Session Overview
Covered this week: 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.
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
 Video Chapters

Watch Lecture 26: Response of 2DOF 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 2DOF 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
 Video Chapters

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: Flowinduced vibration of long strings and beams in fluid flow
 Derivation of the “wave equation”
 Natural frequencies and mode shapes of the string under tensionwith 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
 Video Chapters
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 indepth 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.