The secondary flow in a blade row can be defined as any flow, which is not in the direction of the primary or streamwise flow. The classical theories of secondary flow, as developed by Squire & Winter (1951), Hawthorne (1955) and Smith (1955) described the mechanism of the streamwise vorticity formation at blade row exit. Until now, several physical models have been developed to describe the secondary flow vortices in turbine cascade (e.g. Klein, 1966; Langston et al., 1977; Langston, 1980; Yamamoto, 1987; Wang et al., 1997), which help people have more comprehensive and vivid understanding of the secondary flow vortices structure. Review papers by Sieverding (1985), Wang et al. (1997) and Langston (2006) provide comprehensive summaries of the research on secondary flow structure and outline the most significant developments. The physical model of Wang et al. (1997) is depicted in Fig. 9. It can be seen that the passage vortex has the most significant effect on the endwall losses in turbine cascade. The classical secondary flow theories show that the flow overturning in the endwall

A significant amount of research activity has recently been directed towards understanding the effect of secondary vortex unsteadiness on turbine performance. One of the major sources of unsteadiness was found to be the interaction between the streamwise vortices with the downstream blade row (e.g. Binder, 1985; Binder et al., 1986; Sharma et al., 1988). These streamwise vortices are shown to have a major influence on the secondary flow and viscous flow behavior of the downstream blade row. It has been shown, just like the wake, that the transport of streamwise vorticity can have an impact on the mixing losses in the next blade rows (Valkov & Tan, 1999b; Pullan, 2004). At the leading edge of the blade, the streamwise vortex is compressed so that, also by Kelvin's circulation theorem, the velocity deficit must decrease, and with it the rate of entropy production due to mixing. The converse occurs when the streamwise vortex is then chopped, bowed and stretched as it

In some research for multistage turbines, the blade row exit flow underturning near the hub and overturning towards the mid-span, contrary to the classical model of overturning in the endwall regions and underturning in the mid-span regions, indicating that the secondary flow is strongly influenced by the incoming secondary vortices of upstream blade row (e.g. Sharma et al., 1988; Hobson & Johnson, 1990; Chaluvadi et al., 2004). This unsteady effect raised substantial interest, as it had not been observed in the tests carried out by the other researchers (Boletis & Sieverdin, 1991; Hodson et al., 1993). It shows that the unsteady interactions on the steady performance must be related to the strength of the incoming

greater than the reduction when it is moving backward.

secondary flows. This is especially true for high-pressure turbines.

regions and underturning in the midspan regions at the exit of the blade row.

**4.3 Secondary flow interaction**

passes through the blade row.

secondary flow of the turbines.

Fig. 9. Secondary flow model by Wang et al. (1997)

## **4.4 Unshrouded tip leakage flow interaction**

The tip leakage flow is important in most turbomachinery, where a tip clearance with a height of about 1-2% blade span exists between the stationary endwall and the rotating blades. An unshrouded tip design is widely employed for a low stress and/or a better cooling in modern high-pressure turbines. Pictorial representation of the tip leakage flow in unshrouded blades is given in Fig. 10. The leakage flow over unshrouded blades occurs as a result of the pressure difference between the pressure and suction surfaces and is dominated by the vortex shed near the blade tip.

The tip leakage flow has significant effects on turbomachinery in loss production, aerodynamic efficiency, turbulence generation, heat protection, vibration and noise. As a consequence of the viscous effects, significant losses are generated by the tip leakage flow in regions inside and outside the tip gap. And the entropy creation is primarily due to the mixing processes that take place between the leakage flow and the mainstream flow. Denton (1993) gave a simple prediction model for the tip leakage loss of unshrouded blades.

So far, there are many researches about the leakage flow unsteady interactions in compressor. For example, Sirakov & Tan (2003) investigated the effect of upstream unsteady wakes on compressor rotor tip leakage flow. It was found that strong interaction between upstream wake and rotor tip leakage vortex could lead to a performance benefit in the rotor tip region during the whole operability range of interest. The experimental result of Mailach et al. (2008) revealed a strong periodical interaction of the incoming stator wakes and the compressor rotor blade tip clearance vortices. As a result of the wake influence, the tip clearance vortices are separated into different segments with higher and lower velocities and flow turning or subsequent counter-rotating vortex pairs. The rotor performance in the tip region periodically varies in time.

Fig. 10. Flow over an unshrouded tip gap (Zhou & Hodson, 2009, after Denton, 1993)

Compared with in compressor, very little published literature is available on the unsteady interactions between leakage flows and adjacent blade rows in turbine. Behr et al. (2006) indicated that the pressure field of the second stator has an influence on the development of the tip leakage vortex of the rotor. The vortex shows variation in size and relative position when it stretches around the stator leading edge. The present author (Qi, 2010) performed an experimental and numerical investigation of unsteady flow in the tip region of turbine with the effect of upstream wakes. The study objects included a linear turbine cascade and a turbine rotor, respectively. It shows that the presence of the upstream periodic wakes can reduce the strength and the loss of the tip leakage vortex, which is favorable to the turbine performance. In addition, due to the unsteady effect of the wakes, counter-rotating vortex pairs appear within the tip leakage vortices. These vortex pairs move downstream along the mean tip leakage vortex trajectory, which cause a significant pressure fluctuation and unsteady force on the rotor suction side.

#### **4.5 Coolant injection and rim seal flow interactions**

High-pressure gas turbines present considerable challenges to the designer because of the high aeromechanical loads at elevated temperatures. In order to cool the rotor disk and to avoid hot gas injection into the wheel space interface, cold flow is usually ejected from the cavity between the stator rim and the rotor disk. Fig. 11 illustrates a typical high-pressure gas turbine stage showing the rim seal and the wheel-space between the stator and the rotating turbine disc. Cooling air is supplied through the inner seal, and air leaves the wheel-space through the rim seal.

It was mentioned above that for high-pressure turbines, the secondary flow losses are a major source of the total aerodynamic losses in a blade row. The cooling air is injected into the mainstream and thereafter interacts with the secondary flow; consequently the turbine aerodynamic performance may be changed. So it is important to consider the coolant injection effect to the endwall region flow in unsteady environment.

tip region during the whole operability range of interest. The experimental result of Mailach et al. (2008) revealed a strong periodical interaction of the incoming stator wakes and the compressor rotor blade tip clearance vortices. As a result of the wake influence, the tip clearance vortices are separated into different segments with higher and lower velocities and flow turning or subsequent counter-rotating vortex pairs. The rotor performance in the

Fig. 10. Flow over an unshrouded tip gap (Zhou & Hodson, 2009, after Denton, 1993)

Compared with in compressor, very little published literature is available on the unsteady interactions between leakage flows and adjacent blade rows in turbine. Behr et al. (2006) indicated that the pressure field of the second stator has an influence on the development of the tip leakage vortex of the rotor. The vortex shows variation in size and relative position when it stretches around the stator leading edge. The present author (Qi, 2010) performed an experimental and numerical investigation of unsteady flow in the tip region of turbine with the effect of upstream wakes. The study objects included a linear turbine cascade and a turbine rotor, respectively. It shows that the presence of the upstream periodic wakes can reduce the strength and the loss of the tip leakage vortex, which is favorable to the turbine performance. In addition, due to the unsteady effect of the wakes, counter-rotating vortex pairs appear within the tip leakage vortices. These vortex pairs move downstream along the mean tip leakage vortex trajectory, which cause a significant pressure fluctuation and

High-pressure gas turbines present considerable challenges to the designer because of the high aeromechanical loads at elevated temperatures. In order to cool the rotor disk and to avoid hot gas injection into the wheel space interface, cold flow is usually ejected from the cavity between the stator rim and the rotor disk. Fig. 11 illustrates a typical high-pressure gas turbine stage showing the rim seal and the wheel-space between the stator and the rotating turbine disc. Cooling air is supplied through the inner seal, and air leaves the

It was mentioned above that for high-pressure turbines, the secondary flow losses are a major source of the total aerodynamic losses in a blade row. The cooling air is injected into the mainstream and thereafter interacts with the secondary flow; consequently the turbine aerodynamic performance may be changed. So it is important to consider the coolant

tip region periodically varies in time.

unsteady force on the rotor suction side.

wheel-space through the rim seal.

**4.5 Coolant injection and rim seal flow interactions**

injection effect to the endwall region flow in unsteady environment.

Fig. 11. Typical high-pressure turbine stage showing rim seal and wheel-space (Owen, 2009)

For subsonic turbines, the effects of rim seal purge ejection have been extensively studied in literature. It was found that even a small quantity (1 percent) of cooling air can have significant effects on the performance character and exit conditions of the high-pressure stage (McLean et al., 2001). Leakage flows from upstream cavities on engines generally emerge with lower momentum and less swirl than the mainstream flow, thus reducing the incidence at the inlet of the downstream blade row. The leakage flows may strengthen the endwall secondary flow on the downstream blade row (Anker & Mayer, 2002; Hunter & Manwaring, 2000; Paniagua et al., 2004; Pau et al., 2008). In addition, the ejection swirl angle can have a considerable effect to the efficiency of turbine stage, but the gain was restricted to the rotor due to a reduction in viscous dissipation and secondary losses (Ong et al., 2006).

In transonic turbines, trailing edge loss is a main source of losses. Denton (1993) indicates that trailing edge loss contributes even up to about 50 percent of the total two-dimensional loss in supersonic flow. The intensity of the trailing edge loss is related to the vane outlet Mach number, which is greatly affected by the rim seal. Hence, a complete study of the stator rim purge flow on transonic turbines requires not only the analysis on the downstream blade row (mixing losses and secondary flow), but the effect on the upstream transonic nozzle guide vane. Compared with in subsonic turbines, there is very little published research on transonic vane losses with rim seal. The effects of purge flow on a transonic nozzle guide vane have been studied numerically and experimentally by Pau and Paniagua (2009). In the research, a numerically predicted loss breakdown is presented, focusing on the relative importance of the trailing edge losses, boundary layer losses, shock losses and mixing losses, as a function of the purge rate ejected. Contrary to the experience in subsonic turbines, results in a transonic model demonstrate that ejecting purge flow improves the vane efficiency due to the shock structures modification downstream of the stator.

As stated previously, a small quantity of cooling air can have significant effects on the performance of the high-pressure stage. It means that the coolant injection effect should not be negligible. However, there is little research about the coolant injection effect to the performance of high-pressure gas turbines in real unsteady environment. It needs further studies in the future.

## **4.6 Hot streaks**

Another significant unsteady phenomenon in high-pressure turbines is the interaction of hot streaks (also called "hot spots"), which refer to the radial and circumferential temperature gradients at the gas turbine combustor exit. These pronounced temperature nonuniformities in the combustor exit flow field are caused by circumferentially discrete fuel and dilution air injection within the combustor. The hot streaks can cause significant unsteadiness due to the relative motion of the blade rows. They interact with the different turbine parts and can locally cause increased blade heat transfer leading to reduced blade life and significant risks (see Fig. 12).

Fig. 12. Hot streaks migration (LaGraff et al., 2006)

In practical engine, combustor hot streaks can typically have stagnation temperatures twice of the free stream stagnation temperature. The hot streaks can lead to high heat loads and potentially catastrophic failure of the blades. It has been shown both experimentally and numerically that temperature gradients, in absence of total pressure non-uniformities, do not alter the flow (pressure field) within the first-stage turbine stator but do have a significant impact on the secondary flow and wall temperature of the first stage rotor (Butler et al., 1989).

