7. Elementary inertial forces#

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

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

  • a resultant moment equal to the opposite of the absolute speed of elementary angular momentum \(H\).

To express angular velocity, let’s proceed as in [§6.2] and derive the relationship:

\({t}_{i}={RE}_{i},\)

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

\({\dot{t}}_{i}=\dot{R}{R}^{T}{t}_{i}\text{.}\) eq 7-1

Let’s say:

\(\dot{R}{R}^{T}=\stackrel{ˆ}{\omega }\text{.}\) eq 7-2

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

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

So \(\omega\) is the angular velocity vector of the x-axis beam section \(s\) at time \(t\).

The expression for elementary angular momentum is:

\(H=\mathrm{ds}{I}_{\rho }\omega =\mathrm{ds}R{J}_{\rho }{R}^{T}\omega\) eq 7-3

where \({J}_{\rho }\) is the inertia tensor in the reference configuration:

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

By drifting with respect to time:

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

But:

\(\dot{R}{J}_{\rho }{R}^{T}\omega =\dot{R}{R}^{T}R{J}_{\rho }{R}^{T}\omega =\stackrel{ˆ}{\omega }{I}_{\rho }\omega =\omega \wedge {I}_{\omega }\omega\)

and:

\(R{J}_{\rho }{\dot{R}}^{T}\omega =R{J}_{\rho }{R}^{T}R{\dot{R}}^{T}\omega =0\)

because, according to equation [éq A1-2]:

\(R{\dot{R}}^{T}\omega =-\stackrel{ˆ}{\omega }\omega =0\).

Hence the moment of the inertial forces of the element:

\(-\dot{H}=-\mathrm{ds}{I}_{\rho }\dot{\omega }-\mathrm{ds}\omega \wedge {I}_{\rho }\omega\)

The virtual work of inertial forces is therefore expressed as:

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