Patent Publication Number: US-2021188008-A1

Title: Non-pneumatic tire and other annular devices

Description:
This is a Continuation of application Ser. No. 15/594,024 filed Aug. 4, 2017, which in turn is a National Phase of PCT Application No. PCT/US2016/016630 filed Feb. 4, 2016, which claims the benefit of U.S. Provisional Application No. 62/111,872 filed Feb. 4, 2015. The disclosure of the prior applications is hereby incorporated by reference herein in its entirety. 
    
    
     FIELD 
     The invention generally relates to non-pneumatic tires (NPTs), such as for vehicles (e.g., industrial vehicles such as construction vehicles; all-terrain vehicles (ATVs); agricultural vehicles; automobiles and other road vehicles; etc.) and/or other machines, and to other annular devices. 
     BACKGROUND 
     Wheels for vehicles and other machines may comprise non-pneumatic tires (sometimes referred to as NPTs) instead of pneumatic tires. 
     Pneumatic tires are market leaders across a wide variety of size, speed, and load requirements. For example, radial pneumatic tires are found on automotive tires of 0.6 meter diameter that carry 0.5 metric tons, and also on tires used in mining operations of 4 meter diameter that carry 50 metric tons. Pneumatic tires are thus scalable. 
     Pneumatic tires offer high load capacity per unit mass, along with a large contact area and relatively low vertical stiffness. High contact area results in the ability to both efficiently generate high tangential forces and obtain excellent wear characteristics. However, pneumatic tires are also prone to flats. 
     Non-pneumatic tires offer flat-free operation, yet generally contain some compromise. For various reasons, non-pneumatic tires do not have a predominant market share in various industries because they tend to be expensive, heavy, have a poor ride quality, have limited speed capability under heavy load, and/or have lower traction potential, compared to pneumatic tires. For example, in construction and other field with large tires, in the common dimension 20.5 inch×25 inch (20.5 inches wide, 25 inch diameter wheel), currently available non-pneumatic tires weighs around 2000 lbs., whereas a pneumatic tire and steel wheel only weigh around 650 lbs. 
     Non-pneumatic tires in this size are usually solid, with the addition of circular cutouts in the tire sidewall to reduce the compressive stiffness of the structure. Because of this solid construction, heat build-up is problematic. Elastomers are generally good insulators, and therefore such structures tend to retain heat. This reduces their utility in practical use in some cases. 
     Other annular devices, such as, for instance, tracks for vehicles and/or conveyor belts, may in some cases be affected by similar considerations. 
     For these and other reasons, there is a need to improve non-pneumatic tires and other annular devices. 
     SUMMARY 
     According to an aspect of the invention, there is provided a non-pneumatic tire comprising an annular beam. The annular beam comprises a plurality of layers of different elastomeric materials. The annular beam is free of a substantially inextensible reinforcing layer running in a circumferential direction of the non-pneumatic tire. 
     According to another aspect of the invention, there is provided a wheel comprising a hub and a non-pneumatic tire. The non-pneumatic tire comprises an annular beam. The annular beam comprises a plurality of layers of different elastomeric materials. The annular beam is free of a substantially inextensible reinforcing layer running in a circumferential direction of the non-pneumatic tire. 
     According to another aspect of the invention, there is provided an annular beam comprising a plurality of layers of different elastomeric materials. The annular beam is free of a substantially inextensible reinforcing layer running in a circumferential direction of the annular beam. 
     According to another aspect of the invention, there is provided a method of making a non-pneumatic tire. The method comprises providing a plurality of different elastomeric materials and forming an annular beam of the non-pneumatic tire such that the annular beam comprises a plurality of layers of the different elastomeric materials and is free of a substantially inextensible reinforcing layer running in a circumferential direction of the non-pneumatic tire. 
     According to another aspect of the invention, there is provided a method of making an annular beam. The method comprises providing a plurality of different elastomeric materials and forming the annular beam such that the annular beam comprises a plurality of layers of the different elastomeric materials and is free of a substantially inextensible reinforcing layer running in a circumferential direction of the annular beam. 
     According to another aspect of the invention, there is provided a non-pneumatic tire comprising an annular beam. The annular beam comprises a plurality of layers of different elastomeric materials. The annular beam comprises a plurality of openings distributed in a circumferential direction of the non-pneumatic tire. 
     According to another aspect of the invention, there is provided a wheel comprising a hub and a non-pneumatic tire. The non-pneumatic tire comprises an annular beam. The annular beam comprises a plurality of layers of different elastomeric materials. The annular beam comprises a plurality of openings distributed in a circumferential direction of the non-pneumatic tire. 
     According to another aspect of the invention, there is provided an annular beam. The annular beam comprises a plurality of layers of different elastomeric materials. The annular beam comprises a plurality of openings distributed in a circumferential direction of the annular beam. 
     According to another aspect of the invention, there is provided a method of making a non-pneumatic tire. The method comprises providing a plurality of different elastomeric materials and forming an annular beam of the non-pneumatic tire such that the annular beam comprises a plurality of layers of the different elastomeric materials and a plurality of openings distributed in a circumferential direction of the non-pneumatic tire. 
     According to another aspect of the invention, there is provided a method of making an annular beam. The method comprises providing a plurality of different elastomeric materials and forming the annular beam such that the annular beam comprises a plurality of layers of the different elastomeric materials and a plurality of openings distributed in a circumferential direction of the annular beam. 
     According to another aspect of the invention, there is provided a wheel comprising a hub and a non-pneumatic tire. A ratio of a width of the non-pneumatic tire over an outer diameter of the non-pneumatic tire is no more than 0.1 and a ratio of a diameter of the hub over the outer diameter of the non-pneumatic tire is no more than 0.5. 
     According to another aspect of the invention, there is provided a wheel comprising a hub and a non-pneumatic tire. A ratio of a length of a contact patch of the non-pneumatic tire at a design load over an outer radius of the non-pneumatic tire is at least 0.4 
     According to another aspect of the invention, there is provided a non-pneumatic tire comprising an annular beam and a tread. The annular beam is free of a substantially inextensible reinforcing layer running in a circumferential direction of the non-pneumatic tire. The tread comprises elastomeric material and a reinforcing layer disposed within the elastomeric material and extending in the circumferential direction of the non-pneumatic tire. 
     These and other aspects of the invention will now become apparent to those of ordinary skill in the art upon review of the following description of embodiments of the invention in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A detailed description of embodiments is provided below, by way of example only, with reference to the accompanying drawings, in which: 
         FIG. 1  shows an example of a vehicle comprising wheels that comprises non-pneumatic tires in accordance with an embodiment of the invention; 
         FIG. 2  shows a perspective view of a wheel comprising a non-pneumatic tire; 
         FIG. 3  shows a side-elevation view of the wheel and a contact patch of the wheel; 
         FIG. 4  shows a perspective view of an annular beam of the non-pneumatic tire; 
         FIG. 5  shows a cross section of the annular beam; 
         FIGS. 6 to 9  show a side-elevation view of various embodiments of an annular support of the non-pneumatic tire; 
         FIG. 10  shows an example of a spin casting process that may be used to make the non-pneumatic tire; 
         FIG. 11  shows a cross section view of an example of a straight beam that comprises a laminate configuration of elastomer materials; 
         FIG. 12  shows a side elevation view of the straight beam of  FIG. 11  when simply supported by two parallel contact surfaces and subjected to a constant pressure P; 
         FIG. 13  shows a graph showing an example of a relationship between a ratio of beam deflections due to shear and due to bending and a modulus of elasticity of an elastomeric material; 
         FIG. 14  shows a finite-element model of an embodiment of the annular beam loaded between two parallel contact surfaces; 
         FIG. 15  shows analytical solutions of a contact pressure distribution along a contact length of a contact patch of an embodiment of the annular beam comprising the laminate configuration and an embodiment of an annular beam made of an isotropic elastomer; 
         FIG. 16  shows a finite-element model of an embodiment of the non-pneumatic tire comprising the annular beam of  FIG. 14  and subjected to a vertical load on a rigid contact surface; 
         FIG. 17  shows analytical solutions of a contact pressure distribution along the contact length of the contact patch of an embodiment of the annular beam of the non-pneumatic tire of  FIG. 16  comprising different laminate configurations and an embodiment of an annular beam comprising an isotropic elastomer; 
         FIG. 18  shows an example of a thermoplastic polyurethane exhibiting non-linear stress vs. strain characteristics; 
         FIG. 19  shows a perspective view of the wheel comprising the non-pneumatic tire in accordance with another embodiment of the invention; 
         FIG. 20  shows a finite-element model of the non-pneumatic tire of  FIG. 19  subjected to a vertical load on a deformable contact surface; 
         FIG. 21  shows analytical solutions of a contact pressure distribution along a contact length of the contact patch of the non-pneumatic tire of  FIG. 20 ; 
         FIG. 22  shows a finite-element model of the non-pneumatic tire of  FIG. 20 ; 
         FIG. 23  shows a partial cross-sectional view of the non-pneumatic tire that comprises a tread comprising a reinforcing layer in accordance with another embodiment of the invention; 
         FIGS. 24 and 25  show an example of another vehicle comprising wheels that comprise non-pneumatic tires in accordance with another embodiment of the invention; and 
