Engineering Tripos Part IB, 2P6: Linear Systems and Control, 2017-18
Lecturer
Timing and Structure
Weeks 1-4 and 7-8, 2 lectures/week. Weeks 5-6, 1 lecture/week. 14 lectures.
Aims
The aims of the course are to:
- Introduce and motivate the use of feedback control systems.
- Introduce analysis techniques for linear systems which are used in control, signal processing, communications, and other branches of engineering.
- Introduce the specification, analysis and design of feedback control systems.
- Extend the ideas and techniques learnt in the IA Mechanical Vibrations course.
Objectives
As specific objectives, by the end of the course students should be able to:
- Develop and interpret block diagrams and transfer functions for simple systems.
- Relate the time response of a system to its transfer function and/or its poles.
- Understand the term 'stability', its definition, and its relation to the poles of a system.
- Understand the term 'frequency response' (or 'harmonic response'), and its relation to the transfer function of a system.
- Interpret Bode and Nyquist diagrams, and to sketch them for simple systems.
- Understand the purpose of, and operation of, feedback systems.
- Understand the purpose of proportional, integral, and derivative controller elements, and of velocity feedback.
- Possess a basic knowledge of how controller elements may be implemented using operational amplifiers, software, or mechanical devices.
- Apply Nyquist's stability theorem, to predict closed-loop stability from open-loop Nyquist or Bode diagrams.
- Assess the quality of a given feedback system, as regards stability margins and attenuation of uncertainty, using open-loop Bode and Nyquist diagrams.
Content
|
|
Section numbers in books |
||
|
|
(1) |
(2) |
(3) |
|
Examples of feedback control systems. Use of block diagrams. Differential equation models. Meaning of 'Linear System'. |
1.1-1.11, 2.2-2.3 |
1.1-1.3, 2.1-2.6.1 |
1.1-1.8, 3.1-3.5, 3.18 |
|
Review of Laplace transforms. Transfer functions. Poles (characteristic roots) and zeros. Impulse and step responses. Convolution integral. Block diagrams of complex systems. |
2.4-2.6 |
3.1-3.2 |
3.8-3.14, 4.1-4.8, 6.1-6.2, 7.1-7.8 |
|
Definition of stability. Pole locations and stability. Pole locations and transient characteristics. |
5.6, 6.1 |
3.3-3.4, 4.4.1 |
5.1-5.2, 6.4 |
|
Frequency response (harmonic response). Nyquist (polar) and Bode diagrams. |
8.1-8.3 |
6.1 |
6.5, 11.2, 11.5, 15.1-15.5 |
|
Terminology of feedback systems. Use of feedback to reduce sensitivity. Disturbances and steady-state errors in feedback systems. Final value theorem. |
4.1-4.2, 4.4-4.5 |
4.1 |
9.2, 9.5 |
|
Proportional, integral, and derivative control. Velocity (rate) feedback. Implementation of controllers in various technologies. |
10.6, 12.6 |
4.2 |
|
|
Nyquist's stability theorem. Predicting closed-loop stability from open-loop Nyquist and Bode plots. |
9.1-9.3 |
6.3 |
11.10 |
|
Performance of feedback systems: Stability margins, speed of response, sensitivity reduction. |
6.3,8.5, 9.4, 9.6, 12.5, 12.8-12.9 |
6.4, 6.6, 6.9 |
10.4, 11.11, 13.2, 15.6-15.7 |
REFERENCES
(1) DISTEFANO, J.J., STUBBERUD, A.R. & WILLIAMS, I.J. FEEDBACK AND CONTROL SYSTEMS
(2) FRANKLIN, G.F., POWELL; J.D. & EMAMI-NAEINI, A. FEEDBACK CONTROL OF DYNAMIC SYSTEMS
(3) OPPENHEIM, A.V., WILLSKY, A.S. & NAWAB, S.H. SIGNALS AND SYSTEMS
(4) ÅSTRÖM, K.J. & MURRAY, R.M. FEEDBACK SYSTEMS: AN INTRODUCTION FOR SCIENTISTS AND ENGINEERS
(5) DORF, R.C. & BISHOP, R.H. MODERN CONTROL SYSTEMS
Booklists
Please see the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 28/09/2017 10:45
Engineering Tripos Part IB, 2P1: Mechanics, 2021-22
Courese Leader
Lecturer
Lecturer
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
- Show how to model complex mechanics problems with constraints and multiple degrees of freedom.