An important concept is the "clocking effect of hot streaks". Adjusting the positions of hot streaks with respect to inlet guide vanes can be used to help control blade temperatures in gas turbines. When the hot streaks impinge on the leading edge of the blade, the impact to the blade is the maximum; on the contrary the impact decrease. Although a thermal barrier coating or cooling can reduce the local heat load to a certain degree, they will obviously increase the cost. For this reason, the method of using hot streak clocking effect is proposed. Takahashi et al. (1996) and Gundy-Burlet & Dorney (1997) showed that "clocking" the hot streaks so that it is positioned at the leading edge of the vane results in a

be negligible. However, there is little research about the coolant injection effect to the performance of high-pressure gas turbines in real unsteady environment. It needs

Another significant unsteady phenomenon in high-pressure turbines is the interaction of hot streaks (also called "hot spots"), which refer to the radial and circumferential temperature gradients at the gas turbine combustor exit. These pronounced temperature nonuniformities in the combustor exit flow field are caused by circumferentially discrete fuel and dilution air injection within the combustor. The hot streaks can cause significant unsteadiness due to the relative motion of the blade rows. They interact with the different turbine parts and can locally cause increased blade heat transfer leading to reduced blade

In practical engine, combustor hot streaks can typically have stagnation temperatures twice of the free stream stagnation temperature. The hot streaks can lead to high heat loads and potentially catastrophic failure of the blades. It has been shown both experimentally and numerically that temperature gradients, in absence of total pressure non-uniformities, do not alter the flow (pressure field) within the first-stage turbine stator but do have a significant impact on the secondary flow and wall temperature of the first stage rotor (Butler

An important concept is the "clocking effect of hot streaks". Adjusting the positions of hot streaks with respect to inlet guide vanes can be used to help control blade temperatures in gas turbines. When the hot streaks impinge on the leading edge of the blade, the impact to the blade is the maximum; on the contrary the impact decrease. Although a thermal barrier coating or cooling can reduce the local heat load to a certain degree, they will obviously increase the cost. For this reason, the method of using hot streak clocking effect is proposed. Takahashi et al. (1996) and Gundy-Burlet & Dorney (1997) showed that "clocking" the hot streaks so that it is positioned at the leading edge of the vane results in a

further studies in the future.

life and significant risks (see Fig. 12).

Fig. 12. Hot streaks migration (LaGraff et al., 2006)

**4.6 Hot streaks**

et al., 1989).

diminishing of the effect of the hot streak on the downstream rotor. This was attributed to the deceleration and increased mixing of the hot streak as it interacted with wake of the vane. Shang & Epstein (1997) found the rotor-stator interactions between the rotor and stator can also generate significant non-uniformity of the time-averaged relative total temperature at the rotor inlet. And an optimum NGV-rotor blade count may exist that minimizes the influence of hot streaks on rotor blade life.

Another important concept is the "Kerrebrock-Mikolajczak effect" (Kerrebrock & Mikolajczak, 1970). It was a name associated with hot streak surface heating. These researchers were the first to describe why unsteady flow effects associated with compressors lead to increased pressure side heating. This effect is even more pronounced in turbines due to larger circumferential temperature variations. The description of the effect is based on the fact that the hot streak is moving at speeds significantly different from the surrounding fluid.

## **5. Numerical methods of unsteady flows in turbomachinery**

## **5.1 Intruduction**

Computational fluid dynamics (CFD) is now more and more used to assess unsteady effects in turbomachinery flow. The application has also been extended from original academic use to industrial design application. Most of the flows encountered in turbomachinery are turbulent. Based on the Navier-Stokes equations, the numerical methods to predicting turbulent flows can be traditionally divided into three categories (Pope, 2000): Reynoldsaveraged Navier-Stokes (RANS) equations, large-eddy simulation (LES) and direct numerical simulation (DNS).

### **5.2 Reynolds-averaged Navier-Stokes (RANS) method**

The solution of the (unsteady) Reynolds-averaged Navier-Stokes (RANS) equations is a tool that is most commonly applied to the solution of turbulent flow problems, especially in engineering applications. The Reynolds-averaged method was proposed by Osborne Reynolds in 1895. In RANS method, the equations are obtained by time- (in steady flow) or ensemble-averaging (in unsteady flow) the Navier-Stokes equations to yield a set of transport equations for the averaged momentum. In the averaging processes of the equations, the presence of the Reynolds stresses causes the equations not closed. To close the equations we must introduce a turbulence model. The turbulence models have been the object of much study over the last 30 years, but no model has emerged that gives accurate results in all flows without adjustments of the model constants. This may be due to the fact that the large, energy-carrying eddies are much affected by the boundary conditions, and universal models that account for their dynamics may be impossible to develop.

Turbulence model is a key issue in CFD simulations of turbomachines. According to the decision manner of the Reynolds stresses, turbulence model can be divided into two main categories (Wilcox, 1994; Chen & Jaw, 1998): eddy-viscosity model based on the Boussinesq eddy viscosity assumption and Reynolds stress model (RSM). By the number of turbulence model equations, the Boussinesq eddy-viscosity model in numerical simulation of turbomachinery can be divided again into: algebraic model (or zero-equation model) like the mixing length model or the Baldwin-Lomax model, one-equation model like Spalart-Allmaras model and two-equation model like various k−ε model and k−ω model. The RSM includes: algebraic stress model and Reynolds stress transport equation model. However, because of the complexity of turbulence model, so far there is not a universal turbulence model which can accurately simulate all types of complex flow in turbomachinery. So the formulation and proper application of the turbulence model is a key factor in the accurate prediction of the turbomachinery characteristics.

In addition, an accurate prediction of separation and transition flows is one of the important challenges to develop RANS method. Due to the complexity of separation and transition phenomenon, it is very difficult to make an accurate estimate. Many researchers have made an effort to develop and improve models for transition (Praisner & Clark, 2007; Praisner et al., 2007; Cutrone et al., 2007; Cutrone et al., 2008) in order to solve this problem. Anyhow, in the near future, practical calculations will continue to be carried out mainly by RANS methods with various turbulence models, which is still the most common and effective numerical method to solve engineering problems.

## **5.3 Large eddy simulation (LES)**

Large eddy simulation (LES) is a technique intermediate between the solution of RANS and DNS. It is to simulate explicitly the larger-scale turbulent motions by solving the 3D timedependent Navier-Stokes equations while model the small-scale motion that cannot be resolved on a given grid. LES can be more accurate than the (U)RANS approach and spend a lot less computational time than DNS.

The advantages of LES are as follows: firstly, compared with the large scales motion, the small scales are less affected by the boundary conditions and tend to be more isotropic and homogeneous, so are most likely to subject the universal models (Piomelli & Balaras, 2002); secondly, the large scales motion which is solved directly concludes most of the turbulent kinetic energy and can describe the main characteristics of the flow field better, so compared with RANS method, LES has some advantages (Rodi, 2000).

There are, however, many problems in actual application of LES that is the use of a subgridscale (SGS) model. Research indicates that the quality of the SGS models is of course an important issue in LES, which have a significant impact on the accuracy of the computational result. There are still difficulties in obtaining ideal SGS models. And also, there is the problem of huge calculated amount in LES, so it cannot become the major method in engineering practice at present.

#### **5.4 Direct numerical simulation (DNS)**

The direct numerical simulation (DNS) is a complete time-dependent solution of the Navier-Stokes equations. In DNS, all the scales of motion are resolver accurately, and no modeling is used. It is the most accurate numerical method available at present but is limited by its cost: because all scales of motion must be resolved, the number of grid points in each direction is proportional to the ratio between the largest and the smallest eddy in the flow. Although DNS has huge calculated amount, it also has many advantages: firstly, DNS method is the most accurate method of all, because it directly solves the Navier-Stokes

equations and need not introduce any models and assumptions; secondly, DNS can obtain all the details of the spatial-temporal evolution of the complex flow field, and can provide the most comprehensive numerical databases for the research of flow mechanisms; thirdly, the DNS database can be used to test other numerical simulation results, and can be of value in developing new turbulence models and subgrid-scale (SGS) models. Besides, DNS can be

includes: algebraic stress model and Reynolds stress transport equation model. However, because of the complexity of turbulence model, so far there is not a universal turbulence model which can accurately simulate all types of complex flow in turbomachinery. So the formulation and proper application of the turbulence model is a key factor in the accurate

In addition, an accurate prediction of separation and transition flows is one of the important challenges to develop RANS method. Due to the complexity of separation and transition phenomenon, it is very difficult to make an accurate estimate. Many researchers have made an effort to develop and improve models for transition (Praisner & Clark, 2007; Praisner et al., 2007; Cutrone et al., 2007; Cutrone et al., 2008) in order to solve this problem. Anyhow, in the near future, practical calculations will continue to be carried out mainly by RANS methods with various turbulence models, which is still the most common and effective

Large eddy simulation (LES) is a technique intermediate between the solution of RANS and DNS. It is to simulate explicitly the larger-scale turbulent motions by solving the 3D timedependent Navier-Stokes equations while model the small-scale motion that cannot be resolved on a given grid. LES can be more accurate than the (U)RANS approach and spend

The advantages of LES are as follows: firstly, compared with the large scales motion, the small scales are less affected by the boundary conditions and tend to be more isotropic and homogeneous, so are most likely to subject the universal models (Piomelli & Balaras, 2002); secondly, the large scales motion which is solved directly concludes most of the turbulent kinetic energy and can describe the main characteristics of the flow field better, so

There are, however, many problems in actual application of LES that is the use of a subgridscale (SGS) model. Research indicates that the quality of the SGS models is of course an important issue in LES, which have a significant impact on the accuracy of the computational result. There are still difficulties in obtaining ideal SGS models. And also, there is the problem of huge calculated amount in LES, so it cannot become the major

The direct numerical simulation (DNS) is a complete time-dependent solution of the Navier-Stokes equations. In DNS, all the scales of motion are resolver accurately, and no modeling is used. It is the most accurate numerical method available at present but is limited by its cost: because all scales of motion must be resolved, the number of grid points in each direction is proportional to the ratio between the largest and the smallest eddy in the flow. Although DNS has huge calculated amount, it also has many advantages: firstly, DNS method is the most accurate method of all, because it directly solves the Navier-Stokes equations and need not introduce any models and assumptions; secondly, DNS can obtain all the details of the spatial-temporal evolution of the complex flow field, and can provide the most comprehensive numerical databases for the research of flow mechanisms; thirdly, the DNS database can be used to test other numerical simulation results, and can be of value in developing new turbulence models and subgrid-scale (SGS) models. Besides, DNS can be

compared with RANS method, LES has some advantages (Rodi, 2000).

prediction of the turbomachinery characteristics.

numerical method to solve engineering problems.

**5.3 Large eddy simulation (LES)**

a lot less computational time than DNS.

method in engineering practice at present.

**5.4 Direct numerical simulation (DNS)**

used to carry out some "virtual" numerical experimentation to study the influence of single parameter on the flow field.

In the last ten years, benefiting from the improvement of computer technology and numerical method, the applications of DNS present some new characteristic features. Firstly, to some "simple" flow problems, the computational Reynolds number is becoming higher and higher, such as the simulations of channel flow (Kaneda et al., 2003; Ishihara et al., 2007). Secondly, by DNS methods, important progress was made in mechanism research on some complex flow phenomena. For example, Krishnan and Sandham has performed the direct numerical simulations to the evolution of turbulent spots in supersonic boundary layers. It reveals the basic law of turbulent spots merging, spreading and the turbulent spotseparation bubble interactions (Krishnan & Sandham, 2006a; Krishnan & Sandham, 2006b). Finally, there are a great number of mechanism researches that is about complicated engineering flow at intermediate Reynolds number condition. For example, Wu and Durbin performed the direct numerical simulation of the flow field in low-pressure turbine cascade for the first time. They studied the interactions between upstream periodic wakes and cascade flow (Wu & Durbin, 2001; Kalitzin et al., 2003).

It can be expected that a study on the complicated engineering flow by DNS method will continually go on in-depth. But it is not practical to directly use DNS to study the engineering flow. On the one hand, DNS method need for too high computer performance; On the other hand, even the computer level can meet the requirement, the mass data produced by the DNS will not be needed by engineering researchers. Hence at present and also in the foreseeable future, DNS is not a tool for engineering calculations, but a tool for mechanism researches.

#### **5.5 Conjugate heat transfer (CHT)**

With the development of modern turbine engines, the turbine inlet temperature will be further increased in order to enhance the thermal efficiency. The high temperature of hot gas far exceeds the permissible material temperature of turbine blades. Currently, using a high efficient cooling system is the common and primary way to cool blades and reduce their thermal loads. Therefore, the precondition of designing a high efficient cooling system is the accurate prediction of the temperature field of turbine blades. With the fast development of CFD and computer technology, conjugate heat transfer (CHT) method has become an effective method for the temperature prediction of turbine and other hot components.

For the traditional method of predicting temperature field, the thermal boundary conditions must be specified and it requires tedious and costly iteration that involves sequentially performing numerical predictions of flow/temperature field of the hot gas and temperature field of blades. This method is a "decoupled" method and thus the results are unreliable. Instead, CHT method method is a more efficient and accurate way to predict temperature field. The flow/temperature field of the fluid (hot gas) and temperature field of solid (blades) are solved simultaneously and the temperature and heat flux are exchanged at fluid/solid interface during each iteration step. The CHT method is a "coupled" method and thus the results are more reliable.

