Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_25

⚡ Mecademy AIENG정역학 · ch7  Problem Statement A horizontal force is applied to the four-bar mechanism shown. If the weight of the bars is negligible compared with the applied force , determine the magnitude of the couple needed to hold the mechanism in equilibrium at the orientation shown. (Note: For simplicity, leave your answer in terms of , , and .) Problem 7/25 (a) Magnitude of the couple 1. Formula: The principle of virtual work states that for a system in equilibrium, the total virtual work done by all active forces and couples is zero for any virtual displacement consistent with the constraints: For this single-degree-of-freedom mechanism, considering a virtual rotation of the link , the virtual work equation is: where is the couple applied at and is the horizontal force applied at joint . 2. Substitution: Define a coordinate system with origin at . The coordinates of point are: The virtual displacement of point is: The virtual work done by force is: The total virtual work equation becomes: P PM θφψ M δU=F ⋅ ∑ i δr + i M ⋅ ∑ j δθ = j 0 δθ OA δU=Mδθ+P⋅δr = B 0 MOP=Pi B C B x = B L cosψ,y = 3B Lsinψ 3 B δr = B δx i+ B δy j= B (−L sinψδψ)i+ 3 (L cosψδψ)j 3 P δU = P P⋅δr = B (Pi)⋅(−L sinψδψi+ 3 L cosψδψj)= 3 −PL sinψδ 3 Mδθ−PL sinψδψ= 3 0⟹M=PL sin ψ 3 δθ δψ 3. Calculation: To find , we use the loop-closure equations based on the geometry shown in the diagram: Differentiating these equations with respect to the virtual displacements : (1) (2) Eliminate by multiplying (1) by