BSc: Theoretical Mechanics.previous version
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Theoretical Mechanics
- Course name: Theoretical Mechanics
- Course number:
- Knowledge area: Mechanics, mathematical modeling and calculating of mechanical systems.
Course Characteristics
Key concepts of the class
- Mechanics: Physical principles and methods for calculating kinematic, static and dynamic problems of mechanics
What is the purpose of this course?
Control of robots as mechanical systems requires knowledge of the physical principles and methods of mathematical description of the laws of motion. Robotics specialists should be able to describe the trajectories of objects in different spaces, calculate static, dynamic and shock loads on the system’s bodies, investigate equilibrium conditions of systems, investigate vibrations, and be able to create mathematical models of mechanical systems.
The purpose of the course is to give basic and advanced knowledge on theoretical mechanics. The course covers kinematics of a particle and a rigid body, statics of rigid bodies, particle dynamics, dynamics of a system, analytical mechanics. The objective of the course is to give knowledge and skills which can be used further for calculating of kinematics, statics and dynamics of mechanical parts of robots and studying advanced courses on robotics.
Prerequisites
- CSE203 — Mathematical Analysis II: Linear algebra, vectors and matrices, partial derivatives.
Course Objectives Based on Bloom’s Taxonomy
What should a student remember at the end of the course?
By the end of the course, the students should be able to remember and recognize
- Methods for describing the laws of motion of a particle and a solid,
- Methods for calculating the speeds and accelerations of points and bodies included in a mechanical system,
- Methods for studying the equilibrium of mechanical systems,
- Methods for creating differential equations of motion of a particle and a solid,
- Methods for creating differential equations of motion of a mechanical system based on the classical approach,
- Methods for creating differential equations of motion of a mechanical system based on methods of analytical mechanics.
What should a student be able to understand at the end of the course?
By the end of the course, the students should be able to describe and explain
- How to draw up and use calculation schemes,
- What calculation methods can be used to solve a specific problem,
- What calculation methods are appropriate to use when solving a specific problem,
- What limitations and errors are imposed by a specific method when solving a problem.
What should a student be able to apply at the end of the course?
By the end of the course, the students should be able to
- Analyze and explain mechanical phenomena based on the laws and theorems of theoretical mechanics,
- Apply the basic laws and methods of theoretical mechanics to solving technical problems,
- Create mathematical models, evaluate their value and the relativity of their limits of application.
Course evaluation
| Proposed points | ||
|---|---|---|
| Labs/seminar classes | 20 | 10 |
| Interim performance assessment | 30 | 50 |
| Exams | 50 | 40 |
The course grades are given according to the following rules: Homework assignments (4) = 50 pts, Quizzes (5) = 10 pts, Midterm exam = 20 pts, Final exam 20 pts.
Grades range
| Proposed range | ||
|---|---|---|
| A. Excellent | 90-100 | |
| B. Good | 75-89 | |
| C. Satisfactory | 60-74 | |
| D. Poor | 0-59 |
Resources and reference material
Main textbook:
- S. Targ Theoretical Mechanics. A short course, 1968
- D. Deleanu Theoretical mechanics. Theory and applications / Dumitru Deleanu – Constanta: Nautica, 2012
- Stephen T. Thornton and Jerry B. Marion Classical Dynamics of Particles and Systems. 5th edition, 2004
Other reference material:
- Meshchersky I.V. Collection of Problems in Theoretical Mechanics 2014
- Prof . Dr. Ing. Vasile Szolga Theoretical Mechanics, 2010
- S.M. Targ Kratki kurs teoreticheskoi mechaniki, 1986 - in Russian
- A.I. Lurie Analiticheskaya mechanika, 1961 - in Russian
- Sbornik kursovych rabot po teoreticheskoi mechanike. A.A.Yablonski, 2000 - in Russian
- Meshchersky I.V. Sbornik zadach po teoreticheskoi mechanike, 1986 - in Russian
Course Sections
The main sections of the course and approximate hour distribution between them is as follows:
| Section | Section Title | Teaching Hours |
|---|---|---|
| 1 | Kinematics | 8 |
| 2 | Statics | 4 |
| 3 | Dynamics | 12 |
| 4 | Analytical mechanics | 4 |
Section 1
Section title:
Kinematics
Topics covered in this section:
- Introduction to theoretical mechanics
- Kinematics of a particle
- Translatory and rotational motion of a rigid body
- Plane motion of a rigid body
- Spherical motion of a rigid body
- Motion of a free rigid body
- Resultant motion
What forms of evaluation were used to test students’ performance in this section?
