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SPACE STRUCTURES Volume 28 · Number 3 & 4 · 2013 Generating Smooth Curves in 3 Dimensions by Minimizing Higher Order Strain Energy Measures by Sigrid Adriaenssens, Samar Malek, Masaaki Miki and Chris J K Williams Reprinted from INTERNATIONAL JOURNAL OF SPACE STRUCTURES Volume 28 · Number 3 & 4 · 2013 MULTI-SCIENCE PUBLISHING CO. LTD. 5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom Generating Smooth Curves in 3 Dimensions by Minimizing Higher Order Strain Energy Measures Sigrid Adriaenssens, Samar Malek, Masaaki Miki and Chris J K Williams* Princeton University, University of Bath, University of Tokyo, University of Bath (Submitted on 18/01/2013, Reception of revised paper 28/06/2013, Accepted on 11/09/2013) ABSTRACT: Elastic beams or rods in two and three dimensions produce smooth curves by minimizing the strain energy due to classical Euler-Bernoulli bending and Saint-Venant torsion subject to constraints on the arc length and end positions and orientations. However it is possible to postulate other strain energy measures. In this paper we describe a strain energy measure which is the sum of the square of the rate of change of curvature and the square of the product of the curvature and the Frenet torsion. Key Words: curvature, torsion, elastica, euler spiral, clothoid, cornu spiral, railway transition curve 1. INTRODUCTION 2. A TRUSS ANALOGY AND PROBLEM Traditionally smooth curves were constructed on a STATEMENT drawing board using a flexible wooden spline which Figure 1 shows a two dimensional pin-jointed truss in was pinned to the board at intervals. The spline its bent and unbent forms. The thick members are automatically minimizes the strain energy due to inextensible and the thin members are elastic. bending and forms sections of elastica between the The bending from straight is caused entirely by a pinned constraints where there are discontinuities in change in length in the four red members at the ends the rate of change of curvature. The ‘best’ curve is the of the truss. There are no loads or end forces or one with the fewest pins, allowing the spline itself to moments applied to the truss. Thus the truss is a form define the geometry. of mechanism in which modifying only its end There are many possible strain energy measures, but curvature causes the entire structure to bend. The end it could be argued that a ‘good’ measure only requires curvature is controlled ‘internally’ by the red one dimensional constant, whose actual value does not members rather than ‘externally’ by applying a influence the final shape. Here we will consider the moment. minimization of the line integral of the sum of the Now let us imagine that we apply end forces and square of the rate of change of curvature and the square moments to the truss. Then close examination of the of the product of the curvature and the Frenet torsion. truss shows that the sagging bending moment between The rate of change of curvature and the product of the the (i - 1)th and the i th diamond shaped elements is curvature and torsion both have dimensions 1/length2. proportional to *Corresponding author e-mail: [email protected] International Journal of Space Structures Vol. 28 No. 3&4 2013 119 Generating Smooth Curves in 3 Dimensions by Minimizing Higher Order Strain Energy Measures curvature. Following [1–4] and [5] we will investigate curves which minimize the integral of the square of the rate of change of curvature. The curve will ‘want’ to form an Euler spiral, but end constraints may prevent this. Ordinary rods in three dimensions may have strain energy due to both bending and ‘twisting’. The ‘twisting’ is the twist per unit length associated with a material frame, that is a frame which rotates with the material of the rod. The Saint-Venant torsional moment is proportional to the twist per unit length Figure 1. Truss forming Euler spiral. and sections such as ‘I’ beams also have a Vlasov torsional moment due to warping which is proportional to the second derivative of the twist per unit length. – (λ + –2λ +λ ) + (λ – 2λ + λ ) i 1 i i –1 i i –1 i –2 (1) Harary and Tal [6] minimize the sum of the square = – (λ + – 3λ + 3λ –λ ) i 1 i i–1 i –2 of the rate of change of curvature and the square of the th rate of change of Frenet torsion. However we will where λi is the anticlockwise rotation of the i element and the angle changes between successive replace the rate of change of Frenet torsion by the elements is assumed to be small. Thus the bending of product of the curvature and the Frenet torsion. the truss is controlled by the discrete version of the The Frenet torsion is the