A nonsymmetric state-variable decomposition for modal analysis

Feeny, Brian F.; Farooq, Umar

doi:10.1016/j.jsv.2007.11.012

Cited by 19 publications

(13 citation statements)

References 32 publications

(36 reference statements)

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“…Shortly after SOD, and recognizing that it cannot provide complex modes, S-VD was proposed to identify complex vibration modes and the associated damping ratios and frequencies of the state-variable model of a multi-degree-of-freedom linear system. 5 In this approach, S-VD is accomplished by solving an asymmetric generalized eigenvalue problem,…”

Section: State-variable Decompositionmentioning

confidence: 99%

Toward a unified interpretation of the “proper”/“smooth” orthogonal decompositions and “state variable”/“dynamic mode” decompositions with application to fluid dynamics

2020

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A common interpretation is presented for four powerful modal decomposition techniques: “proper orthogonal decomposition,” “smooth orthogonal decomposition,” “state-variable decomposition,” and “dynamic mode decomposition.” It is shown that, in certain cases, each technique can be interpreted as an optimization problem and similarities between methods are highlighted. By interpreting each technique as an optimization problem, significant insight is gained toward the physical properties of the identified modes. This insight is strengthened by being consistent with cross-multiple decomposition techniques. To illustrate this, an inter-method comparison of synthetic hypersonic boundary layer stability data is presented.

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Section: State-variable Decompositionmentioning

confidence: 99%

Toward a unified interpretation of the “proper”/“smooth” orthogonal decompositions and “state variable”/“dynamic mode” decompositions with application to fluid dynamics

2020

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“…In SVMD [22] the outputs of the freely vibrating, lightly damped beam were used to estimate the mode shapes, natural frequencies, and in some cases modal damping of the beam. When applying SVMD one must first create a state-variable ensemble matrix, Y = [V T X T ] T , where V is the velocity ensemble matrix and Xis the displacement ensemble matrix [22]. As such Y = [y( Once the two correlation matrices are computed then an eigenvalue problem is cast as λRφ φ φ = Nφ φ φ.…”

Section: State Variable Modal Decompositionmentioning

confidence: 99%

Output-Only Modal Identification of a Nonuniform Beam by Using Decomposition Methods

Caldwell

Feeny

2014

Journal of Vibration and Acoustics

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Reduced-order mass weighted proper orthogonal decomposition (RMPOD), smooth orthogonal decomposition (SOD), and state variable modal decomposition (SVMD) are used to extract modal parameters from a nonuniform experimental beam. The beam was sensed by accelerometers. Accelerometer signals were integrated and passed through a high-pass filter to obtain velocities and displacements, all of which were used to build the necessary ensembles for the decomposition matrices. Each of these decomposition methods was used to extract mode shapes and modal coordinates. RMPOD can directly quantify modal energy, while SOD and SVMD directly produce estimates of modal frequencies. The extracted mode shapes and modal frequencies were compared to an analytical approximation of these quantities, and to frequencies estimated by applying the fast Fourier transform to accelerometer data. SVMD is also applied to estimate modal damping, which was compared to the estimate by logarithmic decrement applied to modal coordinate signals, with varying degrees of success. This paper shows that these decomposition methods are capable of extracting lower modal parameters of an actual experimental beam.

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“…Very similar to SOD is state-variable modal decomposition (SVMD) [28,29], which also solves a generalized eigenvalue problem of a matrix of state variables and their time derivatives. In its formulation, a nonsymmetric correlation matrix of state variables and approximations of time derivatives is used as a constraint which closely relates to the formulation of SOD.…”

Section: Introductionmentioning

confidence: 99%

On the Inclusion of Time Derivatives of State Variables for Parametric Model Order Reduction for a Beam on a Nonlinear Foundation

Segala

2017

Journal of Dynamic Systems, Measurement, and Control

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The computational burden of parameter exploration of nonlinear dynamical systems can become a costly exercise. A computationally efficient lower dimensional representation of a higher dimensional dynamical system is achieved by developing a reduced order model (ROM). Proper orthogonal decomposition (POD) is usually the preferred method in projection-based nonlinear model reduction. POD seeks to find a set of projection modes that maximize the variance between the full-scale state variables and its reduced representation through a constrained optimization problem. Here, we investigate the benefits of an ROM, both qualitatively and quantitatively, by the inclusion of time derivatives of the state variables. In one formulation, time derivatives are introduced as a constraint in the optimization formulation—smooth orthogonal decomposition (SOD). In another formulation, time derivatives are concatenated with the state variables to increase the size of the state space in the optimization formulation—extended state proper orthogonal decomposition (ESPOD). The three methods (POD, SOD, and ESPOD) are compared using a periodically, periodically forced with measurement noise, and a randomly forced beam on a nonlinear foundation. For both the periodically and randomly forced cases, SOD yields a robust subspace for model reduction that is insensitive to changes in forcing amplitudes and input energy. In addition, SOD offers continual improvement as the size of the dimension of the subspace increases. In the periodically forced case where the ROM is developed with noisy data, ESPOD outperforms both SOD and POD and captures the dynamics of the desired system using a lower dimensional model.

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A nonsymmetric state-variable decomposition for modal analysis

Cited by 19 publications

References 32 publications

Toward a unified interpretation of the “proper”/“smooth” orthogonal decompositions and “state variable”/“dynamic mode” decompositions with application to fluid dynamics

Toward a unified interpretation of the “proper”/“smooth” orthogonal decompositions and “state variable”/“dynamic mode” decompositions with application to fluid dynamics

Output-Only Modal Identification of a Nonuniform Beam by Using Decomposition Methods

On the Inclusion of Time Derivatives of State Variables for Parametric Model Order Reduction for a Beam on a Nonlinear Foundation

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