An Improved Harmonic Analyser and Apparatus connected therewith.
Year: 1892

page_text: Date of Application, 13th Oct., 1892
Complete Specification Left, 12th Aug., 1893-Accepted, 23rd Sept., 1893
An Improved Harmonic Analyser and Apparatus connected
therewith.
I, ARCHIBALD SHARP, Engineer, 2 Ravenscroft Road, Fairlawn Park,
Chiswick, in the County of Middlesex do hereby declare the nature of this
invention to be as follows:-
Let the curve A, A A A₂ (Fig. 1) be the graphic representation of the
equation y f(x). Then with certain conditions which are discussed in mathema-
tical treatises, f(x)-which may represent any single valued function-can be
represented by the well known Fourier expansion.
+ a. sinn + as sin 2 r
+ a sin nx
Also do
=
1
2π
S
y
It can be shewn that
+b₁ cos a + by cos 2 x .... + b₁ cos nx
da, and can therefore be determined by a planimeter.
2πT
y den
sin nx dx ==
S
an ==
У
cos na dy
7T
Nπ
2πT
1
by deri
1
bp
sin na dy
Nπ
=
cos na da
-
TT
°
the right hand integrals being taken over the same part of the curve as the left
hand integrals.
A harmonic analyser is an instrument for determining the values of the
an, 1, 2, ... b... for any given curve.
coefficients do, a1, agr
If a wheel B (Fig. 2) with horizontal axis rest on a flat surface, and be moved a
distance dy in the direction shewn, while its axis makes an angle 2 with this
direction, the circumferential movement of the wheel will be sin x dy. If another
wheel A be fixed with its axis at right angles to that of wheel B, the circum-
ferential movement of its edge will be cos a dy.
Fig. 3 shews a frame carrying two wheels A and B, the frame being fixed to the
spindle S. The planes of contact of the wheels A and B are at right angles, and
should pass through the axis of the spindle. The spindle S can only move
backwards and forwards along the line S S (Fig. 3). If a tracing point P which
can be moved over the curve A, A A A₂ (Fig. 1) be connected to the spindle S in
such a manner that the displacements of P and S in the direction OY (Fig. 1) are
the same, while the angle which the plane of wheel A, or the axis of wheel B
(Fig. 3) makes with OY is equal to n times the variable a (Fig. 1); then if it be
moved over the complete curve from A, to A,, the wheels A and B will register
numbers proportional to a and b, respectively.
Figs. 4, 5 and 6 are plan, elevation and end elevation respectively of the
apparatus used to produce the required motion. The tracing point P is rigidly
fixed to a rack R, which is free to move longitudinally in the frame T. A
bracket B, fixed to the frame T, carries a swinging bracket B₁, which can be
clamped in any position. The bracket B₁ carries the spindle S₁; which may be
fixed in any. position in the slot of the bracket. Near the upper end of the
1 Price 8d7

page_text: spindle S₁ is fixed the wheel H₁, which gears with the rack R. Near the lower
end of the spindle S₁-but not shewn in Figs. 4, 5 and 6-the frame carrying the
registering wheels A and B (Fig. 3) is fixed. The brackets bb carry, by pivot
bearings, a pair of equal wheels and axle W W. The wheels and carriage can
therefore move only in one straight line, provided the wheels do not slip on
the paper.
Additional spindles S,, S3
may, if desired, be carried by
brackets B2, B3, . . . each spindle having a pair of registering wheels like those
described above. These spindles are geared together by the pairs of toothed
wheels H2, H3, H4, H. The bracket B, is prolonged and carries a third wheel w,
which gives stability to the carriage.
.
•
While the tracer P moves over the curve from A₁ to A2, the carriage moves
forwards or backwards (according as the ordinate to the curve is increasing or
diminishing) in the direction OY. But during this motion, the motion of the rack
relative to the carriage is always forwards in the direction OX. A small
weight L (or spring) and cord may therefore be arranged so as to tend to drive
the rack-and with it the wheels-in this direction. The weight L should be just
sufficient to overcome the frictional resistances of the rack and wheels. The force
to be applied by the operator, to move the tracer P along the curve, will then be
only that required to overcome the pivot friction and rolling friction of the
wheels W.
