me397 intermediate dynamics fall 2015 class notes 04

ME397: Intermediate Dynamics Lecture #04

Copyright © Hilgad Montelo. All rights reserved.

ME397: Intermediate Dynamics

Department of Mechanical Engineering

Lecture #04

September 8, 2015

Disclaimer: The contents of this document are scribe notes for The University of Texas at Austin ME397 Fall 2015, Intermediate Dynamics.

Calculus Review

Note Taker: Hilgad Montelo ([email protected])



Summary

Finish review about vectors;

Rotation Matrices

Vector Derivatives

Angular Velocity

Angular Acceleration

Example

Vectors

Some important definitions:

1) Reference Frame: A rigid 3D object that can hold points, curves, bodies, etc.

2) Coordinate System: Embedded in a reference frame. It is a set of scalar

quantities, (eq. angles, distances, etc.) used to specify points.

Typical Coordinate Systems:

a) Rectangular or Cartesian: , ,

b) Cylindrical (2D or Polar): , ,

c) Spherical: , ,Φ

3) Vector Bases: 3 linearly independent unit vectors that provide a sense of direction in 3D space.

a1

a2

a3



Difference between a Cartesian coordinate system and a Vector Basis (Text

Book, 1)

j

i

k

X

Z

YO

Cartesian Coordinate System

Vector Basis

Vectors functions of Scalar Variables

W

b3b1

b2

a1

a2

a3

h

r

W Wall

D Door

Particle over the door

c

is fixed in D

is fixed in D

is a function of in W

is a function of in D

is a function of and in W

is fixed in W

both magnitude and direction do not change



Derivatives

0 The vector r has variation with respect to W when theta changes

0

The vector r has no variation with respect to D when theta changes

0 The vector r has variation with respect to W when S changes

0

The vector r has variation with respect to D when S changes

Simple Rotation Matrices About one Axis (Text Book, Chapter 5)

bx

by

Θ

X

C

A

NB

ny

nx

N0



ny

nx

Θ

Θ

Rotation Matrix Form

0

0

0 0 1

Rewrite:

Note:

3D rotation:

∗ ∗



Vector Derivatives (Text Book, Chapter 6)

ay

az

ax

ny

nx

nz

A

Ax

Ay

Az

N

A

A

Is a rotating reference frame with angular velocity defined by , ,

Let

Let

me397 intermediate dynamics fall 2015 class notes 04

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