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lec21 (1)

# lec21 (1) - Governing Equation 1 2.003J/1.053J Dynamics and...

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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 Governing Equation 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Peacock 5/7/2007 Lecture 21 Vibrations: Second Order Systems - Forced Response Governing Equation Figure 1: Cart attached to spring and dashpot subject to force, F ( t ). Figure by MIT OCW. mx ¨ + cx ˙ + kx = F ( t ) x ¨ + 2 ζω n x ˙ + ω n 2 x = F ( t ) (1) m ζ : Damping Ratio ω n : Natural Frequency Forced Response - Particular Solution x p ( t ) Can use Fourier Series or Laplace Transforms Start with a simple case F ( t ) = f =constant

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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 2 Forced Response - Particular Solution x p ( t ) F ( t ) is constant The complementary solution below requires ζ < 1. x c = Ce ζω n t cos( ω d φ ) Subscript c for complementary solution. x p =? Try x = At + B . 2 ζω n A + ω n 2 ( At + B ) = f m f f A = 0, B = 2 = k n ω n = k m Therefore: f x = Ce ζω n t cos( ω d t φ ) + k x c = Ce ( ζω n t ) cos( ω d t φ ): unknown constants set by initial conditions x p = f k : determined by
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lec21 (1) - Governing Equation 1 2.003J/1.053J Dynamics and...

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