In using this harmonic analyser the curve to be analysed must be drawn on a flat
sheet of paper which can be pinned down on a drawing board. The wheel H₁ must
be chosen so that its pitch circumference is equal to X₁ (Fig. 1) the abscissa of one
complete period of the curve. The wheels H2, H., H, H are taken so that the
velocity ratios of the spindles S1, S2, S3 are 1:2:3. The axis of the wheels W W
(Fig. 4) is placed parallel to OX (Fig. 1). When the tracer P is on the point A₁
of the curve, the axes of the registering wheels A on the spindles S1, S2, S, should
be all parallel to OX. The tracer P is then moved over the curve A₁ A A A₂
The differences of the readings of the wheels A and B corresponding to the first
and last positions of P, are proportional to the coefficients a1, a2, α, b₁, b₂ b3
required. If now the wheel H₁ be taken so that its pitch circumference is equal to
one fourth of X1, and the wheels H2, H3, H., H, be taken to give the velocity ratios
of the three spindles 4: 5: 6, the coefficients a, a, a, b, b, be will be obtained.
3
The first term a, can be obtained, as already mentioned above, by a planimeter.
Fig. 7 shews a form of planimeter specially adapted for use with the harmonic
analyser. The registering wheel C (Fig. 7) is similar to that described in Fig. 2.
The frame which carries it is attached to the end of a spindle mounted on the
carriage of the instrument. A long lever Sp, fixed to the upper end of the
spindle, has its end p secured in a slot S fixed at right angles to the rack R of
the analyser. The longitudinal motion of the end p of the lever is therefore the
same as that of the tracer P. It can easily be demonstrated that the difference of
the readings of the wheel C, for the initial and final tracer positions A, and A
(Fig. 1) is proportional to the mean ordinate of the .curve A, A. A A, measured
from the line A, A,; and is therefore directly proportional to a
Fig. 8 shews a pantagraph to be used with the harmonic analyser for a curve
whose abscissa X₁ (Fig. 1) is different from the pitch circumference of the
wheel H₁ (Fig. 4). In a jointed-parallelogram pantagraph, the points L, M, N.
are on the same straight line. If the point L be fixed to any convenient point in
the carriage (Fig. 4), while the point M is jointed to the rack R, the longitudinal
motion of the rack is to that of the "tracer" N as L M is to L N. Relative to
the carriage the movements of M and N will be along the straight lines m m and n n
respectively (Fig. 8). Relative to the paper the motion of L is along the straight
line L. Hence the curves described on the paper by the "pencil" M and the
"
tracer N will have their dimensions in the direction Ll equal, and their
dimensions in the direction m m or n n in the ratio of L M to LN. Two such
curves are shewn in Fig. 8.
66
The analyser, with pantagraph, described above is what might be called a

page_text: universal harmonic analyser; that is, a curve of any length of abscissa X₁
(Fig. 1) can be analysed by it; and, if suitable wheels be supplied, any pair of
coefficients b. can be obtained.
an
An analyser to give only one pair of coefficients at a time may be used, and
might have the rack and toothed wheel replaced by a smooth rod and friction
wheel pressed together by a spring.
Again, an instrument to give only the first six, eight, or ten pairs of coefficients,
and to give them for one tracing of the curve, would have its gear wheels and
spindles put in position once for all, and therefore the frame would be of simpler
design.
m
101
UJ
1
1000
COMPLETE SPECIFICATION.
An Improved Harmonic Analyser and Apparatus connected
therewith.
I, ARCHIBALD SHARP, Engineer, 2 Ravenscroft Road, Fairlawn Park, 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 the following statement :-
(The following description refers to the drawings filed with the Provisional
Specification.)
Let the curve A A A (Fig. 1) be the graphic representation of the
equation y f(x). Then with certain conditions which are discussed in mathe-
matical treatises, f(x)-which may represent any single-valued function of x-can
be expressed by the well-known Fourier expansion.
ƒ(x) = a。 + α₁ sin x + a₂ sin 2x +
Also, it can be shewn that
+b₁ cos x + b₂ cos 2 x +
,,:༼",
an
1
TT
S
277
=
nx dx
y si no decas uz ilg
dy
2277
2π
bn
y na
=
-
næ
dy
T
NTT
b = cos mix de sin na dg
the right hand integrals being taken over the same part of
hand integrals.
.
The term a, can be determined by a planimeter.
curve as the left
A harmonic analyser is an instrument for determining the values of the
coefficients a₁, ag, an, b1, b2, bn, for any given curve.
If a wheel B (Fig. 2.) with horizontal axis rest on a flat surface and be moved
a distance dy in the direction shewn, while its axis makes an angle with this
direction, the circumferential movement of the wheel will be sin x dy.
If another wheel A be fixed with its axis at right angles to that of wheel B, the
circumferential movement of its edge will be cos x dy.
