2025 (Current Year) Faculty Courses School of Engineering Undergraduate major in Mechanical Engineering

Mechanical Vibrations

Academic unit or major: Undergraduate major in Mechanical Engineering
Instructor(s): Motoki Shino
Class Format: Lecture/Exercise (Face-to-face)
Media-enhanced courses: -
Day of week/Period (Classrooms): 5-8 Mon (WL1-301（レクチャーシアター）)
Class: -
Course Code: MEC.D201
Number of credits: 1.50.50
Course offered: 2025
Offered quarter: 3Q
Syllabus updated: Sep 17, 2025
Language: Japanese

Syllabus

Course overview and goals

The course teaches on the following contents as the basis of measures against vibration problems.
1. Equation of motion
2. Vibration characteristics (natural frequency, frequency response) of one-degree-of-freedom systems
3. Vibration analysis methods of two-degree-of-freedom systems

Course description and aims

By the end of this course, students will be able to:
1) Understand the natural frequency of one-degree-of-freedom vibration systems, frequency response, resonance, transmissibility (vibration isolation), etc., and understand and apply them to actual vibration problems．
2) Understand coupled natural frequencies and natural modes of two-degree-of-freedom vibration systems and explain the concept of modal analysis.
3) Understand principles of dynamic absorber and how to derive their optimum parameters with the fixed points theory.

Keywords

Free vibration and forced vibration for one-degree-of-freedom systems, Response characteristics of one-degree-of-freedom vibration systems subjected to harmonic excitation, Coupled natural frequencies and natural modes of two-degree-of-freedom systems, Dynamic absorber

Competencies

Specialist skills
Intercultural skills
Communication skills
Critical thinking skills
Practical and/or problem-solving skills
This subject corresponds to the acquisition of basic technical skills in the learning objectives of 1.

Class flow

At the beginning of class, overview and highlights of the previous class are reviewed. To allow students to get a good understanding of the course contents and practice application, exercise problems related to the contents of this course are provided in each of the two lectures.

Course schedule/Objectives

	Course schedule	Objectives
Class 1	Introduction to vibration phenomena	Give some examples of vibration problems.
Class 2	One-degree-of-freedom systems - Equation of motions and free vibrations.	Derive equation of motion, free vibration solution and natural angular frequency for undamped one-degree-of-freedom systems.
Class 3	Damped one-degree-of-freedom systems - Equation of motions and free vibrations.	Derive equation of motion and its solution for damped one-degree-of-freedom systems. Explain critical damping and damped natural angular frequency.
Class 4	Equations of motion and free vibration of a one-degree-of-freedom damped vibration system (exercises and explanations of the problems)	Exercises on the equations of motion of one-degree-of-freedom damped vibration systems. The exercises will provide a clue to the students' level of understanding.
Class 5	Response of one-degree-of-freedom systems to harmonic excitation force	Derive response of one-degree-of-freedom systems to harmonic excitation force. Explain resonance phenomena for one-degree-of-freedom systems.
Class 6	Response of one-degree-of-freedom systems to displacement excitation	Derive frequency response of one-degree-of-freedom systems to displacement excitation.
Class 7	Midterm Examination	Midterm Examination
Class 8	Derivation of frequency response function. Properties of frequency response curve and vibration isolation.	Derive transmissibility of one-degree-of-freedom systems subjected to harmonic excitation. Eplain principles of vibration isolation.
Class 9	Response of one-degree-of-freedom systems to arbitrary excitation force	Derive response of one-degree-of-freedom systems to arbitrary excitation force.
Class 10	Equations of motion and free vibration of two-degree-of-freedom systems	Derive coupled natural frequencies from equations of motion of two-degree-of freedom systems.
Class 11	Modal analysis: Natural modes of two-degree-of-freedom systems	Derive natural modes of two-degree-of-freedom systems.
Class 12	Time response and Frequency response of two-degree-of-freedom systems	Express equations of motion of two-degree-of-freedom system using modal coordinate.　Derive time responses of two-degree-of-freedom systems. Derive time responses and frequency response of two-degree-of-freedom systems.
Class 13	Derivation of the equations of motion and modal analysis for 2-DOF vibrating systems (exercises and explanations)	Exercises on the equations of motion of two-　degree-of-freedom damped vibration systems and Natural modes of two-degree-of-freedom systems.
Class 14	Principle of dynamic absorber and derivation of optimum parameters	Explain principles of dynamic absorber, and derive their optimum parameters with the fixed points theory.

Study advice (preparation and review)

To enhance the effectiveness of learning, students are encouraged to refer to the textbook and handouts, review the contents of the class, and complete exercises in approximately 90 minutes after each lecture.

Textbook(s)

For example;
Mechanical Vibrations, J. P. Den Hartog
Schaum's Outline of Mechanical Vibrations (Schaum's Outlines), S Kelly

Reference books, course materials, etc.

The Japan Society of Mechanical Engineers，『JSME Text Series (6)　Mechanical Vibration』，ISBN-13: 978-4888981286　(Japanese).
Course materials are provided during class.

Evaluation methods and criteria

・Report (every two lectures)
・Midterm Examination and Final Examination（It may be replaced with assignment, if attendance is restricted.）

Related courses

MEC.A201 ： Engineering Mechanics
MEC.B211 ： Ordinary Differential Equations
MEC.B212 ： Complex Function Theory
MEC.P211 ： Basic Experiments for Mechanical Engineering

Prerequisites

Not required.