When angle 9 is very small, proper functioning demands an extremely high precision in the dimensions of the parts. However, at small angles, tremendous forces are exerted on the links and the resulting deformation of the linkage will produce a condition in which the linkage will not act as effectively as theoretical analysis would indicate. For practical applications, the initial value of angle 0 should not he much smaller than from 5 to 10 degrees.
The analysis of the holt motion in a retarded blowback gun employing a toggle linkage can not be conducted bv the same methods as are described in this publication for other blowback guns. This is so because the mass effect of the linkage is not constant but varies as the parts of the linkage rotate, with the result that the motion can not be interpreted in terms of a simple relationship between impulse and momentum.
The differential equation expressing the free recoil conditions in the linkage may be derived from fig. 1-47. F is the force exerted on the bolt by the powder gas pressure at any instant of time. (F is an empirical function of time which may be derived from a graph of chamber pressure versus time of the type shown in fig. 1-9.) Considering mass M-. and arm BC at angle 0, the torque resulting from the force G applied to M> by arm AB will be equal to the moment of inertia of arm BC times the angular acceleration of this arm about point G. That is:
Since the entire mass of the linkage is assumed to be concentrated at point B, the moment of inertia is given by the relation
Making this substitution and solving for G gives:
sin 2 6 dt2
Now considering the equilibrium of the horizontal forces on the bolt at point A gives the relation
Substituting the expression for G previously obtained gives:
Angle 0 may be eliminated from this expression as follows: If L is the initial distance between point A and point C and x is the linear displacement of the bolt along line AC :
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