The early studies of flow/temperature field coupled problem focused on basic studies, such as the work of Perelman (1961) and Luikov et al. (1971). With the fast development of CFD technology and computer hardware, it is possible to carry out full 3D conjugate heat transfer (CHT) calculation. At present, the relatively mature CHT methods or programs are: the CHTflow solver developed by Bohn et al. (1995) at Aachen University, the Glenn-HT code developed by Rigby & Lepicovsky (2001) at NASA Glenn Research Center, the Glenn-HT/BEM code developed by Heidmann et al. (2003) and Kassab et al. (2003) at NASA Glenn Research Center, the HybFlow-Solid\_CHT solver developed by Montomoli et al. (2004) at Florence University, the 2-D and 3-D conjugate heat transfer prediction codes from Han et al. (2000) at University of Texas the fully coupled conjugate heat transfer (CHT) code from Croce (2001) at University of Udine, and so on.

## **6. References**

CHTflow solver developed by Bohn et al. (1995) at Aachen University, the Glenn-HT code developed by Rigby & Lepicovsky (2001) at NASA Glenn Research Center, the Glenn-HT/BEM code developed by Heidmann et al. (2003) and Kassab et al. (2003) at NASA Glenn Research Center, the HybFlow-Solid\_CHT solver developed by Montomoli et al. (2004) at Florence University, the 2-D and 3-D conjugate heat transfer prediction codes from Han et al. (2000) at University of Texas the fully coupled conjugate heat transfer (CHT) code from

Adami, P., Martelli, F. & Cecchi, S. (2007). Analysis of the shroud leakage flow and

Arnone, A., Marconcini, M. & Pacciani, R. (2000). On the use of unsteady methods in

Arnone, A., Marconcini, M. , Scotti Del Greco, A. & Spano, E. (2003). Numerical

Behr, T., Kalfas, A. l. & Abhari, R. S. (2006). Unsteady flow physics and performance of a one-and-1/2-stage unshrouded high work turbine, ASME Paper GT2006-90959. Benzakein, M. J. (1972). Research on fan noise generation, *J. Acoustical Society of America* Vol.

Binder, A. (1985). Turbulent production due to secondary vortex cutting in a turbine rotor,

Binder, A., Forster, W., Mach, K. & Rogge, H. (1986). Unsteady flow interaction caused by stator secondary vortices in a turbine rotor, ASME Paper 86-GT-302. Binder, A., Schroeder, T. & Hourmuziadis, J. (1989). Turbulence measurements in a multistage low pressure turbine, *ASME J. Turbomachinery* Vol. 111: 153-161. Bohn, D., Bonhoff, B. & Schönenborn, H. (1995). Combined aerodynamic and thermal

Boletis, E. & Sieverding, C. H. (1991). Experimental study of the three-dimensional flow field in a turbine stator preceded by a full stage, *ASME J. Turbomachinery* Vol. 113: 1-9. Borislav, T. S. & Choon, S. T. (2002). Effect of upstream unsteady flow conditions on rotor

Butler, T. L., Sharma, O. P., Joslyn, H. D. & Dring, R. P. (1989). Redistribution of an inlet

Casciaro, C., Treiber, M. & Sell, M. (2000). Unsteady transport mechanisms in an axial

Chaluvadi, V. S. P., Kalfas, A. I. & Hodson, H. P. (2003). Blade row interaction in a high-

Chaluvadi, V. S. P., Kalfas A. I. & Hodson H. P. (2004). Vortex transport and blade interactions in high pressure turbines, *ASME J. Turbomachinery* Vol. 126: 395-405.

pressure steam turbine, *ASME J. Turbomachinery* Vol. 125: 14-24.

Chen, C. J. & Jaw, S. Y. (1998). *Fundamentals of turbulence modeling*, Taylor & Francis.

temperature distortion in an axial flow turbine stage, *AIAA J. Propulsion and Power*

*J.Engineering for Gas Turbines Power* Vol. 107: 1039-1046.

analysis of a turbine nozzle guide vane, IGTC-paper-108.

tip leakage flow, ASME Paper GT2002-30358.

turbine, *ASME J. Turbomachinery* Vol. 122: 604-612.

fluid dynamics approach, *IMechE J. Power and Energy* Vol. 221: 837-848. Anker, J. E. & Mayer, J. F. (2002). Simulation of the interaction of labyrinth seal leakage flow

and main flow in an axial turbine, ASME Paper GT2002-30348.

mainflow interactions in high-pressure turbines using an unsteady computational

predicting stage aerodynamic performance, *Proceedings of Conference on Unsteady Aerodynamics, Aeroacustics and Aeroelasticity of Turbomachines, pp. 24-36, Lyon, France.*

investigation of three-dimensional clocking effects in a low pressure turbine, ASME

Croce (2001) at University of Udine, and so on.

Paper GT2003-38414.

51(No. 5A): 1427-1438.

Vol. 5(No. 1): 64-71.

**6. References**

Heidmann, J. D., Kassab, A. J., Divo, E. A., Rodriguez, F. & Steinthorsson, E. (2003).

Hobson, D. E. & Johnson, C. G. (1990). Two stage model turbine performance characteristics,

Hodson, H. P., Baneighbal, M. R. & Dailey, G. M. (1993). Three-dimensional interactions in

Hodson, H. P. & Dawes, W. N. (1998). On the interpretation of measured profile losses in

Hodson, H. P. & Howell, R. J. (2005). Bladerow interactions, transition, and high-lift aerofoils in low-pressure turbines, Annu. Rev. Fluid Mech. Vol. 37: 71-98. Hughes, W. F. & Gaylord, E. W. (1964). Basic equations of engineering science, *Schaum's*

Hunter, S. D. & Manwaring, S. R. (2000). Endwall cavity flow effects on gaspath

Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. (2007). Small-scale statistics in

Kalitzin, G., Wu, X. & Durbin, P. A. (2003). DNS of fully turbulent flow in a LPT passage,

Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. (2003). Energy dissipation rate

Kassab, A. J., Divo, E. A., Heidmann, J. D., Steinthorsson, E. & Rodriguez, F. (2003).

Kerrebrock, J. L. & Mikolajczak, A. A. (1970). Intra-stator transport of rotor wakes and its effect on compressor performance, *ASME J. Engineering for Power* Vol. 92: 359-370. Klein, A. (1966). Investigation of the entry boundary layer on the secondary flows in the

Krishnan, L. & Sandham, N. D. (2006a). Effect of Mach number on the structure of turbulent

Krishnan, L. & Sandham, N. D. (2006b). On the merging of turbulent spots in a supersonic

LaGraff, J. E., Ashpis, D. E., Oldfield, M. L. G. & Gostelow, J. P. (2006). Unsteady flows in

Lakshminarayana, B. (1991). An assessment of computational fluid dynamic techniques in

Langston, L. S., Nice, M. L. & Hooper, R. H. (1977). Three-dimensional flow within a turbine

Langston, L. S. (1980). Crossflows in a turbine cascade passage, *ASME J. Engineering Power*

the analysis and design of turbomachinery—the 1990 freeman scholar lecture, *J.*

boundary-layer flow, *Int. J. Heat and Fluid Flow* Vol. 27: 542-550.

cascade passage, *ASME J. Engineering Power* Vol. 99: 21-28.

GT2003-38553.

276-284.

Powergen Report RD/M/1888/RR90.

*Outline Series*, McGraw Hill.

investigation, ASME Paper 2000-GT-651.

*Int. J. Heat and Fluid Flow* Vol. 24: 636-644.

blading of axial turbines, BHRAT1004.

spots, *J. Fluid Mech.* Vol. 566: 225-234.

turbomachinery, NASA/CP-2006-214484.

*Fluids Engineering* Vol. 113(No. 3): 315-352.

Vol. 102: 866-874.

in a periodic box, *Physics of Fluids* Vol. 15(No. 2): L21-24.

the rotor of an axial turbine, AIAA Paper 93-2255.

Conjugate heat transfer effects on a realistic film-cooled turbine vane, ASME Paper

unsteady wake-turbine blade interaction studies, *ASME J. Turbomachinery*, Vol. 120:

aerodynamics in an axial flow turbine: part 1 – experimental and numerical

high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics, *J. Fluid Mech.* Vol. 592: 335-366.

and energy spectrum in high resolution direct numerical simulations of turbulence

BEM/FVM Conjugate heat transfer analysis of a three-dimensional film cooled turbine blade, *Int. J. Numerical Methods for Heat & Fluid Flow* Vol. 13(No. 5): 581-610.

Paniagua, G., Yasa, T., Andres de la Loma, Castillon, L. & Coton, T. (2007). Unsteady strong

Pau, M., Paniagua, G., Delhaye, D., De la Loma, A. & Ginibre, P. (2008). Aerothermal impact

Pau, M. & Paniagua, G. (2009). Investigation of the flow field on a transonic turbine nozzle guide vane with rim seal cavity flow ejection, ASME Paper GT2009-60155. Perelman, T. L. (1961). On conjugated problems of heat transfer, *Int. J. Heat and Mass Transfer*

Peters, P., Breisig, V., Giboni, A., Lerner, C. & Pfost, H. (2000). The influence of the clearance

Piomelli, U. & Balaras, E. (2002). Wall-layer models for large-eddy simulations, *Annual*

Praisner, T. J. & Clark, J. P. (2007). Predicting transition in turbomachinery, part I: a review and new model development, *ASME J. Turbomachinery* Vol. 129: 1-13. Praisner, T. J., Grover, E. A., Rice, M. J. & Clark, J. P. (2007). Predicting transition in

Pullan, G. (2004). Secondary flows and loss caused by row interaction in a turbine stage,

Pullan, G., Denton, J. D. & Curtis E. (2006). Improving the performance of a turbine with low aspect ratio stators by aft-loading, *ASME J. Turbomachinery* Vol. 128: 492-499. Qi, L. (2010). Investigations of unsteady interaction mechanism and flow control in the

Rigby, D.L. & Lepicovsky, J. (2001). Conjugate heat transfer analysis of internally cooled

Rodi, W. (2000). Simulation of turbulence in practical flow calculations, *European Congress on*

Schaub, U. W. & Krishnappa, G. (1977). The stepped stator concept: Aerodynamic and

Shapiro, A. H. (1953). *The dynamics and thermodynamics of compressible fluid flow*, Wiley, New

Sharma, O. P. & Butler, T. L. (1987). Predictions of endwall losses and secondary flows in axial flow turbine cascades, *ASME J. Turbomachinery* Vol. 109: 229-236. Sharma, O. P., Butler, T. L., Dring, R. P., Joslyn, H. D. & Renaud, E. (1988). Rotor-stator

interaction in multistage axial flow turbines, AIAA Paper 88-3013.

*in America USA*, American Institution of Aeronautics and Astronautics. Schulte, V. & Hodson, H. P. (1998). Unsteady Wake-induced boundary layer transition in high lift LP turbines, *ASME J. Turbomachinery* Vol. 120(No. 1): 28-35. Shang, T. & Epstein, A. H. (1997). Analysis of hot streak effects on turbine rotor heat load,

*Computational Methods in Applied Sciences and Engineering,* Spanish Association for

acoustic evaluation of a thrust fan, *Proceedings of 4th AIAA - Aeroacoustics Conference*

turbine endwall regions, *D. Phil. Thesis*, Beihang University.

Numerical Methods in Engneering, Barcelona, Spain.

*ASME J. Turbomachinery* Vol. 119(No. 3): 544–553.

configurations, ASME Paper 2001- GT-0405.

ISABE Paper ISABE-2007-1218.

stage, ASME Paper GT2008-51295.

subsequent stator, ASME Paper 2000-GT-478.

*Review of Fluid Mechanics* Vol. 34: 349-374. Pope, S. B. (2000). *Turbulent flows*, Cambridge University Press.

*Turbomachinery* Vol. 129: 14-22.

ASME Paper GT2004-53743.

York.

Vol. 3(No. 4): 293-303.

shock interactions in a transonic turbine: experimental and numerical analysis,

of stator-rim purge flow and rotor-platform film cooling on a transonic turbine

of shrouded rotor blades on the development of the flowfield and losses in the

turbomachinery, part II: model validation and benchmarking, *ASME J.*

**Part 3**

**Experimental Tests**

98 Noise Control, Reduction and Cancellation Solutions in Engineering

Yamamoto, A. (1987). Production and development of secondary flows and losses in two

Zhang, X. F. & Hodson, H. P (2004). Experimental study of unsteady wake-induced

Zhou, C. & Hodson, H. P. (2009). The tip leakage flow of an unshrouded high pressure

turbine blade with tip cooling, ASME Paper GT2009-59637.