         FIG. 26  shows an example of another vehicle comprising wheels that comprises non-pneumatic tires in accordance with another embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
       FIG. 1  shows an example of a vehicle  10  comprising a plurality of wheels  100   1 - 100   4  in accordance with an embodiment of the invention. In this embodiment, the vehicle  10  is an industrial vehicle. The industrial vehicle  10  is a heavy-duty vehicle designed to travel off-road to perform industrial work using a work implement  44 . In this embodiment, the industrial vehicle  10  is a construction vehicle for performing construction work using the work implement  44 . More particularly, in this embodiment, the construction vehicle  10  is a loader (e.g., a skid-steer loader). The construction vehicle  10  may be a bulldozer, a backhoe loader, an excavator, a dump truck, or any other type of construction vehicle in other embodiments. In this example, the construction vehicle  10  comprises a frame  12 , a powertrain  14 , a steering system  16 , a suspension  18 , the wheels  100   1 - 100   4 , and an operator cabin  22 , which enable a user, i.e., an operator, of the construction vehicle  10  to move the vehicle  10  on the ground and perform work using the work implement  44 . The construction vehicle  10  has a longitudinal direction, a widthwise direction, and a height direction. 
     In this embodiment, as further discussed later, the wheels  100   1 - 100   4  are non-pneumatic (i.e., airless) and may be designed to enhance their use and performance and/or use and performance of the construction vehicle  10 , including, for example, by having a high load-carrying capacity while being relatively lightweight. 
     The powertrain  14  is configured for generating motive power and transmitting motive power to respective ones of the wheels  100   1 - 100   4  to propel the construction vehicle  10  on the ground. To that end, the powertrain  14  comprises a prime mover  26 , which is a source of motive power that comprises one or more motors. For example, in this embodiment, the prime mover  26  comprises an internal combustion engine. In other embodiments, the prime mover  26  may comprise another type of motor (e.g., an electric motor) or a combination of different types of motor (e.g., an internal combustion engine and an electric motor). The prime mover  26  is in a driving relationship with one or more of the wheels  100   1 - 100   4 . That is, the powertrain  14  transmits motive power generated by the prime mover  26  to one or more of the wheels  100   1 - 100   4  (e.g., via a transmission and/or a differential) in order to drive (i.e., impart motion to) these one or more of the wheels  100   1 - 100   4 . 
     The steering system  16  is configured to enable the operator to steer the construction vehicle  10  on the ground. To that end, the steering system  16  comprises a steering device  28  that is operable by the operator to direct the construction vehicle  10  along a desired course on the ground. The steering device  28  may comprise a steering wheel or any other steering component (e.g., a joystick) that can be operated by the operator to steer the construction vehicle  10 . The steering system  16  responds to the operator interacting with the steering device  28  by turning respective ones of the wheels  100   1 - 100   4  to change their orientation relative to part of the frame  12  of the construction vehicle  10  in order to cause the vehicle  10  to move in a desired direction. In this example, a front frame member  23   1  carrying front ones of the wheels  100   1 - 100   4  is turnable in response to input of the operator at the steering device  28  to change its orientation and thus the orientation of the front ones of the wheels  100   1 - 100   4  relative to a rear frame member  23   2  of the construction vehicle  10  in order to steer the construction vehicle  10  on the ground. 
     The suspension  18  is connected between the frame  12  and the wheels  100   1 - 100   4  to allow relative motion between the frame  12  and the wheels  100   1 - 100   4  as the construction vehicle  10  travels on the ground. For example, the suspension  18  may enhance handling of the construction vehicle  10  on the ground by absorbing shocks and helping to maintain traction between the wheels  100   1 - 100   4  and the ground. The suspension  18  may comprise an arrangement of springs and dampers. A spring may be a coil spring, a leaf spring, a gas spring (e.g., an air spring), or any other elastic object used to store mechanical energy. A damper (also sometimes referred to as a “shock absorber”) may be a fluidic damper (e.g., a pneumatic damper, a hydraulic damper, etc.), a magnetic damper, or any other object which absorbs or dissipates kinetic energy to decrease oscillations. In some cases, a single device may itself constitute both a spring and a damper (e.g., a hydropneumatic, hydrolastic, or hydragas suspension device). 
     The operator cabin  22  is where the operator sits and controls the construction vehicle  10 . More particularly, the operator cabin  22  comprises a user interface  70  including a set of controls that allow the operator to steer the construction vehicle  10  on the ground and operate the work implement  44 . The user interface  70  also comprises an instrument panel (e.g., a dashboard) which provides indicators (e.g., a speedometer indicator, a tachometer indicator, etc.) to convey information to the operator. 
     The wheels  100   1 - 100   4  engage the ground to provide traction to the construction vehicle  10 . More particularly, in this example, the front ones of the wheels  100   1 - 100   4  provide front traction to the construction vehicle  10  while the rear ones of the wheels  100   1 - 100   4  provide rear traction to the construction vehicle  10 . 
     Each wheel  100   i  comprises a non-pneumatic tire  110  for contacting the ground and a hub  120  for connecting the wheel  100   i  to an axle of the vehicle  10 . The non-pneumatic tire  110  is a compliant wheel structure that is not supported by gas (e.g., air) pressure and that is resiliently deformable (i.e., changeable in configuration) as the wheel  100   i  contacts the ground. With additional reference to  FIG. 2 , the wheel  100   i  has an axial direction defined by an axis of rotation  180  of the wheel  100   i  (also referred to as a “Y” direction), a radial direction (also referred to as a “Z” direction), and a circumferential direction (also referred to as a “X” direction). These axial, radial and circumferential directions also apply to components of the wheel  100   i , including the non-pneumatic tire  110 . The wheel&#39;s equatorial plane is that plane defined by the x-z axes, while the wheel&#39;s cross section is that plane defined by the y-z axes. The wheel  100   i  has an outer diameter D W  and a width W W . It comprises an inboard lateral side  147  for facing a center of the vehicle in the widthwise direction of the vehicle and an outboard lateral side  149  opposite the inboard lateral side  147 . As shown in  FIG. 3 , when it is in contact with the ground, the wheel  100   i  has an area of contact  125  with the ground, which may be referred to as a “contact patch” of the wheel  100   i  with the ground. The contact patch  125  of the wheel  100   i , which is a contact interface between the non-pneumatic tire  110  and the ground, has a length L C  in the circumferential direction of the wheel  100   i  and a width W C  in the axial direction of the wheel  100   i    
     The non-pneumatic tire  110  comprises an annular beam  130  and an annular support  140  that is disposed between the annular beam  130  and the hub  120  of the wheel  100  and configured to support loading on the wheel  100  as the wheel  100  engages the ground. In this embodiment, the non-pneumatic tire  110  is tension-based such that the annular support  140  is configured to support the loading on the wheel  100  by tension. That is, under the loading on the wheel  100   i , the annular support  140  is resiliently deformable such that a lower portion  127  of the annular support  140  between the axis of rotation  180  of the wheel  100  and the contact patch  125  of the wheel  100  is compressed and an upper portion  129  of the annular support  140  above the axis of rotation  180  of the wheel  100   i  is in tension to support the loading. 
     The annular beam  130  of the non-pneumatic tire  110  is configured to deflect under the loading on the wheel  100  at the contact patch  125  of the wheel  100   i  with the ground. In this embodiment, the annular beam  130  is configured to deflect such that it applies a homogeneous contact pressure along the length L C  of the contact patch  125  of the wheel  100   i  with the ground. 
     More particularly, in this embodiment, the annular beam  130  comprises a shear band  131  configured to deflect predominantly by shearing at the contact patch  125  under the loading on the wheel  100   i . That is, under the loading on the wheel  100   i , the shear band  131  deflects significantly more by shearing than by bending at the contact patch  125 . The shear band  131  is thus configured such that, at a center of the contact patch  125  of the wheel  100   i  in the circumferential direction of the wheel  100   i , a shear deflection of the annular beam  130  is significantly greater than a bending deflection of the annular beam  130 . For example, in some embodiments, at the center of the contact patch  125  of the wheel  100   i  in the circumferential direction of the wheel  100   i , a ratio of the shear deflection of the annular beam  130  over the bending deflection of the annular beam  130  may be at least 1.2, in some cases at least 1.5, in some cases at least 2, in some cases at least 3, in some cases at least 5, in some cases at least 7, and in some cases even more. For instance, in some embodiments, the annular beam  130  may be designed based on principles discussed in U.S. Patent Application Publication 2014/0367007, which is hereby incorporated by reference herein, in order to achieve the homogeneous contact pressure along the length L C  of the contact patch  125  of the wheel  100   i  with the ground. 
     In this embodiment, the shear band  131  of the annular beam  130  comprises a plurality of layers  132   1 - 132   N  of different elastomeric materials M 1 -M E . The layers  132   1 - 132   N  of the different elastomeric materials M 1 -M E  extend in the circumferential direction of the wheel  100  and are disposed relative to one another in the radial direction of the wheel  100   i . As further discussed later, in some embodiments, this laminate construction of the different elastomeric materials M 1 -M E  may enhance performance of the wheel  100   i , including behavior of its contact patch  125  and may also help the annular beam  130  to have a high load to mass ratio, yet keep the simplicity of an elastomer structure, with no need for inextensible membranes or other composites or reinforcing elements. In this example, the layers  132   1 - 132   N  of the different elastomeric materials M 1 -M E  are seven layers, namely the layers  132   1 - 132   7  and the different elastomeric materials M 1 -M E  are two different elastomeric materials, namely the elastomeric materials M 1 , M 2 . The layers  132   1 - 132   N  and/or the elastomeric materials M 1 -M E  may be present in any other suitable numbers in other examples. 
