# Vibration and Dynamics

Paper, Order, or Assignment Requirements

Learning Outcomes

On successful completion of this assignment, the student should be able to:

Show in depth understanding and ability to evaluate the Kinematics and Dynamics in the component
Show in depth understanding and ability to evaluate the Vibration

Assignment:

A weather balloon moves in the x–y plane so that its co-ordinates measured from a reference point are given by x = 4t and y = 0.04 x2, where t is the time in seconds and x and y are in meters. When t = 3s, determine

a). the magnitude of the velocity,
b). the direction of the balloon’s motion;
c). the magnitude of the acceleration.
Figure 1. Representation of question 1.

An engine box is shown in Figure 2. The gear ratios and the radii of gears on the wheels and engine shafts are given in table 1. If the engine shaft run with a speed of 2500 rev/min, determine

a). the angular speed of each gear;
b). the radius of the gears on the layshaft.

Figure 2. Arrangement of engine box of question 2.

Gear
Gear ratio
Gear radius (r, mm)

1
3.5
28

2
2.5
24

3
1.5
20

4
1.12
18

5
1

6
2
16

Table 1. Engine gearbox configuration of question 2.

Two gears are in mesh as shown in Figure 3. Gear 1 has a radius of 60 mm a radius of gyration of 46 mm and a mass of 2.5 kg; gear 2 has a radius of 180 mm, a radius of gyration 105 mm and a mass of 9.5 kg. If gear 1 has an angular acceleration of 2.5 rad/s2, determine:

a). the angular acceleration of gear 2;
b). the contact force between the two gears;
c). the external moment M1 applied to gear 1.
Figure 3. Representation of question 3.

A shaft carries two masses at planes 1 and 2 as shown in Figure 4. Determine the magnitude and the angular position of two masses that should be added at planes 3 and 4, each at a radial position of 0.25 m in order to achieve complete balance. The planes have the characteristics given in table (where m is the mass, r is the radial position, Θ is the angular position and the reference plane is taken at plane 3)

Figure 4. Representation of question 4.

Three masses are attached along a shaft at the locations in Figure 5, which also summarises the value of the masses and their radial angular positions. The shaft is supported at its ends by two bearings, A and B. In order to dynamically balance the shaft, an additional mass of 1.5 kg at a radial position of 0.5 m is attached to the shaft.

Determine the location along the shaft measured from bearing A and the angular position of this additional mass.
Figure 5. Representation of question 5.

The tower of the wind turbine has a height of 50 m and a circular hollow cross-section that has an inner diameter of 0.3 m and an outer diameter of 0.5 m. The rotor and hub mass is 1.6 × 103 kg and the tower is made of steel that has a Yong’s modulus of 200 GPa. Considering the cases when the tower’s weight is ignored and when it is taken into account, determine:
a) the natural frequency of transverse vibration of the system;
b) the time response due to initial transverse displacement x0 = 0.1 m;
c) the maximum values of velocity and accelerations.

Figure 6. Tower of wind turbine of question 6.

A mass of m = 10 kg is suspended on a spring and set oscillating. The amplitude reduced to 5% of its initial value after 2 oscillations. It takes 0.5 seconds to do them, determine

a). the damping ratio;
b). the natural frequency and actual frequency;
c). the critical damping coefficient and the actual damping coefficient.
Figure 7. Schematic drawing of vibration system of question 7.

A platform of mass 4.2 ×103 kg is supported by three springs, each of stiffness k as shown in Figure 8. Determine:

a). the spring’s stiffness, k, so that the natural frequency of the platform equals 2.8 Hz;
b). the natural frequency of the platform when a truck of mass 42 × 103 kg is loaded onto it.
Figure 8. Platform supported by three springs in parallel of question 8.

An unbalanced centrifugal pump of mass 50 kg is supported by four springs each of 8000 N/m as shown Figure 9. If the pump operates at 1000 rev/min and the steady-state amplitude shouldn’t exceed 4 mm, determine the maximum allowable out-of-balance force and plot the relationship steady-stage amplitude vs frequency using Matlab software.

Figure 9. Unbalanced centrifugal pump supported by springs of question 9.

A harmonic force of F = 100 sin 3t, where F is in newtons and t is the time in seconds, is acting on a machine of mass 200 kg. It is supported by a spring and a damper as shown in Figure 10. If the spring has a stiffness of 125 kN/m and the damping ratio is ζ = 0.20, determine:
a) the undamped and damped natural frequency of the system in Hz;
b) the amplitude of the steady-state vibration.
c) use the Matlab software to plot the steady-state amplitude vs frequency relationship.

Figure 10. Representation of question 10.

Module lecture and support notes.

http://liblists.derby.ac.uk/lists/97C96EAB-332F-B652-B195-580E1A2945FE.html

Note:    These sources are guides only to commonly available material. Students will also be expected to consult other relevant source material according to the nature of their project.

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