109: 186-193.

Cambridge.

types of straight turbine cascades: part I-a stator case, *ASME J. Turbomachinery* Vol.

boundary layer transition in turbomachinery, Whittle Laboratory, University of

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

Wei Cheng1,2, Zhousuo Zhang1,\* and Zhengjia He1 *1State Key Laboratory for Manufacturing Systems Engineering, Xi'an, 2Department of Mechanical Engineering, University of Michigan, Ann Arbor, 1PR China 2USA*

## **1. 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

<sup>\*</sup> Corresponding author

from the mixed signals, and quantitatively evaluates the source contributions. Firstly the vibration signals at the different positions are measured, and the radiated noise is evaluated according to the signal energy. Secondly the signals which radiate the noise significantly are selected as the mixed signals, and the source signals contained in the mixed signals are extracted by an improved ICA method. Thirdly, the vibration signals on the engine bases are measured as the source signals, and the vibration sources are identified according to correlation analysis. Lastly, the contributions of each source are quantitatively calculated according to the mixing mode of the independent components. Therefore, the vibration sources which have big contributions can be online identified and controlled if necessary, which provides a novel approach for active control over the vibration and noise.

Compared with other signal processing method, the independent component analysis can reveal the basic sources contained in the mixed signals. ICA is firstly proposed (Jutten & Herault, 1991) and applied in blind source separation (Comon, 1994). A well known ICA algorithm called fixed-point algorithm based on fourth order cumulant is proposed (Hyvarinen & Oja, 1997), and latter fast fixed-point algorithm based on negentropy is proposed (Hyvarinen, 1999), and then is further improved (Hyvarinen, et al., 2001). Currently, ICA is widely used in image feature extraction and recognition (Hu, 2008; Correa et al., 2007), biological signal analysis and feature extraction (Ye et al., 2008; Xie et al., 2008), fault feature extraction (Zuo et al., 2005), astronomical data analysis (Moussaoui et al., 2008), and data compression (Kwak et al., 2008). However, as a statistical signal processing method, a major problem of ICA is that the reliability of the estimated independent components is not known, and the separated components may be different in the repeatedly calculations. Therefore, the independent components are difficult to be explained. A good way to solve this problem is introduced according to clustering evaluation, and the stability of the separated components can be significantly enhanced.

This paper is organized as follows. In section 1, the motivation and research status are introduced. In section 2, the basic theory of ICA is introduced. In section 3, the quantitative calculation of source contributions method based on the enhanced ICA algorithm is proposed. In section 4, the stability of the enhanced ICA algorithm is validated by a comparative numeric study. In section 5, the proposed method is applied to quantitatively calculate the source contributions of a thin shell structure. In section 6, we give the conclusions and discussions.

#### **2. Independent component analysis**

#### **2.1 Basic theory of BSS and ICA**

Assume that *n* sources 1 2 [ , ,..., ]*<sup>T</sup> S ss s <sup>n</sup>* exist at the same time, and *m* mixed signals 1 2 [ , ,..., ]*<sup>T</sup> X xx x <sup>m</sup>* which are composed by these sources are obtained in different places. And thus each mixed signals can be described as:

$$\mathbf{x}\_{i} = \sum\_{i=1}^{n} a\_{ij}\mathbf{s}\_{j} + n\_{i} \text{ i } = 1, 2, \dots, m, \quad j = 1, \dots, n \tag{1}$$

Where *<sup>i</sup> x* is the *i*th mixed signal observed in the position *i*, *<sup>j</sup> s* is the *j*th source signal, *ij a* is the mixing coefficient, and *ni* is the noise of *i*th mixed signal. The mixed signal can be also described as:

$$X = AS + N \tag{2}$$

Where *A* is the mixing matrix, and *N* is the noise matrix.

Blind source separation (BSS) can be described as follows: in the condition that the mixing matrix and source signals are unknown, BSS obtains the estimates of sources 1 2 [ , ,..., ]*<sup>T</sup> <sup>Y</sup> <sup>n</sup> yy y* according to separating matrix *W* and mixed signals. That is

$$Y = \mathsf{VVA} = \mathsf{VANS} = \mathsf{GS} \tag{3}$$

Where G is a global matrix.

102 Noise Control, Reduction and Cancellation Solutions in Engineering

from the mixed signals, and quantitatively evaluates the source contributions. Firstly the vibration signals at the different positions are measured, and the radiated noise is evaluated according to the signal energy. Secondly the signals which radiate the noise significantly are selected as the mixed signals, and the source signals contained in the mixed signals are extracted by an improved ICA method. Thirdly, the vibration signals on the engine bases are measured as the source signals, and the vibration sources are identified according to correlation analysis. Lastly, the contributions of each source are quantitatively calculated according to the mixing mode of the independent components. Therefore, the vibration sources which have big contributions can be online identified and controlled if necessary,

Compared with other signal processing method, the independent component analysis can reveal the basic sources contained in the mixed signals. ICA is firstly proposed (Jutten & Herault, 1991) and applied in blind source separation (Comon, 1994). A well known ICA algorithm called fixed-point algorithm based on fourth order cumulant is proposed (Hyvarinen & Oja, 1997), and latter fast fixed-point algorithm based on negentropy is proposed (Hyvarinen, 1999), and then is further improved (Hyvarinen, et al., 2001). Currently, ICA is widely used in image feature extraction and recognition (Hu, 2008; Correa et al., 2007), biological signal analysis and feature extraction (Ye et al., 2008; Xie et al., 2008), fault feature extraction (Zuo et al., 2005), astronomical data analysis (Moussaoui et al., 2008), and data compression (Kwak et al., 2008). However, as a statistical signal processing method, a major problem of ICA is that the reliability of the estimated independent components is not known, and the separated components may be different in the repeatedly calculations. Therefore, the independent components are difficult to be explained. A good way to solve this problem is introduced according to clustering evaluation, and the stability

This paper is organized as follows. In section 1, the motivation and research status are introduced. In section 2, the basic theory of ICA is introduced. In section 3, the quantitative calculation of source contributions method based on the enhanced ICA algorithm is proposed. In section 4, the stability of the enhanced ICA algorithm is validated by a comparative numeric study. In section 5, the proposed method is applied to quantitatively calculate the source contributions of a thin shell structure. In section 6, we give the

Assume that *n* sources 1 2 [ , ,..., ]*<sup>T</sup> S ss s <sup>n</sup>* exist at the same time, and *m* mixed signals 1 2 [ , ,..., ]*<sup>T</sup> X xx x <sup>m</sup>* which are composed by these sources are obtained in different places.

*x as n i m j n*

Where *<sup>i</sup> x* is the *i*th mixed signal observed in the position *i*, *<sup>j</sup> s* is the *j*th source signal, *ij a* is

12 1 , ,..., , ,...,

(1)

which provides a novel approach for active control over the vibration and noise.

of the separated components can be significantly enhanced.

conclusions and discussions.

**2. Independent component analysis**

The mixed signal can be also described as:

And thus each mixed signals can be described as:

1

the mixing coefficient, and *ni* is the noise of *i*th mixed signal.

*n i ij j i i*

**2.1 Basic theory of BSS and ICA**

#### **2.2 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:

1. Each source signal ( ,..., ) 1 *<sup>j</sup> s j n* is zero mean and real random variable, and all the source signals are mutually statistic independent at any time.

$$p\_s(\mathbf{s}) = \prod\_{j=1}^n p\_j(\mathbf{s}\_j) \tag{4}$$

Where ( ) *<sup>s</sup> p s* is the probability density of source signals *S* , and ( ) *j j p s* is the probability density of source signal *<sup>j</sup> s* .

#### **2.3 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:

$$N\text{g(s)} = H(\text{s}\_{\text{Gauss}}) - H(\text{s}) \tag{5}$$

Where *gauss s* is a signal of Gaussian distribution and has the same variance with *s*. *H*( ) is the information entropy of a signal.

$$H(\mathbf{s}) = -\int p(\mathbf{s}) \lg p(\mathbf{s}) d\mathbf{s} \tag{6}$$

As the probability density distribution function is unknown, the independence of each separated signal is commonly measured by an approximate equation

$$\log(\text{s}) \propto \left[ E\{\text{G}(\text{s})\} - E\{\text{G}(\text{s}\_{\text{gauss}})\} \right]^2 \tag{7}$$

Where *E*( ) is a mean function, and *G*( ) is a nonlinear function. The commonly used *G*( ) are logarithmic function and exponential function

$$\mathcal{L}G(u) = \frac{1}{a} \text{lg } \cosh(au) \tag{8}$$

$$G(\mu) = -\exp(-\mu^2 \,/\, \mathcal{D})\tag{9}$$

Where 1 2 *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.

#### **2.4 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:

$$\mu\_p(k+1) = \mathbb{E}[\mathbb{Z}\mathbb{g}(\boldsymbol{\mu}\_p^T(k)\boldsymbol{z})] - \mathbb{E}[\mathbb{g}'(\boldsymbol{\mu}\_p^T(k)\boldsymbol{z})]\mu\_p(k) \tag{10}$$

where *g* is the differential of *G*

5. Orthogonal calculation

$$
\mu\_p(k+1) = \mu\_p(k+1) - \sum\_{j=1}^{p-1} < \mu\_p(k+1), \mu\_j > \mu\_j \tag{11}
$$

6. Normalizing calculation

$$
\mu\_p(k+1) \leftarrow \frac{\mu\_p(k+1)}{\left\| \mu\_p(k+1) \right\|\_2} \tag{12}
$$

#### **3. 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).

#### **3.1 Framework of fast fixed-point algorithm**

104 Noise Control, Reduction and Cancellation Solutions in Engineering

Where *E*( ) is a mean function, and *G*( ) is a nonlinear function. The commonly used *G*( )

<sup>1</sup> *G u*() lg cosh( ) *au*

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 non-

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

1. Mean and whiten the mixed signals *X* , remove the redundant information, and thus

1

1

( ) ( ) ( ) *p*

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

*p*

*u k*

*u k*

( ) ( ) ( ), 11 1 *p p p p jj j uk uk uk u u*

1

*p*

7. Determine whether *up* is convergent. If not, return to step (4).

**3. Quantitatively calculate source contributions**

8. Repeat step (3), and else stop calculation.

*u k*

<sup>2</sup> ( ) 0 1 *up* .

( ) [ ( ( ) )] [ ( ( ) )] ( ) 1 *T T u k Ez p p pp g u kz E g u kz u k* (10)

(11)

(12)

2

1

1

2. Set the number *p* of independent component esxtracted for each time.

*<sup>a</sup>* (8)

<sup>2</sup> *Gu u* ( ) exp( / ) 2 (9)

are logarithmic function and exponential function

gaussian feature of the separated components.

**2.4 Framework of fast fixed-point algorithm**

algorithm can be described as eight steps:

4. Iterative calculation

where *g* is the differential of *G* 5. Orthogonal calculation

6. Normalizing calculation

obtain the preprocessed signals *Z* .

3. Set the initial iteration value ( ) 0 *up* , and let

Where 1 2 *a* .

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:

Assume that the mutual correlation coefficient between estimated components *<sup>i</sup> y* and *<sup>j</sup> y* is *ij* , and the cluster validity index *qI* can be defined as follows:

$$I\_q(k) = \frac{1}{\left|\mathbb{C}\_k\right|^2} \sum\_{i,j \in \mathbb{C}\_k} \rho\_{ij} - \frac{1}{\left|\mathbb{C}\_k\right| \left|\mathbb{C}\_{-k}\right|} \sum\_{i \in \mathbb{C}\_k} \sum\_{j \in \mathbb{C}\_{-k}} \rho\_{ij} \tag{13}$$

where *C* is all the IC set, *Ck* is the IC set in the *k*th cluster, *Ck* is its complementary set, and *Ck* is the size of the *k*th 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.

#### **3.2 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 1 2 [, ]*<sup>T</sup> S ss* , it can be inferred that the joint distribution 1 2 , 1 2 (, ) *s s <sup>p</sup> s s* of the two source signals must be included in a rectangle as

1 2 1 2 , 12 1 2 (, ) () ( ) *s s s s p s s p s p s* (14)

Fig. 1. Joint distributions of different signals

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

$$X = AS = \begin{bmatrix} \mathbf{x}\_1, \mathbf{x}\_2 \end{bmatrix}^T \tag{15}$$

The joint distribution of *X* is shown in Fig. 1(b). Nonlinear mixed signals 1 2 [, ]*<sup>T</sup> X xx e ee* are obtained by nonlinear mixing mode, and their joint distribution is shown in Fig. 1(c). Theorem 1: for transformation

$$\begin{cases} p\_1 = h\_1(e\_1) \\ p\_2 = h\_2(e\_2) \end{cases} \tag{16}$$

Where 1*h* and <sup>2</sup> *h* is an analytic function.

