Abstract:
An apparatus and method for attenuating noise and vibration in a propeller aircraft comprises a tuned vibration absorber adapted to be mounted to the skin of a fuselage of a propeller aircraft. The tuned vibration absorber may be tuned to the second harmonic of a blade passage frequency of a propeller of the aircraft. The tuned vibration absorber may be connectable to a separately formed mount, which can be mounted to the fuselage skin independently of the tuned vibration absorber. The apparatus may include a cover to prevent interference between the tuned vibration absorber and other components in the aircraft. The apparatus may be used in combination with other attenuation systems to attenuate noise and vibration over a broad range of frequencies.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    In comparison to turbojet transportation, propeller driven aircraft, such as piston-engine and turboprop aircraft, have historically been considered less comfortable from a noise and vibration standpoint. In a propeller driven aircraft, the propellers tend to contribute noise and vibration at a frequency referred to as the “blade passage frequency” and at harmonics of that frequency. As referred to herein, the blade passage frequency or “BPF” is the product of the propeller shaft rotational speed times the number of blades on a propeller. For example, an aircraft having a 3-blade propeller on a shaft turning at 2000 revolutions per minute has a BPF of 3×2000 or 6000/minute, i.e., 100 Hz. Most commonly, all of the propellers on a multi-propeller aircraft have the same number of blades and operate at the same rotational speed, so that there is only a single BPF for the entire aircraft. Typical aircraft have at least one preferred cruise setting, at which the propellers will operate at a particular rotational speed. Thus, there will typically be a single BPF for the entire aircraft corresponding to that preferred cruise setting. 
         [0002]      FIG. 3  is a graph showing representative interior noise levels in one type of propeller aircraft at high speed cruise. As shown in the graph, the sound pressure level spikes at the BPF and successive harmonics thereof. This aircraft has a BPF of approximately 100 Hz. The interior sound pressure level spikes at a frequency equal to the BPF, also referred to herein as the “first harmonic of the BPF.” The sound pressure level also spikes at the second harmonic of the BPF, or approximately 200 Hz, and at the third harmonic of the BPF, or approximately 300 Hz, and so on. As the graph shows, the highest two spikes in sound pressure level are at the first and second harmonics of the BPF. 
         [0003]    Over the past several decades, considerable effort has been expended developing systems to attenuate some of the undesirable noise and vibration in propeller driven aircraft. Major considerations in the development of noise and vibration control systems include keeping added weight and cost to a minimum, while maximizing attenuation of unwanted noise and vibration. 
         [0004]    Such noise and vibration control systems are generally classified as either active or passive. Active systems comprise using secondary control sources to add additional energy to a vibrating system to cancel out the primary excitation. For example, active noise control comprises using acoustic sources, such as loudspeakers, to cancel targeted sound within the aircraft coming from the propellers. Active structural acoustic control, on the other hand, comprises using vibration inputs, such as shakers or piezoelectric materials, to modify the sound field in the aircraft. Another technique includes “synchrophasing,” which includes adjusting the relative rotational phase of the propellers in a multiple-propeller aircraft to reduce interior noise. 
         [0005]    Passive systems, on the other hand, do not require a power source to provide energy to the system. Passive techniques include providing damping material, such as thermal/acoustical insulation blankets, along the interior of the aircraft fuselage to muffle sound transmission into the interior of the aircraft. Other passive systems include providing vibration absorbers to attenuate vibration of the fuselage structure. For example, one such prior art system (as shown in  FIGS. 1 and 2 ) includes mounting tuned vibration absorbers (“TVAs”) to the frames of the fuselage to attenuate vibration of the fuselage structure. 
         [0006]      FIGS. 1 and 2  illustrate portions of the interior of the fuselage of the King Air 350 model turboprop aircraft manufactured by Hawker Beechcraft Corporation. The fuselage comprises a series of frames  10 . Each frame  10  is generally in the form of a ring that extend around the fuselage in the circumferential direction. The frames  10  are spaced apart along the longitudinal extent of the aircraft and are interconnected by a series of stringers  12 , which run along the longitudinal direction of the aircraft, transverse to the frames  10 . The frames  10  and stringers  12  are connected to the skin  14  of the aircraft, which forms the exterior surface of the aircraft fuselage and which encloses the interior volume of the aircraft. 
