1 / 30

Part B Tensors

Part B Tensors. Or, inverting. The graph above represents a transformation of coordinates when the system is rotated at an angle  CCW. 2B1 Tensors.

gavril
Télécharger la présentation

Part B Tensors

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Part B Tensors Or, inverting The graph above represents a transformation of coordinates when the system is rotated at an angle  CCW

  2. 2B1 Tensors Let be two vectors in a Cartesian coordinate system. If is a transformation, which transforms any vector into some other vector, we can write where and are two different vectors. If For any arbitrary vector and and scalar is called a LINEAR TRANSFORMATION and a Second Order Tensor. (1.1a) and (1.2a) can be written as,

  3. 2B1 Tensors Questions: Yes

  4. 2B1 Tensors Questions: Yes

  5. 2B2 Components of a Tensor Scalar has 1 component Vector has 3 component Tensor has ? component?

  6. 2B2 Components of a Tensor In general The components of Tij can be written as

  7. 2B2 Components of a Tensor (cont.) Example 2B2.3 Q: Let R correspond to a righ-hand rotation of a rigid body about the x3 axis by an angle . Find the matrix of R Thus Q: Let R correspond to a 90° right-hand rigid body rotation about the x3 axis. Find the matrix of R

  8. 2B3 Components of a transformed vector The transformation is linear if is then called a LINEAR TRANSFORMATION and is a second-order tensor.

  9. 2B3 Components of a transformed vector Now what is b? what are the components of b?

  10. 2B2 Components of a Tensor (cont.)

  11. 2B4 Sum of Tensors Definition:

  12. 2B5 Product of Two Tensors Components: or In general, the product of two tensors is not commutative:

  13. 2B5 Product of Two Tensors Q: Let R correspond to a 90° right-hand rigid body rotation about the x3 axis. Find the matrix of R Q: Let R correspond to a 90° right-hand rigid body rotation about the x1 axis. Find the matrix of S Q: Let W correspond to a 90° right-hand rigid body rotation about the x3 axis, then a 90° right-hand rigid body rotation about the x1 axis. Find the matrix of W

  14. r ~ ~ a T T T 2B6 Transpose of a Tensor Transpose of is denoted by and . If then Note

  15. 2B7 Dyadic Product Definition , dyadic product of two vectors is also a tensor. It transforms a vector into a vector parallel to

  16. 2B8 Trace of a Tensor

  17. 2B9 Identity Tensor A linear transformation which transforms every vector into itself is an identify tensor

  18. 2B9 Inverse Tensor Note that

  19. 2B10 Orthogonal Tensor and Thus

  20. ~ ~ ~ = T Q Q I ~ ~ - = 1 Q Q I 2B10 Orthogonal Tensor (cont.) Thus Also, we have , So Thus for an orthogonal matrix, the transpose is also its inverse. Example: A rigid body rotation is an ORTHOGONAL tensor,

  21. 2B11 Transformation Matrix Between Two Coordinate Systems We see that,

  22. 2B11 Transformation Matrix Between Two Coordinate Systems(cont.) Q: Let be obtained by rotating the basis about the x3 axis by an angle . Find the transformation matrix of R from to Thus

  23. 2B12 Transformation laws for Vectors.

  24. 2B12 Transformation laws for Vectors. (cont.) Answer

  25. 2B13 Transformation law for tensor Let and Recall that the components of a tensor are:

  26. 2B13 Transformation law for tensor (cont.)

  27. 2B13 Transformation law for tensor (cont.) Find the matrix of T with respect to the basis

  28. 2B14 Tensors by Transformation laws

  29. 2B14 Tensors by Transformation laws (cont.) The components transform as follows:

  30. 2B14 Tensors by Transformation laws (cont.)

More Related