Abstract:
The system and method for measuring rolling resistance provide for the measurement of various properties associated with pre-rolling resistance. A cruciform pendulum is formed from a rigid rod having opposed upper and lower ends and a horizontal support extending orthogonal thereto. A pair of substantially hemispherical samples formed from a first material are mounted on opposing ends of the horizontal support. Flat, planar samples of a second test material are placed upon spaced apart supporting surfaces. The cruciform pendulum is suspended between the supporting surfaces with the hemispherical first material resting on the planar second material. The rigid rod is deflected from vertical and released to induce pendulum oscillations with the first material rolling on the second material. Based upon the measured angular deviations and periods of pendulum oscillations, coefficients of rolling friction, moments of rolling friction, hysteresis losses, adhesion and moments of elastic rolling resistance may be easily calculated.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to testing and measurement devices, and particularly to a system and method for measuring rolling resistance during pre-rolling, and more particularly, to a system and method for measuring the moment of forces of resistance, dimensionless coefficients of rolling friction, and the hysteretic losses on both uncoated and coated surfaces. 
     2. Description of the Related Art 
     Rolling resistance, sometimes referred to as “rolling friction” or “rolling drag”, is the resistance that occurs when a round object, such as a ball, tire, or wheel, rolls on a surface. The rolling resistance is primarily caused by the deformation of the object, the deformation of the surface, and movement below the surface. Additional contributing factors include wheel diameter, forward speed, the load on the wheel, surface adhesion, sliding, relative micro-sliding between the surfaces of contact, and their roughness. The rolling resistance greatly depends on the material of the wheel or tire and the type of ground or other surface. What might be termed “basic rolling resistance” is steady velocity and straight line motion on a level surface, but there also exists rolling resistance when accelerating, when on curves, and when on a grade. 
     Rolling resistance may be defined as the moment a rolling force) needed to overcome resistance to rotation and to move forward. The rolling resistance is much smaller than the sliding friction between two surfaces under equal loads, typically by a factor of at least one hundred. During the rolling process, it is possible for micro-slip to occur in a region within the contact area, inducing loss of mechanical energy through friction, thus leading to hysteresis loss and non-local memory, as is observed in pre-rolling. Rolling slowly from rest exhibits increasing rolling resistance, which starts from zero to steady-state rolling with constant rolling resistance. In this range of pre-rolling, the rolling resistance has a non-linear behavior. The pre-rolling stage induces hysteresis, which is typically difficult to measure due to its non-linearity. 
     In rolling friction, two separate stages must be considered. The first stage is the pre-rolling stage, in which the deformation forces are dominant and the patch contact includes sub-regions of adhesion and slip. The second stage is the steady rolling stage in which the rolling resistance has been fully developed to its maximum value and has more pronounced rotation. 
     The linear and large-scale steady rolling stage is relatively easy to describe and measure. The pre-rolling stage, however, not only includes non-linear considerations, but occurs only on a very small scale of pre-movement. Thus, it would be desirable to be able to easily make measurements of rolling resistance during the pre-rolling stage. 
     Thus, a system and method for measuring rolling resistance solving the aforementioned problems is desired. 
     SUMMARY OF THE INVENTION 
     The system and method for measuring rolling resistance provide for the measurement of various properties associated with pre-rolling resistance. A cruciform pendulum is formed from a rigid rod having opposed upper and lower ends, and a horizontal support mounted on a central portion of the rigid rod. The upper and lower ends of the rod are both free to rotate. The horizontal support extends along an axis orthogonal to the axis of the rigid rod, forming the cruciform shape. A pair of retainers are respectively secured to horizontally-opposed ends of the horizontal support and extend downward therefrom. The retainers hold a pair of substantially hemispherical samples (or spherical or ball-shaped samples having a hemispherical portion extending below the retainer) formed from a first material. The hemispherical samples bear upon, and are balanced on, a corresponding pair of flat surface samples formed from a second material. The weight of the cruciform pendulum is supported by the hemispherical samples depending from opposite ends of the horizontal support, and the instantaneous axis of rotation of the pendulum is through the points where the hemispherical (or ball-shaped) samples bear upon the flat surface samples. 
     An angular deviation φ of the axis of the rigid rod with respect to the vertical is optically measured. Preferably, the pendulum achieves an angular deviation φ in the range 
               ϕ   ≤     0.1   ⁢     a   R         ,         
where a is a radius of a contact spot between each hemispherical sample and the corresponding one of the flat surface samples, and R is a radius of each hemispherical sample, A current time t and a period of oscillation T i  for each cycle of oscillation of the pendulum are then measured by a timer. Each period T i  corresponds to a time t i , where i is an integer ranging between zero and n, where n represents a final measurement.
 
