Difference between revisions of "BSc: Theoretical Mechanics"

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= Theoretical Mechanics =
  
= Theoretical Mechanics =
  +
* '''Course name''': Theoretical Mechanics
  +
* '''Code discipline''':
  +
* '''Subject area''': Mechanics, mathematical modeling and calculating of mechanical systems.
   
  +
== Short Description ==
−
* <span>'''Course name:'''</span> Theoretical Mechanics
  
  +
This course covers the following concepts: Mechanics: Physical principles and methods for calculating kinematic, static and dynamic problems of mechanics.
−
* <span>'''Course number:'''</span>
  
−
* <span>'''Knowledge area:'''</span> Mechanics, mathematical modeling and calculating of mechanical systems.
  
   
  +
== Prerequisites ==
   
  +
=== Prerequisite subjects ===
−
== Course Characteristics ==
  
  +
* CSE203 — Mathematical Analysis II: Linear algebra, vectors and matrices, partial derivatives.
  +
* CSE205 — Differential Equations: ODE.
   
−
=== Key concepts of the class ===
 +
=== Prerequisite topics ===
   
−
* Mechanics: Physical principles and methods for calculating kinematic, static and dynamic problems of mechanics
  
   
  +
== Course Topics ==
−
=== What is the purpose of this course? ===
  
  +
{| class="wikitable"
−  
  +
|+ Course Sections and Topics
−
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.
  
  +
|-
  +
! Section !! Topics within the section
  +
|-
  +
| Kinematics ||
  +
# 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
  +
|-
  +
| Statics ||
  +
# 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
  +
|-
  +
| Dynamics ||
  +
# 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
  +
|-
  +
| Analytical mechanics ||
  +
# 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
  +
|}
  +
== Intended Learning Outcomes (ILOs) ==
   
  +
=== What is the main purpose of this course? ===
 
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.
  
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.
   
  +
=== ILOs defined at three levels ===
−
== Prerequisites ==
  
−  
−
* [https://eduwiki.innopolis.university/index.php/BSc:MathematicalAnalysisI Mathematical Analysis I]: Linear algebra, vectors and matrices, partial derivatives.
 
−  
−
* [https://eduwiki.innopolis.university/index.php/BSc:DifferentialEquations CSE205 — Differential Equations]: ODE.
  
−  
−
=== 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
  
   
  +
==== Level 1: What concepts should a student know/remember/explain? ====
  +
By the end of the course, the students should be able to ...
 
* Methods for describing the laws of motion of a particle and a solid,
  
* 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 calculating the speeds and accelerations of points and bodies included in a mechanical system,
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* Methods for creating differential equations of motion of a mechanical system based on methods of analytical mechanics.
  
* 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? ===
 +
==== Level 2: What basic practical skills should a student be able to perform? ====
  +
By the end of the course, the students should be able to ...
−  
−
By the end of the course, the students should be able to describe and explain
  
−  
 
* How to draw up and use calculation schemes,
  
* How to draw up and use calculation schemes,
 
* What calculation methods can be used to solve a specific problem,
  
* What calculation methods can be used to solve a specific problem,
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* What limitations and errors are imposed by a specific method when solving a 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? ===
 +
==== Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios? ====
  +
By the end of the course, the students should be able to ...
−  
−
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,
  
* 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,
  
* 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.
 +
* Create mathematical models, evaluate their value and the relativity of their limits of application.
  +
== Grading ==
−  
−
=== Course evaluation ===
  
   
  +
=== Course grading range ===
−
{|
  
  +
{| class="wikitable"
−
|+ Course grade breakdown
  
  +
|+
−
!
  
−
!
  
−
!align="center"| '''Proposed points'''
  
 
|-
  
|-
  +
! Grade !! Range !! Description of performance
−
| Labs/seminar classes
  
−
| 20
  
−
|align="center"| 10
  
 
|-
  
|-
  +
| A. Excellent || 90-100 || -
−
| Interim performance assessment
  
−
| 30
  
−
|align="center"| 50
  
 
|-
  
|-
  +
| B. Good || 75-89 || -
−
| Exams
  
−
| 50
 +
|-
  +
| C. Satisfactory || 60-74 || -
−
|align="center"| 40
  
  +
|-
  +
| D. Poor || 0-59 || -
 
|}
  
|}
   
  +
=== Course activities and grading breakdown ===
−
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.
  
  +
{| class="wikitable"
−  
  +
|+
−
=== Grades range ===
  
−  
−
{|
  
−
|+ Course grading range
  
−
!
  
