Analysis and synthesis of a cam-actuated variable-length crank for frame saw drives with perpendicular springs

This paper develops and evaluates a cam-actuated variable-length crank for reciprocating frame-saw drives equipped with springs placed perpendicular to the slider path. An invariant formulation is used to synthesize a periodic slider motion in which the spring force projected along the stroke cancels the slider inertia force. This yields a second-order differential equation with a non-dimensional stiffness parameter for the perpendicular-spring layout; the corresponding cam (variable crank length) is then recovered from the synthesized law of motion. Numerical studies for a representative frame-saw geometry confirm that the proposed synthesis eliminates inertial loading of the slider and produces a pronounced quasi-constant-velocity segment in mid-stroke (about 40 % of the stroke within a 5 % tolerance). The theoretical cam profile is well approximated by an ellipse (semi-axes 0.4996 and 0.4106 in non-dimensional form) with a maximum radial deviation of 0.00235; the associated pressure angles are about 13°, i.e., comfortably within typical limits. These results demonstrate that combining a cam-modulated variable crank length with a perpendicular-spring environment enables kinematic balancing with feasible pressure angles and motion shaping favorable for wood and stone cutting.