If the parallelogram border on the plane ( <sup>1</sup>*e* , <sup>2</sup>*e* ) is transformed into the plane ( <sup>1</sup>*p* , <sup>2</sup> *p* ), and these parallelogram borders are not parallel to the related coordinate axis, there have real constants 1*a* 、 <sup>2</sup> *a* 、 <sup>1</sup>*b* and <sup>2</sup> *b*

$$\begin{cases} h\_1(\mu) = a\_1 \mu + b\_1 \\ h\_2(\mu) = a\_2 \mu + b\_2 \end{cases} \tag{17}$$

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.

#### **3.3 Quantitative evaluation of source contributions**

From the definition of whitening, the covariance matrix *Rxx* can be described as

$$R\_{xx} = \mathbf{E}[\mathbf{x}\mathbf{x}^T] = \mathbf{E}[\mathbf{Q}\mathbf{x}\mathbf{x}^T\mathbf{Q}^T] = \mathbf{Q}R\_{xx}\mathbf{Q}^T = I\_n \tag{18}$$

Where *Q* is a whitening matrix, and *nI* is a unit covariance matrix. From the equation (18), it can be seen that the mixed signals 1 2 [ , ,..., ]*<sup>T</sup> X xx x <sup>n</sup>* are transformed into unrelated signals 1 2 [ , ,..., ]*<sup>T</sup> Z zz z <sup>n</sup>* by whitening.

According to the definition of orthogonal transformation, there exists follow equation

$$\mathbf{U}\mathbf{U}\mathbf{U}^{T} = \mathbf{U}^{T}\mathbf{U} = I\_{\text{n}} \tag{19}$$

$$Y = \mathsf{LIZ} \tag{20}$$

1 2 [ , ,..., ]*<sup>T</sup> Y yy y <sup>n</sup>* are independent components with unit variances. From the mathematical model of ICA, mixed signals *X* can be obtained by *Y* .

$$X = AS = AG^{-1}Y = \hat{A}Y\tag{21}$$

Each components of *X* are composed by all the components of *Y* according to the mixing matrix ˆ *A* . Each combination coefficients of the mixing matrix ˆ *A* 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 ˆ *A* effectively.

### **4. Simulation experiment analysis**

106 Noise Control, Reduction and Cancellation Solutions in Engineering

0 0 0

(a) (b) (c)

The joint distribution of *S* is shown in Fig. 1(a). Mixed signals are obtained according to *S*

The joint distribution of *X* is shown in Fig. 1(b). Nonlinear mixed signals 1 2 [, ]*<sup>T</sup> X xx e ee* are

1 11 2 22

If the parallelogram border on the plane ( <sup>1</sup>*e* , <sup>2</sup>*e* ) is transformed into the plane ( <sup>1</sup>*p* , <sup>2</sup> *p* ), and these parallelogram borders are not parallel to the related coordinate axis, there have real

> 1 11 2 22

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

[ ][ ] *T TT <sup>T</sup> R E xx E Qxx Q QR Q I xx xx n* (18)

( ) ( ) *h u au b h u au b*

From the definition of whitening, the covariance matrix *Rxx* can be described as

*p h e p h e*

( ) ( )

obtained by nonlinear mixing mode, and their joint distribution is shown in Fig. 1(c).

1*s*

Fig. 1. Joint distributions of different signals

and mixing matrix *A*, which can be shown as

Where 1*h* and <sup>2</sup> *h* is an analytic function.

of the proposed method is further enhanced.

**3.3 Quantitative evaluation of source contributions**

Theorem 1: for transformation

constants 1*a* 、 <sup>2</sup> *a* 、 <sup>1</sup>*b* and <sup>2</sup> *b*

2 *s*

1 2 1 2 , 12 1 2 (, ) () ( ) *s s s s p s s p s p s* (14)

<sup>2</sup> *x <sup>e</sup>*<sup>2</sup> *x*

1 2 [, ]*<sup>T</sup> X AS x x* (15)

(16)

(17)

*e*1 *x*

1 *x*

#### **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*s t*( ) is a white noise signal, 2*s t*( ) is a frequency modulated signal, 3*s t*( ) is an amplitude modulated signal, and 4*s t*( ) 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.

$$\begin{aligned} S(t) &= \begin{bmatrix} s\_1(t) \\ s\_2(t) \\ s\_3(t) \\ s\_4(t) \end{bmatrix} = \begin{bmatrix} n(t) \\ \sin(0.2 \times t) \times \cos(15 \times t) + \sin(2 \times t) \\ \sin(0.3 \times t) \times \sin(5 \times t + \sin(t)) \\ \sin(0.3 \times t \times \sin(0.5 \times t)) \end{bmatrix} \\ A &= \begin{bmatrix} 0.63 \, 0.77 \, 0.54 \, 0.65 \\ 0.94 \, 0.72 \, 0.78 \, 0.83 \\ 0.88 \, 0.93 \, 0.84 \, 0.32 \\ 0.98 \, 0.62 \, 0.54 \, 0.95 \end{bmatrix} \end{aligned} \tag{22}$$

Fig. 2. Waveforms of the source signals

Fig. 3. Waveforms of the mixed signals

Fig. 3. Waveforms of the mixed signals

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Fig. 2. Waveforms of the source signals

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Waveforms of the sources are shown in Fig.2, which indicates that each source is of obviously different waveform features. The mixed signals are composed of four sources by mixing matrix with linear superposition, and the mixed signals are shown in Fig. 3.

Fig. 4. Waveforms of principal components

Fig. 5. Waveforms of independent components by fast fixed-point algorithm

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.

Fig. 6. Waveforms of the separated components by the enhanced ICA

From Fig. 4 to Fig. 6, it can be clearly seen that principal component analysis only separate <sup>3</sup>*s* , and the other sources are not well separated. The independent components separated by the other two algorithms are obvious, and the waveform information of the sources is well separated. Correlation analysis is employed to quantitatively validate the separating performance, and the correlation coefficient *sy* is defined as follows.

$$\rho\_{sy} = \frac{\sum\_{k=1}^{n} s(k)y(k)}{\sqrt{\sum\_{k=1}^{n} s^2(k)\sum\_{k=1}^{n} y^2(k)}}\tag{23}$$

Where *s k*( ) 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:

1. *sy* of principal component analysis

### 2. *sy* of the fast fixed-point algorithm

110 Noise Control, Reduction and Cancellation Solutions in Engineering

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

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From Fig. 4 to Fig. 6, it can be clearly seen that principal component analysis only separate <sup>3</sup>*s* , and the other sources are not well separated. The independent components separated by the other two algorithms are obvious, and the waveform information of the sources is well separated. Correlation analysis is employed to quantitatively validate the separating

*n*

1

*k k*

Where *s k*( ) is the source signal, *y*( ) *k* is the independent component, and *k* is the data

In the numeric studies, the correlation matrices *sy* between the separated components and

*k sy n n*

2 2 1 1

*sk yk*

() ()

()()

*skyk*

*sy* is defined as follows.

(23)

Fig. 6. Waveforms of the separated components by the enhanced ICA

respectively.

EIC1

EIC2

EIC3

EIC4

sequence.

Amplitude

Amplitude

Amplitude

Amplitude

performance, and the correlation coefficient

the sources are listed as follows:

1. *sy* of principal component analysis

0

5

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5

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5

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2

## 3. *sy* of the enhanced ICA algorithm

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.

#### **4.2 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:

$$\begin{aligned} S(t) = \begin{bmatrix} s\_5(t) \\ s\_6(t) \\ s\_7(t) \\ s\_8(t) \end{bmatrix} = \begin{bmatrix} n(t) \\ \sin(0.4 \times \pi \times t) \\ \sin \pi \times \text{coth}(0.5 \times \pi \times t, 0.5) \\ \text{savt} \text{coth}(0.5 \times \pi \times t, 0.5) \\ \text{square}(0.6 \times \pi \times t, 50) \end{bmatrix} \quad A = \begin{bmatrix} 0.65 \, 0.75 \, 0.65 \, 0.60 \\ 0.95 \, 0.70 \, 0.75 \, 0.85 \\ 0.88 \, 0.90 \, 0.80 \, 0.32 \\ 0.90 \, 0.40 \, 0.42 \, 0.95 \end{bmatrix} \end{aligned} \tag{24}$$

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.

Fig. 7. Waveforms of the source signals

Fig. 8. Waveforms of the mixed signals

the results are different from the source signals obviously. One of the inaccurate results is

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Data sequence

shown in Fig. 9, which indicates that the source information is not well separated.

10

10

Fig. 8. Waveforms of the mixed signals

0

0

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10

Fig. 7. Waveforms of the source signals

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Amplitude

Fig. 9. Waveforms of one inaccurate separation by fast fixed-point algorithm

Fig. 10. Waveforms of separated components by the enhanced ICA algorithm

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.

### **4.3 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 ˆ *A* .

$$A = \begin{bmatrix} 0.63 \ 0.77 \ 0.54 \ 0.65 \\ 0.94 \ 0.72 \ 0.78 \ 0.83 \\ 0.88 \ 0.93 \ 0.84 \ 0.32 \\ 0.98 \ 0.62 \ 0.54 \ 0.95 \end{bmatrix} \leftrightarrow \hat{A} = \begin{bmatrix} 0.65 \ 0.75 \ 0.55 \ 0.62 \\ 0.95 \ 0.70 \ 0.75 \ 0.85 \\ 0.88 \ 0.90 \ 0.80 \ 0.32 \\ 0.94 \ 0.60 \ 0.52 \ 0.95 \end{bmatrix}$$

Compared the mixing matrix *A* with the mixing matrix ˆ *A* calculated by the proposed method, the real proportion of each source to <sup>1</sup> *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 <sup>2</sup> *x* , <sup>3</sup> *x* and <sup>4</sup> *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.

## **5. Source contribution evaluation of a thin shell structure**

#### **5.1 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.

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.

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

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

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

method, the real proportion of each source to <sup>1</sup> *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 <sup>2</sup> *x* , <sup>3</sup> *x* and <sup>4</sup> *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

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

*A*

*A* .

*A* calculated by the proposed

0.65 0.75 0.55 0.62 0.95 0.70 0.75 0.85 ˆ 0.88 0.90 0.80 0.32 0.94 0.60 0.52 0.95

10, which indicates that the source information is well extracted.

optimization is of high precision and stability.

*A*

**5.1 Introductions of the thin shell structure**

from the mixed signals.

**4.3 Quantitative evaluation of source contributions**

contributions can be obtained according to the mixing matrix.

with the related sources, and thus get a modified mixing matrix ˆ

0.63 0.77 0.54 0.65 0.94 0.72 0.78 0.83 0.88 0.93 0.84 0.32 0.98 0.62 0.54 0.95

Compared the mixing matrix *A* with the mixing matrix ˆ

**5. Source contribution evaluation of a thin shell structure**

shown in Fig.11, and the photo of the entire system is shown in Fig.12.

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* (1.0*g*=9.8*m/s2*). The rotational speeds of motor 1, motor 2, and motor 3 are respectively 1350 rpm, 1470 rpm, and 1230 rpm.

## **5.2 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.

Fig. 13. Waveforms of the mixed signals on the inside shell

Fig. 14. Waveforms of the ICs by the enhanced ICA algorithm

Fig. 15. Waveforms of the source signals from three motor bases

The correlation matrix *sy* is as follows.

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.

0.4 0.42 0.44 0.46 0.48 0.5

0.4 0.42 0.44 0.46 0.48 0.5

0.4 0.42 0.44 0.46 0.48 0.5

Time (sec)

0.4 0.42 0.44 0.46 0.48 0.5

0.4 0.42 0.44 0.46 0.48 0.5

0.4 0.42 0.44 0.46 0.48 0.5

Time (sec)

5

5

0

0

0

EIC1

EIC2

EIC3

Amplitude (g)

Amplitude (g)

Amplitude (g)

5

Fig. 13. Waveforms of the mixed signals on the inside shell

Fig. 14. Waveforms of the ICs by the enhanced ICA algorithm

0

X1

X2

X3

Amplitude (g)

Amplitude (g)

Amplitude (g)

0.2

*sy* 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.

