Ch10sol - 10.1 Derive the Lam coefficients Eq. (10.2.2)...

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10.1 Derive the Lamé coefficients Eq. (10.2.2) from Fig. 10-1. Solution A surface is defined by where are rectangular coordinates and x, y are parameters called surface coordinates (they are in general curvilinear coordinates). If i, j, k are unit vectors along the X -, Y -, Z -axes, the point is located by the vector r X i Y j Z k . Accordingly, the surface is defined by the vector equation rr ( x , y ). A line on the surface on which only x varies (or only y varies) is called an x -coordinate line (or a y -coordinate line). Since, the vectors and are tangent to the x and y coordinate lines, respectively. Hence, the x - and y -coordinate lines are orthogonal to each other if, and only if, In this case the surface coordinates are said to be orthogonal. From the onset (Fig. 10-1), the x - and y -coordinate lines are orthogonal. The distance between two points with the surface coordinates and is determined by. Hence, for orthogonal surface coordinates, (a) where (b) 10.2 Derive Eq. (10.2.4). 1
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Solution Fig. PS10-2 Since () and () are tangent to and, respectively and andare orthogonal, and are also orthogonal. Hence, is perpendicular to and as a result of vector cross product. Since the lengths of and are A and B , respectively, is a unit normal vector. 10.3 Derive Eq. (10.2.5). Solution Note that (a) since eq. (a) is equivalent to the identity (b) Also, by definition (c) 2
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where are orthogonal unit vectors. Hence, and (d) Substituting eq. (d) into eq. (a) gives (e) Now consider and its components along the axes . Sinceand , is perpendicular to . The projection of on is given by (f) where as they are orthogonal. Rewriting eq. (e) gives (g) Substituting eq. (g) into eq. (f) yields (h) 3
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where and has been shown above. Consider two adjacent points on the coordinate line in x and construct two unit normal vectors at each point as shown in Fig. PS10-3. The normals intersect at the center of curvature O. The corresponding principal radius of curvature of the surface is denoted by . From the similar triangle relation where and . Hence, one finds (j) Fig. PS10-3 Note that the direction of is in the negative direction of at . In a similar manner, one may calculate the derivative of with respect to y (k) 4
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Equations (j) and (k) are called Rodrigues’ formulas. When the unit normal vector is directed outward (toward the convex side away from the center of curvature), the sense of and is reversed from those shown in eqs. (j) and (k). 10.4 Derive Eq. (10.2.7). Solution Taking scalar products of eq. (j) of Problem 10.3, one has (a) Thus, all three components of the derivative have been found. In a similar manner, one may find the components of the .
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Ch10sol - 10.1 Derive the Lam coefficients Eq. (10.2.2)...

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