−
!
  
−
!align="center"| '''Proposed range'''
  
 
|-
  
|-
  +
! Activity Type !! Percentage of the overall course grade
−
| A. Excellent
  
−
| 90-100
  
−
|align="center"|
  
 
|-
  
|-
  +
| Labs/seminar classes || 10
−
| B. Good
  
−
| 75-89
  
−
|align="center"|
  
 
|-
  
|-
  +
| Interim performance assessment || 50
−
| C. Satisfactory
  
−
| 60-74
  
−
|align="center"|
  
 
|-
  
|-
−
| D. Poor
 +
| Exams || 40
−
| 0-59
  
−
|align="center"|
  
 
|}
  
|}
   
  +
=== Recommendations for students on how to succeed in the course ===
−
=== Resources and reference material ===
  
  +
   
  +
== Resources, literature and reference materials ==
−
Main textbook:
  
   
  +
=== Open access resources ===
 
* S. Targ Theoretical Mechanics. A short course, 1968
  
* S. Targ Theoretical Mechanics. A short course, 1968
 
* D. Deleanu Theoretical mechanics. Theory and applications / Dumitru Deleanu – Constanta: Nautica, 2012
  
* 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
  
* 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
  
* Meshchersky I.V. Collection of Problems in Theoretical Mechanics 2014
 
* Prof . Dr. Ing. Vasile Szolga Theoretical Mechanics, 2010
  
* Prof . Dr. Ing. Vasile Szolga Theoretical Mechanics, 2010
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* Meshchersky I.V. Sbornik zadach po teoreticheskoi mechanike, 1986 - in Russian
  
* Meshchersky I.V. Sbornik zadach po teoreticheskoi mechanike, 1986 - in Russian
   
−
== Course Sections ==
 +
=== Closed access resources ===
   
−
The main sections of the course and approximate hour distribution between them is as follows:
  
   
  +
=== Software and tools used within the course ===
−
{|
  
  +
−
|+ Course Sections
  
  +
= Teaching Methodology: Methods, techniques, & activities =
−
!align="center"| '''Section'''
  
  +
−
! '''Section Title'''
  
−
!align="center"| '''Teaching Hours'''
 +
== Activities and Teaching Methods ==
  +
{| class="wikitable"
  +
|+ Activities within each section
 
|-
  
|-
  +
! Learning Activities !! Section 1 !! Section 2 !! Section 3 !! Section 4
−
|align="center"| 1
  
−
| Kinematics
  
−
|align="center"| 8
  
 
|-
  
|-
  +
| Homework and group projects || 1 || 1 || 1 || 1
−
|align="center"| 2
  
−
| Statics
  
−
|align="center"| 4
  
 
|-
  
|-
  +
| Midterm evaluation || 1 || 1 || 0 || 0
−
|align="center"| 3
  
−
| Dynamics
  
−
|align="center"| 12
  
 
|-
  
|-
  +
| Testing (written or computer based) || 1 || 1 || 1 || 1
−
|align="center"| 4
  
  +
|}
−
| Analytical mechanics
  
  +
== Formative Assessment and Course Activities ==
−
|align="center"| 4
  
−
|}
  
   
−
=== Section 1 ===
 +
=== Ongoing performance assessment ===
   
−
==== Section title: ====
 +
==== Section 1 ====
  +
{| class="wikitable"
−  
  +
|+
−
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? ===
  
−  
−
{|
  
−
!
  
−
!align="center"| '''Yes/No'''
  
 
|-
  
|-
  +
! Activity Type !! Content !! Is Graded?
−
| Development of individual parts of software product code
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Calculate of the kinematic parameters of the particle according to the given laws of motion, it is required to determine:<br>particle trajectory,<br>particle velocity,<br>particle acceleration and its normal and tangential components,<br>radius of curvature of the trajectory. || 1
−
| Homework and group projects
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Calculate of the kinematic parameters of the planar mechanism, determine:<br>velocity of specific points of the mechanism and angular velocity of the links of the mechanism using the method of instantaneous velocity centers,<br>velocity of specific points of the mechanism and angular velocity of the links of the mechanism using the analytical method,<br>acceleration of specific points of the mechanism and angular accelerations of the links of the mechanism. || 1
−
| Midterm evaluation
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Calculate of the kinematics of the complex motion of a point, determine:<br>transport, relative and absolute velocity of the point,<br>transport, relative, Coriolis and absolute acceleration of the point. || 1
−
| Testing (written or computer based)
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || 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:<br>gearbox with fixed axles,<br>planetary gearbox with parallel axes,<br>planetary gearbox with intersecting axes. || 1
−
| Reports
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Make a synthesis of the laws of motion of a point and a solid, taking into account given conditions and restrictions. || 0
−
| Essays
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Do a kinematic analysis of complex planar mechanisms with a large number of links. || 0
−
| Oral polls
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Do a kinematic analysis of complex planar mechanisms with several degrees of freedom. || 0
−
| Discussions
  
−
|align="center"| 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? ===
  
−  
−
{|
  
−
!
  
