Applying a lowpass Butterworth filter to data using a single pass filtering approach will result in potentially smoother data, but the data will also be delayed in time by the filter. A simple way to avoid this undesirable distortion is to utilize a bidirectional implementation of the filtering process. With this technique, the data is first passed through the filter (producing a time delay), then this filtered result is reversed and filtered a second time (causing a negative time delay which cancels to first-pass time delay), and then finally this result is reversed again and returned as the filtered result. For the three software products being investigated, MATLAB® and Kornucopia® offer native functions that filter the data bi-directionally. For Mathcad®, the user must manually implement such an algorithm using the single-pass filters offered. Before the data was filtered, one trimming operation was done to help improve results. The raw force vs time data from Fig. 19.3 has a severe and nearly immediate drop in force when the specimen failed. At this point and beyond, there is no interest in that section of the data. In fact, leaving the post-failed section of data in the dataset that is to be filtered would create undesirable filter distortions around the failure onset location, meaning that it will distort part of the force vs time curve just prior to the onset of failure. To avoid this, the data (both force and displacement) was trimmed at the point where failure begins. The red colored raw data curve in Fig. 19.6 shows the trimmed data to be filtered as described below. To apply a DSP filter, the user must specify a filter cutoff frequency. For some types of data and problems, the cutoff frequency might be specified by standard procedures or they might be guided by modal analysis or similar. For the data being discussed here, no such information was available. Another common approach to determine cutoff frequency is to compute Discrete Fourier Spectra (so-called FFT) of the force vs time and displacement vs time raw data to see if any obvious spikes or frequency bands appear that might be related to known causes of noise. For this example, these analyses (not shown) were inconclusive and so the cutoff frequency was determined by a simple trial and error process. Different cutoff frequencies were tried, starting with 1/10 of the sampling frequency, and then lowering the cutoff value until a plausibly smooth result was obtained from the filtering process. This is not an exact approach, meaning that no one specific perfect cutoff frequency will be found. For the data analyzed here, a cutoff frequency between 10 and 3 kHz appeared to remove the noise without removing too much of the actual physics believed to be in the data. The first attempt to clean the data applied a manually created bi-directional filter in Mathcad®. Both the force vs time Fig. 19.6 (left) and the displacement vs time Fig. 19.6 (right) signals were filtered (dashed lines) using a 10 kHz cutoff frequency and are presented on the secondary y-axis while the raw data (solid lines) are on the primary y-axis. The first two thirds of the filtered response appear reasonable, but the filtered result near the right end of the data (beginning around 5e-4 s) shows a highly undesirable behavior. In both the force and displacement responses, the filtered result near the right end shows a Gibbs phenomenon (the rising bump) and then dives down to a zero amplitude value. This distortion is caused by the way the underlying filtering algorithm in Mathcad®works. In short, it has no end-effect minimization scheme and simply assumes the data begins and ends with zero values. More detailed discussion on the general topic of filter end-effect minimization and filter pre-charging can be found in [7]. Bottom line—this filter-induced distortion is especially bad if the data is to be used to create a material law for finite element analysis. 0.1 0.05 0 400 300 200 100 400 Raw Filtered Raw Filtered 300 200 100 0 0 0 2x10−4 4x10−4 Time [s] Time [s] Raw Displacement [in] Filtered Force [lfb] Filtered Displacement [in] Raw Force [lfb] 6x10−4 8x10−4 0 2x10 −4 4x10−4 6x10−4 8x10−4 0 0.05 0.1 Fig. 19.6 Force–time (left) and displacement–time (right) responses with a simple, manually created, bi-directional filter using Mathcad ® (dashed lines) 19 Overcoming Challenges in Material Characterization of Polymers at Intermediate Strain Rates 157
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