Fig. 3 shews a frame carrying two wheels A. and B, the frame being fixed to the
spindle S. The planes of the wheels' A and B are at right angles. The spindle S

page_text: can only be moved backwards and forwards along the line S S (Fig. 3). If a
tracing point P which can be moved over the curve A, A A A, (Fig. 1) be
connected to the spindle S in such a manner that the displacements of P and S in
the direction OY are the same, while the angle which the plane of wheel A or the
axis of wheel B (Fig. 3) makes with OY is equal to 'n times the variable x
(Fig. 1); then if it be moved over the complete curve from A₁ to A, the
wheels A and B will register numbers proportional to a and b, respectively. If
the planes of wheels A and B do not pass through the axis, either a correction
must be made, or the tracing point P must be moved straight back from A₂ to A.
Figs. 4, 5, and 6 are plan, elevation, and end elevation respectively of the apparatus
used to produce the required motion. The tracing point P is rigidly fixed to a
rack R, which is free to move longitudinally in the frame T. A bracket B, fixed
to the frame T carries a swinging bracket B, which can be clamped in any position.
The bracket B₁ carries the spindle S₁, which may be fixed in any position in the
slot of the bracket. Near the upper end of the spindle S₁ is fixed the wheel H,
which gears with the rack R. Near the lower end of the spindle S₁-but not
shewn in Figs. 4, 5, and 6-the frame carrying the registering wheels A and B
(Fig. 3) is fixed. The brackets bb carry by pivot bearings a pair of equal wheels
and axle W W. The wheels and carriage can therefore move only in one straight
line, provided the wheels do not slip on the paper. Additional spindles S., S....
may, if desired, be carried by brackets B2, B3 each spindle having a pair
of registering wheels like those described above. These spindles are geared
together by the pairs of toothed wheels H., H, and H., H. The bracket B₂ is
prolonged and carries a third wheel w which gives stability to the carriage.
...
While the tracer P moves over the curve from A₁ to A2, the carriage moves
forwards or backwards in the direction OY, according as the ordinate to the curve
is increasing or diminishing. But during this motion the motion of the rack
relative to the carriage is always forwards in the direction O X. A small
weight L (or spring) and cord may therefore be arranged so as to tend to drive the
rack and with it the wheels-in this direction. The weight L should be just
sufficient to overcome the frictional resistance of the rack and wheels. The force
to be applied by the operator to move the tracer along the curve will then be
only that required to overcome the pivot friction and rolling friction of the
wheels W W.
In using the harmonic analyser the curve to be traced must be drawn on a flat
sheet of paper which can be pinned down on a drawing board. The wheel H, must
be chosen so that its pitch circumference is equal to ✗₁ the abscissa of the
curve. The wheels H, H3, H., H, are taken so that the velocity ratios of the
spindles S1, S2, S, are 1:2:3. The axis of the wheels W W (Fig. 4) is placed
parallel to OX (Fig. 1). When the tracing point P is on the point A₁ of the
curve (Fig. 1) the axis of the registering wheels A on the spindles S1, S2, S3
should be all parallel to O X. P is then moved over the curve A, A A A₂, till it
coincides with the point A,. The differences of the reading of the wheels A and B
corresponding to the first and last positions of P are proportional to the
coefficients a1, a2, a3, b₁, b₂, b3 required. If now the wheel H, be taken so that its
pitch circumference is equal to one fourth of X, and the wheels H₂, H, H, H. be
taken to give the velocity ratios of the three spindles 4:5:6, the coefficients a, a, a
b, b, be can be obtained.
The first term
can be obtained, as already mentioned above, by a planimeter.
Fig. 7 shews a form of planimeter specially adapted for use with the harmonic
analyser. The registering wheel C (Fig. 7) is similar to that described in Fig. 2.
The frame which carries it is attached to the end of a spindle S mounted on the
carriage of the instrument. A long lever S p fixed to the upper end of the spindle
has its end secured in a slot S fixed at right angles to the rack R of the analyser.
The longitudinal motion of the end p of the lever is therefore the same as that of
the tracing point P. It can easily be demonstrated that the differences of the
readings of the wheel C, for the initial and final tracing point positions A, and A,

page_text: (Fig. 1) is proportional to the mean ordinate of the curve A₁ A. . . A₂, measured
from the line A₁ A₂.
Fig. 8 shews a pantagraph to be used with the harmonic analyser for a curve
whose abscissa X₁ (Fig. 1) is different from the pitch circumference of the
wheel H, (Fig. 4).