−
!align="center"| '''Yes/No'''
  
 
|-
  
|-
  +
| Question || Do a kinematic analysis of spatial mechanisms. || 0
−
| Development of individual parts of software product code
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Do a kinematic analysis of the complex motion of a solid body. || 0
−
| Homework and group projects
  
  +
|}
−
|align="center"| 1
  
  +
==== Section 2 ====
  +
{| class="wikitable"
  +
|+
 
|-
  
|-
  +
! Activity Type !! Content !! Is Graded?
−
| Midterm evaluation
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Derive equilibrium equations for a system of concurrent forces. || 1
−
| Testing (written or computer based)
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Derive equilibrium equations for a solid in 2D. || 1
−
| Reports
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Derive equilibrium equations for a system of two or more solids in 2D. || 1
−
| Essays
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Derive equilibrium equations for a solid in 3D. || 1
−
| Oral polls
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Apply equilibrium equations to calculate the reactions of supports and forces in the rods of a truss in 2D. || 0
−
| Discussions
  
−
|align="center"| 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? ===
  
−  
−
{|
  
−
!
  
−
!align="center"| '''Yes/No'''
  
 
|-
  
|-
  +
| Question || Apply equilibrium equations to calculate the reactions of supports of a solid body in 2D. || 0
−
| Development of individual parts of software product code
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Apply equilibrium equations for calculating the reactions of supports of a system of bodies in 2D. || 0
−
| Homework and group projects
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Apply equilibrium equations to calculate the reactions of supports of a solid body in 3D. || 0
−
| Midterm evaluation
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Investigate the equilibrium of a system of bodies taking into account friction. || 0
−
| Testing (written or computer based)
  
  +
|}
−
|align="center"| 1
  
  +
==== Section 3 ====
  +
{| class="wikitable"
  +
|+
 
|-
  
|-
  +
! Activity Type !! Content !! Is Graded?
−
| Reports
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Derive and solve the differential equations of rectilinear and curvilinear motion of a particle. || 1
−
| Essays
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Derive and solve differential equations based on the theorem on the motion of the center of mass of a system. || 1
−
| Oral polls
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Derive and solve differential equations based on the theorem on the change in the angular momentum of a system. || 1
−
| Discussions
  
  +
|-
−
|align="center"| 0
  
  +
| Question || Derive and solve the differential equations of rectilinear and curvilinear motion of bodies that form a system with one degree of freedom. || 1
−
|}
  
−  
−
=== 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? ===
  
−  
−
{|
  
−
!
  
−
!align="center"| '''Yes/No'''
  
 
|-
  
|-
  +
| Question || Derive and solve differential equations of motion based on the D’Alembert’s principle. || 1
−
| Development of individual parts of software product code
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Derive and solve the differential equation based on the theorem on the change in kinetic energy. || 1
−
| Homework and group projects
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || 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. || 0
−
| Midterm evaluation
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Apply the differential equations of a particle to study oscillations. || 0
−
| Testing (written or computer based)
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || 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. || 0
−
| Reports
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Apply the theorem on the change in the angular momentum of a system to study the gyroscopic effect. || 0
−
| Essays
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Apply the D’Alembert’s principle to determine the dynamic reactions of the supports of a mechanical system || 0
−
| Oral polls
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Apply the kinetic energy change theorem to determine the velosity of bodies of a mechanical system. || 0
−
| Discussions
  
  +
|}
−
|align="center"| 0
  
  +
==== Section 4 ====
−
|}
  
  +
{| class="wikitable"
−  
  +
|+
−
=== Typical questions for ongoing performance evaluation within this section ===
  
  +
|-
−  
  +
! Activity Type !! Content !! Is Graded?
−
# What are generalized coordinates?
  
  +
|-
−
# What are cyclic coordinates?
  