     More particularly, in this embodiment, the layers  132   1 ,  132   3 ,  132   5  and  132   7  are made of the elastomeric material M 1  while the layers  132   2 ,  132   4  and  132   6  are made of the elastomeric material M 2  and are disposed between respective ones of the layers  132   1 ,  132   3 ,  132   5  and  132   7  made of the elastomeric material M 1 . The layers  132   1 - 132   7  of the annular beam  130  are thus arranged such that the different elastomeric materials M 1 , M 2  alternate in the radial direction of the wheel  100   i . 
     For instance, in this embodiment, the shear band  131  comprises the layer  132   1 , composed of elastomeric material M 1 , lying on a radially inward extent of the shear band  131 . The shear band  131  comprises the layer  132   2 , composed of elastomeric material M 2 , lying on a radially outward extent of the layer  132   1 . The shear band  131  comprises the layer  132   3 , composed of elastomeric material M 1 , lying on a radially outward extent of the layer  132   2 . In this embodiment, a laminate configuration of the elastomeric material of the shear band  131  is M 1 /M 2 /M 1 . In other embodiments, the laminate configuration of the elastomeric material of the shear band  131  may be repeated any number of times. For example, in  FIGS. 4 and 5 , the laminate configuration of the elastomeric material of the shear band  131  from an inward to an outward extent of the shear band  131  is M 1 /M 2 /M 1 /M 2 /M 1 /M 2 /M 1 . Each one of the layers  132   1 - 132   7  is composed of a homogeneous elastomer in this example. 
     The different elastomeric materials M 1  and M 2  may differ in any suitable way. For example, in some embodiments, a stiffness of the elastomeric material M 1  may be different from a stiffness of the elastomeric material M 2 . That is, the elastomeric material M 1  may be stiffer or less stiff than the elastomeric material M 2 . For instance, a modulus of elasticity E 1  (i.e., Young&#39;s modulus) of the elastomeric material M 1  may be different from a modulus of elasticity E 2  of the elastomeric material M 2 . A modulus of elasticity herein is Young&#39;s tensile modulus of elasticity measured per ISO 527-1/-2, and “Young&#39;s Modulus,” “tensile modulus,” and “modulus” may be used interchangeably herein. For example, in some embodiments, the modulus of elasticity E 1  of the elastomeric material M 1  may be greater than the modulus of elasticity E 2  of the elastomeric material M 2 . For instance, in some embodiments, a ratio E 1 /E 2  of the modulus of elasticity E 1  of the elastomeric material M 1  over the modulus of elasticity E 2  of the elastomeric material M 2  may be at least 2, in some cases at least 3, in some cases at least 4, in some cases at least 5, in some cases at least 6, in some cases at least 7, in some cases at least 8, and in some cases even more. 
     For example, in some embodiments, the modulus of elasticity E 1  of the elastomeric material M 1  may be at least 150 MPa, and in some cases at least 200 MPa or even more, while the modulus of elasticity E 2  of the elastomeric material M 2  may be no more than 50 MPa, and in some cases no more than 30 MPa or even less. As will be disclosed, such a modulus definition can be engineered to give a beam particular bending and shear properties that are favorable for use in the non-pneumatic tire  110 . 
       FIG. 5  shows a cross section AA of the shear band  131  of the annular beam  130  where the layers  132   1 - 132   7  of the annular beam  131  are shown. In some embodiments, such as the embodiment of  FIGS. 4 and 5 , the innermost layer  132   1  and the outermost layer  132   7  of the shear band  131  may be composed of the elastomeric material M 1  with the modulus of elasticity E 1  higher than the modulus of elasticity E 2  of the elastomeric material M 2 . That is, in this embodiment, the elastomeric material with the higher modulus of elasticity may be used at the inner and outer radial extents of the shear band  131  of the annular beam  130 . 
     In other embodiments, other repeating or non-repeating laminate configurations of the elastomeric material of the shear band  131  comprising the elastomeric material with the higher modulus of elasticity at the inner and outer radial extents of the shear band  131  may be used. That is, in these embodiments, multiple layers composed of multiple elastomeric materials may be used with or without symmetry of the laminate configuration of the elastomeric material of the shear band  131  and the shear band  131  may comprise at least three elastomeric materials in a laminate configuration. For example, the laminate configuration of the elastomeric material of the shear band  131  from an inward to an outward extent of the shear band  131  may be of the type M 1 /M 2 /M 3 /M 2 /M 1  or M 1 /M 2 /M 3 /M 1  or any other combination thereof, where M 3  is an elastomeric material having a modulus of elasticity E 3  different from the modulus of elasticity E 1  of the elastomeric material M 1  and different from the modulus of elasticity E 2  of the elastomeric material M 2 . 
     In some embodiments, and with further reference to  FIGS. 4 and 5 , each one of the layers  132   1 - 132   7  of the shear band  131  extends from the inboard lateral side  147  to the outboard lateral side  149  of the non-pneumatic tire  110 . That is, each one of the layers  132   1 - 132   7  of the shear band  131  extends laterally through the shear band  131  in the axial direction of the wheel  100 . 
     The different elastomeric materials M 1 -M E  may include any other suitable elastomers in various embodiments. For example, in some embodiments, suitable elastomeric materials include thermoplastic and thermoset polyurethane and thermoplastic and thermoset rubbers. 
     In this embodiment, the annular beam  130  is free of (i.e., without) a substantially inextensible reinforcing layer running in the circumferential direction of the wheel  100  (e.g., a layer of metal, composite (e.g., carbon fibers, other fibers), and/or another material that is substantially inextensible running in the circumferential direction of the wheel  100   i ). In that sense, the annular beam  130  may be said to be “unreinforced”. Thus, in this embodiment, useful behavior of the wheel  100   i , including deflection and behavior of its contact patch  125 , may be achieved without any substantially inextensible reinforcing layer running in the circumferential direction of the wheel  100   i , which may help to reduce the weight and cost of the wheel  100   i . 
     In this embodiment, the non-pneumatic tire  110  comprises a tread  150  for enhancing traction between the non-pneumatic tire  110  and the ground. The tread  150  is disposed about an outer peripheral extent  146  of the annular beam  130 , in this case about the outermost layer  132   7  of the shear band  131  composed of the elastomeric material M 1 . More particularly, in this example the tread  150  comprises a tread base  151  that is at the outer peripheral extent  146  of the annular beam  130  and a plurality of tread projections  152   1 - 152   T  that project from the tread base  151 . The tread  150  may be implemented in any other suitable way in other embodiments (e.g., may comprise a plurality of tread recesses, etc.). 
     The annular support  140  is configured to support the loading on the wheel  100   i  as the wheel  100   i  engages the ground. As mentioned above, in this embodiment, the annular support  140  is configured to support the loading on the wheel  100   i  by tension. More particularly, in this embodiment, the annular support  140  comprises a plurality of support members  142   1 - 142   T  that are distributed around the non-pneumatic tire  110  and resiliently deformable such that, under the loading on the wheel  100   i , lower ones of the support members  142   1 - 142   T  in the lower portion  127  of the annular support  140  (between the axis of rotation  180  of the wheel  100   i  and the contact patch  125  of the wheel  100   i ) are compressed and bend while upper ones of the support members  142   1 - 142   T  in the upper portion  129  of the annular support  140  (above the axis of rotation  180  of the wheel  100   i ) are tensioned to support the loading. As they support load by tension when in the upper portion  129  of the annular support  140 , the support members  142   1 - 142   T  may be referred to as “tensile” members. 
     In this embodiment, the support members  142   1 - 142   T  are elongated and extend from the annular beam  130  towards the hub  120  generally in the radial direction of the wheel  100   i . In that sense, the support members  142   1 - 142   T  may be referred to as “spokes” and the annular support  140  may be referred to as a “spoked” support. 
     More particularly, in this embodiment, each spoke  142   i  extends from an inner peripheral surface  148  of the annular beam  130  towards the hub  120  generally in the radial direction of the wheel  100   i  and from a first lateral end  155  to a second lateral end  157  in the axial direction of the wheel  100   i . In this case, the spoke  142   i  extends in the axial direction of the wheel  100   i  for at least a majority of a width W T  of the non-pneumatic tire  110 , which in this case corresponds to the width W W  of the wheel  100   i . For instance, in some embodiments, the spoke  142   i  may extend in the axial direction of the wheel  100   i  for more than half, in some cases at least 60%, in some cases at least 80%, and in some cases an entirety of the width W T  of the non-pneumatic tire  110 . Moreover, the spoke  142   i  has a thickness T S  measured between a first surface face  159  and a second surface face  161  of the spoke  142   i  that is significantly less than a length and width of the spoke  142   i . 
     When the wheel  100   i  is in contact with the ground and bears a load (e.g., part of a weight of the vehicle), respective ones of the spokes  142   1 - 142   T  that are disposed in the upper portion  129  of the spoked support  140  (i.e., above the axis of rotation  180  of the wheel  100   i ) are placed in tension while respective ones of the spokes  142   1 - 142   T  that are disposed in the lower portion  127  of the spoked support  140  (i.e., adjacent the contact patch  125 ) are placed in compression. The spokes  142   1 - 142   T  in the lower portion  127  of the spoked support  140  which are in compression bend in response to the load. Conversely, the spokes  142   1 - 142   T  in the upper portion  129  of the spoked support  140  which are placed in tension support the load by tension. 