| Yes/No | |
|---|---|
| Development of individual parts of software product code | 0 |
| Homework and group projects | 1 |
| Midterm evaluation | 1 |
| Testing (written or computer based) | 1 |
| Reports | 0 |
| Essays | 0 |
| Oral polls | 0 |
| Discussions | 0 |
Typical questions for ongoing performance evaluation within this section
- Calculate of the kinematic parameters of the particle according to the given laws of motion, it is required to determine:
- particle trajectory,
- particle velocity,
- particle acceleration and its normal and tangential components,
- radius of curvature of the trajectory.
- Calculate of the kinematic parameters of the planar mechanism, determine:
- velocity of specific points of the mechanism and angular velocity of the links of the mechanism using the method of instantaneous velocity centers,
- velocity of specific points of the mechanism and angular velocity of the links of the mechanism using the analytical method,
- acceleration of specific points of the mechanism and angular accelerations of the links of the mechanism.
- Calculate of the kinematics of the complex motion of a point, determine:
- transport, relative and absolute velocity of the point,
- transport, relative, Coriolis and absolute acceleration of the point.
- Calculate of the kinematics of gears, determine the gear ratio, angular velocities and angular accelerations of links, velocities and accelerations of specific points of links for:
- gearbox with fixed axles,
- planetary gearbox with parallel axes,
- planetary gearbox with intersecting axes.
Typical questions for seminar classes (labs) within this section
- Make a synthesis of the laws of motion of a point and a solid, taking into account given conditions and restrictions.
- Do a kinematic analysis of complex planar mechanisms with a large number of links.
- Do a kinematic analysis of complex planar mechanisms with several degrees of freedom.
- Do a kinematic analysis of spatial mechanisms.
- Do a kinematic analysis of the complex motion of a solid body.
Test questions for final assessment in this section
- Describe the vector, coordinate, and natural methods of specifying particle motion. Show the transition from one method to another. Find the velocity and acceleration of the particle in various methods.
- Define the angular velocity vector and the angular acceleration vector of the body. Prove the independence of these vectors from the choice of the pole. Use the Euler vector formula to find the velocities and accelerations of points of a rotating rigid body and a rigid body making plane motion.
- Describe ways to set the orientation of a solid in space, including Euler angles, Tight-Brian angles, quaternions. Show the methodology for determining the angular velocity vector in these cases.
- Show the methodology of kinematic analysis of planar mechanisms, including the method of composing the equations of motion for the points of the mechanism, the theorems on the velocities and accelerations of body points in plane motion, and the instantaneous center of velocity method.
- Show the methodology of kinematic analysis of the complex motion of a particle, the theorems on the addition of velocities and accelerations for complex motion of a particle.
Section 2
Section title:
Statics
Topics covered in this section:
- Basic concepts and principles of Statics
- Parallel forces and couples
- Equilibrium of a rigid body system in 2D
- Equilibrium of a rigid body system in 3D
- Friction
- Center of gravity
What forms of evaluation were used to test students’ performance in this section?
| Yes/No | |
|---|---|
| Development of individual parts of software product code | 0 |
| Homework and group projects | 1 |
| Midterm evaluation | 1 |
| Testing (written or computer based) | 1 |
| Reports | 0 |
| Essays | 0 |
| Oral polls | 0 |
| Discussions | 0 |
Typical questions for ongoing performance evaluation within this section
- Derive equilibrium equations for a system of concurrent forces.
- Derive equilibrium equations for a solid in 2D.
- Derive equilibrium equations for a system of two or more solids in 2D.
- Derive equilibrium equations for a solid in 3D.
Typical questions for seminar classes (labs) within this section
- Apply equilibrium equations to calculate the reactions of supports and forces in the rods of a truss in 2D.
- Apply equilibrium equations to calculate the reactions of supports of a solid body in 2D.
- Apply equilibrium equations for calculating the reactions of supports of a system of bodies in 2D.
- Apply equilibrium equations to calculate the reactions of supports of a solid body in 3D.
- Investigate the equilibrium of a system of bodies taking into account friction.
Test questions for final assessment in this section
- Explain the basic axioms of statics.
- Demonstrate the methods for determining the moment of force about a point and about an axis.
