1 PSSC 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 qIntroduction m Definition of mode localization m Literature Survey m Objectives qTheoretical Background m Multispan Beams m Simple Structure m Occurrence of Mode Localization m Conditions of Significant Mode Localization qNumerical Examples m Mode Localization in Two-Span Beam qConclusions

3 q INTRODUCTION m Definition of Mode Localization –Under conditions of weak internal coupling, the mode shapes undergo dramatic changes to become strongly localized when small disorder is introduced into periodic structures. (C. Pierre, 1988, JSV) Trouble by Mode Localization –When mode localization occurs, the modal amplitude of a global mode becomes confined to a local region of the structure, with serious 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 Mode Localization of Parabola Antenna Figure 5. Non-Localized Mode Shape Figure 6. Localized Mode Shape

7 Introduction m Literature Survey Localization of electron eigenstates in a disordered solid –P. W. Anderson (1958) 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 m 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 q THEORETICAL BAGROUND m Multispan Beams 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 m Simple Structure Analysis Figure 9. Simple model with two-substructures and a coupler. Subject : Qualitative analysis of influences of stiffness and mass of coupler on mode localization using simple model

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

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

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

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

15 q NUMERICAL EXAMPLES m Mode Localization in Two-Span Beam Assumptions –All spans have identical span lengths initially. –Length disturbances are introduced into the first span only. 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

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

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

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

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

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

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

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