Expressin g Mech anical Efficiency an d Energy Ma rgin as ratio s rather than as pe rcentag es, a simple  relation ship betwee n the two can be formulated.  1Efficiency  Mechanical11     Margin Energy    −=     ( 26) 

For example,  if the ratio of  Drive  Ene rgy to Fri ction Energy i s 4:1, the Me chanical Efficie ncy is 0.75  (75  percent ) and the Energy M argin is 3.0 (3 00 perce nt).  

Virtual Work  and Virtual Displacem ents 

In textbooks, the co ncepts of virtual di splacem ents and virtual work are commonly develope d for a nalysi s  of mech anical system s and stru ctures in static equili brium. A virtu al displaceme nt is defin ed as a fi ctitious  or infinitesi mal displ acement – som etimes a s a ve ry small di splacem ent. Th e mechani cal system is  considered to undergo virtual displ acements, and t he net virtual work that results fro m the force s actin g  throug h their respe ctive virtual displ acements i s equal to zero.  

The type of mech anical system bein g considered here typically has ju st bee n rele ased from a laun ch  restraint or is being en ergized by a curre nt pulse to a stepp er moto r. The system  is in a state of imminent  motion with some finite i nitial a cceleration, an d an initial velocity equal  to zero . To ap ply th e metho d of  virtual work, this sy stem ca n be con sidered to be in a state of dynamic equili brium, with the inertial force s  due to the i nitial accele ration bein g in equilib rium with the ap plied force s and resi sting  force s (in  accordan ce with D'Alem bert's p rinciple). The sum of  the net virtual work don e by the virtual displa cements  acting throu gh the origi nal force system and th e virtual kin etic ene rgy resulting fro m the virtual  displacement s actin g throu gh the re spective inertial force s, is e qual to zero. 

From a nothe r viewpoint, d uring a te st to mea sure fo rce m argin, the me chani sm is som etimes restraine d  from m otion by a lo ad or torque gage, and th e me asured fo rce or torque  is gradually reduced by slowly  withdrawi ng the gage. The  measu red fo rce o r torque at the point of imminent motion is used to calculate   the ma rgin. In this case, there is no acceleration  until the point of rel ease. Here, the m echanism and the  test appa ratus are in static equilibri um. 

The te chnical approach using virtual displacement s is essentially th e same a s for the precedin g approach  using sm all displ acement s. The terms “sm all displ acements” a nd “e nergy” can b e replaced by “virtu al  displacement s” an d “virtua l work”. T he argument s are similar an d the conclusion is the same. The  net  virtual wo rk done by the in ertial force s and the su m of the drive forces an d the re sistin g force s, after bein g  referen ced to  a commo n point as eq uivalent or virtua l force s, is ze ro; and the system re main s in a state of  equilib rium. T hus, the e quivalent or virtu al force sy stem is eq ual in effect to the o riginal fo rce system, and  the method d escrib ed he rein for cal culating force and t orques margins is the oretically valid. 

If there i s a conceptual advantage in  describin g this method of eq uivalent force s in te rms of virtual work, it  is that virtu al displaceme nts, bein g considered infinitesimal, do not involve an y approximation du e to  small changes in mechanical configuration (if li nkag es are involv ed). Li kewise, there i s no nee d for a  conceptual di stinctio n between stat ic friction and dyna mic frictio n.  

The virtu al work ap proach was not employed expli citly in the ex ample, p artly be cause the mechani cal  device i s in a state of imminent motio n, and re al displacements can be considere d appropriate for the   analysi s. Moreover, the increa sed le vel of abstra ction should n ot be nece ssary to the underst andin g and  accepta nce of these co ncepts.  

Deriv ative Nota tion 

If a function al relatio nship can be established for th e displacement s, this relatio nship can b e differentiated   and the displacem ent rati os ca n be calcul ated as der ivatives of  the di splacements to the  refe rence  117