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Modal Analysis-hw-01 A uniform cylinder of mass m rests on a flat, horizontal, smooth surface as shown below. The cylinder is attached to ground by two springs of equal stiffness and free length. The cylinder is rotated through a small angle q0 in the direction shown and allowed to come to rest before being released. Determine a) the equation of motion and natural frequency of free vibration and b) the equation describing the position of the center of the disk as a function of time. The cylinder rolls without slipping;
Modal Analysis-hw-02 A simply supported beam of length l and mass m2 carries a component of mass m1at its midpoint as shown. Find the dynamic mass of the beam if a) the dynamic deflection shape is the same as the static deflection shape due to a load at the center of the beam and b) the dynamic deflection shape is approximated by y = y0 sin (px/l). Determine the lowest natural frequency of transverse vibration for the mass and beam as a unit for each dynamic mass case. If case a) is taken as the exact value, what is the percent error obtained using case b)?
Modal Analysis-hw-03 An underdamped shock absorber assembly is to contain both a spring and damper and is to be designed for a motorcycle of mass 200 kg which is shown below in Figure (a). When the shock absorber is subjected to an initial vertical velocity due to a road bump, the resulting displacement-time curve is to be as shown in Figure (b). Determine the necessary stiffness and damping coefficients of the shock absorber if the damped period of vibration is to be 2 s and the amplitude x1 is to be reduced to one-fourth in one-half cycle (i.e., x1.5 = x1/4). Also determine the minimum initial velocity that produces a maximum displacement of 250 mm and the length of time required for the amplitude of oscillation to decay to 0.1 x1.
