MACHINERY OF INSECT FLIGHT 137 



The contractile element is in series with a very stiff spring. To describe 

 the theoretical implications of the tension-length loops in terms of these 

 two elements, a simple model can be used. In this model the elastic element 

 is represented by a spring and the contractile element by one's arm muscles. 

 The muscles support the spring, to the end of which a weight is attached. 

 If the weight is set in motion by an external force its movements will soon 

 be damped out. One is instructed, however, to keep the weight in motion 

 at the same amplitude and therefore must contrive to put into the system 

 just enough energy to overcome the damping forces. One can do this most 

 efficiently by shortening and lengthening his muscle in the same sinusoidal 

 movement as the spring and weight. If the muscle moves in phase with the 

 spring, no energy is transferred. But by moving in the same rhythm, and 

 slightly out of phase with the spring, one can maintain the motion. The 

 movements of the muscle must be slightly ahead of the movements of the 

 spring. This is only possible because of the inertia of the weight. If the 

 movement of the weight is more heavily damped, one's movements must be 

 more out of phase with the spring to maintain the system in motion. The 

 tension-length diagrams of the above model would show loops similar to 

 those of Fig. 10. The area of the loop would be greater with greater phase 

 shifts. Tension would be greater during shortening and the movement of 

 the beam around the loop would be counterclockwise, the area representing 

 work the muscle does against damping forces. 



With spontaneous vibrations the loops are always counterclockwise. 

 When, however, a mechanical system is used to make the excited muscle 

 shorten and lengthen at different frequencies, the movement is either 

 clockwise, the system doing work on the muscle, or counterclockwise, the 

 muscle doing work on the system. The phase angle is a function of fre- 

 quency and shifts from a minus value, length reaching its maximum before 

 tension, to a plus value, tension reaching its maximum before length. In 

 the former case, area represents work done by the muscle, and in the latter, 

 work done on the muscle. 



The significance of the loop can be summarized as follows : ( 1 ) the 

 slope of the major axis of the loop is determined by the compliance of the 

 excited muscle, (2) the area of the loop is a measure of the work done by 

 the muscle against external damping forces and is a fvmction of the maxi- 

 mum tension, the maximum length, and the phase angle between tension 

 and length. (3) the position of the loop in the tension-length area depends 

 upon the load and the change in the force the muscle can exert as velocity 

 increases. When velocity is zero the muscle shortens until it attains the 

 minimum length at which it can just sustain the load. In vibration as 

 velocity increases, the muscle, because of the force-velocity relation, cannot 

 exert at this short length a tension equal to the load. The muscle must 



