Four-bar motion generation with elasticity constraints and optimization

Abstract

In conventional planar four-bar motion generation, all mechanism links are assumed rigid or non-deforming. Although the assumption of link rigidity in kinematic synthesis may be generally appropriate and often practiced, a statically loaded planar four-bar mechanism will undergo a degree of elastic deflection, particularly the crank and follower links. In this work, a nonlinear optimization problem is formulated for planar four-bar motion generation that considers an applied coupler force and corresponding crank static torque, crank transverse deflection, and follower buckling. The output from the non-linear optimization problem -mechanism fixed and moving pivot loci -are input for a search algorithm that down selects a mechanism solution that satisfies transmission angle conditions, Grashof conditions, and a mechanism compactness condition. The final output of the presented method is planar four-bar motion generator that approximates prescribed coupler poses with satisfactory crank deflection and without follower buckling and also satisfies conditions for link rotatability, transmission angle and compactness.

Planar Four-Bar Motion and Path Generation with Order and Branching Conditions

“…al. considered prescribed coupler loads (3)(4) (5) , prescribed coupler loads in braking system design (6) as well as the elastic deflection of the crank and follower links in braking system design (7) . Peng and Sodhi (8) presented an optimal continuous path generation method for adjustable planar four-bar linkages.…”

On the synthesis of spatial revolute—revolute—spherical—cylindrical motion generators with prescribed coupler loads

“…For the simplified non-linear optimization problem, equation (23) is the objective function, equations (10) and (11) are the equality constraints, and inequality (21) is the inequality constraint. The reader could refer to the work of Al-Smadi et al [21] for the specific SQP computation steps…”

“…From sheer trial and error, one could eventually calculate an RRSC solution that produces satisfactory static torques under a prescribed coupler load. What inequality constraint (21) ensures is that a solution producing satisfactory static torques is calculated on every attempt. Inequality (21), when included in the RRSC motion generation models, offers efficiency when computing satisfactory RRSC motion generator solutions.…”

“…What inequality constraint (21) ensures is that a solution producing satisfactory static torques is calculated on every attempt. Inequality (21), when included in the RRSC motion generation models, offers efficiency when computing satisfactory RRSC motion generator solutions.…”

“…Like the works of Shen et al [17,18], the authors employ sequential quadratic programming (SQP) for solving the non-linear optimization problems. Equation (24), the general SQP merit function used by Han [19] and Powell [20], includes equation ( 22) as the objective function, equations (1) to (4) as equality constraints (variable m e ), and inequality (21) as an inequality constraint (variable m) for the first non-linear optimization problem. For the simplified non-linear optimization problem, equation (23) is the objective function, equations ( 10) and ( 11) are the equality constraints, and inequality (21) is the inequality constraint.…”