v0.6.0 updates

Author: dprohe

docs/source/modal_tutorials/Modal_05_Experimental_Modal_Analysis/Modal_05a_Computing_FRF_from_Time_Data.ipynb

@@ -574,6 +574,24 @@
     "\n",
     "One interesting feature of this technique is that it should be computed *per output* degree of freedom, so matrices $\\mathbf{P}_{XX}$, $\\mathbf{P}_{XF}$, and $\\mathbf{P}_{FX}$ should be constructed using one $X$ at a time.  Otherwise, the responses will affect one another if the entire matrix is assembled at once and the eigenvectors are computed.\n",
     "\n",
+    "### Coherence\n",
+    "\n",
+    "When constructing frequency response functions, we are computing the relationship between the outputs of our system (i.e. the responses) due to the inputs applied to the system (i.e. the forces).  The **coherence** is an important metric that shows us how much of the response can be attributed to the forces being applied.  The coherence can take a value between 0 and 1, where a value of 1 means that all of the responses are due to the inputs, and a value of 0 means that the responses are not related to the inputs at all.\n",
+    "\n",
+    "There are many reasons poor coherence may occur in a test, and a few common reasons.  If the excitation level is very low, noise in the measurement may contribute significantly to the response, and this noise will not be related to the input applied.  For this reason, we often see coherence drops at anti-resonances in the frequency response function: the response is very small, so noise in the measurement can be signficant.\n",
+    "\n",
+    "If the forces into the system are not measured entirely, the coherence can also suffer; for example if the force is measured using a uniaxial force sensor, but there is also some side loading or bending moment applied to the structure, these unmeasured forces and moments will still contribute to measurable responses of the system, and these responses may not correlate to the measured input.  Finally if a force is applied to a system and the entire response is not measured, for example if we impact a system and stop the measurement while the system is still responding from the impact, the coherence will be lower due to a portion of the response being missing.\n",
+    "\n",
+    "Coherence between an input $F$ and response $X$ can be computed as\n",
+    "\n",
+    "$$\\gamma^2 = {{\\mathbf{P}_{XF}{\\mathbf{P}_{XF}}^*}\\over{\\mathbf{P}_{XX}\\mathbf{P}_{FF}}} = {{\\|\\mathbf{P}_{XF}\\|^2}\\over{\\mathbf{P}_{XX}\\mathbf{P}_{FF}}}$$\n",
+    "\n",
+    "There will be one value of $\\gamma^2$ for each output/input pair in the test at each frequency line.\n",
+    "\n",
+    "This formulation is often used for single-input testing, because all response is expected to be due to that single input.  However, if this formulation is used for multiple-input testing, any given value of $\\gamma^2$ for an output-input pair is expected to be less than 1.  This is because any given response is expected to have some contribution from input 1, some contribution from input 2, ... some contribution from input $n$.  No response can be expected to be 100% correlated to a single input when multiple inputs are present.  In the multiple-input testing case, we instead compute the **Multiple Coherence** function, which shows how much a given response is correlated to *all* inputs.  There will therefore be one value of the multiple coherence function for each response in the test at each frequency line.\n",
+    "\n",
+    "$$Mcoh = {\\mathbf{P}_{XF}{\\mathbf{P}_{FF}}^{-1}{\\mathbf{P}_{XF}}^* \\over {\\mathbf{P}_{XX}}}$$\n",
+    "\n",
     "## Computing Frequency Response Functions using SDynPy\n",
     "\n",
     "Frequency response functions can be computed in SDynPy using either code or interactive windows.  The primary functions that will be useful for computing frequency response functions are `TransferFunctionArray.from_time_data` and `SignalProcessingGUI`.\n",