By pre-selecting a number of engine orders and running the Least Squares Fit of a single blade response on each EO, the Campbell and Data Fit chart are used to determine which EO (or multiple EO’s) are define the response.

Figure 1

LSMF Analysis

Least Squares Model Fitting (LSMF) is an analysis package for curve fitting sine wave models such as a single or multiple engine order sine wave to a data set derived from a set of probes. Typically the input data set is obtained from a single revolution of the engine, although multi-revolution data sets can be useful for non-integral vibration analysis. The input data set is assumed to be properly zeroed and filtered to remove any noise or un-modeled non-integral vibrations (Data Conditioned). A model or set of possible models is created using the Campbell Diagram. These models are fit to the input data using a linear least squares analysis to determine the amplitude and phase for all the orders included in the model. This analysis is then repeated for each revolution on a single blade range or an all blade range. The phase value is the phase relative to the top dead center (TDC). If multiple models are defined, the analysis returns the amplitude, phase and data fit for the model that best fits the data. There can be up to 256 models in a model set. The output data is then displayed in many graphical formats such as bar charts, a blade waterfall plot, a nodal diameter waterfall plot, a real vs. imaginary plot etc. This is the primary analysis technique used with NSMS data sets.

Difference Blade Waterfall

The Difference Blade Waterfall plot Figure 2 is a graphical display of the input data set for an all blades analysis. As with many other plots in the AATAS analysis package, the x axis of the plot can be revolution #, rpm, time or 1/rpm. The y axis is scaled in deflection, although it is not labeled to keep the user from making incorrect amplitude estimations. Each trace is the probe difference envelope for a blade. The probe difference envelope is the span between the highest positive probe difference and the lowest negative probe difference. The difference envelope was chosen for this display since it is a representation of the input data set and it displays an integral response in a form similar to an over sampled strain gage time trace. The data for this plot is generated from zeroed and filtered deflection data. The primary use for this plot is to provide a graphical means, using the mouse, to select the regions for the all blade analysis, which are displayed as the blue regions on the plot.

Figure 2

Difference Blade Waterfall with Data-Fit

If an analysis has been previously performed and the envelop color has been set to data fit, the display color will reflect the data fit from the previous analysis. This is useful in identifying underlying simultaneous responses. If a model does not contain an order for the underlying response, the data fit will be poor in the region of simultaneous responses.

Figure 3

Amplitude Bar Chart

The Amplitude bar chart is used to display the peak amplitude associated with each blade. The range of the maximum amplitude search is defaulted to the full range of the analysis.
This range can be altered from the output blade waterfall or the output single blade plot in order to eliminate erroneously selected data points. The y axis can be displayed in deflection or stress.
Units for deflection can be either mils p-p, mm p-p or um p-p. Stress is displayed in the units of the conversion factor. The red trace indicates the current selected blade. Maximum, Minimum, Average, Maximum/Average Ratio and the Highlighted blade are displayed in below figure 4.

Figure 4

Figure 5

RPM Bar Chart

The RPM bar chart displays the rpm values of the peak responses. For an uncoupled resonant response, this is proportional to the natural resonant frequency of the blade. Maximum, minimum, average, highlighted blade and %variation values are displayed in figure 5. The highlighted blade is shown in red. Frequency is display on the right y axis.

Damping Bar Chart

The damping bar chart displays the output of the LSMF damping analysis. The damping analysis is a non-linear least squares curve fit of a Single Degree of Freedom model to the amplitude data. This analysis is only appropriate for uncoupled resonant vibrations. The results of the analysis can be overlaid on the Output Single Blade plot for comparison with the input data. The data range of the analysis is determined from a multiplier of an estimate of the half power width. The range can be adjusted from the Output Single Blade plot or the Output Blade Waterfall. The maximum, minimum, average and highlight blade values are displayed below the plot. The damping bar chart can use a data fit color scheme to help identify poor fitting blades. The data fit legend will appear upon selection of the color scheme.

Figure 6

Output Blade Waterfall

The Output Blade Waterfall (figure 8) is a display of the amplitude data for each blade. Its primary purpose is an overview of an entire response. By displaying the data from all the blades onto single plot, the user can quickly see blade mistuning, blade coupling interactions, phase interactions, data fit patterns etc. The plot can be displayed in an envelope mode or an amplitude trace. In the envelope mode the color can be set to indicate absolute phase, delta phase, data fit etc. The absolute phase is relative to TDC and for forced responses can be used as an indication of a driver orientation relative to the case. Since a blade’s response is in phase with the driver at low excitation frequencies, the driver phase relative to the case is equal to the blade’s response phase prior to (low frequency) a resonance.

Figure 8

Output Single Blade Plot

The LSMF Output Single Blade plot (figure 9) provides a detail view of the selected blade. There are many graphical options associated with the display. The default settings show the amplitude, phase and data fit. The amplitude, either deflection or stress, is shown as a black trace with the scale on the left y axis. The phase is shown as a blue trace. Do to the sometimes erratic behavior of the phase outside responses or in a non-integral response, the phase is plotted as dots instead of a connected line. The phase scale is on the right y axis. Additionally, the phase display can be toggled with a display of the condition number. The data fit is shown as a green line. The scale is always zero at the base of the graph and 100% at the top. In this example the data the fit is near 100% during the resonance but varies significantly when the amplitude drops to near zero.

Figure 9

Nodal Diameter Waterfall

The Nodal Diameter Waterfall plot (figure 10) is used to show the nodal diameter content of a vibratory response. The amplitude and phase information for all blades is recombined into a time domain data set that used to produce the FFT waterfall plot of the nodal diameter content. The waterfall plot spans the range of the all blade data set where there exists data for all blades

Figure 10

Figure 11

Nodal Diameter Peak Tracing

The Nodal Diameter Tracking is a display of the amplitudes of the individual nodal diameters within the selected region. The graph color legend is on the left hand side. Multiple ranges can be displayed on the plot. This is useful when trying to compare the response of multiple nodal diameters.

Real vs. Imaginary

The Real vs. Imaginary plot (figure 12) displays a resonance response as a circle whose diameter is the peak amplitude. This graphical display is useful in determining the modal content of a resonant response. If there are multiple modes in the response from either adjacent resonant responses or multiple modes caused by system mistuning, there will be multiple circles in the plot.

Figure 12

Amplitude Histogram

The Amplitude Histogram (figure 13) is a plot of the number of cycle that a response accumulated at a specific level of deflection. The plot shows the selected blade as well as the all blade average. The histogram is created from data found in the maximum amplitude search region.

Figure 13