#### **5.3 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:

$$
\Omega\_{\rm sy} = \begin{bmatrix}
0.2907 & \underline{0.8903} & 0.0795 \\
\underline{0.8397} & 0.1041 & 0.3083 \\
\underline{0.2717} & 0.1094 & \underline{0.9539}
\end{bmatrix}
$$

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 ˆ *A* is as follow:

$$
\hat{A} = \begin{bmatrix}
\text{-0.1299 - 0.0097 & 0.0440 \\
\text{-0.1971 } & \text{0.0358 } & \text{0.1656} \\
\text{0.1424 } & \text{0.0318 } & \text{0.0708} \\
\text{-0.0313 } & \text{-0.1375 } & \text{0.0275}
\end{bmatrix}
$$

Fig. 16. Waveforms of the mixed signals

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

*sy*

*A* is as follow:

*A*

algorithm. The mixing matrix ˆ

0.5

0

Fig. 16. Waveforms of the mixed signals

0

2

0

1

X4

X5

X6

X7

Amplitude (g)

Amplitude (g)

Amplitude (g)

Amplitude (g)

0

0.5

0.2907 0.8903 0.0795 0.8397 0.1041 0.3083 0.2717 0.1094 0.9539

0.4 0.42 0.44 0.46 0.48 0.5

0.4 0.42 0.44 0.46 0.48 0.5

0.4 0.42 0.44 0.46 0.48 0.5

0.4 0.42 0.44 0.46 0.48 0.5

Time (sec)

Fig. 17. Waveforms of ICs by the enhanced ICA algorithm with priori information

The mixing matrix ˆ *A* 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.

Table 1. Percentage contributions

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.

## **5.4 Discussions**

## **6. 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.

## **7. Acknowledgement**

This work was supported by the key project of National Nature Science Foundation of China (No. 51035007), the project of National Nature Science Foundation of China (No. 50875197) and National S&T Major Project (No. 2009ZX04014-101).

## **8. References**

120 Noise Control, Reduction and Cancellation Solutions in Engineering

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

1. 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

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

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

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

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

proposed method has been greatly enhanced by priori information.

enhanced ICA algorithm are enhanced.

that the proposed method is of high accuracy.

reliable referances for vibration reduction and monitoring.

**5.4 Discussions**

priori information.

**6. Conclusions**

control.

## **Advanced Vibro-Acoustic Techniques for Noise Control in Helicopters**

Emiliano Mucchi1, Elena Pierro2 and Antonio Vecchio<sup>3</sup> <sup>1</sup>*Engineering Department, University of Ferrara* <sup>2</sup>*DIMeG, Politecnico di Bari* <sup>3</sup>*ARTEMIS Joint Undertaking, Brussels* 1,2*Italy* <sup>3</sup>*Belgium*

#### **1. Introduction**

122 Noise Control, Reduction and Cancellation Solutions in Engineering

Comon. P. (1994). Independent component analysis, a new concept? *Signal Processing* ,

Hyvarinen. A, Oja. E. (1997). A fast fixed-point algorithm for independent component analysis. *Neural Computation*, vol.9, no.7 : pp.1486-1492, ISBN: 0-262-58168-X Hyvarinen. A. (1999). Fast and robust fixed-point algorithm for independent component

Hyvarinen. A, Karhunen. J, Oja. E. (2001). In : *Independent component analysis*. John Wiley and

Hu. HF. (2008). ICA-based neighborhood preserving analysis for face recognition. *Computer Vision and Image Understanding*, vol.112, no.3: pp.286-295, ISSN: 1077-3142 Correa. N, Adali. T, Vince. Calhoun. VD. (2007). Performance of blind source separation

Ye. YL, Zhang. ZL, Zeng. JZ. (2008). A fast and adaptive ICA algorithm with its application

Xie. L, Wu. J. (2006). Global optimal ICA and its application in MEG data analysis.

Zuo. MJ, Lin. J, Fan. XF. (2005). Feature separation using ICA for a one-dimensional time

Moussaoui. S, Hauksdottir. H, Schmidt. F. (2008). On the decomposition of Mars hyper-

Kwak. N, Kim. C, Kim. H. (2008). Dimensionality reduction based on ICA for regression problems. *Neurocomputing*, vol.71, no.13-15: pp.2596– 2603, ISSN: 0925-2312 Himberg. J, Hyvarinen. A. (2003). Icasso: Software for investigating the reliability of ICA

*for Signal Processing*, pp. 259-268, ISBN: 0-7803-8177-7, Toulouse, France Himberg. J, Hyvarinen. A, Esposito. F. (2004). Validating the independent components of

Ma. JC, Niu. YL, Chen. HY. (2006). In : *Blind Signal Processing*. National Defense Industry

He. ZJ, Zi. YY, Zhang. XN. (2006). In : *Modern signal processing technology and its application*, Xi'an Jiaotong University Press, pp.4-9, ISBN:<PHONE_NUMBER>464, Xi'an, China

*Neurocomputing*, vol.69, no.16-18: pp.2438-2442, ISSN: 0925-2312

analysis. *IEEE Transactions On Neural Network*, vol.10, no.3 : pp.626-634, ISSN: 1045-

algorithms for fMRI analysis using a group ICA method. *Magnetic Resonance*

to fetal electrocardiogram extraction. *Applied Mathematics and Computation*, vol.205,

series and its application in fault detection. *Journal of Sound and Vibration*, vol.287,

spectral data by ICA and Bayesian positive source separation. *Neurocomputing*,

estimates by clustering and visualization. *IEEE XIII Workshops On Neural Networks*

neuroimaging time-series via clustering and visualization. *Neuroimage*, vol.22, no.3:

vol.36, no.3: pp.287-314, ISSN: 0165-1684

Sons. ISBN : 0-471-40540-X, New York, USA

*Imaging*, vol.25, no.5: 684–694, ISSN : 0730-725X

vol.71, no10-12: pp.2194–2208, ISSN : 0925-2312

Press, pp.150-160, ISBN:<PHONE_NUMBER>079, BeiJing, China

no.2: pp.799-806, ISSN : 0096-3003

no.3: pp.614-624, ISSN: 0022-460X

pp.1214-1222, ISSN: 1053-8119

9227

Originally, aircraft noise was not much of an issue because of the overarching requirement of improving vehicle performance in critical operational conditions. For many years, the design of aircraft and helicopter has been challenged by the need to improve aeromechanical performance, leaving little rooms for considering additional design parameters such as noise, vibrations and passenger comfort. More recently, a paradigm shift is being recorded that includes vibro-acoustics comfort in the set of critical design parameters. In the particular case of helicopters, this is mostly a consequence of a shift in the definition of the typical helicopter mission: in the past helicopters were mainly used in military context where flight mechanics, maneuverability and avionics are the key performance indicators; todays a trend is visible in the related industry that targets more and more civil applications, where the payload and environmental impact assume a much more significant role. In this respect, vibroacoustic comfort and more in general noise performance are becoming one of the key market differentiators: passenger comfort is included in the key performance indicators (ref. the US101 - Marine One case). In order to cope with this new and more stringent set of performance requirements, specific tools and techniques are required that allow better identifying the noise sources, the mechanics of transfer from source to target (pilot and/or passenger area) and defining an effective noise control strategy. In this scenario, the authors want to present a modus operandi to tackle these issues by several advanced experimental methodologies, since the requirement of a quieter helicopter needs a systematic study of its Noise Vibration and Harshness (NVH) behaviour. This chapter collects the results of several investigations carried out on two different helicopters: EUROCOPTER EC-135 and Agusta Westland AW-109. The proposed methodology faces these issues considering that the noise is a result of the contribution of source(s) reaching the receiver via various transmission paths. Therefore, in a noise control strategy , the designer should consider both the excitation phenomena (i.e. sources) as well as the transfer functions between the source(s) and receiver(s) (i.e. transmission paths). The sections of this chapter discuss both sources and transmission paths. In particular, Sections 2 addresses the deep analysis of transmission paths, taking into account the vibro-acoustic interaction that in cumbersome systems as helicopters is of pivotal importance; Sections 2 and 3 deal with noise and vibration sources, while Section 5 presents a technique for gaining information about sources (operational forces) and transmission paths in operational conditions. Eventually, Section 6 presents an original approach based on sound synthesis allowing designers to listen and judge the sound quality impact of structural design modifications in a process of noise control and mitigation. It is interesting to note that although the presented methodologies concern particular helicopters, they have a general meaning, since they can be applied in a large variety of fields, as in aircraft fuselage, car or track interiors , etc. This work has to be intended as an analytical review of the authors' research (Mucchi & Vecchio, 2009; 2010; Pierro et al., 2009). Due to confidentiality reasons, several vibro-acoustical quantities reported in this chapter are presented without numerical values.

## **2. Noise control in coupled vibro-acoustical systems**

Real-life enclosures, such as aircraft fuselage and car interiors, are very often characterized by a strong fluid-structure coupling (Desmet et al., 2003; Kronast & Hildebrandt, 2000). Therefore, the acoustical response in the cavity is related to the structural excitation and a relation exists between the acoustical loading and the structural responses as well (Everstine, 1997; 1981). Two methods are generally used to characterize the vibro-acoustical behaviour of such systems. In the first, the classical Experimental Modal Analysis (EMA), the modal parameters (natural frequencies, damping factors and mode shapes) are evaluated in laboratory, and consists in to put in vibration the system with a known excitation, out of its normal service environment. The common practice, indeed, is to make the peaks of the input force spectrum coincident with the valleys of the structural frequency response functions (FRFs). However, it is not so straightforward to get these tasks for vibro-acoustical systems because a modal analysis has a high number of theoretical and practical problems: due to the coupling, indeed, the system matrix is nonsymmetrical, yielding different right and left eigenvalues (but closely related). For this reason, care must be taken during the FRFs synthesis, but it has been proven that no matter involves in the modal model derivation itself (Wyckaert et al., 1996). The latter and more recent technique, is the Operational Modal Analysis (OMA), which does not require the knowledge of the operating forces on the system, but considers the assumption of white noise as input. Some investigations have been already made on vibro-acoustical operational modal analysis (Abdelghani et al., 1999; Hermans & Van der Auweraer, 1999), hereafter the modal decomposition in case of vibro-acoustical operational modal analysis is reviewed, in order to point out the influence on the modal parameter estimation of the references choice, i.e. acoustical or structural, used for the Cross power spectra calculation. The complete knowledge of the vibro-acoustical response of an helicopter cabin is a crucial tool for noise control purpose, and in the OMA context, it is strongly related to the reference choice rather than the input forces. For simplicity, only an acoustical input will be considered and the cross power between acoustical and structural responses will be derived and compared with structural cross power responses. The formulation in case of structural excitation can be derived in a similar way. The theoretical results have been verified through an experimental testing on the helicopter EUROCOPTER EC-135.

#### **2.1 Vibro-acoustical systems**

The dynamical properties of vibro-acoustical systems are presented hereafter. In particular the transfer function considering as excitation both the acoustical loading and the structural force and considering as response both the structural displacement and the acoustical pressure 2 Will-be-set-by-IN-TECH

while Section 5 presents a technique for gaining information about sources (operational forces) and transmission paths in operational conditions. Eventually, Section 6 presents an original approach based on sound synthesis allowing designers to listen and judge the sound quality impact of structural design modifications in a process of noise control and mitigation. It is interesting to note that although the presented methodologies concern particular helicopters, they have a general meaning, since they can be applied in a large variety of fields, as in aircraft fuselage, car or track interiors , etc. This work has to be intended as an analytical review of the authors' research (Mucchi & Vecchio, 2009; 2010; Pierro et al., 2009). Due to confidentiality reasons, several vibro-acoustical quantities reported in this chapter are presented without

Real-life enclosures, such as aircraft fuselage and car interiors, are very often characterized by a strong fluid-structure coupling (Desmet et al., 2003; Kronast & Hildebrandt, 2000). Therefore, the acoustical response in the cavity is related to the structural excitation and a relation exists between the acoustical loading and the structural responses as well (Everstine, 1997; 1981). Two methods are generally used to characterize the vibro-acoustical behaviour of such systems. In the first, the classical Experimental Modal Analysis (EMA), the modal parameters (natural frequencies, damping factors and mode shapes) are evaluated in laboratory, and consists in to put in vibration the system with a known excitation, out of its normal service environment. The common practice, indeed, is to make the peaks of the input force spectrum coincident with the valleys of the structural frequency response functions (FRFs). However, it is not so straightforward to get these tasks for vibro-acoustical systems because a modal analysis has a high number of theoretical and practical problems: due to the coupling, indeed, the system matrix is nonsymmetrical, yielding different right and left eigenvalues (but closely related). For this reason, care must be taken during the FRFs synthesis, but it has been proven that no matter involves in the modal model derivation itself (Wyckaert et al., 1996). The latter and more recent technique, is the Operational Modal Analysis (OMA), which does not require the knowledge of the operating forces on the system, but considers the assumption of white noise as input. Some investigations have been already made on vibro-acoustical operational modal analysis (Abdelghani et al., 1999; Hermans & Van der Auweraer, 1999), hereafter the modal decomposition in case of vibro-acoustical operational modal analysis is reviewed, in order to point out the influence on the modal parameter estimation of the references choice, i.e. acoustical or structural, used for the Cross power spectra calculation. The complete knowledge of the vibro-acoustical response of an helicopter cabin is a crucial tool for noise control purpose, and in the OMA context, it is strongly related to the reference choice rather than the input forces. For simplicity, only an acoustical input will be considered and the cross power between acoustical and structural responses will be derived and compared with structural cross power responses. The formulation in case of structural excitation can be derived in a similar way. The theoretical results have been verified through an experimental testing on the helicopter EUROCOPTER

The dynamical properties of vibro-acoustical systems are presented hereafter. In particular the transfer function considering as excitation both the acoustical loading and the structural force and considering as response both the structural displacement and the acoustical pressure

numerical values.

