Angular Velocity of a Rigid body

Section 3.1 Angular Velocity of a Rigid body

A rigid body \(\cB\) is in motion relative to a referential \(\cA\text{,}\) as shown in Figure 3.1.1. Consider an arbitrary vector \(\bvv\) attached to \(\cB\text{.}\) The components \((v_1, v_2, v_3)\) of \(\bvv\) on basis \(b_B =(\bhb_1,\bhb_2,\bhb_3)\) of \(\cB\) are constant. However, its components \((V_1, V_2, V_3)\) on basis \(b_A =(\bha_1,\bha_2,\bha_3)\) of \(\cA\) are necessarily (scalar) functions of time due to the motion of \(\cB\) relative to \(\cA\text{.}\)

Figure 3.1.1. Rigid body \(\cB\) in motion relative to \(\cA\)

We seek an expression for the derivative \((d\bvv /dt)_\cA\) relative to \(\cA\) by a method more tractable than that which consists of differentiating \(\bvv= {V}_1 \bha_1 + {V}_2 \bha_2 + {V}_3 \bha_3\) resolved on basis \((\bha_1,\bha_2,\bha_3)\) of \(\cA\text{.}\) To this end, we introduce the following operator

\begin{equation} \left({ d \over dt} \right)_\cA: \quad \bvv \mapsto \left( {d\bvv \over dt} \right)_\cA \tag{3.1.1} \end{equation}

defined on the set of vectors fixed in \(\cB\text{.}\) This operator is clearly linear. Furthermore, it is skew-symmetric:

\begin{equation*} \buu \cdot \left( {d \bvv \over dt } \right)_{\cA} = - \bvv \cdot \left( {d \buu \over dt } \right)_{\cA} \end{equation*}

since the scalar product \(\buu \cdot \bvv\) is constant for any two vectors \(\buu\) and \(\bvv\) fixed in \(\cB\text{.}\)

The matrix representation of this operator on basis \((\bhb_1 , \bhb_2 , \bhb_3)\) (or on any other basis attached to \(\cB\)) must then be skew-symmetric ¹ given that

\begin{equation*} \bhb_i \cdot \left( {d \bhb_j \over dt } \right)_{\cA} = - \bhb_j \cdot \left( {d \bhb_i \over dt } \right)_{\cA} \end{equation*}

for \(i,j = 1, 2, 3\text{.}\) Hence we can write the matrix form of \((d\bvv /dt)_\cA\) as follows:

\begin{equation*} \left( {d \bvv \over dt } \right)_{\cA} = \begin{pmatrix} 0 \amp - \omega_3 \amp \omega_2 \\ \omega_3 \amp 0 \amp - \omega_1 \\ - \omega_2 \amp \omega_1 \amp 0 \\ \end{pmatrix}_{b_B} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \\ \end{pmatrix} \end{equation*}

In expanded form, this gives

\begin{align*} \left( {d \bvv \over dt } \right)_{\cA} \amp = (- \omega_3 v_2 + \omega_2 v_3 ) \bhb_1 + ( \omega_3 v_1 - \omega_1 v_3 ) \bhb_2 + (- \omega_2 v_1 + \omega_1 v_2 ) \bhb_3 \\ \amp = ( \omega_1 \bhb_1 + \omega_2 \bhb_2 + \omega_3 \bhb_3 ) \times ( v_1 \bhb_1 + v_2 \bhb_2 + v_3 \bhb_3) \end{align*}

We can now state the following theorem:

Theorem 3.1.2. Angular Velocity.

There exists a unique vector denoted \(\bom _{\cB/ \cA}\text{,}\) called angular velocity of \(\cB\) relative to \(\cA\text{,}\) such that

\begin{equation} \left( \frac{d \bvv}{dt} \right)_{\cA} = \bom _{\cB/ \cA} \times \bvv \tag{3.1.2} \end{equation}

for any vector \(\bvv\) fixed in \(\cB\text{.}\) Vector \(\bom_{\cB/ \cA}\) is independent of \(\bvv\) and of the choice of basis of \(\cB\text{.}\)

Proof.

