Frequency-Domain Identification of Linear Time-Periodic Systems Using LTI Techniques

Allen, Matthew S.

doi:10.1115/1.3187151

Cited by 46 publications

(36 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The constants L , L 3 x ∞ and γ are determined by the fminsearch routine in MATLAB. 1 But in general only a local minimum is found by fminsearch, therefore we combined it with a Global Search strategy of the Global Optimization Toolbox in MATLAB. The computed values for L , L 3 x ∞ and γ are given in Table 3.…”

Section: Overview and Numerical Resultsmentioning

confidence: 99%

“…It can be modeled as a linear time-periodic system (1) where A(t) is entire with system dimension n = 4. The same parameter values are chosen as in [1]. This is an asymptotically stable system since the maximal Lyapunov exponent is ν[L] = −0.002000131812440 < 0.…”

Section: Fig 4 Solution Formentioning

confidence: 99%

“…As the second example, we chose a Jeffcott rotor on an anisotropic shaft supported by anisotropic bearings [1]. It can be modeled as a linear time-periodic system (1) where A(t) is entire with system dimension n = 4.…”

Section: Fig 4 Solution Formentioning

confidence: 99%

See 2 more Smart Citations

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Benner

Denißen

Kohaupt

2016

J. Appl. Math. Comput.

View full text Add to dashboard Cite

Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Examples include anisotropic rotor-bearing systems and parametrically excited systems. The structure of the solution to linear time-periodic systems is known due to Floquet's Theorem. We use this information to derive a new norm which yields two-sided bounds on the solution and in this norm vibrations of the solution are suppressed. The obtained results are a generalization for linear time-invariant systems. Since Floquet's Theorem is non-constructive, the applicability of the aforementioned results suffers in general from an unknown Floquet normal form. Hence, we discuss trigonometric splines and spectral methods that are both equipped with rigorous bounds on the solution. The methodology differs systematically for the two methods. While in the first method the solution is approximated by trigonometric splines and the upper bound depends on the approximation quality, in the second method the linear time-periodic system is approximated and its solution is represented as an infinite series. Depending on the smoothness of the time-periodic system, we formulate two upper bounds which incorporate the approximation error of the linear time-periodic system and the truncation error of the series representation. Rigorous bounds on the solution are necessary whenever reliable results are needed, and hence they can support the analysis and, e.g., stability or robustness of the solution may be proven or falsified. The theoretical results are illustrated and compared to trigonometric spline bounds and spectral bounds by means of three examples that include an anisotropic rotor-bearing system and a parametrically excited Cantilever beam.

show abstract

Section: Overview and Numerical Resultsmentioning

confidence: 99%

Section: Fig 4 Solution Formentioning

confidence: 99%

See 1 more Smart Citation

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Benner

Denißen

Kohaupt

2016

J. Appl. Math. Comput.

View full text Add to dashboard Cite

show abstract

“…Other notable contributions include the work by Vanlanduit et al [10], who presented a CSLDV method that uses multisine excitation (periodic broadband excitation). Allen & Sracic explored the use of higher scan frequencies together with impact excitation, presenting the lifting technique that allows conventional modal analysis curve fitting methods and tools such as the CMIF to be applied to CSLDV measurements [11][12][13]. They also presented a method for mass-normalizing the mode vectors obtained by CSLDV when the input force has been measured [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…Since the mode shapes of a linear system are functions of position, the CSLDV measurement appears to be from a time-periodic system when the laser spot moves in a periodic, closed scan path. A few system identification strategies have been proposed for linear time-periodic systems, as discussed in [11]. This work utilizes a method recently presented by Allen et al in [15,16] that is based on the spectra of the output of a linear time periodic system when it is excited by a broadband random input.…”

Section: Introductionmentioning

confidence: 99%

Output-Only Modal Analysis Using Continuous-Scan Laser Doppler Vibrometry and Application to a 20kW Wind Turbine

