Patent Publication Number: US-2015078934-A1

Title: Split fluidic diaphragm

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 61/879,298, filed Sep. 18, 2013, which is hereby incorporated by reference in its entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     1) Field of Invention 
     This invention relates generally to diaphragms for the pumping of fluids in positive displacement pumping devices, otherwise known as fluid movers, such as liquid pumps, gas compressors and synthetic jets and in general to the transfer of energy to fluids. 
     2) Description of Related Art 
     When compared to rotary, piston, centrifugal and other pumping approaches, diaphragms provide a lower profile means for creating a cyclic positive displacement for small fluid movers such as pumps, compressors and synthetic jets. It is an advantage for all sizes of fluid movers to increase their pumping power density as defined by pumping power divided by the fluid mover size. To increase pumping power requires an increase in either displacement per stroke or pressure lift or both. A common limitation of diaphragms is that they do not provide large volumetric displacements due to their small strokes which are limited by the bending and tensile stresses of the diaphragm materials such as metals or plastics. If more resilient or stretchable materials such as common elastomers are used that permit larger strokes, then the diaphragm will typically flex during a stroke in response to increasing pressure, thus preventing larger pressure lifts and preventing higher power densities. 
     One particular diaphragm issue for miniature fluid movers pertains to high power synthetic jets. Synthetic jets can provide significant energy savings when used for cooling high power density and high power dissipation electronics products such as for example servers, computers, routers, laptops, HBLEDs and military electronics. U.S. Pat. No. 8,272,851 and PCT application PCT/US2011/055196 describe various arrangements for synthetic jet systems and other fluid mover systems, and are both hereby incorporated by reference in their entirety. 
     SUMMARY OF THE INVENTION 
     However, the compression chamber of a synthetic jet actuator must provide relatively large strokes and high dynamic pressures in order to drive large, multi-port manifolds while at the same time the actuator must be small enough to fit within many space-constrained products. Conventional diaphragm technologies that are stiff enough to create large fluid pressures, such as those that use a single piece metal diaphragm, cannot provide the required volume displacement to drive multi-port manifolds. Conversely, elastomeric diaphragms that are flexible enough to provide large displacements cannot create high dynamic pressures needed for some applications. These same limitations of conventional diaphragms also make it difficult to downsize other fluid movers such as liquid pumps and gas compressors without significant loss of pumping power performance. 
     There is therefore a need for positive displacement diaphragms that are compliant with respect to relatively large axial strokes, but at the same time are stiff enough to create relatively large dynamic pressures, thereby enabling increased pumping power density for miniature fluid movers including synthetic jets, liquid pumps and gas compressors. 
     Aspects of the invention provide a diaphragm for a fluid mover, such as a synthetic jet generator, that is suitably compliant to accommodate relatively large axial strokes and volumetric displacements, but is suitably stiff to create relatively large dynamic pressures. For example, in one embodiment, a diaphragm substrate includes two or more separate substrate sections which are connected to each other by a resilient material junction (such as an elastomeric over molding). Each diaphragm substrate section, which may be formed from a metal sheet, may include a plurality of cantilever tabs that are closely adjacent to, and are intermeshed with, counter facing cantilever tabs of the other diaphragm section. For example, the cantilever tabs may extend like fingers from each diaphragm substrate section and intermesh with the finger-like tabs of the adjoining diaphragm substrate section. The counter facing tabs are bonded together by the resilient material in such a way that they must follow a nearly identical displacement distribution as the diaphragm is moved. Once bonded together, the counter facing tabs work synergistically so as to respond very differently to axial diaphragm displacements as opposed to pressure deformation, whereby the cantilever tabs present a low spring stiffness (spring k rate) to axial displacements with acceptable material stress levels and conversely present a much higher spring stiffness to pressure deformations resulting in very small pressure deformations. The reason that different spring k rates occur is that axial displacement and pressure deformation encounter very different cantilever end constraints. The resilient elastomeric connection between the diaphragm sections may serve to connect the finger-like tabs together, allowing the cantilever tabs to support each other in bending, and yet allow the tabs to move or slip relative to each other in the plane of the diaphragm, thereby relieving large in-plane stresses that can occur if the diaphragm substrate sections were to bend while being unable to move or slip relative to each other. As such, the diaphragm may accommodate relatively large axial deflections, but since adjoining cantilever tabs of the two diaphragm sections may support each other in bending and slip relative to each other, the diaphragm may remain relatively stiff. That is, once coupled by the elastomeric over molding or other resilient junction material, the adjoining cantilever tabs may work synergistically together to provide a diaphragm that: (1) allows large axial displacements (i.e., in directions transverse to the plane of the diaphragm) without excessive stresses which enables large volumetric displacements in a low-profile small-footprint diaphragm assembly, (2) presents a high stiffness in resistance to fluid pressure induced deformation of the diaphragm to minimize ballooning or yielding of the flexing cantilever section which enables the diaphragm to create large cyclic pressures, and (3) provides a diaphragm design capable of providing a wide range of axial spring k values that can enable the resonant operation of fluid movers at commercially desirable frequencies. 
