The University of Southampton

FEEG2002 Mechanics, Machines & Vibration

Module Overview

This module will help the students to understand the fundamental concepts in Kinematics and Dynamics of multi-body systems. It provides an understanding of the application of simple mathematical models to vibration problems in engineering using different levels of approximation/complexity and some practical experience of vibration measurement and the interpretation of vibration effects. By completing this module an acoustical engineering student will appreciate how simple mechanisms forming a complex machine can be the source of vibration and a mechanical student can appreciate the response of a machine to such excitations. The course assumes knowledge of elementary vector algebra and the concepts of time and partial derivatives. An elementary Physics course covering Newton's laws or course(s) on Statics and Dynamics can be very helpful in understanding the material covered as the course reviews these topics and then applies them to more complex problems. There will be equal emphasis placed on gaining both analytical understanding and insight/intuition on the subject. The material presented in the lectures will emphasize the analytical component of the subject, while with the assignment (through coding and modelling) and the experimental laboratory work will encourage students to see beyond equations. This module will provide a pre-requisite understanding for more advanced topics such as vibration measurement and analysis with numerical tools such as Finite Element Method Analysis and will prepare the students for automotive modules.

Aims and Objectives

Module Aims

• Provide an understanding of the kinematics and kinetics of simple mechanisms and devices. • Instil some practical appreciation of the purpose of simple mechanisms within a modern complex machine by carrying out a design synthesis and analysis assignment. • Introduce some practical experience of vibration measurement. • Develop a competence in the theoretical analysis of free and forced vibration of both discrete and continuous elastic systems. • Convey, through theory and practical examples, an understanding of the usefulness of normal mode descriptions of a vibrating system. • Instil an appreciation for the significance of vibration behaviour of boundary conditions and mechanical coupling between substructures.

Learning Outcomes

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.
Subject Specific Practical Skills

Having successfully completed this module you will be able to:

  • Perform mathematical analysis of displacement, velocity (via instant centres and vector polygons), and acceleration of Mechanisms.
  • Produce a formal technical report
  • Perform kinematic synthesis and analysis of linkage mechanisms
  • Use Working Model 2D software for kinematic design of a linkage mechanism and carry out motion simulation.
  • 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
  • Use a matrix approach for the solution and understanding of the solutions produced
  • Conduct vibration analysis of uniform continuous systems and understand the solutions for axial vibration of rods and flexural vibration of beams
  • Apply approximate methods of solution for non-uniform continuous systems
  • Carry out experimental work and formulate analytical models and solutions for simple systems.
Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

  • Provide critical analysis and conclusions.


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 (20%): 1. Moment of Inertia Measurement of a Connecting Rod (5%) 2. Vibration measurement and control of a multi-storey tower (15%) Design Assignment (10%) 1. Dimensional Synthesis of a Quick Return Mechanism and its Efficiency in Torque Transmission (Using Working Model 2D).

Special Features


Learning and Teaching

Teaching and learning methods

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.

Independent Study112
Supervised time in studio/workshop6
Total study time150

Resources & Reading list

Printed notes issued. 

W T Thomson. Theory of Vibration with Applications. 

Numerous texts under TJ 170, TJ 175, TJ 230. 

Meirovitch, L.. Elements of Vibration Analysis. 

Tse, Morse & Hinkle. Mechanical Vibrations. 

R.L. Norton. Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines. 

Lecture notes, PowerPoint slides, tutorial sheets and solutions available from Blackboard. 

Rao, S.S.. Mechanical Vibrations. 


Assessment Strategy



MethodPercentage contribution
Coursework 10%
Exam  (120 minutes) 70%
Laboratory Report 5%
Laboratory Report 15%


MethodPercentage contribution
Exam  (120 minutes) 100%

Repeat Information

Repeat type: Internal & External

Linked modules


To study this module, you will need to have studied the following module(s):

FEEG1002Mechanics, Structures and Materials
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