Vibration Source Contribution Evaluation of a Thin Shell Structure Based on ICA

As submarines navigate underwater, it is difficult for satellites, anti-submarine aircrafts and warships to detect them. However, the radiated noises produced by the equipment of submarines pose a serious threat to the concealment, and directly influence the operational performance even survivability. Therefore, the reduction and control of vibration and noise is an important work to improve the survival capability and operational performance. As the influences of structural transmission, some components of vibration sources will be changed as they go through mechanical structures, which means the measured vibration signals on the shell are the mixed signals of all the sources. Therefore, it is a challenge but important work to effectively identify the sources and evaluate the source contributions. The radiated noise of submarines is mainly produced by the diesel engines, and it transmits from the engine bases to the shell according to the hull. One basic method to reduce the radiated noise is based on improving the hull structure to reduce vibration transmission. In the past decades, much research work is devoted to vibration transmission characteristic analysis of different structures, such as beams (Lee et al., 2007), girders (Senjanovic et al., 2009), rafts (Niu et al., 2005), casings (Otrin et al., 2005), panels (Lee et al., 2009), plates (Xie et al., 2007; Bonfiglio et al., 2007) and shells (Efimtsov & Lazarev, 2009). Some studies dedicated to the responses of whole ship hull, such as free vibration analysis of thin shell (Lee, 2006), insertion loss prediction of floating floors (Cha & Chun, 2008) and structural responses of ship hull (Iijima et al., 2008). Another method based on the active control over vibration and noise is also deeply studied in recent years, such as controlling high frequencies of vibration signals by structure modification (Tian et al., 2009), active vibration control using delayed position feedback method (Jnifene, 2007), high frequency spatial vibration control for complex structures (Barrault et al., 2008), and active vibration isolation of floating raft system (Niu, et al., 2005). However, all these techniques are static analysis method, and the radiated noise can be reduced limitedly as the strength requirements of hulls and indispensability of diesel engines. Aiming at the active control over vibration and noise, a novel approach based on independent component analysis (ICA) is proposed in this paper, which identifies the vibration sources


Introduction
As submarines navigate underwater, it is difficult for satellites, anti-submarine aircrafts and warships to detect them. However, the radiated noises produced by the equipment of submarines pose a serious threat to the concealment, and directly influence the operational performance even survivability. Therefore, the reduction and control of vibration and noise is an important work to improve the survival capability and operational performance. As the influences of structural transmission, some components of vibration sources will be changed as they go through mechanical structures, which means the measured vibration signals on the shell are the mixed signals of all the sources. Therefore, it is a challenge but important work to effectively identify the sources and evaluate the source contributions. The radiated noise of submarines is mainly produced by the diesel engines, and it transmits from the engine bases to the shell according to the hull. One basic method to reduce the radiated noise is based on improving the hull structure to reduce vibration transmission. In the past decades, much research work is devoted to vibration transmission characteristic analysis of different structures, such as beams (Lee et al., 2007), girders (Senjanovic et al., 2009), rafts (Niu et al., 2005), casings (Otrin et al., 2005), panels (Lee et al., 2009), plates (Xie et al., 2007;Bonfiglio et al., 2007) and shells (Efimtsov & Lazarev, 2009). Some studies dedicated to the responses of whole ship hull, such as free vibration analysis of thin shell (Lee, 2006), insertion loss prediction of floating floors (Cha & Chun, 2008) and structural responses of ship hull (Iijima et al., 2008). Another method based on the active control over vibration and noise is also deeply studied in recent years, such as controlling high frequencies of vibration signals by structure modification (Tian et al., 2009), active vibration control using delayed position feedback method (Jnifene, 2007), high frequency spatial vibration control for complex structures (Barrault et al., 2008), and active vibration isolation of floating raft system (Niu, et al., 2005). However, all these techniques are static analysis method, and the radiated noise can be reduced limitedly as the strength requirements of hulls and indispensability of diesel engines. Aiming at the active control over vibration and noise, a novel approach based on independent component analysis (ICA) is proposed in this paper, which identifies the vibration sources And thus each mixed signals can be described as: Where i x is the ith mixed signal observed in the position i, j s is the jth source signal, ij a is the mixing coefficient, and i n is the noise of ith mixed signal. The mixed signal can be also described as: Where G is a global matrix.