     In one aspect of the invention, a fluid mover includes a chamber having an outlet opening, and a fluidic diaphragm having a portion movable in the chamber to cause fluid to move at the outlet opening. The fluidic diaphragm may include two or more diaphragm substrate sections including first and second diaphragm substrate sections having cantilever tabs that are joined together by a resilient junction material. The cantilever tabs of the first and second diaphragm substrate sections may be at least partially intermeshed such that one or more cantilever tabs of the first diaphragm substrate section is positioned between two cantilever tabs of the second diaphragm substrate section. An actuator may be coupled to the fluidic diaphragm to move the portion of the fluidic diaphragm in the chamber. 
     In another aspect of the invention a fluid mover includes a chamber having an outlet opening, and a fluidic diaphragm having a portion movable in the chamber to cause fluid to move at the outlet opening. The fluidic diaphragm may include first and second diaphragm substrate sections that are flat and concentric such that the first diaphragm substrate section is located inside of the second diaphragm substrate section, and the first and second diaphragm substrate sections may be joined together by a resilient junction material. An actuator may be coupled to the fluidic diaphragm to move the portion of the fluidic diaphragm in the chamber. 
     In some embodiments, the fluidic diaphragm is flat, has a center, and has cantilever tabs with portions that extend radially relative to the center. The cantilever tabs of each diaphragm section may form a directionally alternating and length-wise overlapping array of cantilever tabs that define a primary flexing portion of the diaphragm, and alternating cantilever tabs may provide mutual support of each of their adjacent cantilever tabs. For example, the cantilever tabs of the first and second diaphragm substrate sections may be interdigitated such that the cantilever tabs of the first diaphragm substrate section are each positioned between a respective pair of cantilever tabs of the second diaphragm substrate section. The cantilever tabs of the first diaphragm substrate section may extend radially outwardly from a main body of the first diaphragm substrate section, and the cantilever tabs of the second diaphragm substrate section may extend radially inwardly from the second diaphragm substrate section. The cantilever tabs may have a variety of different shapes, such as a triangular shape that extends from a main body of the corresponding diaphragm substrate section. 
     In some embodiments, the resilient junction material may include an overmolding layer that bonds the cantilever tabs of the first and second diaphragm substrate sections together, e.g., the resilient junction material may be injection molded over and around portions of the cantilever tabs. The resilient junction material may be positioned in gaps between adjacent cantilever tabs, and/or over a top or bottom surface of cantilever tabs of the fluidic diaphragm. 
     In some embodiments, the fluidic diaphragm may have a periphery which is fixed relative to the chamber, and the actuator may be arranged to move portions of the fluidic diaphragm located inward of the periphery relative to the chamber. 
     In some arrangements, the fluidic diaphragm may have a spring stiffness in relation to pressure deformation that is 34 to 64 times greater than a spring stiffness of the fluidic diaphragm in relation to axial displacement. 
     It should also be noted that aspects of the invention include a fluidic diaphragm configured as described herein, and need not be coupled with an actuator and/or positioned in a chamber of a fluid mover. 
     These and other aspects of the invention will be apparent from the following description. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and form a part of the specification, illustrate select embodiments of the present invention and, together with the description, serve to explain the principles of the inventions. In the drawings: 
         FIG. 1  shows a sectional view of a fluid mover in an illustrative embodiment; 
         FIG. 2  shows a top view of a diaphragm of the fluid mover of  FIG. 1  in accordance with an embodiment of the invention; 
         FIG. 3  shows the outer of two diaphragm substrate sections of the  FIG. 2  embodiment; 
         FIG. 4  shows the inner of two diaphragm substrate sections of the  FIG. 2  embodiment; 
         FIG. 5  shows the diaphragm of  FIG. 2  with a two-sided elastomeric over molding resiliently connecting the two diaphragm substrate sections; 
         FIG. 6  shows how the counter facing cantilever tabs of the two diaphragm section are constrained by the resilient over molding (not shown) to bend with the same displacement distribution; 
         FIG. 7  shows a sectional view of a pump or compressor fluid mover with a diaphragm undergoing pressure deformation or (aka ballooning); 
         FIG. 8  illustrates beam deflection for a free end, fixed end beam; 
         FIG. 9  illustrates beam deflection for a free end, simply supported beam; 
         FIG. 10  illustrates beam deflection for a fixed end, fixed end beam; 
         FIG. 11  illustrates another embodiment in which each cantilever tab has a narrow, distal extension that is supported at an opposing diaphragm clamp circle; 
         FIG. 12  illustrates another embodiment of a diaphragm having three diaphragm substrate sections; 
         FIG. 13  illustrates a diaphragm with a non axi-symmetric arrangement, e.g., having a rectangular shape; and 
         FIG. 14  shows a resilient junction material in the form of an elastomeric over molding with ribs to provide additional support while minimizing cyclic elastomeric damping. 
     
