the equation y = f(x),
the scale of abscissa being such that A, A, = 2π. The integrator of the harmonic
analyser consists of a wheel W rolling on a horizontal flat surface, and connected
with the tracing-point, P, of the instrument in such a manner that as P moves
uniformly in the direction OX (Fig. 1) the axis of the wheel W turns uniformly.
in a horizontal plane, and the amount of rolling of the wheel at any instant is equal
to the corresponding displacement of P in the direction OY. Then, if p and pl
(Fig. 2) be the points of contact of the wheel W on the said flat surface corre-
sponding to the tracer positions P and P¹ (Fig. 1), pp¹ is equal to dy (Fig. 1)
sindy, and p q = = cos x dy.
If the axis of the integrating wheel makes one complete turn while P moves
over the curve from A, to A., its point of contact will describe a curve p₁ p p¹
(Fig. 2). Let OR be the initial direction of the integrating wheel; draw p₂
perpendicular to OR; then, evidently, på C = ƒ sin x dy = p₁ C = ƒ
and therefore pi pa =1, and the angle p, pi ¸C = a1.
cos x dy,
and
If the axis of the integrating wheel makes n revolutions while P' moves from
A1 to A,, the corresponding values of pipa and the angle p, p₁ C will be nπ c ад
respectively.
Various arrangements are suggested for connecting the integrating wheel with
the tracer so as to satisfy the conditions described above; the following seems
suitable, as it can be adapted for an instrument to give more than one simple
harmonic constituent term for one tracing of the curve.
The motion of the integrating wheel relative to the flat surface is compounded of
two movements (a) a pure rolling, the distance rolled being equal to dy the element
traced out by the tracer P; (6) a motion of rotation, the angle turned through
being proportional to the abscissa of P. The wheel may therefore be rolled
along a fixed straight line, while the flat surface is made to turn, the centre of
rotation of the said flat surface being the point of contact of the wheel with it,
which is continually varying in position.
Fig. 3 represents diagramatically one form of mechanism. The carriage FF
which forms the base of the instrument is supported by an axle with two equal
wheels w, and a third wheel v, which roll on the paper on which the curve A, AA,
is drawn, the direction of motion of the carriage being OX. The axle drives by
suitable gearing the disc d₁. A long key on the upper surface of this disc fits into
a groove on the under surface of disc d,, which is then free to move in a straight
line relative to disc d₁. A groove on the upper surface of disc d, at right angles
to that on its lower surface, has a key from the lower surface of disc da resting on
it. Thus the disc d, always turns with disc di. Any point on disc d, may be made
the centre of rotation, the three discs being kinematically equivalent to Oldham's
coupling for the transmission of motion between two parallel shafts. The keys
and groove would be replaced in the actual instrument by wheels and rails (as
described hereinafter) in order to diminish frictional resistance.
The tracing point P is mounted on a smaller carriage f, which is free to run in
the direction OY relative to the main carriage. This smaller carriage carries also
the integrating wheel W, which rolls on disc ds.

page_text: The integrating wheel W should be preferably spherical in form, and of as small
diameter as possible, so that its surface of contact with the paper on disc d,
approximates to a point. The friction between the wheel W and disc d, is great
enough to prevent any sliding. As the tracer P moves over the curve A, A A,
the point of contact of the wheel W will describe on the disc ds a curve pip p¹ pr.
(Fig. 2).
A. needle or pencil n, carried by the carriage f, is made to mark on the disc d
when the tracer P is in its initial and final position A, and A.. The direction of
the initial line OR will be recorded by the pencil n on the disc d, by making the
tracer P move over the line O Y.
The instrument gives by direct measurement the amplitude and epoch of each
separate simple harmonic constituent. The gearing must be such that the disc d
turns once while the tracer P moves from A₁ to A,, the equal wheels w₁ w₁ rolling
on the paper. If the wheels w, w, be half the diameter of w, and roll on flat rails
lying on the paper (wheels w₁ w, being clear) the disc d, will turn twice while the
tracer moves over the curve; C, and a, will be obtained; and similarly by having
the diameters w, w
respectively, t that of the coefficients of the
3rd, 4th
terms of the expansion may be obtained.
•
To ensure that as the tracer P moves in the direction OY the wheel W will
roll on the disc d₂, and in case the said disc d, should slide relatively to discs d,
and d₁ a second wheel W' is fixed on the same axle as the integrating wheel W,
the wheel W' rolling on the fixed portion of the carriage F.
If the curve described by the integrating wheel be too large for the disc d., it
may be drawn in portions p, a, a¹h, b¹p, (Fig. 4); the tracer P being fixed while
the disc is displaced so that the point of contact of the integrating shifts from a to
a more convenient position a', and again from 6 to b¹. The final point P, of the
complete curve is got by drawing a B parallel to a¹ b, and B P, parallel to b¹ ps.
An analyser to give a number of pairs of coefficients c and a may be constructed
by having a number of sets of discs turning with the velocity ratios 1, 2, 3, 4,
driven by suitable gearing from a pair of axles, on each of which is fixed a pair of
equal wheels w₁ w₁ which will roll on the paper.
Referring now to the sheet of drawings filed with my Complete Specification,
Figs. 5 and 6 are plan and sectional elevation respectively of a kinematic equivalent
for the intermediate disc above described. In this modification the frame of the
intermediate disc d, has pivoted to it a set of two or more parallel spindles b on
which are formed wheels B and B'. The wheels B run in a straight groove on
the lower disc d, of the integrator above described; thus the frame d, (Fig. 5)
can only move in a straight line relative to the disc d₁ (Fig. 3). The frame d,
has also another set of spindles a pivoted to it at right angles to spindles 6;
wheels A A' are formed on these of exactly the same shape as those on spindle b.
A groove on the under surface of disc d, rests on the top of the wheels A,
and thus the disc ds can move relative to the disc d, in a straight line.
In order to reduce pivot friction, and to keep the dimensions of the integrator
within smaller limits, the following modification may be used:
On the lower disc d₁ rolls a pair of spindles a which are pivoted to a light frame d'
(Figs. 7 and 8); a pair of equal wheels A are formed on each spindle a, and on
the top of these wheels rolls disc d,. A second pair of spindles b (pivoted to a
light frame d) roll on disc d,, and have formed on them a pair of equal wheels B,
which in turn support the disc d₁. Fig. 9 is an enlarged view shewing the guidance
for the rolling spindles on the discs, and also the guidance of the discs on the
rolling wheels.
I prefer, however, the following arrangement of the mechanism to that shewn in
Fig. 3 of the sheet of drawings filed with my Provisional Specification: Fig. 10 is
a front elevation, and Fig. 11 a side elevation partly in section, of an instrument to
give two pairs of coefficients for one tracing of the curve to be analysed. The
carriage FF which forms the base of this instrument is supported by an axle with

