Various Computer Algebra Systems (CAS) are providing (in add-ons, Packs, Packages, ...) tables of engineering data. One example is the tabulation of torsion functions and torsional rigidities for different cross-sections D. The CAS are excellent tools for manipulating closed-form solutions. Progress at implementing in Mathematica some closed-form torsion functions is described.
It remains true that, for most cross-sections, there will be no closed-form solution. Such cases can be solved numerically. We describe implementations involving external C/Fortran called via MathLink from Mathematica .
(1) u = (a^2 - r^2)/ 4
where r denotes the distance from the centre of the disk. There are also easy exact solutions for various other shapes. For example, the torsion function for the equilateral triangle is a cubic polynomial in x and y. There are also some domains where closed-form solution is harder. While exact infinite series solutions exist for the rectangular cross-section, and the sector, no simple, short 'explicit closed-form' formulae have been obtained for their torsion functions. The semi-circle is a shape which has a closed- form solution in terms of elementary functions, and a simple formula for its torsional rigidity, a the radius, is
(2) SSemicircle = (Pi/8 - 1/Pi) a^4
For these nineteenth century results, and more exact solutions from earlier this century, see Higgins [1], Milne-Thomson [2] and Polya and Szego [3]. Polya and Szego [3] have some favorite families of domains including sectors and lenses. For the torsional rigidity of sectors they give a short formula - derived in 1880 - in terms of PolyGamma functions. We will have more to say about the lens domains later.
Torsion functions, and torsional rigidities, are of more than academic interest. We have mentioned, in the Abstract, their uses in elasticity. There are other uses. The same Dirichlet problem describes steady laminar flow through a long pipe of uniform cross-section. S then measures the volume flow rate. For example the symmetric lens solutions we treat below apply to fluid flow along a partially-filled circular pipe.
A further influence on the acceptablility of publication in CAS of engineering data is the fact that many engineering undergraduates now use CAS to help them do sums. (See Keady, Fowkes and Ward [5] for some thoughts on this. Briefly, "keep it simple", and, for the 1990s, "2nd year undergraduate is soon enough".)
(Of course, CAS technology, and the genre - tables of known facts written-up in CAS -, won't be just for engineers. Other tables, e.g. tables of conformal mappings, Kober [6], are likely to find their way into CAS Packs.)
Different CAS vendors have different marketing methods. Wolfram Research, makers of Mathematica , are producing, or encouraging the production of third-party, "Packs" for engineers. Early examples, released in 1994 and 1995, are: the "Electrical Engineering Pack", produced by Wolfram Research, and the third-party "Mechanical Systems Pack".
Advantages of the CAS presentation of the solutions include the following.
The structure, and timetable, for GK's Notebook contributions is as follows. There is room for others to contribute - over the net - to the project of collecting the exact solutions. This paper definitely describes "work in progress". We have Mathematica Notebooks on solutions in lens and lune domains, etc.. The present versions are available from Keady's homepage. When Wolfram Research release the relevant pack (this year, presumably), we will produce, in a format compatible with the Pack, Notebooks just giving the solutions, and these we will offer these for consideration to MathSource. PA's expertise and contacts with WRI will be heavily involved at this stage. The plan is to leave the longer - derivation, checking Notebooks - accessible via the keady homepage, and we will write them up as well as we can. There might be a few different organisations at the front of these. One can imagine "Web-enabled" versions of survey papers. (Added in this html version. We are not allowed to put up [1]. The copyright owner, AIP, does not give permission for papers to go up on the Web. However, they might accept a present, say in 2001, when it a good completed item is prepared, to place at their own Web site.) GK has a longer-term interest in this area, and expects to continue to make progress over the next decade. During this time we will have subdirectories with names suggesting "explorations" where the write-up, while readable by a person very interested in the results, is less polished. We have our doubts about if, in the next decade, derivations as well as solutions will go out with commercial items like the Packs under discussion. With CD-ROM, space isn't a problem: writing with a nice uniform readable style remains a difficulty, and most users - engineers - are more interested in the answers than the verifications.
