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Advanced Dynamics Homework

This document contains an optional homework assignment from the Fall 2007 Dynamics and Control I course at MIT. It involves deriving equations of motion for the lateral vibration of a 100-story building modeled as a 1D spring-mass system. Students are asked to generate MATLAB code to calculate the vibration modes of the building and plot the eigenvectors of selected modes.

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Vaibhav Gupta
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56 views2 pages

Advanced Dynamics Homework

This document contains an optional homework assignment from the Fall 2007 Dynamics and Control I course at MIT. It involves deriving equations of motion for the lateral vibration of a 100-story building modeled as a 1D spring-mass system. Students are asked to generate MATLAB code to calculate the vibration modes of the building and plot the eigenvectors of selected modes.

Uploaded by

Vaibhav Gupta
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Massachusetts Institute of Technology

Department of Mechanical Engineering

2.003J/1.053J Dynamics & Control I

Fall 2007
Homework 9 (OPTIONAL)

Issued: Dec.5. 2007 Due: Dec.12. 2007

Problem 9.1 : Equation of the lateral vibration of a 100 story building.


Consider the lateral vibration of a 100 story building. The mass are assumed to be concentrated
at the floors. The walls constitutes springs. This can be models as a 1-D system. (See figure)

100th floor  M100 X100

K100 K100
99th floor  M99 X99

K99 K99
98th floor  M98 X98

2nd floor  M2 X2

K2 K2
1st floor  M1 X1

K1 K1

i) Derive the equation of motion of 1st floor.

ii) Derive the equation of motion of 100th floor.

iii) Derive the generalized equation of motion of nth floor ( 2 ≤ n ≤ 99 )

Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare
(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Problem 9.2 : Generating code to calculate vibration modes of a 100 story building
In this problem, your task is to generate MATLAB code to calculate vibration mode of a 100 story
building. In order to simplify the problem, it is assumed that M 1 = M 2 = " = M 99 = M 100 = 3kg

and k1 = k2 = " = k99 = k100 = 8 N / m are given. ‘Building_your_kerberos_name’

should be function name (as well as m-file name) and upload it to 2.003 MIT Server site. You
also submit the hardcopy of your code with appropriate comments. Function has following
structure.
function v= Building_your_kerberos_name(nvm)
nvm: vibration mode number you want to generate
v: eigen vector for nvm-th vibration mode

(Hint: Define the mass and the stiffness matrices first, and then solve eigenvalue problem.)

Problem 9.3 : Plotting some vibration modes of a 100 story building


With m-code you generated in problem 9.2, print a single figure for each sub-question where x
axis represents the eigenvector value, and y axis represents the building floor. Use different line
types for different vibration modes. Legend for the vibration modes should be provided.
i) The first four vibration modes
ii) The 25th, 50th, 75th, and 100th vibration modes

Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare
(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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