  +
| Question || What are generalized coordinates? || 1
−
# 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
  
  +
| Question || What are cyclic coordinates? || 1
−
# Derive the differential equations of a mechanical system based on Lagrange’s equations
  
  +
|-
−  
  +
| Question || Derive the differential equations of a mechanical system based on the principle of virtual work || 1
−
==== 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.
 +
| Question || Derive the differential equations of a mechanical system based on the General Equation of dynamics || 1
  +
|-
−
# 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.
 +
| Question || Derive the differential equations of a mechanical system based on Lagrange’s equations || 1
  +
|-
−
# Apply the Lagrange’s equations to study the oscillations of a mechanical system with two degrees of freedom.
  
  +
| Question || Apply the principle of virtual work to study the laws of motion of a mechanical system with one degree of freedom. || 0
−  
  +
|-
−
==== Test questions for final assessment in this section ====
  
  +
| Question || Apply the General Equation of dynamics to study the laws of motion of a mechanical system with one degree of freedom. || 0
−  
  +
|-
  +
| Question || Apply the Lagrange’s equations to study the laws of motion of a mechanical system with several degrees of freedom. || 0
  +
|-
  +
| Question || Apply the Lagrange’s equations to study the oscillations of a mechanical system with two degrees of freedom. || 0
  +
|}
  +
=== Final assessment ===
  +
'''Section 1'''
  +
# 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'''
  +
# 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'''
  +
# 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'''
 
# Formulate the the principle of virtual work. Show for what problems the calculation method based on this principle is effective.
  
# 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.
  
# Formulate the the D’Alembert-Lagrange’s principle. Show for problems tasks the calculation method based on this principle is effective.
Line 442: Line 285:
 
# Show the different forms of writing the Lagrange equations and explain in which cases each of these forms will be more convenient.
  
# 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.
  
# Demonstrate the principles of choosing the most convenient method for solving a given specific problem of mechanics.
  +
  +
=== The retake exam ===
  +
'''Section 1'''
  +
  +
'''Section 2'''
  +
  +
'''Section 3'''
  +
  +
'''Section 4'''

Latest revision as of 12:42, 12 July 2022

Theoretical Mechanics

  • Course name: Theoretical Mechanics
  • Code discipline:
  • Subject area: Mechanics, mathematical modeling and calculating of mechanical systems.

Short Description

This course covers the following concepts: Mechanics: Physical principles and methods for calculating kinematic, static and dynamic problems of mechanics.

Prerequisites

Prerequisite subjects

  • CSE203 — Mathematical Analysis II: Linear algebra, vectors and matrices, partial derivatives.
  • CSE205 — Differential Equations: ODE.

Prerequisite topics

Course Topics

Course Sections and Topics
Section Topics within the section
Kinematics
  1. Introduction to theoretical mechanics
  2. Kinematics of a particle
  3. Translatory and rotational motion of a rigid body
  4. Plane motion of a rigid body
  5. Spherical motion of a rigid body
  6. Motion of a free rigid body
  7. Resultant motion
Statics
  1. Basic concepts and principles of Statics
  2. Parallel forces and couples
  3. Equilibrium of a rigid body system in 2D
  4. Equilibrium of a rigid body system in 3D
  5. Friction
  6. Center of gravity
Dynamics
  1. Particle dynamics
  2. Theorem of the motion of the center of mass of a system
  3. Theorem of the change in the linear momentum of a system
  4. Theorem of the change in the angular momentum of a system
  5. Some cases of rigid body motion.
  6. D’Alambert’s principle
  7. Mechanical work and power
  8. Theorem of the change in the kinetic energy of a system
  9. The theory of impact
  10. Oscillations
Analytical mechanics
  1. Constraints and their classification
  2. Generalized coordinates
  3. Generalized forces
  4. The D’Alembert-Lagrange’s principle
  5. The principle of virtual work
  6. The General Equation of dynamics
  7. Lagrange’s equations
  8. The Hamilton’s equations

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

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.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

  • 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.

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

  • 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.

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

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.

