Engineering Dynamics

Vibration of Multi-Degree-of-Freedom Systems

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Session Overview

Covered this week: This week we introduce the concept of mode superposition. The vibration of N-DOF linear systems can be modeled as the superposition of the response of N Single DOF oscillators. The ripples on this pond are an example of superposed waves that will soon disperse. Image courtesy of John Martin on Flickr.

Assignments

Problems and Concept Questions

Each problem set has concept questions paired with most problems. Answers to the concept questions are handed in before the overall problem set is due. You should therefore answer the concept questions in each problem set while watching the video lectures for the week. Then, use the Concept Question Answer Key (below in the Check Yourself section) to check your work before continuing to work on the problem set.

Please note that if your answer to the concept question is incorrect, it is likely that you may start down an unproductive pathway to a solution. Try to fully understand the answer to the concept question before you begin the regular problem.

• Problem Set 10: Problems and Concept Questions (PDF)

Videos

Lecture Videos

• Watch Lecture 23: Vibration by Mode Superposition

  • Video Chapters
    • Review of the concept questions with Homework 10
    • Overview of the vibration material to be covered in the course
    • Linearization of a 2-DOF system
    • Free vibration of linear multi-DOF systems-intro to mode superposition
    • Finding natural frequencies and mode shapes of multi-dof systems
    • Mode superposition analysis of a 2-DOF system

• Watch Lecture 24: Modal Analysis: Orthogonality, Mass Stiffness, Damping Matrix

  • Video Chapters
    • The modal expansion theorem
    • Vibrating string demonstration
    • Demonstration of modes with the 2 DOF mechanical model
    • Orthogonality and the computation of the mass and stiffness matrices
    • Response to initial conditions using modal analysis
    • Obtaining uncoupled equations of motion and determination of modal initial conditions
    • Dealing with damping in modal analysis
    • Rayleigh damping and experimental fitting of damping ratios

Recitation Video and Notes

• Watch Recitation 11: Double Pendulum System
  • This recitation introduces modal analysis and looks at a double pendulum problem.
• Recitation 11 Notes: Double Pendulum System (PDF)
  • These recitation notes were compiled by Prof. David Gossard, another instructor for the course. His notes go over concepts and problems covered during the recitation sections he taught, and may not precisely correspond to the content covered in the above recitation video.

Check Yourself

Concept Question Answer Key and Problem Set Solutions

The solutions are presented in two files, one with the answers to the concept questions, and one with solutions and in-depth explanations for the problems. Work the problems on your own and check your answers when you’re done.

• Problem Set 10: Concept Question Answer Key (PDF)
• Problem Set 10: Problem Solutions and Explanations (PDF - 1.1MB)

Looking for something specific in this course? The Resource Index compiles links to most course resources in a single page.

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