ME 6590 Multibody Dynamics
Time Derivative of the (Coordinate) Transformation Matrices
Matrix Form of the Derivative of a Vector Fixed in a Rigid Body
Consider a body B : (e1, e2 , e3 ) moving in a fixed reference frame R : ( N1, N2 , N3 ) . If r is
a vec
ME 6590 Multibody Dynamics
Conversion of Direction Cosines to Euler Parameters
(Reference: H. Baruh, Analytical Dynamics, McGraw-Hill, 1999)
Given the coordinate transformation matrix [C ] , the Euler parameters may be computed
as follows. First, recall t
ME 6590 Multibody Dynamics
Generalized Coordinates, Quasi-Coordinates, and Generalized Speeds
Recall that generalized coordinates are a set of physical coordinates that define the
degrees of freedom of the system. For a rigid body in three dimensions, we
ME 6590 Multibody Dynamics
Orientation Angles of a Rigid Body in Three Dimensions
To describe the general orientation of a rigid body in
three dimensions, consider the rigid body shown in the
figure at the right. Here there are two reference frames
the b
ME 6590 Multibody Dynamics
Orientation of a Rigid Body Using Euler Parameters
Euler's Theorem on Rotation
Consider the rigid body shown in the figure at the
right. Let R : N1, N2 , N3 represent the base reference
frame and B : e1, e2 , e3 represent the bo
ME 6590 Multibody Dynamics
Conversion of Direction Cosines to 1-2-3 Body-Fixed Angle Sequence
Given the coordinate transformation matrix [C ] , the orientation angles for a 1-2-3
body-fixed angle sequence may be computed as follows. First, recall that [C
ME 6590 Multibody Dynamics
Euler Parameters and Angular Velocity Components
Earlier we derived relationships between angular velocity components and various sets
of orientation angles and their derivatives. Here, we will derive a similar relationship
betw
Derivation of Expressions for the Angular Velocity Components
in Terms of the Euler Parameters
> restart;
Derivation of the Expression for Omega1
> a1 := 2*(epsilon1*epsilon2 + epsilon3*epsilon4);
a1 := 2 1 2 + 2 3 4
> b1 := 2*(e1dot*epsilon3 + epsilon1*e
ME 6590 Multibody Dynamics
Angular Velocity and Orientation Angles
When using angle sequences to describe the orientation of a rigid body, the
addition rule for angular velocities is used to find the angular velocity of the body.
Consider the case where t
ME 6590 Multibody Dynamics
Coordinate Transformation (Rotation) Matrices
Relationships Between Unit Vectors in Different Reference Frames
The unit vectors of two different mutually perpendicular unit vector sets
(e1, e2 , e3 ) and (n1, n2 , n3 ) can be re
ME 6590 Multibody Dynamics
Representation of Vector Operations as Matrix Operations
Dot (or Scalar) Product
Given two vectors
a a1e1 a2e2 a3e3
b b1e1 b2e2 b3e3
then the dot (or scalar) product of the two vectors is defined to be
3
a b aibi .
i 1
This prod
ME 6590 Multibody Dynamics
Angular Velocity & Partial Angular Velocity Using Absolute Coordinates
Angular Velocity: 1-2-3 Rotation Sequence , ,
Given a rigid body B : e1, e2 , e3 moving in a fixed reference frame R : N1, N2 , N3 , we
define a set of in