Consider the rigid rod of mass m and length L shown. For small oscillations, complete the following: (a) Draw the degree of freedom clearly on an appropriate figure. (b) Determine the equation of motion. What type of motion is this? (c) Determine the natural frequency and the damping ratio. Determine the displacement at the right end of the rod as a function of time. (d) Determine the damping ratio value if the vibrations are measured to decrease by 35% from one cycle to the next. 3 о 3

1. A thin disk attached to the end of an elastic beam has three uncoupled modes of vibration. The longitudinal motion, transverse motion and the torsional oscillations are all kinematically independent. Calculate the following for the system shown. a. Longitudinal stiffness b. Transverse stiffness c. Torsional stiffness mumm 65 cm r= 10 mm E-200 x 10° N/m² G=80 x 10º N/m²

In this lab, we will observe the free response of a first order mechanical system: a rotating bicycle wheel. As shown in the figure, suppose that the bicycle wheel is suspended above the ground, and is free to rotate on its axle. Assume that the only source of damping is the axle bearing, and that after time t=0 no external torques act on the wheel. ✈0(1) a) Derive the differential equation describing the angu- lar velocity of the wheel (t), in terms of the moment of inertia J and the damping B. b) Assume that a time t= 0 the wheel has initial an- gular velocity 22o. Solve the differential equation for response (1). c) Normally, we plot the response as (t) versus t. In- stead, suppose we plot the response as log, (n(t)/Ng) versus t. What type of curve is this plot? If you mea- sured experimentally logg ((t)/no) as a function of t, could you determine the system time constant 7 from the curve? d) What is the time constant 7 of the system expressed in terms of the moment of inertia J and…

A rigid block of mass ? is mounted on four elastic supports, as shown in the figure. A mass ? drops from a height ? and adheres to the rigid block without rebounding. If the spring constant of each elastic support is ?, derive the equation of motion and find the natural frequency of vibration of the system (a) without the mass ?, and (b) with the mass ?. Also, find the resulting motion of the system in case (b). By using lagrange energy method