- Develop skills for analyzing these complex mechanical systems, including stability, vibrations and numerical integration.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body using > graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Apply Lagrange's equation to the motion of particles and rigid bodies under the action of conservative forces
- Identification of equilibrium points, and linearization around equilibrium points
- Linearization around equilibrium points to extract stability information, vibrational frequencies and growth rates.
- Use of the "Effective potential'' when J_z is conserved.
- Understand chaotic motion as observed in simple non-linear dynamics systems
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
Lagrange's Equation
- Introduction to Lagrange's Equation (without derivation)
- Concept of conservative forces
- Application to the motion of particles and rigid bodies under the action of conservative forces
Non-linear dynamics
- Solution of equations of motion for a double pendulum
- Illustration of motion on a phase plane
- Concept of chaos and the sensitivity to initial conditions
Gyroscopic Effect
- Introduction to gyroscopic motion (2: pp 564-571)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please refer to the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 20/05/2021 07:25
Engineering Tripos Part IB, 2P1: Mechanics, 2019-20
Lecturer
Lecturer
Leader
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
- Show how to model complex mechanics problems with constraints and multiple degrees of freedom.
- Develop skills for analyzing these complex mechanical systems, including stability, vibrations and numerical integration.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body using > graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Apply Lagrange's equation to the motion of particles and rigid bodies under the action of conservative forces
- Identification of equilibrium points, and linearization around equilibrium points
- Linearization around equilibrium points to extract stability information, vibrational frequencies and growth rates.
- Use of the "Effective potential'' when J_z is conserved.
- Understand chaotic motion as observed in simple non-linear dynamics systems
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
Lagrange's Equation
- Introduction to Lagrange's Equation (without derivation)
- Concept of conservative forces
- Application to the motion of particles and rigid bodies under the action of conservative forces
Non-linear dynamics
- Solution of equations of motion for a double pendulum
- Illustration of motion on a phase plane
- Concept of chaos and the sensitivity to initial conditions
Gyroscopic Effect
- Introduction to gyroscopic motion (2: pp 564-571)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please see the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 16/05/2019 10:19
Engineering Tripos Part IB, 2P1: Mechanics, 2023-24
Course Leader
Lecturer
Lecturer
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
- Show how to model complex mechanics problems with constraints and multiple degrees of freedom.
- Develop skills for analyzing these complex mechanical systems, including stability, vibrations and numerical integration.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body using > graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Apply Lagrange's equation to the motion of particles and rigid bodies under the action of conservative forces
- Identification of equilibrium points, and linearization around equilibrium points
- Linearization around equilibrium points to extract stability information, vibrational frequencies and growth rates.
- Use of the "Effective potential'' when J_z is conserved.
- Understand chaotic motion as observed in simple non-linear dynamics systems
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
Lagrange's Equation
- Introduction to Lagrange's Equation (without derivation)
- Concept of conservative forces
- Application to the motion of particles and rigid bodies under the action of conservative forces
Non-linear dynamics
- Solution of equations of motion for a double pendulum
- Illustration of motion on a phase plane
- Concept of chaos and the sensitivity to initial conditions
Gyroscopic Effect
- Introduction to gyroscopic motion (2: pp 564-571)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please refer to the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 15/09/2023 14:20
Engineering Tripos Part IB, 2P1: Mechanics, 2022-23
Course Leader
Lecturer
Lecturer
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
- Show how to model complex mechanics problems with constraints and multiple degrees of freedom.
- Develop skills for analyzing these complex mechanical systems, including stability, vibrations and numerical integration.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body using > graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Apply Lagrange's equation to the motion of particles and rigid bodies under the action of conservative forces
- Identification of equilibrium points, and linearization around equilibrium points
- Linearization around equilibrium points to extract stability information, vibrational frequencies and growth rates.
- Use of the "Effective potential'' when J_z is conserved.
