Multibody Mechanics/Printable version

Multibody Mechanics

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Notational Conventions

Throughout this text a consistent notation is maintained, which is essentially that promoted by Kane (and presented to the author with modification by Anderson). This notation eliminates the ambiguity associated with the multiple reference frames inherent in multibody analysis. It has been the author's experience in industry that too many problems are brought on by poor notation, and the notion that for some topics this notation is too rigorous has been dismissed.

Scalar quantities are represented by numbers, letters and/or symbols. The following are all scalars:

   x, $\phi$, X, $\Phi$, 3, and 5.5.

Arbitrary vectors are denoted with over-arrows as

   ${\vec {x}}$, ${\vec {\phi }}$, ${\vec {X}}$, ${\vec {\Phi }}$.

and unit vectors are given the usual hat notation

   ${\hat {i}}$, ${\hat {e}}_{\theta }$, ${\hat {a}}_{1}$.

Reference frames are labeled with a single capital letter, with 'N' being reserved for the inertial frames (also called the 'Newtonian' frame).

Each reference frame is assumed to have a set of right handed basis vectors labeled with the same lowercase letter. Basis vectors for the frame 'B' would be:

   ${\hat {b}}_{1}$, ${\hat {b}}_{2}$, ${\hat {b}}_{3}$  such that ${\hat {b}}_{1}\times {\hat {b}}_{2}={\hat {b}}_{3}$.

Angular Velocity

Need help with the notation? Go to the Notational Conventions Section.

Angular velocity is a vector which describes the rotation rate of a reference frame relative to another reference frame.

Typically the Greek letter $\omega$ (omega) is used exclusively to represent angular velocities (and many authors also use $\Omega$). In this text the angular velocity of frame "B" relative to "A" is denoted as

   ${}^{A}{\vec {\omega }}^{B}$.

The angular velocity vector is defined such that any vector ${\vec {\zeta }}$ which is fixed (constant) in the local frame "B" (reference frame) has its time derivative in another frame "A" expressed with a cross product as

   ${\frac {{}^{A}d}{dt}}{\vec {\zeta }}={}^{A}{\vec {\omega }}^{B}\times {\vec {\zeta }}$.

This relationship is not difficult to prove, however doing so is of little practical use and is deferred to the end of this discussion. It is useful to note that since vectors can be expressed as linear combinations of basis vectors, this relationship finds its way into of the expression for a the derivative of a general vector (including angular velocity itself).

Angular velocity is not

Euler Parameters

Euler parameters (also known as Euler-Rodrigues parameters or unit quaternions) are an alternative to Euler angles as a means to describe the relative orientation of reference frames in three dimensions.

The significant difference between Euler parameters and Euler angles is that where there are only three Euler angles, there are four Euler parameters. The Euler parameters are not independent, and a valid set of parameters must satisfy the constraint that the sum of their squares is constant.

The Euler parameters are defined by an axis and angle, i.e. the relative orientation of frames is described by a rotation about a common axis. If the axis defined by the vector ${\vec {u}}$ decomposed as $u_{1},u_{2},u_{2}$ in both frames (the axis is common), and the rotation by an angle $\phi$ (right-handed about the vector) to arrive at orientation of the second frame, then the Euler parameters are

$e_{0}=\cos \left({\frac {\phi }{2}}\right)|{\vec {u}}|$

$e_{i}=\sin \left({\frac {\phi }{2}}\right)u_{i},i=1...3$

Note that it is common for one to consider Euler parameters in normalized form where the vector ${\vec {u}}$ is a unit vector ${\hat {u}}$, in which case the Euler parameters will, by definition, satisfy the constraint that their squares sum to one.

Euler Parameters

Euler parameters (also known as Euler-Rodrigues parameters or unit quaternions) are an alternative to Euler angles as a means to describe the relative orientation of reference frames in three dimensions.

The significant difference between Euler parameters and Euler angles is that where there are only three Euler angles, there are four Euler parameters. The Euler parameters are not independent, and a valid set of parameters must satisfy the constraint that the sum of their squares is constant.

The Euler parameters are defined by an axis and angle, i.e. the relative orientation of frames is described by a rotation about a common axis. If the axis defined by the vector ${\vec {u}}$ decomposed as $u_{1},u_{2},u_{2}$ in both frames (the axis is common), and the rotation by an angle $\phi$ (right-handed about the vector) to arrive at orientation of the second frame, then the Euler parameters are

$e_{0}=\cos \left({\frac {\phi }{2}}\right)|{\vec {u}}|$

$e_{i}=\sin \left({\frac {\phi }{2}}\right)u_{i},i=1...3$

Note that it is common for one to consider Euler parameters in normalized form where the vector ${\vec {u}}$ is a unit vector ${\hat {u}}$, in which case the Euler parameters will, by definition, satisfy the constraint that their squares sum to one.

Reference Frame

Need help with the notation? Go to the Notational Conventions Section.