Uniqueness is easily shown: consider two vectors \(\bom_1\) and \(\bom_2\) satisfying (3.1.2), then \((\bom_1 -\bom_2) \times \bvv = \bze\) for all vectors \(\bvv\) fixed in \(\cB\text{.}\) This necessarily implies \(\bom_1= \bom_2\text{.}\) Furthermore, formula (3.1.2) is independent of the choice of basis of \(\cB\text{.}\)

Remark 3.1.3.

It can be shown that all skew-symmetric operators \(\cU\) of 3-dimensional oriented vector spaces must be of the type \(\cU (\bvv) = \bU \times \bvv\text{.}\)

Remark 3.1.4.

Vector \(\bom _{\cB/ \cA}\) is not expected to be a constant vector in \(\cB\) nor in \(\cA\text{.}\)

We can generalize formula (3.1.2) for arbitrary vector functions \(\bvv \text{,}\) that is, varying relative to both \(\cA\) and \(\cB\text{.}\)

\begin{equation*} \left( \frac{d \bvv}{ dt } \right)_{\cA} = \dot{v}_{1}\bhb_{1} + \dot{v}_{2} \bhb_{2} + \dot{v}_{3}\bhb_{3} + v_1 \left( \frac{d \bhb_1}{ dt } \right)_{\cA} + v_2\left( \frac{d \bhb_2}{ dt } \right)_{\cA} + v_3\left( \frac{d \bhb_3}{ dt } \right)_{\cA} \end{equation*}

We recognize that \(( {d \bhb_{i} / dt })_{\cA} = \bom _{\cB/ \cA} \times \bhb_{i}\text{,}\) and that \(\dot{v}_{1}\bhb_{1} + \dot{v}_{2} \bhb_{2} + \dot{v}_{3}\bhb_{3}= ( d \bvv / dt )_{\cB}\text{.}\)

Theorem 3.1.5. Time-derivatives from \(\cB\) to \(\cA\).

The time-derivatives of an arbitrary vector \(\bvv\) relative to referentials \(\cA\) and \(\cB\) are related by

\begin{equation} \left( \frac{d \bvv}{ dt } \right)_{\cA} = \left( \frac{d \bvv}{ dt } \right)_{\cB} + \bom _{\cB/ \cA} \times \bvv \tag{3.1.3} \end{equation}

This fundamental formula shows that the time-derivative of \(\bvv\) relative to \(\cA\) can be found from that relative to \(\cB\) by evaluating the term \(\bom _{\cB/ \cA} \times \bvv\) which accounts for the change of orientation of \(\cB\) relative to \(\cA\text{.}\) The consequence of this formula is that it offers a practical way to determine the time-derivatives of vectors. If angular velocities can be determined for arbitrary motions, we have an efficient means of calculating \((d\bvv/dt)_\cA\) by simple cross-product evaluations of vectors, without having to resolve vector \(\bvv\) into components on the chosen basis of referential \(\cA\text{.}\)

Note that the components of angular velocity \(\bom_{\cB/ \cA}\) on basis \((\bhb_{1},\bhb_{2},\bhb_{3})\) of \(\cB\) can be expressed as, according to matrix representation of operator \((d\cdot /dt)_\cA\) on the same basis,

\begin{equation} \omega_{1} = \left( \frac{d \bhb_{2}}{ dt } \right)_{\cA} \cdot \bhb_{3}, \;\;\; \omega_{2} = \left( \frac{d \bhb_{3}}{ dt } \right)_{\cA} \cdot \bhb_{1}, \;\;\; \omega_{3} = \left( \frac{d \bhb_{1}}{ dt } \right)_{\cA} \cdot \bhb_{2} \tag{3.1.4} \end{equation}

Unfortunately these expressions do not offer a practical way of calculating \(\bom_{\cB/ \cA}\) since the time derivatives \((d \bhb_{i} / dt )_{\cA}\) (\(i=1,2,3\)) are not easily found. They could be evaluated by expressing each basis vector \(\bhb_i\) on a basis of \(\cA\text{.}\) this is not only unpractical, but we would lose all benefits of formula (3.1.2). In fact, the simplest representation of \(\bom_{\cB/ \cA}\) is most often resolved on neither a basis of \(\cA\) nor a basis of \(\cB\).

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