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CA-AA242B-Ch7

# CA-AA242B-Ch7 - AA242B MECHANICAL VIBRATIONS AA242B...

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AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Direct Time-Integration Methods 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 Stability and Accuracy of Time-Integration Operators 2 Newmark’s Family of Methods AA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Multistep Time-Integration Methods General multistep time-integration method for first-order systems of the form ˙ u = Au u n +1 = m X j =1 α j u n +1 - j - h m X j =0 β j ˙ u n +1 - j where h = t n +1 - t n is the computational time-step, u n = u ( t n ), and u n +1 = q n +1 ˙ q n +1 is the state-vector calculated at t n +1 from the m preceding state vectors and their derivatives as well as the derivative of the state-vector at t n +1 β 0 6 = 0 leads to an implicit scheme — that is, a scheme where the evaluation of u n +1 requires the solution of a system of equations β 0 = 0 corresponds to an explicit scheme — that is, a scheme where the evaluation of u n +1 does not require the solution of any system of equations and instead can be deduced directly from the results at the previous time-steps AA242B: MECHANICAL VIBRATIONS

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AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Multistep Time-Integration Methods General multistep integration method for first-order systems (continue) u n +1 = m X j =1 α j u n +1 - j - h m X j =0 β j ˙ u n +1 - j trapezoidal rule u n +1 = u n + h 2 u n + ˙ u n +1 ) backward Euler formula (implicit) u n +1 = u n + h ˙ u n +1 forward Euler formula (explicit) u n +1 = u n + h ˙ u n AA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Numerical Example: the One-Degree-of-Freedom Oscillator Consider an undamped one-degree-of-freedom oscillator ¨ q + ω 2 0 q = 0 with ω 0 = π rad/s, submitted to an initial displacement q (0) = 1 , ˙ q (0) = 0 exact solution q ( t ) = cos ω 0 t associated first-order system ˙ u = Au where A = » 0 - ω 2 0 1 0 u = [˙ q , q ] T , and initial condition u (0) = » 0 1 AA242B: MECHANICAL VIBRATIONS

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AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Numerical Example: the One-Degree-of-Freedom Oscillator Numerical solution T = 3 s , h = T 32 0 0.5 1 1.5 2 2.5 3 -4 -3 -2 -1 0 1 2 3 t q Exact solution Trapezoidal rule Euler backward Euler forward AA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Stability Behavior of Numerical Solutions Analysis of the characteristic equation of a time-integration method consider the first-order system ˙ u = Au for this problem, the general multistep method can be written as u n +1 = m X j =1 α j u n +1 - j - h m X j =0 β j ˙ u n +1 - j m X j =0 [ α j I - h β j A ] u n +1 - j = 0 , α 0 = - 1 let μ r be the eigenvalues of A and X be the matrix of associated eigenvectors the characteristic equation associated with m P j =0 [

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CA-AA242B-Ch7 - AA242B MECHANICAL VIBRATIONS AA242B...

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