scholarly journals Natural Vibrations of Repetitive Structures

2000 ◽  
Vol 16 (2) ◽  
pp. 85-95 ◽  
Author(s):  
Dajun Wang ◽  
C.-C. Wang
Keyword(s):  
Eigenvalue Problem ◽  
Natural Vibration ◽  
Eigenvalue Problems ◽  
Vibration Frequencies ◽  
Natural Vibrations

ABSTRACTNatural vibration frequencies and modes of repetitive structures, including symmetric, periodic, linking structures, are considered in this work. By using the repetition of the identical parts, we reduce the eigenvalue problem of the structure to a set of eigenvalue problems of lower dimensions associated with the parts. Special forms and properties of the modes of natural vibrations are observed.

scholarly journals Research into the dynamics of radio telescope foundationsusing laser vibration measuring equipment

2020 ◽  
Vol 10 (3) ◽  
pp. 420-427
Author(s):  
Vladimir I. Sobolev ◽  
◽  
Tatiana N. Chernigovskaya ◽  
Keyword(s):  
Radio Telescope ◽  
Measuring Equipment ◽  
Natural Vibration ◽  
Fourier Transforms ◽  
Process Analysis ◽  
Laser Vibrometer ◽  
Contact Method ◽  
Vibration Frequencies ◽  
Natural Vibrations ◽  
Vibration Measuring

This study presents a technology for determining the dynamic parameters of the foundation structures of radio telescopes by a non-contact method based on the use of laser vibration measuring equipment. The main results of a dynamic process analysis are described. The research was carried out in the form of microdynamic tests of foundation structures with the determination of natural oscillation periods based on lower vibrational forms. Measurements of natural vibrations of radio telescope foundations under applied shock were performed, followed by spectral mapping of vibrational processes. The tests were carried out using a tachometric laser vibrometer RSV-150 included in the register of vibration measuring equipment in Russia. Spectral mappings were obtained using Fourier transforms implemented in software packages of the hardware developers. The dynamic features of radio tele-scope foundations are formalized in the form of spectral maps of the initial records of vibrational proc-esses. The method of microdynamic tests using a vibration meter is presented, along with the principles of its operation and the sequence of numerical processing of records of dynamic processes imple-mented on a personal computer using frequency displays of natural vibrations of structures. The values of natural vibration frequencies obtained as a result of measurements confirmed the rigidity and integ-rity of foundation piles. Pronounced solitary peaks of natural vibration frequencies point to the form of natural vibrations of structures as single-mass systems with the absence of deformational forms of vi-brations in the foundation body.

The Natural Vibration of Fluid-Structure Interaction Systems Subject to the Sommerfeld Radiation Condition

2006 ◽  
Author(s):  
Jing T. Xing
Keyword(s):  
Eigenvalue Problem ◽  
Degrees Of Freedom ◽  
Natural Vibration ◽  
Natural Frequencies ◽  
Fluid Structure Interaction ◽  
Complex Conjugate ◽  
Natural Vibrations ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Natural Modes

A fluid-structure interaction system subject to a Sommerfeld condition is defined as a Sommerfeld system in this paper. It is well known that the natural vibration of a dynamic system is defined by the eigenvalue problem of the corresponding idealized system with no material damping assumed and external forces. From the defined eigenvalue problem, the real natural frequencies and the corresponding natural modes of the system can be derived. What are the characteristics of natural vibrations of a Sommerfeld system? This paper intends to address this problem by investigating three selected fluid-structure interaction systems. The systems chosen involve the solid structures with one, two and infinite degrees of freedom coupling to an infinite fluid domain subject to a Sommerfeld condition, respectively. The governing equations describing these coupled systems are presented using the theory of continuum mechanics. The theoretical solution for each problem is derived and discussed. The analysis demonstrates that a Sommerfeld system undergoing a natural vibration behaves energy dissipative characteristics although there is no material damping in solid and fluid of the system. The natural vibrations of a Sommerfeld system are governed by a complex eigenvalue problem which has only pairs of complex conjugate natural frequencies. The number of the complex conjugate natural frequencies and corresponding natural modes of this Sommerfeld system equals to the number of the degrees of freedom of the dry solid structure in the system and it is independent of the infinite fluid domain. The natural vibration forms of the solid structure in natural vibrations do not satisfy the orthogonal relationship. The findings in this research reveal some common dynamic characteristics of Sommerfeld systems. An approach for the dynamic response analysis of a Sommerfeld system is proposed based on the orthogonal natural modes of the dry structure in the system which is more efficient for engineering analysis.

