## External loading

The external loading objects are listed in the following table.

Point and distributed loads | Dead load | Edge load | Face load |

Gravity and rigid rotation | Gravity | Pressure table | Rigid rotation |

- @INSTRUMENTATION_NAME {InstrumentationName}
**\} {**- @INSTRUMENTATION_TYPE {InstrumentationType}
**\}}** **@MEASUREMENT_NAME {MeasurementName}****\}}****@TIME {sca}****\}}****@SELECTED_MODES {Mode1, Mode2, ..., ModeN}***\}}*

**\}}** - @INSTRUMENTATION_TYPE {InstrumentationType}

**\}} \subsection{NOTES} \label{InstTable: NOTES}**- This section defines an
**instrumentation} object. This instrumentation is used in conjuction with measurements (see section~\ref{Measurement}) and signals (see section~\ref{Signal}) in order to identify the loads history in form of Fourier coefficients. It has to be used in combination with experimental results that represents the goal of the iterative process.** **The corresponding measurement object has to be specified in***MeasurementName}.**A time step (*- @TIME}) at which all data are stored has to be selected. It is suggested to choose a step-time at which regime-conditions are closely reached.
- Not all computed eigenvectors are significant for the identification of loads, therefore the identification numbers (
- @SELECTED_MODES}) of the significant modes need to be defined. This definition automatically defines the total number of considered eigenvectors.
- The purposes of an instrumentation are different depending of the considered analysis: \begin{description}
- [Static Analysis] This object generates data files that are necessary for the iterative process: information regarding eigenvalues and relative eigenvectors are stored in file
*Eigenvectors.dat}; the product of the mass matrix of the model and the matrix of eigenvectors is stored in file**InstMP.dat}; from the user-defined sensors and signals, instrumentation generates corresponding eigensersors and eigensignals (for a total amount of N*_{modes}*Nsignals} eigensensors and eigensignals, see fig.~\ref{fInstSensCycle}). This then generates the matrix that relates the measured quantity and the eigenvectors (matrix B}) and its pseudo-inverse (B^+}) are also stored in a file. The singular values of B^+} are also stored in order to analyse its conditioning (optimization purposes). \begin{figure}[htb] \centering \includegraphics[width=0.3\textwidth]{ExternalLoading/figures/InstSensCycle} \caption{Loop to generate eigensensros and eigensignals.} \label{fInstSensCycle} \end{figure} }**[Dynamic Analysis] Instrumentation generates corresponding sensors and signals to be related with experimental measurements (see fig.~\ref{fLoadId_dyn}). \begin{figure}[htb] \centering \includegraphics[width=0.4\textwidth]{ExternalLoading/figures/LoadId_dyn} \caption{Dynamic Analysis iteration cycle.} \label{fLoadId_dyn} \end{figure} \end{description}****The Load Identification Algorithm can be summarized as follows. Consider the following linear structural dynamics problem \begin{equation} \label{SIT: general linear system} M \ddot{\Vx} + K \Vx =**__F__(t), \end{equation} where M and K are the mass and stiffness matrices, respectively, array \Vx (t) stores the N degrees of freedom of the system, and__F__the array of externally applied forces. The eigenvalues of the system, \omega_{i}, i = 1, 2,\ldots N, and the corresponding eigenvectors, \bar{\Vu}i}, are readily obtained and the following matric is defined \begin{equation} P = \left[ \bar{\Vu}1}, \bar{\Vu}2},\ldots \bar{\Vu}N} \right]. \end{equation} A modal expansion of the system is performed as \Vx (t) = P \Vq (t), where \Vq is the array of modal participation factors. Similarly, the externally applied loads are expanded as F (t) = M P \Vlambda (t). After premultiplication by P^{T}, the governing equations of the system now become \begin{equation} P^{T} M P \ddot{\Vq} + P^{T} K P \Vq = P^{T} M P \Vlambda (t). \end{equation} The eigenmodes of the system are normalized in the space of the mass matrix, and hence, P^{T} M P = I, whereas P^{T} K P = \Diag(\omegai}^{2}), where \Diag(\omegai}^{2}) is a diagonal matrix storing the squares of the natural frequencies of the system. The N decoupled equations of the system now become \begin{equation} \label{SIT: decoupled equations} \ddot{\Vq} + \Diag(\omegai}^{2}) \Vq = \Vlambda (t). \end{equation} The system is now assumed to be periodic with a period \Omega, and all variables are expanded in terms of Fourier series, using a total of M harmonics. \begin{equation} \label{SIT: Fourier expansion of q} \Vq = \Vq0} + \sumj=1}^{M} (\Vqj} \cos j \Omega t + \Vrj} \sin j \Omega t). \end{equation} Similarly, the applied loading is also expanded in Fourier series as \begin{equation} \label{SIT: Fourier expansion of lambda} \Vlambda = \Vlambda0} + \sumj=1}^{M} (\Vlambdaj} \cos j \Omega t + Muj} \sin j \Omega t). \end{equation} Introducing the above expansions into the decoupled equations, eqs.~\eqref{SIT: decoupled equations}, and using harmonic balancing yields the following results \begin{subequations} \label{SIT: load to displacement relationship} \begin{align} \Vlambda0} & = \Diag(\omegai}^{2}) \Vq0}, \\ \Vlambdaj} & = \Diag(\omegai}^{2} - j^{2} \Omega^{2}) \Vqj}, \quad j = 1, 2,\ldots, M, \\ Muj} & = \Diag(\omegai}^{2} - j^{2} \Omega^{2}) \Vrj}, \quad j = 1, 2,\ldots, M. \end{align} \end{subequations} The above equations imply that if the Fourier components of the modal displacements are known, the corresponding components of loading can be readily evaluated. Next, the strain components at specific location of the structure are computed as \begin{equation} \label{SIT: strain-displacement relationship} \Vepsilon = B \Vq, \end{equation} where matrix B contains the appropriate derivatives of the eigenmodes of the structure, and \Vepsilon^{T} = \lfloor \epsilon1}, \epsilon2},\ldots \epsilons} \rfloor stores the s strain components. The above relationship can be inverted as \begin{equation} \label{SIT: displacement-strain relationship} \Vq = B^{+} \Vepsilon, \end{equation} where B^{+} denotes the pseudo-inverse of the rectangular matrix B. Of course, it is possible to expand the strain components in Fourier series as was done for the other variables of the problem, \begin{equation} \label{SIT: Fourier expansion of epsilon} \Vepsilon = \Vepsilon0} + \sumj=1}^{M} (\Vepsilonj} \cos j \Omega t + \Vkappaj} \sin j \Omega t). \end{equation} Finally, harmonic balancing yields the Fourier components of the loads expressed in terms strains as \begin{subequations} \label{SIT: load to strain relationship} \begin{align} \Vlambda0} & = \Diag(\omegai}^{2}) B^{+} \Vepsilon0}, \\ \Vlambdaj} & = \Diag(\omegai}^{2} - j^{2} \Omega^{2}) B^{+} \Vepsilonj}, \quad j = 1, 2,\ldots, M, \\ Muj} & = \Diag(\omegai}^{2} - j^{2} \Omega^{2}) B^{+} \Vkappaj}, \quad j = 1, 2,\ldots, M. \end{align} \end{subequations} If strains are experimentally measured, the following procedure is used to estimate the applied loads. Given the experimentally measured strains, \hat{\Vepsilon} (t), perform a Fourier analysis to find \hat{\Vepsilon}0}, \hat{\Vepsilon}j}, j = 1, 2,\ldots, M, and \hat{\Vkappa}j}, j = 1, 2,\ldots, M. Use eqs.~\eqref{SIT: load to strain relationship} to find estimates of the externally applied loads, \hat{\Vlambda}0}, \hat{\Vlambda}j}, j = 1, 2,\ldots, M, and \hat{Mu}j}, j = 1, 2,\ldots, M. Reconstruct the time history of the loading using eq.~\eqref{SIT: Fourier expansion of lambda} to find \begin{equation} \hat{F} (t) = M P \left[ \hat{\Vlambda}0} + \sumj=1}^{M} (\hat{\Vlambda}j} \cos j \Omega t + \hat{Mu}j} \sin j \Omega t ) \right]. \end{equation} Since the procedure described above is linear, it can also be used to relate changes in loads, \Delta \hat{F} (t) to changes in measured strains, \Delta \hat{\Vepsilon} (t). }- @TIME}) at which all data are stored has to be selected. It is suggested to choose a step-time at which regime-conditions are closely reached.