- Understand chaotic motion as observed in simple non-linear dynamics systems
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
Lagrange's Equation
- Introduction to Lagrange's Equation (without derivation)
- Concept of conservative forces
- Application to the motion of particles and rigid bodies under the action of conservative forces
Non-linear dynamics
- Solution of equations of motion for a double pendulum
- Illustration of motion on a phase plane
- Concept of chaos and the sensitivity to initial conditions
Gyroscopic Effect
- Introduction to gyroscopic motion (2: pp 564-571)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please refer to the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 22/11/2022 15:00
Engineering Tripos Part IB, 2P1: Mechanics, 2018-19
Lecturer
Leader
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
- Show how to model complex mechanics problems with constraints and multiple degrees of freedom.
- Develop skills for analyzing these complex mechanical systems, including stability, vibrations and numerical integration.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body using > graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Apply Lagrange's equation to the motion of particles and rigid bodies under the action of conservative forces
- Identification of equilibrium points, and linearization around equilibrium points
- Linearization around equilibrium points to extract stability information, vibrational frequencies and growth rates.
- Use of the "Effective potential'' when J_z is conserved.
- Understand chaotic motion as observed in simple non-linear dynamics systems
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
Lagrange's Equation
- Introduction to Lagrange's Equation (without derivation)
- Concept of conservative forces
- Application to the motion of particles and rigid bodies under the action of conservative forces
Non-linear dynamics
- Solution of equations of motion for a double pendulum
- Illustration of motion on a phase plane
- Concept of chaos and the sensitivity to initial conditions
Gyroscopic Effect
- Introduction to gyroscopic motion (2: pp 564-571)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please see the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 12/11/2018 21:13
Engineering Tripos Part IB, 2P1: Mechanics, 2017-18
Lecturer
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body in cartesian, polar and intrinsic co-ordinates, using graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Determine the centre of mass and moment of inertia of a plane lamina.
- Understand and apply the perpendicular and parallel axes theorems.
- Recognise whether a body is in static or dynamic equilibrium.
- Understand the concepts of energy, linear momentum and moment of momentum of a rigid body, and recognise when they are conserved.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Understand the concepts of static and dynamic balance of rotors and the methods for balancing rotors.
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a particle Data book p2
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics I - Inertia Forces and Energy
- Centre of mass, moments of inertia Data book Section 4
- D'Alembert force for a particle (3: p 101)
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Kinetic energy of a rigid body in plane motion (2: p 461)
- Conservation of energy for conservative systems (3: pp 453-458)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
- Balancing simple rotors (1: pp 180-182)
Rigid Body Dynamics II - Conservation of Momentum
- Momentum of a rigid body in plane motion (2: pp 267-271)
- Moment of momentum about G in plane motion (3: pp 555-558)
- Moment of momentum about a fixed point (4: p 894)
- Impact problems in plane motion (3: pp 487-493)
- Introduction to gyroscopic motion (2: pp 564-571)
- Lamina rotating about an axis in its own plane (1: pp 185-187)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please see the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 31/05/2017 10:02
Engineering Tripos Part IB, 2P1: Mechanics, 2020-21
Courese Leader
Lecturer
Lecturer
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
- Show how to model complex mechanics problems with constraints and multiple degrees of freedom.
- Develop skills for analyzing these complex mechanical systems, including stability, vibrations and numerical integration.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body using > graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Apply Lagrange's equation to the motion of particles and rigid bodies under the action of conservative forces
- Identification of equilibrium points, and linearization around equilibrium points
- Linearization around equilibrium points to extract stability information, vibrational frequencies and growth rates.
- Use of the "Effective potential'' when J_z is conserved.
- Understand chaotic motion as observed in simple non-linear dynamics systems
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
Lagrange's Equation
- Introduction to Lagrange's Equation (without derivation)
- Concept of conservative forces
- Application to the motion of particles and rigid bodies under the action of conservative forces
Non-linear dynamics
- Solution of equations of motion for a double pendulum
- Illustration of motion on a phase plane
- Concept of chaos and the sensitivity to initial conditions
Gyroscopic Effect
- Introduction to gyroscopic motion (2: pp 564-571)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please refer to the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 26/08/2020 09:22
Engineering Tripos Part IB, 2P1: Mechanics, 2024-25
Course Leader
Lecturer
Lecturer
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
- Show how to model complex mechanics problems with constraints and multiple degrees of freedom.