         [0007]    Vibration attenuation systems  16  are attached to the frames  10  for lessening vibration of the fuselage. Each attenuation system  16  comprises first and second TVAs  18 ,  20  connected to a mounting bracket  22 . Each mounting bracket  22  is secured to a fuselage frame  10  such that the TVAs  18 ,  20  are positioned adjacent to the frame  10 . The attenuation systems  16  are generally arranged in pairs at each frame  10 , as shown in  FIG. 2 , with attenuation systems  16  being attached to both the forward and aft sides of the frames  10 . The TVAs  18 ,  20  of each attenuation system  16  are arranged to attenuate vibration of the frame  10  in the direction normal to the aircraft fuselage. Each TVA  18 ,  20  is a mass and spring system. The mass is configured to move towards and away from the central longitudinal axis of the fuselage to attenuate vibration in that direction. Specifically, each of the TVAs  18 ,  20  includes a spring in the form of an elongated plate  24  connected to the mounting bracket  22  at approximately the center of the plate  24  and having masses  26  connected at each end of the plate  24 . The plate  24  is flexible and permits the masses  26  to move towards and away from the central longitudinal axis of the fuselage in response to vibration of the frame  10  along that direction, which vibration is transmitted to the TVAs  18 ,  20  through the mounting bracket  22 . Each of the two TVAs  18 ,  20  is designed to be tuned to a different frequency. In particular, the first TVA  18  is tuned to 100 Hz (i.e., the first harmonic of the BPF) and the second TVA  20  is tuned to 200 Hz (i.e., the second harmonic of the BPF). 
         [0008]    Despite the above progress in the art, further improvement is still desirable. 
       BRIEF SUMMARY OF THE INVENTION 
       [0009]    One aspect of the present invention provides an apparatus for attenuating noise and vibration in a propeller aircraft. The apparatus according to this aspect of the invention desirably includes a mount adapted to be mounted to the skin of the aircraft fuselage. The apparatus also desirably includes a tuned vibration absorber which is formed separately from the mount and which is adapted to connect to the mount. 
         [0010]    Another aspect of the present invention provides a method for attenuating noise and vibration in a propeller aircraft. The method according to this aspect of the invention desirably includes mounting a mount to the skin of the aircraft fuselage. The method also desirably includes connecting a tuned vibration absorber to the mount. 
         [0011]    The mount may be mounted to the skin before cables or other equipment are provided. Preferably the tuned vibration absorber is connected to the mount after the cables and other equipment are provided. The tuned vibration absorber is also preferably connected to the mount before thermal or acoustical insulation is provided. According to one aspect of the invention, the tuned vibration absorber is encased in a cover, which is adapted to prevent interference between the insulation and the tuned vibration absorber. 
         [0012]    Although the present invention is not limited by any theory of operation, it is believed that, even when the fuselage frames are damped, the skin continues to exhibit significant localized skin vibrations at particular harmonics of the BPF. Mounting to the skin a TVA tuned to that harmonic of the BPF is believed to attenuate such localized skin vibrations, and hence interior noise. In particular, a TVA tuned to the second harmonic of the BPF, or a higher harmonic, provides more effective attenuation when mounted to the skin than when mounted to the frame. By contrast, a TVA tuned to the first harmonic (i.e., fundamental) of the BPF provides effective attenuation when mounted to the frame, since the fuselage vibration at that low frequency is believed to be dominated by the frames. Thus, a particularly effective system uses TVAs tuned to the first harmonic mounted to the frame in combination with TVAs tuned to the second harmonic mounted to the skin. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0013]      FIG. 1  is a perspective view of portions of a prior art propeller aircraft and a prior art system for attenuating noise and vibration in such aircraft. 
           [0014]      FIG. 2  is an elevational view depicting an arrangement of the components shown in  FIG. 1 . 
           [0015]      FIG. 3  is a representative graph of interior noise in an aircraft. 
           [0016]      FIG. 4  is a sectional view of certain components of a system for attenuating noise and vibration in a propeller aircraft in accordance with one embodiment of the invention. 
           [0017]      FIG. 5  is a perspective view of the components of  FIG. 4 . 
           [0018]      FIG. 6  is a perspective view depicting a plurality of the components of  FIG. 4 , together with additional components of the vibration attenuation system and aircraft. 
