lec22 (1) - Motivation 1 2.003J/1.053J Dynamics and Control...

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Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 1 Motivation 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Peacock 5/9/2007 Lecture 22 Vibrations: Free Response of Multi-Degree-of-Freedom Systems Motivation Figure 1: 2 degrees of freedom. Spring with mass on end. The mass rotates in both counterclockwise and clockwise directions. The spring extends and retracts by varying amounts. Sometimes only the spring moves with the mass relatively still; sometimes only the mass rotates. What is happening? Figure by MIT OCW. Two Degrees of Freedom: System of Three Springs and Two Masses Consider the following: Figure 2: System of three springs and two masses. Figure by MIT OCW.
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Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. ± ² ³ Two Degrees of Freedom: System of Three Springs and Two Masses2 2 degrees of freedom 2 generalized coordinates M 1 = M 2 = m x 1 and x 2 are the coordinates that describe displacement from equilibrium Lagrangian 1 1 T = mx ˙ 1 2 + mx ˙ 2 2 2 2 V = 1 k 1 x 2 1 + 1 k 1 x 2 2 + 1 k 2 ( x 2 x 1 ) 2 2 2 2 L = T V Equations of Motion mx ¨ 1 + k 1 x 1 k 2 ( x 2 x 1 ) = 0 mx ¨ 2 + k 2 x 2 + k 2 ( x 2 x 1 ) = 0 Can solve by elimination of x 2 , for example k 1 + k 2 2 k 1 + 2 k 2 x 1 + 2 x ¨ 1 + k 1 m 2 x 1 = 0 m System is equivalent to a single 4th
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lec22 (1) - Motivation 1 2.003J/1.053J Dynamics and Control...

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