### 2019　Fundamentals of Analytical Dynamics (Mechanical Engineering)

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Instructor(s)
Takahara Hiroki
Course component(s)
Lecture
Day/Period(Room No.)
Wed3-4(S011)
Group
-
Course number
MEC.D231
Credits
1
2019
Offered quarter
4Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
Japanese
Access Index ### Course description and aims

【Course description】
The course teaches the method for deriving the equations of motion using scalar quantities (the kinetic energy, the potential energy, virtual work).

【Aims】
The basic equations and equations of motion are required in various engneering field, and the ability to derive these equations based on general approach is vital. Students learn the method for deriving the equations of motion using Lagrangian equation of motion. This will allow them to understand mechanical phenomena deeply and they will be able to solve general problems in mechanics.

### Student learning outcomes

By the end of this course, students will be able to:
1) Explain the constraint in mechanical systems and generalized coordinates.
2) Explain the principle of virtual work and D'Alembert's principle.
3) Derive Lagrangian equations of motion using the principle of virtual work and D'Alembert's principle.
4) Derive the equations of motion using Lagrangian equations of motion.
5) Explain the relationship between variational principle and Euler-Lagrange equations.
6) Explain the relationship between Hamilton's principle and Lagrangian equations of motion.

### Keywords

Generalized coordinates，Principle of virtual work，D'Alembert's principle，Lagrangian equations of motion，Variational principle

### Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
- - -

### Class flow

At the beginning of each class, overview and highlights of the previous class are reviewed. Towards the end of class, students are given exercise problems related to the lecture given that day to solve.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Characteristics and advantages of analytical dynamics Review work and the law of the conservation of energy.
Class 2 Principle of virtual work and D'Alembert's principle Understand the principle of virtual work and D'Alembert's principle.
Class 3 Derivation of Lagrangian equations of motion Derive Lagrangian equations of motion using the principle of virtual work and D'Alembert's principle.
Class 4 Derivation of equations of motion using Lagrangian equations of motion Derive the equations of motion using Lagrangian equations of motion for multiple degrees of freedom system.
Class 5 Lagrangian equations of motion with constraints Derive the Lagrangian equations of motion with constraints using Lagrangian undetermined factors.
Class 6 Variational principle and Euler-Lagrange equations Understand the relationship between variational principle and Euler-Lagrange equations.
Class 7 Hamilton's principle and Lagrangian equations of motion Understand the relationship between Hamilton's principle and Lagrangian
Class 8 Summary

None required.

### Reference books, course materials, etc.

Dover Publications，『The variational principles of mechanics / by Cornelius Lanczos』，ISBN-13: 9780486650678，
Course materials are provided during class.

### Assessment criteria and methods

Learning achievement is evaluated by exercise problems and a final exam.

### Related courses

• MEC.A201 ： Engineering Mechanics
• MEC.B211 ： Ordinary Differential Equations
• MEC.D201 ： Mechanical Vibrations

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed Mechanical Vibrations (MEC.D201.R) or have equivalent knowledge. 