If the point L be fixed to any convenient point in the carriage (Fig. 4) while the
point M is jointed to the rack; the longitudinal movement of the rack R is to that
of the tracer N as L M is to L N. Relative to the carriage, the movements
of M and N will be along the straight lines m m and n n respectively (Fig. 8).
Relative to the paper the motion of L is along the straight line I.. Hence
the curves described on the paper by the pencil M and the tracer N will have
their dimensions in the direction Lequal, and their dimensions in the
directions mm or nn in the ratio of L M to L N (Fig. 8). Two such curves
are shewn in Fig. 8.
The analyser with pantagraph described above is what might be called a
universal one; that is a curve of any length of abseissa X1, can be analysed by
it. Provided suitable wheels be supplied with the analyser any pair of
coefficients "n and can be obtained.
Referring now to Figs. 9 and 10 of the sheet of drawings filed with my Complete
Specification, Fig. 9 is a side view, partly in section, of an analyser to give only
one pair of coefficients at a time. A bar R, supported by antifriction rollers (not
shewn in the drawing), carries the tracing point P. Near the ends of the bar
two studs t project, a wire or string is stretched from them round a pulley s on an
intermediate horizontal shaft h. A wheel w, with narrow projecting teeth, can
be shifted along the shaft h, so as to gear with any set of the narrow teeth
projecting from the disc d on the spindle S carrying the registering wheels A
and B.
3 3
In the instrument shewn in Fig. 9 the shaft h turns three times while the
tracer P moves from A₁ to A, over the complete curve to be analysed (Fig. 1).
The wheel to may have only one set of teeth, as at 3, in which case the ratios of the
mean diameters of the sets of teeth projecting from disc d to the diameter of wheel w
would be 3, 3, 3, 4, }, In order to reduce the necessary diameter of the
disc d, the wheel w shewn in Fig. 9 is provided with two sets of teeth of smaller
diameter; the corresponding sets of teeth of disc d will also be of smaller diameter,
and must be at different levels, causing the disc d to assume the saucer shape shewu
in Fig. 9. The disc d is shewn with 9 sets of projecting teeth (only one set being in
gear at a time) so that the first 9 pairs of coefficients a and b can be obtained by
the instrument.
The frame carrying the registering wheels is attached to the spindle S by
the pivotted spindle g, which is at right angles to the line joining the points of
contact of wheels A and B with the paper; so that the pressures of the two wheels
on the paper are practically equal.
The instrument is mounted on an axle and pair of equal wheels W, the third
support being provided by the registering wheels A and B. The balance of the
instrument should be such, that there should be only sufficient pressure on wheels A
and B to ensure their rolling properly on the paper. A counterweight may
slide along the horizontal shaft h to compensate for the movement of the wheel w.
A clamp and tangent screw t. s., for getting a delicate adjustment of the initial
direction of the axis of the registering wheel B, is shewn in Figs. 9 and 10.
Fig. 10 is a diagrammatic view shewing the arrangement of the pivotted
spindle g.
An analyser which is only required to give a certain number of pairs of
coefficients for one tracing round the curve, may have its gear wheels and spindles
adjusted in position once for all; the frame would then be of simpler design.
I wish to state that I do not claim the element or combination described in
Fig. 3; but,

page_text: Having now particularly described and ascertained the nature of my said inveu-
tion and in what manner the same is to be performed, I declare that what I
claim is
1. In a harmonic analyser the combination of a frame or carriage (mounted on
wheels so as to move in a straight line) carrying spindles with registering wheels
which roll or slide on the paper, the said spindles being turned (by suitable
gearing) by a rack or bar which can be moved relative to the aforesaid carriage
in a direction at right angles to the motion of the aforesaid carriage, the said
rack or bar carrying a tracing point which is moved over the curve to be
analysed.
2. The planimeter described and illustrated in Fig. 7.
3. The pantagraph for enlarging or diminishing in one direction, leaving the
dimensions at right angles to this said direction unaltered, substantially as described
and illustrated in Fig. 8.
4. In a harmonic analyser the pivotting of the spindle to the frame carrying the
pair of registering wheels, substantially as described in Fig. 10.
5. In a single element harmonic analyser the combination, for varying the velocity
ratio, of a disc with projecting teeth fixed to one spindle, and a toothed wheel which
can be moved along a spindle at right angles to the aforesaid spindle, substantially
as described and illustrated in Fig. 9.
Dated this Twelfth day of August 1893