# Impact Load Sparse Recognition Method Based on Mc Penalty Function
## 4 Experimental Analysis of Recognition Methods
### Single Vibration and Shock Test  
This experimental data set collects data from the motorized spindle test stand. The motorized spindle test stand is composed of motorized spindle, hydraulic loading device, a dynamometer, and oil-gas lubrication device. The test bed is shown in Figure 10. The acceleration sensor information of the simulated tool holder position (loading bar) of the spindle is collected to obtain the radial and axial position signals of the spindle. Because of the clamping mechanism of the test bed, the radial run out is less than the axial run out, so the axial vibration signal is used in the analysis. We applied different instantaneous loads to the spindle to simulate the vibration shock of the main shaft bearing. In the test, the sampling frequency \(f_{s}\) is 2000 Hz, the collection condition is 1000 no-load, the spindle speed is constant at 2000 rpm, the vibration signal is recorded by the Donghua test and acquisition analyzer, and the acquisition channel is selected as 1 channel. The acquisition equipment is shown in Figure 11. The time domain signal of the test bench data results is shown in Figure 12.  
Figure 10: The main part of the bearing vibration and impact fatigue testing machine. Figure 9: Sparse decomposition simulation amplitude/envelope results. (**a**) sparse decomposition amplitude spectrum, and (**b**) sparse decomposition envelope spectrum. It can be seen from the time domain diagram in Figure 12 that the time domain waveform is disordered and contains a large amount of noise components. The vibration shock pulse is not obvious. In order to extract the characteristics of the bearing vibration shock signal and verify the effectiveness of the method, the L1-regularized sparse decomposition method and MC penalty function sparse decomposition algorithm are used to perform sparse decomposition, respectively. The results are shown in Figure 13.  
It can be seen from the time domain diagram processed by the two methods in Figure 13 that the sparsity decomposition of MC penalty function is better than the L1 regularization in retaining the high amplitude components, which are 1.8 ms\({}^{-2}\) and 0.96 ms\({}^{-2}\), respectively. The noise control range is also smaller than that of L1 regularization. Next, the amplitude and envelope spectrum of the above two results are analyzed, and the results are shown in Figure 14.  
Figure 11: Donghua test acquisition analyzer. Figure 12: Time domain diagram of test bearing vibration signal. Figure 13: Time domain diagram of sparse decomposition results. It can be clearly seen from Figure 14a that the vibration and impact load signals are retained, and the fault characteristic frequencies are all around 320 Hz, but at the same time, it was found that there are great differences between the two in terms of amplitude and noise. The high amplitude is reserved to 1.1 ms\({}^{-2}\)  based on the MC penalty function algorithm, while the amplitude is reserved to 0.27 ms\({}^{-2}\) based on the L1-regularization algorithm. There is a significant difference between the two algorithms, in terms of noise; the amplitude fluctuation between 400 and 900 Hz appears in the L1-regularized amplitude diagram, which is not in line with the test expectations for single vibration and impact load in this experiment. However, this phenomenon does not appear in the amplitude diagram of MC penalty function, and the noise control effect is significantly improved. Figure 14b of the envelope spectrum analysis reflects the difference between the two: the amplitude of L1 regularization is 0.007 ms\({}^{-2}\), and the amplitude fluctuation of noise is obvious around 670 Hz. The amplitude of MC penalty function is 4.1 ms\({}^{-2}\), and there is no obvious amplitude fluctuation at other frequencies.