Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_35

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The cylinder of mass and radius rolls without slipping on the circular surface of radius . Attached to the cylinder is a small body of mass . Determine the required relationship between and if the body is to be stable in the equilibrium position shown. Problem 7/35 (a) Stability analysis of the rolling cylinder system 1. Formula: The system is stable if the second derivative of the total potential energy with respect to the displacement coordinate is positive at the equilibrium position: For a cylinder of radius rolling inside a bowl of radius , the absolute rotation is related to the center's angular displacement by: 2. Substitution: Given and , the absolute rotation is . Let the mass be at a distance from the center (as interpreted from the diagram where is the larger internal dimension). The heights of the center of mass of the cylinder and the small body relative to the track's center of curvature are: The total potential energy is: 3. Calculation: Differentiate with respect to : MR 3Rm Mm Vθ > dθ 2 dV 2 θ=0 0 rR T ψ θ ψ= −1θ( r R T ) R = T 3Rr=Rψ= (3R/R−1)θ=2θmh=3R/4 O3R/4 Om y = O −(3R−R)cosθ=−2Rcosθ y = m y + O hcosψ=−2Rcosθ+ cos(2 θ ) 4 3R V=Mgy + O mgy = m −Mg(2Rcosθ)+ m g−2Rcosθ+ cos(2( 4 3R V=−2(M+m)gRcosθ+ m gR cos(2 θ ) 4 3 Vθ First derivative: At , , confirming equilibrium. Second derivative: Stability condition at : 4. Result: ● Final Conclusion: For the system to be stable in the equilibrium position shown, the mas