Mode Localization in Multispan Beams with Massive and Stiff Couplers on Supports

1. PSSC 1998. 10. 13. Mode Localization in Multispan Beams with Massive and Stiff Couplers on Supports Dong-Ok Kim and In-Won Lee Department of Civil Engineering in Korea Advance Institute of Science and Technology

2. CONTENTS • Introduction • Definition of mode localization • Literature Survey • Objectives • Theoretical Background • Multispan Beams • Simple Structure • Occurrence of Mode Localization • Conditions of Significant Mode Localization • Numerical Examples • Mode Localization in Two-Span Beam • Conclusions

3. Definition of Mode Localization Under conditions of weak internal coupling, the mode shapes undergo dramatic changes to become strongly localizedwhen small disorder is introduced into periodic structures. (C. Pierre, 1988, JSV) INTRODUCTION • Trouble by Mode Localization • When mode localization occurs, the modal amplitude of a global mode becomes confined to a local region of the structure, withserious implication for the control problem. (O. O. Bendiksen, 1987, AIAA)

4. Introduction • Example : Mode Localization of Two-Span Beam Figure 1. Weakly Coupled Simply Supported Two-Span Beam Figure 2. Non-Localized First Two Mode Shapes Figure 3. Localized First Two Mode Shapes

5. Introduction • Example : Mode Localization of Parabola Antenna Figure 4. Simple Model of Space Parabola Antenna

6. Introduction Figure 5. Non-Localized Mode Shape Mode Localization of Parabola Antenna Figure 6. Localized Mode Shape

7. Introduction • Localization of electron eigenstates in a disordered solid • P. W. Anderson (1958) • Literature Survey • Mode localization in the disordered periodic structures • C. H. Hodges (1982) • Localized vibration of disordered multispan beams • C. Pierre (1987) • Influences of various effects on mode localization • S. D. Lust (1993) • Mode localization up to high frequencies • R. S. Langley (1995)

8. Introduction • Objective: To study influences of the stiffness and mass of coupler on mode localization • Scope • Theoretical Background: Qualitative analysis using simple model • Numerical Examples: Verifications of results of the theoretical background using multispan beams • Conclusions

9. Multispan Beams THEORETICAL BAGROUND Figure 7. Simply supported multispan beam with couplers. - Periodically rib-stiffened plates or - Rahmen bridges • Two-span beam : Two substructures and one coupler Figure 8. Simply supported two-span beam with a coupler.

10. Theoretical Background • Subject : Qualitative analysis of influences of stiffness and mass of coupler on mode localization using simple model • Simple Structure Analysis Figure 9. Simple model with two-substructures and a coupler.

11. Theoretical Background • Eigenvalue Problem (1) • Equation for ratio of and , and (2) where The ratio represents degree of mode localization corresponding mode.

12. Theoretical Background • Equation for Degree of Mode Localization • Occurrence of Mode Localization (3) where and (4) • In Equation (3) Left-hand side : Parabolic curve Right-hand side : Line passing origin with slop

13.   Significant mode localization   No mode localization Theoretical Background • Steep line • Graphical Representation • Identical substructures: Figure 10. Two curves.

14. Theoretical Background • Becoming and , under the condition of results in significant mode localization, . • Conditions for Significant Mode Localization (5) (6) • Classical condition ( ) : • Delocalization condition or (7)

15. Mode Localization in Two-Span Beam NUMERICAL EXAMPLES Figure 11. Simply Supported Two-Span Beam with a Coupler • Subjects to Discuss • Influences of length disturbance of a span • Influences of the stiffness and the mass of coupler • Assumptions • All spans have identical span lengths initially. • Length disturbances are introduced into the first span only.

16. Mode Localization in Multispan Beams • Selected Mode Shapes of Two-Span Beam Figure 12. First ten mode shapes:

17. Mode Localization in Multispan Beams Classical Measure • Span response ratio • Measure of Degree of Mode Localization (8) where : Maximum amplitude of span : Maximum amplitude of span Note ! Classical measure is good for analysis but not for practice.

18. Mode Localization in Multispan Beams Proposed Measure • Normalized number of spans having no vibration (9) where : Total number of spans : Number of spans in which vibration is confined (10) : Maximum amplitude of span Note ! Proposed measure is good for practice but not for analysis.

19. Mode Localization in Multispan Beams • Coupler with Stiffness • Stiffness makes the system sensitive to mode localization Figure 13. Influence of the stiffness

20. Mode Localization in Multispan Beams • Coupler with Mass • Mass makes the system sensitive to mode localization in higher modes. Figure 14. Influence of the mass

21. Mode Localization in Multispan Beams • Coupler with Stiffness and Mass • Stiffness governs sensitivities of lower modes. • Mass governs sensitivities of higher modes. • Delocalized modes can be observed. Figure 15. Influences of stiffness and mass

22. Influences of the Coupler are Discovered. Thestiffness of coupler makes the structures sensitive to mode localization. Themassof coupler makes the structures sensitive to mode localization in higher modes. The coupler withstiffnessandmassis a cause of delocalization*in some modes. CONCLUSIONS • The mass as well as the stiffness of coupler give significant influences on mode localization especially in higher modes. * Delocalization is that mode localization does not occur or is very weak in certain modes although structural disturbances are severe.