1. Yeong-Jong Moon 1), Sun-Kyu Park Lee 2) and In-Won Lee 3) 1) Graduate Student, Department of Civil Engineering, KAIST 2) Professor, Department of Civil Engineering, Sungkyunkwan Univ. 3) Professor, Department of Civil Engineering, KAIST Modified Modal Methods for Eigenderivative Analysis of Asymmetric Damped System

2. - The right eigenvector derivatives - The left eigenvector derivatives  Adhikari’s Modal Method Disadvantage of Adhikari’s modal method - For accurate result, all eigenvalues and eigenvectors of system are needed - In large systems, only a few lower modes are available - The truncated error may become significant

3. Differentiate the above equation with a design parameter The general equation of motion for asymmetric systems  Multiple Modal Acceleration Method (MMA) where Separate the response into and

4.  Multiple Modal Acceleration Method(MMA) - The right eigenvector derivatives - The left eigenvector derivatives

5. is expanded in Taylor’s series at the position  Multiple Modal Accelerations with Shifted Poles - The right eigenvector derivatives - The left eigenvector derivatives

6.  Numerical Example Design parameter = L

7. DOF No. MA (%) MMA (%) SP (%) 123456123456 14.856 0 66.975 11.578 0 48.401 6.736 0 31.353 8.122 0 33.950 0.389 0 1.845 0.377 0 1.546 - 4 mode use, = eigenvlaue - 1 DOF No. 4 mode (%) 3 mode (%) 2 mode (%) 123456123456 0.389 0 1.845 0.377 0 1.546 2.884 0 1.661 4.483 0 1.392 7.148 0 2.211 7.070 0 1.870 - = eigenvlaue - 1 MA : Modal Acceleration Method (first order) MMA : Multiple Modal Acceleration Method (second order) SP : Multiple Modal Acceleration Method with Shifted Poles  Numerical Results Compare the proposed methods Effectiveness of SP

8.  Conclusions The modified modal methods for the eigenpair derivatives of asymmetric damped systems is derived - using only a few modes - applicable to large systems