Engineering Tripos Part IIA, 3C6: Vibration, 2024-25
Module Leader
Lecturers
Prof D Cole, Dr T Butlin
Lab Leader
Timing and Structure
Michaelmas term. Vibration of Continuous Systems: 1 lecture/week, weeks 1-8 Michaelmas term (Dr T Butlin), Vibration of Lumped Systems: Rayleigh's method, 1 lecture/week, weeks 1-8 Michaelmas term (Prof D Cebon). 16 lectures.
Aims
The aims of the course are to:
- Introduce the central ideas and tools for the analysis of small (linear) vibration in mechanical systems.
- Introduce simple continuous systems which may be combined as components of larger systems.
- Introduce the general approach to lumped systems via mass and stiffness matrices, and the resulting properties of vibration modes and their frequencies.
Objectives
As specific objectives, by the end of the course students should be able to:
- Derive the partial differential equations governing the forced or free motion of uniform one-dimensional systems.
- Use these equations and appropriate boundary conditions to obtain vibration modes and natural frequencies.
- Analyse continuous systems using modal methods.
- Compute impulse and harmonic response by modal and direct methods.
- Be able to derive the dispersion relation for wave propagation in 1D structures.
- Understand that vibration can be expressed in terms of wave propagation or superposition of modes.
- Calculate the response of a coupled system from a knowledge of the responses of the separate subsystems.
- Apply Rayleigh's method to continuous systems.
- Take advantage of the link between Lagrange's equations and small vibration.
- Explain the concept of a vibration mode, and be able to find the modes and their natural frequencies by an eigenvector/eigenvalue calculation.
- Understand orthogonality of modes, modal damping, modal density and modal overlap factor.
- Express the frequency response functions or the impulse response functions of a system in terms of the normal modes, and be familiar with the concepts of resonances and antiresonances.
- Recognise and apply the reciprocal theorem for responses.
- Use the stationary property of normal mode frequencies to estimate frequencies given assumed mode shapes, using minimisation with respect to any free parameters.
Content
This course aims to present a systematic approach to the study of small vibration of engineering components and structures. The course picks up where Part IA Linear Systems and Vibration left off. Concepts which were barely discussed (e.g. reciprocity and the orthogonality of vibration modes) are important for building up qualitative insights into vibration behaviour. Alongside the mathematical tools for quantitative analysis the course offers vital ingredients for an engineer's education.
Vibration of Continuous Systems (8L)
- Vibration of strings; axial and transverse vibration of beams, torsional vibration of circular shafts; 1D acoustic vibration in a duct;
- Modal analysis of simple systems;
- Wave-based analysis of vibration, including D'Alembert's solution;
- Dispersion relation for travelling waves;
- Response to impulse and harmonic excitation;
- Transfer functions and the meaning of poles and zeros;
- Coupling of systems;
- Rayleigh's method for continuous systems.
Vibration of Lumped Systems (8L)
- Application of Lagrange's equations to small vibrations; undamped vibration of systems with N degrees of freedom;
- Matrix methods and modal analysis;
- Response functions in frequency and time domains; properties of frequency-response functions; reciprocal theorems;
- Modal damping and modal overlap;
- Rayleigh's method for discrete systems.
Coursework
A data-logging and FFT analysis system is used to measure the frequency response of a vibrating system by three different methods, to compare their merits and disadvantages.
[Coursework]
Learning objectives:
- To investigate alternative methods of determining calibrated frequency response transfer functions of a mechanical vibrating system, using a digital measuring system;
- To predict the response of a system from measured responses of its decoupled subsystems, and to compare with the measured response of the coupled system.
Practical information:
- Sessions will take place in the South Wing Mechanics Laboratory, throughout Lent term.
- This activity doesn't involve preliminary work.