twist per unit length of equation (31) which we shall derive using virtual the Frenet frame, one of whose axes is parallel to the work. curvature vector. The Frenet frame and therefore the Because the trusses in the figure are unloaded, the Frenet torsion is undefined when the curvature is bottom truss is approximately circular and it would zero, but in our case that does not matter since the be exactly circular but for non-linearity caused by product of the curvature and the Frenet torsion will the finite angle change between truss elements. be zero. Similarly the middle truss would be exactly an Euler Langer and Singer [7] describe three possible spiral, also known as a Cornu spiral or the clothoid, frames, the Frenet frame, a material frame, and a but for the finite angle change. The Euler spiral is natural frame. A natural frame is one which does not defined as the plane curve whose curvature varies rotate about the tangent to the curve as we move along linearly with arc length and it includes the circle as the curve. We could add a fourth, the superelevation, a special case. cant or banking of a railway or road, particularly given In this paper we will consider the ‘active bending’ the use of Euler spirals as railway transition curves. of beams or rods with the property of this truss, except We shall see that the deformed Euler spiral could also that they will produce exactly the Euler spiral when be used as a transition curve with the ability to give free to move and rotate at their ends. We will also continuity to the rate of change of curvature as well as apply end forces and moments which will deform the just curvature. Euler spiral elastically, enabling a far larger range of Bertails-Descoubes [8] has used the Euler spiral as curvature/arc length relationships. a finite element shape function for the simulation of The fact that we could make a physical truss with curves whose strain energy is proportional to the these properties shows that we can use engineering square of the change in curvature from some pre-bent concepts such as bending moment, axial force and state. However here we are not primarily concerned shear force in discussion of this type of truss and the with shape functions for approximating curves, but equivalent continuous beam-like structure. with the properties of the deformed Euler spiral itself. Our ‘twisting’ strain energy measure will be 3. PREVIOUS WORK associated with the classical Frenet frame [9–10] and The elastica in two dimensions minimize the strain therefore we do not need to consider other frames. The energy in bending of elastic rods. In the Euler- two dimensional truss in figure 1 could be made Bernoulli theory curvature is proportional to bending physically, but it should be emphasised that it is moment, and therefore the strain energy is doubtful whether a real system could be constructed proportional to the integral of the square of the with our twisting strain energy measure in three 120 International Journal of Space Structures Vol. 28 No. 3&4 2013 Sigrid Adriaenssens, Samar Malek, Masaaki Miki and Chris J K Williams dimensions. This is because real work must be dT associated with a real frame, namely the material = κN frame. If one were to attempt to construct a real model, ds dN one could start by thinking about a very thin, narrow =−κτT+ B (6) metal ribbon for which the Frenet frame and the ds material frame coincide and which has the interesting dB = τN. property that it can be bent without twisting, but ds cannot be twisted without being bent [11]. However the strain energy of a ribbon is given by the Sadowsky functional [12] which is different to that which we 4. KINEMATICS OF RODS AND THE propose. VIRTUAL WORK THEOREM If we are going to use engineering concepts like We will now summarise the geometric notation to bending moment, then we need some way of linking be used. We can write a curve in the parametric form them with geometry and the best way to do this is through virtual work. The virtual work method was = + + r (u) x (u) i y (u) j z (u) k developed by Jean Bernoulli, D’Alembert and Lagrange. The French, principe des puissances in which i, j and k are unit vectors in the directions of virtuelles (principal of virtual power), is a better the Cartesian x, y and z axes and u is the parameter. description since the virtual displacement has to be The arc length, infinitesimal and it is therefore better to use a virtual velocity and the corresponding virtual drrd power. s =⋅∫ du Let us imagine a rod that is moving with time t so du du that the position of a typical point on the rod is r (u, t).
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