Assumptions of independent component analysis
As a result of unknown source signals and mixing mode, information that can be used by ICA is only the mixed signals observed by the sensors, which will give multiple solutions of the problem. Therefore, some assumptions are necessary to get a definite solution: is zero mean and real random variable, and all the source signals are mutually statistic independent at any time. 2. The number of sources are less than the number of mixed signals( nm  ). 3. Only one source is allowed to have a guassian distribution.

Separating criterion based on negentropy
Assume that one random variable is composed of some independent variables according to superposition. According to the central limit theorem, the superposed variable will tend to a Gaussian distribution if the independent variables have the same limited means and variances. Therefore, Gaussian feature are always used to determine whether the separating components are independent or not. Assume that a source signal s has a probability density () p s , and its negentropy is defined as follow equation: Where gauss s is a signal of Gaussian distribution and has the same variance with s . () H  is the information entropy of a signal.
As the probability density distribution function is unknown, the independence of each separated signal is commonly measured by an approximate equation Where () E  is a mean function, and () G  is a nonlinear function. The commonly used () G  are logarithmic function and exponential function Where 12 a .
The independent component analysis method based on negentropy has obvious advantages, such as simple concept, fast computing speed and good stability. It can be also used to determine whether the separating process should be stopped or not according to nongaussian feature of the separated components.

Framework of fast fixed-point algorithm
Fast fixed-point algorithm based on the negentropy criterion is a typical independent component analysis algorithm. Its calculation process includes two steps: signal preprocessing and extracting components one by one. The projection pursuit method is applied to extracting the independent components, and the framework of fast fixed-point algorithm can be described as eight steps: 1. Mean and whiten the mixed signals X , remove the redundant information, and thus obtain the preprocessed signals Z . 2. Set the number p of independent component esxtracted for each time.
where g is the differential of G 5. Orthogonal calculation 7. Determine whether p u is convergent. If not, return to step (4). 8. Repeat step (3), and else stop calculation.

Quantitatively calculate source contributions
To quantitatively calculate source contributions, a novel method based on an enhanced ICA algorithm and priori information is proposed in this paper. The enhanced ICA algorithm extracts independent components by running a single ICA algorithm for many times, and selects the optimal components as the optimal independent components according to clustering analysis (Hamberg & Hyvarinen, 2003;Hyvarinen et al., 2004). The proposed method separates mixed signals into independent components by the enhanced ICA algorithm, and calculates source contributions according to the mixing matrix. Priori information was employed to further enhance the separating performance, because priori information is proved to be able to weaken the uncertainty of problems (Ma, et al., 2006).

Framework of fast fixed-point algorithm
For linear superposition model, fast fixed-point algorithm has good separating performance. However, most algorithms based on ICA are stochastic, and their results may be different in repeatedly calculations, so the outputs of a single run can not be trusted (Hamberg & Hyvarinen, 2003;Hyvarinen et al., 2004). One reason for this problem is that engineering data is not strictly complied with the blind source separation model, and the actual calculation algorithm may converge to local minima rather than the overall minimum value. The other reason is the statistical error of estimated components which is caused by the finite sample size. To enhance the stability of ICA, an enhanced ICA algorithm is constructed based on clustering evaluation. At first the independent components (ICs) are extracted according to a single ICA algorithm for many times, and then the reliability of each IC is estimated by clustering analysis and the optimal Ics are selected as the best solutions. Basic framework of the enhanced ICA algorithm is as follows: 1. Parameters of a single algorithm are set, such as orthogonalization approach and the nonlinearity function. 2. The single ICA algorithm is executed for certain times with different initial values to produce more results in the different condition. 3. All the ICs are clustered according to their mutual similarities, and then these ICs are divided into several clusters. According to the average-linkage clustering criterion, the clustering center which has the largest relevances with other components is selected as the optimal results. Assume that the mutual correlation coefficient between estimated components i y and j y is ij  , and the cluster validity index q I can be defined as follows: where C is all the IC set, k C is the IC set in the kth cluster, k C  is its complementary set, and k C is the size of the kth cluster. By the clustering evaluation method, the ICs produced by the enhanced ICA algorithm are clustered into different clusters, and the separating performance can be indicated by the tightness of ICs. The more tightness they are, the better the separating performance will be. After clustering, the cluster center is selected as the optimal ICs.

