Advanced Nonlinear Strategies for Vibration Mitigation and by Alexander F. Vakakis

By Alexander F. Vakakis

This booklet examines complex nonlinear themes in vibration mitigation and process id, together with analytical tools for strongly nonlinear dyanmical structures; passive vibration mitigation in keeping with nonlinear detailed power move (TET) and more.

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Extra resources for Advanced Nonlinear Strategies for Vibration Mitigation and System Identification (CISM International Centre for Mechanical Sciences) (Volume 518)

Sample text

34 Lawrence A. 9 H56@9 5B8 6 5 89H5=@ C: H<9 /# (, 36 Lawrence A. 9 G=;B=75BH @9J9@G C: G9=GA=7 9B9F;M 5F9 GDF958 :FCA H<9 FGH HC H<9 G97CB8 5B8 H<=F8 @=B95F=N98 AC89G C: H<9 GMGH9A K<=@9 :CF H<9 (CFH

Targeted Energy Transfer in Systems with Periodic Excitations 55 After substitution of (2) and (3) into (1) and subsequent averaging aver the fast oscillations of frequency unity, we obtain the following slow–flow (complex modulation) equations: M1  jHV M1  HM2 H A jH M1 M2  2(1 H ) 2(1 H ) 2 M2  O(1 H ) M2 2  jHV M1  HM2 j(1 H ) 2 HA j  M2 M2 M2 M1  2(1 H ) 2(1 H ) 2 2 (4) The system of equations (4) has a complicated structure and cannot be solved analytically. The first step towards analyzing its steady state solutions is to perform local analysis of its equilibrium (fixed points).

Starosvetsky M10 (1  HV )M 2  j (1  H ) A § 1  HV · M 20 cos T 20 Ÿ M10 ¨ ¸ 1V © 1  V ¹ cos T10 ª º (1  H ) A T10 sgn(1  V ) tan 1 « tan T 20  » (1  HV ) M 20 cos T 20 ¼ ¬ (6) The polynomial in the first of equations (6) can be brought into the following compact form, D1Z  D 2 Z 2  D 3 Z 3  D 4 0 (7) where, M20 D2 2 Z , D1 O2  V2 , 2 1  V 2V , D 3 1, D 4 (1  V )  A2 (1  V ) 2 (8) Depending on the system parameters and the coefficients (8), the generic polynomial (7) can have one or three positive (real) solutions.

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