Vibration analysis and vibration reduction for mechanical systems are prominent problems in numerous research fields. Although the vibrations analysis is an intensively studied topic in recent decades, many problems still remain open. While the case without external excitation leads to the study of homogeneous systems, presence of an external forcing leads to the study of nonhomogeneous systems. Depending on the presence of an external excitation and applications, we will consider four different research themes.
Within the first research theme, we will study theoretical results that are relevant for vibration reduction. We plan to develop theoretical results that characterize important properties of the quadratic eigenvalue problem (QEP) arising from vibration analysis of mechanical systems. Within the second research theme we will develop new methods for frequency isolation and utilize methods which are based on algorithms for non-smooth optimization. For this case we will derive new algorithms that preserve the structure of the matrices and structural properties of the considered QEP. In the third research theme, we will consider vibration reduction based on criteria that use system norms (e.g. H2 and Hinf) for Multiple-Input Multiple-Output case. We will also study approaches for approximating the full-order model with a reduced-order model that retains the structure of parametric dependence. The new approaches will be well suited for computationally efficient parameter optimization and the study of important system properties. In the fourth research theme we will consider integrating research themes I-III and applications in real world examples.
Moreover, we will apply obtained new approaches and algorithms in various academic examples, but also in real life examples that arise, e.g., in car industry (such as disc brake problem) and civil engineering (such as beams, civil buildings), etc. Therefore, the results from this project could have wide applications.
Research themes
I - IV
RT I: homogeneous case
We develop theoretical results that characterize important properties of the Quadratic Eigenvalue Problem (QEP) that arise from vibration analysis of mechanical systems. In relative perturbation bounds typically relative gaps appear, which makes such bounds demanding to compute. We will approximate these perturbation bounds and develop new bounds that can be used for efficient vibration reduction.
RT II: case with given external force
We develop criterion that considers vibration isolation in order to avoid resonance in vibrational system. For this case we will derive new algorithms that preserve the structure of the matrices and structural properties of the considered QEP.
RT III: MIMO case
We consider a new measures that can ensure better robustness, stability and other important system properties. Within this case, to make the optimization algorithm computationally feasible, we will employ a novel approach to parametric model order reduction. We will also study approaches for approximating the full-order model with a reduced-order model that retains the structure of parametric dependence. The new approach will be well suited for computationally efficient parameter optimization and the study of important system properties.
RT IV: integrating research themes and applications
This is integrating research themes I-III and applications in real world examples. Firstly, we will employ the theoretical results obtained for the homogeneous case in the setting of the nonhomogeneous case as well. We propose new approaches that are based on criteria that are nonsmooth, such as Hinf criterion or criterion based on frequency isolation. We will test our new approaches and algorithms in various academic examples, but also in real life examples that arise, for example, in car industry (such as disc break problem) and civil engineering (such as beams, civil buildings), etc.
Publications
Sampling-free parametric model reduction for structured systems
Christopher Beattie, Serkan Gugercin, Zoran Tomljanović
Advances in Computational Mathematics 46/6, 1-34, (2020)
Perturbation bounds for stable gyroscopic systems in motion about an unstable equilibrium position
Ivana Kuzmanović Ivičić, Suzana Miodragović
submitted
Relative perturbation tan Θ theorems for definite matrix pairs
Suzana Miodragović, Ninoslav Truhar, Ivana Kuzmanović Ivičić
submitted
Dissemination
COST Action 18232 Zagreb meeting, 24-27 February 2020
Z. Tomljanović, Sampling-free parametric model reduction for structured systems
ApplMath20, Brijuni, 14-18 September 2020
S. Miodragović, Frequency isolation for the hyperbolic guadratic eigenvalue problem
