me397 intermediate dynamics fall 2015 class notes 04
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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])
ME397: Intermediate Dynamics Lecture #04
Copyright © Hilgad Montelo. All rights reserved.
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
ME397: Intermediate Dynamics Lecture #04
Copyright © Hilgad Montelo. All rights reserved.
Difference between a Cartesian coordinate system and a Vector Basis (Text
Book, Page 31)
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
ME397: Intermediate Dynamics Lecture #04
Copyright © Hilgad Montelo. All rights reserved.
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
ME397: Intermediate Dynamics Lecture #04
Copyright © Hilgad Montelo. All rights reserved.
ny
nx
Θ
Θ
Rotation Matrix Form
0
0
0 0 1
Rewrite:
Note:
3D rotation:
∗ ∗
ME397: Intermediate Dynamics Lecture #04
Copyright © Hilgad Montelo. All rights reserved.
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