Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_13

⚡ Mecademy AIENG정역학 · ch7  Problem Statement Determine the couple required to maintain equilibrium at an angle . The mass of the uniform bar of length is , while that of the uniform bar of length is . Problem 7/13 (a) Determination of the couple 1. Formula: The principle of virtual work states that for a system in equilibrium, the total virtual work is zero for any virtual displacement: where is the virtual work done by gravity and is the virtual work done by the couple . 2. Substitution: Let be the angle of the long bar with the horizontal (as indicated by the arc at the bottom pivot). The long bar has length and mass . Its center of mass is at its midpoint, distance from the pivot. The hinge is also located at distance from the pivot. Thus, . The short bar has length and mass . It is hinged at its midpoint . Thus, its center of mass is also at hinge , so . The total potential energy of the system is . Let be the angle of the short bar with the horizontal. Since the distance from hinge to the roller at the floor is , we have the geometric constraint: , which implies . The virtual work done by gravity is . The virtual work done by the couple (assuming it is applied to the short bar in the direction of ) is . From , differentiating gives , so . Mθ 2l2mlm M δU δU=δW + g δW = M 0 δW = g −δVδW M M θ 2l2mG 1 lBl y = G1 y = B lsinθ lmB G 2 By = G2 y = B lsinθ V=m gy + 1G1 m gy = 2G2 (2m)g(lsinθ)+(m)g(lsinθ)=3mglsinθ φ Bl/2y = B sinφ= 2 l lsinθsinφ=2sinθ δW = g − δθ = dθ dV −3mglco