page_text: two equal wheels w and a third wheel v (or pair of wheels) which roll on the paper
(or on two flat rails R of convenient thickness placed on the paper) in the
direction OY (Fig. 1). The carriage FF carries a vertical spindle s with the
discs: d; d, dz, as already described in Fig. 3. The small carriage f, carrying
the tracing point P, is free to move in the direction OX (Fig. 1) relative to the
main carriage F; and a cord or wire has its ends fixed to the carriage and passes
uver a pulley 9 rigidly fixed to the disc di. The integrating wheel W is formed
on the axle of the wheels ww, its axis and that of the vertical spindle s inter-
secting. Thus the rolling of the wheel W on the disc d is proportional to the
displacement dy of the tracing point P, and since the top of the disc d, being.
driven by the pulley 9 and wire from the tracer carriage turns through an angle
proportional to the abscissa of the tracing-point P, the point of contact of the
integrating wheel W on the top disc d3 evidently traces out the curve described in
Fig. 2 of the sheet of, drawing left with the Provisional Specification. A needle
or pencil n (not shewn in, Figs. 10 and 11) carried by the carriage F, is made to
mark on the disc da when the tracer P is in its initial and final positions, and by
making the tracer P move over the line OY (Fig. 1) the initial direction of OR
can also be recorded.
In an instrument to give two or more pairs of coefficients for one tracing of the
curve, the integrating wheels W, W, if all mounted on the same spindle, may
be conveniently made, .
8.
the diameter of the wheel W.
By fixing two graduated scales on the intermediate disc or its equivalent the
frame d, (Fig. 5), parallel to spindles a and b respectively, and corresponding
index marks on discs d₁ and d, respectively, it is evident that the coefficients a
and b, may be read off by the same instrument. In this case when the tracer P is
at: A₁ (Fig. 1) the instrument must be adjusted so that the spindles a and b are
parallel to OX and OY respectively (Fig. 1).
It may be noted here that no delicate adjustments have to be made before using
this instrument, and no part of it requires to be of excessive accuracy of
construction. For example, in an instrument to give only the coefficients c, and an
the sets of spindles a and b of the integrator need not necessarily be exactly at
right angles. Again, the initial direction of these spindles corresponding to the
initial position of the tracing-point P evidently has no influence on the form of the
curve described by the integrating wheel W (Fig. 2).
Having now particularly described and ascertained the nature of my said inven-
tion, and in what manner the same is to be performed, I declare that what I
claim is :
(1.) The construction of an harmonic analyser in which the integrator consists of
a wheel rolling on a flat surface, and connected with the tracing point of the
instrument in such a manner that the amount of rolling of the said wheel at any
instant is proportional to the increment of the ordinate of the said tracing-point;
substantially as described and illustrated in Figs. 1 and 2.
⋅
(2.) The particular arrangement in the integrator of an harmonic analyser of
three discs, all turning together, one of the said discs having only a motion of
rotation relative to the frame of the instrument, but each of the others having 4
freedom of translation in one direction relative to the adjacent disc-and a wheel
rolling in contact with one of the said three discs; substantially as described and
illustrated hereinbefore.
(3.) In an harmonic analyser having for its integrator the combination of rolling
wheel and three discs, the arrangement of wheels pivoted to (or rolling between 5
any of the two other discs and) the intermediate disc, so that the motion of the
other discs relative to the said intermediate disc may be in a straight line with
only rolling and pivot frictions; substantially as described in Figs. 5, 6, 7, 8.
(4.) In an harmonic analyser having for its integrator the combination of three
discs and rolling wheel, the general arrangement of the mechanism so that the 5

page_text: centre of the integrating wheel always lies on the axis of rotation of that disc
which has only a motion of rotation relative to the frame of the instrument;
substantially as described in Figs. 10 and 11.
Deted this Twenty-ninth day of May 1894
ARCHD. SHARP.