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
Last modified: 31/05/2024 07:28
Engineering Tripos Part IIA, 3C6: Vibration, 2022-23
Module Leader
Lecturers
Prof D Cebon, Dr T Butlin
Lab Leader
Timing and Structure
Lent term. Vibration of Continuous Systems: 1 lecture/week, weeks 1-8 Lent term (Dr T Butlin), Vibration of Lumped Systems: Rayleigh's method, 1 lecture/week, weeks 1-8 Lent term (Prof D Cebon). 16 lectures.
Aims
The aims of the course are to:
- Introduce the central ideas and tools for the analysis of small (linear) vibration in mechanical systems.
- Introduce simple continuous systems which may be combined as components of larger systems.
- Introduce the general approach to lumped systems via mass and stiffness matrices, and the resulting properties of vibration modes and their frequencies.
Objectives
As specific objectives, by the end of the course students should be able to:
- Derive the partial differential equations governing the forced or free motion of uniform one-dimensional systems.
- Use these equations and appropriate boundary conditions to obtain vibration modes and natural frequencies.
- Analyse continuous systems using modal methods.
- Compute impulse and harmonic response by modal and direct methods.
- Be able to derive the dispersion relation for wave propagation in 1D structures.
- Understand that vibration can be expressed in terms of wave propagation or superposition of modes.
- Calculate the response of a coupled system from a knowledge of the responses of the separate subsystems.
- Apply Rayleigh's method to continuous systems.
- Take advantage of the link between Lagrange's equations and small vibration.
- Explain the concept of a vibration mode, and be able to find the modes and their natural frequencies by an eigenvector/eigenvaluecalculation.
- Understand orthogonality of modes, modal damping, modal density and modal overlap factor.
- Express the frequency response functions or the impulse response functions of a system in terms of the normal modes, and be familiar with the concepts of resonances and antiresonances.
- Recognise and apply the reciprocal theorem for responses.
- Use the stationary property of normal mode frequencies to estimate frequencies given assumed mode shapes, using minimisation with respect to any free parameters.
Content
This course aims to present a systematic approach to the study of small vibration of engineering components and structures. The course picks up where Part IA Linear Systems and Vibration left off. Concepts which were barely discussed (e.g. reciprocity and the orthogonality of vibration modes) are important for building up qualitative insights into vibration behaviour. Alongside the mathematical tools for quantitative analysis the course offers vital ingredients for an engineer's education.
Vibration of Continuous Systems (8L)
- Vibration of strings; axial and transverse vibration of beams, torsional vibration of circular shafts; 1D acoustic vibration in a duct;
- Modal analysis of simple systems;
- Electrical transmission line analogy of 1D mechanical wave propagation;
- D'Alembert's solution;
- Dispersion relation for travelling waves;
- Response to impulse and harmonic excitation;
- Transfer functions and the meaning of poles and zeros;
- Coupling of systems;
- Rayleigh's method for continuous systems.
Vibration of Lumped Systems (8L)
- Application of Lagrange's equations to small vibrations; undamped vibration of systems with N degrees of freedom;
- Matrix methods and modal analysis;
- Response functions in frequency and time domains; properties of frequency-response functions; reciprocal theorems;
- Modal damping and modal overlap;
- Rayleigh's method for discrete systems.
Coursework
A data-logging and FFT analysis system is used to measure the frequency response of a vibrating system by three different methods, to compare their merits and disadvantages.
[Coursework]
Learning objectives:
- To investigate alternative methods of determining calibrated frequency response transfer functions of a mechanical vibrating system, using a digital measuring system;
- To predict the response of a system from measured responses of its decoupled subsystems, and to compare with the measured response of the coupled system.
Practical information:
- Sessions will take place in the South Wing Mechanics Laboratory, throughout Lent term.
- This activity doesn't involve preliminary work.
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
Last modified: 21/02/2023 15:01
Engineering Tripos Part IIA, 3C6: Vibration, 2019-20
Module Leader
Lecturers
Prof D Cebon, Dr T Butlin
Lab Leader
Timing and Structure
Lent term. Vibration of Continuous Systems: 1 lecture/week, weeks 1-8 Lent term (Dr T Butlin), Vibration of Lumped Systems: Rayleigh's method, 1 lecture/week, weeks 1-8 Lent term (Prof D Cebon). 16 lectures.