EC-135.

**2.1 Vibro-acoustical systems**

**2. Noise control in coupled vibro-acoustical systems**

is derived. It is well known that due to the fluid-structure interaction such systems are nonsymmetrical. Moreover, the FRF has a different form with respect to the decoupled systems. It is worth to stress that the non-symmetry is due to the choice of the variables **x**, **p**, **f** and · **q**, being **x** the structural displacement, **p** the sound pressure, **f** the structural force and **q** the volume velocity. As stated above, due to the non-symmetry of the system, the right and the left eigenvectors, Ψ*<sup>r</sup>* and Ψ*l*, are different. Moreover the following relations can be proven (Ma & Hagiwara, 1991):

$$\left\{ \begin{array}{c} \Psi\_{sl} \\ \Psi\_{fl} \end{array} \right\}\_{\lambda\_{\hbar}} = \left\{ \begin{array}{c} \Psi\_{sr} \\ \frac{1}{\lambda\_{\hbar}^2} \Psi\_{fr} \end{array} \right\}\_{\lambda\_{\hbar}} \tag{1}$$

where subscript *s* is for the structural response location, *f* for the fluid (acoustical) response location, and *λ<sup>h</sup>* are the roots of the system characteristic equation. In particular the following transfer functions obtained considering a structural force excitation *fj* at location *j*, have been derived in (Wyckaert et al., 1996):

$$\frac{\varphi\_{\rm m}}{f\_{\rm j}} = \sum\_{h=1}^{n} \frac{\left(P\_{\rm h}\right) \left(\psi\_{\rm shm}\right) \left(\psi\_{\rm shj}\right)}{\left(z - \lambda\_{\rm h}\right)} + \frac{\left(P\_{\rm h}\right)^{\*} \left(\psi\_{\rm shm}\right)^{\*} \left(\psi\_{\rm shj}\right)^{\*}}{\left(z - \lambda\_{\rm h}^{\*}\right)} \tag{2}$$

$$\frac{p\_p}{f\_{\bar{f}}} = \sum\_{h=1}^{n} \frac{(P\_h) \left(\psi\_{fhp}\right) \left(\psi\_{slij}\right)}{(z-\lambda\_h)} + \frac{(P\_h)^\* \left(\psi\_{fhp}\right)^\* \left(\psi\_{slij}\right)^\*}{\left(z-\lambda\_h^\*\right)}\tag{3}$$

where *xm* is the structural displacement at location *m*, *pp* the acoustical pressure response inside the cavity at location *p*, *Ph* the modal scaling factors and *n* the number of modes in the frequency band of interest. Similar transfer functions can be obtained considering an acoustical excitation · *q* at location *k*:

$$\frac{\chi\_m}{\dot{q}\_h} = \sum\_{h=1}^n \frac{\left(P\_h\right) \left(\psi\_{shm}\right) \left(\psi\_{fhk}\right)}{\lambda\_h^2 \left(z - \lambda\_h\right)} + \frac{\left(P\_h\right)^\* \left(\psi\_{shm}\right)^\* \left(\psi\_{fhk}\right)^\*}{\lambda\_h^2 \left(z - \lambda\_h^\*\right)}\tag{4}$$

$$\frac{p\_p}{q\_k} = \sum\_{h=1}^n \frac{\left(P\_h\right)\left(\psi\_{fhp}\right)\left(\psi\_{fhk}\right)}{\lambda\_h^2 \left(z-\lambda\_h\right)} + \frac{\left(P\_h\right)^\*\left(\psi\_{fhp}\right)^\*\left(\psi\_{fhk}\right)^\*}{\lambda\_h^2 \left(z-\lambda\_h^\*\right)}\tag{5}$$

Considering *p* = *k* and *j* = *m* the vibroacoustical reciprocity, i.e. (*pk*/ *fm*) = − ·· *xm*/ · *qk* , can be derived from (3) and (4). This modal description (Eqs.2-5) can be used for the parameter estimation (mode shapes, modal frequencies and damping factors) by means of standard algorithms.

Operational Modal Analysis (Peeters et al., 2007) is also called output-only modal analysis, and consists in a modal analysis without the knowledge and/or control of the input excitation. Assuming white noise as input, it enables to estimate the modal parameters from the cross power spectra (CPs) between the outputs and some responses chosen as references. In practice, the main difference with respect to the classical modal analysis is that the cross power spectra between the outputs are used to estimate the modal parameters, instead of the genuine FRFs (ratio between the system responses and the input forces). In vibro-acoustical systems the fluid-structure interaction yields both the acoustical and the structural outputs.

The power spectrum of the outputs, hence, can be obtained in three different forms, i.e. with acoustical outputs only, with structural output only and with both acoustical and structural outputs. Moreover, the input signal, supposed to be white noise, can be either structural or acoustical (or both of them simultaneously). In this section the results of the formulation of the operational modal analysis for vibro-acoustical systems are reported and discussed. For simplicity, only an acoustical input will be considered and the cross power between structural and the acoustical responses will be shown.

Let define the cross-correlation function *Rmpk* (*T*) as the expected value **E** of the product of two stationary responses *xm* (*t*) and *pp* (*t*) due to the single · *qk* acoustical input at the *k*th degree of freedom (d.o.f.) evaluated at a time separation of *T*, i.e. *Rmpk* (*T*) = **E** � *xm* (*t* + *T*) *pp* (*t*) � . Assuming white noise (i.e. **E** � · *qk* (*τ*) · *qk* (*σ*) � = *αkδ* (*τ* − *σ*), with *α<sup>k</sup>* a constant value related to the input and *δ* (*t*) the Dirac delta function) and performing some calculations, one obtains the cross power spectrum between *acoustical f* and *structural s responses*:

$$\left[\mathbf{G}\_{fs}\left(i\omega\right)\right] = \sum\_{h=1}^{n} \left[\frac{\left\{\mathbf{Y}\_{s}\right\}\_{h}\left\{\mathbf{Q}\_{fs}\right\}\_{h}^{T}}{\left(i\omega-\lambda\_{h}\right)} + \frac{\left\{\mathbf{Y}\_{s}\right\}\_{h}^{\*}\left\{\mathbf{Q}\_{fs}\right\}\_{h}^{H}}{\left(i\omega-\lambda\_{h}^{\*}\right)} + \frac{\left\{\mathbf{Q}\_{fs}\right\}\_{h}\left\{\mathbf{Y}\_{f}\right\}\_{h}^{T}}{\left(-i\omega-\lambda\_{h}\right)} + \frac{\left\{\mathbf{Q}\_{fs}\right\}\_{h}^{\*}\left\{\mathbf{Y}\_{f}\right\}\_{h}^{H}}{\left(-i\omega-\lambda\_{h}^{\*}\right)}\right] \right] \tag{6}$$

with:

$$
\left(Q\_{fs}\right)\_{ph} = \sum\_{j=1}^{n} -a\_k \left[ \frac{\left(P\_h\right)\left(\boldsymbol{\upmu}\_{fhk}\right)\left(\boldsymbol{P}\_j\right)\left(\boldsymbol{\upmu}\_{fjk}\right)\left(\boldsymbol{\upmu}\_{fjk}\right)}{\lambda\_h^2 \lambda\_j^2 \left(\lambda\_h + \lambda\_j\right)} + \frac{\left(P\_h\right)\left(\boldsymbol{\upmu}\_{fhk}\right)\left(\boldsymbol{P}\_j\right)^\*\left(\boldsymbol{\upmu}\_{fjk}\right)^\*\left(\boldsymbol{\upmu}\_{fjk}\right)}{\lambda\_h^2 \lambda\_j^2 \left(\lambda\_h + \lambda\_j^\*\right)}\right] \tag{7}
$$

$$
\left(Q\_{fs}\right)\_{mh} = \sum\_{j=1}^{n} -a\_k \left[ \frac{\left(P\_j\right)\left(\boldsymbol{\upmu}\_{sjm}\right)\left(\boldsymbol{\upmu}\_{fjk}\right)\left(\boldsymbol{P}\_h\right)\left(\boldsymbol{\upmu}\_{fhk}\right)}{\lambda\_h^2 \lambda\_j^2 \left(\lambda\_h + \lambda\_j\right)} + \frac{\left(P\_j\right)^\*\left(\boldsymbol{\upmu}\_{sjm}\right)^\*\left(\boldsymbol{\upmu}\_{fjk}\right)^\*\left(\boldsymbol{P}\_h\right)\left(\boldsymbol{\upmu}\_{fhk}\right)}{\lambda\_h^2 \lambda\_j^2 \left(\lambda\_h + \lambda\_j^\*\right)}\right] \tag{8}
$$

The unstable poles −*λ<sup>h</sup>* and *λ*<sup>∗</sup> *<sup>h</sup>* derive from the negative time lags of the cross-correlation function *Rmpk* (*T*). Consequently they can be deleted by means of the weighted correlogram approach determining the following half spectrum:

$$\left[\mathbf{G}\_{fs}\left(i\omega\right)^{++}\right] = \sum\_{h=1}^{n} \left[\frac{\left\{\mathbf{Y}\_{s}\right\}\_{h}\left\{\mathbf{Q}\_{fs}\right\}\_{h}^{T}}{\left(i\omega - \lambda\_{h}\right)} + \frac{\left\{\mathbf{Y}\_{s}\right\}\_{h}^{\*}\left\{\mathbf{Q}\_{fs}\right\}\_{h}^{H}}{\left(i\omega - \lambda\_{h}^{\*}\right)}\right] \tag{9}$$

The cross power spectrum between two *structural responses s* is:

$$\left[\mathbf{G}\_{\rm ss}\left(i\omega\right)\right] = \sum\_{h=1}^{n} \left[\frac{\left\{\mathbf{V}\_{\rm s}\right\}\_{\hbar}\left\{\mathbf{Q}\_{\rm ss}\right\}\_{\hbar}^{T}}{\left(i\omega-\lambda\_{\hbar}\right)} + \frac{\left\{\mathbf{V}\_{\rm s}\right\}\_{\hbar}^{\*}\left\{\mathbf{Q}\_{\rm ss}\right\}\_{\hbar}^{H}}{\left(i\omega-\lambda\_{\hbar}^{\*}\right)} + \frac{\left\{\mathbf{Q}\_{\rm ss}\right\}\_{\hbar}\left\{\mathbf{V}\_{\rm s}\right\}\_{\hbar}^{T}}{\left(-i\omega-\lambda\_{\hbar}\right)} + \frac{\left\{\mathbf{Q}\_{\rm ss}\right\}\_{\hbar}^{\*}\left\{\mathbf{V}\_{\rm s}\right\}\_{\hbar}^{H}}{\left(-i\omega-\lambda\_{\hbar}^{\*}\right)}\right] \tag{10}$$

with:

$$(\mathcal{Q}\_{ss})\_{ph} = \sum\_{j=1}^{n} -a\_k \left[ \frac{(P\_h) \left(\psi\_{fhk}\right) \left(P\_j\right) \left(\psi\_{sjp}\right) \left(\psi\_{fjk}\right)}{\lambda\_h^2 \lambda\_j^2 \left(\lambda\_h + \lambda\_j\right)} + \frac{(P\_h) \left(\psi\_{fhk}\right) \left(P\_j\right)^\* \left(\psi\_{sjp}\right)^\* \left(\psi\_{fjk}\right)^\*}{\lambda\_h^2 \lambda\_j^2 \left(\lambda\_h + \lambda\_j^\*\right)}\right] \tag{11}$$

