Vectors_Tensors_05_Coordinate_Transformation_Vectors

Vectors_Tensors_05_Coordinate_Transformation_Vectors -...

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Section 1.5 Solid Mechanics Part III Kelly 24 1.5 Coordinate Transformation of Vector Components Very often in practical problems, the components of a vector are known in one coordinate system but it is necessary to find them in some other coordinate system. For example, one might know that the force f acting “in the 1 x direction” has a certain value, Fig. 1.5.1 – this is equivalent to knowing the 1 x component of the force, in an 2 1 x x coordinate system. One might then want to know what force is “acting” in some other direction – for example in the 1 x direction shown – this is equivalent to asking what the 1 x component of the force is in a new 2 1 x x coordinate system. Figure 1.5.1: a vector represented using two different coordinate systems The relationship between the components in one coordinate system and the components in a second coordinate system are called the transformation equations . These transformation equations are derived and discussed in what follows. 1.5.1 Rotations and Translations Any change of Cartesian coordinate systems can be split up into a translation of the base vectors and a rotation of the base vectors. A translation of the base vectors does not change the components of a vector. Mathematically, this can be expressed by saying that the components of a vector a are a e i , and these three quantities do not change under a translation of base vectors. 1.5.2 Components of a Vector in Different Systems Vectors are mathematical objects which exist independently of any coordinate system. Introducing a coordinate system for the purpose of analysis, one could choose, for example, a certain Cartesian coordinate system with base vectors i e and origin o , Fig. 1.5.2. In that case the vector can be written as 3 3 2 2 1 1 e e e u u u u + + = , and 3 2 1 , , u u u are its components. 1 x component of force 1 x 2 x f 1 x 2 x 1 x component of force

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Section 1.5 Solid Mechanics Part III Kelly 25 Now a second coordinate system can be introduced (with the same origin), this time with base vectors i e .
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Vectors_Tensors_05_Coordinate_Transformation_Vectors -...

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