Coordinate 3, Cran kshaft L oads (Fig. 4)   The inte rmediate sh aft driv es a cran kshaft with an gular Coordinate θ3, having  two b earing drag to rques  T31B and T 32B. These two drag torques could be referenced directly to the drive shaft, skipping the  intermediate shaft ( θ2), by their di splacement ratio t o the drive  shaft (θ3/θ1). However,  this task is more  system aticall y organized by first collecting the  loads from th e crank and pisto n, and th en referencing their  sum from the crankshaft ( θ3) to the driveshaft ( θ1).  

Coordinates 4, 5, and 6, Linka ge Bea ring Load s and Piston Loa ds (Fig. 4)   Although th e friction to rque from th e inte rmedi ate crank bearing T4B also acts directly on the crankshaft,  this be aring’s displ acemen t, Φ4, is relative to its two mating crank linkages. This relative displacement, Φ4,  is gre ater that  the displa cement θ3 of the lower link relative to the cran kshaft, in the crank position shown.   

Φ4 = θ3 + θ5       (7) 

The fri ction t orque of this cra nk bearing acts on both the upp er crank and th e lower crank. Thu s, the   combi ned effect of this fri ction to rque on the  crankshaft i s greater th an th e reactio n torque of only thi s  beari ng on the shaft. The reason can be seen from t he free body diagram of th e upper link i n Figure 4.  The rea ction of this be aring friction to rque on th e upper crank link results in a moment o n this u pper link  with an asso ciated couple and a reaction force a gainst the lo wer link. Thu s the energy dissi pated by thi s  friction  torque is in creased by the relativ e rotation of the upper crank linkage. This is one example  of the  simplification  inhe rent in  this en ergy metho d. It avoids the  need fo r separating  the cran k linkage,  crankshaft and piston b earing into three separate free  body diagra ms.  

The effect of t he frictio n torque, T 4B, of this be aring on the cran kshaft is determi ned by multip lying it by the  angul ar displacement ratio  (Φ4/θ3).  

T4B/3 = T 4B (Φ4/θ3)       (8) 

The d rag torque T 5B on the pi ston bearing  is referenced to th e cran kshaft i n the same way, u sing th e  angul ar displacement ratio  of the upper crank linkage to the cran k, θ5/θ3.  

T5B/3 = T 5B (θ5/θ3)      (9) 

Since th e direction and magnitud e of Φ4 and θ5 are varia ble with po sition, th eir di splacement ratio s are  also vari able.  Note that for some p ositions of the cran k, Φ4 is less in magnitud e than θ3. 

The fluid pressure force on  the piston i s equal to the pressu re, P, times the pi ston area, A. The equivale nt  load could be directly ref erenced to the motor dr iveshaft, Coordinate 1, by  the ratio of the linear  displacement  of the pisto n, ∆6, divided by the angul ar displacement  of the motor shaft, θ1. However, to be  consistent with the p revious treatment of the two crank bearings, this force will first be  referenced to the  crankshaft by the ratio b etwee n ∆6 and θ3, and then referen ced to the θ1 coordinate by the ratio between   θ3 and θ1. Likewi se, the drag load fro m the friction, F 6P, between the pisto n and its cylinder i s referenced  to Co ordinate  θ3 by thi s same di splacement ratio. T he equatio n for the forces and to rques referen ced to  the crankshaft is as follows.  

T3R = (T 31B + T 32B) + T 4B (θ3 + θ5)/θ3 + T 5B (θ5/θ3) + (P ·A + F 6P)(∆6/θ3)   ( 10) 

If these were referenced directly to the motor drive shaft, the result would be:  

T3/1R = (T 31B + T 32B)(θ3/θ1) + T4B (θ3 + θ5)/θ1 + T 5B (θ5/θ1) + (P ·A + F 6P)(∆6/θ1)  (11)     The ratio ∆6/θ3 constantly varie s throug hout the cyclical moti on of the pi ston, a s does the fluid  pressure P.   Note that the algeb raic sign of the friction force, F P, betwee n the pi ston an d cylin der would al so cha nge a s  the pisto n change s direction at the to p and botto m of  its stroke. Li kewise, the contrib ution of the se  torque s and forces to the fi nal torqu e margin al so vari es thro ughout the cycle of motion.  

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