     The spokes  142   1 - 142   T  may be implemented in any other suitable way in other embodiments. For example,  FIGS. 6 to 9  show various embodiments of the design of the spokes  142   1 - 142   T . In the embodiment of  FIG. 6 , each spoke  142   i  extends generally along a straight line in the radial direction of the wheel  100   i . In the embodiment of  FIG. 7 , each spoke  142   i  extends generally along a straight line in the radial direction of the wheel  100   i , a spoke connector  143  being located between every other pair of successive spokes  142   i  and connecting two successive spokes  142   i . The spoke connector  143  is substantially perpendicular to the radial direction of the wheel  100   i  and may be positioned at any distance from the hub  120 . along the radial direction of the wheel  100   i . In some embodiment, the spoke connector  143  extends in the axial direction of the wheel  100   i  for at least a majority of the width W T  of the non-pneumatic tire  110 , which in this case corresponds to the width W W  of the wheel  100   i . For instance, in some embodiments, the spoke connector  143  may extend in the axial direction of the wheel  100   i  for more than half, in some cases at least 60%, in some cases at least 80%, and in some cases an entirety of the width W T  of the non-pneumatic tire  110 . Moreover, the spoke connector  143  has a thickness T SC  measured between a first surface face  163  and a second surface face  165  of the spoke connector  143  that is significantly less than a length and width of the spoke connector  143 . In other embodiments, the spoke connector  143  may not be substantially perpendicular to the radial direction of the wheel  100   i . In other embodiments, there may be a plurality of spoke connectors  143  connecting two spokes  142   i . In the embodiment of  FIG. 8 , each spoke  142   i  extends generally along a straight line at an angle α or −α in the radial direction of the wheel  100   i  such that two successive spokes  142   i  do not extend generally along a straight line at the same angle in the radial direction of the wheel  100   i . In the embodiment of  FIG. 9 , each spoke  142   i  extends generally as a curved line along the radial direction of the wheel  100   i . Other designs may be possible in other embodiments. 
     The non-pneumatic tire  110  has an inner diameter D TI  and an outer diameter D TO , which in this case corresponds to the outer diameter D W  of the wheel  100 . A sectional height H T  of the non-pneumatic tire  110  is half of a difference between the outer diameter D TO  and the inner diameter D TI  of the non-pneumatic tire  110 . The sectional height H T  of the non-pneumatic tire may be significant in relation to the width W T  of the non-pneumatic tire  110 . In other words, an aspect ratio AR of the non-pneumatic tire  110  corresponding to the sectional height H T  over the width W T  of the non-pneumatic tire  110  may be relatively high. For instance, in some embodiments, the aspect ratio AR of the non-pneumatic tire  110  may be at least 70%, in some cases at least 90%, in some cases at least 110%, and in some cases even more. Also, the inner diameter D TI  of the non-pneumatic tire  110  may be significantly less than the outer diameter D TO  of the non-pneumatic tire  110  as this may help for compliance of the wheel  100   i . For example, in some embodiments, the inner diameter D TI  of the non-pneumatic tire  110  may be no more than half of the outer diameter D TO  of the non-pneumatic tire  110 , in some cases less than half of the outer diameter D TO  of the non-pneumatic tire  110 , in some cases no more than 40% of the outer diameter D TO  of the non-pneumatic tire  110 , and in some cases even a smaller fraction of the outer diameter D TO  of the non-pneumatic tire  110 . In this embodiment, the non-pneumatic tire  110  therefore comprises different tire materials that make up the tire  110 , including the elastomeric materials M 1 -M E  of the shear band  131  of the annular beam  130  and a spoke material  145  that makes up at least a substantial part (i.e., a substantial part or an entirety) of the spokes  142   1 - 142   T . The hub  120  comprises a hub material  172  that makes up at least a substantial part of the hub  120 . In some embodiments, the hub material  172  may be the same as one of the tire materials, namely one of the elastomeric materials M 1 -M E  of the shear band  131  of the annular beam  130  and the spoke material  145 . In other embodiments, the hub material  172  may be different from any of the tire materials, i.e., different from any of the elastomeric materials M 1 -M E  of the shear band  131  of the annular beam  130  and the spoke material  145 . For instance, in some embodiments, the spoke material  145  and the hub material  172  may be any one of the elastomeric material M 1 , M 2 , M 3  or any other elastomeric material that may be comprised in the shear band  131  of the annular beam  130 . 
     In this embodiment, any given material of the wheel  100   i , such as any given one of the tire materials (i.e., the elastomeric materials M 1 -M E  of the shear band  131  of the annular beam  130  and the spoke material  145 ) and/or the hub material  172  may exhibit a non-linear stress vs. strain behavior. For instance, the spoke material  145  may have a secant modulus that decreases with increasing strain of the spoke material  145 . A secant modulus herein is defined as a tensile stress divided by a tensile strain for any given point on a tensile stress vs. tensile strain curve measured per ISO 527-1/-2. The spoke material  145  may have a high Young&#39;s modulus that is significantly greater than the secant modulus at 100% strain (a.k.a. “the 100% modulus”). Such a non-linear behavior of the spoke material  145  may provide efficient load carrying during normal operation and enable impact loading and large local deflections without generating high stresses. For instance, the spoke material  145  may allow the non-pneumatic tire  110  to operate at a low strain rate (e.g., 2% to 5%) during normal operation yet simultaneously allow large strains (e.g., when the wheel  100   i  engages obstacles) without generating high stresses. This in turn may be helpful to minimize vehicle shock loading and enhance durability of the non-pneumatic tire  110 . 
     The non-pneumatic tire  110  may comprise any other arrangement of materials in other embodiments (e.g., different parts of the annular beam  130 , different parts of the tread  150 , and/or different parts of the spokes  142   1 - 142   T  may be made of different materials). For example, in some embodiments, different parts of the tread  150 , and/or different parts of the spokes  142   1 - 142   T  may be made of different elastomers. 
     In this embodiment, the hub material  172  constitutes at least part of the hub  120 . More particularly, in this embodiment, the hub material  172  constitutes at least a majority (e.g., a majority or an entirety) of the hub  120 . In this example of implementation, the hub material  172  makes up an entirety of the hub  120 . 
     In this example of implementation, the hub material  172  is polymeric. More particularly, in this example of implementation, the hub material  172  is elastomeric. For example, in this embodiment, the hub material  172  comprises a polyurethane (PU) elastomer. For instance, in some cases, the PU elastomer may be PET-95A commercially available from COIM, cured with MCDEA. 
     The hub material  172  may be any other suitable material in other embodiments. For example, in other embodiments, the hub material  172  may comprise a stiffer polyurethane material, such as COIM&#39;s PET75D cured with MOCA. In some embodiments, the hub material  172  may not be polymeric. For instance, in some embodiments, the hub material  172  may be metallic (e.g., steel, aluminum, etc.). 
     The hub  120  may comprise one or more additional materials in addition to the hub material  172  in other embodiments (e.g., different parts of the hub  120  may be made of different materials). 
     For example, in some embodiments, for the spoked support  140  and the hub  120 , various cast polyurethanes of either ether or ester systems may be used when appropriate (e.g. with alternative cure systems such as MOCA). In some examples, a shore hardness in the range of 90 A to 75 D and/or a Young&#39;s modulus between 40 MPA to 2000 MPa may be appropriate. 
     In some embodiments, the spoked support  140  and the hub  120  may comprise different materials. For example, the spoked support  140  may comprise a softer material (e.g., with a Young&#39;s modulus between 40 MPA to 100 MPA) and the hub  120  may comprise a harder material (e.g., with modulus between 300 to 2000 MPA). 
     The tread  150  may comprise an elastomeric material  160 . In some examples of implementation, the elastomeric material  160  of the tread  150  may be different from the elastomeric materials M 1 -M E  of the annular beam  130 . For example, in some embodiments, the elastomeric material  160  of the tread  150  may be rubber. In other embodiments, the elastomeric material  160  of the tread  150  may be polyurethane or another elastomer. For instance, in some embodiments, the tread  150  may comprise rubber, cast polyurethane or any other suitable elastomer, and may have a Shore hardness of between 60 A to 85 A, with a Young&#39;s modulus between 3 MPa and 20 MPa. The tread  150  may be provided in any suitable way, such as by molding and/or adhesively bonding the elastomeric material  160  of the tread  150  about the annular beam  130 . 
     The wheel  100   i  may be manufactured in any suitable way. For example, in some embodiments, the non-pneumatic tire  110  and/or the hub  120  may be manufactured via centrifugal casting, a.k.a. spin casting, which involves pouring one or more materials of the wheel  100   i  into a mold  200  that rotates about an axis  202  as shown in  FIG. 10 . The material(s) is(are) distributed within the mold  200  via a centrifugal force generated by the mold&#39;s rotation. In some cases, vertical spin casting, in which the mold&#39;s axis of rotation  202  is generally vertical, may be used. In other cases, as shown in  FIG. 10 , horizontal spin casting, in which the mold&#39;s axis of rotation  202  is generally horizontal, may be used. In some embodiments, horizontal spin casting may be useful for casting the layers  132   1 - 132   N  of the different elastomeric materials M 1 -M E  of the annular beam  130  in a more controlled manner. The wheel  100  may be manufactured using any other suitable manufacturing processes in other embodiments. 
     The wheel  100   i  may be lightweight. That is, a mass M W  of the wheel  100   i  may be relatively small. For example, in some embodiments, a ratio M normalized  of the mass M W  of the wheel  100   i  in kilograms over the outer diameter D W  of the wheel  100   i  normalized by the width W W  of the wheel  100   i , 
     
