Check the principle of reciprocity

modal-analysis/measurement.org

@@ -138,21 +138,24 @@ All the stages are moved to their zero position (Ty, Ry, Rz, Slip-Ring, Hexapod)
 All other elements have been remove from the granite such as another heavy positioning system.
 
 ** Test Planing
-The goal is to identify the full $N \times N$ FRF matrix (where $N$ is the number of degree of freedom of the system).
+The goal is to identify the full $N \times N$ FRF matrix $H$ (where $N$ is the number of degree of freedom of the system):
+\begin{equation}
+  H_{jk} = \frac{X_j}{F_k}
+\end{equation}
 
-However, the principle of reciprocity states that:
-\[ H_{jk} = \frac{X_j}{F_k} = H_{kj} = \frac{X_k}{F_j} \]
-Thus, only one column or one line of the matrix has to be identified.
+However, from only one column or one line of the matrix, we can compute the other terms thanks to the principle of reciprocity.
 
 Either we choose to identify $\frac{X_k}{F_i}$ or $\frac{X_i}{F_k}$ for any chosen $k$ and for $i = 1,\ ...,\ N$.
 
 We here choose to identify $\frac{X_i}{F_k}$ for practical reasons:
-- it is easier to glue the accelerometers on some stages than to excite this particular stage with the Hammer
+- it is easier to glue the accelerometers on all the stages and excite only a one particular point than doing the opposite
 
 The measurement thus consists of:
 - always excite the structure at the same location with the Hammer
 - Move the accelerometers to measure all the DOF of the structure
 
+We will measured 3 columns (3 impacts location) in order to have some redundancy.
+
 ** Location of the Accelerometers
 4 tri-axis accelerometers are used for each solid body.
 
@@ -561,7 +564,7 @@ The coherence associated is shown on figure [[fig:frf_result_example]].
   plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Mod);
   set(gca, 'Yscale', 'log');
   set(gca, 'XTickLabel',[]);
-  ylabel('Magnitude');
+  ylabel('Magnitude [$\frac{m/s^2}{N}$]');
 
   ax2 = subplot(2, 1, 2);
   plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Phas);
@@ -606,7 +609,7 @@ The frequency response matrix is an $n \times p \times q$:
 - $q$ is the number of frequency points $\omega_i$
 
 Thus, the FRF matrix is an $69 \times 3 \times 801$ matrix.
-
+We do the same thing for the coherence matrix.
 
 #+begin_important
 For each frequency point $\omega_i$, we obtain a 2D matrix:
@@ -672,6 +675,7 @@ And we save the obtained FRF matrix and Coherence matrix in a =.mat= file.
 #+begin_src matlab
   save('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
 #+end_src
+
 * Plot showing the coherence of all the measurements
 Now that we have defined a Coherence matrix, we can plot each of its elements to have an idea of the overall coherence and thus, quality of the measurement.
 The result is shown on figure [[fig:all_coherence]].
@@ -755,8 +759,8 @@ This is illustrated on figure [[fig:aligned_accelerometers]].
     \coordinate[] (p1) at (-1.5,  1.5);
     \coordinate[] (p2) at ( 1.5,  1.5);
 
-    \draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0)node[above]{$v_{x1}$};
-    \draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(1, 0)node[above]{$v_{x2}$};
+    \draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0)node[above]{$\delta p_{x1}$};
+    \draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(1, 0)node[above]{$\delta p_{x2}$};
 
     \draw[dashed] ($(p1)+(-1, 0)$) -- ($(p2)+(2, 0)$);
   \end{tikzpicture}
@@ -767,23 +771,23 @@ This is illustrated on figure [[fig:aligned_accelerometers]].
 #+RESULTS:
 [[file:figs/aligned_accelerometers.png]]
 
-The motion of the rigid body of figure [[fig:aligned_accelerometers]] is defined by the velocity $\vec{v}$ and rotation $\vec{\Omega}$ with respect to the reference frame $\{O\}$.
+The motion of the rigid body of figure [[fig:aligned_accelerometers]] is defined by its displacement $\delta p$ and rotation $\vec{\Omega}$ with respect to the reference frame $\{O\}$.
 