     It is useful to calculate amplitudes of oscillation α i  as 
                 α   i     =       ϕ   ⁡     (     t   i     )       ⁢     sec   ⁡     (         2   ⁢   π     T     ⁢     t   i       )           ,         
where T is a mean value of the set T i , and where an initial amplitude is given as α φ  and a final amplitude of oscillation is given as α n . From this, a dimensionless coefficient of rolling friction f between the pair of hemispherical samples and stationary flat surface samples, formed from a second material, may be calculated as
 
     
       
         
           
             f 
             = 
             
               
                 
                   
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       α 
                       n 
                     
                   
                   - 
                   
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       α 
                       0 
                     
                   
                 
                 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           α 
                           0 
                         
                         + 
                         
                           α 
                           n 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     4 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         α 
                         i 
                       
                     
                   
                 
               
               . 
             
           
         
       
     
     In addition to the calculation of the dimensionless coefficient of rolling friction, which is time-independent, an instantaneous coefficient of rolling friction, as well as a moment of rolling friction, may be calculated as a function of the angular deflection φ and an instantaneous moment of rolling friction. Further, hysteresis losses for each cycle of the pendulum oscillation, the pressure of adhesion attraction between the pair of hemispherical samples and stationary flat surface samples, and the moment of elastic rolling resistance may also be calculated. 
     These and other features of the present invention will become readily apparent upon further review of the following specification and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a perspective view of a system for measuring rolling resistance according to the present invention. 
         FIG. 2  is a partial side view of the system for measuring rolling resistance of  FIG. 1 , diagrammatically illustrating rolling contact between a hemispherical sample and a flat test substrate. 
     
    
    