       
         
           
             
                 
             
              
             
               
                 M 
                 normalized 
               
               = 
               
                 
                   ( 
                   
                     
                       M 
                       w 
                     
                     
                       D 
                       w 
                     
                   
                   ) 
                 
                 
                   W 
                   w 
                 
               
             
           
         
       
     
     may be no more than 0.00035 kg/mm 2 , in some cases no more than 0.00030 kg/mm 2 , in some cases no more than 0.00025 kg/mm 2 , in some cases no more than 0.00020 kg/mm 2 , in some cases no more than 0.00015 kg/mm 2 , in some cases no more than 0.00013 kg/mm 2 , in some cases no more than 0.00011 kg/mm 2 , and in some cases even less (e.g., no more than 0.0001 kg/mm 2 ). 
     For instance, in some embodiments, the outer diameter of the wheel  100   i  may be 1.5 m, the width of the wheel  100   i  may be about 0.5 m, and the mass M W  of the wheel  100   i  may be about 336 kg. The load capacity of the wheel  100   i  may be about 10,000 kg at 15 kph. The wheel  100   i  may be a replacement for a 20.5″×25″ pneumatic tire. 
     The wheel  100   i , including the non-pneumatic tire  110  and the hub  120 , may thus be designed to enhance its use and performance. Notably, in some embodiments, the structure of the shear band  131  of the annular beam  130  comprising the different elastomeric materials M 1 -M E  in a laminate configuration may be related to the deflection properties of the annular beam  130  so as to enhance behavior of the contact patch  125  of the wheel  100   i . When connected to the hub  120  via the spokes  142   1 - 142   T , the annular beam  130  has a high load to mass ratio, yet keeps the simplicity of an elastomer structure, with no need for inextensible membranes or other composites or reinforcing elements. 
     For example, in some embodiments, a tire contact pressure may be substantially constant along the length L C  of the contact patch  125 . To achieve this, the annular beam  130  having a radius of curvature R may be designed such that it develops a relatively constant pressure along the length L C  of the contact patch  125  when the annular beam  130  is deformed to a flat surface. With reference to  FIGS. 11 and 12 , this is analogous to designing a straight beam which deforms to a circular arc of radius R when subjected to a constant pressure which is equal to the contact pressure of the annular beam  130  along the length L C  of the contact patch  125 . The inventor has found that a homogeneous beam of solid cross section does not behave like this. To create this desired performance, beam bending stiffness and beam shear stiffness can be designed using a laminate of elastomer materials, such that the beam deforms primarily in shear. An example of a method for doing so will now be discussed, using standard nomenclature (e.g. see for example Muvdi, B. B., McNabb, J. W., (1980). Engineering Mechanics of Materials, Macmillan Publishing Co., Inc., New York, N.Y., “Shear and Bending Moment in Beams,” pp 23-31, and “Deflections of Beams”, pp 266-333, which is hereby incorporated by reference herein). 
     Without wishing to be bound by any theory, it may be useful to consider certain aspects of the physics of elastomers. The relationship of shear force variation to an applied distributed load on a differential beam element can be expressed as follows: 
     
       
         
           
             
               
                 
                   
                     - 
                     
                       dV 
                       dx 
                     
                   
                   = 
                   W 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     Where: 
     
         
         
           
             V=transverse shear force 
             W=Constant distributed load per unit length 
             x=beam length coordinate 
           
         
       
    
     The deflection of the differential beam element due to shear deformation alone can be estimated by combining Equation 1 with other known relationships. Adding relations between shear force, shear stress, shear modulus, and cross-sectional area, Equation 2 can be derived: 
     
       
         
           
             
               
                 
                   
                     
                       
                         d 
                         2 
                       
                        
                       z 
                     
                     
                       
                         d 
                         2 
                       
                        
                       x 
                     
                   
                   = 
                   
                     W 
                     GA 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Where: 
     
         
         
           
             G=beam shear modulus 
             A=effective beam cross sectional area 
             z=transverse beam deflection 
           
         
       
    
     Shear modulus means the shear modulus of elasticity and is calculated according to Equation 10 below. For small deflections, 
     
       
         
           
             
               
                 d 
                 2 
               
                
               z 
             
             
               
                 d 
                 2 
               
                
               x 
             
           
         
       
     
     is equal to the inverse of the deformed beam radius of curvature. Making this substitution and considering a beam of unit depth, one obtains Equation 3: 
     
       
         
           
             
               
                 
                   P 
                   = 
                   
                     GA 
                     R 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Where: 
     
         
         
           
             G=beam shear modulus 
             R=deformed beam radius of curvature 
             A=effective beam cross sectional area, with unit depth 
             P=Constant distributed pressure, with unit depth 
           
         
       
    
     According to equation 3, a straight beam of shear modulus G and effective cross sectional area A, such as the straight beam of  FIG. 11 , will deform into the shape of an arc of radius R when subjected to homogeneous pressure P, provided shear deflection predominates. 
     Similarly, the annular beam  130  having radius of curvature R, designed such that shear deformation predominates, will develop a homogeneous contact pressure P along the contact patch  125  having the length L C  when deflected against a flat contact surface. 
     A constant pressure along the contact patch  125  having the length L C  may be a highly desired performance attribute. It may be particularly useful when embodied in the non-pneumatic tire  110  of  FIGS. 1 to 3 . With further reference to  FIG. 3 , when a design load is applied at the hub  120 , for instance when the wheel  100   i  supports the weight of the vehicle  10 , the annular beam  130  deforms over the contact patch  125  having the length L C  and develops a homogeneous contact pressure over the length L C  of the contact patch  125 . The design load is a usual and expected operating load of the non-pneumatic tire  110 . Lower ones of the support members  142   1 - 142   T  in the lower portion  127  of the annular support  140  (between the axis of rotation  180  of the wheel  100  and the contact patch  125  of the wheel  100 ) are compressed and bend while upper ones of the support members  142   1 - 142   T  in the upper portion  129  of the annular support  140  (above the axis of rotation  180  of the wheel  100 ) are tensioned to support the loading, such that the annular beam  130  passes the load to the central hub  120  via tension in annular support  140 . 
     In some embodiments, a homogeneous contact pressure over the length L C  of the contact patch  125  may be achieved through an appropriate laminate configuration of the shear band  131  of the annular beam  130  that comprises elastomers, such as the layers  132   1 - 132   N  of the different elastomeric materials M 1 -M E . The material properties of the laminate configuration of the shear band  131  may be designed such that shear deflection can be larger than bending deflection at a center of the contact patch  125 . 
     Analysis of a straight beam may be less cumbersome than the analysis of an annular beam such as the annular beam  130 ; therefore a first part of an example of a design process may employ a straight beam geometry such as the one shown in  FIG. 12  subjected to a constant pressure, in order to design the laminate configuration of the annular beam  130  and the thickness of each one of the layers  132   1 - 132   N  of the different elastomeric materials M 1 -M E  in the laminate configuration of the annular beam  130 . Final design verification may then include a complete tire model, as will be discussed. 
     Accordingly, in this example, the first step in developing a design process is to calculate the deflection due to bending and the deflection due to shear of a simply supported straight beam subjected to a constant pressure, as shown in  FIG. 12 . Equation 4 gives the center deflection due to bending; Equation 5 gives the center deflection due to shear; Equation 6 solves for shear deflection divided by bending deflection: 
     
       
         
           
             
               
                 
                   
                     z 
                     b 
                   
                   = 
                   
                     
                       5 
                       384 
                     
                      
                     
                       
                         PL 
                         4 
                       
                       EI 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                     s 
                   
                   = 
                   
                     
                       1 
                       4 
                     
                      
                     
                       
                         PL 
                         2 
                       
                       GA 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       z 
                       s 
                     
                     
                       z 
                       b 
                     
                   
                   = 
                   
                     19.2 
                      
                     
                       EI 
                       
                         L 
                         2 
                       
                     
                      
                     
                       1 
                       GA 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Where: 
     
         
         
           
             z b =beam center deflection due to bending 
             z s =beam center deflection due to shear 
             L=beam length, which is about equal to the length L c  of the contact patch  125   
             E=beam tensile modulus 
             I=beam moment of inertia 
           
