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unit23

# unit23 - MIT 16.20 Fall 2002 Unit 23 Vibration of...

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MIT - 16.20 Fall, 2002 Unit 23 Vibration of Continuous Systems Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001

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MIT - 16.20 Fall, 2002 The logical extension of discrete mass systems is one of an infinite number of masses. In the limit, this is a continuous system . Take the generalized beam-column as a generic representation: 2 d 2 EI d w dx 2 dx 2 d F d w = p z (23-1) dx dx Figure 23.1 Representation of generalized beam-column d F = p x dx This considers only static loads. Must add the inertial load(s). Since the concern is in the z-displacement (w): ˙˙ Inertial load unit length = mw (23-2) where: m(x) = mass/unit length Paul A. Lagace © 2001 Unit 23 - 2
w MIT - 16.20 Fall, 2002 Use per unit length since entire equation is of this form. Thus: 2 d 2 EI d w dx 2 dx 2 d F d w = p z m ˙˙ dx dx or: 2 d 2 EI d w ˙˙ dx 2 dx 2 d F d w + mw = p z (23-3) dx dx Beam Bending Equation often, F = 0 and this becomes: 2 d 2 2 EI d w + mw = p z dx dx 2 ˙˙ --> This is a fourth order differential equation in x --> Need four boundary conditions --> This is a second order differential equation in time --> Need two initial conditions Paul A. Lagace © 2001 Unit 23 - 3

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w i t MIT - 16.20 Fall, 2002 Notes : Could also get via simple beam equations. Change occurs in: d S = p z m ˙˙ dx If consider dynamics along x, must include mu ˙˙ in p x term: ( p x mu ˙˙ ) Use the same approach as in the discrete spring-mass systems: Free Vibration Again assume harmonic motion. In a continuous system, there are an infinite number of natural frequencies (eigenvalues) and associated modes (eigenvectors) so: ω w x ( , t ) = w ( x ) e separable solution spatially (x) and temporally (t) Consider the homogeneous case (p z = 0) and let there be no axial forces (p x = 0 F = 0) Paul A. Lagace © 2001 Unit 23 - 4
i t i t MIT - 16.20 Fall, 2002 So: 2 d 2 EI d w + mw = 0 dx 2 dx 2 ˙˙ Also assume that EI does not vary with x: 4 ˙˙ EI d w + mw = 0 (23-5) dx 4 Placing the assumed mode in the governing equation: 4 EI d w e ω m ω 2 w e ω = 0 dx 4 This gives: 4 EI d w m ω 2

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unit23 - MIT 16.20 Fall 2002 Unit 23 Vibration of...

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