4 Will-be-set-by-IN-TECH

The power spectrum of the outputs, hence, can be obtained in three different forms, i.e. with acoustical outputs only, with structural output only and with both acoustical and structural outputs. Moreover, the input signal, supposed to be white noise, can be either structural or acoustical (or both of them simultaneously). In this section the results of the formulation of the operational modal analysis for vibro-acoustical systems are reported and discussed. For simplicity, only an acoustical input will be considered and the cross power between structural

Let define the cross-correlation function *Rmpk* (*T*) as the expected value **E** of the product of two

the input and *δ* (*t*) the Dirac delta function) and performing some calculations, one obtains

� <sup>+</sup>

� <sup>+</sup>

function *Rmpk* (*T*). Consequently they can be deleted by means of the weighted correlogram

*<sup>h</sup>* {**Q***ss*}*<sup>H</sup> h*

� <sup>+</sup>

⎡ ⎢ ⎣

> � *iω* − *λ*<sup>∗</sup> *h*

{Ψ*s*}*<sup>h</sup>* � **Q***f s* �*<sup>T</sup> h* (*i<sup>ω</sup>* <sup>−</sup> *<sup>λ</sup>h*) <sup>+</sup>

� **Q***f s* � *h* � Ψ*f* �*<sup>T</sup> h*

(*Ph*) � *ψf hk* � � *Pj* �∗ � *ψfjp* �∗ � *ψf jk* �∗

� *Pj* �∗ � *ψsjm* �∗ � *ψf jk* �∗ (*Ph*) � *ψf hk* �

*<sup>h</sup>* derive from the negative time lags of the cross-correlation

{Ψ*s*}<sup>∗</sup> *h* � **Q***f s* �*<sup>H</sup> h*

� *iω* − *λ*<sup>∗</sup> *h* �

� <sup>+</sup> {**Q***ss*}*<sup>h</sup>* {Ψ*s*}*<sup>T</sup>*

(*Ph*) � *ψf hk* � � *Pj* �∗ � *ψsjp* �∗ � *ψf jk* �∗

*qk* acoustical input at the *k*th degree

�

� **Q***f s* �∗ *h* � Ψ*f* �*<sup>H</sup> h*

�

= *αkδ* (*τ* − *σ*), with *α<sup>k</sup>* a constant value related to

(−*i<sup>ω</sup>* <sup>−</sup> *<sup>λ</sup>h*) <sup>+</sup>

*λ*2 *hλ*2 *j* � *λ<sup>h</sup>* + *λ*<sup>∗</sup> *j* �

*λ*2 *hλ*2 *j* � *λ<sup>h</sup>* + *λ*<sup>∗</sup> *j* �

*h* (−*i<sup>ω</sup>* <sup>−</sup> *<sup>λ</sup>h*) <sup>+</sup> {**Q***ss*}<sup>∗</sup>

> *λ*2 *hλ*2 *j* �

⎤ ⎥

�

*λ<sup>h</sup>* + *λ*<sup>∗</sup> *j* �

<sup>⎦</sup> (9)

*<sup>h</sup>* {Ψ*s*}*<sup>H</sup> h*

−*iω* − *λ*<sup>∗</sup> *h* � �

*xm* (*t* + *T*) *pp* (*t*)

−*iω* − *λ*<sup>∗</sup> *h* � � .

> ⎤ ⎥ ⎦

(6)

⎤ ⎥ ⎦

(7)

(8)

(10)

⎤ ⎥ ⎦

(11)

⎤ ⎥ ⎦

and the acoustical responses will be shown.

Assuming white noise (i.e. **E**

⎡ ⎢ ⎣ {Ψ*s*}*<sup>h</sup>* � **Q***f s* �*<sup>T</sup> h* (*i<sup>ω</sup>* <sup>−</sup> *<sup>λ</sup>h*) <sup>+</sup>

⎡ ⎢ ⎣

> ⎡ ⎢ ⎣

� *Pj* � � *ψsjm* � � *ψf jk* � (*Ph*) � *ψf hk* �

approach determining the following half spectrum:

{Ψ*s*}*<sup>h</sup>* {**Q***ss*}*<sup>T</sup>*

++� = *n* ∑ *h*=1

The cross power spectrum between two *structural responses s* is:

*λ*2 *hλ*2 *j* � *λ<sup>h</sup>* + *λ<sup>j</sup>*

*h* (*i<sup>ω</sup>* <sup>−</sup> *<sup>λ</sup>h*) <sup>+</sup> {Ψ*s*}<sup>∗</sup>

**G***f s* (*iω*)

(*Ph*) � *ψf hk* � � *Pj* � � *ψfjp* � � *ψf jk* �

�

with:

� *Qf s* � *ph* <sup>=</sup>

� *Qf s* � *mh* <sup>=</sup>

**G***f s* (*iω*)

� = *n* ∑ *h*=1

> *n* ∑ *j*=1 −*α<sup>k</sup>*

*n* ∑ *j*=1 −*α<sup>k</sup>*

The unstable poles −*λ<sup>h</sup>* and *λ*<sup>∗</sup>

�

*n* ∑ *h*=1

*n* ∑ *j*=1 −*α<sup>k</sup>*

�

⎡ ⎢ ⎣ (*Ph*) � *ψf hk* � � *Pj* � � *ψsjp* � � *ψf jk* �

[**G***ss* (*iω*)] =

(*Qss*)*ph* =

with:

stationary responses *xm* (*t*) and *pp* (*t*) due to the single ·

� · *qk* (*τ*) ·

*λ*2 *hλ*2 *j* � *λ<sup>h</sup>* + *λ<sup>j</sup>*

*λ*2 *hλ*2 *j* � *λ<sup>h</sup>* + *λ<sup>j</sup>*

of freedom (d.o.f.) evaluated at a time separation of *T*, i.e. *Rmpk* (*T*) = **E**

the cross power spectrum between *acoustical f* and *structural s responses*:

*qk* (*σ*) �

> {Ψ*s*}<sup>∗</sup> *h* � **Q***f s* �*<sup>H</sup> h*

� *iω* − *λ*<sup>∗</sup> *h* � <sup>+</sup>

Fig. 1. Measured points of the cabin external surface.

$$(Q\_{ss})\_{mh} = \sum\_{j=1}^{n} -a\_k \left[ \frac{\left(P\_j\right)\left(\psi\_{jfm}\right)\left(\psi\_{fjk}\right)\left(P\_h\right)\left(\psi\_{fhk}\right)}{\lambda\_h^2 \lambda\_j^2 \left(\lambda\_h + \lambda\_j\right)} + \frac{\left(P\_j\right)^\*\left(\psi\_{sjm}\right)^\*\left(\psi\_{fjk}\right)^\*\left(P\_h\right)\left(\psi\_{fhk}\right)}{\lambda\_j^2 \lambda\_h^2 \left(\lambda\_j^\* + \lambda\_h\right)}\right] \tag{12}$$

By looking at Eqs. (6),(10) it is clear that the stable pole terms of the sum (i.e. the positive *λ<sup>h</sup>* and *λ*<sup>∗</sup> *<sup>h</sup>*) refer to the same mode shapes {Ψ*s*}*h*. The only differences between the two formulations are present in the output-only reference vectors {**Q**}*h*. Moreover, rewriting Eqs. (4) in a matrix form:

$$\left[\mathbf{H}\_{fs}\left(\dot{\mathbf{u}}\omega\right)\right] = \sum\_{h=1}^{n} \left(\frac{\left\{\mathbf{V}\_{s}\right\}\_{h}\left\{\mathbf{L}\_{f}\right\}\_{h}^{T}}{\left(\dot{\mathbf{u}}\omega - \boldsymbol{\lambda}\_{h}\right)} + \frac{\left\{\mathbf{V}\_{s}\right\}\_{h}^{\*}\left\{\mathbf{L}\_{f}\right\}\_{h}^{H}}{\left(\dot{\mathbf{u}}\omega - \boldsymbol{\lambda}\_{h}^{\*}\right)}\right) \tag{13}$$

it is evident that Eqs. (6),(10),(13), i.e. respectively the modal decompositions of OMA with an acoustical reference, OMA with a structural reference and vibro-acoustical EMA, depend on the same modal vector {Ψ*s*}*<sup>h</sup>* and system poles *λh*. This means that in order to estimate the modal parameters (i.e. natural frequency and modal damping) it is possible to consider acoustical references instead of the structural ones as used in the common practice of OMA. This is useful for experimental campaigns aimed to a modal analysis of high scale vibro-acoustical structures for which it is common to take a microphone as reference inside the cavity, near the acoustical excitation. No additional reference sensors, hence, have to be located on the surface. In order to verify the practical effectiveness of the modal model obtained considering an acoustical reference (Eq.(6)), an operational modal analysis on a helicopter has been carried out, comparing the results with a classical EMA (Eq.(13)) where the input is the measured excitation (volume acceleration source).

#### **2.2 Experimental application**

A case study has been performed employing the data acquired from a test conducted on the helicopter EUROCOPTER EC-135. The EC-135 is a light twin-engine, multi-purpose helicopter with up to eight seats for pilot/s and passengers. Helicopters are typical examples of coupled vibro-acoustical systems related to a cavity environment. The experimental tasks about these systems mainly regard noise sources identification and modal parameter extraction. In this context, experimental and operational modal analyses are typically used to make the spaced peaks of the input force spectrum coincident with the valleys of the structural frequency response functions, for noise reduction purpose. These two analyses have been

Fig. 2. First mode shape; bottom view of the helicopter cabin.

performed on the cabin external surface of the helicopter while the cavity was excited by means of an acoustical loading. The setup and the results will be shown in the following.

### **2.3 Test setup**

For these applications, acoustical excitation is often preferred since it is usually more practicable than placing a shaker. Moreover, the acoustics of the cavity is excited in a direct way and the measurements are more efficient. In this testing campaign, indeed, a low frequency volume acceleration source (frequency range 20 − 400 Hz) has been located inside the cabin near the pilot's seat, exciting the structure with a random signal. A microphone near the loudspeaker has been used as reference. The structural responses of 1158 points on the right, left and roof part of the cabin external surface (Figure 1) have been measured through a 4x4 PU sensors array, with a 10 cm spacing between the sensors both horizontally and vertically. The PU probes, integrate in a unique casing a hot-wire particle velocity sensor and a very small pressure transducer. They are, hence, specially conceived sensors for measuring acoustic intensity. The distance between the PU probes and the surface, indeed, can be so short that the very-near-field assumption is verified, i.e. the particle velocities simply represent the velocities of the cabin surface (in the normal direction) (De Bree et al., 2004). This is due to the particle velocity level, which is just slightly dependent on the distance to the object and not on the frequency. The very-near-field is a sound field very close to a source, where the particle velocity level is elevated compared to the sound pressure level and a phase shift between sound pressure and particle velocity is observed. The two conditions of the very near field are i) the measurement distance should be closer than the structural size of the object divided by 2*π* and ii) the wavelength of the sound should be larger than the structural size of the object. At last, the possibility of getting with just one measurement the needed data for both a modal analysis and an acoustic intensity calculation highlights the usefulness of these PU probes in this field, instead of classical methods such as accelerometers and scanning laser vibrometers. This point will be shown more in deep hereafter.

### **2.4 Test results**

The results obtained by the two analyses, i.e. EMA and OMA, are shown in the following. In particular the aim is to experimentally verify the effectiveness of the parameter estimation obtained considering an acoustical reference in the OMA analysis, as theoretically predicted through the comparison between the formulation derived in Section 2.1 (Eq.(9)) and the vibro-acoustical EMA (Wyckaert et al., 1996) (Eq.(13)). Tab.1 lists the results in terms of natural frequency *Fn* for the modes lying in the 20 − 200 Hz frequency range. In addition, the modal assurance criterion values (MAC) between the two analyses are given in Tab.1.

6 Will-be-set-by-IN-TECH

performed on the cabin external surface of the helicopter while the cavity was excited by means of an acoustical loading. The setup and the results will be shown in the following.