 The motions at points $1$ and $2$ are:
 \begin{align*}
-  v_1 &= v + \Omega \times p_1 \\
-  v_2 &= v + \Omega \times p_2
+  \delta p_1 &= \delta p + \Omega \times p_1 \\
+  \delta p_2 &= \delta p + \Omega \times p_2
 \end{align*}
 
 Taking only the $x$ direction:
 \begin{align*}
-  v_{x1} &= v + \Omega_y p_{z1} - \Omega_z p_{y1} \\
-  v_{x2} &= v + \Omega_y p_{z2} - \Omega_z p_{y2}
+  \delta p_{x1} &= \delta p_x + \Omega_y p_{z1} - \Omega_z p_{y1} \\
+  \delta p_{x2} &= \delta p_x + \Omega_y p_{z2} - \Omega_z p_{y2}
 \end{align*}
 
 However, we have $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$ because of the co-linearity of the two sensors in the $x$ direction, and thus we obtain
 \begin{equation}
-  v_{x1} = v_{x2}
+  \delta p_{x1} = \delta p_{x2}
 \end{equation}
 
 #+begin_important
@@ -825,7 +829,7 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp
     end
 
     if rem(i, 3) == 1
-      ylabel('Amplitude');
+      ylabel('Amplitude [$\frac{m/s^2}{N}$]');
     end
     xlim([1, 200]);
     title(sprintf('Acc %i and %i - X', acc_i(i, 1), acc_i(i, 2)));
@@ -871,7 +875,7 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp
     end
 
     if rem(i, 3) == 1
-      ylabel('Amplitude');
+      ylabel('Amplitude [$\frac{m/s^2}{N}$]');
     end
     xlim([1, 200]);
     title(sprintf('Acc %i and %i - Y', acc_i(i, 1), acc_i(i, 2)));
@@ -890,3 +894,65 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp
 #+begin_important
   From the two figures above, we are more confident about the rigid body assumption in the frequency band of interest.
 #+end_important
+* Verification of the principle of reciprocity
+Because we expect our system to follow the principle of reciprocity.
+That is to say the response $X_j$ at some degree of freedom $j$ due to a force $F_k$ applied on DOF $k$ should be the same as the response $X_k$ due to a force $F_j$:
+\[ H_{jk} = \frac{X_j}{F_k} = \frac{X_k}{F_j} = H_{kj} \]
+
+This comes from the fact that we expect to have symmetric mass, stiffness and damping matrices.
+
+In order to access the quality of the data and the validity of the measured FRF, we then check that the reciprocity between $H_{jk}$ and $H_{kj}$ is of an acceptable level.
+
+We can verify this reciprocity using 3 different pairs of response/force.
+
+#+begin_src matlab :exports none
+  dir_names = {'X', 'Y', 'Z'};
+
+  figure;
+  for i = 1:3
+    subplot(3, 1, i)
+    a = mod(i, 3)+1;
+    b = mod(i-2, 3)+1;
+    hold on;
+    plot(freqs, abs(squeeze(FRFs(3*(11-1)+a, b, :))), 'DisplayName', sprintf('$\\frac{F_%s}{D_%s}$', dir_names{a}, dir_names{b}));
+    plot(freqs, abs(squeeze(FRFs(3*(11-1)+b, a, :))), 'DisplayName', sprintf('$\\frac{F_%s}{D_%s}$', dir_names{b}, dir_names{a}));
+    hold off;
+    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+    if i == 3
+      xlabel('Frequency [Hz]');
+    else
+      set(gca, 'XTickLabel',[]);
+    end
+    if i == 2
+      ylabel('Amplitude [$\frac{m/s^2}{N}$]');
+    end
+    xlim([1, 200]);
+    legend('location', 'northwest');
+  end
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/principle_reciprocity.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+NAME: fig:principle_reciprocity
+#+CAPTION: Verification of the principle of reciprocity
+[[file:figs/principle_reciprocity.png]]
+
+From figure [[fig:principle_reciprocity]], it seems that the principle of reciprocity is valid from 50Hz to 100Hz.
+Above that, the difference between the two could be due to:
+- rotational DOFs? Local flexibility?
+Below, it could be due to:
+- not exciting and measuring the same DOF
+
+#+begin_important
+  One should note here that the excitation is not applied on the same DOF as the measured response.
+  This could be seen on figures [[fig:hammer_x]], [[fig:hammer_y]] and [[fig:hammer_z]] that show the excitation points, and on figure [[fig:accelerometers_ty_overview]] where the accelerometer on the bottom left shows the one used for the principle of reciprocity validation above.
+  Thus, technically we cannot verify the principle of reciprocity here.
+#+end_important
+
+#+begin_warning
+  As it is usually very important to measure point response $\frac{X_j}{F_j}$, that is to say exciting and measuring the *same* DOF, should the measurements be redone?
+  On the modal software that is used to extract modal parameters, it is suppose that we are exciting the *exact* same DOF as the one measured by the accelerometer 11. As is it not the case in practice, could this induce large errors in the modal model?
+#+end_warning