Unit U-A The Language of Engineering

Size: px

Start display at page:

Download "Unit U-A The Language of Engineering"

Benjamin Jefferson
5 years ago
Views:

1 Unit U-A The Language of Engineering Readings: CDL / Unified Engineering Department of Aeronautics and Astronautics Massachusetts Institute of Technology

2 LEARNING OBJECTIVES FOR UNIT U-A Through participation in the lectures, recitations, and work associated with Unit U-A, it is intended that you will be able to.apply the language of engineering systems (units, dimensions, coordinates).describe systems in different coordinate systems using transformations and other concepts Unit U-A - p. 2

3 Need a number of things in order to describe the concepts and items associated with (Unified) Engineering. The first of these is Dimensions and Units --> A dimension is a physical quantity which can be directly measured Fundamental dimensions are: Unit U-A - p.

4 Dimensions of other quantities are derived example: Force use Newton s law F Ma M L T 2 --> A unit is used to quantify dimensions. Note: [ ] bracket means has the dimensions of There are many systems of units (e.g., furlongs/fortnights) but the two used in engineering are Metric or SI (standard international) --> predominant everywhere except US British system (English) Unit U-A - p. 4

5 Fundamental Dimensions (examples) Quantity Dimension SI Unit English Unit Conversion Length L m ft (in) 0.05 m/ft Time S s(ec) sec - Mass M kg slug 14.6 kg/slug Derived Dimensions (examples) Quantity Dimension SI Unit English Unit Conversion Force M L T 2 N lb 4.45 N/lb Pressure M L T 2 L 2 Pa psi 6900 Pa/psi Unit U-A - p. 5

6 Notes: Newton 1 kg m/sec 2 Pound slug ft/sec 2 Pascal N/m 2 psi lb/in 2 *Be careful of weight/mass confusion --> Use units consistently Can use dimensions/units as a check: Question: If you weigh 200 lbs., what is your mass? energy e mc 2 mass speed of light Start of Dimensional Analysis Unit U-A - p. 6

7 Coordinate Systems In order to describe items which are characterized by vectors (e.g., force, velocity, acceleration), need a coordinate system. Most commonly will use rectangular cartesian Figure U.1 Representation of rectangular cartesian system z x i_ k j_ y or i 1 i _ i 2 x i, j, k are unit vectors In the direction of x, y, z i 1, i 2, i are unit vectors In the direction of,, x Note: x --> y --> z --> x Unit U-A - p. 7

8 Other coordinate systems are used when it makes sense to describe a body/form in that manner. Examples? Important Concept: The physical quantity does not change because it is described in a different coordinate system or different unit. It remains the same, only the description changes (like a different language). Unit U-A - p. 8

9 Transformation of Coordinates Oftentimes still want to use a rectangular cartesian system, but one that is oriented differently from the original set of axes. Why? - structural axes (natural form) - loading axes Examples --> consider a swept wing: Figure U.2a describe wing behavior along axis of wing Unit U-A - p. 9

10 Figure U.2b --> consider a space truss consider behavior of component A along axis aligned with its direction A --> other examples? --> consider this formally via the mathematics: Unit U-A - p. 10

11 Figure U. Two rectangular cartesian coordinate systems with the same origin x x tilde > rotated coordinate system point p is located by the vector r in both systems: r i 1 + i 2 + x i and x m i m m1 i m is unit vector in x m r x 1 i 1 + x 2 i 2 + x i n1 x n i n i n is unit vector in x n Unit U-A - p. 11

12 Thus: r m 1 x m i m --> To relate x m to x n, let s take the dot product of both sides with i 1 : i x m i m x 1 i 1 n m 1 n 1 Only non zero term on left hand side is i 1 i 1 x 1 i 1 i 1 + i 1 i 2 + x i 1 i ( ) i n n1 x n i n Recall definition of dot product: ( ) i 1 i 1 i 1 i 1 cos 1 1 cos angle from axis to axis θ _ i 1 _ i 1 Unit U-A - p. 12

13 Generalizing: i m i n cos x m x n l m n l mn Definition: goes with x n goes with x m cos x m x n i m i n Direction Cosine Returning to equation ( ) l 11 x 1 + l 12 x 2 + l 1 and using the direction cosines: Note repeated index, so can write: n1 x l 1n x n For the other two components ( and x ) similar work gives: Unit U-A - p. 1

14 n1 ( i 2 r) l 2n x n ( i r) x l n x n n1 So have equations that can be expressed as: x m n1 l mn x n (m 1, 2, ) --> can also show the reverse x n m1 l nm x m (n 1, 2, ) Unit U-A - p. 14

15 In addition, can transform forces (velocity, acceleration, etc.,) the same way F n m1 l nm F m (n 1, 2, ) Notes: Angle is measured Positive Counterclockwise l mn l nm (+CCW) since cos is an even function: cos (θ) cos (-θ) But l mn l nm since angle from y m to y n differs by 90 from that from y n to y m Unit U-A - p. 15

16 Figure U.4 --> Demonstrate in a 2-D Rotation x, x θ θ θ θ Make a table of direction cosines: x x Unit U-A - p. 16

17 Remember IMPORTANT CONCEPT: Axis system in which we describe a quantity does not change the quantity, only its description Unit U-A - p. 17

US Customary System (USC) Systeme Internationale (SI) Prefixes. Units & Significant Figures

US Customary System (USC) Systeme Internationale (SI) Prefixes. Units & Significant Figures Units & Significant Figures US Customary System (USC) What is the length of this line? Based on things that made sense to people Previously known as English (or British) 1 inch = 3 dry, round, barleycorns