Priori information
In the applications, some mixed signals do not follow the linear superposition model. Therefore, priori information should be used to weaken the uncertain problems. For sources 12 [, ] T Ss s  , it can be inferred that the joint distribution The joint distribution of S is shown in Fig. 1(a). Mixed signals are obtained according to S and mixing matrix A, which can be shown as The joint distribution of X is shown in Fig. 1 Where 1 h and 2 h is an analytic function. If the parallelogram border on the plane ( 1 e , 2 e ) is transformed into the plane ( 1 p , 2 p ), and these parallelogram borders are not parallel to the related coordinate axis, there have real constants 1 a 、 2 a 、 1 b and 2 b The theorem 1 provides a transmission from nonlinear mixed signals to linear mixed signals based on priori information, and this method does not need the assumption of independence. The limited border of source signals provides additional information, which can optimize the nonlinear parts of the mixed signals. Therefore, the separating performance of the proposed method is further enhanced.

Quantitative evaluation of source contributions
Each components of X are composed by all the components of Y according to the mixing matrixÂ . Each combination coefficients of the mixing matrix Â reveal the contribution of the related independent components. Therefore, the quantitative calculation of source contribution problem in essential is how to obtain the mixing matrix Â effectively.

Simulation experiment analysis 4.1 Validation of separating performance
When the rotational parts, such as gears and rolling bearings occur faults (teeth broken, peeling or rubbing), the vibration signals measured on the shell will be frequency and amplitude modulated (He et al., 2001), and thus the source signals(teeth broken, peeling or rubbing) can not be well revealed just by the measured signals. As the composite faults occur, the vibration signal will be coupled together, and thus it is difficult to reveal the conditions of each parts. Therefore, some vibration signals of typical fault features are selected to test the separating performance of different methods. Four source signals are selected: 1 () st is a white noise signal, 2 () st is a frequency modulated signal, 3 () st is an amplitude modulated signal, and 4 () st is a both amplitude and frequency modulated signal. The data length of t is 1000 and the step is 1. The generating functions of sources and mixing matrix are listed as follows.   Three different BSS methods are applied to separate the mixed signals, including principal component analysis based on second-order cumulant, fast fixed-point algorithm based on negentropy, and the enhanced ICA algorithm based on clustering optimization. The waveforms of the separated components are shown in Fig. 4 -Fig. 6 respectively.
Where () sk is the source signal, () y k is the independent component, and k is the data sequence.
In the numeric studies, the correlation matrices sy  between the separated components and the sources are listed as follows: Comparing the three sy  by different methods, the correlation coefficients between each components and related source signals of fast fixed-point algorithm and the enhanced ICA algorithm are more than 0.98, which means waveform information implied in the mixed signals is well extracted. Therefore, these two algorithms have good separating performance. However, the correlation coefficient between the second component and 2 s is only 0.58, but the other correlation coefficients are even up to 0.61, which means the source information is not well separated. Therefore, principal component analysis fails to separate the sources effectively in this case.

Stability validation of the enhanced ICA algorithm
As a statistical signal processing method, fast fixed-point algorithm may produce different results in repeated executions, and thus the separated components will be unreliable. To illustrate this problem clearly, another case is given in this section. The source signals are: 5 s is a white noise signal, 6 s is a sinusoidal signal, 7 s is a triangle wave signal, and 8 s is a square wave signal. The generating function of sources and the mixing matrix A are as follows: The waveforms of source signals and the mixed signals are shown in Fig. 7 and Fig. 8 respectively. The fast fixed-point algorithm is repeatedly executed for more than 30 times, and more than 60% of the results show that the ICs are accurate. However, the other 40% of the results are different from the source signals obviously. One of the inaccurate results is shown in Fig. 9, which indicates that the source information is not well separated. To enhance the stability of traditional ICA, the enhanced ICA algorithm is constructed. The enhanced ICA algorithm is repeatedly executed for 20 times, and all the result is stable. The waveforms of the separated components by the enhanced ICA algorithm are shown in Fig.  10, which indicates that the source information is well extracted. It can be concluded that fast fixed-point algorithm may produce inaccurate components because it is a statistic signal processing method. To enhance the stability, an effective way is to execute the single ICA algorithm for certain times, and evaluate the separated components by clustering analysis. Therefore, the enhanced algorithm based on clustering optimization is of high precision and stability.

