Applied Mechanics Lab

Mechanics of Continua and Structures

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Computing the Pull-back Acceleration Field

After operating both the left and right sides of the first expression in the previous section with \(\boldsymbol{Q}^{T}\), we get

\[\begin{align} \boldsymbol{Q}^{\rm T}(t) \boldsymbol{a}(\boldsymbol{X}, t)&= \boldsymbol{Q}^{\rm T}(t)\ddot{\boldsymbol{Q}}(t)\boldsymbol{X} + \boldsymbol{Q}^{\rm T}(t)\ddot{\boldsymbol{c}}(t) \label{eq:OperatedAccelerationField} \end{align}\]

Defining \(\begin{align} \boldsymbol{\varphi}^{*}(\boldsymbol{a})(\boldsymbol{X},t) &:= \boldsymbol{Q}^{\rm T}(t)\boldsymbol{a}(\boldsymbol{X},t),\tag{Def.1}\label{eq:PullBackAccelerationFieldDef}\\ \boldsymbol{P}(t) &:=\boldsymbol{Q}^{\rm T}(t)\ddot{\boldsymbol{Q}}(t), \tag{Def.2}\label{eq:PDef} \end{align}\)

and \(\begin{align} \boldsymbol{q}(t) &:= \boldsymbol{Q}^{\rm T}(t)\ddot{\boldsymbol{c}}(t), \tag{Def.3} \label{eq:Def3} \end{align}\) equation \(\eqref{eq:OperatedAccelerationField}\) can be written as

\(\begin{align} \boldsymbol{\varphi}^{*}(\boldsymbol{a})(\boldsymbol{X},t)&= \boldsymbol{P}(t)\boldsymbol{X}+\boldsymbol{q}(t). \label{eq:PullBackAccelerationField} \end{align}\) The field \(\boldsymbol{\varphi}^*\left(\boldsymbol{a}\right)\) is the pull-back of the acceleration field \(\boldsymbol{a}\).

Let the accelerometers be glued onto the material points \(\mathcal{P}^{0}\), \(\mathcal{P}^{1}\), \(\mathcal{P}^{2}\), and \(\mathcal{P}^{3}\). We denote the position vectors of the material point \(\mathcal{P}^{\iota}\) in the reference configurationas \(\boldsymbol{X}^{\iota}\), where \(\iota=0,1,2,3\). It follows from \(\eqref{eq:PullBackAccelerationField}\) that the pull-back acceleration vector at the material points, \(\mathcal{P}^{\iota}\) is

\(\begin{align} \boldsymbol{\varphi}^{*}(\boldsymbol{a})(\boldsymbol{X}^{\iota},t) &= \boldsymbol{P}(t)\boldsymbol{X}^{\iota}+\boldsymbol{q}(t), \label{eq:m3} \end{align}\) where \(\iota=0,1,2,3\),

In order to obtain $\boldsymbol{P}$ in a compact form we define the vectors

\(\begin{align} \Delta \boldsymbol{X}^{\mathscr{i}}&=\boldsymbol{X}^{\mathscr{i}}-\boldsymbol{X}^{\mathbb{0}}, \end{align}\) and \(\begin{align} \Delta\boldsymbol{\varphi}^*\left(\boldsymbol{a}\right)^{\mathscr{i}}(t)&= \boldsymbol{\varphi}^*\left(\boldsymbol{a}\right)^{\mathscr{i}}(t)-\boldsymbol{\varphi}^*\left(\boldsymbol{a}\right)^{0}(t),\\ \end{align}\) where $\mathscr{i}=1,2,3$, and \(\begin{align} \boldsymbol{\varphi}^*\left(\boldsymbol{a}\right)^{\iota}(t)&= \boldsymbol{\varphi}^{*}(\boldsymbol{a})(\boldsymbol{X}^{\iota},t), \end{align}\) where \(\iota=0,1,2,3\).

If follows from \(\eqref{eq:m3}\) that \(\begin{align} \Delta\boldsymbol{\varphi}^*\left(\boldsymbol{a}\right)^{\mathscr{i}}(t) &= \boldsymbol{P}(t) \Delta\boldsymbol{X}^{\mathscr{i}}(t), \label{eq:PhiPX} \end{align}\) where $\mathscr{i}=1,2,3$

Solving the equation \(\eqref{eq:PhiPX}\) for $\mathscr{i}=1,2,3$ we get \(\begin{align} \boldsymbol{P}(t)= \Delta\boldsymbol{\varphi}^*\left(\boldsymbol{a}\right)^{\mathscr{i}}(t)\cdot \Delta\boldsymbol{X}^{\mathscr{j}}(t) \Delta\boldsymbol{Y}^{\mathscr{i}}\otimes \Delta\boldsymbol{Y}^{\mathscr{j}}, \label{eq:Psol} \end{align}\) where \(\Delta \boldsymbol{Y}^{\mathscr{i}}\), \(\mathscr{i}=1,2,3\) are the reciprocal vectors corresponding to \(\Delta \boldsymbol{X}^{\mathscr{i}}\), \(\mathscr{i}=1,2,3\) \(\begin{align} \Delta\boldsymbol{Y}^{\mathscr{1}}&= \frac{\Delta\boldsymbol{X}^{\mathscr{2}} \times \Delta\boldsymbol{X}^{\mathscr{3}}}{\Delta\boldsymbol{X}^{\mathscr{1}} \cdot \left(\Delta\boldsymbol{X}^{\mathscr{2}}\times \Delta\boldsymbol{X}^{\mathscr{3}}\right)}\\ \Delta\boldsymbol{Y}^{\mathscr{2}}&= \frac{\Delta\boldsymbol{X}^{\mathscr{3}} \times \Delta\boldsymbol{X}^{\mathscr{1}}}{\Delta\boldsymbol{X}^{\mathscr{1}} \cdot \left(\Delta\boldsymbol{X}^{\mathscr{2}}\times \Delta\boldsymbol{X}^{\mathscr{3}}\right)}\\ \Delta\boldsymbol{Y}^{\mathscr{3}}&= \frac{\Delta\boldsymbol{X}^{\mathscr{1}} \times \Delta\boldsymbol{X}^{\mathscr{2}}}{\Delta\boldsymbol{X}^{\mathscr{1}} \cdot \left(\Delta\boldsymbol{X}^{\mathscr{2}}\times \Delta\boldsymbol{X}^{\mathscr{3}}\right)} \end{align}\)

After computing \(\boldsymbol{P}(t)\) from \(\eqref{eq:Psol}\), the vector \(\boldsymbol{q}(t)\), defined in \(\eqref{eq:Def3}\), can be computed from \(\eqref{eq:m3}\) with, e.g., \(\iota=0\).

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