- Develop skills for analyzing these complex mechanical systems, including stability, vibrations and numerical integration.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body using > graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Apply Lagrange's equation to the motion of particles and rigid bodies under the action of conservative forces
- Identification of equilibrium points, and linearization around equilibrium points
- Linearization around equilibrium points to extract stability information, vibrational frequencies and growth rates.
- Use of the "Effective potential'' when J_z is conserved.
- Understand chaotic motion as observed in simple non-linear dynamics systems
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
Lagrange's Equation
- Introduction to Lagrange's Equation (without derivation)
- Concept of conservative forces
- Application to the motion of particles and rigid bodies under the action of conservative forces
Non-linear dynamics
- Solution of equations of motion for a double pendulum
- Illustration of motion on a phase plane
- Concept of chaos and the sensitivity to initial conditions
Gyroscopic Effect
- Introduction to gyroscopic motion (2: pp 564-571)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please refer to the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 30/07/2024 08:48
Engineering Tripos Part IB, 2P1: Mechanics, 2025-26
Course Leader
Lecturer
Lecturer
Timing and Structure
16 Lectures, 2 lectures/week
Aims
The aims of the course are to:
- Show how the concepts of kinematics are applied to rigid bodies.
- Explain how Newton's laws of motion and the equations of energy and momentum are applied to rigid bodies.
- Develop an appreciation of the function, design and schematic representation of mechanical systems.
- Develop skills in modelling and analysis of mechanical systems, including graphical, algebraic and vector methods.
- Show how to model complex mechanics problems with constraints and multiple degrees of freedom.
- Develop skills for analyzing these complex mechanical systems, including stability, vibrations and numerical integration.
Objectives
As specific objectives, by the end of the course students should be able to:
- Specify the position, velocity and acceleration of a rigid body using > graphical, algebraic and vector methods.
- Understand the concepts of relative velocity, relative acceleration and instantaneous centres of rigid bodies.
- Apply Newton's laws and d'Alembert's principle to determine the acceleration of a rigid body subject to applied forces and couples, including impact in planar motion.
- Determine the forces and stresses in a rigid body caused by its motion.
- Apply Lagrange's equation to the motion of particles and rigid bodies under the action of conservative forces
- Identification of equilibrium points, and linearization around equilibrium points
- Linearization around equilibrium points to extract stability information, vibrational frequencies and growth rates.
- Use of the "Effective potential'' when J_z is conserved.
- Understand chaotic motion as observed in simple non-linear dynamics systems
- Understand simple gyroscopic motion.
Content
Introduction and Terminology
Kinematics
- Differentiation of vectors (4: pp 490-492)
- Motion of a rigid body in space (3: ch 20)
- Velocity and acceleration images (1: p 124)
- Acceleration of a particle moving relative to a body in motion (2: pp 386-389)
Rigid Body Dynamics
- D'Alembert force and torque for a rigid body in plane motion (4: pp 787-788)
- Inertia forces in plane mechanisms (1: pp 200-206)
- Method of virtual power (4: pp 429-432)
- Inertia stress and bending (1) Ch 5
Lagrange's Equation
- Introduction to Lagrange's Equation (without derivation)
- Concept of conservative forces
- Application to the motion of particles and rigid bodies under the action of conservative forces
Non-linear dynamics
- Solution of equations of motion for a double pendulum
- Illustration of motion on a phase plane
- Concept of chaos and the sensitivity to initial conditions
Gyroscopic Effect
- Introduction to gyroscopic motion (2: pp 564-571)
REFERENCES
(1) BEER, F.P. & JOHNSTON, E.R. VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS
(2) HIBBELER, R.C. ENGINEERING MECHANICS – DYNAMICS (SI UNITS)
(3) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(4) PRENTIS, J.M. ENGINEERING MECHANICS
Booklists
Please refer to the Booklist for Part IB 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.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
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.
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.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 05/06/2025 11:16
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