M. Ugrica, Frequency-weighted damping via nonsmooth optimization and fast computation of QEPs with low-rank updates
Magdeburg Lectures on Optimization and Control, Magdeburg, 25 September 2020
Z. Tomljanović, Damping optimization in mechanical systems using sampling-free model reduction
Optimization and application seminar, Department of Mathematics, Osijek, 25 November 2020
Z. Tomljanović, Sampling-free model reduction of systems with low-rank parameterization
Seminar za primijenjenu matematiku i teoriju upravljanja, University of Dubrovnik, 5 February 2021
Z. Tomljanović, Damping optimization in mechanical systems using parametric model reduction
Optimization and application seminar, Department of Mathematics, Osijek, 3 March 2021
M. Ugrica, Frequency-weighted damping via non smooth optimization and fast computation of QEPs
91st GAMM Annual Meeting in Kassel, 15-19 March 2021
Z. Tomljanović, Frequency-weighted damping via non smooth optimization and fast computation of QEPs
Croatian German meeting on analysis and mathematical physics, 22nd to 25th March 2021
Z. Tomljanović, Sampling-free model reduction of systems with low-rank parameterization
8th European Congress of Mathematics, 20 - 26 June 2021, Portorož, Slovenia
Suzana Miodragović, Hyperbolic quadratic eigenvalue problem and frequency isolation
Ivana Kuzmanović Ivičić, Optimization of parameter dependent structured Sylvester and T-Sylvester equations
Workshop on Control of Dynamical Systems 14-16 June 2021, Dubrovnik, Croatia
M. Ugrica, Frequency-weighted damping via nonsmooth optimization and fast computation of QEPs with low-rank updates
Zoran Tomljanović, Optimal damping of vibrational systems using finite time energy criterion
Suzana Miodragović, Frequency isolation problem for hyperbolic QEP
Applied numerical analysis seminar, 17 Sep 2021, Virginia Tech, Blacksburg, USA
Z. Tomljanović, Fast optimization of viscosities for frequency-weighted damping of second-order systems
GAMM Workshop Applied and Numerical Linear Algebra, September 16-17, 2021
Suzana Miodragović, Frequency isolation problem for the hyperbolic QEP applied to gyroscopic systems
Marinela Pilj Vidaković, Optimal damping of vibrational systems using finite time energy criterion
Optimization and application seminar, Department of Mathematics, Osijek, 16 February 2022
M. Pilj Vidaković, Finite time horizon mixed control of vibrational systems
ALAMA 2022 - ALN2gg, 1st to 3rd June 2022, Madrid, Spain
Ivana Kuzmanović Ivičić, Perturbation bounds for a stable gyroscopic systems
Suzana Miodragović, The tan$\Theta$ theorem for definite matrix pairs
ApplMath22, Brijuni, 5-9 September 2022
Z. Tomljanović, Numerically Efficient $H_{infty}$ Analysis of Cooperative Multi-Agent Systems
R. K. Das, Solving polynomial and rational eigenvalue problems through linearizations
M. Pilj Vidaković, Finite time horizon mixed control of vibrational systems
News
Invited lectures & visiting fellows
Dr. sc. Ion Victor Gosea is our guest at the Optimization and application seminar on 14th December 2022. He will give a talk on Data-driven model reduction of second-order systems based on harnessing the advantages of barycentric forms.
Ion Victor GoseaMPI for Dynamics of Complex Technical Systems
Dr. sc. Petar Mlinarić is our guest at the Optimization and application seminar on 14th September 2022. He will give a talk on Interpolatory Conditions for L2-optimal Reduced-order Modeling.
Petar MlinarićVirginia Tech, USA
Dr. sc. Ranjan Kumar Das is a new postdoc on VIMS project. On 26th January 2022 he will give a talk on the Optimization and application seminar. He will give a talk on Solving rational eigenvalue problems through linearizations.
Ranjan Kumar DasDepartment of mathematics, UNIOS
Dr. sc. Nicat Aliyev is our guest at the Mathematical Colloquium in Osijek on 11th December 2021. He will give a talk on Subspace Methods for Large-Scale Control Problems.
Nicat AliyevCharles University, Czech Republic
Professor Christopher Beattie is our guest at the Mathematical Colloquium in Osijek on 20th May 2021. On-line lecture will start at 7pm. He will give a talk on Balancing and the reduction of dynamical systems based on data.
Christopher BeattieVirginia Tech, USA
Professor Ren-Cang Li is our guest at the Mathematical Colloquium in Osijek on 10th December 2020. He will give a talk on SCF Iteration for Orthogonal Canonical Correlation Analysis.
Ren-Cang LiUniversity of Texas at Arlington, USA
Professor Serkan Gugercin is our guest at the Optimization and application seminar on 10th June 2020. He will give a talk on An Introduction to Interpolatory Methods for Model Reduction.
