MT-1620 al.2002 Unit 7 Transformations and other Coordinate Systems Readings R 2-4,2-5,2-7.29 BMP 5.6.57,5.14,64,6.8,69,6.11 T&g 13,Ch.7(74-83) On "other coordinate systems T&G 27,54,55,60,61 Paul A Lagace Ph D Professor of aeronautics Astronautics and engineering systems Paul A Lagace @2001
MIT - 16.20 Fall, 2002 Unit 7 Transformations and Other Coordinate Systems Readings: R 2-4, 2-5, 2-7, 2-9 BMP 5.6, 5.7, 5.14, 6.4, 6.8, 6.9, 6.11 T & G 13, Ch. 7 (74 - 83) On “other” coordinate systems: T & G 27, 54, 55, 60, 61 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001
MT-1620 al.2002 As we've previously noted, we may often want to describe a structure in various axis systems this involves Transformations Axis, Deflection, Stress, Strain, Elasticity Tensors) e.g., loading axes <-- material principal axis Figure 7.1 Unidirectional Composite with Fibers at an Angle fibers Know stresses along loading axes, but want to know stresses (or whatever)in axis system referenced to the fiber Paul A Lagace @2001 nit7-p. 2
MIT - 16.20 Fall, 2002 As we’ve previously noted, we may often want to describe a structure in various axis systems. This involves… Transformations (Axis, Deflection, Stress, Strain, Elasticity Tensors) e.g., loading axes <--> material principal axis Figure 7.1 Unidirectional Composite with Fibers at an Angle fibers Know stresses along loading axes, but want to know stresses (or whatever) in axis system referenced to the fiber. Paul A. Lagace © 2001 Unit 7 - p. 2
MT-1620 Fall 2002 Problem: get expressions for(whatever) in one axis system in terms of (whatever in another axis system (Review from Unified) Recall: nothing is inherently changing, we just describe a body from a different reference Use (tilde) to indicate transformed axis system Figure 7.2 General rotation of 3-d rectangular axis system ar cartesian) g Paul A Lagace @2001 Unit 7-p 3
MIT - 16.20 Fall, 2002 Problem: get expressions for (whatever) in one axis system in terms of (whatever) in another axis system (Review from Unified) Recall: nothing is inherently changing, we just describe a body from a different reference. Use ~ (tilde) to indicate transformed axis system. Figure 7.2 General rotation of 3-D rectangular axis system (still rectangular cartesian) Paul A. Lagace © 2001 Unit 7 - p. 3
MT-1620 Fall 2002 Define this transformation via direction cosines lmn= cosine of angle from ym to y Notes: by convention, angle is measured positive counterclockwise(+ CCw) (not needed for cosine) mn since cos is an even function CoS(0)=CoS(-8 (reverse direction) But ≠ angle differs by 20 The order of a tensor governs the transformation needed an nth order tensor requires an nth order transformation (can prove by showing link of order of tensor to axis system via governing equations) Paul A Lagace @2001 Unit 7-p 4
MIT - 16.20 Fall, 2002 “Define” this transformation via direction cosines ~ l~mn = cosine of angle from ym to yn Notes: by convention, angle is measured positive counterclockwise (+ CCW) (not needed for cosine) l~ mn = lnm ~ since cos is an even function cos (θ) = cos (-θ) (reverse direction) But l~ ~ mn ≠ lmn angle differs by 2θ! The order of a tensor governs the transformation needed. An nth order tensor requires an nth order transformation (can prove by showing link of order of tensor to axis system via governing equations). Paul A. Lagace © 2001 Unit 7 - p. 4
MT-1620 al.2002 Quantity. Transformation Equation Physical Basis Stress mp ng pq equilibrium Strain mp ng pq geometr Axis Wm=lmip xp geometry Displacement geometr Elasticity tensor E Hookes law mnp pt qustus In many cases, we deal with 2-D cases (replace the latin subscripts by greek subscripts) e. g, OaB=a0 Br Bt (These are written out for 2-D in the handout Paul A Lagace @2001 Unit 7-p 5
MIT - 16.20 Thus: Quantity Transformation Equation ˜ Stress σmn = l mp l ˜ nq ˜ σpq Strain ε˜ = l ˜ l ˜ mn ε mp nq pq Axis x˜m = l mp x ˜ p Displacement u˜m = l mp u ˜ p Fall, 2002 Physical Basis equilibrium geometry geometry geometry Elasticity Tensor E˜ mnpq = l lns l pt l mr˜ ˜ ˜ qu ˜ Erstu Hooke’s law In many cases, we deal with 2-D cases (replace the latin subscripts by greek subscripts) e.g., σ˜ αβ = ll αθ˜ βτ˜ σθτ (These are written out for 2-D in the handout). Paul A. Lagace © 2001 Unit 7 - p. 5