Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_31

⚡ Mecademy AIENG정역학 · ch7  Problem Statement For the mechanism shown, the spring is uncompressed when . Determine the angle for the equilibrium position and specify the minimum spring stiffness which will limit to . The rod passes freely through the pivoted collar , and the cylinder of mass slides freely on the fixed vertical shaft. Problem 7/31 (a) Angle for equilibrium 1. Formula: The system reaches equilibrium when the first derivative of the total potential energy with respect to the generalized coordinate is zero: . Total potential energy , where is the gravitational potential energy and is the elastic potential energy of the spring. 2. Substitution: Let the fixed pivot be at the origin . The vertical shaft lies on the y-axis. The coordinates of the mass at are . From the geometry of the isosceles links and , we have . The gravitational potential energy is . The pivoted collar is at a fixed position based on the dimension labels. The distance between and is . The spring is uncompressed at , so its free length (as represented by the distance ) is . The elastic potential energy is . The total potential energy is: 3. Calculation: Differentiate with respect to : θ=0θ kθ30 ∘ DECm θ θ = dθ dV 0 V=V + g V e V g V e A(0,0) D(0,y ) D AB=bBD=by = D 2bcosθ V = g mgy = D 2mgbcosθ C(b,b) DCCD= = b+(2bcosθ−b) 22 b 1+(2cosθ−1) 2 θ=0 CDL = 0 CD∣ = θ=0 b = 1+(2(1)−1) 2 b 2 V = e k(CD− 2 1 b ) 2 2 V=2mgbcosθ+ kb −b 2 1 (1+(2cosθ−1) 2 2) 2 Vθ = dθ dV −2mgbsinθ+k(CD− b ) 2 dθ d(CD) () Sett