- Demonstrate the methods for transformation a couple of forces.
- Demonstrate the method for determining the principal vector and the principal moment of the force system.
- Describe the method for transformation a force system to the simplest possible form.
Section 3
Section title:
Dynamics
Topics covered in this section:
- Particle dynamics
- Theorem of the motion of the center of mass of a system
- Theorem of the change in the linear momentum of a system
- Theorem of the change in the angular momentum of a system
- Some cases of rigid body motion.
- D’Alambert’s principle
- Mechanical work and power
- Theorem of the change in the kinetic energy of a system
- The theory of impact
- Oscillations
What forms of evaluation were used to test students’ performance in this section?
| Yes/No | |
|---|---|
| Development of individual parts of software product code | 0 |
| Homework and group projects | 1 |
| Midterm evaluation | 0 |
| Testing (written or computer based) | 1 |
| Reports | 0 |
| Essays | 0 |
| Oral polls | 0 |
| Discussions | 0 |
Typical questions for ongoing performance evaluation within this section
- Derive and solve the differential equations of rectilinear and curvilinear motion of a particle.
- Derive and solve differential equations based on the theorem on the motion of the center of mass of a system.
- Derive and solve differential equations based on the theorem on the change in the angular momentum of a system.
- Derive and solve the differential equations of rectilinear and curvilinear motion of bodies that form a system with one degree of freedom.
- Derive and solve differential equations of motion based on the D’Alembert’s principle.
- Derive and solve the differential equation based on the theorem on the change in kinetic energy.
Typical questions for seminar classes (labs) within this section
- Apply the differential equations of a particle to study the motion of a body in a field of gravity under the influence of air resistance.
- Apply the differential equations of a particle to study oscillations.
- Apply the theorem on the motion of the center of mass of a system to determine the dynamic reactions of the support of the mechanism.
- Apply the theorem on the change in the angular momentum of a system to study the gyroscopic effect.
- Apply the D’Alembert’s principle to determine the dynamic reactions of the supports of a mechanical system
- Apply the kinetic energy change theorem to determine the velosity of bodies of a mechanical system.
Test questions for final assessment in this section
- Formulate the theorem of the motion of the center of mass of a system. Show for what problems this theorem is effective.
- Formulate the D’Alambert’s principle. Show for what problems the calculation method based on this principle is effective.
- Describe the concept of force field. Show the method for determining the work of a force at a movement of a particle in a potential force field.
- Describe the processes that occur upon impact and methods for calculating the law of motion of the body upon impact.
Section 4
Section title:
Analytical mechanics
Topics covered in this section:
- Constraints and their classification
- Generalized coordinates
- Generalized forces
- The D’Alembert-Lagrange’s principle
- The principle of virtual work
- The General Equation of dynamics
- Lagrange’s equations
- The Hamilton’s equations
What forms of evaluation were used to test students’ performance in this section?
| Yes/No | |
|---|---|
| Development of individual parts of software product code | 0 |
| Homework and group projects | 1 |
| Midterm evaluation | 0 |
| Testing (written or computer based) | 1 |
| Reports | 0 |
| Essays | 0 |
| Oral polls | 0 |
| Discussions | 0 |
Typical questions for ongoing performance evaluation within this section
- What are generalized coordinates?
- What are cyclic coordinates?
- Derive the differential equations of a mechanical system based on the principle of virtual work
- Derive the differential equations of a mechanical system based on the General Equation of dynamics
- Derive the differential equations of a mechanical system based on Lagrange’s equations
Typical questions for seminar classes (labs) within this section
- Apply the principle of virtual work to study the laws of motion of a mechanical system with one degree of freedom.
- Apply the General Equation of dynamics to study the laws of motion of a mechanical system with one degree of freedom.
- Apply the Lagrange’s equations to study the laws of motion of a mechanical system with several degrees of freedom.
- Apply the Lagrange’s equations to study the oscillations of a mechanical system with two degrees of freedom.
Test questions for final assessment in this section
- Formulate the the principle of virtual work. Show for what problems the calculation method based on this principle is effective.
- Formulate the the D’Alembert-Lagrange’s principle. Show for problems tasks the calculation method based on this principle is effective.
- Demonstrate methods for calculating generalized forces.
- Show the different forms of writing the Lagrange equations and explain in which cases each of these forms will be more convenient.
- Demonstrate the principles of choosing the most convenient method for solving a given specific problem of mechanics.