6. Resonance A mass of one slug is hanging at rest on a spring whose constant is 12 lb/ft. At time = 0 an external force of f(t) =16 cos o t lb is applied to the system. (a)What is the frequency of the forcing function that is in resonance with the system? (b)Find the equation of motion of the mass with resonance.

A one-element bar of length L=400 cm clamped at x=0, and simply supported at x=l. The bar is subject to flexural vibration. Its squared natural frequency can be expressed as a function of the material density p, the cross-sectional area is A, the second moment of area I, and the Young's modulus, E, as: ΕΙ @²=b- PA Answer the following: • The number of degrees of freedom of the bar is The value of b is . The units of b are (give your answer to 2 decimal places) (in the international system, SI) Note local mass and stiffness matrices for bars subject to flexural vibrations are given here: 156 22L 54 -13L 4L² 13L -3L² 54 13L 156 -22L -13L-3L² -22L 4L² M= PAL 22L 420 and K = ΕΙ L³ 12 6L -12 6L 6L 4L2 -6L 2L² -12 -6L 12 -6L 6L 2L² -6L 4L²

Consider a mass spring system with an external force ?(?) = ?0sin?0?. Let's assume ? is the mass of the object attached to the spring, ? is the spring constant, and ? is the damping coefficient. Let's also assume that ?(0) and ?′(0) are the location and speed of the object attached to the spring at time ? = 0. a) Find the general solution of the differential equation governing the motion of the spring, in terms of ?,?,?,?(0),?"(0),?0, and ?o. Your solution shall cover all possible cases including 1. the case with no damping and ?! is not equal to natural frequency of the system, ? =sqrt?/?. 2. the case with no damping and ?! is equal to natural frequency of the system, ? =sqrt?/?. 3. the three cases with damping (overdamped, critically damped, and underdamped).