2021 Vibration Analysis A

Font size  SML

Register update notification mail Add to favorite lecture list
Academic unit or major
Undergraduate major in Mechanical Engineering
Takahara Hiroki  Okuma Masaaki  Yamaura Hiroshi  Matsumura Shigeki  Nakano Yutaka 
Class Format
Media-enhanced courses
Day/Period(Room No.)
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
Access Index

Course description and aims

【Course description】
The course teaches vibration analysis of multi degree of freedom system, continuum such as a string, beam and membrane. It also teaches how to analyze nonlinear vibration systems.

The real vibration system is usually multi-degree-of freedom system, but it's necessary to choose an appropriate model to analyze vibration behavior for vibration reduction. The aim of this course is to learn about vibration behavior in a typical model of multi-degree-of freedom system and to understand analysis method and its characteristics of nonlinear vibration.

Student learning outcomes

By the end of this course, students will be able to:
1. Obtain the natural frequencies and the natural vibration modes of multi degree of freedom vibration systems and the fundamental continuous vibration systems (Ex. Strings, Beams).
2. By using modal analysis, obtain the time histories of free vibration and the frequency responses of continuous vibration systems.
3. Explain the characteristics of nonlinear vibrations.
4. By the analytical methods, obtain the backbone curves and frequency responses of nonlinear vibration systems.
5. Explain the characteristics of self-excited vibrations and parametric vibrations.


multi-degree-of-freedom system, distributed parameter system, non-linear vibration, parametric vibration, self excitation system

Competencies that will be developed

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

Class flow

At the beginning of each class, solutions to exercise problems that were assigned during the previous class are reviewed. Towards the end of class, students are given exercise problems related to the lecture given that day to solve. To prepare for class, students should read the course schedule section and check what topics will be covered. Required learning should be completed outside of the classroom for preparation and review purposes.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Overview of vibration problems of mechanical systems Understand how vibration occurs and how vibration becomes a problem.
Class 2 Kinetic equation of multi-degree-of-freedom system, natural frequencies and eigenmodes Build kinetic equation and calculate natural frequencies and their eigenmodes.
Class 3 Forced and base excitation response of multi-degree-of-freedom system Draw simple overview of forced and base excitation response and explain it.
Class 4 Distributed parameter systems: Vibration behavior, natural frequencies and eigenmodes of string Calculate natural frequencies and eigenmodes of a string.
Class 5 Distributed parameter systems: Mode expansion of string and axial vibration of rod Understand mode expansion with string and axial vibration of rod.
Class 6 Distributed parameter systems: Vibration of beam Understand vibration behavior of bending beam.
Class 7 Distributed parameter systems: Vibration of rectangular memblane Understand vibration behavior of rectangular memblane.
Class 8 Distributed parameter systems: Vibration of circlular memblane Understand vibration behavior of circular memblane.
Class 9 Non-linear vibration's characteristics Understand non-linear vibration behaviors.
Class 10 Understand non-linear vibration behaviors. Understand perturbation method.
Class 11 Non-linear vibration analysis: Average method Understand approximate solution with average method.
Class 12 Non-linear vibration analysis: Stability of steady solution and methodo of harmonic balance Understand method of harmonic balance and stability of non-linear vibration.
Class 13 Non-linear vibration analysis: Self excited vibration Understand self excited vibration.
Class 14 Non-linear vibration analysis: Parametric vibration, Review of the whole course. Understand parametric vibration. Review of the whole course.

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.


Not specified.

Reference books, course materials, etc.

Course materials are provided during class.

Assessment criteria and methods

Learning achievement is evaluated by exercise problems(30%) and a final exam(70%). However, if a face-to-face final examination cannot be held due to the spread of COVID-19 infection or other reasons, it will be evaluated by reports assigned for classes.

Related courses

  • MEC.D201 : Mechanical Vibrations

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

Students should have completed Mechanical Vibrations(MEC.D201) or have equivalent knowledge.

Page Top