Extensio n of Force Margi ns to Compl ex Sy stems 

The static force or torque  margi n as stated in  Equati ons 1  and 2 applie s explicitly to the case of a single  drive force o r torque and a singl e resisting force o r torque at the same tra nslating or rotating  point of the   mech anism. The equ ation can  also be directly applie d when there are multiple driving an d resisting fo rces  and torque s applied at different poi nts of the sa me eleme nt. The cal culation then re quire s direct  summ ation o f all the forces a nd to rques on the  comm on el ement, the d riving forces going i n the  nume rator an d the resisting loa ds in th e denominato r. This often  occurs when  driving force s and resisting  loads are collinear on th e same  sliding element, a nd all of the drive and l oad points move i n the same  direction at the same rate.  

1001Forces  ResistingesForc Drive   Margin Force Static ⋅   − =∑∑                       (3) 

The rel ated situation fo r rotation o ccurs when all of the applie d forces a nd resi sting lo ads are on a   comm on shaft. Any linear forces are mult iplied by their individual radii relative to the axis of rotation, and  thus are easily reduced to t orques about  the shaft. In this case, the formula for static to rque m argin would  have the sam e form as Eq uation 3 ab ove. 

If, on the other hand, the mechani sm involves multi ple gear m eshes, or combi ned rotation and transl ation  from a li nkag e or j ackscre w, the inte rpretation of the stand ard force margin fo rmula i s not a s simple. F or a  compl ex mechani sm the multiple drive  force s and resi sting load s at different  points of the mech anism  need to be related to a  single p oint by the use  of multipliers, sometime s cal led influe nce coeffici ents.  These coefficients a re the displacement  ratios o r velocity ratio s betwee n the el ements. To be consistent  with the concept of a st atic force ma rgin, whe re the displacements and velo cities are i nitially zero, it may  be mo re theoretically correct  to speak of vi rtual displacement s, as discussed later.  Howeve r, the   mech anism i s in a state  of imminent  motion if t he ma rgin i s positive, an d the con cept of small   displacement s can be used. If the mechani sm comp rises only ele ments that h ave con stant displacement  ratios,  such as gears and belts, this distinction  is le ss signifi cant. If l inkages are included, th ese ratio s will  change conti nuou sly with  the chan ging geomet ry of the linkag e, and the concept of suf ficiently small  displacement s is impo rtant. In the followin g exam ple, displa cements a re meant to mea n very small  displacement s.  

This di scussion of mo re complex me chanism s assumes a single degree of fre edom, an d that the motion s  or imminent motions do not involve fle xibility of connec ting mem bers. This assumption is implicit in the  basic definiti ons of stati c torque an d force m argins. The rotati on or tra nslation of these  members is  referred  to u sing the te rm “coo rdinates” to d efine displacement or  veloc ity vectors. Assoc iated w ith a  single degree of freed om, the di splacements a nd velo cities are sai d to be constrained. Fo r example, two   spur gears in mesh would have a rotational coordi nate for each gear shaft, and these coordinates are  constrained to rotate at rates relative  to each oth er by the ratio of the n umbers of teeth (gear ratio ). In thi s  case th e angular displa cement o r angular velo city ra tio between  the el ement s is the reciprocal of the  gear  ratio. A 10:1 redu ction ratio would resul t from having ten times a many teeth in the driven gear a s in the  drive pi nion, giving a  displacement or velocity ratio of 1:10, relative to the  pinion. The  torque multiplication   ratio, exclu ding frictio n drag, is eq ual to the re ciprocal of the di splacement ra tio, i.e. 10:1. This ratio is  sometim es called the me chani cal adva ntage. 

Takin g a jackscre w as another exam ple, one coordinate woul d be at the rotation of the drive nut, and the  other coordinate the t ranslatio n of the screw. T hese two  coordin ates are const rained to have  a  displacement  ratio that i s a function of the lead angle and the pitch diamete r of the scre w threads. Ag ain,  the torque -to-force  ratio, e xcludin g friction d rag, is equal to  the recip rocal of the di splacement ratio (i.e.   radians/in = lb/in-lb).  

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