KDWAKD G. B(JKTTR;EK 



CHARACTERISTICS OF VIBRATING SYSTEMS 



105 



The unit of motion in a vibration is a cycle consisting, in the case of a muscle, 

 of a shortening and a lengthening. A vibration always occurs about an equilib- 

 rium point, the movements and forces being measured plus or minus the equi- 

 librium values. A perfect spring and a mass set into vibration will maintain 

 continuous sinusoidal motion in the absence of all frictional forces. All real 

 vibrating systems generate heat in frictional damping and so can be maintained 

 in motion either by an external force, a driven oscillation, or by an internally 

 generated force. If the sustaining force is internal and itself non-oscillatory in 

 nature, the movement is called a self-excited vibration. In such a system fre- 

 quency is determined by the natural oscillating period. The oscillation is main- 

 tained by a sustaining force which, in the steady state condition, is just suf- 

 ficient to overcome all damping during the cycle. The instantaneous value of 

 this force is in some way controlled by the motion, the force being zero when 

 there is no motion. In the simplest case of sinusoidal motion the sustaining force 

 is always equal and opposite to the damping force. Since the damping force in- 

 creases and decreases with movement velocity, the sustaining force must do 

 so also, its action directed to aid motion throughout the cycle. 



If to the simple mass and spring system is added an internal mechanism to 

 generate the sustaining force, we have a model showing many properties of 

 fibrillar muscle. The behavior of this model will be used in discussing the ex- 

 perimental results. The muscle model contributes the elastic and sustaining 

 forces, while the mechanical system with which it operates furnishes the damp- 

 ing and inertia. 



Using the principle of D'Alembert and considering inertia as a force, the 

 static method of analysis may be employed. The sum of all forces acting is equal 

 to zero throughout the cycle and the following relation may be written: 



m.x -f cix = C2X — kx (j) 



where: mx is the inertia force, mass, m times acceleration, x; kx is the apparent 

 elastic force proportional to displacement, x; c,x is the damping force, propor- 

 tional to velocity, .x; and C2X is the sustaining force, proportional to velocity, x. 

 The sustaining force aids the elastic force during the shortening phase when this 

 force is opposed by damping and aids the inertia force when its action in 

 lengthening the spring is also opposed by damping. It aids the inertia force by 

 opposing the elastic force against which the inertia must work. 



The relations between the forces throughout the movement cycle are illus- 

 trated in figure i. Figure lA is a vector diagram of the maximum values of the 

 forces; OA, the apparent elastic force; OB, the damping force; OC, the inertia 

 force; and OD, the sustaining force. The damping force OB is in phase with 

 velocity and so leads the elastic force OA in phase with movement, by 90°. The 

 inertia force is in opposite phase to the elastic force, and so leads by 180°. The 