         
       
    
     The result of Equation (6) is a dimensionless geometrical term that, for homogeneous materials, is independent of modulus. As z s /z b  becomes larger, shear deflection predominates. As shear deflection predominates, Equation (3) becomes valid and the desired performance of a constant pressure through the length L C  of the contact patch  125  is achieved. 
     In usual engineering calculation of transverse deflection of beams, shear deflection may be assumed to be small compared to bending deflection, and shear deflection may be neglected. Consequently, the result of Equation (6) may not be commonly considered. Beam bending stiffness must be relatively high, and beam shear stiffness must be relatively low in order to have z s /z b  be high enough so that Equation (3) becomes approximately valid. 
     The next step of the design process in this example is to define the procedure to relate the design of the elastomer laminate structure to the terms of Equation 6. Analytical solutions for the terms are provided below. 
       FIG. 11  uses a laminate configuration equivalent to the laminate configuration of the shear band  131  of the annular beam  130  as shown in  FIGS. 4 and 5 . For illustrative purposes, this cross section definition will be used to demonstrate an example of a design methodology. Using the same technique, any general laminate elastomer cross section can be analyzed to determine the quantities for Equation 6. 
     With reference to  FIG. 11  an effective beam shear modulus for this cross-section may be estimated to be used as G in Equation 6. This is calculated using Equation 7: 
     
       
         
           
             
               
                 
                   
                     G 
                     = 
                     
                       
                         G 
                         eff 
                       
                       = 
                       
                         1 
                         
                           
                             
                               V 
                               
                                 f 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                             
                               G 
                               1 
                             
                           
                           + 
                           
                             
                               V 
                               
                                 f 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                             
                               G 
                               2 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       v 
                       
                         f 
                          
                         
                             
                         
                          
                         1 
                       
                     
                     = 
                     
                       
                         
                           
                             2 
                              
                             
                               t 
                               3 
                             
                           
                           
                             t 
                             shear 
                           
                         
                          
                         
                             
                         
                          
                         
                           v 
                           
                             f 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                       = 
                       
                         
                           
                             2 
                              
                             
                               t 
                               2 
                             
                           
                           + 
                           
                             t 
                             4 
                           
                         
                         
                           t 
                           shear 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Where Vf1=volume fraction of elastomer 1 across beam radial thickness t shear.
         Vf2=volume fraction of elastomer 2 across beam radial thickness t shear.   G1=shear modulus of elastomer 1   G2=shear modulus of elastomer 2       

     The effective shear modulus calculation is used as the shear modulus G in Equation (5) to calculate z s , the beam center deflection due to shear. For a unit depth assumption the effective beam cross sectional area A for shear deformation calculation equals the beam shear thickness t shear . Thus: 
         A=t   shear   (8)
 
     Physically, this can be visualized as the shear deflection across the web of an “I” beam; the outer bands of the high modulus elastomer act like the flanges of the “I” beam. These flanges add moment of inertia for high bending stiffness, and are very high in shear modulus as well. This forces the shear strain to occur across the thickness t shear . This shear strain is the value used to calculate the transverse beam deflection due to shear. 
     For homogeneous, isotropic materials, the shear modulus and tensile modulus are related by Poisson&#39;s ratio, as given in Equation (10): 
     
       
         
           
             
               
                 
                   G 
                   = 
                   
                     E 
                     
                       2 
                        
                       
                         ( 
                         
                           1 
                           + 
                           v 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Where: 
     
         
         
           
             υ=Poisson&#39;s ratio 
             E=Young&#39;s tensile modulus 
             G=shear modulus 
           
         
       
    
     For elastomeric materials like cast polyurethane, Poisson&#39;s ratio is generally close to 0.45. Therefore, given Young&#39;s tensile modulus, shear modulus can be calculated, and vice versa. 
     The “G” and the “A” for Equation 6 are now defined. The product of the beam moment of inertia “I” and Young&#39;s modulus “E” can be estimated as follows, using variables shown in  FIG. 11 : 
     
       
         
           
             
               
                 
                   EI 
                   = 
                   
                     
                       
                         2 
                         * 
                         
                           ( 
                           
                             
                               EI 
                               
                                 band 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                             + 
                             
                               EI 
                               
                                 band 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                             + 
                             
                               EI 
                               
                                 band 
                                  
                                 
                                     
                                 
                                  
                                 3 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         EI 
                         
                           band 
                            
                           
                               
                           
                            
                           4 
                         
                       
                     
                     = 
                     
                       
                         2 
                         * 
                         
                           ( 
                           
                             
                               
                                 E 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       t 
                                       1 
                                     
                                      
                                     
                                       h 
                                       1 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       1 
                                       12 
                                     
                                      
                                     
                                       t 
                                       1 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 E 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       t 
                                       2 
                                     
                                      
                                     
                                       h 
                                       2 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       1 
                                       12 
                                     
                                      
                                     
                                       t 
                                       2 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 E 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       t 
                                       3 
                                     
                                      
                                     
                                       h 
                                       3 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       1 
                                       12 
                                     
                                      
                                     
                                       t 
                                       3 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           E 
                           2 
                         
                          
                         
                           1 
                           12 
                         
                          
                         
