Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_34

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The uniform bar of mass and length is hinged about a horizontal axis through its end and is attached to a torsional spring which exerts a torque on the rod, where is the torsional stiffness of the spring in units of torque per radian and is the angular deflection from the vertical in radians. Determine the maximum value of for which equilibrium at the position is stable. Problem 7/34 (a) Determination of maximum length for stable equilibrium 1. Formula: To determine stability, we define the total potential energy of the system as the sum of gravitational potential energy and elastic potential energy . Equilibrium at is stable if the second derivative of the potential energy with respect to the degree of freedom is positive: 2. Substitution: Taking the hinge as the datum for gravitational potential energy, the center of mass of the uniform bar is at height . Gravitational potential energy: Elastic potential energy of the torsional spring: The total potential energy is: 3. Calculation: First, find the equilibrium condition by setting the first derivative to zero: At , . This confirms that is an mlO M=k θ T k T θ l θ=0 V V g V e θ=0 θ V=V + g V e > dθ 2 dV 2 0 O h= cos θ 2 l V = g mgh= m g cosθ( 2 l ) V = e k θ 2 1 T 2 V(θ)= m gl cos θ + 2 1 k θ 2 1 T 2 = dθ dV − m gl sin θ + 2 1 k θ T θ=0 = dθ dV − mglsin(0)+ 2 1 k (0)= T 0θ=0 equilibrium position. Next, find the second derivative for the stability test: Evaluate the second derivati