Mechanics, Machines and Vibration

Learning Outcomes

Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• Conduct vibration analysis of uniform continuous systems and understand the solutions for axial vibration of rods and flexural vibration of beams
• Use Working Model 2D software for kinematic design of a linkage mechanism and carry out motion simulation.
• Use a matrix approach for the solution and understanding of the solutions produced
• Validate theoretical models through laboratory experiments such as measuring moment of inertia of a complex component
• Develop and apply the solutions of the equations of motion to problems for free and forced vibration under harmonic excitation
• Perform kinematic synthesis and analysis of linkage mechanisms
• Perform mathematical analysis of displacement, velocity (via instant centres and vector polygons), and acceleration of Mechanisms.
• Produce a formal technical report
• Carry out experimental work and formulate analytical models and solutions for simple systems.
• Apply approximate methods of solution for non-uniform continuous systems

Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Provide critical analysis and conclusions.

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Demonstrate knowledge and understanding of rigid body kinematics of linkages, design of four bar mechanisms, the kinematics and kinetics of simple machine elements and devices
• Develop relationships between mass, forces and the motion of a mechanism and the consequent vibrational response of a system to such forces.

Partial CEng Programme Level Learning Outcomes

Having successfully completed this module you will be able to:

• In two labs, students develop practical skills for tackling intricate problems. One lab focuses on moment of inertia measurements through basic kinematics for complex structures. The other requires analysis of a multi-storey structure, investigating vibration isolation and absorber effects, alongside numerical predictions. They then critically evaluate and compare their experimental findings with theoretical expectations in order to provide explanation and interpretation
• Lectures and tutorials empower students to refine their modelling and analytical skills for complex engineering scenarios. They apply engineering judgment and fundamental principles, like Newton's laws and Lagrange equations to analyse mechanical systems, focussing on interpreting numerical solutions and the accuracy of the predictions based on the principles and assumptions.
• Our lab sessions enhance the kinematics and dynamics principles, fostering practical, essential skills. Students create mathematical models for the assessed lab, analyse variations between predictions and theory, and grasp the physical system behaviour and interpretation. Lectures delve into advanced dynamic analysis relevant to specialised vibration engineering.
• In written exams, students tackle previously unseen problems, crafting intricate models through analytical or numerical methods. They must choose and apply suitable computational and analytical approaches to address complex issues, while remaining mindful of the limitations and assumptions inherent in their chosen techniques.

1) Kinematics and Dynamics as Part of the Design Process (2 Lectures):

Mechanisms & Machines, Four-Bar Linkage Mechanism, Mobility of Mechanisms, Kinematic Chain (closed), Kinematic Pair, Types of Four-Bar Chain, Kinematic Inversion, Grashoff’s Theory, Effect of

Joints on DOF, Grübler’s Formula, Practical Implications.

2) Design of Mechanisms (2 lectures)

Design Considerations (Kinematics Viewpoint), Transmission Angle & Efficiency, Even-Return Mechanism, Quick-Return Mechanism, Design of a Quick-Return Crank-Rocker.

3) Moment of Inertia (1 lecture)

Experimental Methods for Estimating Moment of Inertia: Compound Pendulum & Trifilar Pendulum, Derivation of Natural Frequencies.

4) Kinematic Analysis of Mechanisms (4 lectures)

Position and Velocity Diagrams for linkage mechanism, Instantaneous Centres, Acceleration Diagrams for Crank-Slider Chain and Four-Bar Chain Mechanism Including Coriolis Component of Acceleration.

5) Static and Dynamic Balancing & Gyroscopic Effects (2 lectures)

Gyroscopic Effects, Static Balance (Single-plane balance), Several out of Balance Masses, Measuring and Correcting Imbalance, Dynamic Balance (2-Plane balance), Graphical Method,

Moment and Force Polygons.

6) Introducing the software Working Model 2D (2 lectures)

7) Vibration of a SDOF System (8 lectures)

Free Vibration, Damping (Viscous and Structural), Logarithmic Decrement, Harmonically Forced Vibration, Response to periodic excitation, Impulse response, Convolution, Shock spectra, Force and

motion transmissibility.

8) Vibration of a 2-DOF System (4 lectures)

Free Vibration and Normal Modes, Co-ordinate Coupling and Principal Co-ordinates, Forced Vibration, Damping, Vibration absorber, Torsional Vibration of Geared systems, Degenerate systems.

9) Vibration of multi-degree of freedom systems (3 lectures)

Free and Forced Response by Modal Analysis, Introduction to Orthogonality and Generalised Coordinates, Modal Damping and Normal Mode Summation.

10) Vibration of Continuous systems (3 lectures)

Longitudinal Vibration of Rods, Modes of Vibration: Natural Frequencies and Mode Shapes. Forced Vibration of Continuous Systems: Modes and Resonance, Flexural Vibration of Beams, Derivation of Equation of Motion and Procedure for Obtaining Free Vibration Solutions.

11) Classical Methods (2 lectures)

Rayleigh’s Method for Fundamental Natural Frequency, Applications to Beams with Discrete Masses and Springs Attached, Effect of Rotation and Different Boundary Conditions.

Laboratory Work (10%):

1. Moment of Inertia Measurement of a Connecting Rod (Formative)

2. Vibration measurement and control of a multi-storey tower (10%)

Design Assignment (Formative)

1. Dimensional Synthesis of a Quick Return Mechanism and its Efficiency in Torque Transmission.

This is a one-semester module, three lectures per week (one double and one single) with two laboratory sessions and one assignment. Lecture notes and tutorial sheets are provided and one-toone

assistance and verbal feedback is facilitated through tutorial classes. Past exam papers are supplied to aid personal study, feedback and revision. Blackboard is used to allow the lectures and

additional material to be disseminated (including solutions to selected past exam papers). The students have to write-up two laboratory reports and one assignment report. Students are encouraged to read supporting texts and a booklist is provided.

Learning activities include

• One laboratory session to measure the moment of inertia of a complex component by using a trifilar pendulum and by treating it as a compound pendulum.
• One laboratory session to introduce the measurement of vibration of an isolator.
• Two lectures to introduce a new software called Working Model 2D that can be used for kinematic design of mechanisms and motion simulation.
• One assignment on the synthesis and analysis of a linkage mechanism wherein the study of motions and transmission forces can help in understanding machine kinematics and dynamics.
• Private study.

Feedback and student support during module study (formative assessment)

• Worked examples to provide self-assessment.
• Written feedback on laboratory and design assignment reports and generic feedback through blackboard.
• Interactive feedback given in lab sessions.
• Generic feedback based on the performance of previous cohorts, when introducing the assignment.
• One to one feedback during tutorial sessions.

Relationship between the teaching, learning and assessment methods and the planned learning outcomes

• Lectures, laboratories, assignments and tutorial sheets contribute to knowledge and understanding as well as the development of intellectual and some key skills.
• Problem assignments and assessment methods promote additional key skills.
• The laboratory sessions are designed to complement the analytical contents of the module on kinematics and dynamics and promote practical and key skills.
• The examination tests the understanding and modelling skills of the students.
• Simple mathematical models of real engineering structures and mechanisms will highlight the engineering context of the analysis.

Formative

This is how we’ll give you feedback as you are learning. It is not a formal test or exam.

Lab Report

• Assessment Type: Formative
• Feedback: Interactive feedback given in lab sessions.
• Final Assessment: No
• Group Work: No

Coursework

• Assessment Type: Formative
• Feedback: Feedback during module study.
• Final Assessment: No
• Group Work: No

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.