                           t 
                           4 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Equations (7) and (10) explicitly calculate G and EI for the laminate elastomer beam of  FIG. 11 . However, using engineering principles of area moment of inertia and the rule of mixtures in series, any laminate beam can be calculated in a similar manner. For instance, in some cases, for any number of different elastomers of the annular beam: EI may be determined as ΣE i I i  which is a sum of products of the modulus of elasticity E i  and the moment of inertia I i  of each of the layers of the annular beam; and G may be determined as 1/Σ(v fi /G i ) where v fi  is the volume fraction and G i  is the shear modulus of each of the layers of the annular beam. 
     With EI known from Equation (10) and GA known from Equations (7) and (8), the only unknown in Equation (6) is the length L C  of the contact patch  125 . This is a design parameter which relates to a rated load of the non-pneumatic tire being designed. The length L C  of the contact patch  125  times a width of the contact patch  125  times a contact pressure P along the contact patch  125  will approximately equal the design load of the tire. 
     When the straight beam parameters E, I, G, and A are known and related to the design parameters of the laminate structure of the straight beam of  FIG. 11 , the simply supported beam with boundary conditions shown in  FIG. 12  can be evaluated using Equations (4) and (5). An example of the results of such calculations is shown in  FIG. 13 . Using the laminate configuration of  FIG. 11 , with geometric values of t1, t2, t3, and t4 that are commensurate with a total tire thickness in the radial (z) direction of 100 mm,  FIG. 13  shows that the ratio z s /z b  increases as the difference between E1 and E2 increases. 
     Additional work by the inventor has shown that a homogeneous contact pressure distribution can be obtained along the length L C  of the contact patch  125  of the non-pneumatic tire  110  provided z s /z b  is sufficiently high. For example, in some embodiments, when z s /z b  is at least about 1.2, in some cases at least about 1.5, in some cases at least 2, in some cases at least 3, and in some cases even more (e.g., 4 or more), the contact pressure will be substantially uniform. 
       FIG. 14  shows an example of a finite-element model of an embodiment of the annular beam  130  comprising the shear band  131  loaded between two parallel surfaces and producing the contact patch  125  having the length L C . 
       FIG. 15  shows the contact pressure through the length L C  of the contact patch  125  for the laminate configuration or for an isotropic configuration of the shear band  131  of the annular beam  130  of  FIG. 14 . With an isotropic elastomer cross section of E=80 MPa, the contact pressure is very non-uniform. The contact pressure peaks occur at the entrance and exit of the contact patch  125 , and the contact pressure is at a minimum in the center of the contact patch  125 . With a laminate configuration like that of  FIG. 11 , with E1=205 MPa and E2=16 MPa, the pressure distribution is substantially uniform. 
     The annular beam  130  comprising the shear band  131  of  FIG. 14  can be connected to the hub  120  via support members  142   1 - 142   T  (i.e., spokes) to create the non-pneumatic tire  110 . An example of a corresponding finite-element model of an embodiment of the non-pneumatic tire  110  comprising the annular beam  130  including the shear band  131  of  FIG. 14 , the spokes  142   1 - 142   T  and the hub  120  is shown in  FIG. 16 . In this example, the non-pneumatic tire  110  has dimensions 20.5×25—a size used in the construction industry, with the outer diameter D TO  of around 1.5 meters. The contact patch  125  has the length L C =370 nm when loaded to a design load of 11 metric tons.  FIG. 17  provides the principle strains in the annular beam  130  comprising the shear band  131  of  FIG. 16 . Maximum elastomer strains are about 0.09 (9%) which is well within the allowable cyclic strain capabilities of thermoset polyurethanes. 
       FIG. 17  further shows the contact pressure profile through the length L C  of the contact patch  125  of the non-pneumatic tire of  FIG. 16  for various laminate configurations and for an isotropic configuration of the shear band  131  of the annular beam  130 . As with the beam analysis of  FIGS. 14 and 15 , the results show that the isotropic case gives pressure peaks at the entrance and exit of the contact patch  125 . In this case, pressure peaks of almost 1 MPa (=10 bar=150 psi) occur. When laminate configurations are used, the pressure profile becomes more uniform. As the difference between E1 and E2 increases, the pressure becomes progressively more uniform. 
     In some embodiments, certain elastomeric materials may exhibit favorable non-linear stress vs. strain characteristics. For example, in some embodiments, a choice may be made of a material having a very non-linear material behavior, for which the secant modulus decreases with increasing strain. The “modulus” is the initial slope of the stress vs. strain curve, often termed “Young&#39;s modulus” or “tensile modulus.” In some embodiments, materials can be used that have a high Young&#39;s modulus that is much greater than their secant modulus at 100% strain, which is often termed “the 100% modulus.” The “secant modulus” is the tensile stress divided by the tensile strain for any given point on the tensile stress vs. tensile strain curve measured per ISO 527-1/-2. This nonlinear behavior provides efficient load carrying during normal operation, yet enables impact loading and large local deflections without generating high stresses. 
     Some thermoset and thermoplastic polyurethanes have this material behavior. An example of such a favorable material is shown in  FIG. 18 . The measured stress vs. strain curve of COIM&#39;s PET-95A, with curative MCDEA, has a Young&#39;s modulus of 205 MPa. However, the secant modulus at 100% strain is only 19 MPa. This may be a favorable attribute in some embodiments; when following the design principles earlier disclosed, the tire normally operates in the 5 to 9% strain region. In this region, the material is moderately stiff and the slope of the stress vs. strain curve is fairly constant. However, if local deformation occurs due to road hazards or impacts, the material is capable of large strains, without generation of high stresses. This minimizes vehicle shock loading, and enhances tire durability. 
     Elastomers are often used in areas of high imposed strains. As such, in some application, testing protocol typically focuses on the performance at high strains, such as 100%, 200%, or more. Mechanical designs that carry load in tension and bending typically do not use one homogeneous elastomer—they employ reinforcements as well. Some embodiments of the annular beam  130  opens this new design space by leveraging this material non-linearity with a favorable mechanical design. 
     The wheel  100   i , including its annular beam  130 , may be implemented in various other ways in other embodiments. 
     For example, in some embodiments, the annular beam  130  may be designed based on principles discussed in U.S. Patent Application Publication 2014/0367007, which is hereby incorporated by reference herein, in order to achieve the homogeneous contact pressure along the length L C  of the contact patch  125  of the wheel  100   i  with the ground. The use of multiple elastomers can be combined with a more complex geometry such that the resulting performance is superior to that which could be obtained by using either technology by itself. 
     In this embodiment, and with reference to  FIGS. 19 and 20 , the shear band  130  comprises an outer rim  133 , an inner rim  135 , and a plurality of openings  156   1 - 156   N  between the outer rim  133  and the inner rim  133  in addition to including the layers  132   1 - 132   N  of the different elastomeric materials M 1 -M E . The shear band  131  comprises a plurality of interconnecting members  137   1 - 137   P  that extend between the outer rim  133  and the inner rim  135  and are disposed between respective ones of the openings  156   1 - 156   N . The interconnecting members  137   1 - 137   P  may be referred to as “webs” such that the shear band  131  may be viewed as being “web-like” or “webbing”. In this embodiment, the shear band  131  comprises intermediate rims  151 ,  153  between the outer rim  133  and the inner rim  135  such that the openings  156   1 - 156   N  and the interconnecting members  137   1 - 137   P  are arranged into three circumferential rows between adjacent ones of the rims  133 ,  151 ,  153 ,  135 . The shear band  131 , including the openings  156   1 - 156   N  and the interconnecting members  137   1 - 137   P , may be arranged in any other suitable way in other embodiments. 
     The openings  156   1 - 156   N  of the shear band  131  help the shear band  131  to deflect predominantly by shearing at the contact patch  125  under the loading on the wheel  100   i . In this embodiment, the openings  156   1 - 156   N  extend from the inboard lateral side  147  to the outboard lateral side  149  of the non-pneumatic tire  110 . That is, the openings  156   1 - 156   N  extend laterally though the shear band  131  in the axial direction of the wheel  100   i . The openings  156   1 - 156   N  may extend laterally without reaching the inboard lateral side  147  and/or the outboard lateral side  149  of the non-pneumatic tire  110  in other embodiments. The openings  156   1 - 156   N  may have any suitable shape. In this example, a cross-section of each of the openings  156   1 - 156   N  is circular. The cross-section of each of the openings  156   1 - 156   N  may be shaped differently in other examples (e.g., polygonal, partly curved and partly straight, etc.). In some cases, different ones of the openings  156   1 - 156   N  may have different shapes. In some cases, the cross-section of each of the openings  156   1 - 156   N  may vary in the axial direction of the wheel  100   i . For instance, in some embodiments, the openings  156   1 - 156   N  may be tapered in the axial direction of the wheel  100   i  such that their cross-section decreases inwardly axially (e.g., to help minimize debris accumulation within the openings  156   1 - 156   N ). 
     Therefore, in this embodiment, the shear band  131  of the annular beam  130  comprises both (1) the openings  156   1 - 156   N  and (2) the layers  132   1 - 132   N  of the different elastomeric materials M 1 -M E . By using both geometry and material effects, further optimization is possible. For example, while thermoset polyurethanes and thermoplastic polyurethanes have a wide processing and optimization window (e.g., modulus values between 10 MPa and 300 MPa being readily assessable), in some embodiments, the physics may demand a very large bending stiffness and a very low shear stiffness, if a long contact patch of low, homogenous pressure is desired, and combining the openings  156   1 - 156   N  and the layers  132   1 - 132   N  of the different elastomeric materials M 1 -M E  may allow to achieve desired effects. 
       FIG. 20  shows a finite-element model of an embodiment of the non-pneumatic tire  110  having these combined technologies. In this non-limiting example, a webbing geometry and laminate configuration have been designed to give about a 0.1 MPa contact pressure, through a length of 600 mm. The length Lc of the contact patch  125  of the embodiment of  FIG. 20  represents a large percentage of the radius of the tire, which is 750 mm. 
     The contact pressure profile through the length L C  of the contact patch  125  of the non-pneumatic tire of  FIG. 20  is shown in  FIG. 21 . In this non-limiting example, the inventor has used a deformable ground, corresponding to the stiffness of clay. This more fully represents the actual usage of such a tire in an off-road condition. The pressure distribution is fairly uniform, equal to about 0.105+/−0.05 MPa (=1.05 bar=16 psi). This level of contact pressure may be particularly appropriate in an agricultural tire usage. 
     In some embodiments, the wheel  100   i , including its non-pneumatic tire  110 , may enable a design space that may not be readily possible with pneumatic tires. Notably, in some embodiments, the wheel  100   i  may be designed to be relatively narrow yet have a high load carrying capacity and a long contact patch. 
     For example, in some embodiments, the wheel  100   i  may be such that (1) a ratio W T /D TO  of the width W T  of the non-pneumatic tire  110  over the outer diameter D TO  of the non-pneumatic tire  110  is no more than 0.1 and (2) a ratio D H /D TO  of the diameter of the hub  120  over the outer diameter D TO  of the non-pneumatic tire  110  is no more than 0.5, namely:
         W T /D TO ≤0.15 (15%)   D H /D TO ≤0.50 (50%)       

