Question

Vibrations

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.  Verified

### Question 44800  Vibrations

A mass is attached at one end of a uniform bar of mass whose other end is pivote dat point O as shown in the following figure. Determine the natural frequency of vibration of the resulting pendulum for small angular displacements. (10 marks)2.
I_{1 \text { aroundo }}=m_{1} l^{2}
I_{2 \text { coundmiddle point }}=\frac{1}{12} m_{2} I^{2}

### Question 44799  Vibrations

1. Derive the constants of X, and & for an underdamped system, where 3 is the damping ratio, by using initial conditionsof x(0)=x, and *(0)=x, (5 marks)

### Question 44696  Vibrations

7. To familiarize yourself with the nature of the forced response, plot the solution of a forced response of equation mit kx =Fcos at with @=2 rad /s, for a variety of values of the initial conditions and @,, as given in the following chart:

### Question 44695  Vibrations

Consider the system in Fig.3, write the equation of motion,and calculate the response assuming that the system is initially at rest for the values k = 100N / m k, = 500N / m and9.m = 89 kg.

### Question 44694  Vibrations

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.

### Question 44692  Vibrations

6. The free response of a 1000-kg automobile with stiffness ofk= 400,000N/m is observed to be of the form give in Figure 6. Modeling the automobile as a single-degree-of-freedom oscillation in the vertical direction, determine the damping coefficient if the displacement at is measured to be 2cm and0.22 cm at

### Question 44691  Vibrations

Repeated problem for the system in Figure 5.5.

### Question 44690  Vibrations

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?

### Question 44689  Vibrations

3. Set up the differential equation of motion for the system shown in Fig.2, Determine the expression for (a) the critical damping coefficient, and (b) the natural frequency of the damped oscillation

### Question 44688  Vibrations

2. The pendulum illustrated in Figure.1 consist of a rigid bar of mass ": and length /, Attached to the end is a lumped mass m. Find the pendulum's natural frequency of oscillation.

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