Quantitative evaluation of source contributions
From the basic theory of blind source separation model, it can be explained that some complicate signals are composed of several independent components, and the combination relationship of the independent components reveals their contributions. Therefore, source contributions can be obtained according to the mixing matrix. We take the case in section 4.1 to show the numeric studies of source contribution evaluation. After the separation, a mixing matrix A of the independent components for the mixed signals is obtained. As the source signals and the independent components have different scales. We add a factor to let the independent components have the same energy with the related sources, and thus get a modified mixing matrix Â .
Compared the mixing matrix A with the mixing matrix Â calculated by the proposed method, the real proportion of each source to 1 x are 0.63, 0.77, 0.54 and 0.65, while the contributions calculated by the proposed method are 0.65, 0.75, 0.55 and 0.62, which means the relative error between the calculated values and the real values is less than 4.6%. The calculated results of source contributions to 2 x , 3 x and 4 x are also close to the real values, and the relative errors are all less than 4.8%. Therefore, the comparative results show that it is effective to quantitatively evaluate the source contributions just from the mixed signals.

Introductions of the thin shell structure
In this application, a thin shell structure is set up to validate the effectiveness of the proposed method. Three motors are installed in the thin shell structure to simulate the vibration sources, and blocks are added in each output shaft of the motors to generate eccentric vibration. Magnetic brakes are installed in the output shaft of these motors for a variable load. Rubber springs are applied to support the entire structure, and they can also eliminate the environmental influences. The structural diagram of the thin shell structure is shown in Fig.11, and the photo of the entire system is shown in Fig.12. www.intechopen.com Vibration Source Contribution Evaluation of a Thin Shell Structure Based on ICA 115 Fig. 11. The structural diagram of the thin shell structure Fig. 12. The photo of the entire system Acceleration sensors are used to sample the vibration signals. Three sensors are installed in the bases of each motor to measure the sources, and nine sensors are installed on the left inside shell to measure the mixed signals. In the testing, the sampling frequency is 16384 Hz, the data length is 16384, and the unit of measured signals is g (1.0g=9.8m/s 2 ). The rotational speeds of motor 1, motor 2, and motor 3 are respectively 1350 rpm, 1470 rpm, and 1230 rpm.

Blind source separation of the mixed signals
To calculate the source contributions quantitatively, a direct method is to separate the mixed signals and calculate the contributions by the mixing matrix. Three vibration signals on the inside shell are selected as the mixed signals, and their waveforms are shown in Fig. 13. The enhanced ICA algorithm is applied to separate the mixed signals, and three independent components extracted are shown in Fig. 14. Three source signals are measured on the motor bases, and their waveforms are shown in Fig. 15. From Fig. 15, it can be clearly seen that the waveforms of the independent components are obviously different from the source signals.  shows that the correlation coefficients between independent components and source signals are relatively small, and the maximum coefficient is only 0.4269, which indicates that the source information is not well separated. Therefore, the separated components extracted cannot be regarded as the sources for calculating source contributions.

