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The vector slide-rule
THE VECTOR SLIDE-RULE ." BY
A. F. ZAHM, Ph .D. Bureau of Construction and Repair, U .S .N .
Preface.-In uniplanar analysis one often has to find the
magnitude and line of a vector when given its components and its moment about a point . For example, from the lift and drag of an xrofoil and its moment about a pivot one may have to plot the resultant force upon the rerofoil profile . This can be done expeditiously with the slide-rule here described . Also the moment of the vector about another point, and its shift with changing incidence of the xrofoil, can be read on the rule . Description of the Instrument .-The vector slide-rule, as shown in Fig . i, comprises a straight scale, preferably transparent, pivoted oil a sheet of cross-section paper, and bearing a small slide like a T-square . A fine line is drawn centrally on the bottom of the ruler and passing through the pivot axis ; and a like line is drawn on the blade of the T-square . The paper has at least two scales ; a circular one about the pivot, and a linear one on a base line through the pivot lengthwise of the sheet . Others can be added to facilitate reading . The large circle shown is used for greater precision in reading small angles . Magnitude and Direction of the Resultant_ To find the magnitude t' L2 -t D 2 , of the resultant R of the lift L and drag D, rotate the ruler till its centre line passes through the point L, D, and read the value on the ruler scale ; to find the direction of the resultant, that is the angle y whose tangent is D/L, read where the centre line crosses the graduated circle . Line of the Resultant .-To draw the resultant on a profile page of the xrofoil, when the moment M is known, press the given centre of momentss for the profile upon the steel pivot under the rule, and orient the page as shown in the photograph ; set the rule at the angle y - i to the base line, where i is the angle of attack ; set the edge of the slide at the distance D1/R, and it will coincide with the desired line of the resultant R, as seen in the photograph . This R, it will be observed, is normal to the one mentioned in the preceding article ; because the ruler was first used to compute R, * Communicated by the Author. 525
The Vector slide-rule .
FIG . i .
0'
April,
ton . I
THE VECTOR
Suva-RULE .
5 27
then to guide the T-square slide in drawing R in a more convenient orientation . Should the profile diagram he ii times the size of the .-crofoil section, it M/R may be used with the same ruler scale, or 31/R may he used with a special ii-fold scale on the ruler . Centre of Pressure .-The centre of pressure travel along the chord can, if the page is transparent, he read directly on the underF:e.. 2 .
Bose line through pivot
Slope of ruler when T-square is parallel to R .
neath section paper, at the points where the vector R crosses the chord, allowance being made for n . Also perpendiculars erected at those points, by aid of the slide and an incidence scale on it, have their tops on the centre of pressure curve, which can be drawn on the same page or on a superposed transparent one . 1llomerat About Any Point .-The moment arm r, about any other point than the pivot can, when R is drawn, be read directly on the centre-line scale as the distance of said new point from the slide, with due allowance for it . Then Ri is the moment .
A . F . ZAHM .
528 Multiplication and
Division-To
find the quotient
[J. F . I .
D/L =x
of
any two quantities, lay the centre-litlc of the rule through the point L, D, and read where it crosses a perpendicular to the base line at any decimal distance, such as I, to. For obviously D/Lx/I or Tox/io . To multiply D by L reverse this process .
Alternative Designs.-The ruler may slide over a circular guide having its centre at the pivot, and carry a try-square slide . thus leaving out the pivot and that part of the ruler which covers its immediate neighborhood, where the vectors are massed . But sometimes the given pivot is well away from the terofoil, so that the vectors need not cross the ruler . Again the farther end of the ruler may have crossing it a circular arc with a slide which can be set at the angle i from the base-line, that is y-i from the centre line . Then to draw R the whole is rotated till this slide coincides with the base line, thus setting the ruler at the proper angle y-i to the base . Substantially such an instrument, known as a propeller slide-rule, was described in the JOURNAL OF THE FRANKLTN INSTITUTE for November, T917 . Geometric Proof .-If the foregoing operations have any novelty, it is in the structure and manipulation rather than the principle of the instrument . Still one may explain why the ruler is set at the angle y-i when drawing R . As. seen in Fig . 2, R inclines y to the lift, and the lift i to the vertical . Hence R and the T-square incline y - i to the vertical, and the ruler inclines the same to the horizontal .
Precision .-Easily made formulas give the above values more accurately, and nomogratns accurately enough . Still instruments are needed to draw R. The present method is as precise as the draughtsman can see to plot the results, and is expeditious with little mental effort .
Use as Trigononteter.-The present instrument obviously can be used as an ordinary trigonometer for the solution of plane right triangles . That is if any two of the five elements-the sides and acute angles-be given, the remaining three can be read on the scales . The addition of a second ruler suitably placed would adapt the instrument to the solution of oblique triangles, as shown in the present writer's article, " Steering Aircraft at Sea," in the Scientific American for July 25, 1914 . The reader may also refer to Appleton's Dictionary for a sketch of a plane trigonometer having a rectangular base sheet with angle scale about its sides, and a ruler pivoted at one corner .