Yang

Allen

2011

Conference Proceedings of the Society for Experimental Mechanics Series

Self Cite

View full text Add to dashboard Cite

Abstract:Continuous-scan laser Doppler vibrometry (CSLDV) is a method whereby one continuously sweeps the laser measurement point over a structure while measuring, in contrast to the conventional scanning LDV approach where the laser spot remains stationary while the response is collected at each point. The continuousscan approach can greatly accelerate measurements, allowing one to capture spatially detailed mode shapes along a scan path in the same amount of time that is typically required to measure the response at a single point. The method is especially beneficial when testing large structures, such as wind turbines, whose natural frequencies are very low and hence require very long time records. Several CSLDV methods have been presented that employ harmonic excitation or impulse excitation, but no prior work has performed CSLDV with an unmeasured, broadband random input. This work extends CSLDV to that class of input, developing an outputonly CSLDV method (OMA-CSLDV). This is accomplished by adapting a recently developed algorithm for linear time-periodic systems to the CSLDV measurements, which makes use of harmonic power spectra and the harmonic transfer function concept developed by Wereley. The proposed method is validated on a randomly excited free-free beam, where one-dimensional mode shapes are captured by scanning the laser along the length of the beam. The natural frequencies and mode shapes are extracted from the harmonic power spectrum of the vibrometer signal and show good agreement with the first seven analytically-derived modes of the beam. The method is then applied to identify the shapes of several modes of a 20kW wind turbine using a ground based laser and with only a light breeze providing excitation.

show abstract

Bounds on the Solution of Linear Time‐Periodic Systems

Benner

Denißen

Kohaupt

2013

Proc Appl Math and Mech

View full text Add to dashboard Cite

Linear time-periodic systems have been an active area of research in the last decades. They arise in various applications such as anisotropic rotor-bearing systems and nonlinear systems linearized about a periodic trajectory. Rigorous bounds support the transient analysis of these systems. Optimal constants are determined by the differential calculus for norms of matrix functions. Bounds based on trigonometric spline approximations of the solution are introduced and convergence results for the approximations are stated. Bounds are illustrated by means of an anisotropic rotor-bearing system. Linear time-periodic systemsWe are considering a set of linear ordinary differential equations (ODEs) with time-periodic coefficients with periodicity T and a given initial condition as a linear time-periodic systeṁwhere x ∈ R n and A : R → R n×n . Under well known assumptions a unique solution to (1) is guaranteed. The solution can then be represented by the following theorem due to Floquet [1], but in a different form.Theorem 1.1 A fundamental matrix Φ(t) of (1) can be represented aswhere R, Z(t) ∈ C n×n and Z(t) = Z(t + T ) is nonsingular ∀t ∈ R.The solution x(t) is a linear combination of n linearly independent solutions w.r.t. the initial condition x 0 : x(t) = Φ(t)x 0 = Z(t)e Rt x 0 with Φ(0) = I. Determining Φ(t) in the interval [0, T ] is sufficient due to the semigroup property of the solution. For the sake of simplicity we consider equidistant nodes t i = ih for i = −1, 0, 1, . . . , r with h = T r and we use quadratic trigonometric splines T i (t) [2] to approximate the solution. . .whereT denotes a vector of the i-th coefficients α (j)i of the spline approximation s j (t) for j = 1, . . . , n. Quadratic trigonometric splines with compact support are chosen in order to simplify the computation of the coefficientswhere θ = 1 sin h sin h 2 . Solving ODEs by spline approximations has been investigated by F. R. Loscalzo and T. D. Talbot [3] and many others, e.g. A. Nikolis [4]. The focus on trigonometric splines is due to the periodicity of (1) and furthermore, we will equip the computation with rigorous upper and lower bounds on the solution. Demanding that the approximation s fulfills (1) at the nodes t i , yields a series of r + 1 linear systems A

show abstract

Frequency-Domain Identification of Linear Time-Periodic Systems Using LTI Techniques

Cited by 46 publications

References 39 publications

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Output-Only Modal Analysis Using Continuous-Scan Laser Doppler Vibrometry and Application to a 20kW Wind Turbine

Bounds on the Solution of Linear Time‐Periodic Systems

Contact Info

Product

Resources

About