    
    
     DETAILED DESCRIPTION 
     Aspects of the invention are not limited in application to the details of construction and the arrangement of components set forth in the following description or illustrated in the drawings. Other embodiments may be employed and aspects of the invention may be practiced or be carried out in various ways. Also, aspects of the invention may be used alone or in any suitable combination with each other. Thus, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. 
       FIG. 1  shows a sectional view of a fluid mover  4  in an illustrative embodiment that includes a chamber  6  having an internal volume in which a diaphragm  2  and actuator  12  are located. In this embodiment, the internal volume of the chamber  1  above the diaphragm  2  is closed except for an opening  8  located in a top wall of the chamber  6 . (The opening  8  may be located in other places, such as a sidewall of the chamber  6 , if desired, and in some embodiments such as liquid pumps and gas compressors, two or more openings may be provided and these openings may also include valves to enable fluid compression and/or one-directional flow). The diaphragm  2  is controllable to move cyclically in the chamber  6  so that air or other fluid is alternately drawn into the opening  8  and then driven out of the opening  8  in the direction of an arrow  10 . As is known to those of skill in the art and described in more detail in U.S. Pat. No. 8,272,851, this air movement at the opening  3  can cause the formation of a series of air pulses and vortex rings that move away from the opening  8  in the direction of the arrow  10  so that a synthetic jet is created. Movement of the diaphragm  2  is caused by the actuator  12 , which may be a voice coil, piezoelectric, magnetostrictive, capacitive, variable reluctance, solenoid or other electromagnetic, mechanical concentric or other drive. Details regarding diaphragm actuators are provided in U.S. Pat. No. 8,272,851 and PCT application PCT/US2011/055196 as well. In short, the diaphragm  2  may be moved by any suitable actuator as aspects of the invention are not limited in this regard. 
     Operation of the actuator  12  may be controlled by a controller  14  (e.g., including a suitably programmed general purpose computer or other data processing device) that receives control information (e.g., from one or more sensors, user input devices, etc.) and correspondingly controls operation of the actuator  4  and/or other fluid mover components. The controller  14  may include any suitable components to perform desired control, communication and/or other functions. For example, the controller  14  may include one or more general purpose computers, a network of computers, one or more microprocessors or PICs, etc., for performing data processing functions, one or more memories for storing data and/or operating instructions (e.g., including volatile and/or non-volatile memories such as optical disks and disk drives, semiconductor memory, magnetic tape or disk memories, and so on), communication buses or other communication devices for wired or wireless communication (e.g., including various wires, switches, connectors, Ethernet communication devices, WLAN communication devices, and so on), software or other computer-executable instructions (e.g., including instructions for carrying out functions related to controlling the actuator  12 , and other components), a power supply or other power source (such as a plug for mating with an electrical outlet, batteries, transformers, etc.), relays, other switching devices and/or drive circuitry for driving the actuator  12 , mechanical linkages, one or more sensors or data input devices (such as a sensor to detect movement and/or position of the diaphragm  2  and/or temperature of a device being cooled by a jet stream created by the fluid mover  4 , user-operated buttons or switches, and interface to receive control instructions from another device, and so on), user data input devices (such as buttons, dials, knobs, a keyboard, a touch screen or other), information display devices (such as an LCD display, indicator lights, a printer, etc.), and/or other components for providing desired input/output and control functions. In short, the controller  14  may include any suitable components to perform desired control and communication functions for the fluid mover  4  or for other fluid movers such as liquid pumps, gas compressors or acoustic pumps and compressors. 
     In accordance with an aspect of the invention, the fluid mover may include a fluidic diaphragm that includes first and second diaphragm substrate sections that are flat, separate from each other, and concentric such that the first diaphragm substrate section is located inside of the second diaphragm substrate section. The diaphragm substrate sections may have portions that are intermeshed with each other and that are joined by a resilient junction material so that the diaphragm substrate sections can support each other in bending, yet move or slip relative to each other in the plane of the diaphragm. 
     In another aspect of the invention, the diaphragm may include two or more diaphragm substrate sections arranged so that the diaphragm substrate sections have counter facing cantilever tabs that are joined together by a resilient junction material. In some embodiments, the cantilever tabs of the diaphragm substrate sections may be at least partially intermeshed such that one or more cantilever tabs of a diaphragm substrate section is positioned between two cantilever tabs of another diaphragm substrate section. In some embodiments, the diaphragm substrate sections may be separated from each other by a gap and be joined together by a resilient junction material so that the diaphragm formed is thereby sealed and made capable of moving fluid and has desired pumping power and volume displacement characteristics. In addition, such an arrangement may allow for the diaphragm to have a desired axial spring stiffness so that the diaphragm can serve as a spring in a fluid mover having a spring-mass resonance thereby allowing the fluid mover to be operated at a desired resonant frequency and relatively high efficiency. That is, it is typically more efficient to drive a fluid mover at or near its spring-mass resonant frequency and aspects of the invention may allow for a diaphragm to be constructed that provides the spring stiffness required for a desired resonant frequency. For example cantilever tab shape, size thickness or other characteristics may be defined to provide the diaphragm with desired spring stiffness. 
       FIG. 2  shows a top view of an illustrative diaphragm that may be used in the fluid mover  4  of  FIG. 1  and has first and second diaphragm substrate sections  2   a ,  2   b . When assembled as a diaphragm  2  as shown in  FIG. 2 , the two diaphragm substrate sections  2   a ,  2   b  may be flat, concentric, lie in a common plane, and be separated by a gap or slot  16 . The diaphragm substrate sections  2   a ,  2   b  (which are shown individually in  FIGS. 3 and 4 ) may be formed as a flat sheet, e.g., cut from a metal sheet such as a steel. The diaphragm design of  FIG. 2  has an outer diameter of 2.650 in and a thickness of 0.003 in, but a wide range of other diameters, thicknesses and cantilever tab dimensions can be chosen to meet the needs of specific application requirements. (While the first diaphragm substrate section  2   a  is shown having a central hole, e.g., to facilitate attachment of the section  2   a  to an actuator  12 , the central hole is not necessarily required.) As noted above, the diaphragm substrate sections  2   a ,  2   b  may have cantilever tabs  18  that are intermeshed so that at least one cantilever tab  18  of one section  2   a ,  2   b  is located between two cantilever tabs  18  of the other diaphragm substrate section  2   a ,  2   b . In this embodiment, the cantilever tabs  18  are interdigitated so that each cantilever tab  18  of one diaphragm substrate section  2   a ,  2   b  is positioned between a pair of cantilever tabs  18  of the other diaphragm substrate section  2   a ,  2   b , but other interleaving or intermeshing arrangements are possible. For example, two or more cantilever tabs  18  of one diaphragm substrate section  2   a ,  2   b  may be located between a pair of cantilever tabs  18  of the other diaphragm substrate section  2   a ,  2   b . Other variations are possible. 