     For instance, in some embodiments, the ratio W T /D TO  of the width W T  of the non-pneumatic tire  110  over the outer diameter D TO  of the non-pneumatic tire  110  may be less than 0.1, in some cases no more than 0.08, in some cases no more than 0.06, and in some cases no more than 0.04, and/or the ratio D H /D TO  of the diameter of the hub  120  over the outer diameter D TO  of the non-pneumatic tire  110  may be less than 0.5, in some cases no more than 0.4, and in some cases no more than 0.3. 
     As another example, in some embodiments, the wheel  100   i  may be such that a ratio L c /R TO  of the length L c  of the contact patch  125  of the non-pneumatic tire  110  at the design load over an outer radius R TO  of the non-pneumatic tire  110  (i.e., half of the outer diameter D TO  of the non-pneumatic tire  110 ) is at least 0.4, in some cases at least 0.5, in some cases at least 0.6, in some cases at least 0.7, in some cases at least 0.8, in some cases at least 0.9, and in some cases even more (e.g., 1 or more). 
       FIG. 22  shows an example of a finite-element model of the non-pneumatic tire  110  of  FIG. 20 , having the width W T =120 mm, and the outer diameter D TO =1500 mm. For inflated tires, a small width and a large outer diameter result in the need for a relatively large mounting rim. The equilibrium curve mechanics of both radial and bias tires are such that a width of 120 mm would result in a maximum sidewall height of only about 120 mm. This limits the contact patch length as well as the ability of the tire to absorb energy when traversing uneven terrain. 
     In this example, the length L C  of the contact patch  125  may approach or be larger than the outer radius of the non-pneumatic tire  110  and there is a larger distance between the tire outer diameter D TO  and the hub  120 . As a result, in this example, the load carrying capacity of the non-pneumatic tire  110  can be quite large. With W T =120 mm and D TO =1500 mm, the design load can be about 750 kg, with sustained speeds of 30 kph or more permitted, with a ground contact pressure at the contact patch  125  of about 1 bar. 
     The non-pneumatic tire  110  may comprise other components in other embodiments. For example, in some embodiments, as shown in  FIG. 23 , the tread  150  may comprise a reinforcing layer  170  disposed within its elastomeric material  160  (e.g., rubber) and extending in the circumferential direction of the wheel  100   i . 
     For example, in some embodiments, the reinforcing layer  170  may comprise a layer of reinforcing cables that are adjacent to one another and extend generally in the circumferential direction of the wheel  100   i . For instance, in some cases, each of the reinforcing cables may be a cord including a plurality of strands (e.g., textile fibers or metallic wires). In other cases, each of the reinforcing cables may be another type of cable and may be made of any material suitably flexible along the cable&#39;s longitudinal axis (e.g., fibers or wires of metal, plastic or composite material). 
     As another example, in some embodiments, the reinforcing layer  170  may comprise a layer of reinforcing fabric. The reinforcing fabric comprises thin pliable material made usually by weaving, felting, knitting, interlacing, or otherwise crossing natural or synthetic elongated fabric elements, such as fibers, filaments, strands and/or others, such that some elongated fabric elements extend transversally to the circumferential direction of the wheel  100   i  to have a reinforcing effect in that direction. For instance, in some cases, the reinforcing fabric may comprise a ply of reinforcing woven fibers (e.g., nylon fibers or other synthetic fibers). 
     In some cases, the reinforcing layer  170  of the tread  150  may be substantially inextensible in the circumferential direction of the wheel  100   i . The non-pneumatic tire  110  may thus be such that its annular beam  130  is free of any substantially inextensible reinforcing layer running in its circumferential direction while its tread  150  includes the reinforcing layer  170  that may be substantially inextensible in its circumferential direction. 
     The tread  150  including the reinforcing layer  170  may be provided in any suitable way. For example, in some embodiments, the tread  150  may be manufactured separately from the annular beam  130  and then affixed to the annular beam  130 . For instance, in some embodiments, the tread  150  may be manufactured by arranging one or more layers of its elastomeric material  160  (e.g., rubber) and its reinforcing layer  170  into a mold and molding them (e.g., compression molding them) into an annular configuration of the tread  150 . The tread  150  may then be affixed to the annular beam  130  in any suitable way. For instance, in some embodiments, the tread  150  may be expanded to fit about the annular beam  130  and then contracted to become attached to the annular beam  130 . In some examples, this may be achieved by a coefficient of thermal expansion of the reinforcing layer  170  of the tread  150  allowing the reinforcing layer  170  to expand for stretching the elastomeric material  160  of the tread  150  in order to fit the tread  150  around the annular beam  130  and then to contract for attaching the tread  150  to the annular beam  130 . The tread  150  may be affixed to the annular beam  130  in any other suitable manner in other examples (e.g., including by using an adhesive to adhesively bond the tread  150  and the annular beam  130 ). 
     While in embodiments considered above the wheel  100   i  is part of the construction vehicle  10 , a wheel constructed according to principles discussed herein may be used as part of other vehicles or other machines in other embodiments. 
     For example, with additional reference to  FIGS. 24 and 25 , in some embodiments, an all-terrain vehicle (ATV)  210  may comprise wheels  220   1 - 220   4  constructed according to principles discussed herein in respect of the wheel  100   i . The ATV  210  is a small open vehicle designed to travel off-road on a variety of terrains, including roadless rugged terrain, for recreational, utility and/or other purposes. In this example, the ATV  210  comprises a frame  212 , a powertrain  214 , a steering system  216 , a suspension  218 , the wheels  220   1 - 220   4 , a seat  222 , and a user interface  224 , which enable a user of the ATV  210  to ride the ATV  210  on the ground. 
     The steering system  216  is configured to enable the user to steer the ATV  210  on the ground. To that end, the steering system  216  comprises a steering device  228  that is operable by the user to direct the ATV  210  along a desired course on the ground. In this embodiment, the steering device  228  comprises handlebars. The steering device  228  may comprise a steering wheel or any other steering component that can be operated by the user to steer the ATV  210  in other embodiments. The steering system  216  responds to the user interacting with the steering device  228  by turning respective ones of the wheels  220   1 - 220   4  to change their orientation relative to the frame  212  of the ATV  210  in order to cause the ATV  210  to move in a desired direction. In this example, front ones of the wheels  220   1 - 220   4  are turnable in response to input of the user at the steering device  228  to change their orientation relative to the frame  212  of the ATV  210  in order to steer the ATV  210  on the ground. More particularly, in this example, each of the front ones of the wheels  220   1 - 220   4  is pivotable about a steering axis  230  of the ATV  210  in response to input of the user at the steering device  228  in order to steer the ATV  210  on the ground. Rear ones of the wheels  220   1 - 220   4  are not turned relative to the frame  212  of the ATV  210  by the steering system  216 . 
     The suspension  218  is connected between the frame  212  and the wheels  220   1 - 220   4  to allow relative motion between the frame  122  and the wheels  220   1 - 220   4  as the ATV  210  travels on the ground. For example, the suspension  218  enhances handling of the ATV  210  on the ground by absorbing shocks and helping to maintain traction between the wheels  20   1 - 20   4  and the ground. The suspension  218  may comprise an arrangement of springs and dampers. A spring may be a coil spring, a leaf spring, a gas spring (e.g., an air spring), or any other elastic object used to store mechanical energy. A damper (also sometimes referred to as a “shock absorber”) may be a fluidic damper (e.g., a pneumatic damper, a hydraulic damper, etc.), a magnetic damper, or any other object which absorbs or dissipates kinetic energy to decrease oscillations. In some cases, a single device may itself constitute both a spring and a damper (e.g., a hydropneumatic, hydrolastic, or hydragas suspension device). 
     In this embodiment, the seat  222  is a straddle seat and the ATV  210  is usable by a single person such that the seat  222  accommodates only that person driving the ATV  210 . In other embodiments, the seat  222  may be another type of seat, and/or the ATV  210  may be usable by two individuals, namely one person driving the ATV  210  and a passenger, such that the seat  222  may accommodate both of these individuals (e.g., behind one another or side-by-side) or the ATV  210  may comprise an additional seat for the passenger. For example, in other embodiments, the ATV  210  may be a side-by-side ATV, sometimes referred to as a “utility terrain vehicle” or “utility task vehicle” (UTV). 
     The wheels  220   1 - 220   4  engage the ground to provide traction to the ATV  210 . More particularly, in this example, the front ones of the wheels  220   1 - 220   4  provide front traction to the ATV  10  while the rear ones of the wheels  220   1 - 220   4  provide rear traction to the ATV  10 . 
     Each wheel  220   i  of the ATV  210  may be constructed according to principles described herein in respect of the wheel  100   i , notably by comprising a non-pneumatic tire  234  and a hub  232  that may be constructed according to principles described herein in respect of the non-pneumatic tire  110  and the hub  120 . The non-pneumatic tire  234  comprises an annular beam  236  and an annular support  241  that may be constructed according principles described herein in respect of the annular beam  130  and the annular support  140 . For instance, the annular beam  236  comprises a shear band  239  comprising a plurality of layers  232   1 - 232   N  of different elastomeric materials M 1 -M E  and the annular support  241  comprises spokes  242   1 - 242   J  that may be constructed according to principles described herein in respect of the shear band  131  and the spokes  142   1 - 142   T . 
     As another example, in some embodiments, with additional reference to  FIG. 26 , a motorcycle  410  may comprise a front wheel  420   1  and a rear wheel  420   2  constructed according to principles discussed herein in respect of the wheel  100   i . 
     As another example, in some embodiments, a wheel constructed according to principles discussed herein in respect of the wheel  100   i  may be used as part of an agricultural vehicle (e.g., a tractor, a harvester, etc.), a material-handling vehicle, a forestry vehicle, or a military vehicle. 
     As another example, in some embodiments, a wheel constructed according to principles discussed herein in respect of the wheel  100   i  may be used as part of a road vehicle such as an automobile or a truck. 
     As another example, in some embodiments, a wheel constructed according to principles discussed herein in respect of the wheel  100   i  may be used as part of a lawnmower (e.g., a riding lawnmower or a walk-behind lawnmower). 
     Although embodiments considered above pertain to a non-pneumatic tire, in other embodiments, other annular devices, such as, for instance, tracks for vehicles and/or conveyor belts, may comprise an annular beam constructed according to principles discussed herein in respect of the annular beam  130 . 
     Certain additional elements that may be needed for operation of some embodiments have not been described or illustrated as they are assumed to be within the purview of those of ordinary skill in the art. Moreover, certain embodiments may be free of, may lack and/or may function without any element that is not specifically disclosed herein. 
     Any feature of any embodiment discussed herein may be combined with any feature of any other embodiment discussed herein in some examples of implementation. 
     In case of any discrepancy, inconsistency, or other difference between terms used herein and terms used in any document incorporated by reference herein, meanings of the terms used herein are to prevail and be used. 
     Although various embodiments and examples have been presented, this was for the purpose of describing, but not limiting, the invention. Various modifications and enhancements will become apparent to those of ordinary skill in the art and are within the scope of the invention, which is defined by the appended claims.