Source contribution evaluation based on priori information
In this application, source signals can be obtained from the motor bases. Source contribution based on priori information is proposed and it can be described as follows: four signals are selected as the mixed signals, one of them is measured inside shell, and the other three signals are measured from the motor bases. The waveforms of the mixed signals are shown in Fig.16. The independent components are extracted by the enhanced ICA algorithm. The optimal independent components are selected by clustering evaluation, and waveforms of the optimal independent components are shown in Fig.17.
Obviously three independent components have the similar waveforms with three source signals.
Correlation matrix sy  between independent components and source signals are: The correlation matrix sy  shows that the correlation coefficient between IC 1 and source signal 2 is 0.8397, IC 2 and source signal 1 is 0.8903, and IC 3 and source signal 3 is up to 0.9539, which shows that three source signals are well extracted from the mixed signals. Therefore, with the priori information, the source information can be well separated in the real applications, which means the priori information can further improve the separating performance. After the separation, the mixing matrix is calculated by the enhanced ICA algorithm. The mixing matrix Â is as follow:  The mixing matrix Â shows that the mixed signals are composed of three ICs, and their proportional contributions are 0.1299, 0.0097 and 0.0440. The three ICs represent the source signals from three motors, and the percentage contributions of motor 1, motor 2 and motor 3 are 23.97%, 70.75% and 5.280% respectively. The percentage contribution indicates that IC 2 (motor 3) has the largest vibration contribution, which means that mixed signal 1 mainly comes from the motor 3. Therefore, motor 3 should be controlled or the vibration reduction equipments should be adapted in the transmission path to reduce vibration. In the measuring point on the shell, the real vibration contributions are measured in the condition that only one motor is running at the given speed (motor1 -1350 rpm, motor2 -1470 rpm and motor3 -1230 rpm respectively), and thus the contribution of the related motor to the same location on the shell can be measured one by one. The energy contributions of motor 1, motor 2 and motor 3 are 5.41, 1.82 and 24.98 respectively. The real contributions also show that motor 3 has the largest contribution. This paper also gives the source contributions calculated by the enhanced ICA without priori information. The percentage contributions are listed in Tab .1. Tab. 1 shows that the real contribution of motor 3 is up to 77.55%, which means that motor 3 gives a large vibration contribution to the shell. The contribution of motor 3 calculated by the enhanced ICA method without priori information is 40.88%, which has a relative error of 36.67%. The relative error of the proposed method is 6.8%, and the other two motor contributions also show that the proposed method has high accuracy. Therefore, it can be concluded that the accuracy of the proposed method has been greatly enhanced by priori information.
The key process of quantitative calculation of source contributions is that mixed signals are well separated. Principal component analysis in the simulation does not give a satisfied result, and fast fixed point algorithm has good separating performance for linear superposition but it is not stable for repeatedly calculation. The enhanced ICA algorithm executes the single ICA algorithm for several times, and selects the optimal ICs by clustering analysis. Therefore, separating performance and reliability of the enhanced ICA algorithm are enhanced. 2. Most of the mixed signals in the engineering are nonlinear mixed signals, and thus it is a challenge work for most ICA algorithms to separate the sources accurately. To overcome this challege, priori information is employed to weaken the uncertainty problem of nonlinear mixed signals. By the priori information, the separated signals can be better separated, and thus separating performance can be further enhanced, which can improve the accuracy of calculated contributions. Therefore, it provides another way to enhance the separating performance of ICA according to making full use of priori information.

Conclusions
As the influences of transmission paths, the vibration signals on the shell are influenced significantly, so it is a very challenging task to identify the sources and quantitatively evaluate the source contributions, which is important for vibration reduction and control.
In this paper, a novel method to quantitatively evaluate the source contributions based on the enhanced ICA algorithm and priori information is proposed. The enhanced ICA method provides a powerful way to effectively and reliably separate sources from the mixed signals.
In the simulations, the comparative results show that principal component analysis cannot deal with some typical mechanical signals. Fast fixed point algorithm has strong separating performance for linear superstition signals, but the reliability is not very good because it is a statistical signal processing method. The enhanced ICA algorithm evaluates the separating components by clustering analysis and selects the optimal ICs as the best results. Therefore, the separating performance and reliability are significantly enhanced. The error of contributions calculated by the enhanced ICA algorithm is less than 4.8%, which indicates that the proposed method is of high accuracy. The proposed method is applied to evaluate the source contributions of a thin shell structure. The priori information is discussed in two different separation process. The case that does not use priori information has an obvious error of 36.67%. Compared with the real values by measurement, the comparative results show that the proposed method obtains a high accuracy with priori information, and the relative error is less than 7.18%. Therefore, the proposed method can effectively evaluate source contributions, which can provide reliable referances for vibration reduction and monitoring.