     In accordance with another aspect of the invention, the two or more diaphragm substrate sections may be separated from each other (e.g., by a gap  16 ), yet be attached to each other by a resilient junction material. The resilient junction material may fill the gap  16  between the diaphragm substrate sections  2   a ,  2   b  and/or be arranged on a top and/or bottom side of the diaphragm substrate sections  2   a ,  2   b . The resilient junction material may be positioned only in the area of the cantilever tabs  18  or may be arranged on other portions of the diaphragm substrate sections  2   a ,  2   b . In the illustrative embodiment as shown in  FIG. 5 , the diaphragm substrate sections  2   a ,  2   b  are joined by a resilient junction material  24  that is arranged in the gap  16  between the diaphragm substrate sections  2   a ,  2   b , as well as being arranged as a layer on top and bottom surfaces of the diaphragm substrate sections  2   a ,  2   b . In this embodiment, the resilient junction material  24  is formed as an over molding of a rubber or other resilient material (e.g., by injection molding a rubber from top and bottom sides of the diaphragm  2 ), but other arrangements are possible. For example, a sheet of rubber or other suitable material may be adhered to the top and/or bottom surface of the diaphragm  2  with or without resilient material being located in the gap  16  or resilient material can be located in gap  16  without being adhered to the top and bottom surface of diaphragm  2 . Alternatively, the clearance between sections  2   a ,  2   b  can be removed so that the gap  16  is eliminated with the edges of sections  2   a ,  2   b  being in contact or in some cases overlapping. 
     Once sections  2   a ,  2   b  of diaphragm  2  are coupled together by the resilient junction material, diaphragm  2  exhibits unique characteristics that enable large volumetric displacements while resisting pressure deformation. The respective axial deformation and pressure deformation characteristics in one illustrative embodiment are described as follows. 
     Axial Deformation 
     In operation, section  2   b  of diaphragm  2  may be rigidly clamped along clamp circle  22  (e.g., between portions of the housing of the fluid mover  4  as suggested in  FIG. 1 ) and section  2   a  may be clamped along clamp circle  20  (e.g., between portions of a reciprocating clamp  11  as shown in  FIG. 1 ). When clamped in this manner, only cantilever tabs  18  bend when reciprocating clamp  11  is displaced from its rest position relative to the housing of the fluid mover  4 . 
     Resilient junction material  24  of  FIG. 5  constrains the counter facing tabs  18  of sections  2   a ,  2   b  to bend with a similar displacement distribution y(r), where y is the displacement from the diaphragm&#39;s rest plane at a radial distance r from the clamp circle  22 .  FIG. 6  illustrates how the cantilever tabs of sections  2   a ,  2   b  follow a same displacement distribution, where the resilient junction material  24  is not shown to more clearly illustrate the bending of tabs  18 . The cantilever bending modes shown in  FIG. 6  were generated using finite element software where the model included a rubber over molding material  24  shown in  FIG. 5 . 
     The resilient junction material  24  allows some slip between counter facing cantilever tabs  18  of sections  2   a ,  2   b  and this slippage relieves the displacement-induced stresses that would otherwise occur in a one-piece diaphragm. Consequently, much larger displacements can be achieved with peak stresses being low enough to provide long life. A further advantage of diaphragm  2  of  FIG. 5  is that it provides an effective piston area that is larger than a clamped one-piece diaphragm. For diaphragm  2  of  FIG. 5 , the diameter of clamp circle  20  is ⅔ the diameter of clamp circle  22  and the effective piston diameter of diaphragm  2  is the average of the diameters of clamp circles  20  and  22 . This diameter yields an effective piston area that is 70% of a piston having a diameter equal to clamp circle  22 . By comparison, a simple metal disk clamped at clamp circle  22  would have an effective piston area of only 30% of a piston having a diameter equal to clamp circle  22 . A larger effective piston area provides proportionately larger volumetric displacement for a given diaphragm stroke, and this is a significant advantage when miniaturizing fluid movers without losing pumping performance. Effective piston areas higher than 70% can be achieved simply by using more cantilever tabs of reduced length, which would allow the diameter of clamp circle  20  to be increased. 
     Pressure Deformation 
     While the axial diaphragm deformations described above provide the volumetric displacements needed to do fluid pumping work, that volumetric displacement can be reduced by pressure deformations, thereby reducing the pumping work done.  FIG. 7  illustrates a case of diaphragm pressure deformation that would reduce the pumping work done by the diaphragm  2 . In  FIG. 7 , a diaphragm  28  is clamped into fluid mover  26 , which in this case is a pump or compressor having outlet port  30  and inlet port  32  with respective valves  34  and  36 . If the unsupported span  29  of diaphragm  28  is too compliant, then when plunger  38  moves upward thereby moving the diaphragm  28  upward and increasing the fluid pressure in chamber  40 , the unsupported span  28  will deform or balloon as shown in  FIG. 7 . This ballooning relieves the pressure within chamber  40  and the ballooning volume of unsupported span  28  subtracts from the displaced volume created by the stroke of diaphragm  28  in the absence of ballooning. Consequently, both pressure and volumetric displacement are reduced which in turn reduces pumping power. To more clearly illustrate the ballooning effect, the fluid mover  26  of  FIG. 7  shows the diaphragm at its rest position. During operation, the ballooning deformation is superimposed on the axial deformation, but otherwise has the same disadvantageous effect described above. 
     Aspects of the invention minimize pressure deformation of the diaphragm thereby maximizing pumping power. In order to minimize pressure deformation while still enabling comparatively large axial strokes with low related material stress, the diaphragm design in accordance with aspects of the invention creates a very high stiffness which acts only against pressure deformation, while presenting a much lower stiffness to the desired axial displacements. Since the cantilever tabs are beams in the analytical sense, some insight can be provided into how the diaphragm can present a lower spring stiffness to axial deformation and a higher spring stiffness to pressure deformation by looking at the beam deflection equations for end constraint conditions that resemble these two types of deformation. Here these equations are used to calculate a comparative spring stiffness constant k, for the two cases. The comparison is intended to approximate the ratio of the spring k values, but not to find their absolute values. For each case the cantilever geometry is assumed identical and only the end constraints are changed, which allows the relative comparisons (i.e. k ratio) to be found. Also, making spring k comparisons requires that point loads are used in each case. 
     Solving for k from the equation F=kx, where is F is the applied force, k is the spring constant and x is the spring displacement, gives k=F/x. 
       FIG. 8  schematically illustrates beam bending for a cantilever with a fixed end, a free end and point load at W. This case most closely resembles the bending mode of a cantilever tab when the diaphragm undergoes an axial deformation (displacement). When the perimeter of diaphragm  2  is clamped and the center portion is displaced, the stiffness contributed by a single cantilever is represented by letting a=0 in max deflection equation. Beam equations for bending as shown in  FIG. 8  are provided in Table 1 below (reproduced from  Roark&#39;s Formulas for Stress and Strain,  2 nd  Edition, page 189). 
     
