Module Title: Vibration and Dynamics 

Programmes BEng (Hons) Mechanical Engineering 

Learning Outcomes 

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

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

2. Show in depth understanding and ability to evaluate the Vibration 

This coursework will be composed of two parts It is compulsory to pass each part.  

Part 1 (30%).  

You are asked to assess the vibration performance of TM16 Range Universal vibration apparatus. You need to conduct a series of practical measurement of free vibration or forced vibration in the absence or presence of viscous damper. You also need to give presentations to the whole class based on selected labs.  

Assessment: Individual mark will consist of two components: group mark and individual contribution.  

Group mark will be awarded by lecturer and teaching assistants based on your presentation; individual contribution will be evaluated by your peers within the group.  

Group mark contributes to 60% while peer assessment contributes to 40% for each presentation.  

You will be asked to give group presentations three times. The average of three presentations’ mark will be your final mark for the Part I. 

It is compulsory to pass this part in order to get valid mark for the whole coursework. 

Part 2 (70%). 

Each sub-question carries equal marks. 

1. A ball is fired out at an angle of 30° from the platform of height h = 0.5 m as shown in Figure 1. It is subjected to the gravitational force only thus follows a projectile motion. It falls onto the ground after 0.6 seconds. Determine: a). the ball’s initial velocity V0 when it is fired out. 

b). the distance the ball travels in the horizontal direction when it falls on the ground. 

c). the magnitude and direction of the ball’s velocity at t = 0.3 seconds after being fired out. 

Figure 1. Representation of Question 1. 

2. 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 runs 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 1.2 18 

5 1 – 

6 2 16 

Table 1. Engine gearbox configuration of question 2. 

3. Two engaged gears are as shown in Figure 3. Gear 1 has a radius of 60 mm a radius of gyration of 45 mm and a mass of 2.5 kg; gear 2 has a radius of 150 mm, a radius of gyration 105 mm and a mass of 9.5 kg. If gear 1 has an angular acceleration of 2.0 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. 

4. A shaft carries two masses at planes 1 and 2 as shown in Figure 4. Determine magnitude and angular position of two masses that should be added at planes 3 and 4 in order to achieve complete balance. Assume each mass is at a radial position of 0.25 m 

Figure 4. Representation of question 4. 

5. Three masses are attached along a shaft at the locations 1, 2, 3 in Figure 5, which also summarises the value of the masses, and their radial and 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 added 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. 

6. The tower of the wind turbine (see Figure 6) has a height of 30 m and a circular hollow cross-section that has an inner diameter of 0.4 m and an outer diameter of 1 m. The rotor and hub mass is 0.3 × 103 kg and the tower is made of steel that has a Young’s modulus of 200 GPa and density 7800 kg/m3. When the tower’s weight is ignored, 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. 

When the tower’s weight is taken into account using equivalent weight 

23.57% of its actual weight, repeat the above sub-questions a), b), and c). 

Figure 6. Tower of wind turbine of question 6.  

7. A mass of m = 5 kg is connected on a spring and set oscillating as shown in 

Figure 7. The oscillation amplitude is reduced to 3.0 % of its initial value after 2 cycles.  Each cycle period = 0.25 seconds. 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. 

8. 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 2.5 Hz; 

b). the natural frequency of the platform when a truck of mass 15 × 103 kg is loaded onto it. 

Figure 8. Platform supported by three springs in parallel of question 8. 

9. An unbalanced centrifugal pump of mass 60.0 kg is supported by four springs each stiffness of 8000 N/m as shown Figure 9. If the pump operates at 1000 rev/min and the steady-state displacement amplitude shouldn’t exceed 5 mm, determine the maximum allowable out-of-balance force. 

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

10. 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. 

Figure 10. Representation of question 10. 

 Reading Materials 

Module lecture and support notes. 

See also module reading list 

Aspire list link: 

http://liblists.derby.ac.uk/lists/97C96EAB-332F-B652-B195580E1A2945FE.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. 

Please note that plagiarism is an academic offence and that if any two, or more, submissions for this assignment are considered to be produced as a 

result of plagiarism from an original then this will be investigated further and could result in students, who are proven to have plagiarised, failing this assignment and, therefore, the module. 

Submission Requirements 

This coursework should be organised on a question basis and complied into a electronic document. The report should not exceed the 3000 words in length. There will be 1% penalty on the mark for every 50 words over. The report should be submitted through e-submission in this module blackboard.  

Submission Details 

Exceptional Extenuating Circumstances (EEC) and Late

Submission

Specific details for the submission of assignments will be provided with the assignment brief. There are no automatic right to late submission, with a capped mark of 40%. However, the University acknowledges that there may be circumstances which prevent students from meeting deadlines and there are now three distinct processes in place to deal with differing student circumstances: 

1) Assessed Extended Deadline (AED)  

Students with disabilities or long term health issues are entitled to a Support Plan. 

The Support Plan will outline any adjustments to assessments which are required to accommodate an individual student’s needs. For further details refer to the link below: 

http://www.derby.ac.uk/studentatozHE/support-plans 

2) Exceptional Extenuating Circumstances (EEC) 

The EEC policy applies to situations where serious, unforeseen circumstances prevent the student from completing the assignment on time or to the normal standard. Students who submit a successful EEC claim will usually be required to complete a different assessment to that which was originally set. All EEC claims will be considered by Faculty/UDC panels, which will convene on a monthly basis.  For further details refer to the link below: http://www.derby.ac.uk/campus/support 

3) Late Submission up to One Week 

Covering unexpected and severe disruption to study, where circumstances do not require the additional time allowed for by an EEC, the Late Submission process enables students to complete their existing assessment up to one week late, without a cap on the grade. Requests for late submission will be made to http://www.derby.ac.uk/eec.  

The relevant Deputy Head in the Department will then consider the request and, if acceptable, authorise an extension of up to a maximum of one week. The Deputy Head will expect to see compelling evidence that such an extension is appropriate. 

Submission Dates 

See cover sheet  

Assessment Criteria 

University Post Graduate Assessment Grades will be used. All grades given for assignment work are provisional until confirmed or otherwise by the relevant examinations board. 

Academic Offences: An “academic offence” has been committed when a student tries to gain improper advantage for her/himself by breaking, or not following, the Academic Regulations concerning any part of the assessment process. This procedure applies to all students engaged in any University assessment activity whether on or off site including collaborative programmes. 

Reading Materials: Module lecture and support notes; also see module reading list. 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 the project.