of 3679 lb will be acting when the rods are moving downward at 5 ft sec an upward frictional force of 3679 lb will be applied the friction law used to create the hypothetical data can be written fig6 a shows two pump cards plotted to the same load and position scales and with a common time origin sixty points are used to plot each card with a constant time interval between points an error function is defined by u03b4 i l a t i u2212 l 0 t i 12 wherein the l a t i are actual true pump loads created by the completely stated predictive program and the l o t i are pump loads calculated with the diagnostic technique with zero friction the u03b4 i measure the error caused by using an incorrect friction law zero friction according to the gibbs theorem since rod friction was set to zero and velocity along the rods is essentially the same at a given time shallow well u03b4 i represents the total friction along the length of the rod string fig6 b shows a time history of pump velocity which is taken to be representative of local velocity everywhere along the rod string finally fig6 c shows a time history of u03b4 i and a time history of the friction law equation 12 used to create the hypothetical example the agreement between the two time histories is close but not perfect the imperfections are caused by the fact that even in a shallow well the rod string stretches such that an idealization of equal velocities along its length is not strictly true still the agreement is close enough to indicate that the gibbs theorem can be used to define more complicated friction laws fig7 a and.