       
    
    
     DETAILED DESCRIPTION 
       [0019]    In describing the preferred embodiments of the invention illustrated in the appended drawings, specific terminology will be used for the sake of clarity. However, the invention is not intended to be limited to the specific terms so selected. 
         [0020]    An apparatus  27  for attenuating noise and vibration in a propeller aircraft in accordance with one embodiment of the present invention includes a TVA  28  ( FIG. 4 ). The TVA  28  is comprised of a mass  30  and an elastomeric component  32  attached to a centerpost  34 . The mass  30  is shaped as a generally annular thick-walled metal tube. The elastomeric component  32  also has a generally annular shape sized to fit between the mass  30  and the centerpost  34 . The centerpost  34  is shaped as an elongated, generally cylindrical rod. At one end of the centerpost  34  is a coupling element in the form of a female threaded socket  38 . At the opposite end of the centerpost  34  is another coupling element in the form of a male threaded portion  42 . The elastomeric component  32  is preferably injection molded, and the mass  30 , elastomeric component  32 , and centerpost  34  can thus be bonded together during the molding process. Alternatively, the mass  30 , elastomeric component  32 , and centerpost  34  can be attached together after they are manufactured by, for example, bonding them together with an adhesive or securing them together with mechanical fasteners. The mass  30 , elastomeric component  32 , and centerpost  34  are interconnected such that the elastomeric component  32  acts as a spring permitting the mass  30  to move back and forth along the longitudinal axis of the centerpost  34 . The elastomeric nature of the elastomeric component  32  also provides damping to the TVA  28 . 
         [0021]    The materials for the mass  30  and elastomeric component  32  are selected in conjunction with the sizing and shaping of those components such that the TVA  28  is tuned to a desired frequency. The TVA  28  is preferably tuned to one of the significant vibration frequencies exhibited by the fuselage skin  14  at a particular cruise setting. For example, a preferred tuning frequency is the second harmonic of the propeller BPF at high speed cruise. The relationship between the tuning frequency and the mass and stiffness of a tuned vibration absorber is known per se and is given by the equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                     t 
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                       
                         k 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where f t  is the tuning frequency in Hz, k is the stiffness of the spring, and m is the TVA mass. The parameters relating to the mass  30  are typically designed based on considerations of the modal mass of the vibrating structure that is being attenuated (e.g., a skin panel). An exemplary mass  30  may be constructed of stainless steel and may have a weight of approximately 0.1 pounds (0.045 kg), a height of approximately inch (2.5 cm), an outer diameter of approximately 1 inch (2.5 cm), and an inner diameter of approximately 0.75 inches (1.9 cm). The parameters relating to the elastomeric component  32  are then designed to produce the desired stiffness and damping based on the selected mass. It is noted that increasing the amount of damping will provide a greater bandwidth of attenuation, however such an increase will conversely decrease the amount of attenuation provided at the tuned frequency. The amount of damping can be expressed as a quality factor, or “Q factor,” which characterizes the TVA&#39;s tuned frequency relative to its bandwidth. Preferred Q factors for the skin-mounted TVA  28  may be in the range of 14 to 33. Exemplary materials for the elastomeric component  32  may include silicone, flurosilicone, or any other rubber-like polymer. 
         [0022]    The male threaded portion  42  of the centerpost  34  is configured to engage a cover  44  to secure the cover  44  to the centerpost  34 . The threaded portion  44  may include thread lock to prevent loosening of the cover  44  from the centerpost  34 . The cover  44  is preferably a generally hollow cylindrical component sized and shaped to encase the TVA  28  to prevent contact between the TVA  28  and external objects that could interfere with the movement of the mass  30 . The cover  44  may be made up of a circular top wall  46  and a cylindrical side wall  48 . The top wall  46  and side wall  48  may be integrally formed, such as by casting or machining the cover  44  to the preferred shape, or the top wall  46  and side will  48  may be separately formed components that are joined together by mechanical fasteners, adhesive bonding, welding, or any other appropriate manner. The cover  44  preferably includes a coupling element in the form of a threaded through-hole  50  shaped to securely engage the threaded portion  42  of the centerpost  34 . 