       
         
           
               
               
             
               
                 TABLE 1 
               
               
                   
               
             
            
               
                 
                   
                     
                       
                         
                           R 
                           A 
                         
                         = 
                         
                           
                             0 
                              
                             
                                 
                             
                              
                             
                               M 
                               A 
                             
                           
                           = 
                           
                             
                               0 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 A 
                               
                             
                             = 
                             
                               
                                 
                                   w 
                                    
                                   
                                     ( 
                                     
                                       l 
                                       - 
                                       a 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 EI 
                               
                             
                           
                         
                       
                     
                   
                 
                 Max M = M B ; max possible value = −Wl when a = 0 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           y 
                           A 
                         
                         = 
                         
                           
                             
                               - 
                               w 
                             
                             
                               6 
                                
                               EI 
                             
                           
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   l 
                                   3 
                                 
                               
                               - 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   l 
                                   2 
                                 
                                  
                                 a 
                               
                               + 
                               
                                 a 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Max 
                              
                             
                                 
                             
                              
                             θ 
                           
                           = 
                           
                             θ 
                             A 
                           
                         
                         ; 
                         
                           
                             max 
                              
                             
                                 
                             
                              
                             possible 
                              
                             
                                 
                             
                              
                             value 
                           
                           = 
                           
                             
                               
                                 
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   
                                     l 
                                     2 
                                   
                                 
                                 
                                   2 
                                    
                                   EI 
                                 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               a 
                             
                             = 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 R B  = W  M B  = −W(l − a) 
                 
                   
                     
                       
                         
                           
                             Max 
                              
                             
                                 
                             
                              
                             y 
                           
                           = 
                           
                             y 
                             A 
                           
                         
                         ; 
                         
                           
                             max 
                              
                             
                                 
                             
                              
                             possible 
                              
                             
                                 
                             
                              
                             value 
                           
                           = 
                           
                             
                               
                                 
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   
                                     l 
                                     3 
                                   
                                 
                                 
                                   3 
                                    
                                   EI 
                                 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               a 
                             
                             = 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 θ B  = 0 y B  = 0 
               
               
                   
               
            
           
         
       
     
       FIG. 9  schematically illustrates beam bending for a cantilever beam that is simply supported at its free end (the free end can slide and rotate) and with a point load at W.  FIG. 10  schematically illustrates beam bending for a beam that is fixed at both ends and with a point load at W. These bending modes most closely resemble the bending of a cantilever in response to pressure deformation of the diaphragm, and  FIG. 10  illustrates bending in response to pressure-induced ballooning as in  FIG. 7 . Beam equations for bending as shown in  FIGS. 9 and 10  are provided in Tables 2 and 3 below (reproduced from  Roark&#39;s Formulas for Stress and Strain,  2 nd  Edition, page 190). 
     
       
         
           
               
               
             
               
                 TABLE 2 
               
               
                   
               
             
            
               
                 
                   
                     
                       
                         
                           R 
                           A 
                         
                         = 
                         
                           
                             
                               w 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   l 
                                   3 
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   l 
                                   - 
                                   a 
                                 
                                 ) 
                               
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   l 
                                 
                                 + 
                                 a 
                               
                               ) 
                             
                              
                             
                                 
                             
                              
                             
                               M 
                               A 
                             
                           
                           = 
                           0 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Max 
                             + 
                             M 
                           
                           = 
                           
                             
                               
                                 Wa 
                                 
                                   2 
                                    
                                   
                                     l 
                                     3 
                                   
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     l 
                                     - 
                                     a 
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     l 
                                   
                                   + 
                                   a 
                                 
                                 ) 
                               
                                
                               
                                   
                               
                                
                               at 
                                
                               
                                   
                               
                                
                               x 
                             
                             = 
                             a 
                           
                         
                         ; 
                         
                           
                             max 
                              
                             
                                 
                             
                              
                             possible 
                              
                             
                                 
                             
                              
                             value 
                           
                           = 
                           
                             
                               0.174 
                                
                               
                                 W 
                                  
                                 l 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               a 
                             
                             = 
                             0.3661 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           θ 
                           A 
                         
                         = 
                         
                           
                             
                               
                                 - 
                                 
                                   w 
                                    
                                   a 
                                 
                               
                               
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 EIl 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   l 
                                   - 
                                   a 
                                 
                                 ) 
                               
                               2 
                             
                              
                             
                                 
                             
                              
                             
                               y 
                               A 
                             
                           
                           = 
                           0 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Max 
                             - 
                             M 
                           
                           = 
                           
                             M 
                             B 
                           
                         
                         ; 
                         
                           
                             max 
                              
                             
                                 
