Vectors_Tensors_05_Coordinate_Transformation_Vectors

Vectors_Tensors_05_Coordinate_Transformation_Vectors -...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

Vectors_Tensors_05_Coordinate_Transformation_Vectors -...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online