MT-1620 al.2002 Unit 23 Vibration of continuous systems Paul A Lagace Ph D Professor of aeronautics Astronautics and engineering systems Paul A Lagace @2001
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
MT-16.20 al.2002 The logical extension of discrete mass systems is one of an infinite number of masses In the limit this is a continuous syStem T ake the generalized beam-column as a generic representation d dw El F dx 23-1) Figure 23.1 Representation of generalized beam-column dF dx 和x 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 (232) where: m(x =mass/unit length Paul A Lagace @2001 Unit 23-2
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 dw dx2 dx2 − d F dw = pz (23-1) dx dx Figure 23.1 Representation of generalized beam-column dF = − px 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
MT-16.20 a.2002 Use per unit length since entire equation is of this form. Thus d/d EI dx dx2 dx dx p -mw or ddw El m=p:(233) x Beam Bending Equation often f=0 and this becomes E mw =p dx dx 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
w MIT - 16.20 Fall, 2002 Use per unit length since entire equation is of this form. Thus: 2 d 2 EI dw dx2 dx2 − d F dw = pz − m ˙˙ dx dx or: 2 d 2 EI dw ˙˙ dx2 dx2 − d F dw + mw = pz (23-3) dx dx Beam Bending Equation often, F = 0 and this becomes: 2 d 2 2 EI dw + mw = pz dx dx2 ˙˙ --> 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
MT-1620 Fall 2002 Notes Could also get via simple beam equations Change occurs in p- mw If consider dynamics along x, must include mu in px term: (P, -mi 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) w(x, t)=w(x)e separable solution spatially(x) and temporally (t) Consider the homogeneous case(p, =0) and let there be no axial forces (px=0→F=0) Paul A Lagace @2001 Unit 23-4
w i t MIT - 16.20 Fall, 2002 Notes: • Could also get via simple beam equations. Change occurs in: dS = pz − m ˙˙ dx • If consider dynamics along x, must include mu˙˙ in px term: ( px − 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 (pz = 0) and let there be no axial forces (px = 0 ⇒ F = 0) Paul A. Lagace © 2001 Unit 23 - 4
MT-1620 al.2002 El d x d2/+m=0 Also assume that El does not vary with X E (23-5) dx Placing the assumed mode in the governing equation dw E 100 I 4e mo we This gives a w E m02=0 (23-6) d x which is now an equation solely in the spatial variable(successful separation of t and x dependencies Must now find a solution for W(x) which satisfies the differential equations and the boundary conditions Note: the shape and frequency are intimately linked (through equation 23-6 Paul A Lagace @2001 Unit 23-5
it it MIT - 16.20 Fall, 2002 So: 2 d 2 EI dw + mw = 0 dx2 dx2 ˙˙ Also assume that EI does not vary with x: 4 EI ˙˙ dw + mw = 0 (23-5) dx 4 Placing the assumed mode in the governing equation: 4 EI d w e ω − mω2 w e ω = 0 dx 4 This gives: 4 EI d w − mω2 w = 0 (23-6) dx 4 which is now an equation solely in the spatial variable (successful separation of t and x dependencies) _ Must now find a solution for w(x) which satisfies the differential equations and the boundary conditions. Note: the shape and frequency are intimately linked (through equation 23-6) Paul A. Lagace © 2001 Unit 23 - 5