Prerequisites
3C5 useful (there is one particular result from the Lagrange section of 3C5 which will be quoted without proof)
Aims
The aims of the course are to:
- Introduce the central ideas and tools for the analysis of small (linear) vibration in mechanical systems.
- Introduce simple continuous systems which may be combined as components of larger systems.
- Introduce the general approach to lumped systems via mass and stiffness matrices, and the resulting properties of vibration modes and their frequencies.
Objectives
As specific objectives, by the end of the course students should be able to:
- Derive the partial differential equations governing the forced or free motion of uniform one-dimensional systems.
- Use these equations and appropriate boundary conditions to obtain vibration modes and natural frequencies.
- Analyse continuous systems using modal methods.
- Compute impulse and harmonic response by modal and direct methods.
- Be able to derive the dispersion relation for wave propagation in 1D structures.
- Understand that vibration can be expressed in terms of wave propagation or superposition of modes.
- Calculate the response of a coupled system from a knowledge of the responses of the separate subsystems.
- Apply Rayleigh's method to continuous systems.
- Take advantage of the link between Lagrange's equations and small vibration.
- Explain the concept of a vibration mode, and be able to find the modes and their natural frequencies by an eigenvector/eigenvaluecalculation.
- Understand orthogonality of modes, modal damping, modal density and modal overlap factor.
- Express the frequency response functions or the impulse response functions of a system in terms of the normal modes, and be familiar with the concepts of resonances and antiresonances.
- Recognise and apply the reciprocal theorem for responses.
- Use the stationary property of normal mode frequencies to estimate frequencies given assumed mode shapes, using minimisation with respect to any free parameters.
Content
This course aims to present a systematic approach to the study of small vibration of engineering components and structures. The course picks up where Part IA Linear Systems and Vibration left off. Concepts which were barely discussed (e.g. reciprocity and the orthogonality of vibration modes) are important for building up qualitative insights into vibration behaviour. Alongside the mathematical tools for quantitative analysis the course offers vital ingredients for an engineer's education.
Vibration of Continuous Systems (8L)
- Vibration of strings; axial and transverse vibration of beams, torsional vibration of circular shafts; 1D acoustic vibration in a duct;
- Modal analysis of simple systems;
- Electrical transmission line analogy of 1D mechanical wave propagation;
- D'Alembert's solution;
- Dispersion relation for travelling waves;
- Response to impulse and harmonic excitation;
- Transfer functions and the meaning of poles and zeros;
- Coupling of systems;
- Rayleigh's method for continuous systems.
Vibration of Lumped Systems (8L)
- Application of Lagrange's equations to small vibrations; undamped vibration of systems with N degrees of freedom;
- Matrix methods and modal analysis;
- Response functions in frequency and time domains; properties of frequency-response functions; reciprocal theorems;
- Modal damping and modal overlap;
- Rayleigh's method for discrete systems.
Coursework
A data-logging and FFT analysis system is used to measure the frequency response of a vibrating system by three different methods, to compare their merits and disadvantages.
[Coursework]
Learning objectives:
- To investigate alternative methods of determining calibrated frequency response transfer functions of a mechanical vibrating system, using a digital measuring system;
- To predict the response of a system from measured responses of its decoupled subsystems, and to compare with the measured response of the coupled system.
Practical information:
- Sessions will take place in the South Wing Mechanics Laboratory, throughout Lent term.
- This activity doesn't involve preliminary work.
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please see the Booklist for Part IIA Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
Last modified: 15/05/2019 09:49
Engineering Tripos Part IIA, 3C5: Dynamics, 2023-24
Module Leader
Lecturers
Prof H.E.M. Hunt, Dr J. Talbot
Lab Leader
Timing and Structure
Lent term, 16 lectures. Introduction and Rigid-body Dynamics: 2 lectures/week, weeks 1-5 (Prof J Woodhouse); Lagrangian Mechanics: 2 lectures/week, weeks 6-8 (Prof R S Langley)
Aims
The aims of the course are to:
- Introduce the ideas and methods of 3D dynamics: the motion of rigid bodies in three dimensions under given forces and moments.