         [0023]    The centerpost  34  is configured to releasably secure the TVA  28  to a mount  36 . A coupling element on the mount  36 , in the form of a male threaded stud  40 , is shaped to releasably connect to the female threaded socket  38 . Thread lock may optionally be provided at the threaded interface to prevent loosening of the threaded engagement. The mount  36  may be formed from a generally circular plate element  52  having the stud  40  disposed at the center thereof. The plate element  52  and the stud  40  may be integrally formed, such as by casting or machining, but those components could also be separately formed and joined together by any appropriate manner. The plate element  52  has a substantially flat back side  54  which is shaped to abut the interior surface  56  of the fuselage skin  14 . The plate element  52  is configured to be securely connected to the skin  14 , such as by adhesively bonding the components together. Preferably the adhesive is designed to support both the weight of the TVA apparatus  27  and the loads subjected to the apparatus  27 . In addition, a preferred adhesive should resist stress corrosion and have a set time of less than 2 hours. The adhesive is preferably applied between the back side  54  of the plate element  52  and the interior surface  56  of the fuselage skin  14 , in order to secure the TVA mount  36  to the skin  14 . 
         [0024]    The centerpost  34 , mount  36 , and cover  44  are preferably constructed of materials that provide strength and durability while minimizing weight and increased costs. Preferred materials may include aluminum, but any other suitable materials may be used. 
         [0025]    In an aircraft fuselage, a bay  68  (see  FIGS. 1 and 6 ) is defined as the space defined between two adjacent frames  10  and two adjacent stringers  12 . A skin panel  70  (see  FIGS. 1 and 6 ) is defined as the approximately rectangular portion of fuselage skin  14  defined between adjacent frames  10  and adjacent stringers  12  and adjoining each bay  68 . A skin panel  70  will have minimum flexural rigidity at its center, and thus forced vibration from the propellers at the second harmonic of the BPF will likely result in the highest deflections of the skin panel  70  at its center point. Thus, a preferred location of the skin-mounted TVA apparatus  27 , in accordance with one aspect of the present invention, may include locating one of the apparatuses  27  at approximately the center of the skin panel  70 . A system of such skin-mounted TVA apparatuses  27  may be distributed throughout the aircraft in numerous bays  68 , as shown in  FIG. 6 . It is believed that the noise and vibration attenuation of the skin-mounted TVA apparatuses  27  has a local effect, and therefore, to maximize effectiveness while minimizing weight and cost, it is preferable to locate the apparatuses  27  where they will produce the greatest noise and vibration reduction from the standpoint of the aircraft passengers. In this regard, a preferred arrangement of skin-mounted TVA apparatuses  27  may include locating one apparatus in each bay  68  in the passenger portion of the aircraft (i.e., the portion above the floor line). Furthermore, particularly important locations may include the skin panels  70  just below the passenger windows  72 . 
         [0026]    In conjunction with the system of skin-mounted TVA apparatuses  27  described above, frame mounted TVAs may also be incorporated as part of a system for attenuating noise and vibration in accordance with a preferred embodiment of the present invention. In a preferred system, as shown in  FIG. 6 , the skin-mounted TVA apparatuses  27  containing TVAs tuned to the second harmonic of the BPF are implemented in conjunction with frame-mounted TVAs  18  tuned to the first harmonic. 
         [0027]    A surface damping treatment  58  is also preferably used in conjunction with the system of frame-mounted and skin-mounted TVAs. The surface damping treatment  58  may be applied directly to the fuselage skin  14  to dissipate vibrational energy. The surface damping treatment  58  may be a free-layer damping system or a constrained-layer damping system and may incorporate a stand-off layer. A free-layer system is one in which a layer of damping material, such as a viscoelastic material, is directly adhered to a vibrating surface (e.g., the fuselage skin  14 ). A free-layer system dissipates energy by stretching and compressing as the underlying surface flexes during vibration. A constrained-layer system includes a layer of damping material adhered to the vibrating surface, like a free layer system, and further includes a relatively stiff layer (i.e., a “constraining” layer) overlying the damping material layer. The constraining layer in a constrained-layer system induces shear strains in the damping material layer when the underlying surface flexes during vibration, thus dissipating energy. A stand-off layer is typically used in conjunction with a constrained-layer system. In particular, a stand-off layer is included between the damping material layer and the vibrating surface to magnify shear deformation in the damping material layer, due to the increased distance between the damping material layer and the neutral axis of the vibrating surface. 