     Similar reference characters denote corresponding features consistently throughout the attached drawings. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     As shown in  FIG. 1 , the system for measuring rolling resistance, designated generally as  10 , is based on pendulum motion for measuring rolling resistance parameters between hemispherical samples  12 , formed from a first material, and flat surface samples  14 , which are formed from a second material. It should be understood that both first and second material samples may be varied, thus allowing the system  10  to be used for measurement of rolling resistance between any two desired materials. Additionally, since the point of contact is where the frictional effects take place, it should be understood that the hemispherical samples  12  may be coated with the material of interest, rather than being wholly formed from the material. 
     The system  10  includes a rigid rod  16  having an upper end  18  and a lower end  20 . Both upper end  18  and lower end  20  are free to rotate in oscillatory fashion. In the particular exemplary configuration of  FIG. 1 , the rod  16  oscillates back and forth in the left-right direction (i.e., in the plane of the page). Motion during the pre-rolling stage is of primary importance. Thus, it should be understood that the pendulum motion of the rigid rod  16  is very small. 
     As shown in  FIG. 1 , a horizontal support  26  is mounted to a central portion of rigid rod  16 , forming a cruciform pendulum. A pair of holders  24  extend downward from opposite ends of the horizontal support  26 . The pair of hemispherical samples  12  (or ball-shaped samples having a hemispherical portion protruding from the holders  24 ) are mounted to the respective lower ends of holders  24 . The samples  12  are hemispherical to provide accurate simulation of spherical balls experiencing rolling. It is important to note that the hemispherical samples  12  are fixed with respect to the holders  24 ; i.e., they do not rotate in holders  24 , but with the holders  24 . This is because only pre-rolling considerations are being taken into account. Thus, additional rolling of the samples  12  is not being measured. The samples  12  contact and bear upon the flat surface samples  14 , which are supported by stands  28 , which remain fixed with respect to the pendulum motion. The entire system  10  is balanced on samples  14  at the contacting spots of the hemispherical samples  12 . Thus, the system  10  is a cruciform pendulum balanced on these spots, and the instantaneous axis of rotation of the entire pendulum is through an axis O-O′ extending through a tangent to the hemispherical samples  12  extending through the spots that the hemispherical samples  12  bear upon. The center of mass of the cruciform pendulum is located centrally between upper ends of the hemispherical portions. 
     As shown in  FIG. 2 , as the rigid rod  16  rotates very slightly, the hemispherical sample  12  enters the pre-rolling stage, tipping slightly such that its axis (corresponding to the axis A of rigid rod  16 ) is angled with respect to the vertical V by an angle φ. The angle φ is also the angular position of the oscillation of rigid rod  16  with respect to the vertical. It should be noted that the angle shown in  FIG. 2  is exaggerated for purposes of illustration. It should be noted that the cruciform pendulum is balanced so that the center of mass coincides with a cross-point of an axis O-O′ and an axis A of rigid rod  16  when the cruciform pendulum is vertical; i.e., when the angle φ is zero. This prevents sliding of the cruciform pendulum with respect to samples  12  in the presence of vibrations or other external forces. 
     The value of angle φ varies over time, thus we may consider a time-dependent angular displacement φ(t). As will be seen in the calculations below, it is useful to define a time-dependent amplitude of oscillation α(t), such that 
                 ϕ   ⁡     (   t   )       =       α   ⁡     (   t   )       ⁢   cos   ⁢       2   ⁢   π     T     ⁢   t       ,         
where T is the mean period of oscillation. In order to measure φ(t), a fixed laser  30  generates a beam  36 , which is reflected from a planar reflector  34  mounted on the rigid rod  16 . As shown, the planar reflector  34  extends vertically along the axis A of rigid rod  16 , and extends horizontally in the direction of horizontal support  26 . The axis of rotation O-O′ of the pendulum bisects the reflector  34 . As the rigid rod  16  rotates through angle φ with respect to the horizontal, the planar reflector  34  also rotates by angle φ with respect to the horizontal, and the angular deflection of the beam  36  is picked up and measured by a photodetector  32 . It should be understood that any suitable light source may be utilized for generating the light beam  36 , and that any suitable type of photodetector  32 , such as a charge-coupled device, may be used for measuring the angular deflection φ. The laser  30  may also be used in combination with any suitable optics for focusing or the like, as is conventionally known. It should be noted that the actual angle of reflection between the source  30  and the detector  32  is 2φ. Thus, the actual measured angle is simply halved to produce φ.
 
     With the accurate measurement of angular deflection φ by the photodetector  32 , a mean value of a dimensionless coefficient of rolling friction between the hemispherical sample  12  and the flat surface sample  14  may be calculated as 
               f   =         cos   ⁢           ⁢     α   n       -     cos   ⁢           ⁢     α   0             2   ⁢     (       α   0     +     α   n       )       +     4   ⁢       ∑     i   =   1       n   -   1       ⁢     α   i               ,         
where α 0  is an initial angular amplitude of pendulum oscillation, α n  is a final angular amplitude of pendulum oscillation, and α i  is an angular amplitude in an intermediate cycle of pendulum oscillation i. For each small oscillation of the pendulum, the angular deflection φ is measured for each full cycle, ranging from φ(t 0 ) to φ(t n ), where t 0  is the time of initial measurement (i.e., the greatest value of φ) and t n  being the time of final measurement, such that an instantaneous time t i  is defined with i=0, 1, 2, 3, . . . , n. The period of each full cycle T i  is measured by a timer  40 . Thus, α i  is calculated as
 
                 ϕ   ⁡     (     t   i     )       ⁢     sec   ⁡     (         2   ⁢   π     T     ⁢     t   i       )         ,         
α 0  is calculated as
 
               ϕ   ⁡     (     t   0     )       ⁢     sec   ⁡     (         2   ⁢   π     T     ⁢     t   0       )             
and α n  is calculated as
 
                 ϕ   ⁡     (     t   n     )       ⁢     sec   ⁡     (         2   ⁢   π     T     ⁢     t   n       )         ,         
where T is the measurement average of all of the T i .
 
