EDWARD G. BOETTIGER 



107 



As the elastic force is in phase with motion it is also maximum at this instant 

 so OA lies on line ()L. 



For a muscle, the apparent elastic and sustaining forces are components of 

 the muscle tension, which in case of the model is represented by OP, the re- 

 sultant of OD and OA. The resultant OP makes the angle ^ with OA, the elastic 

 force, and so also with the length. This angle is the phase angle, the existence 

 of which accounts for the sustaining force. In ligure i.l, tension OP lags behind 

 length OA. To obtain the fraction of cycle completed at any instant by the ten- 

 sion, the angle if> must be subtracted from the angle cot completed by the 

 length. If the tension rotates ahead of length, the angle ^ must be added. In 

 the first case the phase angle is considered negative, in the second case positive. 

 Only when the phase angle is negative is there a sustaining force component 

 to oppose damping and so a self-excited vibration. 



Tension and length are the parameters usually measured. The change in 

 tension during a lo-millisecond cycle of shortening and lengthening is 

 shown in ligure iB. For a negative phase angle the tension is given by the full 

 line, for a positive phase angle by the dotted line. As length change is in phase 

 with apparent elastic force (fig. i5), the relative values of this force are also 

 represented by the line showing length. The tension in the muscle is above that 

 of the elastic component during shortening and below during lengthening. The 

 difference between the total tension and that of the elastic component de- 

 termines the value and direction of the sustaining force. 



Vibrations may be conveniently studied by plotting tension as a function 

 of length. This can be done experimentally, using a cathode ray oscilloscope. 

 A tension transducer is connected to give a vertical deflection of the beam, and 

 a length transducer to give horizontal deflection. In figure \C the solid line is 

 a tension-length plot of the model system in sinusoidal motion with a phase 

 angle of 30° and a period of 10 milliseconds. The force of gravity on the mass 

 shifts the equilibrium length and tension to the values L' and T'. The tension 

 varies from T' -f Pq to T' - Po; and the length from L' -f x„ to \J - xq. The 

 line EF gives the apparent elastic force at each length, the slope of the line 

 being a measure of the elastic constant. Points on the loop are values of the 

 total tension OP. The direction of movement of the beam in drawing out the 

 loop is clockwise for a positive phase angle, and counterclockwise for a negative 

 phase angle. The difference between the elastic and total forces at any length 

 is a measure of the sustaining force; OH and OH' represent the maximum values 

 of this force at the two points of the cycle when length equals the equilibrium 

 length L' and velocity is maximum. 



The area of the loop is the external work done by the muscle model against 

 damping each cycle. For sinusoidal motion it can be shown that: 



work per cycle = ttPoXs Sin <p (2) 



Power output per second = tt/PoXq Sin v? (j) 