Others may choose more "commercial" orientations to their Engineering Tables efforts, i.e. just key easily found entries in and leave it at that. GK's interest in the torsion problem is as much driven by mathematical interests as CAS possibilities or engineering use. Checking approximations against known inequalities is an interest: we have found that some entries in Roark's Tables violate the well- known St Venant inequality [3] - even though [4] reminds the engineering users of the inequality. We report items like this in our Notebooks. We see our efforts - writing in a CAS - as representative of an increasingly important part of engineering mathematics culture: we return to this theme in our conclusions.
(3) S = (1/4) integral_{boundary_of_D} (z.n) (u_n)^2 ds
where n denotes the outward normal, and z=(x,y) denotes the coordinate vector for points on the boundary of D. We denote the area of D by A.
and a corresponding lune would be
{{{-1,0},{0,1}},{Sin[xi1],-Cos[xi1/2]}} ,
A semicircle might be specified by
{{{-1,0},{0,1}},{0,1}} .
We have a Mathematica Notebook finding various geometric functionals, e.g.
Area[CircularArcPolygon[itsParameters]].
Nearly all the shapes in Roarke's Tables where approximations to torsional rigidity are given are circular-arc polygons. One can imagine future software with good numerical solvers, as discussed in the final section on Numerics in this paper, allowing users to call up for
TorsionFunction[CircularArcPolygon[itsParameters], otherParameters]
One can also imagine Mathematica putting the shape into a standard form, centroid at the origin, principal axes aligned with the x,y-axes, and analysing if a particular circular-arc polygon is in some family of special shapes - a lens or lune, for example.
The conformal mapping of the unit disk to D can be used to find the torsion function. Series expansions for the torsional rigidity in terms of the coefficients of the conformal map are given in [3] and [9]. Circular-arc polygons are amenable to this approach as analogues of Schwarz-Christoffel are available for them. Particular examples for which we have prepared Notebooks, using this comformal- mapping approach are the symmetric lens (dumbbell), and the equilateral equiangular circular-arc triangles: see [10], p383 and [6] p201 for the conformal mappings. The latter family includes the ordinary equilateral triangle with straight sides. It also includes the Reuleaux triangle, which is used in various engineering mechanisms where a noncircular curve of constant width is needed.
The sector shape is already treated in the materials to be distributed in the WRI Pack. (Using the correct formula we told the developers shows that, even in the restricted range claimed for the approximation in Roark's tables, the relative error in Roark's entry is as high as 10%. Roark's approximation fails disastrously for thin sectors: relative error goes to infinity.) Other circular-arc polygons in Roark's tables [4] include the circular segmental section. (See end of Section 1.) The lens domains which are a major example in our Notebooks can be regarded as a generalisation of segments .
We haven't attempted to find solutions for general lenses, lunes or dumbells. We expect the situation to be similar to that for sectors in the sense that it will only be in special cicumstances where the solutions can be expressed in terms of elementary functions. Two families - OR and CC - where we have solutions in terms of elementary functions are treated in following sections.
OR: In this family the circles intersect orthogonally.
CC: In this family, the centre of one circle is on the circumference of the other. We always normalise so that the circles intersect at (-c,0) and (c,0), usually with c=1.
For the CC lenses, the solutions we have are, to the best of our knowledge (except for the special case of semicircle), not published by other authors. However, the subject is old, and there is a literature other than that in English - Russian and German, notably - which we haven't really made any serious efforts to search. We gave the torsion functions in [7], but only did numerical work for S there. The easiest new result for us to mention here, to the best of our knowledge unpublished prior to this paper, is that for two identical disks (both radii being 2/ , the centre of each on the circumference of the other, the torsional rigidity is
(4) S = -Sqrt[3]/2 + (11 Pi)/36
We have separate Notebooks, one or more for each of the next subsections.