     The dependence of α with respect to time can be approximated using the analytic function of regression, where b and p are the parameters of regression, as 
               α   ⁡     (   t   )       =           α   0     ⁡     (     1   -     4   ⁢     b     α   0     1   -   p         ⁢         1   -   p       1   +   p       ·     t   T           )         1     1   -   p         .           
Further, the moment of rolling friction, as a function of the angular deflection φ, M fr  (φ) between hemispherical sample  12  and planar surface sample  14  can be calculated as
 
                   M   fr     ⁡     (   ϕ   )       =       -   mgRb     ⁢           ⁢       ϕ   p     ·     sign   ⁡     (       ⅆ   ϕ       ⅆ   t       )             ,         
where in represents the overall mass of the pendulum apparatus, g is the gravitational acceleration, and R is the radius of the hemispherical sample  12 . The function “sign” is determined by the direction of oscillation; i.e., by the sign of
 
     
       
         
           
             
               
                 ⅆ 
                 ϕ 
               
               
                 ⅆ 
                 t 
               
             
             . 
           
         
       
     
     Further, the hysteresis losses W(α i ) for each cycle of the pendulum oscillation with amplitude α i  may be calculated as 
               W   ⁡     (     α   i     )       =     4   ⁢   mgRb   ⁢         α   i     1   +   p         1   +   p       .             
Above, the mean value of the dimensionless coefficient of rolling friction f between hemispherical sample  12  and flat surface sample  14  was calculated as a function of α 0  and α n . The instantaneous value f φ  may be calculated as
 
               f   ϕ     =                M   fr     ⁡     (   ϕ   )            mgR     =     b   ⁢           ⁢       ϕ   p     .               
Additionally, the approximation of the dependence T i  on α i  with the analytic function of regression may be calculated as:
 
                 T   ⁡     (   α   )       =         T   0     [     1   -       π     ⁢   γ   ⁢         a   2     ⁢     α   q       gm     ⁢         Γ   ⁡     (       q   2     +     3   2       )         Γ   ⁡     (       q   2     +   2     )         ·     (     1   -     0.55   ⁢       R   ⁢           ⁢   α     a         )           ]       -   1         ,         
where a is the radius of the contact spot between the sample  12  and the sample  14 , T 0 , γ and q are the parameters of regression (determined by experiment of cycling time intervals versus rolling body displacement), and Γ is the gamma function. The parameter γ is a pressure of an adhesion force between hemispherical samples  12  and the flat surface samples at points of contact therebetween.
 
     Additionally, the moment of elastic rolling resistance M el (φ) may be calculated as 
                 M   el     ⁡     (   ϕ   )       =     2   ⁢   γ   ⁢           ⁢     a   2     ⁢   R   ⁢          ϕ          q   +   1       ⁢       (       π   2     -       R   a     ⁢   ϕ       )     ·       sign   ⁡     (   ϕ   )       .               
The full moment of rolling resistance is then, simply, M(φ)=M fr (φ)+M el (φ). As noted above, since pre-rolling is the stage of consideration, the angular displacement is preferably within the limit of
 
     
       
         
           
             ϕ 
             ≤ 
             
               0.1 
               ⁢ 
               
                 
                   a 
                   R 
                 
                 . 
               
             
           
         
       
     
     As shown in  FIG. 1 , the system  10  is symmetric about the vertical axis. A pair of holders  24  is provided for retaining the pair of hemispherical samples  12 , which contact identical flat surface samples  14 . This arrangement prevents any friction-based torque from being introduced into the experiment (i.e., unwanted rotation about the vertical axis). Further, as shown in  FIG. 1 , upper and lower adjustable weights  42 ,  44  may be provided on the rod  16  for large-scale adjustment of the oscillation of rod  16 , and additional smaller weights  46  may be provided for fine-scale adjustment. As shown, the smaller weights  46  are preferably adjustably mounted on a rod  48  that extends orthogonal to the axis of rod  16  and also to the axis of rotation O-O′ of the pendulum. 
     It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.