## Analysis of Dynamic Systems

• Model Reduction
• Calculation of Eigensystem Derivatives
• Isospectral Flows and Topics from Algebra
• Analysis of Damped Systems

### Model Reduction

Model reduction whereby the number of degrees of freedom in a model is reduced is applied to large Finite element models to give faster computation of the natural frequencies and mode shapes of a structure. Model reduction also has a role to play in experimental modal analysis since the reduced mass and stiffness matrices may also be used to compare the analytical and experimental models by using orthogonality checks. The transformation inherent in the model reduction schemes may also be used to expand the measured mode shapes to the full size of the finite element model and these mode shapes may then be used in test analysis correlation or model updating exercises.

One of the oldest and most popular reduction methods is static or Guyan reduction. In this process the inertia terms associated with the discarded degrees of freedom are neglected. However whilst exact for a static model, when applied to a dynamic model the reduced model generated is not exact and often lacks the required accuracy. O'Callahan proposed a modified method which he called the Improved Reduced System (IRS) method, where an extra term is added to the static reduction transformation to make some allowance for the inertia forces. Improvements to this reduction method are the major thrust of the model reduction research.

MI Friswell, SD Garvey & JET Penny, The Convergence of the Iterated IRS Method. Journal of Sound and Vibration, 211(1), March 1998, 123-132.

MI Friswell, SD Garvey & JET Penny, Model Reduction using Dynamic and Iterated IRS Techniques. Journal of Sound and Vibration, 186(2), October 1995, 311-323.

MI Friswell & DJ Inman, Reduced Order Models of Structures with Viscoelastic Components. AIAA Journal, 37(10), October 1999, 1318-1325.

### Calculation of Eigensystem Derivatives

The sensitivity of eigenvalues and eigenvectors to changes in system parameters has considerable importance in finite element model updating and design optimisation. Many techniques exist to calculate these sensitivities for systems with distinct eigenvalues. Nelson proposed a very efficient algorithm for systems with distinct eigenvalues that only requires the eigenvector and eigenvalue of interest to calculate their derivatives. Nelson's method has been extended to systems with repeated eigenvalues and to second and higher order derivatives. The problems that occur in systems with repeated eigenvalues do not end with the calculation of the first order sensitivities. The use of the calculated derivatives in model updating and design optimisation often requires the variation of many parameters simultaneously.

S Adhikari & MI Friswell, Eigenderivative Analysis of Asymmetric Non-Conservative Systems. International Journal for Numerical Methods in Engineering, 51(6), June 2001, 709-733.

MI Friswell & S Adhikari, Derivatives of Complex Eigenvectors using Nelson's Method. AIAA Journal, 38(12), December 2000, 2355-2357.

MI Friswell, The Derivatives of Repeated Eigenvalues and their Associated Eigenvectors. ASME Journal of Vibration and Acoustics, 118(3), July 1996, 390-397.

MI Friswell, On the Calculation of Second and Higher Order Eigenvector Derivatives. AIAA Journal of Guidance, Control and Dynamics, 18(4), July-August 1995, 919-921.

U Prells & MI Friswell, On the Partial Derivatives of Repeated Eigenvalues and their Eigenvectors. AIAA Journal, 35(8), August 1997, 1363-1368.

U Prells & MI Friswell, Calculating Derivatives of Repeated and Non-Repeated Eigenvalues without Explicit Use of their Eigenvectors. AIAA Journal, 38(8), August 2000, 1426-1436.

### Isospectral Flows and Topics from Algebra

SD Garvey, U Prells, MI Friswell & Z Chen, Isospectral Flows for Second Order Systems. Linear Algebra and its Applications, 385C, July 2004, 335-368.

SD Garvey, MI Friswell & U Prells, Coordinate Transformations for Second-Order Systems: Part I General Transformations. Journal of Sound and Vibration, 258(5), December 2002, 885-909.

SD Garvey, MI Friswell & U Prells, Coordinate Transformations for Second-Order Systems: Part II Elementary Structure Preserving Transformations. Journal of Sound and Vibration, 258(5), December 2002, 911-930.

SD Garvey, MI Friswell & JET Penny, Clifford Algebraic Perspective on Linear Second-Order Systems. AIAA Journal of Guidance, Control, and Dynamics, 24(1), January-February 2001, 35-45.

SD Garvey, MI Friswell & JET Penny, Some Further Insight into Self-Adjoint Second-Order Systems. Journal of Vibration and Control, 5(2), March 1999, 237-252.

MI Friswell, SD Garvey & JET Penny, Extracting Second Order Systems from State Space Representations. AIAA Journal, 37(1), January 1999, 132-135.

SD Garvey, JET Penny & MI Friswell, The Relationship between the Real and Imaginary Parts of Complex Modes. Journal of Sound and Vibration, 212(1), April 1998, 75-83.

U Prells, MI Friswell & SD Garvey, Utilisation of Geometric Algebra: Compound Matrices and the Determinant of the Sum of two Matrices. Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, 459(2030), February 2003, 273-285

### Analysis of Damped Systems

MI Friswell, U Prells & SD Garvey, Low Rank Modifications of Classical Damping and Defective Systems. Journal of Sound and Vibration, 279(3-5), January 2005, 757-774.

U Prells & MI Friswell, A Measure of Non-proportional Damping. Mechanical Systems and Signal Processing, 14(2), March 2000, 125-137.

U Prells & MI Friswell, A Relationship Between Defective Systems and Unit-Rank Modification of Classical Damping. Journal of Vibration and Acoustics, 122(2), April 2000, 180-183.

Last updated February 2009 by M I Friswell