Modified Modal Methods in Asymmetric Systems Yeong-Jong Moon1), Jong-Heon Lee2) and In-Won Lee3) 1) Graduate Student, Department of Civil Engineering, KAIST 2) Professor, Department of Civil Engineering, Kyungil Univ. 3) Professor, Department of Civil Engineering, KAIST Thank you Mr. Chairman. The title of my presentation is ‘modified modal methods in asymmetric systems’
Contents 1. Introduction 2. Previous Study 3. Proposed Methods 4. Numerical Example 5. Conclusions The contents of this presentation are as follows; Introduction, previous study, proposed methods, numerical example and conclusions.
Introduction Many real systems have asymmetric mass, damping and stiffness matrices. - moving vehicles on roads - ship motion in sea water - offshore structures Recently Adhikari and M. I. Friswell proposed a modal method for asymmetric damped systems. It is well known that many real systems have asymmetric mass, damping and stiffness matrices. For example, moving vehicles on roads, ship motion in sea water and offshore structures. To calculate the eigenpair derivatives in these cases, Adhikari and Friswell proposed a modal method for asymmetric damped systems.
Previous Study Modal Method – Expand and as complex linear combinations of and Modal Method (1) (2) In the modal method, the basic idea is as follows. To get the eigenvector derivatives, we can expand the eigenvector derivatives as complex linear combination of eigenvectors as shown equations (1) and (2). where : the j-th right eigenvector : the j-th left eigenvector : the derivatives of j-th right eigenvector : the derivatives of j-th left eigenvector
From this idea, Adhikari has calculated the eigenvector derivatives - The derivatives of right eigenvectors (3) - The derivatives of left eigenvectors From this idea, Adhikari has calculated the derivatives of eigenvectors. The equations (3) and (4) are the derivatives of right eigenvectors and left eigenvectors, respectively. (4)
Disadvantage of conventional 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 But conventional modal method has following disadvantages. For accurate result, all eigenvalues and eigenvectors of system are needed. But In large systems, only a few lower modes are available. So the truncated error may become significant.
- Develop the effective sensitivity techniques for Objective - Develop the effective sensitivity techniques for asymmetric large systems The objective of our study is to develop the effective sensitivity techniques for asymmetric large systems.
Proposed Methods 1. Modal Acceleration Method (MA) • The general equation of motion for asymmetric systems (5) (6) • Differentiate the Eq. (5) with a design parameter (7) We modify the Adhikari’s modal method to obtain the proposed method. The eigenproblem of asymmetric systems is given as equations (5) and (6). First, we consider the right eigenvector derivatives. Differentiate the equation (5) with respect to design parameter alpha. We can get equation (7).
• Separate the response into and (8) where (9) (10) And separate the right eigenvector derivatives into d_sub_s0 and d_sub_d0. where d_sub_s0 is (B inverse f) and d_sub_d0 is this. This equation is transformed into equation (10). where Z and Y is the modal matrix to be formed by the right and left eigenvector, respectively.
• Substituting the Eq. (9) and (10) into the Eq. (8) (11) • By the similar procedure, the left eigenvector derivatives can be obtained The derivatives of right eigenvector are obtained by substituting the equations (9) and (10) into the equation (8). By the similar procedure, the left eigenvector derivatives can be obtained. When k is larger than j, these term are smaller than 1. So the convergence of modal acceleration method is better than that of conventional modal method. Therefore the effects of truncated higher modes are reduced. (12)
• Separate the response into and 2. Multiple Modal Acceleration Method (MMA) • Separate the response into and (13) where (14) For more high convergence rate, separate the eigenvector derivatives into d_sub_s1 and d_sub_d1. Where d_sub_s1 is equation (14) and d_sub_d1 is this. This equation is transformed into equation (15). (15)
• Therefore the right eigenvector derivatives are given as (16) • By the similar procedure, Therefore the right eigenvector derivatives are given as equation (16). By the similar procedure, we can obtain the left eigenvector derivatives. These equations converge faster than the modal acceleration method because this term is second order. (17)
• Based on the similar procedure, we can obtain the higher order equations (18) Based on the similar procedure, we can obtain the higher order equations. (19)
Multiple Modal Acceleration with Shifted-Poles (SP) • For more high convergence rate, the term is expanded in Taylor’s series at the position (20) For more high convergence rate, this term (s_sub_j A+B inverse) is expanded in Taylor’s series at the position beta
• Using the Eq. (20), we can obtain the following equation (21) Using the previous equation, the derivatives of right eigenvectors are given as following equations. If beta is close to s_sub_j, the convergence of equation (21) is further speeded up.
• By the similar procedure (22) By the similar procedure, left eigenvector derivatives are given as equation (22).
Numerical Example Design parameter = L To verify the effectiveness of proposed method, following example is considered. This example is whirling beam whose system matrices are asymmetric. This example is a gyroscopic system rotating with high speed and has a lumped mass in center of beam as figure. We choose the length L as the design parameter The material properties are as follows. Design parameter = L
1. The first right-eigenvector and their derivative DOF Number Eigenvector Derivative 1 2 3 4 5 6 -4.138e-2 - 5.596e-5i -1.404e-1 + 7.721e-4i 5.446e-5 - 4.221e-2i -7.804e-4 - 1.433e-1i 7.416e-3 - 4.927e-5i 5.578e-3 + 8.632e-5i 4.942e-5 + 7.736e-3i -8.601e-5 + 6.276e-3i This table shows the exact first right eigenvector and their derivative with respect to design parameter.
2. Proposed Methods 4 mode use, = eigenvlaue - 1 DOF Number MA (%) MMA SP 1 2 3 4 5 6 14.856 66.975 11.578 48.401 6.736 31.353 8.122 33.950 0.389 1.845 0.377 1.546 To demonstrate the effectiveness of proposed methods, right eigenvector derivative is calculated using four modes. This table shows errors of results. As you can see in the table, the multiple modal acceleration method with shifted poles is very effective. • MA : Modal Acceleration Method (first order) • MMA : Multiple Modal Acceleration Method (second order) • SP : Multiple Modal Acceleration Method with Shifted Poles
3. Multiple Modal Acceleration Method with Shifted Poles = eigenvlaue - 1 DOF Number 4 mode (%) 3 mode 2 mode 1 2 3 4 5 6 0.389 1.845 0.377 1.546 2.884 1.661 4.483 1.392 7.148 2.211 7.070 1.870 To demonstrate the effectiveness of multiple modal acceleration method with shifted poles, right eigenvector derivatives are calculated using a fewer modes. We can see that the maximum error is about 7% when by using only two modes. The eigenvalues of this system are very close each other. The convergence of proposed methods depend on the distance between eigenvalues. So if the distance of eigenvalues of system is relatively far, we can obtain more accurate results.
Conclusions The modified modal methods for the eigenpair derivatives of asymmetric damped systems is derived - using only a few modes - applicable to large systems The modified modal methods for the eigenpair derivatives of asymmetric damped system has been proposed. In the proposed methods, the eigenvector derivatives of asymmetric systems can be calculated by using only a few lower modes. So the proposed method can be used in the large systems.