Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_38

⚡ Mecademy AIENG정역학 · ch7  Problem Statement Each of the two gears carries an eccentric mass and is free to rotate in the vertical plane about its bearing. Determine the values of for equilibrium and identify the type of equilibrium for each position. Problem 7/38 (a) Determination of the potential energy function 1. Formula: Let the centers of the gears define the datum for potential energy (). The total gravitational potential energy of the system is the sum of the potential energies of the two eccentric masses: 2. Substitution: From the geometry shown in the diagram, the height of mass relative to the center of Gear 1 is . Since Gear 1 has radius and Gear 2 has radius , they are meshed such that Gear 2 rotates by in the opposite direction. The height of mass relative to the center of Gear 2 is . 3. Calculation: For equilibrium, the first derivative of the potential energy with respect to must be zero: Using the double-angle identity : 4. Result: Setting each factor to zero within the range : m θ y=0 V V=mgy + 1 mgy 2 G 1 y = 1 2acosθ2r rφ=(2r/r)θ=2θ G 2 y = 2 acos2θ V(θ)=mg(2acosθ)+mg(acos2θ)=amg(2cosθ+cos2θ) θ = dθ dV amg(−2sinθ−2sin2θ)=−2amg(sinθ+sin2θ)=0 sin2θ=2sinθcosθ sinθ+2sinθcosθ=0⟹sinθ(1+2cosθ)=0 0≤ ∘ θ<360 ∘ sinθ=0⟹θ=0,180 ∘∘ 1+2cosθ=0⟹cosθ=−1/2⟹θ=120,240 ∘∘ ● Final Conclusion: The system has equilibrium positions at . (b) Stability analysis of equilibrium positions 1. Formula: The stability of an equilibrium position is determined by the sign of the second deri