Grading

Course grading range

Grade Range Description of performance
A. Excellent 90-100 -
B. Good 75-89 -
C. Satisfactory 60-74 -
D. Poor 0-59 -

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Labs/seminar classes 10
Interim performance assessment 50
Exams 40

Recommendations for students on how to succeed in the course

Resources, literature and reference materials

Open access resources

  • 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
  • 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

Closed access resources

Software and tools used within the course

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section
Learning Activities Section 1 Section 2 Section 3 Section 4
Homework and group projects 1 1 1 1
Midterm evaluation 1 1 0 0
Testing (written or computer based) 1 1 1 1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type Content Is Graded?
Question 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.		 1
Question 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.		 1
Question 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.		 1
Question 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.		 1
Question Make a synthesis of the laws of motion of a point and a solid, taking into account given conditions and restrictions. 0
Question Do a kinematic analysis of complex planar mechanisms with a large number of links. 0
Question Do a kinematic analysis of complex planar mechanisms with several degrees of freedom. 0
Question Do a kinematic analysis of spatial mechanisms. 0
Question Do a kinematic analysis of the complex motion of a solid body. 0

Section 2

Activity Type Content Is Graded?
Question Derive equilibrium equations for a system of concurrent forces. 1
Question Derive equilibrium equations for a solid in 2D. 1
Question Derive equilibrium equations for a system of two or more solids in 2D. 1
Question Derive equilibrium equations for a solid in 3D. 1
Question Apply equilibrium equations to calculate the reactions of supports and forces in the rods of a truss in 2D. 0
Question Apply equilibrium equations to calculate the reactions of supports of a solid body in 2D. 0
Question Apply equilibrium equations for calculating the reactions of supports of a system of bodies in 2D. 0
Question Apply equilibrium equations to calculate the reactions of supports of a solid body in 3D. 0
Question Investigate the equilibrium of a system of bodies taking into account friction. 0

Section 3

Activity Type Content Is Graded?
Question Derive and solve the differential equations of rectilinear and curvilinear motion of a particle. 1
Question Derive and solve differential equations based on the theorem on the motion of the center of mass of a system. 1
Question Derive and solve differential equations based on the theorem on the change in the angular momentum of a system. 1
Question Derive and solve the differential equations of rectilinear and curvilinear motion of bodies that form a system with one degree of freedom. 1
Question Derive and solve differential equations of motion based on the D’Alembert’s principle. 1
Question Derive and solve the differential equation based on the theorem on the change in kinetic energy. 1
Question 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. 0
Question Apply the differential equations of a particle to study oscillations. 0
Question 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. 0
Question Apply the theorem on the change in the angular momentum of a system to study the gyroscopic effect. 0
Question Apply the D’Alembert’s principle to determine the dynamic reactions of the supports of a mechanical system 0
Question Apply the kinetic energy change theorem to determine the velosity of bodies of a mechanical system. 0

Section 4

Activity Type Content Is Graded?
Question What are generalized coordinates? 1
Question What are cyclic coordinates? 1
Question Derive the differential equations of a mechanical system based on the principle of virtual work 1
Question Derive the differential equations of a mechanical system based on the General Equation of dynamics 1
Question Derive the differential equations of a mechanical system based on Lagrange’s equations 1
Question Apply the principle of virtual work to study the laws of motion of a mechanical system with one degree of freedom. 0
Question Apply the General Equation of dynamics to study the laws of motion of a mechanical system with one degree of freedom. 0
Question Apply the Lagrange’s equations to study the laws of motion of a mechanical system with several degrees of freedom. 0
Question Apply the Lagrange’s equations to study the oscillations of a mechanical system with two degrees of freedom. 0

Final assessment

Section 1

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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

  1. Explain the basic axioms of statics.
  2. Demonstrate the methods for determining the moment of force about a point and about an axis.
  3. Demonstrate the methods for transformation a couple of forces.
  4. Demonstrate the method for determining the principal vector and the principal moment of the force system.
  5. Describe the method for transformation a force system to the simplest possible form.

Section 3

  1. Formulate the theorem of the motion of the center of mass of a system. Show for what problems this theorem is effective.
  2. Formulate the D’Alambert’s principle. Show for what problems the calculation method based on this principle is effective.
  3. 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.
  4. Describe the processes that occur upon impact and methods for calculating the law of motion of the body upon impact.

Section 4

  1. Formulate the the principle of virtual work. Show for what problems the calculation method based on this principle is effective.
  2. Formulate the the D’Alembert-Lagrange’s principle. Show for problems tasks the calculation method based on this principle is effective.
  3. Demonstrate methods for calculating generalized forces.
  4. Show the different forms of writing the Lagrange equations and explain in which cases each of these forms will be more convenient.
  5. Demonstrate the principles of choosing the most convenient method for solving a given specific problem of mechanics.

The retake exam

Section 1

Section 2

Section 3

Section 4

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