**\clearpage \section{Measurement definition} \label{Measurement}**

**@MEASUREMENT_DEFINITION {**- @MEASUREMENT_NAME { MeasurementName }
**\} {**- @SIGNAL_LIST {Signal1, Signal2, ..., SignalN}
**\}}** **@OMEGA { \Omega }****\}}****@NUMBER_OF_FREQUENCIES { sca }****\}}****@LIST_OF_FILE_NAMES { FileName1, .., FileNameN }****\}}**

**\}}** - @SIGNAL_LIST {Signal1, Signal2, ..., SignalN}

**\}} \subsection{NOTES} \label{MeasTable: NOTES}**- This section defines a
**measurement} type. During a dynamic analysis, this object reads an FFT for each signal definition and uses all data generated by an instrumentation to compute the Fourier's coefficients of the load, and then computes the load history on the body.** **Only the location and type of sensors (and corresponding signals) should be defined as an input. These signals have to be listed in**- @SIGNAL_LIST }. The data flow of measurement is shown if fig.~\ref{fLoadId}). \begin{figure}[htb] \centering \includegraphics[width=0.4\textwidth]{ExternalLoading/figures/LoadId_mod} \caption{Purpose of measurement.} \label{fLoadId} \end{figure}
- The main frequency at which FFT has to be performed is specified in
- @OMEGA}. If a rotor is considered, \Omega usually refers to its angular velocity in regime conditions.
- The number of considered harmonics has to be defined in
- @NUMBER_OF_HARMONICS}. FFT coefficients are evaluated for these harmonics only.
- @FILE_NAME } are the names of the files in which Fourier's coefficients of experimental measurements are stored.
- @SIGNAL_LIST }. The data flow of measurement is shown if fig.~\ref{fLoadId}). \begin{figure}[htb] \centering \includegraphics[width=0.4\textwidth]{ExternalLoading/figures/LoadId_mod} \caption{Purpose of measurement.} \label{fLoadId} \end{figure}

**%%% MC END**