- Introduce the Lagrange and Hamiltonian formulations of mechanics.
- To show how to apply these methods in a straightforward way to a wide range of problems.
Objectives
As specific objectives, by the end of the course students should be able to:
- Represent the inertia of a rigid body by an inertia matrix, be able to calculate the moments and products of inertia for simple shapes, be able to find the principal axes of inertia.
- Derive Euler's equations for the motion of a rigid body under prescribed moments.
- Apply these equations to the motion of symmetrical rotors, to explain the phenomena of precession, nutation and the rate gyroscope.
- Analyse simple problems involving the rolling of rigid bodies, for example a spinning penny on a table.
- Explain the concepts of generalised coordinates and generalised forces.
- Express the kinetic and potential energies of a system in term of the generalised coordinates, and to use these to obtain Lagrange's equations of motion.
- Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of a system about its equilibrium position.
- Explain the concept of generalized momentum and show how the Hamilton's equations can be used to find the equations of motion.
- Explain the concepts of Poisson brackets, conserved quantities, and canonical transformations.
Content
This module aims to present a systematic approach to the study of dynamics. Once the main techniques have been grasped, a very wide range of problems can be tackled with confidence. The first part of the course presents the tools required to analyse rigid-body motion in three dimensions. These are necessary for a proper understanding of gyroscopic systems, inertial navigation, satellites in space and the stability of high-speed rotating systems such as turbines and compressors.
The second part of the course deals with Lagrangian and Hamiltonian mechanics, a systematic way to formulate dynamical problems using energy functions.
Introduction and Rigid-body Dynamics (10L)
- Equations of motion of a rigid body in three dimensions.
- The inertia tensor; principal axes.
- Gyroscopes and their application.
- Problems involving rolling bodies.
Lagrangian Mechanics (6L)
- Lagrange's equations; connection to Newton's laws; generalised coordinates and generalised forces.
- Applications to a range of problems.
- Hamilton's equations, Poisson brackets, conserved quantities, canonical transformations.
- Example applications.
Coursework
Gyroscopic Phenomena
Learning objectives:
Practical information:
- Sessions will take place in [Location], during week(s) [xxx].
- This activity [involves/doesn't involve] preliminary work ([estimated duration]).
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 13/10/2023 11:08
Engineering Tripos Part IIA, 3C5: Dynamics, 2025-26
Module Leader
Lecturers
Prof H.E.M. Hunt, Dr A Cicirello
Lab Leader
Timing and Structure
Lent term, 16 lectures. Introduction and Rigid-body Dynamics: 2 lectures/week, weeks 1-5 (Prof H E M Hunt); Lagrangian Mechanics: 2 lectures/week, weeks 6-8 (Dr A Cicirello)
Aims
The aims of the course are to:
- Introduce the ideas and methods of 3D dynamics: the motion of rigid bodies in three dimensions under given forces and moments.
- Introduce the Lagrange and Hamiltonian formulations of mechanics.
- To show how to apply these methods in a straightforward way to a wide range of problems.
Objectives
As specific objectives, by the end of the course students should be able to:
- Represent the inertia of a rigid body by an inertia matrix, be able to calculate the moments and products of inertia for simple shapes, be able to find the principal axes of inertia.
- Derive Euler's equations for the motion of a rigid body under prescribed moments.
- Apply these equations to the motion of symmetrical rotors, to explain the phenomena of precession, nutation and the rate gyroscope.
- Analyse simple problems involving the rolling of rigid bodies, for example a spinning penny on a table.
- Explain the concepts of generalised coordinates and generalised forces.
- Express the kinetic and potential energies of a system in term of the generalised coordinates, and to use these to obtain Lagrange's equations of motion.
- Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of a system about its equilibrium position.
