Mechanics of Continua and Structures
Let \(\boldsymbol{X}^{\iota}\) and \(\boldsymbol{x}^{\iota}(t)\) be the Lagrangian and Eulerian position vectors of the material point $\mathcal{P}^{\iota}$. The acceleration of the material particle \(\mathcal{P}^{\iota}\) is given as \(\boldsymbol{A}(\boldsymbol{X}^{\iota},t)\), or alternately as \(\boldsymbol{a}(\boldsymbol{x}^{\iota}(t),t)\). Let’s call this acceleration \(\boldsymbol{a}^{\iota}(t)\). Employing the Lagrangian description, the acceleration \(\boldsymbol{a}^{\iota}(t)\) can be computed as
\[\begin{align} \boldsymbol{a}^{\iota}(t)&=\ddot{\boldsymbol{Q}}(t)\boldsymbol{X}^{\iota}+\ddot{c}(t)\\ \end{align}\]Let the accelerometers $a^{\iota}$, $b^{\iota}$, and $c^{\iota}$ be glued at the material point \(\mathcal{P}^{\iota}\). In the reference configuration, i.e., at $t=0$, let these accelerometers point in the directions characterized by the unit vectors \(\hat{\boldsymbol{N}}^{\iota}_{i}\), $i=1,~2,~3$. These vectors are mutually perpendicular to each other, i.e, \(\hat{\boldsymbol{N}}^{\iota}_{i}\cdot\hat{\boldsymbol{N}}^{\iota}_{j}=\delta_{ij}\). When $t>0$, these accelorometers will point in the directions characterized by the unit vectors \(\hat{\boldsymbol{n}}^{\iota}_{i}(t)=\boldsymbol{Q}(t)\hat{\boldsymbol{N}}^{\iota}_{i}\). Note that given any time $t$, the vectors \(\{\hat{\boldsymbol{n}}^{\iota}_{i}\}_{i=1,2,3}\) form an orthonormal set. Let the measurements of these accelorometers be \(\alpha^{\iota}_i(t)\). These measurements give the components of the acceleration vector of the material point \(\mathcal{P}_{\iota}\) in the directions \(\hat{\boldsymbol{n}}^{\iota}_{i}\). That is, \(\begin{align} \boldsymbol{a}^{\iota}(t) &= \alpha^{\iota}_i(t)\hat{\boldsymbol{n}}^{\iota}_{i}(t)\\ \end{align}\)
It then follows that \(\begin{align} \boldsymbol{a}^{\iota}(t) &= \alpha^{\iota}_i(t)\boldsymbol{Q}(t)\hat{\boldsymbol{N}}^{\iota}_{i}\\ &=\boldsymbol{Q}(t)\left( \alpha^{\iota}_i(t)\hat{\boldsymbol{N}}^{\iota}_{i}\right)\\ \end{align}\) which implies that \(\begin{align} \boldsymbol{Q}^{\rm T}(t)\boldsymbol{a}^{\iota}(t) &= \alpha^{\iota}_i(t)\hat{\boldsymbol{N}}^{\iota}_{i}\\ \end{align}\)
The vector \(\alpha^{\iota}_i(t)\hat{\boldsymbol{N}}^{\iota}_{i}\) is called the pull-back of the acceleration vector \(\boldsymbol{a}^{\iota}(t)\). Therefore, we denote it as \(\varphi_{*}(\boldsymbol{a})^{\iota}(t)\). Since both \(\alpha_i\) and \(\hat{\boldsymbol{N}}^{\iota}_{i}\) are known for all $t$, we know the vector \(\varphi_{*}(\boldsymbol{a})^{\iota}\) for any given $t$.
In the following, we discuss an arrangement of accelerometers that will allow us to determine the acceleration \(\varphi_{*}(\boldsymbol{a})^{\iota}(t)\) of any material point $\mathcal{P}^{\iota}$. The acceleration \(\varphi_{*}(\boldsymbol{a})^{\iota}(t)\) is of course not the true acceleration of the particle \(\mathcal{P}^{\iota}\). It is the pull-back of the material point $\mathcal{P}^{\iota}$’s true acceleration, which is \(\boldsymbol{a}^{\iota}(t)\). However, what is quite remarkable is that the magnitude of \(\varphi_{*}(\boldsymbol{a})^{\iota}(t)\) equals the magnitude of \(\boldsymbol{a}^{\iota}(t)\). Thus, on knowing \(\varphi_{*}(\boldsymbol{a})^{\iota}(t)\) of any material point $\mathcal{P}^{\iota}$ will allow us to compute the magnitude of any material point \(\mathcal{P}^{\iota}\)’s acceleration.