The Pre-Test Calculator employs the Effective Independence Method (EIM) to identify the optimum location to place sensors. The method is based on the assumption that the optimum location of sensors is the one that ensures that the measured mode shapes are distinguishable from each other and provides good signal strength.
The algorithm has the following parameters:
SN={xn}N – Set of candidate nodes, with NN being its size (the total number of candidate nodes after filtering).
Nmax – Maximum number of sensors defined by the user in the Sensor Definition section of Pre-Test Calculator Details.
{Φi} – Set of modes considered in the analysis. Only the modes that are checked in the Frequency Worksheet inside the Pre-Test Worksheet are employed.
α – Percentage used to remove nodes after each iteration. This is dependent on the number of nodes available at each iteration:
2.5% if the number of nodes is less than or equal to 1000
5% if the number of nodes is greater than 1000 and less than 5000
10% if the number of nodes is greater than or equal to 5000
EIM maximizes the Fisher Information Matrix (FIM) determinant, defined as follows:
Where Φxi is the target modal matrix of
the ith node. The sensor type affects the
size of Φxi and the method is followed to
maximize FIM. The value of x is 1 for uniaxial
sensors 3 for triaxial sensors.
If the sensors are uniaxial, only the displacement degree of freedom in the Z direction of the Coordinate System defined at the sensor is employed. Thus, the mode shape is first projected in the local Z direction for each node (reducing from 3N0 to N0 components) and Φxi contains only one row.
If the sensors are triaxial, the three displacement DOFs defined at each node are employed in the calculation, and Φxi contains three rows.
The implemented EIM algorithm removes a set of nodes after each iteration until the specified number of sensors requested by the user is reached. The metric used to assess which sensors are removed is the determinant of the Fisher Information Matrix. The Effective Independence value corresponding to the ith sensor is given by:
If the sensors are uniaxial, E1i is a scalar value that lies in the range
[0,1].Nodes are ranked and sorted in a descending order of E1i. Sensors with a value of this metric closer to 0.0 contribute less to the determinant of FIM, so they can be deleted.
If the sensors are triaxial, E3i is a 3x3, fully populated matrix and its metric is calculated as:
Where I3 is a 3-dimensional identity matrix.
Nodes are ranked and sorted in a descending order of E3i. Sensors with a value of this metric closer to 0.0 contribute less to the determinant of FIM, so they can be deleted.
Sensors remaining after each iteration k are calculated as:
This set of nodes is sorted based on its effective independence value.
Additionally, nodes must comply with the Proximity between Sensors property.
At each iteration, after removing sensors at a rate of α and sorting the nodes based on the E1i or E3i value, nearby nodes are removed as follows:
The top Nmax nodes with the highest value of E1i or E3i are considered in turn and nearby nodes that do not meet the Proximity between Sensors property are removed. Removal of nodes stops if the distance of the remaining sensors reaches the Nmax value.
Iteration continues until the Maximum Number of Sensors value is reached, either at a rate of α or using the Proximity between Sensors property. If the Respect Maximum Number of Sensors property in Pre-Test Calculator Details is set to , you will always get the Maximum Number of Sensors. Otherwise, removal of nearby nodes at the last iteration could result in less than the Maximum Number of Sensors.
This final set of sensors gives the best configuration for sensor placement, based on the Effective Independence value. It must be noted, however, that these locations offer a degree of manual modification. If a location offered by the Pre-Test Calculator is inaccessible, it is possible to place the sensor near to that location instead.
Additionally, an AutoMAC matrix is computed and displayed in the Sensors AutoMAC Panel for the final set of sensors, Sf.
The AutoMAC matrix is calculated using the following equation:
