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curvilinear - UNIVERSITY OF CALIFORNIA BERKELEY Structural...

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UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering, Department of Civil Engineering Mechanics and Materials Fall 2002 Professor: S. Govindjee A Quick Overview of Curvilinear Coordinates 1 Introduction Curvilinear coordinate systems are general ways of locating points in Eu- clidean space using coordinate functions that are invertible functions of the usual x i Cartesian coordinates. Their utility arises in problems with obvious geometric symmetries such as cylindrical or spherical symmetry. Thus our main interest in these notes is to detail the important relations for strain and stress in these two coordinate systems. Shown in Fig. 1 are the definitions of the coordinate functions. Note that while the definition of the cylindrical co- ordinate system is rather standard, the definition of the spherical coordinate system varies from book to book. Both systems to be studied are orthogonal. The precise definitions used here are: Cylindrical x 1 = r cos( θ ) x 2 = r sin( θ ) x 3 = z (1) r = x 2 1 + x 2 2 θ = tan 1 ( x 2 /x 1 ) z = x 3 (2) Spherical x 1 = r sin( ϕ ) cos( θ ) x 2 = r sin( ϕ ) sin( θ ) x 3 = r cos( ϕ ) (3) r = x 2 1 + x 2 2 + x 2 3 ϕ = cos 1 ( x 3 x 2 1 + x 2 2 + x 2 3 ) θ = tan 1 ( x 2 /x 1 ) (4) 2 Basis Vectors For convenience in some of the equations to be given later we will denote our curvilinear coordinates as z k where ( z 1 , z 2 , z 3 ) = ( r, θ , z ) in the cylindrical case and ( z 1 , z 2 , z 3 ) = ( r, ϕ , θ ) in the spherical case. For the basis vectors we will introduce for two types of basis vectors. The natural basis vectors and 1
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r ϕ z θ r θ x3 x2 x1 x1 x2 x3 Figure 1: Definition of the cylindrical and spherical coordinate systems. the physical basis vectors. Both bases are orthogonal but the physical basis has the additional property of orthonormality. The basic definitions are g k = x i z k e i (5) e k = g k g k . (6) To di ff erentiate between the physical basis vectors and the usual Cartesian ones we typically write e r , e θ , · · · etc. For the cylindrical coordinate system one has: g 1 cos( θ ) sin( θ ) 0 g 2 r sin( θ ) r cos( θ ) 0 g 3 0 0 1 (7) and e r = g 1 e θ = 1 r g 2 e z = g 3 . (8)
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