Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_40

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The handle is fastened to one of the spring-connected gears, which are mounted in fixed bearings. The spring of stiffness k connects two pins mounted in the faces of the gears. When the handle is in the vertical position, θ = 0 and the spring force is zero. Determine the force P required to maintain equilibrium at an angle θ. Problem 7/40 (a) Force P required for equilibrium 1. Formula: The principle of virtual work for a conservative system in equilibrium states that the virtual work done by external forces must equal the change in the system's potential energy for any virtual displacement: For a single degree of freedom , this can be expressed as: where is the external torque and is the potential energy of the spring. 2. Substitution: First, we determine the geometry. Let the gears have radius . Their centers are at a distance . Let the center of the top gear be and the bottom gear be . The pins are at a distance from their respective centers. At , the spring force is zero, meaning its length equals the distance between pins: At a general angle , the top gear rotates clockwise by and the bottom gear rotates counter-clockwise by due to meshing. The coordinates of the pins are: The current length of the spring is: δW=δV θ M = ext dθ dV M ext V r 2rO (0,2r) 1 O (0,0) 2 bθ=0 l 0 l = 0 (2r−b)−b=2r−2b θθ θ Pin 1 (top): (bsinθ,2r−bcosθ) Pin 2 (bottom): (−bsinθ,bcosθ) l l= 2 (bsinθ−(−bsinθ))+ 2 (2r−bcosθ−bcosθ) 2 The potential energy is