CA-AA242B-Ch3 - AA242B MECHANICAL VIBRATIONS AA242B...

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AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Damped Vibrations of n-DOF Systems These slides are based on the recommended textbook: M. G´ eradin and D. Rixen, “Mechanical Vibrations: Theory and Applications to Structural Dynamics,” Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Outline 1 Modal Superposition 2 Space-State Formulation & Analysis of Viscous Damped Systems AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Modal Superposition Damped Oscillations in Terms of Undamped Natural Modes Assume damping mechanism can be described by a viscous dissipation quadratic in the generalized velocities D = 1 2 ˙ q T C ˙ q 0 where C is a symmetric and non-negative damping matrix Lagrange’s equations = M ¨ q + C ˙ q + Kq = p ( t ) Consider again the eigenpairs ( ω 2 i , q 0 i ) , i = 1 , · · · , n of the undamped system Look for a solution of the damped equations of dynamic equilibrium of the form q = n X i =1 y i ( t ) q 0 i AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Modal Superposition Damped Oscillations in Terms of Undamped Natural Modes M ¨ q + C ˙ q + Kq = p ( t ); q = n X i =1 y i ( t ) q 0 i = Qy = Q T MQ ¨ y + Q T CQ ˙ y + Q T KQy = Q T p ( t ) = I ¨ y + Q T CQ ˙ y + Ω 2 y = Q T p ( t ) In general, Q T CQ = [ β ij ] is a full matrix Hence, ¨ y i + n X j =1 β ij ˙ y j + ω 2 i y i = q T 0 i p ( t ) , i = 1 , · · · , n where β ij = q T 0 i Cq 0 j The above equation shows that unless some assumptions are introduced, the method of modal superposition is not that interesting for solving the dynamic equations of equilibrium, because the resulting modal equations are coupled in the presence of general damping AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Modal Superposition Damped Oscillations in Terms of Undamped Natural Modes If a small number of modes k n suffices to compute an accurate solution, the modal superposition technique can still be interesting because in that case the size of the modal equations is much smaller than that of the original equations (reduced-order modeling) C n × n -→ Q T k × n C n × n Q n × k k × k AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Modal Superposition Damped Oscillations in Terms of Undamped Natural Modes It will be shown that if the structure is lightly damped, a diagonal matrix Q T CQ is a consistent even though not physical assumption consider M ¨ q + C ˙ q + Kq = 0 search for a solution of the form q = z e λ t = ( λ 2 k M + λ k C + K ) z k = 0 without damping, one would have λ k = ± i ω k and z k = q 0 k if the system is lightly damped, it can be assumed that λ k and z k differ only slightly from ω k and q 0 k , respectively λ k = i ω k + Δ λ z k = q 0 k + Δ z substituting in the characteristic equation and neglecting the second-order terms gives ( K - ω 2 k M z + (2 i ω k M + C ) q 0 k Δ λ + i ω k C ( q 0 k + Δ z ) 0 the light-damping assumption allows one to neglect the terms in C Δ λ and C Δ z = ( K - ω 2 k M z + i ω k ( C + 2Δ λ M ) q 0 k 0 AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Modal Superposition
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