         [0028]    A preferred surface damping treatment  58  may be a stand-off constained-layer system, such as those manufactured by Damping Technologies Incorporated of Mishawaka, Ind. A stand-off constrained-layer system is illustrated in  FIG. 4 , having a constrained-layer system  62  overlying a stand-off layer  60  that is attached to the fuselage skin  14 . The stand-off layer has properties which are weak in bending and stiff in shear. Grooves  64  may be distributed along the layer to decrease the bending stiffness and mass of the stand-off layer  60 . The surface damping treatment  58  is preferably adhered to the skin  14  by a pressure sensitive adhesive. The surface damping treatment  58  also preferably includes a die-cut hole  66  therethrough sized and shaped to receive the plate element  52  of the TVA mount  36  so that the mount  36  may be directly bonded to the fuselage skin  14  through the hole  66 , as shown in  FIG. 4 . 
         [0029]    An appropriately designed surface damping treatment  58 , such as a stand-off constrained-layer system as illustrated in  FIG. 4 , is believed to provide effective attenuation of noise and vibration at the third and higher harmonics of the BPF. Additionally, the frame-mounted TVAs  18  are believed to provide effective attenuation at the first harmonic of the BPF. Thus, one preferred configuration in accordance with the present invention, as shown in  FIG. 6 , may include: frame-mounted TVAs  18  tuned to the first harmonic of the BPF and mounted to the frames  10 , skin-mounted TVA apparatuses  27  tuned to the second harmonic of the BPF and mounted to the skin  14 , and a stand-off constrained-layer surface damping treatment  58  attached to the skin  14 . 
         [0030]    As discussed above, the centerpost  34  of the skin-mounted TVA apparatus  27  may be removably coupled to the mount  36  via coupling elements, such as threaded female socket  38  and threaded male stud  40  (see  FIG. 4 ). In one preferred installation method in accordance with the present invention, the mount  36  is secured to the fuselage skin  14  independently of the remainder of the apparatus  27 . In this way, the remaining components of the skin-mounted TVA apparatus  27  do not interfere with the installation of other components near the fuselage, such as cables  74  and other equipment. The remaining components of the skin-mounted TVA apparatus  27  can then be provided as a sub-assembly  76  which is attached to the already mounted TVA mount  36 , such as by connecting the socket  38  of the TVA sub-assembly  76  to the stud  40  on the mount  36 . Preferably the sub-assembly  76  is attached after other equipment has been installed, but before the thermal/acoustical insulation blankets are installed. The thermal/acoustical insulation blankets are typically installed on the fuselage after the other equipment running along the fuselage has been run and before the interior trim is installed. The insulation blankets typically overlie the other equipment, such as cables, to provide a generally continuous layer of thermal and acoustic insulation. The above-described cover  44  of the skin-mounted TVA apparatus  27  preferably prevents other components, such as the cables  74  and the thermal/acoustical insulation blankets, from interfering with the movement of the components of the TVA  28 . 
         [0031]    If the skin-mounted TVA apparatus  27  is being used in conjunction with other noise and vibration attenuation systems, such as a surface damping treatment  58 , the skin-mounted TVA apparatus  27  and the other attenuation systems may be installed in any order. For example, a surface damping treatment  58  having a hole  66  may be installed before the TVA apparatus  27  or its mount  36 . In that case, the mount  36  may be subsequently attached to the skin  14  through the hole  66  in the surface damping treatment  58 . Alternatively, the TVA mount  36  may be installed before the surface damping treatment  58 , which can then be installed by fitting the hole  66  over the already installed mount  36  and adhering the treatment  58  to the skin  14 . If the surface damping treatment  58  includes a pressure sensitive adhesive for adhering to the skin  14 , the adhesive may be protected by a release liner. In such a case, the release liner may be peeled away before installing the surface damping treatment  58 . 