                             
                              
                             possible 
                              
                             
                                 
                             
                              
                             value 
                           
                           = 
                           
                             
                               
                                 - 
                                 0.1924 
                               
                                
                               
                                 W 
                                  
                                 l 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               a 
                             
                             = 
                             
                               0.5773 
                                
                               l 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           R 
                           B 
                         
                         = 
                         
                           
                             wa 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 l 
                                 3 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   3 
                                    
                                   
                                     l 
                                     2 
                                   
                                 
                                 - 
                                 
                                   
                                     a 
                                     
                                       ) 
                                       2 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     B 
                                   
                                 
                               
                               = 
                               0 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           Max 
                            
                           
                               
                           
                            
                           y 
                         
                         = 
                         
                           
                             
                               
                                 - 
                                 wa 
                               
                               
                                 6 
                                  
                                 EI 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   l 
                                   - 
                                   a 
                                 
                                 ) 
                               
                               2 
                             
                              
                             
                               
                                 ( 
                                 
                                   a 
                                   
                                     
                                       2 
                                        
                                       l 
                                     
                                     + 
                                     a 
                                   
                                 
                                 ) 
                               
                               
                                 1 
                                 / 
                                 2 
                               
                             
                              
                             
                                 
                             
                              
                             at 
                              
                             
                                 
                             
                              
                             x 
                           
                           = 
                           
                             
                               
                                 
                                   l 
                                    
                                   
                                     ( 
                                     
                                       a 
                                       
                                         
                                           2 
                                            
                                           l 
                                         
                                         + 
                                         a 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   1 
                                   / 
                                   2 
                                 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               a 
                             
                             &gt; 
                             
                               0.414 
                                
                               l 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           M 
                           B 
                         
                         = 
                         
                           
                             
                               
                                 - 
                                 wa 
                               
                               
                                 2 
                                  
                                 
                                   l 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   l 
                                   2 
                                 
                                 - 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                                 
                             
                              
                             
                               y 
                               B 
                             
                           
                           = 
                           0 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Max 
                              
                             
                                 
                             
                              
                             y 
                           
                           = 
                           
                             
                               
                                 
                                   - 
                                   
                                     
                                       wa 
                                        
                                       
                                         ( 
                                         
                                           
                                             l 
                                             2 
                                           
                                           - 
                                           
                                             a 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     3 
                                   
                                 
                                 
                                   3 
                                    
                                   
                                     
                                       EI 
                                        
                                       
                                         ( 
                                         
                                           
                                             3 
                                              
                                             
                                               l 
                                               2 
                                             
                                           
                                           - 
                                           
                                             a 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     2 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               at 
                                
                               
                                   
                               
                                
                               x 
                             
                             = 
                             
                               
                                 
                                   
                                     l 
                                      
                                     
                                       ( 
                                       
                                         
                                           l 
                                           2 
                                         
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       3 
                                        
                                       
                                         l 
                                         2 
                                       
                                     
                                     - 
                                     
                                       a 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 when 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               &lt; 
                               
                                 0.414 
                                  
                                 l 
                               
                             
                           
                         
                         ; 
                         
                           
                             max 
                              
                             
                                 
                             
                              
                             possible 
                              
                             
                                 
                             
                              
                             y 
                           
                           = 
                           
                             - 
                             0.0098 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               wl 
                               3 
                             
                             EI 
                           
                            
                           
                               
                           
                            
                           when 
                            
                           
                               
                           
                            
                           x 
                         
                         = 
                         
                           a 
                           = 
                           
                             0.414 
                              
                             l 
                           
                         
                       
                     
                   
                 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                 TABLE 3 
               
               
                   
               
             
            
               
                 
                   
                     
                       
                         
                           R 
                           A 
                         
                         = 
                         
                           
                             w 
                             
                               l 
                               3 
                             
                           
                            
                           
                             
                               ( 
                               
                                 l 
                                 - 
                                 a 
                               
                               ) 
                             
                             2 
                           
                            
                           
                             ( 
                             
                               l 
                               + 
                               
                                 2 
                                  
                                 a 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Max 
                             + 
                             M 
                           
                           = 
                           
                             
                               
                                 
                                   2 
                                    
                                   
                                     wa 
                                     2 
                                   
                                 
                                 
                                   l 
                                   3 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     l 
                                     - 
                                     a 
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                   
                               
                                
                               at 
                                
                               
                                   
                               
                                
                               x 
                             
                             = 
                             a 
                           
                         
                         ; 
                         
                           
                             max 
                              
                             
                                 
                             
                              
                             possible 
                              
                             
                                 
                             
                              
                             value 
                           
                           = 
                           
                             
                               
                                 wl 
                                 8 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               a 
                             
                             = 
                             
                               l 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           M 
                           A 
                         
                         = 
                         
                           
                             
                               - 
                               wa 
                             
                             
                               l 
                               2 
                             
                           
                            
                           
                             
                               ( 
                               
                                 l 
                                 - 
                                 a 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Max 
                             - 
                             M 
                           
                           = 
                           
                             
                               
                                 M 
                                 A 
                               
                                
                               
                                   
                               
                                
                               if 
                                
                               
                                   
                               
                                
                               a 
                             
                             &lt; 
                             
                               1 
                               2 
                             
                           
                         
                         ; 
                         
                           
                             max 
                              
                             
                                 
                             
                              
                             possible 
                              
                             
                                 
                             
                              
                             value 
                           
                           = 
                           
                             
                               
                                 - 
                                 0.1481 
                               
                                
                               Wl 
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               a 
                             
                             = 
                             
                               l 
                               3 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 θ A  = 0 y A  = 0 
                 
                   
                     
                       
                         
