Engineering Dynamics

Vibration Analysis by Mode Superposition

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Session Overview

Covered this week: This week we apply modal superposition to the analysis of the response of multi-dof systems to initial conditions and to excitation by steady state harmonic forces. Piano strings are each chosen and tuned such that they have a specific natural frequency. Image courtesy of Maruku San 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 11: Problems and Concept Questions (PDF)

Videos

Lecture Videos

• Watch Lecture 25: Modal Analysis: Response to IC’s and to Harmonic Forces

  • Video Chapters
    • Outline of the final three lectures
    • Response to initial conditions and the use of Rayleigh damping to model modal damping ratios
    • Steady state response to harmonic excitation by the method of modal analysis
    • Demonstration of the response of a string to harmonic excitation
    • Demonstration of the response of a string to initial deflection
    • Direct method for assembling the stiffness of an N DOF system

• Watch Lecture 26: Response of 2-DOF Systems by the Use of Transfer Functions

  • Video Chapters
    • Discussion of problem set concept questions
    • Review of the use of a transfer function to analyze the response of a single DOF system to harmonic input
    • Analysis of the response of a 2-DOF system to harmonic excitation by means of transfer functions and the introduction of the application of the dynamic absorber to mitigate problem vibration
    • Demonstration of a dynamic absorber using a strobe and a vibrating beam
    • Questions and answers: discussion of helicopter video and the 300 ton dynamic absorber in the Hancock building

• Watch Lecture 27: Vibration of Continuous Structures: Strings, Beams, Rods, etc.

  • Video Chapters
    • Demonstration of wave propagation on a long string
    • My personal research area: Flow-induced vibration of long strings and beams in fluid flow
    • Derivation of the “wave equation”
    • Natural frequencies and mode shapes of the string under tension-with demonstration
    • Application of the wave equation to rods, organ pipes, shower stalls with demonstrations
    • The vibration of beams: natural frequencies as a function of boundary conditions, dispersion in wave propagation

Recitation Video and Notes

• Watch Recitation 12: Modal Analysis of a Double Pendulum System
  • This recitation takes an in depth look at modal analysis for a double pendulum system.
• Recitation 12 Notes: Modal Analysis (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 11: Concept Question Answer Key (PDF)
• Problem Set 11: Problem Solutions and Explanations (PDF)

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

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