- Explain the concept of generalized momentum and show how the Hamilton's equations can be used to find the equations of motion.
- Explain the concepts of Poisson brackets, conserved quantities, and canonical transformations.
Content
This module aims to present a systematic approach to the study of dynamics. Once the main techniques have been grasped, a very wide range of problems can be tackled with confidence. The first part of the course presents the tools required to analyse rigid-body motion in three dimensions. These are necessary for a proper understanding of gyroscopic systems, inertial navigation, satellites in space and the stability of high-speed rotating systems such as turbines and compressors.
The second part of the course deals with Lagrangian and Hamiltonian mechanics, a systematic way to formulate dynamical problems using energy functions.
Introduction and Rigid-body Dynamics (10L)
- Equations of motion of a rigid body in three dimensions.
- The inertia tensor; principal axes.
- Gyroscopes and their application.
- Problems involving rolling bodies.
Lagrangian and Hamiltonian Mechanics (6L)
- Lagrange's equations; connection to Newton's laws; generalised coordinates and generalised forces.
- Applications to a range of problems.
- Hamilton's equations, Poisson brackets, conserved quantities, canonical transformations.
- Example applications.
Coursework
Gyroscopic Phenomena
Learning objectives:
- To observe gyroscopic effects and to compare these observations with theory;
- To discover some of the engineering applications of gyroscopes;
- To develop a physical understanding of gyroscopic phenomena.
Practical information:
- Sessions will take place in the South Wing Laboratory, during weeks 1-7.
- This activity doesn't involve preliminary work.
- The lab can be completed in two hours
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 05/06/2025 18:53
Engineering Tripos Part IIA, 3C5: Dynamics, 2022-23
Module Leader
Lecturers
Prof H.E.M. Hunt, Dr J. Talbot
Lab Leader
Timing and Structure
Michaelmas term, 16 lectures. Introduction and Rigid-body Dynamics: 2 lectures/week, weeks 1-5 (Prof J Woodhouse); Lagrangian Mechanics: 2 lectures/week, weeks 6-8 (Prof R S Langley)
Aims
The aims of the course are to:
- Introduce the ideas and methods of 3D dynamics: the motion of rigid bodies in three dimensions under given forces and moments.
- Introduce the Lagrange and Hamiltonian formulations of mechanics.
- To show how to apply these methods in a straightforward way to a wide range of problems.
Objectives
As specific objectives, by the end of the course students should be able to:
- Represent the inertia of a rigid body by an inertia matrix, be able to calculate the moments and products of inertia for simple shapes, be able to find the principal axes of inertia.
- Derive Euler's equations for the motion of a rigid body under prescribed moments.
- Apply these equations to the motion of symmetrical rotors, to explain the phenomena of precession, nutation and the rate gyroscope.
- Analyse simple problems involving the rolling of rigid bodies, for example a spinning penny on a table.
- Explain the concepts of generalised coordinates and generalised forces.
- Express the kinetic and potential energies of a system in term of the generalised coordinates, and to use these to obtain Lagrange's equations of motion.
- Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of a system about its equilibrium position.
- Explain the concept of generalized momentum and show how the Hamilton's equations can be used to find the equations of motion.
- Explain the concepts of Poisson brackets, conserved quantities, and canonical transformations.
Content
This module aims to present a systematic approach to the study of dynamics. Once the main techniques have been grasped, a very wide range of problems can be tackled with confidence. The first part of the course presents the tools required to analyse rigid-body motion in three dimensions. These are necessary for a proper understanding of gyroscopic systems, inertial navigation, satellites in space and the stability of high-speed rotating systems such as turbines and compressors.
The second part of the course deals with Lagrangian and Hamiltonian mechanics, a systematic way to formulate dynamical problems using energy functions.
Introduction and Rigid-body Dynamics (10L)
- Equations of motion of a rigid body in three dimensions.
- The inertia tensor; principal axes.
- Gyroscopes and their application.
- Problems involving rolling bodies.