                           
                             Max 
                              
                             
                                 
                             
                              
                             y 
                           
                           = 
                           
                             
                               
                                 
                                   
                                     - 
                                     2 
                                   
                                    
                                   
                                     
                                       w 
                                        
                                       
                                         ( 
                                         
                                           l 
                                           - 
                                           a 
                                         
                                         ) 
                                       
                                     
                                     3 
                                   
                                    
                                   
                                     a 
                                     3 
                                   
                                 
                                 
                                   3 
                                    
                                   
                                     
                                       EI 
                                        
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             2 
                                              
                                             a 
                                           
                                         
                                         ) 
                                       
                                     
                                     2 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               at 
                                
                               
                                   
                               
                                
                               x 
                             
                             = 
                             
                               
                                 
                                   
                                     2 
                                      
                                     al 
                                   
                                   
                                     l 
                                     + 
                                     
                                       2 
                                        
                                       a 
                                     
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               &gt; 
                               
                                 l 
                                 2 
                               
                             
                           
                         
                         ; 
                         
                           
                             max 
                              
                             
                                 
                             
                              
                             possible 
                              
                             
                                 
                             
                              
                             value 
                           
                           = 
                           
                             
                               
                                 
                                   - 
                                   
                                     wl 
                                     3 
                                   
                                 
                                 
                                   192 
                                    
                                   EI 
                                 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               x 
                             
                             = 
                             
                               a 
                               = 
                               
                                 l 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           R 
                           B 
                         
                         = 
                         
                           
                             
                               - 
                               
                                 wa 
                                 2 
                               
                             
                             
                               l 
                               3 
                             
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 l 
                               
                               - 
                               
                                 2 
                                  
                                 a 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                   
               
               
                   
               
               
                 
                   
                     
                       
                         
                           M 
                           B 
                         
                         = 
                         
                           
                             
                               - 
                               
                                 wa 
                                 2 
                               
                             
                             
                               l 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               l 
                               - 
                               a 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                   
               
               
                   
               
               
                 θ B  = 0 y B  = 0 
               
               
                   
               
            
           
         
       
     