scholarly journals Numerical algorithm for searching for layouts of electroelastic bodies with external electric circuits for obtaining the best damping properties

2020 ◽  
pp. 108-124
Author(s):  
N. A Iurlova ◽  
D. A Oshmarin ◽  
N. V Sevodina ◽  
M. A Iurlov
Keyword(s):  
Natural Vibration ◽  
Mathematical Formulation ◽  
Piezoelectric Element ◽  
Electric Circuit ◽  
Problem Solution ◽  
Vibration Frequencies ◽  
Damping Properties ◽  
Natural Vibrations ◽  
Electric Circuits ◽  
Continuous Frequency

This paper presents an algorithm which allows finding such layouts of electromechanical systems that provide the best vibration damping whether for one mode or a set of vibration modes within some continuous frequency ranges. The basis for the algorithm is the problem solution about natural vibrations. An elastic structure with a piezoelectric element located on its surfaces, which electrodes are connected to a passive external electric circuit, is treated as an electromechanical system. The piezoelectric elements shunted with an electric circuit are the devices where energy dissipation occurs, thus leads to damping of vibrations. A change in damping properties of such systems can be reached by a proper choice of parameters of the electric circuits and corresponding location of the piezoelectric element, which provides the highest energy withdrawal into the electric circuit. The paper presents the mathematical formulation of the natural vibrations problem for piecewise-homogeneous electroelastic bodies shunted with passive external electric circuits. Within the proposed mathematical statement, the problem solution of natural vibrations for such objects is based on values of complex natural vibration frequencies. Real parts of the complex natural vibration frequencies are the circular frequency of vibrations, and the imaginary parts are the damping indices of vibrations. We proposed techniques aimed at determining the location of the piezoelectric element and selecting parameters of shunting the external electric circuit. These approaches are based on values of the complex natural vibration frequencies obtained as results of solving the natural vibrations problem. The proposed approach is demonstrated using a specimen of a thin-walled shell in the semi-cylinder form. The piezoelectric element has a form of a segment of a ring made of PZT-4 piezoceramics. Electrodes of the piezoelectric element are connected to the series resonant RL -circuit. The formulated problem is solved numerically using the finite element method and ANSYS commercial package.

Transparent boundary conditions for discretized non-Hermitian eigenvalue problems

2021 ◽  
pp. 2150138
Author(s):  
Jonathan Heinz ◽  
Miroslav Kolesik
Keyword(s):  
Quantum Mechanics ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Complex Scaling ◽  
Transparent Boundary Conditions ◽  
Drastic Reduction ◽  
Optical Cavities ◽  
Energy Dependent ◽  
Lossy Waveguides

A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.

Simplified characteristics of active magnetic bearings

2002 ◽  
Vol 216 (5) ◽  
pp. 623-628 ◽  
Author(s):  
D J Peel ◽  
C M Bingham ◽  
Y Wu ◽  
D Howe
Keyword(s):  
Natural Vibration ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Viscous Damping ◽  
Vibration Frequencies ◽  
Active Magnetic Bearings ◽  
Complex Dynamic ◽  
Speed Range ◽  
Constant Coefficients ◽  
System Characteristics

Traditionally, active magnetic bearing (AMB) systems are designed as an integral component of machines having generally complex dynamic characteristics. An AMB supported rotor has been tested over a speed range that included system natural vibration frequencies. A linear stiffness and viscous damping AMB characteristic with constant coefficients was identified which was independent of the overall system characteristics and which can thus provide simple and transferable data for a machine designer.

scholarly journals Linearization schemes for Hermite matrix polynomials

2015 ◽  
Author(s):  
Nikta Shayanfar ◽  
Heike Fassbender
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Matrix Polynomial ◽  
Multiple Input Multiple Output ◽  
Matrix Pencil ◽  
Least Square ◽  
Coupled Systems ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
The Matrix