c = 1;
uLune[X_,Y_,xi1_] := Module[
{a = c/Sin[xi1], b = c/Cos[xi1/2], r2},
r2= X^2 + (Y+(b^2)/(2*a))^2;
(b^2 - r2)*(1 - 2*a*(Y+(b^2)/(2*a))/r2)/4];
SLune[a_,b_]:= Module[
{ta = ArcCos[b/(2*a)],sa = (1 - b^2/(4*a^2))^(1/2)},
(a*b*sa*(2*a^2 + 7*b^2) + 2*(2*a^4 - 4*a^2*b^2 - b^4)*ta)/16];
These results were known about 1920, if not earlier: see [1] for the history. In particular, when the disks are identical (xi1 = Pi/3, the disks' radii a, b both being 2/sqrt(3), the centre of each on the circumference of the other), the torsional rigidity is
(5) S = Sqrt[3]/2 - (2 Pi)/9
c = 1;
x[xi_,eta_] := c*(Exp[eta]-Exp[-eta])/
((Exp[eta]+Exp[-eta])-2*Cos[xi]);
y[xi_,eta_] := 2*c*Sin[xi]/
((Exp[eta]+Exp[-eta])-2*Cos[xi]);
(* Next a plot of a lens, D. Pi + xi1 gives lower curve *)
xi1t = Pi/4;
ParametricPlot[{{x[Pi+xi1t,eta],y[Pi+xi1t,eta]},
{x[(Pi+xi1t)/2,eta],y[(Pi+xi1t)/2,eta]}},
{eta,-10,10},
AspectRatio -> Automatic,
PlotLabel -> Evaluate[xi1t]]
xi1t = .;
psi[xi_,eta_,xi1_]:= Module[
{U = -c/((Pi+xi1)*Sin[xi1]),
re = Pi*(Pi+xi1-xi)/(Pi+xi1),
im = -Pi*eta/(Pi+xi1)},
U*Pi*c*2*Sin[2*re]/(
(Exp[2*im]+Exp[-2*im])-2*Cos[2*re])];
uLens[xi_,eta_,xi1_]:=
uLune[x[xi,eta],y[xi,eta],xi1] - psi[xi,eta,xi1];
We then expressed the torsional rigidity as the boundary integral given in equation (3) above, and note that in the special cases discussed before, e.g. (2), (4), can be evaluated in closed form. See also equation (6) for the torsional rigidity for nearly circular lenses in this family.
(6) See Mathematica Notebook.
With further work, it may be possible to find asymptotics to all orders. Results, if obtained, might guide conjectures concerning the form of S for general lenses in the CC lens family. The work in this area is ongoing, equation (6) is due to GK: future work is likely to be joint with Nev Fowkes - watch the URL for references.
[2] L.M. Milne-Thomson, Antiplane Elastic Systems, Springer-Verlag, Berlin, 1962.
[3] G. Polya and G. Szego, Isoperimetric Inequalities of Mathematical Physics, Princeton U.P., Princeton, 1951.
[4] Roark's Tables of Stress and Strain.
[5] G.Keady, N. Fowkes and J.Ward, "Interactive maths texts from Computer Algebra Systems: use in teaching engineering students", Proc. of the Australian Computers in Education Conference, Perth, Vol 2, July 1995, pp 137-144. (Ed. R. Oliver and M. Wyld, ECAWA: 1995).
[6] H. Kober, Dictionary of Conformal Transformations, Dover, 1957.
[7] G.Keady and A. McNabb, "The torsion problem in a lens", Uni. Waikato Maths Res. Rept, 1989.
[8] G.Keady and A. McNabb, "The elastic torsion problem: solutions in convex domains", N.Z. Jnl of Maths, 22, 1993, 43-64.
[9] P. Henrici, Applied and Computational Complex Analysis, Vol 1, Wiley, 1974.
[10] E. Hille, Analytic Function Theory, Vol 2, Chelsea, 1962.
[11] I.S. Sokolnikoff and E.S. Sokolnikoff, "Torsion of regions bounded by circular arcs", Bull. American Math. Soc., 4, 1938, 384-387.
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