     However, in the subject invention the resilient junction material constrains the counter facing cantilever tabs to follow a nearly identical deflection curve, so it is clear that the supported end of the tabs is not free to rotate and not completely free to slip. That is, the resilient junction material allows some slippage between the counter facing cantilever tabs, but since the counter facing tabs are constrained to a nearly identical deflection curve and bonded to each other, the slippage that could otherwise occur is reduced. Consequently, the spring k value related to pressure deformation of the subject diaphragm is likely to lie somewhere between the k value of the bending mode of  FIG. 9  and the k value of the bending mode of  FIG. 10 . 
     To make the axial spring k vs. pressure spring k comparison, the max deflection equation for the bending modes of  FIGS. 8-10  is solved for k which then provides equations for k a , k c  and k d , where k a , corresponds to the bending mode of  FIG. 8 , k c  corresponds to the bending mode of  FIG. 9 , and k d  corresponds to the bending mode of  FIG. 10 . For bending modes for  FIGS. 9 and 10 , the load is applied at midpoint along the cantilever&#39;s length L (i.e. a=L/2). The resulting ratios are k c /k a =34 and k d /k a =64. Consequently, the spring k which resists pressure deformation is somewhere between 34 to 64 times higher than the spring k that resists axial displacements. 
     In general, stiffness is a trend-wise indicator of a beam&#39;s stress at a given deflection (i.e., a higher stiffness will be accompanied by a higher stress for a given beam deflection). From the above beam analysis it can be seen that embodiments in accordance with aspects of the invention successfully provide a single set of cantilever springs that respond very differently to axial diaphragm displacements and pressure deformation, whereby axial displacements are accompanied by a lower spring k rate that enables axial displacements within acceptable material stress levels and pressure deflections are minimized by a much higher spring k rate that resists ballooning and yet both spring k values are created by the same cantilever springs. The different spring k rates occur because the single set of cantilever springs can present one unique set of beam end constraints to the axial deflections and a second unique set of beam end constraints to ballooning deflections. 
     For clarity of illustration, the beam deflection cases of  FIGS. 9 and 10  show a load being applied while the diaphragm is at its rest position (i.e. no axial deflection), while in an operating oscillating diaphragm pressure deformation would be superimposed on an axial deformation. In other words, the actual deformation curve of the cantilevers would include the superposition of axial deflection and pressure deformation. Nevertheless, the diaphragm&#39;s unique characteristics described previously hold for either the stationary or dynamic case. 
     The diaphragm design represented in  FIGS. 2-6  was designed using FEA software and was then built and tested. Diaphragm specifications were: spring steel substrate thickness of 0.003 in, steel substrate outer diameter 2.650 in, cantilever dimensions were exactly as shown in  FIGS. 2-6 , the resilient junction material was 0.020 in thick  40 A durometer silicone which was injection molded from both sides of the diaphragm and the gap between inner and outer diaphragm pieces was 0.020 in. This diaphragm could provide 0.25 in axial strokes with a stress safety factor of 2 (yield divided by peak), and a spring stiffness k for axial displacements of 10,440N/m. 
     The FEA model was also used to find the spring stiffness k that resists pressure deformation. In this model both the center and perimeter of the diaphragm were constrained so that only the cantilever tabs could move in response to an applied pressure of 25 psi. For this test the ballooning displacement (i.e., distance from the diaphragm plane to the peak of the pressure deformation of the cantilever tabs) was 0.014 in, with a corresponding ballooning metal stress safety factor of 4.4. To calculate the spring stiffness, the load was found by multiplying the applied pressure by the area of the cantilever tabs (i.e., the area bounded by the two clamp circles  20  and  22 ). The resulting force of 88.8N results in a spring k of 250,000 N/m which is a factor of 60× higher than the axial k value. 
     While in the embodiment of  FIG. 2  the distal ends of the cantilever tabs  18  were not rigidly connected to a clamp plate  11  or to a housing of the fluid mover  4 , the diaphragm  2  may be configured in other ways. For example, the diaphragm  2  of  FIG. 11  is arranged to have the distal end of each cantilever tab  18  extend beyond the opposite clamp line  42 ,  44  so that the distal ends of the cantilever tabs  18  may be directly supported by the opposing clamp plate  11  or housing. Such an arrangement may provide for a stiffer diaphragm  2  that relies less on the junction material  24  for interconnecting cantilever tabs  18  to each other at their distal ends. Note that the cantilever tab distal ends need not be secured at the opposing clamp line in such a way that the distal ends are prevented from sliding movement relative to the clamp line. Instead, the distal ends may be permitted to slide by virtue of the junction material in the plane of the diaphragm at the opposing clamp line as in the  FIG. 11  embodiment yet have axial force of the clamp plate  11  or housing applied more directly to the distal end. 
     Other diaphragm configurations are possible, and variations may provide for different diaphragm operating characteristics. For example, the diaphragm  2  shown in  FIG. 12  includes three diaphragm substrate sections  2   a ,  2   b ,  2   c  which all have counter-facing cantilever tabs  18 . When the diaphragm  2  is over molded or otherwise provided with a junction material  24 , the counter-facing cantilever tabs  18  of the diaphragm substrate sections  2   a ,  2   b ,  2   c  function in much the same manner and provide at least some of the same advantages of the diaphragm arrangement of  FIG. 2 . This approach could be extended to include any suitable number of diaphragm substrate sections and will be appreciated by those of skill in the art. 
     While embodiments above have a diaphragm with a circular shape, aspects of the invention are not limited in this regard. Instead, a diaphragm may have any suitable shape, such as rectangular (as shown in  FIG. 13  for example), triangular, oval, pentagonal, etc. The diaphragm  2  of  FIG. 13  also illustrates that cantilever tabs  18  in a diaphragm need not all have the same shape and/or size, but rather may vary as desired. In addition, the aspect ratio of cantilever tabs  18  may vary as desired, e.g., along their length, width or in other ways. 
     As mentioned above, the junction material  24  may function to help transmit force from one cantilever tab  6  to another adjacent cantilever tab  6  and so constrain the counter facing cantilever tabs to a nearly identical displacement distribution. As such, the junction material  24  may be arranged to help transmit force more efficiently and/or reliably. For example, a diaphragm  2  shown in  FIG. 14  is arranged like that in  FIG. 5 , except that the resilient junction material  24  includes one or more concentric ribs  46  that are formed to provide additional mechanical support to the counter-facing cantilever tabs  18  to help keep the tabs  18  in the same displacement distribution. However, some radial stretching of the junction material  24  may still occur with diaphragm displacement and damping may increase with increasing thickness and/or durometer of the junction material  24 . The ribs  46  in  FIG. 14  may provide support for the cantilever tabs  18  while allowing for radial stretching to occur in the thinner portions of junction material between the ribs  46  to minimize cyclic damping energy losses. Other rib geometries could be used such as honey comb, etc. 
     Many improvements and/or other changes to the embodiments described above will occur to those skilled in the art. For example, the embodiments above mostly include isosceles triangle-shaped cantilever tabs. Many other cantilever tab shapes can used within the scope of the present invention including, for example, sawtooth, “T” shaped, “I” shaped, trapezoid-shaped, claw-shaped, or hooked tabs. In general, there are many counter-facing cantilever tab shapes with widths, thicknesses and/or moments of inertia that become progressively smaller when traversing from clamped or base end to the tip or distal end, and any of these may be employed. The spacing or gap  16  between the diaphragm sections need not be constant along its extent but may vary as required by various cantilever tab shapes. To improve elastomeric-to-diaphragm bonding, flow through holes could be added to the diaphragm sections to allow the junction material and/or adhesive to flow through from both sides during the injection molding or other bonding process. Diaphragm substrate materials could be metal, plastic or any material that meets the design requirements of a given application. In some applications a single one-sided over molding layer could be used. 
     Applications for the diaphragm of the present invention can be found wherever energy is transferred to fluids by means of positive volumetric displacement. Applications include, for example, fluid movers such as pumps, compressors and synthetic jets; applying fluidic energy to fluid filled acoustic resonators for applications such as acoustic compressors or thermoacoustic engines, buzzers and as speaker cone elements in sound reproduction. 
     The embodiments provided herein are not intended to be exhaustive or to limit the invention to a precise form disclosed, and many modifications and variations are possible in light of the above teachings. The embodiments were chosen and described in order to best explain the principles of the invention and its practical application to thereby enable others skilled in the art to best utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated. Although the above description contains many specifications, these should not be construed as limitations on the scope of the invention, but rather as an exemplification of alternative embodiments thereof. 
     The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.” 
     The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. 
     The use of “including,” “comprising,” “having,” “containing,” “involving,” and/or variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items. 
     It should also be understood that, unless clearly indicated to the contrary, in any methods claimed herein that include more than one step or act, the order of the steps or acts of the method is not necessarily limited to the order in which the steps or acts of the method are recited. 
     While aspects of the invention have been described with reference to various illustrative embodiments, such aspects are not limited to the embodiments described. Thus, it is evident that many alternatives, modifications, and variations of the embodiments described will be apparent to those skilled in the art. Accordingly, embodiments as set forth herein are intended to be illustrative, not limiting. Various changes may be made without departing from the spirit of aspects of the invention.