1 ME 302 DYNAMICS OF MACHINERY Dynamic Force Analysis IV Dr. Sadettin KAPUCU © 2007 Sadettin Kapucu

Gaziantep University 2 Preliminary l Coordinate Transformation –Reference coordinate frame OXYZ –Body-attached frame O’uvw O, Point represented in OXYZ: Point represented in O’uvw: Two frames coincide ==> O’

Gaziantep University 3 Preliminary l Mutually perpendicular l Unit vectors Properties of orthonormal coordinate frame Properties: Dot Product Let and be arbitrary vectors in and be the angle from to, then x y

Gaziantep University 4 Preliminary l Coordinate Transformation –Rotation only How to relate the coordinate in these two frames?

Gaziantep University 5Preliminary l Basic Rotation –,, and represent the projections of onto OX, OY, OZ axes, respectively –Since

Gaziantep University 6 Preliminary lBlBasic Rotation Matrix –R–Rotation about x-axis with

7 Preliminary l Is it True? –Rotation about x axis with

Gaziantep University 8 Basic Rotation Matrices –Rotation about x-axis with –Rotation about y-axis with –Rotation about z-axis with

Gaziantep University 9 Preliminary l Basic Rotation Matrix –Obtain the coordinate of from the coordinate of <== 3X3 identity matrix Dot products are commutative!

Gaziantep University 10 Example 2 l A point is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.

Gaziantep University 11 Example 3 l A point is the coordinate w.r.t. the reference coordinate system, find the corresponding point w.r.t. the rotated OU-V- W coordinate system if it has been rotated 60 degree about OZ axis.

Gaziantep University 12 Coordinate Transformations position vector of P in {B} is transformed to position vector of P in {A} description of {B} as seen from an observer in {A} Rotation of {B} with respect to {A} Translation of the origin of {B} with respect to origin of {A}

Gaziantep University 13 Coordinate Transformations l Two Special Cases 1. Translation only –Axes of {B} and {A} are parallel 2. Rotation only –Origins of {B} and {A} are coincident

Gaziantep University 14 Homogeneous Representation Coordinate transformation from {B} to {A} Homogeneous transformation matrix Position vector Rotation matrix Scaling

Gaziantep University 15 Homogeneous Transformation l Special cases 1. Translation 2. Rotation

Gaziantep University 16 O, O’ Example 5 l Translation along Z-axis with h: O, O’ h

Gaziantep University 17 Example 6 l Rotation about the X-axis by

Gaziantep University 18 Homogeneous Transformation l Composite Homogeneous Transformation Matrix l Rules: –Transformation (rotation/translation) w.r.t (X,Y,Z) (OLD FRAME), using pre- multiplication –Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post- multiplication

Gaziantep University 19 Example 7 l Find the homogeneous transformation matrix (T) for the following operation:

Gaziantep University 20 Homogeneous Representation l A frame in space (Geometric Interpretation) Principal axis n w.r.t. the reference coordinate system

Gaziantep University 21 Homogeneous Transformation l Translation