Lagrangian Mechanics (6L)
- Lagrange's equations; connection to Newton's laws; generalised coordinates and generalised forces.
- Applications to a range of problems.
- Hamilton's equations, Poisson brackets, conserved quantities, canonical transformations.
- Example applications.
Coursework
Gyroscopic Phenomena
Learning objectives:
Practical information:
- Sessions will take place in [Location], during week(s) [xxx].
- This activity [involves/doesn't involve] preliminary work ([estimated duration]).
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 24/05/2022 12:55
Engineering Tripos Part IIA, 3C5: Dynamics, 2020-21
Module Leader
Lecturers
Dr H E M Hunt
Lab Leader
Timing and Structure
Michaelmas term, 16 lectures. Introduction and Rigid-body Dynamics: 2 lectures/week, weeks 1-5 (Prof J Woodhouse); Lagrangian Mechanics: 2 lectures/week, weeks 6-8 (Prof R S Langley)
Aims
The aims of the course are to:
- Introduce the ideas and methods of 3D dynamics: the motion of rigid bodies in three dimensions under given forces and moments.
- Introduce the Lagrange and Hamiltonian formulations of mechanics.
- To show how to apply these methods in a straightforward way to a wide range of problems.
Objectives
As specific objectives, by the end of the course students should be able to:
- Represent the inertia of a rigid body by an inertia matrix, be able to calculate the moments and products of inertia for simple shapes, be able to find the principal axes of inertia.
- Derive Euler's equations for the motion of a rigid body under prescribed moments.
- Apply these equations to the motion of symmetrical rotors, to explain the phenomena of precession, nutation and the rate gyroscope.
- Analyse simple problems involving the rolling of rigid bodies, for example a spinning penny on a table.
- Explain the concepts of generalised coordinates and generalised forces.
- Express the kinetic and potential energies of a system in term of the generalised coordinates, and to use these to obtain Lagrange's equations of motion.
- Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of a system about its equilibrium position.
- Explain the concept of generalized momentum and show how the Hamilton's equations can be used to find the equations of motion.
- Explain the concepts of Poisson brackets, conserved quantities, and canonical transformations.
Content
This module aims to present a systematic approach to the study of dynamics. Once the main techniques have been grasped, a very wide range of problems can be tackled with confidence. The first part of the course presents the tools required to analyse rigid-body motion in three dimensions. These are necessary for a proper understanding of gyroscopic systems, inertial navigation, satellites in space and the stability of high-speed rotating systems such as turbines and compressors.
The second part of the course deals with Lagrangian and Hamiltonian mechanics, a systematic way to formulate dynamical problems using energy functions.
Introduction and Rigid-body Dynamics (10L)
- Equations of motion of a rigid body in three dimensions.
- The inertia tensor; principal axes.
- Gyroscopes and their application.
- Problems involving rolling bodies.
Lagrangian Mechanics (6L)
- Lagrange's equations; connection to Newton's laws; generalised coordinates and generalised forces.
- Applications to a range of problems.
- Hamilton's equations, Poisson brackets, conserved quantities, canonical transformations.
- Example applications.
Coursework
Gyroscopic Phenomena
Learning objectives:
Practical information:
- Sessions will take place in [Location], during week(s) [xxx].
- This activity [involves/doesn't involve] preliminary work ([estimated duration]).
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 28/08/2020 10:58
Engineering Tripos Part IIA, 3C5: Dynamics, 2018-19
Module Leader
Lecturers
Dr H Hunt and Prof R Langley
Lab Leader
Dr H Hunt
Timing and Structure
Michaelmas term, 16 lectures. Introduction and Rigid-body Dynamics: 2 lectures/week, weeks 1-5 (Prof J Woodhouse); Lagrangian Mechanics: 2 lectures/week, weeks 6-8 (Prof R S Langley)
Aims
The aims of the course are to:
- Introduce the ideas and methods of 3D dynamics: the motion of rigid bodies in three dimensions under given forces and moments, and the Lagrangian formulation.
