Elementary inertial forces
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The inertial forces applied to an element :math:`\mathrm{ds}` form a torsor which admits, at the center of gravity:


* a general result, :math:`-\rho A\mathrm{ds}\ddot{{x}_{0}}(s,t)`;


* a resultant moment equal to the opposite of the absolute speed of elementary angular momentum :math:`H`.

To express angular velocity, let's proceed as in [:ref:`§6.2 <§6.2>`] and derive the relationship:

:math:`{t}_{i}={RE}_{i},`

compared to :math:`t`, taking into account that :math:`{E}_{i}` does not depend on time. We get:

 :math:`{\dot{t}}_{i}=\dot{R}{R}^{T}{t}_{i}\text{.}` **eq 7-1**

Let's say:

 :math:`\dot{R}{R}^{T}=\stackrel{ˆ}{\omega }\text{.}` **eq 7-2**

By deriving the relationship [:ref:`éq 4.2-7 <éq 4.2-7>`] with respect to :math:`t`, we see that the matrix :math:`\stackrel{ˆ}{\omega }` is antisymmetric. If we denote the axial vector of this matrix by :math:`\omega`, the relationship [:ref:`éq 7.1 <éq 7.1>`] is written as:

:math:`{\dot{t}}_{i}(s,t)=\omega \wedge {t}_{i}(s,t)\text{.}`

So :math:`\omega` is the **angular velocity vector** of the x-axis beam section :math:`s` at time :math:`t`.

The expression for elementary angular momentum is:

 :math:`H=\mathrm{ds}{I}_{\rho }\omega =\mathrm{ds}R{J}_{\rho }{R}^{T}\omega` **eq 7-3**

where :math:`{J}_{\rho }` is the inertia tensor in the reference configuration:

:math:`{J}_{\rho }={R}_{0}\text{DIAG}\left[\rho {I}_{1},\rho {I}_{2},\rho {I}_{3}\right]{R}_{0}^{T}`

By drifting with respect to time:

:math:`\dot{H}=\mathrm{ds}{I}_{\rho }\dot{\omega }+\mathrm{ds}(\dot{R}{J}_{\rho }{R}^{T}+R{J}_{\rho }{\dot{R}}^{T})\omega`

But:

:math:`\dot{R}{J}_{\rho }{R}^{T}\omega =\dot{R}{R}^{T}R{J}_{\rho }{R}^{T}\omega =\stackrel{ˆ}{\omega }{I}_{\rho }\omega =\omega \wedge {I}_{\omega }\omega`

and:

:math:`R{J}_{\rho }{\dot{R}}^{T}\omega =R{J}_{\rho }{R}^{T}R{\dot{R}}^{T}\omega =0`

because, according to equation [:ref:`éq A1-2 <éq A1-2>`]:

:math:`R{\dot{R}}^{T}\omega =-\stackrel{ˆ}{\omega }\omega =0`.

Hence the moment of the inertial forces of the element:

:math:`-\dot{H}=-\mathrm{ds}{I}_{\rho }\dot{\omega }-\mathrm{ds}\omega \wedge {I}_{\rho }\omega`

The virtual work of inertial forces is therefore expressed as:

 :math:`{W}_{\mathrm{iner}}=-{\int }_{{S}_{1}}^{{S}_{2}}\left\{\begin{array}{}\rho A\ddot{{x}_{0}}\\ {I}_{\rho }\dot{\omega }+\omega \wedge {I}_{\rho }\omega \end{array}\right\}\mathrm{.}\left\{\begin{array}{}\delta {\omega }_{0}\\ \delta \Psi \end{array}\right\}\mathrm{ds}` **eq 7-4**