One of the tail rotor blades of a helicopter has an unbalance mass of m=0.5 kg at a distance ofe=0.15 m from the axis of rotation, as shown in the figure below. The tail section has a length of 4 m, a mass of 240 kg, a flexural stiffness (EI) of 2.5 x 106 N-m², and a damping ratio of 0.15. The mass of the tail rotor blades, including their drive system is 20 kg.Determine the amplitude of steady state response of the tail section when the blade rotates at 1500 rpm.

Two masses are connected via three springs to each other and to the wall at left and right. The equations of motion are -k_{1} x_{1}-k\left(x_{1}-x_{2}\right)=m_{1} \ddot{x}_{1} -k_{2} x_{2}-k\left(x_{2}-x_{1}\right)=m_{2} \ddot{x}_{2} Your job is to calculate expressions giving the position of each mass as a function of time using the diagonalization method discussed in class. This classic problem can be solved in a number of different ways, but I will not give you credit unless you solve using the diagonalization method from the course. Suppose the force constants of the left and right springs are k1= k2 =2 and that of the center spring is k=1 let the masses be m1=m2=1 the two masses are given initial displacements at t =0 of x1 ==1 and x2=0 the initial velocity of each mass at t=0 is zero

A variable speed electric motor, having an unbalance, is mounted on an isolator. As the speed of the motor is increased from zero, the amplitudes of the vibration of the motor have been observed to be 0.55 in. at resonance and 0.15 in. beyond resonance (i.e. faraway from resonance). Find the damping ratio of the suspension system.

4. Calculate the natural frequency and damping ratio for the100system in Figure 4. It is given the values m =10 kg, c =kg/s, = 4000 N/m, = 200 N/m, = 1000 N/m. Ignore the friction between the cart and the ground. Is the system overdamped, critically damped, or underdamped?

8. Consider the system in Fig. 2, write the equation of motion,and calculate the response assuming (a) that the system is initially at rest, and (b) that the system has an initial displacement of 0.05 m.

A steel shaft of diameter d and length I is fixed at one end and carries a propeller of mass m and aradius of gyration of r, at the other end (see Figure 1). Data: d = 5 cm,l=1m, m = 100 kg, r, = 10 cm 1- Determine the fundamental natural frequeney of vibration of the shaft in axial vibration Determine the fundamental natural frequency of vibration of the shaft in torsional Task 1.3- Is it possible to change the system parameters (d, l, m, r,) such that the fundamentalfrequencies for both axial and torsional vibration become identical?[30%] If yes, what are the new parameter values? Explain the method and report your detailedanalysis leading to the new values. If no, explain in detail why it is impossible. I.Note 1: In Tasks 1.1 and 1.2, derive boundary conditions, start from the relevant equation of motionand apply the boundary conditions, find characteristic equation, solve it for natural frequencies, andfind the fundamental frequency.

A variable speed compressor operates between 400 and 600 rpm. The compressor has a mass of 450 kg and is mounted on springs having stiffness of 2000 kN/m. a.) Determine the maximum unbalance (me) allowed if the maximum allowable displacement within the operating speed range is 0.5 mm assuming no damping. b.) Repeat part a, if damping is added with c=2000 N*s/m.

During an earthquake, the one-story frame structure shown below is subjected to a ground vibration amplitude of 0.1 m at 10 Hz. a) Determine the amplitude of motion of the girder.Assume the girder is rigid and the structure has a damping ratio of 0.03. b) What is the amplitude of acceleration of the structure?

1- A vibrating mass of 300 kg, mounted on a massless support by a spring of stiffness 40,000 N/m and a damper of unknown damping coefficient, is observed to vibrate with a 10-mm amplitude while the support vibration has a maximum amplitude of only 2.5 mm (at resonance). Calculate the damping constant and the amplitude of the force on the base.

1) Sec. 4.2,P.8 : Find the total response of a viscously damped single degree of freedom system subjected to a harmonic base excitation for the following data: m=10 \mathrm{~kg}, \quad c=20 \mathrm{~N}-\mathrm{m} / \mathrm{s}, \quad k=4000 \mathrm{~N} / \mathrm{m} y(t)=0.05 \sin 5 t \mathrm{~m}, \quad x_{0}=0.02 \mathrm{~m}, \quad \dot{x}_{0}=10 \mathrm{~m} / \mathrm{s}