- To show how to apply these methods in a straightforward way to a wide range of problems in mechanics.
Objectives
As specific objectives, by the end of the course students should be able to:
- Represent the inertia of a rigid body by an inertia matrix, be able to calculate the moments and products of inertia for simple shapes, be able to find the principal axes of inertia.
- Derive Euler's equations for the motion of a rigid body under prescribed moments.
- Apply these equations to the motion of symmetrical rotors, to explain the phenomena of precession, nutation and the rate gyroscope.
- Analyse simple problems involving the rolling of rigid bodies, for example a spinning penny on a table.
- Explain the concept of generalised coordinates, and their associated generalised forces.
- Express the kinetic and potential energies of simple systems in term of their generalised coordinates, and to use these to obtain Lagrange' equations of motion for the system.
- Explain the link between conservation laws and the form of the energy
- Explain the concept of a constraint, and be able to calculate the force of constraint from Lagrange's equations.
- Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of system about its equilibrium position.
Content
This module aims to present a systematic approach to the study of dynamics. Once the main techniques have been grasped, a very wide range of problems can be tackled with confidence. The first part of the course presents the tools required to analyse rigid-body motion in three dimensions. These are necessary for a proper understanding of gyroscopic systems, inertial navigation, satellites in space and the stability of high-speed rotating systems such as turbines and compressors.
The second part of the course deals with Lagrangian mechanics, a systematic way to formulate dynamical problems using energy functions.
Introduction and Rigid-body Dynamics (10L)
- Equations of motion of a rigid body in three dimensions.
- The inertia tensor; principal axes.
- Gyroscopes and their application.
- Problems involving rolling bodies.
Lagrangian Mechanics (6L)
- Lagrange's equations; connection to Newton's laws; generalised coordinates and generalised forces.
- Applications; calculation of forces of constraint.
Coursework
Gyroscopic Phenomena
Learning objectives:
Practical information:
- Sessions will take place in [Location], during week(s) [xxx].
- This activity [involves/doesn't involve] preliminary work ([estimated duration]).
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please see the Booklist for Part IIA Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 13/09/2018 14:36
Engineering Tripos Part IIA, 3C5: Dynamics, 2024-25
Module Leader
Lecturers
Prof H.E.M. Hunt, Dr A CIcirello
Lab Leader
Timing and Structure
Lent term, 16 lectures. Introduction and Rigid-body Dynamics: 2 lectures/week, weeks 1-5 (Prof J Woodhouse); Lagrangian Mechanics: 2 lectures/week, weeks 6-8 (Prof R S Langley)
Aims
The aims of the course are to:
- Introduce the ideas and methods of 3D dynamics: the motion of rigid bodies in three dimensions under given forces and moments.
- Introduce the Lagrange and Hamiltonian formulations of mechanics.
- To show how to apply these methods in a straightforward way to a wide range of problems.
Objectives
As specific objectives, by the end of the course students should be able to:
- Represent the inertia of a rigid body by an inertia matrix, be able to calculate the moments and products of inertia for simple shapes, be able to find the principal axes of inertia.
- Derive Euler's equations for the motion of a rigid body under prescribed moments.
- Apply these equations to the motion of symmetrical rotors, to explain the phenomena of precession, nutation and the rate gyroscope.
- Analyse simple problems involving the rolling of rigid bodies, for example a spinning penny on a table.
- Explain the concepts of generalised coordinates and generalised forces.
- Express the kinetic and potential energies of a system in term of the generalised coordinates, and to use these to obtain Lagrange's equations of motion.
- Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of a system about its equilibrium position.
- Explain the concept of generalized momentum and show how the Hamilton's equations can be used to find the equations of motion.
- Explain the concepts of Poisson brackets, conserved quantities, and canonical transformations.
Content
This module aims to present a systematic approach to the study of dynamics. Once the main techniques have been grasped, a very wide range of problems can be tackled with confidence. The first part of the course presents the tools required to analyse rigid-body motion in three dimensions. These are necessary for a proper understanding of gyroscopic systems, inertial navigation, satellites in space and the stability of high-speed rotating systems such as turbines and compressors.
