Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_43

⚡ Mecademy AIENG정역학 · ch7  Problem Statement One end of the torsion spring is secured to the ground at A, and the other end is fastened to the shaft at B. The torsional stiffness of the elastic spring is the torque required to twist the spring through an angle of one radian. The spring resists the moment about the shaft axis caused by the tension in the cable wrapped around the drum of radius . Determine the equilibrium value of measured from the dashed position, where the spring is untwisted. Problem 7/43 (a) Equilibrium displacement 1. Formula: For a conservative system in equilibrium, the first derivative of the total potential energy with respect to the coordinate of interest (here, the vertical displacement ) must be zero: The total potential energy is the sum of elastic potential energy and gravitational potential energy : 2. Substitution: The elastic potential energy of a torsion spring is , where is the angle of rotation in radians. The gravitational potential energy of mass relative to the untwisted dashed position () is . From the geometry of the cable wrapped around the drum, the relationship between rotation and downward displacement is: Substituting these into the expression for total potential energy: k T mgr h h V h = dh dV 0 V e V g V=V + e V g V = e k θ 2 1 T 2 θ mh=0V = g −mgh θ h h=rθ⟹θ= r h V= k − 2 1 T ( r h ) 2 mgh= − 2r 2 k h T 2 mgh 3. Calculation: Differentiate the total potential energy with respect to and set the expression to zero to find the equi