A. F. ZAHM, Ph .D. Bureau of Construction and Repair, U .S .N .
Preface.-In uniplanar analysis one often has to find the
magnitude and line of a vector when given its components and its moment about a point . For example, from the lift and drag of an xrofoil and its moment about a pivot one may have to plot the resultant force upon the rerofoil profile . This can be done expeditiously with the slide-rule here described . Also the moment of the vector about another point, and its shift with changing incidence of the xrofoil, can be read on the rule . Description of the Instrument .-The vector slide-rule, as shown in Fig . i, comprises a straight scale, preferably transparent, pivoted oil a sheet of cross-section paper, and bearing a small slide like a T-square . A fine line is drawn centrally on the bottom of the ruler and passing through the pivot axis ; and a like line is drawn on the blade of the T-square . The paper has at least two scales ; a circular one about the pivot, and a linear one on a base line through the pivot lengthwise of the sheet . Others can be added to facilitate reading . The large circle shown is used for greater precision in reading small angles . Magnitude and Direction of the Resultant_ To find the magnitude t' L2 -t D 2 , of the resultant R of the lift L and drag D, rotate the ruler till its centre line passes through the point L, D, and read the value on the ruler scale ; to find the direction of the resultant, that is the angle y whose tangent is D/L, read where the centre line crosses the graduated circle . Line of the Resultant .-To draw the resultant on a profile page of the xrofoil, when the moment M is known, press the given centre of momentss for the profile upon the steel pivot under the rule, and orient the page as shown in the photograph ; set the rule at the angle y - i to the base line, where i is the angle of attack ; set the edge of the slide at the distance D1/R, and it will coincide with the desired line of the resultant R, as seen in the photograph . This R, it will be observed, is normal to the one mentioned in the preceding article ; because the ruler was first used to compute R, * Communicated by the Author. 525
The Vector slide-rule .
FIG . i .
0'
April,
ton . I
THE VECTOR
Suva-RULE .
5 27
then to guide the T-square slide in drawing R in a more convenient orientation . Should the profile diagram he ii times the size of the .-crofoil section, it M/R may be used with the same ruler scale, or 31/R may he used with a special ii-fold scale on the ruler . Centre of Pressure .-The centre of pressure travel along the chord can, if the page is transparent, he read directly on the underF:e.. 2 .
Bose line through pivot
Slope of ruler when T-square is parallel to R .
neath section paper, at the points where the vector R crosses the chord, allowance being made for n . Also perpendiculars erected at those points, by aid of the slide and an incidence scale on it, have their tops on the centre of pressure curve, which can be drawn on the same page or on a superposed transparent one . 1llomerat About Any Point .-The moment arm r, about any other point than the pivot can, when R is drawn, be read directly on the centre-line scale as the distance of said new point from the slide, with due allowance for it . Then Ri is the moment .
A . F . ZAHM .
528 Multiplication and
Division-To
find the quotient
[J. F . I .
D/L =x
of
any two quantities, lay the centre-litlc of the rule through the point L, D, and read where it crosses a perpendicular to the base line at any decimal distance, such as I, to. For obviously D/Lx/I or Tox/io . To multiply D by L reverse this process .
Alternative Designs.-The ruler may slide over a circular guide having its centre at the pivot, and carry a try-square slide . thus leaving out the pivot and that part of the ruler which covers its immediate neighborhood, where the vectors are massed . But sometimes the given pivot is well away from the terofoil, so that the vectors need not cross the ruler . Again the farther end of the ruler may have crossing it a circular arc with a slide which can be set at the angle i from the base-line, that is y-i from the centre line . Then to draw R the whole is rotated till this slide coincides with the base line, thus setting the ruler at the proper angle y-i to the base . Substantially such an instrument, known as a propeller slide-rule, was described in the JOURNAL OF THE FRANKLTN INSTITUTE for November, T917 . Geometric Proof .-If the foregoing operations have any novelty, it is in the structure and manipulation rather than the principle of the instrument . Still one may explain why the ruler is set at the angle y-i when drawing R . As. seen in Fig . 2, R inclines y to the lift, and the lift i to the vertical . Hence R and the T-square incline y - i to the vertical, and the ruler inclines the same to the horizontal .
Precision .-Easily made formulas give the above values more accurately, and nomogratns accurately enough . Still instruments are needed to draw R. The present method is as precise as the draughtsman can see to plot the results, and is expeditious with little mental effort .
Use as Trigononteter.-The present instrument obviously can be used as an ordinary trigonometer for the solution of plane right triangles . That is if any two of the five elements-the sides and acute angles-be given, the remaining three can be read on the scales . The addition of a second ruler suitably placed would adapt the instrument to the solution of oblique triangles, as shown in the present writer's article, " Steering Aircraft at Sea," in the Scientific American for July 25, 1914 . The reader may also refer to Appleton's Dictionary for a sketch of a plane trigonometer having a rectangular base sheet with angle scale about its sides, and a ruler pivoted at one corner .