A constrained eigenvalue problem and nodal and modal control of vibrating systems

Ram, Yitshak M.

doi:10.1098/rspa.2009.0426

Cited by 3 publications

(4 citation statements)

References 3 publications

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“…In this case, the uncertainty of the system synthesis problem arises: to build a modal controller, it is necessary to know the parameters of the reference channel с0 and с1, and to build the reference channel, it is necessary to know the characteristic polynomial of the in such a way that the inequality   (-а0,M/а1, M) holds [27][28][29]. Restricting ourselves to the case of negative real roots that is important for this problem, we prove the following theorem.…”

Section: Synthesis Methodsmentioning

confidence: 99%

Modal synthesis of precision control systems

2020

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The problem of synthesis of precision modal control systems is considered. It is noted that a common approach to solving this problem is to consistently meet the requirements for the nature of the transient process and for the indicators of its accuracy. This approach to synthesis is faced with the need to make design decisions under incomplete conditions. In practice, this circumstance leads to obtaining synthesis results with undesirable deviations from technical requirements. When designing precision control systems, such deviations are unacceptable. To eliminate the difficulties that arise, a transition to interval methods for formulating and solving modal synthesis problems is proposed. The theoretical possibility of the interval approach is based on the excessive variety of possible placement of eigenvalues in the spectrum of the characteristic matrix of the system. An example of an interval synthesis of a system with a modal controller and additional output feedback is considered. The restrictions on the spectrum of the specified matrix are formed, which determine the fulfillment of the requirements for the monotonicity of the transient process, the regulation time and the accuracy of the response to harmonic influences. It is noted that the variety of solutions obtained creates the preconditions for a multi-alternative approach to modal synthesis of systems.

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Section: Synthesis Methodsmentioning

confidence: 99%

Modal synthesis of precision control systems

2020

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“…In an attempt to generalize such an approach to allow for higher order formulas, and differential constraints which are arbitrary in form and number, we might consider the problem [15],…”

Section: Eigenfunctions Of Linear Differential Operatorsmentioning

confidence: 99%

“…with the additional requirement ∥⃗ y∥ 2 2 = 1. While this does appear to be a standard problem of linear algebra, the fact is that while it may be a consistent discretization, it may yield a grand total of zero eigenvalueeigenvector pairs [15]. The formulation of the discretized eigenfunction problem in Eqn.…”

Section: Eigenfunctions Of Linear Differential Operatorsmentioning

confidence: 99%

“…The method is demonstrated on the following problems: Solving the vibrating string problem with Robin boundary conditions; solving two forms of singular Bessel equations; and computing the modes of an overconstrained vibrating beam, as well as fitting these mode shapes to real-world data of a cantilever beam subject to a load. Some applications of the proposed method include the solution of variational problems, such as optimal control [13], a discrete Rayleigh-Ritz method [5,14], the control of distributed parameter systems [15,16], or the design of beams with desired mode shapes [17]. Code for the proposed method is available freely on the MATLAB file-exchange at: http://www.mathworks.com/matlabcentral/fileexchange/authors/321598…”

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Approximation of Physical Measurement Data by Discrete Orthogonal Eigenfunctions of Linear Differential Operators

Harker

O’Leary

2017

Transactions of the Canadian Society for Mechanical Engineering

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We present a new method for computing the eigenfunctions of a linear differential operator such that they satisfy a discrete orthogonality constraint, and can thereby be used to efficiently approximate physical measurement data. Classic eigenfunctions, such as those of Sturm-Liouville problems, are typically used for approximating continuous functions, but are cumbersome for approximating discrete data. We take a matrix-based variational approach to compute eigenfunctions based on physical problems which can be used in the same manner as the DCT or DFT. The approach uses sparse matrix methods, so it can be used to compute a small number of eigenfunctions. The method is verified on common eigenvalue problems, and the approximation of real-world measurement data of a bending beam by means of its computed mode-shapes.

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Numerical Methodology of Tuning a System to Target Frequencies by Adding Mass

Chen¹,

Gu²

2019

SAE Technical Paper Series

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A constrained eigenvalue problem and nodal and modal control of vibrating systems

Cited by 3 publications

References 3 publications

Modal synthesis of precision control systems

Modal synthesis of precision control systems

Approximation of Physical Measurement Data by Discrete Orthogonal Eigenfunctions of Linear Differential Operators

Numerical Methodology of Tuning a System to Target Frequencies by Adding Mass

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