The second part of the course deals with Lagrangian and Hamiltonian mechanics, a systematic way to formulate dynamical problems using energy functions.
Introduction and Rigid-body Dynamics (10L)
- Equations of motion of a rigid body in three dimensions.
- The inertia tensor; principal axes.
- Gyroscopes and their application.
- Problems involving rolling bodies.
Lagrangian Mechanics (6L)
- Lagrange's equations; connection to Newton's laws; generalised coordinates and generalised forces.
- Applications to a range of problems.
- Hamilton's equations, Poisson brackets, conserved quantities, canonical transformations.
- Example applications.
Coursework
Gyroscopic Phenomena
Learning objectives:
Practical information:
- Sessions will take place in [Location], during week(s) [xxx].
- This activity [involves/doesn't involve] preliminary work ([estimated duration]).
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 31/05/2024 07:28
Engineering Tripos Part IIA, 3C5: Dynamics, 2019-20
Module Leader
Lecturers
Prof R Langley and Prof J Woodhouse
Lab Leader
Timing and Structure
Michaelmas term, 16 lectures. Introduction and Rigid-body Dynamics: 2 lectures/week, weeks 1-5 (Prof J Woodhouse); Lagrangian Mechanics: 2 lectures/week, weeks 6-8 (Prof R S Langley)
Aims
The aims of the course are to:
- Introduce the ideas and methods of 3D dynamics: the motion of rigid bodies in three dimensions under given forces and moments.
- Introduce the Lagrange and Hamiltonian formulations of mechanics.
- To show how to apply these methods in a straightforward way to a wide range of problems.
Objectives
As specific objectives, by the end of the course students should be able to:
- Represent the inertia of a rigid body by an inertia matrix, be able to calculate the moments and products of inertia for simple shapes, be able to find the principal axes of inertia.
- Derive Euler's equations for the motion of a rigid body under prescribed moments.
- Apply these equations to the motion of symmetrical rotors, to explain the phenomena of precession, nutation and the rate gyroscope.
- Analyse simple problems involving the rolling of rigid bodies, for example a spinning penny on a table.
- Explain the concepts of generalised coordinates and generalised forces.
- Express the kinetic and potential energies of a system in term of the generalised coordinates, and to use these to obtain Lagrange's equations of motion.
- Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of a system about its equilibrium position.
- Explain the concept of generalized momentum and show how the Hamilton's equations can be used to find the equations of motion.
- Explain the concepts of Poisson brackets, conserved quantities, and canonical transformations.
Content
This module aims to present a systematic approach to the study of dynamics. Once the main techniques have been grasped, a very wide range of problems can be tackled with confidence. The first part of the course presents the tools required to analyse rigid-body motion in three dimensions. These are necessary for a proper understanding of gyroscopic systems, inertial navigation, satellites in space and the stability of high-speed rotating systems such as turbines and compressors.
The second part of the course deals with Lagrangian and Hamiltonian mechanics, a systematic way to formulate dynamical problems using energy functions.
Introduction and Rigid-body Dynamics (10L)
- Equations of motion of a rigid body in three dimensions.
- The inertia tensor; principal axes.
- Gyroscopes and their application.
- Problems involving rolling bodies.
Lagrangian Mechanics (6L)
- Lagrange's equations; connection to Newton's laws; generalised coordinates and generalised forces.
- Applications to a range of problems.
- Hamilton's equations, Poisson brackets, conserved quantities, canonical transformations.
- Example applications.
Coursework
Gyroscopic Phenomena
Learning objectives:
Practical information:
- Sessions will take place in [Location], during week(s) [xxx].
- This activity [involves/doesn't involve] preliminary work ([estimated duration]).
Full Technical Report:
Students will have the option to submit a Full Technical Report.
Booklists
Please see the Booklist for Part IIA Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 12/09/2019 12:18
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