The polynomial eigenvalue problem is to find the eigenpair of $(\lambda,x) \in \mathbb{C}\bigcup \{\infty\} \times \mathbb{C}^n \backslash \{0\}$ that satisfies $P(\lambda)x=0$, where $P(\lambda)=\sum_{i=0}^s P_i \lambda ^i$ is an $n\times n$ so-called matrix polynomial of degree $s$, where the coefficients $P_i, i=0,\cdots,s$, are $n\times n$ constant matrices, and $P_s$ is supposed to be nonzero. These eigenvalue problems arise from a variety of physical applications including acoustic structural coupled systems, fluid mechanics, multiple input multiple output systems in control theory, signal processing, and constrained least square problems. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Such methods convert the eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploit and preserve the structure and properties of the original eigenvalue problem. The linearizations have been extensively studied with respect to the basis that the matrix polynomial is expressed in. If the matrix polynomial is expressed in a special basis, then it is desirable that its linearization be also expressed in the same basis. The reason is due to the fact that changing the given basis ought to be avoided \cite{H1}. The authors in \cite{ACL} have constructed linearization for different bases such as degree-graded ones (including monomial, Newton and Pochhammer basis), Bernstein and Lagrange basis. This contribution is concerned with polynomial eigenvalue problems in which the matrix polynomial is expressed in Hermite basis. In fact, Hermite basis is used for presenting matrix polynomials designed for matching a series of points and function derivatives at the prescribed nodes. In the literature, the linearizations of matrix polynomials of degree $s$, expressed in Hermite basis, consist of matrix pencils with $s+2$ blocks of size $n \times n$. In other words, additional eigenvalues at infinity had to be introduced, see e.g. \cite{CSAG}. In this research, we try to overcome this difficulty by reducing the size of linearization. The reduction scheme presented will gradually reduce the linearization to its minimal size making use of ideas from \cite{VMM1}. More precisely, for $n \times n$ matrix polynomials of degree $s$, we present linearizations of smaller size, consisting of $s+1$ and $s$ blocks of $n \times n$ matrices. The structure of the eigenvectors is also discussed.

Influence of Gravity and Taper on the Vibration of a Standing Column

2012 ◽  
Vol 4 (04) ◽  
pp. 483-495 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Cross Section ◽  
Numerical Stability ◽  
Natural Vibration ◽  
Vibration Frequencies ◽  
Stability Criteria ◽  
Initial Value ◽  
Vertical Column ◽  
Stability Boundaries ◽  
Section Shape ◽  
The Stability

AbstractThe stability and natural vibration of a standing tapered vertical column under its own weight are studied. Exact stability criteria are found for the pointy column and numerical stability boundaries are determined for the blunt tipped column. For vibrations we use an accurate, efficient initial value numerical method for the first three frequencies. Four kinds of columns with linear taper are considered. Both the taper and the cross section shape of the column have large influences on the vibration frequencies. It is found that gravity decreases the frequency while the degree of taper may increase or decrease frequency. Vibrations may occur in two different planes.

scholarly journals Finite elements apparatus in thin-walled rods dynamics problems

2018 ◽  
Vol 245 ◽  
pp. 08007 ◽  
Author(s):  
Vladimir Rybakov ◽  
Stanislav Dyakov ◽  
Daniil Sovetnikov ◽  
Artur Azarov ◽  
Sergey Ivanov
Keyword(s):  
Natural Vibration ◽  
Steel Structures ◽  
Vibration Frequencies ◽  
Construction Engineering ◽  
Shear Theory ◽  
Mass Matrices ◽  
Element System ◽  
Mathematical Problems ◽  
Thin Walled ◽  
Dynamics Problems

The calculation of thin-walled rods is extremely relevant problem of structural mechanics and not only from the scientific standpoint, but also due to the widespread use of so-called lightweight thin-walled steel structures for construction engineering sector. Regardless of a sufficiently large number of studies connected with the statics of thin-walled rods, the dynamics of such systems have not been thoroughly studied yet. Based on one of the forward-looking theories of calculation i.e. the semi-shear theory by Slicker, the paper provides a technique for solving the dynamics problems of thin-walled rods. The stiffness and mass matrices of the finite element system are obtained for linear approximation of the form functions, and the natural vibration frequencies of the rods are calculated. The obtained solution is accomplished by the extrapolation method of estimating the accuracy of numerical methods for solving mathematical problems.

scholarly journals High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter eigenvalue problem

2020 ◽  
Vol 96 ◽  
pp. 102639
Author(s):  
Carolin Penke ◽  
Andreas Marek ◽  
Christian Vorwerk ◽  
Claudia Draxl ◽  
Peter Benner
Keyword(s):  
Eigenvalue Problem ◽  
High Performance ◽  
Eigenvalue Problems ◽  
Symmetric Eigenvalue Problems
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