         [0032]    Among the benefits believed to be provided by the present invention is better attenuation performance over a wider range of frequencies than simply using frame-mounted TVAs alone. In particular, it is believed that the increased damping effect caused by the elastomeric material in the elastomeric component  32  provides a wider bandwidth of attenuation than the plate-spring  24  and mass  26  system of the frame-mounted TVAs  18 ,  20 . This preferably leads to better performance over many flight conditions, including aircraft climb and travel with a variety of propeller speeds. This design also preferably leads to greater tolerances with respect to tuning frequency (e.g., 4%, compared to approximately 0.25% for the frame-mounted TVAs  18 ,  20 ). Furthermore, the skin-mounted TVA apparatus  27  may also preferably be made smaller and lighter than a corresponding frame-mounted TVA tuned to the same frequency. Thus, to reduce weight, skin-mounted TVA apparatuses  27  tuned to a particular frequency, such as the second harmonic of the BPF, may be used instead of the frame-mounted TVAs  20  tuned to that frequency. However, in an alternative embodiment of a system in accordance with the present invention, frame-mounted TVAs tuned to the second harmonic of the BPF may be used in conjunction with skin-mounted apparatuses  27  containing TVAs tuned to the same harmonic. 
         [0033]    Many variations of the above described embodiments are possible within the scope of the present invention. For example, the present invention is not limited to the above-described shapes of the components. For instance, the TVA  28 , the centerpost  34 , and the cover  44  need not be cylindrical and the mount  36  need not be circular. Any other appropriate shapes for those components may be utilized. Moreover, other means for connecting the various components together can be utilized. For example, instead of the centerpost  34  having a coupling element in the form of a male threaded portion  42  for connecting to the cover  44 , the coupling element could include a female threaded socket in the centerpost  34 . In such a configuration, a separate threaded fastener (e.g., a bolt) configured to couple to the socket could be provided to secure the cover  44  to the centerpost  34 . In an alternative, the cover  44  may include a male threaded member for directly coupling to the socket. However, threaded fastening engagement is not required. The cover  44  can be secured to the centerpost  34  by any appropriate fastener. The cover  44  could also be glued or welded to the centerpost  34 . In another example, the corresponding male and female coupling elements of the centerpost  34  and the mount  36  may be interchanged, and a male member on the centerpost may engage a female member on the mount. Again, threaded fastening engagement is not required, and the centerpost  34  and mount  36  may be connected by any appropriate mechanism that preferably allows the TVA sub-assembly  76  to be easily mounted to the mount  36  after the mount  36  has already been secured to the fuselage skin  14 . The TVA sub-assembly  76  is preferably removable from the mount  36  after it has been mounted thereto, however that is not necessary. 
         [0034]    In other variations, the TVA apparatus  27  may be attached to the fuselage skin  14  by other than adhesive bonding. For example, the mount  36  may be mechanically fastened to the skin  14 . However, the use of adhesives is preferred because of issues which can arise from mechanical fastening of the apparatus  27  to the skin  14 , including increased weight and additional difficulty manufacturing and installing the relevant components, as well as stress concentrations. 
         [0035]    In still further variations, the TVA apparatus  27  may be attached to the skin  14  via other components. For example, the mount  36  of the apparatus  27  may be adhered to the surface damping treatment  58 , rather than to the skin  14  exposed through a hole  66  provided in the surface damping treatment  58 . Furthermore, other variations may include a different TVA structure than that described above. Any TVA structure that can be appropriately configured in an apparatus secured to the skin of an aircraft fuselage may be used. For example, the TVA could be a simple mass-and-spring system, with or without a dashpot. The TVA could also be structured according to any known elastomeric TVA design. 
         [0036]    Additional variations could include TVAs tuned to other frequencies. For example, in an aircraft other than the exemplary King Air 350 discussed above, the fuselage skin panels may vibrate at other frequencies than those discussed above. For instance, after providing damping systems to the fuselage frames, the skin panels may continue to vibrate at frequencies other than 200 Hz or the second harmonic of the BPF. In such a case, the significant frequencies of vibration may be determined, and then TVAs tuned to such frequencies may be provided and secured to the skin panels. Similarly, the TVAs may be secured to different locations or in different quantities on the skin panels. For example, in a particular aircraft it may be determined that a skin panel exhibits significant vibration with a particular mode shape, having one or more antinodes that are located at particular locations. In such a case, one or more skin-mounted TVAs may be provided for attachment at the appropriate locations, such as at each of the antinodes of vibration. 
         [0037]    Although the invention herein has been described with reference to particular embodiments, it is to be understood that these embodiments are merely illustrative of the principles and applications of the present invention. It is therefore to be understood that numerous modifications may be made to the illustrative embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims.