Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_51

⚡ Mecademy AIENG정역학 · ch7  Problem Statement A control mechanism consists of an input shaft at A which is turned by applying a couple M and an output slider B which moves in the x-direction against the action of force P. The mechanism is designed so that the linear movement of B is proportional to the angular movement of A, with x increasing 60 mm for every complete turn of A. If M = 10 N · m, determine P for equilibrium. Neglect internal friction and assume that all mechanical components are ideally connected rigid bodies. Problem 7/51 (a) Determination of the equilibrium force 1. Formula: The principle of virtual work states that for a system in equilibrium, the total virtual work done by all external active forces during an infinitesimal virtual displacement is zero: 2. Substitution: The work done by the input couple for a virtual angular displacement is . The work done by the output force for a virtual linear displacement is (the negative sign indicates that the force opposes the displacement). The problem states that increases by for every complete turn ( ). Thus, the relationship between virtual displacements is: Substituting these into the virtual work equation: 3. Calculation: P δU δU=Mδθ + ∑F δs= ∑0 Mδθ δU = M Mδθ Pδx δU = P −PδxP x60 mm 2π radians δx= δθ = 2π rad 60×10 m −3 δθ 2π 0.06 Mδθ−Pδx=0⟹10⋅δθ−P δθ = ( 2π 0.06 )0 Dividing the equation by : Rearranging to solve for : Performing the numerical calculation: 4. Result: or . ● Final Conclusion: The force required